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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

15

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

20

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.

28

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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