# Cross-Trained Worker Assignment Problem in Cellular Manufacturing System Using Swarm Intelligence Metaheuristics

Authors: Lang Wu; Fulin Cai; Li Li; Xianghua Chu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Cross-trained worker assignment has become increasingly important for manufacturing efficiency and flexibility in cellular manufacturing system because of the recent increase in labor cost. Researchers mainly focused on assigning skilled workers to tasks for favorable capacity or cost. However, few of them have recognized the need for skill level enhancement through cross-training to avoid excessive training, especially for workload balance across multiple cells. This study presents a new mathematical programming model aimed at minimum training and maximum workload balance with economical labor utilization, to address the worker assignment problem with a cross-training plan spanning multiple cells. The model considers the trade-off between training expenditure and workload balance to achieve a more flexible solution based on decision-maker’s preference. Considering the computational complexity of the problem, the classical swarm intelligence optimizers, i.e., particle swarm optimization (PSO) and artificial bee colony (ABC), are implemented to search the problem landscape. To improve the optimization performance, a superior tracking ABC with an augmented information sharing strategy is designed to address the problem. Ten benchmark problems are employed for numerical experiments. The results indicate the efficiency and effectiveness of the proposed models as well as the developed algorithms.


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## Body

## 1. Introduction

Following the recent emerging Industrial Revolution 4.0, many manufacturers are engaged in finding new ways to increase the productivity and flexibility of their manufacturing systems so they can cope with varied production environments, such as multiproduct and small-batch production. One of such solutions is to use cellular manufacturing system (CMS), which implements group technology to classify families of parts produced and allocate machine groups to part families. CMS is a hybrid system that highlights the strengths of job shop (flexibility in producing a wide variety of products) and flow line (efficient flow and high production rate) [1].Previous studies on CMS have mainly focused on the cell formation problem, which refers to the technology of forming appropriate part families and their corresponding machine groups [2–4]. However, CMS includes not only parts and machines, but also groups of tasks and workers. Owing to the rapid increase in labor cost recently, the assignment of suitable workers to handle various tasks in each manufacturing cell becomes an important factor in the implementation of CMS.CMS comprises multiple manufacturing cells, where each cell includes multiple tasks in real-world manufacturing situation. Each worker is proficient in one or more skills at different levels. The standard task time is fixed generally. In practice, the actual task time varies due to the differences of the skill levels of the workers. Mutlu et al. [5] noted that skill types and skill levels of workers should be considered during worker assignment. Cross-training can assist workers in obtaining more skills and enhance their skill levels. Cross-trained workers are more flexible than specialized workers since there is more opportunity to balance workload and relieve overloaded stress among workers, though there would be significant additional costs to train multifunctional workers [6]. The question is, to what extent should the labor force be cross-trained? And, more precisely, who should be cross-trained for which machine or task [7]? This raises further questions, such as how to assign the cross-trained workers to various tasks so as to optimize some output measure [8]. In previous studies, the cross-training problem and the worker assignment problem were always handled separately. However, the associated trade-offs between the two problems have not been answered yet. In this study, we aim to propose a framework to solve the worker assignment problem in CMS incorporating cross-training.A major challenge arising in this context concerns the principle of cross-trained worker assignment in CMS, where worker-task configuration among multiple cells is explored based on customer demand to seek workload balance of assigned workers with minimum cross-training and labor cost. Specifically, two questions should be answered:( 1 ) Which worker should be assigned to which task in multiple manufacturing cells?( 2 ) Which worker should be trained to reach which skill level for the assigned task in multiple manufacturing cells?Previous studies related to the above questions mainly have two limitations.(1) They only answered the first question by assuming that the skill levels of workers are fixed, while neglecting the need for skill level enhancement via cross-training. Skill level enhancement denotes that the assigned worker should be trained not only for higher efficiency in its existing skills but also for the acquisition of new skills. And (2) they mainly paid attention to flow-line system or a single manufacturing cell and ignored the complex interaction among multiple cells during cross-trained worker assignment in CMS.To address this gap, this paper presents a novel mathematical programming model that aims to minimize training cost, workload imbalance, and the number of assigned workers based on customer demand. The proposed model not only takes into account the problem of cross-trained worker assignment for multiple cells but also determines the training requirements considering the trade-off between training expenditure and workload balance in the following scenarios:( 1 ) Training is recommended strongly for more exact workload balance, which results in more expenditure.( 2 ) Training is not recommended for less expenditure, which results in less exact workload balance.In view of the complexity and computational burden of the presented model, we implement swarm intelligence optimizers for the global optimization. Swarm intelligence (SI) algorithms, motivated by collective behaviors in natural system, have achieved great successful applications in various areas [9, 10]. As a result, a number of swarm-based metaheuristics have been developed recently, among which particle swarm optimization (PSO) [11] and artificial bee colony (ABC) [12] are two representative algorithms. PSO was proposed for global optimization by emulating the velocity adjusting of fish schooling and bird flocking; ABC was developed by simulating the information exchange through bee dancing. PSO and ABC have shown high efficiency and effectiveness in solving real-world global optimization problems [9]. In this study, the global topology PSO (GPSO), local topology PSO (LPSO), and ABC are implemented to solve the proposed cross-trained worker assignment in CMS. To improve the optimization performance, a superior tracking artificial bee colony (STABC) with an augmented information sharing strategy is developed. In STABC, there are two main contributions compared to the canonical ABC: (1) instead of learning one dimension from neighbors, in STABC a bee can learn from others in all dimensions; (2) instead of chasing the random individual, in STABC each bee either learns from its previous information or move towards other superior bees. Computational experiments and comparisons are conducted to evaluate the efficiency of the swarm intelligence algorithms.Comparing with the previous research, the contribution of this study lies in three aspects:( 1 ) A new model is presented to solve the worker assignment problem for multiple manufacturing cells incorporating cross-training planning.( 2 ) The skill enhancement is integrated to avoid excessive training, and the trade-off between training expenditure and workload balance is combined to obtain more flexible solution in the model.( 3 ) Swarm intelligence optimizers are developed and improved to solve the proposed mathematical model efficiently.Remainder of this paper is organized as follows. Section2 reviews the literaturerelating to worker assignment. Section 3 presents model for effective cross-trained worker assignment. Section 4 describes the swarm intelligence metaheuristics used and the algorithmic implementation for solving the problem. Section 5 describes experiments with computational results to justify the proposed model and algorithms. Section 6 summarizes the conclusions and future research directions.


## 2. Literature Review

The approaches regarding the worker assignment problem are briefly reviewed in this section. The literature can be classified by considering two types of manufacturing systems: flow-line manufacturing system (FLMS) and cellular manufacturing system (CMS). The limitations in previous studies are summarized for the motivation of our work.


### 2.1. Worker Assignment in Flow-Line Manufacturing System

Hopp et al. [13] considered two cross-training strategies: cherry picking and skill chain. In zoned work sharing, some machines/tasks on the line are shared between workers to seek capacity balance. Inman et al. [6] argued that cross-training should be used judiciously since it is costly and is limited by learning capacity and can confound the search for quality problems. They presented a training strategy called chaining in which workers are trained to perform a second task, and the assignments of task types to workers are linked in a chain. Their research has shown that cross-training in chaining is a practical and effective strategy to compensate for absenteeism on assembly lines. Moreira et al. [14] proposed simple heuristics for solving the assembly line worker assignment and balancing problem. Their idea was to use task and worker priority rules to define which worker and which set of tasks should be assigned to each workstation by constructive heuristic framework. Parvin et al. [15] introduced a new canonical model of worker cross-training, called a Fixed Task Zone Chain. A new heuristic worker control policy was presented to design a zone structure that can be balanced by assigning worker to work station based on his or her skill set for maximum throughput. Mutlu et al. [5] solved the assembly line worker assignment and balancing problem when task times differ depending on operator skills and concerns with the assignment of tasks and operators to stations in order to minimize the cycle time. Saidi-Mehrabad et al. [16] presented a novel integer linear programming model for dynamic manufacturing systems in the presence of system configurations, worker assignment, and production plan for each part type at each period. The objective of worker assignment is to minimize training and salary of worker costs. Sungur and Yavuz [17] considered qualification requirements and levels of workers to achieve assembly line balance and worker assignment problem. They suggested that the workers should be ranked hierarchically according to their qualification requirements and levels. From the standpoint of the paper, a higher qualified worker implies a higher cost and lower process time.


### 2.2. Worker Assignment in Cellular Manufacturing System

The proposed strategies of worker assignment in CMS can be divided into two categories:(1) simultaneous formation of manufacturing cells and worker assignment and (2) assignment of workers to cells after cell formation.


#### 2.2.1. Simultaneous Formation of Manufacturing Cells and Worker Assignment

The strategy of forming manufacturing cells and worker assignment simultaneously was achieved first by Aryanezhad et al. [18]. The first part of the model objective function proposed in their study sought to minimize production cost, intercell material handling cost, and machine costs in the planning horizon. The second part involved human issues including hiring cost, firing cost, training cost, and salary into worker assignment in cells. However, this study ignored efficiency of workers’ skills in different tasks. Mahdavi et al. [19] determined optimal cell configurations, worker assignments, and process plans using an integer mathematical programming model. The purpose was to minimize holding and backorder, intercell material handling, machine and reconfiguration, and workers hiring, firing, and salary costs. Their model was solved by branch-and-bound (B&B) method using Lingo 8.0 software. Mahdavi et al. [20] presented a fuzzy goal programming for grouping machines, parts, and workers simultaneously and determining production planning in dynamic virtual CMS. Their model aimed to minimize inventory holding and backorder costs, as well as the number of exceptional elements in a cubic space of machine-part-worker incidence matrix under constraints of machine capacity, worker capacity, and customer demand. Bagheri and Bashiri [21] proposed a mathematical model to simultaneously solve the cell formation, operator assignment, and intercell layout problems. Salary, hiring, firing, and training cost were considered while optimizing worker assignment. Their results indicated that consideration of the operator assignment problem has significant impact on the overall system efficiency. Niakan et al. [22] proposed a new biobjective mathematical model to solve dynamic cell formation and skill-based worker assignment problem. Environmental and social criteria were considered in their research. Due to the NP-hardness of the problem, they merged an efficient hybrid metaheuristic based on the nondominated sorting genetic algorithm with multiobjective simulated annealing. Liu et al. [1] built an integrated model to solve the problems of machine grouping, part scheduling, worker assignment for minimizing material handling costs, and the fixed and operating costs of machines and workers. They also developed a discrete bacteria foraging algorithm combining with priority rule based parallel schedule generation scheme for the intractable model. Liu et al. [23] built another integrated model to solve worker assignment and production planning problem again for CMS. In the new model, they assumed that all workers had the characteristics of learning or forgetting.


#### 2.2.2. Assignment of Workers to Cells after Cell Formation

More previous studies focused on the problem of assigning workers to tasks in cells after cell formation. Askin and Huang [24] presented a mixed integer goal programming model for solving the worker assignment problem and coming up with a training plan for technical and administrative skills. The model aimed to maximize team synergy between worker abilities and task requirements, while minimizing training cost. Greedy heuristic, filtered beam search, and simulated annealing techniques were developed and tested to solve the problem. Norman et al. [25] considered productivity, output quality, and training costs in assigning workers to manufacturing cells with the objective of maximizing the effectiveness of the organization. Comparing with traditional worker assignments, the model in their research did not include not only technical skills but also human skills. Ertay and Ruan [26] noted a data envelopment analysis to determine the most efficient number of operators and efficient measurement of labor assignment in CMS. Fitzpatrick and Askin [27]) developed mathematical models for forming effective human teams by selecting suitable interpersonal construction and considering technical skill requirements from initial labor pools. Meanwhile, extensive cross-training policies were also attained to optimize team performance, and a balanced placement heuristic was proposed and evaluated. Cesani and Steudel [28] simultaneously considered concepts of workload sharing, workload balancing, and the presence of bottleneck operations to classify labor strategies according to type of machine-operator assignments including dedicated, shared, and combined assignment. Simulation modeling was used to test suitability of these concepts in an actual cell implementation. Süer and Tummaluri [7] developed mathematical models to tackle three problems: finding alternative cell configurations, loading cells and finding crew sizes, and assigning operators to operations. They also proposed two heuristic approaches for operator assignment. McDonald et al. [29] presented a mathematical model to solve the worker assignment arising in a lean manufacturing cell. The model sought minimum cost of training, inventory and poor quality to determine skill training necessary for workers to meet tasks and customer demand. Their study also proposed the addition of pay-for-skills reward in evaluating the costs and benefits of cross-training. However, their study had only focused on worker assignment for a single cell, although multiple cells exist in the CMS. Murali et al. [30] developed an artificial neural network model for assigning workforce in virtual cells under dual resource constrained context where the number of available workers is less than total number of available machines. Egilmez et al. [31] developed a four-phased hierarchical methodology to address the stochastic skill-based manpower allocation problem. In their study, worker’s expected processing time and the standard deviation for each operation were considered individually to optimize the worker assignment to cells and maximize the production rate of manufacturing system. Table 1 presents a classification of the previous studies.Table 1
The classification of related literature.
Author Manufacturing system type Number of cells Skill level Cross-Training Workload balance Data nature Solving method Hopp et al. [13] FLMS   √ √ Stochastic Heuristic Policies Inman et al. [6] FLMS   √  Stochastic Simulation Moreira et al. [14] FLMS    √ Certain HGA Mutlu et al. [5] FLMS    √ Certain IGA Saidi-Mehrabad et al. [16] FLMS  √ √  Stochastic LINGO Sungur and Yavuz et al. [17] FLMS    √ Certain CPLEX Parvin et al. [15] FLMS   √ √ Stochastic Heuristic Policies Aryanezhad et al. [18] CMS Multiple √ √  Certain LINGO Mahdavi et al. [19] CMS Multiple    Certain LINGO Mahdavi et al. [20] CMS Multiple    Certain LINGO Bagheri and Bashiri [32] CMS Multiple  √  Stochastic LINGO Niakan et al. [22] CMS Multiple √ √  Stochastic NSGA II-MOSA Liu et al. [1] CMS Multiple    Stochastic DBFA Liu et al. [23] CMS Multiple √   Certain HBFA Askin and Huang [24] CMS Multiple √ √  Stochastic Greedy heuristic Norman et al. [25] CMS Single √ √  Stochastic CPLEX Ertay and Ruan [26] CMS Single    Stochastic DEA Fitzpatrick and Askin [27] CMS Single √   Stochastic Balanced placement heuristic Cesani and Steudel [28] CMS Single   √ Certain Simulation Süer and Tummaluri [33] CMS Single √   Stochastic Max and MaxMin heuristic McDonald et al. [29] CMS Single √ √  Certain CPLEX Murali et al. [30] CMS Multiple √ √  Certain ANN Egilmez et al. [31] CMS Multiple √   Certain LINGOThe aforementioned studies had pointed out that worker assignment plays an essential role in both flow-line and cellular manufacturing system. To explore the problem of workload balance among assigned workers, the cross-training strategy has been examined widely. However, most of previous studies just determined which worker should be trained for which task. The question of an effective strategy of need for cross-training by prioritizing exact skill level of worker to avoid excessive training was ignored. That is, very few achievements specifically indicated how many workers and in which skills they should be trained to which level with minimum training and labor cost as well as maximum workload balance. Moreover, the question of trade-off between training expenditure and workload balance could lead to more flexible worker assignment and cross-training policies which did not attract significant interest in previous study. Besides, for the question of balancing workload of assigned workers during cross-trained worker assignment, most of previous studies paid attention to flow-line manufacturing system or a single manufacturing cell. The solution of this question for multiple manufacturing cells should receive more concern because it is more complex and practical due to the interplay among cells.Considering the high complexity of the NP-hard problem, optimization software (LINGO, CPLEX) were accompanied with long runtime consumption. Consequently, efficient metaheuristics for the models have been attracting growing attention in recent years, especially swarm intelligence metaheuristics. For example, ABC was employed to address the uncovering community problem in complex networks [42] and a novel PSO algorithm was utilized for power system optimization [43]. However, few researchers used the swarm intelligence metaheuristics to solve the problem of cross-trained worker assignment for multiple cells.To fill the gaps in the literature, this study aims to build a new model to address the worker assignment problem among multiple manufacturing cells while incorporating cross-training planning for workload balance and develop solutions based on the swarm intelligence algorithms.


## 2.1. Worker Assignment in Flow-Line Manufacturing System

Hopp et al. [13] considered two cross-training strategies: cherry picking and skill chain. In zoned work sharing, some machines/tasks on the line are shared between workers to seek capacity balance. Inman et al. [6] argued that cross-training should be used judiciously since it is costly and is limited by learning capacity and can confound the search for quality problems. They presented a training strategy called chaining in which workers are trained to perform a second task, and the assignments of task types to workers are linked in a chain. Their research has shown that cross-training in chaining is a practical and effective strategy to compensate for absenteeism on assembly lines. Moreira et al. [14] proposed simple heuristics for solving the assembly line worker assignment and balancing problem. Their idea was to use task and worker priority rules to define which worker and which set of tasks should be assigned to each workstation by constructive heuristic framework. Parvin et al. [15] introduced a new canonical model of worker cross-training, called a Fixed Task Zone Chain. A new heuristic worker control policy was presented to design a zone structure that can be balanced by assigning worker to work station based on his or her skill set for maximum throughput. Mutlu et al. [5] solved the assembly line worker assignment and balancing problem when task times differ depending on operator skills and concerns with the assignment of tasks and operators to stations in order to minimize the cycle time. Saidi-Mehrabad et al. [16] presented a novel integer linear programming model for dynamic manufacturing systems in the presence of system configurations, worker assignment, and production plan for each part type at each period. The objective of worker assignment is to minimize training and salary of worker costs. Sungur and Yavuz [17] considered qualification requirements and levels of workers to achieve assembly line balance and worker assignment problem. They suggested that the workers should be ranked hierarchically according to their qualification requirements and levels. From the standpoint of the paper, a higher qualified worker implies a higher cost and lower process time.


## 2.2. Worker Assignment in Cellular Manufacturing System

The proposed strategies of worker assignment in CMS can be divided into two categories:(1) simultaneous formation of manufacturing cells and worker assignment and (2) assignment of workers to cells after cell formation.


### 2.2.1. Simultaneous Formation of Manufacturing Cells and Worker Assignment

The strategy of forming manufacturing cells and worker assignment simultaneously was achieved first by Aryanezhad et al. [18]. The first part of the model objective function proposed in their study sought to minimize production cost, intercell material handling cost, and machine costs in the planning horizon. The second part involved human issues including hiring cost, firing cost, training cost, and salary into worker assignment in cells. However, this study ignored efficiency of workers’ skills in different tasks. Mahdavi et al. [19] determined optimal cell configurations, worker assignments, and process plans using an integer mathematical programming model. The purpose was to minimize holding and backorder, intercell material handling, machine and reconfiguration, and workers hiring, firing, and salary costs. Their model was solved by branch-and-bound (B&B) method using Lingo 8.0 software. Mahdavi et al. [20] presented a fuzzy goal programming for grouping machines, parts, and workers simultaneously and determining production planning in dynamic virtual CMS. Their model aimed to minimize inventory holding and backorder costs, as well as the number of exceptional elements in a cubic space of machine-part-worker incidence matrix under constraints of machine capacity, worker capacity, and customer demand. Bagheri and Bashiri [21] proposed a mathematical model to simultaneously solve the cell formation, operator assignment, and intercell layout problems. Salary, hiring, firing, and training cost were considered while optimizing worker assignment. Their results indicated that consideration of the operator assignment problem has significant impact on the overall system efficiency. Niakan et al. [22] proposed a new biobjective mathematical model to solve dynamic cell formation and skill-based worker assignment problem. Environmental and social criteria were considered in their research. Due to the NP-hardness of the problem, they merged an efficient hybrid metaheuristic based on the nondominated sorting genetic algorithm with multiobjective simulated annealing. Liu et al. [1] built an integrated model to solve the problems of machine grouping, part scheduling, worker assignment for minimizing material handling costs, and the fixed and operating costs of machines and workers. They also developed a discrete bacteria foraging algorithm combining with priority rule based parallel schedule generation scheme for the intractable model. Liu et al. [23] built another integrated model to solve worker assignment and production planning problem again for CMS. In the new model, they assumed that all workers had the characteristics of learning or forgetting.


### 2.2.2. Assignment of Workers to Cells after Cell Formation

More previous studies focused on the problem of assigning workers to tasks in cells after cell formation. Askin and Huang [24] presented a mixed integer goal programming model for solving the worker assignment problem and coming up with a training plan for technical and administrative skills. The model aimed to maximize team synergy between worker abilities and task requirements, while minimizing training cost. Greedy heuristic, filtered beam search, and simulated annealing techniques were developed and tested to solve the problem. Norman et al. [25] considered productivity, output quality, and training costs in assigning workers to manufacturing cells with the objective of maximizing the effectiveness of the organization. Comparing with traditional worker assignments, the model in their research did not include not only technical skills but also human skills. Ertay and Ruan [26] noted a data envelopment analysis to determine the most efficient number of operators and efficient measurement of labor assignment in CMS. Fitzpatrick and Askin [27]) developed mathematical models for forming effective human teams by selecting suitable interpersonal construction and considering technical skill requirements from initial labor pools. Meanwhile, extensive cross-training policies were also attained to optimize team performance, and a balanced placement heuristic was proposed and evaluated. Cesani and Steudel [28] simultaneously considered concepts of workload sharing, workload balancing, and the presence of bottleneck operations to classify labor strategies according to type of machine-operator assignments including dedicated, shared, and combined assignment. Simulation modeling was used to test suitability of these concepts in an actual cell implementation. Süer and Tummaluri [7] developed mathematical models to tackle three problems: finding alternative cell configurations, loading cells and finding crew sizes, and assigning operators to operations. They also proposed two heuristic approaches for operator assignment. McDonald et al. [29] presented a mathematical model to solve the worker assignment arising in a lean manufacturing cell. The model sought minimum cost of training, inventory and poor quality to determine skill training necessary for workers to meet tasks and customer demand. Their study also proposed the addition of pay-for-skills reward in evaluating the costs and benefits of cross-training. However, their study had only focused on worker assignment for a single cell, although multiple cells exist in the CMS. Murali et al. [30] developed an artificial neural network model for assigning workforce in virtual cells under dual resource constrained context where the number of available workers is less than total number of available machines. Egilmez et al. [31] developed a four-phased hierarchical methodology to address the stochastic skill-based manpower allocation problem. In their study, worker’s expected processing time and the standard deviation for each operation were considered individually to optimize the worker assignment to cells and maximize the production rate of manufacturing system. Table 1 presents a classification of the previous studies.Table 1
The classification of related literature.
Author Manufacturing system type Number of cells Skill level Cross-Training Workload balance Data nature Solving method Hopp et al. [13] FLMS   √ √ Stochastic Heuristic Policies Inman et al. [6] FLMS   √  Stochastic Simulation Moreira et al. [14] FLMS    √ Certain HGA Mutlu et al. [5] FLMS    √ Certain IGA Saidi-Mehrabad et al. [16] FLMS  √ √  Stochastic LINGO Sungur and Yavuz et al. [17] FLMS    √ Certain CPLEX Parvin et al. [15] FLMS   √ √ Stochastic Heuristic Policies Aryanezhad et al. [18] CMS Multiple √ √  Certain LINGO Mahdavi et al. [19] CMS Multiple    Certain LINGO Mahdavi et al. [20] CMS Multiple    Certain LINGO Bagheri and Bashiri [32] CMS Multiple  √  Stochastic LINGO Niakan et al. [22] CMS Multiple √ √  Stochastic NSGA II-MOSA Liu et al. [1] CMS Multiple    Stochastic DBFA Liu et al. [23] CMS Multiple √   Certain HBFA Askin and Huang [24] CMS Multiple √ √  Stochastic Greedy heuristic Norman et al. [25] CMS Single √ √  Stochastic CPLEX Ertay and Ruan [26] CMS Single    Stochastic DEA Fitzpatrick and Askin [27] CMS Single √   Stochastic Balanced placement heuristic Cesani and Steudel [28] CMS Single   √ Certain Simulation Süer and Tummaluri [33] CMS Single √   Stochastic Max and MaxMin heuristic McDonald et al. [29] CMS Single √ √  Certain CPLEX Murali et al. [30] CMS Multiple √ √  Certain ANN Egilmez et al. [31] CMS Multiple √   Certain LINGOThe aforementioned studies had pointed out that worker assignment plays an essential role in both flow-line and cellular manufacturing system. To explore the problem of workload balance among assigned workers, the cross-training strategy has been examined widely. However, most of previous studies just determined which worker should be trained for which task. The question of an effective strategy of need for cross-training by prioritizing exact skill level of worker to avoid excessive training was ignored. That is, very few achievements specifically indicated how many workers and in which skills they should be trained to which level with minimum training and labor cost as well as maximum workload balance. Moreover, the question of trade-off between training expenditure and workload balance could lead to more flexible worker assignment and cross-training policies which did not attract significant interest in previous study. Besides, for the question of balancing workload of assigned workers during cross-trained worker assignment, most of previous studies paid attention to flow-line manufacturing system or a single manufacturing cell. The solution of this question for multiple manufacturing cells should receive more concern because it is more complex and practical due to the interplay among cells.Considering the high complexity of the NP-hard problem, optimization software (LINGO, CPLEX) were accompanied with long runtime consumption. Consequently, efficient metaheuristics for the models have been attracting growing attention in recent years, especially swarm intelligence metaheuristics. For example, ABC was employed to address the uncovering community problem in complex networks [42] and a novel PSO algorithm was utilized for power system optimization [43]. However, few researchers used the swarm intelligence metaheuristics to solve the problem of cross-trained worker assignment for multiple cells.To fill the gaps in the literature, this study aims to build a new model to address the worker assignment problem among multiple manufacturing cells while incorporating cross-training planning for workload balance and develop solutions based on the swarm intelligence algorithms.


## 2.2.1. Simultaneous Formation of Manufacturing Cells and Worker Assignment

The strategy of forming manufacturing cells and worker assignment simultaneously was achieved first by Aryanezhad et al. [18]. The first part of the model objective function proposed in their study sought to minimize production cost, intercell material handling cost, and machine costs in the planning horizon. The second part involved human issues including hiring cost, firing cost, training cost, and salary into worker assignment in cells. However, this study ignored efficiency of workers’ skills in different tasks. Mahdavi et al. [19] determined optimal cell configurations, worker assignments, and process plans using an integer mathematical programming model. The purpose was to minimize holding and backorder, intercell material handling, machine and reconfiguration, and workers hiring, firing, and salary costs. Their model was solved by branch-and-bound (B&B) method using Lingo 8.0 software. Mahdavi et al. [20] presented a fuzzy goal programming for grouping machines, parts, and workers simultaneously and determining production planning in dynamic virtual CMS. Their model aimed to minimize inventory holding and backorder costs, as well as the number of exceptional elements in a cubic space of machine-part-worker incidence matrix under constraints of machine capacity, worker capacity, and customer demand. Bagheri and Bashiri [21] proposed a mathematical model to simultaneously solve the cell formation, operator assignment, and intercell layout problems. Salary, hiring, firing, and training cost were considered while optimizing worker assignment. Their results indicated that consideration of the operator assignment problem has significant impact on the overall system efficiency. Niakan et al. [22] proposed a new biobjective mathematical model to solve dynamic cell formation and skill-based worker assignment problem. Environmental and social criteria were considered in their research. Due to the NP-hardness of the problem, they merged an efficient hybrid metaheuristic based on the nondominated sorting genetic algorithm with multiobjective simulated annealing. Liu et al. [1] built an integrated model to solve the problems of machine grouping, part scheduling, worker assignment for minimizing material handling costs, and the fixed and operating costs of machines and workers. They also developed a discrete bacteria foraging algorithm combining with priority rule based parallel schedule generation scheme for the intractable model. Liu et al. [23] built another integrated model to solve worker assignment and production planning problem again for CMS. In the new model, they assumed that all workers had the characteristics of learning or forgetting.


## 2.2.2. Assignment of Workers to Cells after Cell Formation

More previous studies focused on the problem of assigning workers to tasks in cells after cell formation. Askin and Huang [24] presented a mixed integer goal programming model for solving the worker assignment problem and coming up with a training plan for technical and administrative skills. The model aimed to maximize team synergy between worker abilities and task requirements, while minimizing training cost. Greedy heuristic, filtered beam search, and simulated annealing techniques were developed and tested to solve the problem. Norman et al. [25] considered productivity, output quality, and training costs in assigning workers to manufacturing cells with the objective of maximizing the effectiveness of the organization. Comparing with traditional worker assignments, the model in their research did not include not only technical skills but also human skills. Ertay and Ruan [26] noted a data envelopment analysis to determine the most efficient number of operators and efficient measurement of labor assignment in CMS. Fitzpatrick and Askin [27]) developed mathematical models for forming effective human teams by selecting suitable interpersonal construction and considering technical skill requirements from initial labor pools. Meanwhile, extensive cross-training policies were also attained to optimize team performance, and a balanced placement heuristic was proposed and evaluated. Cesani and Steudel [28] simultaneously considered concepts of workload sharing, workload balancing, and the presence of bottleneck operations to classify labor strategies according to type of machine-operator assignments including dedicated, shared, and combined assignment. Simulation modeling was used to test suitability of these concepts in an actual cell implementation. Süer and Tummaluri [7] developed mathematical models to tackle three problems: finding alternative cell configurations, loading cells and finding crew sizes, and assigning operators to operations. They also proposed two heuristic approaches for operator assignment. McDonald et al. [29] presented a mathematical model to solve the worker assignment arising in a lean manufacturing cell. The model sought minimum cost of training, inventory and poor quality to determine skill training necessary for workers to meet tasks and customer demand. Their study also proposed the addition of pay-for-skills reward in evaluating the costs and benefits of cross-training. However, their study had only focused on worker assignment for a single cell, although multiple cells exist in the CMS. Murali et al. [30] developed an artificial neural network model for assigning workforce in virtual cells under dual resource constrained context where the number of available workers is less than total number of available machines. Egilmez et al. [31] developed a four-phased hierarchical methodology to address the stochastic skill-based manpower allocation problem. In their study, worker’s expected processing time and the standard deviation for each operation were considered individually to optimize the worker assignment to cells and maximize the production rate of manufacturing system. Table 1 presents a classification of the previous studies.Table 1
The classification of related literature.
Author Manufacturing system type Number of cells Skill level Cross-Training Workload balance Data nature Solving method Hopp et al. [13] FLMS   √ √ Stochastic Heuristic Policies Inman et al. [6] FLMS   √  Stochastic Simulation Moreira et al. [14] FLMS    √ Certain HGA Mutlu et al. [5] FLMS    √ Certain IGA Saidi-Mehrabad et al. [16] FLMS  √ √  Stochastic LINGO Sungur and Yavuz et al. [17] FLMS    √ Certain CPLEX Parvin et al. [15] FLMS   √ √ Stochastic Heuristic Policies Aryanezhad et al. [18] CMS Multiple √ √  Certain LINGO Mahdavi et al. [19] CMS Multiple    Certain LINGO Mahdavi et al. [20] CMS Multiple    Certain LINGO Bagheri and Bashiri [32] CMS Multiple  √  Stochastic LINGO Niakan et al. [22] CMS Multiple √ √  Stochastic NSGA II-MOSA Liu et al. [1] CMS Multiple    Stochastic DBFA Liu et al. [23] CMS Multiple √   Certain HBFA Askin and Huang [24] CMS Multiple √ √  Stochastic Greedy heuristic Norman et al. [25] CMS Single √ √  Stochastic CPLEX Ertay and Ruan [26] CMS Single    Stochastic DEA Fitzpatrick and Askin [27] CMS Single √   Stochastic Balanced placement heuristic Cesani and Steudel [28] CMS Single   √ Certain Simulation Süer and Tummaluri [33] CMS Single √   Stochastic Max and MaxMin heuristic McDonald et al. [29] CMS Single √ √  Certain CPLEX Murali et al. [30] CMS Multiple √ √  Certain ANN Egilmez et al. [31] CMS Multiple √   Certain LINGOThe aforementioned studies had pointed out that worker assignment plays an essential role in both flow-line and cellular manufacturing system. To explore the problem of workload balance among assigned workers, the cross-training strategy has been examined widely. However, most of previous studies just determined which worker should be trained for which task. The question of an effective strategy of need for cross-training by prioritizing exact skill level of worker to avoid excessive training was ignored. That is, very few achievements specifically indicated how many workers and in which skills they should be trained to which level with minimum training and labor cost as well as maximum workload balance. Moreover, the question of trade-off between training expenditure and workload balance could lead to more flexible worker assignment and cross-training policies which did not attract significant interest in previous study. Besides, for the question of balancing workload of assigned workers during cross-trained worker assignment, most of previous studies paid attention to flow-line manufacturing system or a single manufacturing cell. The solution of this question for multiple manufacturing cells should receive more concern because it is more complex and practical due to the interplay among cells.Considering the high complexity of the NP-hard problem, optimization software (LINGO, CPLEX) were accompanied with long runtime consumption. Consequently, efficient metaheuristics for the models have been attracting growing attention in recent years, especially swarm intelligence metaheuristics. For example, ABC was employed to address the uncovering community problem in complex networks [42] and a novel PSO algorithm was utilized for power system optimization [43]. However, few researchers used the swarm intelligence metaheuristics to solve the problem of cross-trained worker assignment for multiple cells.To fill the gaps in the literature, this study aims to build a new model to address the worker assignment problem among multiple manufacturing cells while incorporating cross-training planning for workload balance and develop solutions based on the swarm intelligence algorithms.


## 3. Proposed Model for Cross-Trained Worker Assignment

This section elaborates the mathematical model for the problem of cross-trained worker assignment in CMS. We assume that part families have already been formed and the machines have been allocated to each part family. The type of manufacturing cell is considered a divisional cell where tasks are divided among several workers in accordance with their skill levels and task time. Workers cooperate and shuttle among the workstations to complete one or several discrete or successive tasks [44]. Based on the feature of cellular manufacturing system, several part types with similar operations form a part family which are processed by a group of machines and workers in each manufacturing cell. That means tasks for processing different part types in a cell are similar. In this paper, we consider skill as worker’s capability on assigned task including machine and hand operations. Some other operational assumptions and statements are stated as follows.(i) The initial skill level of each worker is given.(ii) The processing time for each task for each part type is known.(iii) Each worker is assigned just to one cell.(iv) Each task is assigned just to one worker.(v) Machine breakdown and absenteeism are not considered.(vi) All materials are delivered on time.(vii) Training time is not considered.(viii) The training cost of each worker for each task depends on the task type and the skill level, but not on the part types. For instance, part types 1 and 2 include task 1 which is handled by worker 1 in a cell. Training cost of worker 1 just depends on task 1 but not on the part type.(ix) Each worker can handle all part types that arrive at his or her task stations.


### 3.1. Model Notations

Table2 summarizes the index sets, parameters, and variables used in the proposed model.Table 2
Model notations and definitions.
Index sets Definitions c Index of cellsc=1,2,…,C p Index of partsp=1,2,…,P j Index of tasksj=1,2,…,J k Index of workersk=1,2,…,K l Index of skill levelsl=1,2,…,L  Parameters Definitions  C Number of cells P Number of parts J Number of tasks K Number of workers L Number of skill levels T C j Training cost of taskj T c p j Standard processing time of part typep at task j in cell c d c p j Required throughput of part typep at task j in cell c in one shift W k j Initial skill level weight of workerk at task j w l Weight of skill levell H Available working time in one shift V Maximum permissible training cost α Penalty cost of workload imbalance M Large value π 1 , π 2 Weight factors  Variables Definitions  Z c k l j 1, if workerk with skill level l will be assigned to task j in cell c; 0, otherwise y c k j 1, if workerk will be assigned to task j in cell c; 0, otherwise N c k 1, if workerk will be assigned into cell c; 0, otherwise W L k Workload of workerk W B c Workload of bottleneck worker in cellc


### 3.2. Mathematical Formulation

The cross-trained worker assignment problem for multiple cells in CMS can be formulated as a nonlinear 0–1 integer programming model as follows:(1)Min=π1∗∑c=1C∑k=1K∑j=1J∑l=1Lwl-Wkj∗TCj∗Zcklj(2)+π2∗α∗∑c=1CWBc∗∑k=1KNck∑k=1KWLk∗Nck(3)+∑c=1C∑k=1KNck∗Msubject to(4)∑k=1Kyckj=1for∀c,jsatisfying∑p=1PTcpj>0(5)∑k=1Kyckj=0for∀c,jsatisfying∑p=1PTcpj=0(6)Nck=1for∀c,ksatisfying∑j=1Jyckj≥1(7)Nck=0for∀c,ksatisfying∑j=1Jyckj=0(8)∑c=1CNck≤1∀k(9)∑l=1LZcklj=yckj∀c,k,j(10)∑j=1J∑l=1L∑p=1PZcklj∗Tcpj∗dcpjwl≤H∀c,k(11)WLk=∑c=1C∑j=1J∑l=1L∑p=1PTcpj∗Zcklj∗dcpjwl∗H∀k(12)WBc=maxk∈1,2,…K⁡WLk∗Nck∀c(13)∑l=1LZcklj∗wl≥yckj∗Wkj,∀c,k,j(14)∑c=1C∑k=1K∑j=1J∑l=1Lwl-Wkj∗TCj∗Zcklj≤V(15)yckj,Nck,Zcklj=0or1∀c,k,l,jThe first part of the above objective function is to minimize total skill training cost of the needed workers. The model set skill level varies according to standard task time. A higher skill level refers to the worker needing less time to finish the assigned task. The cost increases when the worker is trained from a lower skill level to a higher one.The second part of the objective function is to minimize the penalty cost of workload imbalance, i.e., to maximize workload balance of assigned workers in each cell. The penalty cost forces workload balance by reasonable worker assignment and cross-training.π1 andπ2 display the trade-off between training expenditure and workload balance. A relatively lower value ofπ1 and a higher value ofπ2 stand for the situation that training is recommended strongly for more exact workload balance, but is accompanied with more expenditure. Conversely, a relatively higher value ofπ1 and a lower value ofπ2 stand for the situation that training is not recommended for less expenditure, but is accompanied with less exact workload balance. The decision-maker could set the value ofπ1 andπ2 according to the recommendation of training for skill level enhancement.The third part focuses on minimizing the number of assigned workers. Considering the current economic environment where labor cost is rapidly increasing, this research considers the minimum number of workers as the top priority in the model.Constraints (4) and (5) ensure that all tasks should be allocated to the workers. Constraints (6) and (7) guarantee that one worker can handle more than one task. Constraint (8) ensures that one worker could not be assigned to more than one cell. Constraint (9) guarantees that worker assigned to a task with one and only skill level. Constraint (10) ensures that the customer demand is met. Constraint (11) calculates workload of each assigned worker. Constraint (12) defines workload of the bottleneck worker in cell. Constraint (13) ensures that the skill level of the assigned worker is not less than its initial level. Constraint (14) ensures that the total work training cost does not exceed budget.


## 3.1. Model Notations

Table2 summarizes the index sets, parameters, and variables used in the proposed model.Table 2
Model notations and definitions.
Index sets Definitions c Index of cellsc=1,2,…,C p Index of partsp=1,2,…,P j Index of tasksj=1,2,…,J k Index of workersk=1,2,…,K l Index of skill levelsl=1,2,…,L  Parameters Definitions  C Number of cells P Number of parts J Number of tasks K Number of workers L Number of skill levels T C j Training cost of taskj T c p j Standard processing time of part typep at task j in cell c d c p j Required throughput of part typep at task j in cell c in one shift W k j Initial skill level weight of workerk at task j w l Weight of skill levell H Available working time in one shift V Maximum permissible training cost α Penalty cost of workload imbalance M Large value π 1 , π 2 Weight factors  Variables Definitions  Z c k l j 1, if workerk with skill level l will be assigned to task j in cell c; 0, otherwise y c k j 1, if workerk will be assigned to task j in cell c; 0, otherwise N c k 1, if workerk will be assigned into cell c; 0, otherwise W L k Workload of workerk W B c Workload of bottleneck worker in cellc


## 3.2. Mathematical Formulation

The cross-trained worker assignment problem for multiple cells in CMS can be formulated as a nonlinear 0–1 integer programming model as follows:(1)Min=π1∗∑c=1C∑k=1K∑j=1J∑l=1Lwl-Wkj∗TCj∗Zcklj(2)+π2∗α∗∑c=1CWBc∗∑k=1KNck∑k=1KWLk∗Nck(3)+∑c=1C∑k=1KNck∗Msubject to(4)∑k=1Kyckj=1for∀c,jsatisfying∑p=1PTcpj>0(5)∑k=1Kyckj=0for∀c,jsatisfying∑p=1PTcpj=0(6)Nck=1for∀c,ksatisfying∑j=1Jyckj≥1(7)Nck=0for∀c,ksatisfying∑j=1Jyckj=0(8)∑c=1CNck≤1∀k(9)∑l=1LZcklj=yckj∀c,k,j(10)∑j=1J∑l=1L∑p=1PZcklj∗Tcpj∗dcpjwl≤H∀c,k(11)WLk=∑c=1C∑j=1J∑l=1L∑p=1PTcpj∗Zcklj∗dcpjwl∗H∀k(12)WBc=maxk∈1,2,…K⁡WLk∗Nck∀c(13)∑l=1LZcklj∗wl≥yckj∗Wkj,∀c,k,j(14)∑c=1C∑k=1K∑j=1J∑l=1Lwl-Wkj∗TCj∗Zcklj≤V(15)yckj,Nck,Zcklj=0or1∀c,k,l,jThe first part of the above objective function is to minimize total skill training cost of the needed workers. The model set skill level varies according to standard task time. A higher skill level refers to the worker needing less time to finish the assigned task. The cost increases when the worker is trained from a lower skill level to a higher one.The second part of the objective function is to minimize the penalty cost of workload imbalance, i.e., to maximize workload balance of assigned workers in each cell. The penalty cost forces workload balance by reasonable worker assignment and cross-training.π1 andπ2 display the trade-off between training expenditure and workload balance. A relatively lower value ofπ1 and a higher value ofπ2 stand for the situation that training is recommended strongly for more exact workload balance, but is accompanied with more expenditure. Conversely, a relatively higher value ofπ1 and a lower value ofπ2 stand for the situation that training is not recommended for less expenditure, but is accompanied with less exact workload balance. The decision-maker could set the value ofπ1 andπ2 according to the recommendation of training for skill level enhancement.The third part focuses on minimizing the number of assigned workers. Considering the current economic environment where labor cost is rapidly increasing, this research considers the minimum number of workers as the top priority in the model.Constraints (4) and (5) ensure that all tasks should be allocated to the workers. Constraints (6) and (7) guarantee that one worker can handle more than one task. Constraint (8) ensures that one worker could not be assigned to more than one cell. Constraint (9) guarantees that worker assigned to a task with one and only skill level. Constraint (10) ensures that the customer demand is met. Constraint (11) calculates workload of each assigned worker. Constraint (12) defines workload of the bottleneck worker in cell. Constraint (13) ensures that the skill level of the assigned worker is not less than its initial level. Constraint (14) ensures that the total work training cost does not exceed budget.


## 4. Swarm Intelligence Metaheuristics for Worker Assignment

In this section, two classical PSO variants, i.e., GPSO and LPSO, and ABC are introduced briefly. Next, we present the STABC with enhanced learning strategy to improve computational performance. This is followed by the designed implementations of the algorithms for the mathematical model.


### 4.1. Particle Swarm Optimization

Particle swarm optimization (PSO) is a stochastic swarm-based metaheuristic inspired by the group behaviors of bird flocking and fishing schooling [11]. In PSO, a group of particles are employed to collaboratively explore the problem landscape. Each particle has velocity and position information and moves towards its historical best position in conjunction with population best position found so far. Supposing that searching space isD-dimensional, them-th particle’s velocity and position are updated as follows:(16)pvmd=pvmd+r1∗rand1md∗pbestmd-Xmd+r2∗rand2md∗gbestmd-Xmd(17)Xmd=Xmd+pvmdwhere r1 and r2 are two positive constriction coefficients; rand1d and rand2d are two random numbers between [0,1]. Xm=(Xm1,Xm2…,XmD) is the position of particlem; pvm=(pvm1,pvm2,…,pvmD) is the velocity of particlem; pbestm=(pbestm1,pbestm2,…,pbestmD) is the best previous position for particlem; and gbestm=(gbestm1,gbestm2…,gbestmD) is the best position discovered by the population so far.Many variants of PSO have been studied since the inception [45, 46]. A standard version is presented in [32]. The standard PSO employs constricted factorχ, as shown in the following, to guide population search:(18)χ=22-φ-φ2-4φ,φ=r1+r2where r1 and r2 are set to be constants to ensure the speed of convergence [45]. With the constriction factor, the original velocity equation is given as follows:(19)pvmd=χpvmd+r1∗rand1md∗pbestmd-Xmd+r2∗rand2md∗LEmd-XmdwhereLEd represents the previous best information in population. It is GPSO when a neighbor could be anyone within the population; otherwise, it is LPSO. The structures of the two patterns are shown in Figure 1. The general procedures of PSO are given as follows:(1)
Initialize population size of particle swarm, maximum number of iterations and the constants (r1 and r2).(2)
Randomly generate a swarm of particles and initialize the velocity and position of each particle in the search boundary.(3)
Evaluate each particle’s fitness value.(4)
Renew particle’s personal best by comparing current best information with its historical best information.(5)
Update the swarm best information using the best neighbors’ information in population.(6)
Each particle updates its velocity by (19).(7)
Each particle updates its position by (17).(8)
Go to (3) if the stop criteria are not met; otherwise, terminate the iteration.Figure 1
Global and local topology structure.


### 4.2. Artificial Bee Colony Algorithm

Artificial bee colony (ABC) algorithm is inspired by the foraging behaviors in a honey bee swarm for real-parameter optimization [12]. In ABC, there are three types of bees: employed bees, onlooker bees, and scout bees. The goal of the whole bees is to collaboratively find the largest source of nectar. The actual location of the food source represents a possible solution for the optimization problems in ABC, and the density of nectar corresponds to a fitness value of solution. The number of food sources represents the swarm sizePS.In the iteration, each employed bee explores the food source using the following.(20)bvmd=xmd+ϕmdxmd-xqd,m≠q.where bvmd denotes a candidate food source positionm,m = 1, 2, …, PS,ϕmd is a randomly produced value within [-1,1], xmd refers to the corresponding food source, and xqd is a randomly selected food source.Each onlooker bee randomly selects its food source,m, for further refinement with a probability, pm, which is determined as follows: (21)pm=fitnessm∑s=1Sfitnessswhere fitnessm is the quality of them-th food source. For the minimization problem, the fitnessm can be calculated using the following.(22)fitnessm=11+fmiffm≥01+absfmiffm≤0where fm is them-th objective function value. The value of“limit” to abandon the poor food source is suggested to be set toPS∗D[47]. The main steps of ABC are given below:(1)
Initialize the number of food sources, thelimit and the maximum number of iterations.(2)
Randomly generate the positions of food sources in problem landscape.(3)
Evaluate the fitness value of food sources found currently.(4)
Each employed bee explores between a random exemplar and their own food source using (20).(5)
Apply greedy selection process to select the bestPS positions and record the updated food sources.(6)
Calculate the probability of the food sources with which onlooker bees are going to follow by (21).(7)
Produce new solution using (20) for the onlooker bees depending on pi.(8)
Conduct greedy selection and update food source information for onlooker bees.(9)
Abandon the food sources with poor nectar and let scouts search for new food sources.(10)
Terminate if the stop criteria are met; otherwise go to (4).


### 4.3. Superior Tracking Artificial Bee Colony Algorithm

As a newly proposed algorithm, while promising, ABC still suffers from two weaknesses. First, the convergence speed is relatively slow compared to many existing swarm intelligence algorithms. Second, the capability of exploring global optima is still underperforming on many multimodal problems [48]. In the original search behavior of ABC, only one dimension of a bee is selected to learn and renew in each round of information updating. This search rule may result in the slow convergence or inferior exploring capability [49]. To address this issue, the STABC is introduced to improve the search capability and convergence speed for global optimization.In STABC, the employed bees and onlooker bees are updated in accordance with the following equation:(23)bvm=xm+rmxm-SNmwhere bvm represents the food position of them-th (m =1, 2, …,PS) food source to be updated, rm refers to a randomly generated D-dimensional vector, andSNis theD-dimensional constructed exemplar for tracking, in which the elements are either from itself or from the other superior food sources randomly chosen by roulette selection and tournament selection.The detailed procedure of constructingSNm is presented as follows:(1)
Initialize a given probabilityPr to decide which exemplar the current bee should chase, itself or the other superior neighbors in the population.(2)
For each dimension ofSNmd (d = 1 toD), a value within [0,1] is randomly generated.(3)
If the generated value is larger thanPr, two food positions are randomly chosen for comparison, and thed-th dimension of the better one will be learned; otherwise, SNmd will be itself.(4)
Repeat the above steps till all dimensions ofSNm are determined.In STABC, all elements of an individual will be likely to update at each iteration. As a result, each bee either exploits the local optima by learning from itself or explores the global landscape by chasing the other individuals in the population. This search strategy accelerates the convergence speeds of the population. In addition, it remains from ABC original properties that every individual could be learning exemplars. Figure2 displays the flow chart of the STABC algorithm.Figure 2
Flowchart of the STABC algorithm.


### 4.4. Implementation of the SI-Based Optimizers for Cell Formation

In the proposed mathematical model, the computational complexity is dependent on the variable size ofZcklj which is equal to c∗k∗l∗j. The search dimension of solutionZcklj will be too large to optimize using the traditional mathematical methods. Our preliminary experiment indicates that it is extremely hard to obtain a suboptimum if the variables are without proper treatment. Meanwhile, a feasible solutionZ would be hard to locate due to the complexity and discontinuity of constraints. To address these issues, a SI-based metaheuristic is proposed to explore the global optima. While implementing the swarm-based optimizers for cross-trained worker assignment problem, three important issues should be considered: encoding, decoding, and constrains handling.An effective encoding method is developed in this study. The original 0-1 planning problem is transformed into an integer programming problem, in which the targeting skill level and the worker are allocated to each task to meet the production requirements. For the population initialization, them-th individual is represented as xm=[xm,1,xm,2,…,xm,j],j = 1, 2, …,J, whereJ denotes the value of tasks. The range of the algorithmic representation xm,1 is [1,k+l].In the decoding process, the worker and skill level indices need to be determined through the algorithmic representation. A cross-reference matrix is generated here for converting individualxm to the indices of worker and skill level. Details of procedures involved in the decoding process for xm are shown in Figure 3.Figure 3
Decoding procedure.For the constraints handling process, the basic penalty function method is adopted [50]. Once the objective function value of xm and the corresponding constraint violation have been obtained, the penalty value is added to the objective function value of xm.


## 4.1. Particle Swarm Optimization

Particle swarm optimization (PSO) is a stochastic swarm-based metaheuristic inspired by the group behaviors of bird flocking and fishing schooling [11]. In PSO, a group of particles are employed to collaboratively explore the problem landscape. Each particle has velocity and position information and moves towards its historical best position in conjunction with population best position found so far. Supposing that searching space isD-dimensional, them-th particle’s velocity and position are updated as follows:(16)pvmd=pvmd+r1∗rand1md∗pbestmd-Xmd+r2∗rand2md∗gbestmd-Xmd(17)Xmd=Xmd+pvmdwhere r1 and r2 are two positive constriction coefficients; rand1d and rand2d are two random numbers between [0,1]. Xm=(Xm1,Xm2…,XmD) is the position of particlem; pvm=(pvm1,pvm2,…,pvmD) is the velocity of particlem; pbestm=(pbestm1,pbestm2,…,pbestmD) is the best previous position for particlem; and gbestm=(gbestm1,gbestm2…,gbestmD) is the best position discovered by the population so far.Many variants of PSO have been studied since the inception [45, 46]. A standard version is presented in [32]. The standard PSO employs constricted factorχ, as shown in the following, to guide population search:(18)χ=22-φ-φ2-4φ,φ=r1+r2where r1 and r2 are set to be constants to ensure the speed of convergence [45]. With the constriction factor, the original velocity equation is given as follows:(19)pvmd=χpvmd+r1∗rand1md∗pbestmd-Xmd+r2∗rand2md∗LEmd-XmdwhereLEd represents the previous best information in population. It is GPSO when a neighbor could be anyone within the population; otherwise, it is LPSO. The structures of the two patterns are shown in Figure 1. The general procedures of PSO are given as follows:(1)
Initialize population size of particle swarm, maximum number of iterations and the constants (r1 and r2).(2)
Randomly generate a swarm of particles and initialize the velocity and position of each particle in the search boundary.(3)
Evaluate each particle’s fitness value.(4)
Renew particle’s personal best by comparing current best information with its historical best information.(5)
Update the swarm best information using the best neighbors’ information in population.(6)
Each particle updates its velocity by (19).(7)
Each particle updates its position by (17).(8)
Go to (3) if the stop criteria are not met; otherwise, terminate the iteration.Figure 1
Global and local topology structure.


## 4.2. Artificial Bee Colony Algorithm

Artificial bee colony (ABC) algorithm is inspired by the foraging behaviors in a honey bee swarm for real-parameter optimization [12]. In ABC, there are three types of bees: employed bees, onlooker bees, and scout bees. The goal of the whole bees is to collaboratively find the largest source of nectar. The actual location of the food source represents a possible solution for the optimization problems in ABC, and the density of nectar corresponds to a fitness value of solution. The number of food sources represents the swarm sizePS.In the iteration, each employed bee explores the food source using the following.(20)bvmd=xmd+ϕmdxmd-xqd,m≠q.where bvmd denotes a candidate food source positionm,m = 1, 2, …, PS,ϕmd is a randomly produced value within [-1,1], xmd refers to the corresponding food source, and xqd is a randomly selected food source.Each onlooker bee randomly selects its food source,m, for further refinement with a probability, pm, which is determined as follows: (21)pm=fitnessm∑s=1Sfitnessswhere fitnessm is the quality of them-th food source. For the minimization problem, the fitnessm can be calculated using the following.(22)fitnessm=11+fmiffm≥01+absfmiffm≤0where fm is them-th objective function value. The value of“limit” to abandon the poor food source is suggested to be set toPS∗D[47]. The main steps of ABC are given below:(1)
Initialize the number of food sources, thelimit and the maximum number of iterations.(2)
Randomly generate the positions of food sources in problem landscape.(3)
Evaluate the fitness value of food sources found currently.(4)
Each employed bee explores between a random exemplar and their own food source using (20).(5)
Apply greedy selection process to select the bestPS positions and record the updated food sources.(6)
Calculate the probability of the food sources with which onlooker bees are going to follow by (21).(7)
Produce new solution using (20) for the onlooker bees depending on pi.(8)
Conduct greedy selection and update food source information for onlooker bees.(9)
Abandon the food sources with poor nectar and let scouts search for new food sources.(10)
Terminate if the stop criteria are met; otherwise go to (4).


## 4.3. Superior Tracking Artificial Bee Colony Algorithm

As a newly proposed algorithm, while promising, ABC still suffers from two weaknesses. First, the convergence speed is relatively slow compared to many existing swarm intelligence algorithms. Second, the capability of exploring global optima is still underperforming on many multimodal problems [48]. In the original search behavior of ABC, only one dimension of a bee is selected to learn and renew in each round of information updating. This search rule may result in the slow convergence or inferior exploring capability [49]. To address this issue, the STABC is introduced to improve the search capability and convergence speed for global optimization.In STABC, the employed bees and onlooker bees are updated in accordance with the following equation:(23)bvm=xm+rmxm-SNmwhere bvm represents the food position of them-th (m =1, 2, …,PS) food source to be updated, rm refers to a randomly generated D-dimensional vector, andSNis theD-dimensional constructed exemplar for tracking, in which the elements are either from itself or from the other superior food sources randomly chosen by roulette selection and tournament selection.The detailed procedure of constructingSNm is presented as follows:(1)
Initialize a given probabilityPr to decide which exemplar the current bee should chase, itself or the other superior neighbors in the population.(2)
For each dimension ofSNmd (d = 1 toD), a value within [0,1] is randomly generated.(3)
If the generated value is larger thanPr, two food positions are randomly chosen for comparison, and thed-th dimension of the better one will be learned; otherwise, SNmd will be itself.(4)
Repeat the above steps till all dimensions ofSNm are determined.In STABC, all elements of an individual will be likely to update at each iteration. As a result, each bee either exploits the local optima by learning from itself or explores the global landscape by chasing the other individuals in the population. This search strategy accelerates the convergence speeds of the population. In addition, it remains from ABC original properties that every individual could be learning exemplars. Figure2 displays the flow chart of the STABC algorithm.Figure 2
Flowchart of the STABC algorithm.


## 4.4. Implementation of the SI-Based Optimizers for Cell Formation

In the proposed mathematical model, the computational complexity is dependent on the variable size ofZcklj which is equal to c∗k∗l∗j. The search dimension of solutionZcklj will be too large to optimize using the traditional mathematical methods. Our preliminary experiment indicates that it is extremely hard to obtain a suboptimum if the variables are without proper treatment. Meanwhile, a feasible solutionZ would be hard to locate due to the complexity and discontinuity of constraints. To address these issues, a SI-based metaheuristic is proposed to explore the global optima. While implementing the swarm-based optimizers for cross-trained worker assignment problem, three important issues should be considered: encoding, decoding, and constrains handling.An effective encoding method is developed in this study. The original 0-1 planning problem is transformed into an integer programming problem, in which the targeting skill level and the worker are allocated to each task to meet the production requirements. For the population initialization, them-th individual is represented as xm=[xm,1,xm,2,…,xm,j],j = 1, 2, …,J, whereJ denotes the value of tasks. The range of the algorithmic representation xm,1 is [1,k+l].In the decoding process, the worker and skill level indices need to be determined through the algorithmic representation. A cross-reference matrix is generated here for converting individualxm to the indices of worker and skill level. Details of procedures involved in the decoding process for xm are shown in Figure 3.Figure 3
Decoding procedure.For the constraints handling process, the basic penalty function method is adopted [50]. Once the objective function value of xm and the corresponding constraint violation have been obtained, the penalty value is added to the objective function value of xm.


## 5. Computational Experiment

To validate effectiveness and efficiency of the proposed mathematical model and optimizers, computational experiments are performed in this section. Ten problems are randomly generated based on previous research (can be accessed athttps://drive.google.com/open?id=0B18FVuMxAffXLW1mRzVLczlyem8). The experiments are implemented using MATLAB R2012a, and the desktop is with Intel Core i5 CPU 3.20 GHz and 8 GB memory.


### 5.1. Experiment Settings

The cross-trained worker assignment problem is based on the formed manufacturing cells in this research; that is, part and tasks types had been fixed after cell formation in CMS. Ten cell formation problems are selected from previous literature (see Table3). As the problems are generated with different sizes, they are not directly correlated to each other. The different types of problem size can comprehensively evaluate the scalable performance of the proposed method on this mathematical model. The additional parameters are randomly generated as shown in Table 4. To allow one-to-one correspondence between workers and tasks in extreme situation, we assumed that the number of available workers is equal to the number of tasks in CMS. The available working time of each worker in a shift is calculated as 8 h∗60m∗85%=408m. The training budget in the factory is sufficient. Penalty cost of workload imbalance α is assumed to be 100. With respect to the trade-off between training expenditure and workload balance, “π1=1,π2=10” stands for the first situation, which refers to the pursuit of more exact workload balance although more training cost should be spent, while “π1=10,π2=1” stands for the second situation, which refers to saving training cost although workload balance may not be enforced very well.Table 3
Dimension of test problems.
Problem No. Number of Parts Tasks Cells Workers P1 [34] 5 5 2 5 P2 [35] 7 5 2 5 P3 [36] 6 6 2 6 P4 [37] 8 9 2 9 P5 [38] 10 8 3 8 P6 [36] 8 10 3 10 P7 [39] 10 10 3 10 P8 [35] 12 10 3 10 P9 [40] 18 10 3 10 P10 [41] 22 11 3 11Table 4
Pattern of data generation.
Parameters Values T C j ∗ DU(500, 1000) T c p j DU(0.1, 1.5) d c p j DU(50, 150) W k j Randomly generated in set (0, 0.7, 0.8, 0.9, 1, 1.1) w l w 1 =0.7, w2=0.8, w3=0.9, w4=1.0, w5=1.1 ∗ DU(m, n) means discrete uniform distribution varying between m and n. The unit of time is minute.The maximum number of iterations is set as 20000 where all the candidate algorithms are found to converge fast. For the standard PSO, the constantφ=4.1 and the valuesχ=0.7298 andr1=r2=2.05 are gathered (Bratton and Kennedy 2007). For the STABC, the range of rm is empirically set as [0, 1.49], which could be a time-varying function as well [49]. With respect to the threshold probabilityPr of STABC, the method introduced in Liang et al.’s research [51] is adopted. The population size is set at 30 for all algorithms [47].


### 5.2. Experimental Results

Each problem is run independently 20 times with the four algorithms (GPSO, LPSO, ABC, and STABC). The CPU time for each run does not exceed 90s. In Table5, OFVb, OFVw, and OFVa represent the best, worst, and average objective function values (OFV) of the ten problems after 20 runs under the two situations (π1=1,π2=10) and (π1=10,π2=1). The best solutions of OFVb, OFVw, and OFVa found by the four algorithms are shown in bold. Based on experimental results, three measures to evaluate the effectiveness of STABC algorithm are defined as follows:Table 5
Results from GPSO, LPSO, ABC, and STABC.
GPSO LPSO ABC STABC O F V b O F V w O F V a O F V b O F V w O F V a O F V b O F V w O F V a O F V b O F V w O F V a P1 π1=1, π2=10 33183 33307 33212 33183 33344 33285 32852 42983 35125 32839 43122 34597 π1=10, π2=1 41163 41958 41590 41258 42753 42152 38955 48235 44105 38105 48837 42540  P2 π1=1, π2=10 32607 32667 32638 32596 32714 32648 32259 32959 32370 32259 33204 32499 π1=10, π2=1 35260 35865 35293 35216 35860 35330 32602 40752 35266 32602 40907 33392  P3 π1=1, π2=10 33497 33782 33607 33453 33769 33619 32772 33116 32885 32772 32864 32802 π1=10, π2=1 42745 44415 43450 42745 44344 43563 36861 41403 38321 36811 39207 37424  P4 π1=1, π2=10 53681 57009 55124 53277 56045 54768 63133 76480 67822 52493 64168 60957 π1=10, π2=1 66237 80183 75246 65879 81657 76064 55113 91142 79567 52515 77852 67482  P5 π1=1, π2=10 54720 55644 55089 54771 55589 55163 53669 55151 54324 53607 54088 53879 π1=10, π2=1 59274 67264 63857 59871 69167 64942 52293 63298 57622 51536 57031 53826  P6 π1=1, π2=10 55149 66403 60150 55108 66335 56981 46219 63596 54613 53296 54392 53768 π1=10, π2=1 70131 84437 79483 72363 86381 80558 55765 64943 61086 51664 61076 55726  P7 π1=1, π2=10 65351 68255 66758 65730 69136 67249 64034 74700 65529 63464 64682 64020 π1=10, π2=1 82555 99389 92564 87731 103806 96119 66279 87778 76697 63026 74477 68798  P8 π1=1, π2=10 75074 78381 76041 75630 78120 76446 73638 85692 80359 73637 83820 78977 π1=10, π2=1 84676 97413 91203 85879 102599 93956 72901 90037 81322 72557 81016 77905  P9 π1=1, π2=10 75750 78238 77138 76227 78917 77378 74097 85694 76667 73342 74960 74237 π1=10, π2=1 97288 117201 105674 95973 120132 108403 81629 105846 89569 71176 92650 79304  P10 π1=1, π2=10 86208 89887 88252 85905 98570 89542 85065 95448 87622 83881 86995 84821 π1=10, π2=1 94291 132599 121452 110404 139528 123986 87300 121262 103559 85200 101088 91675( 1 )   B G S T A B C ∗ denotes the gap of OFVb between STABC and any other algorithms (GPSO, LPSO, or ABC). For instance, BGSTABCGPSO means the rising percentage of OFVbby GPSO over OFVb by STABC.( 2 )   W G S T A B C ∗ denotes the gap of OFVwbetween STABC and any other algorithms (GPSO, LPSO, or ABC). For instance, WGSTABCLPSO means the rising percentage of OFVw by LPSO over OFVw by STABC.( 3 )   A G S T A B C ∗ denotes the gap of OFVa between STABC and any other algorithms (GPSO, LPSO, or ABC). For instance, AGSTABCABC means the rising percentage of OFVa by ABC over OFVa by STABC.With respect to the above measures, the presented STABC is employed as the basis to compare the performance of the involved algorithms. A positive value indicates that STABC outperforms the other algorithms on the optimized problem.Table6 shows the comparison results on the ten problems under the two situations (π1=1,π2=10) and (π1=10,π2=1). We see that most of BGSTABC∗, WGSTABC∗, AGSTABC∗, and all average values are positive. Thus, better results with respect to OFVb, OFVw, and OFVa can be attained by STABC compared to GPSO, LPSO, and ABC under both situations (π1=1,π2=10) and (π1=10,π2=1)Table 6
Performance comparison among GPSO, LPSO, ABC, and STABC.
B G S T A B C G P S O W G S T A B C G P S O A G S T A B C G P S O B G S T A B C L P S O W G S T A B C L P S O A G S T A B C L P S O B G S T A B C A B C W G S T A B C A B C A G S T A B C A B C P1 π1=1, π2=10 1.0% -22.8% -4.0% 1.0% -22.7% -3.8% 0.0% -0.3% 1.5% π1=10, π2=1 8.0% -14.1% -2.2% 8.3% -12.5% -0.9% 2.2% -1.2% 3.7%  P2 π1=1, π2=10 1.1% -1.6% 0.4% 1.0% -1.5% 0.5% 0.0% -0.7% -0.4% π1=10, π2=1 8.2% -12.3% 5.7% 8.0% -12.3% 5.8% 0.0% -0.4% 5.6%  P3 π1=1, π2=10 2.2% 2.8% 2.5% 2.1% 2.8% 2.5% 0.0% 0.8% 0.3% π1=10, π2=1 16.1% 13.3% 16.1% 16.1% 13.1% 16.4% 0.1% 5.6% 2.4%  P4 π1=1, π2=10 2.3% -11.2% -9.6% 1.5% -12.7% -10.2% 20.3% 19.2% 11.3% π1=10, π2=1 26.1% 3.0% 11.5% 25.4% 4.9% 12.7% 4.9% 17.1% 17.9%  P5 π1=1, π2=10 2.1% 2.9% 2.2% 2.2% 2.8% 2.4% 0.1% 2.0% 0.8% π1=10, π2=1 15.0% 17.9% 18.6% 16.2% 21.3% 20.7% 1.5% 11.0% 7.1%  P6 π1=1, π2=10 3.5% 22.1% 11.9% 3.4% 22.0% 6.0% -13.3% 16.9% 1.6% π1=10, π2=1 35.7% 38.2% 42.6% 40.1% 41.4% 44.6% 7.9% 6.3% 9.6%  P7 π1=1, π2=10 3.0% 5.5% 4.3% 3.6% 6.9% 5.0% 0.9% 15.5% 2.4% π1=10, π2=1 31.0% 33.4% 34.5% 39.2% 39.4% 39.7% 5.2% 17.9% 11.5%  P8 π1=1, π2=10 2.0% -6.5% -3.7% 2.7% -6.8% -3.2% 0.0% 2.2% 1.7% π1=10, π2=1 16.7% 20.2% 17.1% 18.4% 26.6% 20.6% 0.5% 11.1% 4.4%  P9 π1=1, π2=10 3.3% 4.4% 3.9% 3.9% 5.3% 4.2% 1.0% 14.3% 3.3% π1=10, π2=1 36.7% 26.5% 33.3% 34.8% 29.7% 36.7% 14.7% 14.2% 12.9%  P10 π1=1, π2=10 2.8% 3.3% 4.0% 2.4% 13.3% 5.6% 1.4% 9.7% 3.3% π1=10, π2=1 10.7% 31.2% 32.5% 29.6% 38.0% 35.2% 2.5% 20.0% 13.0%  Mean 11.4 % 7.8 % 11.1 % 13.0 % 9.9 % 12.0 % 2.5 % 9.1 % 5.7 %Figures4 and 5 compare the OFVbobtained by GPSO, LPSO, ABC, and STABC (see Table 5). They demonstrate that the best solutions of STABC are better than the others. For the relatively simple small-size problems, such as problems 1, 2, and 3, most of the algorithms can easily identify the best results. With the increase of problem size, STABC outperforms the other algorithms on the relatively large-size problems. The improvement of STBAC’s search capability can be attributed to two properties: (1) each individual’s learning speed for exploration is improved by learning from other bees in all dimensions and (2) the individual’s learning sources for exploitation are enriched by learning from both its historical information and the superior individuals. The novel strategies can significantly enhance the global exploration capability and local exploitation capability especially in large-size problems which are proved by the experimental results.Figure 4
O F V b results for the case of π1=1, π2=10.Figure 5
O F V b results for the case of π1=10, π2=1.The CPU time of all test problems by the four algorithms fluctuates between 10s and 90s. Although the advantage of computational efficiency of STABC is not significant among the four algorithms, it is efficient to obtain solution of the problem. In contrast, the computational software Lingo is unable to obtain any feasible results in half an hour for most of the experiments.In addition, we compare the training cost and workload balance associated with the ten problems between the two scenarios (π1=1,π2=10) and (π1=10,π2=1); see Figures 6 and 7. All the training cost and workload balance in scenario (π1=10,π2=1) is equal to or lower than that of scenario (π1=1,π2=10). This indicates that less training effort which is in the scenario (π1=10,π2=1) will lead to less balanced workload in all the test problems though corresponding worker assignment is reasonable. It indicates that the cross-training strategy obtained by our proposed approach is effective while seeking workload balance.Figure 6
Training costs obtained by STABC.Figure 7
Workload balance results obtained by STABC.


## 5.1. Experiment Settings

The cross-trained worker assignment problem is based on the formed manufacturing cells in this research; that is, part and tasks types had been fixed after cell formation in CMS. Ten cell formation problems are selected from previous literature (see Table3). As the problems are generated with different sizes, they are not directly correlated to each other. The different types of problem size can comprehensively evaluate the scalable performance of the proposed method on this mathematical model. The additional parameters are randomly generated as shown in Table 4. To allow one-to-one correspondence between workers and tasks in extreme situation, we assumed that the number of available workers is equal to the number of tasks in CMS. The available working time of each worker in a shift is calculated as 8 h∗60m∗85%=408m. The training budget in the factory is sufficient. Penalty cost of workload imbalance α is assumed to be 100. With respect to the trade-off between training expenditure and workload balance, “π1=1,π2=10” stands for the first situation, which refers to the pursuit of more exact workload balance although more training cost should be spent, while “π1=10,π2=1” stands for the second situation, which refers to saving training cost although workload balance may not be enforced very well.Table 3
Dimension of test problems.
Problem No. Number of Parts Tasks Cells Workers P1 [34] 5 5 2 5 P2 [35] 7 5 2 5 P3 [36] 6 6 2 6 P4 [37] 8 9 2 9 P5 [38] 10 8 3 8 P6 [36] 8 10 3 10 P7 [39] 10 10 3 10 P8 [35] 12 10 3 10 P9 [40] 18 10 3 10 P10 [41] 22 11 3 11Table 4
Pattern of data generation.
Parameters Values T C j ∗ DU(500, 1000) T c p j DU(0.1, 1.5) d c p j DU(50, 150) W k j Randomly generated in set (0, 0.7, 0.8, 0.9, 1, 1.1) w l w 1 =0.7, w2=0.8, w3=0.9, w4=1.0, w5=1.1 ∗ DU(m, n) means discrete uniform distribution varying between m and n. The unit of time is minute.The maximum number of iterations is set as 20000 where all the candidate algorithms are found to converge fast. For the standard PSO, the constantφ=4.1 and the valuesχ=0.7298 andr1=r2=2.05 are gathered (Bratton and Kennedy 2007). For the STABC, the range of rm is empirically set as [0, 1.49], which could be a time-varying function as well [49]. With respect to the threshold probabilityPr of STABC, the method introduced in Liang et al.’s research [51] is adopted. The population size is set at 30 for all algorithms [47].


## 5.2. Experimental Results

Each problem is run independently 20 times with the four algorithms (GPSO, LPSO, ABC, and STABC). The CPU time for each run does not exceed 90s. In Table5, OFVb, OFVw, and OFVa represent the best, worst, and average objective function values (OFV) of the ten problems after 20 runs under the two situations (π1=1,π2=10) and (π1=10,π2=1). The best solutions of OFVb, OFVw, and OFVa found by the four algorithms are shown in bold. Based on experimental results, three measures to evaluate the effectiveness of STABC algorithm are defined as follows:Table 5
Results from GPSO, LPSO, ABC, and STABC.
GPSO LPSO ABC STABC O F V b O F V w O F V a O F V b O F V w O F V a O F V b O F V w O F V a O F V b O F V w O F V a P1 π1=1, π2=10 33183 33307 33212 33183 33344 33285 32852 42983 35125 32839 43122 34597 π1=10, π2=1 41163 41958 41590 41258 42753 42152 38955 48235 44105 38105 48837 42540  P2 π1=1, π2=10 32607 32667 32638 32596 32714 32648 32259 32959 32370 32259 33204 32499 π1=10, π2=1 35260 35865 35293 35216 35860 35330 32602 40752 35266 32602 40907 33392  P3 π1=1, π2=10 33497 33782 33607 33453 33769 33619 32772 33116 32885 32772 32864 32802 π1=10, π2=1 42745 44415 43450 42745 44344 43563 36861 41403 38321 36811 39207 37424  P4 π1=1, π2=10 53681 57009 55124 53277 56045 54768 63133 76480 67822 52493 64168 60957 π1=10, π2=1 66237 80183 75246 65879 81657 76064 55113 91142 79567 52515 77852 67482  P5 π1=1, π2=10 54720 55644 55089 54771 55589 55163 53669 55151 54324 53607 54088 53879 π1=10, π2=1 59274 67264 63857 59871 69167 64942 52293 63298 57622 51536 57031 53826  P6 π1=1, π2=10 55149 66403 60150 55108 66335 56981 46219 63596 54613 53296 54392 53768 π1=10, π2=1 70131 84437 79483 72363 86381 80558 55765 64943 61086 51664 61076 55726  P7 π1=1, π2=10 65351 68255 66758 65730 69136 67249 64034 74700 65529 63464 64682 64020 π1=10, π2=1 82555 99389 92564 87731 103806 96119 66279 87778 76697 63026 74477 68798  P8 π1=1, π2=10 75074 78381 76041 75630 78120 76446 73638 85692 80359 73637 83820 78977 π1=10, π2=1 84676 97413 91203 85879 102599 93956 72901 90037 81322 72557 81016 77905  P9 π1=1, π2=10 75750 78238 77138 76227 78917 77378 74097 85694 76667 73342 74960 74237 π1=10, π2=1 97288 117201 105674 95973 120132 108403 81629 105846 89569 71176 92650 79304  P10 π1=1, π2=10 86208 89887 88252 85905 98570 89542 85065 95448 87622 83881 86995 84821 π1=10, π2=1 94291 132599 121452 110404 139528 123986 87300 121262 103559 85200 101088 91675( 1 )   B G S T A B C ∗ denotes the gap of OFVb between STABC and any other algorithms (GPSO, LPSO, or ABC). For instance, BGSTABCGPSO means the rising percentage of OFVbby GPSO over OFVb by STABC.( 2 )   W G S T A B C ∗ denotes the gap of OFVwbetween STABC and any other algorithms (GPSO, LPSO, or ABC). For instance, WGSTABCLPSO means the rising percentage of OFVw by LPSO over OFVw by STABC.( 3 )   A G S T A B C ∗ denotes the gap of OFVa between STABC and any other algorithms (GPSO, LPSO, or ABC). For instance, AGSTABCABC means the rising percentage of OFVa by ABC over OFVa by STABC.With respect to the above measures, the presented STABC is employed as the basis to compare the performance of the involved algorithms. A positive value indicates that STABC outperforms the other algorithms on the optimized problem.Table6 shows the comparison results on the ten problems under the two situations (π1=1,π2=10) and (π1=10,π2=1). We see that most of BGSTABC∗, WGSTABC∗, AGSTABC∗, and all average values are positive. Thus, better results with respect to OFVb, OFVw, and OFVa can be attained by STABC compared to GPSO, LPSO, and ABC under both situations (π1=1,π2=10) and (π1=10,π2=1)Table 6
Performance comparison among GPSO, LPSO, ABC, and STABC.
B G S T A B C G P S O W G S T A B C G P S O A G S T A B C G P S O B G S T A B C L P S O W G S T A B C L P S O A G S T A B C L P S O B G S T A B C A B C W G S T A B C A B C A G S T A B C A B C P1 π1=1, π2=10 1.0% -22.8% -4.0% 1.0% -22.7% -3.8% 0.0% -0.3% 1.5% π1=10, π2=1 8.0% -14.1% -2.2% 8.3% -12.5% -0.9% 2.2% -1.2% 3.7%  P2 π1=1, π2=10 1.1% -1.6% 0.4% 1.0% -1.5% 0.5% 0.0% -0.7% -0.4% π1=10, π2=1 8.2% -12.3% 5.7% 8.0% -12.3% 5.8% 0.0% -0.4% 5.6%  P3 π1=1, π2=10 2.2% 2.8% 2.5% 2.1% 2.8% 2.5% 0.0% 0.8% 0.3% π1=10, π2=1 16.1% 13.3% 16.1% 16.1% 13.1% 16.4% 0.1% 5.6% 2.4%  P4 π1=1, π2=10 2.3% -11.2% -9.6% 1.5% -12.7% -10.2% 20.3% 19.2% 11.3% π1=10, π2=1 26.1% 3.0% 11.5% 25.4% 4.9% 12.7% 4.9% 17.1% 17.9%  P5 π1=1, π2=10 2.1% 2.9% 2.2% 2.2% 2.8% 2.4% 0.1% 2.0% 0.8% π1=10, π2=1 15.0% 17.9% 18.6% 16.2% 21.3% 20.7% 1.5% 11.0% 7.1%  P6 π1=1, π2=10 3.5% 22.1% 11.9% 3.4% 22.0% 6.0% -13.3% 16.9% 1.6% π1=10, π2=1 35.7% 38.2% 42.6% 40.1% 41.4% 44.6% 7.9% 6.3% 9.6%  P7 π1=1, π2=10 3.0% 5.5% 4.3% 3.6% 6.9% 5.0% 0.9% 15.5% 2.4% π1=10, π2=1 31.0% 33.4% 34.5% 39.2% 39.4% 39.7% 5.2% 17.9% 11.5%  P8 π1=1, π2=10 2.0% -6.5% -3.7% 2.7% -6.8% -3.2% 0.0% 2.2% 1.7% π1=10, π2=1 16.7% 20.2% 17.1% 18.4% 26.6% 20.6% 0.5% 11.1% 4.4%  P9 π1=1, π2=10 3.3% 4.4% 3.9% 3.9% 5.3% 4.2% 1.0% 14.3% 3.3% π1=10, π2=1 36.7% 26.5% 33.3% 34.8% 29.7% 36.7% 14.7% 14.2% 12.9%  P10 π1=1, π2=10 2.8% 3.3% 4.0% 2.4% 13.3% 5.6% 1.4% 9.7% 3.3% π1=10, π2=1 10.7% 31.2% 32.5% 29.6% 38.0% 35.2% 2.5% 20.0% 13.0%  Mean 11.4 % 7.8 % 11.1 % 13.0 % 9.9 % 12.0 % 2.5 % 9.1 % 5.7 %Figures4 and 5 compare the OFVbobtained by GPSO, LPSO, ABC, and STABC (see Table 5). They demonstrate that the best solutions of STABC are better than the others. For the relatively simple small-size problems, such as problems 1, 2, and 3, most of the algorithms can easily identify the best results. With the increase of problem size, STABC outperforms the other algorithms on the relatively large-size problems. The improvement of STBAC’s search capability can be attributed to two properties: (1) each individual’s learning speed for exploration is improved by learning from other bees in all dimensions and (2) the individual’s learning sources for exploitation are enriched by learning from both its historical information and the superior individuals. The novel strategies can significantly enhance the global exploration capability and local exploitation capability especially in large-size problems which are proved by the experimental results.Figure 4
O F V b results for the case of π1=1, π2=10.Figure 5
O F V b results for the case of π1=10, π2=1.The CPU time of all test problems by the four algorithms fluctuates between 10s and 90s. Although the advantage of computational efficiency of STABC is not significant among the four algorithms, it is efficient to obtain solution of the problem. In contrast, the computational software Lingo is unable to obtain any feasible results in half an hour for most of the experiments.In addition, we compare the training cost and workload balance associated with the ten problems between the two scenarios (π1=1,π2=10) and (π1=10,π2=1); see Figures 6 and 7. All the training cost and workload balance in scenario (π1=10,π2=1) is equal to or lower than that of scenario (π1=1,π2=10). This indicates that less training effort which is in the scenario (π1=10,π2=1) will lead to less balanced workload in all the test problems though corresponding worker assignment is reasonable. It indicates that the cross-training strategy obtained by our proposed approach is effective while seeking workload balance.Figure 6
Training costs obtained by STABC.Figure 7
Workload balance results obtained by STABC.


## 6. Conclusion and Future Research

A novel mathematical programming model for solving the worker assignment problem in CMS while incorporating cross-training is presented in this paper. Then, the swarm intelligence optimizers are introduced to obtain promising solution.The proposed model defines the objective function as minimum training cost, optimized workload balance, and the minimum number of assigned workers based on considering customer demand, available time, task time, limited training cost, and initial skill level of worker. This model not only assigns suitable workers with the desired skill levels to the various tasks in multiple cells, but also determines necessity of skill enhancement by quantifying skill level needed to avoid excessive training. As for the trade-off between training expenditure and workload balance of workers, decision-maker has more flexible selections on solution based on the preference to cross-training or workload balance. The experimental results demonstrate that the proposed model is effective and feasible to produce interaction among cells for solving worker assignment problem with reasonable cross-training policy.In addition, the efficient swarm intelligence metaheuristics including GPSO, LPSO, and ABC, are employed to solve the CMS model. To avoid premature convergence and low exploitation, STABC with the enhanced information learning strategy is proposed for the complex cross-trained worker assignment problem. In this strategy, an all-dimension-wise learning mechanism is proposed to enable each individual to both exploit itself and track the promising individuals. To compare the performances of the four algorithms, we execute ten computational experiments with respect to the several defined measures. The results reveal that STABC significantly outperforms the comparison algorithms in terms of the best, worst, and average solutions of repeated experiments. With increase in problem scale, the performance differences between STABC and PSO are more remarkable. In addition, the convergence capability of the STABC is observed more efficiently than the others over the experiments.In future research, we will consider more useful and practical issues to allocate cross-trained workers into tasks in CMS, such as quality level of worker, learning and forgetting capability, and work-sharing mechanism. Additionally, we intend to extend this problem from divisional cell to rotating cell which is the other primary type of manufacturing cell.



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*Source: 4302062-2018-11-04.xml*