Abstract

This paper investigates a two-machine flow shop problem with release dates in which the job processing times are variable according to a learning effect. The bicriterion is to minimize the weighted sum of makespan and total completion time subject to release dates. We develop a branch-and-bound (B&B) algorithm to solve the problem by using a dominance property, several lower bounds, and an upper bound to speed up the elimination process of the search tree. We further propose a multiobjective memetic algorithm (MOMA), enhanced by an initialization strategy and a global search strategy, to obtain the Pareto front of the problem. Computational experiments are also carried out to examine the effectiveness and the efficiency of the B&B algorithm and the MOMA algorithm.

1. Introduction

In many industrial systems, logistics, and service settings, the processing times of jobs can decrease due to the learning effects, i.e., firms and employees perform a job (task) over and over and they learn how to perform more efficiently [1, 2]. This is especially observed in seru production system, consisting of some equipment and workers who produce one or more part types. After undertaking cross-training, the learning effects greatly affect their efficiencies [3]. “As there is a significant involvement of humans in scheduling environments, the number of activities subject to learning is high, too. Hence it seems reasonable to consider learning in scheduling environments” [2]. On the other hand, the importance of flow shop scheduling optimization is widely recognized in many manufacturing and assembling processes, e.g., the internet connectivity problem in 3-tiered client-server databases is a two-machine flow shop problem (see [4]); the scheduling of multimedia date objectives for WWW applications reduces to a two-machine flow shop problem (see [5–7]; see also the reviews given in [6] and [8–10]).

More recently, Xu et al. [11], Eren [12], Wu and Lee [13], Rudek [14], Kuo et al. [15], Lee and Chung [16], Sun et al. [17, 18], Cheng et al. [19], Li et al. [20], Wang and Zhang [21], J. B. Wang and J. J. Wang [22], Lai et al. [23], Liu and Feng [24], Wang and Zhang [21], Wu et al. [25], Shiau et al. [26], Lu [27], Qin et al. [28], He [29], and Wang et al. [30] considered flow shop scheduling problems with learning effects, but without release dates. Xu et al. [11] considered the flow shop scheduling problems with position-dependent learning effect, i.e., if job is in position of a schedule, the actual processing time of job on machine is and , where denotes the normal processing time of job on and , , and are the learning rates. For several regular objective functions, they presented approximate algorithms. Cheng et al. [19] considered the model . For the maximum lateness minimization, they gave a mathematical programming model. Wang et al. [31] considered the model . For the total completion time minimization, they gave a branch-and-bound algorithm and several well-known heuristics. Lai et al. [23] considered the model . For the total tardiness minimization, they gave a branch-and-bound algorithm and two heuristics. He [29] and Li et al. [20] considered the flow shop scheduling problems with time-dependent learning effect, i.e., if job is in position of a schedule, the actual processing time of job on machine is , where is the learning rate and denotes the job that occupies the th position in a sequence. For several regular objective functions, they presented approximate algorithms. Shiau et al. [26] considered the flow shop scheduling problems with general position-dependent learning effect, i.e., if job is in position of a schedule, the actual processing time of job on machine is , where is a nonincreasing function. Sun et al. [18] considered the total weighted completion time minimization flow shop scheduling problem. For three position-dependent learning effects, they gave heuristic algorithms. Cheng et al. [19], Lai et al. [23], Wang et al. [31], Wu and Lee [13], and Wang et al. [30] considered the flow shop scheduling with truncated learning effects. J.-B. Wang and J.-J. Wang [22] considered the flow shop scheduling with a general exponential learning effect. For five regular objective functions, they presented heuristic algorithms. Wang and Zhang [21] considered the flow shop scheduling with position-weighted learning effects. For the weighted sum of makespan and total completion time minimization, they gave a branch-and-bound algorithm and some heuristic algorithms. Qin et al. [28] considered the flow shop group scheduling with position-based learning effect. For four objectives (i.e., the makespan, total completion time, total weighted completion time, and maximum lateness), they gave several heuristics and metaheuristics. For new trends in flow shop scheduling with learning effects, we refer the reader to Rudek [14], Liu and Feng [24], Shiau et al. [26], Lu [27], and He [29].

However, in fact, the scheduling with release dates is interesting and closer to real problems [32–34]. With modeling a realistic production system in mind, Yin et al. [35] considered single-machine scheduling with release dates and position-dependent learning effects. To the best of our knowledge, the flow shop scheduling problems with a learning effect and release dates (i.e., the ready times) simultaneously have barely been investigated. Bai et al. [36] is the only identifiable exception. Hence, in this paper, we consider the two-machine flow shop scheduling problem with position-dependent learning effect and release dates, and the objective is to minimize the weighted sum of makespan and total completion time. Obviously, the problem under study is NP-hard [34]; thus, the branch-and-bound algorithm might be a good way to obtain the optimal schedule. We further design an evolutionary multiobjective optimization algorithm to obtain the Pareto front of high quality. This contribution makes a clear distinction between this paper and Bai et al. [36], which solely focuses on a single aggregated objective.

The remaining of the paper is organized as follows. In Section 2, we formalize the problem. Dominance conditions for the problem are presented in Section 3. Branch-and-bound algorithm is discussed in Section 4. Numerical experiments are conducted in Section 5. The last section concludes the paper.

2. Notations and Assumptions

The problem may be stated as follows. We are given jobs and two machines and . Each machine is available at time zero and the job becomes available at its release date . Let represent the set of jobs which are to be processed on 2 machine permutation flow shop settings. Each job is required to be processed on machine and then . As in Biskup [2], Lee and Chung [16], Lee and Wu [37], and Lee et al. [38], if job is in position of a schedule, then the actual processing times and of job on machines and are respectively, where denotes the normal processing time (i.e., the basic processing time before learning effects happened) of job on and is the learning index for both machines and depending on the learning rate LR. For a given schedule , let be the completion time of job on machine . The objective is to minimize , where is the makespan and is the total completion time, . Using the conventional three-field notation for describing scheduling problems (Graham et al. [39]), the problem under consideration can be described as , where denotes the learning effect [1, 2].

3. Dominance Conditions

In this section, we give some dominance rules for . “Dominance rules are basically necessary conditions for any optimal schedule that can be generated. Applying such rules results in a set of precedence relations between jobs. These precedence relations are then used to reduce the number of branches in a branch-and-bound search tree” (Yin et al. [35]). Let and be obtained from by only interchanging jobs and , where and are partial sequences, and there are jobs in . To show that dominates , it is sufficient to show that , , and . Let denote the last completion time prior to jobs and on machine in ; obviously, and remain unchanged in .

Proposition 1. If the jobs and satisfy the following conditions: (i)Either or or (ii)Either or or (iii)Either or or (iv)Either or (v)Either or or (vi)Either or (vii)Either or or or or or (viii)Either or or or or or (ix)Either or or or or or (x)Either or or or or or (xi)Either or or or or or (xii)Either or or or or or then dominates .

Proof 1. Under schedules and , the completion times of jobs and are If the cases , , , , , , , , , , , and can be satisfied, we have the following: first term in (7) ≤ first term in (3), second term in (7) ≤ first term in (3), third term in (7) ≤ first term in (3), first term in (8) ≤ first term in (4), second term in (8) ≤ first term in (4), third term in (8) ≤ first term in (4), first term in (9) ≤ first term in (5), second term in (9) ≤ first term in (5), third term in (9) ≤ first term in (5), fourth term in (9) ≤ first term in (5), fifth term in (9) ≤ first term in (5), and sixth term in (9) ≤ first term in (5).
Hence, , , , and . Similarly to this case, the other cases can be obtained.