A hybrid simulated annealing for scheduling in dual-resource cellular manufacturing system considering worker movement

ABSTRACT This paper presents a novel linear mathematical model for integrated cell formation and task scheduling in the cellular manufacturing system (CMS). It is suitable for the dual-resource constrained setting, such as garment process, component assembly, and electronics manufacturing. The model can handle the manufacturing project composing of some tasks with precedence constraints. It provides a method to assign the multi-skilled workers to appropriate machines. The workers are allowed to move among the machines such that the processing time of tasks might be reduced. A hybrid simulated annealing (HSA) is proposed to minimize the makespan of manufacturing project in the CMS. The approach combines the priority rule based heuristic algorithm (PRBHA) and revised forward recursion algorithm (RFRA) with conventional simulated annealing (SA). The result of extensive numerical experiments shows that the proposed HSA outperforms the conventional SA accurately and efficiently.


Introduction
With rapidly changing customer expectation and global competition, cellular manufacturing system (CMS) has been an important way of producing goods in the last several decades. It shows many advantages such as an effective response to the rising product variety, a reduction in material handling cost and production lead time, streamlined production control, and enhanced productivity. Cell formation including grouping machines and tasks, and task scheduling involving decisions on task dispatching rules and timetable, are two main issues in the CMS. Consequently, these problems have attracted much investigating interest from researchers and practitioners.
For the basic cell formation problem with machine assignment, Rabbani et al. [1] used a two-phase fuzzy linear programming approach to solve a bi-objective cell formation problem with stochastic production quantities. Arkat et al. [2] proposed a branch and bound algorithm to minimize the total number of movements between each pair of machines locating in two different cells. Saraç and Ozcelik [3] used a genetic algorithm to maximize the grouping efficacy. Chung et al. [4] proposed an efficient tabu search algorithm to solve the cell formation problem with alternative routings and machine reliability considerations. Rafiei et al. [5] designed a dynamic cellular manufacturing system to pursue fundamentals of Just-In-Time production philosophy. A nonlinear programming model is proposed with two conflicting objective functions: minimizing the sum of cost, and minimizing the work-in-process. Mar-Ortiz et al. [6] presented a mathematical programming model to minimize the sum of the machine amortization cost, the machine relocation cost, and the intercellular material handling cost. By a reconfigurable approach, the cells are rearranged periodically to deal with demand variability in a multi-period planning horizon. Jayakumar and Raju [7] presented a nonlinear mixed-integer mathematical model for the cell formation problem with the uncertainty of the product mix for a single period. A solution methodology for best possible cell formation using simulated annealing (SA) is presented in order to minimize the total sum of the machine constant cost, the operating cost, the intercell material handling cost, and the intra-cell material handling cost. Some other methods have emerged for cell formation problems, such as particle swarm optimization method [8], clustering method [9] and scatter search algorithm [10]. It should be pointed out that the results in the aforesaid references only consider machine constrained setting.
For the cell formation problem with worker and machine assignment, Mahdavi et al. [11] presented a fuzzy goal programming-based approach for solving a multi-objective mathematical model of cell formation problem and production planning in a dynamic virtual cellular manufacturing system. Bagheri and Bashiri [12] proposed a new mathematical model to solve the worker assignment and inter-cell layout problems. The objective function of the proposed model consists of two main cost categories. The preferred solution is obtained by a LP-metric approach. Mahdavi et al. [13] investigated an integer mathematical programming model for the design of CMSs in a dynamic environment. The aim of the proposed model is to minimize holding and backorder costs, inter-cell material handling cost, machine and reconfiguration costs, and hiring, firing and salary costs. Azadeh et al. [14] presented a simulation approach for optimization of operator allocation in the CMS. Süer et al. [15] proposed a three-phase methodology to deal with the problem of cell loading and product sequencing in labour-intensive cells. Bootaki et al. [16] designed a robust method to configure cells in a dynamic CMS to minimize the intercell movements and maximize the machine and worker utilisation.
In contrast with the cell formation problem, there are only a small quantity of articles addressing the problem of scheduling in the CMS. Venkataramanaiah [17] developed a SA for scheduling of parts within a cell for the objective of minimizing weighted sum of makespan, flowtime and idle time. Tavakkoli-Moghaddam et al. [18] designed a scatter search method for a multicriteria group scheduling problem in the CMS. Halat and Bashirzadeh [19] suggested a heuristic for scheduling operations of manufacturing cells considering sequence-dependent family setup times and intercellular transportation times. Arkat et al. [20] presented a mathematical model to concurrently identify the formation of cells, cellular layout and the operations sequence with the objective of minimizing total transportation cost of parts as well as minimizing makespan. Liu et al. [21] developed a discrete bacteria foraging algorithm to solve the model of CMS with the objective of minimizing the material handling costs as well as the fixed and operating costs of machines and workers.
Because of the high complexity of CMS which is subject to dual-resource constrained conditions, the cell formation and group scheduling problems are often analyzed independently. To the best of the authors' knowledge, few related research has involved the CMS problem simultaneously considering multi-functional machines, multi-skilled workers and task sequence yet. Moreover, the impact of worker movement on task scheduling is also desired to be discussed.
The remainder of this paper is organized in the following: In Section 2, the proposed problem is stated and formulated as a mathematical model integrating cell formation and task scheduling. In Section 3, the PRBHA algorithm is suggested to obtain an initial solution with high quality. In Section 4, the HSA algorithm is designed for further search to get a global optimum. In Section 5, the performance of the proposed HSA is validated in comparison with the conventional SA by computational experiments. Finally, conclusions are drawn followed by some potential research directions in Section 6.

Problem statement and formulation
In this section, the problem of cell formation is formulated as a linear integer programming mathematical model. The objective is minimizing the makespan of the project which is composed of n tasks (i.e. maximum completion time of all tasks). The following hypotheses are made for the proposed problem.
(1) Machine and worker hypotheses: The number of machines and workers are known in advance. The number of machines is more than the number of workers.
(2) Task hypothesis: For each task, at least one machine has the ability to process it. For each machine, any worker has the ability to operate it. The processing of each task is not allowed to be interrupted, which implies that each task is processed on only one machine by only one worker. The processing time of task depends on the assigned machine and worker. There exists precedence relationship among tasks.
(3) Cell size hypothesis: The number of machines in each cell can not exceed a specified maximum, because redundant machines in a cell may generate cluttered flows in many routes.
(4) Worker movement hypothesis: The workers are permitted to move among different machines, and the movement time is known in advance.

Subscripts w
Index for worker. c Index for cell. The proposed problem is formulated as the following linear integer programming model: The objective function (1) is to minimize the makespan C max . Constraint (2) ensures that each machine should be located in one and only one cell. Constraint (3) shows that the number of machines in each cell can not exceed the upper limit of cell size. Constraint (4) guarantees that each task is processed on one of the machine that can process the task. Constraint (5) implies that each task is processed on only one machine by only one worker, and each task should be processed in the worker's only one continuous time interval. Constraint (6) guarantees each worker operates no more than one machine and processes no more than one task in his/her continuous time interval. Constraint (7) guarantees only if certain worker processes one task in a continuous time interval, the worker must have processed another task in the previous continuous time interval. Constraint (8) shows the relationship of the completion time of two tasks which are processed in one worker's consecutive time intervals. Constraint (9) ensures the precedence relationship between consecutive tasks. Constraint (10) ensures the type of the decision variables.
The proposed model has some advantages and characteristics. (1) It is suitable for the CMS with dualresource constrained setting, such as garment process, component assembly, and electronics manufacturing.
(2) The model can handle the manufacturing project composing of some tasks with precedence order. For example, Figure 1 shows the manufacturing process of professional road bike. The task of standard wheel subassembly can not start until the tasks of wheel spokes, wheel tire and wheel rim are finished. (3) In many labour intensive companies, the workers are crosstrained with multiple skills in order to increase flexibility and reduce salary cost. The model can provide a method to assign the workers to appropriate machines. (4) In this model, each worker is permitted to move from one machine to another for performing another task, which might be able to decrease processing time.

Priority rule based heuristic algorithm
In this section, we develop a priority rule based heuristic algorithm (PRBHA) that is embedded in SA for determining an initial feasible schedule. The PRBHA consists of n iterations (n is the total number of takes). At each iteration, a prior task is selected according to the EFT (earliest finishing time first) rule. Assuming there is a dummy task s with 0 unit of processing time at the beginning of the project. Moreover, assuming the machine that processes the dummy task s is a dummy machine 0, and the cell where the dummy machine 0 is located is a dummy cell 0.
The variables used for the PRBHA are listed in the following:

D
The decision task-set, i.e. the unscheduled tasks whose immediate predecessor tasks have been completed. The completed task-set, i.e. the tasks that have been completed.

ft jwm
The hypothesis completion time of task j which is processed by worker w on machine m.
The prior triple form, where j * denotes the prior task, m * denotes the prior machine, and w * denotes the prior worker.

FT j
The finish time of task j.

ST j
The start time of task j.

7:
FT j * = ft j * m * w * 8: Step 1, some variables , IW w , IM m , and M w are initialized. Let Y 0cm c = 0, FT s = 0, C 0 = 0. The machines are randomly assigned to the cells. In Step 2, each job is scheduled at each iteration. In each iteration, firstly, compute the decision task-set D in Step 3, and then compute the hypothesis completion time of each task j in D with different workers and different machines in M j in Steps 4-5. Secondly, according to the EFT rule, select a prior task j * , a prior machine m * , and a prior worker w * in Step 6. Finally, record the finish time of task j * , and update the start idle time of machine m * , the start idle time of worker w * , the machine operated by the worker w * , and the completed task-set in Steps 7-8. In Step 10, the makespan of the project is computed according to the finish time of each task.

Hybrid simulated annealing algorithm
Since SA was introduced by Kirkpatrick et al. [22]. It has become one of the most popular metaheuristic methods to solve complex optimization problems in manufacturing systems [23,24]. The name of SA and its inspiration comes from annealing in metallurgy. The main mechanism is that SA decreases the probability of accepting worse solutions as the temperature drops gradually. Accepting worse solutions allows for a more extensive search in the solution space, and provide the chances to jump out the local optima. In this section, we propose a hybrid simulated annealing (HSA) which combines the PRBHA approach with conventional SA algorithm.

Initial solution
By the PRBHA, the value of the C m , FT j , C max and (j * , m * , w * ) are generated. From the prior triple form (j * , m * , w * ), we can see that task j * is processed on machine m * by worker w * . From the values of C m , FT j and C max , we know the location cell C m for machine m, the finish time of task j, and the makespan C max . Therefore, the PRBHA generates a feasible initial solution in the HSA.

Neighborhood generation strategy
It is important to design superior solution mutation(SM) operators for the search of HSA. In this research, three different mutation strategies are provided in the following: (2) Task-machine mutation(SM2): A task which can be processed by more than one machine is randomly selected, and then randomly reassigned to another machine that can process the task. (3) Task-worker mutation(SM3): A task is randomly selected, and then randomly reassigned to another worker.
The objective function values (i.e. makespan C max ) of the neighbourhood solutions can be calculated by the revised forward recursion algorithm (see Algorithm 2). Randomly select a job j from U.

4:
Allocate task j to machine m operated by worker w (from solution schedule, the values m and w are known).

5:
If the finish time of predecessor i of task j has not been determined, recursively execute Step 4 for predecessor i.

Cooling schedule
(1) Parameters of HSA algorithm: Initial temperature T 0 , final temperature T f , cooling rate α and Markov chain length L max are set to 200,0.5, 0.95 and 200, respectively. The temperature T is decreased by using the following common equation: T := αT.
(2) Termination condition: Let the best schedule up to now is x best , the HSA algorithm is stopped if x best is not changed after three consecutive temperature levels or the final temperature T f is reached.

The pseudo code of the HSA
Algorithm 3 provides the pseudo-code of the HSA. The major optimization procedure is that: Generate an initial solution by the PRBHA. If C max of neighbourhood solution x l is less than C max of current solution x c , accept x l as current solution x c , otherwise neighbourhood solution x l is accepted as current solution x c by certain probability, which can escape from local optima to reach a global optimum. At the start, the probability of accepting nonimproving solutions is high, but as the search continues (i.e. the temperature drops), the probability of accepting nonimproving solutions decreases. if r 1 < 1/3 then 7: Generate neighborhood x l by SM1, compute C max (x l ) by algorithm 2 8: else if 1/3 ≤ r 1 < 2/3 then 9: Generate neighborhood x l by SM2, compute C max (x l ) by algorithm 2 10: else 11: Generate neighborhood x l by SM3, compute C max (x l ) by algorithm 2 12: end if 13: if C max (x l ) ≤ C max (x c ) then 14: x c = x l 15: if C max (x c ) ≤ C max (x best ) then 16: x best = x c , Change := Change + 1, Unchange = 0 17: end if 18: else 19: Randomly generate a number   [5,20], N m ∼ DU [1,3], B u = 4).  [5,20], N m ∼ DU [1,10], B u = 4).  Table 3. Comparison between SA and HSA with different number of worker types (W) (M = 50, J = 200, T pkmw ∼ DU [5,20], N m ∼ DU [1,10], B u = 4).

Computational experiments
In order to evaluate the performance of the HSA and SA algorithms for the problem, extensive numerical experiments are conducted.  [5,10], T pkmw ∼ DU [5,20], T pkmw ∼ DU [5,30] and T pkmw ∼ DU [5,40], and four different distributions of N m are used, including N m ∼ DU [1,4], N m ∼ DU [1,6], N m ∼ DU [1,8] and N m ∼ DU [1,10], where DU[a, b] represents a discrete uniform distribution with an integer range from a to b.
In the experiments, we use the approach presented in [25] for generating the precedence constraints of tasks. To do this, let P ij = Pr{arc(i, j) exists in the immediate precedence graph}, and let D represent the target density of the precedence constraint graph, that is D = Pr{arc(i, j) exists in the precedence constraint graph}, for 1 ≤ i < j ≤ J. The D and P ij satisfy: where D ∈ (0, 1). Randomly generate a number r ij from the uniform distribution over the interval [0, 1]. If r ij < D, then arc(i, j) exists in the immediate precedence graph. Given   [5,20], N m ∼ DU [1,3]   The six experiment results are presented in Tables 1-6, respectively. Let HSA-C max and SA-C max denote the objective function values (i.e. makespan) of the problem using the HSA and SA, respectively. Each table entry represents the minimum, maximum and average of its associated 10 instances. D-C max denotes the declining percentage of average HSA-C max over average SA-C max . Let HSA-CPU and SA-CPU denote the mean CPU time of the HSA and SA algorithms without including input and output time,

Conclusions
This paper gives a new optimization model of cellular manufacturing system (CMS) under dual resources and task precedence constrained setting. The objective of the problem is to minimize the makespan. The PRBHA, which schedules a prior task on a prior machine by a prior worker according to the priority rule at each iteration, is embedded to the HSA for initial feasible solution that can be improved in further stages. Computational experiments are conducted to show that the quality of results obtained by the HSA is better than the SA regardless of the variation of some important parameters.
A valuable future research direction is to consider the impact of learning and forgetting effects of workers on their assignment and movement. The other possible extension to this research would investigate various efficient priority rules and corresponding heuristics for SA. It is also desired to linearize the proposed model in the future, so that the HSA can be compared with the branch-and-bound approach (B&B) under the ILOG CPLEX software for small or medium sized instances. Moreover, some state-of-the-art heuristics, such as firefly algorithm, league championship algorithm, and migrating birds optimization, can be developed from various aspects on the basis of CMS characteristics. Therefore, we wish to design these heuristic based HSA approaches, and compare them with the proposed HSA in this paper.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This research was supported by the Zhejiang Provincial Natural Science Foundation of China under Grants LY19A010007 and LY19G020015, the Humanities and Social Sciences Foundation of the PRC Ministry of Education under Grants 19YJA630078 and 17YJC630093, and the National Natural Science Foundation of China under Grant 71871076.