行政院國家科學委員會專題研究計畫成果報告

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行政院國家科學委員會專題研究計畫成果報告

粒子群最佳化於多目標排程問題之應用(第 3 年) 研究成果報告(完整版)

計畫類別：個別型

計畫編號： NSC 96-2221-E-216-052-MY3

執行期間： 98 年 08 月 01 日至 99 年 07 月 31 日執行單位：中華大學工業管理學系

計畫主持人：沙永傑

計畫參與人員：博士班研究生-兼任助理人員：林信宏

報告附件：出席國際會議研究心得報告及發表論文

處理方式：本計畫可公開查詢

中華民國 99 年 09 月 24 日

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行政院國家科學委員會補助專題研究計畫成果報告行政院國家科學委員會補助專題研究計畫成果報告

粒子群最佳化於多目標排程問題之應用粒子群最佳化於多目標排程問題之應用粒子群最佳化於多目標排程問題之應用粒子群最佳化於多目標排程問題之應用

計畫類別計畫類別計畫類別

計畫類別：：：： _■ 個別型計畫個別型計畫個別型計畫個別型計畫 □整合型計畫整合型計畫整合型計畫整合型計畫計畫編號

計畫編號計畫編號

計畫編號：：：：NSC 96－－－－2221－－－－E－－－－216－－－－052－－－－MY3 執行期間

執行期間執行期間

執行期間：：：： 96 年年年年 8 月月月月 1 日日日日至至至至 99 年年年年 7 月月月月 31 日日日日

計畫主持人計畫主持人計畫主持人

計畫主持人：：：：沙永傑沙永傑沙永傑沙永傑

執行單位執行單位執行單位

執行單位：：：：中華大學工業工程與管理學系中華大學工業工程與管理學系中華大學工業工程與管理學系中華大學工業工程與管理學系

中中中

中華華華華民民民民國國國國九十九九十九九十九九十九年年年年九九九九月月月月十十十十日日日日

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行政院國家科學委員會專題研究計畫成果報告行政院國家科學委員會專題研究計畫成果報告行政院國家科學委員會專題研究計畫成果報告行政院國家科學委員會專題研究計畫成果報告

粒子群最佳化於多目標排程問題之應用粒子群最佳化於多目標排程問題之應用粒子群最佳化於多目標排程問題之應用粒子群最佳化於多目標排程問題之應用

A Particle Swarm Optimization for Multi-Objective Scheduling Problems

計畫編號：NSC 96-2221-E-216-052MY3 執行期限：96 年 8 月 1 日至 99 年 7 月 31 日

主持人：沙永傑中華大學工業工程與管理學系

一一一

一、、、、中文摘要中文摘要中文摘要中文摘要

粒子群最佳化 (Particle Swarm Optimization, PSO)是一種以群體搜尋為基礎的最佳化演算法，其群體是由各自獨立的粒子所組成。PSO 模擬空間中的粒子運動，以搜尋空間(search space)表達出問題的解集合空間(solution space)。每個搜尋空間中的位置 (position)對應到一個該問題解集合空間中的解。粒子群於搜尋空間中(解集合空間中)合作搜尋出最佳位置(最佳解)。PSO 最初應用於求解連續性最佳化問題。然而近年來也被應用於求解組合最佳化問題。

多數與排程問題有關的研究都只探討如何使單一目標最佳化，如：總完工時間、總延遲成本、延遲工件數…等。在現實情況下，決策者必需作出能夠將這些目標最佳化的決策，卻又面臨這些目標之間互相衝突的問題。

若只追求單一目標的最佳化排程，則會造成另一目標的損失。本研究的主要目的在架構出能求解多目標排程問題之 PSO，幫助決策者在面對如此複雜排程問題時能夠做出合理決策。

本研究以三年時間分階段進行。第一年以流程型排程問題(flow shop scheduling problem) 為研究對象；第二年以零工式排程問

題(job shop scheduling problem)為研究對象；

第三年以開放型排程問題 (open shop

scheduling problem)為研究對象。在這三種最基本的排程問題中，我們以完工時間 (makespan)、總流程時間(total flow time)、總延遲時間(total tardiness)作為目標函數。本研究所提出的 PSO，應用於標竿測試問題與過去的啟發式演算法，在求解的速度與品質上有 較佳的表現。

關鍵詞關鍵詞關鍵詞

關鍵詞：：：：：：：：粒子群最佳化粒子群最佳化粒子群最佳化粒子群最佳化、、、、流程型排程流程型排程流程型排程流程型排程、、、、零工零工零工零工型

型型

型排程排程排程、排程、、開放型排程、開放型排程開放型排程開放型排程、、、、多目標排程多目標排程多目標排程多目標排程

Abstract

Particle Swarm Optimization (PSO) is a population-based optimization algorithm. Each particle is an individual and the swarm is composed of particles. PSO mimics the particle movement in a space. In PSO, the problem solution space is formulated as a search space.

Each position in the search space is a correlated solution of the problem. Particles cooperate to find out the best position (best solution) in the search space (solution space).

The original intention of PSO is to solve continuous optimization problems. However, it has been implemented to solve many combinatorial optimization problems in recent years.

In most previous research about scheduling,

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2 example, total complete time, total tardiness, and maximum makespan. In fact, the decision makers have to simultaneously optimize these objectives, but there are conflicts between these objectives. If we only optimize one of these objectives, we will loose another objective. In this research, we will construct a PSO to solve multi-objective scheduling problems. That can help decision makers to make a strategy to handle such complex scheduling problems.

We will execute this research in three years.

In the first year, we focus on the flow shop scheduling problem. In the second year, we focus on the job shop scheduling problem. In the third year, we focus on the open shop scheduling problem. In these three scheduling problems, the objective functions are maximum completion time, total weighted completion time, and total weighted tardiness. Moreover, we will add some constraints into the scheduling problems, such as limited intermediate storage and dependent setup time.

Keywords: Particle swarm optimization, Multi-objective scheduling problem

二二二

二、、、、研究目的研究目的研究目的研究目的

多數與排程問題有關的研究都只探討如何使單一目標最佳化，如：總完工時間、總延遲成本、延遲工件數…等。在現實情況下，決策者必需作出能夠將這些目標最佳化的決策，卻又面臨這些目標之間互相衝突的問題。

若只追求單一目標的最佳化排程，則會造成另一目標的損失。

大多數排程問題皆為 NP 問題，也就是說，此問題無法在合理的計算時間內求出最佳解。因此，學者們發展出各類型啟發式解法試圖求出近似最佳解，而演化式計算(Evolution Computation)則是近十年來最被廣泛應用，也

計算中又以粒子群最佳化 (Particle Swarm Optimization, PSO)為最新、且為近 3 年來最被廣泛研究及討論的演算法。另於一般單目標的排程問題中，我們已將 PSO 應用於 JobShop 及 Open Shop 問題上。研究發現應用 PSO 求解會得到比其它方法更好的結果(Sha & Hsu, 2006a, 2006b)。因此，本研究的主要目的在架構出能求解多目標排程問題之 PSO，幫助決策者在面對此複雜的排程問題時能夠做出合理的決策。

因此本研究的主要目的是架構出適合求解多目標排程問題的 PSO。除此之外，由於 PSO 原本是用來求解連續性最佳化問題 (continues optimization problems)，在組合最佳化方面的應用尚未成熟，還有許多可供研究及改良的空間。排程問題也是組合最佳化問題之一，因此本研究所發展之 PSO 將為後續 PSO 應用於組合最佳化問題研究之基礎。我們也將以本研究為基礎，將本研究所發展之 PSO 應用於求解其它組合最佳化問題。

三三三

三、、、粒子群、粒子群粒子群粒子群演算法演算法演算法演算法相關文獻相關文獻相關文獻相關文獻

粒子群最佳化 (Particle Swarm Optimization, PSO) 由 Kennedy 和 Eberhart 於 1995 年提出。它是由模仿粒子於空間中運動的行為，而發展出的最佳化方法。PSO 和基因演算法(Genetic Algorithm,GA)相同，都是以群體為基礎的(population-based)最佳化演算法。在 PSO 中，群體(swarm)是由粒子(particle) 所組成。群體(swarm)和粒子(particle)的關係，

和在 GA 中母體 (population) 和染色體 (chromosome)的關係相似。

PSO 以搜尋空間(search space)表達出問題的解集合空間(solution space)。每個搜尋空間中的位置(position)對應到一個該問題解集合空間中的解。粒子群於搜尋空間中(解集合空間中)合作搜尋出最佳位置(最佳解)。而其中粒子運動主要受到三個因素影響：慣性

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3 (inertia)、個體最佳位置(pbest position)、群體最佳位置(gbest position)。慣性(inertia)為粒子於上一循環 (iteration) 所殘留下來的速度 (velocity)，它能夠由慣性權重(inertia weight) 控制。慣性(inertia)的用意在於防止粒子停留在同一範圍內，而能跳出區域最佳解(local optima)。個體最佳位置(pbest solution)為粒子本身到目前為止所搜尋出的最佳位置(或最佳解)，因此每個粒子有它自已的個體最佳位置 (pbest position)。群體最佳位置(gbest position) 則是到目前為止群體(swarm)所搜尋出的最佳位置(最佳解)，而整個群體(swarm)只會有一個群體最佳位置(gbest position)。

在 PSO 中粒子速度(velocity)由一向量表示，而在每個循環(iteration)中粒子根據它所擁有的速度移動它的位置。在每個循環(iteration) 裡，粒子朝著個體最佳位置(pbest position)以及群體最佳位置(gbest position)移動，而其速度也是根據個體最佳位置(pbest position)及群體最佳位置(gbest position)隨機求得。

四四四

四、、、、研究成果研究成果研究成果研究成果

本研究已完成求解流程型排程問題 (Flow-shop Scheduling Problem)之 PSO，為了證明所發展的 PSO 的適用性、求解品質與效率，我們將本研究的 PSO 與 TSP-GA(Ponnambalam, 2004)同時測試 21 個問題進行比較。目標式有三個，分別為最小完工時間(Makespan)、平均流程時間(Mean Flow Time)與機器閒置時間(Machine Idle Time)。運算的結果顯示，以平均相對誤差而言，針對最小完工時間，PSO 有 17 個問題表現優於 TSP-GA；平均流程時間，PSO 有 18 個問題表現優於 TSP-GA；機器閒置時間，PSO 有 19 個問題表現優於 TSP-GA。綜合而言，PSO 在 21 個案例中的 19 個，同時達到三個目標且優於 TSP-GA 。此結果已發表於 The 38^th International Conference on Computers and Industrial Engineering (2008).

另一方面，亦將 PSO 與傳統的啟發式演算法 CDS 與 NEH 進行比較。同樣以最小完工時間(Makespan)、平均流程時間(Mean Flow Time)與機器閒置時間(Machine Idle Time)作為目標式。此次共測試 161 個標竿問題，包含 Rec01 至 Rec41，Tai20×5 至 Tai500×20。

求解的結果顯示 PSO 比傳統的啟發式演算法 CDS 與 NEH 具備明顯的優勢。結果發表於 The 9th Asia Pacific Industrial Engineering &

Management Systems Conference (2008).

PSO 應用於求解多目標流程型排程問題的結果已發表於 International Journal of Advanced Manufacturing Technology Vol. 45, No. 7, pp. 749-762. Vol. 37, No. 2, pp.

1065-1070, 2009 (SCI)(如附錄一).

本研究已完成求解零工式排程問題 (Job-shop Scheduling Problem)之 PSO，同樣的，我們選擇三個目標式，最小完工時間 (Makespan)、總延遲時間(Total Tardiness)與機器閒置時間(Machine Idle Time)。比較的對象則是 MOGA (Ponnambalam, 2001)，在 23 個標竿問題中，PSO 在最小完工時間與總延遲時間，完全領先 MOGA；至於機器閒置時間則於 22 個問題中具競爭力。

PSO 應用於求解多目標零工型排程問題的結果已發表於 Expert Systems with Applications, Vol. 37, No. 2, pp. 1065-1070, 2010 (SCI) (如附錄二).

對於開放式排程問題，本研究亦完成 PSO 設計，我們以最小完工時間(Makespan)、總流程時間 (Total Flow Time) 與機器閒置時間 (Machine Idle Time) 作為目標式。由於開放式排程問體文獻相對較少，本研究另外設計並撰寫基因演算法作為比較，並以 Guéret and Prins (1999)標竿問題進行測試，求解結果顯示本驗就提出的 PSO 整體績效大於 GA。此結果已投稿於 Journal of Industrial and Management Optimization.

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五、、、、參考文獻參考文獻參考文獻參考文獻

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Hu, X., & Eberhart, R.C. (2002). “Multiobjective optimization using dynamic neighborhood particle swarm optimization,” Proceedings of the 2002 Congress on Evolutionary Computation, Vol. 2,1677-1681.

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六、、、附錄、附錄附錄附錄

1. A particle swarm optimization for multi-objective flow-shop scheduling

2. A multi-objective PSO for job-shop scheduling problems

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ORIGINAL ARTICLE

A particle swarm optimization for multi-objective flowshop scheduling

D. Y. Sha_&Hsing-Hung Lin

Received: 5 September 2008 / Accepted: 6 February 2009

# Springer-Verlag London Limited 2009

Abstract The academic approach of single-objective flowshop scheduling has been extended to multiple objectives to meet the requirements of realistic manufacturing systems.

Many algorithms have been developed to search for optimal or near-optimal solutions due to the computational cost of determining exact solutions. This paper provides a particle swarm optimization-based multi-objective algorithm for flowshop scheduling. The proposed evolutionary algorithm searches the Pareto optimal solution for objectives by considering the makespan, mean flow time, and machine idle time. The algorithm was tested on benchmark problems to evaluate its performance. The results show that the modified particle swarm optimization algorithm performed better in terms of searching quality and efficiency than other traditional heuristics.

Keywords PSO . Multi-objective . Flowshop scheduling . Pareto optimal

1 Introduction

Production scheduling in real environments has become a significant challenge in enterprises maintaining their com- petitive positions in rapidly changing markets. Flowshop

scheduling problems have attracted much attention in academic circles in the last five decades since Johnson’s initial research. Most of these studies have focused on finding the exact optimal solution. A brief overview of the evolution of flowshop scheduling problems and possible approaches to their solution over the last 50 years has been provided by Gupta and Stafford [5]. That survey indicated that most research on flowshop scheduling has focused on single-objective problems, such as minimizing completion time, total flow time, or total tardiness. Numerous heuristic techniques have been developed for obtaining the approx- imate optimal solution to NP-hard scheduling problems. A complete survey of flowshop scheduling problems with makespan criterion and contributions, including exact methods, constructive heuristics, improved heuristics, and evolutionary approaches from 1954 to 2004, was offered by Hejazi et al. [7]. Ruiz et al. [24] also presented a review and comparative evaluation of heuristics and meta-heuristics for permutation flowshop problems with the makespan criterion. The NEH algorithm [17] has been shown to be the best constructive heuristic for Taillard’s benchmarks [28] while the iterated local search [27] method and the genetic algorithm (GA) [23] are better than other meta-heuristic algorithms.

Most studies of flowshop scheduling have focused on a single objective that could be optimized independently.

However, empirical scheduling decisions might not only involve the consideration of more than one objective, but also require minimizing the conflict between two or more objectives. In addition, finding the exact solution to scheduling problems is computationally expensive because such problems are NP-hard. Solving a scheduling problem with multiple objectives is even more complicated than solving a single-objective problem. Approaches including meta-heuristics and memetics have been developed to reduce the complexity and improve the efficiency of solutions.

Int J Adv Manuf Technol DOI 10.1007/s00170-009-1970-6

The English in this document has been checked by at least two professional editors, both native speakers of English. For a certificate, see:http://www.textcheck.com/cgi-bin/certificate.cgi?id=emRe2r D. Y. Sha

Department of Industrial Engineering and System Management, Chung Hua University,

Hsinchu, Taiwan, Republic of China H.-H. Lin (*)

Department of Industrial Engineering and Management, National Chiao Tung University,

Hsinchu, Taiwan, Republic of China e-mail: hsinhung@gmail.com

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Hybrid heuristics combining the features of different methods in a complementary fashion have been a hot issue in the fields of computer science and operational research [15]. Ponnambalam et al. [19] considered a weighted sum of multiple objectives, including minimizing the makespan, mean flow time, and machine idle time as a performance measurement, and proposed a multi-objective algorithm using a traveling salesman algorithm and the GA for the flowshop scheduling problem. Rajendran et al. [21]

approached the problem of scheduling in permutation flowshop using two ant colony optimization (ACO) approaches, first to minimize the makespan, and then to minimize the sum of the total flow time. Yagmahan [30]

was the first to apply ACO meta-heuristics to flowshop scheduling with the multiple objectives of makespan, total flow time, and total machine idle time.

The literature on multi-objective flowshop scheduling problems can divided into two groups: a priori approaches with assigned weights of each objective, and a posteriori approaches involving a set of non-dominated solutions [18]. There is also a multi-objective GA (MOGA) called PGA-ALS, designed to search non-dominated sequences with the objectives of minimizing makespan and total flow time. The multi-objective solutions are called non- dominated solutions (or Pareto optimal solutions in the case of Pareto optimality). Eren et al. [4] tackled a multi- criteria two-machine flowshop scheduling problem with minimization of the weighted sum of total completion time, total tardiness, and makespan.

Particle swarm optimization (PSO) is an evolutionary technique for unconstrained continuous optimization problems proposed by Kennedy et al. [10] The PSO concept is based on observations of the social behavior of animals such as birds in flocks, fish in schools, and swarm theory.

To minimize the objective of maximum completion time (i.

e., the makespan), Liu et al. [15] invented an effective PSO- based memetic algorithm for the permutation flowshop scheduling problem. Jarboui et al. [9] developed a PSO algorithm for solving the permutation flowshop scheduling problem; this was an improved procedure based on simulated annealing. PSO was recommended by Tasgetiren et al. [29] to solve the permutation flowshop scheduling problem with the objectives of minimizing makespan and the total flow time of jobs. Rahimi-Vahed et al. [22] tackled a bi-criteria permutation flowshop scheduling problem where the weighted mean completion time and the weighted mean tardiness were minimized simultaneously.

They exploited a new concept called the ideal point and a new approach to specifying the superior particle’s position vector in the swarm that is designed and used for finding the locally Pareto optimal frontier of the problem. Due to the discrete nature of the flowshop scheduling problem, Lian et al. [14] addressed permutation flowshop scheduling

with a minimized makespan using a novel PSO. All these approaches have demonstrated the advantages of the PSO method: simple structure, immediate applicability to prac- tical problems, ease of implementation, quick solution, and robustness.

The aim of this paper is to explore the development of PSO for elaborate multi-objective flowshop scheduling problems. The original PSO was used to solve continuous optimization problems. Due to the discrete solution spaces of scheduling optimization problems, we modified the particle position representation, particle movement, and particle velocity in this study.

The remainder of this paper is organized as follows.

Section 2 contains a formulation of the flowshop scheduling problem with two objectives. Section 3 describes the algorithm of the proposed PSO approach. Section 4 contains the simulated results of benchmark problems. Sec- tion5 provides some conclusions and future directions.

2 Problem formulation

The problem of scheduling in flowshops has been the subject of much investigation. The primary elements of flowshop scheduling include a set of m machines and a collection of n jobs to be scheduled on the set of machines.

Each job follows the same process of machines and passes through each machine only once. Each job can be processed on one and only one machine at a time, whereas each machine can process only one job at a time. The processing time of each job on each machine is fixed and known in advance. We formulate the multi-objective flowshop scheduling problem using the following notation:

& n is the total number of jobs to be scheduled,

& m is the total number of machines in the process,

& t(i, j) is the processing time for job i on machine j (i=1, 2,…n) and (j=1,2,…m), and

& {π1,π2,…, πn} is the permutation of jobs.

The objectives considered in this paper can be calculated as follows:

& Completion time (makespan) C p; jð Þ:

Cðp1; 1Þ ¼ t pð 1; 1Þ

Cðpi; 1Þ ¼ C pð i1; 1Þ þ t pð i; 1Þi ¼ 2; . . . ; n Cðp1; jÞ ¼ C pð 1; j 1Þ þ t p; jð Þj ¼ 2; . . . ; m Cðpi; jÞ ¼ max C pf ð i1; jÞ; C pð i; j 1Þg þ tðpi; jÞ

i¼ 2; . . . ; n; j ¼ 2; . . . ; m

& Makespan, fC max¼ C pð n; mÞ,

& Mean flow time, fMFT ¼ Pⁿ

i¼1Cðpi; mÞ

n,

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& Machine idle time, and

& fMIT ¼ fC pð 1; j 1Þ þPⁿ

i¼2fmax Cf ðpi; j 1Þ C pð i1; jÞ; 0ggjj ¼ 2:::mg

3 Basic PSO concept

PSO is an evolutionary technique (Kennedy et al. [10]) for solving unconstrained continuous optimization problems.

The PSO concept is based on observations of the social behavior of animals. The population consisting of individ- uals (particles) is assigned a randomized initial velocity according each individual’s own movement experience and that of the rest of the population. The relationship between the swarm and the particles in PSO is similar to the relationship between the population and the chromosomes in the GA.

The PSO problem solution space is formulated as a search space. Each position of the particles in the search space is a correlated solution of the problem. Particles cooperate to determine the best position (solution) in the search space (solution space).

Suppose that the search space is D-dimensional and there are m particles in the swarm. Each particle is located at position Xi={xi1, xi2,…, xiD} with velocity Vi={vi1, vi2,…, viD}, where i=1, 2,…, m. In the PSO algorithm, each particle moves toward its own best position (pbest) denoted as Pbesti={pbesti1, pbesti2,…, pbestin}. The best position of the whole swarm (gbest) denoted as Gbest={gbest1, gbest2,…, gbestn} with each iteration. Each particle changes its position according to its velocity, which is randomly generated toward the pbest and gbest positions. For each particle r and dimension s, the new velocity vrsand position xrsof particles can be calculated by the following equations:

v^t_rs¼ w v^t1_rs þ c1 rand1 pbest _rs^t1 x^t1_rs þ c2

rand2 gbest _s^t1 x^t1_rs

ð1Þ

x^t_rs¼ x^t1_rs þ v^t1_rs ð2Þ

where t is the iteration number. The inertial weight w is used to control exploration and exploitation. A large value of w keeps particles at high velocity and prevents them from becoming trapped in local optima. A small value of w maintains particles at low velocity and encourages them to exploit the same search area. The constants c1and c2are acceleration coefficients that determine whether particles prefer to move closer to the pbest or gbest positions. The rand1 and rand2 are independent random numbers uni- formly distributed between 0 and 1. The termination

criterion of the PSO algorithm includes the maximum number of generations, the designated value of pbest, and no further improvement in pbest. The standard PSO process outline is as follows.

Step 1: initialize a population of particles with random positions and velocities on D dimensions in the search space.

Step 2: update the velocity of each particle according to Eq. (1).

Step 3: update the position of each particle according to Eq. (2).

Step 4: map the position of each particle into the solution space and evaluate its fitness value according to the desired optimization fitness function. Simulta- neously update the pbest and gbest positions if necessary.

Step 5: loop to Step 2 until an exit criterion is met, usually a sufficient goodness of fitness or a maximum number of iterations.

The original PSO was designed for a continuous solution space. We modified the PSO position representation, particle velocity, and particle movement as described in the next section to make PSO suitable for combinational optimization problems.

4 Formation of the proposed PSO

There are two different representations of particle position associated with a schedule. Zhang [31] demonstrated that permutation-based position representation outperforms priority-based representation. While we have chosen to implement permutation-based position representation, we must also adjust the particle velocity and particle movement as described in Sections4.2and4.3. We have also included the maintenance of Pareto optima and local search procedures to achieve better performance.

4.1 Position representation

In this study, we randomly generated a group of particles (solutions) represented by a permutation sequence that is an ordered list of operations. The following example is a permutation sequence for a six-job permutation flowshop scheduling problem, where jnis the operation of job n.

Index : 1 2 3 4 5 6

Permutation : j₄ j₃ j₁ j₆ j₂ j₅

An operation earlier in the list has a higher priority of being placed into the schedule. We used a list with a length Int J Adv Manuf Technol

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of n for an n-job problem in our algorithm to represent the position of particle k, i.e.,

X^k¼ x ^k₁x^k₂. . . x^k_n

;

x^k_i is the priority of jiin particle k:

Then, we convert the permutation list to a priority list.

Thex^k_iis a value randomly initialized to some value between (p–0.5) and (p + 0.5). This means x^k_i p þ rand 0:5, where p is the location (index) of jiin the permutation list, and rand is a random number between 0 and 1. Conse- quently, the operation with smaller x^k_ihas a higher priority for scheduling. The permutation list mentioned above can be converted to

X^k¼ 2:7 5:2 1:8 0:6 6:3 3:9½

4.2 Particle velocity

The original PSO velocity concept is that each particle moves according to the velocity determined by the distance between the previous position of the particle and the gbest (pbest) solution. The two major purposes of the particle velocity are to move the particle toward the gbest and pbest solutions, and to maintain the inertia to prevent particles from becoming trapped in local optima.

In the proposed PSO, we concentrated on preventing particles from becoming trapped in local optima rather than moving them toward the gbest (pbest) solution. If the priority value increases or decreases with the present velocity in this iteration, we maintain the priority value increasing or decreasing at the beginning of the next iteration with probability w, which is the PSO inertial weight. The larger the value of w is, the greater the number of iterations over which the priority value keeps increasing or decreasing, and the greater the difficulty the particle has returning to the current position. For an n-job problem, the velocity of particle k can be represented as

V^k¼ v ^k₁ v^k₂. . . v^k_n

; v^k_i 2 1; 0; 1f g where v^k_i is the velocity of jiof particle k:

The initial particle velocities are generated randomly.

Instead of considering the distance from x^k_i to pbest^k_iðgbest_iÞ, our PSO considers whether the value of x^k_i is larger or smaller than pbest^k_iðgbest_iÞ If x^k_i has decreased in the present iteration, this means that pbest^k_iðgbest_iÞ is smaller than x^k_i, and x^k_i is set moving toward pbest^k_iðgbest_iÞ by letting v^k_i 1. Therefore, in the next iteration, x^k_i is kept decreasing by one (i.e., x^k_i x^k_i 1) with probability w.

Conversely, if x^k_i has increased in this iteration, this means that pbest^k_iðgbest_iÞ is larger than x^k_i, and x^k_i is set moving toward pbest^k_iðgbest_iÞ by letting v^k_i 1. Therefore, in the

next iteration, x^k_i is kept increasing by one (i.e. x^k_i x^k_i þ 1) with probability w.

The inertial weight w influences the velocity of particles in PSO. We randomly update velocities at the beginning of each iteration. For each particle k and operation ji, if v^k_i is not equal to 0, v^k_i is set to 0 with probability (1–w). This ensures that x^k_i stops increasing or decreasing continuously in this iteration with probability (1–w).

4.3 Particle movement

The particle movement is based on the insertion operator proposed by Sha et al. [25, 26]. The insertion operator is introduced to the priority list to reduce computational complexity. We illustrate the effect of the insertion operator using the permutation list example described above. If we wish to insert j4 into the third location of the permutation list, we must move j6to the sixth location, move j1to the fifth location, move j2to the fourth location, and then insert j4in the third location. The insertion operation comprising these actions costs O(n/2) on average. However, the insertion operator used in this study need only set x^k_i 3þ rand 0:5 when we want to insert j5 in the third location of the permutation. This requires only one step for each insertion. If the random number rand equals 0.1, for example, after j4 is inserted into the third location, then X^kbecomes X^k ¼ 2:7 5:2 1:8 0:6 2:6 3:9½ .

If we wish to insert ji into the pth location in the permutation list, we could set x^k_i p þ rand 0:5. The location of operation ji in the permutation sequence of the kth pbest and gbest solutions are pbest^k_i and gbesti, respectively. As particle k moves, if v^k_i equals 0 for all ji, then x^k_i is set to pbest^k_i þ rand 0:5 with probability c1and set to gbesti+ rand−0.5 with probability c2, where rand is a random number between 0 and 1, c1and c2are constants between 0 and 1, and c1þ c2 1. We explain this concept by assuming specific values for V^k, X^k, pbest^k, gbest, c1, and c2.

V^k¼ 1 0 0 1 0 0½ ;

X^k¼ 2:7 5:2 1:8 0:6 6:3 3:9½ ;

pbest^k¼ 5 1 4 6 3 2½ ;

gbest¼ 6 3 4 5 1 2½ ; c1¼ 0:8; c2¼ 0:1:

– For j1, since v^k₁ 6¼ 0 and x^k₁ x^k₁þ v^k₁, then x^k₁¼ 1:7.

– For j2, since v^k₂¼ 0, the generated random number rand1=0.6. Since rand1 c1, then the generated random number rand2=0.3. Since pbest^k₂ x^k₂, set v^k₂

1 and x^k₂ pbest^k₂þ rand2 0:5, i.e., x^k₂¼ 0:8.

– For j3, since v^k₃¼ 0, the generated random number rand1=0.93. Since rand₁> c1þ c₂, x^k₃ and v^k₃ do not need to be changed.

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– For j4, since v^k₄¼ 1, then x^k₄ x^k₄þ v^k₄, i.e., x^k₄¼ 1:6.

– For j5, since v^k₅ ¼ 0, the generated random number rand1=0.85. Since c1 < rand1 c1þ c2, the generated random number rand2= 0.7. Since gbest₅ x^k₅, set v^k₅ 1. Then x^k₅ gbest₅þ rand2 0:5, i.e., x^k₅ ¼ 1:2.

– For j6, since v^k₆ ¼ 0, the generated random number rand1=0.95. Since rand1> c1þ c2, x^k₆ and v^k₆ do not need to be changed.

Therefore, after particle k moves, the V^kand X^kare

V^k¼ ½1 1 0 1 1 0

X^k¼ ½1:6 0:8 1:8 1:7 1:2 3:9

In addition, we use a mutation operator in our PSO algorithm. After moving a particle to a new position, we randomly choose an operation and then mutate its priority value x^k_i in accordance with v^k_i. If x^k_i ðn=2Þ, we randomly set x^k_i to a value between (n/2) and n, and set v^k_i 1. If x^k_i > ðn=2Þ, we randomly set x^k_i to a value between 0 and (n/2), and set v^k_i 1.

4.4 Pareto optimal set maintenance

Real empirical scheduling decisions often involve not only the consideration of more than one objective at a time, but also must prevent the conflict of two or more objectives.

The solution set of the multi-objective optimization problem with conflicting objective functions consistent with the solutions so that no other solution is better than all other objective functions is called Pareto optimal. A multi-objective minimization problem with m decision variables and n objectives is given below to describe the concept of Pareto optimality.

Minimize F xð Þ ¼ fð1ð Þ; fx 2ð Þ; . . . ; fx nð Þx Þ where; x 2 <^m; F xð Þ 2 <ⁿ

A solution p is said to dominate solution q if and only if fkð Þ fp kð Þ 8k 2 1; 2; . . . ; nq f g

fkð Þ < fp kð Þ 9k 2 1; 2; . . . ; nq f g

Non-dominated solutions are defined as solutions that dominate the others but do not dominate themselves.

Solution p is said to be a Pareto optimal solution if there exists no other solution q in the feasible space that could dominate p. The set including all Pareto optimal solutions is referred to as the Pareto optimal or Pareto optimalPareto optimal set. A graph plotted using collected Pareto optimal solutions in feasible space is referred to as the Pareto front.

The external Pareto optimal set is used to produce a limited size of non-dominated solutions (Knowles et al., [11]; Zitzler et al. [32]). The maximum size of the archive

set is specified in advance. This method is used to avoid missing fragments of the non-dominated front during the search process. The Pareto optimal front is formed as the archive is updated iteratively. When the archive set is sufficiently empty and a new non-dominated solution is detected, the new solution enters the archive set. As the new solution enters the archive set, any solution already there that is dominated by this solution will be removed.

When the maximum archive size reaches its preset value, the archive set must decide which solution should be replaced. In this study, we propose a novel Pareto archive set update process to preclude losing non-dominated solutions when the Pareto archive set is full. When a new non-dominated solution is discovered, the archive set is updated when one of the following situations occurs: either the number of solutions in the archive set is less than the maximum value, or if the number of solutions in the archive set is equal to or greater than the maximum value, then the one solution in the archive set that is most dissimilar to the new solution is replaced by the new solution. We measure the dissimilarity by the Euclidean distance. A longer distance implies a higher dissimilarity.

The non-dominated solution in the Pareto archive set with the longest distance to the newly found solution is replaced.

For example, the distance (dij) between X¹ and X² is calculated as

X¹¼ 2:7 5:2 1:8 0:6 6:3 3:9½ X²¼ 1:6 0:8 1:8 1:7 1:2 3:9½

dij¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:7 1:6

ð Þ²þ 5:2 0:8ð Þ²þ 0:6 1:7ð Þ²þ 6:3 1:2ð Þ² q

¼ 6:91

The Pareto archive set is updated at the end of each iteration in the proposed PSO.

4.5 Diversification strategy

If all the particles have the same non-dominated solutions, they will be trapped in the local optimal. To prevent this, a diversification strategy is proposed to keep the non- dominated solutions different. Once any new solution is generated by the particles, the non-dominated solution set is updated according to one of three situations.

1. If the solution of the particle is dominated by the gbest solution, assign the particle solution to gbest.

2. If the solution of the particle equals any solution in the non-dominated solution set, replace the non-dominated solution with the particle solution.

3. If the solution of the particle is dominated by the worst non-dominated solution and not equal to any non- dominated solution, set the worst non-dominated solution equal to the particle solution.

Int J Adv Manuf Technol

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5 Computational results

The proposed PSO algorithm was verified by benchmark problems obtained from the OR-Library that were contrib- uted by Carlier [2], Heller [8], and Reeves [23]. The test program was coded in Visual C++ and run 20 times on each problem using an Intel Pentium 4 3.0-GHz processor with 1 GB of RAM running Windows XP. We used four swarm sizes N (10, 20, 60, and 80) to test the algorithm during a pilot experiment. A value of N=80 was best, so it was used in all subsequent tests. The algorithm parameters were set as follows: c1and c2were tested over the range 0.1–0.7 in increments of 0.2, and the inertial weight w was reduced from wmax to wmin during the iterations. Parameter wmax

was set to 0.5, 0.7, and 0.9 corresponding to wminvalues of 0.1, 0.3, and 0.5. Settings of c1=0.7, c2=0.1, wmax=0.7, and wmin=0.3 worked best.

The proposed PSO algorithm was compared with five heuristic algorithms: CDS[1], NEH[17], RAJ[20], GAN- RAJ[6] and Laha[13]. We also coded these methods in Visual C++. The CDS heuristic [1] takes its name from its three authors and is a heuristic generalization of Johnson’s algorithm. The process generates a set of m−1 artificial two-machine problems, each of which is then solved by Johnson’s rule. In this study, we modified the original CDS and compared the makespan, mean flow time, and machine idle time of all m−1 generated problems. The non- dominated solution was selected to compare with the solutions obtained from our PSO algorithm. The other comparison was based on solutions determined by the NEH algorithm introduced by Nawaz et al. [17]. The NEH investigates n(n+1)/2 permutations to find near-optimal solutions. As we did for CDS, we modified the original NEH and compared the three objectives of all n(n+1)/2

sequences. We compared the non-dominated solution from these sequences with the solutions from our PSO.

The following two performance measures are used in this study: average-relative percentage deviation (ARPD) and maximum percentage deviation (MPD) where MS stands for makespan, TFT represents total flow time, MIT stands for machine idle time, H is the heuristic.

ARPD_MS¼100 10

X¹⁰

i¼1

MS_H;i BestMS_i BestMSi

ð3Þ

MPDMS¼ MAXi¼1::10 MSH;i BestMSi

BestMS_i

100 ð4Þ

ARPDTFT¼100 10

X¹⁰

i¼1

TFTH;i BestTFTi

BestTFT_i

ð5Þ

MPDTFT¼ MAX_i¼1::10 TFT_H;i BestTFTi

BestTFTi

100 ð6Þ

ARPDMIT¼100 10

X¹⁰

i¼1

MIT_H;i BestMITi

BestMITi

ð7Þ

MPDMIT¼ MAX_i¼1::10 MIT_H;i BestMITi

BestMITi

100 ð8Þ

We tested our PSO on nine different problem sizes (n=

20, 50, 100 and m = 5, 10, 20) from Taillard’s [28]

benchmarks. Table 1 compares the six methods using the

Table 1 Comparison of makespan(MS) for different heuristics

Problem size NEH [17] CDS [1] RAJ [20] GAN-RAJ [6] Laha [13] PSO

n m ARPD MPD ARPD MPD ARPD MPD ARPD MPD ARPD MPD ARPD MPD

20 5 1.84 0.25 0.76 0.15 0.44 0.12 0.63 0.14 1.55 0.2 0.00 0.00

10 1.78 0.23 0.71 0.12 0.85 0.17 0.83 0.14 1.50 0.20 0.00 0.00

20 1.27 0.17 0.44 0.06 0.88 0.14 0.82 0.12 1.06 0.15 0.00 0.00

50 5 1.24 0.17 0.83 0.14 0.26 0.05 0.37 0.08 1.29 0.22 0.02 0.02

10 1.28 0.19 0.59 0.08 0.48 0.09 0.53 0.10 1.29 0.18 0.01 0.01

20 1.08 0.17 0.07 0.02 0.35 0.07 0.39 0.07 1.02 0.16 0.06 0.03

100 5 1.04 0.19 0.46 0.12 0.36 0.07 0.23 0.07 1.05 0.16 0.07 0.07

10 0.28 0.06 0.47 0.07 0.29 0.06 0.24 0.04 0.89 0.13 0.01 0.01

20 0.65 0.11 0.16 0.04 0.21 0.05 0.18 0.04 0.72 0.10 0.01 0.01

NEH Nawaz et al. [17], CDS Campbell et al. [1], RAJ Rajendran C [20], GAN-RAJ Gangadharan and Rajendran [6], Laha Laha and Chakraborty [12,13], PSO proposed PSO)

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