Aisle Designs for Unit-Load Warehouses with
Multiple Pickup and Deposit Points
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
Goran Ivanovi?c
Certificate of Approval:
Russell D. Meller
Professor
Industrial Engineering
University of Arkansas
Kevin R. Gue, Chair
Associate Professor
Industrial and Systems Engineering
Emmett J. Lodree, Jr.
Assistant Professor
Industrial and Systems Engineering
George T. Flowers
Interim Dean, Graduate School
Aisle Designs for Unit-Load Warehouses with
Multiple Pickup and Deposit Points
Goran Ivanovi?c
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
December 17, 2007
Aisle Designs for Unit-Load Warehouses with
Multiple Pickup and Deposit Points
Goran Ivanovi?c
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Goran Ivanovi?c, son of Stoi?sa and Milanka Ivanovi?c, was born on April 9th, 1979, in
?Cuprija, Serbia. Soon after, his family relocated to a nearby town of Jagodina, where he
spent his childhood and teenage years. He studied at University of Belgrade, School of
Traffic and Transportation Engineering, and graduated from Department of Logistics in
June 2004. Prior to entering Auburn University in May 2006, he worked as a logistics
consultant.
iv
Thesis Abstract
Aisle Designs for Unit-Load Warehouses with
Multiple Pickup and Deposit Points
Goran Ivanovi?c
Master of Science, December 17, 2007
(B.S., University of Belgrade, 2004)
77 Typed Pages
Directed by Kevin Gue
This thesis focuses on unit load warehouses having multiple pickup and deposit (P&D)
locations on the warehouse dock(s) ? i.e., a P&D region. Our aim is to design a cross aisle
structure that will reduce the cost of operating a unit load warehouse. We concentrate on
the labor cost of material handling, or more specifically, on the travel cost associated with
unit load putaway and retrieval. We define two ?candidate shapes.? Namely, the Flying-V
shape and ? shape. We state a nonlinear optimization problem to find an improved cross
aisle configuration by minimizing the expected cost of a single pickup and retrieval task. We
separately carry out optimization for each candidate shape and choose the more economical
one. Our results are a sort of worst case analysis for the designs we consider and the benefits
from our solutions range typically from 3% to 6.5%.
The second part of this thesis is concerned with applications of the same design prin-
ciples to floor storage. A less rigorous, but effective approach is developed. We devise a
number of aisle and lane designs and apply them to a realistic setting.
v
Acknowledgments
Serbian saying goes: ?Man without a woman is like a rotten tree stump by the side of
the road.? I thank my wife Anica, for all her love, trust and patience.
If I try to write a ?real acknowledgement? to my adviser, Dr. Gue, it might end up
either being too formal, or too saccharine. I am most grateful for his help and support.
vi
Style manual or journal used Latex: A Document Preparation System : User?s Guide
and Reference Manual, by Leslie Lamport (together with the style known as ?aums?).
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file aums.sty.
vii
Table of Contents
List of Figures ix
1 Aisle Designs for Unit Load Warehouses with Selective Pallet Racks 1
1.1 Unit load warehouse and aisle design in practice and literature . . . . . . . 3
1.2 Model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Upper and lower bounds for a single pickup and retrieval travel task . . . . 8
1.3.1 Continuous and discrete upper bounds ? Rectilinear travel . . . . . 9
1.3.2 Continuous and discrete lower bounds ? Travel-by-flight . . . . . . 11
1.4 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Flying-V cross aisle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 ?-shaped cross aisle . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.3 The most economical cross aisle . . . . . . . . . . . . . . . . . . . . . 27
1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.6 A flow-through crossdock . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 Duality of representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.8 Some practical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Application of aisle design principles to floor storage ? a case study 42
2.1 Problem background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Simple floor storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.1 Side putaway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.2 Two P&D points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 Alternative layout of the facility . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Bibliography 65
Appendix ? Tables of savings over orthogonal aisle design 66
viii
List of Figures
1.1 Model assumptions ? warehouse layout with a cross aisle . . . . . . . . . . 6
1.2 Continuous vs. discrete upper and lower travel bound . . . . . . . . . . . . 9
1.3 Flying-V shape - scheme of travel patterns and related picking regions . . . 15
1.4 ? shape - scheme of travel patterns and related picking regions . . . . . . . 22
1.5 ? shape ? a rarely used and complicated travel pattern . . . . . . . . . . . 24
1.6 Typical Flying-V (n = 39, w = 2, a = 5, h = 100) and ? (n = 21, w = 2,
a = 5, h = 100) shape cross aisles . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7 Example comparison of savings over orthogonal design for Flying-V and ?-
shape layouts, and Travel-by-flight . . . . . . . . . . . . . . . . . . . . . . . 30
1.8 Savings over orthogonal design for Flying-V and ?-shape layouts . . . . . . 31
1.9 Example of warehouse configurations (w = 1.5, n = 11?35) for which it is
better not to have a cross aisle . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.10 Comparisson of savings over orthogonal design for crossdock configuration,
Flying-V and ?-shape layouts . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1 Current layout. Arrows pointing outward from the storage space represent
P&D points with retrieval activity. Arrow pointing into the storage space
denotes inflow of goods and putaway activity. Letters, or combinations of
letters and numbers represent ?labels? of the storage blocks. . . . . . . . . . 44
2.2 One P&D point with putaway and picking from the same side . . . . . . . . 48
2.3 One P&D point with putaway and picking at different sides . . . . . . . . . 50
2.4 Two P&D point with putaway and picking from the same side . . . . . . . 53
2.5 Two P&D points with putaway and picking at different sides . . . . . . . . 54
2.6 Improved Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.7 Travel patterns in a floor storage with diagonal lanes . . . . . . . . . . . . . 59
2.8 Travel patterns in a floor storage with diagonal lanes and cut-through aisles 62
ix
Chapter 1
Aisle Designs for Unit Load Warehouses
with Selective Pallet Racks
Very rarely is it possible to transport, sell, consume or utilize goods immediately after
they have been produced. Instead, often times it is necessary to store the goods temporar-
ily and wait for the right moment to introduce them for consumption or utilization. This
interruption of the natural product flow through the supply chain usually appears to be a
result of interaction of numerous factors. Production, logistics and distribution systems are
becoming more and more complex, and at an unprecedented rate. Even the operations tak-
ing place in basic building blocks of logistics systems, such as warehouses, have been raised
to a new level by numerous technological advances over the course of the last two decades.
Notwithstanding, RFID, sophisticated order-picking systems, warehouse management sys-
tems, on-board AGV computers and other technological novelties, have not changed the
need to put a pallet on the floor (or in a rack) and let it dwell there for some time. Apart
from the technological changes warehouses have undergone, there is a constantly increasing
pressure from both upstream and downstream supply chain structures to adopt numerous
value-added activities: final product assembly, thermal processing, labeling and relabeling,
etc. However, value-added activities do not change the fundamentally unproductive nature
of the warehouse rationale ? they only make opportune use of the unfavorable environ-
ment, that is a warehouse. That being the case, we can freely say that reasons to have a
warehouse have remained unchanged for decades, if not centuries.
1
Due to physical, chemical or nutritional properties of the product it is often desirable or
inevitable to store and therefore preserve the product for future use. Examples may include
various perishable goods, such as agricultural and pharmaceutical products. Secondly, tim-
ing and mismatch between production and consumption processes dictate that commodities
be stored. This is almost mandatory for goods with highly pronounced seasonal production
or consumption characteristics. The former are impossible to sell right after being finalized
and have to be stored since they are usually consumed at a slower rate. As for the latter,
it is impossible to meet seasonal peaks of demand without keeping significant quantity of
a product in stock. At last, uncertainty in demand or supply usually leads to buildup of a
so-called safety stock. This is ?just-in-case? inventory, held in order to be able to respond
to unexpected changes in demand or sudden interruptions in supply. It is usually caused by
the inability to forecast market caprices and uncertainty inherent in the outcome of even
the most carefully planned human activities.1
These are all but not the only reasons to store goods in a warehouse. Nevertheless,
they are sufficient to sketch the most important characteristics of warehousing: Usually one
does not garner and store because he wants, but because he must. Hence, warehousing is in
essence an unproductive endeavor. Modern times might have changed the technology of, but
not the rationale for warehousing. In addition, the last two decades have witnessed several
strategies in industry that attempt to reduce the amount of the unproductive inventory. In
the realm of warehousing, crossdocking is the most prominent one. By means of coordination
1Note that the order in which we present motives to store the goods is very likely to correspond chrono-
logically to the evolution of warehousing. In the ancient times people stored goods to protect them from
decay. With development of markets and trading it became desirable to store, say in the Fall, in order to
trade in the Spring. At last, with emergence of the first supply chains (e.g., silk road) it became prudent
for the traders to protect themselves from the highly risky ?supply chain failures? of those times.
2
and aggregation, crossdocking aims to cut the transportation and inventory holding cost of
a traditional supply chain.
Prior to the last two decades of the Twentieth Century, warehousing costs have not
raised significant concern with the industry. Since then however, decreasing profit margins
of western economies have imposed a strong incentive to deal with many of the costs that
were previously considered peripheral. With time, this became the case with a large class
of warehousing and material handling problems. Consequently, we believe attention is
being given to warehouse design issues that would have until recently been regarded as
unimportant.
1.1 Unit load warehouse and aisle design in practice and literature
The main purpose of this work is to explore the opportunities for reduction of opera-
tional costs in a widely encountered class of warehouses. More specifically, we propose and
evaluate a number of alternative cross aisle designs for unit load warehouses. Our analysis
focuses on warehouses having multiple pickup and deposit (P&D) locations on the ware-
house dock(s) ? i.e., a P&D region. The motivation for this thesis stems from Gue and
Meller (2006), in which they explore a similar problem with only one P&D point.
Unit load storage is characterized by homogenous dimensions of the stored skus. A
typical unit of handling and storage is a pallet. Storage equipment commonly includes one-
deep and two-deep selective pallet racks, and the most common material handling device
is the counterbalance forklift truck. This type of warehouse is usually considered to be the
simplest, because the cost of operation is straightforwardly described by the overall cost of
stowing, keeping and retrieving the pallets. The unit load warehouse takes on a number of
3
forms depending on the logistics system it is supporting. Physically, it can be a stand-alone
facility or a part of a larger warehouse that houses other forms of storage in addition to
the unit load module. Functionally, the unit load warehouse serves as a finished goods
warehouse, a 3rd party transshipment warehouse, or a distribution center (DC).
Theoretical work on unit load storage design and operations can be traced back to
the 1960?s. In fact, Roberts and Reed (1972) claim that Francis (1967) was the first to
address warehouse design analytically. Traditional unit load design with orthogonal aisle
and rack structure has been discussed by Francis (1967). This work gives sufficient con-
ditions for optimal facility layout, under certain assumptions. Roberts and Reed (1972),
develop a comprehensive cost scheme for warehouse optimization. Their approach includes
both operational and construction cost of employing a certain bay structure. Yet, they
only describe one P&D point configurations with orthogonal rack and aisle structure. It is
suggested that their model can be used for warehouses with a P&D region, provided that
individual P&D points serve a dedicated set of racks and aisles.
First discussions on non-orthogonal aisle and rack layout come from Berry (1968) and
White (1972). Berry considered floor storage in particular, rather than unit load warehouses
in general. He devised a design with a diagonal aisle structure and showed that layouts which
minimize travel distance and maximize space utilization are different ones. White examined
radial aisle designs with a single P&D point dock, but neglected the issue of radial aisle
width. Also, picking aisles needed to access items deeper in storage space, i.e., ones not
directly accessible from the radial aisles, are not modeled.
4
Finally, Gue and Meller (2006) address the aisle design problem from a different per-
spective. They utilize a prescriptive modeling approach and formulate a non-linear opti-
mization function to find an improved non-orthogonal aisle structure for a single P&D point
problem. In doing so, they relax two ?unspoken design rules:?
1. Picking aisles must be straight, and they must be parallel.
2. If present, cross aisles must be straight, and they must meet picking aisles at right
angles.
The authors provide solutions for a range of warehouse sizes and compare them with tradi-
tional, orthogonal designs.
The challenge we are facing is more demanding, since Gue and Meller only deal with a
single P&D point. Geometry and operational procedures of P&D regions suggest different
shapes and more complicated travel patterns than those of single P&D point. We also
explore the potential for redesign of unit-load warehouses and DC?s by introducing a non-
standard cross aisle and thus relaxing the second ?unspoken rule.? We essentially generalize
the initial single P&D point model to provide solution for a more complicated case. In
addition, we address some practical aspects of new aisle designs.
1.2 Model assumptions
The P&D region appears in variety of forms across a number of warehouses and DC?s
we have visited. In our model (Figure 1.1), n P&D points are equally spaced along the
?bottom? of the warehouse. We assume n to be an odd number. It is important to note
that n equals both the number of picking aisles and the number of doors along the bottom of
the warehouse rectangle. Picking aisles are orthogonal to the bottom of the warehouse. For
5
Figure 1.1: Model assumptions ? warehouse layout with a cross aisle
the purpose of this work we define ?bottom? as the side of the rectangle with the shipping
dock. Strictly speaking, this is adequate only for those facilities where the entire volume
of shipping and receiving is concentrated along a single side of the warehouse rectangle. In
all other cases, for ease of representation, we would rotate the rectangle so that shipping
assumes the ?southern,? bottom part.
Besides equal spacing of dock doors, which is common in practice, we assume that
incoming and outgoing trailers choose among dock doors with equal probabilities. This is
not likely to be the case in real life since centrally located doors are closer to more good
storage locations than doors farther from the center (see Bartholdi and Hackman, 2007,
chap. 6). The reason for taking this approach is that trailer assignment can be considered
a form of optimization, and we wish to investigate only the impact of aisle structure on
travel cost. As we will show, our results are a sort of worst case analysis for the designs we
consider.
The storage policy is random, which means that the probability of making a pick (or a
stow) from (to) a randomly chosen storage location is equal to the probability of making a
pick from any other storage location. In other words, picking is uniformly distributed inside
6
the storage space. This assumption greatly simplifies the mathematical model. Similar to
the case with trailer assignment, more elaborate analysis of storage policies would lead to
loss of precision in judging the effect of new aisle structure on travel costs. It should be
recognized that few warehouses strictly employ random or dedicated storage. It is more
common to assign one storage policy to portions of a warehouse. Also, changes in storage
policy can be a matter of adaptation and evolution. According to some managers we spoke
with, random storage policy is common in the incipient stages of warehouse life. After
a ?warm up? period, material flow reaches ?steady state? and it becomes much easier to
decide upon the right storage policy.
Forklift drivers utilize P&D points as bases for their travels into the storage space and
back. All of our models assume a single command cycle. For incoming trailers this means
that a forklift unloads the pallet from the trailer, travels to a random part of storage area,
unloads the pallet into an empty rack cell and travels back to the same P&D point. For
outgoing trailers the process is reversed: the forklift travels to the randomly chosen storage
location, loads the pallet, travels back to the P&D point and unloads the pallet into a
trailer. An alternative is the dual command cycle which reduces deadheading but brings in
complications in forklift fleet supervision and guidance.
Our aim is to design an aisle structure that will reduce the cost of operating a unit load
warehouse. We concentrate on the labor cost of material handling, or more specifically, on
the travel cost associated with unit load putaway and retrieval. This cost is defined as the
average distance traveled to make a pick or a stow. Because of the single command cycle,
the same distances are traveled into the storage space and out of it, which enables us to
7
consider only the half of the actual distance crossed. In our case, that is going to be the
travel into the storage space, for both activities.
Unit load length is the adopted measure of distance. For easier representation, our
units have a square-shaped footprint. All activities in a warehouse revolve around the unit
load and we found that adopting it as a basic measure provides easier description.
1.3 Upper and lower bounds for a single pickup and retrieval travel task
Traditional orthogonal aisle and rack structures impose rectilinear travel patterns.
Travel distances resulting from these patterns are one way to describe the cost of making a
stow or a pick in such a warehouse. Gue and Meller (2006) claim that it is often inefficient
to use rectilinear travel patterns and that such travels can be made more economical.
Consider the following example: Two picks need to be made from a P&D point lo-
cated in the central bottom part of the warehouse to two storage locations in the left
warehouse half-space. Figure 1.2 depicts two possible travel patterns available to a picker.
Dashed lines represent the first ? rectilinear, whereas continuous lines connect the P&D
point and picking locations in a straight fashion ? with a Travel-by-flight path. In the
Euclidean geometry, a line connecting two points represents the shortest distance between
them. Therefore, minimum travel cost for any single pick up and retrieval task is achieved
if the travel path to the pick location is a straight line. Such is the case of a hypothetical
warehouse in which an order picker could fly directly to the desired pick location. While
travel-by-flight design is implausible, the expected distance to make a pick remains an ideal
lower bound for the cost of a single pickup and retrieval task. Traditional rectilinear travel
is the upper bound for the cost of a single pickup and retrieval travel task.
8
Figure 1.2: Continuous vs. discrete upper and lower travel bound
Both upper and lower bounds can be modeled as discrete or continuous. In the previous
example we concentrated on the left warehouse half-space which does not display any aisle,
rack or lane structure at all, while right half-space features a discrete, aisle-like layout.
Discrete modeling assumes picking activity in a clearly defined number of racks only, and is
modeled through sums of costs. Because there is no rack or aisle division, a continuous model
theoretically allows for picking of pallets with infinitesimally small dimensions, where travel
cost is modeled as an integral. Despite the apparent formal difference, our experiments
show that bounds perform almost identically.
1.3.1 Continuous and discrete upper bounds ? Rectilinear travel
Upper bounds result from travel paths where a picker is first moving along the bottom
of the rectangle and then up along the picking aisle until he reaches the desired pick location.
This pattern forms a rectilinear travel and holds for both continuous and discrete bound.
9
For a continuous bound, let A denote the width and let h be the height of a given
warehouse. If x1 is the x coordinate of a randomly chosen P&D point, and x2 and y2 are
the coordinates of a randomly chosen point in the storage space, we express the average
cost of a rectilinear single pickup or deposit travel as:
Uc = 1A2h
integraldisplay h
0
integraldisplay A
0
integraldisplay A
0
(|x1 ?x2|+y2)dx1dx2dy2
= A3 + h2.
For a discrete bound, let n and i be the total number of picking aisles in a warehouse,
and the index of a randomly chosen P&D point, respectively. Let a be the horizontal
distance between two consecutive picking aisles. We can express the average cost of a single
pickup or deposit travel as:
Ud = 1n
bracketleftBigg nsummationdisplay
i=2
i?1summationdisplay
j=1
1
nh
integraldisplay h
0
bracketleftBig
(i?j)a+x
bracketrightBig
dx+ 1nh
nsummationdisplay
i=1
integraldisplay h
0
xdx +
+
n?1summationdisplay
i=1
nsummationdisplay
k=i+1
1
nh
integraldisplay h
0
bracketleftBig
(k ?i)a+x
bracketrightBig
dx
bracketrightBigg
= a(n
2 ?1)
3n +
h
2 .
The two expressions have similar structure. Knowing that for the length of the warehouse
rectangle we have that A = a/2 + (n ? 1)a + a/2 = na, let us hypothesize a discrete
warehouse with an infinite number of picking aisles. In that case we obtain:
limn?? a(n
2 ?1)
3n =
A
3 .
10
So, for an infinitely wide discrete warehouse, discretization of picking space is irrelevant.
Our tests show that for realistic warehouse sizes this inaccuracy is less than one tenth of a
percent.
1.3.2 Continuous and discrete lower bounds ? Travel-by-flight
Lower travel bounds arise from a straight travel-by-flight pick. The average cost of
such a pick is the expected distance from a randomly chosen point on the bottom side of
the rectangle to a randomly chosen point somewhere in the storage space. Using the same
notation we obtain both bounds:
Lc = 1A2h
integraldisplay h
0
integraldisplay A
0
integraldisplay A
0
radicalbig
(x1 ?x2)2 +y22 dx1dx2dy2
= 112A2h
bracketleftBigg
2h4 +3A2h
radicalbig
A2 +h2 ?2h3
radicalbig
A2 +h2?
?2Ah3 ln(?A+
radicalbig
A2 +h2)+2Ah3 ln(A+
radicalbig
A2 +h2)?
?A4 ln Ah+?A2 +h2
bracketrightBigg
Ld = 1n
bracketleftBigg nsummationdisplay
i=2
i?1summationdisplay
j=1
1
nh
integraldisplay h
0
radicalbig
a2(i?j)2 +x2dx+ 1nh
nsummationdisplay
i=1
integraldisplay h
0
xdx +
+
n?1summationdisplay
i=1
nsummationdisplay
k=i+1
1
nh
integraldisplay h
0
radicalbig
a2(k ?i)2 +x2dx
bracketrightBigg
11
= 12n2h
bracketleftBigg nsummationdisplay
i=2
i?1summationdisplay
j=1
(h
radicalbig
h2 +a2(i?j)2 +a2(i?j)2 ln(h+
radicalbig
h2 +a2(i?j)2)?
?a2(i?j)2 ln(a(i?j)))+
nsummationdisplay
i=1
h2 +
+
n?1summationdisplay
i=1
nsummationdisplay
k=i+1
(h
radicalbig
h2 +a2(k ?i)2 +a2(k ?i)2 ln(h+
radicalbig
h2 +a2(k ?i)2)?
?a2(k ?i)2 ln(a(k ?i)))
bracketrightBigg
.
A similar but more complicated problem is faced by electrical engineers and power network
designers. The expected wire distance between two randomly chosen terminals in a rectangle
is equal to the problem of travel-by-flight distance between two randomly chosen storage
locations inside a warehouse. This problem is more complicated than ours in that the y-
coordinate of the first terminal point is randomized, as opposed to the y-coordinate of a
P&D point always being zero. This seemingly trivial added complexity produces a very
difficult quadruple integral whose closed form solution was only obtained in 1994 by Lazoff
and Sherman.
1.4 The models
We have approximated the cost of operating a warehouse in terms of expected distance
to make a pick or a stow. Hence, our goal is to design a rack and aisle structure that
minimizes this cost. This can be achieved in two ways:
1. Choose a number of cross aisle shapes. For each shape, route pickers in the most
efficient way, and then choose the most economical shape.
2. Define a number of sets of routing rules. For each set of rules, let the model choose
the aisle structure, and then choose the most economical shape.
12
For the designs we consider, there is a one-one correspondence between a set of routing
rules and a cross aisle shape. At this point in our discussion, we opt for the first approach.
However, after laying the ground and providing some insight into more intuitive issues of
the aisle designs, we explain the routing rules paradigm in a separate section.
To begin, we define two ?candidate shapes.? Namely, the Flying-V shape and ? shape
(Figures 1.3 and 1.4). Since we choose warehouses with an odd number of picking aisles, we
also assume symmetry about the central picking aisle. Based on geometrical features of the
shapes, we define the most efficient routing of pickers. Subsequently, for a given warehouse,
defined by its design parameters ? number of aisles (n), height of a warehouse (h), width of
picking (a?2) and cross (2w) aisles ? we state a nonlinear optimization problem to find an
improved cross aisle configuration by minimizing the expected cost of a single pickup and
retrieval task. We separately carry out optimization for each candidate shape and choose
the more economical one. We solve the optimization problems with Mathematica.
1.4.1 Flying-V cross aisle
We examine the Flying-V shape first, depicted in Figure 1.3. Because we assume
the cross aisle is symmetric about the center aisle, we consider only the right half of the
warehouse. We number picking aisles in the right half-space starting from the center and
ending at the rightmost aisle. Thus, i = 0,...,m, where m = (n ? 1)/2. We describe
a V-shaped cross aisle as a vector ??b = {b0,b1,...,bm}. Every point bi is a y coordinate
of the intersection of the cross aisle and the picking aisle i. In order to provide pickers
with sufficient maneuvering space we require the cross aisle to have a width of 2w at the
intersection points. That is, below and above the bi point, the cross aisle consumes w units
13
of storage space. A more realistic alternative, where one would define a constant cross-
aisle width independent from the intersection points, would lead to a more complicated
mathematical model. We believe that possible increase in precision would not be sufficient
to justify the added complication. For the same reason, Gue and Meller (2006) refer to the
2w convention as a ?judicious modeling choice.?
Introducing the cross aisle penalty width decreases available storage space. Therefore,
any possible travel savings from the insertion of a cross aisle comes at the cost of this wasted
space. To illustrate (Figure 1.3), assume that a pick is being made from the P&D point i
into the picking aisle i. In other words, the travel path is a simple straight line. If a cross
aisle is present, every time the pick is above it, there will be an increase of 2w in the travel
distance.2
We assume that P&D points are located exactly at the bottoms of the picking aisles
(Figure 1.3). This is an idealization, because in reality, they would be w units below, in
keeping with the bottom cross aisle. The effect of our assumption is a slight overstatement
of the estimated improvements.
Next, we define the cost structure of a V-shaped cross aisle. We divide the picking space
according to appropriate travel patterns (Figure 1.3). We define the following notation
? E[Ri] is the expected cost of making a pick to the right of aisle i;
? E[Si] is the expected cost of making a pick in aisle i;
2If, however, there is no cross aisle, all the picks that were previously above it will be 2w units closer,
which will result in a more economic travel. For some warehouses with smaller h this is the reason why
rectilinear travel, and therefore the upper travel limit too, might perform better than any cross-aisle-based
improvement.
14
Figure 1.3: Flying-V shape - scheme of travel patterns and related picking regions
? E[Li] is the expected cost of making a pick in an aisle located left of aisle i, but no
farther than the center picking aisle; and
? E[Lmosti ] is the expected cost of making a pick in an aisle to the left of aisle i, from
the first aisle after the center aisle to the leftmost aisle.
Now, we express the expected cost of picking from a randomly chosen P&D point i located
right from the central picking aisle, and denote it by E[Ci]:
E[Ci] = pRiE[Ri]+pSiE[Si]+pLiE[Li]+pMiE[Lmosti ],
where pRi = (m?i)/n,pSi = 1/n,pLi = i/n, and pMi = m/n are the probabilities of making
a pick in the respective regions.
The total expected picking cost for all P&D points i = 0,...,n is:
E[C] =
nsummationdisplay
i=0
piE[Ci],
15
where pi = 1/n is the probability of choosing a random P&D point. Because of the sym-
metry we assume, the total picking cost becomes:
E[C] = p0E[C0]+2
msummationdisplay
i=1
piE[Ci].
The cost structure above conveys basic information about the regions of a warehouse
and their relationship to the picking cost. To be able to derive detailed mathematical
expressions for all of the above cost components, we need to learn more about the routing
of workers and its relationship with mathematical formulation of the picking cost. In that
sense, especially important is the case of E[Ri]. Consider aisles i and r, such that i < r,
and assume that a pick is being made from P&D point i into a random picking point in the
aisle r. Depending on the location of the random pick in the aisle r, the picker should take
different routes to access the pick location. If the desired pick is above the cross aisle, the
picker will travel straight ahead in aisle i, along the cross aisle, and then up aisle r ? the
upper path. If the pick is somewhere between the bottom of the aisle r and the intersection
of the aisle r with the cross aisle (br?w), there are two possible routes from which to chose:
In the part closer to the br?w, straight ahead the aisle i, along the cross aisle and down the
aisle r ? the middle path. In the part closer to the bottom of the aisle r, along the bottom
of the warehouse and up the aisle r ? the lower path. There exists a delimiter point ? qir
(q-delimiter henceforth), where middle and lower paths are equidistant. For a given set of
parameters i,r,a,h,w and appropriate vector ??b , we have that cross aisle length between
16
two consecutive picking aisles and any two picking aisles respectively, is:
dj =
radicalBig
a2 +(bj ?bj?1)2
dir =
rsummationdisplay
j=i+1
dj,
also q-delimiter between any two picking aisles is derived as:
qir +(r ?i)a = bi +dir +br ?qir
qir = 12
bracketleftbigg
bi +dir +br ?(r ?i)a
bracketrightbigg
.
For the picks above the q-delimiter, the middle path is preferred and for those below, the
lower path is best. Yet, there are cases of an anomaly that renders the middle travel path
unusable. Knowing that every bi, and hence every br point, has an associated 2w penalty,
we impose the constraint w ? br ? h ? w. On the other hand, constraints that govern
behavior of the q-delimiter have to be chosen very carefully. Since we defined qir as a point
of indifference for the middle and lower paths, we expect w ? qir ? br ?w to hold. So, for
every i and r such that i < r we have that:
qir ? br ?w
1
2(bi +dir +br ?(r ?i)a) ? br ?w
dir ?(br ?bi) ? (r ?i)a?2w
rsummationdisplay
j=i+1
radicalBig
a2 +(bj ?bj?1)2 ?(br ?bi) ? (r ?i)a?2w.
17
Adjacent aisles are especially sensitive to this constraint. In this case,
radicalBig
a2 +(bj ?bj?1)2 ?(bj ?bj?1) ? a?2w
bj ?bj?1 ? 2w(a?w)a?2w ,
which effectively imposes a lower bound on the slope of the consecutive cross aisles segments.
The constraint tends to be tight whenever height h is relatively small compared to the width
A = na. In extreme cases, the result is a straight V-shaped cross aisle with the rightmost
bi?s either being very close to, or reaching h?w. If the latter occurs, i.e. br = h?w and
r < m, all aisles with indices greater than r do not have a cross aisle.
The previously discussed constraint is in no way imposed by the needs of a warehouse;
rather, it is an artifact of the model. If the constraint is simply removed, a cross aisle
with a very low slope might cause E[Ri] to become negative. If the constraint is held as
it is, whenever the height to width ratio is small, the cross aisle will be limited by the
lower bound slope. To address this, we force the model to choose a cross aisle slope that
excludes the middle travel pattern whenever it is prudent to do so. That is, we define
qir = min(qir,br ? w). In the case of qir > br ? w, the middle travel path vanishes and
qir = br ?w.
Finally, we derive expressions for each component of the total picking cost:
E[Ri] =
msummationdisplay
r=i+1
1
(m?i)(h?2w)
bracketleftbiggintegraldisplay qir
0
[(r ?i)a+x]dx +
integraldisplay br?w
qir
[bi +dir +br ?x]dx +
+
integraldisplay h
br+w
[bi +dir +x?br]dx
bracketrightbigg
18
=
msummationdisplay
r=i+1
1
(m?i)(h?2w)
bracketleftBigg
(r ?i)aqir + qir
2
2 +(br ?w ?qir)
bracketleftbigg
bi +dir +br?
? 12(br ?w +qir)
bracketrightbigg
+(h?br ?w)
bracketleftbigg
bi +dir ?br + 12(h+br +w)
bracketrightbiggbracketrightBigg
=
msummationdisplay
r=i+1
1
(m?i)(h?2w)
bracketleftBigg
qir
bracketleftbigg
(r ?i)a+ qir2
bracketrightbigg
+(br ?w ?qir)
bracketleftbigg
bi +dir+
+ 12(br +w ?qir)
bracketrightbigg
+(h?br ?w)
bracketleftbigg
bi +dir + 12(h?br +w)
bracketrightbiggbracketrightBigg
,
where
dj =
radicalBig
a2 +(bj ?bj?1)2
dir =
rsummationdisplay
j=i+1
dj
qir = 12
bracketleftbigg
bi +dir +br ?(r ?i)a
bracketrightbigg
qir = min(qir,br ?w).
E[Si] = 1(h?2w)
bracketleftbiggintegraldisplay h
0
xdx ?
integraldisplay bi+w
bi?w
xdx
bracketrightbigg
= 1(h?2w)
bracketleftbiggh2
2 ?2biw
bracketrightbigg
,
E[Li] =
i?1summationdisplay
l=0
1
i(h?2w)
bracketleftbiggintegraldisplay h
0
[(i?l)a+x]dx ?
integraldisplay bl+w
bl?w
[(i?l)a+x]dx
bracketrightbigg
=
i?1summationdisplay
l=0
1
i(h?2w)
bracketleftbigg
a(i?l)(h?2w)+ h
2
2 ?2blw
bracketrightbigg
, and
19
E[Lmosti ] =
msummationdisplay
lm=1
1
m(h?2w)
bracketleftBiggintegraldisplay q
0lm
0
bracketleftbigg
ia+lma+x
bracketrightbigg
dx +
+
integraldisplay bl
m?w
q0lm
bracketleftbigg
ia+b0 +d0lm +blm ?x
bracketrightbigg
dx +
integraldisplay h
blm+w
bracketleftbigg
ia+b0 +d0lm +x?blm
bracketrightbigg
dx
bracketrightBigg
=
msummationdisplay
lm=1
1
m(h?2w)
bracketleftBigg
(i+lm)aq0lm + q0lm
2
2 +(blm ?w ?q0lm)
bracketleftbigg
ia+b0 +d0lm+
+blm ? 12(blm ?w +q0lm)
bracketrightbigg
+(h?blm ?w)
bracketleftbigg
ia+b0 +d0lm ?blm + 12(h+blm +w)
bracketrightbiggbracketrightBigg
=
msummationdisplay
lm=1
1
m(h?2w)
bracketleftBigg
q0lm
bracketleftbigg
(i+lm)a+ q0lm2
bracketrightbigg
+(blm ?w ?q0lm)
bracketleftbigg
ia+b0 +d0lm+
+ 12(blm +w ?q0lm)
bracketrightbigg
+(h?blm ?w)
bracketleftbigg
ia+b0 +d0lm + 12(h?blm +w)
bracketrightbiggbracketrightBigg
.
Finally let us consider the special case of the central picking aisle. Since costs of
traveling to the right and left from the central aisle are identical, we can simplify the
regular cost expression to:
E[C0] = 2pR0E[R0]+pS0E[S0],
where pR0 = m/n and p0 = 1/n are the probabilities of picking to the right of aisle zero
and the probability of picking in aisle 0.
E[R0] =
msummationdisplay
r=1
1
m(h?2w)
bracketleftbiggintegraldisplay q0r
0
[ra+x]dx +
integraldisplay br?w
q0r
[b0 +d0r +br ?x]dx +
+
integraldisplay h
br+w
[b0 +d0r +x?br]dx
bracketrightbigg
=
msummationdisplay
r=1
1
m(h?2w)
bracketleftBigg
raq0r + q
20r
2 +(br ?w ?q0r)
bracketleftbigg
b0 +d0r +br ? 12(br ?w +q0r)
bracketrightbigg
+
+(h?br ?w)
bracketleftbigg
b0 +d0r ?br + 12(h+br +w)
bracketrightbiggbracketrightBigg
20
=
msummationdisplay
r=1
1
m(h?2w)
bracketleftBigg
q0r(ra+ q0r2 )+(br ?w ?q0r)
bracketleftbigg
b0 +d0r + 12(br +w ?q0r)
bracketrightbigg
+
+(h?br ?w)
bracketleftbigg
b0 +d0r + 12(h?br +w)
bracketrightbiggbracketrightBigg
,
and the last component ? the cost of picking straight ahead, in the central aisle:
E[S0] = 1(h?2w)
bracketleftbiggintegraldisplay h
0
xdx ?
integraldisplay b0+w
b0?w
xdx
bracketrightbigg
= 1(h?2w)
bracketleftbiggh2
2 ?2b0w
bracketrightbigg
.
1.4.2 ?-shaped cross aisle
We refer to our second candidate shape as a ? shape, because it is a concave structure
(Figure 1.4) with a single saddle point at b0. This shape imposes different travel patterns
and results in a more complicated cost function. Comparison with the Flying-V shape
reveals similarity in some travel paths. If we choose a random P&D point in the right half
of the dock, picking to the right in a ? structure is equivalent to picking to the left in the
Flying-V structure. The same relationship holds for picking to the left (no further than
the central aisle) in the ? structure and picking to the right in a Flying-V structure. Both
shapes share similar q-delimiters and constraints. For the ? shape, we also assume that
P&D points are located exactly at the bottoms of picking aisles.
We follow the same steps applied to the Flying-V cross aisle, and establish cost expres-
sion:
E[Ci] = pRiE[Ri]+pSiE[Si]+pLiE[Li]+pMiE[Lmosti ].
21
Figure 1.4: ? shape - scheme of travel patterns and related picking regions
Development of the first three components is as before; the fourth is slightly more compli-
cated and we discuss it in detail.
E[Ri] =
msummationdisplay
r=i+1
1
(m?i)(h?2w)
bracketleftbiggintegraldisplay h
0
[(r ?i)a+x]dx ?
integraldisplay br+w
br?w
[(r ?i)a+x]dx
=
msummationdisplay
r=i+1
1
(m?i)(h?2w)
bracketleftBigg
a(r ?i)(h?2w)+ h
2
2 ?2brw
bracketrightBigg
E[Si] = 1(h?2w)
bracketleftbiggintegraldisplay h
0
xdx ?
integraldisplay bi+w
bi?w
xdx
bracketrightbigg
= 1(h?2w)
bracketleftbiggh2
2 ?2biw
bracketrightbigg
E[Li] =
i?1summationdisplay
l=0
1
i(h?2w)
bracketleftbiggintegraldisplay qil
0
[(i?l)a+x]dx +
integraldisplay bl?w
qil
[bi +dli +bl ?x]dx+
+
integraldisplay h
bl+w
[bi +dli +x?bl]dx
bracketrightbigg
=
i?1summationdisplay
l=0
1
i(h?2w)
bracketleftBigg
(i?l)aqil + qil
2
2 +(bl ?w ?qil)
bracketleftbigg
bi +dil +bl ? 12(bl ?w +qil)
bracketrightbigg
+
+(h?bl ?w)
bracketleftbigg
bi +dil ?bl + 12(h+bl +w)
bracketrightbiggbracketrightBigg
22
=
i?1summationdisplay
l=0
1
i(h?2w)
bracketleftBigg
qil((i?l)a+ qil2 )+(bl ?w ?qil)
bracketleftbigg
bi +dil + 12(bl +w ?qil)
bracketrightbigg
+
+(h?bl ?w)
bracketleftbigg
bi +dil + 12(h?bl +w)
bracketrightbiggbracketrightBigg
When picking from some P&D points in the right half-space into the left storage half-space,
one can chose between two travel patterns: the simpler one, depicted in Figure 1.4 and the
more complicated, given in Figure 1.5. The illustrations reveal that picks at the top and
bottom are always picked in the same way. For picks in the middle region, Figure 1.5 shows
that it is possible to use the cross aisle. In the next paragraph we explain when and how
the cross aisle is used.
For a P&D point i in the right and picking aisle lm in the left warehouse half-space,
we introduce bel(i,lm) = (i+lm)a+blm and ab(i,lm) = bi +di0 +dlm0, which are the lower
and upper shortest paths from the ith P&D point to the blm point. If ab(i,lm) < bel(i,lm),
it is cheaper to access blm from above, via the ? cross aisle. In this situation, it is better
to use the cross aisle, which would produce the pattern given in Figure 1.5. Note that this
pattern creates two q-delimiters in aisle lm, one above and one below blm. This holds for
every aisle lm such that ab(i,lm) < bel(i,lm) and 1 ? lm < i.
For the q-delimiter below the cross aisle:
(lm +i)a+qLilm = bi +di0 +dlm0 +blm ?qLilm
qLilm = 12
bracketleftbigg
bi +di0 +dlm0 +blm ?(lm +i)a
bracketrightbigg
.
23
Figure 1.5: ? shape ? a rarely used and complicated travel pattern
and for the q-delimiter above the cross aisle:
dlm0 +qUilm ?blm = h?b0 +lma+h?qUilm
qUilm = 12
bracketleftbigg
blm +2h?b0 +lma?dlm0
bracketrightbigg
.
In order to avoid imposing a bound for the slope of the ? cross aisle and the resulting cost
anomalies, we define, as before:
qLilm = max(w,min(qLilm,blm ?w))
qUilm = max(blm +w,min(qUilm,h?w)).
The other possibility is that bel(i,x) < ab(i,x). Now it becomes better not to use the cross
aisle and the outcome is the pattern depicted in Figure 1.4. The travel is less complicated
24
and features only one q-delimiter above the cross aisle:
(i+lm)a+qlmilm = bi +di0 +h?b0 +lma+h?qlmilm
qlmilm = 12
bracketleftbigg
bi +di0 +2h?b0 ?ia
bracketrightbigg
.
As before, we need to let the model decide freely on the slope of the aisle, so we define:
qlmilm = max(blm +w,min(qlmilm,h?w)).
For a fixed P&D point i, we choose the cheaper of the two arising travel patterns for every
aisle lm in the left half-space. Travel cost for the travel pattern with two q-delimiters is
given by:
Lmosti,lm (ab) = 1m(h?2w)
bracketleftbiggintegraldisplay qL
ilm
0
[(i+lm)a+x]dx +
integraldisplay bl
m?w
qLilm
[bi +di0 +dlm0 +blm ?x]dx+
+
integraldisplay qU
ilm
blm+w
[bi +di0 +dlm0 +x?blm]dx+
integraldisplay h
qUilm
[bi +di0 +h?b0 +lma+h?x]dx
bracketrightbigg
= 1m(h?2w)
bracketleftBigg
(i+lm)aqLilm + q
L
ilm
2
2 +(blm ?w ?q
L
ilm)
bracketleftbigg
bi +di0 +dlm0 +blm?
? 12(blm ?w +qLilm)
bracketrightbigg
+(qUilm ?blm ?w)
bracketleftbigg
bi +di0 +dlm0 ?blm + 12(qUilm +blm +w)
bracketrightbigg
+
+(h?qUilm)
bracketleftbigg
bi +di0 +2h?b0 +lma? 12(h+qUilm)
bracketrightbiggbracketrightBigg
= 1m(h?2w)
bracketleftBigg
qLilm((i+lm)a+ q
L
ilm
2 )+(blm ?w ?q
L
ilm)
bracketleftbigg
bi +di0 +dlm0+
+ 12(blm +w ?qLilm)
bracketrightbigg
+(qUilm ?blm ?w)
bracketleftbigg
bi +di0 +dlm0 + 12(qUilm ?blm +w)
bracketrightbigg
+
+(h?qUilm)
bracketleftbigg
bi +di0 +2h?b0 +lma? 12(h+qUilm)
bracketrightbiggbracketrightBigg
.
25
For the travel pattern with a single q-delimiter above the cross aisle we proceed in a similar
manner:
Lmosti,lm (bel) = 1m(h?2w)
bracketleftBiggintegraldisplay b
lm?w
0
bracketleftbigg
(i+lm)a+x
bracketrightbigg
dx +
integraldisplay qlm
ilm
blm+w
bracketleftbigg
(i+lm)a+x
bracketrightbigg
dx +
+
integraldisplay h
qlmilm
bracketleftbigg
bi +di0 +h?b0 +lma+h?x
bracketrightbigg
dx
bracketrightBigg
= 1m(h?2w)
bracketleftBigg
(i+lm)a(blm ?w)+ (blm ?w)
2
2 +(q
lm
ilm ?blm ?w)
bracketleftbigg
(i+lm)a+
+ 12(qlmilm +blm +w)
bracketrightbigg
+(h?qlmilm)
bracketleftbigg
bi +di0 +2h?b0 +lma? 12(h+qlmilm)
bracketrightbiggbracketrightBigg
.
Finally, we arrive at the expression that favors cheaper travel pattern:
E[Lmosti ] =
msummationdisplay
lm=1
min(Lmosti,lm (bel),Lmosti,lm (ab)).
Cost for picking from a central P&D point is calculated similar to the V-shaped cross
aisle cost.
E[C0] = 2pR0E[R0]+pS0E[S0]
E[R0] =
msummationdisplay
r=1
1
m(h?2w)
bracketleftbiggintegraldisplay h
0
(ra+x)dx ?
integraldisplay br+w
br?w
(ra+x)dx
=
msummationdisplay
r=1
1
m(h?2w)
bracketleftBigg
ar(h?2w)+ h
2
2 +2brw
bracketrightBigg
E[S0] = 1n
integraldisplay
S
= 1h?2w
bracketleftbiggintegraldisplay h
0
xdx ?
integraldisplay b0+w
b0?w
xdx
bracketrightbigg
= 1h?2w
bracketleftbiggh2
2 ?2b0w
bracketrightbigg
.
26
1.4.3 The most economical cross aisle
The most economical cross aisle enables a single pickup and retrieval travel into the
storage space at a minimum cost. We optimize candidate shapes one at a time, compare
them and choose a more cost-effective solution. Due to the complexity and nonlinearity of
the cost function, it is not possible to apply any calculus based methods for finding minima.
Being unable to generate closed form solutions, we resort to nonlinear optimization. Both
optimization problems are of the following form:
Min E[C]
S.T. w ? bi ? h?w, i ? [0,m].
We make use of Mathematica?s NMinimize function, which employs a number of standard
algorithms for solving nonlinear problems. In the series of experiments we carried out, it
never took more than fifteen minutes to solve problems for warehouses with a large number
of P&D points. Smaller instances of the problem are solved in a matter of seconds.
1.5 Results
We carry out the experiment on a variety of warehouse configurations. We perform the
following changes in basic design parameters:
? n = 11?39 in increments of 4,
? w = 1?3 in increments of 0.5, and
? h = 50?125 in increments of 25.
27
Figure 1.6: Typical Flying-V (n = 39, w = 2, a = 5, h = 100) and ? (n = 21, w = 2, a = 5,
h = 100) shape cross aisles
28
We keep a at value of 5 because our designs assume use of counterbalance forklifts. For this
type of fork truck picking aisles are usually 12 feet wide, which corresponds to a?2 pallet
units in our notation. Among the obtained population of 160 instances, not all layouts are
realistic. In fact, one could say that those with n = 15?30,w = 1.5?2,h = 50?100 are
perhaps closest to existing facilities, while others diverge mostly because of unusual width
to height ratio or cross aisle width. Nevertheless, a larger experiment, encompassing the
applicable and less pragmatic alike, helps build some intuition about laws governing the
quality of a layout. We stipulate that configurations being compared have the same overall
length of picking aisles. For example, ? and Flying-V shape with w = 1.5,h = 100 together
with traditional no-cross-aisle layout of h = 97 represent a valid comparison triplet.
Before discussing the results we note a few physical and visual properties of typical
Flying-V and ? shapes given in Figure 1.6. Our experiments confirm the expectation that
whenever it is advantageous to have a V-shaped cross aisle, b0 is equal to w. Gue and
Meller (2006) explain this observation in detail. Suffice it to say that having a b0 point as
low as possible shortens all right and left travel traversing P&D point zero or originating at
it. Similarly, for all justifiable ? layouts, travel originating at the leftmost and rightmost
P&D points are shortened with bm = w. For 39 P&D points (aisles), 100 units of height,
and 4 units wide cross aisle, the Flying-V shape yields a 5.64% travel savings over the
traditional design. For the same height and cross aisle width, a ? warehouse with 23 P&D
points (aisles) does not confer any benefit. Step-by-step increase in number of P&D points
induces small benefits. Yet, a ? warehouse with 39 P&D points, 100 units of height, and 4
units wide cross aisle results in only 1.09% savings.
29
15 20 25 30 35 #aisles
5
10
15
20
Savings for h?50,w?1.5
?shape
Vshape
Travel?by?flight
Figure 1.7: Example comparison of savings over orthogonal design for Flying-V and ?-shape
layouts, and Travel-by-flight
Neither of the shapes comes close to the Travel-by-flight savings (Figure 1.7), which
are about 20% for all configurations we consider. In Figure 1.8 we present the summary of
savings for applicable layouts for ? shape and Flying-V shape. A comprehensive comparison
is given in Table 1 of the Appendix. A glimpse at Figure 1.8 or Table 1 makes it clear that
the V shape always performs better than ? shape.
For ? shapes with up to n = 31 aisles, heights of h = 50,75 or 100, benefits do not
exceed 2.16%. For cross aisle width of w = 2 (or more), a ? aisle provides no benefit and
the optimization model returns a vector ??b = {h?w,h?w,...,h?w}, which corresponds
to a traditional orthogonal design.
With respect to the Flying-V shape, narrow aisles (of w = 1) offer savings as high as
7.24%. Regrettably, small increases in cross aisle width cause considerable drop in savings.
A large percentage of travel crosses the 2w space and increases the average distance to make
a pick. This disadvantage accumulates rather quickly, thereby making the overall average
30
15 20 25 30 35
1
2
3
4
5
Savings for h?100,w?2
15 20 25 30 35
1
2
3
4
5
6
Savings for h?125,w?2
15 20 25 30 35
0.5
1
1.5
2
2.5
3
3.5 Savings for h?50,w?2
15 20 25 30 35
1
2
3
4
Savings for h?75,w?2
15 20 25 30 35
1
2
3
4
5
6
Savings for h?100,w?1.5
15 20 25 30 35
1
2
3
4
5
6
Savings for h?125,w?1.5
15 20 25 30 35 # aisles
1
2
3
4
Savings for h?50,w?1.5
? shape
Flying ?V
15 20 25 30 35
1
2
3
4
5
Savings for h?75,w?1.5
Figure 1.8: Savings over orthogonal design for Flying-V and ?-shape layouts
31
cost very sensitive to it. Consequently, Flying-V shape layouts with realistic cross aisle
widths carry benefits not higher than 6.07% for w = 2, or 6.65% for w = 1.5.
In the worst case, for some warehouses it is less costly to keep the traditional design.
The break-even point for a warehouse with a fixed number of P&D points can be reached
in two ways: by reducing h or by increasing w. Both of those have the same meaning ?
the picking space is shrinking. When decreasing h we diminish the ability of a cross aisle
to provide savings. When h becomes small enough, economies of scale that were enabling
V-shape savings vanish. The number of farther locations that were more prudent to access
via the cross aisle is reduced to insignificance. There is a similar effect with an increase in
cross aisle width. Very often a travel into the storage space includes the cross aisle width,
which is not the case with the traditional design. An increasing width adds to the travel
distance more than before, while a smaller traditional warehouse with height of h?2w does
not have this handicap. In Figure 1.9 we give an example of break-even configurations for a
fixed value of w = 1.5. As the number of picking aisles in a warehouse increases, break-even
heights decrease.
1.6 A flow-through crossdock
A flow-through crossdock receives pallets on one side and ships from the opposite. In
keeping with our terminology, receiving occurs on the top and shipping at the bottom. If a
V-shaped cross aisle is inserted, a picker on the shipping side will perceive it as a V shape,
whereas his coworker on the receiving side will see a ? shape. Therefore, a flow-through
crossdock is a natural composition of Flying-V and ? shape travel patterns and hence, cost
32
15 20 25 30 #aisles
12
14
16
18
20h
h
Figure 1.9: Example of warehouse configurations (w = 1.5, n = 11 ? 35) for which it is
better not to have a cross aisle
functions. The optimal cross aisle is found with respect to optimizing the sum of Flying-V
and ? shape travel costs divided by two.
To determine the potential advantage of having a single cross aisle in a crossdock, we
conducted an experiment over the same combinations of n,w, and h as in the Section 1.5.
Detailed results of the experiment are given in the Table 1 of the Appendix. In Figure 1.10
we present the crossdock savings for applicable combinations of h and w and graphically
compare them with savings of ? and Flying-V shapes. Benefits, if any, are relatively close
to the arithmetic mean of the ? and Flying-V shape savings. Additionally, for instances of
? shape that incur loss (compared to the upper travel bound) part of the Flying-V shape
gains is wasted on making up for the deficit.
Knowing that Flying-V shape savings dominate those of ?, we would also expect to
see b0 = w, which indeed occurs in all instances of our experiment. As regards the ? shape,
not even once does the bm equal h?w.
33
15 20 25 30 35
1
2
3
4
5
Savings for h?100,w?2
15 20 25 30 35
1
2
3
4
5
6
Savings for h?125,w?2
15 20 25 30 35
0.5
1
1.5
2
2.5
3
3.5 Savings for h?50,w?2
15 20 25 30 35
1
2
3
4
Savings for h?75,w?2
15 20 25 30 35
1
2
3
4
5
6
Savings for h?100,w?1.5
15 20 25 30 35
1
2
3
4
5
6
Savings for h?125,w?1.5
15 20 25 30 35 # aisles
1
2
3
4
Savings for h?50,w?1.5
? shape
X?dock
Flying ?V
15 20 25 30 35
1
2
3
4
5
Savings for h?75,w?1.5
Figure 1.10: Comparisson of savings over orthogonal design for crossdock configuration,
Flying-V and ?-shape layouts
34
1.7 Duality of representation
In this section we return to the alternative modeling paradigm we defined in Section
1.4. The methodology we have used so far can be summarized as choosing a number of
cross aisle shapes, routing workers to determine shortest travels into the storage space, and
then choosing the best aisle shape. Alternatively, we could first specify a number of sets of
rules governing the travel, let the model choose the shape resulting from each set of rules,
and then choose the best aisle shape. Let us demonstrate by fully defining one set of rules:
1. If the P&D point is centrally located
(a) for a ?right? pick above the cross aisle, travel to the cross aisle along picking
aisle zero, along the cross aisle and then up the destination aisle.
(b) for a ?right? pick below the cross aisle and above the q-delimiter (point of indif-
ference), travel to the cross aisle along picking aisle zero, along the cross aisle,
and down the destination aisle.
(c) for a ?right? pick below the point of indifference, travel right along the bottom
and up the destination aisle.
(d) for a ?straight? pick, travel up the picking aisle zero.
(e) for a ?left? pick ?mirror? the ?right? pick travel rules 1(a)-1(d).
2. P&D point is in the right half-space.
(a) for a ?right? pick above the cross aisle, travel up along the current picking aisle,
along the cross aisle and up the destination aisle.
35
(b) for a ?right? pick below the cross aisle and above the point of indifference, travel
up along the current picking aisle, along the cross aisle, and down the destination
aisle.
(c) for a ?right? pick below the point of indifference, travel right along the bottom
and up the destination aisle.
(d) for a ?straight? pick, travel up the current aisle.
(e) for a left pick in the right half-space including the central aisle, travel left along
the bottom, and up the destination aisle.
(f) for a left pick in the left half-space not-including the central aisle, travel left
along the bottom to reach central P&D point and then follow the rule 1(e).
3. P&D point is in the left half-space.
(a) ?mirror? the rules 2(a)-2(f).
Flying-V cross aisle depiction (Figure 1.3) reveals analogy between the optimum routing
of workers illustrated, and the routing rules we define here. The difference is that in the first
case cross aisle shape is known a priori, and routing rules are inferred from shortest path
travel within the V shape. Conversely, in the second approach routing rules are established
first and cross aisle shape is therefore a consequence of routing rules.
To demonstrate the validity of an alternative representation we conduct the following
test. Let us define a set of routing rules:
1. any P&D point
(a) for a ?right? pick above the cross aisle, travel up, along the cross aisle and up
the destination aisle.
36
(b) for a ?right? pick below the cross aisle and above the point of indifference, travel
up, along the cross aisle, and down the destination aisle.
(c) for a ?right? pick below the point of indifference, travel right along the bottom
and up the destination aisle.
(d) for a ?straight? pick, travel up the current aisle.
(e) for a left pick ?mirror? the rules 1(a)-1(d).
The rules state that for each aisle a pick can be made to the right, straight and left from
the current aisle, which in terms of cost structure can be stated as:
E[Ci] = pRiE[Ri]+pSiE[Si]+pLiE[Li].
The rules further translate into cost expressions below. Because derivation of final formulae
is straightforward and has been presented before, we include only the initial expressions:
E[Ri] =
nsummationdisplay
r=i+1
1
(n?i)(h?2w)
bracketleftbiggintegraldisplay qir
0
[(r ?i)a+x]dx +
integraldisplay br?w
qir
[bi +dir +br ?x]dx +
+
integraldisplay h
br+w
[bi +dir +x?br]dx
bracketrightbigg
E[Si] = 1(h?2w)
bracketleftbiggintegraldisplay h
0
xdx ?
integraldisplay bi+w
bi?w
xdx
bracketrightbigg
E[Li] =
i?1summationdisplay
l=1
1
(i?1)(h?2w)
bracketleftbiggintegraldisplay qil
0
[(i?l)a+x]dx +
integraldisplay bl?w
qil
[bi +dil +bl ?x]dx +
+
integraldisplay h
bl+w
[bi +dil +x?bl]dx
bracketrightbigg
.
37
The total cost of picking is expressed with:
E[C] =
nsummationdisplay
i=1
piE[Ci]
= 1n
nsummationdisplay
i=1
(pRiE[Ri]+pSiE[Si]+pLiE[Li]).
To find the most economic cross aisle shape for this set of routing rules, we state an
optimization problem:
Min E[C]
S.T. w ? bi ? h?w, i ? [1,n].
Observe that this set of rules has no mention of central aisle or central P&D point. For
any pick above the point of indifference (which also includes the points above the cross aisle)
the picker travels directly to the cross aisle and then along the cross aisle. An ?educated
guess? that neither Flying-V nor ? shape would perform well is actually correct. We ran the
experiment on the same sample of 160 warehouses and the result was always a traditional,
orthogonal layout with the ?cross aisle? at ??b = {h?w,h?w,...,h?w}, which defines
no cross aisle at all.
We have demonstrated two examples where there is a complete analogy between pre-
scribing a cross aisle shape, and choosing a set of routing rules to serve as a basis for
optimization. However, we believe that the latter approach is more general and more ap-
propriate from the methodological point of view. The former is easier to grasp and especially
easier to present in a written form, but we stress that the shapes we dealt with result in
a relatively simple travel and cost structure. We allow the possibility that future work in
38
this area might benefit from the travel rules paradigm, since there may exist sets of travel
rules where it would not be possible to predict the aisle shape ahead of optimization.
1.8 Some practical issues
Through several contacts with engineers, logistics analysts and managers we received
valuable feedback on practical issues of cross aisle implementation. In addition to the cross
aisle width problem, worker orientation and adaptation to new designs were among most
cited concerns. To quote a supervisor who has been suggested to implement our designs:
?Whatever we do, we must keep it as simple as possible.? We set out to examine the
implications of ?simple.? When making a right pick in the right half-space of a Flying-
V-shaped warehouse, a travel pattern of the form ?up, along the cross aisle and down the
destination aisle? is the most confusing. It is much easier to instruct workers to use the cross
aisle only for the picks above it and eliminate the problematic travel pattern instead. In
connection with this, several managers have broached the subject of traffic safety and forklift
stability when making a sharp right turn down into the picking aisle. We proposed the same
solution for both problems. Since we have simplified the accompanying travel pattern, we
call this new shape V-simple. Results show that the new shape offers less benefit, but
only slightly so ? maximum reduction in savings measured across all configurations is only
0.85%. For convenience, we give complete results of this experiment in the Table 2 of the
Appendix.
Another potential problem in practice is the curvature of cross aisles. Most of our
shapes are curved slightly, and this was perceived as an obstacle to visibility and traffic
safety. There were concerns about perception, suspecting that workers would not have a
39
?feel? for the warehouse space and that non-orthogonal curved aisles might cause them
to lose orientation. We evaluated straight instead of the curved, Flying-V-shaped cross
aisle. This straight V shape changes the field of view and improves visibility at a maximum
increase in travel distance of 0.33% across all configurations, and 0.21% in realistic ones.
Again, we give complete results in the Table 2 of the Appendix.
1.9 Conclusions
In this chapter we focused on optimization of picking cost in unit load warehouses
with rack storage. We presented two cross aisle designs, ? and Flying-V, discussed their
combination in a flow-through crossdock and quantified the potential for cost reduction.
To summarize, ? shape savings generally are nonexistent or too small to justify imple-
mentation. The same applies to crossdock, where associated benefits can be roughly thought
of as the averaged savings of the two aisle shapes. Regarding the Flying-V shape, benefits
are small and considerably less than those of Flying-V shape with a single P&D point (Gue
and Meller, 2006), but may be worth implementing. A simple explanation exists: With a
single P&D point a greater portion of travel uses the cross aisle, whereas the region does
not have this advantage. As P&D points get farther away from the center, travel gains less
and less advantage from the cross aisle. It becomes necessary to travel greater distances up
the aisle i just to reach the cross aisle.
Flying-V shape is superior to ? , which is a consequence of their respective geometries
and characteristics of the material flow we hypothesize. Assuming uniform pick density,
material flows along a linear region will be greatest in the center. The Flying-V shape
facilitates such flows by providing a ?launching point? from which to enter the picking
40
space after travel along the bottom aisle. On the other hand, ? shape provides no such
advantage. The lowest bi points of a ? shape, the ones with highest ?launching potential,?
are located in the corners of the warehouse rectangle where only a small fraction of material
flow can benefit from them.
As we have stated, our results are a sort of worst case analysis for the designs we
consider. This particularly holds for the assumption about uniform flows across the entire
shipping region. Wise implementation would concentrate more flows into the central doors
and results should improve. Moreover, decreasing the number of P&D points should provide
more benefit and is expected to come closer to the results of Gue and Meller (2006). Also,
one should be aware that all benefits discussed in this work represent a relative measure and
are tied to the expected cost of making a single pick. When deciding on implementation of
the cross aisle, the designer should use the performance metrics such as daily or monthly flow
through the warehouse, total number of picks per shift and similar to realize the absolute
financial benefit.
41
Chapter 2
Application of aisle design principles to floor storage ? a case study
In the process of developing the cross aisle designs (Chapter 1), we made a number
of assumptions that help alleviate the complexity associated with the model. They do not
significantly affect the accuracy of our results, but do not fully represent the factual state
either. For example, we model individual rack locations as continuous, as opposed to them
being discrete in reality. Strictly speaking, insertion of a w units wide cross aisle into a
continuous rack space results in unrealistic storage space allocation. For example, a bi
at 12.35, with the w = 1.5, means that the storage space below the aisle numbers 10.85
locations. Theoretically, it is not possible to utilize 0.85 units of storage space in a unit
load warehouse. Similar problem occurs with the space above the aisle. Let us recall that
we integrate across this nonexisting storage space and therefore increase the actual travel
cost. Even though we account for the increase by calculating the average cost, we indeed
alter the real cost, which although slight, is an inaccurate representation of reality per se.
When one tries to implement a theoretical result or principle in practice, it is necessary
to adopt an even more more flexible approach. In this chapter, we demonstrate a case of a
large home appliance manufacturer in the Southeast. We present a less rigorous modeling
approach and apply the main ideas of our designs to a new problem ? unit load storage
using floor stacking. We discuss several possible amendments to the general floor storage
layouts with different positions of P&D points. We develop rather simple estimates of the
savings associated with our designs. Finally, we propose a new layout for the manufacturer?s
warehouse and calculate the approximate savings in the putaway and picking costs.
42
2.1 Problem background
The company operates a large unit-load warehouse, with floor storage area that is
used to keep the finished goods being delivered from the adjacent manufacturing facility.
Figure 2.1 depicts the current layout. As can be inferred from the figure, finished goods
are delivered from the left side of the warehouse rectangle and the loading dock is located
on the bottom side. Upon order receipt, pickers travel into the storage space from two
P&D points located in close proximity of the loading dock. In reality, picking activity is
not uniformly distributed since every effort is made to reserve the most convenient picking
locations A1, A2, A3, A4 and G, for the fastest moving skus. Less frequent items are stored
in blocks B1, B2, C, D, H, I, L and K, whereas slow movers are in blocks E, F, J and M.
This storage policy is best described in terms of dedicated storage. After being picked, the
item is transported to the same P&D point from which the picker started the trip (single
command cycle) for consolidation. In the end, it is loaded onto one of the outgoing trailers.
The loading dock houses twelve doors where each P&D point is serving only the closer half
of the doors. An alternative is a centrally located P&D and consolidation point serving all
the doors. The former solution provides significant savings in the average distance traveled
from P&D points to the doors and is preferred by the warehouse management.
Before proceeding to aisle design implementation, let us acknowledge several properties
of alternative aisle designs as they are seen from the practical point of view. In our model,
we express expected cost of a single pick or a stow as average distance traveled to complete
these tasks. It is only advisable to do so if we assume constant or close to constant traveling
speed. Several factors might impair the soundness of this assumption. First of all, pickers
will have to turn at least once in order to reach the destination aisle. In our opinion,
43
Figure 2.1: Current layout. Arrows pointing outward from the storage space represent P&D
points with retrieval activity. Arrow pointing into the storage space denotes inflow of goods
and putaway activity. Letters, or combinations of letters and numbers represent ?labels? of
the storage blocks.
44
this will not affect significantly the average speed. More importantly, we assume ideal
traffic conditions, i.e. congestion does not occur in the cross and picking aisles. Therefore,
our model does not predict any negative effects caused by possible traffic impediment. A
separate model is needed to represent the reduction in savings due to congestion.
Cost estimate by distance only includes reaching the point in the center of the picking
aisle and opposite to the unit to be picked from the rack, not the actual travel needed
to reach and load the item. If one needs to include the cost of reaching and loading the
pallet onto a material handling device (most often a forklift), distance ceases to be an
accurate estimator, because the half-width of the picking aisle is very likely to be crossed
with speed significantly different from the average traveling speed. After a forklift reaches
the pallet rack, only the pallet handling device is moving and is doing so in a vertical plane
with a speed that is again different from the forklift traveling speed. This means that
reaching distance and elevation cannot be simply added to the original distance traveled.
In situations where exact assessment is needed, it is more suitable to use time as the cost
estimator ? distance to reach the center of the picking aisle is divided by the average speed
and the time interval needed to reach and load the pallet is added. Usually, this interval is
broken up into time to rotate the truck for 90?, elevate the forks, position them, load the
pallet, lower the forks and again rotate for -90?. Unfortunately, all the loading components
may vary significantly due to different forklift model performance, driver experience, pallet,
rack and floor properties. Hence, quantifying the loading cost may be very tedious and
prescribing a generic formula not appropriate. In the end, if distance is the chosen cost
estimator and one needs to compare the cost of two alternative layout solutions, he can
only do so accurately for layouts with the same rack depths (single deep with single deep
45
and double deep with double deep) and same heights (number of levels). However, we
believe that for most applications of rack layout, distance is the appropriate estimator and
that loading/extraction issues become more important in floor storage.
One generally opts for floor storage when it is necessary to achieve greater density
and better floor space utilization, or when access to each individual unit is not required at
all times. The classical problem pertaining to this warehousing technology is determining
the optimal lane depth, one that would minimize the loss from honeycombing. Although
Bartholdi and Hackman (2007) provide a straightforward and elegant solution, in reality it
is often not possible to calculate optimal lane depths, because it is difficult to maintain an
appropriate database of sku inventory information.
The essential advantage of floor storage is the improved footprint utilization, but an
apparent disadvantage is the lack of access to individual items. Double deep selective pallet
racks suffer from this shortcoming too, but in a less severe form. Since floor storage is
arranged in blocks and lanes within them, a picker might travel significant distances through
the empty part of the lane in order to reach the intended item (extraction distance). As
Bartholdi and Hackman note, deeper lanes increase the space utilization but also increase
insertion/extraction times. If there is a single lane depth or a small number of lane depths
throughout the warehouse, cost can be safely expressed in terms of reaching the center of the
picking aisle. Nonetheless, when floor storage features a variety of lane depths, extraction
distance needs to be included. If there is a substantial difference in traveling vs. extraction
speed, slower extraction should be accounted for. In our case, on-site observations suggested
approximately equal travel and extraction speeds, and we made this assumption throughout.
46
Depending on the layout, replenishment and picking in a lane can be performed from
different directions, and consequently lead to different storage queueing disciplines. When
replenishment is done from the back and picking from the front, FIFO is preserved; both
replenishment and picking at the front lead to LIFO discipline.
2.2 Simple floor storage
The practical problem we discuss brings in a number of special considerations. One is
putaway from the side (Figure 2.1), a question we did not deal with in Chapter 1. To
get acquainted with and better build our understanding of the floor storage and potential
improvements, we first discuss a layout analogous to one of the rack storage, with a single
P&D point and both putaway and picking at the bottom of the warehouse rectangle.
In Figure 2.2(a) we present a classical floor storage while in Figures 2.2(b), 2.2(c) and
2.2(d), we give a schematic depiction of some alternative improvements. Note that the
numbering in storage locations in all of Figures 2.2 represents the order in which individual
pallet positions are being restocked and subsequently picked from ? that is to say, a storage
discipline. Upon emptying the lane of a rectilinear (classical) floor layout the restocker first
takes care of the positions in the upper part of the warehouse rectangle. Once the whole lane
has been restocked, the picker will start from the lowermost positions, therefore establishing
the LIFO storage discipline.
Figure 2.2(b) shows a layout similar to Flying-V cross aisle of rack storage. The left
half-space features all three travel patterns present in a Flying-V. It can be inferred that
q-delimiters (depicted with curvy irregular line) govern the storage discipline. If one wishes
to utilize all three travel patterns, the storage discipline in the positions below the aisle will
47
Figure 2.2: One P&D point with putaway and picking from the same side
48
be neither FIFO nor LIFO. In rack storage, a so called ?middle travel pattern? complicates
picking. We showed in Chapter 1 that using only rectilinear travel below the cross aisle
results in lesser savings, but simpler travel patterns. Same is the case with floor storage,
though with an additional benefit ? simpler, LIFO storage discipline. In such a layout
(Figure 2.2(b), blue half-space), all positions, below and above the aisle, have LIFO storage.
TranslatingfishboneprinciplestothefloorstorageresultsinthelayoutofFigure2.2(c).1
It confers more travel savings but changes the lane orientation in the space below the aisle.
For a fishbone and chevron layout (Figure 2.2(d)) where the fishbone aisle is parallel to the
chevron storage lanes, chevron lanes are deeper since they represent the hypotenuse and
fishbone lanes resemble the cathetus. Therefore, the choice between the two might depend
on the lane depth preferences. Note that other practical issues might complicate the designs
and the choice to be made among them. For example, various safety concerns might impose
an additional central aisle in the fishbone structure. Such an aisle would cut through the
existing storage space and facilitate evacuation from the storage area.
2.2.1 Side putaway
A more complicated task is designing a layout where the inflow of skus occurs at an
entry point located on the left side of the warehouse rectangle. Unlike the previous example,
we now encounter conflicting flows in some parts of the warehouse, i.e., in the same portions
of the storage space, travel patterns resulting from a new design might offer advantage for
picking and induce cost for replenishment, and vice versa. Figure 2.3 represents some of
the choices available to the designer faced with this problem.
1For more details on fishbone design see Gue and Meller (2006)
49
Figure 2.3: One P&D point with putaway and picking at different sides
50
In Figures 2.3(b) and 2.3(c) the right storage half-space is the region with conflicting
flows. In the left half-space in both designs, the picker and restocker will simultaneously
enjoy the benefit of the diagonal aisle. Unfortunately, in the right half-space restocker will
be forced to take on the route that is inferior to his original rectilinear travel, in order for
picker to benefit from the diagonal aisle. To resolve this situation one must take a closer
look at the relationship between picking and replenishing and possibly investigate for daily
peaks in both of the activities. If the number of picks per day is considerably greater than
number of replenishments per day, the layout in Figures 2.3(b) and 2.3(c) should remain
intact. In that case, loss on the putaway side will be offset by picking savings. If there exists
a period during the day where surge in picking activity occurs, new aisle designs should
be given serious thought. Since an average pick would take less time, the system would be
more responsive and the peak period would not last as long. If one cannot judge about the
picking to putaway ratio or daily peaks, or simply finds that there is no difference between
picking and putaway, it is safest to employ traditional rectilinear lane and aisle structure in
the right half-space.
Another way to support a disproportional increase in picking activity is proposed in
Figure 2.3(d), where a cut-through diagonal aisle is inserted in the right half-space. On
the picking side, the aisle is simply treated as a ?no-storage space?. In other words, once
a picker has emptied the locations in a specific lane that are below the aisle, he proceeds
to those in the same lane above the aisle. For a picker, a cut-through aisle does not bear
any meaning except that he needs to traverse its empty space on his way to the locations
above it. On the putaway side, however, things are different. The cut-through aisle is used
to avoid the losses from other alternative designs. If a cut-through aisle and a storage lane
51
intersect at approximately a right angle, the putaway travel pattern becomes rectilinear
and resembles that of traditional storage. Provided that lane depths are acceptable, this is
a safe way to keep the picking savings and as we will show later, gain small savings on the
putaway side.
2.2.2 Two P&D points
Incentive to increase the number of P&D points might stem from a number of op-
portunities to realize savings, or from legitimate constraints imposed on the warehouse
operations. So far, we have treated P&D points as simple origins of travel into the storage
space. They might involve much more in reality: stretch wrappers or other consolidation
devices, inspection stations, labeling and repackaging points and so on. What is usually
tied to these small hubs within the warehouse are space requirements. For example, in a
busy unit load warehouse items may be subject to a detailed and lengthy inspection taking
place after the picking. In the case of a single P&D point at the loading dock, congestion is
very likely to occur. A logical solution is to increase the number of P&D points. Simplified
communication on the dock, less traffic, improved safety; all those might be motives for
having more P&D points. It is worth remarking that many of the benefits obtained are not
possible to quantify. While it provides unquestionable operational improvements, separat-
ing the points makes the problems with conflicting flows more severe. In Figures 2.4 and
2.5, we depict some of the design alternatives available to the decision maker.
Figure 2.4 involves putaway and picking from the bottom of the warehouse, whereas
Figure 2.5 moves the putaway to the left side. In Figure 2.4 we see that conflicting flows
can occur without inflow from the side. We assume that picking is evenly distributed inside
52
Figure 2.4: Two P&D point with putaway and picking from the same side
53
Figure 2.5: Two P&D points with putaway and picking at different sides
54
the whole warehouse space and that P&D points are chosen with equal probability. We
designate the middle (blue) segment of the Figures 2.4(c) and 2.4(d) is designated to keep
the traditional lane structure. A small improvement is achieved in 2.4(d), where we insert
a diagonal aisle in the middle region to facilitate travels from the left to the right region
and vice versa. Figure 2.4(b) features a cut-through aisle that defines the ?neutral,? middle
region in a different way.
The same principles apply to the configurations with two P&D points and side putaway.
The only difference is that all of the alternative designs suffer from conflicting flows in two of
the three regions. Note that now there are three conflicting flows: two coming from picking
and one from the putaway. This leads us to mirror the same middle region designs as in
Figure 2.4. As for the rightmost region, we depict the alternative, diagonal lanes, and note
that they should be given preference only if picking is more labor intensive than putaway.
2.3 Alternative layout of the facility
Having presented the basic modeling choices and underlying principles, we now tackle
the manufacturer?s layout. The management identified three major groups of skus ? fast
movers, less frequent and slow movers, and assigned them to appropriate regions in the
warehouse. The topmost regions contain slow movers and we do not optimize its lane and
aisle structure. For the two faster regions we devise a new aisle and lane structure and
propose the layout in Figure 2.6.
In our proposition, we mostly use diagonal lanes and cut-through aisles, but we retain
classical floor storage where necessary. In the regions A1 and B1 we are free to use diagonal
lanes, but in the areas A2, B2 and B3 we resort to cut-through aisles due to the conflicting
55
Figure 2.6: Improved Layout
56
flows. For the region A3 which is the intersection of conflicting flows, and B4 which is not,
we keep the traditional storage.
Next, we estimate potential savings from the new layout. For this purpose, we develop
a small computational aid. Improved storage blocks feature diagonal lanes and cut-through
aisles, so our assessment of benefits needs to address these novelties. In doing so, we make
one simplifying assumption ? we disregard the width of the cut-through aisle. Although
this leads to an overstatement of the anticipated savings, lost storage locations account for
less than 1% of the total storage capacity of this warehouse. Therefore, we do not expect
our assumption to significantly reduce the precision of the estimate.
We begin by expressing the cost of picking in a rectangular half-space with diagonal
lanes and one P&D point in the lower left corner (Figure 2.7). The diagonal divides the
storage space into two equal regions and we denote the average cost of picking in the lower
and upper region as E[L] and E[U], respectively. In the region L, the picker is moving
along the bottom of the rectangle and then parallel to the diagonal. In the region U, travel
is along the left edge of the rectangle and then parallel to the diagonal. Because we assume
that picking is uniformly distributed throughout the space, probabilities of choosing a pick
in upper or lower regions will be equal, i.e., pl = pu = 1/2. From the mean value theorem
for multiple integrals, expected cost of a single pick is:
E[C] = plE[L]+puE[U] = pl 1m
l
integraldisplay integraldisplay
L+pu 1m
u
integraldisplay integraldisplay
U .
57
Specifying the measures of the areas of integration,
ml =
integraldisplay A
0
integraldisplay A?x
0
1 dydx = A
2
2
mu =
integraldisplay h
0
integraldisplay h?x
0
1 dydx = h
2
2 ,
and from similarity of triangles, we obtain the following expressions:
integraldisplay integraldisplay
L =
integraldisplay A
0
integraldisplay A?x
0
(x+ y
?A2 +h2
A )dydx =
1
12A
2(A+radicalbigA2 +h2)
integraldisplay integraldisplay
U =
integraldisplay A
0
integraldisplay h?x
0
(x+ y
?A2 +h2
h )dydx =
1
12h
2(h+radicalbigA2 +h2).
By means of elementary substitution we arrive at the average cost of a single pick:
E[C] = 16(A+h+2
radicalbig
A2 +h2)
= A+h6 + D3 .
This formula can be interpreted as traveling (A+h)/6 units along the bottom of the longer
side of the warehouse rectangle and then D/3 units parallel to the diagonal.
In the case of the cut-through aisle, it is possible to develop the cost expression in a
similar manner. A diagonal divides the warehouse rectangle into two equal triangle-spaces.
Triangle altitudes (ht) further divide the lower and upper triangles into four cost regions:
L1, L2, U1 and U2. For simplicity, in Figure 2.8 we show only the structure of regions
L1 and L2. A picker travels along the diagonal of the rectangle, and then parallel to the
altitude into the space of the lower (or upper) triangle. Expected cost of making a pick is
58
Figure 2.7: Travel patterns in a floor storage with diagonal lanes
given with:
E[C] = pl1E[L1]+pl2E[L2]+pu1E[U1]+pu2E[U2]
= pl1 1m
l1
integraldisplay integraldisplay
L1 +pl2 1m
l2
integraldisplay integraldisplay
L2 +pu1 1m
u1
integraldisplay integraldisplay
U1 +pu2 1m
u2
integraldisplay integraldisplay
U2 .
Geometric properties of the warehouse layout lead to basic geometric relations below:
D =
radicalbig
A2 +h2
ht = AhD = Ah?A2 +h2
dB = A
2
D =
A2?
A2 +h2
Al1 = Au2 = A
3h
2(A2 +h2)
Al2 = Au1 = Ah
3
2(A2 +h2) ,
59
where dB simply stands for the length of the diagonal from the vertex to the foot of the
altitude, and other variables are self-explanatory.
Probabilities of choosing one among four regions are ratios of storage space in the
picking region and total storage space available:
pl1 = pu2 = Al1Ah = Au2Ah = A
2
2(A2 +h2)
pl2 = pu1 = Al2Ah = Au1Ah = h
2
2(A2 +h2).
The first part of the pick travel is done along the diagonal, but the second part, along
the line parallel to the altitude, depends on the length of the first part. We denote these
functions as d1(x), d2(x), d3(x) and d4(x). They pertain to the regions L1, L2, U1 and U2
respectively:
d1(x) = xhA
d2(x) = A(D ?x)h
d3(x) = xAh
d4(x) = h(D ?x)A .
Cost integrals for four picking regions are given with:
integraldisplay integraldisplay
L1 =
integraldisplay dB
0
integraldisplay d1(x)
0
(x+y)dydx = A
4h(2A+h)
6(A2 +h2)3/2integraldisplay integraldisplay
L2 =
integraldisplay D
dB
integraldisplay d2(x)
0
(x+y)dydx = Ah
3(3A2 +Ah+h2)
6(A2 +h2)3/2
integraldisplay integraldisplay
U1 =
integraldisplay D?dB
0
integraldisplay d3(x)
0
(x+y)dydx = Ah
4(A+2h)
6(A2 +h2)3/2integraldisplay integraldisplay
U2 =
integraldisplay D
D?dB
integraldisplay d4(x)
0
(x+y)dydx = A
3h(A2 +Ah+3h2)
6(A2 +h2)3/2 .
60
Measures of areas of integration are:
ml1 =
integraldisplay dB
0
integraldisplay d1(x)
0
1 dydx = A
3h
2(A2 +h2)
ml2 =
integraldisplay D
dB
integraldisplay d2(x)
0
1 dydx = Ah
3
2(A2 +h2)
mu1 =
integraldisplay D?dB
0
integraldisplay d3(x)
0
1 dydx = Ah
3
2(A2 +h2)
mu2 =
integraldisplay D
D?dB
integraldisplay d4(x)
0
1 dydx = Ah
3
2(A2 +h2) .
Since we derived all the necessary expressions, we proceed to formulate the final picking
cost:
E[C] = 3A
2 +2ah+3h2
6?A2 +h2
= D2 + Ah3D
= D2 + ht3 ,
which is conveniently interpreted as first traveling one half of the diagonal?s length, and
then one third of the altitude length along a line parallel to the altitude.
The natural use of these formulae is to calculate the expected cost of a pick from the
P&D point that is located at or near a vertex of a rectangular region. In the layout we
propose, not all travels have the convenience of direct access to the storage space. Most of
the time, a picker will travel a certain straight or rectilinear distance and then gain access
to the picking locations through one of the points (not necessarily the P&D point) at the
vertices of a rectangular storage space. Therefore, the approximate average cost of picking
61
Figure 2.8: Travel patterns in a floor storage with diagonal lanes and cut-through aisles
in the block Bi from the P&D point j is expressed as:
E[Bji] = SiS (Rji +E[Ci]),
where Si is the storage space in block i, S is the total storage space, Rji is the distance
from P&D point j to the vertex of a block i, and E[Ci] is the expected cost of picking in a
block from its vertex. Multiplying every expected cost Bji by probability of choosing the
P&D point j and summing across all P&D points and blocks, gives the total expected cost
of picking in a number of blocks.
62
2.4 Results and conclusions
In Chapter 1 we assumed uniformly distributed activity throughout the entire storage
space. Yet, we have stated that the manufacturer?s operations indicate the contrary ?
dedicated storage policy, with separate regions reserved for skus with different activity.
Therefore, we believe that the costs should be separated region by region, with random
picking assumed inside each region, and the cost of operation measured with the formulae
we developed. More than that, we believe that for practical purposes, picking and putaway
costs should be separated in a warehouse with conflicting flows. Dividing the costs by region
and type (picking or putaway) helps to assess accurately the importance of the regions from
the cost standpoint and possible prevalence of a single cost type.
We calculated the expected cost and savings for picking and putaway in the regions
holding fast movers (lower region) and less frequent skus (middle region). Picking savings
for the lower and upper region are 10.54% and 9.18%, respectively. When it comes to the
putaway cost, the new aisle and lane structure saves 10.57% in the lower region and 9.84%
in the upper region. The numbers do look promising but should be put in the right context.
Because of a number of assumptions we made, we think of this result as an approximation
or even as a proof that significant savings are possible, rather than the guarantee that the
savings are going to amount exactly to what we have calculated. However, the principles
we presented, if applied correctly, promise improvement.
In addition to the quantitative aspect of the layout we propose, there exist other, more
practical issues that need to be addressed. Similar to our cross aisle designs for rack storage,
worker orientation and routing might become an obstacle. A very deep diagonal lane might
cause a worker to lose orientation. Frequent visits to lanes with different orientations might
63
have a similar effect. Additional clearance on the exit of a cut-through aisle may be required
to prevent interference with a worker?s field of vision.
Although models to determine optimal lane depth for each sku exist (Bartholdi and
Hackman (2007)), it is our impression that most warehouse managers prefer having only a
small number of lane depths. Such a solution is robust with respect to day-to-day operations,
but it utilizes the floor space optimally only if skus have similar optimal lane depths. On
the other hand, a design with diagonal lanes features a wide variety of lane depths. If the
lane depths are uniform across the sku spectrum, there is not much sense in insisting on a
diagonal design. However, if that is not the case, and the warehouse keeps track of its sku
data in a way that enables easy calculation of the optimal lane depths, diagonal designs are
a promising choice. Another potential problem is that the deepest diagonal lanes might be
much deeper than any lane in the traditional design. If the new lane depths are not suitable
for the skus currently present in the warehouse, one possible solution is to design smaller
blocks with shorter diagonals.
64
Bibliography
Bartholdi, J. J. and Hackman, S. T. (2007). Warehouse and Distribution Science. Version
0.85.
Berry, J. R. (1968). Elements of warehouse layout. International Journal of Production
Research, 7(2):105?121.
Francis, R. L. (1967). Sufficient conditions for some optimum property facility designs.
Operations Research, 15(3):448?466.
Gue, K. R. and Meller, R. D. (2006). Aisle configurations for unit load warehouses. Technical
report, Auburn University, Department of Industrial and Systems Engineering, Auburn,
AL 36849.
Lazoff, D. M. and Sherman, A. T. (1994). An exact formula for the expected wire length
between two randomly chosen intervals. Technical Report TR CS-94-08, University of
Maryland Baltimore County, Computer Science Department, Baltimore, MD 21228-5398.
Roberts, S. D. and Reed, R. (1972). Optimum warehouse bay configurations. AIIE Trans-
actions, 4(3):178?185.
White, J. A. (1972). Optimum design of warehouses having radial aisles. AIIE Transactions,
4(4):333?336.
65
Appendix ? Tables of savings over orthogonal aisle design
66
h=50 h=75 h=100 h=125
n/w ? X V LB ? X V LB ? X V LB ? X V LB
11/1 0 2.21 4.78 22.10 0 2.32 4.85 20.16 0 2.23 4.60 18.05 0 2.09 4.28 16.20
15/1 0.66 2.72 5.41 22.50 0.94 2.92 5.76 21.81 1.01 2.86 5.64 20.32 1.02 2.72 5.38 18.75
19/1 1.06 2.97 5.62 22.10 1.36 3.28 6.23 22.46 1.45 3.28 6.27 21.61 1.45 3.17 6.10 20.41
23/1 1.31 3.08 5.62 21.36 1.65 3.49 6.44 22.55 1.76 3.56 6.65 22.28 1.77 3.49 6.58 21.46
27/1 1.48 3.12 5.51 20.50 1.85 3.62 6.50 22.31 1.99 3.75 6.85 22.55 2.02 3.73 6.90 22.10
31/1 1.59 3.11 5.35 19.60 1.99 3.68 6.47 21.90 2.16 3.87 6.94 22.56 2.21 3.89 7.09 22.45
35/1 1.67 3.07 5.17 18.73 2.09 3.70 6.38 21.38 2.28 3.95 6.96 22.40 2.36 4.00 7.20 22.58
39/1 1.72 3.02 4.99 17.89 2.16 3.69 6.26 20.82 2.37 3.99 6.93 22.13 2.47 4.09 7.24 22.57
11/1.5 0 0.93 3.06 22.15 0 1.36 3.46 20.25 0 1.46 3.43 18.13 0 1.44 3.28 16.27
15/1.5 0 1.59 3.95 22.49 0 2.05 4.55 21.87 0 2.15 4.60 20.38 0 2.12 4.47 18.81
19/1.5 0 1.96 4.33 22.03 0.30 2.48 5.14 22.48 0.51 2.62 5.33 21.65 0.61 2.61 5.27 20.46
23/1.50.23 2.16 4.47 21.26 0.68 2.76 5.46 22.54 0.90 2.95 5.79 22.30 1.00 2.97 5.82 21.50
27/1.50.47 2.27 4.48 20.37 0.96 2.93 5.60 22.28 1.19 3.17 6.06 22.56 1.30 3.23 6.19 22.13
31/1.50.69 2.32 4.41 19.45 1.17 3.04 5.64 21.85 1.42 3.33 6.21 22.55 1.53 3.42 6.32 22.46
35/1.50.85 2.34 4.31 18.57 1.33 3.09 5.61 21.31 1.59 3.42 6.27 22.38 1.72 3.56 6.57 22.58
39/1.50.97 2.33 4.18 17.72 1.45 3.12 5.55 20.74 1.72 3.49 6.28 22.10 1.87 3.66 6.65 22.56
11/2 0 0 1.40 22.20 0 0.39 2.10 20.33 0 0.68 2.29 18.21 0 0.80 2.29 16.33
15/2 0 0.45 2.52 22.47 0 1.18 3.35 21.92 0 1.44 3.58 20.45 0 1.52 3.57 18.87
19/2 0 0.93 3.08 21.96 0 1.68 4.07 22.50 0 1.96 4.40 21.70 0 2.04 4.45 20.51
23/2 0 1.23 3.35 21.15 0 2.02 4.49 22.52 0 2.33 4.93 22.33 0.25 2.43 5.06 21.54
27/2 0 1.42 3.47 20.23 0.13 2.24 4.72 22.24 0.43 2.59 5.27 22.56 0.60 2.72 5.48 22.15
31/2 0 1.54 3.49 19.30 0.28 2.39 4.82 21.79 0.70 2.78 5.48 22.54 0.88 2.94 5.76 22.47
35/2 0.10 1.61 3.46 18.40 0.51 2.49 4.86 21.25 0.92 2.91 5.59 22.36 1.10 3.1 5.95 22.59
39/2 0.25 1.65 3.40 17.55 0.71 2.54 4.84 20.65 1.09 3.00 5.64 22.06 1.28 3.23 6.07 22.55
11/2.5 0 0 0 22.25 0 0 0.78 20.42 0 0 1.16 18.29 0 0.14 1.31 16.40
15/2.5 0 0 1.14 22.44 0 0.29 2.17 21.96 0 0.72 2.56 20.51 0 0.91 2.68 18.93
19/2.5 0 0 1.87 21.88 0 0.87 3.02 22.51 0 1.29 3.48 21.74 0 1.48 3.64 20.56
23/2.5 0 0.30 2.27 21.03 0 1.27 3.53 22.51 0 1.71 4.09 22.35 0 1.9 4.31 21.57
27/2.5 0 0.56 2.49 20.09 0 1.55 3.84 22.21 0 2.01 4.49 22.57 0 2.22 4.78 22.18
31/2.5 0 0.74 2.60 19.14 0 1.74 4.02 21.73 0.02 2.23 4.75 22.53 0.24 2.46 5.11 22.48
35/2.5 0 0.87 2.65 18.23 0 1.87 4.11 21.17 0.27 2.39 4.91 22.33 0.50 2.65 5.34 22.59
39/2.5 0 0.95 2.65 17.37 0.12 1.97 4.14 20.57 0.48 2.50 5.00 22.03 0.71 2.79 5.49 22.55
11/3 0 0 0 22.29 0 0 0 20.50 0 0 0.07 18.37 0 0 0.36 16.47
15/3 0 0 0 22.41 0 0 1.03 22.01 0 0 1.57 20.57 0 0.30 1.81 18.99
19/3 0 0 0.71 21.80 0 0.05 1.99 22.52 0 0.62 2.58 21.78 0 0.90 2.83 20.61
23/3 0 0 1.23 20.91 0 0.52 2.60 22.49 0 1.08 3.26 22.37 0 1.36 3.56 21.61
27/3 0 0 1.55 19.94 0 0.85 2.99 22.16 0 1.42 3.72 22.57 0 1.71 4.08 22.20
31/3 0 0 1.75 18.98 0 1.09 3.23 21.67 0 1.67 4.04 22.52 0 1.98 4.46 22.50
35/3 0 0.13 1.86 18.05 0 1.26 3.37 21.10 0 1.85 4.24 22.31 0 2.19 4.72 22.59
39/3 0 0.26 1.92 17.19 0 1.39 3.45 20.48 0 2.00 4.37 21.99 0.16 2.35 4.91 22.54
Table 1: Comparison of savings over orthogonal aisle design ? ? shape, crossdock, Flying-V
shape and Travel-by-flight (Lower bound for cost)
67
h=50 h=75 h=100 h=125
n/w Smp Str V Smp Str V Smp Str V Smp Str V
11/1 4.33 4.73 4.78 4.55 4.83 4.85 4.38 4.58 4.60 4.12 4.27 4.28
15/1 4.81 5.35 5.41 5.32 5.73 5.76 5.31 5.62 5.64 5.12 5.36 5.38
19/1 4.91 5.53 5.62 5.66 6.19 6.23 5.83 6.24 6.27 5.75 6.08 6.10
23/1 4.84 5.52 5.62 5.78 6.39 6.44 6.11 6.61 6.65 6.14 6.56 6.58
27/1 4.69 5.40 5.51 5.77 6.44 6.50 6.23 6.81 6.85 6.38 6.87 6.90
31/1 4.51 5.23 5.35 5.68 6.40 6.47 6.26 6.90 6.94 6.50 7.06 7.09
35/1 4.32 5.05 5.17 5.56 6.30 6.38 6.22 6.91 6.96 6.55 7.16 7.20
39/1 4.14 4.85 4.99 5.41 6.18 6.26 6.14 6.87 6.93 6.54 7.20 7.24
11/1.5 2.67 2.98 3.06 3.20 3.42 3.46 3.24 3.41 3.43 3.14 3.26 3.28
15/1.5 3.40 3.85 3.95 4.14 4.50 4.55 4.30 4.57 4.60 4.24 4.45 4.47
19/1.5 3.68 4.22 4.33 4.62 5.08 5.14 4.92 5.30 5.33 4.95 5.25 5.27
23/1.5 3.75 4.34 4.47 4.84 5.39 5.46 5.28 5.74 5.79 5.41 5.79 5.82
27/1.5 3.71 4.33 4.48 4.91 5.52 5.60 5.47 6.01 6.06 5.69 6.15 6.19
31/1.5 3.63 4.26 4.41 4.90 5.55 5.64 5.55 6.15 6.21 5.79 6.28 6.32
35/1.5 3.51 4.14 4.31 4.83 5.52 5.61 5.56 6.21 6.27 5.95 6.53 6.57
39/1.5 3.39 4.01 4.18 4.73 5.44 5.55 5.53 6.21 6.28 5.98 6.60 6.65
11/2 1.05 1.27 1.4 1.87 2.04 2.10 2.12 2.25 2.29 2.16 2.26 2.29
15/2 2.02 2.37 2.52 2.98 3.28 3.35 3.30 3.53 3.58 3.36 3.54 3.57
19/2 2.49 2.92 3.08 3.59 3.99 4.07 4.02 4.35 4.40 4.15 4.42 4.45
23/2 2.69 3.18 3.35 3.91 4.40 4.49 4.46 4.88 4.93 4.67 5.02 5.06
27/2 2.76 3.28 3.47 4.07 4.62 4.72 4.72 5.21 5.27 5.01 5.44 5.48
31/2 2.76 3.30 3.49 4.12 4.72 4.82 4.85 5.41 5.48 5.23 5.72 5.76
35/2 2.72 3.26 3.46 4.11 4.74 4.86 4.91 5.51 5.59 5.35 5.90 5.95
39/2 2.66 3.19 3.40 4.07 4.71 4.84 4.92 5.55 5.64 5.42 6.01 6.07
11/2.5 0 0 0 0.57 0.68 0.78 1.01 1.10 1.16 1.20 1.27 1.31
15/2.5 0.70 0.93 1.14 1.84 2.08 2.17 2.31 2.51 2.56 2.49 2.64 2.68
19/2.5 1.33 1.65 1.87 2.57 2.92 3.02 3.13 3.42 3.48 3.35 3.59 3.64
23/2.5 1.66 2.05 2.27 3.00 3.42 3.53 3.65 4.02 4.09 3.94 4.26 4.31
27/2.5 1.84 2.25 2.49 3.23 3.72 3.84 3.97 4.42 4.49 4.34 4.73 4.78
31/2.5 1.93 2.35 2.60 3.36 3.89 4.02 4.16 4.67 4.75 4.60 5.05 5.11
35/2.5 1.96 2.39 2.65 3.40 3.97 4.11 4.27 4.82 4.91 4.76 5.27 5.34
39/2.5 1.95 2.38 2.65 3.41 3.99 4.14 4.31 4.90 5.00 4.86 5.41 5.49
11/3 0 0 0 0 0 0 0 0 0.07 0.27 0.30 0.36
15/3 0 0 0 0.73 0.90 1.03 1.34 1.49 1.57 1.63 1.75 1.81
19/3 0.22 0.42 0.71 1.58 1.85 1.99 2.25 2.50 2.58 2.57 2.78 2.83
23/3 0.68 0.94 1.23 2.10 2.46 2.60 2.84 3.17 3.26 3.22 3.50 3.56
27/3 0.96 1.25 1.55 2.42 2.83 2.99 3.23 3.63 3.72 3.66 4.02 4.08
31/3 1.12 1.43 1.75 2.60 3.07 3.23 3.48 3.94 4.04 3.97 4.39 4.46
35/3 1.22 1.54 1.86 2.71 3.20 3.37 3.63 4.13 4.24 4.18 4.65 4.72
39/3 1.28 1.60 1.92 2.76 3.27 3.45 3.71 4.26 4.37 4.31 4.83 4.91
Table 2: Comparison of savings over orthogonal aisle design ? Simple-V, Straight-V and
Flying-V shapes
68