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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
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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.
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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
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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.
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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
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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.
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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