A material handling scheduling method for mixed-model automotive assembly lines based on an improved static kitting strategy

Journal Pre-proofs A material handling scheduling method for mixed-model automotive assembly lines based on an improved static kitting strategy Binghai Zhou, Zhaoxu He PII: DOI: Reference:

S0360-8352(20)30002-4 https://doi.org/10.1016/j.cie.2020.106268 CAIE 106268

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Computers & Industrial Engineering

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12 February 2019 8 December 2019 3 January 2020

Please cite this article as: Zhou, B., He, Z., A material handling scheduling method for mixed-model automotive assembly lines based on an improved static kitting strategy, Computers & Industrial Engineering (2020), doi: https://doi.org/10.1016/j.cie.2020.106268

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A material handling scheduling method for mixed-model automotive assembly lines based on an improved static kitting strategy Binghai Zhou *, Zhaoxu He School of Mechanical Engineering, Tongji University, Shanghai 201804, PR China *Corresponding author. E-mail: [email protected]; Tel: +86 21 69589598. Present address: Caoan Road 4800, Mechanical Building A444, Shanghai City, PR China, Code: 201804

Binghai Zhou was born in 1965, in Zhejiang Province, China. He received his Master degree in Industrial Engineering from Shanghai Jiaotong University in 1989 and Doctor degree in 1992, respectively. He is currently a professor in the Mechanical Engineering school of Tongji University, Shanghai, China. His research interests cover scheduling and simulation of discrete systems (manufacturing/logistics Systems), preventive maintenance modeling of equipment and artificial intelligent algorithm.

Zhaoxu He was born in 1997, in Hebei Province, China. He received his Bachelor degree in Industrial Engineering from Tongji University, Shanghai, China. Currently, He is a senior student. His current research interests include evolutionary algorithm, manufacturing system modeling and simulation.

Acknowledgement: This work was partly supported by the National Natural Science Foundation of China (Grant No.71471135)

A material handling scheduling method for mixed-model automotive assembly lines based on an improved static kitting strategy Abstract: Since the diversification of customer demands poses a great challenge for manufacturing enterprises and the scheduling problem of material handling affects the efficiency of assembly lines, this paper proposes a novel scheduling method, an improved static kitting strategy, to solve the scheduling problems of the material handling for automotive mixed-model assembly lines (MMALs) based on lineintegrated supermarkets. Firstly, an integer programming mathematical model is established with the objective of minimizing the number of logistic workers. Then, an improved static kitting strategy is presented to solve the problem and a model based on graph theories is constructed to transform the scheduling problem to a mathematical one. Afterwards, a Kuhn-Munkres algorithm and an elite opposition-based learning adaptive dynamic differential evolution algorithm, named EOADDE algorithm, is developed to solve the scheduling problem. The elite opposition-based learning (EOL) and self-adaptive operators are applied to the proposed EOADDE algorithm to enhance the local search ability and the convergence speed. Finally, computational experiments of the proposed algorithm are carried out compared with benchmark algorithms, and the feasibility and effectiveness of proposed methods are verified by results. Key words: material handling; mixed-model assembly lines; improved static kitting strategy; line-integrated supermarkets; scheduling

1 Introduction Under the comprehensive influence of energy-saving, environmental protection requirements and highly-diversified consumer demands in the past few decades, the automobile industry has been facing serious challenges. Improving productivity and

reducing production costs have become important issues for enterprises. Consequently, mixed-model assembly lines (MMALs) have been widely employed by automobile manufacturers to satisfy customers’ diversified demands (Jainury et al. (2014)). However, the scheduling problem of material handling is a complicated mission. In many automobile manufacturers, the handling cost of in-house parts accounts for 15%30% of the total production cost. The operating cost of the manufacturing systems can be greatly reduced by optimizing the strategy of material handling (Kilic & Durmusoglu (2015)). Therefore, it is of great theoretical significance and practical value to investigate the scheduling problem of material handling for MMALs. In recent years, the scheduling problem of material handling for MMALs has attracted the attention of quite some researchers. Johansson et al. (2006) classified a strategy of material handling into batch supply, continuous supply and kitting. Battini et al. (2009) believed that the material handling strategy can be divided into pallets to workstations, trolleys to workstations and kits to assembly lines. Subsequently, by summarizing the previous studies, Caputo and Pelagagge (2011) divided into line storage, kitting and just-in-time kanban-based continuous supply strategies. Now, most of the literatures refer to two main delivery strategies, namely, line stocking and kitting (Hua & Johnson (2008)). In the former concept, a line stocking strategy, also named as continuous supply or bulk feeding (Limère et al. (2015)), indicates that homogenous parts are stored directly in large containers at assembly stations. Large containers are replenished by forklifts, trolleys etc. from a material warehouse. In this case, the major advantages are that there are free of double-handling operations and much flexibility in case of unexpected events (Bozer &McGinnis (1992)). When there is an alteration of material demands or an occurrence of a defective part, assembly workers can quickly withdraw spare parts. However, these material containers occupy the limited space of stations at the line (Limère (2012)). Also, assembly workers always need to retrieve and select required parts from multiple containers when assembling several models, which will aggravate the workload of workers and increase the assembly time. On the other hand, the kitting strategy can be classified into two types, namely the traveling kitting and the static kitting strategies. In the traveling

kitting strategy, kits are delivered to the first station and move together with assembling products (Caputo et al. (2015)). The static kitting strategy means that materials are sorted into small kits at the decentralized logistic areas which are loaded in the wagons of vehicles and delivered to each station. Thus, the kitting system can quickly respond to flexible demands in the assembly process compared to the line stocking strategy to fulfill the diversity of models to be assembled. Also, using small kits can reduce the line-side inventory in all stations. Nevertheless, Faccio (2014) pointed out that each part of kits is managed through an initial picking activity, which increases the handling costs, and picking a defective part or failure in preparation of kits will significantly affect the assembly process. Moreover, a new concept named the line-integrated supermarket was introduced by Boysen and Emde (2014) and successfully employed in a North-American plant of a German automobile enterprise. In this paper, a novel concept of the improved static kitting strategy is presented which unifies the advantages of kitting and line stocking strategies by applying the layout of line-integrated supermarkets. It integrates decentralized supermarkets of the kitting strategy into the assembly line and materials are supplied in bulk to every station by logistic workers, so called the improved static kitting strategy. Compared to traditional line stocking and kitting strategies, the strategy has the advantages of stable supply, saving labors and less inventory at the line. In the line-integrated supermarket, the storage area for materials to be assembled by the respective assembly workers is located directly next to the assembly line. Logistic containers named JIS-bins (Just-insequence bins) are used to accommodate required parts and to deliver them from the logistic area of a station to the moving conveyor. From the JIS-bin, an assembly worker withdraws the part required by the car having currently entered the station and assembles it during the production cycle. The JIS-bin is prepared by a group of logistic workers, who pick the parts according to the predefined production sequence and pack the picked parts into the JIS-bin. In this case, it is clear that the improved static kitting strategy makes do without vehicles to deliver kits to stations, so that it saves labors and decreases the amount of double handling compared to the traditional kitting strategy. Moreover, in case of unforeseen events, spare parts are readily available at

stations and the effort to replace the defective parts is reduced to a minimum. These properties make the improved static kitting strategy an attractive managerial concept for in-house material handling not only in the automotive industry but also in many other high-variety circumstances. The proposed novel delivery system of the improved static kitting strategy gives rise to some important decision problems to be solved. This paper focuses on the scheduling problem of the refilling of JIS-bins on the assembly line with the improved static kitting strategy. Given the number of stations and their deterministic material demands over time, a schedule for the refilling of JIS-bins should be completed to ensure that there are no stock-outs and the number of logistic workers should be minimized on the premise of a certain number of stations. To cope with the scheduling problem of the material handling for MMALs, this paper employs two methods, the exact approach and the meta-heuristic algorithm. The contributions of this paper are summarized as the following three points. 

A novel material handling scheduling method for MMALs based on the improved static kitting strategy in the layout of a line-integrated supermarket with the handling resource consideration is proposed in this paper. The objective of this problem is to minimize the total number of logistic workers over the planning horizon with the constraints of handling resources of logistic workers and to meet the operational performance requirement for the batch supply.



The graph theory is introduced to illustrate the scheduling problem and a Kuhn-Munkres algorithm is proposed to obtain exact solutions.



A meta-heuristic algorithm, namely EOADDE algorithm with an elite opposition-based learning (EOL) and self-adaptive operators is adopted to solve the scheduling problem and its effectiveness and competitiveness is proved by comparison with other two algorithms.

The remainder of this paper is organized as follows. Section 2 provides a brief literature review. In Section 3, the problem description and model formalization are provided for the material handling of automotive MMALs with detailed assumptions

and notations. Section 4 discusses the bipartite graph model of the scheduling problem, presents a Kuhn-Munkres algorithm to solve the optimal matching problem and illustrates an example to detail the steps of the algorithm. The EOADDE algorithm is introduced in Section 5. Computational experiments are carried out to validate the performance of the proposed algorithm in Section 6. Finally, Section 7 draws conclusions and provides the prospect of future research.

2 Literature review In the area of in-house logistics, the concepts of the line stocking and kitting strategies are extensively discussed and applied for material handling in the automobile industries. Due to the concept of line-integrated supermarkets unifying the advantages of the line stocking and kitting strategies, a brief review of the two strategies is necessary. Many researchers have made different kinds of studies on the comparison between line stocking and kitting strategies. Bozer and Mcginnis (1992) presented the fundamental differences between line stocking and kitting strategies, and proposed a descriptive model for selecting optimal material handling strategy by trading off storage and retrieval of the component containers, component container flows, space requirements and the average WIP (work in process). Hua and Johnson (2008) gave methods to choose which system to implement in a given environment by comparing line stocking and kitting strategies on product characteristics, storage and material handling, production control, performance impact and implementation. Hanson et al. (2012) focused on the difference of time spent by the assembly workers fetching parts in the line stocking and kitting strategies and provided a case study of the automobile industry for quantitative studies. Moreover, Hanson and Brolin (2013) further studied the impact of line stocking and kitting strategies on human consumption, product quality, flexibility, inventory levels and space requirements in an automobile manufacturing case. Finally, Sali et al. (2016) proposed an optimization model which assigned each individual part to the material handling strategies among line stocking, kitting and

sequencing modes, which aimed at minimizing the average total cost mainly including labor costs of part preparation, transportation costs, picking operation costs and part storage costs. The above literatures have studied the performance of the line stocking and kitting strategies from various aspects to select the optimal material handling strategy or combine them. However, the benefits and drawbacks of the line stocking and kitting strategies are related to the characteristics of the production and material handling environment. It is difficult to predict the performance of the two strategies through these characteristics and select the optimal strategy. In addition, the converting between the two strategies also takes a long time. A new material handling scheduling method based on line-integrated supermarkets was first proposed by Boysen and Emde (2014). This literature related to the scheduling problem for refilling the JIS-bins and presented an exact approach and a heuristic algorithm to solve the problem. Nowadays, there are a few studies on lineintegrated supermarkets, and it is of great theoretical and practical significance for further research. Boysen and Emde (2014) described the material handling of MMALs in line-integrated supermarkets using the bipartite graph and applied CPLEX to solve the scheduling model. However, the CPLEX only obtained the solutions of small-scale problems within the given 30 minutes due to longer runtime and it is failed to compare with the other three algorithms proposed by the authors in most of the instances. Therefore, in order to obtain exact solutions for the scheduling problem of refill events, a bipartite graph model is presented to transform the scheduling problem into a mathematical problem. The methods for solving the bipartite graph model are the Hungarian algorithm and the Kuhn-Munkres algorithm. The result obtained by the Hungarian algorithm may contain many different scheduling schemes, and only one of them is the most realistic. Therefore, the Kuhn-Munkres algorithm which can obtain optimal and pratical scheduling scheme is adopted to solve the exact solutions of the small and middle scale problems. Generally, exact approaches are extensively used to address small scale problems. However, due to the NP-hardness nature, when solving medium or large scale problems, the run-time will considerably long due to the rapid growth in search space.

Therefore, meta-heuristic algorithms have been widely used to solve complicated combination optimal problems because of their excellent computational efficiency. In recent years, meta-heuristic algorithms have been greatly developed, such as Genetic Algorithm (GA), Differential Evolution (DE), Particle Swarm Optimization (PSO), Simulated Annealing (SA), Ant Colony Optimization (ACO), etc., and many effective modified algorithms have been produced. For instance, Peng and Zhou (2018) developed a Hybrid Ant Colony Optimization algorithm (HACO) integrating basic Ant Colony Optimization (ACO) and local optimizers to solve a multiple server scheduling problem in the JIT part supply. And later, a hybrid bi-objective gray wolf optimization (HBGWO) combining the gray wolf optimizer (GWO) and the decomposition framework was proposed by Peng and Zhou (2019). Xie and Chen (2018) proposed an Elitism Genetic Algorithm (EGA) combining an elitism strategy to address uncertainty job shop scheduling problem with interval grey processing time. Zhou and Shen (2018) developed a Taboo Enhanced Particle Swarm Optimization algorithm (TEPSO) to solve a scheduling problem of traditional material handling in MMALs considering energy consumption. In addition to some common meta-heuristic algorithms, Fathollahi-Fard et al. (2018) developed a Social Engineering Optimizer (SEO) that started with two initial solutions, the attacker and the defender. Mohammadzadeh et al. (2018) employed three novel algorithms, Red Deer Algorithm (RDA), Virus Colony Search (VCS) and Water Wave Optimization (WWO). Furthermore, many literatures applied hybrid meta-heuristics based on the advantages of recent and traditional algorithms. Hajiaghaei-Keshteli et al. (2018) proposed two new hybrid meta-heuristic algorithms, Hybrid of Keshtel Algorithm and Simulated Annealing (HKSA), Hybrid of Interior Search Algorithm and Genetic Algorithm (HISGA). Fathollahi-Fard et al. (2018) introduced a Hybrid of Salp swarm algorithm and Simulated annealing (H-SS). Even combine RDA, SA and GA to obtain a more powerful RDSAGA which can be found in Fathollahi-Fard et al. (2018). Among the above algorithms, because of the strong global search ability, high convergence speed and strong robustness, the differential evolution algorithm (DE) is widely discussed in recent years. The DE algorithm is a swarm intelligent algorithm

presented by Storn and Price (1997), originally used to solve the Chebyshev polynomial problem. It is simple to code and has low space complexity, so the DE algorithm has a higher convergence speed and is suitable for solving more complex problems. Therefore, it has been employed in many fields, including flow shop scheduling problems (Wang et al. (2010)), resource allocation (Tsai et al. (2013)) etc. In order to improve performance in solving large-scale problems, many modified DE algorithms have been generated. Myszkowski et al. (2017) presented a hybrid Differential Evolution and Greedy Algorithm (DEGA) and proved that the DEGA is very robust and effective compared with other benchmark algorithms. Zhang and Sanderson (2009) developed a novel DE algorithm, JADE, with optional external archive and adaptive updating control parameters increasing the convergence performance and robustness of the algorithm. Piotrowski (2018) added a populationwide inertia term (PWI) to the successful history-based adaptive DE variants with linear population size reduction (L-SHADE), which improved the performance of the algorithm. In this paper, an elite opposition-based learning adaptive dynamic differential evolution algorithm (EOADDE) is developed to solve the scheduling problem of material handling for MMALs, which adopts the DE algorithm as a global search algorithm and introduces the elite opposition-based learning (EOL) as a local search algorithm. EOL has strong local search capability, which can improve the convergence speed and the quality of solutions. Moreover, self-adaptive mutation and crossover operators can trade off the population diversity and local optimization ability. This research differs from existing literature in the following three ways. 

The scheduling problem of material handling for MMALs based on the improved static kitting strategy is studied in this paper.



The Kuhn-Munkres algorithm which can obtain optimal and pratical scheduling scheme is introduced to solve the exact solutions.



The novel EOADDE algorithm is first adopted to solve the scheduling problem based on line-integrated supermarkets.

3 Problem description and mathematical model 3.1 Problem description Fig.1 schematically depicts the concept of a line-integrated supermarket. In a lineintegrated supermarket, the storage area for parts to be assembled by the respective assembly workers is located directly next to the assembly line. Logistic containers named JIS-bins are used to accommodate required parts and to deliver them from the logistic area of a station to the moving conveyor. From the JIS-bin, an assembly worker withdraws the part required by the car having currently entered the station and assembles it during the production cycle. The JIS-bin is prepared by a group of logistic workers, who pick the parts according to the predefined production sequence and pack the picked parts into the JIS-bin. Compared to the traditional decentralized supermarket, the layout of the lineintegrated supermarket has two core advantages. First of all, the application of the lineintegrated supermarket can greatly improve the stability of material handling in an assembly line. When emergencies such as stock-outs or a defective part occur during the assembly process, this strategy can quickly and directly replenish spare parts nearby to avoid line-stoppage due to the shortage of parts. Next, the material preparation and distribution in the line-integrated supermarket are integrated and assigned to the logistic workers. Compared to the traditional supermarket where the logistic workers prepare and deliver the materials by tractors, it is labor saving, convenient and fast. In order to effectively illustrate the problem, the following assumptions are introduced: 1) A cycle time in assembly line ranges between 70 and 90 seconds. 2) For each station s  S and all production cycles t  1,, T ， the deterministic part demand d st is assumed to be given. 3) The production sequence over a planning horizon is fixed in advance. 4) For each station s  S

and all production cycles

t  1,, T , all parts to be assembled on this station during the production cycle are packed into a single JIS-bin. 5) For each station s  S , one JIS-bin filled with parts

is consumed for each production cycle. At the beginning of any production cycle

t  1,, T , up to N JIS-bins filled with parts can be stored at any stations, of which N defines the replenishment batch of JIS-bin by a logistic worker at one time, and the size is determined in previous planning step. [Insert Fig.1]

3.2 The improved static kitting strategy In the improved static kitting strategy, the supply process of logistic workers can be depicted as follows. For production cycle t  T , firstly, a logistic worker w  W s ' S

walks from station

to station s  S , consuming time  s ' s . Next, the logistic worker along the

fixed routing picks a number of a parts required in the next N production cycles from the part containers at that station, consuming time p(a, s) . In the end, the logistic worker packs the picked parts respectively into N corresponding JIS-bins just in the right assembly sequence, consuming time

 (a, s) . After completing the refill event at

the station, the logistic worker will stay at this station until the end of the production cycle. At the beginning of the next production cycle, the logistic worker w will start a new refill event and walk to the next station. In the improved static kitting strategy, the replenishment batch N of logistic workers will affect the time p(a, s) taken by the workers to pick parts and the time

 (a, s) taken by the workers to pack them. The larger the batch, the more time it takes. Therefore, when the batch is large, it is difficult for a single logistic worker to complete a refill event containing three steps independently. At this point, two logistic workers are assigned to complete the refill event, so that the time of picking parts and packing parts can be reduced. When the batch size is too large and two workers cannot cooperate to complete a refill event in one production cycle, it is considered that the batch is too large and is not conducive to the scheduling and assigning of refill events. 3.3 Mathematical model A formal definition of the material handling scheduling problem for automotive

MMALs in line-integrated supermarkets is given in the following: a schedule  consists of a set of refill events of logistic workers. Each refill event is represented by a quadruple (t, s, a, w)  and defines that a logistic worker w  W

completes a

refill event of the JIS-bin at station s  S with a parts according to the given retrieval sequence at time t  T . Based on the above problem description, a mathematical model of the material handling scheduling problem for automotive MMALs based on line-integrated supermarkets is established. For the sake of convenience, the notations and explanations are summarized in Table 1. [Insert Table 1] According to the above problem descriptions, model assumptions and problem notations, the mathematical model of the material handling scheduling problem is as follows:

Min Z | W |

(1)

Subject to



qts 



xtsw , t T ; s  S

(2)

cs  dst  N   qt ' s  dst , t T ; s  S

(3)

tT , sS

tT , sS , wW t

t '1

(t  CT )  (2  xtsw  xt ' s ' w )  M  qt ' s '  (t ' CT   ss '  p (a, s )   (a, s )),

wW; s, s '  S; t, t ' T ; t  t '

(4)

( st   s ' s  p (a, s )   (a, s ))  xtsw  CT , w  W ; s, s '  S ; t  T

(5)

t  CT  qts  ( p (a, s )   (a, s ))  1  (1  xtsw )  M , w  W ; s  S ; t  T

(6)

xtsw  {0， 1}，w  W ; s  S ; t  T

(7)

qts  {0， 1}，s  S ; t  T

(8)

In the model, objective function (1) minimizes the number of logistic workers. Constraints (2) represents that each refill event must be completed by at least one worker. Constraints (3) indicates that the number of parts delivered by logistics

workers causes neither stock-outs nor overloaded bins. Constraints (4) represents that if the same worker performs two refill events in different production cycle, the time of the two events must be not overlapped. Constraints (5) indicates that the logistic workers must complete the refill event before the end of production cycle. Constraints (6) represents that the start time of any refill event cannot be earlier than 1. Constraints (7) and (8) define binary variables.

4 Kuhn-Munkres Algorithm 4.1 Bipartite graph model of scheduling problem A bipartite graph is a special model in graph theories, which has significant applications in some assigning and scheduling problems, and in many cases, it can obtain an algorithm for solving out accurate and efficient optimal solutions. According to the properties and concepts of bipartite graph, this paper will transfer the scheduling problem of material handling to the problem of finding a maximum matching in a bipartite graph. The specific formal expression is listed as follows: Given the material handling scheduling problem (MHSP) in line-integrated supermarkets, construct a bipartite graph G  (V , E) with bipartition ( P, Q ) where nodes of both sets in schedule  represent refill events (t , s, a,?) to be assigned and P  Q   . Find a maximum matching M in the bipartite graph

(t, s, a, w)  P and

G  (V , E) and let nodes

(t ', s ', a ', w ')  Q connected by each edge in M satisfy the

followings: (1) The node (t, s, a, w)  P and (t ', s ', a ', w ')  Q represent different refill events. (2) The refill event (t ', s ', a ', w') can be processed after the node (t, s, a, w) by the same worker. (3) The start time of the refill event (t, s, a, w)  P and (t ', s ', a ', w ')  Q are the

beginning of the production cycle t and t ' . According to the above-mentioned problem transformation, the correspondence between the maximum matching M in G  (V , E) and a feasible solution to MHSP is given in this paper. We can construct a feasible solution to the MHSP with W workers by a set of W chains of refill events and each chain constitutes a set of refill events successively processed by a specific logistic worker. Every edge ((t, s, a, w),(t ', s ', a ', w))  M indicates a pair ((t, s, a, w),(t ', s ', a ', w)) of consecutive refill events by the same logistic worker, namely that after the worker w completes the refill event (t, s, a, w) in t, the next refill event is (t ', s ', a ', w) in t ' . When a refill event makes it difficult for any worker to complete independently in one production cycle, the event is divided into two events and the result indicates that the event is given to two workers to fulfill. Following properties are derived based on characters of the problem. Property 1: If M is the maximum matching in the bipartite graph G  (P, Q) for a given MHSP, then | M ||  |  | W | , and M is a feasible solution to the MHSP, where  is the set of nodes.

Proof: For each worker w  W , only one refill event has no predecessors, namely the first event in  that is executed by the worker, and only one event has no successors, namely the last event in  . Therefore, each worker will make two nodes in G not be connected by other nodes, so the bipartite graph G  (P, Q) that contains | W | workers has a total of 2  2 | W | nodes that need to be connected in pairs, namely the maximum matching M in G should contain |  |  | W | edges. Once finding the maximum matching M in G, the feasible scheduling method can be obtained according to the following method: Select the node in Q that is not covered by M as the starting node of a chain. Add each successive event

((t, s, a, w),(t ', s ', a ', w))  M to the end of the chain from the starting node. Repeat the

last step until all event nodes are added to  . Property 1 is proved. Property 2: If M is not the maximum matching in the bipartite graph G  (P, Q) for the MHSP, then M isn’t a feasible solution to MHSP. Proof: For any matching M, if M is not the maximum matching in G for MHSP, then

| M ||  |  | W | , namely that M cannot saturate 2 2 | W | nodes and M cannot contain all event nodes, so the scheduling method is not feasible. Property 2 is proved. Corollary 1: Given that M is a maximum matching in G for a given MHSP, we can construct a feasible solution to MHSP by M with minimal workforce. Proof: From M is the maximum matching in G, according to property 1,

| M ||  |  | W | . And according to property 2, the matching M ' in G corresponding to another feasible solution to MHSP must also be a maximum matching. According to the property of maximum matching in bipartite graph, for another maximum matching

M ' M

in

any

G  (P, Q) ,

if

| M '|| M | ,

then

|  |  | W M ' ||  |  | W M || W M || W M ' | , namely that we construct a feasible solution to MHSP by M with minimal logistic workers. 4.2 KM algorithm From corollary 1, if only a maximum matching M in the bipartite graph

G  (P, Q) for a given MHSP is acquired, then the feasible solution with minimum logistic workers to MHSP can be constructed by M, that is, the optimal objective function can be obtained by the proposed scheduling method. The Hungarian algorithm can be used to solve the bipartite graph maximum matching problems. However, the maximum matching in G is usually more than one. The scheduling schemes constructed by different maximum matching corresponding to the MHSP contain the same number of logistic workers, but the specific assignment methods of refill event are different. Only one of the maximum matchings can construct the most realistic scheduling scheme. The Hungarian algorithm cannot solve the maximum matching M stably. In this paper, according to the nature of the problem, the Kuhn-Munkres algorithm for solving optimal matching in a bipartite graph is used

to replace the Hungarian algorithm. The reasonability of the Kuhn-Munkres algorithm to replace the Hungarian algorithm to solve the bipartite graph model corresponding to the MHSP is based on the following two points. (1) The perfect matching of the bipartite graph must be the maximum matching of the bipartite graph at the same time, and the optimal matching solved by KuhnMunkres algorithm must be the perfect matching. Therefore, the solution obtained by the Kuhn-Munkres algorithm can construct a scheduling scheme with the least number of logistic workers. (2) The optimal matching solution obtained by the Kuhn-Munkres algorithm is the matching containing the maximum edge weight sum, that is, the distance of workers’ walking is minimal. Therefore, the scheduling scheme constructed by the optimal matching does not have the situation that logistic workers seek far and neglect what lies close at hand, which is in line with a practical significance. When solving the MHSP problem with the Kuhn-Munkres algorithm, first weights are assigned to each edge of the bipartite graph. If the refill event (t ', s ', a ', w') can be processed after (t, s, a, w) by the same worker, then the edge weight cij is the opposite number of the distance between the stations where the two refill events are located. Otherwise, the edge weight cij   inf  100000000 . The specific steps of the Kuhn-Munkres algorithm are listed as follows: Step 1: Set the initial vertex labelling l ( xi )  max(cij ), l ( yi )  0 . j

Step 2: Find the edge set El  {( xi , y j ) | l ( xi )  l ( y j )  cij } , determine the equality subgraph Gl  ( X , Y , El ) corresponding to the vertex labelling l, and select any of the matching M in Gl . Step 3: If X is M-saturated, then M is an optimal matching solution and the algorithm stops. Otherwise, proceed to the next stop. Step 4: Let x be an M-unsaturated vertex. Set S  {x}, B   . Step 5: If | N G ( S ) || S | , go to step (8); otherwise, choose a vertex y  N G ( S )  B

where N G ( S ) is the node set adjacent to the node in S. Step 6: If y is M-saturated, go to step (7); otherwise, there is an M-augmenting(x, y)-path P, namely M  M E(P) and go to step (3). Step 7: Because of y is M-saturated, then let edge

yu  M , and set

S  S {u}, B  B {y} . Go to step (5). Step 8: Calculate  l  min {l ( x)  l ( y )  w( xy )} , and the vertex labelling l is xS , yB

updated to lˆ:  l (v)   l , if v  S  lˆ(v)  l (v)   l , if v  B  l (v), otherwise 

Apparently, l  0 and N G ( S )  B . Replace l by lˆ, and go to step (2). l

4.3 An illustrative example Next, take an example to further illustrate the steps of the Kuhn-Munkres algorithm. A given partially fixed schedule  is shown in Table 2 [Insert Table 2] Also given C T  90s , logistic workers spend 3s in walking through two consecutive stations. The specific steps to solve the MHSP are as follows: (1) According to the partially fixed set of event nodes, the bipartite graph model is established as shown in Fig.2. [Insert Fig.2] All of the xi in the figure constitute P set and all of yi constitute Q set. x1, y1 represent the event node 1, x2 , y2 represent the event node 2 etc. First, x1, y1 are the same refill event, so the two nodes are not connected. Event node x1 means the refill event of station s  1 at production cycle t  1 , and event node y2 means the refill event of station s  2 at production cycle t  2 . It takes

p(6,2)   (6,2)  40  32  72s to complete the refill event y2 . While if the identical logistic worker wants to finish the refill event y2 after completing x1 , it will take  12  3s to walk from the station s  1 to station s  2 . Because of

12  p(6, 2)   (6, 2)  3  40  32  75s
y1 , the first edge

x1 y2

in the optimal matching M '

represents the refill event 2 can be processed after the refill event 1 and refill event 3 can be processed after the refill event 2 by the identical worker. Therefore, the final

scheduling scheme is   {{1,2,3},{4},{5}} and the 5 refill events are completed by 3 logistic workers.

5. The proposed EOADDE algorithm In the last section, the KM algorithm was proposed to solve small-scale problems. It will take considerably long time to solve medium or large-scale problems. Consequently, we introduce a meta-heuristic algorithm which can randomly search optimal solution and get a suboptimal solution in a limited time. We use differential evolution algorithm (DE) as a main framework of an optimization algorithm. However, due to the complexity of the proposed problem in this paper, standard DE is difficult to obtain good solutions. Therefore, we introduce the elite opposition-based learning adaptive dynamic differential evolution algorithm (EOADDE). 5.1 Encoding and decoding As to the problem of this paper, it is a task assignment problem in which we assign refill events to logistic workers. The proposed EOADDE algorithm employs a twolevel encoding to present solutions as shown in Fig. 4. The length of encoding equals the total station numbers |S| multiplying production cycles T. Moreover, the first level presents all the station numbers in each production cycle and it is fixed in the iteration process. It consists of T segments and every one starts with 1 and ends with |S| which is established only in order to facilitate understanding of the coding. The second level presents the movement of logistic workers in an integer code. When the kth station needs to be replenished in the ith production cycle, then (𝑈) 𝑛𝑖,𝑘 = ⌊𝑟𝑛𝑑𝑢𝑛𝑖(𝑛(𝐿) 𝑘 ,𝑛𝑘 )⌉

(9)

where ⌊ ∙ ⌉ presents rounding to the nearest integer, nk( L ) , nk(U ) denote the lower and upper bounds of the kth station respectively. ni , k is an integer that is uniformly distributed between lower and upper bounds and represents that the refill event for this station in the current cycle is fulfilled by the worker of ni , k th station of the last cycle.

Otherwise, if the station does not need replenishment, then ni ,k  0 . As the time for a worker to refill parts satisfies  s ' s  p (a, s )   (a, s )  CT , when the parts a  0 , we can solve the maximum distance d M that logistic workers can move. Therefore, the lower and upper bound satisfy as follows: nk( L )  max{1, k  d M } nk(U )  min{S , k  d M }

(10) (11)

When the code of kth station is out of the range of [ n k( L ) , n k(U ) ] , it is impossible for the logistic workers to replenish the station. [Insert Fig.4] Because the above coding procedures will produce infeasible solutions in the process of crossover and mutation, a two-stage technique called Random-Key (RK) is employed in this paper. The RK technique can produce discrete solutions in the continuous space. The RK technique utilizes some random continuous numbers and transforms this solution to a feasible discrete one, and the algorithm doesn’t require the repairing mechanism. The two-stage technique can be found in SadeghiMoghaddam et al. (2019). A numerical example is given as follows. Consider the instance where | S | 5 and T  3 , as shown in Fig. 5. [Insert Fig.5] We can see the decoding of the instance as shown in Fig. 6. The first worker primarily replenishes the first station, then to the third station and finally to the second station. The second worker is in the similar way. Consequently, the final scheduling scheme is   {{1,3,2},{4,5,4}} and we need two logistic workers. [Insert Fig.6] 5.2 The elite opposition-based learning Opposition-based learning (OBL) was presented by Rahnamayan et al. (2008) and applying to DE improved the convergence speed and the quality of solutions. On this basis, Wang et al. (2010) introduced an enhanced DE algorithm. They presented a

generalized OBL and it expanded the searching space of OBL. In order to search the entire feasible solution space, we define the opposite of x ij , G as follows: xij* ,G  r1  xij ,G  r 2  ( x (jU )  x (j L ) )

(12)

where x ij , G denotes the ith individual of the Gth generation population, x (jU ) and x (j L ) denotes the upper and lower bounds of the jth dimension respectively, and r1

and r 2 are random numbers between 0 and 1. The EOADDE algorithm has an elite pool compared with the standard DE. We establish the elite pool which contains the optimal individual of every generation. When starting iteration, if uniformly distributed random number between 0 and 1

rnd uni (0,1) is less than learning probability (LP), then it will execute formula (12) on one of the elite individuals, which is called the elite opposition-based learning (EOL). After operating all the individuals in a generation, update the elite pool, that is, add the optimal individual of current population to the elite pool. EOL only operates on the vectors in the elite pool. Since the elite pool consists of an optimal solution of each generation, each time we choose an elite solution which may be the local optimal solution or the optimal solution of previous generations. If it is a local optimal solution, EOL can enhance the local search. If it is a previous optimal solution of previous generations, EOL can avoid falling into a local optimal solution and increase the diversity of solutions. 5.3 Mutation operation When rnd uni (0,1) is greater than LP, the algorithm will perform the DE mutation and crossover on the population. Mutation is the main operation of DE and we designed a self-adaptive mutation operator as follows: F  F0  2 

(13) 1

 e

Gm Gm 1G

(14)

where F0 denotes the initial value, Gm denotes the maximum generation, G

denotes the current generation. At the beginning of the algorithm, the mutation operator is large and the population diversity is maintained. In the process of iteration, the mutation operator decreases gradually and the convergence speed and local optimization ability increase. 5.4 Crossover operation Similar to mutation operation, a self-adaptive crossover operator is also employed in the algorithm. However, contrary to the mutation operation, crossover operator gradually increases with evolutionary generation. The smaller crossover operator can avoid the destruction of the mutated individuals while the larger one in the later evolutionary stage can avoid the algorithm falling into the local optimum and increasing the population diversity. Therefore, the self-adaptive mutation operator is shown as follows: CR  CR0  2 1

 e

Gm G 1

(15) (16)

where CR0 denotes the initial value. Then execute the crossover operation. In standard DE, mutation and crossover operation will produce a new population of the same size as the initial one and select the optimal individuals one-to-one by the fitness function. Instead, when producing a new individual by mutation and crossover operation or EOL, if it is superior to the individual of the last generation, then replace the bad individual to update population. Each newly generated individual can participate in the mutation and crossover of other individuals earlier, which enhances the convergence speed. 5.5 Overall procedure of the EOADDE algorithm Based on detailed descriptions in subsections 5.1-5.4, the overall procedure of the EOADDE algorithm proposed to solve the MHSP problem is illustrated in Fig. 7. [Insert Fig.7]

6 Computational experiments

When carrying out computational experiments, taking the number of logistic workers as an optimization performance index and considering the characteristics of MMALs, we select two main parameters to present the characteristics, factor A that is the number of assembly stations and factor B that is the number of production cycles. We implement our algorithms in Matlab2018a and solve them through a set of test instances on a PC with an Intel Core i7-8550U 1.8 gigahertz CPU and 8GB RAM. 6.1 Experimental settings In order to assess the proposed EOADDE algorithm, the test data are firstly established referring to the other literature (Boysen et al. (2014)). In the literature, the sequence of models to be assembled at a station determines the material demands. We simplify the material demands of each stations and assume that it is randomly generated as 𝑑𝑠𝑡 = ⌊𝑟𝑛𝑑𝑢𝑛𝑖(0,4)⌉, that is, material demands are uniformly distributed in the interval [0,4]. Moreover, the number of JIS-bins at a station is not certain and we assume it is within the range of [2,6]. Each JIS-bin only contains materials of one production cycle. Considering the characters of the improved static kitting strategy, the picking and packing time can be described as linear functions, which both contain a constant time and a variable time. Constant time is the walking time of logistic workers when picking or packing parts, consumption of pc  15s ( c  10s) . Variable time means the time of picking or packing one part, consumption of pv  2s ( v  2s) . Therefore, logistic workers pick a parts consuming p (a, s )  pc  pv  a and pack a parts consuming  (a, s )   c   v  a . Furthermore, it takes 3s for logistic workers to pass through two consecutive stations. Finally, the test examples (|S|, T) are constructed as follows: the number of stations | S |{16,30,50,80,120,150} , in each case corresponds to 4 different production cycles, and the production cycles

T {12,30,60,100,200}. At small and medium scale problems, the production cycles

T {12,30,60,100} , and at large scale problems, the production cycles T {30,60,100,200}. There are 24 instances in total.

6.2 Parameters settings The proposed EOADDE algorithm contains three important parameters, including the initial mutation operator F0 , the initial crossover operator CR0 and the learning probability LP, which have significant effect on the performance of EOADDE algorithm in solving the MHSP. To choose the optimal combination of parameter values, the DOE is carried out at different scale problems. The values of the three parameters with four levels are presented in Table 3. Since the intensive computational effort of full-factotial experiments, the Taguchi’s orthogonal array

L16 (43 )

is

implemented, as shown in Table 4. Each group of experiment runs 20 times. [Insert Table 3] [Insert Table 4] The analysis results obtained by Minitab are presented in Table 5. It can be known from Table 5 that the optimal combination of parameters is different at different scale problems. The initial mutation operator is the most significant parameter among the three parameters according to the rank of major factors and the impact of the initial mutation operator is greater with the increasing of problem scale. At small-scale problems, the optimal combination of parameters is F0  0.6 , CR0  0.1 and LP  0.2 . At medium-scale problems, the optimal combination is F0  0.6 , CR0  0.2 and

LP  0.05 . At large-scale problems, the optimal combination is F0  0.6 , CR0  0.2 and LP  0.2 . [Insert Table 5] In addition to the EOADDE algorithm, we also compared it with other two famous meta-heuristics, the discreet particle swarm optimization algorithm (DPSO) (Fathollahi-Fard and Hajaghaei-Keshteli (2018)) and the genetic simulated annealing algorithm (GSAA) (Torkaman et al. (2018)). The two algorithms are widely used as benchmark algorithms in the MHSP and they perform well. The detailed parameters of three algorithms are summarized in Table 6. The parameters of the two benchmark algorithms are selected from the above literatures respectively.

[Insert Table 6] 6.3 Runtime analysis of the Kuhn-Munkres algorithm In order to investigate the performance and runtime of the exact approach constructed in this paper, computational experiments are firstly implemented to solve small and medium scale problems and the isogram of the algorithm runtime is given in Fig.8. [Insert Fig.8] Figure 9 shows that the increase in the number of stations and production cycles has a significant impact on the algorithm runtime. When the number of stations and production cycles is both below 50, the runtime is minute-level, which meets the practical scheduling requirements. As the number of stations and cycles increase, the isogram of the algorithm runtime becomes more and more dense. When the number of stations and cycles is over 80, the algorithm runtime begins to grow distorted. Therefore, it is impracticable to solve complex problems using the Kuhn-Munkres algorithm and meta-heuristic algorithm can be used to solve the small, medium and large scale problems. 6.4 Computational results of the EOADDE compared with the DPSO and the GSAA First, we tested the performance of the proposed EOADDE algorithm by comparing with the GSAA, the DPSO and the Kuhn-Munkres algorithm. According to the test examples as described above, we generated 6  4  2 4 combinations of the number of stations and production cycles. For each combination, we set the cycle time to one of the three values,

CT {70,80,90} seconds. Therefore, there are

6  4  3  72 instances in total. These instances are divided into three segments based

on the problem scale (small, medium and large) and each segment has 24 instances. In different problem scales | S |  | T | , the computational time of the meta-heuristic algorithm is limited to 300 seconds, and each optimization algorithm repeats for 30 times to obtain the performance of the algorithm. Due to the long runtime of the KM algorithm when the problem scale is too large, the EOADDE algorithm is only

compared to the other two meta-heuristic algorithms at the large-scale instances. The computational results for different problem scales are shown in Table 7-9, where the hit rate on the bottom row of each table presents the ratio of dominant performance for a selected algorithm at a certain scale. Moreover, the one-way ANOVA test is applied to each instance to verify if the performance of the EOADDE algorithm is significantly superior to the other algorithms. Apparently, the null hypothesis is that there is no significant difference in the mean of results obtained by different algorithms. At the 0.05 level, if the P-value is less than 0.05, it presents the performance of the EOADDE algorithm is superior to the others. [Insert Table 7] [Insert Table 8] [Insert Table 9] As can be seen from the above three tables, the performance of the EOADDE algorithm outperforms the other algorithms in most instances. At small-scale problems, namely | S |{16,30} , the hit rates of the three algorithms are 15, 7 and 4 respectively. The performance of EOADDE algorithm is superior to the GSAA and DPSO algorithm while only 5 out of 24 instances are significantly better than the other algorithms according to the one-way ANOVA test. At medium-scale problems, namely

| S |{50,80} , the number of optimal solutions obtained by EOADDE algorithm is 20, which is larger than benchmark algorithms, and 13 of them are significantly better than the other two algorithms. Also, the GAP value between the proposed algorithm and the KM algorithm is smaller in most instances. At large-scale problems, namely

| S |{120,150}, the EOADDE algorithm obtains the optimal solutions in all 24 instances and the performance of the proposed algorithm is significantly better than the other two on 23 out of 24 instances. Furthermore, we established the boxplots shown in Fig. 9-11 to visualize the performance of different algorithms at different scale problems. It can be seen that the results obtained by EOADDE have smaller mean values and lower variability in most instances. [Insert Fig.9]

[Insert Fig.10] [Insert Fig.11] In addition, the best value of generation for each algorithm at different scales is given in Fig. 12. Each group of experiments employing the three algorithms runs 30 times independently at the number of stations |S|=30, 80 and 120, and the best value of each generation is obtained. It can be known from the three figures that although the convergence speed of EOADDE algorithm is not the highest, the quality of solution is the best in contrast to GSAA and DPSO. The convergence speed of GSAA algorithm outperforms the other algorithms, however, it is easy to fall into local optimum and the quality of solution is poor, while the performance of DPSO is the worst. Consequently, from analyzing the experimental results we can conclude that the performance of the EOADDE algorithm is better and more robust than the DPSO and GSAA algorithms in solving scheduling problems of the material handling. There are two main reasons why the EOADDE algorithm is superior. First, we adopted the self-adaptive mutation and crossover operators to balance the population diversity and the ability of local search. Second, the application of EOL enhances the local search ability and the convergence speed. [Insert Fig.12] 6.5 The effect of batch size on objective function In the improved static kitting strategy, the replenishment batch N of logistic workers will affect picking time p(a, s) and packing time

 (a, s) , and the larger the

batch is, the more time it takes and the more spare parts are prepared at one time. However, when the batch size becomes too large, it takes too long to pick and pack parts, and it is difficult for two logistic workers to cooperate to complete a refill event within a production cycle, which is not conducive to schedule and assign refill events. In order to study the optimal batch of refill events by a logistic worker at one time, the batches of 2bin, 3bin, 4bin, 5bin and 6bin materials were selected to experiments. In each replenishment batch size, 24 instances were repeatedly run for 30 times with the EOADDE algorithm with a CT value of 90 seconds. Then, all experiment results of

the same instances were averaged. The experiment results were shown in Table 10. In order to better illustrate the relationship between the number of logistic workers and the batch size, average the number of logistic workers of the same number of stations |S| and the line chart was presented in Fig. 13. It can be seen clearly from the figure that the batch size has a significant impact on the number of logistic workers. For the small-scale instances, where the number of stations |S| is 16 and 30, the number of logistic workers required gradually decreases as the batch size increases. The reason is that the smaller batch size makes the logistic workers still have a large amount of free time after completing the refill event, which increases the number of logistic workers and wastes the labor resources. However, as the problem scale increases, that is, in medium and large scales, the number of logistic workers required at the batch size of 4bin is the lowest. That is because too large batch size makes it more likely that two or more workers will be required for a refill event, thus increasing the number of logistic workers. According to the experiment results, although in the small scale, the number of logistic workers is not the lowest, it is close to the minimum. Therefore, if the line-side space allows, the optimal batch of the refill event of logistic workers is 4bin. [Insert Table 10] [Insert Fig.13] 6.6 Computational complexity analysis The computational complexity of meta-heuristic algorithms is usually measured by the big-O-notations. The runtime complexity of the standard DE is O (n) (Das et al. (2011)) while the EOL and self-adaptive operators of EOADDE algorithm don’t increase the algorithm complexity, so the runtime complexity of EOADDE algorithm is O (n) , which can be proved by the actual runtime of algorithm. We compare the total runtime of three algorithms and the results are presented in Table 11. In addition, in order to better illustrate the difference in the total runtime of the three algorithms, the runtime is presented in Fig. 14. [Insert Table 11]

[Insert Fig.14] It can be seen from the figure that for the first 16 instances, namely small-scale and medium-scale instances, the computational time of the EOADDE, GSAA and DPSO algorithms is all lower than the time limit of 300s. However, as the problem scale increases, the computational time of the meta-heuristic algorithms increases rapidly at large-scale instances. Although the EOADDE algorithm proposed in this paper is more complicated, the runtime of the EOADDE algorithm is between two benchmark algorithms and close to the other algorithms. Therefore, due to the excellent convergence speed, quality of solutions and local search ability, the performance of the EOADDE algorithm is better than the GSAA and DPSO algorithms on solving the scheduling problem. 6.7 Managerial application In this section, we will employ the proposed improved static kitting strategy to get managerial insights. In particular, the problem of selecting which material handling strategy is to be solved for the decision makers. The relevant parameters of material handling study in literatures (Hua & Johnson (2008), Hanson & Brolin (2013)) are cited to study the total labor hours of the improved static kitting strategy in the number of different stations and production cycles, which is compared with the total labor hours of line stocking or kitting strategies with the same parameters.  ik ,  line ,  kitting represent the total labor hours of the improved static kitting, line stocking and kitting strategies respectively.  ik / line / kitting  t preparation  ttransportation  t assembler  t replenishment

(17)

t preparation , ttransportation , tassembler , treplenishment represent material preparation time, material

transportation time, material handling time of assembler and material replenishment time respectively. GAP is used to compare the total labor hours of the improved static kitting strategy with the line stocking and kitting strategies, and its formula is

GAP 

min{ line ,  kitting }   ik  ik

100%

(28)

The larger GAP value indicates the advantage of total labor hours of the improved

static kitting strategy is greater than the line stocking and kitting strategies. The experiment results are shown in Fig. 15. The experiment results represent that the improved static kitting strategy of lineintegrated supermarkets has some advantages in the total labor hours compared with the line stocking and kitting strategies. In different problem scales, the improved static kitting strategy averagely saves an average of 2.6% of total labor hours contrasting with line stocking and kitting strategies. [Insert Fig.15] In a real-world applications, the production sequence and material demands are usually fixed three to four days before production starts (Emde & Boysen (2012)). When these are determined, the number of logistic workers required and the sequence of material handling tasks performed by each worker can be determined. At least one logistic worker is responsible for one refill event. Logistic workers deliver sufficient material to the JIS-bins without overlapping the refill events performed by the same worker and complete the material replenishment between time 1 and the end of production cycle. This section provides referable managerial insights for real-world applications. When the managers need to decide which material handling strategy to employ in MMALs, the results of this section show that in the case of sufficient line-side space, the results of the improved static kitting strategy are minimal in terms of the total labor hours. According to the scheduling problem of this paper, the total cost is mainly the employment cost of the logistic workers in the material handling process. Therefore, the number of workers required at different number of stations and production cycles can help managers make better decisions.

7 Conclusions In this paper, we study the scheduling problem of the material handling for MMALs based on an improved static kitting strategy in the layout of the lineintegrated supermarket. First, considering the application background of the line-

integrated supermarket, we formally describe the scheduling problem of the material handling for MMALs, determine some hypotheses and establish an integer programming mathematical model. Subsequently, an improved static kitting strategy is introduced to solve the problem of line-integrated supermarkets. In order to solve the scheduling problem of material feeding of logistic workers, a model based on graph theories are constructed to transform the scheduling problem into a mathematical problem. Since the problem is NP-hard, the EOADDE algorithm is proposed, which adopts the elite opposition-based learning (EOL) and self-adapted mutation and crossover operator. Finally, we design the experimental study and verify that the performance of the EOADDE algorithm outperforms the other meta-heuristic algorithms with the one-way ANOVA test. Furthermore, we verify that when the batch size of the refill events of logistic workers is 4bin at one time, the required number of logistic workers is minimum. Comparing the total labor hours of the improved static kitting strategy with the traditional line stocking and kitting strategies, the computational experiments prove that the total labor hours of the improved static kitting strategy is efficiently reduced. As a novel concept of material handling strategy, the line-integrated supermarket requires further research in the future. First, this paper is based on the assumptions of deterministic material requirements and production sequence over a planning horizon. But unforeseen events, like the change of production planning or the failure of picking and packing materials of logistic workers, may occur in the real world. Therefore, the dynamic scheduling problem of material handling deserves further exploration. Moreover, how to make the improved static kitting strategy conform to the JIT material handling is another problem that needs to be studied.

Declaration of conflicting interests: The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix: An overview of DE algorithm

The DE is a swarm intelligent algorithm, originally used to solve the Chebyshev polynomial problem. Unlike traditional evolutionary algorithm (EA), the variants of DE perturb the current generation population with the multiple randomly selected individuals. Because of the properties of a few parameters, fast convergence and strong robustness, it has been used in many fields. The DE algorithm mainly includes mutation and crossover operations. Mutation is the main operation of DE, and the most basic form of mutation as follows: vi ,G 1  xr1,G  F  ( xr 2,G  xr 3,G )

(1)

where 𝑥𝑟1, 𝐺, 𝑥𝑟2, 𝐺 and 𝑥𝑟3, 𝐺 are three different individuals in the Gth generation, 𝐹 is mutation operator. The mutation operator affects the quality of solutions and the convergence speed and is generally constant in standard DE. The general form of the crossover is defined as follows:  v ji ,G 1 if rand ij [0,1]  CR or j  jrand u ji ,G 1   otherwise  x ji ,G 1

(2) where randij is a uniformly distributed random number and

jrand is a random

integer among the numbers [1,| S |*T ] . Opposition-based learning (OBL) was presented by Rahnamayan et al. (2008) and applying to DE improved the convergence speed and the quality of solutions. They defined the opposite point: P  ( x1 , x2 ,  , xD ) is a point of the D-dimensional space, and the opposite point is P *  ( x1* , x2* ,  , x D* ) . The component can be described as follows: xi*  ai  bi  xi

(3)

where ai , bi are the lower and upper bound of xi . The opposite point has a 50% chance of being better than the original point. If the fitness function value of greater than P , replace P with

P* .

P* is

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Figure List Fig.1 The layout of the line-integrated supermarket Fig.2 The bipartite graph corresponding to PSSP Fig.3 The weight matrix of bipartite graph Fig.4 The general encoding representation for a PSSP Fig.5 The encoding representation of an instance Fig.6 The decoding illustration of Fig. 5 Fig.7 Flowchart of the EOADDE algorithm Fig.8 The isogram of the algorithm runtime Fig.9 Boxplots at small scales Fig.10 Boxplots at medium scales Fig.11 Boxplots at large scales Fig.12 The best value of generation for each algorithm at small (a), medium (b) and large (c) scale Fig.13 The number of logistic workers in different batch sizes Fig.14 The computational time for each algorithm at small (a), medium (b) and large (c) scale Fig.15 Comparison of the total labor hours of three strategies

Fig.1 The layout of the line-integrated supermarket

Fig. 2 The bipartite graph corresponding to PSSP

y1 x1  -inf x2  -inf C  x3   -inf x4  -inf x5   -inf

y2

y3

y4

y5

1 -inf -inf -inf -inf

2 1 -inf -inf -inf

3 2 -inf -inf -inf

4 3 -inf -inf -inf

       

Fig. 3 The weight matrix of bipartite graph

Fig. 4 The general encoding representation for a MHSP

Fig. 5 The encoding representation of an instance

Fig. 6 The decoding illustration of Fig. 5

Fig. 7 Flowchart of the EOADDE algorithm

Fig. 8 The isogram of the algorithm runtime

Fig. 9 Boxplots at small scales

Fig. 10 Boxplots at medium scales

Fig. 11 Boxplots at large scales

(a)

(b)

(c) Fig. 12 The best value of generation for each algorithm at small (a), medium (b) and large (c) scale

Fig. 13 The number of logistic workers in different batch sizes

(a)

(b)

(c) Fig. 14 The computational time for each algorithm at small (a), medium (b) and large (c) scale

Fig. 15 Comparison of the total labor hours of three strategies

Table List Table 1 Notations and Explanations Table 2 Parameters of algorithm example Table 3 Parameter values of different levels Table 4 Orthogonal table Table 5 Analysis results of Taguchi’s experiment Table 6 Parameters of the metaheuristic algorithms Table 7 Computational results for small-scale instances Table 8 Computational results for medium-scale instances Table 9 Computational results for large-scale instances Table 10 Computational results for different batch size Table 11 Computational times (in seconds) and results

Notations

T t S s W w

cs

 s 's p ( a, s )

 ( a, s ) d st CT

qts xtsw

M

Table 1 Notations and Explanations Explanations Production horizon, set of production cycles The index of production cycles, t  T Set of stations The index of stations, s  S Set of logistic workers The index of logistic workers, w  W Capacity of the JIS-bin at station s Walking time from station s to s ' Time for picking a parts at station s Packing time for putting a parts into the JIS-bin at station s Cumulated material demands up to time t at station s Cycle time of production 1, if a parts are replenished at stations s in cycle t 0, otherwise 1, if worker w performs a refill event at station s in cycle t 0, otherwise Big integer

Event node 1 2 3 4 5

Table 2 Parameters of algorithm example p ( a, s ) (t , s, a,?) (1,1,5,?) 35s (2, 2, 6,?) 40s (3,3,5,?) 38s (3, 4, 4,?) 32s (3,5,5,?) 38s

 ( a, s ) 30s 32s 30s 29s 32s

Table 3 Parameter values of different levels Parameters Level 1 2 3 4

F0

CR0

0.4 0.5 0.6 0.7

0.05 0.1 0.2 0.3

LP 0.05 0.1 0.2 0.3

Table 4 Orthogonal table Parameters Group 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

F0

CR0

0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7

0.05 0.1 0.2 0.3 0.05 0.1 0.2 0.3 0.05 0.1 0.2 0.3 0.05 0.1 0.2 0.3

LP 0.05 0.1 0.2 0.3 0.1 0.05 0.3 0.2 0.2 0.3 0.05 0.1 0.3 0.2 0.1 0.05

Problem scale Small

Medium

Large

Table 5 Analysis results of Taguchi’s experiment Parameters Level F0 CR0 1 2 3 4 Range R Rank of major factors Optimal level Optimal combination 1 2 3 4 Range R Rank of major factors Optimal level Optimal combination 1 2 3 4 Range R Rank of major factors Optimal level Optimal combination

13.40 12.85 12.50 12.55 0.90 1 3 0.6 32.05 31.25 30.90 31.40 1.15 1 3 0.60 46.45 45.60 44.40 45.30 2.05 1 3 0.6

12.85 12.65 13.10 12.70 0.45 2 2 0.1 31.45 31.40 31.25 31.50 0.25 3 3 0.20 45.55 45.65 45.15 45.40 0.50 3 3 0.2

LP 13.00 12.75 12.65 12.90 0.35 3 3 0.2 31.10 31.25 31.35 31.90 0.80 2 1 0.05 45.70 45.90 44.90 45.25 1.00 2 3 0.2

Table 6 Parameters of the metaheuristic algorithms Metaheuristi c algorithm EOADDE

GSAA

Parameters Number of population Initial mutation operator Initial crossover operator Learning probability Maximum generation Number of population Initial temperature Cooling rate

Low 30 0.6 0.1

Parameter levels Medium 40 0.6 0.2

High 50 0.6 0.2

0.2 200 30 75 0.945

0.05 300 40 100 0.963

0.2 400 50 125 0.971

DPSO

Step Maximum generation Number of population Learning factor Inertia weight range Velocity range Maximum generation

0.1 200 30 1.5 [0.2,0.65] [-1.2,1.2] 200

0.2 300 40 1.5 [0.2,0.8] [-1.5,1.5] 300

0.3 400 50 1.5 [0.2,0.9] [-2,2] 400

Table 7 Computational results for small-scale instances (|S|, T)

CT (s)

KM

EOADDE

GAP (%)

GSAA

GAP (%)

DPSO

GAP (%)

P

Whether significantly

(16, 12)

70

6.3

6.35

0.79

6.55

3.97

6.60

4.76

0798

N

(16, 12)

80

6.1

6.25

2.46

6.45

5.74

6.40

4.92

0.763

N

(16, 12)

90

6.3

6.40

1.59

6.50

3.17

6.90

9.52

0.416

N

(16, 30)

70

6.9

7.05

2.17

7.15

3.62

7.35

6.52

0706

N

(16, 30)

80

6.7

7.00

4.48

6.95

3.73

7.40

10.45

0.296

N

(16, 30)

90

6.8

7.10

4.41

6.85

0.74

7.30

7.35

0.390

N

(16, 60)

70

7.1

7.20

1.41

7.25

2.11

7.55

6.34

0.383

N

(16, 60)

80

7

7.25

3.57

7.15

2.14

7.95

13.57

0.022

Y

(16, 60)

90

7.3

7.50

2.74

7.45

2.05

7.45

2.05

0.986

N

(16, 100)

70

7.6

7.95

4.61

7.85

3.29

7.70

1.32

0.740

N

(16, 100)

80

7.4

7.70

4.05

8.10

9.46

7.70

4.05

0.346

N

(16, 100)

90

7.2

7.55

4.86

7.70

6.94

7.90

9.72

0.603

N

(30, 12)

70

10.4

11.10

6.73

10.60

1.92

10.95

5.29

0.468

N

(30, 12)

80

10.3

10.60

2.91

10.45

1.46

10.60

2.91

0.924

N

(30, 12)

90

10.2

10.60

3.92

10.55

3.43

10.45

2.45

0.896

N

(30, 30)

70

10.6

11.10

4.72

11.60

9.43

12.10

14.15

0.035

Y

(30, 30)

80

10.5

11.30

7.62

11.60

10.48

11.85

12.86

0.551

N

(30, 30)

90

11.4

10.80

3.85

11.05

6.25

11.45

10.10

0.425

N

(30, 60)

70

11.6

12.45

7.33

12.40

6.90

12.70

9.48

0.642

N

(30, 60)

80

11

11.50

4.55

12.20

10.91

13.05

18.64

0.000

Y

(30, 60)

90

11.3

11.65

3.10

11.75

3.98

11.85

4.87

0.824

N

(30, 100)

70

12

13.05

8.75

13.50

12.50

13.80

15.00

0.049

Y

(30, 100)

80

11.9

12.85

7.98

13.10

10.08

13.65

14.71

0.143

N

(30, 100)

90

12.1

12.30

1.65

13.45

11.16

12.75

5.37

0.011

Y

Hit rate

15/24

7/24

4/24

Table 8 Computational results for medium-scale instances (|S|, T)

CT (s)

KM

EOADDE

GAP (%)

GSAA

GAP (%)

DPSO

GAP (%)

P

Whether significantly

(50, 12)

70

16.

16.60

1.22

17.05

3.96

17.80

8.54

0.111

N

17.15

4.57

17.20

4.88

16.70

1.83

0.614

N

16.20

1.89

16.90

6.29

16.40

3.14

0.315

N

18.85

3.57

18.65

2.47

18.45

1.37

0.516

N

18.25

4.29

18.75

7.14

19.00

8.57

0.263

N

17.80

5.33

18.40

8.88

18.45

9.17

0.250

N

19.55

3.99

20.40

8.51

20.90

11.17

0.000

Y

19.20

3.78

19.60

5.95

20.85

12.70

0.000

Y

18.85

1.89

20.25

9.46

20.45

10.54

0.000

Y

20.75

6.96

22.55

16.24

22.65

16.75

0.000

Y

20.60

5.64

21.90

12.31

22.40

14.87

0.000

Y

4 (50, 12)

80

16. 4

(50, 12)

90

15. 9

(50, 30)

70

18. 2

(50, 30)

80

17. 5

(50, 30)

90

16. 9

(50, 60)

70

18. 8

(50, 60)

80

18. 5

(50, 60)

90

18. 5

(50, 100)

70

19. 4

(50, 100)

80

19. 5

(50, 100)

90

19

20.15

6.05

20.75

9.21

22.85

20.26

0.000

Y

(80, 12)

70

26.

27.05

0.56

27.15

0.93

27.15

0.93

0.980

N

27.30

2.63

27.10

1.88

27.20

2.26

0.946

N

9 (80, 12)

80

26. 6

(80, 12)

90

26

27.25

4.81

26.45

1.73

26.85

3.27

0.399

N

(80, 30)

70

28.

30.05

5.44

30.25

6.14

30.35

6.49

0.798

N

5 (80, 30)

80

28

28.40

1.43

29.50

5.36

29.25

4.46

0.088

N

(80, 30)

90

26.

28.25

5.81

29.35

9.93

29.80

11.61

0.022

Y

7 (80, 60)

70

29

31.60

8.97

33.10

14.14

34.35

18.45

0.000

Y

(80, 60)

80

28.

31.00

8.77

31.70

11.23

32.35

13.51

0.002

Y

5 (80, 60)

90

28

30.65

9.46

32.55

16.25

32.70

16.79

0.000

Y

(80, 100)

70

30

34.20

14.00

35.40

18.00

36.80

22.67

0.000

Y

(80, 100)

80

29.

32.40

10.96

34.30

17.47

35.00

19.86

0.000

Y

32.45

13.07

35.05

22.13

36.00

25.44

0.000

Y

2 (80, 100)

90

28.

7 Hit rate

20/24

2/24

2/24

Table 9 Computational results for large-scale instances (|S|, T)

CT (s)

EOADDE

GSAA

GAP (%)

DPSO

GAP (%)

P

Whether significantly

(120, 30)

70

44.50

45.35

1.91

45.80

2.92

0.104

N

(120, 30)

80

42.15

44.55

5.69

45.30

7.47

0.002

Y

(120, 30)

90

42.45

44.65

5.18

44.35

4.48

0.000

Y

(120, 60)

70

48.70

50.05

2.77

50.40

3.49

0.000

Y

(120, 60)

80

47.55

49.25

3.58

49.95

5.05

0.000

Y

(120, 60)

90

45.85

48.45

5.67

49.10

7.09

0.000

Y

(120, 100)

70

51.85

53.20

2.60

53.65

3.47

0.000

Y

(120, 100)

80

49.70

52.95

6.54

52.50

5.63

0.000

Y

(120, 100)

90

49.35

52.50

6.38

53.00

7.40

0.000

Y

(120, 200)

70

54.75

59.40

8.49

58.30

6.48

0.000

Y

(120, 200)

80

54.10

57.30

5.91

57.30

5.91

0.000

Y

(120, 200)

90

52.45

57.60

9.82

55.80

6.39

0.000

Y

(150, 30)

70

54.90

60.15

9.56

57.45

4.64

0.000

Y

(150, 30)

80

53.00

56.35

6.32

55.05

3.87

0.000

Y

(150, 30)

90

52.10

55.60

6.72

54.75

5.09

0.000

Y

(150, 60)

70

60.00

64.25

7.08

64.90

8.17

0.000

Y

(150, 60)

80

58.85

63.25

7.48

63.95

8.67

0.000

Y

(150, 60)

90

56.40

61.70

9.40

63.15

11.97

0.000

Y

(150, 100)

70

65.55

70.10

6.94

67.90

3.59

0.000

Y

(150, 100)

80

63.90

68.60

7.36

66.20

3.60

0.000

Y

(150, 100)

90

63.00

66.40

5.40

64.70

2.70

0.000

Y

(150, 200)

70

68.95

74.90

8.63

72.35

4.93

0.000

Y

(150, 200)

80

66.95

73.05

9.11

71.80

7.24

0.000

Y

(150, 200)

90

65.55

71.25

8.70

70.70

7.86

0.000

Y

Hit rate

(|S|, T) (16,12) (16,30) (16,60) (16,100) (30,12) (30,30) (30,60) (30,100) (50,12) (50,30) (50,60) (50,100)

24/24

0/24

0/24

Table 10 Computational results for different batch size The number of logistic workers 2bin 3bin 4bin 5bin 10.3 8.4 6.4 5.9 10.6 8.3 6.9 6.4 11.3 9.0 7.4 7.9 12.0 9.4 7.8 7.5 17.6 12.6 10.6 9.1 19.1 14.1 11.1 10.5 19.7 15.1 11.9 11.2 20.4 15.2 12.6 12.3 28.5 21.4 16.1 15.3 30.1 21.7 17.6 17.6 30.9 24.4 19.2 20.5 32.3 23.9 20.3 22.9

6bin 5.1 7.6 7.3 7.8 8.8 11.3 12.5 13.4 14.4 18.4 22.5 24.1

(80,12) (80,30) (80,60) (80,100) (120,30) (120,60) (120,100) (120,200) (150,30) (150,60) (150,100) (150,200)

Instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

45.1 48.1 48.0 48.8 68.7 71.1 72.1 72.8 84.0 86.3 89.5 90.7

31.9 35.9 36.5 37.5 52.2 54.4 56.4 58.3 64.7 67.3 69.5 73.0

27.6 28.6 30.9 33.0 42.7 46.5 50.1 53.0 52.9 57.2 63.8 66.4

25.5 29.7 33.5 36.9 43.1 52.1 57.1 61.0 54.1 63.0 71.4 76.6

Table 11 Computational times (in seconds) and results EOADDE GSAA (|S|, T) 4.3 5.1 (16,12) 10.3 11.3 (16,30) 16.9 20.8 (16,60) 25.9 35.4 (16,100) 7.4 7.5 (30,12) 14.5 16.4 (30,30) 28.8 34.2 (30,60) 47.4 60.5 (30,100) 22.3 22.4 (50,12) 49.6 57.2 (50,30) 97.6 109.8 (50,60) 166.8 192.9 (50,100) 31.0 32 (80,12) 76.2 83.3 (80,30) 143.1 168.5 (80,60) 259.7 298.4 (80,100) 191.5 211.6 (120,30) 407.6 435.4 (120,60) 762.4 770 (120,100) 1243.7 1527.9 (120,200) 235.6 235.1 (150,30) 459.5 473.4 (150,60) 764.2 957.9 (150,100) 1522.8 1864.2 (150,200)

26.3 32.3 37.6 41.1 45.4 55.0 59.4 64.2 55.8 67.1 75.0 78.2

DPSO 3.5 8.7 13.1 22.8 6.8 12.4 25.1 41.3 17.2 44.6 83.1 154.2 25.9 64.7 122.4 246.5 172.6 357.4 604.5 1083.9 184.6 402.7 642.2 1346.8

Highlights: 1. Introduce an improved static kitting strategy for MMALs. 2. Apply the graph theory to describe the problem and propose a Kuhn-Munkres algorithm. 3. Propose an EOADDE algorithm to solve the scheduling problem.

Author Contributions Binghai Zhou: Conceptualization, Validation, Writing - Review & Editing, Supervision, Project administration, Funding acquisition. Zhaoxu He: Methodology, Software, Formal analysis, Investigation, Data Curation, Writing Original Draft, Visualization.