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Space-efficient Layouts for Block Stacking Warehouses
Shahab Derhamia,∗, Jeffrey S. Smithb, Kevin R. Guec
aSchool of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
bDepartment of Industrial and Systems Engineering, Auburn University, Auburn, AL 36849, USA
cDepartment of Industrial Engineering, University of Louisville, Louisville, KY 40292, USA
Abstract
In block stacking warehouses, pallets of stock keeping units (SKUs) are stacked on top of one
another in lanes on the warehouse floor. A conventional layout consists of multiple bays of lanes
separated by aisles. The depths of the bays and the number of aisles determine the storage
space utilization. Using an analytical model, we show that the traditional lane depth model
underestimates accessibility waste and therefore does not provide an optimal lane depth. We
propose a new model of wasted storage space and embed it in a mixed integer program to find
the optimal bay depths. The model improves space utilization by allowing multiple bay depths
and allocating SKUs to appropriate bays. Our computational study shows the proposed model is
capable of solving large-scale problems with a relatively small optimality gap. We use simulation
to evaluate performance of the proposed model on small to industrial-sized warehouses. We also
include a case study from the beverage industry.
Keywords: Facility layout, Warehouse design, Block stacking, Space utilization, Optimal lane
depth
1. Introduction
Storing pallets of Stock Keeping Units (SKUs) on top of one another on a warehouse floor is
known as block stacking. This storage system does not generally require storage racks and can be
inexpensively implemented in any open area. Hence, it is widely used as the main storage system
in many manufacturing and non-manufacturing operations as well as distribution centers. Block
stacking is also economical and prevalent where storage items are heavy and large pallets, boxes,
∗Corresponding author
Email address: shahab.derhami@isye.gatech.edu (Shahab Derhami)
Preprint submitted to IISE Transactions December 5, 2018
or containers. Examples of such environments are bottled beverage companies, food industries,
major home appliance producers/distribution centers, and maritime container terminals.
Block stacking is mainly used with a shared or dedicated policy (Bartholdi & Hackman, 2008).
In the dedicated policy, lanes are dedicated to SKUs, and each SKU is allowed to be stored only
in its assigned lanes, whereas in the shared (random) policy empty lanes are available to all SKUs.
Hence, the shared policy is more efficient in utilizing storage space and therefore widely employed,
but is generally less efficient for order picking. However, when the variety of SKUs is much more
than the storing quantities, like maritime container terminals, stacking different items in the same
lane is inevitable. In such situations, the goal is to allocate storage space such that relocation costs
are minimized (Kim & Hong, 2006; Yang & Kim, 2006; Jang et al., 2013). For a detailed review
of the research on container stacking refer to Carlo et al. (2014).
To prevent lane blockage or pallet relocations, a lane in the shared policy is temporarily dedi-
cated to the SKU that occupies its first pallet position, making unoccupied pallet positions of the
lane unavailable to other SKUs. This effect is called honeycombing, and waste associated with it
remains in the system until the lane becomes fully occupied or emptied. The focus of this paper
is on the shared policy. As explained by Bartholdi & Hackman (2008), aisles also contribute to
the waste of space, because they are not used for pallet storage but are required to access lanes.
To enhance utilization of the storage space, the warehouse must be designed such that both of
these wastes are minimized. However, there is a trade-off. Layouts with shallow lanes generate less
honeycombing waste but require more aisles, whereas the opposite is true for deep lanes.
Various researchers have studied layout design for conventional rack storage systems (Baker &
Canessa, 2009; Gu et al., 2010). These studies mostly considered designing the layout with respect
to transportation costs for order picking (Gue & Meller, 2009; Gue et al., 2012; ¨
Ozt¨urko˘glu et al.,
2012, 2014; Thomas & Meller, 2014; Cardona et al., 2015). Further details can be found in de Koster
et al. (2007). Other researchers investigated this problem from the perspectives of operational cost
(Thomas & Meller, 2015; Mowrey & Parikh, 2014; Zhou et al., 2016), space utilization (Derhami
et al., 2017), product allocation (Moshref-Javadi & Lehto, 2016; Ramtin & Pazour, 2015; Li et al.,
2016), operating policies (Ramtin & Pazour, 2014; Roodbergen et al., 2015; Guo et al., 2016), and
warehouse throughput (Pazour & Meller, 2011; Lamballais et al., 2017; Zaerpour et al., 2017).
A few researchers have studied the design of block stacking storage systems. The conventional
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models of block stacking focus exclusively on finding the lane depth that minimizes honeycombing
and accessibility waste by considering the trade-off between the depth and width of the storing
block. The lane depth here is defined as the depth of the lanes used to store a batch of pallets.
These models are hereafter called traditional lane depth models. Kind (1975) is a seminal study of
this kind. He proposed to consider the trade-off between a block width and depth to find the lane
depth minimizing waste of floor space. However, he did not provide any derivations for his formula.
Later, Marsh (1979) used simulation to evaluate space utilization on alternative lane depths and
storage policies in this storage system.
Matson (1982) extended Kind’s model (Kind, 1975) and proposed a model to approximate the
lane depth when lanes are replenished instantaneously (i.e., replenishment rate is infinity). She
also developed a model to find the common lane depth for multiple SKUs. Her models are suitable
for warehouses storing products received from suppliers, in which a truck unloads a batch of pallets
at once (infinite replenishment rate).
Goetschalckx & Ratliff (1991) showed if multiple lane depths are allowed, then the optimal lane
depths follow a triangular pattern. They developed a dynamic programming algorithm to select
multiple optimal lane depths from a set of finite allowable lane depths so that the occupied floor
space is minimized. They used a heuristic to form the warehouse layout by selecting at most five
depths that form an approximately geometric series and then calculating the required number of
lanes for each product based on the selected lane depths. The algorithm then rounds up or down
the aggregated number of required lanes for each depth to the nearest multiple of lanes in an aisle.
Larson et al. (1997) proposed a heuristic to design a class-based layout that maximizes floor
space utilization and minimizes material handling cost. Their algorithm consists of three phases.
In the first phase, the aisle directions (layout) and storage zone dimensions are determined. Then,
storage types (rack storage or floor storage) are determined for all SKUs, and the required storage
space for each storage type is calculated. Finally, the floor space is allocated for the storage zones
(types) based on their types, required number of storage locations, and throughput. Accorsi et al.
(2017) developed a decision support system to solve storage allocation problem in block stacking.
Their proposed mixed integer programming model assigns the lane depth among the available lanes
to the upcoming SKUs on a daily basis.
Derhami et al. (2017) extended Matson’s model (Matson, 1982) with two models to minimize
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waste of storage volume instead of floor space. They developed two finite production (replenish-
ment) rate models: one for continuous demand less than the production rate, and the other for
demand greater than the production rate. They showed using an infinite production rate model
in a finite production rate system results in lane depths about twice as deep as they should be.
However, the resulting waste of volume is not significant because the space utilization curve, as a
function of lane depth, is quite flat as the lane depth increases.
The traditional lane depth model (Derhami et al., 2017; Matson, 1982) is the only analytical
model in the literature, to the best of our knowledge, that considers the lane depth. It is mostly
suitable to find the optimal block sizes for temporary storage in a wide area because it computes
accessibility waste only for the period that a lane is occupied. That is, it treats aisle space as
wasted space only for the period that a lane is occupied and considers the dedicated space to the
aisles as available storage space otherwise. Hence, it is not appropriate for a fixed layout where
the dedicated space to the aisles is always used for transportation. Many warehouses use block
stacking as the main storage media. To facilitate storage operations and material handling, these
warehouses use fixed layouts like the one presented in Figure 1b. A typical block stacking layout
consists of bays with different depths formed by a group of adjacent lanes. Bays are separated by
aisles. The dedicated space to the aisles is always used for accessibility (i.e., not storage space) no
matter if the adjoining lanes are occupied or not. Hereafter, we refer to the problem of determining
the number of aisles and depth of bays for such layouts as the layout design problem. The main
research questions in the layout design problem are:
•How many aisles and bays should the layout have?
•How deep should the bays be?
•What is the most space-efficient storage space allocation for block stacking?
The traditional lane depth model does not provide optimal bay depth for a layout design problem.
This is mainly because its objective function underestimates the accessibility waste of this problem
as it computes waste of the dedicated space to the aisles only for the period that lanes are occupied.
The other limitation of this model is that it does not take into account warehouse dimensions and
enforces the same depth on all bays. Hence, space utilization cannot be as good as it could be
because only some SKUs are stored in their preferred depths. To the best of our knowledge, no
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analytical model exists to answer these design questions comprehensively. In this paper, we develop
a model to address all of the above questions with the same model. Our model finds the optimal
number of aisles and bay depths maximizing utilization of the storage space in the entire layout
while allowing multiple bay depths in the layout. Our paper contributes to the warehouse design
literature in three key ways:
1. We analyze waste of storage volume in block stacking from a layout design perspective. Our
approach relaxes the immediate replenishment assumption of the traditional lane depth model
and considers the volume dedicated to aisles as wasted storage volume for the entire planning
horizon. This leads to a new model to estimate waste of storage space in the entire layout as
a function of bay depths.
2. We propose the first mathematical model to find the optimal bay depths minimizing the new
waste function we developed for the layout design problem. Our model is a mixed integer
program that finds the optimal number of bays and bay depths maximizing utilization of the
storage space. It maximizes utilization of the storage space by allowing various bay depths
in the layout and assigning SKUs to their preferred bay depths.
3. We develop analytical solution techniques to reduce computational efforts of the model and
produce optimal or near-optimal solutions. This includes various inequalities to break the
problem symmetry, tightening the bounds on the influential decision variables, and tightening
the lower bound of the LP-relaxation.
The remainder of this paper is organized as follows. First, in section 2, the total waste of storage
space is formulated for the layout design problem, and its difference with the waste function of the
traditional lane depth model is discussed. Then, a nonlinear mathematical model and its linearized
version are presented in section 3 to find the optimal number of aisles and bay depths minimizing
the new waste function. Next, analytical solution approaches are formulated in section 4. Finally,
an experimental study is presented in section 5 to investigate the computational efficiency of the
proposed models, and simulation results are analyzed to evaluate performance of the model under
stochastic conditions.
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(a) (b)
Figure 1: A layout generated by the proposed model (a) and after adding cross-aisles and an extra aisle (b).
2. Waste of the storage volume
The warehouse layout is defined as the number, shape, and arrangement of bays, aisles, cross-
aisles, and bay depths for the given warehouse area. A typical warehouse layout and its elements
are shown in Figure 1b.
Cross-aisles are used to facilitate access to lanes and to reduce travel distances inside the
warehouse. So, unlike picking aisles, they are not necessary for pallet storage and consequently their
space is considered a pure waste of storage space. Because the objective of our model is to maximize
utilization of the storage volume, including the number of cross-aisles as a decision variable in
the model would lead to zero cross-aisles. Hence, they are not considered in the modeling, and
we assume the number of cross-aisles is given based on the warehouse width, material handling
system, operational practices, and traffic congestion. To better utilize the storage volume, we
assume each aisle is shared between two bays. Therefore, any additional aisles and cross-aisles (for
ease of transportation) must be added afterward. This is presented in Figure 1. Figure 1a presents
a layout generated by our algorithm, and Figure 1b shows the same layout after adding cross-aisles
and an extra aisle. The extra aisle provides enough space to have travel paths next to both short
sides of the layout for ease of transportation and flexibility in distributing loading/unloading docks.
We assume:
•The production schedule and production sequences are not known in advance.
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•The warehouse size is sufficiently large, and the production sequences will be such that the
warehouse can accommodate all produced SKUs under a shared storage policy.
•All lanes in the same bay have the same depth.
•Lanes are accessible from one side and depleted in the Last-In-First-Out (LIFO) order.
•Lanes are perpendicular to short sides of the layout (labeled “Effective width” in Figure 1).
•Aisles are bidirectional.
•No lot splitting.
To simplify modeling, we represent dimensions in units of floor space (pallets) rather than units
of distance like feet or meters. In the next section, we calculate the total waste of storage volume
and describe a model to minimize this waste. We develop the model for finite production (re-
plenishment) rates where the production rates are greater than the demand rates, and demand is
continuous (i.e., a warehouse is located in a manufacturing system). This model can be converted
to instantaneous replenishment by letting the production rate, in the limit, approach infinity (i.e.,
a warehouse is replenished by suppliers).
2.1. Waste of storage space
Assume a batch of Qipallets of SKU iis produced and stored at the rate of Pipallets per
unit of time. Pallets are retrieved from the storage lanes at the rate of λipallets per unit of time,
where Pi> λi. Assume pallets of this SKU are Hifeet high and can be stacked up to Zipallets.
We define Hiin units of distance rather than pallets to distinguish different pallet heights in the
waste calculations. The change in the inventory of this SKU over its cycle time is shown in Figure
2 where Tiis the inventory cycle time of SKU i,T1
iand T2
iare replenishment and retrieval periods,
respectively, and Viis the maximum inventory level of SKU iin Ti. Three types of waste are
generated in the warehouse:
1. Honeycombing waste: Pallet positions in a partially occupied lane that are unoccupied
but unavailable to all SKUs. This waste is generated until a lane is partially occupied.
2. Unoccupied volume at the top of stacks: Unoccupied space between the top of lanes
and the clear height of the warehouse. This waste is incurred as a result of different pallet
heights or stackable heights, which lead to various stack heights for different SKUs.
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Figure 2: Changes in the inventory of SKU iover time, Pi> λi.
3. Accessibility waste: The volume of space dedicated to aisles is not directly used for pallet
storage and therefore considered as wasted space. Unlike the traditional lane depth model,
the accessibility waste is considered a permanent waste in the layout design problem.
In the next section, we formulate these three types of waste for the layout design problem. The first
two components are similar to the waste function of the traditional lane depth model (Derhami
et al., 2017).
2.1.1. Honeycombing waste
As described in Derhami et al. (2017) the total honeycombing waste incurred to replenish a
lane with depth xipallets with SKU iis
Hi
Pi−λi((Zixi−1) + (Zixi−2) + · · · + (Zixi−(Zixi−1))) ,(1)
and the honeycombing waste generated when the lane is being emptied is
Hi
λi(1 + 2 + · · · + (Zixi−1)) .(2)
The total honeycombing waste generated by replenishing and retrieving a batch of Qipallets is
obtained by summing (1) and (2), and multiplying the result by the number of lanes the SKU
occupies. That is,
WH
i≈1
2λi(HiQi(Zixi−1)) .(3)
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2.1.2. Waste of unoccupied volume at the top of stacks
We define the clear height of the warehouse as the greatest stackable height. So, unoccupied
waste at the top of a stack is zero if a SKU can be stored to the maximum stackable height.
Consequently, this waste is removed from the model if all SKUs have the same stackable height.
This waste is incurred to the system for the entire time that a lane is partially or fully occupied.
As shown by Derhami et al. (2017), the total time that all lanes are occupied in T1
iis
1
Pi−λi((Vi−1) + (Vi−Zixi−1) + (Vi−2Zixi−1) + · · · + (Vi−Ki(xi)Zixi)) ,(4)
where
Vi≈Qi(Pi−λi)
Pi
,(5)
and Ki(xi) is the number of storage lanes SKU ioccupies if it is stored in lanes with depth xi
pallets. It is obtained by
Ki(xi)≈Qi(Pi−λi)
PiZixi
.(6)
Note that both expressions (5) and (6) are estimates because we assume pallet positions are filled
at the rate of Pi−λirather than being filled at the rate Piand emptied simultaneously at the rate
λi. The total time that lanes are occupied in T2
iis
1
λi(Vi+ (Vi−Zixi)+(Vi−2Zixi) + · · · + (Vi−Ki(xi)Zixi)) .(7)
The total waste of unoccupied volume at the top of lanes is obtained by adding (4) and (7) and
multiplying the result by the volume wasted at the top of a lane, which is (Sh−ZiHi)xi, where
Shis the clear height of the warehouse in units of distance. That is,
WU
i≈Qi(Sh−ZiHi)
2PiλiZi(Qi(Pi−λi) + PiZixi−2λi).(8)
2.1.3. Waste of the dedicated volume to the aisles
As opposed to the traditional lane depth models in which the waste of the dedicated volume to
aisles is computed only for the period that a lane is occupied (Derhami et al., 2017; Bartholdi &
Hackman, 2008), we consider the volume devoted to aisles as a permanent waste. Thus, the total
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accessibility waste is
WA=AShSwn, (9)
where nis the number of aisles in the layout, Ais the aisle width, and Swis the warehouse width.
2.1.4. Total waste of the storage volume in the warehouse
The total waste of storage volume in the warehouse is the sum of honeycombing waste, un-
occupied volume at the top of stacks, and the volume dedicated to the aisles. Denote the least
common multiple of the cycle times of all SKUs by TL. It also can be considered as a long period
of time (steady state) in which all SKUs will have sufficient inventory turns. Given that the cycle
time of SKU iis Qi/λi, the number of inventory turns for this SKU in TLis TLλi/Qi. Therefore,
the total WH
iand WU
igenerated by SKU iin TLis
WHU
i=TLλi(WH
i+WU
i)
Qi
.(10)
The total waste in the warehouse is given by summing WHU
ifor all SKUs and adding the aisle
volume to the result. Note that the aisle volume remains as a waste for the entire TL. Hence, the
total storage volume wasted in the warehouse in TLis given by
W=TLAShSwn+X
i∈ITLλi(Wi
H+Wi
U)
Qi,(11)
where Iis the set of all SKUs stored in the warehouse. Dividing (11) by TLgives the average
waste of storage volume in the warehouse. That is,
¯
W=AShSwn+Sh
2X
i∈I
xi+X
i∈I1
2PiZi(Qi(Sh−ZiHi)−ZiHi)(Pi−λi)−λi(2Sh−ZiHi).
(12)
Expression (12) depends on the following decision variables: set of assigned lane depths to
SKUs (xi), and the number of aisles in the layout (n). Hence, optimizing it requires considering
only the first two terms. It is also restricted to the following constraint: the sum of bay depths
and aisle widths must equal the warehouse length. This causes a trade-off between bay depths and
the number of aisles. A layout with deep bays has fewer aisles but generates greater honeycombing
waste while the reverse is true for a layout with shallow bays. In the next section, we develop a
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model to optimally address this trade-off and minimize the total waste of storage volume in the
warehouse.
3. Designing the warehouse layout
We minimize (12) with a mixed integer programming model that finds the optimal values for
nand xi. We call it MBD to distinguish it from the common lane depth model, which we call
CLD. The MBD finds the optimal number of aisles and also bay depths for the given warehouse
dimensions. It allows multiple depths in a layout. Hence, to measure waste of space, it assigns
SKUs to bays considering the number of lanes they require with respect to the assigned depths.
Because the warehouse is intended to operate under the shared policy, it is not spacious enough
to dedicate the required number of lanes to all SKUs. That is, the warehouse cannot accommodate
all SKUs at their maximum inventory levels at the same time, but the sequence of SKU replenish-
ment and retrieval operations are such that there is enough storage space for all incoming pallets
(see assumptions in Section 2). Notice that this would not be an issue in a layout operated un-
der the dedicated policy. Therefore, the model conceptually expands the layout width to provide
sufficient space to dedicate the required number of lanes to all SKUs (i.e., sufficient hypothetical
space to allow switching the operating policy to the dedicated policy). That is, the same number
of hypothetical lanes are added to all bays.
The model then assigns SKUs to bays ensuring all SKUs have been assigned to the exact number
of lanes they require with respect to their assigned bay depths, and no bay is over-assigned. The
resulting SKU assignment is a space-efficient operating policy used to prioritize bays with empty
lanes for assignment to incoming SKUs.
To provide a clear view to the readers, we first present the initial version of the MBD, which
is a nonlinear model. The linearized model will be described next. The definition of the sets,
parameters, and decision variables used in the mathematical models are presented next.
3.1. Nonlinear model
Sets and parameters:
Bset of bays, B={1, . . . , bmax}
Eexpansion ratio
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Slwarehouse length, must be integer (in units of pallets)
Swwarehouse width (in units of pallets)
Lminminimum allowable bay depth
Lmaxmaximum allowable bay depth
bmin minimum number of bays in a layout
bmax maximum number of bays in a layout
Decision variables:
yib 1 if SKU iis assigned to bay b, 0 otherwise
rbdepth of bay b(in units of pallets)
eb1 if bay bexists in the optimal layout (i.e., rb>0), 0 otherwise
Minimize ASwX
b∈B
eb+X
i∈IX
b∈B
yibrb(13)
Subject to
eb−1=eb∀b∈ {2,4, . . . , bmax}(14)
X
b∈B
rb+A
2X
b∈B
eb=Sl(15)
Lmineb≤rb≤Lmax eb∀b∈B(16)
dKi(rb)eyib ≤Sweb∀i∈I, b ∈B(17)
Sweb≤X
i∈I
dKi(rb)eyib ≤ESw∀b∈B(18)
yib ∈ {0,1} ∀i∈I, b ∈B(19)
eb∈ {0,1} ∀b∈B(20)
rb∈Z+∀b∈B(21)
The objective function (13) minimizes the total waste of storage volume in the warehouse. It
takes into account the variable parts of (12). Note that xi=Pb∈Byibrb, and considering constraint
(14), n=Pb∈Beb/2. We removed the common factor Sh/2 from both terms in (13).
Constraint (14) guarantees the number of bays to be twice the number of aisles. It pairs the
existence of two subsequent bays and hence forces the total number of existing bays to be even.
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As a result, Pb∈Beb/2 gives the number of aisles in the layout. Note that bmax must be even.
Because bmin and bmax have an influential impact on the decision space, in section 4.2, we propose
an analytical approach to find appropriate values of these two parameters that tighten the lower
bounds of the LP-relaxation. Constraint (15) ensures the sum of all bay depths and aisle widths
equals the warehouse length.
Constraint (16) relates eband rbtogether in addition to setting lower and upper bounds on rb.
If bay bexists in the solution (i.e. eb= 1), rbis forced to be between Lmin and Lmax. Otherwise,
it is forced to be zero. Note that bmax is sufficiently large to allow the model to select the optimal
number of bays. Hence, the optimal solution may have fewer bays than bmax. In this case, a zero
bay depth (i.e. rb= 0) implies the respective bay does not exist in the optimal layout.
Constraint (17) ensures the total number of assigned lanes from a bay to a SKU does not
exceed the number of lanes in a bay. Ki(rb) is the maximum number of lanes a SKU occupies if it
is assigned to a bay whose depth is rbpallets and calculated by (6).
Constraint (18) restricts SKU assignments and aims to balance the assignment loads among
different bays. The left-hand side inequality ensures all lanes of an existing bay are assigned to
at least one SKU. The right-hand side restricts the total number of assigned lanes of a bay to all
SKUs to be fewer than or equal to the warehouse width (the original number of lanes in a bay)
multiplied by the expansion ratio, E. The expansion ratio conceptually expands the number of
lanes in all bays to provide enough space to dedicate the required number of lanes to all SKUs.
We conceptually expand the layout because the warehouse is operated under the shared policy and
does not have sufficient storage space to keep all SKUs to their maximum inventory levels at the
same time. Based on our production scheduling assumption, the assignment will not violate the
physical constraints even though the model conceptually expands the facility.
Eis the minimum expansion of the number of lanes in bays such that there is enough storage
space to store all SKUs at the same time. It must be determined carefully because a large value of
Eallows the model to assign many SKUs to the shallow bays (to minimize Pi∈Ixi) and a small E
results in an infeasible solution. We compute Efor bmin and bmax and then select the largest ratio.
An analytical procedure to assign appropriate values to bmin and bmax is described in section 4.2.
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The procedure is as follows. First, a common bay depth is calculated for a layout with bmin bays:
¯xmin ="Sl−1
2bminA
bmin #.(22)
Then, dKi(¯xmin)eis calculated using (6). Summing Kis for all SKUs and dividing the result by
the total number of lanes in the layout gives
Emin =1
bminSw X
i∈IQi(Pi−λi)
PiZi¯xmin !.(23)
Similarly, Emax is calculated for bmax . The expansion ratio is
E=Max{Emin , Emax}+, (24)
where is a small number to compensate for the approximation error due to the common bay depth
assumption we made to derive a closed-form solution for Emin and Emax. After a preliminary
experiments, we found that setting it equal to 0.05 compensates for the approximation error.
The model described in this section is a nonlinear integer program with different types of
nonlinearity appeared in the objective function and constraints (17) and (18), making it extremely
hard to solve in a reasonable time for industrial-sized problems. In the next section, we propose a
linearized version of this model.
3.2. Linearized model
We linearize the model by introducing the following two sets of decision variables in addition
to the ones introduced for the nonlinear model:
xid 1 if SKU iis assigned to a bay whose depth is Ldpallets, 0 otherwise
zib number of lanes of bay bassigned to SKU i
The following set and data are used in the new model in addition to the ones defined before.
Dset of allowable depths, D={1, . . . , dmax}
dmax size of the set of allowable depths
Lddth allowable bay depth (in units of pallets), L={Lmin , Lmin + 1 , . . . , Lmax}
Rid number of lanes SKU ioccupies when stored in a bay whose depth is Ldpallets
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Mjarbitrarily large number, (j={1,...,3}).
Minimize ASwX
b∈B
eb+X
i∈IX
d∈D
Ldxid (25)
Subject to
X
d∈D
xid = 1 ∀i∈I(26)
X
b∈B
zib =X
d∈D
Ridxid ∀i∈I(27)
zib ≤Sweb∀i∈I, b ∈B(28)
Sweb≤X
i∈I
zib ≤ESw∀b∈B(29)
yib ≤zib ≤M1yib ∀i∈I, b ∈B(30)
rb≤X
d∈D
Ldxid +M2(1 −yib)∀i∈I , b ∈B(31)
rb≥X
d∈D
Ldxid −M3(1 −yib)∀i∈I , b ∈B(32)
X
b∈B
yib = 1 ∀i∈I(33)
constraints (14)+(15)+(16) (34)
yib ∈ {0,1} ∀i∈I, b ∈B
eb∈ {0,1} ∀b∈B(35)
rb∈Z+∀b∈B(36)
xid ∈ {0,1} ∀i∈I, d ∈D(37)
zib ∈Z+∀i∈I, d ∈D(38)
The objective function (13) is linearized to (25) by introducing xid to the model and replacing
Pi∈IPb∈Dyibrbwith Pi∈IPd∈DLdxid . Constraint (26) limits SKUs to only one depth. Con-
straints (27) and (28) linearize constraint (17). Constraint (27) forces the total number of assigned
lanes to a SKU to be exactly equal to the number of lanes it requires with respect to the assigned
bay depth. Rid is the equivalent of Ki(xi) in (6). It is the maximum number of lanes that SKU
irequires if assigned to a bay with depth Ldpallets. It is given to the model as an input and
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calculated by
Rid =Qi(Pi−λi)
PiZiLd∀d∈D. (39)
Constraint (28) ensures the total number of lanes assigned to a SKU from one bay does not
exceed the warehouse width. Constraint (29) is a linearization of constraint (18). Constraint (30)
relates zib and yib. The right hand side of (30) forces yib = 1 if a lane from bay bis assigned to
SKU i(i.e., zib >0). Otherwise, yib is forced to zero by the left hand side of (30). Because always
zib ≤Sw,M1must be greater than or equal to Sw.
Constraints (31) and (32) assign bay depths to the SKUs. If SKU iis assigned to bay b
(i.e., yib = 1), 1 −yib becomes zero, forcing both constraints to work as an equality constraint
setting Pd∈DLdxid =rb. Considering constraint (26), xid will be one for Ld=rb. If the SKU is
not assigned to the bay, the right hand side of constraint (31) becomes a large positive number,
making the constraint a loose constraint. Similarly, constraint (32) becomes non-binding as its
right hand side becomes less than zero. M2and M3must be large enough to prevent violating
these constraints when a SKU is not assigned to a bay. M2≥Lmax −Lmin and M3≥Lmax satisfy
this condition.
Constraint (33) ensures each SKU is assigned to only one bay. It reduces the computational
efforts of the model. Removing this constraint allows the model to assign a SKU to multiple
bays with the same depth. However, this increases the search space and also interferes with the
inequality we will describe in section 4.3 to solve the model. If this constraint is included in the
model, Ri1must be less than Swfor all SKUs; otherwise, the model will be infeasible. If this
condition does not hold for SKU i, the SKU must be broken down into |J|SKUs (i.e., lot splitting)
with the same production and demand rates, and stackable heights, but smaller batch quantities
such that Pj∈JQj=Qiand Rj1< Swfor all j∈J.
4. Solution techniques
The solution of the MBD model includes the optimal number of aisles, SKU assignments to
bays and bay depths. The search space is highly symmetric as all possible combinations of bay
depths with the same SKU assignments result in degenerate solutions with the same objective value.
We introduce symmetry-breaking constraints to remove these symmetric solutions from the search
space. We develop another class of inequalities to tighten the lower bound of the LP-relaxation
16
and reduce the search space by developing tight lower and upper bounds on the number of bays.
4.1. Reducing problem symmetry
For every feasible solution in the MBD, there exist multiple degenerate solutions that have
different depth orders but the same depth assignments to SKUs. For example, for a warehouse
with three bays and three SKUs all following solutions provide identical layouts regarding space
utilization.
Sol 1: r= (10,15,20) ,(y1,1= 1, y2,2= 1, y3,3= 1) , x = (10,15,20),
Sol 2: r= (15,10,20) ,(y1,2= 1, y2,1= 1, y3,3= 1) , x = (10,15,20),
Sol 3: r= (20,10,15) ,(y1,2= 1, y2,3= 1, y3,1= 1) , x = (10,15,20).
The following inequality prevents such symmetric solutions by forcing the bay depths to a
non-decreasing lexicographic order.
rb≤rb+1 ∀b∈B− {bmax}.(40)
The order of bays in the layout does not impact space utilization. Hence, once the optimal
layout is found, one can arrange bays in the layout based on the total retrieval and replenishment
operations of the assigned SKUs to minimize transportation costs.
4.2. Tightening the LP-relaxation lower bounds
Set Lcontains the set of allowable depths bounded by Lmin and Lmax. Solving the model
with many allowable depths increases the computational burden, so the bounds must be selected
carefully. From the space utilization perspective, assigning a large value to Lmax is preferable
because it provides the model with more depth choices. However, retrieving and replenishing deep
lanes are more laborious from the transportation and safety perspectives because forklifts have to
travel longer distances inside narrow lanes. So, forklift restrictions, safety requirements, response
times, and other technical restrictions limit Lmax. We arbitrarily set Lmax equal to 30 pallets in
our computational experiments to show the computational performance of the model under an
extreme case scenario. We also tested the model for a more realistic case where Lmax = 15.
17
Derhami et al. (2017) showed the space utilization curve, as a function of lane depth, drops
significantly when lanes are too shallow. This loss is significant enough to prevent the model
from selecting very small bay depths. So, we set Lmin equal to 5 in our experiment. Therefore,
L={5,6, , . . . , 30}.
Setting tight lower and upper bounds on the number of bays significantly reduces the search
space. Lmax can be used to find a tight lower bound on the number of bays, as follows
bmin = 2 Sl
2Lmax +A.(41)
Lmin does not provide a tight upper bound on the number of bays because it is generally
too small. We use the trade-off between honeycombing and accessibility waste to find bmax. As
expression (3) shows, honeycombing waste depends on the lane depth. The deeper the lane,
the more honeycombing waste is generated. It also depends on the frequency of retrievals and
replenishments. The MBD aims to optimize the trade-off between honeycombing and accessibility
waste. If honeycombing waste is low, then the model makes bays deeper to decrease the number
of aisles in the layout. But if the honeycombing waste is significant, then the model minimizes the
total waste by decreasing the bay depths and consequently increasing the number of aisles. Hence,
the maximum possible honeycombing waste forces the maximum number of bays to the layout.
For the sake of simplicity in modeling assume all SKUs have the same stack height denoted by Z.
The honeycombing waste generated to retrieve and replenish a lane with xpallets deep is obtained
from
WH
l=1
λi
+1
Pi−λiZx(Zx −1)
2.(42)
The time it takes to retrieve and replenish this lane is
tl=1
λi
+1
Pi−λiZx. (43)
The maximum honeycombing cost is generated in this lane when it never remains fully occupied
or empty (it is replenished immediately after it becomes empty and emptied immediately after it
becomes fully occupied). So, the maximum honeycombing waste this lane generates in TL, a long
18
period, will be WH
lTL/tL. It follows
WH
l−max =1
2TL(Zx −1) .(44)
Similarly, the maximum honeycombing waste in a bay with rbpallets deep is obtained from
WH
b−max =1
2TLSw(Zrb−1) ,(45)
and the maximum honeycombing waste in the entire warehouse would be
WH
max =1
2TLSw(ZX
b∈B
rb−bmax).(46)
Adding the accessibility waste, the total waste of storage volume in the warehouse is given by
W=1
2TLSw ZX
b∈B
rb−bmax +ZAbmax!.(47)
Assuming a common bay depth, ¯x, yields Pb∈Brb= ¯xbmax, and the total cost appears as
W=1
2TLSwbmax (Z¯x−1 + ZA),(48)
subject to the following constraint:
bmax ¯x+A
2=Sl.(49)
Solving (49) for ¯xand substituting the result in (48) converts the total cost to a function of
bmax. Taking the derivative of the new Wwith respect to bmax, setting it to zero, and then solving
for bmax gives the optimal value for bmax :
bmax ≈sSLZ
AZ −1.(50)
Expression (50) provides a continuous approximation for bmax. Our preliminary experiments
on relaxing the common bay depth and stackable height assumptions show rounding up bmax to its
19
next even integer plus two (the number of bays should be even) will provide a tight upper bound on
the number of bays while covering the largest bmax required after relaxing these two assumptions.
The following inequality restricts the number of bays in the model:
bmin ≤n≤bmax (51)
Because bays are arranged in nondecreasing order of their depths, the depth of the last bmin
bays will be always nonzero. Hence, the following equality tightens the lower bound of the LP-
relaxation:
eb= 1 ∀b∈ {bmax −bmin + 1, ..., bmax}.(52)
Constraint (29) is a loose constraint when a bay depth is zero. Replacing it with the following
constraint, which is binding when a bay depth is zero, tightens the lower bound of the LP-relaxation.
Sweb≤X
i∈I
zib ≤ESweb∀b∈B. (53)
4.3. Extra cut to reduce solution efforts
Derhami et al. (2017) proposed the following formula to find the lane depth:
x∗
i≈sA(Qi(Pi−λi)−2λi)
2ZiPi
.(54)
Taking advantage of non-decreasing bay depths imposed by inequality (40), we use (54) to assign
SKUs to bays based on the magnitudes of their lane depths. We calculate x∗
ifor all SKUs and
sort them in a non-decreasing order. Let fibe the index of the SKU located at the ith position
of the sorted list. Then, the following inequality ensures SKUs are assigned to bays based on the
ascending order of their lane depths:
X
b∈B
bybfi≤X
b∈B
bybfi+1 ∀i∈ {1, . . . , Ns−1},(55)
where Nsis the number of SKUs stored in the warehouse. Inequality (55) allows SKU ito be
assigned to bay bonly if all SKUs whose lane depths are smaller than or equal to the lane depth of
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SKU iare assigned to bays band before. That means it assigns SKUs with smaller lane depths to
the shallower, initial bays. Although this inequality reduces the feasible region by restricting SKU
assignments, it may remove some valid feasible integer solutions from the solution space. However,
this might not considerably deteriorate the objective function as the order of the SKU assignments
is still based on their optimal lane depths. We study the effects of this cut on the computational
time of the model and quality of the solutions in the next section.
5. Experimental analysis
The experimental framework is as follows. First, the characteristics of the test problems are
described in the next section. Then, computational difficulty of the model and effectiveness of the
proposed cuts and bounds are analyzed on the test problems. Next, the simulation model used
for the layout evaluation is described, and finally, the layouts obtained by the MBD model are
evaluated by the simulation model and compared with the ones obtained by the CLD model.
5.1. Test problems
We generated test problems that vary from small to industrial-sized to analyze the performance
of the proposed model on different warehouse sizes. First, a pool of 4000 different SKUs was
randomly generated using characteristics of real SKUs from an industrial partner. The parameters
of the SKUs were sampled from the uniform random distributions whose parameters are shown in
Table 1. cp
iin the table is the cost of producing one pallet of SKU i. The Qis were obtained using
the Economic Order Quantity (EOQ) model as follows (Nahmias, 2005):
Qi=
v
u
u
t
2cs
iλi
ch
i1−λi
Pi
,(56)
where ch
iis the monthly holding cost and was set to cp
i/4, and cs
iis the set-up cost to produce SKU
iand was set to 5cp
i.
We designed 19 test problems with 10 to 1000 SKUs. SKUs in each test problem were randomly
sampled from the pool of random SKUs. We used disproportionate stratified random sampling
based on the SKUs’ lane depths as described in the following. This is to ensure multiple SKUs
with a wide range of desirable lane depths exist in each test problem. The lane depths were
21
Table 1: Parameters of the uniform distributions used to generate the pool of random SKUs
Parameter Min Max
Pi(pallets/month) 7200 36000
λi(pallets/month) 30 3000
Zi(pallet) 2 4
Hi(feet) 3 5
cp
i(dollars) 50 500
obtained by (54). We divided the pool of SKUs into four groups such that the lane depths for
the SKUs in the groups were less than or equal to 13 pallets, between 14 and 18 pallets, between
19 and 24 pallets, and greater than or equal to 25 pallets, respectively 30% of the SKUs in each
test problem were randomly selected from the first group, 35% from the second group, 20% from
the third group, and 15% from the last group. We considered two cross-aisles (assuming one next
to each long side of the warehouse) for the test problems containing 50 SKUs or fewer and three
cross-aisles (the additional one at the middle of the warehouse) for the remaining problems. The
clear height of the warehouse was set to 16 feet for all test problems, and the aisle and cross-aisle
widths were set to three pallets. We also assumed pallet sizes are 42 by 42 inches.
Warehouse dimensions must be determined such that there will be sufficient space (storage
and aisle) to accommodate the maximum possible inventory. To find the maximum possible in-
ventory for each test problem, we developed an event-based simulation model only to keep track
of the SKU inventories over the simulation time. We used the event log of the main simulation
model for this purpose. We calculated the required floor space for the maximum inventory level
recorded by simulation. Using (57), we then approximated the number of aisles in the warehouse
for any given warehouse length. We assumed the warehouse layout has a rectangular shape and
its length is almost twice its width. Hence, we determined the warehouse length and width such
that the available floor space for storage (warehouse area subtracted by the space dedicated to the
aisles and cross-aisles) is 10% greater than the maximum required floor space (to account for the
underestimated waste by the traditional lane depth model).
5.2. Computational experiment
The proposed model was coded with Python 2.7.11 and solved using Gurobi 6.0.5. The model
was run on the Auburn University Hopper Cluster on Intel Xeon processors E5-2660 (2.6GHz) with
22
128 GB of RAM. We ran all experiments on 20 cores. We tested three scenarios to evaluate the
effectiveness of the proposed cuts and bounds:
•MBD, which only includes the MBD model without any of the developed cuts and bounds.
•MBDC, which includes the MBD model with cuts (40), (51), (52), and (53) and excludes
(29).
•MBDCE, which includes the MBD model with cuts (40), (51), (52), (53), and extra cut (55)
and excludes (29).
To have a fair comparison, we disabled the built-in symmetry detection function in Gurobi but
kept the other parameters of the solver to their default values. Also, a time limit of 10 hours was
forced on the optimization process. Table 2 compares the computational efforts for all scenarios.
As the results show, using the developed cuts and bounds reduces the solution time. The MBDC
model found optimal solutions for the first two small test problems and feasible solutions within
reasonable GAPs (up to 7.6% for the large problems) for the remaining problems that it did
not solve optimally. While the MBD model, which does not use any of the developed cuts and
bounds, resulted in solving only one problem optimally in a significantly longer computational
time (comparing to the MBDC) and no feasible solutions for the problems containing 250 or more
SKUs. Also, the MBDC model obtained smaller GAPs than the MBD model for all test problems
(20-200 SKUs).
Comparing the MBDC with MBDCE shows, as we expected, the extra constraint significantly
reduced computational efforts. The MBDCE model reached solutions (not global optimal) for
problems with 150 or fewer SKUs and obtained feasible solutions with relatively small GAPs (up
to 4.01%) for the remaining problems that it could not solve. However, it did not find feasible
solutions for test problems with 600, 800, and more SKUs. This is because cut (55) forces an
additional limitation on SKU assignments and adds extra complexity to the model as the problem
size increases.
From the computational point of view, the MBDCE performed faster than MBDC. This is
because it explored fewer nodes to find the optimal solutions. The objective values of the solutions
obtained by the MBDCE are close to that of the MBDC. Both models obtained the same solutions
for the 10 and 30 SKUs test problems. MBDCE did not perform as well as the MBDC model on the
23
Table 2: Computational efforts with/without the developed cuts and bounds.
Problems MBD MBDC MBDCE
GAP Obj. Ex. nodes Time GAP Obj. Ex. nodes Time GAP Obj. Ex. nodes Time
10 SKUs 0.00 11642.7 31811598 2421 0.00 11642.7 21934 2 0.00 11642.7 161 1
20 SKUs 5.30∗17268.6 179684803 36000 0.00 17268.6 239845458 22136 0.00 17276.6 9270 2
30 SKUs 9.25∗31259.9 96154756 36000 3.77∗31219.9 153487249 36000 0.00 31219.9 18915 17
40 SKUs 3.56∗35449.9 84169226 36000 2.37∗35329.9 72580693 36000 0.00 35353.9 30580 23
50 SKUs 3.86∗45108.3 71044945 36000 3.34∗45044.3 119435452 36000 0.00 45028.3 38253 31
100 SKUs 5.28∗92386.1 29715077 36000 5.01∗92282.1 62559424 36000 0.00 92194.1 331882 818
150 SKUs 3.77∗140945.8 15799077 36000 2.83∗140401.8 47609432 36000 0.00 140441.8 514334 2516
200 SKUs 5.74∗186478.3 736440 36000 5.48∗186318.3 1703775 36000 0.24∗185750.3 22742389 36000
250 SKUs NIF — — 36000 5.37∗232791.4 25157771 36000 2.25∗232535.4 7277756 36000
300 SKUs NIF — — 36000 4.32∗279102.9 19094755 36000 2.63∗279038.9 9910535 36000
350 SKUs NIF — — 36000 4.64∗325601.5 7765123 36000 1.31∗324945.5 5070159 36000
400 SKUs NIF — — 36000 7.50∗374733.0 5596467 36000 3.72∗369837.0 1477441 36000
450 SKUs NIF — — 36000 5.05∗421758.6 8519992 36000 3.79∗421406.6 1638156 36000
500 SKUs NIF — — 36000 5.32∗456491.3 8252609 36000 4.01∗455459.3 562694 36000
600 SKUs NIF — — 36000 5.02∗531191.4 2917775 36000 NIF — — 36000
700 SKUs NIF — — 36000 6.63∗650150.5 7047738 36000 3.76∗643254.5 — 36000
800 SKUs NIF — — 36000 7.05∗732353.8 2289581 36000 NIF — — 36000
900 SKUs NIF — — 36000 7.56∗822260.9 2494914 36000 NIF — — 36000
1000 SKUs NIF — — 36000 7.04∗920743.6 1507563 36000 NIF — — 36000
∗Optimization prematurely terminated after 10 hours of computation.
NIF: no integer solution found after 10 hours of computation.
20, 40, 150 SKUs test problems. Its solutions are 0.05%, 0.07%, and 0.03% larger than the MBDC,
respectively. However, it improved the best feasible solutions obtained by the MBDC between
0.02% to 1.31% in the remaining test problems. The small differences between the results of these
two models show although the MBDCE model is a heuristic algorithm and does not guarantee an
optimal solution, it provides good quality solutions in a short time.
From the computational perspective, the solution GAP increases, for all three models, as the
problem size increases. Among the three models, however, the MBDCE model obtained smaller
GAPs on problems that it could solve (small to medium-sized problems). The MBDC model was
capable of finding feasible solutions with relatively small GAPs for the industrial-sized problems.
The results presented in Table 2 were obtained by setting Lmin and Lmax to 5 and 30 pallets,
respectively. This is an extreme case used to study the computational performance of the proposed
models under a worst-case scenario. A large gap between Lmin and Lmax increases solution efforts
as the model has to find the optimal bay depths among a large set of possible options. We tested
the model for more realistic bay depths with Lmin and Lmax set to 5 and 15 pallets, respectively.
As we expected, the new model is solved faster. It could find optimal solutions for test problems
up to 40 SKUs (it was 20 SKUs for the extreme case scenario) and for the remaining test problems
it ended up with smaller gaps under the same termination criteria. In all test problems with more
24
than 40 SKUs, the model with realistic lane depths obtained smaller gaps than the extreme case
scenario. The maximum gap among the test problems reduced from 7.5% to 2.9% and the average
gap among all test problems decreased from 4.6% to 1.68%.
5.3. Analyzing performance of the layouts
We evaluated the layouts obtained by the MBD model using the simulation model of Derhami
et al. (2016). They developed an event-oriented simulation model to evaluate a given warehouse lay-
out with respect to multiple performance metrics pertinent to space utilization and transportation
cost. Their model simulates lane replenishment and retrieval operations under stochastic variations
on the production times, demand, retrieval quantities, and production line set-up times. We tuned
variations of these parameters in our experiments as follows. The production times were sampled
from symmetric triangular distributions with parameters (0.8/Pi,1/Pi,1.2/Pi) hours. Similarly, the
outbound load times and production line set-up times were sampled from analogous distributions
with parameters (0.5/λi,1/λi,1.5/λi) hours and (10,20,30) minutes, respectively. The retrieval
quantities were sampled from a discrete uniform distribution with parameters [1,5] pallets.
We disabled the transportation module in the simulation because our analytical model does not
take transportation into account. We set the warm-up period to one month, start-up inventories to
zero, number of replications to 8, and simulation time to 8 months as described in Derhami et al.
(2016). We ran simulations on the Auburn University Hopper Cluster on Intel Xeon processors
E5-2660 (2.6GHz) with 128 GB of RAM memory. We ran replications of the simulation on parallel
processors and therefore used 8 cores for each experiment. To reduce the variance, we used common
random numbers (CRN) across the replications for all scenarios. The layouts obtained from the
following models were simulated:
1. MBDC: The layouts obtained by the MBDC model, described in section 5.2, are compared,
as baselines, with the ones obtained by the following models.
2. MBDC-RAN: The optimal SKU assignments produced by the MBDC model, described in
section 5.2, are used to prioritize bay assignment when more than one bay has empty lanes.
That is, when a new pallet of a SKU requires a new empty lane, the storage lane is chosen
among bays with empty lanes through the following process. First, the assigned bay to the
SKU is checked for any empty lanes. If no lane is available in that bay, bays whose depths
are equal to the assigned depth to the SKU are checked for an empty lane. If such an empty
25
lane is not available then the bay with the closest depth to the assigned depth to the SKU
that has an empty lane is selected.
It may be costly (time or labor) for non-automated material handling systems to follow SKU
assignments. Such warehouses may prefer employing a random SKU assignment rather than
the optimal assignment. In the random assignment, an empty lane from a randomly selected
bay is assigned to the incoming SKU. However, this decision imposes storage waste to the
system. We are interested in first, examining the significance of the optimal SKU assignments
on the volume utilization and second, estimating the loss in storage volume incurred by
ignoring the optimal SKU assignments. For this reason, we tested the layouts obtained by
the MBDC model under random SKU assignment policy and called it MBDC-RAN.
3. MBDCE: We showed the MBDCE model, described in section 5.2, is computationally faster
than the MBDC model for small to medium-sized problems. However, this model removes
some valid feasible solutions. We simulate the layouts obtained by this model to study the
impact of the extra cut (55) on the quality of the solution.
4. CLD: To have a baseline for performance comparison and also evaluate the layouts obtained
by the common lane depth model proposed in Derhami et al. (2017), we developed a simple
algorithm to design the warehouse layout using the common lane depth. The CLD algorithm
works as follows. First, the common lane depth is calculated as follows:
x∗
c="sA
2NsX
i∈I1
ZiPi(Qi(Pi−λi)−2λi)#.(57)
Then, the layout is divided into evenly deep bays whose depths are x∗
cpallets. Because this
approach does not take the warehouse length into account, it is possible that the number of
bays becomes an odd integer with the last bay depth smaller than x∗
c. This means one aisle
is used to access only one bay instead of two. We remove this inefficiency by splitting the last
bay to two equally deep bays if its depth is greater than 10 pallets. Otherwise, the last bay
is equally split between the other bays and removed from the layout. The layouts produced
by invoking this adjustment are marked in Table 5 for the respective test problems.
Table 3 presents average waste of the storage volume (yd3), volume utilization (%), and floor
utilization (%) for the four scenarios. Interested readers are referred to Derhami et al. (2016)
26
Table 3: Average waste of storage volume, volume utilization, and floor utilization obtained by simulation.
Problems MBDC MBDC-RAN MBDCE CLD
Waste Vol. Util. Fl. util. Waste Vol. Util. Fl. util. Waste Vol. Util. Fl. util. Waste Vol. Util. Fl. util.
10 SKUs 10674±16 42.85±0.05 65.05±0.07 10837±17 42.48±0.06 64.49±0.09 10674±16 42.85±0.05 65.05±0.07 10733±15 42.72±0.05 64.85±0.07
20 SKUs 14574±18 50.69±0.04 70.61±0.04 14836±12 50.25±0.04 69.99±0.05 14564±15 50.71±0.04 70.63±0.03 15282±13 49.51±0.04 68.96±0.04
30 SKUs 23526±18 46.03±0.02 72.67±0.02 23912±26 45.62±0.02 72.04±0.03 23532±19 46.02±0.01 72.66±0.02 23660±17 45.89±0.02 72.45±0.03
40 SKUs 26404±18 52.19±0.01 75.36±0.01 26946±18 51.68±0.01 74.63±0.02 26411±16 52.18±0.01 75.35±0.02 27407±16 51.26±0.01 74.01±0.02
50 SKUs 32573±15 52.04±0.02 76.65±0.02 33278±36 51.51±0.03 75.86±0.04 32504±19 52.10±0.02 76.72±0.01 33566±20 51.29±0.02 75.54±0.01
100 SKUs 69034±40 50.58±0.02 76.91±0.02 69648±40 50.36±0.02 76.58±0.02 68730±35 50.69±0.02 77.08±0.02 70617±40 50.02±0.02 76.05±0.02
150 SKUs 95514±24 51.96±0.01 77.47±0.01 96655±30 51.66±0.01 77.03±0.01 95118±26 52.06±0.01 77.62±0.01 98033±32 51.31±0.01 76.50±0.01
200 SKUs 125965±40 52.66±0.01 79.23±0.01 127823±54 52.29±0.01 78.68±0.01 125673±42 52.72±0.01 79.31±0.01 129212±37 52.02±0.01 78.27±0.01
250 SKUs 152994±24 52.39±0.00 79.94±0.00 156673±36 51.80±0.01 79.04±0.01 152842±23 52.42±0.00 79.98±0.01 155664±26 51.96±0.01 79.28±0.01
300 SKUs 181800±29 52.94±0.01 80.58±0.01 186578±67 52.30±0.01 79.59±0.01 181730±26 52.95±0.01 80.59±0.00 185261±33 52.47±0.01 79.86±0.00
350 SKUs 207845±39 54.09±0.00 81.17±0.00 211176±49 53.69±0.00 80.58±0.01 207531±44 54.12±0.00 81.23±0.00 212269±47 53.56±0.00 80.39±0.00
400 SKUs 236940±44 53.97±0.01 80.99±0.00 239286±27 53.73±0.01 80.62±0.01 234514±34 54.23±0.01 81.38±0.00 241280±42 53.52±0.01 80.32±0.00
450 SKUs 267527±82 54.30±0.01 81.95±0.00 273618±123 53.75±0.01 81.10±0.01 267348±74 54.32±0.00 81.97±0.01 272822±83 53.82±0.01 81.21±0.00
500 SKUs 287833±37 55.42±0.00 82.16±0.00 295697±56 54.75±0.01 81.17±0.01 287333±27 55.46±0.00 82.23±0.00 294926±33 54.82±0.00 81.27±0.00
600 SKUs 332589±47 56.81±0.00 82.51±0.00 339560±81 56.30±0.01 81.77±0.01 — — — 339243±44 56.32±0.00 81.80±0.00
700 SKUs 382820±72 54.76±0.01 81.59±0.01 387210±71 54.48±0.01 81.17±0.01 379576±78 54.97±0.01 81.90±0.00 387753±90 54.45±0.01 81.11±0.01
800 SKUs 428758±43 55.77±0.00 81.76±0.00 436664±111 55.32±0.01 81.10±0.01 — — — 436859±48 55.31±0.00 81.08±0.00
900 SKUs 479724±70 55.64±0.00 81.84±0.00 485179±100 55.37±0.00 81.43±0.00 — — — 486855±64 55.28±0.00 81.30±0.00
1000 SKUs 531729±86 55.60±0.00 82.13±0.00 541648±79 55.14±0.01 81.46±0.01 — — — 536423±96 55.38±0.00 81.81±0.00
for more details on the definition and calculation of these parameters. We used a paired t-test
to evaluate significant differences among the alternatives. Table 4 shows the test statistics and
p-values for all comparisons. The paired t-test relies on the normality assumption among the pairs
(replications). We used the Shapiro-Wilk test (Shapiro & Wilk, 1965) to examine normality of the
differences between the pairs. The test statistics and p-values for the Shapiro-Wilk test are also
shown in Table 4. The results of the normality tests show the null hypothesis (i.e., samples are
taken from a normal population) cannot be rejected for any of the comparisons at the significant
level of 5%. The following alternatives are analyzed pairwise:
5.3.1. MBDC vs. CLD
The layouts produced by the CLD model imposed greater waste of storage volume than the
MBDC’s in all test problems. The paired t-test detects significant differences between the two
alternatives and rejects the null-hypothesis for all comparisons at the significant level of 5%. The
layouts produced by the MBDC model generated, on average, 0.5% to 4.6% less waste of storage
volume than the ones produced by the CLD model. This shows, as explained in section 1, the CLD
model cannot find the optimal bay depth for the layout design problem. However, the improvement
in volume utilization is between 0.3% to 2.4%. This is because, as mentioned in Derhami et al.
(2017), the volume utilization curve, as a function of bay depths, becomes flat around the optimal
solution and therefore changes in the bay depth vector in the vicinity of the optimal solution do
not result in significant changes in volume utilization.
27
Table 4: Statistical results of the the pairwise comparisons, α= 0.05.
Problems
MBDC vs. CLD MBDC vs. MBDC-RAN MBDC vs. MBDCE
Paired t-test Shapiro-Wilk Paired t-test Shapiro-Wilk Paired t-test Shapiro-Wilk
Stat. p-value Stat. p-value Stat. p-value Stat. p-value Stat. p-value Stat. p-value
10 SKUs 47.95 4.49 ×10−10 0.91 0.3766 30.4 1.07 ×10−08 0.93 0.5532 — — — —
20 SKUs 203.71 1.81 ×10−14 0.92 0.4607 43.0 9.57 ×10−10 0.93 0.5180 2.44 0.0447 0.91 0.3608
30 SKUs 65.37 5.15 ×10−11 0.95 0.6670 74.4 2.08 ×10−11 0.95 0.6772 4.34 0.0034 0.96 0.8247
40 SKUs 320.37 7.62 ×10−16 0.87 0.1547 106.0 1.75 ×10−12 0.94 0.6277 2.03 0.0814 0.91 0.3362
50 SKUs 323.77 7.08 ×10−16 0.98 0.9811 52.9 2.25 ×10−10 0.99 0.9986 32.77 6.37 ×10−09 0.93 0.4792
100 SKUs 240.16 5.72 ×10−15 0.90 0.3132 114.8 1.00 ×10−12 0.86 0.1151 43.99 8.19 ×10−10 0.96 0.8066
150 SKUs 427.89 1.00 ×10−16 0.94 0.6390 105.6 1.80 ×10−12 0.91 0.3742 83.89 9.00 ×10−12 0.94 0.6506
200 SKUs 512.03 2.86 ×10−17 0.89 0.2431 110.3 1.32 ×10−12 0.91 0.3388 59.51 9.92 ×10−11 0.84 0.0727
250 SKUs 391.05 1.88 ×10−16 0.92 0.4178 172.4 5.84 ×10−14 0.86 0.1280 40.91 1.35 ×10−09 0.95 0.7459
300 SKUs 397.88 1.67 ×10−16 0.87 0.1356 215.0 1.24 ×10−14 0.86 0.1259 17.26 5.38 ×10−07 0.84 0.0766
350 SKUs 446.36 7.48 ×10−17 0.94 0.5812 217.2 1.15 ×10−14 0.97 0.9022 42.56 1.03 ×10−09 0.83 0.0533
400 SKUs 411.10 1.33 ×10−16 0.96 0.8317 164.9 7.94 ×10−14 0.97 0.9062 223.14 9.58 ×10−15 0.95 0.7363
450 SKUs 531.50 2.20 ×10−17 0.91 0.3855 215.3 1.23 ×10−14 0.93 0.4770 20.44 1.68 ×10−07 0.96 0.7963
500 SKUs 980.40 3.03 ×10−19 0.89 0.2579 301.6 1.16 ×10−15 0.83 0.0561 65.56 5.04 ×10−11 0.93 0.5123
600 SKUs 778.94 1.51 ×10−18 0.92 0.4067 310.9 9.41 ×10−16 0.91 0.3790 — — — —
700 SKUs 252.64 4.01 ×10−15 0.94 0.6079 366.6 2.96 ×10−16 0.98 0.9819 266.51 2.76 ×10−15 0.91 0.3585
800 SKUs 696.36 3.32 ×10−18 0.90 0.2833 187.0 3.29 ×10−14 0.99 0.9985 — — — —
900 SKUs 489.30 3.93 ×10−17 0.92 0.4359 249.0 4.45 ×10−15 0.88 0.1716 — — — —
1000 SKUs 304.96 1.07 ×10−15 0.86 0.1244 364.8 3.07 ×10−16 0.91 0.3684 — — — —
5.3.2. MBDC vs. MBDC-RAN
The layouts obtained by the MBDC model wasted less storage volume than the MBDC-RAN
in all problems. The paired t-test rejects equivalence of results between the two alternatives for
all problems. Discarding the optimal SKU assignments increased the average waste of storage
volume between 0.9% to 2.7%. This shows the SKU assignments found by the MBDC model are
statistically significant in better utilizing the storage volume.
5.3.3. MBDC vs. MBDCE
Both models obtained the same solution for the 10 SKUs problem, and hence their simulation
results are the same for this test problem. The MBDC obtained slightly better solutions for 30
and 40 SKUs problems (0.03% improvement). For the remaining problems, the MBDCE obtained
slightly smaller waste of storage volume. It achieved 0.04% to 0.44% improvement over the MBDC
results. The results of the paired t-test show it has explored significant differences between the
two alternatives for all problems. This shows the extra cut (52) not only did not deteriorate the
solution quality but it also slightly improved feasible solutions obtained by the MBDC model.
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5.3.4. Analysis of optimal layouts
The layouts obtained by the CLD, MBDC, and MBDCE models are displayed in Table 5.
They are presented in pairs of ab, which represent bbays with apallets depth. Parameter βis the
number of aisles to warehouse length ratio, which is calculated by dividing the number of aisles in
the layout by the warehouse length.
As Table 5 shows, CLD finds the same bay depth for almost all problems, while the other two
approaches use a diverse set of depths. CLD computed the common lane depth of 20 pallets for
the first test problem and 18 pallets for the rest.
The CLD model underestimates the optimal bay depth in almost all test problems. On average
bay depths in the layouts produced by the CLD are 28% and 26% shallower than that of the MBDC
and MBDCE models, respectively. This is because CLD underestimates the accessibility waste and
consequently imposes unnecessary aisles in the layout.
The total average bay depth across the problems are 24.3 and 23.2 pallets for the MBDC and
MBDCE models, respectively. Their layouts contain bays with 11 pallets and deeper (except two
problems). This shows the honeycombing waste is not as costly as the accessibility waste; therefore,
an optimal model tends to choose deeper bays to prevent having many aisles in the layout. This
is because honeycombing waste is incurred only when lanes are being filled or depleted (i.e., no
honeycombing waste when lanes are entirely occupied or emptied), and it also depends on the
frequency of retrievals and replenishments, whereas the accessibility waste is permanent. Therefore,
a layout with deep lanes better utilizes the storage space.
Using the βratio, we suggest a rule of thumb to determine the number of aisles for a warehouse
similar to our test problems. The βratio equals 0.020 ±0.001 for both MBDC and MBDCE models
and remained almost steady for all test problems. Hence, a near optimal solution can be found by
calculating the number of aisles by n=βSland then dividing the layout into 2nevenly deep
bays whose depths are (Sl−An)/2npallets.
5.4. A case study in the beverage industry
We used our proposed model (MBDC) to design a warehouse layout for a leading supplier of
bottled water in North America whose name remains anonymous in this paper. The company
produces different types of bottled beverages and has more than 40 plants and warehouse facilities
across the U.S.A, Canada, and Mexico. The storage type in all these facilities is block stacking. For
29
Table 5: Layouts obtained by the proposed models (in units of pallets).
Problems Warehouse dimensions CLD MBDC MBDCE
Length Width Layout # aisles Avg. depth βLayout # aisles Avg. depth βLayout # aisles Avg. depth β
10 SKUs 82 37 (161,203) 2 19.0 0.024 (122,221,301) 2 19.0 0.024 (122,221,301) 2 19.0 0.024
20 SKUs 97 47 (82,184) 3 14.7 0.031 (141,201,271,301) 2 22.8 0.021 (131,221,261,301) 2 22.8 0.021
30 SKUs 122 57 (184,192)∗3 18.3 0.025 (121,131,151,211,262) 3 18.8 0.025 (131,151,161,211,221,261) 3 18.8 0.025
40 SKUs 132 62 (62,186)∗4 15.0 0.030 (111,171,191,201,261,301) 3 20.5 0.023 (131,171,191,221,231,291) 3 20.5 0.023
50 SKUs 142 72 (41,187) 4 16.3 0.028 (111,161,231,271,282) 3 22.2 0.021 (141,181,211,231,281,291) 3 22.2 0.021
100 SKUs 212 97 (72,1810)∗6 16.2 0.028 (191,242,252,271,282) 4 25.0 0.019 (151,201,231,251,271,303) 4 25.0 0.019
150 SKUs 257 127 (21,1813) 7 16.9 0.027 (202,213,221,281,291,302) 5 24.2 0.019 (111,171,211,221,262,291,303) 5 24.2 0.019
200 SKUs 297 142 (31,1815) 8 17.1 0.027 (141,201,212,221,232,242,281,291,301) 6 23.3 0.020 (131,161,181,201,221,231,241,251,281,303) 6 23.3 0.020
250 SKUs 317 162 (1814,192)∗8 18.1 0.025 (121,162,242,271,306) 6 24.9 0.019 (131,161,191,231,251,261,271,305) 6 24.9 0.019
300 SKUs 357 172 (1815,193)∗9 18.2 0.025 (111,121,161,171,231,251,261,281,292,304) 7 24.0 0.020 (121,151,171,191,211,231,241,261,291,305) 7 24.0 0.020
350 SKUs 377 187 (51,1819) 10 17.4 0.027 (141,222,232,264,292,303) 7 25.4 0.019 (141,171,201,211,231,251,271,291,306) 7 25.4 0.019
400 SKUs 407 202 (72,1820)∗11 17.0 0.027 (244,251,261,291,307) 7 27.6 0.017 (131,151,171,181,212,231,241,251,261,306) 8 23.9 0.020
450 SKUs 427 217 (161,1821) 11 17.9 0.026 (141,151,161,171,191,261,281,292,307) 8 25.2 0.019 (131,171,201,212,231,251,261,271,307) 8 25.2 0.019
500 SKUs 447 227 (71,81,1822)∗12 17.1 0.027 (91,112,152,161,231,272,294,305) 9 23.3 0.020 (111,141,182,203,211,253,297) 9 23.3 0.020
600 SKUs 497 242 (81,1825) 13 17.6 0.026 (151,161,171,181,253,291,3010) 9 26.1 0.018 — — — —
700 SKUs 527 262 (81,91,1826)∗14 17.3 0.027 (141,262,288,307) 9 27.8 0.017 (161,181,203,221,233,243,308) 10 24.9 0.019
800 SKUs 577 287 (101,1829) 15 17.7 0.026 (121,211,221,231,241,261,291,3013) 10 27.4 0.017 — — — —
900 SKUs 617 297 (111,1831) 16 17.8 0.026 (141,254,264,275,283,293,302) 11 26.5 0.018 — — — —
1000 SKUs 627 317 (1831,211) 16 18.1 0.026 (61,131,141,211,3018) 11 27.0 0.018 — — — —
Average — — — — 17.2 0.027 — — 24.3 0.020 — — 23.2 0.020
∗Shows bays have been adjusted.
30
Table 6: Results of a case study in the beverage industry.
Current layout Optimal layout
Bay depths (pallets) (32,72,82,93,101,112,124,133,151) (111,121,152,171,191,206)
Number of aisles 11 7
Number of cross-aisles 4 4
Average space utilization (%) 48.18±0.02 50.10±0.02
Average floor utilization (%) 73.26 76.18
Average waste of space (yd3) 75979±36 70381±39
our study, we select one of the high-density facilities near an urban area in the U.S. The selected
facility produces and stores 105 different SKUs. The size of the facility is more than 267,000 ft2.
The characteristics of the current layout are presented in Table 6. The material handling in the
facility is handled by Automated Guided Vehicles (AGVs). Considering the limitations of the firm’s
material handling system in serving deep lanes, we restricted Lmin and Lmax to 5 and 20 pallets in
the MBDC model, respectively. The results of the optimal layout along with performance metrics
measured by simulation are compared with the current layout in Table 6. The optimal layout has
fewer bays and aisles than the current one. It has improved the current layout in all performance
metrics. The improvement in space utilization might seem marginal, but the impact will last for a
long period as the layout design process is a medium to long term decision. Hence, the firm benefits
from an optimal design for a long period of time. Considering the direct and indirect (e.g. utilities,
insurance, taxes, etc.) costs of the storage space in this facility, the improvement in utilization of
the storage space is worth more than a million dollars over a period of eight years.
6. Conclusions
We have developed a model to design a space-efficient layout for block stacking warehouses.
We showed the common lane depth (CLD) model is not appropriate to find bay depths for a layout
and developed a new waste function to estimate the total waste of storage volume in the layout.
We optimized this function with a mixed integer programming model to find the optimal bay
depths and the number of aisles in the layout. We developed various cuts to reduce the problem
symmetry and effectively bounded the decision variables to tighten the lower bound of the LP-
relaxation (MBDC and MBDCE models). While both models produce good quality solutions,
we found the MBDCE model more likely performs better and faster on small to medium-sized
31
problems (less than 700 SKUs), and the MBDC model performs better on large-sized problems.
The simulation experiments showed the layouts produced by our models always generate less
waste of storage volume than the layouts obtained by the CLD model. The improvement varied
between 0.5% to 4.7% in our experimental study. We found that the CLD model underestimates
the accessibility waste; as a result, its bay depths were on average 27% shallower than the optimal
solution. In other words, the layouts obtained by the CLD model devoted up to 50% more space to
the aisles. The common bay depth produced by this model is insensitive to the number of SKUs.
In our experiments, the resulting common bay depth was 18 pallets for all test problems with more
than 20 SKUs. However, the volume utilization of the layouts produced by this model are close
to the optimal solutions. This is because the volume utilization curve, as a function of bay depth,
is almost flat around the optimal solution, and changes in bay depths will not result in significant
changes in volume utilization. Hence, the CLD model produces near-optimal solutions with respect
to volume utilization.
We found the accessibility waste outweighs honeycombing waste in their trade-off and an op-
timal model tends to choose deep bays. So, a layout with deep bays and fewer aisles utilizes the
storage volume better than a layout with shallow bays. The average bay depth across the optimal
solutions was 23.9 pallets in our experiments. We introduced a new ratio, the number of aisles
to warehouse length, to estimate the number of aisles for a given warehouse length. This ratio
remained steady around 0.020 for all problems in our experiments. This can be used as a rule of
thumb to find a near optimal solution for warehouses similar to the ones tested in our experiments.
The results of our experiments also indicate the optimal SKU assignments proposed by our
model are statistically significant in better utilizing the storage volume, and implementing them
decreased the wasted volume approximately 2% on average.
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