Multiobjective Cell Formation with Routing Flexibility

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Multiobjective Cell Formation with Routing Flexibility: A Graph Partitioning Approach Menouar Boulif1 and Kar im Atif2 1 Département d'Informatique. Faculté des Sciences. Université M'hamed Bouguerra à Boumerdès. Avenue de l'indépendance. 35000. Boumerdès, ALGERIE. 2 Département d'Informatique. Faculté Génie Electrique et Informatique. Université des Sciences et de la Technologie Houari Boumediene. BP 32. El-Alia. 16111. Babezzouar. Alger. ALGERIE.

Abstract—This paper deals with the manufacturing cell formation (MCF) problem using a multiobjective graph-theoretic approach. In cellular manufacturing related recent works, several research paths have been taken such as the multi-periodic and the routing flexibility approaches. Our paper falls into the latter by proposing a multiobjective approach. First, the problem is formulated using a graph theoretic model, afterwards two resolution methods are proposed: the first one is a Genetic Algorithm (GA) whereas the second uses a combination between GA and Tabu Search. The two methods are applied on a set of medium-sized instances and the obtained results are reported. Keywords—Cellular manufacturing, multiple routing, graph partitioning, multiobjective optimization, genetic algorithm, Tabu search

The remainder of our paper is organized as follows: In section 2, the problem is formulated using a graph theoretic approach. The next section presents two resolution methods: the first one is a GA whereas the second uses a combination between the GA and Tabu search (TS). In section 3, the solving methods are applied on a set of instances and the obtained results are reported. We conclude the paper by drawing our conclusions as well as our recommendations for further research.

II. FORMULATION In this section, we propose an extension to the model presented in [1] in order to deal with the multiple routings aspect.

A. Nomenclature

I. INTRODUCTION

M P m p k Pk Rk t sk Opkt MOpkt qkt i,q Mi tfkti C j,u Cj J

This paper deals with the manufacturing cell formation (MCF) problem, one of three major steps to be conducted in Cellular Manufacturing Systems (CMS) design. MCF step devides the manufacturing facility into cells so that similar parts are essentially processed in the same cell. The success of CMS is rooted in their proven ability to reduce set-up times, in-process inventories, lot sizes and production equipment while improving productivity and production system mastery [1]. The MCF problem has been the subject of a wide investigation in the literature that yield to the identification of the problem NP-hardness. Indeed, since the seventies, when the first mathematical approaches have been published [2,3], several models have been proposed to tackle it. King [4] proposed a clustering technique he called rank order clustering to generate diagonalized blocks of the part-machine incidence matrix. Vannelli and Kumar [5] proposed a bipartite part-machine graph based approach that minimizes the number of bottleneck cells. Sofianopoulou [6] proposed a linear integer programming model to minimize inter cell movement. However, a gap has been reported between the proposed approaches and the actual problem due to the non consideration of several real features of CMS such as the cost of part revisiting of machines, the unfeasibility of duplicating expensive machines and the changes in the product mix. Recent works try to fill in this gap by considering such above-mentioned features and others like the fluctuations in the product demand [7,8], the ability of manufacturing the products according to different routings [9] and the multicriteria aspect of the problem [8]. Our paper falls into these two latter categories by proposing, for the first time to the best of our knowledge, an extension to a graph partitioning approach to handle simultaneously the multiple routings and the Pareto multiobjective aspects.

αkti α

vkiq E G N Jmin Jmax SC SN rk ci 1

set of available machines. set of parts. number of machines. number of parts. part index. kth part. operation sequence of part Pk. operation index. number of operations in Rk. tth operation of Rk. subset of machines that can do Opkt. number of machines in MOpkt. machine indexes. ith machine. processing time for doing Opkt on Mi partition of M. cell indexes jth cell. number of cells. decision variable that equals one if Op kt is done on Mi, zero otherwise. cubic matrix of decision variables αkti. number of times Mi and Mq are contiguous in the routing of part Pk. set of edges that link every non ordered couple of machines that can have a direct flow of parts. (M,E) undirected flow graph. upper bound for cell size. lower bound for the number of cells. upper bound for the number of cells. set of cohabiting couples of machines. set of non cohabiting couples of machines. production volume of part Pk. capacity of machine Mi.

edge weight function. subset of intercellular edges associated with partition C. intercellular traffic objective function. vertex weight function. workload of cell Cj.

W() E(C) T V() Chj

Ch

mean of cell workloads. variance of cell workloads objective function. set of all feasible machine partitions. set of all feasible operation-to-machine assignments.

CH S A

B. Graph construction (1) We consider M={M1,M2,...,Mm} a set of m machines and P = {P1,P2,...,Pp} a set of p part types. Parts in P are to be manufactured on machines in M. (2) For each part type Pk (k=1,2,..., p), we suppose given a single sequence of operations, Rk=, ,…,< Op ks > where: is the tth (t=1,2,..., sk) k

of the sk operations in the sequence of Pk; (3) For each operation Opkt we suppose that the followings are given: (i) MOpkt Ž M, the subset of machines able to achieve Opkt: MOpkt={,,…., Mktqkt ! } where •M for i=1,2,…,qkt. (ii) tfkti, the processing time for doing Opkt on Mi •MOpkt. If Mi •MOpkt then tfkti =0. (4) Let C = {C1, C2, ..., CJ} be a partition of M in J cells, such that: J

Cj

M and C u ˆ C j

u, j • ^1,2,..., J`, u z j

‡

j 1

(5)

We

D

D kti

D kti

define

the

k 1 p t 1 s k ,i 1 m

three-dimension as follows:

-1 if the t th operation ® ¯ 0 otherwise

of Pk is assigned

variable to M i

(6) For each (Pk,Mi,Mq) • PuMuM, we denote as vkiq the number of times in which Mi follows Mq or inversely (i,q=1,…,m) in the routing of Pk. vkiq is defined as follows : sk 1

v kiq

¦

D kti ˜ D k, t

1, q

D ktq ˜ D k, t

where Card(Cj) is the cardinal of the cell Cj (number of Cj assigned machines). (2) Given an upper bound Jmax and a lower bound Jmin for the number of cells allowed, J must verify: Jmin d J d Jmax (3) Given a set of cohabiting machine couples that must be in the same cell, SC, we have to respect: Cj • C : Mi,Mq • Cj where i,q • (Mi,Mq)•SC , {1,…,m} and j • {1,…,J} Cohabitation constraints are added to the model in order to, for instance, take into account those machines whose dangerous or particular energy sources require that they be placed in close proximity so as to facilitate their supervision and satisfy their energy needs. (4) Given a set of non-cohabiting machine couples that must be in different cells, SN, we have to respect: (Mi,Mq)•SN , Cu, Cj • C : Mi • Cu and Mq • Cj where i,q • {1,…,m} and u,j • {1,…,J}, u • M. Non-cohabitation constraints are added so that, for example, high precision machines could be placed far from those that generate a high level of vibration. (5) An operation must be assigned to only one machine: m

¦D

1 k = 1,2,….p and t = 1,2,…,sk.

kti

i 1

(6) The designated machine must be able to achieve its assigned operation: m

¦D

kti

˜ tf kti ! 0 k = 1,2,….p and t = 1,2,….sk.

i 1

(7) The machine capacities must be respected: p

sk

¦¦D

kti

˜ rk ˜ tf kti d c i i=1..m,

k 1 t 1

where rk is the volume per unit of time to be manufactured for part Pk and ci is the capacity of machine Mi. The space of feasible solutions is given by the set of (C,.) couples that verify these seven constraint sets. D. Objective functions (1) First objective, (i) An edge weight function, W, defines for each edge the amount of part flow between its extremities: p

1, i

t 1

(7) We define the undirected flow graph G=(M,E), where the set of vertices M is the set of machines and the set of edges E is the set of non-ordered machine couples that can be connected by a part flow: E = {ei,q /(Mi,Mq)• MuM , i,q = 1,…,m ; i z q and k•{1, … ,p}, t•{1, ... ,sk}: Mi•MOpkt and ( Mq•MOpk,t+1)} The solution space is given by the set of (C . FRXSOHV.

W (eiq )

¦r

k

˜ vkiq

eiq • E

k 1

(ii) We define E(C) as the subset of intercellular edges of the partition C: E(C) = { eiq • E / (Mi,Mq) • Cu × Cj ; u,j = 1,2,..., J ; u z j and i,q = 1,2,…,m } (iii) The first objective is defined by the total intercellular traffic of the partition C given by:

T (C , D )

¦ W (e

iq

)

eiq • E ( C )

C. Constraints (1) Given the maximum number of machines allowed in a cell, N, we must have Cj • C (j = 1,2,...,J), Card(Cj) d N,

Minimizing intercellular traffic helps to meet one of the most important CMS design objectives, namely, having cells as independent as possible. However, if it is considered alone, it can lead to unbalanced cells. To prevent this bad side effect,

can lead to unbalanced cells. To prevent this bad side effect, requires the following objective to be considered too. (2) Second objective, (i) A vertex weight function, V, is defines for each machine its workload: sk

p

V (M i ) 

M4MOpkt and M2MOpk,t+1) is true for the first operation of the fourth part (k=4, t=1). The vertex weights are V(M1)=10, V(M2)=10, V(M3)=6α133+9α223+4α313+3α333+2α413+2α433, V(M4)=6α134+9α224+4α314+3α334+2α414+2α434 and V(M5)=9 whereas the edge weights are W(e13)=6α232+α331+α333, W(e15)=3, W(e14)=6α242+α343, W(e23)=2α431+2α433, W(e24)=2α441+2α443, W(e25)=2, W(e35)=2α133+α333 and W(e45)=4α143+α343.

 

kti

 rk  tf kti i = 1,…,m.

k  1 t 1

(ii) The workload of each cell Cj, we note Chj, is calculated as follows:

III. SOLVING METHODS

Ch j   M C V ( M i ) i

j

Two solving methods are proposed. In the first one that extends the edge-coded GA [1], an enhancement of the coding structure enables handling the multiple routings aspect. Furthermore, the selection operator has been modified to tackle the multiobjective case. The second solving method is a combination of the former GA with a Tabu Search method inspired from [10]. This hybridization is a well-known approach that combines the exploration capabilities of the evolutionary algorithm with the exploitation skills of the local search method.

(iii) The second objective is defined by the variance of intercellular workloads given by: J



CH (C , )   Chj  Ch



2

J

j 1

where Ch is the mean of the cells' workloads. Therefore, assuming that S is the set of all feasible machine partitions and A is the set of all feasible operation assignments, the problem amounts to searching the (C, α)* couples that optimize in the sense of Pareto both objectives. That is,

A. Genetic Algorithm

Min T (C ,  ), CH (C ,  ) 

. C S ,  A Optimizing a set of objectives in the sense of Pareto means to seek for the non-dominated solutions. A solution is non-dominated if there is no other solution that is better in all the objectives. Example: Four parts are to be manufactured on five machines. The operation sequences, their times and the production volumes are presented in table 1.

The Genetic Algorithm is a heuristic inspired by the biological phenomenon of natural selection. A chromosome structure provides a way to representing the solutions of the problem. Using this structure, an initial population of solutions is generated. Afterwards, members of this population are selected, based on an evaluation function, called fitness that associates a value to each member according to its objective functions. The higher a member's fitness value, the more likely it is to be selected. Thus, the less fit individuals give up their places to those who perform better. Genetic operators are then applied to the selected members to produce a new population. This process is repeated until reaching a certain number of iterations.

TABLE I

PART INFORMATION Part #

1 2 3 4

Operation sequence, time Op1 Op2 Op3 M2,2 M5,1 M3, M4,3 M1,1 M3, M4,3 M1,2 M3, M4,4 M1,1 M3, M4,3 M3, M4,1 M2,3 M3, M4,3

Op4 M5,1 M5,5 M5,2 -

Volume

A.1 Chromosome structure

2 3 1 2

The code is formed of two parts. The first part has bivalent alleles that indicate whether or not the edges of the flow graph are intra- or intercellular. The second part contains alleles with integer values that indicate to which machine the operations have been assigned. Example: In the example of section 2, we consider the solution defined by the cells C1={M1,M3,M5} and C2={M2,M4} and the operation assignment that assigns the first three operations to M3 and the remainder operations to M4 (we enumerate the operation sequences from left to right, begin from the first sequence to the last one and we consider only the operations with multiple candidate machines. Doing so, we get Op13 first, then Op22 and so on till arriving to Op43). The solution will be represented by the following string:

The flow graph is defined by the set of vertices M={M1, …, M5} and the set of edges E={e13, e14, e15, e23, e24, e25, e35, e45}, see figure 1. For instance, there is an edge e24 linking M2 and M4 because the proposition (  k{1, …,p},  t{1, ... ,sk}: M4

M1 M5 M3

M2

e13 e14 e15 e23 e24 e25 e35 e45 Op13 Op22 Op31 Op33 Op41 Op43

Fig. 1.

0 1 0 1 0 1 0 1

First example flow graph

0

0

0

1

1

1

The interpretation of the first part of the string is straightforward: The edges e14, e23, e25 and e45 are 3

This yields to a lack of efficiency. To tackle this shortcoming we need to address the following question: Is it possible to the solving method to at least maintain the same performances in such situations? This thought leads to another question: since this deficiency stems from the operation-to-machine assignment decision, is it possible to undertake it without resorting to a compromising two-step decision? An answer to this question is given by the following idea: the model highlights the fact that with the multiple-routings aspect, the vertex and edge weights can fluctuate according to the decision variable α. By considering the interval of fluctuation entirely, we permit to the solving method to consider the underlined aspect implicitly. Example: In the example in section 3.A.1, the fluctuation of the vertex and edge weights of the flow graph can be bounded by the intervals represented in figure 2.

the value of the rank of the machine designated to achieve this operation from MOpkt according to the machine indexes. Thus, in our example, the sets MOpkt associated to the six operations with alternate machines are equal to {M3,M4}. Therefore, every choice of M3 is represented by 0, whereas the choice of M4 is represented by 1. A.2 Initial population The initial population is randomly generated but without repetition. Furthermore, in order to avoid frequent unicellular solutions, the probability of generating an intercellular edge is set higher than that of an intracellular one. For further explanation, see [1]. A.3 Selection The population is classified according to the number of violated constraints. Afterwards, we apply on each class the ranking procedure proposed in [11]. This former is achieved by considering the transformed objective function defined in [1]. Thus, if the ranks of the last class (eventually the solutions that violate all constraints) range from 1 to k, the ranks of the non dominated individuals of the precedent class begin from k+1 and so on. This enables a lexicographical ranking according to the feasibility then to the dominance. After the identification of the fitness, we use the roulette wheel selection procedure [12]. On this wheel, each individual in the population has a slot proportional to its fitness. Spining the wheel as many as required defines the individuals eligible to the crossover.

10 M1 3

[0,8]

[0,5]

M3

21 M5 [0,4]

[0,26]

Fig. 2.

[0,26]

[0,8]

M4 [0,5] 2

[0,4] M2 10

Non parametric flow graph.

B.2 Implementation a. The GA proceeds with the non parametric flow graph To approach the dynamic values of the flow graph, we use the mean values of the non parametric flow graph intervals. Therefore, instead of proceeding with the bipartite chains, the GA uses chromosomes build up by the machine-to-cell part solely without the long operation-to-machine tail. b. Tabu involvement Tabu is intended to determine the real solutions corresponding to the space explored by the GA. To achieve this objective, one can consider Tabu to support the GA with different manners. In our implementation, we appeal to Tabu after the GA had terminated its search as follows: b.1 Initial solutions The best solutions of the GA are extended with random but feasible tails. These solutions are confided to Tabu in order to initialize the set of current solutions. b.2 Neighborhood definition The neighborhood is generated essentially by using the following transformation: • T1 : Changing the assignment of an operation. However, since the cell compositions have been defined according to approximate mean values, it will be worthwhile to let Tabu changing the cells too. Thus, the following transformations are allowed but with lesser probability (see figure 3): • T2 : Migration of a machine to another cell. • T3 : Swapping between two machines from different cells. • T4 : Migration of a machine to a new created cell.

A.4 Crossover For simplicity and due to the bipartite feature of the chromosomes, we have opted for a two-cutting-points crossover. One random point in the machine-to-cell part with a probability Pc1 and another one in the operation-to-machine part with a probability Pc2. The ratio of individuals that undergo this operator is defined by the parameter Pc3. A.5 Mutation For this operator, we choose one part with the probability Pm1 for the first and 1-Pm1 for the second. Then the value of a random allele of the chosen individual is modified. In the machine-to-cell part, a modification yields to the transformation of an intracellular edge to an intercellular one or inversely. In other part, a modification enables us to change the assignment of the corresponding operation to another alternate machine. B. The hybrid AG-Tabu method with implicit alternation B.1 Motivation When the number of operations with alternate machines is great, the length of the genetic code becomes too important.

4

a) Routing:

b)

T1: Machine changing New cell

T2: Migration

T3: Machine swapping

T4: Cell creation

Fig. 3. Basic transformations: a) Operation assignment perturbations. b) Composition perturbations. The geometric shapes (star, triangle, square, …) represent machines. Machine with dotted lines refers to an alternate machine (a) or a removed one (b). The existence of identical shapes refers to machines of the same type. b.3 Tabu lists For each current solution we use three attributive lists. The first, concerns changing the machine that achieves a randomly chosen operation. The second is associated to the migration of machines to another existing or new cell. The last is associated to the machine swapping. b.4 Stopping criterion Two termination criteria are considered. In the first one, we set a number of maximum iterations. In the second, we consider a number of maximum iterations without modification of the best solutions archive.

fifth operation of the second part is missing in the manuscript, …). The routing information of the first example have been already described in section 3.A.1. The remaining inputs concern the constraints. The maximum number of machines in each cell is set to 3. The machines M3 and M4 cannot cohabitate. Finally, the maximum workload of the five machines is 10, 10, 20, 20 and 25 respectively. In the second example, we have 9 machines and 6 parts. The part information is gathered in table 2.

IV. RESULTS

SECOND EXAMPLE PART INFORMATION

TABLE II Part #

A. Illustrative examples

1

The two solving methods, say GA and GAT, in addition to an adaptation of the method SPEA II [13] have been applied on a set of mainly hypothetical examples ranging from small to medium size. We used a Celeron cpu, 1.8 Ghz system with 224 Mb of RAM and a C++ compiler. For the small-sized instances, an enumerative algorithm gives the true Pareto front. For SPEA II, the adaptation concerns the modification of the fitness to take into account the constrained feature of the problem. For this purpose, if a solution is feasible, its strength is calculated only upon the dominated feasible solutions, else it is severely penalized. However, the diversification principle of SPEA II, has been implemented as described by the author. Among the different examples, we chose a sample of nine examples. The first one is hypothetical and has been used along the paper for illustrative purposes. The rest is from [14] (we have modified some of the examples’ inputs: for instance, the maximum workloads of example 2 yield an empty search space. In addition, the operating time of the

Operation sequence, time Op1 M5, M6, 1.23 M7, M9, 1.23 M2, M7, 3.85 M4, M9, 3.35 M1, M3, 4.78 M2, M3, 1.91

2 3 4 5 6

Op2 M5, M9, 4.13 M3, M4, 3.55 M2, M7, 2.90 M1, M9, 3.07 M1, M4, 14.10 M3, M8, 3.81

Vol.

Op3 M3, M7, 4.69

Op4 M4, M6, 3.65

-

-

0.90

-

-

1.07

M6, M7, 2.68 M2, M6, 2.14 M4, M6, 1.95

M5, M8, 3.11

0.91

-

0.84

0.94

0.80

The number of machines should not exceed three per cell. Machines M2 and M7 should not be put in a close vicinity. The workload capacities of the nine machines are presented in table 3. TABLE III

EXAMPLE 2 MACHINE CAPACITIES Mach. # Capa.

5

1 15

2 7.5

3 15

4 20.5

5 7.5

6 7.5

7 10.5

8 6.5

9 7.5

The third example deals with 10 machines and 12 parts. The operation sequences and the related information are presented in table 4.

5 6 7 8 9 10

TABLE IV

THIRD EXAMPLE PART INFORMATION Operation sequence, time

Part #

Op1 M2, M3, 3.52 M1, M9, 3.53

1 2

Op2 M9, 5.40 M7, 5.44 M3, M8, 4.64 M3, M7, 2.26 M1, 2.35 M5, M7, 4.13 M6, M10, 5.71

M10, 4.18

3 4

M7, 3.68

5

M10, 5.82

6

M1, 2.25

7

M7, 5.78 M5, M8, 3.58 M5, M9, 5.92

8 9

M5, 5.57

M2, 5.04

M6, 5.67

11

M7, 2.51

12

M9, 2.42

M8, 3.68 M3, M10, 4.52

10

-

0.88

M5, M9, 4.85

M10, 4.18

0.89

TABLE VII

-

-

0.81

EXAMPLE 4 MACHINE CAPACITIES

0.92 0.87

-

-

0.91

M1, 3.93

-

-

0.94

M1, M5, 2.45 M4, 5.94 M4, M7, 4.33

M9, M10, 3.13 -

-

0.93

-

-

0.82

Mach. # Capa.

0.98

Part#

3

4

5

6

7

8

9

10

95.8

73.9

94.1

78.8

91.9

84.9

100

71.5

1 2 3 4

Op3

M 2 ,M 5 ,2 M 1 ,M 3 ,5 M 1 ,3 M 4 ,M 5 ,8

M 3 ,M 4 ,1 M 2 ,M 4 ,1 M 3 ,2 M 2 ,2

M 1 ,3 M 1 ,M 5 ,2 M 1 ,M 2 ,6 M 3 ,3

M1 ,2

M9 ,4

5

M9 ,1

M8 ,7

M8 ,1

M7 ,8

M3 ,M7 , M9 ,5 M2 ,M5 , M6 ,7

Mach. # Capa.

Vol.

Operation sequence, time Op4 Op5 M3 ,2 M2 ,2 M1 ,6 M8 ,5 M7 ,8 M1 ,9 M9 ,1 M6 ,M7 ,M8 ,8 M2 ,M7 ,M9 ,6 M1 ,M4 ,M6 ,4

Vol.

Op3

6 10 5 6 8 5

M6 ,1

10

M5 ,7

7

1

2

3

4

5

6

7

8

9

150

100

50

70

100

100

220

130

150

The sixth example deals with eight machines and nine parts. The operation sequences and the related information are presented in table 10.

Op4

M 4 ,7

5 150

TABLE IX EXAMPLE 5 MACHINE CAPACITIES

TABLE VI

Op2

4 120

The number of machines should not exceed 3 per cell. Machines M1 and M9 should not be gathered in the same cell. The machine capacities are presented in table 9.

FOURTH EXAMPLE PART INFORMATION Op1

4

6

The fourth example deals with five machines and ten parts. The operation sequences and the related information are presented in table 6.

Operation sequence, time

Op2 M4 ,9 M9 ,3 M3 ,1

8

2

Part#

1 2 3

Op1 M1 ,M5,3 M1 ,M5 ,5 M2 ,M8 ,1

7

88.8

3 120

TABLE VIII

TABLE V EXAMPLE 3 MACHINE CAPACITIES 1

2 100

FIFTH EXAMPLE PART INFORMATION

0.80

83.1

1 150

The fifth example deals with nine machines and eight parts. The operation sequences and the related information are presented in table 8.

0.93

The number of machines should not exceed 4 per cell. Machines M1 and M10 should be put in the same cell; whereas M5 and M7 should not be gathered in the same cell. The machine capacities are presented in table 5.

Mach. # Capa.

M 3 ,9

-

M8, 5.12 M2, 5.91

M9, M10, 3.61

M3, 2.67

2 4 5 6 3 4

Op5

-

M9, 2.22

M 3 ,M 4 ,3 M 5 ,3 M 2 ,M 5 ,2 M 4 ,1 M 2 ,M 3 ,3 M 5 ,2

Op4

M1, M5, 3.66 M7, 5.17 M5, M8, 5.39 M4, M10, 2.89

M4, 3.20

M 1 ,M 2 ,7 M 2 ,M 4 ,1 M 1 ,M 3 ,7 M 3 ,M 5 ,9 M 4 ,7 M 1 ,M 2 ,8

The number of machines should not exceed 3 per cell. Machines M1 and M5 should not be gathered in the same cell. The machine capacities are presented in table 7.

Vol.

Op3 M7, M8, 5.50 M6, M7, 3.03 M1, M2, 4.63 M9, M10, 3.17 M3, 4.48

M 3 ,5 M 1 ,6 M 2 ,M 4 ,8 M 1 ,M 2 ,2 M 1 ,M 5 ,9 M 4 ,M 5 ,7

2 1 5 3

6

TABLE X

SIXTH EXAMPLE PART INFORMATION Part#

1 2 3 4 5 6 7 8 9

Vol.

Operation sequence, time Op3 Op4

Op1

Op2

M 2 ,M 6 ,1 M 2 ,4 M 1 ,M 2 ,3 M 3 ,M 4 ,M 5 ,5 M 1 ,M 2 ,M 6 ,5 M 2 ,6 M 5 ,9 M 1 ,M 2 ,M 3 ,7 M 1 ,M 3 ,M 8 ,6

M 5 ,1 M 1 ,M 3 ,3 M 8 ,8 M 1 ,3 M 4 ,5 M 8 ,4 M 3 ,M 6 ,M 7 ,5 M 8 ,4 M 7 ,1

M 3 ,6 M 7 ,7 M 4 ,M 7 ,7 M 8 ,2 M 3 ,2 M 5 ,M 6 ,M 7 ,5

No more than 3 machines are accepted per cell. Machines M1 and M8 should be put in different cells. Table 11 presents the machine capacities.

Op5

M 1 ,9 M 5 ,9 M 5 ,7

Op6

M 7 ,6 M 4 ,M 8 ,2

M 4 ,M 7 ,10

10 9 5 10 10 10 2 6 6

The seventh example deals with eleven machines and twelve parts. The operation sequences and the related information are presented in table 12.

TABLE XI

EXAMPLE 6 MACHINE CAPACITIES Mach. # Capa.

1 250

2 200

3 220

4 180

5 220

6 100

7 250

8 150

TABLE XII

SEVENTH EXAMPLE PART INFORMATION Part#

Op1 M1 ,M4 ,M6 ,5 M3 ,M9 ,3 M5 ,M11 ,1 M5 ,M10 ,7 M1 ,M5 ,7 M1 ,M3 ,M10 ,2 M8 ,M10 ,M11 ,7 M2 ,M8 ,M10 ,7 M3 ,M4 ,M6 ,6 M2 ,M5 ,M9 ,7 M7 ,M9 ,M11 ,5 M1 ,M9 ,M11 ,3

1 2 3 4 5 6 7 8 9 10 11 12

Operation sequence, time Op2 Op3 Op4 M2 ,8 M11 ,5 M8 ,9 M6 ,6 M4 ,9 M2 ,M7 ,9 M8 ,9 M7 ,2 M3 ,M4 ,8 M8 ,6 M5 ,4 M1 ,M4 ,4 M10 ,6 M8 ,8 M6 ,M7 ,1 M2 ,4 M11 ,5 M6 ,5 M4 ,6 M11 ,9 M9 ,5 M4 ,6 M2 ,9 M10 ,6 M6 ,9 M8 ,3

The number of machines should not exceed 4 per cell. Machines M6 and M11 should not be gathered in the same cell. The machine capacities are presented in table 13.

Vol.

Op5 M7 ,7

Mach. # Capa.

7 100

Op6 M10 ,2

8 250

10 1 8 8 1 8 9 1 6 9 9 7 9 100

10 200

TABLE XIII 1 100

2 200

3 80

4 200

5 100

6 180

TABLE XIV

EIGHTH EXAMPLE PART INFORMATION Part#

1 2 3 4 5 6 7 8 9 10 11 12 13

Op1 M4 ,M11 , 0.5 M4 ,M11 ,0.55 M5 ,M8 ,0.45 M2 ,M6 ,M10 ,0.23 M3 ,M7 ,M 9 ,0.37 M3 ,M11 ,0.4 M1 ,M9 ,0.3 M6 ,M8 ,0.5 M4 ,M8 ,M10 ,0.4 M5 ,M6 ,M9 ,0.55 M2 ,M6 ,M11 ,0.6 M3 ,M5 ,M7 ,0.54 M3 ,M5 ,M8 ,0.46

Op2 M2 ,0.68 M8 ,0.7 M6 ,0.25 M9 ,0.19 M6 ,0.23 M9 ,0.39 M11 ,0.18 M4 ,0.67 M9 ,0.53 M4 ,0.74 M11 ,0.61 M10 ,0.58 M7 ,0.8

-

The eighth example deals with eleven machines and therteen parts. The operation sequences and the related information are presented in table 14.

EXAMPLE 7 MACHINE CAPACITIES Mach. # Capa.

11 200

Operation sequence, time Op3 Op4 M10 ,0.71 M9 ,0.42 M6 ,0.42 M4 ,0.5 M4 ,0.42 M2 ,0.44 M6 ,0.76 M4 ,0.4 M4 ,0.21 M3 ,0.76 M6 ,M7 ,0.3 M7 ,M8 ,0.6 M1 ,M2 ,0.85

7

Vol. Op5 M6 ,0.72 M1 ,M2 ,0.4 M7 ,M10 ,0.68

Op6 M1 ,M2 ,0.60

1 9 10 5 10 7 1 5 6 7 1 1 8

The number of machines should not exceed 4 per cell. Machines M2 and M11 should not be gathered in the same cell. The machine capacities are presented in table 15.

Mach. # Capa.

1 9

2 10

3 15

4 25

5 13

8 15

9 10

10 10

11 7

-

The ninth example deals with fourteen machines and twenty parts. The operation sequences and the related information are presented in table 16.

TABLE XV EXAMPLE 8 MACHINE CAPACITIES Mach. # Capa.

7 15

6 20

TABLE XVI

NINTH EXAMPLE PART INFORMATION Part# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Operation sequence, time Op1 M 8,0.89 M 7,0.21 M 9,0.79 M 2,0.14 M 3,M 9,0.66 M 8,M 14,0.73 M 6,M 9,0.97 M 1,M 2,M 13,0.96 M 2,0.64 M 11,0.11 M 6,M 12,0.49 M 11,M 14,0.72 M 1,0.66 M 10,0.4 M 11,0.48 M 10,0.77 M 13,0.6 M 11,M 12,M 14,0.64 M 7,M 10,M 12,0.85 M 2,M 5,M 11,0.41

Op2 M 11,M 13,0.53 M 3,0.47 M 5,0.61 M 5,0.14 M 5,0.85 M 2,0.22 M 3,0.83 M 4,0.34 M 13,0.91 M 14,0.18 M 14,0.9 M 2,0.34 M 2,M 11,0.75 M 12,0.17 M 2,M 7,M 13,0.99 M 6,M 9,M 12,0.5 M 2,M 7,M 9,0.75 M 8,0.36 M 7,0.39 M 5,0.88

Op3 M 7,0.59 M 5,0.71 M 5,M 8,M 13,0.63 M 1,0.68 M 7,M 8,0.32 M 13,0.93 M 1,M 5,0.88 M 14,0.55 M 1,M 6,M 14,0.36 M 2,M 4,M 7,0.37 M 8,M 11,0.56 M 10,M 12,0.53 M 10,M 13,0.18 M 1,M 3,M 8,0.33

The number of machines should not exceed 5 per cell. Machines M7 and M9 should be put in the same cell whereas M5 and M2 should not be gathered in the same cell. The machine capacities are presented in table 17.

2 20 9 20

3 21 10 15

4 16 11 20

5 32 12 15

6 15 13 10

Op6 M 4,0.85

1 8 5 10 6 1 7 1 1 8 1 9 9 5 1 7 7 8 7 8

TABLE XVIII Solution # 0 1 2 3 4 5 6 7

TABLE XVII

1 20 8 10

Op5 M 6,M 8,0.58 M 4,M 12,M 14,0.92 M 6,0.44

PARETO FRONT OF EXAMPLE ONE

EXAMPLE 9 MACHINE CAPACITIES Mach. # Capa. Mach. # Capa.

Vol.

Op4 M 9,0.9 M 1,0.36 M 4,0.95 M 3,M 6,M 9,0.74 M 4,0.3 M 1,M 11,0.21

7 15 14 15

B. Discussions

Code 12211-110100 12121-001011 12122-100011 12212-011100 11122-100100 11212-011011 11122-101100 11212-010011

Traffic 0.136364 0.136364 0.181818 0.181818 0.272727 0.272727 0.318182 0.318182

Workload 0.214079 0.214079 0.050123 0.050123 0.010916 0.010916 0.000223 0.000223

When applied on this data set, the three methods gave the results presented in table 19.

The first example small size makes it possible to determine, by a naïve enumerative method, the optimal Pareto set which is composed of the eight solutions presented in table 18 with their associated normalized objectives.

TABLE XIX

PERFORMANCES FOR EXAMPLE ONE Number of sol. Run time (sec.) Generational Dist. Hyperarea

8

GA

SPEA II

GAT

6 73.26 0 0.151299

3 62.83 0 0.209370

7 2.31 0 0.151299

The obtained approximate Pareto front sets are depicted in table 20.

For the third example, the performances of the three methods are presented in table 23.

TABLE XX

TABLE XXIII

APPROX. PARETO FRONTS FOR EXAMPLE I Method

GA

SPEA II

So. # 0 1 2 3 4 5 0 1 2 0 1 2 3 4 5 6

Code 12121-001011 12211-110100 12122-100011 12212-011100 11122-100100 11212-010011 12212-011000 12213-011100 12213-011000 12211-110100 12122-100011 12212-011100 11212-011011 11122-100100 11212-010011 11122-101100

Traffic 0.136364 0.136364 0.181818 0.181818 0.272727 0.318182 0.181818 0.454545 0.500000 0.136364 0.181818 0.181818 0.272727 0.272727 0.318182 0.318182

PERFORMANCES OF EXAMPLE III

Workload 0.214079 0.214079 0.050123 0.050123 0.010916 0.000223 0.098240 0.006906 0.000891 0.214079 0.050123 0.050123 0.010916 0.010916 0.000223 0.000223

Number of sol. Running time Hyper a r ea

GA

SPEA II

GAT

18 302.75 0.344757

2 889.96 0.397586

13 36.45 0.335223

The approximate Pareto fronts are reported in table 24. TABLE XXIV

APPROX. PARETO FRONTS FOR EXAMPLE III Meth.

Sol.#

Traf.

Workl.

0

12132213210101100011011001101011

0.339574

0.068945

1

12123312310101100011011001101011

0.365068

0.062091

2

12123312310101100011011001111011

0.365068

0.062091

3

12132213210101100011011001101010

0.367268

0.051286

4

12132213210101100011011001111000

0.367268

0.051286

For the second example the three methods gave the results depicted in table 21.

5

12132213210101000001011001101000

0.367268

0.051286

6

12132213210101100011011001101000

0.367268

0.051286

7

12123312310101100011011001111000

0.392762

0.049232

TABLE XXI PERFORMANCES FOR EXAMPLE II

8

12123312310101100011011001101000

0.392762

0.049232

9

12121233210011000011001001101011

0.420320

0.012974

10

12121233210011010011001001101011

0.420320

0.012974

11

12121233210011010111001001101000

0.447879

0.008285

12

12121233210011010111001001101011

0.447879

0.006373

13

12121233210111100011001001101011

0.450554

0.005650

14

12321233210011010111001001100010

0.475438

0.000186

15

12321233210011010011001000001100

0.479229

0.000793

16

12121233210011010111111001001011

0.511900

0.000200

17

12122133210011000011011001101011

0.546873

0.000168

0

12321232311011001101101101000101

0.335071

0.003872

1

12321232310011001101101101001101

0.364864

0.000915

2

12321232310011001101101101000101

0.364864

0.000915

3

12321232310011001111101101001101

0.364864

0.000915

4

12321232310011011101101101000101

0.364864

0.000915

5

12321232310011011111101101001101

0.364864

0.000915

6

12321232310011001101101101001100

0.392558

0.000060

7

12321232310011011101101101001100

0.392558

0.000060

8

12321232311011001111001011100100

0.425094

0.000032

9

12321232311011001111101011100100

0.425094

0.000032

10

12321232311011001111001011101100

0.425094

0.000032

11

12321232310010010101000001100000

0.574601

0.000009

12

12321232310010010101100001100000

0.574601

0.000009

0 1

12321132331110011100101110001101 12321132441110011100101110001101

0.388699 0.569760

0.047328 0.000740

GAT

Number of sol. Running time Hyperarea

GA 9 118.19 0.249859

SPEA II 4 62.87 0.251654

GA

GAT 14 4.72 0.000463

The obtained Pareto sets are presented in table 22. TABLE XXII APPROX. PARETO FRONTS FOR EXAMPLE II Meth.

GA

SPEA II

GAT

Sol. # 0 1 2 3 4 5 6 7 8 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13

Code

112233312-000011110111001100 112233312-000011111111001100 112233312-000011111111001011 112233312-000111111111001011 112321323-000011111111001011 112322331-000111111111001111 112233312-000111111111001111 112213323-000111111111001111 112312332-000011111111001100 122344324-001001001100001100 122344324-011001001100001100 122344324-001010111100001100 122344324-011011011100001100 112322313-100101111110001101 112322313-100101110110001101 112322313-100111110110000100 112322313-000101110110000100 112322313-000101111110000100 112322313-000111111110000100 112322313-000111110110010110 112322313-100001111011001101 112322313-100011110011001101 112322313-000011110011001101 112322313-100011111011001101 112322313-000001110011001101 112322313-100001110011001101 122322313-100001110100001101

Traffic

Workload

0.245344 0.245344 0.322894 0.409617 0.410451 0.481794 0.487168 0.503382 0.577689 0.248587 0.248587 0.332345 0.431761 0.000000 0.000000 0.161864 0.161864 0.161864 0.161864 0.239414 0.319652 0.319652 0.319652 0.319652 0.319652 0.319652 0.329658

0.042290 0.042290 0.010002 0.003842 0.003827 0.001308 0.000401 0.000300 0.000131 0.006198 0.006198 0.005547 0.003513 0.002371 0.002371 0.000498 0.000498 0.000498 0.000498 0.000424 0.000028 0.000028 0.000028 0.000028 0.000028 0.000028 0.000009

GAT

SPEAII

Code

For the fourth example, the performances of the three methods are presented in table 25. TABLE XXV

PERFORMANCES OF EXAMPLE IV # of sol. Running time Hyperarea

GA 37 80.02 0.308952

SPEA II 16 54.14 0.302975

GAT 5 3.47 0.275420

The approximate Pareto fronts are reported in table 26.

9

TABLE XXVI The approximate Pareto fronts are reported in table 28.

APPROX. PARETO FRONTS FOR EXAMPLE IV Meth.

GA

GAT

SPEA II

Sol.# 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 0 1 2 3 4

Code 121121101001010110110111 121120111001010110110111 121121111001010110110111 121120101001010110110111 121120101001000110110111 121120011001010110110111 121120111001010111110111 121121111001000110110111 121120101001000111110111 121120011001010111110111 121120111001000111110111 121121111001010111110111 121120111001000110110111 121121101001010111110111 121121010001010110110111 121120110001010110110111 121121010001010111110111 121120010001010110110111 121120101001010111110011 121121101001010111110011 121121111001010111110011 121120101001110111110111 121120101001110110110111 121120111001010011110111 121120101001000011110111 121121111001010011110111 121120111001000011110111 121120011001010011110111 121121101001010011110111 121120101001010011110111 121121101001000011110111 121121111001000011110111 121120100001010011110111 121120010001010011110111 121120110001010011110111 122121101000100010110100 121221111001000011110100 121120100101010111000111 121220010100001111111011 122120110001101101110011 122120110001101101110111 122120110100101101110011

0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 329114 0. 329114 0. 329114 0. 329114 0. 341772 0. 341772 0. 341772 0. 354430 0. 354430 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 392405 0. 392405 0. 392405 0. 405063 0. 556962 0. 240506 0. 367089 0. 392405 0. 392405 0. 417722

0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 074935 0. 074935 0. 074935 0. 074935 0. 059511 0. 059511 0. 059511 0. 050772 0. 050772 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 015567 0. 015567 0. 015567 0. 001529 0. 000031 0. 262251 0. 061342 0. 006412 0. 006412 0. 000003

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

121120000001010111000011 121120101001000100001011 121120111001000110001011 121121001001000100001011 121120111001010110110011 121121001001000100110011 121121000001000100110011 121120101001000100111011 121120101001010000110011 121121000001000000110011 121121001001010010111011 121220100001001100001011 121220110000000100001011 121220111001000100001011 121220111001001100001011 121221110001000110001011

0. 291139 0. 303797 0. 303797 0. 303797 0. 341772 0. 341772 0. 367089 0. 379747 0. 405063 0. 430380 0. 443038 0. 455696 0. 455696 0. 455696 0. 455696 0. 455696

0. 236229 0. 151476 0. 151476 0. 151476 0. 059511 0. 059511 0. 057707 0. 020560 0. 009020 0. 008326 0. 000031 0. 000003 0. 000003 0. 000003 0. 000003 0. 000003

Traf.

TABLE XXVIII APPROX. PARETO FRONTS FOR EXAMPLE V

Wd.

Meth.

GA

GAT

SPEAII

Workl. 0.087216 0.043643 0.043643 0.016452 0.000879 0.000634 0.073578 0.023527 0.023527 0.015429 0.015429 0.014074 0.014074 0.014074 0.014074 0.014074 0.014074 0.014074 0.014074 0.009744 0.009744 0.004911 0.004911 0.004562 0.004562 0.000905 0.000905 0.000726 0.000726 0.000215 0.000215 0.000189 0.000189 0.000027 0.000027 0.000023 0.000023 0.000010 0.000010 0.086346 0.045228 0.001221

TABLE XXIX

# of sol. cpu Hyperarea

TABLE XXVII

SPEA II 3 55.76 0.437561

Traf. 0.387097 0.451613 0.451613 0.459677 0.508065 0.814516 0.306452 0.338710 0.338710 0.379032 0.379032 0.387097 0.387097 0.387097 0.387097 0.387097 0.387097 0.387097 0.387097 0.395161 0.395161 0.403226 0.403226 0.419355 0.419355 0.443548 0.443548 0.459677 0.459677 0.491935 0.491935 0.508065 0.508065 0.516129 0.516129 0.596774 0.596774 0.653226 0.653226 0.427419 0.475806 0.596774

PERFORMANCES OF EXAMPLE VI GA 14 104.35

SPEA II 4 62.74

GAT 36 22.61

0.348601

0.409300

0.345499

The approximate Pareto fronts are reported in table 30.

PERFORMANCES OF EXAMPLE V GA 6 103.27 0.394259

Code 11122233301012012 11122233301010022 11122233301010012 12113322301010022 11213322301001020 12133452401010022 11123223301022211 11122323310002121 11122323310022121 11122323310002021 11122323310022021 11122233310002102 11122233310001202 11122233310001102 11122233310002202 11122233310002101 11122233310001101 11122233310001201 11122233310002201 11122323310022120 11122323310002120 11122323310020121 11122323310000121 11122323310002101 11122323310022101 11122323310020021 11122323310000021 11122323310022001 11122323310002001 11122323300020021 11122323300000021 11122323300002001 11122323300022001 11123223301010010 11123223301000010 11123223301000002 11123223301010002 11123223300102002 11123223300112002 12112233301012210 12112233301002210 12112234401012210

For the sixth example, the performances of the three methods are presented in table 29.

For the fifth example, the performances of the three methods are presented in table 27.

# of sol. Running time Hyperarea

Sol.# 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 1 2

GAT 33 31.59 0.310330

10

TABLE XXX APPROX. PARETO FRONTS FOR EXAMPLE VI Meth.

Sol.# 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

GA

GAT

Code

12132312010110020110 12133312111010020110 12133312111010020111 12133312110110020110 12133312110110020111 12133312111110020110 12133312111110020111 12133312101110020111 12332312010110022120 12332312011110022220 12132332011110022220 12332213110100222121 12322313001001001001 11213232111010020111 12132312011110020111 12132312011110020110 12333312110010022010 12333312110010222010 12333312110010122010 12333312110010122212 12333212010010011010 12333212010010211010 12333212010010111010 12333212010010111210 12333212010010011210 12333212010010121210 12333212110010121210 12333212010010021210 12333212010010011002 12333212110010011002 12333212011011011010 12333212111011011010 12333212011011211010 12333212111011211010 12333212010001211111

Traf.

Workl.

0.342857 0.360000 0.360000 0.411429 0.411429 0.411429 0.411429 0.468571 0.491429 0.542857 0.577143 0.645714 0.777143 0.891429 0.342857 0.342857 0.360000 0.360000 0.360000 0.405714 0.417143 0.417143 0.417143 0.428571 0.428571 0.428571 0.428571 0.428571 0.485714 0.485714 0.497143 0.497143 0.497143 0.497143 0.520000

0.096235 0.029296 0.029296 0.026336 0.026336 0.026336 0.026336 0.013589 0.007367 0.003950 0.001551 0.000975 0.000139 0.000071 0.096235 0.096235 0.019245 0.019245 0.019245 0.009117 0.000171 0.000171 0.000171 0.000057 0.000057 0.000057 0.000057 0.000057 0.000048 0.000048 0.000043 0.000043 0.000043 0.000043 0.000037

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0 1 2 3

SPEAII

12333212110001011111 12333212110001211111 12333212010001011111 12333212010001111111 12333212010001221111 12333212110001111111 12323312110010020210 12323312110110020210 12323312110110021210 12323312110110221210 12323312110010021210 12323312110110220210 12323312110010221210 12323312110010220210 12223313110111010222 12133312111010021212 12133312111110021212 12133312101010021212 12133312101110021212

0.520000 0.520000 0.520000 0.520000 0.520000 0.520000 0.537143 0.537143 0.537143 0.537143 0.537143 0.537143 0.537143 0.537143 0.554286 0.405714 0.457143 0.462857 0.514286

0.000037 0.000037 0.000037 0.000037 0.000037 0.000037 0.000009 0.000009 0.000009 0.000009 0.000009 0.000009 0.000009 0.000009 0.000002 0.016244 0.015175 0.008157 0.004621

For the seventh example, the performances of the three methods are presented in table 31. TABLE XXXI

PERFORMANCES OF EXAMPLE VII

# of sol. Running time Hyperarea

GA 2 125.46 0.336192

SPEA II 4 56.34 0.271642

GAT 9 9.07 0.271336

The approximate Pareto fronts are reported in table 32.

TABLE XXXII

APPROX. PARETO FRONTS FOR EXAMPLE VII Meth. GA

GAT

SPEAII

Sol.#

0 1 0 1 2 3 4 5 6 7 8 0 1 2 3

Code 1 1 2 2 3 2 3 3 23 1 0 0 1 1 0 1 1 01 0 2 0 1 1 1 0 1 2 3 4 3 4 3 1 43 2 0 0 1 1 0 1 1 01 0 2 0 1 1 1 0 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 0 0 11 0 1 0 1 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 0 0 11 0 1 2 1 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 1 0 0 0 1 00 0 1 2 1 2 1 1 1 1 2 1 2 3 2 2 33 1 0 1 0 0 0 0 0 01 0 1 2 2 2 1 1 1 1 2 1 2 3 2 2 33 1 2 1 0 0 0 0 0 00 0 1 0 1 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 1 1 11 0 1 0 0 2 1 1 1 1 2 1 2 3 2 2 33 1 0 1 0 0 0 1 1 11 0 1 0 0 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 1 0 11 0 1 2 0 2 1 1 1 1 2 1 2 3 2 2 33 1 0 1 0 0 0 1 1 11 0 1 2 0 2 1 1 1 2 2 2 1 3 1 1 33 2 1 1 0 0 1 0 0 11 1 1 1 2 2 1 0 1 2 2 2 1 3 1 1 33 2 1 1 1 0 1 0 0 11 1 1 1 2 2 1 0 1 2 2 2 1 3 1 1 33 2 1 1 0 0 1 0 0 11 2 1 1 2 2 1 0 1 2 2 2 1 3 1 1 33 2 2 1 0 0 1 0 0 11 1 1 1 2 2 1 0

For the eighth example, the performances of the three methods are presented in table 33. TABLE XXXIII GA 10 129.57 0.332264

SPEA II 2 132.38 0.416271

Traf. 3 33 3 3 3 5 59 3 2 2 2 71 1 8 6 2 76 8 3 6 2 88 1 3 6 3 10 7 3 4 3 33 3 3 3 3 50 2 8 2 3 50 2 8 2 3 55 9 3 2 3 55 9 3 2 2 71 1 8 6 2 76 8 3 6 3 16 3 8 4 3 27 6 8 4

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

Workl. 009829 001447 004322 003331 003184 000386 000185 000037 000037 000006 000006 002437 002370 001007 000501

The approximate Pareto fronts are reported in table 34.

PERFORMANCES OF EXAMPLE VIII # of sol. cpu Hyperarea

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

GAT 42 20.64 0.329853

11

TABLE XXXIV

APPROX. PARETO FRONTS FOR EXAMPLE VIII Meth.

GA

GAT

SPEAII

Sol.#

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 0 1

Code 121121233230110010001110011221 121121233230010012001111011221 121121233230110010001111011221 121121233230110010001110010221 121121233230110012001110010221 121121233230110012001111111221 123121233230110010001110011221 123121233230010012001110010221 123121233230110010001110010221 123121233230110012001111110221 121131332231010012011100121212 121131332231110112011100122222 121131332231110012011100122222 121131332231110112011100122221 121131332231110012211100122222 121131332231110012211100122212 121131332231110112211100122222 121131332231010012211011122201 121131332231110012211101122222 121131332231011010111110122222 121131332231011010111100122222 121131332231011012111100122222 121131332231011012111110122222 121131332231111112211110120212 121131332231111112211110120222 121131332231011012211011122211 112232331131110012211001122212 112132323130011101011101022201 112132323130111101011101022201 112132323130111101011100022201 112132323130111101011100002201 112132323130011101011100020201 112132323130011101011101020201 112132323130111101011101020201 112132323130111101011100020201 112132323130111101011101122201 112132323130111101011100122201 112132323130111101011100120201 112132323130011101011100022201 112132323230011100011100100001 112132323230110100011100100001 112132323230011100011100002001 112132323230011100011101000001 112132323230011100011101002001 112132323230010100011100002001 112132323230011100011101100001 112132323230011100011101102001 112132323230011100011100102001 112132323230010100011101102001 112132323230011100011100000001 112132323230110100011101102001 112132323230010100011100102001 123333212210011010001111001122 123133212210011010001111001122

For the last example, the performances of the three methods are presented in table 35. TABLE XXXV GA 11 185.12 0.331120

SPEA II 5 64.57 0.343895

Workl. 0.094107 0.077839 0.074103 0.071735 0.071735 0.052036 0.016175 0.006650 0.005384 0.000169 0.046334 0.024235 0.024235 0.024235 0.012294 0.012294 0.012294 0.009096 0.005963 0.005131 0.005131 0.005131 0.005131 0.004071 0.004071 0.002375 0.001870 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.075851 0.011346

The approximate Pareto fronts are reported in table 36.

PERFORMANCES OF EXAMPLE IX Number of sol. Running time Hyperarea

Traf. 0.325714 0.348571 0.354286 0.365714 0.365714 0.382857 0.400000 0.434286 0.440000 0.497143 0.325714 0.371429 0.371429 0.371429 0.428571 0.428571 0.428571 0.451429 0.457143 0.474286 0.474286 0.474286 0.474286 0.480000 0.480000 0.502857 0.508571 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.394286 0.628571

GAT 57 24.64 0.298304

12

TABLE XXXVI

APPROX. PARETO FRONTS FOR EXAMPLE IX Meth.

GA

GAT

SPEAII

Sol.# 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 0 1 2 3 4

Code 12131312123233011100011112110110112100101 12131312123233011100001012110110112100101 12131312123233011120001012110110112100101 12131312123233011120001012110110112100100 12131312123233011100001012110110112100100 11232321213133011100011112110110112100101 11232321213133011100001012110110112100101 11232321213133011100011012110110112100101 12113123223323000120111011121010101102101 12113123223323000120111011121010110100101 12113123223323000120111011121010110100100 12111331332222000100111110201101110100101 12111331332222000100111102201101110100101 12111331332222102100111110201101110200101 12111331332222102100111110101101110200101 12111331332222000120111112201101111100101 12111331332222000120111112201101110100101 12131331332222000100011012201111012210001 12131331332222000100011012200101112210001 12131331332222000100011012200111112210001 12131331332222000100011012200101111210001 12131331332222000100011012200111111210001 12131331332222000100011012200111012210001 12131331332222000200011012201111012210001 12131331332222100110011102201101112200201 12131331332222100110011101201101112000201 12131331332222000110011101201101112000201 12131331332222000110011102201101112000201 12131331332222000110011102201101112200201 12131331332222000110011101201101112200201 12131331332222000210011112201101110210011 12131331332222000220011101201101112200201 12131331332222000210011112201101110200201 12131331332222100210011112201101110210011 12131331332222000210011101201101112200201 12131331332222102011011102201101112200201 12131331332222002011011102201101112200201 12131331332222002011011102101101112200201 12131331332222002011011102101101112000201 12131331332222100210011112201101010200200 12131331332222100210011111201101010200200 12131331332222100210011111201101010200202 12131331332222000210011111201101111210210 12131331332222000210011112201101111210210 12131331332222000210011111201101111010212 12131331332222100210011112201101110200200 12131331332222100210011111201101110200200 12131331332222100210011111201101110200202 12131331332222000210011101201101111010212 12131331332222000210011111201101110200010 12131331332222100210011112201101110200210 12131331332222000210011112201101110210210 12131331332222000210011111201101110210210 12131331332222000210011112201101110200100 12131331332222100210011112201101110210210 12131331332222000220011012201101010200200 12131331332222000210011111201101012200200 12131331332222100210011111201101012200200 12131331332222000210011111201101110210010 12131331332222000210011112201101110200210 12131331332222000210011112201101110200200 12131331332222000210011111201101111010210 12131331332222000210011112201101010200200 12131331332222000210011112201101012200200 12131331332222102221011111120111111210101 12131331332222002011011111120101111210102 12131331332222102021011111120111111210102 12131331332222102221111112100001110212100 12131314132422110100011111201111112000100 12311134312422110100011111201111112000100 12341431342122110100011111201111112000100 12341435342522110100011111201111112000100 12341435342522 110100011111101111112000100

13

Traf. 0.324561 0.350877 0.350877 0.385965 0.385965 0.412281 0.438596 0.442982 0.627193 0.627193 0.662281 0.293860 0.298246 0.324561 0.328947 0.342105 0.342105 0.350877 0.355263 0.355263 0.355263 0.355263 0.355263 0.372807 0.394737 0.394737 0.394737 0.394737 0.394737 0.394737 0.416667 0.416667 0.416667 0.416667 0.416667 0.421053 0.421053 0.425439 0.425439 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.486842 0.500000 0.500000 0.521930 0.315789 0.464912 0.478070 0.500000 0.504386

Workl. 0.069115 0.043785 0.043785 0.032855 0.032855 0.023710 0.008749 0.008144 0.001717 0.001717 0.000345 0.078672 0.075150 0.057872 0.057109 0.043011 0.043011 0.015681 0.015661 0.015661 0.015661 0.015661 0.015661 0.010275 0.005131 0.005131 0.005131 0.005131 0.005131 0.005131 0.001783 0.001783 0.001783 0.001783 0.001783 0.001482 0.001482 0.001356 0.001356 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000051 0.000036 0.000036 0.000001 0.112403 0.052780 0.025785 0.020634 0.020164

These results were derived using the values 0.8, 0.8 and 0.02 for the rates of crossover, selection and mutation respectively. The rest of parametrizing values is defined in table 37.

these performance tables. In the first example, the possession of the true Pareto front enables to use, in addition, the generational distance [16]. The presented results broadly show the superiority of GAT over GA and SPEA II. Indeed, in the first example, in spite of the fact that none of the methods was able to determine the entire Pareto set (in the decision space), they gave all Pareto optimal solutions as indicated by the null value of the generational distance (see table 19). However, only GAT was able to discover the entire front as depicted by its least Hyperarea value (see figure 4 for a graphical representation of the true Pareto

TABLE XXXVII PARAMETRIZING

2

3

4

5

6

7

8

Pop. size

GA

120 100 +20 (archive)

SPEA II GAT

40

GA

120 100 +20 (archive)

SPEA II GAT

40

GA

120 100 +20 (archive)

SPEA II GAT

40

GA

120 120 +20 (archive)

SPEA II GAT

40

GA

120 120 +20 (archive)

SPEA II GAT

80

GA

120 120 +20 (archive)

SPEA II GAT

80

GA

120 120 +20 (archive)

SPEA II GAT

40

GA

120 120 +20 (archive)

SPEA II GAT GA

9

SPEA II GAT

40 120 120+20 (archive) 80

# of generations

600 100 100 + 50 (Tabu) 1100

0,25 0,2

100 Traffic

1

Method

100 + 50 (Tabu) 600

0,15 0,1 0,05

1100

0 0

600 + 50 (Tabu) 1100

0,05

0,1

0,15

0,2

0,25

0,3

0,35

Workload

Fig. 4.

600

The true Pareto Front of example one

front). Regarding running time, still GAT displays the best result. The parametrizing used for each method (see table 37) informs that GAT was able to reach the true front with the least amount of effort concerning the number of iterations and the population size. For the second example, GA gives results that are slightly more interesting than SPEA II, as reported by the Hyperarea value. However, the solutions found by GAT are clearly better than both of its competitors (see table 21). Indeed, GA and SPEA II's solutions are all dominated by at least one solution from GAT (see table 22). The efficacy of GAT is palpable from SPEA II and GA running times that are far from GAT own time.

100 + 50 (Tabu) 1100 100 1100 + 50 (Tabu) 1100 100 600 + 50 (Tabu) 1100 100 600 + 50 (Tabu) 1100

0,6

600

0,55

100 + 50 (Tabu) 1100

0,5 Traffic

Example

GAT SPEA GA

0,45

100

0,4

100 + 50 (Tabu)

0,35 0,3 0

As depicted in the performance tables (i.e., 19, 21, …, 35), we use the running time and a metric equivalent to Hyperarea [15] in order to evaluate the three methods. Indeed, the objective functions being normalized so that their values are included in the interval [0,1], the greatest surface of the objective space cannot exceed 1. Hence, it suffices to take one minus Zitzler's Hyperarea to have a metric that decreases if the approximate front gets better. After all, we refer to the so obtained metric as Hyperarea in

Fig. 5.

0,02

0,04 Workload

0,06

0,08

Pareto Front of example 3

In the third example, GA gives the most sized front: 18 solutions in front of 13 and 2 solutions of GAT and SPEA II respectively (see tables 23,24). However, the position of GAT's front shows that each solution found by GA is dominated by at least one solution of GAT (see figure 5).

14

improving the prospecting domain of the algorithm, and last we'd like to tackle the Manufacturing Cell Formation problem by considering non-deterministic inputs.

Similar conclusions can be drawn from the results related to the rest of examples. 0,5 0,45

REFERENCES

0,4 0,35 Hyperarea

0,3

[1] M. Boulif and K. Atif (2006). A new branch-&-bound-enhanced GA for the manufacturing cell formation problem. Computers & Operations research, 33:2219-2245. [2] R. Rajagopalan and J. L. Batra (1975). Design of cellular production systems: A graph theoretic approach. International Journal Production Research, 13(6):567-579. [3] G. F. K. Purcheck (1975). A linear programming method for the combinatorial grouping of an incomplete power set. Journal of Cybernetics, 5:51–76. [4] J. King (1980). Machine-Component Grouping in Production Flow Analysis: An Approach Using a Rank Order Clustering Algorithm. International Journal of Production Research, 18(2):213-32. [5] A. Vanelli and R. Kumar (1986). A Method For Finding Minimal Bottleneck Cells For Grouping Part-Machine Families, International Journal of Production Research, 24:387-400. [6] S. Sofianopoulou (1997). Application of simulated annealing to a linear model for the formulation of machine cells in group technology. International Journal of Production Research, 35:501-11. [7] J. Balakrishnan and C. Cheng (2007). Multi-period planning and uncertainty issues in cellular manufacturing: A review and future directions. European Journal of Operational Research, 177:281-309. [8] M. Boulif and K. Atif (2008). A new Fuzzy genetic algorithm for the dynamic manufacturing cell formation problem considering Passive and active strategies, International Journal of Approximate Reasoning, 47(2):141-165. [9] T. Wu, S. Chung and C. Chang ( 2009 ). Hybrid simulated annealing algorithm with mutation operator to the cell formation problem with alternative process routings. Expert Systems with Applications , 36:3652-3661. [10]A. Souilah (1995). Simulated annealing for manufacturing systems layout design. European Journal of Operational Research, 82:592-614. [11]N. Srinivas and K. Deb (1994). Multiobjective optimization using non-dominated sorting in Genetic Algorithms. Evolutionary Computation, 2(3):221-48. 1994. [12]G. E. Goldberg (1989). Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley. [13]E. Zitzler, M. Laumanns and L. Thiele (2001). SPEA2: Improving the Strength Pareto Evolutionary Algorithm. Technical Report 103, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology, Switzerland. [14]A. Mungwatanna (2000). Design of Cellular Manufacturing Systems for Dynamic and Uncertain Production requirements with Presence of Routing Flexibility, PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg. [15]E. Zitzler and L. Thiele (1998). An evolutionary algorithm for multiobjective optimization: the Strength Pareto approach, TIK-Report 43, Swiss Federal Institute of Technology, Switzerland. [16]D. A. Van Veldehuizen (1999). Multiobjective Evolutionary Algorithms: Classifications Analyses and New Innovations, PhD Thesis AFIT/DS/ENG/99-01. Air Force Institute of Technology, Wright-Patterson, AFB.

GA

0,25

SPEAII GAT

0,2 0,15 0,1 0,05 0 ex1

Fig. 6.

ex2

ex3

ex4

ex5

ex6

ex7

ex8

ex9

Quality comparison

Overall, the hybrid method GAT gave the best performances concerning the quality (see figures 6) and the efficiency (see figures 7). GA arrives in the second position before SPEA II in the majority of examples. However, none of the methods was able to achieve its performance with acceptable stability for medium sizes: the Pareto front differs from one performance to another. For SPEA II, this lack of stability extends to the running time that considerably differs during the tests, even by maintaining the same parameter values. This stems from the diversification procedure that can have running times that depends on the solution repartition in the objective space. 300

Run time (sec.)

250 200 GA

150

SPEAII GAT

100 50 0 ex1 ex2 ex3 ex4 ex5 ex6 ex7 ex8 ex9

Fig. 7.

Efficiency comparison

V. CONCLUSION Cell formation is one of the main problems to be solved in the design of a cellular manufacturing system. In this paper, we propose a graph partitioning formulation of this problem and consider the multiple routings and multiobjective aspects simultaneously. To solve the problem, we mainly propose a combination between a Genetic Algorithm and a Tabu Search method. The results obtained show that this combination was worthy. As future recommendations the following directions seems to be promising. First, we hope to extend our model to take the multiperiodic aspect along with the multiple-routings one. Second, we would like to explore how the parallelization capabilities of the Genetic Algorithm can be used to reduce the processing time, thus

15

M4

M1 M5 M3

M2

Fig. 1.

10 M1 [0,8] M3

[0,26]

[0,8] 21

3 [0,5]

M5

M4 [0,5] 2

[0,4]

M2 10

[0,26]

Fig. 2.

[0,4]

Non parametric flow graph.

a) Routing:

b)

T1: Machine changing New cell

T2: Migration

T3: Machine swapping

T4: Cell creation

Fig. 3. Basic transformations: a) Operation assignment perturbations. b) Composition perturbations. The geometric shapes (star, triangle, square, …) represent machines. Machine with dotted lines refers to an alternate machine (a) or a removed one (b). The existence of identical shapes refers to machines of the same type.

0,25

Traffic

0,2 0,15 0,1 0,05 0 0

0,05

0,1

Fig. 4.

0,15

0,2

Workload

0,25

0,3

0,35

The true Pareto Front of example one

0,6 0,55

GAT SPEA

Traffic

0,5

GA

0,45 0,4 0,35 0,3 0

Fig. 5.

0,02

0,04 Workload

0,06

Pareto Front of example 3

0,08

0,5 0,45 0,4 0,35 Hyperarea

0,3

GA

0,25

SPEAII GAT

0,2 0,15 0,1 0,05 0 ex1

Fig. 6.

ex2

ex3

ex4

ex5

ex6

ex7

ex8

ex9

Quality comparison

300

Run time (sec.)

250 200 GA

150

SPEAII GAT

100 50 0 ex1 ex2 ex3 ex4 ex5 ex6 ex7 ex8 ex9

Fig. 7.

Efficiency comparison

TABLE I PART INFORMATION Operation sequence, time Op1 Op2 Op3 M2,2 M5,1 M3, M4,3 M1,1 M3, M4,3 M1,2 M3, M4,4 M1,1 M3, M4,3 M3, M4,1 M2,3 M3, M4,3

Part #

1 2 3 4

Volume

Op4 M5,1 M5,5 M5,2 -

2 3 1 2

TABLE II

SECOND EXAMPLE PART INFORMATION Part # 1 2 3 4 5 6

Operation sequence, time Op1 M5, M6, 1.23 M7, M9, 1.23 M2, M7, 3.85 M4, M9, 3.35 M1, M3, 4.78 M2, M3, 1.91

Op2 M5, M9, 4.13 M3, M4, 3.55 M2, M7, 2.90 M1, M9, 3.07 M1, M4, 14.10 M3, M8, 3.81

Vol.

Op3 M3, M7, 4.69

Op4 M4, M6, 3.65

-

-

0.90

-

-

1.07

M6, M7, 2.68 M2, M6, 2.14 M4, M6, 1.95

0.94

0.80 M5, M8, 3.11

0.91

-

0.84

TABLE III

EXAMPLE 2 MACHINE CAPACITIES Mach. # Capa.

1 15

2 7.5

3 15

4 20.5

5 7.5

6 7.5

7 10.5

8 6.5

9 7.5

TABLE IV THIRD EXAMPLE PART INFORMATION Part # 1 2 3

Op1 M2, M3, 3.52 M1, M9, 3.53 M10, 4.18

4

M7, 3.68

5

M10, 5.82

6

M1, 2.25

7

M7, 5.78

8 9

Vol.

Operation sequence, time

M5, M8, 3.58 M5, M9, 5.92

Op2 M9, 5.40 M7, 5.44 M3, M8, 4.64 M3, M7, 2.26 M1, 2.35 M5, M7, 4.13 M6, M10, 5.71

Op3 M7, M8, 5.50 M6, M7, 3.03 M1, M2, 4.63 M9, M10, 3.17 M3, 4.48 M4, 3.20 M9, 2.22

Op4

Op5

-

-

0.88

M5, M9, 4.85

M10, 4.18

0.89

-

-

0.81

-

0.92

M1, M5, 3.66 M7, 5.17 M5, M8, 5.39 M4, M10, 2.89

M8, 5.12 M2, 5.91

0.87 0.98 0.93

M3, 2.67

M9, M10, 3.61

-

-

0.91

M5, 5.57

M1, 3.93

-

-

0.94

M1, M5, 2.45 M4, 5.94 M4, M7, 4.33

M9, M10, 3.13 -

-

0.93

-

-

0.82

10

M2, 5.04

M6, 5.67

11

M7, 2.51

12

M9, 2.42

M8, 3.68 M3, M10, 4.52

0.80

.TABLE V

EXAMPLE 3 MACHINE CAPACITIES Mach. # Capa.

1

2

3

4

5

6

7

8

9

10

83.1

88.8

95.8

73.9

94.1

78.8

91.9

84.9

100

71.5

.TABLE VI

FOURTH EXAMPLE PART INFORMATION Vol.

Operation sequence, time

Part# 1 2 3 4 5 6 7 8 9 10

Op1

Op2

Op3

M 2 ,M 5 ,2 M 1 ,M 3 ,5 M 1 ,3 M 4 ,M 5 ,8 M 3 ,5 M 1 ,6 M 2 ,M 4 ,8 M 1 ,M 2 ,2 M 1 ,M 5 ,9 M 4 ,M 5 ,7

M 3 ,M 4 ,1 M 2 ,M 4 ,1 M 3 ,2 M 2 ,2 M 1 ,M 2 ,7 M 2 ,M 4 ,1 M 1 ,M 3 ,7 M 3 ,M 5 ,9 M 4 ,7 M 1 ,M 2 ,8

M 1 ,3 M 1 ,M 5 ,2 M 1 ,M 2 ,6 M 3,3 M 3 ,M 4 ,3 M 5 ,3 M 2 ,M 5 ,2 M 4 ,1 M 2 ,M 3 ,3 M 5 ,2

Op4 2 1 5 3 2 4 5 6 3 4

M 4 ,7

M 3 ,9

TABLE VII

EXAMPLE 4 MACHINE CAPACITIES Mach. # Capa.

1 150

2 100

3 120

4 120

5 150

TABLE VIII

FIFTH EXAMPLE PART INFORMATION Part#

Operation sequence, time

1 2 3

Op1 M1 ,M5,3 M1 ,M5 ,5 M2 ,M8 ,1

Op2 M4 ,9 M9 ,3 M3 ,1

4

M1 ,2

M9 ,4

5

M9 ,1

M8 ,7

M8 ,1

M7 ,8

6

M3 ,M7 , M9 ,5 M2 ,M5 , M6 ,7

7 8

Op3

Op4 M2 ,2 M7 ,8 M9 ,1

M3 ,2 M8 ,5 M1 ,9 M6 ,M7 ,M8 ,8 M2 ,M7 ,M9 ,6 M1 ,M4 ,M6 ,4

Vol.

Op5

M1 ,6

6 10 5 6 8 5

M6 ,1

10

M5 ,7

7

TABLE IX

EXAMPLE 5 MACHINE CAPACITIES Mach. # Capa.

1

2

3

4

5

6

7

8

9

150

100

50

70

100

100

220

130

150

TABLE X SIXTH EXAMPLE PART INFORMATION Part#

1 2 3 4 5 6 7 8 9

Operation sequence, time Op3 Op4

Op1

Op2

M 2 ,M 6 ,1 M 2 ,4 M 1 ,M 2 ,3 M 3 ,M 4 ,M 5 ,5 M 1 ,M 2 ,M 6 ,5 M 2 ,6 M 5 ,9 M 1 ,M 2 ,M 3 ,7 M 1 ,M 3 ,M 8 ,6

M 5 ,1 M 1 ,M 3 ,3 M 8 ,8 M 1 ,3 M 4 ,5 M 8 ,4 M 3 ,M 6 ,M 7 ,5 M 8 ,4 M 7 ,1

M 3 ,6 M 7 ,7 M 4 ,M 7 ,7 M 8 ,2 M 3 ,2 M 5 ,M 6 ,M 7 ,5

Vol.

Op5 Op6

M 1 ,9 M 5 ,9 M 5 ,7

M 7 ,6 M 4 ,M 8 ,2

M 4 ,M 7 ,10

10 9 5 10 10 10 2 6 6

TABLE XI

EXAMPLE 6 MACHINE CAPACITIES Mach. # Capa.

1 250

2 200

3 220

4 180

5 220

6 100

7 250

8 150

TABLE XII SEVENTH EXAMPLE PART INFORMATION Part#

1 2 3 4 5 6 7 8 9 10 11 12

Operation sequence, time Op2 Op3 Op4 M2 ,8 M11 ,5 M8 ,9 M6 ,6 M4 ,9 M2 ,M7 ,9 M8 ,9 M7 ,2 M3 ,M4 ,8 M8 ,6 M5 ,4 M1 ,M4 ,4 M10 ,6 M8 ,8 M6 ,M7 ,1 M2 ,4 M11 ,5 M6 ,5 M4 ,6 M11 ,9 M9 ,5 M4 ,6 M2 ,9 M10 ,6 M6 ,9 M8 ,3

Op1 M1 ,M4 ,M6 ,5 M3 ,M9 ,3 M5 ,M11 ,1 M5 ,M10 ,7 M1 ,M5 ,7 M1 ,M3 ,M10 ,2 M8 ,M10 ,M11 ,7 M2 ,M8 ,M10 ,7 M3 ,M4 ,M6 ,6 M2 ,M5 ,M9 ,7 M7 ,M9 ,M11 ,5 M1 ,M9 ,M11 ,3

Vol.

Op5 M7 ,7

Op6 M10 ,2

10 1 8 8 1 8 9 1 6 9 9 7

TABLE XIII

EXAMPLE 7 MACHINE CAPACITIES Mach. # Capa.

1 100

2 200

3 80

4 200

5 100

6 180

7 100

8 250

9 100

10 200

11 200

TABLE XIV

EIGHTH EXAMPLE PART INFORMATION Part#

1 2 3 4 5 6 7 8 9 10 11 12 13

Op1 M4 ,M11 , 0.5 M4 ,M11 ,0.55 M5 ,M8 ,0.45 M2 ,M6 ,M10 ,0.23 M3 ,M7 ,M 9 ,0.37 M3 ,M11 ,0.4 M1 ,M9 ,0.3 M6 ,M8 ,0.5 M4 ,M8 ,M10 ,0.4 M5 ,M6 ,M9 ,0.55 M2 ,M6 ,M11 ,0.6 M3 ,M5 ,M7 ,0.54 M3 ,M5 ,M8 ,0.46

Op2 M2 ,0.68 M8 ,0.7 M6 ,0.25 M9 ,0.19 M6 ,0.23 M9 ,0.39 M11 ,0.18 M4 ,0.67 M9 ,0.53 M4 ,0.74 M11 ,0.61 M10 ,0.58 M7 ,0.8

Operation sequence, time Op3 Op4O M10 ,0.71 M9 ,0.42 M6 ,0.42 M4 ,0.5 M4 ,0.42 M2 ,0.44 M6 ,0.76 M4 ,0.4 M4 ,0.21 M3 ,0.76 M6 ,M7 ,0.3 M7 ,M8 ,0.6 M1 ,M2 ,0.85

Vol. p5 M6 ,0.72 M1 ,M2 ,0.4 M7 ,M10 ,0.68

Op6 M1 ,M2 ,0.60

1 9 10 5 10 7 1 5 6 7 1 1 8

-

TABLE XV

EXAMPLE 8 MACHINE CAPACITIES Mach. # Capa.

1 9

2 10

3 15

4 25

5 13

6 20

7 15

8 15

9 10

10 10

11 7

TABLE XVI NINTH EXAMPLE PART INFORMATION Part# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Operation sequence, time Op1 M 8,0.89 M 7,0.21 M 9,0.79 M 2,0.14 M 3,M 9,0.66 M 8,M 14,0.73 M 6,M 9,0.97 M 1,M 2,M 13,0.96 M 2,0.64 M 11,0.11 M 6,M 12,0.49 M 11,M 14,0.72 M 1,0.66 M 10,0.4 M 11,0.48 M 10,0.77 M 13,0.6 M 11,M 12,M 14,0.64 M 7,M 10,M 12,0.85 M 2,M 5,M 11,0.41

Op2 M 11,M 13,0.53 M 3,0.47 M 5,0.61 M 5,0.14 M 5,0.85 M 2,0.22 M 3,0.83 M 4,0.34 M 13,0.91 M 14,0.18 M 14,0.9 M 2,0.34 M 2,M 11,0.75 M 12,0.17 M 2,M 7,M 13,0.99 M 6,M 9,M 12,0.5 M 2,M 7,M 9,0.75 M 8,0.36 M 7,0.39 M 5,0.88

Op3 M 7,0.59 M 5,0.71 M 5,M 8,M 13,0.63 M 1,0.68 M 7,M 8,0.32 M 13,0.93 M 1,M 5,0.88 M 14,0.55 M 1,M 6,M 14,0.36 M 2,M 4,M 7,0.37 M 8,M 11,0.56 M 10,M 12,0.53 M 10,M 13,0.18 M 1,M 3,M 8,0.33

Vol.

Op4 M 9,0.9 M 1,0.36 M 4,0.95 M 3,M 6,M 9,0.74 M 4,0.3 M 1,M 11,0.21

Op5 M 6,M 8,0.58 M 4,M 12,M 14,0.92 M 6,0.44

TABLE XVII EXAMPLE 9 MACHINE CAPACITIES Mach. # Capa. Mach. # Capa.

1 20 8 10

2 20 9 20

3 21 10 15

4 16 11 20

5 32 12 15

6 15 13 10

7 15 14 15

TABLE XVIII PARETO FRONT OF EXAMPLE ONE Solution # 0 1 2 3 4 5 6 7

Code 12211-110100 12121-001011 12122-100011 12212-011100 11122-100100 11212-011011 11122-101100 11212-010011

Traffic 0.136364 0.136364 0.181818 0.181818 0.272727 0.272727 0.318182 0.318182

Workload 0.214079 0.214079 0.050123 0.050123 0.010916 0.010916 0.000223 0.000223

Op6 M 4,0.85

1 8 5 10 6 1 7 1 1 8 1 9 9 5 1 7 7 8 7 8

TABLE XIX

PERFORMANCES FOR EXAMPLE ONE Number of sol. Run time (sec.) Generational Dist. Hyperarea

GA

SPEA II

GAT

6 73.26 0 0.151299

3 62.83 0 0.209370

7 2.31 0 0.151299

TABLE XX

APPROX. PARETO FRONTS FOR EXAMPLE I Method

GA

SPEA II

GAT

So. # 0 1 2 3 4 5 0 1 2 0 1 2 3 4 5 6

Code 12121-001011 12211-110100 12122-100011 12212-011100 11122-100100 11212-010011 12212-011000 12213-011100 12213-011000 12211-110100 12122-100011 12212-011100 11212-011011 11122-100100 11212-010011 11122-101100

Traffic 0.136364 0.136364 0.181818 0.181818 0.272727 0.318182 0.181818 0.454545 0.500000 0.136364 0.181818 0.181818 0.272727 0.272727 0.318182 0.318182

Workload 0.214079 0.214079 0.050123 0.050123 0.010916 0.000223 0.098240 0.006906 0.000891 0.214079 0.050123 0.050123 0.010916 0.010916 0.000223 0.000223

TABLE XXI

PERFORMANCES FOR EXAMPLE II Number of sol. Running time Hyperarea

GA 9 118.19 0.249859

SPEA II 4 62.87 0.251654

GAT 14 4.72 0.000463

TABLE XXII APPROX. PARETO FRONTS FOR EXAMPLE II Meth.

GA

SPEA II

GAT

Sol. # 0 1 2 3 4 5 6 7 8 0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13

Code

112233312-000011110111001100 112233312-000011111111001100 112233312-000011111111001011 112233312-000111111111001011 112321323-000011111111001011 112322331-000111111111001111 112233312-000111111111001111 112213323-000111111111001111 112312332-000011111111001100 122344324-001001001100001100 122344324-011001001100001100 122344324-001010111100001100 122344324-011011011100001100 112322313-100101111110001101 112322313-100101110110001101 112322313-100111110110000100 112322313-000101110110000100 112322313-000101111110000100 112322313-000111111110000100 112322313-000111110110010110 112322313-100001111011001101 112322313-100011110011001101 112322313-000011110011001101 112322313-100011111011001101 112322313-000001110011001101 112322313-100001110011001101 122322313-100001110100001101

Traffic

Workload

0.245344 0.245344 0.322894 0.409617 0.410451 0.481794 0.487168 0.503382 0.577689 0.248587 0.248587 0.332345 0.431761 0.000000 0.000000 0.161864 0.161864 0.161864 0.161864 0.239414 0.319652 0.319652 0.319652 0.319652 0.319652 0.319652 0.329658

0.042290 0.042290 0.010002 0.003842 0.003827 0.001308 0.000401 0.000300 0.000131 0.006198 0.006198 0.005547 0.003513 0.002371 0.002371 0.000498 0.000498 0.000498 0.000498 0.000424 0.000028 0.000028 0.000028 0.000028 0.000028 0.000028 0.000009

TABLE XXIII

PERFORMANCES OF EXAMPLE III Number of sol. Running time Hyper a r ea

GA

SPEA II

GAT

18 302.75 0.344757

2 889.96 0.397586

13 36.45 0.335223

TABLE XXIV

APPROX. PARETO FRONTS FOR EXAMPLE III Meth.

GA

GAT

SPEAII

Sol.#

Traf.

Workl.

0

12132213210101100011011001101011

Code

0.339574

0.068945

1

12123312310101100011011001101011

0.365068

0.062091

2

12123312310101100011011001111011

0.365068

0.062091

3

12132213210101100011011001101010

0.367268

0.051286

4

12132213210101100011011001111000

0.367268

0.051286

5

12132213210101000001011001101000

0.367268

0.051286

6

12132213210101100011011001101000

0.367268

0.051286

7

12123312310101100011011001111000

0.392762

0.049232

8

12123312310101100011011001101000

0.392762

0.049232

9

12121233210011000011001001101011

0.420320

0.012974

10

12121233210011010011001001101011

0.420320

0.012974

11

12121233210011010111001001101000

0.447879

0.008285

12

12121233210011010111001001101011

0.447879

0.006373

13

12121233210111100011001001101011

0.450554

0.005650

14

12321233210011010111001001100010

0.475438

0.000186

15

12321233210011010011001000001100

0.479229

0.000793

16

12121233210011010111111001001011

0.511900

0.000200

17

12122133210011000011011001101011

0.546873

0.000168

0

12321232311011001101101101000101

0.335071

0.003872

1

12321232310011001101101101001101

0.364864

0.000915

2

12321232310011001101101101000101

0.364864

0.000915

3

12321232310011001111101101001101

0.364864

0.000915

4

12321232310011011101101101000101

0.364864

0.000915

5

12321232310011011111101101001101

0.364864

0.000915

6

12321232310011001101101101001100

0.392558

0.000060

7

12321232310011011101101101001100

0.392558

0.000060

8

12321232311011001111001011100100

0.425094

0.000032

9

12321232311011001111101011100100

0.425094

0.000032

10

12321232311011001111001011101100

0.425094

0.000032

11

12321232310010010101000001100000

0.574601

0.000009

12

12321232310010010101100001100000

0.574601

0.000009

0 1

12321132331110011100101110001101 12321132441110011100101110001101

0.388699 0.569760

0.047328 0.000740

TABLE XXV

PERFORMANCES OF EXAMPLE IV # of sol. Running time Hyperarea

GA 37 80.02 0.308952

SPEA II 16 54.14 0.302975

GAT 5 3.47 0.275420

TABLE XXVI

APPROX. PARETO FRONTS FOR EXAMPLE IV Meth.

GA

GAT

SPEA II

Sol.# 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 0 1 2 3 4

Code 121121101001010110110111 121120111001010110110111 121121111001010110110111 121120101001010110110111 121120101001000110110111 121120011001010110110111 121120111001010111110111 121121111001000110110111 121120101001000111110111 121120011001010111110111 121120111001000111110111 121121111001010111110111 121120111001000110110111 121121101001010111110111 121121010001010110110111 121120110001010110110111 121121010001010111110111 121120010001010110110111 121120101001010111110011 121121101001010111110011 121121111001010111110011 121120101001110111110111 121120101001110110110111 121120111001010011110111 121120101001000011110111 121121111001010011110111 121120111001000011110111 121120011001010011110111 121121101001010011110111 121120101001010011110111 121121101001000011110111 121121111001000011110111 121120100001010011110111 121120010001010011110111 121120110001010011110111 122121101000100010110100 121221111001000011110100 121120100101010111000111 121220010100001111111011 122120110001101101110011 122120110001101101110111 122120110100101101110011

0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 303797 0. 329114 0. 329114 0. 329114 0. 329114 0. 341772 0. 341772 0. 341772 0. 354430 0. 354430 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 367089 0. 392405 0. 392405 0. 392405 0. 405063 0. 556962 0. 240506 0. 367089 0. 392405 0. 392405 0. 417722

0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 076988 0. 074935 0. 074935 0. 074935 0. 074935 0. 059511 0. 059511 0. 059511 0. 050772 0. 050772 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 016510 0. 015567 0. 015567 0. 015567 0. 001529 0. 000031 0. 262251 0. 061342 0. 006412 0. 006412 0. 000003

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

121120000001010111000011 121120101001000100001011 121120111001000110001011 121121001001000100001011 121120111001010110110011 121121001001000100110011 121121000001000100110011 121120101001000100111011 121120101001010000110011 121121000001000000110011 121121001001010010111011 121220100001001100001011 121220110000000100001011 121220111001000100001011 121220111001001100001011 121221110001000110001011

0. 291139 0. 303797 0. 303797 0. 303797 0. 341772 0. 341772 0. 367089 0. 379747 0. 405063 0. 430380 0. 443038 0. 455696 0. 455696 0. 455696 0. 455696 0. 455696

0. 236229 0. 151476 0. 151476 0. 151476 0. 059511 0. 059511 0. 057707 0. 020560 0. 009020 0. 008326 0. 000031 0. 000003 0. 000003 0. 000003 0. 000003 0. 000003

Traf.

Wd.

TABLE XXVII

PERFORMANCES OF EXAMPLE V # of sol. Running time Hyperarea

GA 6 103.27 0.394259

SPEA II 3 55.76 0.437561

GAT 33 31.59 0.310330

TABLE XXVIII APPROX. PARETO FRONTS FOR EXAMPLE V Meth.

GA

GAT

SPEAII

Sol.# 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 0 1 2

Code 11122233301012012 11122233301010022 11122233301010012 12113322301010022 11213322301001020 12133452401010022 11123223301022211 11122323310002121 11122323310022121 11122323310002021 11122323310022021 11122233310002102 11122233310001202 11122233310001102 11122233310002202 11122233310002101 11122233310001101 11122233310001201 11122233310002201 11122323310022120 11122323310002120 11122323310020121 11122323310000121 11122323310002101 11122323310022101 11122323310020021 11122323310000021 11122323310022001 11122323310002001 11122323300020021 11122323300000021 11122323300002001 11122323300022001 11123223301010010 11123223301000010 11123223301000002 11123223301010002 11123223300102002 11123223300112002 12112233301012210 12112233301002210 12112234401012210

Traf. 0.387097 0.451613 0.451613 0.459677 0.508065 0.814516 0.306452 0.338710 0.338710 0.379032 0.379032 0.387097 0.387097 0.387097 0.387097 0.387097 0.387097 0.387097 0.387097 0.395161 0.395161 0.403226 0.403226 0.419355 0.419355 0.443548 0.443548 0.459677 0.459677 0.491935 0.491935 0.508065 0.508065 0.516129 0.516129 0.596774 0.596774 0.653226 0.653226 0.427419 0.475806 0.596774

Workl. 0.087216 0.043643 0.043643 0.016452 0.000879 0.000634 0.073578 0.023527 0.023527 0.015429 0.015429 0.014074 0.014074 0.014074 0.014074 0.014074 0.014074 0.014074 0.014074 0.009744 0.009744 0.004911 0.004911 0.004562 0.004562 0.000905 0.000905 0.000726 0.000726 0.000215 0.000215 0.000189 0.000189 0.000027 0.000027 0.000023 0.000023 0.000010 0.000010 0.086346 0.045228 0.001221

TABLE XXIX PERFORMANCES OF EXAMPLE VI # of sol. cpu Hyperarea

GA 14 104.35

SPEA II 4 62.74

GAT 36 22.61

0.348601

0.409300

0.345499

TABLE XXX

APPROX. PARETO FRONTS FOR EXAMPLE VI Meth.

GA

GAT

SPEAII

Sol.# 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 0 1 2 3

Code

12132312010110020110 12133312111010020110 12133312111010020111 12133312110110020110 12133312110110020111 12133312111110020110 12133312111110020111 12133312101110020111 12332312010110022120 12332312011110022220 12132332011110022220 12332213110100222121 12322313001001001001 11213232111010020111 12132312011110020111 12132312011110020110 12333312110010022010 12333312110010222010 12333312110010122010 12333312110010122212 12333212010010011010 12333212010010211010 12333212010010111010 12333212010010111210 12333212010010011210 12333212010010121210 12333212110010121210 12333212010010021210 12333212010010011002 12333212110010011002 12333212011011011010 12333212111011011010 12333212011011211010 12333212111011211010 12333212010001211111

12333212110001011111 12333212110001211111 12333212010001011111 12333212010001111111 12333212010001221111 12333212110001111111 12323312110010020210 12323312110110020210 12323312110110021210 12323312110110221210 12323312110010021210 12323312110110220210 12323312110010221210 12323312110010220210 12223313110111010222 12133312111010021212 12133312111110021212 12133312101010021212 12133312101110021212

Traf.

Workl.

0.342857 0.360000 0.360000 0.411429 0.411429 0.411429 0.411429 0.468571 0.491429 0.542857 0.577143 0.645714 0.777143 0.891429 0.342857 0.342857 0.360000 0.360000 0.360000 0.405714 0.417143 0.417143 0.417143 0.428571 0.428571 0.428571 0.428571 0.428571 0.485714 0.485714 0.497143 0.497143 0.497143 0.497143 0.520000

0.096235 0.029296 0.029296 0.026336 0.026336 0.026336 0.026336 0.013589 0.007367 0.003950 0.001551 0.000975 0.000139 0.000071 0.096235 0.096235 0.019245 0.019245 0.019245 0.009117 0.000171 0.000171 0.000171 0.000057 0.000057 0.000057 0.000057 0.000057 0.000048 0.000048 0.000043 0.000043 0.000043 0.000043 0.000037

0.520000 0.520000 0.520000 0.520000 0.520000 0.520000 0.537143 0.537143 0.537143 0.537143 0.537143 0.537143 0.537143 0.537143 0.554286 0.405714 0.457143 0.462857 0.514286

0.000037 0.000037 0.000037 0.000037 0.000037 0.000037 0.000009 0.000009 0.000009 0.000009 0.000009 0.000009 0.000009 0.000009 0.000002 0.016244 0.015175 0.008157 0.004621

TABLE XXXI

PERFORMANCES OF EXAMPLE VII

# of sol. Running time Hyperarea

GA 2 125.46 0.336192

SPEA II 4 56.34 0.271642

GAT 9 9.07 0.271336

TABLE XXXII

APPROX. PARETO FRONTS FOR EXAMPLE VII Meth. GA

GAT

SPEAII

Sol.#

0 1 0 1 2 3 4 5 6 7 8 0 1 2 3

Code 1 1 2 2 3 2 3 3 23 1 0 0 1 1 0 1 1 01 0 2 0 1 1 1 0 1 2 3 4 3 4 3 1 43 2 0 0 1 1 0 1 1 01 0 2 0 1 1 1 0 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 0 0 11 0 1 0 1 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 0 0 11 0 1 2 1 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 1 0 0 0 1 00 0 1 2 1 2 1 1 1 1 2 1 2 3 2 2 33 1 0 1 0 0 0 0 0 01 0 1 2 2 2 1 1 1 1 2 1 2 3 2 2 33 1 2 1 0 0 0 0 0 00 0 1 0 1 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 1 1 11 0 1 0 0 2 1 1 1 1 2 1 2 3 2 2 33 1 0 1 0 0 0 1 1 11 0 1 0 0 2 1 1 1 1 2 1 2 3 2 2 33 1 1 1 0 0 0 1 0 11 0 1 2 0 2 1 1 1 1 2 1 2 3 2 2 33 1 0 1 0 0 0 1 1 11 0 1 2 0 2 1 1 1 2 2 2 1 3 1 1 33 2 1 1 0 0 1 0 0 11 1 1 1 2 2 1 0 1 2 2 2 1 3 1 1 33 2 1 1 1 0 1 0 0 11 1 1 1 2 2 1 0 1 2 2 2 1 3 1 1 33 2 1 1 0 0 1 0 0 11 2 1 1 2 2 1 0 1 2 2 2 1 3 1 1 33 2 2 1 0 0 1 0 0 11 1 1 1 2 2 1 0

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

Traf. 3 33 3 3 3 5 59 3 2 2 2 71 1 8 6 2 76 8 3 6 2 88 1 3 6 3 10 7 3 4 3 33 3 3 3 3 50 2 8 2 3 50 2 8 2 3 55 9 3 2 3 55 9 3 2 2 71 1 8 6 2 76 8 3 6 3 16 3 8 4 3 27 6 8 4

TABLE XXXIII

PERFORMANCES OF EXAMPLE VIII # of sol. cpu Hyperarea

GA 10 129.57 0.332264

SPEA II 2 132.38 0.416271

GAT 42 20.64 0.329853

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

Workl. 009829 001447 004322 003331 003184 000386 000185 000037 000037 000006 000006 002437 002370 001007 000501

TABLE XXXIV

APPROX. PARETO FRONTS FOR EXAMPLE VIII Meth.

GA

GAT

SPEAII

Sol.#

0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 0 1

Code 121121233230110010001110011221 121121233230010012001111011221 121121233230110010001111011221 121121233230110010001110010221 121121233230110012001110010221 121121233230110012001111111221 123121233230110010001110011221 123121233230010012001110010221 123121233230110010001110010221 123121233230110012001111110221 121131332231010012011100121212 121131332231110112011100122222 121131332231110012011100122222 121131332231110112011100122221 121131332231110012211100122222 121131332231110012211100122212 121131332231110112211100122222 121131332231010012211011122201 121131332231110012211101122222 121131332231011010111110122222 121131332231011010111100122222 121131332231011012111100122222 121131332231011012111110122222 121131332231111112211110120212 121131332231111112211110120222 121131332231011012211011122211 112232331131110012211001122212 112132323130011101011101022201 112132323130111101011101022201 112132323130111101011100022201 112132323130111101011100002201 112132323130011101011100020201 112132323130011101011101020201 112132323130111101011101020201 112132323130111101011100020201 112132323130111101011101122201 112132323130111101011100122201 112132323130111101011100120201 112132323130011101011100022201 112132323230011100011100100001 112132323230110100011100100001 112132323230011100011100002001 112132323230011100011101000001 112132323230011100011101002001 112132323230010100011100002001 112132323230011100011101100001 112132323230011100011101102001 112132323230011100011100102001 112132323230010100011101102001 112132323230011100011100000001 112132323230110100011101102001 112132323230010100011100102001 123333212210011010001111001122 123133212210011010001111001122

Traf. 0.325714 0.348571 0.354286 0.365714 0.365714 0.382857 0.400000 0.434286 0.440000 0.497143 0.325714 0.371429 0.371429 0.371429 0.428571 0.428571 0.428571 0.451429 0.457143 0.474286 0.474286 0.474286 0.474286 0.480000 0.480000 0.502857 0.508571 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.542857 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.548571 0.394286 0.628571

Workl. 0.094107 0.077839 0.074103 0.071735 0.071735 0.052036 0.016175 0.006650 0.005384 0.000169 0.046334 0.024235 0.024235 0.024235 0.012294 0.012294 0.012294 0.009096 0.005963 0.005131 0.005131 0.005131 0.005131 0.004071 0.004071 0.002375 0.001870 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000068 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.075851 0.011346

TABLE XXXV

PERFORMANCES OF EXAMPLE IX Number of sol. Running time Hyperarea

GA 11 185.12 0.331120

SPEA II 5 64.57 0.343895

GAT 57 24.64 0.298304

TABLE XXXVI

APPROX. PARETO FRONTS FOR EXAMPLE IX Meth.

GA

GAT

SPEAII

Sol.# 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 0 1 2 3 4

Code 12131312123233011100011112110110112100101 12131312123233011100001012110110112100101 12131312123233011120001012110110112100101 12131312123233011120001012110110112100100 12131312123233011100001012110110112100100 11232321213133011100011112110110112100101 11232321213133011100001012110110112100101 11232321213133011100011012110110112100101 12113123223323000120111011121010101102101 12113123223323000120111011121010110100101 12113123223323000120111011121010110100100 12111331332222000100111110201101110100101 12111331332222000100111102201101110100101 12111331332222102100111110201101110200101 12111331332222102100111110101101110200101 12111331332222000120111112201101111100101 12111331332222000120111112201101110100101 12131331332222000100011012201111012210001 12131331332222000100011012200101112210001 12131331332222000100011012200111112210001 12131331332222000100011012200101111210001 12131331332222000100011012200111111210001 12131331332222000100011012200111012210001 12131331332222000200011012201111012210001 12131331332222100110011102201101112200201 12131331332222100110011101201101112000201 12131331332222000110011101201101112000201 12131331332222000110011102201101112000201 12131331332222000110011102201101112200201 12131331332222000110011101201101112200201 12131331332222000210011112201101110210011 12131331332222000220011101201101112200201 12131331332222000210011112201101110200201 12131331332222100210011112201101110210011 12131331332222000210011101201101112200201 12131331332222102011011102201101112200201 12131331332222002011011102201101112200201 12131331332222002011011102101101112200201 12131331332222002011011102101101112000201 12131331332222100210011112201101010200200 12131331332222100210011111201101010200200 12131331332222100210011111201101010200202 12131331332222000210011111201101111210210 12131331332222000210011112201101111210210 12131331332222000210011111201101111010212 12131331332222100210011112201101110200200 12131331332222100210011111201101110200200 12131331332222100210011111201101110200202 12131331332222000210011101201101111010212 12131331332222000210011111201101110200010 12131331332222100210011112201101110200210 12131331332222000210011112201101110210210 12131331332222000210011111201101110210210 12131331332222000210011112201101110200100 12131331332222100210011112201101110210210 12131331332222000220011012201101010200200 12131331332222000210011111201101012200200 12131331332222100210011111201101012200200 12131331332222000210011111201101110210010 12131331332222000210011112201101110200210 12131331332222000210011112201101110200200 12131331332222000210011111201101111010210 12131331332222000210011112201101010200200 12131331332222000210011112201101012200200 12131331332222102221011111120111111210101 12131331332222002011011111120101111210102 12131331332222102021011111120111111210102 12131331332222102221111112100001110212100 12131314132422110100011111201111112000100 12311134312422110100011111201111112000100 12341431342122110100011111201111112000100 12341435342522110100011111201111112000100 12341435342522 110100011111101111112000100

Traf. 0.324561 0.350877 0.350877 0.385965 0.385965 0.412281 0.438596 0.442982 0.627193 0.627193 0.662281 0.293860 0.298246 0.324561 0.328947 0.342105 0.342105 0.350877 0.355263 0.355263 0.355263 0.355263 0.355263 0.372807 0.394737 0.394737 0.394737 0.394737 0.394737 0.394737 0.416667 0.416667 0.416667 0.416667 0.416667 0.421053 0.421053 0.425439 0.425439 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.451754 0.486842 0.500000 0.500000 0.521930 0.315789 0.464912 0.478070 0.500000 0.504386

Workl. 0.069115 0.043785 0.043785 0.032855 0.032855 0.023710 0.008749 0.008144 0.001717 0.001717 0.000345 0.078672 0.075150 0.057872 0.057109 0.043011 0.043011 0.015681 0.015661 0.015661 0.015661 0.015661 0.015661 0.010275 0.005131 0.005131 0.005131 0.005131 0.005131 0.005131 0.001783 0.001783 0.001783 0.001783 0.001783 0.001482 0.001482 0.001356 0.001356 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000159 0.000051 0.000036 0.000036 0.000001 0.112403 0.052780 0.025785 0.020634 0.020164

TABLE XXXVII PARAMETRIZING Example

1

2

3

4

5

6

7

8

9

Method

Pop. size

GA

120 100 +20 (archive)

SPEAII GAT

40

GA

120 100 +20 (archive)

SPEAII GAT

40

GA

120 100 +20 (archive)

SPEAII GAT

40

GA

120 120 +20 (archive)

SPEAII GAT

40

GA

120 120 +20 (archive)

SPEAII GAT

80

GA

120 120 +20 (archive)

SPEAII GAT

80

GA

120 120 +20 (archive)

SPEAII GAT

40

GA

120 120 +20 (archive)

SPEAII GAT

40

GA

120 120+20 (archive)

SPEAII GAT

80

# of generations 600

100 100 + 50 (Tabu) 1100 100 100 + 50 (Tabu) 600 1100 600 + 50 (Tabu) 1100 600 100 + 50 (Tabu) 1100 100 1100 + 50 (Tabu) 1100 100 600 + 50 (Tabu) 1100 100 600 + 50 (Tabu) 1100 600 100 + 50 (Tabu) 1100 100 100 + 50 (Tabu)