Elsevier

Expert Systems with Applications

Volume 65, 15 December 2016, Pages 28-39
Expert Systems with Applications

A two-stage adaptive fruit fly optimization algorithm for unrelated parallel machine scheduling problem with additional resource constraints

https://doi.org/10.1016/j.eswa.2016.08.039Get rights and content

Highlights

  • A two-stage knowledge-based FOA for RCUPMSP.

  • Several properties for RCUPMSP.

  • Search manners with the guidance of knowledge.

  • Parameter setting and effectiveness demonstration.

Abstract

In this paper, an unrelated parallel machine scheduling problem with additional resource constraints (UPMSP_RC) from the real world manufacturing systems is studied. With the objective of minimizing the makespan, a mixed integer linear programming model is presented and several properties are analyzed. Furthermore, a two-stage adaptive fruit fly optimization algorithm (TAFOA) is proposed to solve the UPMSP_RC. At the first stage, a heuristic is proposed to generate an initial solution with high quality. At the second stage, the initial solution is adopted as the initial swarm center for further evolution. During the evolution, the search manners are selected adaptively with the guidance of the problem-specific knowledge, which is a sufficient condition of the best schedule under a given job-to-machine assignment. Moreover, the effect of parameters on the performance of the TAFOA is investigated by using the two-factor analysis of variance (ANOVA). Finally, extensive numerical comparisons are carried out to show the effectiveness of the TAFOA in solving the UPMSP_RC.

Introduction

The parallel machine scheduling problem (PMSP) is one of the most general scheduling problems. According to the machine environment, the PMSPs can be classified into three main categories: the identical PMSP, the uniform PMSP, and the unrelated PMSP (UPMSP) (Pinedo, 2012). As a generalization of the other two types (Edis, Oguz, & Ozkarahan, 2013), the UPMSP has gained much attention from many researchers. In the traditional UPMSP, a common assumption is that the processing can be executed by machines without accessory resources (Caniyilmaz et al., 2015, Edis et al., 2013, Lin and Hsieh, 2014, Sels et al., 2015, Strohhecker et al., 2016). However, in some realistic manufacturing systems, additional resources are also needed in the production procedure. For example, dies and machine operators are required to produce plastic parts for shipment in the injection-molding department of an electrical appliance plant (Edis & Oguz, 2012). A specific mask (also called reticle) is needed to shape the pattern on the wafer in the photolithography workshop of a semiconductor plant (Bitar, Dauzère-Pérès, Yugma, & Roussel, 2014). In the final testing stage of semiconductor manufacturing, boards besides the ovens are needed to hold the integrated circuits in the burn-in operations and the absence of necessary boards prevents the lot of circuits from being loaded into the ovens (Ventura & Kim, 2003). Therefore, it is of great practical significance to study the UPMSP with additional resource constraints (UPMSP_RC).

Due to its wide applications, the UPMSP_RC has attracted increasing attention (Edis et al., 2013) during recent years. Daniels, Hua, & Webster, (1999) considered the worker as an additional resource to formulate a static and dynamic RCPMSP and develop two efficient heuristics. In the static case, resource allocation decisions remain fixed throughout the scheduling horizon. In the dynamic case, resources can be reassigned to jobs once an operation is completed. Chen and Wu (2006) proposed an effective heuristic based on threshold-accepting methods, tabu lists, and improvement procedures for the UPMSP with auxiliary equipment constraints. Ruiz-Torres, Lopez, and Ho (2007) analyzed two versions of the UPMSP_RC. It was assumed that jobs were pre-assigned to machines in the first version while jobs can be assigned freely to machines in the second one. An integer programming formulation and a set of heuristics were proposed for two versions respectively. Su and Lien (2009) proposed a heuristic for the parallel machines resource-dependent processing time scheduling problem. Edis and Ozkarahan (2011) presented a combined integer programming (IP) / constraint programming (CP) for the resource-constrained identical parallel machine scheduling problem with machine eligibility restrictions. Edis and Oguz (2012) proposed a relaxed IP based CP approach for the RCPMSP and considered an additional resource, i.e. machine operators. Meanwhile, the processing time of each job was a non-increasing function of the associated amount of the allocated resource. Edis and Ozkarahan (2012) modeled the real-life scheduling problem in an injection-molding department of the electrical appliance plant as a UPMSP_RC with machine eligibility restrictions. Bitar et al. (2014) proposed a metaheuristic to solve the UPMSP with auxiliary resources in a photolithography workshop of a semiconductor plant. Joo and Kim (2015) presented hybrid genetic algorithms with dispatching rules for the unrelated parallel machine scheduling with production availability. Hsieh, Yang, and Yang (2015) developed an IP method and a heuristic method to solve the UPMSP with discrete controllable processing time, which can be controlled by the amount of an indivisible resource allocated.

Although the UPSMP_RC were widely studied, most of them considered only one additional resource with limited supply and assumed that the processing time of an operation depended on the amount of the additional resource. However, in a real world manufacturing factory, it is common that the limited resource affects the starting time of an operation instead of its processing time. For example, in a photolithography workshop, the machine will not start the processing of transferring an integrated circuit pattern on a wafer until the right reticle is available (Bitar et al., 2014). In addition, the procedure cannot be speeded up even if more reticles are provided for the machine. As for the indispensable board in the burn-in operations during the final testing stage of semiconductor manufacturing and the machine operator in the injection-molding department of the electrical appliance plant, the cases are similar. Thus, in this paper we focus on the UPMSP with limited common shared additional resources, where the availability of resources affects the starting time of an operation instead of the processing time. It can be defined as Rm|R|Cmax using the classification scheme expanded by Blazewicz, Lenstra, and Rinnooy (1983). Since Rm||Cmax is NP-hard (Pinedo, 2012), the UPMSP_RC is also NP-hard.

The fruit fly optimization algorithm (FOA) (Pan, 2012) is a newly proposed swarm-based evolutionary algorithm, which is inspired by the food finding procedure of fruit fly swarm using sensitive olfactory and vision. Compared with other existing metaheuristics, FOA has an easy understanding principle and fewer parameters. It has been applied to a variety of optimization problems, including power load forecasting (Li, Guo, Zhao, Su, & Wang, 2012), web auction logistics service (Lin, 2013), neural network parameter optimization (Li, Guo, Li, & Sun, 2013), proportional integral derivative controller parameter tuning (Sheng & Bao, 2013), multi-dimensional knapsack problem (Wang, Zheng, & Wang, 2013), semiconductor final testing scheduling problem (Zheng, Wang, & Wang, 2014), steelmaking casting problem (Li, Pan, Mao, & Suganthan, 2014), and joint replenishment problem (Wang, Shi, & Liu, 2015). The FOA has the potential to well solve the complex problems especially when the problem-oriented knowledge is embedded to guide the search. Therefore, we will propose a two-stage adaptive fruit fly optimization algorithm (TAFOA) for the UPMSP_RC in this paper. To be specific, an effective heuristic is proposed to generate an initial solution, which is adopted as the initial swarm center to perform further adaptive evolution. Moreover, a sufficient condition is derived to identify the optimality of a schedule under a given machine assignment. In addition, the effect of parameters on the performance of the TAFOA is investigated by using the two-factor analysis of variance (ANOVA). Finally, extensive numerical comparisons are carried out to show the effectiveness and efficiency of the TAFOA in solving the UPMSP_RC.

The remaining of the paper is organized as follows: In Section 2, the mixed integer linear programming (MILP) model of the UMPSP_RC is presented. In Section 3, the basic FOA is briefly introduced. In Section 4, the detailed design of the TAFOA for solving the UPMSP_RC is proposed. In Section 5, extensive numerical tests and comparisons are provided. Finally, the paper ends with some conclusions and future work in Section 6.

Section snippets

MILP model

The UPMSP_RC can be described as follows: n jobs (J1, J2,…, Jn) are to be processed on m unrelated machines (M1, M2,…, Mm) and the processing time of Ji on Mj is denoted as Pij. A common shared additional renewable resource is considered. To process a job, one unit of resource is required per unit time. The available amount of the renewable resource is R<m at any time. The processing procedure of a job cannot be started until both the assigned machine and the allocated resource are available.

Fruit fly optimization algorithm

The FOA (Pan, 2012) is a nature inspired swarm based intelligent technique, which mimics the foraging behaviors of the fruit fly. In real world, a fruit fly finds food sources using its sensitive osphresis and vision. During the foraging, the fruit fly can smell the scents of food source from far away, and the swarms fly towards the possible location of the food source with the largest smell concentration. After the fruit fly gets close to the food source, it can find the accurate location of

TAFOA for the UPMSP_RC

The FOA is first proposed for continuous optimization. To adopt the FOA for solving the UPMSP_RC, a discrete version of FOA is proposed in this section. First, the permutation-based representation and the decoding scheme are presented. Then, the detailed implementations of each phase of the TAFOA are introduced respectively, including a heuristic for the initialization, knowledge guided smell-based search, and vision-based search.

Experimental results

The TAFOA is coded in C++ and run on a PC with 2.3 GHz CPU. Since there are no available benchmark instances for the UPMSP_RC, we extend the widely used test instances for the UPMSP (Fanjul-Peyro & Ruiz, 2010) to test the performance of the algorithm. Considering that the performance of a solution method is greatly affected by the processing time matrix (Fanjul-Peyro & Ruiz, 2010), two sets of instances with uniform distributions U(1100) (denoted as set1) and U(100,200) (denoted as set2) are

Conclusion

The main contribution of this paper is to present an MILP model and propose an effective two-stage knowledge based fruit fly optimization algorithm for the UPMSP_RC. At the first stage, a simple effective heuristic is developed to generate the initial solution. At the second stage, fruit fly swarms perform search guided by the problem-oriented knowledge, and problem specific operators are designed. The effect of key parameters of the TAFOA is investigated by using the two-way ANOVA method. In

Acknowledgments

This research is partially supported by the Key R&D Project of China (No. 2016YFB0901901), and the National Science Fund for Distinguished Young Scholars of China (No. 61525304), and the National Natural Science Foundation of China (No. 61174189).

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