Scientific Programming

Volume 2016, Article ID 5127253, 13 pages

http://dx.doi.org/10.1155/2016/5127253

## Robust Parallel Machine Scheduling Problem with Uncertainties and Sequence-Dependent Setup Time

^{1}Logistics Engineering College, Shanghai Maritime University, Shanghai, China^{2}Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

Received 6 September 2016; Accepted 11 October 2016

Academic Editor: Si Zhang

Copyright © 2016 Hongtao Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A parallel machine scheduling problem in plastic production is studied in this paper. In this problem, the processing time and arrival time are uncertain but lie in their respective intervals. In addition, each job must be processed together with a mold while jobs which belong to one family can share the same mold. Therefore, time changing mold is required for two consecutive jobs that belong to different families, which is known as sequence-dependent setup time. This paper aims to identify a robust schedule by min–max regret criterion. It is proved that the scenario incurring maximal regret for each feasible solution lies in finite extreme scenarios. A mixed integer linear programming formulation and an exact algorithm are proposed to solve the problem. Moreover, a modified artificial bee colony algorithm is developed to solve large-scale problems. The performance of the presented algorithm is evaluated through extensive computational experiments and the results show that the proposed algorithm surpasses the exact method in terms of objective value and computational time.

#### 1. Introduction

Parallel machine systems are widely adopted in a variety of manufacturing environments, such as the semiconductor manufacturing industry [1] and the electronics industry [2]. Minimizing makespan is one of the commonly used objectives in manufacturing scheduling problems [3, 4]. Parallel machine scheduling problems have both theoretical and practical importance. Many literatures assume that the parameters of the problem (i.e., the processing time and job release time of jobs) are known in advance precisely before production process begins. However, the results of the solution derived under the deterministic assumption may deviate considerably in real situations [5]. In practice, it is difficult to get the exact parameters before production process begins due to some uncertainties, such as machine conditions, production environments, and jobs’ characteristics [6–9]. To characterize and overcome the impact of uncertainties, several robust scheduling approaches are proposed to enhance the quality and stability of the derived solution in real situations.

Unlike the typical parallel machine system, a parallel machine system for plastic product involves mold changes during the process of production, which is also known as setup requirements. In a plastic production system, jobs are allocated to one of parallel injection machines and the corresponding injection mold is required to be installed onto the injection machine before the injection process. Jobs belonging to one family can be processed with the same mold and the mold changing process costs a period of time. The production planner would try to arrange the jobs such that jobs that belong to the same family are processed together in order to avoid extra mold change process and improve efficiency as well. For two jobs that are scheduled to be processed consecutively but belong to different families, mold change must be conducted before the next job can be performed. Therefore, reducing mold changing time is crucial in plastic production scheduling.

This paper investigates the uniform parallel machine problem in the plastic production system, which involves uncertain processing time, job release time, and setup time for mold. It is an extended problem studied by [5]. The preliminary objective is to minimize the makespan. A scenario is firstly defined as a possible realization of processing times and job release time for all jobs. Due to the uncertainty in job processing time and release time, the makespan of a given solution can be different under various scenarios. It is proposed to identify an optimal solution with the strongest stability across all scenarios. The performance (or stability) of a given solution under its worst-case scenario is the major concern in robust scheduling. For generality, it is assumed that the processing times and job release time lie in their respective intervals with lower bound and upper bound value. The intervals of the corresponding parameters are considered to be known but not fall into any statistical distribution due to asymmetric information [5, 9]. And then the robust deviation criterion is adopted to evaluate the robustness of each candidate schedule. The robust deviation criterion is also known as min–max regret criterion, which aims to find out a solution with a minimum maximal deviation across all possible scenarios. This criterion has its advantages in the highly competitive environment for the reason that the robust decision performs well in any set of potential realizable scenarios [5]. The robust deviation criterion confines the magnitude of missed opportunities by identifying a schedule that possesses a performance close to that of the optimal (or near optimal) decision in any scenarios.

To the best knowledge of the authors, parallel machine scheduling problem with uncertain processing time, ready time, and mold change consideration has not been covered by the other researchers in the parallel machine scheduling problem. In this paper, a mixed integer linear programming (MILP) formulation is introduced to identify a robust schedule with minimum maximal regret across all scenarios. The concept of critical machine proposed in [5] is revised and adopted in the problem to eliminate the worst-case scenario into a finite number of extreme point scenarios in order to evaluate the robust deviation of given solution by calculating the maximal regret. Exact algorithm based iterative relaxation procedures are presented, and a modified artificial bee colony algorithm is proposed for the research problem. To demonstrate the effectiveness and efficiency of the proposed heuristic algorithm, a set of testing problem is carried out.

The contribution of this paper is shown as follows:(1)A parallel machine scheduling problem with consideration of uncertain processing time, job release time, and mold changing time is studied. For generality, the uncertain data is assumed to lie in intervals, which capture the situation in real environment.(2)Swarm intelligence algorithm is proposed to solve the NP-hard problem. The computational results demonstrate the stability and effectiveness of the proposed algorithm. To the best of our knowledge, this research is the first to adopt artificial bee colony algorithm in robust optimization for a parallel machine scheduling problem.The rest of this paper is organized as follows. Section 2 provides the literature review of works related to parallel machine as well as robust scheduling approaches. Section 3 presents the description and the mathematical model of the proposed problem. The exact algorithm based on iterative relaxation is provided in Section 4. The modified artificial bee colony algorithm for large-scale problem is proposed in Section 5. Results of extensive computational experiment and comparison are provided in Section 6. Section 7 provides the conclusion and direction for further works in related fields.

#### 2. Literature Review

Stochastic approaches for tackling scheduling problem under uncertainties are available [10], but some of these approaches have their limitation due to the strict prerequisite and assumption [5]. For example, stochastic approaches require certain information on probability distribution of processing time or release time of each job, which can be inferred on the condition that a substantial amount of historical data is available [9, 11]. However, such amount of historical data is unavailable in highly uncertain environment and the only information is an educated guess of the lower bound and upper bound of some parameters, such as processing time and ready time [12]. For some one-time jobs, decision-makers are more interested in obtaining a robust schedule, which is against the worst-case performance across all scenarios, rather than obtaining an expected optimal performance under an expected situation. Robust deviation approach, which is known as min–max regret [13], is suitable for these circumstances to obtain such solution [14, 15]. The robust deviation approach is widely adopted in various combinational optimization problems when the input data are presented as intervals, such as the shortest path [16, 17], spanning tree [14, 17–19], and production problems [5, 20]. This approach yields a satisfactory result in an uncertain environment. Moreover, robust scheduling approaches are also adopted in other problems, such as maritime transportation problem [21, 22], routing problem [23], and scheduling problem in public health service department [24]. Most recently, a min–max regret makespan minimization in an identical parallel machine scheduling environment with interval data is studied [5]; in particular, they considered the processing time of jobs lies in respective intervals. To solve this nondeterministic polynomial-time hard (NP-hard) problem, the concept of critical machine and extreme point scenarios and two properties are proposed to avoid visiting an infinite number of possible scenarios and eliminating worst-case scenario into finite number of extreme point scenarios. A makespan minimization problem with interval job processing time on identical parallel machines is addressed in [5] for the first time in related field. Furthermore, the approach of robust deviation on a uniform parallel machine scheduling problem is adopted in [12] to minimize the total flow time with uncertain processing time which lies in respective intervals. The concept of worst-case scenario used to identify the maximal regret for a feasible solution is adopted in [5, 12] and then the exact and heuristic algorithms are proposed to find the robust schedule across all scenarios.

Scheduling problem with sequence-dependent setup time is a very active research area [25–28]. In the plastic manufacturing process, injection operation is a typical single-stage manufacturing process, which requires the plastic to be injected into specified mold to produce plastic products in different shape. Several jobs are manufactured by a single machine with their respective molds [29]. A setup of mold is required for a product to be produced in injection operation, while setup is a sequence-depending operation and parameters such as time for mold change and install time vary depending on two consecutive jobs [30]. If two consecutive jobs are processed by different mold, cost of setup is induced during the manual mold change operation. Therefore, the time for mold change as well as reinstall cannot be ignored in a mass production situation and planner would try to arrange those jobs by the same mold to be processed together in order to reduce the number of mold changes during production. The sequence-dependent family setup time represented by interval data on a single machine scheduling problem to minimize the total flow time is considered in [20].

Exact and heuristic algorithms have been proposed to solve min–max regret problem. A min–max regret problem of minimizing the total flow time on a single machine in [6] and then this problem are investigated further by the other researchers. In order to reduce the computational effort, different approaches have been proposed for the min–max regret model. It is proved that the optimal schedule under the mid-point scenario guarantees a 2-approximation of the optimal solution [15]. Heuristics method is an approximation algorithm to obtain near optimal result for min–max regret model. Typical example is shown in [5]. Some heuristic algorithms based on job swap moves and insert moves are presented by researchers. A hybrid tabu search algorithm for batching and sequencing decision-making in a single machine scheduling problem is proposed by Suppiah and Omar [31]. Job swapping and insertion approaches are applied to generate neighborhood solution and employed arcs in which the solution appears in the form of arcs in the tabu list to represent the sequence of job on a single machine; in addition, they also implemented a search depth strategy in the process of neighborhood generation to eliminate noneffective moves so as to reduce the computational burden while obtaining final schedule with outperformed quality. The work of Bilge et al. [32] points out that, for the situation where neighborhood of solution is in large number or its elements are expensive to evaluate, it is essential to restrict the number of solutions examined on a given iteration to screen the neighborhood so as to concentrate on promising moves at each iteration. Three candidate list strategies are proposed to confine the number of neighborhood solution for calculation efficiency.

Various studies on parallel machine problem have, respectively, considered uncertain processing time, arbitrary ready time, and mold changing time in the model development stage while there are no studies concerning these factors altogether in an integrated robust scheduling model. Therefore, this paper investigates these factors altogether and formulates a novel model in a plastic production environment. The presented paper is an extended version of the previous paper, containing more uncertain factors and features in a practical manufacturing system.

#### 3. Problem Formulation and Mathematical Model

It is started by describing the problem and the definition of the maximal regret of a feasible schedule. Then the MILP is presented.

##### 3.1. Problem Description

The problem under consideration deals with the scheduling of parallel machine over an assigned planning horizon in order to minimize the makespan. The job is processed by machine and mold and each job can be processed only once. Under the previous assumptions, the problem can be modelled as a mixed integer linear programming formulation as stated below.

##### 3.2. MILP Formulation

*Notations* : job, , where is the total number of jobs, : machine, , where is the total number of machines, : mold installation time for job : mold removal time for job : job ’s processing time under scenario , : job ’s arrival time under scenario , : processing speed of machine : scenario which is a possible realization of the processing times and arrival time of jobs, : a large positive number

*Decision Variables* : 1 if job is scheduled to be processed on machine and 0 otherwise : 1 if job is scheduled to be processed after job on the same machine and 0 otherwise : the beginning time of job to be processed on machine in schedule under scenario : the completion time of job scheduled on machine in schedule under scenario : the completion time of machine in schedule under scenario A feasible solution should satisfy and , , . That is, each job should be processed exactly once in parallel machine system and there exists a processing sequence for jobs to be processed on the same machine. Let be the set of feasible schedules.

Completion time of job in schedule under scenario can be defined as follows:Calculation of completion time of machine under scenario :

The completion time of machine in schedule under scenario is equal to the completion time of the job that is scheduled to be processed by the last machineThat is,The starting of job on machine is equal to or greater than the arrival time of job under scenario :In addition, if job and its preceding job are processed by different mold, the time for mold change is involved before job starts to be processed:The makespan of schedule under scenario is equal to the completion time of the machine that finished last in parallel machine system:orWe define the minimum makespan under scenario as and the corresponding optimal schedule as :For a given schedule , its regret under scenario is defined asThe maximum regret of schedule isThe robust parallel machine scheduling with interval processing time and arrival time and mold change can be formulated asThe proposed problem is a generalization of classical schedule problem and the optimal solution derived from (11) corresponds to the robust schedule across all possible scenarios.

The parallel machine robust scheduling with uncertain processing time and arrival time (RS) can be formulated as follows:We note that the above formulation is nonlinear due to the two operators in the objective function. However, the original model of RS can be transformed into a mixed integer linear programming model:After linearization of the original model of RS problem, the formulation still cannot be solved directly due to an infinite number of possible scenarios. To tackle this problem, the primal goal is to identify the maximal regret of each feasible solution and then select the solution with the minimum maximal regret across all solutions. Therefore, two properties are proposed to identify and confine worst-case scenario to a finite number of extreme point scenarios, which means that we do not have to visit all possible scenarios to obtain maxima regret of a given solution. In order to capture the uncertainties in real manufacturing environment, we assume that the processing time and arrival time of each job are uncertain by lying in their respective intervals. Therefore, the number of possible scenarios is infinite under such assumption and it is impractical to enumerate all possible scenarios to get the worst-case scenario for a given solution. The derived properties are used to narrow down the worst-case scenario to several extreme point scenarios.

*Definition 1. *A machine is said to be critical in a schedule under scenario , if is the machine with maximum completion time in under :

*Definition 2. *An extreme point scenario for schedule is defined as follows:

*Property 1. *For any schedule , let be a worst-case scenario for in which machine is critical. Then a scenario exists for schedule such that(a)machine is also critical in under ;(b)scenario is a worst-case scenario for .Assume that a given solution has a worst-case scenario that satisfies conditions (a) and (b) of Property 1. Since machine is critical in under scenario , the makespan and maximal regret for solution can be calculated as follows.

Note that the set containing all the jobs on machine in schedule is denoted as , . If , then .

*Proof. *Worst-case scenario can be transformed into scenario by decreasing to and to for all and by increasing to and to for all , and is also the critical machine in under . Let denote the increase in the makespan under . Let be the optimal schedule for scenario . Because the makespan in cannot increase by more than under scenario in comparison with , we getFrom the abovementioned inequality, we have , indicating that the maximal regret cannot decrease if we replace with , which is also the worst-case scenario for .

*Property 2. *The maximal regret of a given solution can be expressed as follows:

*Proof. *Equation in Property 2 shows thatSuppose by contradiction that there exists a machine such thatSince , where ,

#### 4. Exact Algorithm

RS is a min–max problem that can be solved using a general iterative relaxation (IR) procedure proposed by [33–35]. First, the set of all possible scenarios is replaced in RS with a finite set of scenarios , resulting in relaxed mixed integer program (RS-relaxed): *Note*. In this iterative relaxation approach, we replace the set of all possible scenarios () with , and the scenario in is the worst-case scenario for the feasible solution that is identified in the last iteration.

The minimal makespan can be solved by solving a mixed integer programming if the scenario is fixed.

We refer to the constraints and associated with as regret cuts.

*Iterative Relaxation Algorithm*. The IR procedures stop when , where is the maximum regret for current relaxed solution .

*Step 0*. Set ( is lower bound for regret value) and ( is upper bound for regret value); choose an initial solution .

*Step 1*. Identify solution ’s worst-case scenario and its respective through Property 1 and Property 2. If , then , . If , go to Step .

*Step 2*. Add regret cuts and ; to RS-relaxed scenario .

*Step 3*. and are identified by solving RS-relaxed. Set ; go to Step .

*Step 4*. Stop.

#### 5. Swarm Intelligence Approach

##### 5.1. Modified Artificial Bee Colony Algorithm

###### 5.1.1. Description of Modified Artificial Bee Colony Algorithm

ABC algorithm has been well-studied in parallel machine scheduling [36–38]. ABC algorithm is a promising swarm intelligence algorithm for optimization problems, which simulates the foraging behavior of honeybees. Three types of bees serve different functions to contribute to the forage acquisition for the bee social structure in the hive. The collected nectar will be stored as a food supply during the shortage of nectar. Employed bees and onlooker bees respond to obtain nectar from the nearby plants, while the main force of scout bee is to gather nectar information to guide the foraging behavior of the employed bees and onlooker bees. The design of the ABC algorithm follows a recursive search procedure with four major phases, including initialization phase, employed bee phase, onlooker bee phase, and scout bee phase. The process flowchart is shown in Figure 1.