New Warehouse Designs: Angled Aisles and Their E ects on Travel
Distance
by
 Omer  Ozt urko glu
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
August 6, 2011
Keywords: warehouse, distribution center, aisle design, pickup and deposit point
Copyright 2011 by  Omer  Ozt urko glu
Approved by
Kevin R. Gue, Chair, Associate Professor of Industrial and Systems Engineering
Alice E. Smith, Professor of Industrial and Systems Engineering
Robert L. Bul n, Professor of Industrial and Systems Engineering
Russell D. Meller, Professor of Industrial Engineering, University of Arkansas
Abstract
In supply chain and logistics systems, warehouses and third-party logistics cen-
ters have become the backbone of distribution networks that store and deliver goods.
However, these facilities are still built to look much as they have been for the past
 fty years (Gue and Meller, 2009). This dissertation builds on some recent research
that challenges traditional ways of organizing picking aisles and cross aisles in unit-
load warehouses. The designs in this dissertation, and the models that produced
them, o er to the research and practicing communities a new and fundamentally
di erent way to extract cost and improve the performance of industrial warehouses.
For warehouses with one, two and three cross aisles, we develop closed form,
single-command travel distance functions in a continuous warehouse space. We use
these functions to minimize expected travel distance in a unit-load warehouse with
a centrally located pickup and deposit (P&D) point under randomized storage. We
propose three optimal unit-load warehouses with respect to the inserted cross aisles,
the Chevron, the Leaf and the Butter y. In order to provide a more accurate model
and to analyze the wasted space for angled aisle designs, we build a discrete model
of aisles and pallet locations that better represents warehouse designs.
We develop a general warehouse design tool that models aisles, pallet locations
and dock doors as discrete objects. The tool implements a network representation of
the warehouse, a shortest path algorithm to determine travel distances, and a particle
swarm meta-heuristic to search for the best arrangement of aisles. To illustrate the
use of model, we de ne  ve design problems that are di erentiated by the locations
of multiple P&D points. We propose improved non-traditional aisle designs for each
design problem with one and two inserted cross aisles.
ii
Last, we introduce the idea of robustness in non-traditional aisle designs with
respect to a varying number of P&D points. For two commonly found  ow patterns
in industry, we use our models to determine optimal designs for an increasingly large
block of P&D points to  nd the point at which the structure \breaks." The results
suggest that the Chevron design, in particular, is robust over a wide range of P&D
points.
iii
Acknowledgments
If the pages of this dissertation could talk, they would say that this dissertation
is far more than the culmination of years of study. This dissertation has been a
big portion of my life in the past  ve years. Today, I stand on my own two feet
with the honor of completing my doctoral study with the support of many generous,
admirable, inspiring and honorable people.
First, I would like to express my deepest gratitude to my advisor, Dr. Kevin
R. Gue, for his excellent guidance, support, and for providing me with an excellent
atmosphere for doing research. His knowledge, wisdom and commitment to the
highest standards of academic excellence contributed to my development as a scholar
and teacher.
I would also like to thank my committee members, Dr. Alice E. Smith, Dr. Robert
L. Bul n and Dr. Russell D. Meller for guiding my research and for their thoughtful
criticism, time and attention. I am grateful to the National Science Foundation,
which supported this research under Grants DMI-0600374 and DMI-0600671, the
Material Handling Education Foundation, which honored me with a scholarship,
and the Auburn University Industrial and Systems Engineering Department and
Graduate School, which funded my graduate work.
I would probably have not completed my dissertation without the support of
several people: my grandparents, parents, sister, my wife and, of course, my daugh-
ter. I would like to thank my grandparents, Sevim and Abit, for their endless support
since my childhood. I would like to express my deepest gratitude and love to my
parents, _Ifakat and Halil, for their presence, endless support, love and understand-
ing during my whole life. I would like thank my sister,  Ozlem, for her best wishes
iv
and for supporting my parents when I am far away from them. I would also like to
apologize to her for missing her gorgeous wedding and thank for her understanding.
I would like to thank my wife, Y ucel, for her understanding, patience and love
during the last seven years. I thank her being with me and loving me during both
good days and bad, and also further dedication through hard times that we faced in
the past  ve years. She deserves a medal.
Last, but not least, of course, I would like to thank to my little \pure light",
Duru Nur for her patience, good behavior, especially when I am not with her. I still
remember her saying \do not leave, daddy!", when I needed to go to the o ce at
nights. I also feel sorry for missing her third birthday and not being with her for
 ve months. My little daisy, I will always love you.
I dedicate this dissertation to my parents. I aspire to guide my daughter.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1 Warehousing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . 5
2 Warehouse Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Receiving and Shipping . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Designing Warehouses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Non-Traditional Aisle Designs . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Material Flow Designs in Nature and Other Structures . . . . . . . . 21
4 Optimal Unit-Load Warehouse Designs for Single Command Operations 25
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 One Cross Aisle Warehouse Designs . . . . . . . . . . . . . . . . . . . 30
4.3.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.2 One Cross Aisle Designs for Non-Square Half-Warehouses . . 35
4.4 A Two Cross Aisle Design . . . . . . . . . . . . . . . . . . . . . . . . 36
vi
4.5 Three Cross Aisle Design . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.6 Comparison of Continuous Warehouse Designs . . . . . . . . . . . . . 45
4.7 A Continuum of Designs . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.8 Implications for Practice . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Non-Traditional Unit-Load Warehouse Designs with Multiple Pickup-
and-Deposit Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 The Warehouse Network Model . . . . . . . . . . . . . . . . . . . . . 61
5.5 Optimization Methodology: Particle Swarm Optimization (PSO) . . . 65
5.5.1 Formulation of the problem . . . . . . . . . . . . . . . . . . . 66
5.5.2 PSO Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.6 Optimization of Layouts for Unit-Load Warehouses . . . . . . . . . . 74
5.6.1 Design Problem A . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6.2 Design Problem B . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6.3 Design Problem C . . . . . . . . . . . . . . . . . . . . . . . . 88
5.6.4 Design Problem D . . . . . . . . . . . . . . . . . . . . . . . . 94
5.6.5 Design Problem E . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.7 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . 101
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Robustness of Non-Traditional Aisle Designs with respect to a Varying
Number of P&D Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 Model and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Unit-Load Warehouse Designs with P&D Points at the Bottom of the
Warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.1 One Cross Aisle Designs . . . . . . . . . . . . . . . . . . . . . 108
vii
6.3.2 Two Cross Aisle Designs . . . . . . . . . . . . . . . . . . . . . 111
6.4 Unit-Load Warehouse Designs with P&D Points at the Bottom and
the Top of the Warehouse . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4.1 One Cross Aisle Designs . . . . . . . . . . . . . . . . . . . . . 115
6.4.2 Two Cross Aisle Designs . . . . . . . . . . . . . . . . . . . . . 118
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . . . 122
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.1 Dynamics of inserted P&D points . . . . . . . . . . . . . . . . 126
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.1 Closed form expressions for E[WR] . . . . . . . . . . . . . . . . . . . 139
A.2 Proof of Observation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.3 Expected travel distances in a three cross aisle model . . . . . . . . . 140
A.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.1 Two Cross Aisle Model Cases . . . . . . . . . . . . . . . . . . . . . . 144
B.2 Computational Results in Design Problem A . . . . . . . . . . . . . . 146
B.2.1 Results for Design A1 . . . . . . . . . . . . . . . . . . . . . . 146
B.2.2 Results for Design A2 . . . . . . . . . . . . . . . . . . . . . . 149
B.3 Computational Results in Design Problem B . . . . . . . . . . . . . . 152
B.3.1 Results for Design B1 . . . . . . . . . . . . . . . . . . . . . . . 152
B.4 Computational Results in Design Problem C . . . . . . . . . . . . . . 158
B.4.1 Results for Design C1 . . . . . . . . . . . . . . . . . . . . . . . 158
B.4.2 Results for Design C2 . . . . . . . . . . . . . . . . . . . . . . . 161
B.5 Computational Results in Design Problem D . . . . . . . . . . . . . . 164
B.6 Computational Results in Design Problem E . . . . . . . . . . . . . . 167
viii
B.6.1 Results for Design E1 . . . . . . . . . . . . . . . . . . . . . . . 167
B.6.2 Results for Design E2 . . . . . . . . . . . . . . . . . . . . . . . 167
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
C.1 One Cross Aisle Designs with P&D Points at the Bottom of the Ware-
house . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
C.2 Two Cross Aisle Designs with P&D Points at the Bottom of the Ware-
house . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.3 One Cross Aisle Designs with P&D Points at the Bottom and the Top
of the Warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
C.4 Two Cross Aisle Designs with P&D Points at the Bottom and the
Top of the Warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . 178
ix
List of Figures
1.1 Traditional warehouse designs. . . . . . . . . . . . . . . . . . . . . . . . 3
3.1 A traditional warehouse layout with design elements. . . . . . . . . . . . 16
3.2 Two main variations of traditional warehouses. . . . . . . . . . . . . . . 16
3.3 Non-rectangular warehouses with radial aisles (White, 1972). N is the
number of regions that are de ned by radial cross aisles. Travel in these
regions is along the cross aisles  rst and then parallel to either x or y axis. 19
3.4 Proposed designs for a single P&D point by Gue and Meller (2009). . . . 20
3.5 Proposed designs for multiple P&D points at the bottom of the ware-
house by Gue et al. (2010). . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 A continuum of veins in a leaf that is developed and represented by
Runions et al. (2005). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.7 A developed orchid leaf by Runions et al. (2005). . . . . . . . . . . . . . 23
3.8 (a) The picture shows the Palis de Chailot and Jardins du Trocadero. It is
taken from the Ei el Tower by \http://www.planetware.com/picture/paris-
palais-de-chaillot-f-f1037.htm." (b) By Rolland et al. (2011). . . . . . . . 23
4.1 Traditional rectangular warehouse. . . . . . . . . . . . . . . . . . . . . . 26
4.2 The Flying-V (Left) and Fishbone (Right). . . . . . . . . . . . . . . . . . 28
4.3 One cross aisle warehouse design. . . . . . . . . . . . . . . . . . . . . . . 30
x
4.4 Example travel paths based on some possible  R?s. . . . . . . . . . . . . 32
4.5 The representation of case 1 (left) and case 2 (right) for the one cross
aisle model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 The optimal one cross aisle design. . . . . . . . . . . . . . . . . . . . . . 35
4.7 Change in optimal  R for non-square half-warehouses. Solid lines rep-
resent optimal angles of picking aisles; dashed lines with circle markers
represent angles of the diagonal on the right side. . . . . . . . . . . . . . 36
4.8 A two cross aisle warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.9 (a) An infeasible symmetric warehouse design with two cross aisles when
 RC < 90?, (b) sample travel paths to the point P with di erent  RC. . . . 37
4.10 Cases in the two cross aisle model. . . . . . . . . . . . . . . . . . . . . . 38
4.11 Travel paths according to the de ned areas in two cross aisle design.. . . 39
4.12 The Leaf: Optimal two cross aisle design for square half-warehouses. . . 41
4.13 Travel paths in regions on the right side of the three cross aisle design. . 41
4.14 Cases in the three cross aisle model. . . . . . . . . . . . . . . . . . . . . 42
4.15 Representation of the Butter y design. . . . . . . . . . . . . . . . . . . . 46
4.16 Equally distributed imperfections in designs. . . . . . . . . . . . . . . . . 47
4.17 Equal travel distance in dual designs shown via parallelogram. . . . . . . 48
4.18 The Chevron design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.19 The Leaf design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
xi
4.20 The Butter y design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.21 Comparison of several approximately square half-warehouses with an
equivalent traditional warehouse. (Data for the plots are in Appendix A.4.) 52
4.22 Chevron versus Traditional in an equivalent small size warehouse: Chevron
has lower expected distance, but requires more space. . . . . . . . . . . . 52
4.23 An implementation of non-traditional aisles at Generac Power Systems. . 55
5.1 Traditional rectangular warehouses. . . . . . . . . . . . . . . . . . . . . . 56
5.2 The Modi ed Flying-V and the Inverted-V designs for multiple P&D
points. From Gue et al. (2010). . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 The intersecting and non-intersecting cross aisles in a warehouse. . . . . 60
5.4 Description of the warehouse design tool . . . . . . . . . . . . . . . . . . 61
5.5 The procedure to represent a discrete space warehouse design for given
set of parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.6 Network representation of a small, example warehouse. (Each storage
location is represented by an access node in a real network.) . . . . . . . 63
5.7 Possible arrangements of one cross aisle. (Numbers on the cross aisles
represent cases in the one cross aisle model.) . . . . . . . . . . . . . . . . 68
5.8 Possible directions of cross aisles . . . . . . . . . . . . . . . . . . . . . . 68
5.9 Possible region de nitions and angles of aisles in these region. . . . . . . 69
5.10 Issues show that the variable goes out of its boundary if vtisd is applied,
when 0 <vtisd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
xii
5.11 Issues show that the variable goes out of its boundary if vtisd is applied,
when vtisd < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.12 Pseudo-code for the PSO algorithm . . . . . . . . . . . . . . . . . . . . . 75
5.13 Changing number of available storage locations in a discrete warehouse
space by angle of aisles. The dark squares represent the disappearing
pallet locations because they overlap at the appropriate cross aisle. . . . 76
5.14 Design problems di erentiated by the locations of P&D points. . . . . . 77
5.15 Designs with one cross aisle in design problem A. . . . . . . . . . . . . . 78
5.16 Comparison of Design A1 and Trad-A. . . . . . . . . . . . . . . . . . . . 79
5.17 Comparison of modi ed Design A1 and Trad-A. . . . . . . . . . . . . . . 80
5.18 Design A2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.19 Comparison of Design A2 and Trad-A. . . . . . . . . . . . . . . . . . . . 82
5.20 The modi ed Flying-V design with two, inserted linear cross aisles. . . . 83
5.21 Comparison of Design A2 and the modi ed Flying-V. (See Appendix B.10
for detailed results.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.22 Design B1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.23 Comparison of Design B1 and Trad-B. . . . . . . . . . . . . . . . . . . . 86
5.24 Designs with two cross aisles in design problem B. . . . . . . . . . . . . . 87
5.25 Comparison of Design B2 and Trad-B. . . . . . . . . . . . . . . . . . . . 87
5.26 Comparison of modi ed Design B2 and Trad-B. . . . . . . . . . . . . . . 88
xiii
5.27 Designs with one cross aisle in design problem C. . . . . . . . . . . . . . 90
5.28 Comparison of Design C1 and Trad-A. . . . . . . . . . . . . . . . . . . . 91
5.29 Comparison of modi ed Design C1 and Trad-A. . . . . . . . . . . . . . . 91
5.30 Designs with two cross aisles in design problem C. . . . . . . . . . . . . . 92
5.31 Comparison of Design C2 and Trad-A. . . . . . . . . . . . . . . . . . . . 93
5.32 Comparison of modi ed Design C2 and Trad-A. . . . . . . . . . . . . . . 94
5.33 Designs with one and two cross aisles in design problem D. . . . . . . . . 95
5.34 Design D0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.35 Comparison of Design D0 and Trad-A. . . . . . . . . . . . . . . . . . . . 97
5.36 Designs with one cross aisle in design problem E. . . . . . . . . . . . . . 99
5.37 Designs with two cross aisles in design problem E. . . . . . . . . . . . . . 100
5.38 Comparison of Design E2 and Trad-B. . . . . . . . . . . . . . . . . . . . 100
6.1 One cross aisle designs with a varying number of P&D points at the
bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Comparison of the Chevron and the improved designs. . . . . . . . . . . 110
6.3 (a) Comparison of improved designs with a  oating and a  xed cross
aisle. (b) Angles of picking aisles on the right side of improved designs
with a  xed cross aisle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.4 The Chevron and the improved design with 13 P&D points at the bottom.111
6.5 Comparison of the Chevron and improved designs for equally capacitated
warehouses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xiv
6.6 Two cross aisle designs with a varying number of P&D points at the
bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.7 Comparisons of the Leaf and the improved designs over equally sized and
equally capacitated warehouses. . . . . . . . . . . . . . . . . . . . . . . . 114
6.8 Designs with centrally located P&D points at the bottom and the top of
the warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.9 Improved designs with one cross aisle and P&D points at the bottom
and the top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.10 Comparison of Design C1 and improved designs with one cross aisle and
multiple P&D points at the bottom and the top. . . . . . . . . . . . . . 117
6.11 Improved designs with two cross aisles and multiple P&D points at the
bottom and the top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.12 Comparison of Design C2 and improved designs with two cross aisles and
multiple P&D points at the bottom and the top. . . . . . . . . . . . . . 120
7.1 An improved design with a  xed cross aisle and the locations of P&D
points are marked based on the closeness to the center. . . . . . . . . . . 127
7.2 Costs of P&D points at di erent location in di erent designs developed
for a speci c number of P&D points. . . . . . . . . . . . . . . . . . . . . 128
7.3 Expected travel distance and marginal cost of an inserted P&D point at
locations in improved designs. . . . . . . . . . . . . . . . . . . . . . . . . 128
B.1 The de ned main cases and their transformations for two cross aisle model.145
xv
List of Tables
4.1 The relative expected travel distance comparison among equivalent ware-
house designs based on the continuous model. . . . . . . . . . . . . . . . 47
5.1 Upper and lower limits of start and end points of cross aisles in the two
cross aisle model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Performance of Trad-A when P&D points are placed at the 1=3 and 2=3
point of the bottom side of the warehouse. . . . . . . . . . . . . . . . . . 77
5.3 Performance of Design A1 for di erent warehouse sizes. . . . . . . . . . . 79
5.4 Performance of Design A2 for di erent warehouse sizes. . . . . . . . . . . 82
5.5 Performance of Trad-B when P&D points are placed at the mid-left and
mid-bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 The lowest heuristic results of one cross aisle Design B for equal sizes of
Trad-B designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Performance of Design B2 for di erent warehouse sizes. . . . . . . . . . . 87
5.8 Performance of Trad-A when P&D points are placed at the mid-bottom
and mid-top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.9 Performance of Design C1 for di erent warehouse sizes. . . . . . . . . . . 90
5.10 Performance of Design C2 for di erent warehouse sizes. . . . . . . . . . . 93
5.11 Performance of Trad-A when P&D points are placed at the lower-left
and upper-right corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.12 Performance of Design D0 for di erent warehouse sizes. . . . . . . . . . . 97
5.13 Performance of Trad-B when P&D points are placed in the middle of
each side of the warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.14 Performance of Design E1 for di erent warehouse sizes. . . . . . . . . . . 99
5.15 Performance of Design E2 for di erent warehouse sizes. . . . . . . . . . . 100
xvi
5.16 Average running time in design problems for di erent warehouse sizes. . 102
A.1 Improvements in expected travel distance over an equivalent traditional
design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.2 Number of storage locations in the proposed designs and in the equivalent
traditional design when the shape ratio is approximately 2:1. \Locations"
refers to two-dimensional space|the number of columns of pallet rack
or stacks of pallets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B.1 Computational results for Design A1 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
B.2 Computational results for Design A1 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.3 Computational results for Design A1 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.4 Performance of the expanded Design A1 for di erent warehouse sizes. . . 148
B.5 Performance of modi ed Design A1 for di erent warehouse sizes. . . . . 148
B.6 Computational results for Design A2 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.7 Computational results for Design A2 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.8 Computational results for Design A2 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.9 Performance of the expanded Design A2 for di erent warehouse sizes. . . 151
B.10 Performance of the expanded, our modi ed Flying-V design for di erent
warehouse sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.11 Computational results for Design B1 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B.12 Computational results for Design B1 in a medium-size warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.13 Computational results for Design B1 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
B.14 Performance of the expanded Design B1 for di erent warehouse sizes. . . 154
xvii
B.15 Computational results for Design B2 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.16 Computational results for Design B2 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.17 Computational results for Design B2 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.18 Performance of the expanded Design B2 for di erent warehouse sizes. . . 157
B.19 Performance of modi ed Design B2 for di erent warehouse sizes. . . . . . 157
B.20 Computational results for Design C1 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.21 Computational results for Design C1 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.22 Computational results for Design C1 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.23 Performance of the expanded Design C1 for di erent warehouse sizes. . . 160
B.24 Performance of modi ed Design C1 for di erent warehouse sizes. . . . . 160
B.25 Computational results for Design C2 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B.26 Computational results for Design C2 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
B.27 Computational results for Design C2 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
B.28 Performance of the expanded Design C2 for di erent warehouse sizes. . . 163
B.29 Performance of modi ed Design C2 for di erent warehouse sizes. . . . . 163
B.30 Computational results of Design D0 for di erent warehouse sizes. . . . . 164
B.31 Performance of the expanded Design D0 for di erent warehouse sizes. . . 164
B.32 Computational results for Design D2 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.33 Computational results for the Design D2 in a medium-sized warehouse
(W=150, H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
xviii
B.34 Computational results for Design D2 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
B.35 Performance of the expanded Design D2 for di erent warehouse sizes. . . 166
B.36 Computational results for Design E1 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.37 Computational results for Design E1 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.38 Computational results for Design E1 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B.39 Computational results for Design E2 in a small-sized warehouse (W=100,
H=50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.40 Computational results for Design E2 in a medium-sized warehouse (W=150,
H=75). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
B.41 Computational results for Design E2 in a large-sized warehouse (W=200,
H=100). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B.42 Performance of the expanded Design E2 for di erent warehouse sizes. . . 170
C.1 The heuristic results calculated by the PSO algorithm for small size ware-
houses with one  oating cross aisle and multiple P&D points at the bot-
tom of the warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
C.2 The expected travel distance in a small-sized traditional and equally sized
and equally capacitated Chevron designs with multiple P&D points at
the bottom of the warehouse. The equally capacitated Chevron has a
width of 105 and height of 52. . . . . . . . . . . . . . . . . . . . . . . . . 171
C.3 The heuristic results calculated by the PSO algorithm for small size ware-
houses with a  xed inserted cross aisle and multiple P&D points at the
bottom of the warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . . 172
C.4 The heuristic results calculated by the PSO algorithm for small size ware-
houses with two inserted cross aisles and multiple P&D points at the
bottom of the warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . . 173
C.5 The expected travel distances in Leaf designs. They provide 1575 storage
locations in a small size, and 1875 storage locations in an expanded small
size warehouse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
C.6 The expected travel distance in the two cross aisle proposed designs for
equally capacitated warehouses with multiple P&D points at the bottom
of the warehouse (W=108, H=54). . . . . . . . . . . . . . . . . . . . . . 175
xix
C.7 The heuristic results calculated by the PSO algorithm for small size ware-
houses with two cross aisles and multiple P&D points located both at
the bottom and the top sides of the warehouse. . . . . . . . . . . . . . . 176
C.8 The expected travel distances in the Design C1 for varying number of
P&D points at the bottom and the top sides of the warehouse. . . . . . . 177
C.9 The heuristic results calculated by the PSO algorithm for small size ware-
houses with two inserted cross aisles and multiple P&D points at the
bottom and the top sides of the warehouse. . . . . . . . . . . . . . . . . 178
C.10 The expected travel distances in the Design C2 for varying number of
P&D points at the bottom and the top sides of the warehouse. They
provide 1539 storage locations in a small size, and 1876 storage locations
in an expanded small size warehouse. . . . . . . . . . . . . . . . . . . . . 179
xx
Chapter 1
Warehousing
1.1 Introduction
In a global economy, supply chain management plays a critical role in maintain-
ing the  ow of goods and services. A well-managed supply chain transfers the right
items to customers at the right time in the right quantity at a low cost. Warehouses
are important links in this chain, and therefore their e cient design and operation
is essential to e ective supply chain processes. In a world where customer satisfac-
tion and quick delivery are two of the most important competitive advantages of
companies, warehouses become even more important. For example, if a company
produces goods in China, then assembles them in the US for sales in Europe, the
company needs multiple warehouses to deliver items quickly and to get the bene t
of economies scale in transportation.
Because of long supplier lead times, many companies serve customers from in-
ventory rather than making to order. Relentless e orts to reduce supply chain and
inventory costs have led to a documented reduction in shipment size and its concomi-
tant increase in material handling cost. According to the Bureau of Transportation
Statistics, smaller sized shipments have increased by around 56% since 1993, based
on the Commodity Flow Survey shipments in 2002 (RITA, 2009). Also, during times
of economic growth or stability, stocks must be processed daily or weekly instead of
monthly or quarterly, because some warehouses o er same-day or next-day shipping
to customers from inventory (Baker, 2004). For example, Amazon o ers same-day
delivery if an eligible item is ordered before 10 a.m. or 1 p.m. depending on the
1
location. Therefore, quick receiving, storing, retrieving and shipping have become
important to respond quickly to customer orders and increase customer satisfaction.
On another front, manufacturers returning to their core competencies or unwill-
ing to use space to store inventory has led to an increase of 3rd party warehousing.
In the future, other small and medium manufacturers as well as big companies are
expected to continue this trend of utilizing 3rd party logistic centers. Many smaller
warehouses are also being replaced by fewer large warehouses so as to get the bene-
 ts of economies of scale. In addition to the trend of using warehousing, a dramatic
increase in imported goods, approximately 140% more in 2006 than a decade ear-
lier, is leading industry to build larger warehouses and distribution centers in the US
(Hudgins, 2006). Hudgins noted that 300,000 square feet or more was the de nition
of a large warehouse almost a decade ago, but today it is upwards of 1 million square
feet. As the size of the warehouse increases, costs related to managing warehouses
or distribution centers, such as operation and maintenance costs, also increase. Of
all operational costs, labor cost, which is caused by traveling to store and retrieve
items, is very signi cant in most warehouses. Additionally, since 3rd party logistics
(3PLs) centers pay their forklift drivers for work-hours and bill their customers for
two handles (receiving/storing and retrieving/shipping) for each pallet, both reduc-
ing required receiving and retrieving time for a pallet and increasing the number of
handles per hour are two of the cornerstones of running a warehouse e ciently and
pro tably (Bartholdi and Hackman, 2008). Therefore, our purpose in this disser-
tation is to explore opportunities for reducing operational costs, in terms of travel
cost, for a widely encountered class of warehouses.
Although the importance of warehouses and travel distances to store and re-
trieve items in warehouses is known, warehouses are still built to look much as they
have been for the past  fty years (Gue and Meller, 2009). In our experience, the
most common layout in these facilities, both implemented in industry and studied
in academia, is comprised of a number of parallel aisles with an orthogonal cross
2
Figure 1.1: Traditional warehouse designs.
aisle to allow more e cient travel between storage locations and pickup and deposit
(P&D) points (Figure 1.1).
In this dissertation, we focus on unit-load warehouses because nearly every
industrial warehouse has a signi cant portion of its available space devoted to pallet
storage, and many warehouses store and handle pallets exclusively. Because pallet
operations are also labor-intensive, much of a worker?s time is spent traveling its
aisles. Therefore, in our research, our main purpose is to reduce traveling time in
aisles to make workers more productive. Our main approach is to design cross and
picking aisles for given locations of P&D points in order to reduce travel distances
for workers. In the next section, we discuss the main problems of the dissertation.
1.2 Problem Statement
Our dissertation considers ways in which aisles can be oriented to facilitate
material  ow between established pickup-and-deposit (P&D) points and storage
locations, in order to minimize expected travel distance to visit a location.
Problem 1. What is the optimal angle of aisles for minimizing expected travel dis-
tance in unit-load warehouses with a centrally located P&D point?
We develop a closed form travel distance function, which we use to search for the
optimal aisle angles by minimizing expected travel distance in a unit-load warehouse
with a centrally located P&D point under randomized storage and single-command
3
operations. We consider both the one and two cross aisle cases. Next, we address
another problem: what percentage of space is wasted in new aisle designs due to
angled aisles, or how much additional space do we need to have in new aisle designs
so as to provide the same number of storage locations as in traditional designs?
In order to analyze the wasted space for angled aisle designs, we develop a more
accurate model in which discrete locations of available pallet positions and the P&D
point are represented in a network.
Problem 2. What is the best arrangement of aisles in unit-load warehouses with
multiple P&D points?
Our discussions with warehouse and distribution center managers triggered our
consideration of several alternative locations for multiple P&D points in unit-load
warehouses. Consequently, we look for the best possible arrangement of aisles in
a warehouse with pre-speci ed P&D points in order to minimize expected travel
distance. In order to model a warehouse with multiple P&D points and angled
aisles, we develop a warehouse network model and a shortest path algorithm, which
 nds the shortest distance between a storage location and a P&D point. We use
a constructive meta-heuristic algorithm, Particle Swarm Optimization (PSO), to
search for aisle angles in the warehouse space in order to  nd the best aisle structure
for given P&D locations.
Problem 3. How robust are non-traditional aisle designs with respect to a varying
number of P&D points?
To answer this question, we  rst focus on two con gurations of P&D points:
(1) P&D points are placed at the bottom, and (2) P&D points are placed both at
the bottom and the top sides of the warehouse. We considered the non-traditional
aisle designs developed for centrally located P&D point(s) in these con gurations as
primary designs, and we show the changes in improvement of the primary designs
as the number of P&D points increases. We then increase the number of P&D
4
points in these con gurations gradually and search for improved designs with one
and two cross aisles. The primary designs are compared to the improved designs
in order to investigate their robustness in terms of changes in aisle structure and
improvement. Hence, we show speci c improved designs and the e ectiveness of
the primary designs when the material  ow is distributed along the side(s) of the
warehouse.
1.3 Organization of the Dissertation
The remainder of this dissertation is organized as follows. In the next chapter,
we discuss the main operations encountered in warehouses and present the underly-
ing reasons behind our model assumptions. In Chapter 3, we review traditional and
non-traditional warehouse designs and their similarities with some natural struc-
tures and material  ow systems. In Chapter 4, we introduce three optimal designs,
the Chevron, the Leaf, and the Butter y, for unit-load warehouses with a single
P&D point. In Chapter 5, we present a warehouse network model and PSO algo-
rithm for  nding the best aisle structures in a unit-load warehouse with pre-located
P&D points. In Chapter 6, we introduce the concept of grouping a number of P&D
points and representing them with a centrally located P&D point. Lastly, we o er
conclusions and plans for future work in Chapter 7.
5
Chapter 2
Warehouse Operations
2.1 Introduction
Manufacturing plants often have a warehouse to store raw material, work-in-
process items or  nished goods. The term distribution center is often used to di er-
entiate a place that is a transshipment node in a supply chain network. De Koster
et al. (2007) used the term warehouse as a place \for storing or bu ering prod-
ucts" and distribution center as a place \for transshipment or cross-docking" in
addition to storage. Rouwenhorst et al. (2000) and Frazelle (2002) used the term
warehouse in a wider sense to address a place handling transshipment as well as
storage. Hence, in a broader sense, we need warehouses: (1) to bu er products for
a certain time to better match supply with customer demand in a shorter time,
(2) to consolidate or accumulate shipments from suppliers into combined shipments
for downstream customers, and (3) to provide value-added activities such as pric-
ing, product customization, labeling or kitting (Bartholdi and Hackman, 2008). In
addition to these reasons, Lambert et al. (1998) mentioned that the advantage of
purchase discounts, transportation economies and supporting just-in-time programs
of suppliers and customers are other reasons why warehouses exist.
The basic operations in warehouses are to receive items from suppliers, stow
them in storage locations, retrieve them to ful ll customers? orders, sort them if
required, and ship the completed orders to customers. Receiving starts with an
arrival truck to a dock. Products usually arrive in larger units, such as pallets, and
sometimes in cases. Even though the warehouse receives pallets, it might need to
break pallets out into separate cartons. After products are unloaded to a pickup
6
and deposit point or input/output point, they are taken by workers to be put away
in storage locations. The storage area might be comprised of two main sections: a
reserve area and a forward area. Whereas stock keep units (SKUs) are stored as
pallets or unit-loads in a reserve area, they are stored in smaller units such as cases
in a forward picking area. Pallets are moved in a single trip from and to the reserve
area. However, in a forward picking area, orderpickers perform multiple picks in a
trip to ful ll customer orders. On the other hand, while many warehouses allocate
an important portion of their space to a reserve storage area, there are also some
warehouses that only handle pallets at a time such as 3PLs centers and import
warehouses. Once picking is completed, customers? orders are gathered together at
the pickup and deposit point, then packed and loaded into trucks to ship them to
customers.
2.2 Receiving and Shipping
The  rst process encountered in a warehouse is receiving, which is initiated by
the noti cation of the arrival of goods. After a truck or an internal transport (in
a manufacturing plan) arrives to the docks, products are unloaded and transported
to a place for inspection or depalletizing before being put away. In our research, we
call this place a \pickup and deposit (P&D) point."
Once an order arrives to the warehouse management system, products are re-
ceived from storage locations and transferred to the P&D points in the shipping area.
After a customer order is sorted and palletized, if necessary, it is loaded to trucks
at the docks for shipping them to the customers. In some warehouses, docks are
assigned for both receiving and shipping operations; for others, they are separate.
Hence, a P&D point in a warehouse could be a place that comprises an inspection
area, a palletizing or shrink wrap machine, or that provides instructions for workers.
In our research, we assume that travel starts and ends at P&D points.
7
The literature on receiving and shipping has mainly focused on truck-to-dock
assignment policies or problems for cross-docking warehouses (see Gu et al., 2007,
for details). In a cross-docking warehouse, arriving products to the warehouse are
sorted and transferred to the shipment area without being put away to storage
locations. Generally, dock doors in the shipping area are speci cally assigned to
customers. Hence, the assignment problem is to specify which incoming trucks and
outgoing trucks are scheduled for which dock doors. Additionally, Gue et al. (2010)
considered multiple dock doors on one side of the warehouse and presented di erent
aisle orientations to reduce travel distance from docks.
2.3 Storage
Storing and picking are the most time consuming activities in most warehouses
(Tompkins et al., 2003). Storage locations can be distinguished as unit-load storage
and less-than unit-load storage. In unit-load storage, large products are stored either
as block stacking or in pallet racks. Pallet racks can be single deep or multi-deep. We
focus on unit-load storage areas in our research, because they are almost ubiquitous
in warehousing. For example, most retail distribution centers include a reserve area
in which items are moved in pallets, and replenishment is done in unit-loads from
reserve areas to forward picking areas. Additionally, unit-load warehousing is very
common in warehouses such as 3rd party transshipment warehouses and import
warehouses.
The main storage policies used to allocate pallets to the storage locations are
randomized, closest-open-location, dedicated, class-based, and full turnover-based.
In a randomized storage policy, each open storage location has an equal probability
of being selected to store pallets. Because tracking which locations are available
for storage requires computerized management, this policy can only work with an
appropriate technology. The randomized storage policy is widely used in industry
because of its simplicity and higher utilization of storage locations (Petersen, 1999;
8
Petersen and Aase, 2004). In the closest-open-location policy, open locations are
tracked by a management system and once a pallet arrives, it is assigned to the
available storage location closest to the P&D point. This method is also widely
used in industry. Hausman et al. (1976), Schwarz et al. (1978), and Graves et al.
(1977) used the randomized storage policy to approximate the closest-open-location
policy.
If storage locations are reserved for speci c products, the policy is called ded-
icated storage. Order pickers know where to store pallets in advance, hence they
become familiar with their locations in time, and this decreases searching time for
locations. Dedicated storage has the disadvantage of low utilization of storage loca-
tions. Bartholdi and Hackman (2008) note that on average, storage utilization is only
50% in dedicated storage. This policy might be useful when applied to forward pick
areas whereas randomized storage is applied to the reserve area (De Koster et al.,
2007). Class-based storage is a hybrid between randomized and dedicated storage.
In this policy, products are partitioned into classes based on their turnover rate, and
each class occupies some set of dedicated storage locations. Higher turnover items
are assigned to pallet locations closest to the P&D point. Pallets are stored ran-
domly within each class. The advantage of this policy is that fast moving items are
stored close to the P&D point, and space utilization is relatively high because of ran-
domized storage in a class. Last, in full turnover-based storage, the highest-turnover
item is assigned to the location closest to the P&D point. The next higher-turnover
item is assigned to the next closest location and so on. Hence, this policy represents
the limit of class-based storage policy to obtain a bound for the improvement of
potential of class-based storage (Schwarz et al., 1978). One of the most well known
implementations of full turnover-based storage is cube-per-order index (COI). The
COI of a product is the ratio of the product?s total required space to the number
of trips required to satisfy its demand per period. The lower the COI of a product,
9
the closer to the P&D point. The detailed literature about COI and other storage
policies are seen in De Koster et al. (2007) and Gu et al. (2007).
The literature about storage policies that we have reviewed up to this point has
focused exclusively on conventional warehouse designs. However, some recent non-
traditional aisle design studies open a new area in the warehouse literature. Gue
and Meller (2009) presented two non-traditional aisle designs in unit-load ware-
houses under the randomized storage policy. Pohl et al. (2009b) studied one of
the proposed designs by Gue and Meller (2009). They also assumed a randomized
storage policy because of its advantage of high space utilization. Pohl et al. (2010)
also investigated the e ect of turnover-based storage on the travel in non-traditional
unit-load warehouses. They concluded that the non-traditional aisle designs devel-
oped for randomized storage still o er improvement in expected travel distance for
turnover-based storage.
Because new aisle designs require larger warehouses than traditional ware-
houses, we assume the randomized storage policy in our models. Hence, the random-
ized storage policy is expected to reduce the e ect of losing some storage locations
in new aisle designs, because it o ers higher storage utilization than other storage
policies. With this policy, pallets are approximately stored in the closest available
location that leads to the most e cient use of storage space. Also, if there is inad-
equate information about the characteristics of arriving items, which is needed to
manage dedicated and class-based storage policies, a randomized storage policy can
be easily implemented. Hence, a warehouse management system can easily direct
workers to appropriate locations.
2.4 Picking
Picking of items in a warehouse involves traveling to the storage location(s),
extracting items, and taking them for shipping to customers. Because these activi-
ties are time consuming and require a large workforce in many warehouses, picking
10
is one of the major cost components among operational costs (Tompkins et al.,
2003). There are two main picking systems that involve workers in di erent roles in
a warehouse: part-to-picker and picker-to-part systems. In part-to-picker systems,
workers are not involved in the picking actively; instead, automated storage and
retrieval systems (AS/RS) or carousels, are used to extract items from storage loca-
tions. Then, workers handle the remaining operations such as sorting (if necessary),
inspection and transferring to the next station. In picker-to-part systems, workers
are actively involved in picking from traveling and  nding items to extracting and
transferring them to the next station. De Koster (2004) mentioned that the picker-
to-part systems are the most common systems where order pickers spend most of
their time on walking or driving along the aisles. De Koster (2004) also pointed out
that 80% of all order-picking systems in Western Europe, as well as the majority
of the warehouses worldwide, are comprised of picker-to-part systems. Therefore,
we consider picker-to-part systems in our research. See Gu et al. (2007) and Gu
et al. (2010) for detailed literature on picking activities and travel models in AS/RS
systems or carousels.
A picking activity can be done in a single trip, or done in multiple trips. In a
picking environment, as well as a storing environment, single-command operation is
the simplest way to transfer items to and from storage locations. In single-command
operations, one item, usually a pallet, is picked or stored in one trip by an opera-
tor. Therefore, workers travel empty either when they return to a P&D point or
go to a storage location for picking (dead-heading). This operation is common in
unit-load warehousing and unit-load replenishment in a reserve area (van den Berg
et al., 1998). Francis (1967a) and Bassan et al. (1980) are two of the earliest studies
that have presented a single-command travel distance model in conventional ware-
houses. Most studies have mainly focused on modeling expected travel time for
single-command operations in AS/RS systems. For example, Hausman et al. (1976)
modeled single-command travel in unit-load AS/RS for randomized and dedicated
11
storage policies. Goetschalckx and Ratli (1990) also simulated single-command
operations in a unit-load warehouse for shared and dedicated storage policies.
In order to eliminate the dead-heading in single-command operations, some
unit-load warehouses work with dual-command operations. In a dual-command op-
eration, an operator handles two transactions in one trip, stowing and picking. On
the trip from a P&D point to a storage location, an operator performs a deposit
transaction. Then, the operator performs a withdrawal transaction and returns to
the P&D point. Hence, the dual-command operation minimizes the empty travel in
manual picking and dead-head travel of the crane in AS/RS. However, in this opera-
tion, matching storage and retrieval locations might require a level of computerized
management. One of the earliest papers on modeling dual-command travel in con-
ventional warehouses is Malmborg and Krishnakumar (1987). Some other research
has focused on dual-command operations in AS/RS systems. We refer Gu et al.
(2010) for a list of references about single-command and dual-command operations
in unit-load AS/RS.
If a number of goods are retrieved from a number of predetermined storage lo-
cations in one trip, this operation called multi-command operation or order-picking.
Order-pickers visit multiple aisles to pick items given as a list that shows the SKUs,
their required quantities and locations. According to Bartholdi and Hackman (2008),
travel time in aisles is a non-value added activity in order-picking operations, and it
has the  rst priority to be improved. De Koster et al. (2007) discussed that travel
is the most dominant component of order-picking time, even though some other ac-
tivities might have an e ect on it. Hence, minimization of the travel distance in a
tour has been extensively considered in order-picking. Kunder and Gudehus (1975)
and Jarvis and McDowell (1991) developed a model for expected tour length using a
traversal strategy under randomized storage in a single-block warehouse. Jarvis and
McDowell used their model to optimally allocate products to storage locations in
12
symmetric and non-symmetric warehouses. Hall (1993) considered randomized stor-
age policy in a single-block warehouse with an even number of aisles to derive travel
models. He applied several routing algorithms in their models such as traversal, mid-
point, largest-gap and double traversal. Caron et al. (1998) proposed travel models
considering COI-based storage policies in a two-block warehouse. Roodbergen and
Vis (2006) also derived an average travel distance as a function of a number of layout
elements in a similar way to Kunder and Gudehus (1975) and Hall (1993). Ratli 
and Rosenthal (1983) formulated a picking tour as a Traveling Salesman Problem
(TSP) and developed a dynamic programming based optimal algorithm to solve the
TSP problem in polynomial time. Goetschalckx and Ratli (1988a,b), De Koster
and van der Poort (1998) and Roodbergen and de Koster (2001) also proposed an
optimal algorithm to solve the sequence of the pick lists under some varieties of
order-picking policies. For a detailed review of order picking, we refer to De Koster
et al. (2007).
The references that we have mentioned so far focused on conventional warehouse
layouts. Recently, some studies also focused on picking in non-traditional unit-load
warehouses. Gue and Meller (2009) developed a single-command expected travel
distance expression for two non-traditional layouts, the Flying-V and the Fishbone,
in order to optimize some design parameters. Pohl et al. (2009a) developed expected
travel distance equations in traditional designs with a middle cross aisle for dual-
command operations. Pohl et al. (2009b) also modeled a dual-command expected
travel distance expression in Fishbone designs. Gue et al. (2010) derived single-
command expected travel distance expressions from multiple P&D points in two
non-traditional aisle designs, the modi ed Flying-V and the Inverted-V.
In this dissertation, we model our problems for single-command operations,
because it is common in unit-load warehouses and bulk storage areas, as well as
in crossdocking operations. Although dual-command operations reduce dead-head
travel, single-command operations are still appropriate and common in warehouses
13
where required information systems to manage dual-command operations are in-
adequate. Gu et al. (2010) mentioned that single-command operation is the most
common assumption for storage department layout problems in warehouse design.
14
Chapter 3
Designing Warehouses
3.1 Introduction
The adoption of new technologies and management philosophies bring both op-
portunities for warehouses to improve shipment accuracy and to reduce inventory
and lead times. For example, lean warehousing, Just-In-Time, information technolo-
gies such as labeling, bar coding, radio frequency communications, and warehouse
management systems provide opportunities to improve warehouse operations. Also,
operational e ciency in warehouses is strongly a ected by design decisions, although
these can be expensive or impossible to change once the warehouse is built (Gu et al.,
2010). However, Gu et al. also note that most warehouse design research has fo-
cused on analysis instead of synthesis, which might be accomplished by combining,
say, layout problems with operational problems, such as receiving or order picking.
Analyzing a warehouse design is quantifying the e ectiveness of its layout, stor-
age policies, order picking strategies, etc. Designing a warehouse is an act of creation,
in which the task is to select and arrange available or new technologies to accomplish
some operational objective. Gu et al. (2010) divide warehouse design decisions into
 ve main categories: selecting an aisle structure and orientation, a number of aisles,
door locations, storage policies and sizes and dimensions of departments. Most of
the existing research on warehouse design focuses on some variations of the design
in Figure 3.1, in which storage racks are arranged to form straight and parallel pick-
ing aisles, and cross aisles (if present) are perpendicular to the picking aisles. The
variations of this design mainly include one middle cross aisle (see in Figure 3.2).
15
P&D
cross aisle
top cross aisle
bottom cross aisle
pick
ing aisle
storage location (pallet position)
single-deep rack
pick
ing aisle
Figure 3.1: A traditional warehouse layout with design elements.
midlle cr
oss aisle
(a)
midlle cross aisle
(b)
Figure 3.2: Two main variations of traditional warehouses.
Gue and Meller (2009) mentioned two unspoken design rules for these traditional
warehouses:
? All picking aisles must be straight and parallel to each other.
? Cross aisles must be orthogonal and must form a right angle with picking
aisles.
Interestingly, to the best of our knowledge, most warehouses in industry and
also in warehouse literature meet these rules. In traditional warehouses, researchers
used di erent combinations of P&D locations. Kunder and Gudehus (1975) and
Hall (1993) considered a centrally located P&D point to derive their travel models
in manual order-picking. Roodbergen and Vis (2006) considered two di erent P&D
16
locations for di erent order-picking problems with one middle cross aisle, at the
lower-left and in the middle of the bottom cross aisle. Roodbergen and Vis also
showed that the middle P&D point is the best location for multi-command opera-
tions. Caron et al. (1998) also proposed travel models in a warehouse with a middle
cross aisle and a centrally located P&D point. On the other hand, some studies
located a P&D point at the corner of the warehouse instead of placing it centrally.
Chew and Tang (1999), Goetschalckx and Ratli (1990) and Hausman et al. (1976)
located a single P&D point at the lower-left corner of the warehouse. Hsieh and
Tsai (2006) considered two di erent locations of input and output points, and they
are located at the lower-left and the lower-right corner of the warehouse. Hence, the
picker starts traveling from the input point and  nishes it at the output point.
Because Roodbergen and Vis (2006) showed that the middle P&D point is the
best, we  rst consider a centrally located P&D point in our research. Because the
locations of P&D points play an important role on the travel distance, we consider
multiple P&D points at several prede ned locations of di erent sides of the ware-
house. We then search for better aisle designs than traditional designs for unit-load
warehouses for given P&D locations, if present.
Although many studies have focused on a single-block layout without any cross
aisles, some studies have also considered one or multiple middle cross aisles in ware-
houses. Petersen (2002) investigated the e ects of a number of aisles and the aisle
length on the average travel distance of a tour. Guenov and Raeside (1992) investi-
gated the optimum aisle width in an AS/RS system for multi-command operations.
Roodbergen and de Koster (2001) compared the average travel time in traditional
layouts with and without a middle cross aisle, and proved that the warehouse with a
cross aisle has a shorter travel in order-picking. Vaughan and Petersen (1999) inves-
tigated the e ects of cross aisles on order picking e ciency. They inserted a number
of lateral cross aisles into a warehouse that consists of one bottom and one top cross
aisle. They showed that increasing the number of cross aisles reduces the average
17
picking distance in order-picking. In our research, we also investigate the e ects
of additional inserted cross aisles on expected travel distance for single-command
operations, as well as the wasted space by cross aisles in unit-load warehouses.
Up to now, we have reviewed studies on design characteristics in traditional
warehouse layouts. In the next section, we also examine non-traditional warehouse
designs and some of their design characteristics.
3.2 Non-Traditional Aisle Designs
Non-traditional aisle designs propose a di erent aisle structure or aisle orienta-
tion than that in the traditional layouts so that the new designs o er improvement in
travel distance or some other objectives. For example, Moder and Thornton (1965)
introduced a variable, which is \slant angle of the pallets" in order to measure the
 oor space utilization. They developed a mathematical model to evaluate how  oor
space e ciency changes with the angle of placement of the pallets and width of the
aisle. Francis (1967a,b) investigated the shape of optimal warehouse designs with a
single dock. They considered rectilinear travel along \the presupposed orthogonal
network of aisles parallel to the x and y axis", between the dock and the storage
space. Berry (1968) proposed two types of warehouse layouts to investigate space
utilization and traveling cost of handling a unit. The  rst layout assumes that there
are rectangular pallet blocks with the same depth arranged around a main orthog-
onal gangway. The second layout assumes that  oor-stored pallets are arranged in
di erent depths around a \single diagonal gangway providing access to all stacks."
Although these studies seemed to introduce the idea of placement of cross aisles
di erently, we think White (1972) is the  rst study that shows the impact of angled
aisles on travel distance.
White (1972) proposed \radial aisles" in order to reduce travel distance in a
non-rectangular warehouse design. White showed that travel distance from a P&D
point to any point in a storage area gets closer to the Euclidean distance, when a
18
P&D P&D
N = 4 N = 8
Figure 3.3: Non-rectangular warehouses with radial aisles (White, 1972). N is the
number of regions that are de ned by radial cross aisles. Travel in these regions is
along the cross aisles  rst and then parallel to either x or y axis.
number of radial aisles increases (see Figure 3.3). White modeled the warehouse
as a continuous space. Gue and Meller (2009) extended this idea to propose two
non-traditional warehouse designs for unit-load operations using single-command
operations (Figure 4.2). They designed their model in a discrete warehouse space
with continuous picking on aisles. In the Flying-V design, they inserted two nonlin-
ear cross aisle segments into a traditional layout with vertical picking aisles. (Gue
and Meller refer to the Flying-V and Fishbone designs as having one cross aisle.
For clarity of exposition, we describe them as having two cross aisle segments.) The
Fishbone design features picking aisles at di erent angles with two diagonal cross
aisle segments. The authors showed that the Flying-V design o ers about 10%
improvement over the traditional design; the Fishbone o ers about 20%. In the
Fishbone design, Gue and Meller identi ed and (partially) relaxed the conventional
rule of parallel picking aisles, but the Fishbone design had its own design rule that
picking aisles should be horizontal or vertical.
Pohl et al. (2009a) showed that the optimal placement of a \middle" cross
aisle in traditional designs should be slightly aft of the middle. Pohl et al. (2009b)
examined the Fishbone design for dual-command operations (also called task inter-
leaving). They showed that Fishbone designs o er a decrease in expected travel
distance, but require slightly more space compared to several common traditional
designs. They also o ered a modi ed Fishbone design for dual-command operations
19
(a) The Flying-V
 
(b) The Fishbone
Figure 3.4: Proposed designs for a single P&D point by Gue and Meller (2009).
called Fishbone-Triangle (see in Figure 3.5a). In this design, Pohl et al. (2009b)
improved average travel-between distance in Fishbone designs by inserting an ad-
ditional cross aisle segment between two diagonal cross aisles for dual-command
operations. Pohl et al. (2010) analyzed Fishbone, Flying-V and three traditional
designs under turnover-based storage policy and both single- and dual-command
operations. Gue et al. (2010) investigated the expected travel distance from multi-
ple P&D points on the bottom side of the warehouse, and showed that the Flying-V
still has some bene t, but not as much as with a centrally located, single P&D point.
They also developed another non-traditional aisle design called the Inverted-V (see
in Figure 3.5b) to investigate the e ciency of single-command operation from mul-
tiple P&D points; however, this design o ers much less bene t than the Flying-V.
In their designs, they adhered to their own design rules:(1) P&D points are at  xed
locations and placed on one side; (2) picking aisles are straight and parallel to each
other.
In this dissertation, we take these non-traditional aisle design ideas even further
with the goal of developing new aisle designs in unit-load warehouses that are oper-
ated under randomized storage and single-command operations with either a single
P&D point or multiple P&D points. Additionally, we help to identify the magnitude
of the impact of newly developed aisle designs by relaxing design rules such that:
? Picking and cross aisles can take any angle.
20
 (a) The Fishbone-Triangle (b) The Inverted-V
Figure 3.5: Proposed designs for multiple P&D points at the bottom of the ware-
house by Gue et al. (2010).
? Cross aisles can originate from any location along the side of the warehouse.
? P&D points can be placed at any location on the periphery of the warehouse.
We acknowledge our own design constraints, such as rectangular shape of the
warehouse and P&D points on the periphery.
3.3 Material Flow Designs in Nature and Other Structures
Warehouses are not the only places where materials  ow in the world. From
veins in our body to river basins, they all have a main common point,  ow of ma-
terials. While these materials are sometimes tangible goods and products, they are
sometimes liquid or gas. Bejan (1996) discusses the common structures in material
 ow with his \constructal theory." Constructal theory is the view that the genera-
tion of  ow structures in nature is a physics phenomenon, which is described by a
constructal law (Bejan and Lorente, 2008): \For a  nite-size  ow system to persist
in time (to live), its con guration must change in time such that it provides easier
and easier access to its currents ( uid, energy, species, etc.)." Bejan and Lorente
(2008) show how to formulate the  ow structures we see in nature with constructal
theory, such as circulatory systems in the body, lungs, river basins and leaves as well
as mechanical and  uid  ow systems.
21
a
b
c d
Figure 3.6: A continuum of veins in a leaf that is developed and represented by
Runions et al. (2005).
Runions et al. (2005) studied on generating leaf venation patterns in order
to investigate the development of veins corresponding to several leaf patterns and
sources. Runions et al. introduced a biologically-motivated algorithm that relies on
Voronoi diagrams. Using this algorithm, they simulated how vein patterns change
with a given hormone source to the leaf blade. In Figure 3.6, leaves (a), (b) and
(c) show di erent vein patterns generated for di erent kill distances (de ning the
length of sub-veins or branches) according to a single source node at the bottom of
the leaf with a same growth rate. When the growth rate of veins is slow, the leaf
generates a vein pattern seen in (d) in Figure 3.6. The similarities between these leaf
patterns and warehouse designs are the single source node (a P&D point) and kill
distances (di erent aisle lengths). These patterns also show some similar structures
with Chevron aisle designs that we develop in Chapter 4. When the main shape of
the leaf is changed, the model generates two main veins for each side and auxiliary
veins from each of them (see Figure 3.7) . This also looks like Leaf aisle designs we
develop in Chapter 4.
It is not surprising to  nd literature about  ows in street and parking lot designs.
Arlinghaus and Nystuen (1991) separated  ows of cars, pedestrians and bicycles in
an urban street design. Arlinghaus and Nystuen introduced diagonal elements for
travel in consideration of square-block street structures including several obstacles.
Figure 3.8a might be an interesting example for street designs such that it resembles
three cross aisles from a single P&D point as in Butter y aisle designs we develop
22
Figure 3.7: A developed orchid leaf by Runions et al. (2005).
(a) Streets in Paris. (b) A parking lot design.
Figure 3.8: (a) The picture shows the Palis de Chailot and Jardins du Trocadero. It
is taken from the Ei el Tower by \http://www.planetware.com/picture/paris-palais-
de-chaillot-f-f1037.htm." (b) By Rolland et al. (2011).
in Chapter 4. Rolland et al. (2011) discussed how to design the most e cient
parking lot considering tra c circulation or an its overall capacity based on several
design constraints using a market software. The parking lot design they developed
in Figure 3.8b also shows some similar structures with non-traditional aisle designs
such as angled aisles with storage locations.
Airport design is also another  ow system that involves the movement of pas-
sengers and baggages from one location to another location. Robust e (1991) and
Robust e and Daganzo (1991) modeled average walking distance for several airport-
hub designs assuming that each passenger has the same probability to go to a random
gate. Then, Robust e and Daganzo (1991) proposed optimal shapes for airport-hub
designs that minimize total passenger walking distance and baggage travel costs.
23
Bandara and Wirasinghe (1992a,b) focused on determining the con guration of an
airport terminal that minimizes average passenger walking distance. They have
considered three di erent airport con gurations: centralized radial, centralized par-
allel, and semi-centralized parallel. Airport designs generally model average travel
distance either between a hub and a gate (single-command) or between hubs (dual-
command) that show some similarities with travel in warehouses.
As a result, generating the best structures for  ow systems in di erent envi-
ronments re ects some similarities with material  ows in warehouses. Therefore,
a unit-load warehouse is also such a  ow system that acts to facilitate the  ow of
pallets from an open space to a pickup and deposit point via picking and cross aisles.
24
Chapter 4
Optimal Unit-Load Warehouse Designs for Single Command Operations
The contents of this chapter have been accepted for publication in IIE Trans-
actions, with co-authors Kevin R. Gue and Russell D. Meller.
25
Figure 4.1: Traditional rectangular warehouse.
4.1 Introduction
Unit-load warehouses, such as import distribution centers and third-party trans-
shipment warehouses, are widespread in industry and integral to the global economy.
Unit-load operations are also common in many warehouses in the form of bulk stor-
age areas. A dramatic increase in imported goods, approximately 140% more in
2006 than that of a decade earlier, is leading industry to build larger industrial
warehouses and distribution centers (Hudgins, 2006). Total storage space of the
top-20 North American third-party logistics (3PLs) reached 528 million square feet
in 2009 (Rogers, 2009). Hudgins (2006) says that \300;000 square feet was the de -
nition of a large warehouse almost a decade ago, but today it is upwards of 1 million
square feet."
Larger warehouses, of course, impose greater travel distances for the items in-
side, and greater travel distances lead to higher labor costs. Unit-load warehouses
can, for slow-moving items, be rarely visited, but many have several workers actively
storing and picking pallets. The largest operations, such as in import distribution
centers, employ dozens of workers in these spaces. All to say, the labor cost associ-
ated with unit-load warehousing can be signi cant, especially for large retailers and
other importers.
26
In a traditional warehouse design, storage racks are arranged to form parallel
picking aisles, with a cross aisle along the bottom (Figure 4.1). If there are addi-
tional cross aisles in a traditional design to facilitate travel between picking aisles,
they are arranged at right angles to the picking aisles. Cross aisles are appropriate
for order picking operations, in which more than one location is visited per trip,
but are inappropriate for the single-command operations we consider in this paper
(Roodbergen and de Koster, 2001). Due to the aisle structure in the traditional
design, workers travel rectilinear paths to store and receive pallets. White (1972)
proposed \radial aisles" to reduce travel distance in a non-rectangular warehouse
design. He showed that when the number of radial aisles increases, travel distance
from the pickup and deposit (P&D) point to any point in the storage area is close
to the Euclidean distance. His model assumed the warehouse as a continuous space.
Gue and Meller (2009) extended this idea to propose two non-traditional ware-
house designs for unit-load operations using single-command cycles (Figure 4.2). In
the Flying-V design, they inserted two nonlinear cross aisle segments into a tradi-
tional layout with vertical picking aisles. (Gue and Meller refer to the Flying-V and
Fishbone designs as having one cross aisle. For clarity of exposition, we describe
them as having two cross aisle segments.) The Fishbone design features picking
aisles at di erent angles. The authors show that the Flying-V design o ers about
10% improvement over the traditional design; the Fishbone o ers about 20%. In the
Fishbone design, Gue and Meller identi ed and (partially) relaxed the conventional
rule of parallel picking aisles, but the  shbone design had its own design rule: that
picking aisles should be horizontal or vertical.
In this study we make three improvements to the work of Gue and Meller (2009).
First, we allow picking aisles to take on any angle, in an e ort to facilitate more direct
travel than that in the Fishbone design. Second, Gue and Meller restricted their
models to consider two inserted cross aisle segments. We consider the one and three
cross aisle cases as well, and show that there is a continuum of designs appropriate
27
 
Figure 4.2: The Flying-V (Left) and Fishbone (Right).
for increasingly large warehouses. Last, we develop a more precise model of expected
distance by assuming discrete pallet locations. The model allows better estimates
of lost space due to inserted cross aisles. The main contribution of our work is to
propose to the academic and practicing communities optimal warehouse designs for
operations conforming to three main assumptions: uniform picking activity, a single
pickup and deposit point, and single-command cycles.
In the next section we review the literature in non-traditional warehouse de-
signs. In Sections 4.3{4.5 we introduce three optimal, non-traditional warehouse
designs in continuous space, which we call the Chevron, Leaf, and Butter y. Sec-
tion 4.6 presents a comparison of these proposed designs with equivalent traditional
warehouses. In Section 4.7 we develop a discrete model to show how much space
is lost in implementations of these designs. In Section 4.8 we discuss implications
for practice, including a brief description of two warehouses that have implemented
non-traditional designs.
4.2 Related Literature
Moder and Thornton (1965) evaluated how  oor space utilization is a ected
by some dependent and independent variables. One of the independent variables is
the \slant angle of the pallets." They developed a mathematical model to evaluate
how  oor space e ciency changes with the angle of placement of the pallets and
width of the aisle. Francis (1967a,b) investigated the shape of optimal warehouse
28
designs with a single dock considering rectilinear travel along \the presupposed
orthogonal network of aisles parallel to the x and y axis" between the dock and
storage space. Berry (1968) proposed two types of warehouse layouts to investigate
space utilization and the traveling cost of handling a unit. The  rst layout assumes
that there are rectangular pallet blocks with the same depth arranged around a
main orthogonal gangway. The second layout assumes that  oor-stored pallets are
arranged in di erent depths around a \single diagonal gangway providing access to
all stacks."
Pohl et al. (2009a) showed that the optimal placement of a \middle" cross
aisle in traditional designs should be slightly aft of the middle. Pohl et al. (2009b)
examined the Fishbone design for dual-command operations (also called task inter-
leaving). They showed that Fishbone designs o er a decrease in expected travel
distance over several common traditional designs. They evaluated the changes in
storage area of the Fishbone design over equivalent traditional designs and showed
that Fishbone designs require approximately 5% more space. They also o ered a
modi ed Fishbone design for dual-command operations. Pohl et al. (2010) analyzed
three traditional designs, the Fishbone, and the Flying-V designs under turnover-
based storage policies and both single- and dual-command operations. Gue et al.
(2010) investigated the expected travel distance from multiple P&D points on one
side of the warehouse, and showed that the Flying-V still has some bene t, but not
as much as with a centrally-placed, single P&D point.
In addition to these studies, we would like to point out a conceptual relationship
between our work and the \constructal theory" developed by Bejan (1996). Bejan?s
work concerns heat  ow and other mechanical and  uid  ow systems, but the simi-
larities with material  ows in a warehouse are striking (see Bejan and Lorente, 2008,
especially in Chapter 8).
29
P&D w w
h
 R L
 
Figure 4.3: One cross aisle warehouse design.
4.3 One Cross Aisle Warehouse Designs
Suppose we have a rectangular warehouse space with one bottom cross aisle and
an additional inserted cross aisle segment passing through a single, centrally-located
P&D point, as in Figure 4.3. We assume a symmetric design, and therefore the
inserted cross aisle has angle  = 90?. The bottom cross aisle has angle 0?. All travel
in this warehouse is from the single P&D point to any point in the warehouse, either
along the angled cross aisle or bottom cross aisle and then along the appropriate
picking aisle. The question is, what are the angles of the picking aisles on the right
and left side of the angled cross aisle to obtain the lowest expected travel distance
in this design?
Because we assume there is a bottom cross aisle available in all our designs, we
call the design with one inserted cross aisle segment a \one cross aisle" warehouse.
In our models, we assume only one pallet is carried at a time in a single-command
cycle. The P&D point could be a place that has a palletizing or shrink wrap machine,
or that provides instructions for workers.
In our models, we assume a randomized storage policy for the same reasons
described in Gue and Meller (2009). The randomized storage policy is widely used
in industry because of its simplicity and higher utilization of storage locations (Pe-
tersen, 1999). In this policy, the probability of picking or storing any pallet in the
30
warehouse is the same. We assume that picking aisles within the same region are par-
allel to each other, and that cross aisles and picking aisles have zero width. Although
a continuous storage space is not an especially good model of storage locations in
a real warehouse, it is close enough for our purposes, especially when considering
large warehouse spaces typically found in industry. In later sections, we investigate
the e ect of space lost by aisles required in an implementation of our designs.
4.3.1 Model
In this section, we determine the angles of picking aisles that minimize expected
travel distance in a rectangular warehouse. Picking aisles can take on any angle be-
tween 0? and 180?. Parameters and variables for the one cross aisle warehouse model
are shown in Figure 4.3 and described as follows.
Parameters:
P&D single pickup and deposit point placed at (0,0)
 angle of the cross aisle, which is 90?
w half width of the warehouse
h height of the warehouse
(x;y) coordinates of a randomly chosen storage location in the
warehouse space,  w x w and 0 y h
Variables:
 R,  L angles of picking aisles on the right and left sides.
Because a 90? cross aisle naturally divides the warehouse into two equal sides
and we assume symmetry in our designs, we need to model only the right side of
the warehouse.
Proposition 4.1. The optimal  R is less than 90?.
Proof. (See Figure 4.4.) Assume that O represents the single P&D point. Point
W = (w;0) is on the right boundary of the warehouse and point X = (x;y) is a
random storage location in the right half of the warehouse. Let B = (x;0) and
D = (0;y) represent points of access to vertical and horizontal picking aisles, such
31
O
h
 R
 
wA B C
D
E
X
Figure 4.4: Example travel paths based on some possible  R?s.
that 6 OBX = 6 ODX = 90?. Let A = f(z;0) : 0  z < xg, E = f(0;z) : 0  
z < yg and C = f(z;0) : x < z wg be points of access such that 6 WAX < 90?,
(6 OEX  90?) < 90? and 6 WCX > 90?. By the triangle inequality,
jOAj+jAXj<jOBj+jBXj=jODj+jDXj<jOCj+jCXj:
The picking aisles associated with paths OEX and OAX are both less than 90?,
therefore the optimal  R < 90?.
We divide the optimization problem into two cases based on the possible angles
of picking aisles. De ne angle  R = arctan hw, which is the angle de ned by the
bottom aisle and a diagonal line to the upper right corner. There are two cases:
0?   R   R and  R   R < 90?. We include  R in both cases because the
optimal result, which we are about to show, is obtained at the diagonal for square
half-warehouses.
Figure 4.5 illustrates two picking regions constructed by the two cases, along
with example travel paths. The single solid line in each  gure represent the angle of
picking aisles on the right side of the warehouse. We refer to these lines originating
from the P&D point as \median picking aisles." For example, region A is the area
between the bottom cross aisle and the median picking aisle on the right. Travel to
32
P&D w
h
 R
 
B
A
P&D w
h
 R
 
B
A
Figure 4.5: The representation of case 1 (left) and case 2 (right) for the one cross
aisle model.
any point in this region has the same routing rule: travel along the bottom cross
aisle, then travel along a picking aisle with angle  R.
The travel distance functions from the P&D point to a point (x;y) on the right
side are
TA(x;y) = (x ycot R) + ysin 
R
; (4.1)
TB(x;y) = (y xtan R) + xcos 
R
: (4.2)
The  rst terms in (4.1) and (4.2) represent the travel distance along the bottom
cross aisle; the second terms, the distance along the picking aisle. These expressions
are similar to those in White (1972).
Expected travel distance is the ratio of total travel distance to every available
location in a region to its total storage area. So, the expected travel distance to
make a pick on the right side (E[R]), which is the sum of expected travel distance
in regions A and B with the probability of picking in these regions, is:
E[R] = pAE[A] +pBE[B]
= pA 1m
A
Z
x
Z
y
TA(x;y)dydx+pB 1m
B
Z
x
Z
y
TB(x;y)dydx;
33
where mA and mB are the areas of picking regions A and B, respectively. Because we
assume that picking is uniformly distributed throughout the space, the probabilities
of choosing one of two regions are,
pA = mAhw;
pB = mBhw:
Hence, expected travel distances to reach a location on the right side for case 1 and
case 2 are
E[R1] = 1wh
 Z w
0
Z xtan R
0
TA(x;y) dydx+
Z w
0
Z h
xtan R
TB(x;y) dydx
 
= 16h 3h2 +wsec R(3h wtan R) +wtan R( 3h+w +wtan R) ;
E[R2] = 1wh
 Z
h
0
Z w
y=tan R
TA(x;y) dxdy +
Z h
0
Z y=tan R
0
TB(x;y) dxdy
!
= 16w h2 cot2 R +hcot R(h 3w hcsc R) + 3w(w +hcsc R) :
We obtained the results of the integrations with Mathematica. The optimiza-
tion problem is to choose  R such that E[R] = minfE[R1];E[R2]g is a minimum,
which we can solve by taking derivatives and setting them equal to zero.
dE[R1]
d R =
wsec2 R
12h (wsec R ( 3 + cos(2 R)))
+ wsec
2 R
12h (2 ( 3h+w + 3hsin R + 2wtan R)) = 0
dE[R2]
d R =
hcsc R
6w (cot R ( 3w +hcot R))
 hcsc R6w  csc R (h 3w + 2hcot R) +hcsc2  = 0
When h = w for square half-warehouses, the single solution (critical point) to
these equations is the same:  R =  2 arctan(1 p2), which is 45?. The second
partial derivatives are positive at this critical point; therefore, the critical point is
34
P&D w w
h
 R =45? L =135?
 =90?
Figure 4.6: The optimal one cross aisle design.
a local minimum. At the boundaries  R =0? and  R =90?, E[R1] and E[R2] are
greater than the local minimum; therefore, the critical point is the global minimum.
As expected,
Proposition 4.2. The optimal  R = 45? in a square half-warehouse.
Figure 4.6 illustrates the optimal one cross aisle design, which we call the
Chevron.
4.3.2 One Cross Aisle Designs for Non-Square Half-Warehouses
When h6= w, should we expect the optimal angles of picking aisles to equal the
angles of the diagonals? Figure 4.7 shows how the optimal  R (computed numer-
ically, with Mathematica) changes when the ratio of h=w increases. The  gure
shows that the optimal  R increases as the ratio increases, but that it is always less
than or equal to  R, the angle of the diagonal. For warehouses in which w>h, the
optimal  R decreases as w=h increases (the warehouse gets wider), but it remains
above the diagonal.
35
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2 4 6 8 10
h?w
0
20
40
60
80
angle?
Figure 4.7: Change in optimal  R for non-square half-warehouses. Solid lines rep-
resent optimal angles of picking aisles; dashed lines with circle markers represent
angles of the diagonal on the right side.
P&D w w
h
 RC LC
 R L
Figure 4.8: A two cross aisle warehouse.
4.4 A Two Cross Aisle Design
In this section we insert into the warehouse two angled cross aisles, in addition
to the bottom cross aisle. We call this design a two cross aisle warehouse.
We make the same assumptions discussed in the one cross aisle model. There-
fore, the two angled cross aisles are placed on each side, symmetric with respect to
a vertical axis drawn through the P&D point. This axis divides the warehouse into
two equal parts. We use the following variables depicted Figure 4.8.
Variables:
 R,  L angles of the right and left cross aisles
 R,  L angles of picking aisles in the rightmost and leftmost sections
 RC,  LC angles of the central picking aisles on the right and left sides
36
Because the feasible space for angles of picking and cross aisles has many cases
to consider, we prove two propositions to limit the search space. Additionally, it is
enough to search for the optimal  RC, because  RC and  LC are symmetric.
Proposition 4.3. In an optimal two cross aisle design,  LC =  RC = 90?.
Proof. When  RC <90?, we have an infeasible design because some locations in the
central region are unreachable (see Figure 4.9a). Now consider  RC  90? in Fig-
ure 4.9b, and let point P be a random storage location on the right side of the central
region. Point O is the P&D point. Let A, B, and C be points of access to the picking
aisle containing point P corresponding to  RC = 90?, 90? <  RC < 180?,  RC = 180?
respectively. Path OA is common travel for all three paths in Figure 4.9b. From the
triangle inequality,
jAPj<jABj+jBPj<jACj+jCPj:
Therefore  LC =  RC = 90?.
(a)
O w w
h 
R
C 
L
C  R L
A
B
CP
(b)
Figure 4.9: (a) An infeasible symmetric warehouse design with two cross aisles when
 RC < 90?, (b) sample travel paths to the point P with di erent  RC.
Proposition 4.4. In an optimal two cross aisle design, 0?  R  R.
The proof is analogous to the proof of Proposition 4.3.
37
P&D w w
h
 R L
 C  R L
P&D w w
h
 R C L
 R L
P&D w w
h
 R
 C L
 R L
Figure 4.10: Cases in the two cross aisle model.
Because the warehouse design is assumed to be symmetric, the optimization
problem is to minimize expected travel in the right half of the warehouse. There
are three cases determined by possible ranges of  R and  R (see Figure 4.10): (1)
0?  R   R and 0?  R   R, (2)  R   R  90? and 0?  R   R, and (3)
 R  R 90? and  R  R  R.
De ne A, B and C as regions on the right side of the warehouse (see Figure 4.11).
Regions A and B are divided by the median picking aisle with an angle  R. These
regions specify that the travel path to reach any point in the de ned region is the
same. For example, to reach any point in region A, a picker should  rst go along
the bottom cross aisle, then along the appropriate picking aisle. Let TA, TB and TC
be the travel distance functions to random points (x,y) in these regions.
TA(x;y) = (x ycot R) + ysin 
R
; (4.3)
TB(x;y) =
p
x2 +y2
 
cos
 
 R arctan yx
 
 sin
 
 R arctan yx
 
cot( R  R)
 
+
p
x2 +y2
 
sin  R arctan yx 
sin( R  R)
!
;and (4.4)
TC(x;y) = (y xtan R) + xcos 
R
: (4.5)
The  rst terms in these equations are the travel distances along the cross aisle;
the second terms are the distances along the appropriate picking aisles. E[WR1],
E[WR2] and E[WR3] are the expected travel distances on the right half of the
38
P&D w w
h
C
B
A
Figure 4.11: Travel paths according to the de ned areas in two cross aisle design..
warehouse for the speci ed cases.
E[WR1] = 1wh
 Z w
0
Z xtan R
0
TA(x;y) dydx+
Z w
0
Z xtan R
xtan R
TB(x;y) dydx
 
+ 1wh
 Z w
0
Z h
xtan R
TC(x;y) dydx
 
;
E[WR2] = 1wh
 Z
w
0
Z xtan R
0
TA(x;y) dydx+
Z h=tan R
0
Z xtan R
xtan R
TB(x;y) dydx
!
+ 1wh
 Z w
h=tan R
Z h
xtan R
TB(x;y) dydx
 
+ 1wh
 Z
h=tan R
0
Z h
xtan R
TC(x;y) dydx
!
;and
E[WR3] = 1wh
 Z
h
0
Z w
y=tan R
TA(x;y) dxdy +
Z h
0
Z y=tan R
y=tan R
TB(x;y) dxdy
!
+ 1wh
 Z
h=tan R
0
Z h
xtan R
TC(x;y) dydx
!
:
Closed form expressions are in Appendix A.1. The optimization problem is to
 nd  R and  R that minimizes
E[WR] = minfE[WR1];E[WR2];E[WR3]g:
39
Theorem 4.1. An optimal two cross aisle warehouse has  R =  =2 
 
arccos 6+
p6
10
 
and  R =
 
arccos 6+
p6
10
 
.
Proof. Consider case 2  rst. Functions f R and f R are the  rst partial derivatives
of E[WR2] with respect to  R and  R, respectively.
f R( R; R) = @E[WR2]@ 
R
= 0; (4.6)
f R( R; R) = @E[WR2]@ 
R
= 0: (4.7)
The solution to (4.6) and (4.7) (solved with Mathematica) has only one
root within the de ned ranges of variables:  R =  =2 
 
arccos 6+
p6
10
 
and  R = 
arccos 6+
p6
10
 
. To show that this root is a global minimum, it is su cient to show
that it is a local minimum, and that it has a functional value lower than the extreme
points. The critical point is a local minimum if
  
  
 HE[WR2]
 
arccos 6 +
p6
10 ; =2 arccos
6 +p6
10
!  
  
 > 0;
@2E[WR2]
@ 2R
 
arccos 6 +
p6
10 ; =2 arccos
6 +p6
10
!
> 0;and
@2E[WR2]
@ 2R
 
arccos 6 +
p6
10 ; =2 arccos
6 +p6
10
!
> 0:
where H is the Hessian matrix of E[WR2]. We veri ed these inequalities in Math-
ematica. Because the determinant of H is not always greater than or equal to 0
for all possible values of  R and  R in their de ned ranges, E[WR2] is not posi-
tive semi-de nite and not convex. However, we applied the Extreme Value Theo-
rem to check the boundaries of E[WR2]. For this, we solve f R( R; R) = 0 and
f R( R; R) = 0 simultaneously by  xing one of the boundary conditions ( R = 0?,
 R = 45?,  R = 45? and  R = 90?) and solving for the other variable. Based on the
four solutions, we see that the values of E[WR2] at its boundaries are greater than
40
P&D w w
h
 R 32.33? L 147.67?
 C =90?  R 57.67? L 122.33?
Figure 4.12: The Leaf: Optimal two cross aisle design for square half-warehouses.
P&D w w
h
 R L  C
 R
 L
 CR
 CL
B
A
CD
Figure 4.13: Travel paths in regions on the right side of the three cross aisle design.
the local optimum. Similar analysis for case 1 and case 3 also yields local optima
greater than the local optimum for case 2. The single root for case 1 is:  =  =4
and  R = arcsin(p2 1). For case 3,  =  =2 and  R =  =4. Hence, although
E[WR2] is not convex,
 
arccos 6+
p6
10 ; =2 arccos
6+p6
10
 
is the global minimum be-
cause there is an unique root, it is the local optimum, and the extreme values are
larger.
Figure 4.12 illustrates the optimal two cross aisle design, which we call the Leaf
because of its veiny structure and organic appearance.
41
Case 1
P&D w w
h
 R L
 CR CL
 R L
Case 2
P&D w w
h
 R L
 CR CL
 R L
Case 3
P&D w w
h
 R
 CR CL
 L
 R L Case 4
P&D w w
h
 R
 CR CL
 L
 R L
Figure 4.14: Cases in the three cross aisle model.
4.5 Three Cross Aisle Design
Development of the three cross aisle model is as before. There are three inserted
cross aisles, in addition to the bottom cross aisle (see Figure 4.13). Due to the
assumption of symmetry,  C = 90?, and this vertical cross aisle divides the warehouse
into two equal parts. Therefore, we can focus on  nding the optimal angles of aisles
on the right half because the left half is a mirror image.
De ne A, B, C and D as regions on the right side of the warehouse (see Fig-
ure 4.13). Travel in regions A and B is the same as travel to the same regions in two
cross aisle designs (see Figure 4.11). Therefore, travel distance functions in regions
A (TA) and B (TB) are de ned by equations (4.3) and (4.4) given in Section 4.4.
Travel in region D is also the same as travel in region B in a one cross aisle design
(see Figure 4.5), and the travel function in this region (TD) is de ned by equation
(4.2) given in Section 4.3. Therefore, we need only develop a travel distance function
(TC) to a random point (x, y) in region C. The  rst part is the travel distance along
42
the cross aisle; the second part is along the appropriate picking aisle.
TC(x;y) =
p
x2 +y2
 
cos
 
arctan yx  R
 
 sin
 
arctan yx  R
 
cot( R  R)
 
+
p
x2 +y2
 
sin arctan yx  R 
sin( R  R)
!
:
There are four cases determined by possible orientations of the cross and picking
aisles on the right side of the warehouse (see Figure 4.14). According to Proposi-
tion 4.4, these cases are
1. 0?< R  R, 0?< R  R and  R  CR  R,
2. 0?< R  R, 0?< R  R and  R  CR < 90?,
3.  R  R < 90?, 0?< R  R and  R  CR < 90?, and
4.  R  R < 90?,  R < R  R and  R  CR < 90?,
where  R is the angle of the diagonal on the right side of the warehouse. E[WR1],
E[WR2], E[WR3] and E[WR4] are the associated expected travel distances. Closed
form expressions are in Appendix A.3.
E[WR1] = 1wh
 Z w
0
Z xtan R
0
TA(x;y) dydx+
Z w
0
Z xtan R
xtan R
TB(x;y) dydx
 
+ 1wh
 Z w
0
Z xtan CR
xtan R
TC(x;y) dydx
 
+ 1wh
 Z w
0
Z h
xtan CR
TD(x;y) dydx
 
; (4.8)
43
E[WR2] = 1wh
 Z w
0
Z xtan R
0
TA(x;y) dydx+
Z w
0
Z xtan R
xtan R
TB(x;y) dydx
 
+ 1wh
 Z
h=tan CR
0
Z xtan CR
xtan R
TC(x;y) dydx
!
+ 1wh
 Z w
h=tan CR
Z h
xtan R
TC(x;y) dydx
 
+ 1wh
 Z
h=tan CR
0
Z h
xtan CR
TD(x;y) dydx
!
; (4.9)
E[WR3] = 1wh
 Z w
0
Z xtan R
0
TA(x;y) dydx+
Z w
h=tan R
Z h
xtan R
TB(x;y) dydx
 
+ 1wh
 Z
h=tan R
0
Z xtan R
xtan R
TB(x;y) dydx
!
+ 1wh
 Z
h
0
Z y=tan R
y=tan CR
TC(x;y) dxdy
!
+ 1wh
 Z
h
0
Z y=tan CR
0
TD(x;y) dxdy
!
;and (4.10)
E[WR4] = 1wh
 Z
h
0
Z w
y=tan R
TA(x;y) dxdy +
Z h
0
Z y=tan R
y=tan R
TB(x;y) dxdy
!
+ 1wh
 Z
h
0
Z y=tan R
y=tan CR
TC(x;y) dxdy
!
+ 1wh
 Z
h
0
Z y=tan CR
0
TD(x;y) dxdy
!
: (4.11)
Proposition 4.5. An optimal three cross aisle, square half-warehouse has  R =  =4
and  CR = 90?  R.
Proof. Follows from a simple argument of symmetry.
The proposition allows us to focus only on  nding the optimal  R. We derive an
expected travel distance function only on the right side of the warehouse according
to cases 2 and 3, because the optimal design occurs in these cases for square half-
warehouses. E[AB] is the expected travel distance in regions A and B in these cases,
when  R =  =4.
44
E[AB] = 1wh
 Z
w
0
Z xtan R
0
TA(x;y) dydx+
Z w
0
Z xtan =4
xtan R
TB(x;y) dydx
!
= w
3
6
 
tan R + sec R
 
1 +p2 sec R 2 sin
 
 R +  4
 
tan R
  
:
The  rst order condition is
d E[AB]
d R =
w3  2 +p2 +p2 sin R 
6p2 cos2 R = 0:
The solution produces only one root:  R = arcsin(p2 1). The second order con-
ditions show that E[AB] is convex in the range 0  R < 90? (see Proposition 4.1).
d2E[AB]
d 2R =
 w3
12 cos3 R
 
 3 + cos(2 R) + 4( 1 +p2) sin R
 
: (4.12)
That d2E[AB]d 2
R
is strictly positive can be seen by setting cos(2 R) and sin R to
their maximum values of 1, and noting that cos3 R is positive. Therefore, E[AB]
is convex and the solution is a global minimum. In an optimal three cross aisle,
square half-warehouse design:  R =  =4,  C =  =2,  R = arcsin(p2 1) and
 CR =  =2 arcsin(p2 1). We call this design the \Butter y" (Figure 4.15).
4.6 Comparison of Continuous Warehouse Designs
Gue and Meller (2009) give a lower bound on expected travel distance for a
continuous warehouse with a single P&D point. We call this model \Travel by
Flight." In terms of our model parameters, the lower bound for Travel-by-Flight is
E[TF ] = 16wh
 
2hwph2 +w2 +w3 ln
 
h+ph2 +w2
w
!
+h3 ln
 
w +ph2 +w2
h
!!
:
45
P&D w w
h
 R = 45? L = 135?  C = 90?
 R 24:47? L 155:53?
 CR 65:53? CL 114:47?
Figure 4.15: Representation of the Butter y design.
Christo des and Eilon (1969) and Stone (1991) develop similar expressions for Travel
by Flight in square and rectangular areas, respectively.
If travel is rectilinear, as in a traditional warehouse, the expected travel distance
for a warehouse with a single, centrally located P&D point de nes a reasonable upper
bound. In terms of our model parameters, expected value for rectilinear travel TR
is
E[TR] = (w +h)2 :
Table 4.1 shows the relative cost performance of several designs, where the cost
of a traditional design serves as the base 100%. The Chevron and Fishbone designs
o er 19.53% less expected travel distance than an equivalent traditional warehouse.
The Leaf design o ers a 21.72% savings and is only 1.76% worse than the Travel-
by-Flight design. The Butter y design reduces expected travel distance by 22.52%
and is 0.96% worse than Travel-by-Flight. All of these results are for the continuous
space representation of a warehouse.
Observation 1. In an optimal two cross aisle design for a square half-warehouse,
 R + R = 90?.
46
Table 4.1: The relative expected travel distance comparison among equivalent ware-
house designs based on the continuous model.
Warehouse Performance %
Traditional 100
Chevron 80.47
Fishbone 80.47
Leaf 78.28
Butter y 77.48
Travel-by-Flight 76.52
P&D w w
h
A
B
CD
Figure 4.16: Equally distributed imperfections in designs.
The observation follows directly from the optimal values in Theorem 4.1. Un-
fortunately, we are unable to interpret it in terms of warehousing. However, this
observation is likely related to,
Observation 2. In the optimal two cross aisle design for a square half-warehouse,
the expected travel distance in the region below the median picking aisle (E[A] in
Figure 4.16) is the same as expected travel distance in the central region (E[D]).
The expected travel distance in regions above and below the diagonal in the half
space are also the same (E[A] +E[B] = E[C] +E[D]).
See Appendix A.2 for a proof. The observation suggests that optimal solu-
tions exhibit a sort of spatial uniformity of costs, similar in spirit to the \optimal
distribution of imperfection" in Bejan (1996).
47
P&D w w
h
 0
 1 2
 1
 2
 3
P&D w w
h
 1 3
 2
 1 4
 2 3
Figure 4.17: Equal travel distance in dual designs shown via parallelogram.
Observation 3. For any warehouse with an even number k of angled cross aisles
and one bottom cross aisle, there is a \dual" warehouse with k+1 angled cross aisles
having the same expected travel distance, such that the angles of the picking (cross)
aisles in one can be determined directly from the cross (picking) aisles in the other.
The observation can be shown by noting equivalent parallelograms in dual de-
signs (see Figure 4.17) and by an inductive argument. Note also that the Chevron
and Fishbone are dual designs, and that both o er the same savings over traditional
designs. Costs in dual warehouses are not the same in practice, where aisles occupy
space and pallet positions are discrete. We explain why in the next section.
4.7 A Continuum of Designs
With a continuous model of warehouse space, we have shown that two cross
aisles is better than one, and three is better than two. Why not four,  ve, or ten?
The answer, of course, is that the continuous model fails to account for the space
lost to aisles. The greater the number of cross aisles, the more space is lost to
those aisles, and the farther pallet locations are pushed away from the P&D point.
Moreover, because cross aisles meet at the P&D point, increasing their number
eliminates more and more of the best locations, which are those nearest the P&D
point. We have, then, a tradeo : increasing the number of properly placed cross
aisles tends to decrease expected distance to locations, but it also tends to push
48
Figure 4.18: The Chevron design.
those locations farther away. How many cross aisles is best, and for what range of
warehouse sizes?
To investigate these questions, we build a discrete model of aisles and pallet
locations that better represents designs that could be implemented in practice (see
Figures 4.18{4.20). In the discrete model, we assume pallet locations are square,
that picking and cross aisles are the width of three pallets (about 12 feet), and that
there is a bottom cross aisle. We measure travel distance from the P&D point to
the location in a picking aisle from which the pallet can be accessed. The expected
travel distance for a design is the total distance to all pallet locations divided by the
number of locations in that design.
Is it possible to transfer the results of the continuous model directly into dis-
crete space? That is, is a discrete design with angles of cross aisles and picking
aisles taken from the optimal continuous solution also optimal in discrete space? To
assess the robustness of optimal solutions from the continuous space model, we per-
turbed slightly these optimal angles in the discrete design to look for lower expected
distance. In some cases, we were able to improve very slightly on the continuous
result, but the improvements were always less than 0.01%. Part of the discrepancy
is likely due to our method of calculating expected cost in a discrete design, in which
49
Figure 4.19: The Leaf design.
Figure 4.20: The Butter y design.
50
a very slight change of angle could allow a single distant pallet location to  t into
a crevice that previously could not quite contain it (a careful look at Figure 4.20
should make this clear). In any real implementation, of course, there would be many
reasons why neither of our models would be exact: aisle widths may vary because of
vehicle types, pallet locations may not be exactly square, there is clearance between
racks, and so on. Nevertheless, we contend that direct transfer of results from the
continuous model to the discrete model is appropriate, and for our purposes the
resulting discrete designs are \optimal."
Figure 4.21 shows the expected distances for designs having approximately a
2:1 width to depth ratio. (We say \approximately," because the discrete nature
of the designs precludes maintaining an exact ratio.) All designs o er more than
approximately 13% improvement in expected travel distance over equivalent tradi-
tional warehouses of di erent sizes (data for the number of pallet locations in these
warehouses are in Appendix A.4.) Also, as the size of the warehouse increases, the
advantage of inserting angled cross aisles increases, and the percentage of additional
space required for angled aisles decreases.
Among all non-traditional designs, Chevron is the best design for warehouses
with 27 or fewer aisles. As the size of the Chevron increases from 19 aisle-widths to
27 aisle-widths, savings increase from approximately 16.1% to 17.1% and additional
space requirements decrease from 11.2% to 7.3%. Chevron always has less expected
travel distance than the equivalent traditional design, even if the warehouse is small
(see Figure 4.22). Although the Chevron and the Fishbone o er the same bene t
in continuous space, the Chevron has slightly lower expected travel distance in a
discrete design. Nevertheless, the concept of duality does explain why, even in a real
layout, the costs of a Chevron and Fishbone are so close.
For warehouses with more than 27 aisles, the Leaf o ers more savings than the
Chevron, but requires slightly more space due to having two inserted cross aisles.
As the size of the warehouse increases, the additional savings of the Leaf increase
51
# aisles
Percent Improvement in
Expected Travel Distance
Leaf
Butterfly
Chevron
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20 30 40 50 60 70
14
15
16
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19
20
# aisles
Percent Increase in Area
Chevron
Butterfly
Leaf
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20 30 40 50 60 70
5
10
15
20
25
Figure 4.21: Comparison of several approximately square half-warehouses with an
equivalent traditional warehouse. (Data for the plots are in Appendix A.4.)
Figure 4.22: Chevron versus Traditional in an equivalent small size warehouse:
Chevron has lower expected distance, but requires more space.
52
and the additional space requirement decreases compared to the equivalent Chevron
and traditional designs. Even for a huge warehouse with 63 aisles, the Leaf provides
more bene t than the Butter y and requires less space. For a large warehouse with
51 aisles, the Leaf has 19.3% improvement over an equivalent traditional design and
uses 6% more space.
Although the Butter y o ers more improvement than the Leaf in the continuous
space model, in a discrete design it has less savings for most warehouse sizes because
of the wasted space due to more cross aisles. However, the Butter y o ers slightly
more improvement than the Leaf for warehouses larger than 65 aisles.
4.8 Implications for Practice
We have shown that there are new ways to arrange aisles in a unit-load ware-
house such that expected distance to store and retrieve pallets is signi cantly lower.
The goal, of course, is to reduce the labor costs for such operations, but as Figure 4.21
illustrates, reducing expected travel distances comes at a cost of lower storage den-
sity, or equivalently, of larger facilities. This tradeo suggests that  rms facing the
task of designing unit-load storage areas, whether for exclusive pallet-handling op-
erations or for bulk reserve operations in support of order picking, should weigh the
 xed cost of a larger space against the operational savings of lower travel distances.
The details of such problems will vary with the operational situation, of course.
Our models are based on several important assumptions. First, we have as-
sumed uniform picking density, which is roughly equivalent to the randomized stor-
age policy (Schwarz et al., 1978) used in many bulk storage areas. Pohl et al. (2010)
showed that the Fishbone design, which is similar to the designs we describe here,
o ers signi cant bene t even under turnover-based storage, so we expect similar
results would be found for the Chevron, Leaf, and Butter y. A second important
assumption is that travel begins and ends at a single pickup and deposit point,
which is more restrictive. In practice, this assumption would be appropriate when,
53
for example, picks must pass through a single stretch-wrap machine, or when re-
ceived pallets must pass through a processing station (we describe examples below).
Our  nal important assumption is that all travel is for single-command cycles. This
is true for many operations in practice, but even operations capable of executing
dual-commands must execute single commands when there are no stows (picks) in
the queue. Furthermore, as Pohl et al. (2009b) show, designs that reduce single-
command travel also perform well for dual commands because two of the three legs
in a dual command cycle are essentially single command distances.
Our computational results show that Chevron is the best design for warehouses
with fewer than 27 aisles, which, in our experience, is the majority of unit-load
warehouses in practice. The Leaf o ers slightly lower expected travel distances for
larger warehouses, but the penalty cost in terms of space is high. For example, the
Leaf is 0.15% better than Chevron for a 31 aisle-width warehouse, but requires 4%
more space. It is hard to imagine such a small operational bene t overcoming the
 xed cost of such additional space. Even for warehouses equivalent to a traditional
design with 57 aisles (an unrealistic size, in our experience), the Chevron is within
one percent of the bene t o ered by the Leaf. The Butter y o ers no improvement
over the Leaf for warehouses smaller than 65 aisles, and is simply too space-ine cient
to be a viable candidate design. For these reasons, we believe the Chevron design is
the best choice for any realistic scenario that might be encountered in industry.
But will distributors actually use these ideas? There is now evidence that, un-
der the right conditions, they will. At the time of writing, the authors knew of  ve
implementations of non-traditional aisles, each motivated by the results in Gue and
Meller (2009). Generac Power Systems in 2007 implemented a modi ed Fishbone
design at its  nished goods warehouse in Whitewater, Wisconsin (Figure 4.23). The
lower left and right triangles in the design contain  oor stacked pallets and pallet
 ow rack, whereas the central region is single-deep pallet rack. Pallets are taken
by counterbalanced fork truck from receiving doors to a central P&D point, where
54
Figure 4.23: An implementation of non-traditional aisles at Generac Power Systems.
they are transferred to a second, more expensive vehicle capable of vertically lifting
them to their storage locations. A second implementation, in Florida, uses almost
exclusively  oor stacked pallets (goods are extremely heavy, making pallet rack in-
feasible due to  oor loading constraints). All picks are taken to a single stretch-wrap
machine, thus satisfying the single P&D requirement. Unfortunately, for di erent
reasons, neither of these warehouses could establish a before-and-after comparison
of performance, but both have reported satisfaction with the designs.
Of course, there were unanticipated details in the implementations, some good,
some not as good. For example, even though forklift drivers rarely need to make
a 135? turn when coming out of a picking aisle, still they must look both ways
before entering the cross aisle, and that sometimes requires looking back 135?. Semi-
hemispherical mirrors were installed to improve safety at these intersections. On the
other hand, workers at one warehouse reported that they liked the design because
they were \able to take [the 45?] turns at full throttle," a bene t that had not
occurred to us.
Non-traditional aisle designs are not right for every operation. The bene t
for small warehouses, or for those that do not consume much labor, likely would
not overcome the  xed cost of needing a larger storage space. Such warehouses
should use conventional designs, for which storage density is highest. Labor-intensive
operations, however, should consider the long-term bene ts of the Chevron design.
55
Chapter 5
Non-Traditional Unit-Load Warehouse Designs with Multiple Pickup-and-Deposit
Points
5.1 Introduction
In global logistics and transportation systems, warehousing plays a critical role
in assuring high levels of customer service and overall supply chain performance.
Among warehouses in a supply chain network, unit-load receiving and retrieving,
where one item or pallet is carried at a time, often makes up a considerable portion
of warehouse operations, especially in 3rd party distribution centers. According
to Rogers (2009), during times of economic downturn, in addition to increasing
customer and supplier satisfaction, many warehouses are also looking internally for
ways to cut costs. Among operations costs in a warehouse, labor cost is signi cant
because workers in distribution centers spend their time traveling to store or retrieve
items (sometimes traveling empty). Also, because 3rd party distribution centers
pay their forklift drivers for work-hours and bill their customers for two movements
(receiving and retrieving) for each pallet, reducing travel time and increasing the
number of handles per hour are two of the cornerstones of running a warehouse
e ciently and pro tably (Bartholdi and Hackman, 2008).
(a) Trad-A (b) Trad-B (c) Trad-C
Figure 5.1: Traditional rectangular warehouses.
56
In traditional warehouses, forklift drivers travel along the aisles, where cross
aisles are arranged at right angles to parallel picking aisles, to reach a storage location
(see Figure 5.1). Because of the rectilinear travel paths in traditional unit-load
warehouses, inserting an orthogonal cross aisle o ers no advantage in travel distance
to and from a pickup and deposit point when it is placed at the bottom or top
of the warehouse (Roodbergen and de Koster, 2001). However, if there is more
than one P&D location and at least one of them is placed on the right or left side
of the warehouse, a cross aisle is needed to facilitate travel from P&D points in
traditional warehouses (see Figure 5.1b and 5.1c). Additionally, it is not uncommon
that warehouses in industry have multiple entry points to access the storage area.
For example, while one side of the warehouse has receiving docks, the other side
might have shipping docks. Or, receiving and shipping docks might be on the same
side of the warehouse.
Gue et al. (2010) took the idea of non-traditional warehouse designs and investi-
gated unit-load warehouses for multiple P&D points on one side. They inserted two
angled cross aisles that form \modi ed Flying-V" and \Inverted-V" designs, into the
traditional warehouses (Figure 5.2). They showed that the modi ed Flying-V design
can reduce travel by 3-6%, and the Inverted-V o ers less than 1% in savings com-
pared to a traditional warehouse. Additionally, they suggested that if the inserted
P&D points are concentrated toward the center of the bottom of the warehouse, the
bene t of the Flying-V would be increased.
Although Gue et al. (2010) considered multiple P&D points for non-traditional
aisles, the resulting designs still assume that picking aisles should be vertical and
parallel to each other. In Chapter 4, by inserting a number of cross aisles, we
showed that the expected travel distance between a centrally located pickup and
deposit (P&D) point and a storage location can be reduced by 15% to 20%. We
also showed that allowing picking aisles and cross aisles to take any angle provides
additional reduction in expected travel distances in warehouses. In this chapter we
57
(a) The Modi ed Flying-V (b) The Inverted-V
Figure 5.2: The Modi ed Flying-V and the Inverted-V designs for multiple P&D
points. From Gue et al. (2010).
take these ideas further and propose new aisle designs close to the optimal for unit-
load storage spaces that have multiple access locations. Additionally, whereas Gue
et al. (2010) focused on multiple P&D points on only one side of the warehouse,
we consider multiple P&D points both on one side and on di erent sides of the
warehouse to cover more practical cases seen in industry.
In the next section, we review relevant literature in non-traditional warehouse
designs. In Section 5.3, we discuss underlying assumptions of a general warehouse
network model we present in Section 5.4. Then, in Section 5.5, we discuss our
implementation of Particle Swarm Optimization. In Section 5.6, we introduce new
warehouse designs to some speci c cases di erentiated by the locations of P&D
points. In the  nal section, we summarize our  ndings.
5.2 Previous Research
The  rst discussions of non-orthogonal aisle designs came from Moder and
Thornton (1965), Francis (1967a,b), Berry (1968) and White (1972). White (1972)
investigated \Euclidean e ciencies" by inserting radial aisles into a continuous ware-
house space. Then, Gue and Meller (2009) proposed two non-traditional designs,
the Flying-V and the Fishbone. Pohl et al. (2009a) showed that for dual command
operations the optimal placement of a \middle" orthogonal cross aisle in traditional
58
designs is slightly beyond the middle. Pohl et al. (2009b) studied the Fishbone
design for dual-command operations (also called task interleaving). They showed
that the Fishbone design o ers a reduction in expected travel distance compared to
several common traditional designs. They also o ered a modi ed Fishbone design
for dual-command operations. Pohl et al. (2010) analyzed three traditional designs,
as well as Fishbone and Flying-V designs, under turnover-based storage policies and
both single- and dual-command operations.
Some research has been done on multiple depots and their di erent con gura-
tions in a warehouse. Randhawa et al. (1991) evaluated an unit-load AS/RS for two
di erent con gurations of dual-dock layouts on performance such as throughput and
waiting time. Yano et al. (1998) discussed decentralized receiving in a manufactur-
ing assembly facility. They developed an optimization-based procedure to determine
some variables including multiple access points to the building so as to minimize the
cost. Eisenstein (2008) considered dual depots in a discrete order picking system
to minimize required walking distance. In addition to the warehousing literature,
Garces-Perez et al. (1996) examined facility layout problems for pre-speci ed I/O
locations for departments by using Dijkstra?s shortest path algorithm with genetic
programming. Also, Kim and Kim (2000) focused on the facility layout problem
with predetermined shape and input and output points. They placed one input and
one output point on di erent sides of a facility represented by four di erent shapes.
5.3 Assumptions
We assume a randomized storage policy, which is common in unit-load ware-
houses, for the same reasons described in Chapter 4. With this policy, pallets are
stored in the closest available locations, as opposed to reserved locations in a dedi-
cated storage policy. Thus, pallets may be stored at any location in the warehouse
that increases the space utilization. In this policy, the probability of picking or
59
Figure 5.3: The intersecting and non-intersecting cross aisles in a warehouse.
storing any pallet in the warehouse is the same. For easier representation, our stor-
age locations have a square-shaped footprint and one pallet length is the adopted
measure of distance.
From the perspective of warehouse design, we assume that picking aisles, which
are within regions determined by cross aisles, are parallel to each other. Picking
aisles and cross aisles can take on any angle. Contrary to our model in Chapter 4
and the model of White (1972), we take into account aisle width and its e ect on
travel distances in our new model. Each picking and cross aisle is assumed to have
a width of 3 pallet locations. We assume standard, single-deep pallet rack, although
our models could easily be modi ed for double-deep rack or deeper lane storage. The
model allows for no more than two angled cross aisles. Cross aisles are not allowed to
intersect, but they can originate from the same point. This limitation is serious, but
is necessary for computations reasons. The number of cases to consider for the case
of intersecting aisles is much greater than the number of cases we describe below.
To allow for P&D points along the periphery of the warehouse, we assume that
there are no storage racks along the sides of the warehouse. We call these available
spaces on the periphery of the warehouse \side cross aisles" so that they can facilitate
travel between locations. In a warehouse, P&D points could be places which have
palletizing or shrink-wrap machines, or pick lists for workers. All travel is from the
P&D points located on any side of the warehouse to any location in the warehouse
(i.e. there are no zones). We assume only one pallet is carried at a time in a single
60
Warehouse Network 
Model
OptimizerEvaluator
Graph-Based Network
Representation
Warehouse 
Representation
Cross Aisles
P&D Points
Warehouse Size
Picking Aisles
Travel Edges
Travel Nodes
Travel Distance
Calculation
Shortest Path
 Algorithm
Figure 5.4: Description of the warehouse design tool
command cycle, comprised of travel to a random storage location and back to the
P&D point.
Because of the single-command cycle, the same distances are traveled into the
storage space and out of it, which enables us to consider only the half of the actual
distance traveled. Hence, travel distance in our model is the shortest distance from
the P&D point to the location on the picking aisle that accesses the center of a
storage location. Picking aisles can also be traversed to reach a location if the travel
path through that aisle has the shortest distance. The expected travel distance
to pick (store) an item is the sum of the one-way shortest distances to all storage
locations from all P&D points divided by the total number of storage locations and
the number of P&D points in that design. Therefore, we assume that each P&D
point has the same probability of being visited or used.
5.4 The Warehouse Network Model
Our warehouse network model consists of four main modules (Figure 5.4). These
modules are designed as the fundamental classes of our conceptual software.
61
Warehouse representation: In this module, the warehouse as an entity is de ned
the width (W) and length (H) of the space, the width of the cross and picking aisles
(a), the location of P&D points, the number of cross aisles, and the angles of cross
and picking aisles in a suitable coordinate system (see Figure 5.6 for an example
warehouse). P&D points are placed along the side cross aisles that provide direct
travel to the storage area based on the angles of aisles. Based on the aisle width,
cross aisles are de ned by a straight line going through the storage space and whose
start and end points are de ned on di erent sides of warehouse. Single-deep racks
are constructed in a space divided by cross aisles based on the angle and width of
related picking aisles. Picking aisles are automatically arranged in accordance with
valid positions of the pallet racks. Hence, in each pallet rack, storage locations and
their access points on aisles are determined by the described procedure in Figure 5.5.
A sample design is given in Figure 5.6.
Graph-based network representation: We use the warehouse representation to
construct a graph representing the whole warehouse as a network (Figure 5.6). The
nodes of the network include all of the potential P&D points, all intersections of the
aisles, and an access node for each of the storage locations. P&D points are placed on
the center lines of the side cross aisles. The network is constructed with non-negative
edge lengths that represent the distances between the connected nodes. An access
node for a storage location is placed on the center line of the appropriate picking
aisle and to provide access to the center of the corresponding storage locations. The
edge length for two connected neighbor access nodes, representing storage locations,
in the same rack is one pallet (unit) length. Two access nodes on opposite racks,
representing opposite storage locations, are connected and the edge length is zero,
because they are actually served from the same coordinate.
Evaluator: In this module, we calculate the distance between a P&D point
and a node by computing the shortest paths with Dijkstra?s one-to-many algorithm.
Dijkstra?s algorithm searches the graph to  nd the path from the source node (P&D)
62
Warehouse 
Representation
Define warehouse size and the 
aisle width. Locate P&D points.
Specify the number of cross aisles 
and their start and end points.
Specify angles of  picking 
aisles in each region.
Determine regions.
Determine reference points to 
construct the first pallet location 
in each region.
Construct the first pallet location 
in the first rack in each region.
Construct the first pallet locations 
in other possible racks in parallel 
to the first pallet location of the 
preceeding rack. 
Construct other pallet locations 
in each rack according to its first 
pallet location. 
Figure 5.5: The procedure to represent a discrete space warehouse design for given
set of parameters.
W
a
H
Figure 5.6: Network representation of a small, example warehouse. (Each storage
location is represented by an access node in a real network.)
63
to the target node (storage location) with the lowest cost (distance). The worst case
running time for the algorithm on a graph with n nodes and m edges is O(n2)
(Schrijver, 2005).
The expected travel distance to pick an item in a warehouse with k P&D points,
n total storage locations and the shortest distance between ith P&D point and jth
storage location (dij) is
E[C] =
kX
i=1
nX
j=1
dij pi
n =
1
kn
kX
i=1
nX
j=1
dij;
where pi = 1=k is the probability of choosing P&D point, because we assume each
P&D has the same probability to be utilized.
Proposition 5.1. E[C] is the same as two-way expected travel distance if the re-
turning P&D point is selected randomly with equal probability.
Proof. Let I be a set of k P&D points. pi is the probability of choosing the ith P&D
point as a source node. p(rji) is the probability of choosing the returning P&D point
r when the source ith P&D point is known. The expected travel distance to pick an
item in this system is
E[RC] =
kX
i=1
kX
r=1
nX
j=1
1
n
 d
ij +drj
2
 
pip(rji);
where pi = 1=k and p(rji) = 1=k, because returning P&D point might be the same
as the source node. Hence, we can write
E[RC] = 12nkk
nX
j=1
 kX
i=1
kX
r=1
dij +drj
!
= 12nk2
nX
j=1
 kX
i=1
 
kdij +
kX
r=1
drj
!!
= 12nk2
nX
j=1
 kX
i=1
 
kdij +
kX
i=1
dij
!!
;
64
where
kP
i=1
dij =
kP
r=1
drj due to the same probability of visiting P&D points. Finally,
E[RC] = 12nk2
nX
j=1
 
k
kX
i=1
dij +
kX
i=1
kX
i=1
dij
!
= 12nk2
nX
j=1
 
k
kX
i=1
dij +k
kX
i=1
dij
!
= 2k2nk2
kX
i=1
nX
j=1
dij = 1kn
kX
i=1
nX
j=1
dij;
and is equal to E[C].
Therefore, in our model we only calculate one-way expected travel distance to
pick an item from each P&D point.
Optimizer: We use Particle Swarm Optimization algorithm to minimize our
objective function to  nd optimal designs. We discuss the details in the next section.
5.5 Optimization Methodology: Particle Swarm Optimization (PSO)
Because start and end points of cross aisles can be on any side of the warehouse,
there are 12 two cross aisle cases without considering where the P&D points are
located (see in Appendix B.1). Considering possible angles of picking aisles in each
region increases total cases to approximately 192. Even though each case can be
solved, when di erent start and end points of cross aisles and di erent locations of
P&D points are considered, the problem becomes intractable. Additionally, simply
evaluating the cost of a given design requires solving multiple one-to-many shortest
path problems from multiple P&D points, we can not derive a closed form distance
expressions as an objective function need to be minimized. Therefore, we choose a
meta-heuristic solution procedure to search for the optimal values of variables that
65
minimizes the expected travel distance. Speci cally, we use an implementation of
Particle Swarm Optimization (PSO).
PSO, which was  rst introduced to optimize continuous nonlinear functions by
Kennedy and Eberhard (1995), is one of the latest evolutionary algorithms inspired
by nature. Its development was based on observations of social interaction and
communication involved in bird  ocking and  sh schooling. In PSO, each member is
called a particle and each particle searches the multi-dimensional function space with
a speci c velocity. Two cognitive aspects of this initial metaphor, individual learning
and learning from a social group, refer to local and global search, respectively (Banks
et al., 2008). By individual learning, each particle can memorize its best previous
position, called personal best, and move towards it in its restricted neighborhood.
Then, the particles share all the information about their personal best points that
they have received so far. Hence, to accomplish global search, each particle also
moves towards the best particle, called global best, that has the best point among
all particles in the whole swarm. By using personal and global bests, each particle
adjusts direction and the magnitude of its velocity for movement in the space. Hence,
the whole population of particles moves toward the global best as a  ock of birds or
school of  sh. As with other meta-heuristics, PSO is also designed to  nd a unique,
possibly optimum solution.
Advantages of a basic PSO algorithm include its simple structure, ease of imple-
mentation, speed in solving nonlinear, non-di erentiable multi-model problems, and
robustness (Tasgetiren et al., 2007). Additionally, its ability to handle continuous
variables is another advantage for our problem.
5.5.1 Formulation of the problem
We follow three steps to formulate our problem and to search the solution space
quickly. The  rst step is \encoding," which in our case is a simple vector of real
numbers representing a warehouse. It can be easily translated back into a warehouse
66
design. Such an encoding allows rapid search of the solution space. The elements of
this vector are:
? A linear cross aisle is represented with its start (S) and end (E) points, which
de ne the angle, in a given suitable coordinate system. Hence, a set of cross
aisles can be represented as a list of such pairs.
? Picking aisles in any resulting region can be represented simply by their angle
0    in radians.
To simplify the encoding, start and end points of a cross aisle are represented by
real numbers in a single dimensional coordinate system. The origin is at the upper-
left corner and is de ned as 0 and 4 in order to make a closed loop for the perimeter
a rectangular warehouse. Lower-left, lower-right and upper-right corners are de ned
as 1, 2 and 3, respectively (see also in Figure 5.7). Hence, if the appropriate point of
the cross aisle, S or E, is located on the left side2[0;1), on the bottom side2[1;2),
on the right 2[2;3) and on the top 2[3;4]. When evaluating the cost of a design,
we convert the encoding of the cross aisle to an x;y point in the coordinate system.
Additionally, P&D points are pre-located at their given positions in the coordinate
system, which are along the perimeter of the warehouse rectangle.
To narrow the search space, some inappropriate cases, such as having the start
and end points of a cross aisle on the same side of the warehouse, and duplication
(switching the start and end) are avoided. Hence, for the one cross aisle model,
there are mainly 6 cases representing the possible orientations of the cross aisle
(Figure 5.7). Because cases 3, 4 and 6 are symmetric cases of case 1, we only
consider cases 1, 2 and 5 as subproblems. In these subproblems, S 2 [0;1) and
E2(1;2], S2[0;1) and E2[2;3], S2[1;2] and E2[3;4], respectively.
For the two cross aisle model, there are 34 cases (see in Appendix B.1). Twelve
of them are de ned as main cases and others are symmetric. We divide these 12
main cases into 3 subproblems as described in Figure 5.8 to narrow the search space
67
S
E
E
E
0
1 2
34
1
3
2
S
E
E
45 S
E
O
6
Figure 5.7: Possible arrangements of one cross aisle. (Numbers on the cross aisles
represent cases in the one cross aisle model.)
0
1 2
34
S2
S1
E1, E2
E2, E1
E2
E1
Subproblem 1
S1 S2
E1, E2
E2
E1
E1
E2
Subproblem 2
S1 S2
E1
E2
Subproblem 3
Figure 5.8: Possible directions of cross aisles
and control the movement of the cross aisles. In the two cross aisle model, S1 and
E1 are the start and end points of the  rst cross aisle, as well as S2 and E2 of the
second cross aisle. Because of the assumption of non-intersecting cross aisles, we
de ne upper and lower limits for start and end points of cross aisles (see Table 5.1).
Additionally, because our model assumes at most two cross aisles and we assume
non-intersecting cross aisles, there are at most three regions (Figure 5.9). Angles
 R,  L and  C (if present) represent the angles of picking aisles in the right, left
and central regions. For an example, the encoding for a two cross aisle model is
f L; C; R;S1;E1;S2;E2g.
S1 S2 E1 E2
Subproblem 1 (0,1) (0, S1) (1, E2) (1,4)
Subproblem 2 (3,4) (3, S1) (0, E2) (0,3)
Subproblem 3 (0,1) (2, 3) (1, 2) (3,4)
Table 5.1: Upper and lower limits of start and end points of cross aisles in the two
cross aisle model.
68
 R
 L
(a) One cross aisle design
 R
 L
 C
(b) Two cross aisle design
Figure 5.9: Possible region de nitions and angles of aisles in these region.
The second step is \evaluation," in which we evaluate the performance of a
design represented by a speci c encoding (a vector of design parameters) on expected
travel distance of a pick or stow. For this, we use the graph-based network model
of the warehouse discussed in the previous section, and our objective function is to
minimize the expected travel distance E[C].
The third step is to \search" encodings to  nd the lowest possible E[C] for each
de ned subproblem in the one and two cross aisle models. For this task, we use the
PSO algorithm. Finally, we select the aisle design that has the lowest E[C] among
solutions of the subproblems with respect to the one and two cross aisle models.
5.5.2 PSO Algorithm
In the simple PSO algorithm, each particle represents a solution and moves
toward its previous best position and the global best position found in the popula-
tion so far. After initializing the parameters and generating the initial population
randomly, each particle is evaluated by the  tness function. After evaluation, each
particle is updated with its position, velocity and  tness value. If there is an im-
provement in its  tness value, it updates its personal best. Also, the best particle in
the population is used to update the global best. Next, the velocity of the particle
is updated by using its previous velocity, personal best and the global best to move
the particle to a better place in the search space. In doing these steps repeatedly,
the algorithm searches the space until it is terminated by a stopping criterion.
69
The structure of our PSO algorithm is basically the same as the modi ed PSO
by Shi and Eberhart (1998a,b). Hence, the basic elements of our algorithm are
particles, population, particle velocity, personal best, global best, inertia weights,
coe cients and maximum velocity.
Population and Subproblem: F is the set of s subproblems, F =fI1;I2;:::;Isg.
Is is also the set of i particles at iteration t, Its =fXt1s;Xt2s;:::;Xtisg.
Particles: Xtis denotes the ith particle in the sth subproblem at iteration t. It
is represented by d dimensions: Xtis =fxtis1;xtis2;:::;xtisdg, where xtisd is the position
or the value of ith particle in the sth subproblem for the dth dimension. Dimensions
can be thought of as model variables or parameters.
Velocity of particles: Vtis is the set of d velocities of particle i in the sth subprob-
lem at iteration t, Vtis = fvtis1;vtis2;:::;vtisdg. Each dimension of each particle moves
in the search space with a distance vtisd at each iteration.
Personal best: Pis is the best previous position, which gives the best  tness
value, of the ith particle in the sth subproblem, and Pis = fpis1;pis2;:::;pisdg where
pisd is the best value of dimension d of the ith particle in the sth subproblem so far.
Global best: Gs = fgs1;gs2;:::;gsdg is the best particle among all the particles
in the sth subproblem, where gsd is the best value of dimension d in Gs.
Inertia weights: Shi and Eberhart (1998a,b) introduced an inertia weight to the
algorithm. It is denoted as w and is used to balance global and local searches. It
can be a positive constant or a function of time or iteration. A larger w favors the
global search, while a smaller w favors the local.
Coe cients and maximum velocity: Coe cients c1 and c2 are the weights for
\cognitive" and \social" parts in the velocity update equation given in (5.1). To
control the explosion of velocities and provide stability, there are two other mecha-
nisms: maximum velocity Vmax and a constriction coe cient (K). For the quality of
the search, Shi and Eberhart (1998a) showed the e ect of Vmax that determines the
maximum change one particle can undergo in its positional coordinates during an
70
iteration. Clerc and Kennedy (2002) showed that the following value of K provides
rapid convergence:
K = 2j2   p 2 4 j;
 = c1 +c2 > 4:
Also, r1 and r2 are uniformly distributed random numbers [0,1] that in uence
the movement toward personal or global best. Thus, the particles are updated based
on the equations,
vtisd = K wt 1vt 1isd +c1r1(pisd xt 1isd ) +c2r2(gsd xt 1isd ) (5.1)
xtisd = vtisdxt 1isd: (5.2)
Implementation Details
In accordance with the one and two cross aisle models, we have 3 subproblems
for each model (s = 3), and each subproblem is searched by 7 particles. Note
that there is no interaction or communication among particles in these di erent
subproblems. Picking aisles can take any angle between the lower bound of 0? and
the upper bound of 180?. Because each subproblem is de ned based on the cases
of cross aisles, start and end points of cross aisles in each subproblem can take any
real numbers between their lower and upper bounds. In order to keep the variables
in their boundary, we need to adjust the velocity of particles based on their bounds
in addition to Vmax. Therefore, the applied velocity of the dth dimension for the ith
particle in the sth subproblem (misd) is used to update the positions of the particle in
our algorithm. Generally, misd is calculated based on the issues shown in Figure 5.10
and 5.11, and described as the following
71
mtisd =
8
>>>
>>>
>>>>
>>>
>>>>
><
>>>
>>>
>>>>
>>>
>>>>
>:
  (Ud Ld) (xtisd Ud); if Issue 1 or 5 or 6 occurs
 (Ud xtisd); if Issue 2 occurs
 (Ud Ld) + (Ld xtisd); if Issue 3 or 4 or 8 occurs
  (xtisd Ld); if Issue 7 occurs
 Vmax; otherwise if vtisd < Vmax
Vmax; otherwise if Vmax <vtisd
xtisd = mtisdxt 1isd;
where Ld and Ud are lower and upper bounds of the dth dimension, respectively.
Parameter  is a uniformly distributed random variable between [0,1] that speci es
the magnitude of the movement toward the speci ed bound. Additionally, we set
Vmax equal to one third of the length of the warehouse. This value allows enough
space for exploration, but not so much that particles do not converge (Kennedy,
2007).
For the two cross aisle model, in the encoding, the dimensions corresponding to
the start and end points of cross aisles move according to a rule such that: S1 moves
 rst, S2 second, E2 third and E1 last. Thus, we can maintain their movement within
their limits de ned in Table 5.1 using misd. For example, as soon as S1 moves, if it
moves to a lower value than the current value of S2 in subproblem 1 for two cross
aisle model, then the movement of S2 might be adjusted based on issues 1, 5 or 6.
After moving cross aisles with an applied distance mtisd, cross aisles might get
so close to each other that there is not enough space for having storage locations in
a region between cross aisles because of the aisle width. If this happens, we select
the most appropriate cross aisle that has more available space to be moved than the
other cross aisle. We assign  , which equals Vmax in our algorithm, to the velocity of
72
Issue 1: L
d Ud xt 1isd
vtisdmtisd
Issue 2: L
d Udxt 1isd
vtisdmtisd
Issue 3: L
d Udxt 1isd
vtisd
mtisd
Issue 4: L
d Udxt 1isd
vtisd mtisd
Figure 5.10: Issues show that the variable goes out of its boundary if vtisd is applied,
when 0 <vtisd.
the selected cross aisle?s start and end points and move them away by recalculating
mtisd. Thus, this repair function is used to separate cross aisles from each other so
that we can have reasonable warehouse designs for the two cross aisle model.
We de ne inertia weight as a linear function of the number of iterations: wt =
( t=k)+q, where t is the iteration number, q is the allowable maximum value of wt,
and k determines the minimum value of wt. In our algorithm, we chose q = 1:2 in
order to increase global search at the beginning of the iterations and then reduced
it to around 0.8 at the end, because Shi and Eberhart (1998a) mentioned that PSO
had the best chance of  nding the global optimum when 0:8 <w < 1:2. Clerc and
Kennedy (2002) recommended that c1 + c2 = 4:1, sol we selected c1 = c2 = 2:05.
As a termination criterion, we both use the total number of iterations and number
iterations the global best solution for each subproblem was not improved. Hence,
the algorithm runs for the longer time. In our models, if the solutions cannot be
improved in the last 100 iterations, we terminate the search after 1000 iterations.
Pseudo-code for the PSO algorithm is given in Figure 5.12.
73
Issue 5: L
d Ud xt 1isd
vtisdmtisd
Issue 6: L
d Ud xt 1isd
vtisd mtisd
Issue 7: L
d Udxt 1isd
vtisd mtisd
Issue 8: L
d Udxt 1isd
vtisd mtisd
Figure 5.11: Issues show that the variable goes out of its boundary if vtisd is applied,
when vtisd < 0
5.6 Optimization of Layouts for Unit-Load Warehouses
To validate the PSO algorithm, we applied it to a warehouse with a single,
centrally located P&D point and one cross aisle. The warehouse has a width of 100
and height of 50 units. The model terminated after 1000 iterations. As expected,
the model proposed a design that looks like the Chevron, which is the optimal
one cross aisle design with a centrally located P&D point (see Chapter 4). In the
proposed design, while the cross aisle is vertical and originated from the P&D point,
the angles of the picking aisles are slightly di erent from the optimal angles in the
Chevron. The angle of the picking aisles in the right region is approximately 44.4?,
instead of 45?, as in the Chevron. The expected travel distance to pick an item in
this design is approximately 0.2% greater than in the Chevron. Therefore, there
might be a slightly better design than the proposed design, and it could be obtained
by slightly modifying the aisle angles. This is because a few pallet locations might
either disappear or appear in a discrete warehouse space by a slight change in the
angles of the aisles (see in Figure 5.13.) Hence, we might accept to modify the
74
Initialize xisd randomly between its bounds (Ld;Ud)
Initialize visd randomly within [ Vmax, Vmax]
Convert encoding of start and end points of cross aisles to (x,y) points
Evaluate the cost of design for Xis by Evaluator module
Dof8s2F
Dof8i2Is
Find personal best Pis
Find global best Gs
Calculate visd and misd
If needed, use repair function to separate cross aisles
Convert encoding of start and end points of cross aisles to (x,y) points
Update the particle Xis
Evaluate the cost of design for Xis by Evaluator module
g
g Until termination criterion is met.
Figure 5.12: Pseudo-code for the PSO algorithm
angles of aisles slightly for exploring better design than the proposed design. In the
next sections, we will present several example modi ed designs and discuss their
performance, comparing to the proposed designs.
In this section, we look for the optimal aisle structures for warehouse designs
with di erent locations of P&D points. Figure 5.14 shows the locations of the
P&D points for  ve di erent design problems. These problems are nominal, in the
sense that they are intended to explore optimal designs for di erent  ow patterns,
rather than represent actual industrial problems. Nevertheless, the solutions to
these problems do suggest the potential bene t of alternative aisle designs for actual
industrial problems.
We search for the best aisle angles in these design problems for three di erent
warehouse sizes each having a 2:1 width to depth ratio. The small-sized warehouse is
equivalent to a 19-aisle width traditional design, and it has a width of 100 and height
of 50 pallet units. The medium-sized warehouse is equivalent to a 29-aisle width
traditional design that has a width of 150 and height of 75 units. The larger one is
equivalent to a 39 aisle-width, 200-pallet width and a 100-pallet depth traditional
design. We insert one and two angled cross aisles into warehouses with multiple
75
Figure 5.13: Changing number of available storage locations in a discrete warehouse
space by angle of aisles. The dark squares represent the disappearing pallet locations
because they overlap at the appropriate cross aisle.
P&D points that are de ned in Figure 5.14. After running our algorithm three
times, we select the design that has the lowest expected travel distance.
Although meta-heuristics do not guarantee optimality, they do o er feasible and
good solutions. The designs we propose in the next sections are the best designs we
have found so far for the speci ed design problems with an appropriate number of
cross aisles. Hereafter, we call them \improved" designs. We compare the improved
designs with appropriate traditional designs with respect to reductions in expected
travel distance to pick an item in equally sized warehouses. Because the improved
designs often propose angled aisles and include inserted cross aisles, they provide
less storage capacity than an equivalent traditional design. Therefore, we expand
the warehouse space while preserving the aisle structure and shape ratio so that
they provide approximately the same storage capacity as an equivalent traditional
design. We then compare them with respect to the improvement in expected travel
distance to pick an item and the additional space requirement to provide an equal
capacity over traditional designs.
5.6.1 Design Problem A
In design problem A, we locate two P&D points at the 1=3 and 2=3 point of
the bottom side of the warehouse. For an accurate comparison, we take the design
76
Design Problem A Design Problem B Design Problem C
Design Problem D Design Problem E
Figure 5.14: Design problems di erentiated by the locations of P&D points.
Warehouse E[C] # Pallets
Small (19 aisles) 528.3 1880
Medium (29 aisles) 791.7 4320
Large (39 aisles) 1055.8 7760
Table 5.2: Performance of Trad-A when P&D points are placed at the 1=3 and 2=3
point of the bottom side of the warehouse.
Trad-A as a base in this problem because it is the most compact traditional design,
while it allows direct travel from both of the P&D points to the storage area. The
expected travel distance and the number of storage locations in the Trad-A design
for small-, medium- and large-sized warehouses are given in Table 5.2. In the next
sections, we look for the best aisle designs with one and two inserted cross aisles for
this problem and compare them with the Trad-A.
Design A with One Cross Aisle
The improved aisle design proposed by the heuristic model, with one inserted
cross aisle in design problem A, is shown in Figure 5.15a. We call this \Design A1"
(see Appendix B.2.1 for detailed computational results).
For a medium-sized warehouse, the cross aisle originates from the left P&D
point to slightly below the upper-right corner with an angle ( ) of approximately
33?. The model seems to select one of the P&D points and originates the cross aisle
77
(a) Design A1 (b) Modi ed Design A1
Figure 5.15: Designs with one cross aisle in design problem A.
from that P&D point. The angles of the picking aisles in the right ( R) and left ( L)
regions are 88? and 117?, respectively.  R is slightly angled so that it can reduce the
expected travel distance to the right region from the left P&D point, in two ways:
(1) it presents slightly better travel paths than rectilinear travel, and (2) it reduces
the number of storage locations due to the angled aisles.  R also improves travel
from the right P&D point to some storage locations in the right region; however, it
also slightly worsens for some other locations in the same region. For example, only
travel to the right side of the right P&D point to reach the right storage area has a
slightly better travel than the rectilinear travel. Travel from the left P&D point to
the left region has a better travel path than rectilinear travel because the di erence
in the  L and  is less than 90?. Travel from the right P&D point to the left region
is also slightly improved for most storage locations in that region. For this travel,
the picking aisles in the right region are utilized as cross aisles in order to provide
the shortest paths to the furthest locations from the right P&D point.
In Table 5.3, we present the best solutions that we have obtained in each run
for small-, medium- and large-sized warehouses. For medium-sized warehouses, De-
sign A1 o ers 9.6% reduction in expected travel distance, but it provides a 9.1%
lower number of storage locations than Trad-A (see Figure 5.16b). The lost storage
capacity in equally sized, non-traditional aisle designs is caused by the inserted cross
aisles and the angled aisles. In order to have a fair comparison, we expand the size
78
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 475.9 1617 477.8 1620 477.2 1615
Medium 715.8 3910 715.7 3927 717.1 3927
Large 955.3 7232 954.8 7204 957.2 7240
Table 5.3: Performance of Design A1 for di erent warehouse sizes.
20 25 30 35 40
8
10
12
14
numberofaisles
Percen
tage
EqualSize
Improvement
LossinCapacity
(a)
20 25 30 35 404
6
8
10
12
numberofaisles
Percen
tage
EqualCapacity
Improvement
IncreaseinArea
(b)
Figure 5.16: Comparison of Design A1 and Trad-A.
of Design A1, without changing its aisle structure, so that it has approximately the
same storage capacity as Trad-A. For medium-sized, equally capacitated warehouses,
Design A1 presents 5.6% savings in expected travel distance, but requires a 8.9%
larger space compared to Trad-A (see Figure 5.16b and Table B.4 in the Appendix
for details).
As the warehouse size increases, the new, additional storage locations are placed
at the furthest locations to the P&D points. The distance between the P&D points
and these storage locations increases. Therefore, because the expected travel dis-
tance to newly added storage locations is greater than the expected travel distance
in a regular size warehouse, the improvement decreases.
In Design A1, the picking aisles in the right region are slightly slanted by ap-
proximately 2?, compared to vertical aisles. Because of this and the fact that 90?
aisles provide more compact storage than angled aisles, we modify  R to 90? in order
79
20 25 30 35 40
6
8
10
numberofaisles
Percen
tage
EqualSize
Improvement
LossinCapacity
20 25 30 35 40
5
6
7
8
9
numberofaisles
Percen
tage
EqualCapacity
Improvement
IncreaseinArea
Figure 5.17: Comparison of modi ed Design A1 and Trad-A.
to explore the change in the performance. We call the resulting design \modi ed
Design A1" (see Figure 5.15b).
The expected travel distances and the number of pallets in modi ed Design A1
are given in Table B.5 in the Appendix. For an equal, medium-sized warehouse,
modi ed Design A1 o ers 0.5% less improvement than than Design A1. This is an
expected result because Design A1 is the best design we have found so far for equally
sized warehouses. However, for equally capacitated, medium warehouses, modi ed
Design A1 o ers an additional 0.3% improvement and requires 2.1% less warehouse
space than Design A1 (see Figure 5.17 for the performance of modi ed Design A1).
Design A with Two Cross Aisles
As shown in Chapter 4, improvement in expected travel distance increases with
the number of inserted cross aisles. Therefore, in this section, we insert two angled
cross aisles into a warehouse space that represents the problem design A, and we
search for the best aisle design. The improved aisle design proposed by the heuristic
model for this problem is shown in Figure 5.18. We call this design \Design A2."
Design A2 is an approximately symmetric design with respect to a vertical axis
going through the center of the warehouse. The angles of the picking aisles in the
right ( R) and the central ( C) regions are 53? and 180?, respectively. Because there
80
Figure 5.18: Design A2.
are two P&D points and two inserted cross aisles, the model originates each cross
aisle from each P&D point in order to facilitate travel to the storage locations from
both of the P&D points equally. The cross aisles also intersect at the middle of the
top side of the warehouse so that: (1) they can create a  shbone-like travel in the
central region with an appropriate angle of picking aisles, and (2) they can increase
the right and left storage areas where they present a better travel path than the
rectilinear travel, with appropriate angles of picking aisles. Therefore, travel from
both P&D points is better than rectilinear travel.
Table 5.4 presents the best solutions that we have obtained in each run for
small-, medium- and large-sized warehouses. Based on the best result for a medium-
sized warehouse, Design A2 o ers 14.8% reduction in expected travel distance, but
it supplies a 9.6% lower number of storage locations compared to Trad-A (see Fig-
ure 5.19a). While preserving the same aisle structure in Design A2, we expand its
size with an approximate 2:1 width to depth ratio so that it has the same storage
capacity with the equivalent Trad-A. Thus, Design A2 presents an 11% improvement
in travel, but it requires a 9.6% larger space compared to an equally capacitated,
medium-sized Trad-A design (see Figure 5.19b and Appendix B.2.2 for details).
In comparison to Design A1, as expected, Design A2 increases travel e ciency,
but requires more space, mainly due to the additional cross aisle. For an equally
81
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 448.2 1610 455.4 1586 452.6 1616
Medium 674.4 3905 674.5 3906 676.5 3906
Large 899.8 7209 898.4 7204 904.7 7199
Table 5.4: Performance of Design A2 for di erent warehouse sizes.
20 25 30 35 40
8
10
12
14
numberofaisles
Percen
tage
EqualSize
Improvement
LossinCapacity
(a)
20 25 30 35 40
8
10
12
14
numberofaisles
Percen
tage
EqualCapacity
Improvement
IncreaseinArea
(b)
Figure 5.19: Comparison of Design A2 and Trad-A.
capacitated, medium-sized warehouse, while the modi ed Design A1 o ers a 5.9%
improvement, Design A2 o ers an 11% improvement in expected travel distance. On
the other hand, while modi ed Design A1 requires a 6.8% larger warehouse, Design
A2 requires a 9.6% larger warehouse than an equivalent, medium-sized Trad-A.
Gue et al. (2010) also considered multiple P&D points at the bottom of the
warehouse in a Flying-V design. The Flying-V has two nonlinear cross aisle segments
that are symmetric with respect to a vertical axis passing through the central P&D
point. In order to compare Design A2 with the Flying-V, we slightly modify the
cross aisles in the Flying-V to be linear because our model can only accommodate
linear cross aisles. To  nd the points of intersection on the cross aisles and the sides
of the warehouse, we used a central P&D point and used a manual local search to
 nd the best point. Then, we insert two P&D points at the 1=3 and 2=3 point of the
bottom in order to compare with Design A2 (Figure 5.20). For equally capacitated
82
Figure 5.20: The modi ed Flying-V design with two, inserted linear cross aisles.
20 25 30 35 40
5
10
15
numberofaisles
Percen
tage
EqualCapacity
DesignA2
Modi edFlying-V
Figure 5.21: Comparison of Design A2 and the modi ed Flying-V. (See Ap-
pendix B.10 for detailed results.)
warehouses, Figure 5.21 shows the comparison of Design A2 and the modi ed Flying-
V. Because Flying-V is speci cally developed for a central P&D point, Design A2
dramatically o ers more bene t than the modi ed Flying-V when P&D points are
placed at the 1=3 and 2=3 of the bottom.
5.6.2 Design Problem B
In design problem B, the two P&D locations are placed to the middle of the
bottom and the left sides of the warehouse. For this problem, Trad-A does not allow
direct travel from the left P&D point to the storage area. Therefore, Trad-B and
Trad-C are the alternative designs for our comparison (see Figure 5.1). Because
Trad-C requires slightly more space than Trad-B due to the horizontal cross aisle,
83
Warehouse E[C] # Pallets
Small (19 aisles) 562.5 1880
Medium (29 aisles) 844.2 4320
Large (39 aisles) 1125 7760
Table 5.5: Performance of Trad-B when P&D points are placed at the mid-left and
mid-bottom.
we take design Trad-B as a base for comparisons with the proposed designs in this
problem. The performance of Trad-B for small-, medium- and large-sized warehouses
are given in Table 5.5. In the next sections, we look for optimal designs with one
and two inserted cross aisles for this problem and compare them with Trad-B.
Design B with One Cross Aisle
The improved aisle design proposed by the heuristic model, with one inserted
cross aisle in design problem B, is shown in Figure 5.22. We call this \Design B1"
and describe its angles of aisles in a medium-sized warehouse as the following (see
Appendix B.3.1 for the detailed computational results). In Design B1, the cross
aisle originates from the bottom P&D point and intersects with the top side of the
warehouse with an angle of approximately 123?. The model seems to save some
pallet locations closer to the left P&D point by placing the cross aisle further away
from the upper-left corner. The picking aisles in the right region have a right angle
with the cross aisle. This results in improved travel from the left P&D point to the
right storage area, while the worst travel path from the bottom P&D point to the
right is still rectilinear travel. The angle of the picking aisles in the left region is
approximately 180?. Hence, while the travel from the bottom P&D point to the left
region enjoys  shbone-like travel, the worst travel path from the left P&D point to
the left storage area is still rectilinear travel. As a result, the model tries to balance
the improvement in travel paths from both of the P&D points such that there is
a rectilinear travel path for some storage locations and an improved path for some
other locations. We present the best solutions that we have obtained in each run for
84
Figure 5.22: Design B1.
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 533.8 1714 534.9 1717 531.9 1726
Medium 805.6 4072 803.3 4074 804.5 4068
Large 1067.6 7437 1069.0 7423 1070.7 7454
Table 5.6: The lowest heuristic results of one cross aisle Design B for equal sizes of
Trad-B designs.
small-, medium- and large-sized warehouses in the Table 5.6 (see Appendix B.3.1
for details).
For an equal, medium-sized warehouse, Design B1 o ers a 4.9% improvement in
expected travel distance, but it provides 5.7% fewer storage locations than Trad-B.
When its storage area is expanded in order to provide an equal storage capacity with
Trad-B, the improvement reduces to 1.5% for an equally capacitated, medium-sized
warehouse, and it requires 5.3% additional warehouse space. Figure 5.23 shows the
performance of Design B1 over Trad-B for equally sized and equally capacitated
warehouses. Table B.14 in the Appendix also gives the details for the expanded
Design B1.
Design B with Two Cross Aisles
The improved aisle design (Design B2) proposed by the heuristic model, with
two inserted cross aisles in design problem B, is shown in Figure 5.24a. The angles
85
20 25 30 35 40
4
5
6
7
8
numberofaisles
Percen
tage
EqualSize
Improvement
LossinCapacity
20 25 30 35 40
0
2
4
6
8
numberofaisles
Percen
tage
EqualCapacity
Improvement
IncreaseinArea
Figure 5.23: Comparison of Design B1 and Trad-B.
of the picking aisles in the right ( R) and left ( L) regions are approximately 71?
and 8?, respectively. In Design B2, each cross aisle originates from each of the
P&D points, and both intersect at the upper-right corner. This occurs because of
the similar reasons that we discussed for Design A2: (1) they can create a leaf-
like travel in the right and left regions from the bottom and the left P&D points,
respectively, and (2) increase the right and the left storage areas where travel is
better than the rectilinear travel. The angle of the picking aisles in the central ( C)
region is approximately 136? that forms a right angle with the cross aisle originated
from the bottom P&D point. Thus, (1) while travel from the bottom P&D point
to some portion of the central storage area is improved, travel to the other portion
has a rectilinear travel path, and (2) travel from the left P&D point to the central
region is slightly improved. On the other hand, for travel from the left P&D point
to the right region and travel from the bottom P&D point to the left region, central
picking aisles are utilized as cross aisles in order to provide the shortest paths to the
furthest locations from the P&D points. We might also say that this travel worsens
if  R increases or  L decreases. We present the best solutions that we have obtained
in each run for small-, medium- and large-sized warehouses in Table 5.7.
Design B2 o ers an 11.6% savings in expected travel distance, but provides a
13.2% fewer storage locations over an equal, medium-sized Trad-B design. When
86
(a) Design B2 (b) Modi ed Design B2
Figure 5.24: Designs with two cross aisles in design problem B.
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 492.6 1501 495.3 1501 493.5 1491
Medium 746.9 3725 748.2 3716 746.5 3752
Large 999.6 6957 1001.6 6947 999.9 6945
Table 5.7: Performance of Design B2 for di erent warehouse sizes.
we enlarge the storage area in Design B2 by preserving its aisle structure and shape
ratio, the savings reduce to 6.1% with a cost of 14.4% larger space than an equivalent,
medium-sized Trad-B. Figure 5.25 shows the performance of Design B2 over equally
sized and equally capacitated Trad-B designs.
Figure 5.24b shows the more compact modi ed Design B2. As expected, the
modi ed Design B2 has less improvement in expected travel distance than Design
20 25 30 35 40
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12
14
16
18
20
numberofaisles
Percen
tage
EqualSize
Improvement
LossinCapacity
20 25 30 35 40
5
10
15
20
numberofaisles
Percen
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EqualCapacity
Improvement
IncreaseinArea
Figure 5.25: Comparison of Design B2 and Trad-B.
87
20 25 30 35 40
8
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16
numberofaisles
Percen
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Improvement
LossinCapacity
20 25 30 35 40
6
8
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14
16
numberofaisles
Percen
tage
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Improvement
IncreaseinArea
Figure 5.26: Comparison of modi ed Design B2 and Trad-B.
B2 when their warehouse sizes are equal. However, when their warehouse spaces
are expanded, modi ed Design B2 o ers an additional 0.5% improvement and ne-
cessitates 3.5% less space than the equally capacitated, medium-sized Design B2
(see Table B.19 in the Appendix for details). The modi ed Design B2 o ers a little
more savings in expected travel distance than Design B2 because decreasing ne-
cessity of additional space brings some of the storage locations closer to the P&D
points. The performance of modi ed Design B2 over the equivalent Trad-B is given
in Figure 5.26.
For an equally capacitated, medium-sized warehouse, modi ed Design B2 o ers
approximately a 7% improvement in expected travel distance, while Design B1 o ers
an 1.5% improvement. However, as expected, modi ed Design B2 also increases the
additional space requirement to 11% from 5.3% due to an additional cross aisle.
5.6.3 Design Problem C
One of the most likely encountered P&D locations in industry has P&D loca-
tions at the bottom and the top sides of the warehouse. Therefore, in this section we
locate one P&D point at the center of the bottom and the top sides of the warehouse.
In order to have an accurate comparison with the proposed designs, we take Trad-A
as a base design in this section because of its high storage capacity and allowance
88
Warehouse E[C] # Pallets
Small (19 aisles) 500 1880
Medium (29 aisles) 750 4320
Large (39 aisles) 1000 7760
Table 5.8: Performance of Trad-A when P&D points are placed at the mid-bottom
and mid-top.
of direct travel from both of the P&D points. The performance of Trad-A for this
P&D point con guration in small-, medium- and large-sized warehouses are given in
Table 5.8. In the next sections, we insert one and two cross aisles into a warehouse
space that represents the design problem C consecutively, and we search for optimal
designs for this problem.
Design C with One Cross Aisle
The improved aisle design (Design C1) is shown in Figure 5.27a. The cross
aisle originates from the bottom P&D point through the upper-right corner with an
angle of 45?. The angles of the picking aisles on the right and left sides of the cross
aisle are approximately 165? and 98?, respectively. Because our model calculates
the shortest travel path to the storage locations, there are two potential travel paths
to reach the right storage area from the bottom P&D point: (1) travel along the
angled cross aisle, then an appropriate picking aisle, (2) travel along the bottom
cross aisle with an appropriate picking aisle. Hence, there is an indi erence point
in each picking aisle such that it has the same travel distance to be reached from
both of these travel paths (Gue and Meller, 2009). Even though the  rst travel path
o ers a little improvement for some storage locations in an aisle, the second one
proposes a worse travel than a rectilinear travel path to the other locations in the
same aisle. That is why the angle of aisles in the right region seem not to o er fully
improved travel paths from the bottom P&D point to the storage locations in the
right region. Travel from the top P&D point to the right storage area is also better
than rectilinear travel.
89
(a) Design C1 (b) Modi ed Design C1
Figure 5.27: Designs with one cross aisle in design problem C.
Best Solution 1 Best Solution 2 Best Solution 3
Warehosue E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 464.7 1646 465.1 1646 464.9 1642
Medium 701.5 3979 700.5 3968 701.2 3965
Large 937.2 7281 938.8 7295 935.8 7279
Table 5.9: Performance of Design C1 for di erent warehouse sizes.
We present the best solutions that we have obtained in each run for di erent
warehouse sizes in Table 5.9, and details are given in Appendix B.4.1. Compared to
an equal, medium-sized Trad-A, Design C1 o ers 6.5% reduction in expected travel
distance, but provides 8.2% fewer storage capacity due to the angled aisles. In a
warehouse with equal storage capacity, Design C1 presents 2.9% improvement in
expected travel distance, but requires 8.2% more space than a medium-sized Trad-A
(see Table B.23). The performance measures of Design C1 for equally sized and
equally capacitated warehouses are plotted in Figure 5.28.
For the same reasons discussed in Section 5.6.2, we modi ed the angles of picking
aisles in Design C1 as seen in Figure 5.27a. The rectilinear travel path from the top
P&D point to the right region, diminishes the total bene t of the design because
the cross aisle pushes storage locations in the right region a little further away from
the P&D point. All travel paths from both of the P&D points to the left half
of the warehouse are rectilinear. Hence, we might expect that this design o ers
at most 25% of what Fishbone designs o er because only half of the warehouse
is a Fishbone design for only one P&D point. For an equally capacitated, medium
90
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LossinCapacity
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2
4
6
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Improvement
IncreaseinArea
Figure 5.28: Comparison of Design C1 and Trad-A.
20 25 30 35 40
2
4
6
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Improvement
LossinCapacity
20 25 30 35 40
3
4
5
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EqualCapacity
Improvement
IncreaseinArea
Figure 5.29: Comparison of modi ed Design C1 and Trad-A.
warehouse, modi ed Design C1 o ers an additional 0.3% reduction in expected travel
distance and also requires 4.8% less space than Design C1. Although the additional
improvement is inconsiderable in modi ed Design C1, it provides great savings in
the additional space requirement. The performance of the modi ed Design C1 is
plotted in Figure 5.29, and details are given in Table B.24 in the Appendix.
Design C with Two Cross Aisles
Figure 5.30a shows the improved aisle design (Design C2) proposed by the
heuristic model, with two inserted cross aisles in design problem C. As we would
expect, each cross aisle originates from each of the P&D points. Furthermore, Design
91
(a) Design C2 (b) Modi ed Design C2
Figure 5.30: Designs with two cross aisles in design problem C.
C2 is approximately symmetric to the diagonal which passes through the lower-left
corner to the upper-right corner. We might have expected this, because the locations
of the two P&D points, centrally located on the opposite sides of the warehouse,
raise the idea of equal usage of the two cross aisles and the square-half warehouse
space. Because of symmetry, we now focus on the right half of the warehouse. The
angle of the picking aisles in the right region ( R) is approximately 161?. Travel from
the bottom P&D point to the right storage area is the same as in Design C1 that
we discussed in Section 5.6.3. Additionally, the angle of picking aisles in the right
region seems to facilitate travel from the top P&D point to the right storage area
through the picking aisles in the central region. Also, note that the storage locations
in the right region are the furthest locations to the top P&D point. Hence, while the
model loses a little improvement from the bottom P&D point to its closest locations
in the right region, it actually increases the improvement in travel from the top P&D
point to these locations with an appropriate angle in the central ( C) region. The
angle of picking aisles in the central region is approximately 108?. Hence, travel to
the central region from both of the P&D points is also improved by these angles.
The angle of aisles in the left ( L) region is approximately the same as  R, due to
symmetry in the design. We present the best solutions that we have obtained in
Table 5.10 and Appendix B.4.2.
92
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 423.1 1539 425.3 1539 423.4 1541
Medium 639.1 3790 639.8 3788 639.9 3802
Large 855.1 7034 855.9 7052 857.8 7043
Table 5.10: Performance of Design C2 for di erent warehouse sizes.
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Improvement
IncreaseinArea
Figure 5.31: Comparison of Design C2 and Trad-A.
For a, medium-sized warehouse, Design C2 reduces expected travel distance by
14.8%. However, it provides 12.3% less storage capacity than the equivalent medium-
sized Trad-A design. Design C2 with an equal capacity o ers a 9.8% improvement
in expected travel distance, but requires an 11.7% more space than an equivalent
Trad-A design.
In the modi ed Design C2, we make the angle of picking aisles 180? in the right
and the left regions. We preserve the angle of aisles in the central region as appears in
Design C2. In equal space, the modi ed Design C2 reduces expected travel distance
by 12.3%, but provides a 9.1% fewer storage locations over an equal, medium-sized
Trad-A. An equal capacity design o ers 8.6% savings in expected travel distance,
but requires an 8.9% larger space (Figure 5.32 and Table B.29 in the Appendix).
93
20 25 30 35 40
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LossinCapacity
20 25 30 35 40
8
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Improvement
IncreaseinArea
Figure 5.32: Comparison of modi ed Design C2 and Trad-A.
Warehouse E[C] # Pallets
Small (19 aisles) 750 1880
Medium (29 aisles) 1125 4320
Large (39 aisles) 1500 7760
Table 5.11: Performance of Trad-A when P&D points are placed at the lower-left
and upper-right corners.
5.6.4 Design Problem D
In this section, we search for optimal aisle structures in a warehouse that has one
P&D point at the lower-left and one at the upper-right corner. For this problem, we
take Trad-A as a base for our comparisons with the proposed designs. The expected
travel distances and the number of pallets in the Trad-A with these P&D points are
given in Table 5.11. In the next section, we look for the best aisle designs with one
and two inserted cross aisles for this problem and compare them with the Trad-A.
Designs D with One and Two Cross Aisles
We  rst insert one angled cross aisle into a warehouse that represents the design
problem D. Design D1 is the improved aisle design proposed by the heuristic model
(see Figure 5.33a). In Design D1, the model originates the cross aisle from the
lower-left corner and places it as close as to the the left side of the warehouse. The
angles of the picking aisles in the right and the left regions are approximately 33?. It
94
(a) Design D1 (b) Design D2
Figure 5.33: Designs with one and two cross aisles in design problem D.
seems that the main role of the cross aisle is to reduce the storage capacity, rather
than facilitating travel from P&D points to the storage area. Therefore, a single
cross aisle seems to be redundant in this problem. However, for future comparisons,
the expected travel distance in a medium-sized Design D1 is 921.9 units, and there
are 4048 pallet locations. When the size of Design D1 is expanded, it o ers an
15.4% improvement in expected travel distance, and it requires 6.1% more space
than Trad-A.
We then insert two cross aisles into a warehouse that represents the design
problem D. Figure 5.33b presents the improved aisle design (Design D2) proposed
by the heuristic model (see Appendix B.5 for details). In Design D2, as expected,
each cross aisle originates from each of the P&D points. Design D2 is approximately
symmetric with respect to the diagonal that passes through the lower-left and upper-
right corners. The angles of the picking aisles in the right ( R), central ( C) and
the left ( L) regions are approximately 33?, 32? and 34?. Thus, cross aisles facilitate
travel from P&D points to the storage area.
Design D2 o ers an 18.5% improvement, but it provides a 14.7% fewer storage
locations than an equally sized Trad-A design (see details in Appendix B.5). For
equally capacitated warehouses, Design D2 o ers a 12.2% improvement in expected
travel distance with a cost of 15.2% larger space than Trad-A. (see Table B.35 in the
Appendix). Design D2 presents worse performance than Design D1, with respect to
95
improvement in expected travel distance and additional space requirement. Inter-
estingly, this is the  rst time that a non-traditional aisle design does not increase
the improvement in expected travel distance with an additional cross aisle. Because
of this and Design A1 implies a design without any cross aisles, in the next section,
we look for a better aisle design than Design D1 and Design D2, without inserting
any cross aisles.
Design D without Any Cross Aisles
Figure 5.34 shows the improved aisle design (Design D0) proposed by the heuris-
tic model without inserting any additional cross aisles. The angle of the picking aisles
( ) in this design is approximately 33? (see Appendix B.5 for details). Travel from
the P&D points to the storage locations is facilitated by the side cross aisles with an-
gled picking aisles. Hence, Design D0 enjoys with chevron-like travel paths. We also
can see the whole space in the Design D0 as a half space in the Chevron. Therefore,
the Design D becomes a non-square, half Chevron design design with an additional
P&D point at the upper-right corner. Therefore, as in a non-square half-Chevron
design, the angle of the picking aisles in Design D0 is greater than the angle of
the diagonal that passes through the lower-left and upper-right corners, because the
half-width of the warehouse is greater than its height.
Table 5.12 presents the best solutions that we have obtained for Design D0.
Design D0 o ers a 17.8% reduction in expected travel distance with a 4.6% loss
in the storage capacity compared to a medium-sized Trad-A (see Figure 5.35). For
equally capacitated warehouses, Design D0 o ers a 16.1% improvement, but requires
a 4% larger space than Trad-A (see Figure 5.35 and Table B.31 in the Appendix).
As expected, Design D0 dominates Design D1. Therefore, the best design in design
problem D is Design D0.
96
Figure 5.34: Design D0.
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 615.8 1748 616.1 1751 615.7 1747
Medium 925.2 4122 925.6 4125 925.5 4119
Large 1234.7 7492 1234.9 7494 1234.9 7493
Table 5.12: Performance of Design D0 for di erent warehouse sizes.
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0
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20 25 30 35 40
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Improvement
IncreaseinArea
Figure 5.35: Comparison of Design D0 and Trad-A.
97
Warehouse E[C] # Pallets
Small (19 aisles) 562.5 1880
Medium (29 aisles) 844.2 4320
Large (39 aisles) 1125.0 7760
Table 5.13: Performance of Trad-B when P&D points are placed in the middle of
each side of the warehouse.
5.6.5 Design Problem E
In this section, we locate four P&D points at the periphery of a unit-load,
square-half warehouse. Each of the P&D points are placed in the middle of the
each side of the warehouse. For this arrangement of P&D points, we take design
Trad-B as a base because design Trad-C requires slightly more space than Trad-B,
and Trad-B provides direct travel from the P&D points to the storage area. The
performance of Trad-B for di erent size warehouses with these P&D points is given
in Table 5.13.
Design E with One Cross Aisle
Figure 5.36a shows the aisle design (Design E1) proposed by the heuristic model.
The angles of picking aisles in the bottom and the top region are approximately 85?
and 95?. It seems that these picking aisles are slightly slanted so that the model
reduces the expected travel distance by both slightly improving the travel path and
reducing the storage capacity. Design E1 proposes 0.5% reduction in expected travel
distance with a 9.3% loss in the storage capacity compared to an equal, medium-
sized Trad-B. In order to recover the loss capacity in Design E1, we expand its
storage area. Thus, Design E1 does not o er any savings in expected travel distance
over an equally capacitated Trad-B. This is because the travel e ciency gained by
the angled aisles does not compensate the increase in travel distances due to an
increasing warehouse space. The detailed computational results for Design E1 is
given in the Appendix B.6.1.
98
(a) Design E1 (b) Trad-B
Figure 5.36: Designs with one cross aisle in design problem E.
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 559.1 1623 559.4 1634 559.6 1629
Medium 840.2 3917 840.4 3931 841.9 3906
Large 1121.1 7220 1121.9 7224 1122.6 7208
Table 5.14: Performance of Design E1 for di erent warehouse sizes.
Design E with Two Cross Aisles
An improved design proposed with two cross aisles (Design E2) is Figure 5.37a.
Design E2 is approximately symmetric with respect to the diagonal that passes
through the lower-left and the upper-right corners. The angles of the picking aisles
in the right ( R) and left ( L) regions are approximately 6?. The angle of the central
( C) picking aisles is approximately 138?. Because of the symmetry in the design,
we focus on the travel paths from the bottom and the right P&D points.
Travel from the bottom P&D point to the right storage area is improved by
approximately  shbone-like travels. The worst travel path from the bottom P&D
point to the central region is approximately rectilinear travel because  C forms a
right angle with the cross aisle originated from the bottom P&D point. Travel from
the bottom P&D point to the left region seems to be worse than rectilinear travel.
Travel from the right P&D point to most storage locations in the design has worse
travel than the rectilinear because, (1) it does not utilize the cross aisles, and (2)
 C does not o er an improved transit travel to the left region.
99
(a) Design E2 (b) Modi ed Design E2
Figure 5.37: Designs with two cross aisles in design problem E.
Run 1 Run 2 Run 3
Warehouse E[C] # Pallets E[C] # Pallets E[C] # Pallets
Small 543.7 1554 544.7 1548 545.0 1548
Medium 818.7 3825 819.1 3773 819.7 3790
Large 1089.1 7079 1091.1 7111 1090.5 7121
Table 5.15: Performance of Design E2 for di erent warehouse sizes.
Table 5.15 shows the performance of Design E2 (see Appendix B.6.2 for details).
Design E2 o ers a 3% savings in expected travel distance, but it provides an 11.5%
fewer storage locations than an equal, medium-sized Trad-B. An equal capacity
version of Design E2 does not o er any savings in expected travel distance compared
to an equivalent Trad-B (see Figure 5.38 and Table B.42 in the Appendix for details).
A modi ed Design E2 (Figure 5.37b) increases storage e ciency, but it still does
not o er any improvement in expected travel distance over an equally capacitated
20 25 30 35 40
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LossinCapacity
Improvement
20 25 30 35 40
 5
0
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IncreaseinArea
Improvement
Figure 5.38: Comparison of Design E2 and Trad-B.
100
Trad-B for small- and medium-sized warehouses. However, modi ed Design E2 o ers
0.3% improvement in expected travel distance for large-sized warehouses with a cost
of 9% more space than Trad-B. Therefore, even modi ed design E2 is not a good
design for design problem E. This result shows that it is hard to obtain a better
design than the traditional designs by inserting one and two non-intersecting cross
aisles, when the P&D points are distributed all around the warehouse.
5.7 Computational Complexity
We evaluated the shortest path distance from each P&D point to all storage
locations with Dijkstra?s one-to-many algorithm. According to Schrijver (2005), the
worst case running time for Dijkstra?s algorithm is O(n2), where n is the number of
nodes. As the number of P&D points increases, the running time increases approx-
imately linearly.
We run the model on two computers, both running Mac OS X version 10.6.7.
The  rst has 4GB 1067 MHz RAM and a 2.66 GHz Intel Core 2 Duo processor.
The second has a 2GB 667 MHz RAM and 1.83 GHz Intel Core Duo processor.
The average computational time in three runs for small, medium and large size
warehouses in each design problem is given in Table 5.16. In these runs, the best
objective function values obtained in each solution are relatively close to one other,
even though the angles in these solutions of aisles are slightly di erent in these
solutions.
5.8 Conclusion
We have proposed new aisle designs for di erent  ow patterns in unit-load ware-
houses such that the arrangement of aisles suggests a potential bene t in decreasing
expected travel distance. The purpose is to decrease the travel distance to store and
pick pallets in order to make workers more productive. However, reducing travel
distance by angled aisles causes an additional space requirement in warehouses in
101
Average Running Time (min)
Design Problem Inserted cross aisles Small Medium Large
A 1 1592.1 4751.4 11765.92 1573.6 4589.2 11276.8
B 1 1702.4 4942.5 11878.92 1477.5 4249.3 10635.1
C 1 1659.2 5119.2 12381.52 1507.1 4663.1 11559.6
D 0 1762.3 5303.5 12935.12 1392.3 4413.9 10926.9
E 1 2782.9 8619.4 20624.42 2590.0 7929.7 19775.3
Table 5.16: Average running time in design problems for di erent warehouse sizes.
order to provide an equal storage capacity as in an equivalent traditional design.
Therefore, there is a tradeo between reducing variable operational cost by decreas-
ing expected travel distance, or increasing  xed cost by increasing warehouse size
and number of storage racks. The decision of improving one of these con icting
purposes is based on operational characteristics in a speci c warehouse in practice.
We have considered di erent locations of multiple P&D points in order to rep-
resent possible patterns of material  ows in warehouses. For this representation, we
place a single P&D point on di erent sides of a warehouse. A single P&D point on
one side would be appropriate, when there is a single stretch-wrap machine for all
outgoing pallets, or all incoming pallets need to be processed at a place before they
are stored. We also place the single P&D point at the center of an appropriate side
of the warehouse, because, material  ow is most likely to be dense at the center in
practice, and Gue et al. (2010) showed that concentrated P&D points toward center
presents more bene t than distributed P&D points along one side of the warehouse.
We have proposed improved designs with one and two cross aisles in the de ned
design problems A, B and C. In these problems, the improved designs with two cross
aisles o er considerable amount of savings in expected travel distance compared
to one cross aisle designs. This is because each of the cross aisles in two cross
aisle designs facilitate travel from each of the P&D points. However, in design
102
problem D, the improved design does not require any additional cross aisles because
travel from P&D points is facilitated by Chevron-like travel. Additionally, when we
place one P&D point at each side of the warehouse, we observe that it is hard to
obtain a better design than traditional designs with one or two non-intersecting cross
aisles. Improved designs o er better travel paths from P&D points to most storage
locations; however, occasionally they have travel paths worse than rectilinear for a
small number of locations. The maximum turn angle in such cases is approximately
98?.
Our improved designs are developed under two main assumptions, randomized
storage and single command operation. Randomized storage is widely used in unit-
load warehouses in industry because of its higher utilization of storage locations (Pe-
tersen, 1999; Petersen and Aase, 2004). Therefore, randomized storage is expected
to alleviate the negative impact of angled aisles on storage density. The second
important assumption is that all travel is for single-command operations. This op-
eration is common in unit-load warehousing and unit-load replenishment in a reserve
area (van den Berg et al., 1998). The improved designs in this chapter are expected
to perform well for dual command operations because dual-command travel is com-
prised of two times single-command travel distance and one times travel-between
distance.
Because every warehouse is di erent, the improved designs are not right for
every unit-load warehouses. For example, warehouses where storage density is im-
portant do not utilize angled aisles because of the wasted space. The improved
designs also perform well where material  ow is concentrated at the center of the
sides of warehouses.
103
Chapter 6
Robustness of Non-Traditional Aisle Designs with respect to a Varying Number of
P&D Points
6.1 Introduction
It is not uncommon that warehouses in industry have multiple entry points to
access a storage area. In Chapter 5, we have simpli ed this case, and we mainly
placed centrally located P&D points at di erent sides of the periphery of a ware-
house. In doing so, we assume that incoming and outgoing materials  ow through
the center of a side. In this chapter, we relax this assumption and di use more P&D
points on the sides of a warehouse in order to investigate the change in aisle designs
for non-traditional unit-load warehouses.
In Chapter 5, we reviewed some literature that addressed multiple P&D points.
Here, we also acknowledge some facility layout studies that include the location of
I/O points. Several researchers have studied the optimal placement of I/O locations
by considering either a block or a detailed layout. Kane and Nagi (1997) solved the
placement of I/O locations for a given automatic guided vehicle network. Benson
and Foote (1997) determined both the locations of I/O points and the placement of
aisles, after they solved the location of departments and the general aisle structure
for the layout. Kado et al. (1995) found the optimal locations of I/O points in an
integrated facility layout. Arapoglu et al. (2001) found out the best location of an
I/O station for each department in a given block facility layout in order to minimize
material handling costs. Deb and Bahttacharyya (2005) considered a manufacturing
environment with an open  eld rectangular shape facility and pickup and drop-o 
locations along the periphery.
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To the best of our knowledge, Gue et al. (2010) is the only study that considers
a varying number of P&D points in warehouses. However, they consider P&D points
only at the bottom of the warehouse with a vertical picking aisles. In this chapter,
we also consider P&D points at the bottom and the top by allowing cross aisles to
take any angles. Hence, we seek answers to the following questions:
? How good are previously proposed designs when the number of P&D points
increases?
? How robust are their aisle structures for a varying number of P&D points?
6.2 Model and Assumptions
Industrial warehouses often have many dock doors along an exterior side. For
some operations, a single P&D point is an appropriate assumption because travel
begins and ends at a stretch-wrap or palletizing machine, but for many others this
assumption is inappropriate.
In this chapter, we consider two common di erent con gurations: (1) P&D
points are placed at the bottom, and (2) P&D points are placed both at the bottom
and the top sides of the warehouse. These con gurations represent the majority
of industrial warehouses. We use the following primary designs in our comparisons:
the Chevron, Leaf, Design C1, and Design C2, which were all developed for centrally
located P&D point(s). In our investigation, we will permit the aisle design to change
as the number of P&D point increases and we will compare the performance of these
new designs with the appropriate primary design.
In order to seek answers for our questions, we increase the number of P&D points
per side by placing two additional P&D points incrementally. These additional
P&D points are equally distributed around the central one. The distance between
two consecutive P&D points is the distance between center lines of two consecutive
picking aisles in a traditional design. Hence, in our models, this distance is equal to
105
 ve pallet locations. We also assume an equal  ow from each P&D point. However,
when considering the utilization of dock doors in practice, we expect that the designs
developed for centrally located P&D point(s) would perform even better. This is
because, in practice, all P&D points are not occupied all the time because of the
di erent number of arrival trucks to the warehouse. Thus, central P&D points are
most likely to be utilized compared to the P&D points farthest from the center.
We also have the same assumptions that we discussed in Chapter 4 and Chapter 5,
except for the locations of P&D points.
In our  rst question, we increase the number of P&D points in primary designs
and evaluate the change in improvement in expected travel distance compared to
equivalent traditional designs.
In the second question, we  rst increase the number of P&D points per side in
the de ned P&D con gurations. We then search for the best aisle designs for each
change in number of P&D points inserting one and two cross aisles. We use our
warehouse design tool and the PSO algorithm to search for the best aisle designs.
Hence, we call the best designs we found so far \improved designs." Next, we compare
our primary designs with the improved designs in order to investigate the robustness
of our primary designs with respect to a varying number of P&D points. The
robustness of designs in our problem is assessed both by the visual similarity in aisle
structures, allowing for slightly di erent angles of aisles, and the similarity of the
improvements in expected travel distance between primary and improved designs
with respect to a number of P&D points. For the latter criterion, we chose that 2 {
3% loss in improvement in the primary design over an improved design.
We also consider small-sized, square-half warehouses in our evaluations, because
smaller warehouses require less computational e ort than the larger warehouses due
to the number of storage locations that are needed to be evaluated by Dijkstra?s
shortest path algorithm. (A small-sized warehouse has a width of 100 and height
106
of 50 pallet locations. A small-sized, traditional design is comprised of 19 aisles.)
Small-sized warehouses also have other advantages:
? Small changes in the coordinates of the start and end points of a cross aisle
re ect bigger variances in its angle for smaller warehouses than that of larger
warehouses. Hence, smaller warehouses provide easy track in the change of
aisle structure.
? The di erences in expected travel distance due to variances in angles of picking
aisles are larger for smaller size warehouses compared to that of larger size
warehouses. This is because smaller size warehouses have a lower number of
storage locations than that of larger warehouses.
? Based on our experience with the proposed non-traditional aisle designs, the
orientation of aisles does not change with the size of the warehouse even though
there might be a slight change in the angles of aisles.
As we showed in Chapter 4 and Chapter 5, the improvement in expected travel
distance increases in non-traditional aisle designs as the size of the warehouse in-
creases. Because we consider small size warehouses in this chapter, we expect that
the primary and the improved designs with a varying number of P&D points will
most likely provide more improvements for larger warehouses than that of smaller
warehouses.
6.3 Unit-Load Warehouse Designs with P&D Points at the Bottom of
the Warehouse
In this section, we increase the number of P&D points at the bottom of the
warehouse and search for the best designs. We consider both the one and two cross
aisle cases. We then compare the improved designs with the Chevron and Leaf.
107
6.3.1 One Cross Aisle Designs
We insert an angled cross aisle into a warehouse such that it can take on any
angle and can be oriented from any point on the periphery of the warehouse. We
then increase the number of P&D points from 3 to 19 at the bottom on either side of
the center point, incrementally. We run our model three times to search for the best
aisle structure for each problem. We also use the same parameters that we used in
Chapter 5 for the warehouse network model and the PSO algorithm. The improved
aisle structures that we have obtained so far for the nine design problems with one
cross aisle are shown in Figure 6.1.
Figure 6.1 shows that the model generates Chevron-like designs for warehouses
that have less than or equal to 13 P&D points. In these Chevron-like designs, the
cross aisle is approximately vertical, originates from the central P&D point and
divides the warehouse into approximately two equal spaces. The angles of picking
aisles are approximately symmetric to each other with respect to the vertical axis
going through the central P&D point. For the improved designs with more than
13 P&D points, the model generates a completely di erent aisle structure with one
cross aisle. In this aisle structure, the cross aisle is not vertical anymore; instead,
it originates from a P&D point on the left half of the warehouse and approximately
intersects with the upper-right corner of the warehouse. The picking aisles in the
right region are slightly angled, at approximately 87?. The angle of the picking aisles
on the left is approximately 114?. Figure 6.2 shows the percentage improvement in
expected travel distance of the Chevron and the improved designs for equally sized
warehouses (see Appendix C.1 for details).
Because we assume a vertical cross aisle in the Chevron and the improved
designs generate Chevron-like designs, we insert a  xed vertical cross aisle and search
for the best angles of picking aisles. Figure 6.3a shows that the improved designs
with a  oating and a  xed cross aisle are nearly identical for less than or equal to
13 P&D points. Because of this and the aisle structure changes for warehouses with
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3 P&Ds 5 P&Ds
7 P&Ds 9 P&Ds
11 P&Ds 13 P&Ds
15 P&Ds 17 P&Ds
19 P&Ds
Figure 6.1: One cross aisle designs with a varying number of P&D points at the
bottom.
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0 5 10 15 200
5
10
15
20
numberofP&Dpoints
Impro
vemen
t(%)
EqualSize
Chevron
Improveddesigns
Figure 6.2: Comparison of the Chevron and the improved designs.
0 5 10 15 20
5
10
15
20
numberofP&Dpoints
Impro
vemen
t(%)
EqualSize
Fixedcrossaisle
Floatingcrossaisle
(a)
5 10 15 20
45
50
55
60
65
70
numberofP&Dpoints
Angle
(%)
(b)
Figure 6.3: (a) Comparison of improved designs with a  oating and a  xed cross
aisle. (b) Angles of picking aisles on the right side of improved designs with a  xed
cross aisle.
more than 15 P&D points, hereafter, we use designs with the  xed cross aisle as a
basis for comparison. Figure 6.4 shows the aisle structures in the Chevron and the
improved design with 13 P&D points.
We expand the size of the Chevron and the improved designs so that they can
have the equal storage capacity with a small-sized traditional design. Figure 6.5
shows that the 45? aisles in the Chevron loses their power to facilitate travel, as the
number of P&D points increases at the bottom. However, the Chevron design is
robust with respect to a maximum number of 13 P&D points, because the improved
110
(a) The improved design (b) The Chevron
Figure 6.4: The Chevron and the improved design with 13 P&D points at the
bottom.
0 5 10 15 20
0
5
10
15
numberofP&Dpoints
Impro
vemen
t(%)
EqualCapacity
Chevron
Improveddesigns
Figure 6.5: Comparison of the Chevron and improved designs for equally capacitated
warehouses.
designs maintain Chevron-like aisle structure and the di erence in improvement
between the improved and Chevron design is, at most, 2% for equally capacitated,
small-sized warehouses.
6.3.2 Two Cross Aisle Designs
In this section, we locate a number of P&D points at the bottom of the ware-
house with two cross aisles in order to investigate changes in the Leaf aisle structure.
We assume that cross aisles are symmetric with respect to the vertical axis going
through the central P&D point. However, we allow cross aisles to originate from
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any sides of the warehouse and take on any angle. Hence, we consider the sub-
problems de ned in Chapter 5 for two cross aisle models. Here, we also consider
small-sized, square-half warehouses that are comprised of 19 aisles. We solve nine
design problems, each with a di erent number of P&D points that starts from 3 and
ends at 19 P&D points with an increment of two. These two P&D points are equally
distributed around the central P&D point.
The improved aisle structures for these nine design problems are shown in Fig-
ure 6.6 (see Appendix C.2 for details). The model generates Leaf-like designs for
warehouses with 13 P&D points or fewer. Even though the cross aisles are not orig-
inated from the central P&D points and their angles are di erent from the angles
of the cross aisles in the Leaf design, their orientations are still similar to the Leaf.
In the Leaf design, the angle of picking aisles on the right ( R) must be less than
or equal to the angle of the cross aisle on the right ( R). Hence, the cross aisle is
utilized to reach some storage locations in the right region with a shorter distance
than the rectilinear travel. However,  R is greater than  R in the improved designs
with more than 7 P&D points, in contrast to the Leaf aisle structure. Additionally,
 R increases as the P&D points are placed further away from the center.
For warehouses with more than 13 P&D points, the solutions have approxi-
mately vertical picking aisles in all regions with angled cross aisles originated from
the central P&D point, similar to the Flying-V in Gue et al. (2010). This is a
surprising result, because the \modi ed Flying-V" developed by Gue et al. (2010)
was never claimed to be the best structure; the authors assumed vertical picking
aisles, for example. Our results suggest that the Flying-V is the best design if cross
aisles are not allowed to intersect. Designs with intersecting cross aisles might be
investigated for potential bene ts in travel distance.
Figure 6.7 shows improvements in expected travel distance that the Leaf and the
improved designs o er for equally sized and equally capacitated warehouses. Because
the Leaf is speci cally developed for a single P&D point, it loses its e ciency as P&D
112
3 P&Ds 5 P&Ds
7 P&Ds 9 P&Ds
11 P&Ds 13 P&Ds
15 P&Ds 17 P&Ds
19 P&Ds
Figure 6.6: Two cross aisle designs with a varying number of P&D points at the
bottom.
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5 10 15 20
0
5
10
15
20
numberofP&Dpoints
Impro
vemen
t(%
)
EqualSize
Leaf
Improveddesigns
(a)
5 10 15 20
 5
0
5
10
numberofP&Dpoints
Impro
vemen
t(%
)
EqualCapacity
Leaf
Improved designs
(b)
Figure 6.7: Comparisons of the Leaf and the improved designs over equally sized
and equally capacitated warehouses.
points move away from the center. For more than 9 P&D points, it does not o er any
savings over an equivalent, small-sized traditional warehouse. However, as improved
designs suggest, it is possible to increase the bene t of the Leaf by moving cross
aisles away from the center and increasing the angle of picking aisles on the right
(decreasing on the left). Therefore, the Leaf is not as robust as the Chevron. The
Leaf is robust with respect to the maximum of 7 P&D points such that: (1) the
improved designs also provide a similar aisle structure to the Leaf, and (2) the Leaf
o ers, at most, 2% less improvement than the equivalent improved design.
For a large number of P&D points, the Flying-V-like designs seem to present
less savings compared to the results in Gue et al. (2010). This is because;
? The improved designs require more warehouse space than the Flying-V because
of slightly angled aisles.
? The improvements of the modi ed Flying-V in Gue et al. (2010) is investigated
for larger warehouses. However, we work on small-sized warehouses.
114
(a) Design C1 (b) Design C2
Figure 6.8: Designs with centrally located P&D points at the bottom and the top
of the warehouse.
6.4 Unit-Load Warehouse Designs with P&D Points at the Bottom and
the Top of the Warehouse
In this section, we locate a number of P&D points both at the bottom and the
top sides of the warehouse. We then search for the best aisle designs, inserting one
and two cross aisles to investigate the robustness of Design C1 and Design C2 (see
in Figure 6.8).
6.4.1 One Cross Aisle Designs
In this section, we insert one angled cross aisle and increase the number of
P&D points by placing two additional P&D points both at the bottom and the top
of the warehouse. These P&D points are equally distributed around the central P&D
points to preserve a symmetric distribution. Figure 6.9 shows the improved designs
for a varying number of P&D points. The improved designs with up to 9 P&D points
per side have similar aisle structure with Design C1. In these designs, the angles
of aisles in the right and left regions are approximately 168? and 93?, respectively.
However, when the number of P&D points increases, a di erent structure emerges,
especially in the right region. For more than 9 P&D points per side, the angle of
picking aisles on the right becomes approximately vertical so that the picking aisles
allow direct travel to the storage area from the P&D points that are further away
from both the central P&D point and the inserted cross aisle.
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3 P&Ds 5 P&Ds
7 P&Ds 9 P&Ds
11 P&Ds 13 P&Ds
15 P&Ds 17 P&Ds
19 P&Ds
Figure 6.9: Improved designs with one cross aisle and P&D points at the bottom
and the top.
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0 5 10 15 20
2
4
6
numberofP&Dpoints
Impro
vem
ent
(%)
EqualSize
DesignC1
Improved designs
(a)
0 5 10 15 20
 3
 2
 1
0
1
2
numberofP&Dpoints
Impro
vemen
t(%)
EqualCapacity
DesignC1
Improveddesigns
(b)
Figure 6.10: Comparison of Design C1 and improved designs with one cross aisle
and multiple P&D points at the bottom and the top.
Figure 6.10 shows the improvements in expected travel distance that the im-
proved designs and Design C1 have over equally sized and equally capacitated ware-
houses. Design C1 does not provide any improvement for more than 3 P&D points
per side, and the improved designs do not o er any savings in travel for more than 7
P&D points per side. Even though the improved designs have similar aisle structure
with Design C1, the improved designs o er more improvement. This is because the
improved designs place the cross aisle away from the center in order to allow newly
located P&D points to better utilize the cross aisle (see Appendix C.3 for details).
Design C1 is a robust design with respect to a maximum of 9 P&D points
per side such that the aisle structures in improved designs are similar to the aisle
structure in Design C1. Additionally, for 9 P&D points per side, even though the
improved design and Design C1 have worse expected travel distance than the equiv-
alent traditional design, Design C1 provides approximately 1.5% less expected travel
distance than the improved design.
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6.4.2 Two Cross Aisle Designs
Figure 6.11 shows the improved two cross aisle designs with P&D points at
the bottom and the top. Designs with less than or equal to 9 P&D points per side
have similar aisle structure with Design C2. These designs are also approximately
symmetric with respect to the diagonal that passes through the lower-left and the
upper-right corners. For more than 11 P&D points per side, the model generates
di erent aisle structures.
In improved designs with 9 P&D points or fewer, the angles of picking aisles are
similar to the the angles in Design C2, because the model moves the cross aisles away
from the central P&D points instead of changing angles of picking aisles. Thus, travel
is facilitated from the P&D points further away from the central one. For designs
with more than 9 P&D points, the angles of picking aisles in regions, either right
or left regions and the central region, are approximately the same because these
picking aisles facilitate travel from the P&D points to the storage locations further
away from them.
Figure 6.12 shows the comparison of Design C2 and the improved designs with
equally sized and equally capacitated traditional warehouses (see details in Ap-
pendix C.4). The aisle structure in Design C2 is robust for warehouses with up to 9
P&D points per side, because of the similarity in aisle structures of Design C2 and
the improved designs. However, for equally capacitated warehouses, the improved
design with 9 P&D points per side proposes 3.2% more improvement than Design
C2. Even for 3 P&D points per side, the improved design has an additional 2% more
bene t than Design C2. Therefore, up to 9 P&D points per side, we can conclude
that Design C2 is a robust design, and it o ers similar bene t to improved designs,
if cross aisles are slightly moved away from the center.
118
3 P&Ds 5 P&Ds
7 P&Ds 9 P&Ds
11 P&Ds 13 P&Ds
15 P&Ds 17 P&Ds
19 P&Ds
Figure 6.11: Improved designs with two cross aisles and multiple P&D points at the
bottom and the top.
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0 5 10 15 20
5
10
15
numberofP&Dpoints
Impro
vem
ent
(%)
EqualSize
DesignC2
Improved designs
(a)
0 5 10 15 20
 4
 2
0
2
4
6
8
numberofP&Dpoints
Impro
vemen
t(%)
EqualCapacity
DesignC2
Improveddesigns
(b)
Figure 6.12: Comparison of Design C2 and improved designs with two cross aisles
and multiple P&D points at the bottom and the top.
6.5 Conclusion
Many warehouses spaces have multiple P&D points along their sides, we have in-
vestigated non-traditional aisle designs developed for centrally located P&D point(s)
and found them to be relatively robust with respect to a varying number of P&D
points. We  rst investigated changes in performance of the Chevron and Leaf as the
number of P&D points increases at the bottom. Whereas the improvement in the
Chevron is 14% for 3 P&D points, it loses all of its e ciency when there are 15 P&D
points at the bottom of a small-sized warehouse. The Chevron is also very robust
with respect to a maximum of 13 P&D points compared to the improved designs.
When we insert two cross aisles into a warehouse with bottom P&D points, we
have observed that the Leaf preserves its aisle structure for less than or equal to 7
P&D points, and it o ers more than 5% savings in expected travel distance over an
equivalent, small-sized traditional design. Therefore, the Leaf only has, at most, 2%
less improvement than the equivalent improved designs with 7 P&D points or fewer.
Surprisingly, the model generates designs that look like the Flying-V for warehouses
with two cross aisles and more than 13 P&D points at the bottom. Hence, if there
120
is almost one P&D point at the beginning of every aisle in a small-sized warehouse,
the best design we found is the Flying-V (with non-intersecting cross aisles).
We have also considered warehouses with P&D points both at the bottom and
the top sides of the warehouse. Design C1 provides positive improvement in expected
travel distance for less than 5 P&D points per side over equivalent, small-sized
traditional warehouses. The improved designs with one cross aisle propose very
similar aisle structure with Design C1 for 9 P&D points per side or less. Design C1
also has 1.5% higher expected travel distance than the improved design with 9 P&D
points per side. Even though Design C1 o ers worse expected travel distance than
traditional designs for more than 3 P&D points per side, in terms of robustness, it
is a robust design for up to 9 P&D points per side.
When two cross aisles are inserted with P&D points at the bottom and the top,
we have shown that Design C2 is a robust design with respect to change in its aisle
structure for a maximum of 9 P&D points per side. However, the improved designs
o er more improvement than Design C2; for example, 3.2% more improvement for 9
P&D points. Hence, the improved designs suggest that the cross aisles in Design C2
need to be moved slightly away from the center by preserving their orientation, as
the number of P&D points increases. For more than 9 P&D points per side, di erent
aisle structure emerges, because of the spreading P&D points per side.
We have assumed an equal  ow from each P&D point. However, P&D points
close to the center likely would be more utilized than more distant P&D points.
This is because a number of trucks is sometimes less than the number of available
doors; therefore, P&D points farther from the center are not always active. Thus, we
can expect that the improvement from primary designs would be higher than these
results indicate. Additionally, because non-traditional aisle designs o er an increas-
ing improvement in expected travel distance as the size of warehouses increase, we
expect that primary designs would yield better performance for larger warehouses.
121
Chapter 7
Conclusions and Future Research
7.1 Conclusions
In this research, we extended the idea of non-traditional aisle designs and pro-
posed new layouts for a variety of warehouse design problems. We also show their
robustness with respect to a varying number of P&D points.
Contribution 1. We proposed three optimal aisle designs for unit load warehouses
with a single P&D point.
Although Gue and Meller (2009) relaxed some established rules of warehouse
design, they did adhere to one rule: Picking aisles must be either horizontal or
vertical. We relaxed this assumption by allowing cross aisles and picking aisles to
take any angle in our model. We presented travel distance functions in continuous
warehouse space for a given half-width and height of the warehouse. By solving
our model, we introduced an optimal one cross aisle design, called the Chevron, in
square-half, unit-load warehouses. We presented the optimal two cross aisle design,
called the Leaf, which has symmetric cross aisles in square-half, unit-load warehouses.
Additionally, we proposed an optimal three cross aisle design, called the Butter y.
We showed that the maximum savings of the Chevron, the Leaf and the Butter y
are 19.53%, 21.72% and 22.52%, respectively.
In order to assess the real bene t of these designs and to analyze the wasted
space due to angled aisles, we developed a discrete cost analysis that considers travel
to every available storage location. We showed that the Chevron is the best design
for square-half warehouses with a width of 27 aisles or fewer. As the size of the
Chevron increases from 19-aisles to 27-aisles, its savings increase from approximately
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16.1% to 17.1%, and additional space requirements decrease from 11.2% to 7.3%.
The Chevron always has less expected travel distance than the equivalent traditional
design, even if the warehouse is small. For warehouses with more than 27 aisles, the
Leaf o ers more savings than the Chevron, but requires slightly more space due to
having two inserted cross aisles. For a large warehouse with 51 aisles, the Leaf has
19.3% improvement over an equivalent traditional design and uses 6% more space.
The Butter y o ers similar improvement as the Leaf for warehouses larger than 65
aisles, but the Leaf requires less space than the Butter y. Therefore, we concluded
that the Chevron is the best design for the most warehouse sizes in industry with a
centrally located P&D point.
Contribution 2. We developed better aisle designs than traditional for unit-load
warehouses with multiple P&D points.
We allowed picking and cross aisles to take any angle, and the cross aisles to
be oriented freely as long as their start and end points are on di erent sides of the
warehouse. We also permitted multiple prede ned P&D points to be on di erent
sides of the warehouse.
We developed a warehouse network model and used a shortest path algorithm
to e ciently evaluate travel distance between storage locations and P&D points.
Next, we used a Particle Swarm Optimization algorithm to search for aisle angles in
a given warehouse with pre-located P&D points.
We di erentiated  ve design problems based on the location of P&D points.
We inserted one and two cross aisles incrementally, and searched for the best aisle
structures for each of the design problems.
In comparison with an equivalent, medium-sized traditional warehouse, the De-
sign A2 o ers approximately 11% savings with a cost of 9.6% larger space. The
Design B2 provides approximately 6% reduction in expected travel distance but re-
quires 14.4% more space. The Design C2 has approximately 10% less savings but
necessitates a 11.7% bigger warehouse. In design problem D, the best design we
123
found so far does not need any additional cross aisle because of the locations of
P&D points. Hence, it o ers 16% savings with a cost of 4% larger space. When
we distribute one P&D point to each side of the warehouse we could not get any
solution better than the traditional design with a vertical cross aisle.
Last, we modi ed several of these best or nearly best designs in order to increase
their storage capacity as well as to decrease their additional space requirements.
We modi ed their angles of picking aisles to 90? or 180?. Some of these modi ed
designs slightly increase the improvement in expected travel distance for equally
capacitated warehouses. They o er a considerable reduction in additional warehouse
space required to provide a capacity equal to non-traditional designs, 3% on average.
Contribution 3. We investigated the robustness of aisle designs with respect to a
varying number of P&D points.
In some warehouses in industry, incoming and outgoing material  ows to and
from a warehouse are distributed along a number of dock doors. Therefore, we
 rst investigated to see how much improvement is presented by non-traditional aisle
designs, the Chevron, Leaf, Design C1, and Design C2, that are speci cally developed
for centrally located P&D point(s), as the number of P&D points increases.
Next, we search for better aisle designs than these designs by increasing their
number of P&D points. The improved designs show that the Chevron is very robust
for a large number of P&D points, such that its aisle structure is similar to the aisle
structures in improved designs. However, the Leaf is not as robust as the Chevron.
As the P&D points move away from the center, the Leaf loses its power because of
the angle of picking aisles in the right and left regions. Design C1 and Design C2
are also robust designs for warehouses with 9 P&D points or fewer. Hence, the aisle
structure in improved designs and these designs are similar for that number of P&D
points. Because improvements in Design C2 are slightly di erent from the results in
improved designs, the model suggests to move the cross aisles away from the central
124
P&D points. Hence, robustness of non-traditional aisle designs is assessed according
to the change in the aisle structure and the change improvement.
Interestingly, we have also showed that the improved designs we obtain with
two cross aisles and a large number of P&D points at the bottom look like Flying-V
designs. Hence, the Flying-V is the best design we found if there is one P&D point
at the beginning of every aisle. with non-intersecting cross aisles.
7.2 Future Research
This research is one of the  rst attempts to propose a general warehouse de-
sign tool and optimization approach for designing aisles to minimize expected travel
distance to visit locations in a warehouse. However, the tool and the designs we
developed rely on operational and design assumptions: randomized storage, single-
command operations, non-intersecting cross aisles and a rectangular shape ware-
house.
There is a potential for savings in travel distance for other common types of
operations, such as dual-command and order picking. Existing non-traditional aisle
designs could be evaluated for dual-command and order picking e ciency, or en-
tirely new aisle designs could be investigated. Aisle designs for order picking is an
especially important problem.
We assume in our models that each storage location has the same probability of
being visited by workers. This assumption refers to the randomized storage policy
and is also used to approximate the closest-open-location storage policy (Graves
et al., 1977). Pohl et al. (2010) evaluated the performance of Fishbone and Flying-
V designs under turnover-based storage policy and showed that these designs still
o er improvement over traditional designs under turnover-based storage. Therefore,
even though randomized storage policy is common in industry, the performance of
our existing aisle designs could be evaluated for other common storage policies, such
as class-based and turnover-based storage policies.
125
Our existing aisle designs that are developed for multiple, pre-located P&D
points consider travel to all available storage locations from all P&D points with an
equal probability. However, some P&D points might be utilized intensively compar-
ing to some others. Or, the storage space could be divided into zones or could be
partitioned based on customers. Then, P&D points could be distributed to zones or
partitions such that travel is restricted from speci c P&D points to storage locations
in a speci c area. New aisle designs could be developed for this type of problem.
Even though intersecting cross aisles seem to waste more space than the non-
intersecting cross aisles, they might still o er bene ts compared to non-intersecting
aisles, especially when travel between storage locations becomes important. Hence,
intersecting aisles might be allowed in warehouse design models in order to develop
new aisle designs for new problems. Because we assume a rectangular shape ware-
house, non-rectangular shape warehouses might also be considered with a new aisle
designs in order to minimize expected travel distance.
7.2.1 Dynamics of inserted P&D points
As shown in Section 6.3.1, the model generates Chevron-like designs up to 13
P&D points and approximately symmetric designs with a  xed cross aisle. We
observe that the model generates a di erent aisle structure after inserting 13 P&D
points, because the cross aisle needs to be appropriately placed in order to facilitate
travel from P&D points furthest to the center. Therefore, here we quantify the cost
of inserting P&D points and open a discussion for future research about what kind
of dynamics there are between inserting additional P&D points and the change in
aisle structure. In order to investigate the cost of each inserted P&D location in
improved designs, their locations are numbered according to their closeness to the
center (see Figure 7.1). Because P&D points are symmetrically distributed around
the center, P&D points with a same distance to the center on each side have the
same expected travel distance as long as the design is symmetric.
126
#1 #3#2 #4#5#6#7#8#9#10#2#3#4#5#6#7#8#9#10
Figure 7.1: An improved design with a  xed cross aisle and the locations of P&D
points are marked based on the closeness to the center.
We then evaluate the expected travel distance from each numbered location in
improved designs that are developed for a speci c number of P&D points (see Fig-
ure 7.1). For example, the black circled line presents the expected travel distance
of location #1 in improved designs speci cally developed with 1 { 19 P&D points.
Thus, the cost of locations closer to the center increases as the inserted number of
P&D points increases in improved designs, especially the locations #1 and #2 be-
cause the designs are not speci cally developed for central P&D points. In Locations
after location #4, the expected travel distance decreases as the inserted number of
P&D points increases because the improved designs are speci cally developed for a
larger number of P&D points.
The left plot in Figure 7.3 also shows how much cost an inserted P&D point
has at a speci c location if that P&D point is the last inserted one in the design.
For example, as the cost of one P&D point at location #1 is 398 if there is only one
P&D point in the improved design; or, the cost of inserted one P&D point at location
#4 is 479.62 if the improved design is developed for 7 P&D points at the bottom.
Therefore, the right plot in Figure 7.3 shows that the model does not change the
aisle structure until some point that has a big marginal cost of inserting one P&D
point at a location further to the center. When the model provides a di erent aisle
127
0 5 10 15 20
400
500
600
700
number of P&D points in designs
Exp
ected
tra
vel
distance
#1
#2
#3
#4
#5
#6
#7
#8
#9
#10
Figure 7.2: Costs of P&D points at di erent location in di erent designs developed
for a speci c number of P&D points.
2 4 6 8 10
400
500
600
700
locationsofP&Dpoints
Exp
ected
tra
vel
distance
2 4 6 8 10
20
30
40
locationsofP&Dpoints
Marginal
tra
vel
distance
Figure 7.3: Expected travel distance and marginal cost of an inserted P&D point at
locations in improved designs.
structure, the marginal cost of inserting the last P&D point decreases dramatically,
then it starts to increase again for newly inserted P&D points.
As a result, quantifying the cost of inserting a P&D locations might be also
helpful for practitioners to determine the e ciency of dock doors. To  nd an in ec-
tion point of non-traditional warehouses in terms of number of P&D locations also
could be a good research topic.
128
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Appendices
138
Appendix A
A.1 Closed form expressions for E[WR]
E[WR1] = h2 + w
2 tan R (1 + sec R)
6h  
w2 sec R tan R sec2 R cos2  R  R2  
3h
+ w
2 tan2 R
6h  
wtan R
2
+ wsec R (3h+wtan R sec
2 R ( cos (2 R) + cos ( R  R)))
6h
E[WR2] = h
2 cot R (1 + csc R)
6w +
sec  R  R2  (w2 h2 cot2 R) cos  R+ R2  
2w
+ hsec
  
R  R
2
 (w hcot 
R) sin
  
R+ R
2
 
2w +
w2 tan R (1 + sec R)
6h
 sec R tan R (w
3 h3 cot3 R) 3 + cos 3 R+ R
2
 sec  
R  R
2
  
12hw
+ h
2 cot R csc R sec R cos2  R  R
2
 (sec 
R csc R tan R)
3w
E[WR3] = w2 + h
2 cot2 R
6w +
hcsc R
2w
+ h
2 cot R (1 csc R ( 1 + csc2 R (1 + cos ( R  R))))
6w
+ hcot R ( 3w +hcsc R csc
2 R (cos ( R  R) + cos (2 R)))
6w
A.2 Proof of Observation 2
Let us  rst show E[A] = E[D] by using travel distance functions TA and TC
de ned in (4.3) and (4.5) (see also Figure 4.11).
139
E[A] = 2w2 tan 
R
Z w
0
Z xtan R
0
TA(x;y) dydx
E[D] = 2h2 cot 
R
Z h=tan R
0
Z h
xtan R
TC(x;y) dydx
Because w = h and  R +  R = 90? in the optimal design (see Observation 1),
we can substitute and solve in Mathematica. We  nd that
E[A] = E[D] = 16w3 tan R (1 + sec R):
The travel distance function for regions B and C in Figure 4.16 is TB which is
described in (4.4). E[B] and E[C] are
E[B] = 2w(h wtan 
R)
Z w
0
Z xtan =4
xtan R
TC(x;y) dydx;
E[C ] = 2h(w hcot 
R)
Z h
0
Z y=tan =4
y=tan R
TC(x;y) dxdy:
The solution, obtained by Mathematica, produces
E[B] = E[C ] = w
3 ( 3 + 2 tan R + tan( R)2)
6 (cos R + sin R) :
A.3 Expected travel distances in a three cross aisle model
The following expressions were obtained with Mathematica, as solutions to
equations 4.8{4.11.
140
E[WR1] = h+wsec CR wtan CR2  w
2 sec R sin2( R=2) tan R
3h
+ w
2 sec CR tan CR
6h +
w2 sec R tan CR
6h +
w2 tan2 CR
6h +
w2 sec R tan R
6h
+ w
2 sec CR tan R
6h
E[WR2] = h
2 cot CR 1 + csc CR + 2 cos2  CR  R
2
 csc 
CR sec2 R
 
6w
+ 12 cos
  
CR + R
2
 
sec
  
R  CR
2
  
w h
2 cot CR
w
 
+ h2 sin
  
CR + R
2
 
sec
  
R  CR
2
  
1 hcot CRw
 
+ w
2 tan R
6h
 
1 + sec R 2 sec R sec2 R cos
  
R  R
2
  
+ sec R tan R4
 
w2
h +
h2 cot3 CR
w  
4h2 cot CR csc2 CR cos2  CR  R2  
3w
!
+ w
2 sec R tan R
3h
 
sec2 R cos2
  
R  R
2
 
 cos
  
CR+3 R
2
 sec  
R  CR
2
 
4
!
+ h
2
12w tan R sec R cot
3 CR cos
  
CR + 3 R
2
 
sec
  
R  CR
2
 
E[WR3] = h
2 cot CR
6w (1 + csc CR) +
w2 tan R
6h
 
1 sec R2
 
+ h
2 cos2  CR  R
2
 csc 
R
3w
 cot 
R csc2 CR cot CR csc CR csc R + cot R sec2 R
 
+ 12 cos
  
R + R
2
 
sec
  
R  R
2
  
w h
2 cot2 R
w
 
+ 12 sin
  
R + R
2
 
sec
  
R  R
2
  
h h
2 cot2 R
w
 
+ h
2 cot R sec R tan R
12w
 
3 cot2 R 4 csc2 R cos2
  
R  R
2
  
 112 cos
 3 
R + R
2
 
sec
  
R  R
2
 
sec R tan R
 w2
h  
h2 cot3 R
w
 
141
E[WR4] = 16w h2 cot2 R + 3w(w +hcsc R) hcot R(3w + 2hcsc R) 
+ h
2
6w cot CR(1 + csc CR)
+ 2h
2
6w cos
2
  
R  R
2
 
csc2 R csc2 R(cos R cos R)
+ 2h
2
6w cos
2
  
CR  R
2
 
csc2 CR csc2 R(cos R cos CR)
A.4 Data
Chevron Leaf Butter y
# Aisles Improvement Area Improvement Area Improvement Area
19 16.12 11.30 15.59 16.64 13.53 25.44
21 16.50 10.25 15.87 15.07 13.94 23.01
23 16.70 9.38 16.52 13.78 14.72 21.00
25 16.94 8.64 16.84 12.69 15.42 19.31
27 17.10 7.27 17.28 11.76 15.74 17.88
29 17.47 6.77 17.53 10.95 16.45 16.64
31 17.65 6.34 17.80 10.25 16.71 15.56
33 17.76 5.97 17.97 9.63 17.15 14.62
35 17.87 5.63 18.16 9.09 17.38 13.78
37 17.96 5.33 18.37 8.60 17.67 13.03
39 18.05 4.04 18.54 8.16 18.02 12.36
41 18.06 4.82 18.73 7.76 18.10 11.76
43 18.12 4.60 18.85 7.40 18.37 11.21
45 18.18 4.39 18.96 7.08 18.47 10.71
47 18.38 3.78 19.05 6.78 18.68 10.25
49 18.42 3.63 19.19 6.50 18.89 9.83
51 18.47 3.49 19.26 6.25 18.95 9.44
53 18.51 3.36 19.37 6.01 19.12 9.09
55 18.56 3.24 19.46 5.80 19.21 8.76
57 18.59 3.13 19.54 5.59 19.38 8.45
59 18.54 3.36 19.61 5.40 19.50 8.16
61 18.57 3.25 19.68 5.23 19.56 7.89
63 18.59 3.79 19.75 5.06 19.66 7.64
65 18.63 3.05 19.80 4.91 19.84 7.40
67 18.62 2.96 19.87 4.76 19.97 7.18
69 18.65 2.88 19.91 4.62 20.02 6.97
71 18.67 2.80 19.97 4.49 20.08 6.78
Table A.1: Improvements in expected travel distance over an equivalent traditional
design.
142
# Pallet Locations
# Aisles Traditional Chevron Leaf Butter y
19 1880 1904 1868 1900
21 2288 2306 2272 2304
23 2736 2760 2726 2756
25 3224 3240 3208 3248
27 3752 3746 3732 3764
29 4320 4328 4300 4348
31 4928 4928 4908 4948
33 5576 5568 5554 5604
35 6264 6248 6246 6284
37 6992 6972 6974 7004
39 7760 7760 7734 7792
41 8568 8580 8548 8592
43 9416 9428 9390 9456
45 10304 10304 10278 10328
47 11232 11220 11204 11260
49 12200 12180 12174 12240
51 13208 13178 13174 13240
53 14256 14212 14230 14300
55 15344 15284 15308 15368
57 16472 16406 16438 16508
59 17640 17640 17598 17680
61 18848 18832 18816 18884
63 20096 20072 20058 20144
65 21384 21344 21344 21356
67 22712 22752 22670 22695
69 24080 24104 24034 23995
71 25488 25506 25444 25460
Table A.2: Number of storage locations in the proposed designs and in the equivalent
traditional design when the shape ratio is approximately 2:1. \Locations" refers to
two-dimensional space|the number of columns of pallet rack or stacks of pallets.
143
Appendix B
B.1 Two Cross Aisle Model Cases
144
Main Case x axis y axis x and y axis
Figure B.1: The de ned main cases and their transformations for two cross aisle
model.
145
B.2 Computational Results in Design Problem A
B.2.1 Results for Design A1
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (334,500) (1000,25) 116.40 87.18 1617 475.92
2 (332,500 ) (1000,35) 120.99 88.90 1638 480.33
3 (330,500 ) (1000,5) 115.83 85.46 1621 477.71
Subproblem 5
1 (332,500) (970,0) 115.26 84.32 1630 479.38
2 (335,500 ) (1000,0) 118.12 85.46 1620 477.77
3 (333,500 ) (990,0) 114.11 86.03 1615 477.23
Subproblem 2
1 (0,437) (1000,73) 114.11 95.78 1603 503.41
2 (0,480) (1000,25) 117.55 87.18 1575 496.53
3 (0,450) (1000,0 ) 116.40 93.48 1605 501.76
T able B.1: Computational results for Design A1 in a small-sized w arehouse (W=100, H = 50).
146
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (500,750) (1500,25) 116.97 87.18 3910 715.75
2 (500,750 ) (1500,95) 116.40 88.33 3927 715.68
3 (500,750 ) (1500,85) 120.41 84.32 3942 719.19
Subproblem 5
1 (500 ,750) (1485,0) 114.68 86.61 3930 717.53
2 (500,750) (1465,0) 111.25 83.74 3928 719.54
3 (500,750 ) (1447,0) 115.83 87.18 3927 717.14
Subproblem 2
1 (0,722) (1500,53) 115.83 88.90 3874 746.15
2 (0,738) (1500,15) 117.55 86.03 3868 748.22
3 (0,695) (1500,35) 107.23 87.75 3879 752.09
T able B.2: Computational results for Design A1 in a medium-sized w arehouse (W=150, H=75).
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (666,1000) (2000,105) 118.12 88.90 7232 955.33
2 (667,1000) (2000,50) 114.11 88.33 7204 954.80
3 (666,1000) (2000,88) 121.56 86.61 7242 958.23
Subproblem 5
1 (666,1000) (1953,0) 117.55 87.75 7229 956.59
2 (666,1000) (1935,0) 115.26 86.03 7241 958.43
3 (667,1000) (1975,0) 116.40 87.18 7240 957.23
Subproblem 2
1 (0,948) (2000,16) 115.26 88.33 7135 997.98
2 (0,830) (2000,34) 116.40 94.63 7177 1010.66
3 (0,882) (2000,54) 118.69 86.03 7164 1003.73
T able B.3: Computationa l results for Desig n A1 in a large-sized w arehouse (W=200, H=100).
147
W H S E  L ( ? )  R ( ? ) # P allets E [ C ]
Small 106 53 (353,530) (1000,27) 116.40 87.18 1871 503.67
Medium 157 78 (523,780) (1570,100) 116.4 88.33 4316 747.70
Large 207 103.5 (690,1035) (2070,52) 114.11 88.33 7746 988.45
T able B.4: P erfor mance of the expanded Design A1 for di eren t w arehouse sizes.
W H S E  L ( ? )  R ( ? ) # P allets E [ C ]
Small 100 50 (334,500) (1000,25) 116.4 90 1670 481.05105 52 (350,500) (1000,26) 116.4 90 1874 502.05
Medium 150 75 (500,750) (1000,95) 116.4 90 4005 719.28156 77 (520,770) (1560,89) 116.4 90 4312 744.81
Large 200 100 (667,1000) (2000,50) 114.11 90 7324 960.00206 102 (687,1020) (2060,42) 114.11 90 7755 984.73
T able B.5: P erform ance of mo di ed Desig n A1 for di eren t w arehouse sizes.
148
B.2.2 Results for Design A2
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (333 ,500) (500,0) (667,500) (500,0) 127.52 180.00 52.46 1610 448.24
2 (333,500 ) (483, 0) (667,500) (504,0) 129.58 174.75 47.65 1586 455.39
3 (333,500 ) (465,0) (667,500) (478,0) 122.13 180.00 56.81 1616 452.56
Subproblem 1
1 (333 ,500) (0,80) (667,500) (0, 15) 9.36 86.61 100.36 1576 471.62
2 (333,500 ) (0,148) (667,500) (0,13 ) 14.42 87.75 113.54 1562 464.24
3 (333,500 ) (0,126) (667,500) (0,49 ) 171.98 84.89 121.56 1558 469.03
Subproblem 3
1 (333 ,500 ) (0,46) (1000,117 ) (924,0) 155.36 88.90 100.93 1664 489.28
2 (333,500 ) (0,12 ) (1000,97) (916,0) 158.80 87.18 94.63 1655 486.63
3 (333,500 ) (0,21 ) (1000,13) (935 ,0) 157.65 84.89 98.07 1655 484.48
T able B.6: Computational results for Design A2 in a small-sized w arehouse (W=100, H = 50).149
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (500 ,750) (750,0) (1000,750 ) (753,0) 127.52 180.00 52.46 3905 674.35
2 (500,750 ) (733,0) (1000,750) (742,0) 133.59 179.34 47.07 3906 674.45
3 (500,750 ) (737,0) (1000,750) (750,0) 123.28 180.00 57.96 3906 676.49
Subproblem 1
1 (500 ,750) (0,59) (1000,750 ) (0,45) 10.40 83.74 107.23 3800 701.30
2 (500,750 ) (0,127) (1000,750) (0,62 ) 18.43 85.46 103.80 3809 703.14
3 (500,750 ) (0,91 ) (1000,750) (0,26 ) 21.86 86.03 110.67 3808 695.52
Subproblem 3
1 (500 ,750) (0,5) (1500,142 ) (1449,0) 156.51 84.32 180.00 3979 732.02
2 (500,750 ) (0,9) (1500,82) (1433,0) 158.23 87.18 0.66 3981 735.55
3 (500,750 ) (0,2) (1500 ,95) (1427,0) 151.93 82.02 8.11 3983 729.23
T able B.7: Computational results for Design A2 in a medium-sized w arehouse (W=150, H=75).
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (667 ,1000) (988,0) (1333, 1000) (996,0) 125.57 180 49.37 7209 899.85
2 (667,1000) (1005,0) (1333,1000) (1009 ,0) 129.58 180 50.51 7204 898.38
3 (667,1000) (1018,0) (1333,1000) (1019,0) 127.29 179.57 62.54 7199 904.67
Subproblem 1
1 (667 ,1000) (0,60) (1333,1000) (0 ,48 ) 26.45 83.17 98.64 7055 945.96
2 (667,1000) (0,102) (1333,1000) (0,8) 23.58 87.18 107.23 7043 927.88
3 (667,1000) (0,43 ) (1333,1000) (0,2) 31.60 88.90 100.93 7033 935.86
Subproblem 3
1 (667,1000) (0,13) (2000,108) (1941,0) 154.79 79.73 2.38 7303 973.04
2 (667,1000) (0,28 ) (2000,119) (1943,0 ) 157.65465 86.03492 75.14872 7318 983.64
3 (667,1000) (0,22 ) (2000,98) (1939,0) 156.50873 78.58647 90.04563 7317 971.54
T able B.8: Computationa l results for Desig n A2 in a large-sized w arehouse (W=200, H=100).
150
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) P allet E [ C ]
Small 105 54 (350,540) (525,0) (700,540) (525,0) 127.52 180 52.46 1875 478.26
Medium 156 79 (520,790) (708,0) (1040,790) (888,0) 127.52 180 52.46 4317 705.24
Large 206 104 (687,1040) (959,0) (1373,1040) (1138,0) 129.58 180 10.51 7758 929.58
T able B.9: P erfor mance of the expanded Design A2 for di eren t w arehouse sizes.
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) P allet E [ C ]
Small 105 52.5 (525,525) (0,185) (525,525) (1050,185) 90 90 90 1875 513.57
Medium 156 77.5 (780,775) (0,275) (780,775) (1560,275) 90 90 90 4329 756.76
Large 206 102 (1030,1020) (0,400) (1030,1020) (2060,400) 90 90 90 7739 994.52
T able B.10: P erf ormance of the expanded, our mo di ed Flying-V design for di eren t w arehouse sizes.
151
B.3 Computational Results in Design Problem B
B.3.1 Results for Design B1
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 5
1 (500,500) (130,0) 179.81 41.92 1714 533.88
2 (500,500 ) (158,0) 179.67 44.21 1717 534.87
3 (500,500 ) (201,0) 179.92 33.32 1726 531.87
Subproblem 1
1 (0,250) (500,500) 128.43 63.69 1683 562.36
2 (0,263) (500 ,500) 122.13 65.98 1673 568.88
3 (0,255) (500,500) 115.26 61.40 1669 561.92
Subproblem 2
1 (0,250) (1000,45) 85.46 95.20 1601 567.77
2 (0,250) (1000,255 ) 53.38 90.05 1643 551.80
3 (0,250) (1000,262 ) 46.50 93.48 1610 552.54
T able B.11: Computation al results for Design B1 in a small-sized w arehouse (W=100, H=50).
152
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 5
1 (750,750) (233,0) 179.88 29.31 4072 805.62
2 (750,750 ) (257, 0) 179.92 34.47 4074 803.31
3 (750,750 ) (198,0) 179.75 37.33 4068 804.47
Subproblem 1
1 (0,384) (750,750) 111.25 59.11 3985 834.47
2 (0, 375) (763,750) 117.55 65.41 4001 844.42
3 (0,375) (750,750) 119.84 52.23 3986 833.65
Subproblem 2
1 (0,375) (1500,405) 43.64 94.63 3887 830.98
2 (0,375) (1500,379 ) 40.77 90.15 3988 840.09
3 (0,375) (1500,394 ) 44.78 90.05 3975 833.35
T able B.12: Computationa l results for Desig n B1 in a medium-size w arehouse (W=150, H=75).
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 5
1 (1000,1000) (326 ,0) 179.62 33.95 7437 1067.59
2 (1000,1000) (282,0) 179.81 39.63 7423 1069.04
3 (1000,1000) (506 ,0) 179.86 37.33 7454 1070.73
Subproblem 1
1 (0,504) (1016,1000) 110.67 49.94 7307 1113.35
2 (0,501) (1000,1000) 115.83 62.54 7321 1114.93
3 (0,503) (1007,1000) 118.12 57.96 7323 1111.37
Subproblem 2
1 (0,500) (2000,1055) 49.37 92.91 7184 1092.18
2 (0,500) (2000,1126) 47.65 90.05 7302 1093.89
3 (0,500) (2000,1072) 45.93 95.20 7204 1094.81
T able B.13: Computationa l results for Desig n B1 in a large-sized w arehouse (W=200, H=100).
153
W H S E  L ( ? )  R ( ? ) # P allets E [ C ]
Small 108 50 (540,500) (241,0) 180 33.32 1881 560.45
Medium 158 75 (790,750) (297,0) 180 34.47 4314 831.97
Large 208 100 (1040,1000) (366,0) 180 33.95 7756 1096.22
T able B.14: P erfor mance of the expanded Design B1 for di ere n t w arehouse sizes.
154
Results for the Design B2
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (500,500) (1000,0) (0,250) (1000,0) 71.14 135.02 8.11 1501 492.62
2 (500,500 ) (1000,33) (0,250) (975,0) 76.20 134.65 8.02 1501 495.26
3 (500,500 ) (1000,51) (0,250) (982,0) 77.92 132.93 10.89 1494 497.29
Subproblem 1
1 (500,500) (985,0) (0,250) (982,0) 107.72 135.79 23.49 1488 496.49
2 (500,500 ) (984,0) (0,250) (976,0) 84.22 134.65 20.05 1498 495.33
3 (500,500 ) (1000,0) (0,250) (1000,0) 69.99 134.16 6.39 1491 493.47
Subproblem 3
2 (0,250) (1000,260) (500,500) (0,250) 170.74 63.03 50.42 1520 529.93
1 (0,250) (1000,251 ) (500,500) (0,250) 164.44 65.89 53.29 1514 528.45
3 (0,250) (1000,150 ) (500,500) (0,250) 159.86 60.16 52.14 1508 531.30
T able B.15: Computation al results for Design B2 in a small-sized w arehouse (W=100, H=50).155
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (750,750) (1500,33) (0,375) (971,0) 110.58 131.21 5.16 3733 753.85
2 (750,750 ) (1500,22) (0,375) (982,0) 84.22 134.07 6.88 3731 750.08
3 (750,750 ) (1500,0) (0,375) (1000,0) 70.57 135.88 8.11 3717 746.65
Subproblem 1
1 (750 ,750) (1481,0) (0,375) (1478,0) 79.64 138.08 18.33 3725 746.98
2 (750,750 ) (1483,0) (0,375) (1480,0) 76.20 134.07 22.35 3716 748.20
3 (750,750 ) (1500,0) (0,375) (1000,0) 74.00 135.02 7.54 3752 746.50
Subproblem 3
2 (0,375) (1500,295) (750,750) (0,375) 161.00 63.60 55.00 3763 788.12
1 (0,375) (1500,370 ) (750,750) (0,375) 156.42 63.03 48.13 3774 786.92
3 (0,375) (1500,345 ) (750,750) (0,375) 166.16 64.74 52.71 3754 789.20
T able B.16: Computational results for Design B2 in a medium-sized w arehouse (W=150, H=75).
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (1000,1000) (2000,0) (0,500) (2000,0) 74.00 135.02 8.11 6957 999.63
2 (1000,1000) (2000,19) (0,500) (1980,0) 80.79 136.36 10.31 6947 1001.57
3 (1000,1000) (2000,17) (0,500) (1976,0) 107.72 134.07 11.46 6968 1007.57
Subproblem 1
1 (1000,1000) (1484,0) (0,500) (1481,0) 96.26 136.94 17.19 6956 1002.08
2 (1000,1000) (1480,0) (0,500) (1477,0) 81.93 133.50 21.20 6951 1002.36
3 (1000,1000) (2000,0) (0,500) (2000,0) 72.28 134.74 9.26 6945 999.89
Subproblem 3
2 (0,500) (2000,503) (1000,1000) (0,500) 169.02 63.03 59.01 6993 1048.03
1 (0,500) (2000,465 ) (1000,1000) (0,500) 165.58 64.17 54.43 6997 1047.58
3 (0,500) (2000,497 ) (1000,1000) (0,500) 163.29 65.89 50.99 7002 1047.40
T able B.17: Computationa l results for Desig n B2 in a large-sized w arehouse (W=200, H=100).
156
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) P allet E [ C ]
Small 105 57 (525,570) (1050,0) (0,525) (1050,0) 71.14 135.02 8.11 1868 533.00
Medium 157 82 (785,820) (1570,0) (0,410) (1570,0) 74.00 135.02 7.54 4314 793.00
Large 209 106 (1045,1060) (2090,0) (0,530) (2060,0) 74.00 135.02 8.11 7751 1051.59
T able B.18: P erfor mance of the expanded Design B2 for di ere n t w arehouse sizes.
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Small 100 50 (500,500) (1000,0) (0,250) (1000,0) 90 135.02 180 1579 498.12103 56 (515,560) (1030,0) (0,515) (1050,0) 90 135.02 180 1877 528.09
Medium 150 75 (750,750) (1500,0) (0,375) (1000,0) 90 135.02 180 3884 752.27156 80 (780,800) (1560,0) (0,400) (1560,0) 90 135.02 180 4321 789.22
Large 200 100 (1000,1000) (2000,0) (0,500) (2000,0) 90 135.02 180 7129 1005.11208.5 104 (1042.5,1040) (2085,0) (0,520) (2085,0) 90 135.02 180 7758 1048.31
T able B.19: P erform ance of mo di ed Desig n B2 for di eren t w arehouse sizes.
157
B.4 Computational Results in Design Problem C
B.4.1 Results for Design C1
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (500 ,500) (1000,0) 93.48 165.10 1646 464.66
2 (500,500 ) (1000,8) 94.63 165.68 1646 465.14
3 (500,500 ) (1000,15) 92.91 163.96 1642 464.94
Subproblem 5
1 (500 ,500) (983,0) 91.76 135.88 1641 470.00
2 (500,500 ) (997,0) 87.75 5.25 1642 472.32
3 (500,500 ) (980,0) 92.34 6.97 1643 468.69
Subproblem 2
1 (0,256) (1000,253) 92.34 88.33 1645 494.33
2 (0,305) (1000,168 ) 87.75 91.76 1643 489.07
3 (0,150) (1000,265 ) 88.33 92.91 1634 489.15
T able B.20: Computational results for Design C1 in a small-sized w arehouse (W=100, H=50).
158
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (750,750) (1500,50) 94.06 165.10 3979 701.53
2 (750,750 ) (1500,0) 98.07 164.53 3968 700.45
3 (750,750 ) (1500,3) 96.92 167.39 3965 701.17
Subproblem 5
1 (750 ,750) (1480,0) 92.91 6.39 3954 707.22
2 (750,750 ) (1488,0) 92.34 8.69 3954 707.83
3 (750,750 ) (1494,0) 93.48 7.54 3963 707.67
Subproblem 2
1 (0,445) (1500,190) 92.34 87.18 3935 732.83
2 (0,285) (1500,605 ) 86.61 94.06 3903 730.36
3 (0,225) (1500,580 ) 87.75 92.34 3919 731.00
T able B.21: Computationa l results for Desig n C1 in a medium-sized w arehouse (W=150, H=75).
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (1000,1000) (2000,2) 94.06 165.10 7281 937.24
2 (1000,1000) (2000,45) 93.48 163.38 7295 938.80
3 (1000,1000) (2000,6) 96.92 165.68 7279 935.77
Subproblem 5
1 (1000,1000) (1990,0) 93.48 3.53 7293 942.49
2 (1000,1000) (1998,0) 94.06 7.54 7278 945.27
3 (1000,1000) (1975,0) 92.34 5.82 7272 945.49
Subproblem 2
1 (0,845) ( 2000,260) 91.76 87.18 7211 975.32
2 (0,720) (2000,195 ) 93.48 88.33 7187 973.49
3 (0,330) (2000,780 ) 86.61 92.91 7185 974.04
T able B.22: Computation al results for D esign C1 in a large-sized w arehouse (W=200, H=100).
159
W H S E  L ( ? )  R ( ? ) # P allets E [ C ]
Small 105 53 (525,530) (1050,0) 93.48 165.10 1865 490.96
Medium 156 78 (780,780) (1560,3) 96.92 167.39 4328 728.48
Large 205 103 (1025,1030) (2050,6) 96.92 165.68 7755 961.30
T able B.23: P erfo rmance of the expanded Design C1 for di e ren t w arehouse sizes.
W H S E  L ( ? )  R ( ? ) # P allets E [ C ]
Small 100 50 (500,500) (1000,0) 90 180 1750 492.34101 52 (505,520) (1010,0) 90 180 1875 488.40
Medium 150 75 (750,750) (1500,3) 90 180 4120 728.86151 77 (755,770) (1510,3) 90 180 4119 726.08
Large 200 100 (1000,1000) (2000,6) 90 180 7490 966.01201 102 (1005,1020) (2010,6) 90 180 7758 963.81
T able B.24: P erfor mance of mo di ed De s i gn C1 for di eren t w arehouse size s .
160
B.4.2 Results for Design C2
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (500 ,500) (1000,0) (500, 0) (0,500 ) 160.43 108.86 160.43 1539 423.07
2 (500,500 ) (1000,7) ( 500 ,0) (0,495) 162.15 109.43 163.29 1538 426.41
3 (500,500 ) (1000,483 ) (500,0) (0,485) 163.29 110.58 164.44 1539 425.29
Subproblem 1
1 (500,500) (995,0) (500 ,0) (0,492) 165.58 110.01 159.28 1544 425.99
2 (500,500 ) (1000,0) (500,0) (0,500) 162.72 108.29 159.86 1541 423.44
3 (500,500 ) (996,0) (500,0) (0,380) 164.44 110.58 162.72 1552 427.36
Subproblem 3
1 (0,300) (500,500) (0,280) (500,0) 21.77 87.66 151.83 1584 456.28
2 (0,250) (500,500) (0,245) (500,0) 24.06 92.25 155.27 1585 455.66
3 (0,265) (500,500) (0,234) (500,0) 21.77 86.52 153.55 1580 457.83
T able B.25: Computational results for Design C2 in a small-sized w arehouse (W=100, H=50).161
Run S 1 E 1 S 2 E 2  L  C  R # P allets E [ C ]
Subproblem 2
1 (750 ,750) (1500,5) (750,0) (0,750) 161.57 107.72 160.43 3790 639.13
2 (750,750 ) (1500,0) (750,0) (0,750) 160.43 108.86 159.86 3790 640.60
3 (750,750 ) (1500,10) (750,0) (0,735) 159.86 112.30 155.27 3788 639.79
Subproblem 1
1 (750 ,750 ) (1490,0) (750,0) (0,735) 160.43 108.86 161.00 3793 640.96
2 (750,750 ) (1465,0) (750,0) (0,750) 161.57 109.43 163.87 3802 639.98
3 (750,750 ) (1485,0) (750,0) (0,725) 163.29 110.58 162.15 3802 640.59
Subproblem 3
1 (0,375) (750,750) (0,370) (750,0) 22.35 87.09 157.56 3867 688.89
2 (0,400) (750,750) (0,395) (750,0) 25.21 85.37 161.00 3875 691.52
3 (0,360) (750,750) (0,360) (750,0) 23.49 85.94 158.71 3874 690.91
T able B.26: Computationa l results for Desig n C2 in a medium-sized w arehouse (W=150, H=75).
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (1000,1000) (2000,0) (1000,0) (0,1000) 158.71 111.73 157.56 7034 855.07
2 (1000,1000) (2000,45) (1000,0) (0,1000) 156.99 108.29 155.84 7053 858.72
3 (1000,1000) (2000,30) (1000,0) (0,990) 158.14 110.01 160.43 7052 855.98
Subproblem 1
1 (1000,1000) (2000,0) (1000,0) (0,990) 161.00 110.01 159.86 7039 858.29
2 (1000,1000) (1980,0) (1000,0) (0,1000) 163.87 112.87 166.16 7043 857.75
3 (1000,1000) (1995,0) (1000,0) (0, 995) 166.16 107.72 166.73 7047 857.72
Subproblem 3
1 (0,525) (1000,1000) (0,515) (1000,0) 25.78 87.09 156.42 7153 921.89
2 (0,500) (1000,1000) (0,500) (1000,0) 22.92 87.66 158.14 7144 921.05
3 (0,490) (1000,1000) (0,485) (1000,0) 24.06 86.52 156.99 7160 922.17
T able B.27: Computation al results for D esign C2 in a large-sized w arehouse (W=200, H=100).
162
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Small 107 55 (535,550) (1070,0) (535,0) (0,550) 160.43 108.86 160.43 1876 462.68
Medium 159 79 (795,790) (1590,5) (795,0) (0,790) 160.43 107.72 161.57 4312 676,55
Large 208 105 (1040,1050) (2080,0) (1040,0) (0,1050) 157.56 111.73 158.71 7755 893.42
T able B.28: P erfo rmance of the expanded Design C2 for di e ren t w arehouse sizes.
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Small 100 50 (500,500) (1000,0) (500,0) (0,500) 180 108.86 180 1616 438.68104 54 (525,540) (1040,0) (525,0) (0,540) 180 108.86 180 1875 464.47
Medium 150 75 (750,750) (1500,5) (750,0) (0,750) 180 107.72 180 3928 657.59157 78 (785,770) (1570,5) (785,0) (0,770) 180 107.72 180 4319 685.45
Large 200 100 (1000,1000) (2000,0) (1000,0) (0,1000) 180 111.73 180 7202 878.54207 103 (1035,1030) (2060,0) (1035,0) (0,1030) 180 111.73 180 7760 906.47
T able B.29: P erfor mance of mo di ed De s i gn C2 for di eren t w arehouse size s .
163
B.5 Computational Results in Design Problem D
Results for the Design D0
Run  R ( ? ) # P allets E [ C ]
Small
1 32.75 1748 615.77
2 30.46 1751 616.05
3 33.90 1747 615.69
Medium
1 32.75 4122 925.2
2 31.03 4125 925.57
3 33.90 4119 925.48
Large
1 32.18 7492 1234.68
2 33.32 7494 1234.95
3 31.60 7493 1234.97
T able B.30: Computational results of Design D0 for di e ren t w arehouse sizes.
W H  R ( ? ) # P allets E [ C ]
Small 103 51.5 33.9 1876 634.59
Medium 153 76.5 32.75 4317 943.79
Large 203 101.5 32.18 7758 1253.35
T able B.31: P erfo rmance of the expanded Design D0 for di ere n t w arehouse sizes.
164
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (0,500) (1000,300) (0,225) (1000,0) 43.64 45.35 42.34 1449 612.97
2 (0,500) (1000,295 ) (0,325) (1000,0) 44.78 43.64 44.06 1479 612.58
3 (0,500) (1000,335 ) (0,155) (1000,0) 42.92 35.04 43.06 1450 610.22
Subproblem 2
1 (0,500) (95,0) (1000,0) (920 ,500) 44.69 40.68 39.53 1619 613.6
2 (0,500) (140,0) (1000,0) (925 ,500) 40.11 38.96 40.68 1619 612.08
3 (0,500) (220,0) (1000,0) (835,500) 41.83 40.68 42.97 1593 614.57
Subproblem 3
1 (0,150) ( 300 ,500) (1000,400) (280,0) 33.32 29.31 31.03 1555 618.81
2 (0,40) (270,500) (1000,470) (350,0) 30.46 27.59 34.47 1561 622.44
3 (0,95) (360,500) (1000,450) (120,0) 31.60 28.74 30.46 1511 623.18
T able B.32: Computational results for Design D2 in a small-sized w arehouse (W=100, H=50).
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (0,750) (1500,630) (0,145) (1500,0) 37.91 41.92 36.76 3686 920.46
2 (0,750) (1500,675 ) (0,95) (1500,0) 36.19 37.91 35.04 3731 917.74
3 (0,750) (1500,560 ) (0,210) (1500,0) 34.47 31.60 33.32 3687 916.88
Subproblem 2
1 (0,750) (110,0) (1500,0) (1460,750) 41.25 38.96 44.12 3921 921.75
2 (0,750) (60,0 ) (1500,0) (1270,750 ) 45.84 43.54 44.69 3921 930.5
3 (0,750) (180,0) (1500,0) (1350,750 ) 46.41 42.40 46.98 3895 927.39
Subproblem 3
1 (0,130) (395,750) (1500,375) (110,0) 29.31 26.45 28.74 3794 938.42
2 (0,180) (450,750) (1500,405) (70,0 ) 25.30 24.16 27.02 3781 946.07
3 (0,145) (430,750) (1500,430) (95,0 ) 28.17 25.87 31.60 3773 933.52
T able B.33: Computationa l results for th e Design D2 in a medium-sized w arehouse (W=150, H=75).
165
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 1
1 (0,1000) (2000,690) (0,270) (2000,0) 40.06 32.75 39.05 6870 1224.2
2 (0,1000) (2000,550 ) (0,400) (2000,0) 37.48 33.90 36.19 6830 1225.1
3 (0,1000) (2000,875 ) (0,110) (2000,0) 34.61 32.18 33.32 6965 1224.89
Subproblem 2
1 (0,1000) (160,0) (2000,0) (1810,1000) 47.56 46.41 45.26 7196 1250.22
2 (0,1000) (250,0) (2000,0) (1900,1000) 43.54 41.83 44.12 7196 1237.66
3 (0,1000) (130,0) (2000,0) (1905,1000) 44.69 45.84 42.97 7210 1248.49
Subproblem 3
1 (0,100) (410,1000) (2000,400) (80,0) 29.89 28.17 30.46 7030 1241.57
2 (0,125) (400,1000) (2000,360) (95,0 ) 26.45 25.30 28.74 7036 1250.38
3 (0,85) (425,1000) (2000,420) (130,0) 29.31 27.59 31.03 7031 1240.77
T able B.34: Computationa l results for De s ig n D2 in a large-sized w arehouse (W=200, H=100).
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Small 109 56 (0,560) (1090,331) (0,169) (1090,0) 41.92 35.04 43.06 1870 670.31
Medium 162 80 (0,800) (1620,703) (0,281) (1620,0) 34.47 31.6 33.32 4321 987.73
Large 210 106 (0,1060) (2100,674) (0,189) (2100,0) 43.06 32.75 39.05 7753 1289.04
T able B.35: P erfo rmance of the expanded Design D2 for di ere n t w arehouse sizes.
166
B.6 Computational Results in Design Problem E
B.6.1 Results for Design E1
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (0,250) (1000,250 ) 94.63 85.46 1623 559.14
2 (0,250) (1000,250 ) 92.91 86.61 1634 559.36
3 (0,250) (1000,250 ) 94.06 84.32 1629 559.58
Subproblem 5
1 (500,0) ( 500 ,500) 176.56 2.96 1697 560.58
2 (500,0) (500,500) 171.98 6.39 1694 562.05
3 (500,0) (500,500) 5.82 5.25 1689 562.40
Subproblem 1
1 (0,250) (1000,500 ) 93.48 54.52 1602 598.33
2 (0,250) (935, 500) 92.34 55.67 1609 601.32
3 (0,250) (950,500) 94.63 53.38 1602 602.41
T able B.36: Computational results for Desig n E1 in a small-sized w arehouse (W=100, H=50).
B.6.2 Results for Design E2
167
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (0,375) (1500,375 ) 94.06 84.89 3917 840.15
2 (0,375) (1500,375 ) 95.78 84.32 3931 840.42
3 (0,375) (1500,375 ) 99.21 82.60 3906 841.94
Subproblem 5
1 (750 ,0) (750,750) 177.14 2.96 4025 841.25
2 (750,0) (750,750) 175.99 1.81 4041 842.22
3 (750,0) (750,750) 3.53 1.81 4027 842.71
Subproblem 1
1 (0,375) (1495,750 ) 92.34 51.66 3888 897.13
2 (0,375) (1495,750 ) 93.48 55.10 3882 897.33
3 (0,375) (1495,750 ) 94.06 53.95 3886 899.25
T able B.37: Computationa l results for Desig n E1 in a medium-sized w arehouse (W=150, H=75).
Run S E  L ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (0,500) (2000,500 ) 93.48 86.61 7220 1121.08
2 (0,500) (2000,500 ) 95.20 84.32 7224 1121.94
3 (0,500) (2000,500 ) 96.92 83.74 7208 1122.58
Subproblem 5
1 (1000,0) (1000,1000) 176.56 3.53 7390 1121.71
2 (1000,0) (1000,1000) 177.14 2.38 7369 1121.86
3 (1000,0) (1000,1000) 2.96 4.10 7382 1123.60
Subproblem 1
1 (0,500) (2000,1000) 93.48 54.52 7170 1195.13
2 (0,500) (1995,1000) 95.20 52.80 7168 1195.97
3 (0,500) (1995,1000) 94.06 49.37 7176 1199.00
T able B.38: Computation al results for Design E1 in a large-sized w arehouse (W=200, H=100).
168
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (500,0) (50,500) (450, 0) (500 ,500) 5.73 137.51 6.30 1554 543.70
2 (500,0) (35,500) (475,0) (500,500) 6.88 136.36 8.02 1558 545.30
3 (500,0) (20,500) (460, 0) (500,500) 5.73 140.37 173.03 1548 544.71
Subproblem 3
1 (0,15) (500, 500) (1000,470) (500, 0) 173.70 48.79 173.12 1541 543.80
2 (0,25) (500, 500) (1000,475 ) (500, 0) 62.54 48.22 59.68 1548 545.00
3 (0,250) (950, 500) (1000,250 ) (75,0 ) 94.63 42.49 93.48 1459 547.05
Subproblem 1
1 (0,250) (1000,200 ) (0,250) (1000,325) 85.94 86.52 85.37 1503 565.02
2 (0,250) (1000,250 ) (0,250) (1000,385 ) 86.52 84.80 87.09 1503 556.86
3 (0,250) (1000,190 ) (0,250) (1000,405 ) 87.66 88.81 87.09 1484 566.33
T able B.39: Computational results for Desig n E2 in a small-sized w arehouse (W=100, H=50).
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (750,0) (65,750) (430,0) (750 ,750) 5.73 138.08 5.73 3825 818.73
2 (750,0) (100, 750) (410, 0) (750, 750) 174.18 146.10 6.88 3820 823.08
3 (750,0) (90,750) (395, 0) (750 ,750) 6.30 142.67 7.45 3816 821.07
Subproblem 3
1 (0,55) (750, 750) (1500,725 ) (750, 0) 173.70 46.50 174.27 3806 820.22
2 (0,15) (750, 750) (1500,740 ) (750, 0) 174.84 45.35 172.55 3773 819.09
3 (0,0 ) (750, 750) (1500,735 ) (750, 0) 172.55 45.93 173.12 3790 819.74
Subproblem 1
1 (0,375) (1500,365 ) (0,375) (1500,500 ) 86.52 85.94 84.22 3724 839.36
2 (0,375) (1500,376 ) (0,375) (1500,450 ) 85.37 84.80 86.52 3746 906.93
3 (0,375) (1500,180 ) (0,375) (1500,350 ) 85.94 84.80 85.37 3716 841.12
T able B.40: Computationa l results for Desig n E2 in a medium-sized w arehouse (W=150, H=75).
169
Run S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Subproblem 2
1 (1000,0) (110 ,1000) (900, 0) (1000,1000) 4.58 136.94 6.30 7079 1089.08
2 (1000,0) (195, 1000 ) (830 ,0) (1000,1000) 6.30 138.08 6.88 7111 1091.08
3 (1000,0) (245, 1000 ) (800 ,0) (1000,1000) 5.73 140.37 5.16 7121 1090.48
Subproblem 3
1 (0,40) (1000,1000) (2000,1000) (1000,0) 171.98 43.64 173.12 7058 1091.78
2 (0,20) (1000,1000) (2000,990) (1000,0) 172.55 44.78 171.41 7059 1093.03
3 (0,500) (960, 1000 ) (2000,500 ) (55,0 ) 94.63 39.05 95.78 6859 1098.88
Subproblem 1
1 (0,500) (2000,490 ) (0,500) (2000,715) 85.94 85.37 86.52 6930 1112.89
2 (0,500) (2000,400 ) (0,500) (2000,660 ) 87.66 86.52 85.37 6923 1125.6
3 (0,500) (2000,515 ) (0,500) (2000,690 ) 86.52 86.52 84.80 6938 1117.31
T able B.41: Computation al results for Design E2 in a large-sized w arehouse (W=200, H=100).
W H S 1 E 1 S 2 E 2  L ( ? )  C ( ? )  R ( ? ) # P allets E [ C ]
Small 108 54 (540,0) (54,540) (486,0) (540,540) 5.73 137.51 6.30 1854 589.54
Medium 159 79 (795,0) (73,790) (721,0) (795,790) 5.73 138.08 5.73 4290 866.34
Large 206 106 (1030,0) (117,1060) (954,0) (1060,1030) 4.58 136.94 6.30 7779 1134.35
T able B.42: P erf ormance of the expanded Design E2 for di er en t w arehouse sizes.
170
Appendix C
C.1 One Cross Aisle Designs with P&D Points at the Bottom of the
Warehouse
# P&D points E[C] # Pallets S E  L(?)  R(?)
1 399.68 1667 (500,0) (500,500) 135.60 44.40
3 416.39 1667 (500,0) (520,500) 130.39 45.66
5 428.94 1667 (500,0) (510,500) 127.49 52.20
7 443.86 1669 (500,0) (525,500) 126.65 53.37
9 458.89 1674 (500,0) (550,500) 120.29 60.26
11 475.59 1683 (500,0) (520,500) 117.00 63.51
13 492.07 1674 (500,0) (510,500) 116.11 63.89
15 505.24 1623 (400,500) (1000,0) 110.07 86.60
17 520.14 1618 (350,500) (1000,0) 113.99 87.48
19 537.80 1619 (300,500) (1000,0) 116.91 88.06
Table C.1: The heuristic results calculated by the PSO algorithm for small size
warehouses with one  oating cross aisle and multiple P&D points at the bottom of
the warehouse.
# P&D points Traditional Chevron:equal size Chevron:equal capacity
1 500.00 398.81 416.67
3 501.67 416.94 434.82
5 505.00 433.59 451.01
7 510.00 450.95 467.96
9 516.67 468.97 486.05
11 525.00 487.98 504.96
13 535.00 507.66 524.16
15 546.67 527.41 543.86
17 560.00 547.19 563.87
19 575.00 567.01 583.70
Table C.2: The expected travel distance in a small-sized traditional and equally sized
and equally capacitated Chevron designs with multiple P&D points at the bottom
of the warehouse. The equally capacitated Chevron has a width of 105 and height
of 52.
171
(W=100, H=50) (W=105, H=53)
# P&D p oin ts E [ C ] # P allets S E  L ( ? )  R ( ? ) E [ C ] # P allets
1 399 .68 1667 (500,500) (500,0) 135.60 44.40 420.33 1880
3 415 .33 1670 (500,500) (500,0) 130.38 49.62 437.11 1876
5 428 .38 1672 (500,500) (500,0) 127.71 53.06 451.86 1877
7 443 .75 1669 (500,500) (500,0) 124.21 55.79 465.61 1876
9 458 .49 1680 (500,500) (500,0) 120.97 60.86 479.96 1886
11 473.78 1684 (500,500) (500,0) 116.68 63.23 496.53 1872
13 491.13 1674 (500,500) (500,0) 116.72 63.28 513.09 1872
15 509.46 1682 (500,500) (500,0) 111.81 68.19 531.78 1900
17 528.77 1676 (500,500) (500,0) 111.73 68.27 550.06 1895
19 547.44 1680 (500,500) (500,0) 111.89 68.32 569.53 1890
T able C.3: The heuristic results calculated b y the PSO algorithm for small size w arehouses with a  xed inserted cross aisle and
m ultiple P&D p oin ts at the b o ttom of the w arehouse.172
C.2 Tw o Cross Aisle Designs with P&D P oin ts at the Bottom of the W arehouse
# P&D p oin ts E [ C ] # P allets S 1 E 1 S 2 E 2  L ( ? )  R ( ? )  C ( ? )
3 407.7 2 1587 (450,0) (105,500) (550,0) (895,500) 151.32 28.68 90
5 420.6 1 1571 (450,0) (60,500) (550,0) (940,500) 148.78 32.69 90
7 434.3 3 1578 (423,0) (17,500) (577,0) (983,500) 134.03 47.11 90
9 452.9 2 1596 (400,0) (14,500) (600,0) (986,500) 124.36 56.25 90
11 467.18 1590 (388,0) (17,500) (61 2,0) (983,500) 125.08 56.81 90
13 483.55 1600 (375,0) (37,500) (62 5,0) (963,500) 120.69 58.18 90
15 500.73 1550 (500,0) (0,475) (500,0) (1000,475) 92.86 87.81 90
17 516.50 1547 (500,0) (0,470) (500,0) (1000,470) 92.54 88.01 90
19 532.86 1548 (500,0) (0,460) (500,0) (1000,460) 93.03 88.13 90
T able C.4: The heuristic results calculated b y the PSO algorithm for small size w arehouses with t w o inserte d cross aisles and
m ultiple P&D p oin ts at the b o ttom of the w arehouse.
173
(W=100, H=50) (W=108, H=54)
# P&D points E[C] E[C]
1 392.46 421.66
3 416.79 444.68
5 436.62 465.02
7 456.86 484.69
9 476.61 504.86
11 496.88 524.90
13 516.68 544.93
15 536.39 564.92
17 555.66 584.31
19 574.13 603.65
Table C.5: The expected travel distances in Leaf designs. They provide 1575 stor-
age locations in a small size, and 1875 storage locations in an expanded small size
warehouse.
174
# P&D p oin ts E [ C ] # P allets S 1 E 1 S 2 E 2  L ( ? )  R ( ? )  C ( ? )
3 437.1 7 1885 (490,0) (105,540) (590,0) (975,54 0) 151.32 28.68 90
5 455.1 8 1869 (490,0) (60,540) (590,0) (1020,540) 148.78 32.69 90
7 474.4 4 1887 (463,0) (17,540) (617,0) (1063,540) 134.03 47.11 90
9 489.5 7 1879 (440,0) (14,540) (640,0) (1066,540) 124.36 56.25 90
11 505.98 1883 (428,0) (17,540) (65 2,0) (1063 ,540) 125.0 8 56.81 90
13 524.44 1876 (425,0) (37,540) (65 5,0) (1043 ,540) 120.6 9 58.18 90
15 539.18 1879 (540,0) (0 ,475 ) (540,0) (1040,475 ) 92.86 87.81 90
17 554.37 1883 (540,0) (0 ,470 ) (540,0) (1040,470 ) 92.54 88.01 90
19 570.75 1890 (540,0) (0 ,460 ) (540,0) (1040,460 ) 93.03 88.13 90
T able C.6: Th e exp ecte d tra v el distance in the t w o cross aisle prop osed designs for equally capacita ted w arehouses with m ultiple
P&D p oi n ts at the b ottom of the w arehouse (W=108, H=54).
175
C.3 One Cross Aisle Designs with P&D P oin ts at the Bottom and the T op of the W arehouse
(W=100, H=50) (W=105, H=53)
# P&D p oin ts (p er side) E [ C ] # P allets S E  L ( ? )  R ( ? ) E [ C ] # P allets
1 464.66 1646 (500 ,500) (1000,0) 93.47 165.10 490.96 1865
3 469.41 1647 (550 ,500) (1000,0) 93.51 168.80 495.76 1871
5 475.53 1658 (600 ,500) (1000,0) 93.72 169.43 501.24 1878
7 483.88 1657 (600 ,500) (1000,0) 92.79 170.05 509.28 1876
9 493.90 1656 (600 ,500) (1000,0) 93.94 170.62 519.68 1873
11 502.39 1669 (500,500) (1 000,200) 96.69 94.28 528.40 1871
13 513.71 1674 (500,500) (1 000,200) 95.81 94.38 538.78 1879
15 526.87 1661 (550,500) (1 000,150) 95.37 94.78 551.67 1878
17 540.08 1656 (600,500) (1 000,150) 95.95 98.60 566.67 1887
19 553.48 1647 (650,500) (1 000,150) 95.43 98.03 582.37 1896
T able C.7: The heuristic results calculated b y the PSO algorithm for small size w arehou s e s with t w o cross aisles and m ul tiple P&D
p oin ts lo cated b oth at the b ottom and the top sides of the w arehouse.
176
(W=100, H=50) (W=105, H=53)
# P&D points (per side) E[C] E[C]
1 464.66 490.96
3 474.07 500.28
5 481.22 509.08
7 490.45 518.73
9 501.09 528.65
11 512.80 538.70
13 524.62 550.22
15 537.69 562.88
17 551.71 576.34
19 566.46 590.42
Table C.8: The expected travel distances in the Design C1 for varying number of
P&D points at the bottom and the top sides of the warehouse.
177
C.4 Tw o Cross Aisle Designs with P&D P oin ts at the Bottom a nd the T op of the W arehouse
(W=100, H=50) (W=108, H=54)
# P&D p oin ts (p er side) E [ C ] # P allets S 1 E 1 S 2 E 2  L ( ? )  R ( ? )  C ( ? ) E [ C ] # P allets
1 423.07 1539 (0,0 ) (500,500) (1000,500) (500,0) 160.43 160.43 108.86 462.68 1876
3 435.23 1564 (0,50 ) (450,500) (1000,450) (550,0) 159.17 159.17 109.79 471.35 1878
5 448.99 1572 (0,50 ) (400,500) (1000,450) (600,0) 159.71 159.71 114.32 485.38 1877
7 459.99 1572 (0,50 ) (400,500) (1000,450) (600,0) 159.71 159.71 105.49 494.24 1879
9 473.62 1564 (0,25 ) (400,500) (1000,475) (600,0) 160.79 160.79 106.53 509.24 1874
11 484.15 1569 (0,125 ) (500,500) (1000,475) (700 ,0) 114.00 153.82 105.47 520.39 1876
13 498.54 1567 (0 ,45 ) (300 ,500) (1000,400) (440,0) 103.35 162.96 109.17 531.80 1877
15 513.10 1546 (0 ,35 ) (740 ,500) (1000,470) (710,0) 161.39 105.77 107.36 544.97 1867
17 523.99 1537 (0 ,55 ) (715 ,500) (1000,470) (540,0) 134.63 101.42 111.59 557.05 1865
19 540.12 1534 (0 ,25 ) (250 ,500) (1000,410) (155,0) 99.78 1 52.66 107.46 573.77 1865
T able C.9: The heuristic results calculated b y the PSO algorithm for small size w arehouses with t w o inserte d cross aisles and
m ultiple P&D p oin ts at the b o ttom and the top sides of the w are house.
178
# P&D points (per side) (W=100, H=50) (W=107, H=55)
1 423.07 462.68
3 441.49 480.97
5 458.12 497.65
7 472.09 512.89
9 485.05 525.17
11 499.38 539.16
13 513.40 553.52
15 527.02 566.84
17 540.81 580.56
19 554.11 594.03
Table C.10: The expected travel distances in the Design C2 for varying number of
P&D points at the bottom and the top sides of the warehouse. They provide 1539
storage locations in a small size, and 1876 storage locations in an expanded small
size warehouse.
179