A simulated annealing approach to a bi-criteria sequencing problem in a two-stage supply chain

Computers & Industrial Engineering 50 (2006) 105–119 www.elsevier.com/locate/dsw

A simulated annealing approach to a bi-criteria sequencing problem in a two-stage supply chain S. Afshin Mansouri 1 Industrial Engineering Department, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran Received 19 December 2003; received in revised form 5 August 2005; accepted 20 January 2006 Available online 29 March 2006

Abstract In this paper, a multi-objective simulated annealing (MOSA) solution approach is proposed to a bi-criteria sequencing problem to coordinate required set-ups between two successive stages of a supply chain in a flow shop pattern. Each production batch has two distinct attributes and a set-up occurs in each stage when the corresponding attribute of the two successive batches are different. There are two objectives including: minimizing total set-ups and minimizing the maximum number of set-ups between the two stages that are both NP-hard problems. The MOSA approach starts with an initial set of locally non-dominated solutions generated by an initializing heuristic. The set is then iteratively updated through the annealing process in search for true Paretooptimal frontier until a stopping criterion is met. Performance of the proposed MOSA was evaluated using true Pareto-optimal solutions of small problems found via total enumeration. It was also compared against a lower bound in large problems. Comparative experiments show that the MOSA is robust in finding true Pareto-optimal solutions in small problems. It was also shown that MOSA is very well-performing in large problems and that it outperforms an existing multi-objective genetic algorithm (MOGA) in terms of quality of solutions. q 2006 Elsevier Ltd. All rights reserved. Keywords: Bi-criteria sequencing; Multi-objective simulated annealing; Supply chain; Set-up coordination

1. Introduction Supply chain management (SCM) has attracted ever increasing attention over the last two decades as a consequence of the pressure on organizations in competitive marketplace to create and deliver value to customers. A supply chain is an integrated system which synchronizes a series of inter-related business processes in order to: (1) acquire raw materials and parts; (2) transform these raw materials and parts into finished products; (3) add value to these products; (4) distribute and promote these products to either retailers or customers; (5) facilitate information exchange among various business entities, e.g. suppliers, manufacturers, distributors, third-party logistics providers, and retailers (Min & Zhou, 2002). For review of the research issues in this area readers may refer to Ho, Au, and Newton (2002), Min and Zhou (2002). Thomas and Griffin (1996) discuss the need for research dealing with supply chain problems on the operational level using deterministic rather than stochastic models. Supply chain scheduling models represent a recently memerging area for this type of research. Vergara, Khouja, and Mickalewicz (2002) 1

Present address: Department of Industrial Engineering, Faculty of Engineering, University of Tehran, P.O. Box 11365-4563 Tehran, Iran. Fax: +98 21 88013102 E-mail address: [email protected].

0360-8352/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2006.01.002

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emphasize on coordination between different members of supply chains and develop an evolutionary algorithm (EA) for optimal synchronization of the chains using an economic delivery and scheduling model. Moon, Kim, and Hur (2002) propose an integrated process planning and scheduling for a multi-plant supply chain to minimize total tardiness and develop a genetic algorithm (GA) for the model. Lee, Jeong, and Moon (2002) address simultaneous machine selection, sequencing and outsourcing decisions in an integrated model to minimize makespan. They also develop a GA-based solution approach to solve the model. Thoney, Hodjson, King, Taner, and Wilson (2002) consider scheduling of multi-factory supply chains including inter-factory transportation and propose a heuristic solution approach for the solution of the problem. Li, Ganesan, and Sivakumar (2005) study the transportation and assembly scheduling problems in a synchronized model to achieve accurate delivery with minimum cost in an electronic supply chain. Selvarajah and Steiner (in press) study the batch sequencing problem in a two-level supply chain from the supplier’s point of view and present a polynomial algorithm for minimization of the sum of inventory holding and batch delivery costs paid by the supplier. Torabi, Fatemi Ghomi, and Karimi (in press) study both lot sizing and scheduling in a supply chain with a cost minimization objective. They model the problem as a mixed integer non-linear (MINLP) model and develop a hybrid GA for large problems. In this paper, we focus on the sequencing of production batches between two successive stages of a supply chain. In each stage, items are grouped according to different attributes. For instance consider two departments in a kitchen furniture production plant that operates in an integrated supply chain. In the cutting department, parts are grouped according to their shape and material while, in the subsequent painting department parts may be grouped according to their color. In a same way, there may be a third department, say assembly, wherein parts are grouped according to the requirements of the final products. In each department, a set-up occurs when the attribute of a new part changes. It is assumed that the two stages follow the same common sequence. Each stage desires to minimize its own total set-up costs, but the objectives of the two stages may be conflicting. It is also assumed that set-up costs are equal and are sequence independent. It means that say in a painting department set-up change from yellow to green has the same cost as that of set-up changeover from green to yellow. Hence, instead of minimizing set-up costs, simply minimizing the number of set-ups can be considered. The following objectives are considered in this research: † Minimizing the total number of set-ups in the two stages. † Minimizing the maximum number of set-ups in either stage. The first objective corresponds to the maximization of overall utility and the second one is to balance the set-ups between the two stages. Agnetis, Detti, Meloni, and Pacciarelli (2001) show that any of the two above problems is NP-hard, hence attaining optimal solution for large problems is prohibitive. They present a graph theory based heuristic for the first objective, i.e. minimizing total number of set-ups. In this paper, we consider both problems simultaneously as a multi-objective optimization problem (MOP) and propose a MOSA solution approach in search for Pareto-optimal solutions. Quality of the solutions of the MOSA is compared against the results of total enumeration in small problems and against a lower bound for larger problems. Performance of the MOSA is also compared with an existing MOGA approach for the same problem. The remaining parts of the paper are organized as follows. Section 2 gives a brief explanation of multi-objective optimization. Formulation of the MOP for coordination of set-ups between the two stages is given in Section 3. Comprehensive explanation of the proposed MOSA approach is given in Section 4. The experimental design is explained in Section 5 followed by discussion of results in Section 6. Finally in Section 7, concluding remarks are outlined and future research directions highlighted.

2. Multi-objective optimization problem A MOP can be defined as determining a vector of design variables within a feasible region to minimize a vector of objective functions that usually conflict with each other. Such a problem takes the form Minimizeff1 ðXÞ;f2 ðXÞ;.;fm ðXÞg

subject to gðXÞ% 0

(1)

where X is vector of decision variables; fi(X) is the ith objective function; and g(X) is constraint vector. A decision vector X is said to dominate a decision vector Y (also written as X_Y) iff

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Fig. 1. Pareto-optimal or non-dominated solutions.

fi ðXÞ% fi ðYÞ

for all i 2f1;2;.;mg;

(2)

for at least one i 2f1;2;.;mg

(3)

and: fi ðXÞ! fi ðYÞ

All decision vectors that are not dominated by any other decision vector are called non-dominated or Paretooptimal. These are solutions for which no objective can be improved without detracting from at least one other objective. Fig. 1 illustrates Pareto-optimal or non-dominated solutions for a two-objective minimization problem. There are various solution approaches for solving the MOP. Among the most widely adopted techniques are: sequential optimization, 3-constraint method, weighting method, goal programming, goal attainment, distance based method and direction based method. For a comprehensive study of these approaches, readers may refer to Szidarovsky, Gershon, and Dukstein (1986).

3. The MOP for set-up coordination Consider B as a set of batches to be produced. The batches must be processed by two stages of a supply chain, called D1 and D2, in the same order. Each batch is characterized by two attributes, say a1 and a2. Let A1 and A2 denote the sets of all possible attributes in stages D1 and D2, respectively. The attributes in stage D1 are denoted as a1,i, iZ1,.jA1j and the attributes in stage D2 are represented by a2,j, jZ1,.jA2j. Each batch is therefore defined by a pair of (a1,i,a2,j). If batch (a1,i,a2,j) is processed immediately after batch (a1,h,a2,k), a set-up is paid in stage D1 if hsi, and a set-up is paid in stage D2 if ksj. For a given sequence S, the number of set-ups incurred by stages D1 and D2 are denoted by N1(S) and N2(S), respectively. Let xi,h be equal to 1 if ish and zero otherwise, and let a1(Sq) denote the first attribute of the qth batch in the sequence S, and let a2(Sq) denote its second attribute. Hence: N1 ðSÞ Z 1 C

BjK1 jX qZ1

N2 ðSÞ Z 1 C

BjK1 jX qZ1

xa1ðSq Þ ;a1ðS

(4)

xa2ðSq Þ ;a2ðS

(5)

qC1 Þ

qC1 Þ

It is assumed here that an initial set-up is required in each stage at the beginning of the sequence, hence set-up changeovers in each stage is added by 1 to obtain set-ups in that particular stage. That’s the only difference between our formulation and that of Agnetis et al. (2001). The MOP addressed in this paper can be formulated as follows

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Minimizeff1 ðSÞ;f2 ðSÞg

(6)

where S is the sequence at which batches are processed, and: f1 ðSÞ ZN1 ðSÞ C N2 ðSÞ

(7)

f2 ðSÞ Z MaxðN1 ðSÞ;N2 ðSÞÞ

(8)

The problems (7) and (8) are referred to as MinSum and MinMax, respectively, by Agnetis et al. (2001). These two objectives are not equivalent. For more explanations and examples concerning non-equivalence of these problems, see Agnetis et al. (2001) and Mansouri (2005). 3.1. A lower bound An ideal sequence S* for the MOP problem (6) is a solution with the objective vector: ð f1
; f2
Þ, which is described below (
 ; if jBjR ðjA1 j C jA2 jÞ jBj; MaxfdjBj=2e;jA1 j;jA2 jg

ðf1 ;f2 Þ Z (9) ; Otherwise ððjA1 j C jA2 jÞ; MaxfjA1 j;jA2 jgÞ where d$e denotes ‘rounded up to the nearest integer’. Eq. (9) introduces lower bounds on the total number of set-ups and maximum number of set-ups in either stage. To elaborate, consider a situation where number of batches is more than total attributes. In this case, total required set-ups cannot be less than the total number of batches as at least one set-up is required for each batch. In this case, the lowest possible number of maximum set-ups in either stage is the largest of the three values as: half of the total number of set-ups (rounded up to the smallest integer in case jBj is an odd number), the number of attributes in the first stage, and the number of attributes in the second stage. On the other hand, when the number of batches is smaller than total attributes, the lower bound for the total number of set-ups is equal to the total number of attributes due to the fact that one set-up is required to switch from one attribute to another one. In this case, the maximum number of attributes in either department determines the lower bound for the maximum number of set-ups in the two stages. 4. The multi-objective simulated annealing approach Simulated annealing is an approach that can provide near-optimal solutions to combinatorial optimization problems. Kirkpatrick, Gelatt, and Vecchi (1983) and Eglese (1990) do provide fundamental descriptions of simulated annealing in addition to informative examples. Simulated annealing has been applied to a vast number of single objective optimization problems over the past years. It has also been applied as a tool for multi-objective optimization problems in some applications, for instance by Ruiz-Torres, Enscore, and Barton (1997), Karasakal and Ko¨ksalan (2000) and McMullen (2001). Readers for review of approaches to multi-objective optimization by means of simulated annealing may refer to Czyzak and Jaszkiewicz (1998) and Nam and Park (2000). In this section, the algorithmic steps of the MOSA heuristic in search for Pareto-optimal solutions to the multiobjective problem (6) are explained. 4.1. Steps of the MOSA approach The MOSA heuristic is guided by the temperature level, T, and the cooling rate, CR up to the freezing temperature, TF. Fig. 2 shows the pseudocode of the MOSA approach. 4.2. Initializing heuristic For the generation of an initial set of solution we adopt the initializing heuristic proposed in Mansouri (2005). Steps of this heuristic are given in Fig. 3.

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Fig. 2. Pseudocode of the proposed MOSA.

The resultant {Elite Set} will be used then as the Initial Non-dominated Frontier for the annealing process. The underlying idea of the initializing heuristic procedure is to try to minimize the number of dual set-ups in switching from one batch to another one while keeping a balanced level of set-ups paid in each stage. 4.3. Generate neighboring solutions A neighboring solution to a current solution is generated via either inversion or mutation operators, which are described below. 4.3.1. Inversion Inversion is an operator that first chooses two random cut points in a solution. The elements between the cut points are then reversed. An example of the inversion operator is presented below: Before inversion : b7 b1 jb2 b4 b9 b6 jb3 b5 b8 After inversion :

b7 b1 jb6 b9 b4 b2 jb3 b5 b8

Inversion is implemented on a number of individuals according to Inversion Rate. Once an individual is selected for inversion, the operator may be applied for several iterations whose number is determined by Inversion Numbers.

Fig. 3. Pseudocode of the initializing heuristic.

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4.3.2. Mutation Mutation for this research is the simple swapping of two unique elements in the sequence of interest. Consider the following sequence: Before mutation : b1 b3 b9 b4 b2 b8 b5 b7 b6 The two bold elements are randomly selected unique elements that are targeted for swapping. After swapping, or mutation, the sequence is as follows: After mutation : b1 b3 b5 b4 b2 b8 b9 b7 b6 :

4.4. The Metropolis criterion The Metropolis criterion that states the probability of accepting a dominated solution (P(A)) is as follows   KDE PðAÞ Z exp (10) Kb T where 



di;j DE Z Max Min j i fi;j Kb Z

KDE1 ðT1 lnðPðAÞ1 ÞÞ

 (11)

(12)

in which DE is the minimum distance between a test solution and its associated dominating solutions in nondominated frontier, di,j denotes the distance between the ith objective value of test solution and that of the dominating solution j, and fi,j represents the ith objective value of the dominating solution j. The value Kb is referred to as the Boltzman constant and reflects the probability of accepting a dominated solution (P(A)1) by the distance DE1 from the furthest dominating solution in the current non-dominated frontier at the initial temperature T1. This value gives the user some control over the probability of dominated solutions being accepted. The probability of accepting a dominated solution is a function of both temperature of the system (T) and the distance between the dominated solution and the current non-dominated frontier (DE). The distance DE in the MOSA is equivalent to the distance between the worse solution and the current solution in a single objective SA. In fact, current nondominated frontier in the MOSA is used in a similar way as the current solution is used in a single objective SA as a reference point. As a result, the reference point for the calculation of DE changes in the MOSA in a similar way as in a single objective SA. As the temperature decreases, the probability of accepting worse moves decreases. Obviously at TZ0, no worse move is accepted. It should be explained here that(P(A)1), DE1 and T1 are among the parameters of the MOSA and their values need to be determined in the parameter setting stage. This will be discussed later in Section 5. To elaborate on the distance metric used in the proposed MOSA, i.e. minimum distance between a given dominated solution x and the non-dominated frontier, consider an instance in a bi-objective minimization problem depicted in Fig. 4. Solutions a, b, c and d constitute the non-dominated frontier and x represents a dominated solution, which is dominated by b and c. The objective values for x regarding the first and the second objectives are f1,x and f2,x, respectively. The objective values for b and c are also f1,b, f2,b, f1,c and f2,c. Consider the pair of x and b where x is dominated by b. The minimum level of improvement that pushes away x from being dominated by b is the minimum relative distance between x’s objective values and those of b, which is calculated below:    f1;x Kf1;b f Kf Min ; 2;x 2;b (13) f1;b f2;b

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Fig. 4. Distance between a dominated solution and non-dominated frontier.

Therefore, the minimum improvement that revives x from being dominated is considered as the distance between x and the non-dominated frontier, which can be calculated as below:        f1;x Kf1;b f2;x Kf2;b f1;x Kf1;c f2;x Kf2;c DE Z Max Min ; ; Min ; (14) f1;b f2;b f1;c f2;c

5. Design of experiment In order to evaluate efficiency of the proposed MOSA approach, an experiment was designed. For smaller problems, the solutions found by the MOSA were compared to the true Pareto-optimal solutions obtained via complete enumeration. For the larger test problems, the solutions found via the MOSA were compared to the lower bounds introduced in Eq. (9). 5.1. Search parameter values In order to find an efficient set of parameters for the MOSA approach, extensive experiments were conducted on several test problems. There are six parameters whose values need to be determined for the MOSA. These include: DE1, P(A)1, CR, T1, TF and Iterations. Several values were tested for each parameter and the more efficient values selected empirically. Table 1 details these values. Concerning the initializing heuristic, the number of idle iterations Table 1 Search parameters for the MOSA approach Parameter

Value

DE1 P(A)1 Cooling rate (CR) Initial temperature (T1) Freezing temperature (TF) Iterations

0.1 0.1 0.999999 10 1 2

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for the initialization procedure, i.e. the number of successive non-improving iterations was set to 1000 as the stopping criterion. This procedure terminates once no improvement was observed for this number of successive iterations. 5.2. Test problems For the experiments, data sets with different number of production batches were generated at random. In addition to the number of batches, two other features were also considered in generating the test problems. These features, as mentioned in Agnetis et al. (2001) include balance ness and density of the incidence matrix of the problems. A balanced problem is the one wherein the number of attributes is equal in both departments. Density reflects the ratio of total batches in a given incidence matrix (or jBj/(jA1j!jA2j)). Ten small problems were generated with number of batches between 8 and 12. For each number of batches, one balanced and one un-balanced problems were generated. Forty large problems were also generated in two categories as balanced and un-balanced. The maximum number of attributes and the maximum number of batches were set at 200 and 4000, respectively, which provide a solution space of up to (4000)! possible sequences. For each level of total attributes, four levels for density were considered as 0.1, 0.2, 0.3 and 0.4. 5.3. Comparison method To evaluate quality (Q) of a solution to a given problem with respect to the appropriate ideal solution proposed by the lower bounds of Eq. (9), the following metric was employed:  Q Z 1KMax

  f1 Kf1
f2 Kf2
; f1
f2


(15)

It should be noted here that the ideal solution of a given problem is just an ideal destination and hence, a quality index 1.000 may be un-attainable even with total enumeration (if total enumeration were possible). Regarding small problems, true Pareto-optimal solutions found via total enumeration were employed as reference instead of ideal solutions. A second performance measure for small problems was diversity (D), which reflects ratio of disjoint Paretooptimal solutions found by the MOSA compared to the true number of disjoint Pareto-optimal solutions found via total enumeration. A perfect solution regarding the diversity measure (DZ1.0) means that all Pareto-optimal solutions have been identified. The MOSA was executed 1.0 min on each problem for 20 times. The average (AVRG) and standard deviation (STDV) of results of these runs were reported. Quality of the MOSA on large problems was also compared with an existing MOGA approach proposed by Mansouri (2005). The pseudocode and the values selected for the parameters of the MOGA as proposed by Mansouri (2005) is given in Appendix A. 5.4. Hardware and software specifications The solution algorithms were coded in CCC and executed on a Pentium IV Intelw Centrinoe processor at 1.7 GHz under Windowsw XP with 512 MB RAM. 5.5. Research questions There are two questions need to be answered as follows: (i) Is there any significant difference between performance of the MOSA and MOGA approaches? If so, which approach provides more desirable results? (ii) Do problem-specific features (balance ness and density) have any significant effect on performance of the MOSA approach?

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Table 2 A real case with 301 production batches Attribute 1 (Shape)

Attribute 2 (Color) 1

2

3

4

5

6

7

8

9

10

11

12

13

14

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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 0 1 1 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1

1 1 1 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 1 1

1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1

1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1

0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 1 1

1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0

1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0

1 1 1 1 0 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 1 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0

Note that each ‘1’ in the incidence matrix represents a production batch.

6. Experimental results 6.1. Experiment on a real problem In order to examine performance of the MOSA in a real problem, an industrial case was selected from Agnetis et al. (2001). It concerns the production chain of a kitchen furniture manufacturer. There are 32 cutting classes and 14 color classes. The production of the plant follows an assemble-to-order (ATO) strategy. Every week a new production plan is computed on the basis of the actual customer orders. Table 2 shows a real weekly production plan consisting 301 production batches, which means that there are (301)! possible sequences for the problem. As jBjZ301O(jA1jCjA2j)Z(32C14)Z46, the ideal solution for the example problem contains: f1 Z ð301Þ and f2
Z Maxðfd301=2eg;32;14ÞZ 151. Fig. 5 shows the results of sample runs of the MOSA and MOGA heuristics. It shows that both heuristics attain the same level of quality after 1.0 min of execution. However, it can be observed that MOSA converges faster than MOGA. Quality of final solutions found by both heuristics (with f1Z302 and f2Z151) can be calculated using the Eq. (15) as follow:  Q Z 1KMax

  302K301 151K151 ; Z 1KMaxð0:003; 0:000Þ Z 0:997 301 151

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Fig. 5. Sample runs of the MOSA and MOGA for the real case problem.

6.2. Experiments on small problems Each problem was solved 20 times, each time for 1.0 min. The results were compared against true Pareto-optimal solutions found via total enumeration. The average and standard deviation of these runs are reported in Table 3. As it can be observed in Table 3, performance of the MOSA is perfect in small problems regarding both quality and diversity measures in all problem sets. The average quality and diversity measures are equal to 1.000. Moreover, the standard deviation of the test runs regarding both measures is 0.000, which implies that the MOSA is robust in finding true Pareto-optimal solutions in small problems in a short amount of CPU time. 6.3. Experiments on large problems Each problem was solved 20 times by both MOSA and MOGA solution approaches, each time for an equal amount of CPU time, i.e. 1.0 min for problems having up to 1000 batches and 2.0 min for larger problems. The average and standard deviation of these runs on the balanced and un-balanced problems are reported in Tables 4 and 5, respectively. As it can be observed in Tables 4 and 5, the average quality indices for the whole experiments on balanced and un-balanced problems are 0.945 and 0.952, respectively, which show that performance of the MOSA is very satisfactory for large problems. The same indices for the MOGA for the same set of problems are 0.940 and 0.943.

Table 3 Experiments on small problems Un-balanced problems jBj

8 9 10 11 12

jA1j

4 4 4 3 6

jA2j

5 6 7 9 4

Balanced problems Quality

jBj

Diversity

AVRG

(STDV)

AVRG

(STDV)

1.000 1.000 1.000 1.000 1.000

(0.000) (0.000) (0.000) (0.000) (0.000)

1.000 1.000 1.000 1.000 1.000

(0.000) (0.000) (0.000) (0.000) (0.000)

AVRG: average; STDV: standard deviation.

8 9 10 11 12

jA1j

4 5 5 6 6

jA2j

4 5 5 6 6

Quality

Diversity

AVRG

(STDV)

AVRG

(STDV)

1.000 1.000 1.000 1.000 1.000

(0.000) (0.000) (0.000) (0.000) (0.000)

1.000 1.000 1.000 1.000 1.000

(0.000) (0.000) (0.000) (0.000) (0.000)

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Table 4 Comparisons on large, balanced problems Characteristics of problems

Quality of 20 test runs

jBj

jA1j

jA2j

Density

MOSA AVRG

(STDV)

AVRG

(STDV)

40 80 120 160 90 180 270 360 250 500 750 1000 640 1280 1920 2560 1000 2000 3000 4000 Average

20 20 20 20 30 30 30 30 50 50 50 50 80 80 80 80 100 100 100 100

20 20 20 20 30 30 30 30 50 50 50 50 80 80 80 80 100 100 100 100

0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40

0.803 0.973 0.982 0.986 0.958 0.986 0.988 0.989 0.972 0.975 0.972 0.970 0.945 0.942 0.921 0.919 0.866 0.901 0.924 0.930 0.945

(0.011) (0.006) (0.002) (0.004) (0.018) (0.004) (0.006) (0.004) (0.009) (0.006) (0.005) (0.012) (0.009) (0.024) (0.047) (0.035) (0.152) (0.027) (0.021) (0.033)

0.803 0.975 0.981 0.986 0.949 0.981 0.986 0.987 0.967 0.966 0.960 0.929 0.935 0.931 0.914 0.899 0.895 0.907 0.915 0.925 0.940

(0.011) (0.000) (0.005) (0.005) (0.022) (0.008) (0.005) (0.005) (0.009) (0.006) (0.005) (0.056) (0.006) (0.020) (0.033) (0.053) (0.010) (0.023) (0.022) (0.033)

MOGA

AVRG: average; STDV: standard deviation.

Table 5 Comparisons on large, un-balanced problems Characteristics of problems jBj

22 42 48 80 90 160 263 312 233 495 630 968 562 1109 1361 2000 932 1500 2880 3990 Average

jA1j

17 19 7 10 27 20 35 41 37 45 70 41 72 84 42 50 74 150 80 105

Quality of 20 test runs jA2j

13 11 23 20 33 40 25 19 63 55 30 59 78 66 108 100 126 50 120 95

AVRG: average; STDV: standard deviation.

Density

0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40 0.10 0.20 0.30 0.40

MOSA

MOGA

AVRG

(STDV)

AVRG

(STDV)

0.967 0.949 0.958 0.975 0.967 0.984 0.990 0.991 0.976 0.978 0.964 0.967 0.956 0.942 0.921 0.919 0.908 0.915 0.914 0.901 0.952

(0.000) (0.008) (0.000) (0.000) (0.000) (0.006) (0.004) (0.004) (0.007) (0.006) (0.049) (0.004) (0.006) (0.048) (0.056) (0.038) (0.041) (0.045) (0.040) (0.103)

0.967 0.952 0.957 0.975 0.964 0.985 0.988 0.990 0.976 0.972 0.969 0.940 0.944 0.919 0.887 0.905 0.886 0.872 0.904 0.915 0.943

(0.000) (0.000) (0.004) (0.000) (0.013) (0.006) (0.006) (0.004) (0.007) (0.007) (0.008) (0.044) (0.008) (0.088) (0.096) (0.058) (0.031) (0.056) (0.047) (0.044)

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Table 6 t-Test: paired two sample for mean quality of MOSA and MOGA on large problems (aZ0.01)

Mean quality Variance Observations Pearson correlation Hypothesized mean difference Degrees of freedom t Stat t Critical one-tail

MOSA

MOGA

0.949 0.002 40

0.941 0.002 40 0.945 0 39 3.360 2.426

Table 7 Analysis of variance for effect of balance ness and density on quality of MOSA on large problems (aZ0.01) Source of variation

Sum of squares

Degrees of freedom

Mean square

F (Test)

F-critical

Balance ness Density Interaction Within combinations Total

0.000 0.004 0.005 0.050 0.059

1 3 3 32 39

0.000 0.001 0.002 0.002

0.314 0.806 1.097

7.499 4.459 4.459

6.4. Addressing research question i To investigate whether superiority of MOSA over MOGA is significant, a t-test was made at significance level aZ0.01 using the data given in Tables 4 and 5. The result of this test is presented in Table 6. It shows that the mean quality index for the MOSA in all experiments on large problems is equal to 0.949, while the same index for the MOGA is 0.941. As it can be seen, the t Stat (3.360) is greater than t Critical one-tail (2.426) which implies that superiority of MOSA over MOGA with respect to the quality measure is significant. 6.5. Addressing research question ii To investigate if the problem-specific features have significant effect on performance of the MOSA, a two-way analysis of variance (ANOVA) was implemented at aZ0.01 level of significance. Two main features, i.e. balance ness and density of the incidence matrix were examined in this analysis. Table 7 shows the result of this experiment. The F (Test) values, as it can be observed in Table 7, for balance ness, density and interaction are below the critical 1% values. Hence, it is concluded that the data do not give sufficient evidence of the presence of interaction, density or balance ness effect. In other words, performance of the MOSA is not influenced by balance ness nor density of the problems.

7. Conclusion In this paper, a MOSA approach was proposed to a bi-criteria sequencing problem in a two-stage supply chain to coordinate required set-ups between the two successive stages. Production batches are processed in the supply chain in a flow shop pattern. Each of the production batches has two attributes and one set-up is required in the upstream (downstream) stage once the first (second) attribute of a batch is different from that of the immediately previous one. The two criteria considered include minimizing total number of set-ups and minimizing the maximum number of setups in each stage. These problems are NP-hard so exact optimization methods are not applicable for large problems. Quality of the proposed MOSA was compared against Pareto-optimal solutions found via total enumeration in small data sets and against a lower bound in larger problems. Comparative results show that the MOSA is capable of finding perfect solutions in small problems. Regarding larger problems, it was observed that the MOSA is capable of finding very close solutions to the lower bound. Efficiency of the proposed MOSA was also compared with a MOGA solution

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approach in large problems. It was observed that the proposed MOSA performs better than the MOGA in terms of the quality of the solutions. The number of Pareto or near Pareto optimal solutions found in the examined problem sets by means of the MOSA or its comparator MOGA, was either one or two. This observation could be an indication of the Pareto-optimal set in the bi-criteria problem addressed in this paper being too small comprising of very close solutions in the objective space. This could be a reason for superiority of the proposed MOSA over the MOGA which is a population based metaheuristic with inherent capabilities for tackling the MOPs with diverse Pareto-optimal frontiers. Extension of the proposed MOSA approach to the cases where more than two attributes exist for each production batch is a potential direction for more research. Development of more efficient meta-heuristics for the problem addressed in this paper is also a challenging area for further research. The problem addressed in this paper can be regarded as a special case of bi-criteria two-machine flow shop sequencing problem. Application of the proposed MOSA approach to other bi-criteria sequencing problems, and in particular two-machine flow shop case, is another avenue for further research. Acknowledgements The author sincerely thanks the two anonymous referees for their valuable comments on the early version of this paper. Appendix A: Pseudocode and parameter values of the MOGA approach let time counter tZ0 /* initialization */ do execute initializing procedure while (termination criterion is met) Consider resultant set of solutions as {Elite Set} /* evolution */ initialize search parameters do perform non-dominated sorting and niching on current generation /* selection */ for jZ0 to Population Size1 calculate expected number (solutionj) select [expected number] copies from solutionj jZjC1 next j do select solutions from {Elite Set} randomly while (selected solutions!Population Size) shuffle selected solutions /* crossover */ for jZ1 to Population Size If (rand!Crossover Rate2) then perform crossover between solutions j and jC1 else reproduce solutions j and jC1 for next generation end If jZjC2 next j /* inversion */ for jZ1 to Population Size

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if (rand!Inversion Rate3) then for iZ1 to Inversion Numbers4 perform inversion on parent solution (solutioni) compare offspring with its Parent if (offspring is dominated) then if (rand!P(A)) replace parent with offspring end if else replace parent with offspring end if iZiC1 next i end if jZjC1 next j /* mutation */ for jZ1 to Population Size for iZ1 to Mutation Numbers5 perform mutation on parent solution (solutioni) compare offspring with its Parent if (offspring is dominated) then if (rand!P(A)) replace parent with offspring end if else replace parent with offspring end if iZiC1 next i jZjC1 next j /* update {Elite Set} */ compare non-dominated frontier of current generation with {Elite Set} remove dominated solutions from {Elite Set} add new non-dominated solutions to {Elite Set} while (t!T) report resultant {Elite Set} Parameter values used for the MOGA: 1Population SizeZ50, 2Crossover RateZ0.05, 3Inversion RateZ0.80, 4 Inversion NumbersZ5, 5Mutation NumbersZ1.

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