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

多目標演化式演算法理論基礎之探討: 不均衡性, 群族大 小和收斂時間(第 2 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-216-037-MY2

執 行 期 間 ： 97 年 08 月 01 日至 98 年 07 月 31 日 執 行 單 位 ： 中華大學資訊工程學系

計 畫 主 持 人 ： 陳建宏

計畫參與人員： 碩士班研究生-兼任助理人員：李誌堅 碩士班研究生-兼任助理人員：魏良哲 碩士班研究生-兼任助理人員：康志瑋

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 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中 華 民 國 98 年 11 月 03 日

行政院國家科學委員會補助專題研究計畫 ▉ 成 果 報 告

□期中進度報告

多目標演化式演算法理論基礎之探討: 不均衡性, 群族大小和收斂時間

計畫類別： ▉ 個別型計畫 □ 整合型計畫

計畫編號：NSC － － － － －

執行期間： 96 年 8 月 1 日至 98 年 7 月 31 日

計畫主持人： 陳建宏 共同主持人：

計畫參與人員： 康志瑋、陳建宏、盧金榮、魏良哲、李誌堅

成果報告類型(依經費核定清單規定繳交)：□精簡報告 ▉ 完整報告

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

▉ 出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

處理方式：除產學合作研究計畫、提升產業技術及人才培育研究計畫、

列管計畫及下列情形者外，得立即公開查詢

□涉及專利或其他智慧財產權，□一年 ▉ 二年後可公開查詢

執行單位：中華大學資訊工程學系

中 華 民 國 98 年 10 月 31 日

I、

許多現實生活上的最佳化問題常需要最佳化多個不同尺度且互相衝突競爭的目標，這 些問題通常被稱為多目標最佳化問題。相對於單目標最佳化問題只需求解一個單一的最佳 解答，多目標最佳化問題最大的差異處在於其必須求解出許多個最佳解答。近年來，由於 多目標演化式演算法可以在一次的執行中有效地同時搜尋多個解答，因此多目標演化式演 算法被廣泛認為非常適用於求解多目標最佳化問題。儘管多目標演化式演算法已被廣泛認 為適用於求解現實生活中的許多多目標最佳化問題，絕大多數的研究卻僅限於針對該研究 領域的問題來設計有效的多目標演化式演算法。僅僅有非常少數的研究探討多目標演化式 演算法在求解多目標最佳化問題時所需的族群大小和收斂時間。

本研究計畫的主題在於探討多目標演化式演算法在解決具有各種不均衡特性的多目標 最佳化問題之效能。本計劃之研究結果除了在學理上可以提供改良多目標演算法之貢獻 外，並可以提供工業界在應用多目標最佳化演算法之參數決定問題。本計劃之研究成果並 已部份實踐於數篇會議論文之中，並加以延伸投稿期刊論文。

關鍵字：多目標最佳化、演化式演算法

II、

Many real-world optimization problems involve multiple incommensurable and often competent objectives; these problems are known as multi-objective optimization problems (MOOPs). Many MOOPs cannot satisfactorily be characterized by a single performance measure.

Due to the nature of trade-offs involved, MOOPs seldom have a unique solution. Instead of obtaining a single optimal solution, the ultimate goal of solving MOOPs is to find a complete set of Pareto-optimal solutions. Recently, multi-objective evolutionary algorithms (MOEAs) have been recognized to be well-suited for solving MOOPs because their abilities to exploit and explore multiple solutions in parallel and to find a widespread set of non-dominated solutions in a single run. Although MOEAs have been shown to be effective for solving many real-world applications and exploring complex non-linear search spaces as efficient optimizers, but only a few preliminary analysis based on selectorecombinative MOEAs and (1+1)MOEA have been conducted in analyzing the population sizing and convergence time of MOEAs in solving MOOPs.

The main topics of this project are to investigate the performances of MOEAs in solving MOOPs with disequilibrium, and study the important factors that affect the convergence time and population sizing of MOEAs. These models can provide practitioners guidance in choosing key MOEAs parameters, and also assists MOEA practitioners to get maximum mileage on designing their MOEAs. The results of this project have been published in several conference papers, and their extended results have been submitted for the review of journals.

III、Background, Motivation, and Objectives

Multi-objective optimization problems (MOOPs) are common in our real life. A MOOP has a number of objective functions to be maximized or minimized. For example, consider the design of a car. Generally, the cost of such systems is to be minimized, while maximum performance is

desired. Depending on conditions of the application, further objectives may be important such as reliability and energy dissipation. Considering the design of a car, and assuming that the two objectives cheapness (f1) and performance (f2) are to be maximized under speed constraints. Then, an optimal design might be an architecture which achieves maximum performance at minimal cost and does not violate the speed constraint. However, what makes MOOPs difficult is that a solution may be optimal in an objective function, but bad in other objective functions. The objectives are conflicting and cannot be optimized simultaneously. Instead, a satisfactory trade-off has to be found. In the example of designing a car, cheapness (the inverse of cost) and performance are generally competing. High-performance car architectures substantially increase costs, while car architectures with cheap costs usually provide low performance. Depending on the market requirements, an intermediate solution (medium performance, medium cost) may be an appropriate trade-off for decision makers.

There are many industrial applications belong to MOOPs. Take a manufacturing factory for another example, production planning have to consider routing optimization, equipment optimization and machine optimization. Take an IC design application for an example, in the layout processes of an IC, the floorplan process usually seeks to optimize two competing objectives: area and routeability; and the result of the placement and routing depend on the result of floorplanning.

Assume the multi-objective functions are to be minimized. Mathematically, MOOPs can be represented as the following vector mathematical programming problems:

1 2

( ) { ( ), ( ), ..., ( )}._i

Minimize F Y

f Y f Y f Y

where Y denotes a solution and fi(Y) is generally a nonlinear objective function. Pareto dominance relationship and some related terminologies are introduced below. When the following inequalities hold between two solutions Y1 and Y2, Y2 is a non-dominated solution and is said to dominate Y1 (Y2

; Y

1 2 1 2

: ( ) ( ) _{i i} : ( ) ( )._{j j}

i f Y f Y j f Y f Y

∀ > ∧ ∃ > (2)

When the following inequality hold between two solutions Y1 and Y2, Y2 is said to weakly dominate Y₁ (Y₂;

Y

1 2

: ( ) ( ) ._{i i}

i f Y f Y

∀ ≥ (3)

A feasible solution Y * is said to be a Pareto-optimal solution if and only if there does not exist a feasible solution Y where Y dominates Y *, and the corresponding vector of Pareto-optimal solutions is called Pareto-optimal front.

The great success for evolutionary computation techniques, including evolutionary programming (EP), evolutionary strategy (ES), genetic algorithm (GA), came in the 1980s when extremely complex optimization problems from various disciplines were solved, thus facilitating the undeniable breakthrough of evolutionary computation as a problem-solving methodology.

Inspired from the mechanisms of natural evolution, evolutionary algorithms (EAs) utilize a collective learning process of a population of individuals. Descendants of individuals are generated using randomized operations such as mutation and recombination. Mutation corresponds to an erroneous self-replication of individuals, while recombination exchanges information between two or more existing individuals. According to a fitness measure, the

selection process favors better individuals to reproduce more often than those that are relatively worse. Specifically, GAs are used to illustrate the basic framework of EAs. GAs are stochastic, population-based search and optimization algorithms loosely modeled after the paradigms of evolution. GAs guide the search through the solution space by using natural selection and genetic operators, such as crossover, mutation, and the like. EAs have been shown to be effective for solving NP-hard problems and exploring complex non-linear search spaces as efficient optimizers.

The robust capability of EAs to find solutions to difficult problems has permitted them to become a popular optimization and search technique in many industries.

Recently, multi-objective evolutionary algorithms (MOEAs) have been recognized to be well-suited for solving MOOPs because their abilities to exploit and explore multiple solutions in parallel and to find a widespread set of non-dominated solutions in a single run. Several MOEAs based on Pareto dominance relationship are proposed to solve MOOPs directly, and present more promising results than single-objective optimization techniques theoretically and empirically. By making use of Pareto dominance relationship, MOEAs are capable of performing fitness assignment without using a weighted linear combination of all objectives.

The objectives of this project are to study the four important factors that affect the performance of MOEAs and to discover the relationship of these factors with convergence time and population sizing of MOEAs. By making uses of our results, we can further develop efficient multi-objective evolutionary algorithms to solve real-world application more quickly and reliable.

IV、Results

The results of this project have been submitted for possible publication of a journal and published in the following conference papers:

[1]

Computation Conference (GECCO-2008), pp. 2123-2128. (EI)

[2] Jian-Hung Chen, “Memetic Approach for Multi-objective Flexible Process Sequencing

248-254.

[3]

Genetic and Evolutionary Computation Conference (GECCO-2009), pp. 2059-2064. (EI) [4]

Differentiated Sensor Network Deployment.” in Proceeding of 12th IEEE International

Conference on Computational Science and Engineering (CSE-09) ), pp. 187-193. (EI)

[5]

WORLDCOMP Conference (WORLDCOMP-2009).

[6]

IEEE MASS Conference (MASS-2009) (EI)

Multi-objective Memetic Approach for Flexible Process Sequencing Problems

Jian-Hong Chen

Department of Computer Science and Information Engineering 707 Sec. 2 Wu-Fu Road Hsin-Chu 300, Taiwan

Jian-Hung Chen

Department of Computer Science and Information Engineering 707 Sec. 2 Wu-Fu Road Hsin-Chu 300, Taiwan

jh.chen@ieee.org ABSTRACT

This paper describes a multi-objective memetic approach for solving multi-objective flexible process sequencing problems in flexible manufacturing systems (FMSs). FMS can be de- scribed as an integrated manufacturing system consisting of machines, computers, robots, tools, and automated guided vehicles (AGVs).FMSs usually pose complex problems on process sequencing of operations among multiple parts. An efficient multi-objective memetic algorithm with fitness in- heritance mechanism is proposed to solve flexible process problems (FPSs) with the consideration the machining time of operations and machine workload load balancing. The experimental results demonstrate that our approach can ef- ficiently solve FPSs and fitness inheritance can speed up the convergence speed of the proposed algorithm in solving FPSs.

Categories and Subject Descriptors

J.6 [COMPUTER-AIDED ENGINEERING]: Computer- aided manufacturing (CAM)

General Terms

Algorithms, Design, Performance

Keywords

process planning, flexible manufacturing systems, multi-objective optimization, memetic algorithms, fitness inheritance

1. INTRODUCTION

Computer-aided process planning (CAPP) is an automated system for preparation of a plan that specifies machines, ma- chine conditions, operations, operation sequence, and tools required to production these components. Traditionally, the process sequencing has been solved by either the experience of process planners or a fixed and static process plan con- sisting of an ordered sequence of operations. However, the

Copyright is held by the author/owner(s).

GECCO’08,July 12–16, 2008, Atlanta, Georgia, USA.

ACM 978-1-60558-131-6/08/07.

traditional mythologies are not suitable in real flexible en- vironment, because the techniques have a few constraints in order to cope with dynamic situations of the flexible environ- ment [7]. Moreover, as the number of operations increase, it poses more difficulties for decision makers to plan a cost- effective process sequences for manufacturing.

In this paper, a memetic algorithm using fitness inher- itance (MEFI) is proposed to solve multi-objective flexible process sequencing problems (FPSs) having three objectives:

minimizing total machining time, maximum machine work- load and machine workload unbalance. The proposed ap- proach can obtain a set of non-dominated solutions for deci- sion makers in a single run, without the necessary of problem decomposition and relative preferences. Decision makers can easily distinguish between the costs of different process se- quences and choose more than one satisfactory process se- quences at a time. Six benchmark problems with differ- ent complexities are used to evaluate the performance of the proposed approach. A multi-objective genetic algorithm (MOGA) without local search and fitness inheritance is used for performance comparisons. It is shown empirically that MAFI outperforms MOGA in terms of the solution quality.

This paper is organized as follows: Section 2 presents the background of process sequencing problems, multi-objective evolutionary optimization. Section 3 introduces the setup of flexible manufacturing system and the mathematical for- mulation of FPSs. Section 4 presents the multi-objective memetic algorithm for solving FPSs. Section 5 presents the experimental analysis of the proposed algorithm, and Sec- tion 6 summarizes our conclusions.

2. BACKGROUND

2.1 Process Sequencing Problems

Flexible process sequencing problems are well known among the combinatorial optimization problems. Previous research focused on two important key issues of process sequenc- ing problems, described as follows. The first key issue is the objective functions of process sequencing. Several ap- proaches [4, 1] are proposed for process sequencing with various objectives. Another key issue that arises recently is the alternative process sequences. In the view of real time scheduling, alternative process sequences provide additional capability for the decision maker (DM) to cope with unpre- dictable events such as machine failures or rush orders. From the view of off-line scheduling, alternative process sequences may be used to improve the schedule quality by reducing

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the load on bottleneck machines [1]. It is essential but also a challenge for DM to prepare a set of alternative process sequences considering the trade-off between schedule qual- ity and the costs of process sequences. However, traditional techniques are not able to provide such flexibility for DM.

The above issues lead to flexible process sequencing prob- lems (FPSs), which simultaneously considers alternative pro- cess plans with multiple objectives and the flexibility of pro- cess sequences. Over the past decade, a number of models have been developed to solve the process sequencing prob- lems, but only few models [1, 7] have been reported to design the process sequencing problem considering the above issues.

To date, solving the problem of flexible process sequencing with multiple objectives that are conflicting in nature is still a hard task.

2.2 Multi-objective Evolutionary Optimization

Assume all the objective functions Fm are to be mini- mized. Mathematically, multi-objective optimization prob- lems (MOOPs) can be represented as the following vector mathematical programming problems:

M inimize F (X) = {F1(X), F2(X), ..., Fm(X)}, (1) where X denotes a solution and Fm(X) is generally a nonlin- ear objective function. When the following inequalities hold between two solutions X1 and X2, X2 is a non-dominated solution and is said to dominate X1(X2 X1):

∀m : Fm(X1) ≥ Fm(X2) and ∃n : Fn(X1) > Fn(X2). (2) When the following inequality hold between two solutions X1 and X2, X2 is said to weakly dominate X1(X2 X1):

∀m : Fm(X1) ≥ Fm(X2). (3) A feasible solution X^∗ is said to be a Pareto-optimal solu- tion if and only if there does not exist a feasible solution X where X dominates X^∗. The corresponding vector of Pareto-optimal solutions is called Pareto-optimal front.

By making use of Pareto dominance relationship, multi- objective evolutionary algorithms (MOEAs) are capable of performing the fitness assignment of multiple objectives with- out using relative preferences of multiple objectives. Thus, all the objective functions can be optimized simultaneously.

As a result, MOEA seems to be an alternative approach to solving production planning and inspection planning prob- lems on the assumption that no prior domain knowledge is available.

3. PROBLEM STATEMENT 3.1 The FMS Environment

An FMS consists of a set of identical and/or complemen- tary numerically controlled machines and tool systems. All components are connected through an AGV system. Fig- ure 1 shows the layout of a simple FMS with several ma- chines, AGVs and a tool system.

In order to design the production planning of FMSs, the environment within which the FMS under consideration op- erates can be described below.

• The term machine is to describe a machine cell. A ma- chine cell consists of several identical devices/machines.

The types and number of machines are known. There

Figure 1: FMS with several machines, a coordinate measuring machine (CMM), AGVs and a central tool magazine.

is a sufficient input/output buffer space at each ma- chine.

• A part type requires a number of operations. A number of part types will be manufactured simultaneously in batches. Parts can choose one or more machines at each of their operation stages, and the transportation of the parts within different machines is handled by an AGV system.

• A machine can perform several types of operations, and an operation can be performed on alternative ma- chines.

• A machine can only process an operation at one time.

Operations to be performed in the machine are non- preemptive. Operation lot splitting is ignored in this paper.

• A process sequence is a series of machine indices cor- responding to operations of all parts. Based on a pro- cess sequence, each operation is operated on its corre- sponding machine. An illustrative process sequence of 3 parts and 10 operations is presented in Figure 2, and the operations are operated on 3 different machines.

An example of the series of machine indices to be op- timized is Y =[ 1 1 1 3 1 2 2 2 3 3 ].

• Workload on each machine is contributed by those op- erations assigned to a machine.

• A load/unload (L/U) station serves as a distribution center for parts not yet processed and as a collection center for parts finished. All vehicles start from the L/U station initially and return to there after accom- plishing all their assignments. There are sufficient in- put/output buffer spaces at the L/U station.

• The number of AGVs is given and the transportation time of AGVs are known. Some machines may not be linked.

• AGVs carry a limited number of products at a time.

They move along predetermined paths, with the as- sumption of no delay because of congestion. Preemp- tion of trips is not allowed.

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• It is assumed that all the design, layout and set-up issues within FMS have already been resolved.

• Real-time issues, such as traffic control, congestion, machine failure or downtime, scraps, rework, and ve- hicle dispatches for battery changer are ignored here and left as issues to be considered during real-time control.

Part index 1 2 3

Operation index 1 2 3 4 1 2 3 1 2 3 Process Sequence 1 1 1 3 1 2 2 2 3 3

(Machine index)

Figure 2: A process sequence of 3 parts and 10 op- erations, operated on 3 different machines. For ex- ample, the operation 4 of the part 1 is assigned to the machine 3.

3.2 Mathematical Formulation of FPSs

In order to formulate FPSs, the following notations are introduced:

• i : part index, i = 1, 2, 3, ..., I.

• j : operation index for part i, j = 1, 2, 3, ..., Ji.

• k, l : machine index k, l = 1, 2, 3, ..., K.

• Y : process sequence.

• pvi: production volume (unit) for part i.

• ptijk : processing time per unit to perform operation j of part i using machine k.

• mk : maximum workload of machine k.

• twk : workload in machine k, twk= ptijk× pvi.

• rtwk: workload ratio in machine k, rtwk=^tw_m^k

k.

• ew : average workload of machines.

• sikl:

(1, if part i is to transfer from machine k to l ; 0, otherwise.

• xijk :







1, if machine k is selected to perform operation j of part i ;

0, otherwise.

• abl : available capacity of AGV per trip, abl is set to 10 in this chapter.

• nikl: the number of trips between machine k and l for part i,

nikl= sikl× dpvi

able,

where the bracket represents a ceiling operation.

• tmkl: transportation time from machine k to l. If ma- chines k and l are not linked, it is set to be a negative value for constraint handling.

• tikl : total transportation time between machines k and l for part i,

tikl= nikl× tmkl.

3.2.2 Objectives

There are three objectives to be optimized in flexible pro- cess sequencing problems, described below.

1. Minimization of total flow time. This objective is to minimize the processing time and transportation time for producing the parts. The total machine processing time (e1) is defined as Equation 4, the transportation time (e2) is defined as Equation 5, and the total flow time (f1) is defined as Equation 6. Transportation between unlinked machines are penalized in e2.

e1=

I

X

i=1 J_i

X

j=1 K

X

k=1

pvi× ptijk× xijk, (4)

e2=

I

X

i=1 J_i−1

X

j=1 K

X

k=1 K

X

l=1

tikl× xijk× xi(j+1)l, (5)

f1= e1+ e2. (6)

2. Minimization of machine workload unbalance. Balanc- ing the machine workload can avoid creating bottle- neck machines. The objective function (f2) is defined as Equation 7.

f2=

K

X

k=1

(rtwk− ew)². (7)

3. Minimization of greatest machine workload. Pursuing this objective also implies attempting to minimize the total flow time. The objective function (f3) is defined as Equation 8.

f3= max{rtwk}. (8)

3.2.3 Multi-objective Mathematical Model

The overall multi-objective mathematical model of FPSs can be formulated as follows. Given the production vol- ume pvi, the processing time ptijk, the maximum workload mk, the available capacity of AGV per trip abl, the trans- portation time tmkland the tool costs cijk, find a series of machine indices, Y , for operations of all parts such that

minimize f1, f2, f3, (9) subject to

K

X

k=1

xijk= 1, ∀(i, j), (10)

tmkl≥ 0, ∀(k, l), (11)

rtwk≤ 1, ∀i. (12)

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The constraint, Equation 10, ensures that only one ma- chine is selected for each operation of a part. Equation 11 en- sures an AGV path exists between machines k and l. Equa- tion 12 is to ensure the machine workload twkis smaller or equal to its maximum machine workload mk.

If the total number of machines is x and the total number of operations is y, then the complexity of the investigated problem is O(x^y).

4. MULTI-OBJECTIVE MEMETIC ALGO- RITHM WITH FITNESS INHERITANCE MEFI

4.1 Schemata-Guided Local Search Strategy

Based on schema theorem and the niche hypothesis [5], a schemata-guided local search strategy is proposed to be combined with MOGA for improving the convergence speed to the Pareto-front. Extended from the niche hypothesis, it is assumed that, given a MOOP with Q Pareto-optimal solutions, Q Pareto-optimal solutions can be regarded as Q niches of the MOOP. In the worst case, to ensure MOEAs is capable of searching Q Pareto-optimal solutions, it is as- sumed that the population were divided into Q species (sub- populations). Thus, each species is expect to optimize its own niche (Pareto-optimal solution), as shown in Figure 3.

Therefore, the optimal schemata of a species is its Pareto- optimal solution.

Let the schema of species be Hq, where the fixed positions are the maximum common string of all individuals in its species and the others are ”don’t care”(*). Since species are in the same population, a schemata of a species may be disrupted by schemata of the other species due to genetic operators. The disruption between species can be further classified into the following two types:

1. Species disrupt noise: The fixed schemata of Horigin

are altered to ”don’t care” schemata by the correspond- ing positions of the schemata Hother. Thus, a species requires more time for fixing it’s ”don’t care” schemata.

2. Species hitchhiking noise: The ”don’t care” schemata of Horigin are altered to fixed schema by the corre- sponding positions of the schema Hother. If the altered schemata are located in the similarity regions of their optimal schemata, the change is good for the schemata Horigin. On the contrary, the change is bad for the schemata Horigin.

Based on the foregoing inference, it is desired that a species should keep its good schemata (building blocks) while mak- ing good efforts to alter its ”don’t care” schemata to its ideal optimal schemata. As results, a schemata-guided local search strategy is proposed based on this guideline. Infor- mation of fixed and ”don’t care” schemata in species are utilized to guide local search. However, the key question of this local search strategy is that how do we classify popula- tion to different species when true Pareto-optimal solutions of MOOPs are unknown. To deal with this question, it is as- sumed that the best individuals in each objective functions are the pioneers of each species. These pioneers will be used to classify all individuals in population to different species.

Given a maximum local search times M axLS and a tem- porary elite set E⁰, the procedure of the used schemata- guided local search strategy is written as follows:

*00011****

****1011**

f2

f₁

Figure 3: The population were divided into several species, and each species optimizes its own niche (Pareto-optimal solution).

Step 1: (Identification) Identify the best individuals Bq, q = 1, 2, ..., Q, in each objective from the current popula- tion. For FPSs, Q=3.

Step 2: (Classification) Classify the current population into Q species by the best solutions in each objective.

Step 3: (Schemata computation) For each species, compute its schemata Hq. Both fixed and ”don’t care” schemata are identified.

Step 4: (Parameter setting) Let q = 1, counter = 0.

Step 5: (Perturbation) Perturb Bqinto a new solution B_q⁰. Ac- cording to Hq, apply the mutation operator only on

”don’t care” locations of Bq with a mutation probabil- ity pm.

Step 6: (Evaluation) Evaluate the objective functions of Bq⁰. Let counter = counter + 1.

Step 7: (Comparison) There is 3 cases in comparisons of Bq

and Bq⁰. Case 1: If Bq dominates Bq⁰ and counter <

M axLS, go to Step 5. Case 2: If Bq is dominated by B_q⁰, replace Bq by B⁰_q. Case 3: If Bq and B_q⁰ doesn’t dominated each other. Stored B_q⁰ in a temporary elite set E⁰.

Step 8: (Termination test) Let q = q + 1 and counter=0, if q>Q, stop the local search strategy. Otherwise, go to Step 5.

4.2 Fitness Inheritance

An efficiency enhancement techniques called fitness inher- itance [2] is used for speedup of MEFI. During the evolution of EAs, the fitness of some proportion of individuals in the subsequent population is inherited. This proportion is called the inheritance proportion, pi.

Mathematically, for a multi-objective problem with z ob- jective, the used fitness inheritance is defined as

fz =w1fz,p1+ w2fz,p2

w1+ w2

, (13)

where fz is the fitness value in objective z, w1, w2 are the weights for the two parents p1, p2, and f(z, p1), f(z, p2) is

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the fitness values of p1, p2 in objective z, respectively. In this paper, w1 and w2are set to 1.

According the literature of fitness inheritance, the pop- ulation size of FIEA should be bigger than the population size used for MOGA, as shown in the following equation:

Npop,F IEA=Npop,M OGA

1 − p³_i (14)

4.3 MEFI for solving FPSs

A series of machine indices Y for operations of all parts is directly encoded as a integer chromosome. The range of each gene of Y is [1, K]. Each gene of Y stands for a machine index.

The selection operator of MEFI uses a binary tourna- ment selection which works as follows. Choose two indi- viduals randomly from the population and copy the better individual into the intermediate population. The one-point crossover is used in MEFI. A simple mutation operator is used to alter genes. For each gene, randomly generate a real value from the range [0, 1] with the probability pm.

MEFI uses a generalized Pareto-based scale-independent fitness function GPSIFF [6] by the following function:

F (X) = p − q + c, (15)

where p is the number of individuals which can be dominated by the individual X, and q is the number of individuals which can dominate the individual X in the objective space.

c is the number of all participant individuals.

Based on the proposed chromosome representation, Equa- tion 10 is always satisfied. If Equation 11 is violated, the transportation time between machines k and l, tmkl, is set to be a large value, 10⁷. In this way, f2 will be penalized.

For each machine k, if Equation 12 is not satisfied, one is added to rtwk, as follows:

rtwk= (_tw

k

m_k, if twk≤ mk;

tw_k

m_k + 1, otherwise. (16)

4.4 Procedure of MEFI

Since it has been recognized that the incorporation of elitism may be useful in maintaining diversity and improv- ing the performance of multi-objective EAs [3], MEFI se- lects a number of elitists from an elite set E in the selection step. The elite set E with capacity Emaxmaintains the best non-dominated solutions generated so far. In addition, an external set E with no capacity is used to store all the non- dominated solutions ever generated so far. The procedure of MEFI is written as follows:

Step 1: (Initialization) Randomly generate an initial popula- tion of Npop individuals and create two empty elite sets E, E and an empty temporary elite set E⁰. Step 2: (Evaluation) For each individual Y in the population,

excluding the inherited individuals, compute the value of objective functions f1(Y ), f2(Y ), and f3(Y ).

Step 3: (Fitness assignment) Assign each individual a fitness value by using GPSIFF.

Table 1: The parameter settings of MEFI and MOGA.

Parameters MEFI MOGA

Npop 115 100

Emax 115 100

ps 0.25 0.25

pi 0.5 N/A

pc 0.6 0.6

pm 0.05 0.05

M axLS 3 N/A

Step 4: (Local search) Apply the proposed schemata-guided lo- cal search strategy. Non-dominated solutions obtained by the local search strategy will be stored in temporary elite set E⁰.

Step 5: (Update elite sets) Add the non-dominated individu- als in both the population and E⁰ to E, and empty E⁰. Considering all individuals in E, remove the dom- inated ones in E. Add E to E, remove the dominated ones in E. If the number of non-dominated individu- als in E is larger than Emax, randomly discard excess individuals.

Step 6: (Selection) Select Npop−Npsindividuals from the pop- ulation using the binary tournament selection and ran- domly select Nps individuals from E to form a new population, where Nps= Npop× ps and ps is a selec- tion proportion. If Npsis greater than the number NE

of individuals in E, let Nps= NE.

Step 7: (Recombination) Perform the one-point crossover op- eration with a recombination probability pc.

Step 8: (Fitness inheritance) Perform fitness inheritance on the selected Npop× pi individuals. The inherited ob- jective values are calculated according to Equation 13.

Step 9: (Mutation) Apply the mutation operator to each gene in the individuals with a mutation probability pm. Step 10: (Termination test) If a stopping condition is satisfied,

stop the algorithm and output E. Otherwise, go to Step 2.

5. RESULTS AND DISCUSSION

Six benchmark problems: m3o10, m4o20, m5o100, m5o200, m10o100 and m10o200, where mxoy stands for the x ma- chine and y operation problem. A MOGA, MEFI without the local search strategy and fitness inheritance, is imple- mented to solve FPSs as the baseline performance. The pa- rameter settings of MEFI and MOGA are given in Table 1.

Thirty independent runs with the same number of function evaluations 100xy were performed per test problems.

The coverage metric C(A, B) of two solution sets A and B [8] used to compare the performance of two corresponding algorithms considering the six objectives:

C(A, B) =|{a ∈ A, b ∈ B, a  b}|

|B| , (17)

Fig. 4 depicts the coverage metrics of C(M EF I, M OGA) and C(M OGA, M EF I) from 30 runs. In solving the small problem m3o10, Fig. 4 shows that the performance of MEFI

2127

m3o10 m4o20 m5o100 m5o200 m10o100 m10o200 0

0.2 0.4 0.6 0.8 1

(b) C(MOGA, MEFI)

m3o10 m4o20 m5o100 m5o200 m10o100 m10o200 0

0.2 0.4 0.6 0.8 1

(a) C(MEFI, MOGA)

Figure 4: Box plots based on the cover metric. (a) C(MEFI, MOGA), (b) C(MOGA, MEFI).

and MOGA are almost the same. For another small prob- lem m4o20, the non-dominated solutions obtained by MEFI dominates 80% of the solutions obtained by MOGA in aver- age, while the non-dominated solutions obtained by MOGA only dominates 60% of the non-dominated solutions obtained by MEFI in average. As the complexity of problems in- creases, Fig. 4 shows that 80%-90% of the non-dominated solutions obtained by MOGA are weakly dominated by the non-dominated solutions obtained by MEFI in solving the problems m4o20, m5o100, m5o200, m10o100 and m10o200.

On the contrast, the non-dominated solutions of MOGA dominate nearly 3-10% of the non-dominated solutions ob- tained by MEFI. Fig. 5 shows the non-dominated solutions obtained by thirty runs of MEFI and MOGA in solving the m10o200 problem. The results indicate that MEFI can con- verge to better solutions more quickly than MOGA. It re- veals that the proposed schemata-guided local search strat- egy and fitness inheritance plays an important role in obtain- ing good solutions and accelerating the convergence speed.

6. CONCLUSION

In this paper, a novel approach to solve flexible process sequencing problems using an multi-objective memetic al- gorithm MEFI is proposed. A schemata-guided local search strategy and fitness inheritance are integrated in the pro- posed algorithm for enhancing the performance. Experimen- tal results demonstrated that the quality of non-dominated solutions obtained by MEFI is better than that of MOGA in terms of convergence speed and accuracy using the same number of function evaluations. While prior domain knowl- edge for the decomposition of problems or relative prefer- ences of multiple objectives are not available, the proposed approach is an expedient method to solve flexible process sequencing problems. Moreover, the proposed approach can obtain a set of non-dominated solutions for decision mak- ers in a single run. Decision makers can easily distinguish between the costs of different process sequences and choose more than one satisfactory process sequences at a time.

4 4.5

5 5.5

6 6.5

x 10⁵ 0

0.01 0.02 0.03 0.040.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44

f₁ f₂

f3

MOGA MEFI

Figure 5: The non-dominated solutions obtained by MEFI and MOGA in solving the m10o200 problem, merged from 30 runs.

7. ACKNOWLEDGMENTS

This work was supported by the National Science Council of Taiwan, R.O.C. under Contract NSC-96-2221-E-216-037- MY2 and NSC-095-SAF-I-564-616-TMS, and Chung-Hua Uni- versity under Contract CHU-96-2221-E-216-037-MY2.

8. REFERENCES

[1] P. Brandimarte. Exploiting process plan flexibility in production scheduling: A multi-objective approach.

European Journal of Operational Research, (114):59–71, 1999.

[2] J.-H. Chen, D. E. Goldberg, S.-Y. Ho, and K. Sastry.

Fitness inheritance in multi-objective optimization. In GECCO ’02: Proceedings of the Genetic and

Evolutionary Computation Conference, pages 319–326, San Francisco, CA, USA, 2002. Morgan Kaufmann Publishers Inc.

[3] K. Deb. Multi-objective optimization using evolutionary algorithms. Wiley-Interscience series in systems and optimization. John Wiley & Sons, 2001.

[4] M. Gen and R. Cheng. Genetic algorithms and engineering design. John Wiley, New York, 1997. 1944- Mitsuo Gen, Runwei Cheng. ill. ; 24 cm.

[5] D. E. Goldberg. Genetic algorithms in search, optimization, and machine learning. Addison-Wesley Pub. Co., 1989.

[6] S.-Y. Ho, L.-S. Shu, and J.-H. Chen. Intelligent evolutionary algorithms for large parameter optimization problems. IEEE Transaction on Evolutionary Computation, 8(6):522–541, Dec. 2004.

[7] C. Moon, Y.-Z. Li, and M. Gen. Evolutionary algorithm for flexible process sequencing with multiple objectives.

In Proceeding of IEEE International Conference on Computational Intelligence, pages 27–32, 1998.

[8] E. Zitzler and L. Thiele. Multiobjective evolutionary algorithms: A comparative case study and the strengthen Pareto approach. IEEE Transaction on Evolutionary Computation, 4(3):257–271, 1999.

2128

Memetic Approach for Multi-objective Flexible Process Sequencing Problems

Jian-Hung Chen

Department of Computer Science and Information Engineering Chung-Hua University, Hsin-Chu, Taiwan 300

E-mail: jh.chen@ieee.org

Abstract—This paper describes a novel multi-objective memetic algorithm for solving multi-objective flexible pro- cess sequencing problems in flexible manufacturing systems (FMSs). FMS can be described as an integrated manufacturing system consisting of machines, computers, robots, tools, and automated guided vehicles (AGVs). While FMSs give great advantages through the flexibility, FMSs usually pose complex problems on process sequencing of operations among multiple parts. Considering the machining time of operations and machine workload load balancing, the problem is formu- lated as multi-objective flexible process sequencing problems (FPSs). An efficient multi-objective memetic algorithm with fitness inheritance mechanism is proposed to solve FPSs.

The experimental results demonstrate that our approach can efficiently solve FPSs and fitness inheritance can speed up the convergence speed of the proposed algorithm in solving FPSs.

Keywords—process planning, flexible manufacturing sys- tems, multi-objective optimization, memetic algorithms, fitness inheritance

1 Introduction

Computer-aided process planning (CAPP) is an automated system for preparation of a plan that specifies machines, machine conditions, operations, operation sequence, and tools required to production these components [1]. CAPP techniques are being developed in an attempt to overcome some of the problems occurring in manual process planning, such as long turn around times, inconsistent routing or tooling, non-uniqueness in cost and labor requirements and scarcity of skilled process planners. During the past two decades, a number of CAPP systems have been developed for the automated planning and increased efficiency of process plan- ning function. Traditionally, the process sequencing has been solved by either the experience of process planners or a fixed and static process plan consisting of an ordered sequence of operations [2]. However, the traditional mythologies are not suitable in real flexible environment, because the techniques have a few constraints in order to cope with dynamic situations of the flexible environment [3]. Moreover, as the number of op- erations increase, it poses more difficulties for decision makers to plan a cost-effective process sequences for manufacturing.

In this paper, a memetic algorithm using fitness inheritance (MAFI) is proposed to solve multi-objective flexible process

sequencing problems (FPSs) having three objectives: mini- mizing total machining time, maximum machine workload and machine workload unbalance. The fundamental difference of the proposed approach from the traditional approaches is that the problem decomposition and relative preferences are not necessary. In addition, the proposed approach can obtain a set of non-dominated solutions for decision makers in a single run. Decision makers can easily distinguish between the costs of different process sequences and choose more than one satisfactory process sequences at a time. Six benchmark problems with different complexities are used to evaluate the performance of the proposed approach. A multi-objective genetic algorithm (MOGA) without local search and fitness inheritance is used for performance comparisons. It is shown empirically that MAFI outperforms MOGA in terms of the solution quality.

This paper is organized as follows: Section 2 presents the background of process sequencing problems, multi-objective optimization problems and evolutionary algorithms. Section 3 introduces the setup of flexible manufacturing system and the mathematical formulation of FPSs. Section 4 presents the multi-objective memetic algorithm for solving FPSs. Section 5 presents the experimental analysis of the proposed algorithm, and Section 6 summarizes our conclusions.

2 Background

2.1 Process Sequencing Problems

Flexible process sequencing problems are well known among the combinatorial optimization problems. Previous research focused on two important key issues of process sequencing problems, described as follows. The first key issue is the objective functions of process sequencing. Sev- eral approaches are proposed for process sequencing with various objectives. For examples, Kusiak and Finke [2] have developed a model for selecting a set of process plans with the objective of minimizing the makespan. Bhaskaran [4]

provided a model for minimizing the total machine time and the total number of processing steps. Zhang and Huang [5]

presented a fuzzy-based model for the selection of a set of process plans considering the imprecise information of shop floor. Furthermore, various heuristic approaches [6] have been proposed for minimizing the makespan.

Another key issue that arises recently is the alternative process sequences. In the view of real time scheduling, al- ternative process sequences provide additional capability for the decision maker (DM) to cope with unpredictable events such as machine failures or rush orders. From the view of off- line scheduling, alternative process sequences may be used to improve the schedule quality by reducing the load on bottleneck machines [4]. Generally speaking, finding a set of optimal alternative process sequences economically plays an important role in solving the process sequencing problems.

However, it is easier to obtain the alternative process sequences with single objective than that with multiple objectives. It is because, simultaneous optimization of several incommen- surable and conflicting objectives in nature is much more complex and difficult. On the other hand, flexible process sequencing with multiple objectives makes more practical applications in the design phase of industrial manufacturing.

As a result, it is essential but also a challenge for DM to prepare a set of alternative process sequences considering the trade-off between schedule quality and the costs of process sequences.

The above issues lead to flexible process sequencing prob- lems (FPSs), which simultaneously considers alternative pro- cess plans with multiple objectives and the flexibility of pro- cess sequences. Over the past decade, a number of models have been developed to solve the process sequencing problems, but only few models [3], [4] have been reported to design the process sequencing problem considering the above issues. To date, solving the problem of flexible process sequencing with multiple objectives that are conflicting in nature is still a hard task.

2.2 Multi-objective Evolutionary Optimization Assume all the objective functions Fm are to be mini- mized. Mathematically, multi-objective optimization problems (MOOPs) can be represented as the following vector mathe- matical programming problems:

M inimize F (X) = {F₁(X), F₂(X), ..., F_m(X)}, (1) where X denotes a solution and Fm(X) is generally a nonlin- ear objective function. When the following inequalities hold between two solutions X1 and X2, X2 is a non-dominated solution and is said to dominate X1(X2 X1):

∀m : F_m(X₁) ≥ F_m(X₂) and ∃n : F_n(X₁) > F_n(X₂).

(2) When the following inequality hold between two solutions X1

and X2, X2 is said to weakly dominate X1(X2 X1):

∀m : Fm(X1) ≥ Fm(X2). (3) A feasible solution X^∗ is said to be a Pareto-optimal solution if and only if there does not exist a feasible solution X where X dominates X^∗. The corresponding vector of Pareto-optimal solutions is called Pareto-optimal front.

In the past few years, multi-objective evolutionary algo- rithms (MOEAs) have been recognized to be well-suited for

Fig. 1. FMS with several machines, a coordinate measuring machine (CMM), AGVs and a central tool magazine.

solving MOOPs because their abilities to exploit and explore multiple solutions in parallel and to find a widespread set of non-dominated solutions in a single run [7]. By making use of Pareto dominance relationship, MOEAs are capable of performing the fitness assignment of multiple objectives with- out using relative preferences of multiple objectives. Thus, all the objective functions can be optimized simultaneously. One of the recent growing areas in evolutionary algorithms (EAs) research is memetic agorithms (MAs). MAs are population- based meta-heuristic search methods inspired by Darwinian principles of natural evolution and Dawkins notion of a meme defined as a unit of cultural evolution that is capable of local refinements [8]. From an optimization point of view, MAs are hybrid EAs that combine global and local search by using an EA to perform exploration while the local search method performs exploitation. Combining global and local search is known as an efficient strategy in many successful optimization approaches [9], [10].

3 Problem Statement

The aim of flexible process sequencing is to develop a cost- effective and operative process sequences for the assignments of operation to machines over planning phases. With the assignments of operations to machines, three optimization ob- jectives: minimizing total machining time, machine workload unbalance, and greatest machine workload are considered in this paper.

3.1 The FMS Environment

An FMS consists of a set of identical and/or complementary numerically controlled machines and tool systems. All compo- nents are connected through an AGV system. Figure 1 shows the layout of a simple FMS with several machines, AGVs and a tool system.

In order to design the production planning of FMSs, the en- vironment within which the FMS under consideration operates can be described below.

Part index 1 2 3 Operation index 1 2 3 4 1 2 3 1 2 3 Process Sequence 1 1 1 3 1 2 2 2 3 3

(Machine index)

Fig. 2. A process sequence of 3 parts and 10 operations, operated on 3 different machines. For example, the operation 4 of the part 1 is assigned to the machine 3.

• The term machine is to describe a machine cell. A ma- chine cell consists of several identical devices/machines.

The types and number of machines are known. There is a sufficient input/output buffer space at each machine.

• A part type requires a number of operations. A number of part types will be manufactured simultaneously in batches. Parts can choose one or more machines at each of their operation stages, and the transportation of the parts within different machines is handled by an AGV system.

• A machine can perform several types of operations, and an operation can be performed on alternative machines.

• A machine can only process an operation at one time.

Operations to be performed in the machine are non- preemptive. Operation lot splitting is ignored in this paper.

• A process sequence is a series of machine indices corre- sponding to operations of all parts. Based on a process sequence, each operation is operated on its corresponding machine. An illustrative process sequence of 3 parts and 10 operations is presented in Figure 2, and the operations are operated on 3 different machines. An example of the series of machine indices to be optimized is Y =[ 1 1 1 3 1 2 2 2 3 3 ].

• Workload on each machine is contributed by those oper- ations assigned to a machine.

• A load/unload (L/U) station serves as a distribution center for parts not yet processed and as a collection center for parts finished. All vehicles start from the L/U station initially and return to there after accomplishing all their assignments. There are sufficient input/output buffer spaces at the L/U station.

• The number of AGVs is given and the transportation time of AGVs are known. Some machines may not be linked.

• AGVs carry a limited number of products at a time. They move along predetermined paths, with the assumption of no delay because of congestion. Preemption of trips is not allowed.

• It is assumed that all the design, layout and set-up issues within FMS have already been resolved.

• Real-time issues, such as traffic control, congestion, ma- chine failure or downtime, scraps, rework, and vehicle dispatches for battery changer are ignored here and left as issues to be considered during real-time control.

3.2 Mathematical Formulation of FPSs

3.2.1 Notations: In order to formulate FPSs, the following notations are introduced:

• i : part index, i = 1, 2, 3, ..., I.

• j : operation index for part i, j = 1, 2, 3, ..., Ji.

• k, l : machine index k, l = 1, 2, 3, ..., K.

• Y : process sequence.

• pv_i : production volume (unit) for part i.

• pt_ijk: processing time per unit to perform operation j of part i using machine k.

• mk : maximum workload of machine k.

• twk : workload in machine k, twk= ptijk× pvi.

• rtwk : workload ratio in machine k, rtwk= ^tw_m^k

k.

• ew : average workload of machines.

• s_ikl :

(1, if part i is to transfer from machine k to l;

0, otherwise.

• xijk :







1, if machine k is selected to perform operation j of part i;

0, otherwise.

• abl : available capacity of AGV per trip, abl is set to 10 in this chapter.

• nikl : the number of trips between machine k and l for part i,

nikl= sikl× dpvi

able,

where the bracket represents a ceiling operation.

• tmkl : transportation time from machine k to l. If machines k and l are not linked, it is set to be a negative value for constraint handling.

• tikl : total transportation time between machines k and l for part i,

t_ikl= n_ikl× tm_kl.

3.2.2 Objectives: There are three objectives to be optimized in flexible process sequencing problems, described below.

1) Minimization of total flow time. This objective is to minimize the processing time and transportation time for producing the parts. The total machine processing time (e1) is defined as Equation 4, the transportation time (e2) is defined as Equation 5, and the total flow time (f1) is defined as Equation 6. Transportation between unlinked machines are penalized in e2.

e₁=

I

X

i=1 Ji

X

j=1 K

X

k=1

pv_i× pt_ijk× x_ijk, (4)

e2=

I

X

i=1 J_i−1

X

j=1 K

X

k=1 K

X

l=1

tikl× xijk× x_i(j+1)l, (5)

f₁= e₁+ e₂. (6)

2) Minimization of machine workload unbalance. Balanc- ing the machine workload can avoid creating bottleneck

machines. The objective function (f2) is defined as Equation 7.

f2=

K

X

k=1

(rtwk− ew)². (7) 3) Minimization of greatest machine workload. Pursuing this objective also implies attempting to minimize the total flow time. The objective function (f3) is defined as Equation 8.

f₃= max{rtw_k}. (8)

3.2.3 Multi-objective Mathematical Model: The overall multi-objective mathematical model of FPSs can be formulated as follows. Given the production volume pvi, the processing time ptijk, the maximum workload mk, the available capacity of AGV per trip abl, the transportation time tmkl and the tool costs cijk, find a series of machine indices, Y , for operations of all parts such that

minimize f1, f2, f3, (9) subject to

K

X

k=1

x_ijk= 1, ∀(i, j), (10)

tm_kl ≥ 0, ∀(k, l), (11)

rtwk≤ 1, ∀i. (12)

The constraint, Equation 10, ensures that only one machine is selected for each operation of a part. Equation 11 ensures an AGV path exists between machines k and l. Equation 12 is to ensure the machine workload twk is smaller or equal to its maximum machine workload mk.

If the total number of machines is x and the total number of operations is y, then the complexity of the investigated problem is O(x^y).

4 Multi-objective Memetic Algorithm with Fit- ness Inheritance MAFI

The proposed MAFI differs from MOGA in the local search strategy and fitness inheritance. The used schemata- guided local search strategy is presented in Section 4.1. Fitness inheritance is summarized in Section 4.2. MAFI for solving FPSs is presented in Section 4.3, including the representation of chromosomes, genetic operators, constraint handling, and the procedure of MAFI.

4.1 Schemata-Guided Local Search Strategy

Based on schema theorem and the niche hypothesis [11], a schemata-guided local search strategy is proposed to be combined with MOGA for improving the convergence speed to the Pareto-front. Extended from the niche hypothesis, it is assumed that, given a MOOP with Q Pareto-optimal solutions, Q Pareto-optimal solutions can be regarded as Q niches of the MOOP. In the worst case, to ensure MOEAs is capable

*00011****

****1011**

f2

f₁

Fig. 3. The population were divided into several species, and each species optimizes its own niche (Pareto-optimal solution).

of searching Q Pareto-optimal solutions, it is assumed that the population were divided into Q species (sub-populations).

Thus, each species is expect to optimize its own niche (Pareto- optimal solution), as shown in Figure 3. Therefore, the optimal schemata of a species is its Pareto-optimal solution.

Let the schema of species be Hq, where the fixed positions are the maximum common string of all individuals in its species and the others are ”don’t care”(*). Since species are in the same population, a schemata of a species may be disrupted by schemata of the other species due to genetic operators. The disruption between species can be further classified into the following two types:

1) Species disrupt noise: The fixed schemata of Horigin

are altered to ”don’t care” schemata by the correspond- ing positions of the schemata Hother. Thus, a species requires more time for fixing it’s ”don’t care” schemata.

2) Species hitchhiking noise: The ”don’t care” schemata of Horigin are altered to fixed schema by the corre- sponding positions of the schema Hother. If the altered schemata are located in the similarity regions of their optimal schemata, the change is good for the schemata Horigin. On the contrary, the change is bad for the schemata Horigin.

Based on the foregoing inference, it is desired that a species should keep its good schemata (building blocks) while making good efforts to alter its ”don’t care” schemata to its ideal optimal schemata. As results, a schemata-guided local search strategy is proposed based on this guideline. Information of fixed and ”don’t care” schemata in species are utilized to guide local search. However, the key question of this local search strategy is that how do we classify population to different species when true Pareto-optimal solutions of MOOPs are unknown. To deal with this question, it is assumed that the best individuals in each objective functions are the pioneers of each species. These pioneers will be used to classify all individuals in population to different species.

Given a maximum local search times M axLS and a tem- porary elite set E⁰, the procedure of the used schemata-guided

local search strategy is written as follows:

Step 1 : (Identification) Identify the best individuals Bq, q = 1, 2, ..., Q, in each objective from the current popu- lation. For FPSs, Q=3.

Step 2 : (Classification) Classify the current population into Q species by the best solutions in each objective.

Step 3 : (Schemata computation) For each species, com- pute its schemata Hq. Both fixed and ”don’t care”

schemata are identified.

Step 4 : (Parameter setting) Let q = 1, counter = 0.

Step 5 : (Perturbation) Perturb Bq into a new solution B⁰_q. According to Hq, apply the mutation operator only on ”don’t care” locations of Bq with a mutation probability pm.

Step 6 : (Evaluation) Evaluate the objective functions of B⁰_q. Let counter = counter + 1.

Step 7 : (Comparison) There is 3 cases in comparisons of B_q and B_q⁰. Case 1: If B_q dominates B_q⁰ and counter < M axLS, go to Step 5. Case 2: If Bq is dominated by B_q⁰, replace Bq by B⁰_q. Case 3: If Bq

and B_q⁰ doesn’t dominated each other. Stored B_q⁰ in a temporary elite set E⁰.

Step 8 : (Termination test) Let q = q + 1 and counter=0, if q¿Q, stop the local search strategy. Otherwise, go to Step 5.

4.2 Fitness Inheritance

An efficiency enhancement techniques called fitness inheri- tance [12] is used for speedup of MAFI. During the evolution of EAs, the fitness of some proportion of individuals in the subsequent population is inherited. This proportion is called the inheritance proportion, pi.

Mathematically, for a multi-objective problem with z objec- tive, the used fitness inheritance is defined as

f_z=w1fz,p1+ w2fz,p2

w₁+ w₂ , (13)

where fz is the fitness value in objective z, w1, w2 are the weights for the two parents p1, p2, and f₍z, p1), f(z, p2) is the fitness values of p1, p2 in objective z, respectively. In this paper, w1 and w2 are set to 1.

According the literature of fitness inheritance, the popula- tion size of FIEA should be bigger than the population size used for MOGA, as shown in the following equation:

Npop,F IEA=Npop,M OGA

1 − p³_i (14)

4.3 MAFI for solving FPSs

A series of machine indices Y for operations of all parts is directly encoded as a integer chromosome. The range of each gene of Y is [1, K]. Each gene of Y stands for a machine index.

The selection operator of MAFI uses a binary tournament selection which works as follows. Choose two individuals randomly from the population and copy the better individual

into the intermediate population. Crossover is a recombina- tion process in which genes from two selected parents are recombined to generate offspring chromosomes. The one-point crossover is used in MAFI. A simple mutation operator is used to alter genes. For each gene, randomly generate a real value from the range [0, 1]. If the value is smaller than the mutation probability p_m, replace its index with a randomly generated integer among its possible values.

MAFI uses a generalized Pareto-based scale-independent fitness function GPSIFF [13] by the following function:

F (X) = p − q + c, (15)

where p is the number of individuals which can be dominated by the individual X, and q is the number of individuals which can dominate the individual X in the objective space. c is the number of all participant individuals.

Based on the proposed chromosome representation, Equa- tion 10 is always satisfied. If Equation 11 is violated, the transportation time between machines k and l, tmkl, is set to be a large value, 10⁷. In this way, f2will be penalized. For each machine k, if Equation 12 is not satisfied, one is added to rtwk, as follows:

rtwk= (_tw

k

m_k, if twk ≤ mk;

twk

m_k + 1, otherwise. (16)

4.4 Procedure of MAFI

Since it has been recognized that the incorporation of elitism may be useful in maintaining diversity and improving the performance of multi-objective EAs [7], MAFI selects a number of elitists from an elite set E in the selection step.

The elite set E with capacity Emax maintains the best non- dominated solutions generated so far. In addition, an external set E with no capacity is used to store all the non-dominated solutions ever generated so far. The procedure of MAFI is written as follows:

Step 1 : (Initialization) Randomly generate an initial popu- lation of Npopindividuals and create two empty elite sets E, E and an empty temporary elite set E⁰. Step 2 : (Evaluation) For each individual Y in the popu-

lation, excluding the inherited individuals, compute the value of objective functions f1(Y ), f2(Y ), and f3(Y ).

Step 3 : (Fitness assignment) Assign each individual a fit- ness value by using GPSIFF.

Step 4 : (Local search) Apply the proposed schemata- guided local search strategy. Non-dominated solu- tions obtained by the local search strategy will be stored in temporary elite set E⁰.

Step 5 : (Update elite sets) Add the non-dominated indi- viduals in both the population and E⁰ to E, and empty E⁰. Considering all individuals in E, remove the dominated ones in E. Add E to E, remove the dominated ones in E. If the number of non- dominated individuals in E is larger than Emax, randomly discard excess individuals.