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Please cite this article as: Shahab Derhami, Jeﬀrey S. Smith, and Kevin R. Gue, International Journal of

Production Economics, doi:10.1016/j.ijpe.2019.107525

A simulation-based optimization approach to design optimal

layouts for block stacking warehouses

Shahab Derhami1, Jeﬀrey S. Smith2, and Kevin R. Gue3

1School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta,

GA 30332, USA

2Department of Industrial and Systems Engineering, Auburn University, Auburn, AL

36849, USA

3Department of Industrial Engineering, University of Louisville, Louisville, KY 40292, USA

Abstract

Storing pallets of products on top of one another on the ﬂoor of a warehouse is called block stacking.

The arrangement of lanes, aisles, and cross-aisles in this storage system aﬀects both utilization of the

storage space and material handling costs; however, the existing literature focuses exclusively on lane

depths and their impact on space utilization. This paper ﬁlls this gap and studies the optimal layout

design for block stacking, which includes determining the numbers of aisles and cross-aisles, bay depths,

and cross-aisle types. We show that lane depths aﬀect material handling cost in addition to space

utilization and develop a simulation-based optimization algorithm to ﬁnd optimal layouts with respect

to both of these objectives. We also propose a closed-form solution to the optimal number of aisles in

a layout. The results of a case study in the beverage industry show that the resulting layout can save

up to ten percent of the operational costs of a warehouse. We present the computational eﬃciency of

the algorithm and some insights into the problem through an exhaustive experimental analysis based on

various test problems that cover small- to industrial-sized warehouses.

Keyword: block stacking, warehouse design, lane depth, facility layout, space utilization, cross-aisle

1 Introduction

Block stacking refers to stacking pallets of stock keeping units (SKUs) on top of one another in lanes on

a warehouse ﬂoor. This storage system is usually operated without using any storage racks, which makes

1

it inexpensive to implement but challenging in terms of space planning because lower pallets cannot be

reached until the ones stacked on top and in front of them are retrieved. Block stacking is widely used for

warehousing and cross-docking, especially when pallets are heavy and large. Examples of such environments

are bottled beverage and home appliance companies.

Block stacking is mainly operated under the shared (also known as random) storage policy, in which,

unlike the dedicated storage policy, lanes are not dedicated to SKUs and an empty lane is available to all

SKUs. However, to avoid blockage and relocation of pallets, a lane is temporarily dedicated to the SKU

that occupies its ﬁrst pallet position until it is fully depleted. This restricts utilization of the storage space

because some unoccupied pallet positions in a partially occupied lane will not be available to other SKUs.

This eﬀect is called honeycombing, and the waste associated with it accumulates in a lane until it becomes

entirely occupied or emptied [3]. Aisles also contribute to the amount of wasted space because they are used

for accessibility rather than storage. There is a trade-oﬀ between honeycombing and accessibility waste with

respect to space utilization: shallow lanes generate less honeycombing waste but impose more aisles on the

layout, while the opposite is true for deep lanes [3].

The other inﬂuential factor that must be considered when designing a warehouse layout is transportation

cost. The number of cross-aisles and their conﬁgurations (locations, directions, etc.) aﬀect the material han-

dling cost. This eﬀect is escalated in warehouses operated under continuous multi-command pick-up/drop-oﬀ

operations, where vehicles (pickers) perform replenishment or retrieval operations continuously in a working

shift before going to their home/parking—a situation that frequently occurs in block stacking warehouses.

In such warehouses, a replenishment or a retrieval operation starts by moving a vehicle from its last drop-oﬀ

location to the pick location. Hence, vehicles travel between any points of the warehouse rather than only

traveling from/to well-deﬁned P/D points and storage locations. In such environments, it is very diﬃcult

to develop analytical models to formulate and assess the total transportation cost because the distance that

vehicles travel depends on the sequence of the retrieval and replenishment operations and their assignment

to the vehicles. Simulation is an appropriate tool to accurately estimate traveled distance for such cases

while considering the stochastic nature of the problem.

Material handling is usually performed in block stacking warehouses by unit-load forklifts that perform

pallet retrieval and replenishment operations continuously based on a load assignment policy. Cross-aisles

are used to facilitate travel. However, like aisles, they are not used directly for pallet storage and are

subsequently considered a waste of storage space.

We deﬁne bays in a layout as a set of adjoining lanes of the same depth separated by aisles (see Figure

1). To better utilize the storage space, we assume that bays are back to back and that every pair of bays

shares an aisle. Bay depths also aﬀect the travel distances. Consider the two layouts presented in Figure 1.

2

Figure 1: Travel distance vs. space utilization with respect to the number of aisles.

One layout has two bays, each being seven pallet positions deep, and the other has the same dimensions but

consists of six bays, each two pallet positions deep. The total area in both layouts is the same. Assume that

aisles and cross-aisles are one pallet position wide and that pallets are stacked up to one pallet. The total

distance a forklift has to travel from the P/D point to replenish a pallet located in the very ﬁrst position

of a lane in the top bay of the left layout (as highlighted in Figure 1) is 44 pallet positions, which includes

traveling 32 pallet positions in the cross-aisle and 12 pallet positions in the lane (round trip). This distance

for a pallet located in the same position at the bottom bay is 14 pallet positions, which includes traveling

12 pallet positions inside the lane and 2 pallet dispositions to cross the aisle. Similarly, the total vertical

distance to replenish all pallet positions in both layouts is

Layout with deep bays: 9((14 + 12+,...,+2) + (44 + 42+,...,+32)) = 2898 pallet positions.

Layout with shallow bays: 9((4 + 2) + 2(14 + 12) + 2(24 + 22) + (34 + 32)) = 1944 pallet positions.

A longer distance must be traveled to replenish the layout with deeper bays. However, this layout has

more storage positions (i.e., better space utilization). The total horizontal travel distance is the same for

both cases and was disregarded in the comparison. Derhami et al. [11] used simulation to show there is a

trade-oﬀ between space utilization and transportation costs with respect to bay depths. Moreover, studying

cross-aisles is meaningless without considering the material handling cost. Hence, both of these objectives

must be considered when designing a layout.

In this paper, we develop a simulation-based optimization algorithm to ﬁnd the optimal number of aisles,

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cross-aisles, and bay depths in the layout of a block stacking warehouse. Our algorithm optimizes two

objectives: (1) maximizing space utilization; and (2) minimizing material handling costs. We develop a

closed-form solution model to determine the optimal number of aisles in the layout and use it along with an

experimental analysis to restrict the search space for the algorithm. Implementing the algorithm for a case

in the bottled beverage industry shows that signiﬁcant improvement is possible.

The remainder of this paper is organized as follows. In section 2, we review the related research and

depict the gap in the literature along with the contributions of this paper. Then, we present the proposed

algorithm in section 3. We describe the three main components of the algorithm, develop a model to ﬁnd

the optimal number of aisles in the layout, and show how the model is used to tighten the search space for

our algorithm. Finally, we present a case study and extensive numerical analysis in section 4 and evaluate

the results.

2 Related research

Most of the papers studying warehouse layout focus on conventional warehouses with storage racks [37, 17, 2,

18]. They mostly aim to design a layout to minimize the transportation cost (or distance) for order picking

[19, 20, 30, 38, 7, 4, 14]. Interested readers are referred to [9] for further details on this research. The

other objectives considered in warehouse layout design problems are operational costs [39, 28, 43], product

allocation [27, 32, 25], storage space utilization[10, 6], warehouse throughput [29, 31, 23], and operating

policies [8, 41, 36, 21, 1].

There is a stream of research focused on the eﬀect of cross-aisles on transportation costs. Roodbergen and

de Koster [33] developed multiple heuristics to ﬁnd the shortest path for order picking when multiple cross-

aisles exist. Vaughan and Petersen [40] proposed a heuristic to ﬁnd the shortest path for order picking and

studied the eﬀect of adding cross-aisles on transportation costs. They showed that the number of cross-aisles

needed to maximum transportation eﬃciency depends primarily on the length of the storage aisles relative

to the length of the cross-aisles. Roodbergen and Vis [34] proposed an analytical model to approximate the

average length of an order picking route for two routing policies in a layout with one block (two cross-aisles).

Their approximation can be used as an objective function in a nonlinear model to obtain the optimal number

of aisles. Roodbergen et al. [35] developed an analogous approximation for a layout with multiple blocks.

Only a few papers have studied the design of block stacking layouts. Kind [22] considered the trade-oﬀ

between honeycombing and accessibility waste to ﬁnd the optimal lane depth. He proposed a model to

estimate the lane depth that optimizes this trade-oﬀ. However, he did not provide any insights into how

to derive the model. Matson [26] developed another model to approximate the optimal lane depth under

4

Figure 2: Space utilization vs. travel distance [11].

instantaneous replenishment (i.e., inﬁnite storage rates). Her model was appropriate for warehouses receiving

products from suppliers.

Goetschalckx and Ratliﬀ [15] showed that if multiple lane depths are allowed, the set of optimal lane

depths follows a continuous triangular pattern. They developed a dynamic programming algorithm to select

the set of optimal lane depths from a set of ﬁnite allowable lane depths to minimize the occupied ﬂoor

space. Larson et al. [24] proposed a heuristic to design a class-based layout. Their three-phase algorithm

characterizes the aisle directions and storage zone dimensions, determines the storage types, and assigns the

required storage space to each storage zone.

Derhami et al. [12] proposed a closed-form solution to ﬁnd the lane depth that maximizes volume utiliza-

tion under ﬁnite storage rate constraints. Their model can be used for warehouses located in a manufacturing

facility. They showed that using the traditional lane depth model with a ﬁnite replenishment rate produces

lane depths up to twice as deep as they should be, but the resulting loss of storage space might not be

signiﬁcant. This is because the space utilization curve, as a function of lane depth, is relatively ﬂat as the

lane depth increases.

Derhami et al. [13] showed that the traditional lane depth model [12, 26, 22] underestimates accessibility

waste when used to design a layout because it computes accessibility waste only for the time that a lane is

occupied. They developed a new waste function to compute the total waste of storage volume as a function

of bay depth and used it in a mixed integer program to ﬁnd the optimal bay depths. Their model allows

multiple bay depths in a layout and minimizes the waste of space by assigning SKUs to their preferred bays.

Derhami et al. [11] used simulation to study the trade-oﬀ between space utilization and transportation

costs with respect to bay depths. Using a common bay depth assumption, they simulated multiple layouts

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with various numbers of aisles (i.e., diﬀerent bay depths). Figure 2 shows the simulation results for a

warehouse with 50 SKUs. As the graph shows, utilization of the storage volume increases as the number

of aisles increases from two to four (i.e., bay depth decreases), where it reaches its peak. It then starts

declining as the bay depth becomes shallower because accessibility waste increases. On the other hand, the

total travel distance improves as the bay depth decreases. The improvement becomes modest after a certain

point, and then the travel distance remains steady. Hence, transportation costs improve at the cost of lower

space utilization.

Most of the research on the design of block stacking systems focuses exclusively on the utilization of the

storage space [26, 12, 13] rather than transportation costs or both. As discussed, bay depths and cross-

aisles aﬀect both material handling costs and space utilization. Designing a block stacking layout includes

determining bay depths and the number of aisles and cross-aisles; therefore, both space utilization and

transportation costs must be considered. Considering a conventional block stacking layout similar to the one

presented in Figure 3, this research answers the following questions:

•How many cross-aisles should a layout have?

•How many aisles should a layout have?

•How deep should the bays be?

We develop a simulation-based optimization algorithm that ﬁnds optimal layouts with respect to space

utilization and material handling costs. This paper contributes to the current warehouse design literature

in three primary ways:

•We ﬁnd the optimal layouts with respect to two main objectives of the layout design problem (space

utilization and material handling costs). This allows the model to determine the optimal number of

aisles and cross-aisles while accounting for the eﬀects these variables have on one another;

•We develop a closed-form solution for the optimal number of aisles in a layout. In contrast with

the traditional lane depth model, which optimizes the trade-oﬀ between block width and depth, it

minimizes the total wasted storage space in the entire layout rather than a block; and

•The simulation-based modeling allows the algorithm to capture the stochastic variations that exist in a

real world situation. The stochastic variables considered in the simulation model are production rates,

demand rates, production batch quantities, and forklift travel times. Moreover, the simulation model

is an appropriate tool for accurately assessing the transportation cost in continuous multi-command

pick-up/drop-oﬀ operations; analytical models are unable to accurately estimate the total distance

6

Figure 3: Components of a conventional block stacking layout.

traveled by vehicles in this context because it depends on the sequence of retrieval and replenishment

operations, which is stochastic, and their assignments to the vehicles.

3 Designing optimal layouts

The major factors in designing a block stacking layout are the numbers of aisles and cross-aisles, bay depths,

and cross-aisle types (unidirectional vs. bidirectional). Our algorithm takes all these design factors as

decision variables and attempts to ﬁnd the set of layouts optimizing space utilization and transportation

costs. It simulates diﬀerent potential layouts by varying values of the above variables and produces a Pareto

frontier with respect to two objectives: maximizing space utilization and minimizing transportation costs.

It has three steps: layout scenario generation, pre-simulation assessment, and simulation.

The algorithm ﬁrst generates diﬀerent layout scenarios for simulation. The layouts are diﬀerent from

one another in terms of the numbers of aisles and cross-aisles and the cross-aisle types. We use analytical

and empirical approaches to bound the number of aisles and cross-aisles in the layout scenario generation

and to eﬃciently limit the number of layouts being evaluated by the simulation without disregarding any

optimal solutions. In the next step, the layouts are assessed with respect to the total travel distance, and key

measurements, for example, the travel distances between all potential origins and destinations, are computed

and stored to prevent having to re-calculating them when executing any event in the simulation. Finally,

the generated layouts are evaluated by a steady state simulation and compared in terms of space utilization

and material handling costs. Figure 4 demonstrates the ﬂowchart of the proposed algorithm, and Table 1

presents the deﬁnition of parameters and variables used in the algorithm. The components of the algorithm

are described in detail in the following sections.

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Table 1: Table of notation

amin minimum number of aisles allowed in layout generation

amax maximum number of aisles allowed in layout generation

cmin minimum number of cross-aisles allowed in layout generation

cmax maximum number of cross-aisles allowed in layout generation

Qiproduction (storage) batch quantity of SKU i

Piproduction (storage) rate of SKU i

λidemand (retrieval) rate of SKU i

Hiheight of a pallet of SKU i, in units of distance (e.g., meters. feet, etc.)

Zistackable height of SKU i, in units of pallets

Swwarehouse width, in units of pallets

Shwarehouse height, in units of distance (e.g., meters. feet, etc.)

nanumber of aisles

Aaisle width, in units of pallets

xiassigned lane depth to SKU i, in units of pallets

Iset of all SKUs

¯xcommon bay depth, in units of pallets

Nsnumber of SKUs stored in the warehouse

Slwarehouse length, in units of pallets

n∗

aoptimal number of aisles

¯x∗optimal common bay depth, in units of pallets

αparameter to set amin

βparameter to set amax

(xo, yo) coordinate of the origin for a shipment

(xd, yd) coordinate of the destination for a shipment

Nvtotal number of vehicles required

Dutotal distance loaded forklifts traveled in the simulation

Dltotal distance unloaded forklifts traveled in the simulation

Tssimulation period

Twsimulation warm-up period

Vaverage speed of a forklift

ncnumber of cross-aisles

Ccross-aisle width, in units of pallets

Nbnumber of bays in the simulated layout

Nlnumber of lanes in a bay

WH

ij honeycombing waste generated at the jth lane of bay iin simulation

Lmin minimum number of lanes between two subsequent cross-aisles

Lmax maximum number of lanes between two subsequent cross-aisles

8

Figure 4: Flowchart of the proposed model.

3.1 Layout scenario generation

The optimization problem at hand is a multi-objective optimization problem with two objectives: maximizing

space utilization and minimizing transportation costs. The ﬁrst step of our algorithm is generating layout

scenarios for simulation. The number of aisles and cross-aisles and the cross-aisle types are decision variables

and change over diﬀerent layout scenarios. To perform a complete search of the solution space and obtain a

comprehensive Pareto frontier, all potential layouts must be simulated. If the lower and upper bounds on the

numbers of aisles and cross-aisles are denoted by amin,amax,cmin , and cmax, respectively, the total number

of layouts generated for simulation by assuming a common bay depth is (amax −amin + 1)(cmax −cmin + 1).

Hence, the ranges of [amin, amax ] and [cmin, cmax] determine the number of layouts that must be evaluated

by the simulation and aﬀect the computational diﬃculty and solution quality of the algorithm. These ranges

are usually too wide; this imposes too many potential layouts for simulation. The number of potential

layouts grows exponentially as the size of the warehouse increases because more aisles and cross-aisles are

required. However, the simulation process is computationally intensive, and evaluating all possible scenarios

is not computationally practical.

We narrow the search space by bounding the number of aisles and cross-aisles allowed for generating

layout scenarios. The bounds must be tight enough to prevent an extensive computational burden from

simulating too many scenarios but large enough to ensure that no potentially optimal solution is omitted.

Once the bounds are set, the layout scenarios are generated using all possible combinations of the allowed

values for the number of aisles and cross-aisles. In the next sections, we describe an analytical approach to

bound the number of aisles in the layout generation and describe how we do the same for the cross-aisles.

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Figure 5: Changes in the inventory of SKU iover its cycle time, Pi> λi.

3.1.1 Bounding the number of aisles

To avoid simulating too many layout scenarios with only marginal diﬀerences, we use a common bay depth

policy. That is, all bays in the layout should have the same depth.This setup is widely used in practice

because it discards the need for lane depth assignment to SKUs. This policy signiﬁcantly shrinks the search

space by disregarding many layouts that have the same number of aisles but slightly diﬀerent bay depth

proﬁles. For example, if bay depths are allowed to be between 5 to 25 pallets, then for every single layout of

(amax −amin + 1)(cmax −cmin + 1) layouts, all variations of bay depths between 5 to 25 pallets for which

the sum of bay depths and dedicated space to the aisles equals the warehouse length must be considered for

simulation. This results in an enormous number of layouts diﬀer only slightly from one another.

Thus, setting the number of aisles in each scenario gives the bay depths (note that the dimensions of the

layouts are kept ﬁxed in all scenarios). We develop a closed-form solution to obtain the optimal number of

aisles to maximize space utilization under the common bay depth constraint and use it to determine amin

and amax. This analytical model assumes deterministic production and demand rates, but this assumption

is relaxed later in the simulation model.

Consider SKU iproduced in batches of Qipallets and stored at a rate of Pipallets per unit of time.

Assume it is retrieved at a rate of λipallets per unit of time, where Pi> λiand the replenishment starts

when the inventory reaches zero. Pallets of this SKU are Hifeet high and can be stacked up to Zipallets.

The changes in the inventory of this SKU are shown in Figure 5. Assume set Ithat consists of NsSKUs is

stored in a warehouse whose length and width are Sland Sw, respectively, and its aisles are Apallets wide.

The following lemma determines the optimal number of aisles and lane depth for the warehouse.

10

Lemma 1. The optimal number of aisles in the warehouse is obtained by

n∗

a=rSlNs

4SwA.(1)

Proof. As shown in [13], the average wasted storage volume generated in the layout is given by

¯

W=AShSwna+Sh

2X

i∈I

xi+X

i∈I1

2PiZi(Qi(Sh−ZiHi)−ZiHi)(Pi−λi)−λi(2Sh−ZiHi),(2)

where Shis the warehouse clear height (in units of distance, i.e., inches, feet, etc.), nais the number of

aisles, and xiis the assigned lane depth to SKU i. Variables xiand naare the only decision variables in (2);

therefore, the optimal bay depths are obtained by optimizing (2) with respect to xiand na. Note that these

two variables are dependent; that is, the sum of aisle widths and bay depths equals the warehouse length.

The constant part of (2) is trivial in optimization and can be removed. Assuming a common bay depth,

denoted by ¯x, we have

X

i∈I

xi=Ns¯x, (3)

where Nsis the number of SKUs. Replacing (3) in (2), the optimal common bay depth is obtained by solving

the following constrained optimization problem:

Minimize AShSwna+1

2ShNs¯x, (4)

s.t.

2na¯x+naA=Sl,(5)

¯x, na∈Z+.(6)

Constraint (5) guarantees that the sum of bay depths and aisle widths is equal to the warehouse length.

Solving (5) for ¯xand substituting ¯xinto the objective function (4) produces an unconstrained optimization

model whose objective function is

Minimize AShSwna+1

4naSh(Sl−naA).(7)

Diﬀerentiating (7) with respect to na, setting the results equal to zero, and solving for nagives n∗

a. Expression

(7) is continuously diﬀerentiable, has only one extreme point, and its second derivative with respect to nais

non-negative. Hence, it is a unimodal function and n∗

ais its global optimum.

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Variable n∗

amust be an integer. To have an integer value for n∗

a, we evaluate the two nearest integers

smaller and greater than n∗

ain (7) and select the integer that produces a smaller waste of space. Once n∗

ais

determined, the optimal common bay depth is obtained by

¯x∗=Sl−n∗

aA

2n∗

a

.(8)

When ¯x∗is not integer, arranging the layout into exactly 2n∗

aequally deep bays is not possible. In this case,

we create 2n∗

abays with the depth of b¯x∗cpallets and split the remaining Sl−n∗

a(2b¯x∗c+A) pallet positions

equally among all bays.

From an operational standpoint, deep lanes limit storage space availability, as it takes longer to fully

deplete or replenish a deep lane. Storage space availability becomes more important when the SKU variety

is high. In such a situation, it is important to have suﬃcient storage space available (empty) at any time to

provide storage lanes for incoming SKUs and allow ﬂexibility in production planning without any concerns

about storage space availability. The following proposition discusses another property of n∗

afor the layout

design problem.

Lemma 2. n∗

amaximizes storage space availability.

Proof. Proof is provided in Appendix A.

Using n∗

a, we set amin =αn∗

aand amax =βn∗

afor layout scenario generation. Parameters αand βare

set through a numerical experiment presented in section 4.1.

3.1.2 Bounding the number of cross-aisles

For each layout scenario, two alternatives are considered: unidirectional and bidirectional cross-aisles. Bidi-

rectional cross-aisles are twice as wide as the unidirectional cross-aisles. Hence, in the layouts with unidirec-

tional cross-aisles, we put twice as many cross-aisles as the respective layouts with bidirectional cross-aisles.

This allows a fair comparison between the two alternatives in terms of space utilization. For example, a

layout with three bidirectional cross-aisles is compared to the layouts with six unidirectional cross-aisles.

Cross-aisles are evenly spaced from each other, and the distance between any two subsequent cross-aisles is

the same. Unidirectional cross-aisles are added to the layout in pairs with opposite directions.

Cross-aisles facilitate material handling in a warehouse. Developing a closed-form model to estimate the

transportation cost (or total travel distance) as a function of the number of cross-aisles (similar to the method

used for the number of aisles) is not possible because the shipment operations are performed continuously,

and the travel distance for a shipment depends on the sequence of operations assigned to a vehicle (i.e., the

12

N

W E

S

Figure 6: Components of the layout and their relative conﬁgurations.

last drop-oﬀ location of a vehicle must be known). For this reason, we use an experimental approach with

simulation in section 4.1 to analyze the total travel distance with respect to the number of cross-aisles and

determine cmin and cmax.

3.2 Pre-simulation assessment

We compute the rectilinear shortest distances between all potential origin and destination pairs in the layout

and store them for simulation use. This is to avoid recalculating travel distances every time an event is

initiated. For each pair of layout scenarios that only diﬀer in cross-aisle type (i.e., the same numbers of aisles

and cross-aisles but one with unidirectional and the other with bidirectional cross-aisles), we calculate the

total travel distance between all potential pick-up and drop-oﬀ locations and consider only the layout with

the smaller total travel distance for simulation. The total space dedicated to the cross-aisles is the same in

both alternatives; therefore, the layout with a smaller total travel distance is the non-dominated solution.

Figure 6 presents the relative locations of the main components of the layout. The locations of production

lines, vehicle parking, and outbound docks, as well as the number of P/D points, are given and can diﬀer

from those presented in Figure 6. Although the main purpose of this model is not to optimize the location

of these components, one can analyze their arrangements by simulating diﬀerent scenarios while keeping the

remaining design factors ﬁxed. The following distances are required in the simulation:

•Distances between all storage lanes.

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•Distances from storage lanes to the production line and vice versa.

•Distances from storage lanes to outbound docks and vice versa.

•Distances from storage lanes to the vehicle parking and vice versa.

•Distances from outbound docks to the production line.

•Distances from outbound docks to the vehicle parking.

•Distances from the vehicle parking to the production line.

A rectilinear distance between two locations is obtained by adding the distances traveled along the x-axis

and y-axis. Considering both unidirectional and bidirectional cross-aisles adds an extra level of complexity

to the calculations because the directions of the move and the cross-aisle must match. As Figure 6 illustrates,

we use a Cartesian coordinate system and assume that the origin is located at the southwest corner of the

layout. Instead of designing a web of P/D locations to ﬁnd the shortest path, we take advantage of the

special shape of the layout to simplify calculations.

The shortest path between two points is given by the path that travels through one cross-aisle and at

most two aisles (the aisles that the origin and destination points are located in). This is highlighted in Figure

6 by two paths connecting an example shipment between two P/D locations in two diﬀerent bays. All other

alternatives that produce the same travel distance are not evaluated for the sake of computational eﬃciency.

The shortest path for each move is a direct path that passes either through the closest cross-aisle located

north of the origin and allows moving toward the destination or a similar cross-aisle located south of the

origin (see Figure 6). Hence, the distances of the origin and destination points to the eligible cross-aisles (the

closest cross-aisles located north and south of the origin that allow moving toward the destination) must be

taken into account for each move.

Consider a west-to-east move in Figure 6. The travel distance along the shortest path through the

cross-aisle located north of the origin is

DN

W→E=

|yc−yo|+

yc−yd

+

xd−xo

if yc/∈ ∅,

∞otherwise,

(9)

where ycis the y-coordinate of the closest cross-aisle located north of the origin heading east, xoand

yoare the coordinates of the origin, and xdand ydare the coordinates of the destination. If an eligible

cross-aisle does not exist, the distance is set to inﬁnity for this path.

14

Similarly, the travel distance along the shortest path from the southern cross-aisle, DS

W→E, is obtained

by replacing the coordinates of the respective cross-aisle in (9). The shortest path is then obtained by

DW→E=min{DN

W→E, DS

W→E}.(10)

Note that DE→Wis not necessarily equal to DW→Ebecause unidirectional cross-aisles may impose longer

travel distance along the y-axis. Expressions (9) and (10) are valid in the layout in Figure 6 for all distances

between the storage lanes. They are also valid for distances between the storage lanes and pick-up points,

outbound docks, and vehicle parking if they are located on the short sides of the warehouse. Similarly, they

are also valid for the distances between any pair of outbound docks, pick-up points, and vehicle parking if

they are located on the short sides of the layout. If either of these components are on the long sides of the

warehouse, the shortest distance between the storage lane and that component is simply obtained by

DS→N=DN→S=

yo−yd

+

xo−xd

.(11)

This is because aisles are bidirectional (to provide enough space to the forklifts to easily maneuver when

replenishing/retrieving lanes), and the cross-aisles located in front of those locations are bidirectional for the

ease of transportation. Expression 11 is also used to calculate distances between pairs of outbound docks,

pick-up points, and the vehicle parking if either the origin or the destination is located on the long sides of

the layout.

3.3 Simulation model

The core of the model is an event-oriented simulation of pallet storage and retrieval operations along with

material handling for a given layout. The output reports performance metrics pertinent to space utiliza-

tion and transportation costs. The model has four sources of variation: production rates, demand rates,

production batch quantities, and vehicle transportation times. The company that we studied for our case

study uses automated guided vehicles (AGV) for material handling. Accordingly, the simulation model is

built assuming operations are handled by AGVs. If vehicles are operated by humans, human factors such as

learning aﬀect the performance of the system because order picking is a repetitive and cognitive task that

involves human learning [5, 16, 42]. Incorporating the human learning factors in simulation is an extension

to our model that provides more accurate prediction of the performance of the system for such cases.

The simulation model consists of nine procedures: three events to simulate a replenishment operation,

three events for a retrieval operation, two events for a vehicle release, and a warm-up event. In a replenish-

15

ment operation, a vehicle is sent to pick up a produced (or inbound arrival) pallet from the production line

(or inbound dock) and deliver it to a storage lane. In a retrieval operation, a vehicle is sent to pick up a

pallet from a storage lane and deliver it to an outbound dock. The simulation events are as follows:

•Production pick-up: The closest available vehicle to the pick-up point (production line or inbound

dock) is dispatched to pick up a waiting pallet, and a “lane drop-oﬀ” event is scheduled taking into

account the travel distance. The storage lane is determined for the pallet.

•Lane drop-oﬀ: The dispatched vehicle picks up the pallet from the production line or inbound dock

and starts traveling to the assigned storage lane. A “replenishment” event is scheduled considering the

travel distance.

•Replenishment: The pallet is stored in the target lane, and a “release vehicle” event is scheduled at

the simulation time plus epsilon time unit.

•Outbound pick-up: The closest vehicle to the pick-up lane is dispatched to pick up the requested SKU

from its assigned lane. A “retrieval” event is scheduled taking the travel distance into consideration.

An outbound dock is assigned for drop-oﬀ.

•Retrieval: The dispatched vehicle picks up the requested pallet from the lane and starts traveling to

the assigned outbound dock. A “truck drop-oﬀ” event is scheduled considering the travel time.

•Truck drop-oﬀ: The requested SKU is delivered to the assigned outbound dock and a “release vehicle”

event is scheduled at the simulation time plus epsilon time unit if there is no waiting pick-up request.

•Release vehicle: The empty vehicle is sent to parking. A “Park vehicle” event is scheduled considering

the travel distance.

•Park vehicle: The released vehicle is parked and becomes available.

•Warm-up: This event is executed once at a given time and resets all variables used for performance

evaluation (not the control variables) to their initial values.

Figure 7 shows the simulation ﬂowchart and steps. The simulation model uses an event list to queue and

execute all scheduled events based on their execution times. We assume the production and outbound plans

are known at the beginning of each working shift (i.e., which SKUs and how many are produced/shipped),

though the exact times are stochastic. They are added to the simulation event list at the beginning of

each working shift. Outbound events are sequenced based on the truck schedules, truck capacity, and SKU

16

Figure 7: Flowchart of the discrete event-based simulation model.

demands. The “Release vehicle” event is scheduled at the simulation time plus epsilon time unit to allow

awaiting pick-up requests to be processed before an idle vehicle is sent to parking.

Along with many performance metrics pertinent to space utilization and transportation costs, such as

total distance traveled by vehicles and vehicle utilization, the simulation model computes two major perfor-

mance metrics reﬂecting the objective functions: the required number of vehicles (Nv) and the percentage of

wasted space (W). We purposefully deﬁned these two metrics such that they are suﬃciently tangible for the

decision making and easily convertible to a dollar amount. Multiplying Nvand Wby the unit vehicle cost

and unit space cost yields the total transportation cost and space cost for the layout. Space cost includes

direct and indirect space-related costs such as rent, insurance, lights, building amortization and repairs.

The material handling cost includes the costs of operating vehicles such as forklift costs and amortization,

maintenance, and labor.

Nvis obtained by summing the distance traveled by all vehicles in the simulation and then dividing the

result by the total distance that a vehicle can travel. That is,

Nv=Du+Dl

V∗(Ts−Tw),(12)

17

Table 2: Preliminary experimental study to ﬁne tune parameters: parameters used in the algorithm and the

minimum and maximum numbers of aisles and cross-aisles observed in the solutions of the Pareto frontiers.

Test problem Parameter values Pareto frontier solutions

n∗

aamin amax cmin cmax Min(na)M ax(na)M in(nc)Max(nc)

10 SKUs 2 1 3 2 5 2 3 2 5

50 SKUs 3 2 5 2 10 3 5 2 9

100 SKUs 5 4 8 2 12 5 8 2 5

150 SKUs 6 4 9 2 15 6 9 2 9

200 SKUs 7 5 11 2 17 7 11 2 5

300 SKUs 8 6 12 2 20 8 12 2 7

where Du, and Dlare the total loaded and unloaded distances traveled by all vehicles, Tsis the simulation

time, Twis the warm-up period, and Vis the average speed of a vehicle. The percentage of wasted space

is obtained by summing the total wasted space in lanes and dividing it by the total space-time. The total

wasted space in the warehouse is the sum of waste from honeycombing and accessibility [10]. The volume

dedicated to the aisles and cross-aisles comprises the accessibility waste [13] and honeycombing waste is

generated in the lanes as they are being replenished and retrieved[3]. Hence the percentage of wasted space

is obtained by

W=Sh(naASw+ncC Sl)(Ts−Tw) + PNb

i=1 PNl

j=1 WH

ij

(SwSlSh)(Ts−Tw),(13)

where ncis the number of cross-aisles, Cis the cross-aisle width, Nbis the number of bays, Nlis the number

of lanes in a bay, and WH

ij is the honeycombing waste at the jth lane of bay i. All simulation scenarios are

compared with respect to these two objective functions, and non-dominated solutions produce the Pareto

frontier.

4 Numerical experiments

In this section, we perform an experimental analysis to ﬁne tune the parameters of the algorithm. Then, we

test the algorithm on a case in the beverage industry. Using data from the case study, we design various test

problems to analyze the computational eﬃciency of the model and obtain some insights from the generated

Pareto frontiers on small- to industrial-sized test problems.

The proposed model was coded with Python 2.7 and run on a cluster whose nodes are equipped with ten

Intel Xeon processors E5-2660 (2.6GHz) and 128 GB of RAM. We ran the algorithm in parallel on ten cores.

Following the simulation setups proposed in [11], we set the warm-up period to one month, the start-up

inventory to zero, and the simulation time to 8 months. To reduce the number of replications, we used

common random numbers across scenarios.

18

4.1 Tuning the parameters

We conducted a numerical experiment to set amin,amax ,cmin, and cmax such that the computational eﬀort

is minimized while no potentially optimal solution is removed from the search space. We generated random

test problems with various sizes (10 to 300 SKUs) and tested the algorithm by setting wide ranges for the

parameters without considering the resulting computational times. The goal was to analyze the solutions in

the Pareto frontiers and use them to eﬃciently tighten the range of the input parameters.

Deﬁne the minimum and maximum numbers of lanes between two subsequent cross-aisles by Lmin and

Lmax, respectively. Then, cmin = (Sw+Lmax )/(Lmax + 2C) and cmax = (Sw+Lmin )/(Lmin + 2C), where

Cis the width of a unidirectional cross-aisle in units of pallets. We tested the following setups: α= 0.8,

β= 1.5, Lmin = 10, and Lmax = 40. Table 2 shows the resulting amin,amax,cmin , and cmax for each test

problem and the minimum and maximum numbers of aisles and cross-aisles observed in the solutions of the

Pareto frontier for each test problem.

As Table 2 shows, the smallest number of aisles observed in all solutions is n∗

afor all test problems,

although smaller numbers were allowed in the experiment. This is because decreasing the number of aisles

beyond n∗

adeteriorates both objective functions. Therefore, the eﬃcient value for αwill be one for the ﬁnal

experiments. Increasing the number of aisles reduces space utilization but improves the total travel distance.

However, as highlighted in [11] and shown in Figure 2, the improvement rate declines as the number of aisles

grows and becomes negligible once the layout has many aisles. Beyond this point, savings in transportation

costs do not justify the loss of storage space. Taking this into account, we set β= 1.4.

The maximum number of cross-aisles observed in the Pareto frontiers is smaller than cmax for almost

all test problems except the 10 SKU test problem, for which it is equal to cmax. This shows that cmax

can be reduced without risking the removal of potentially optimal solutions. The largest Lmin that yields a

cmax equal to or larger than the maximum number of cross-aisles observed in all Pareto frontiers is obtained

by setting Lmin = (0.1)Sw. This setup limits cmax while preserving all solutions (including the 10 SKUs

problem, for which this condition yields cmax = 5). We set cmin = 2 because we assumed, for ease of

transportation, that one cross-aisle exists next to the long side of the warehouse.

4.2 A case study in the beverage industry

We used the algorithm to design the warehouse layout for one of the production facilities of a leading supplier

of bottled beverages in North America. The company produces various types of bottled beverages and runs

dozens of plants and warehouses across the U.S.A., Canada, and Mexico. The storage system in all of these

facilities is mainly block stacking. The selected facility produces more than 100 SKUs and stores them in a

19

Table 3: 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) (222,218)

Number of aisles 11 5

Number of cross-aisles 4 16

Cross-aisle type Unidirectional Unidirectional

Wasted space (%) 47.69 65.13

Required number of vehicles 38.58 34.43

Figure 8: Pareto frontier of the case study vs. the current layout, solutions format:(na,nc, cross-aisles type).

368,000 ft2storage area. Material handling is performed by unit-load forklifts. The main characteristics of

the current layout and the results from simulating it are presented in Table 3.

The Pareto frontier generated by the algorithm is presented in Figure 8. Quantifying the objectives with

their costs produces an optimal layout presented in Table 3. Like most facilities located in suburban areas,

the material handling cost in our studied facility is greater than the space cost. As a result, the optimal

solution is the layout that most reduces the travel distance by using 16 unidirectional cross-aisles in contrast

with the current layout, which has four cross-aisles of the same type. This solution dedicates four times the

current amount of space to the cross-aisles and consequently has lower space utilization compared to the

current layout, but it improves the total travel distance by approximately 10%. The optimal layout has fewer

than half the aisles of the current layout. This shows that the beneﬁt of shallow lanes on transportation

costs did not justify the cost of lost space utilization. Implementing the proposed layout would save more

than half a million dollars annually on the total operational costs of the studied facility.

20

Table 4: Computational experiments.

Problem Warehouse

size (ft) amin amax cmin cmax

# of simulated

layouts

# of layouts

in Pareto frontier

Avg. Simulation time

per scenario (sec)

Computational

time (sec)

10 SKUs 200 ×400 2 3 2 8 14 2 16 36

50 SKUs 400 ×720 3 5 2 10 27 6 32 123

100 SKUs 400 ×920 5 7 2 10 27 10 54 160

200 SKUs 760 ×1400 7 10 2 10 36 9 237 977

300 SKUs 920 ×1600 8 12 2 11 50 12 337 1746

400 SKUs 1080 ×1840 9 13 2 11 50 12 518 2676

500 SKUs 1200 ×2080 10 14 2 11 50 8 536 3084

600 SKUs 1320 ×2200 11 16 2 11 60 10 544 3421

700 SKUs 1440 ×2400 12 17 2 11 60 8 658 4177

800 SKUs 1600 ×2600 13 19 2 11 70 8 1034 11520

900 SKUs 1720 ×2880 14 20 2 11 70 12 1057 16080

1000 SKUs 1840 ×3000 14 20 2 11 70 14 1159 17155

Figure 9: Pareto frontier of the 500 SKUs test problem, solutions format:(na,nc, cross-aisles type).

4.3 Experimental analysis

We randomly generated 12 test problems using the characteristics of the SKUs in our case study such as

production rates, demand rates, and production batch quantities. The test problems cover small (10 SKUs)

to industrial-sized (1000 SKUs) warehouses and are used to evaluate the computational performance of the

proposed algorithm and obtain some insights into the Pareto frontier of the problem.

Table 4 shows the computational times and the number of scenarios simulated for each test problem.

The simulation times ranged from 16 to 1159 seconds depending on the size of the problem. The total

computational time of the algorithm grows from 36 seconds (for the smallest problem) up to 17155 seconds

for the largest test problem. The total computational times are smaller than the sum of the simulation times

for all scenarios because the simulation scenarios were run in parallel. The computational times show that

although the model simulates multiple scenarios in a steady state, it is capable of ﬁnding the Pareto frontier

in a reasonable time for large-sized test problems.

Figures 9 and 10 demonstrate the Pareto frontiers for test problems with 500 and 1000 SKUs, respectively.

These test problems present extremely large warehouses that store a high number of SKUs and have high

21

Figure 10: Pareto frontier of the 1000 SKUs test problem, solutions format:(na,nc, cross-aisles type).

inventory turnover. For example, forklifts in the 1000 SKUs test problem perform, on average, 3470 picks per

hour. This explains the relatively large number of vehicles in these test problems. The following observations

and insights are inferred:

•Solutions in the Pareto frontiers can be clustered into two groups. The ﬁrst group covers a few solutions

at the top left of the Pareto frontiers that impose a small amount of wasted space at the cost of high

transportation costs. The layouts in this group have few cross-aisles and aisles. The second group

consists of a larger set of solutions located at the bottom of the Pareto frontiers. These solutions

mainly incur lower transportation costs and higher wasted space as a result of including more cross-

aisles and aisles in the layout. The decision maker should select an optimal solution from the ﬁrst

group when the unit space cost is considerably greater than the unit transportation cost. Otherwise,

the optimal solution is among the second group.

•Increasing the number of cross-aisles from two (bidirectional or four unidirectional) to three improves

transportation costs signiﬁcantly with only a marginal loss in space utilization. The transportation

costs continue to decrease but with marginal improvements as the number of cross-aisles increases,

while the utilization of the storage space deteriorates considerably. Therefore, adding too many cross-

aisles not only reduces space utilization but also does not necessarily improve the transportation costs,

as pickers must traverse the cross-aisles as well.

•Fixing the number of cross-aisles, transportation costs (or total travel distance) decrease as the number

of aisles increases (consequently, bay depths decrease); however, the resulting loss in space utilization

may not justify the improvement. This is in line with the ﬁndings of [11].

•The locations of production lines and outbound docks (P/D points) signiﬁcantly aﬀect the material

22

handling cost, especially when the warehouse length-to-width ratio is large. In such cases, lanes are

arranged along the long side of the layout as presented in Figure 6. Our experiments show that

arranging the production lines and outbound docks on the long sides of the warehouse, as presented

in Figure 6, may result in up to 25% savings in material handling costs.

•The improvement gained in transportation costs by adding a new aisle is not as signiﬁcant as those

gained by adding a new cross-aisle.

•Unidirectional cross-aisles appear more frequently in the Pareto frontiers. However, the eﬀects of the

cross-aisle type on traﬃc congestion must also be taken into account.

•The number of solution points in a Pareto frontier increases as the size of the warehouse grows. This

is mainly because the range of the parameters increases.

5 Conclusions

In this paper, we developed a simulation-based optimization algorithm to simultaneously optimize utilization

of the storage space and transportation costs in the layout of a block stacking warehouse. We developed a

closed-form solution to ﬁnd the optimal number of aisles to maximize utilization of the storage space. Our

algorithm ﬁnds the optimal numbers of aisles and cross-aisles and the cross-aisle type.

A case study in the beverage industry shows that considering both space utilization and transportation

costs leads to a diﬀerent layout than when focusing exclusively on only one of these objectives and the

obtained layout signiﬁcantly improves the operational costs. Exhaustive computational experiments show

that the model ﬁnds the Pareto frontier in a reasonable time for large-sized test problems. Although adding

new cross-aisles improves the total travel distance, the improvement rate decreases as more cross-aisles are

added, while the reduction in utilization of the storage space continues. Hence, adding new cross-aisles

beyond some level does not justify the loss of storage space even if the transportation unit cost is higher

than the space unit cost.

The solutions in the Pareto frontiers can be arranged into two clusters: layouts with high space utilization

but less eﬃciency in terms of transportation and those with high transportation eﬃciency but lower space

utilization. This can help the decision makers to identify the optimal layout considering transportation and

space unit costs. Layouts in the ﬁrst group contain two to three cross-aisles, whereas the solutions in the

next group have relatively many cross-aisles (up to ten).

The number of aisles and consequently bay depths aﬀects the material handling cost in addition to space

utilization. We found that increasing the number of aisles in a layout leads to a reduction in travel distance;

23

however, this reduction becomes less signiﬁcant as the number of aisles grows. We also found that adding a

new cross-aisle to the layout results in a greater improvement in material handling cost than adding a new

aisle.

References

[1] E. Ardjmand, H. Shakeri, M. Singh, and O. S. Bajgiran. Minimizing order picking makespan with

multiple pickers in a wave picking warehouse. International Journal of Production Economics, 206:

169–183, 2018. doi: 10.1016/j.ijpe.2018.10.001.

[2] P. Baker and M. Canessa. Warehouse design: A structured approach. European Journal of Operational

Research, 193(2):425–436, 2009. doi: 10.1016/j.ejor.2007.11.045.

[3] J. J. Bartholdi and S. T. Hackman. Warehouse & Distribution Science: Release 0.98. The Supply Chain

and Logistics Institute, Atlanta, GA, 2017.

[4] M. Bortolini, M. Faccio, M. Gamberi, and R. Manzini. Diagonal cross-aisles in unit load warehouses

to increase handling performance. International Journal of Production Economics, 170:838–849, 2015.

doi: 10.1016/j.ijpe.2015.07.009.

[5] J. Boudreau, W. Hopp, J. O. McClain, and L. J. Thomas. On the interface between operations and

human resources management. Manufacturing & Service Operations Management, 5(3):179–202, 2003.

doi: 10.1287/msom.5.3.179.16032.

[6] L. F. Cardona and K. R. Gue. How to determine slot sizes in a unit-load warehouse. IISE Transactions,

2018. doi: 10.1080/24725854.2018.1509159.

[7] L. F. Cardona, D. F. Soto, L. Rivera, and H. J. Mart´ınez. Detailed design of ﬁshbone warehouse layouts

with vertical travel. International Journal of Production Economics, 170, Part C:825–837, 2015. doi:

10.1016/j.ijpe.2015.03.006.

[8] L. Chen, A. Langevin, and D. Riopel. A tabu search algorithm for the relocation problem in a

warehousing system. International Journal of Production Economics, 129(1):147–156, 2011. doi:

10.1016/j.ijpe.2010.09.012.

[9] R. de Koster, T. Le-Duc, and K. J. Roodbergen. Design and control of warehouse order picking: A

literature review. European Journal of Operational Research, 182(2):481–501, 2007. doi: 10.1016/j.ejor.

2006.07.009.

24

[10] S. Derhami. Designing Optimal Layouts for Block Stacking Warehouses. PhD thesis, Auburn University,

Auburn, AL, 8 2017.

[11] S. Derhami, J. S. Smith, and K. R. Gue. A simulation model to evaluate the layout for block stacking

warehouses. In 14th IMHRC Proceedings: Progress in Material Handling Research: 2016, Karlsruhe,

Germany, 2016.

[12] S. Derhami, J. S. Smith, and K. R. Gue. Optimising space utilisation in block stacking warehouses.

International Journal of Production Research, 55(21):6436–6452, 2017. doi: 10.1080/00207543.2016.

1154216.

[13] S. Derhami, J. S. Smith, and K. R. Gue. Space-eﬃcient layouts for block stacking warehouses. IISE

Transactions, 51(9):957–971, 2019. doi: 10.1080/24725854.2018.1539280.

[14] A. H. Gharehgozli, Y. Yu, X. Zhang, and R. de Koster. Polynomial time algorithms to minimize total

travel time in a two-depot automated storage/retrieval system. Transportation Science, 51(1):19–33,

2017. doi: 10.1287/trsc.2014.0562.

[15] M. Goetschalckx and D. H. Ratliﬀ. Optimal lane depths for single and multiple products in block

stacking storage systems. IIE Transactions, 23(3):245–258, 1991. doi: 10.1080/07408179108963859.

[16] E. H. Grosse and C. H. Glock. The eﬀect of worker learning on manual order picking processes.

International Journal of Production Economics, 170:882–890, 2015. doi: 10.1016/j.ijpe.2014.12.018.

[17] J. Gu, M. Goetschalckx, and L. F. McGinnis. Research on warehouse operation: A comprehensive

review. European Journal of Operational Research, 177(1):1–21, 2007. doi: http://dx.doi.org/10.1016/

j.ejor.2006.02.025.

[18] J. Gu, M. Goetschalckx, and L. F. McGinnis. Research on warehouse design and performance evaluation:

A comprehensive review. European Journal of Operational Research, 203(3):539–549, 2010. doi: 10.

1016/j.ejor.2009.07.031.

[19] K. R. Gue and R. D. Meller. Aisle conﬁgurations for unit-load warehouses. IIE Transactions, 41(3):

171–182, 2009. doi: 10.1080/07408170802112726.

[20] K. R. Gue, G. Ivanov´c, and R. D. Meller. A unit-load warehouse with multiple pickup and deposit points

and non-traditional aisles. Transportation Research Part E: Logistics and Transportation Review, 48(4):

795–806, 2012. doi: 10.1016/j.tre.2012.01.002.

25

[21] X. Guo, Y. Yu, and R. B. de Koster. Impact of required storage space on storage policy performance

in a unit-load warehouse. International Journal of Production Research, 54(8):2405–2418, 2016. doi:

10.1080/00207543.2015.1083624.

[22] D. Kind. Elements of space utilization. Transportation and Distribution Management, 15:29–34, 1975.

[23] T. Lamballais, D. Roy, and M. de Koster. Estimating performance in a robotic mobile fulﬁllment system.

European Journal of Operational Research, 256(3):976–990, 2017. doi: 10.1016/j.ejor.2016.06.063.

[24] T. N. Larson, H. March, and A. Kusiak. A heuristic approach to warehouse layout with class-based

storage. IIE Transactions, 29(4):337–348, 1997. doi: 10.1080/07408179708966339.

[25] J. Li, M. Moghaddam, and S. Y. Nof. Dynamic storage assignment with product aﬃnity and ABC

classiﬁcation—a case study. The International Journal of Advanced Manufacturing Technology, 84(9):

2179–2194, 2016. doi: 10.1007/s00170-015-7806-7.

[26] J. O. Matson. The analysis of selected unit load storage systems. PhD thesis, Georgia Institute of

Technology, Atlanta, GA, 1982.

[27] M. Moshref-Javadi and M. R. Lehto. Material handling improvement in warehouses by parts clustering.

International Journal of Production Research, 54(14):4256–4271, 2016. doi: 10.1080/00207543.2016.

1140916.

[28] C. H. Mowrey and P. J. Parikh. Mixed-width aisle conﬁgurations for order picking in distribution

centers. European Journal of Operational Research, 232(1):87–97, 2014. ISSN 0377-2217. doi: 10.1016/

j.ejor.2013.07.002.

[29] I. V. Nieuwenhuyse and R. B. de Koster. Evaluating order throughput time in 2-block warehouses

with time window batching. International Journal of Production Economics, 121(2):654–664, 2009. doi:

10.1016/j.ijpe.2009.01.013.

[30] O. ¨

Ozt¨urko˘glu, K. Gue, and R. Meller. A constructive aisle design model for unit-load warehouses with

multiple pickup and deposit points. European Journal of Operational Research, 236(1):382–394, 2014.

doi: 10.1016/j.ejor.2013.12.023.

[31] J. A. Pazour and R. D. Meller. An analytical model for A-frame system design. IIE Transactions, 43

(10):739–752, 2011. doi: 10.1080/0740817X.2010.549099.

[32] F. Ramtin and J. A. Pazour. Product allocation problem for an AS/RS with multiple in-the-aisle pick

positions. IIE Transactions, 47(12):1379–1396, 2015. doi: 10.1080/0740817X.2015.1027458.

26

[33] K. J. Roodbergen and R. de Koster. Routing methods for warehouses with multiple cross aisles. Inter-

national Journal of Production Research, 39(9):1865–1883, 2001. doi: 10.1080/00207540110028128.

[34] K. J. Roodbergen and I. F. A. Vis. A model for warehouse layout. IIE Transactions, 38(10):799–811,

2006. doi: 10.1080/07408170500494566.

[35] K. J. Roodbergen, G. P. Sharp, and I. F. Vis. Designing the layout structure of manual order picking

areas in warehouses. IIE Transactions, 40(11):1032–1045, 2008. doi: 10.1080/07408170802167639.

[36] K. J. Roodbergen, I. F. Vis, and G. D. T. Jr. Simultaneous determination of warehouse layout and

control policies. International Journal of Production Research, 53(11):3306–3326, 2015. doi: 10.1080/

00207543.2014.978029.

[37] B. Rouwenhorst, B. Reuter, V. Stockrahm, G. van Houtum, R. Mantel, and W. Zijm. Warehouse

design and control: Framework and literature review. European Journal of Operational Research, 122

(3):515–533, 2000. doi: http://dx.doi.org/10.1016/S0377-2217(99)00020-X.

[38] L. M. Thomas and R. D. Meller. Analytical models for warehouse conﬁguration. IIE Transactions, 46

(9):928–947, 2014. doi: 10.1080/0740817X.2013.855847.

[39] L. M. Thomas and R. D. Meller. Developing design guidelines for a case-picking warehouse. International

Journal of Production Economics, 170, Part C:741–762, 2015. doi: 10.1016/j.ijpe.2015.02.011.

[40] T. S. Vaughan and C. G. Petersen. The eﬀect of warehouse cross aisles on order picking eﬃciency.

International Journal of Production Research, 37(4):881–897, 1999. doi: 10.1080/002075499191580.

[41] N. Zaerpour, R. B. de Koster, and Y. Yu. Storage policies and optimal shape of a storage system.

International Journal of Production Research, 51(23-24):6891–6899, 2013. doi: 10.1080/00207543.2013.

774502.

[42] J. Zhang, F. Liu, J. Tang, and Y. Li. The online integrated order picking and delivery considering

pickers learning eﬀects for an o2o community supermarket. Transportation Research Part E: Logistics

and Transportation Review, 123:180–199, 2019. doi: 10.1016/j.tre.2019.01.013.

[43] S. Zhou, Y. Y. Gong, and R. de Koster. Designing self-storage warehouses with customer choice.

International Journal of Production Research, 54(10):3080–3104, 2016. doi: 10.1080/00207543.2016.

1158880.

27

A Proof of lemma 2

Proof. Once a lane is fully occupied, Shxistorage volume becomes unavailable for the period that the lane

is partially or fully occupied. As described in [13], the total lane-time that SKU ioccupies in T1

iis

1

Pi−λi((Imax

i−1) + (Imax

i−Zixi−1) + (Imax

i−2Zixi−1) + · · · + (Imax

i−KiZixi)) ,(A.14)

where Imax

iis the maximum inventory level for SKU iand obtained by

Imax

i≈Qi(Pi−λi)

Pi

,(A.15)

and Kiis the number of required lanes for storage and is given by

Ki≈Qi(Pi−λi)

PiZixi

.(A.16)

Similarly, the total lane-time that SKU ioccupies in T2

iis

1

λi(Imax

i+ (Imax

i−Zixi)+(Imax

i−2Zixi) + · · · + (Imax

i−KiZixi)) .(A.17)

Summing (A.14) and (A.17) produces the total lane-time that SKU ioccupies in its cycle time. Multiplying

the result by Shxigives the total occupied volume-time, and multiplying it by λi/Qi, the cycle time of SKU

i, gives the average unavailable storage space occupied by SKU i

Ui=Shxi

2+Sh(Qi(Pi−λi)−2λi)

2PiZi

.(A.18)

Adding unavailable storage space dedicated to the aisles gives the total unavailable storage space in the

warehouse

¯

U=AShSwna+Sh

2X

i∈I

xi+X

i∈I

Sh(Qi(Pi−λi)−2λi)

2PiZi

.(A.19)

The variable part of (A.19) is equivalent to the variable part of (2). Therefore, assuming a common bay

depth, n∗

aminimizes (A.19).

28