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Citation: Wu, G.-H.; Cheng, C.-Y.;
Liu, M.-H. Two-Stage Metaheuristic
Algorithms for Order-Batching and
Routing Problems. Appl. Sci. 2022,12,
10921. https://doi.org/10.3390/
app122110921
Academic Editors: Krzysztof Stepien
and Numan M. Durakbasa
Received: 9 September 2022
Accepted: 25 October 2022
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applied
sciences
Article
Two-Stage Metaheuristic Algorithms for Order-Batching and
Routing Problems
Gen-Han Wu 1,* , Chen-Yang Cheng 2and Ming-Hong Liu 3
1Department of Industrial Engineering and Management, Yuan Ze University, Taoyuan 32003, Taiwan
2Department of Industrial Engineering and Management, National Taipei University of Technology,
Taipei 10608, Taiwan
3Graduate Institute of Logistics Management, Dong Hwa University, Hualien 974301, Taiwan
*Correspondence: genhanwu@saturn.yzu.edu.tw
Abstract:
Because of time and cost constraints, item picking plays a major role in warehouse opera-
tions. Considering diversified orders and a constant warehouse design, deciding how to combine
each batch and picker route effectively is a challenge in warehouse management. In this study, we
focus on the evaluation of order-batching strategies for a single picker facing multiple orders with
the objective of minimizing the total traveling distance. We propose two-stage simulated annealing
and variable neighborhood search algorithms to solve the combined problem. The orders are first
merged into batches, followed by determining the sequence in each batch. The computational analy-
sis revealed that the best-fit-decreasing (BFD) batch ordering strategy in the two-stage algorithms,
the variable neighborhood search algorithm, obtained superior solutions to those of the simulated
annealing algorithm.
Keywords:
order batching; orderpicking; warehouse; simulated annealing; variable neighborhood search
1. Introduction
Achieving high efficiency in order batching and picking operations has become highly
crucial. To achieve low-cost and large-quantity throughput with fast delivery, batching
and picking operations are the most effective picking activities. Picking time constitutes
approximately 30–40% of the total working hours, and labor cost constitutes 15–20% of the
total cost; moreover, traveling and picking time form 60% and 20%, respectively, of the total
picking time [
1
]. Consequently, effectively reducing operating costs necessitates focusing
on enhancing the efficiency of traveling and picking procedures.
Warehouse management involves several categories, including storage planning,
order batching, order picking, warehouse layout design, and storage policies [
2
]. The major
challenge in warehouse management is the planning required to reduce the time, cost,
and distance in picking operations. Regarding batching and picking problems, the major
objective of this study is to effectively obtain an optimal solution within a limited time frame.
We adopted the model of batching and picking problems proposed by
Kulak et al. [3]
as the
study background and proposed five batch ordering strategies with two-stage simulated
annealing (SA) and variable neighborhood search (VNS) algorithms to improve the solution.
Finally, the results were compared with the same questions to determine which batch
ordering strategy can quickly and effectively obtain solutions.
The rest of this paper is organized as follows. Section 2describes related work, and
Section 3introduces the integrated order-batching and routing problem, and presents the
formulation of a mixed-integer optimization model. The two-stage SA and VNS algorithms
proposed for deriving near-optimal solutions are presented in Section 4. Section 5presents
the performance evaluation of the proposed algorithms, conducted through computa-
tional analysis. The final section provides the conclusion and potential applications of the
proposed algorithms.
Appl. Sci. 2022,12, 10921. https://doi.org/10.3390/app122110921 https://www.mdpi.com/journal/applsci
Appl. Sci. 2022,12, 10921 2 of 18
2. Literature Review
In the literature, most studies on warehouse order picking have focused only on picker
routing or order batching. Articles that only consider order picking can be classified into
certain [
4
–
17
] and uncertain cases [
18
,
19
]. Picker routing can be considered a traditional
traveling-salesman problem (TSP). Through the distance between storage spaces, pickers
plan how to visit each storage space through the shortest path before returning to the
starting point. However, when the number of storage spaces increases, the computation
time required to obtain an optimal solution is long [
20
]. Among the certain cases, many
studies have considered the operating environment of AS/RS [
4
,
5
,
10
–
14
,
17
]. Storage
systems, capacity, and strategies were introduced in [
14
,
17
]. Zone picking was used as the
picking method in [
8
,
15
,
16
]. Petersen [
21
] evaluated the returning characteristic in routing
policies to ensure that pickers enter an aisle to collect the required items, exit through the
path they entered, and then, travel to subsequent aisles in succession. Regarding algorithm
design, the genetic algorithm (GA) can be seen in [9,15].
In the area of order batching, most of the papers are certain cases [
22
–
34
]. Uncertain
issues can only be seen in [
35
,
36
]. Hwang and Kim [
25
] developed an order-batching
algorithm that is based on cluster analysis by considering three outing policies, namely
traversal, return, and midpoint routing. Moreover, De Koster et al. [
37
] applied the midpoint
method in which pickers enter from one end of an aisle to pick goods, and then, turn to
another aisle in succession as each aisle is completed. Henn and Wäscher [
38
] resolved the
question of how to combine customer orders to shorten the total picking distance. They
suggested two metaheuristic algorithms, namely a classic tabu search (TS) algorithm and
an attribute-based hill-climbing approach, and both approaches were superior to existing
methods. Hwang et al. [
39
] solved batch order processing as a clustering problem. They
defined items as attribute vectors and used three similarity measures to develop six cluster
classification algorithms. The experimental results proved that merging orders with higher
degrees of similarity improved the picking efficiency. The heterogeneous pick devices [
34
]
and multiple pickers [
36
] were extended to the batching problem. Regarding the algorithm
design, GA [28,31,33] and VNS [29,34,36] were widely introduced.
Among the integration issues considered together with order picking and order batch-
ing, we can see that most of them are certain issues [
3
,
40
–
50
] and uncertain issues can be
found in [
51
,
52
]. The concept of storage systems, capacity, and strategy was introduced
in [45,51]. Yu and Koster [52] used AS/RS as the operating environment.
Regarding the algorithm design, Won and Olafasson [
40
] solved the order-batching
and picking-sequence problem by using hybrid heuristic algorithms. The first heuris-
tic determines order batching, and the second heuristic identifies the picking sequence.
Kulak et al. al. [3]
and Henn [
44
] used TS to deal with the problem of order batching, while
on the issue of picking, Kulak et al. [
3
] adopted a 2-opt method, and Henn [
44
] proposed
a picking strategy. Tsai et al. [
53
] employed a two-stage GA to solve an order-batching
and picking-sequence problem. The objective was to minimize the total traveling time.
The first stage of the GA involved determining the order batches, and the second stage
entailed determining the shortest picking paths for these batches. Lin et al. [
48
] considered
a picking route based on the Manhattan distance and used particle swarm optimization
(PSO) to solve the joint order-batching and picking-route problem. Cheng et al. [
47
] used
PSO to deal with order batching and used ACO to find the most efficient route for each
batch.
Van Gils [49]
set high customer service level and improved picking efficiency as its
goals and developed an iterated local search algorithm to solve.
3. Problem Statement
This study deals with the order-batching problem and picking-route-sequence prob-
lem in the picker-to-part system; that is, in the conventional rectangular warehouse, the
different orders are first classified and sorted into batches, with group orders with similar
paths grouped together as much as possible. For the items to be picked in order,
Ok
,
i
,
BOl
, and
DOk/BOl
represent the
k
-th order, storage location
i
,
l
-th batch, and the picking
Appl. Sci. 2022,12, 10921 3 of 18
distance of order kor batch
l
, respectively. mrepresents the distance unit.
O1
={3, 6, 8},
indicating that order 1 needs to be picked from storage positions 3, 6, and 8. If there are
five orders,
O1= {3, 6, 8}
;
O2
=
{62, 74, 80}
;
O3
= {2, 3, 4, 6, 7};
O4
= {67, 74, 80}; and
O5
= {61, 62, 76, 80} are to be classified into batches, and then, the distance of the orders
individually calculated. As shown in Figures 1and 2, if we pick the items by following the
number sequence, in order 1, it takes 4 m to travel from the entrance to position 3, 3 m from
position 3 to 6, 2 m from position 6 to 8, and 9 m from position 8 to the entrance and exit.
Hence, the distances of order 1 to order 5,
DO1∼DO5
, are 18 m, 40 m, 16 m, 46 m, and
40 m. Since order 1 and order 3 are both picked in the first aisle, they can be grouped into
batch 1,
BO1={2, 3, 4, 6, 7, 8}
. The distance of batch 1,
DBO1
, is 18 m. Without order
batching, the total distance of these two orders is 34 m. Orders 2, 4, and 5 can be classified
into batch 2,
BOj={61, 62, 67, 74, 76, 80}
, based on a similar situation. If the items are
picked by individual order, the total distance of these three orders is 126 m, but the distance
of batch 2, DBO2, is largely decreased to 46 m.
Figure 1. Calculation Representation of the Distance.
Figure 2. Conventional Warehouse Layout.
Appl. Sci. 2022,12, 10921 4 of 18
After combining the orders into batches, the picker picks items in sequence according
to the sorted picking list of the batch, but if the picking route is different, the picker might
go as far as possible in the warehouse. For example, {37,64,21,19,43} and {19,37,21,43,64}
are picking routes 1 and 2 of the order. The distances of picking routes 1 and 2 are 68 m
and 42 m, respectively.
We adopted the model proposed by Kulak et al. [
3
] to address the batching and picking
problems and minimize the overall route. The problem assumptions, notations, and model
are as follows.
Problem Assumptions:
1. Warehouses have a picker-to-part system and parallel aisles.
2.
The storage strategy is dedicated location, so each storage position has a fixed size.
Only one item can be stored or picked, and the quantity of the same item being picked
is fixed at one.
3.
There is only one picker; that is, there is only one picker on the aisle, and the picker
will take the path with the shortest distance and will not choose the path that bypasses
the aisle.
4.
It is prohibited to split the order into different batches, and the items in the order
cannot be cut into different batches. The size of one order cannot be over the size of
one batch.
5.
The picker can pass up and down the aisle and pick the item on the left or the right.
The picker picks the item in front of the storage position, and the distance and time
from the picker to the item can be ignored.
6.
The warehouse layout plan shown in Figure 2is a conventional rectangular warehouse,
starting from and finally returning to the I/O point to complete the closed loop. There
are 10 storage spaces in a row—two rows on the left and right in one aisle, and five
aisles, with a total of 100 storage positions.
7.
All customers are known in advance, and no orders will be inserted, reduced, or
changed in the middle of the process.
Notations:
b∈B: number of batches
k∈K: number of orders
i,j∈V: number of storages
S⊂V: subset of storage
C: picker capacity
wk: weight of order k
dij : distance from storage ito storage j
sik =1, item in the order klocated at storage i
0, otherwise
Xb
k=1, order kassigned to batch b
0, otherwise
Yb
ij =1, batch btraveled from storage jto storage i
0, otherwise
Zb
i=1, batch bvisit storage i
0, otherwise
minZ=∑b∈B∑i6=j∈VdijYb
ij (1)
Subject to:
∑j∈V,j6=iYb
ij =Zb
i∀b∈B,i∈V(2)
∑i∈V,i6=jYb
ij =Zb
j∀b∈B,j∈V(3)
∑i∈S,j∈V\SYb
ij ≥Zb
i∀b∈B,S⊂V(4)
Zb
i≥sik ·Xb
k∀b∈B,i∈V k ∈K(5)
Appl. Sci. 2022,12, 10921 5 of 18
∑b∈BXb
k=1∀k∈K(6)
∑j∈V,j6=iwk·Xb
k≤C∀b∈B(7)
Xb
k,Yb
ij ,Zb
i∈0, 1 , ∀i,j∈V,b∈B,k∈K(8)
Objective (1) represents the objective function of the total picker route distance of
all batches. Constraints (2) and (3) indicate that in order to avoid repetitive calculations,
storage ican serve as the depot and destination only once. In addition, the constraints
can be used to express whether the batches in storage have been picked. Constraint (4)
represents a subcycle-eliminating constraint that prevents the formation of subcycles. The
limiting function ensures that all picking points are picked and that each storage unit that
requires picking is picked only once. Constraint (5) forces the selected orders and items
in a batch to transform into binary variables. Constraint (6) reveals that an order can be
assigned to only one batch to prevent such an order from being repetitively assigned to
various batches. Constraint (7) indicates the capacity of the picking vehicle, signifying
that the batch weight after order consolidation cannot exceed the maximum vehicle load.
Constraint (8) indicates that all variables are binary.
4. Proposed Algorithm
The concept of the proposed algorithm is to decompose the order-batching and picking-
route problem into two independent algorithms. First, in the first algorithm (order batch-
ing), the encoding solution, obtained from the initial batch order or the order-batching
neighborhood (mutation/swap), is the input of the second algorithm (picking route). In the
second algorithm, the initial picking route is obtained from storage locations with lower
numbers to those with high numbers. The picking-route neighborhood (swap/insert) is
used to obtain the best route of the current batches. Then, the distance of the best route is
the objective value (distance) of the first algorithm. The whole framework of the proposed
algorithm can be seen in Figure 3.
Figure 3. The framework of the proposed algorithm.
4.1. Initial Batch Order
Through order consolidation, each reduced batch can shorten the picking route of an
order. Consequently, the only factor considered in determining the initial batch is capacity
restriction. Moreover, we excluded route proximity from the criteria used to determine
Appl. Sci. 2022,12, 10921 6 of 18
the initial batch. We considered five approaches for obtaining the initial solutions of the
batching problem, and among these approaches, the picking route was not considered.
The single-order (SO) method differs from the other methods, and this method involves
considering one order as one picking batch. The random batch (RB) method was adopted
to enable comparison between random and planned operations.
Gupta and Ho [
54
] focused on a one-dimensional bin-packing problem and proposed
three evaluation methods to minimize the number of batches: the first-fit-decreasing (FFD),
best-fit (BF), and best-fit-decreasing (BFD) approaches. Because the bin-packing problem
is similar to the order-batching problem, we thus adopted these three approaches in this
study. The FFD method is applied to address weight limitations. Because of the capacity
restriction for each batch, on the basis of the FFD method, orders with heavy weights
should be assigned first; however, when such orders are assigned later, they generate a
new batch, thus increasing the total number of batches and adding another picking route.
The batches can be determined on the basis of the principle of determining the average
batch weight (i.e., the BF method). We eventually combined the FFD and BF methods
and denoted the integrated method as BFD. In this hybrid method, orders are assigned
initially according to the FFD method and their weight, and they are subsequently assigned
according to the BF method.
A.
SO method
The SO method was adopted to enable comparison of the results before and after
batch consolidation. In contrast to batching methods, which are used to consolidate several
orders into a single order, the SO method facilitates arranging the order-picking sequence.
Each order comprises a batch, and dissimilar orders are categorized into different batches.
B. RB method
The RB method involves randomly generating each order’s assigned batch. The orders
are randomly selected from the pool of orders and assigned to a batch individually. When
the batch to which an order is assigned cannot accommodate the weight of the order, the
system reassigns another batch. The process continues until all orders are assigned.
C.
FFD method
Compared with light-weight orders, heavy-weight orders tend to be more difficult to
assign to batches. Therefore, the system starts assigning batches from the heaviest orders. In
addition, each batch load is expected to reach the maximum load. The orders are assigned
to batches sequentially according to their weight, from the heaviest to the lightest. The
assigning principle involves minimizing the remaining weight of each batch. When a batch
is fully loaded, the order is reallocated to a new batch.
D.
BF method
Maintaining a balanced batch weight facilitates neighboring solutions to jump from
the local optima when the swap move operator is used. The assignment starts from the
heaviest order, and an order is assigned to the batch with the greatest remaining weight.
In other words, when there are several batches to select from, the system selects the batch
with the lowest remaining capacity (batch total capacity–batch present capacity) after an
order is assigned to it. Specifically, an order is assigned to the least full batch, and when the
batch cannot contain the order, the order is reassigned to a new batch, thus balancing the
weight among batches.
E. BFD method
This method is a combination of the BF and FFD methods. Orders are first prioritized
using the FFD approach and are subsequently assigned to batches through the BF method.
Orders are assigned sequentially from the heaviest to the lightest batch, and during the
assignment, the guiding principle involves minimizing the remaining weight of a batch.
Specifically, when there are several batches to select from, the system selects the batch
with the lowest remaining capacity (batch total capacity–batch present capacity) after an
Appl. Sci. 2022,12, 10921 7 of 18
order is assigned to it. When a batch cannot contain the order, the order is reassigned to a
new batch.
4.2. Initial Picking Route
Because the number of picking items is arranged according to orders, the optimal
order-picking route is constructed on the basis of the number of picking items. In the same
batch, the initial picking route moves from locations with lower numbers to those with high
numbers. The relative shortest distance matrix of these locations can be easily obtained
because in warehouses, storage is arranged from left to right and from bottom to top.
4.3. Encoding
In this study, we encode the batch numbers and number of orders that must be picked.
The batch number of orders is obtained from the phase of the initial batch order, and the
total weight must remain within the vehicle capacity. The number of order-picking items
represents the items from which orders must be picked in a batch.
4.4. Neighborhood Searching
To minimize the number of batches and shorten the total picking-route distance, we
used neighborhood searching to determine the optimal neighboring solution for batching
and picking problems. These two problems require different neighboring solution designs.
First, to effectively reduce the number of batches, a mutation procedure is applied. For
example, when a mutated order is the only order in a batch, the mutation of the order
from the original batch to another batch can reduce the batch by one. In addition, a swap
procedure changes the existing combination of batches in pairs according to a fixed total
number of batches. Second, according to existing picking solutions, we designed a swap
procedure to improve picking routes to a small extent. Moreover, to increase the extent of
improvement, the insertion procedure is adopted for generating neighboring solutions by
inserting one order into another and, hence, moving the remaining items in the order by
one place.
A.
Order Batching
Using the existing number corresponding to each order to conduct neighborhood
searching can reveal whether more effective solutions are available. Mutation and swap
procedures can be applied in the search process; adopting mutation procedures can effec-
tively reduce the total number of batches. The original batch can be emptied, and thus,
eliminated by mutating batches that exhibit only one order to another batch. In contrast to
the mutation procedure, which is used to reconstruct overall batches, the swap procedure
improves the batch arrangement and is the most commonly applied method in neighbor-
hood design. The swap procedure is a type of optimization procedure, and it cannot be
used to reduce the number of batches; however, it facilitates reassigning the orders to
optimal batches. When the aforementioned two methods are used, vehicle capacity must
be considered during neighborhood search processes.
B. Order Picking
When the picking route is arranged, items should be collected according to prioritized
orders. Each batch has a specific picking order, and minor changes can be implemented
on the basis of the existing picking order. Order optimization can be performed within
batches by assessing whether the changed orders are more favorable than the original
orders. Therefore, the shortest picking route can be obtained by conducting neighborhood
searches on the existing picking solutions. The neighboring solutions can be acquired
through swapping, insertion, and random-selection procedures. Because the number of
items remaining to be picked is fixed, new combinations must be sequenced. In design-
ing neighboring solutions, a two-point swap is most commonly used and an insertion
procedure that remains farther from the neighborhood can be used to obtain more varied
neighboring results.
Appl. Sci. 2022,12, 10921 8 of 18
4.5. SA Algorithm
Because this study involved two problems, a single-solution metaheuristic is more
suitable for solving the problems compared with a population-based metaheuristic. If a
population-based metaheuristic is adopted, then multiple solutions are generated at the
first stage, and each solution results in the generation of several solutions at the second
stage, thereby reducing the efficiency of deriving solutions. In addition, the SA algorithm
involves a random mechanism; hence, it was applied for solving the problems in the
two stages.
The SA algorithm can be applied to solve both TSPs and batching and packing prob-
lems. Therefore, we initially combined two types of SA. First, the framework of the
first stage was used when solving the batches through the SA algorithm. Each time the
neighboring solution was substituted, the SA algorithm applied to the picking routes was re-
considered. The distance of the picking routes must be determined to yield the neighboring
solution of the batch. The following parameter values must be established first:
(1)
Starting temperature (
T0
): Temperature setting is crucial in the SA algorithm. If
the value is excessively high, then the convergence is slow; however, if the value is
excessively low, then the solution quality becomes unsatisfactory.
(2)
Maximum chain length (
Lmax
): The chain length is determined to control the search
frequency at each temperature. If the length is too short, then the execution time can
be shortened; however, the quality of solutions cannot be enhanced. If the length is
too long, then the eventual quality and effect of the solution increase. However, the
additional execution time reduces the efficiency of obtaining solutions. The maximum
batching and picking chain lengths are set as LLmax and Lm ax , respectively.
(3)
Annealing parameter (
β
): The variable
s
represents the annealing frequency and
β
represents a parameter whose value is set at 1; moreover, T(s+1)=T/(1+βT(s)).
(4)
Stopping criterion: To avoid falling into infinite loops, a terminal temperature and
computation time are set.
The SA algorithm steps are described as follows:
Step 1: Set the starting temperature as
T0
, the number of maximum order items as
itemmax
, number of minimum order items as
itemmin
, the Boltzmann constant as
K
, the
number of orders as
O
, the maximum location of an item as
Imax
, the minimum location of
an item as
Imin
, the maximum capacity in terms of weight as
capmax
, the minimum capacity
as
capmin
, the annealing parameter as
β
, the maximum picking chain length as
Lmax
, the
maximum batching chain length as
LLmax
, and the starting chain length as
L=
0 and
LL =0.
Step 2: For each order, randomly generate the weight of the order
OW
, and set the
order item numbers as OI and the item location as OL.
Step 3: Assign orders to batches according to the order sequence; specifically, the order
number corresponds to the batch number. Assume that the results acquired are from the
initial batching solution. Moreover, assign
OW
,
OI
, and
OL
to the corresponding batches,
and set the initial batching solution as the current batching solution Bcurr .
Step 4: Using the
OL
sequence as the picking order of the initial batching solution,
start from the minimum and proceed to the maximum. In addition, set the current picking
order as
Pcurr
, calculate the picking distance of each batch
dM
, and sum all batch distances
to obtain the current total distance
Dcurr
. Set the optimal total distance as
Dbest
=
Dcurr
,
the optimal picking order as
Pbest
=
Pcurr
, and the optimal batch as
Bbest
=
Bcurr
, and then,
proceed to Step 6.
Step 5: In
Bcurr
, randomly select the swap or mutation procedure to generate the
neighboring batching solution Bnei gh, and then, proceed to Step 6.
Step 6: In the current picking order, randomly choose the swap or insertion procedure
to generate neighboring solutions for the picking order Pneigh.
Step 7: Calculate the picking distance of each batch according to
Pnei gh
, and sum the
distances of all batches to obtain the total distance of the neighboring picking order
Dneig h
.
Appl. Sci. 2022,12, 10921 9 of 18
Step 8: Calculate the objective value deviation ∆D=Dneigh −Dcurr .
Step 9: Determine whether
∆D<
0. If yes, proceed to Step 10; otherwise, proceed to
Step 13.
Step 10: Substitute Pcurr with Pneigh , and then, proceed to Step 11.
Step 11: When the solutions are substituted, reset the picking chain length as
L=L+
1
and proceed to Step 12.
Step 12: If
Dneig h ≥Dbest
, then proceed to Step 15; otherwise, substitute
Pbest
with
Pcurr
.
Step 13: According to the uniform (0,1), generate a random probability value
U
and
proceed to Step 14.
Step 14: Determine whether
U<exp−∆D
KT
. If yes, return to Step 10; otherwise,
proceed to Step 15.
Step 15: Determine whether
L
is
Lmax
. If yes, then perform Step 16; otherwise, return
to Step 6.
Step 16: Execute the annealing procedure. Set the picking temperature as
T=T/(1+β×T)
and the picking chain length as L=0; subsequently, proceed to Step 17.
Step 17: Determine whether the picking temperature
T<Tb
. If yes, then export
Dneig h
and proceed to Step 18; otherwise, return to Step 7.
Step 18: Calculate the objective value deviation ∆D=Dneigh −Dcurr .
Step 19: Determine whether
∆D<
0. If yes, proceed to Step 22; otherwise, proceed to
Step 20.
Step 20: According to the uniform (0, 1), generate a random probability value U.
Step 21: Determine whether
U<exp−∆D
KT
. If yes, then proceed to Step 22; otherwise,
return to Step 5.
Step 22: Substitute Bcurr with Bneigh, and then, proceed to Step 23.
Step 23: When the solutions are substituted, reset the batching chain length as
LL =LL + 1 and proceed to Step 24.
Step 24: If
Dneig h >Dbest
, then substitute
Bbest
with
Bneigh
and proceed to Step 25.
Otherwise, no substitution procedure is required; proceed to Step 25.
Step 25: Determine whether the batching chain length
LL >Lmax
. If yes, perform
Step 26; otherwise, return to Step 6.
Step 26: Perform the annealing procedure. Set the batching temperature as
TT =TT/(1+β×TT)and the batching chain length as LL =0 and proceed to Step 27.
Step 27: Determine whether the batching terminal temperature
TT
is lower than
Tb
or
the computation time is met. If yes, end the procedure; otherwise, return to Step 5.
4.6. VNS Algorithm
Compared with the SA algorithm, the VNS algorithm involves fewer parameters, thus
rendering it relatively easy to control during its application. In this algorithm, a problem is
divided into two stages: batching and picking. The picking distance must be recalculated
with every batching substitution. Hence, the process consistently proceeds to picking
insertion and swapping. Compared with swapping, insertion can be used to search for
solutions in a wider range; therefore, insertion was adopted as the first neighborhood
when searching neighboring picking routes. After completing the search process in the
first neighborhood, we proceeded to search the second neighborhood, adopting picking
swapping. The eventual export value of the picking distance obtained from a complete
search of all neighborhoods represents the objective value of batching neighboring solu-
tions. Next, we compared the search results with the current solution until all mutated
neighborhoods in the batch were searched. Subsequently, we proceeded with the search
process through batch swapping. Batch mutation was applied first because it can quickly
consolidate orders into batches. By contrast, batch swapping is less effective in reducing
the number of batches.
The VNS algorithm begins to solve the problem from the initial batching solution.
With each substitution of neighboring mutation or swapping solutions, the search process
must proceed to the neighboring picking solutions to calculate the objective distance value.
Appl. Sci. 2022,12, 10921 10 of 18
The search process begins with picking swapping, and all solutions are calculated and
compared with the optimal solutions. Subsequently, the search proceeds with picking
insertion; the solutions are also calculated and compared with the optimal solutions, and
an objective value representing the neighboring batching solution is exported. Batching
mutation is applied to ensure that the solutions change rapidly and to eliminate local
solutions. In comparison, the batch-swapping procedure is less effective.
The steps of the VNS algorithm are as follows:
Step 1: First, identify the objective function value, current optimal solution, current
solution, and number of iterations.
Step 2: Construct the initial batching, examine whether the construction corresponds to
the vehicle capacity. If the construction corresponds to the vehicle capacity, then construct an
initial picking route and proceed to the next step; otherwise, repeat the construction process.
Step 3: Assuming that the iteration number
t=t+
1, use mutation and swapping
procedures to improve the current batching solution. Next, examine whether the new
solution corresponds to the vehicle capacity. If yes, then the new solution is adopted as a
neighboring solution of this step; otherwise, return to Step 3. If the solution is improved,
then update the current solution
B
with
B_neigh
, recalculate
count =count +
1, and re-
execute Step 3; otherwise, proceed to Step 4.
Step 4: Adopt the insertion or swapping procedure to improve the current solution
P
and assess whether the new solution corresponds to the vehicle capacity. If yes, then the
new solution is adopted as a neighboring solution of this step; otherwise, return to Step 4.
If the solution is improved, then update the current solution Pwith P_neigh, recalculate
count =count + 1, and re-execute Step 4; otherwise, proceed to Step 5.
Step 5: Examine whether count = 0. If yes, then proceed to Step 6; otherwise, return to
Step 3.
Step 6: Conduct a perturbation process on the current solution. Specifically,
S_ratio * number of orders = number of perturbations
. Proceed to other neighborhoods,
apply mutation and swapping procedures to improve the current batching solution, and ex-
amine whether the current solution corresponds to vehicle capacity. If yes, then the solution
after the perturbation process is derived; otherwise, repeat the perturbation process.
Step 7: Examine whether the terminating condition of the solution searching time has
been satisfied. If yes, then terminate the algorithm; otherwise, return to Step 3.
5. Computational Experiment
The programs used in this experiment were run on a computer with an i7-3770
CPU@3.40-GHz and 20 GB RAM. Moreover, we adopted the programming language C to
compile the program. According to the aforementioned setting, an analysis was performed
to determine how each set of experimental factors influenced the computation time and
solution quality.
5.1. Parameter Settings
First, the number of orders was set at 10, 20, 40, and 80. Subsequently, the settings
were applied to eight scenarios, which are presented in Table 1. In all eight scenarios related
to the problem scale, for each order, the number of order items (1–5) and capacity of the
orders (1–3) were generated randomly according to a uniform distribution. In addition, the
vehicle capacity was associated with two variations: 10 and 20.
Appl. Sci. 2022,12, 10921 11 of 18
Table 1. Sample problems.
Scenario Number of Items Capacity of the Orders Vehicle Capacity
1 1~3 1~3 10
2 1~5 1~3 10
3 1~3 1~3 20
4 1~5 1~3 20
5 1~3 1~5 10
6 1~5 1~5 10
7 1~3 1~5 20
8 1~5 1~5 20
Each type of question was then matched with random seeds set at 10, 20, and 30.
The problem thus involved eight scenarios
×
four order numbers
×
three random seeds,
equaling 96 independent tests. In the algorithm, the terminating condition of the solution
searching time was 60 s.
5.2. Initial Batch Solution
As described in Section 4.1, five initial solutions were provided; according to the four
order types and eight scenarios, 32 cases were generated in Table 2. Each type of problem
was matched with random seeds set at 10, 20, and 30.
Table 2. Initial solutions.
Number of Orders 10 20 40 80
Scenarios 1~8 1~8 1~8 1~8
Cases 1~8 9~16 17~24 25~32
These five initial solutions were tested in 32 cases (Figure 4). According to the mean
value, the SO and RB methods yielded fewer effective solutions compared with the other
methods. By contrast, the FFD, BF, and BFD methods yielded effective solutions. Figure 4
indicates that the objective value tended to increase from order numbers 10, 20, and 40 to 80
and that the objective values associated with the same order number differed according to
various scenarios, thus explaining the scattered points illustrated in this figure. In general,
the solutions derived using the BF method were less effective than those obtained using
the BFD method, and when the order was small, the FFD method yielded more effective
solutions than did the BFD method.
Figure 4. Comparison of five initial solutions.
Appl. Sci. 2022,12, 10921 12 of 18
The solutions obtained using the FFD, BF, and BFD methods according to the order
size are presented in Table 3. As shown in the table, all the FFD, BF, and BFD methods
returned valuable solutions, with the differences among the solutions being minor. The BFD
method tended to generate the minimum objective value and minimum mean objective
value, followed by the FFD, and then, the BF methods, which yielded the least effective
solutions. The blue shadowed cells show the best results among these three methods.
Table 3.
Comparison of initial objective values obtained using FFD, BF, and BFD methods (unit: m).
Initial Batch
Solution # FFD BF BFD
Scenario # Number of Orders Number of Orders Number of Orders
10 20 40 80 10 20 40 80 10 20 40 80
1 152 341 685 1311 183 325 665 1339 174 332 660 1330
2 213 415 868 1769 216 397 907 1797 206 420 829 1746
3 125 290 554 1066 126 263 526 1082 126 278 547 1117
4 199 399 788 1575 205 363 845 1615 205 362 749 1645
5 193 383 853 1715 205 417 861 1722 217 396 815 1574
6 235 477 1037 2047 251 492 1013 2051 237 457 994 1953
7 151 295 609 1185 159 297 591 1165 147 298 586 1209
8 213 403 865 1667 201 373 810 1723 195 359 855 1713
Table 4shows the mean value and standard deviation of the initial solution approaches.
Because the BFD method yielded the minimum mean value, it was used to test hypotheses
involving other initial solutions. Table 5presents the test statistics. The mean objective
values obtained using the SO and RB methods differed significantly from those obtained
using the BFD method, and the Z value exceeded 1.645. However, the difference between
the mean values obtained using the FFD and BF methods and those derived using the BFD
method was not significant; because 0.08 and 0.1 are less than 1.645, these values do not
belong to the reject region, and H0was thus not rejected.
Table 4. Mean value and standard deviation of initial solutions (unit: m).
SO FFD RB BF BFD
Mean 1603.9 721.1 1197.0 724.6 710.0
Std. 1202.1 557.8 906.5 566.4 542.1
Table 5. Test statistics.
(SO, BFD) (FFD, BFD) (RB, BFD) (BF, BFD)
Z value 3.83 0.08 2.6 0.1
5.3. Algorithm Parameters
Regarding the VNS algorithm, the perturbation frequency equaled the perturbation
factors, and the perturbation frequency was derived as follows: perturbation factor
×
order
number. A test was performed in all scenarios involving 40 orders and various perturbation
factors (Table 6). Regarding the SA algorithm, the starting temperature was set at 10,000
◦
C,
the annealing parameter was 0.001, and the unimproved number of iterations was 100. A
test was performed in all scenarios involving 40 orders and different
Lmax
values (Table 6).
Appl. Sci. 2022,12, 10921 13 of 18
Table 6. Mean objective value of Lma x associated with 40 orders (unit: m).
Scenario Number of Items Capacity of the Orders Vehicle Capacity Objective Value When
Lmax = 10
Objective Value When
Lmax = 15
1 1~3 1~3 10 524.7 544.0
2 1~5 1~3 10 717.3 726.7
3 1~3 1~3 20 416.0 433.3
4 1~5 1~3 20 566.0 584.0
5 1~3 1~5 10 670.0 680.7
6 1~5 1~5 10 808.7 838.0
7 1~3 1~5 20 482.7 487.3
8 1~5 1~5 20 616.0 655.3
First, we focused on two parent groups—those involving chains with lengths of 10
and 15—to test whether the variance was the same.
H0
:
σ1=σ2
;
H1
:
σ16=σ2
; mean value
XL10 =
600.2;
XL15 =
618.7; standard deviation
S2
L10 =
16, 702.9;
S2
L15 =
17, 618.7; rejection
region
C=F0.025(7, 7)or
1
/F0.025(7, 7)
; and F= 0.948
/∈
C; moreover,
H0
:
σ2
10 =σ2
15
.
Subsequently, we continued to test the average of the two parent groups,
H0
:
µ1−µ2≤
0
and
H1
:
µ1−µ2>
0, where
µ1
represents the average of a chain with a length of 10,
µ2
is the average of a chain with a length of 15, and t= 0.291, C{t|t>t
0.05
(14) = 1.761}, and
t/∈C
. Hence,
H0
was not rejected, and consequently, the average influence of the chain
with Lmax = 10 on the objective value was nonsignificant.
The test results revealed that the chain length influenced the objective value of the
solution to only a small degree and that no significant difference in the solving speed
existed under the various perturbation factors. Moreover, Table 6demonstrates no apparent
significant difference. Consequently, we adopted Lm ax =10 as the parameter setting.
When the perturbation factors were 0.2 and 0.4 (Table 7), the factors demonstrated
the same objective value and did not influence the solving value; moreover, the solving
speed did not differ notably. However, in the example of scenario 5, a relative difference
was apparent.
Table 7. Average computation time under different perturbation factors (unit: sec).
Scenario Number of Items Capacity of
the Orders Vehicle Capacity Perturbation Factor = 0.2
(Objective Value/Solving Time)
Perturbation Factor = 0.4
(Objective Value/Solving Time)
1 1~3 1~3 10 462.7 0.44 462.7 0.43
2 1~5 1~3 10 544.7 0.80 544.7 0.84
3 1~3 1~3 20 320.7 0.58 320.7 0.58
4 1~5 1~3 20 380.7 0.84 380.7 0.79
5 1~3 1~5 10 615.3 0.51 615.3 0.63
6 1~5 1~5 10 702.0 0.59 702.0 0.59
7 1~3 1~5 20 408.0 0.47 408.0 0.51
8 1~5 1~5 20 448.0 1.05 448.0 1.08
Regardless of whether the perturbation factor was 0.2 or 0.4, the objective value was
the same. Therefore, we set the parameter of the perturbation factor at 0.2, a value that
reduced the computation time. Moreover, because a few parameters in each scenario, such
as order item, order capacity, and vehicle capacity, were generated randomly through a
uniform distribution, merely comparing the influences and correlations of these items can
hardly yield meaningful results.
5.4. Evaluation Analysis
This experiment was conducted to understand the solving capability and solution
quality of the algorithms applied to medium- and large-scale problems. According to
the types of initial solution, two types of algorithm were applied. This section presents
a comparison of the effectiveness of the VNS and SA algorithms in solving BFD initial
solutions derived with 40 and 80 orders, respectively.
Appl. Sci. 2022,12, 10921 14 of 18
As shown in Figure 5, the computation time trend associated with the 40 orders
revealed that the VNS algorithm converged within 1 s and quickly achieved stability,
whereas the SA algorithm demonstrated a marked convergence tendency within 5 s and
slower converging processes afterward. Regarding the solving trend associated with
the 80 orders, the VNS algorithm converged within 1 s; by contrast, the SA algorithm
converged within 4–5 and 18–20 s. Therefore, the SA algorithm converged more slowly,
and its objective value was less favorable than that of the VNS algorithm.
Figure 5. Computation time trends of 40 and 80 orders.
The SA algorithm converged quickly when the number of orders was low. The higher
the order number was, the slower the solving process achieved stability; specifically, the
convergence proceeded rather slowly. Next, we conducted another comparison between
the VNS and SA algorithms by using the various BFD initial solutions obtained for the 40
and 80 orders in each of the eight scenarios. Seeds 10, 20, and 30 were also used in the
test. The objective value used in this study was the optimal picking distance. We assumed
that the objective value was BV, the optimal mean value of the other algorithm was PV,
and the error ratio formula was
(PV−BV )
BV
. Figure 6illustrates a comparison of the SA and
VND algorithms, revealing that the difference between these algorithms was lower in
Scenarios 1 and 5. For the 80 orders, Scenario 5 exhibited the lowest error ratio, followed
by Scenario 1. Moreover, as shown in Figure 7, Scenarios 4 and 8 demonstrated high
error ratios. According to the aforementioned experimental results, we drew the following
conclusion: In batching and picking problems, the VNS algorithm is more favorable than
the SA algorithm for medium- and small-scale problems. Moreover, for medium- and
large-scale problems, the VNS algorithm demonstrates higher solution quality and a faster
solving speed than does the SA algorithm. In the solving process, the VNS algorithm
can achieve convergence rapidly. In addition, when the problem becomes more complex,
the advantages of the VNS algorithm become more salient, and this is attributable to the
algorithm’s rapid convergence when solving problems and its capability to determine
batches and picking routes swiftly. Therefore, this study confirms that the VNS algorithm
is the optimal method. In Scenario 4, the order capacity was small and the vehicle capacity
was large, enabling each batch to be filled with a higher number of orders and each order
to have a higher number of items. The error ratio of each algorithm differed significantly,
and this is possibly because the quality of the picking-route solution of the SA algorithm is
lower than that of the VNS algorithm. Scenario 5 was not consistent with Scenario 4: In
Scenario 5, because of the small vehicle capacity, fewer orders were consolidated, causing
Appl. Sci. 2022,12, 10921 15 of 18
fewer items to be listed in the orders. Consequently, after order consolidation, the items that
remained to be picked in one batch decreased, as did the number of solutions; therefore,
the error ratio of the objective value decreased.
Figure 6. Error ratio of objective value associated with 40 orders.
Figure 7. Error ratio of objective value associated with 80 orders.
6. Conclusions
In this study, we investigated how warehouses can effectively divide received orders
into batches, obtain efficient picking routes for each batch to achieve the shortest total
distance, and consequently, reduce costs and enhance competitiveness. In small-scale
problems, the exhaustion method may be applied to determine the optimal solution, but
as the number of orders increases and the problems become more complex, an algorithm
becomes indispensable for efficiently obtaining solutions.
The basic limitation of this study is that the same order cannot be divided into different
batches, and the capacity of the picking vehicle will be greater than that of any single order;
therefore, this study is more applicable to retail items purchased by general consumers,
that is, small- and medium-sized online shopping retail warehouses or traditional retail
warehouses with a large number of items. Each order contains several items, and the
quantity of each item is only one to several pieces. Furthermore, all orders are known in
advance and order insertion/removal are not allowed. This means that the algorithm for
static problems is considered. This is mainly because most of the picking is a daily routine,
and therefore, in the general retail industry, there is rarely a need to adjust the order in time.
For this order-batching and picking-route problem, we mainly introduce two-stage SA
and VNS algorithms based on a set of neighborhood search algorithms. The concept is clear
and easy to understand, and the solution speed is excellent. In addition, for such research,
it is easy to extend to various single-solution-based algorithms such as tabu search and
iterated local search because the concept of the algorithm is clear. Moreover, with even a
little adjustment, it is easy to apply methods to swarm intelligence algorithms.
Five initial order-batching methods are proposed; we further tested these solutions
under eight scenarios and with three random seeds. The results revealed that the BFD
method yielded favorable results under all scenarios. In addition, the chain length and
perturbation factors of the solutions were considered. Finally, the SA and VNS algorithms
were incorporated into the solution comparisons. The algorithm processes were divided
into two stages: The first stage involved batching, and the second stage involved optimizing
the route. Under various scenarios, the VNS algorithm returned more favorable solutions
Appl. Sci. 2022,12, 10921 16 of 18
than did the SA algorithm; moreover, the VNS algorithm demonstrated a faster solving
speed. By contrast, the SA algorithm exhibited a slow solving speed in large-scale problems
and converged slowly.
This study takes the conventional rectangular warehouse layout as an example layout.
However, in practical applications, the layouts of almost all warehouses are different.
This does not affect the algorithm design of order batching, but only affects the optimal
picking distance between any two points in the order-picking-route problem. The distance
between any two points can be obtained by calculating the minimum distance between
any two points in advance, that is, it does not affect the implementation of the overall
algorithm. Therefore, in practical applications, for the retail industry, small- and medium-
sized online shopping retail warehouses or traditional retail warehouse can utilize this
two-stage algorithm for their daily routine picking jobs. As for the picking of large objects
in practice, the picking efficiency can be measured over time by increasing the picking time
of some picking points. The objective can be adjusted to the minimal picking time and the
concept of the whole two-stage algorithm can still be applied.
In further studies, this topic could be divided into three aspects to discuss. First, some
restrictions could be loosened in the problem. For example, the same order could be divided
into different batches, the picking time could be used to measure the picking efficiency, and
different types of picking container equipment could be considered. Secondly, more effec-
tive initial batching methods could be developed because a good initial batching solution
can save much computational time. Some metaheuristic algorithms newly developed in
recent years, such as the spotted hyena optimizer (SHO) and dandelion algorithm (DA),
could be developed for this problem because of their simple steps and fast computation.
Finally, more warehouse layouts could be covered in the study in order to test the pros and
cons of the proposed algorithm.
Author Contributions:
Conceptualization, G.-H.W.; methodology, G.-H.W. and M.-H.L.; software,
M.-H.L.; validation, G.-H.W. and M.-H.L.; formal analysis, G.-H.W.; investigation, G.-H.W. and
M.-H.L.; resources, G.-H.W.; data curation, M.-H.L.; writing—original draft preparation, C.-Y.C.;
writing—review and editing, G.-H.W. and C.-Y.C.; visualization, M.-H.L.; supervision, G.-H.W. All
authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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