Int J Adv Manuf Technol (2008) 37:166–177 DOI 10.1007/s00170-007-0950-y
ORIGINAL ARTICLE
The two-stage assembly flowshop scheduling problem with bicriteria of makespan and mean completion time Ali Allahverdi & Fawaz S. Al-Anzi
Received: 17 September 2006 / Accepted: 17 January 2007 / Published online: 22 February 2007 # Springer-Verlag London Limited 2007
Abstract In this paper, we address the two-stage assembly flowshop scheduling problem with a weighted sum of makespan and mean completion time criteria, known as bicriteria. Since the problem is NP-hard, we propose heuristics to solve the problem. Specifically, we propose three heuristics; simulated annealing (SA), ant colony optimization (ACO), and self-adaptive differential evolution (SDE). We have conducted computational experiments to compare the performance of the proposed heuristics. It is statistically shown that both SA and SDE perform better than ACO. Moreover, the experiments reveal that SA, in general, performs better than SDE, while SA consumes less CPU time than both SDE and ACO. Therefore, SA is shown to be the best heuristic for the problem. Keywords Assembly flowshop . Bicriteria . Makespan . Mean competition time . Heuristic
1 Introduction Consider the situation where there are n jobs such that each job has more than two operations. The first m operations of
A. Allahverdi (*) Department of Industrial and Management Systems Engineering, Kuwait University, P.O. Box 5969, Safat, Kuwait e-mail:
[email protected] F. S. Al-Anzi Department of Computer Engineering, Kuwait University, P.O. Box 5969, Safat, Kuwait e-mail:
[email protected]
a job are performed at the first stage in parallel and the final operation is conducted at the second stage. Each of the m operations of a job at the first stage is performed by a different machine and the last operation on the machine at the second stage may start only after all m operations at the first stage are completed. Each machine can process only one job at a time. The described problem is known as a two-stage assembly flowshop scheduling problem with m operations at the first stage and one operation at the second stage. It should be noted that the problem reduces to the two-machine flowshop scheduling problem when there is only one machine at the first stage, i.e., m=1. Makespan and mean completion time are two commonly used performance measures in the scheduling literature. Minimizing makespan is important in situations where a simultaneously received batch of jobs is required to be completed as soon as possible. For example, a multi-item order submitted by a single customer needs to be delivered at the minimal possible time. The makespan criterion also increases the utilization of resources. There are other reallife situations in which each completed job is needed as soon as it is processed. In such situations, one is interested in minimizing the mean completion time of all jobs, rather than minimizing makespan. This objective is particularly important in real-life situations where reducing inventory or holding cost is of primary concern. The assembly flowshop scheduling problem was introduced independently by Lee et al. [22] and Potts et al. [31]. The two-stage assembly scheduling problem has many applications in industry. Potts et al. [31] described an application in personal computer manufacturing where central processing units, hard disks, monitors, keyboards, etc., are manufactured at the first stage, and all of the required components are assembled to customer specification at a packaging station (the second stage). Lee et al. [22]
Int J Adv Manuf Technol (2008) 37:166–177
described another application in a fire engine assembly plant. The body and chassis of fire engines are produced in parallel in two different departments. When the body and chassis are completed and the engine has been delivered (purchased from outside), they are fed to an assembly line where the fire engine is assembled. Another application is in the area of queries scheduling on distributed database systems, as discussed by Allahverdi and Al-Anzi [6]. Lee et al. [22] considered the problem with m=2, while Potts et al. [31] considered the problem with an arbitrary m. Both studies addressed the problem with respect to makespan minimization and both proved that the problem with this objective function is NP-hard in the strong sense for m=2. Lee et al. [22] discussed a few polynomially solvable cases and presented a branch and bound algorithm. Moreover, they proposed three heuristics and analyzed their error bounds. Potts et al. [31] showed that the search for an optimal solution may be restricted to permutation schedules. They also showed that any arbitrary permutation schedule has a worst-case ratio bound of two, and they presented a heuristic with a worst-case ratio bound of 2–1/m. Hariri and Potts [19] also addressed the same problem, developed a lower bound, and established several dominance relations. They also presented a branch and bound algorithm incorporating the lower bound and dominance relations. Another branch and bound algorithm was proposed by Haouari and Daouas [17]. Sun et al. [37] also considered the same problem with the same makespan objective function and proposed heuristics to solve the problem. Allahverdi and Al-Anzi [6] obtained a dominance relation for the same problem when setup times are considered as separate from processing times. They also proposed two evolutionary heuristics (a particle swarm optimization and a tabu search) and proposed a simple and yet efficient algorithm with negligible computational time. Tozkapan et al. [38] considered the two-stage assembly scheduling problem with the total weighted flowtime performance measure. They showed that permutation schedules are dominant for the problem with this performance measure. They developed a lower bound and a dominance relation, and utilized the bound and dominance relation in a branch and bound algorithm. It should be noted that the performance measures of flowtime and completion time are equivalent when jobs are ready at time zero. It should also be noted that the total and mean completion times are equivalent performance measures. Al-Anzi and Allahverdi [3] considered the same problem with total completion time criterion. They obtained optimal solutions for two special cases and proposed a simulated annealing (SA) heuristic, a tabu search heuristic, and a hybrid tabu search heuristic. They compared their heuristics with existing ones and showed that their hybrid tabu search heuristic is the best.
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The research mentioned so far addressed only the single criterion of either makespan or mean completion time, while the majority of real-life problems requires the decision maker to consider both criteria before arriving at a decision. The problem with both makespan and mean completion time has not been addressed for the considered two-stage assembly scheduling problem and is the topic of the current paper. As explained earlier, the problem reduces to the twomachine flowshop scheduling problem when m=1. There has been some effort to address the flowshop problem with multiple criteria. Two different approaches might be distinguished for the multiple-criteria problems: those where all efficient solutions are generated and then tradeoffs are made between the solutions and those where a single objective function is constructed by the integration of all of the relevant criteria, usually by forming a weighted linear combination of them. The former approach is used by many researchers, including Sayin and Karabati [34], Rajendran [32], and Ho and Chang [18], while the latter approach is utilized by some other researchers, including Nagar et al. [27], Sivrikaya-Serifoglu and Ulusoy [35], Yeh [40, 41], Lee and Chou [21], Allahverdi [5], Allahverdi and Aldowaisan [8], and Yeh and Allahverdi [42]. The latter approach is taken in this paper. We, in this paper, address the two-stage assembly flowshop scheduling problem with a weighted sum of makespan and mean completion time. The problem is described in the next section, and the three proposed heuristics are explained in Sect. 3. Comparison of the heuristics on randomly generated problems is performed in Sect. 4, while possible future research directions are provided in Sect. 5.
2 Problem definition We assume that n jobs are simultaneously available at time zero and that preemption is not allowed, i.e., any started operation has to be completed without interruptions. Each job consists of a set of m+1 operations. The first m operations are performed at stage one on m parallel machines, while the last operation is performed at stage two on the assembly machine. We use the following notation: t[i,
j]
p[i] C[i]1
Operation time of the job in position i on machine j, i=1,..., n, j=1,..., m Operation time of the job in position i on the assembly machine Completion time of the job in position i
Note that job k is complete once all of its operations t[k, j] (j=1,..., m) and p[k] are completed, where the operation p[k]
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Int J Adv Manuf Technol (2008) 37:166–177
may start only after all operations t[k, j] (j=1,..., m) have been completed. Potts et al. [31] and Tozkapan et al. [38] showed that permutation schedules are dominant with respect to makespan and total flowtime (completion time) criterion, respectively. Therefore, permutation schedules are also dominant for the problem addressed in this paper. Thus, we restrict our search for the optimal solution to permutation schedules. In other words, the sequence of jobs on all of the machines, including the assembly machine, is the same. It can be shown that the completion time of the job in position j is as follows [3]: ( ( ) ) j X C½ j ¼ max max t½i; k ; C½j1 þ p½ j k¼1;...; m
i¼1
where C½0 ¼ 0
n P The mean completion time is MCT ¼ 1n C½i and the i¼1 makespan is Cmax=C[n]. If the weight given to the makespan is denoted by α, then the objective function (OF) is defined by the following equation:
OF ¼ aCmax þ ð1 aÞMCT where 0
