A multi-objective optimization algorithm for solving the supplier selection problem with assembly sequence planning and assembly line balancing

Accepted Manuscript A multi-objective optimization algorithm for solving the supplier selection problem with assembly sequence planning and assembly line balancing Z.H. Che PII: DOI: Reference:

S0360-8352(16)30514-9 http://dx.doi.org/10.1016/j.cie.2016.12.036 CAIE 4588

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

25 February 2016 26 August 2016 27 December 2016

Please cite this article as: Che, Z.H., A multi-objective optimization algorithm for solving the supplier selection problem with assembly sequence planning and assembly line balancing, Computers & Industrial Engineering (2016), doi: http://dx.doi.org/10.1016/j.cie.2016.12.036

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A multi-objective optimization algorithm for solving the supplier selection problem with assembly sequence planning and assembly line balancing Z.H. Che Department of Industrial Engineering & Management, National Taipei University of Technology, 1, Sec. 3, Chung-Hsiao E. Rd., Taipei 106, Taiwan 

Corresponding author. Tel.: +886-2-2771-2171 ext. 2346; fax: +886-2-7317168. E-mail address: [email protected].

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A multi-objective optimization algorithm for solving the supplier selection problem with assembly sequence planning and assembly line balancing

Abstract Supplier selection is a key strategic decision-making activity for building a competitive advantage at an assembly plant. Quality suppliers can understand a firm’s operational goals and provide high-quality components. Simultaneously, achieving efficient production requires a production plan. Therefore, a superior competitive strategy should consider the suppliers’ availability and the plant’s ability. We apply production line planning to address specific problems associated with supplier selection by constructing a multi-objective optimization model. The proposed model considers both assembly sequence planning and assembly line balancing. In addition, a novel hybrid algorithm is proposed to solve the model. The algorithm combines the guided search algorithm and multi-objective particle swarm optimization (MPSO) algorithm, as well as a metic multi-objective particle swarm optimization (MMPSO) algorithm. A real case of a computer assembly plant is used to verify the performance of the MMPSO. The analysis results show that the proposed algorithm not only identifies more non-dominated solutions, but also obtains higher Pareto-optimal solution ratios. Keywords: Supplier selection; Assembly sequence planning; Assembly line balancing; Multi-objective particle swarm optimization

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1. Introduction In dynamic markets for products with continually shortening life cycles, a business must boost its competitive edge to keep up with changes in the market. In addition to building its core competence, a company should focus on external resource integration. However, various production conditions exist under the supply chain management framework. To increase its competitive advantage, a company should strengthen its productivity and boost its efficiency; moreover, it should integrate its up- and down-stream customers and parts suppliers in the production process. Supplier selection is a key factor in decision-making processes for production and logistics management. Selecting the appropriate suppliers can effectively reduce procurement costs and enhance businesses’ competitive advantages [17, 18]. Sha and Che [44] indicated that a company should select suitable suppliers to decrease risk exposures in the industry by building its core competence. When a buyer establishes a long-term partnership with a supplier, the supply chain can create a robust barrier against potential competitors [10]. To satisfy customer demand and to lower internal cost and risk, companies should select appropriate suppliers to manufacture products that are more competitive and distribute these products to customers, thus, meeting their various demands [6]. Zhang and Chen [47] indicated that buyers select specific suppliers and allocate the ordering quantities among the suppliers to minimize the total cost (including selecting, purchasing, holding, and shortage costs). In addition, Kilic [24] stated that the supplier selection process is a critical stage in supply chain management. Supplier selection is the first stage of supply chain management, and it is a critical process that affects the subsequent stages. In other words, managing suppliers and creating a supply chain has a permanent effect on the competitive advantage of a business.

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In summary, supplier selection is a critical issue for building an effective supply chain system [7]. In the traditional supplier selection problem, the internal operational effectiveness of an assembly plant is not the criterion for supplier evaluation. The assembly time, thus, between parts is not included in the mechanism of supplier selection. In practice, the parts provided by different suppliers have different sizes/tolerances. Unless the high dimensional accuracy of the parts purchased is an exact requirement of the assembly plant, the tools and equipment must be adjusted and different operation times should be applied for parts assembly in the plant. Therefore, the assembly time of parts is affected by the suppliers. Assembly sequence planning (ASP) and assembly line balancing (ALB) are two key considerations for production planning. Boothroyd et al. [5] indicated that ASP can effectively connect production and design information. By considering the demands of the assembly site when designing a product, the duration of actual assembly and the overall production cost can be decreased substantially. Regarding modern flow-line production systems, Nearchou [29] asserted that decisions about solving ALB problems affect the ultimate cost and quality of a product, as well as how the business responds to the market. Recently, numerous scholars, such as Hamta et al. [21], Tuncel and Topaloglu [41], Tseng et al. [38], Marian et al. [26], and Zhu et al. [49], have examined ASP and ALB. However, different assembly times are caused by the differences in part sizes (from different suppliers). This issue has not been considered in traditional ASP and ALB planning. Therefore, this study considers how ASP and ALB problems affect supplier selection problems in the supply chain to enhance the quality of assembly lines and improve the time-to-market for specific products.

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Because ASP and ALB are combinatorial optimization problems [26, 38], relevant studies have employed heuristic algorithms. Senin et al. [34] and Marian et al. [26] applied genetic algorithms (GAs) to address ASP problems. Kim et al. [25] and Purnomo [31] applied GAs to address ALB problems such as large scopes or various types. In addition, Chen et al. [8] proposed a GA-based hybrid approach to solve an integrated ASP and ALB problem, and Tseng and Tang [40] integrated ASP with ALB an the connector concept by applying GAs. However, integrated ASP–ALB–supplier selection problems have not been discussed in extant literature; moreover, the particle swarm optimization (PSO) algorithm has been effectively applied to a combination and optimization of a sequence or assignment, and is superior to GA in regards to speed and quality [9]. Therefore, this study proposes a modified PSO algorithm to solve this integrated problem. Compared with the original PSO, we introduce a guided search algorithm to conform to the assembly constraints during the solution process, develop a calculation mechanism for the initial cycle time and work assignment, and employ the elitism-preservation mechanism to reserve better individuals for the next evolution. This study addresses the following three issues: (a) Proposing a multi-objective optimization model that integrates ASP, ALB, and supplier selection. The considered objectives include minimum assembly line cycle duration, maximum assembly line efficiency, and minimum product delivery time. To the best of our knowledge, no study has proposed a model that integrates ASP, ALB, and supplier selection; (b) Combining the guided search algorithm and the multi-objective particle swarm algorithm with elitism to develop a MMPSO, and subsequently applying the MMPSO to solve the proposed multi-objective optimization model; (c) Comparing the solving performance of the MMPSO,

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non-dominated sorting GA II (NSGA-II), and guided NSGA-II (gNSGA-II) to verify that the MMPSO has excellent capabilities for solving the problems defined in this study. The criteria for the performance comparison are the mean number of non-dominated solutions and the mean ratio of Pareto-optimal solutions. This study is organized as follows: Section 2 reviews extant literature on supplier selection, production planning, multi-objective GAs, and multi-objective PSOs, and Section 3 presents the problem assumptions and multi-objective optimization model. Section 4 outlines the procedure of the proposed MMPSO algorithm, and a case study and the analytical results of the algorithms are discussed in Section 5. Finally, Section 6 offers the conclusions and recommendations for future studies.

2. Literature Review 2.1 Assembly sequence planning and assembly line balancing ASP is designed to determine a certain assembly priority based on the planner’s specific assembly experience in consideration of various other constraint factors such as geometric properties and assembly duration. In previous studies [3, 19] on solutions to ASP problems, scholars have typically employed the liaison graph proposed by De Fazio and Whitney [13] to describe the assembly model of a product, and utilized the graph-based approach and exhaustive searches to identify the optimum feature model. Because the information level is too low and difficult to extend, this study proceeds with ASP using the connector concept rather than applying the liaison graph to describe specific products. Based on the fastener method, Tseng and Li [39] developed a novel means of generating assembly sequences by applying the

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connector concept, which substantially decreases the search complexity. Specific reviews of the connector concept for ASP are available in the literature, such as those by Yin et al. [46] and Tseng et al. [38]. Assembly

line

balancing

problems

(ALBPs)

occur

during

the

decision-making process of assigning work to workstations. ALB minimizes the anticipated cycle time and enables all workstation process times to be as consistent as possible. Tasan and Tunali [36] summarized previous literature and classified the following types of ALBPs objectives: (a) Type-F establishes whether a feasible line balance exists under a given number of workstations and cycle time; (b) Type-1 minimizes the number of stations for a given cycle time; (c) Type-2 minimizes the cycle time for a given number of stations; and (d) Type-E maximizes the line efficiency by simultaneously minimizing the cycle time and the number of stations [33]. Type-3 and Type-4 correspond to the maximization of workload smoothness and work relatedness, and Type-5 corresponds to multiple objectives with Type-3 and Type-4 [25]. This study addresses Type-2 ALBPs. A Type-2 ALBP is applied to decrease the cycle time for a certain number of stations to an approximate value within a specific range [33]. Therefore, the examination of cycle time plays a critical role in the quality of the final solution. 2.2 Applications of GA for ASP and ALB Recently, because ASP and ALB problems are associated with combination and optimization, the majority of scholars have employed algorithms to solve these problems [32, 37, 45]. Among these algorithms, GAs are advantageous because of their speed and reliability in addressing large-scale problems and applying

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combination planning. Tseng et al. [38] and Marian et al. [26] employed GAs to obtain the optimal assembly sequences of specific products. When applying GAs to Type-2 ALBPs, Valente et al. [42] employed GAs to address Type-2 ALBPs for an automobile assembly factory. Their findings showed that the proposed GA could reduce the existing assembly time by 28.5%. Nearchou [29] applied a differential evolution algorithm to solve Type-2 ALBPs targeted at the minimization of cycle time. 2.3 GA for multi-objective optimization Optimization is always a key tool for engineering. A set of Pareto-optimal solutions has been derived for multi-objective optimization, the solutions of which constitute a curve, which is called the Pareto front. Pareto-optimal solutions comprise non-dominated solutions. To solve multi-objective optimization problems, Srinivas and Deb [35] proposed a non-dominated sorting genetic algorithm (NSGA). Initially, a NSGA sorts populations based on the characteristics of non-domination, and subsequently assigns higher fitness values to better non-dominated solutions, thereby forcing solutions toward the Pareto front. Deb et al. [14] proposed an NSGA-II, which is an algorithm that retains elitism

and

specific

variation.

Moreover,

they

incorporated

a

novel

crowded-comparison operator that allows Pareto-optimal solutions to be distributed evenly on the Pareto front. Research on this topic has been conducted by Panda and Yegireddy [30] and Ahmed et al. [1]. 2.4 PSO for multi-objective optimization Kennedy [23] developed the first PSO algorithm to describe the social behaviour of bird flocking or fish schooling. The PSO algorithm is initialized with

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a population of random solutions, and it identifies optima by updating generations. The formulas for updating the velocity vidj 1 and position xidj 1 of each particle are expressed as vidj 1  vidj  1  rand()  ( p Best  xidj )  2  rand()  ( g Best  xidj )

(1)

xidj 1  xidj  vidj 1 ,

(2)

where i is the particle index, d is the dimension index, j is the number of iterations,

1 , 2 are the acceleration constants (also known as cognitive parameters and social parameters), pBest is the best position of particle i, gBest is the best position among all particles in the swarm, and rand() is the independent random variable evenly distributed within [0,1]. Previous studies have shown that PSO is an effective, sound, and simple optimization algorithm [43]. Hu and Eberhart [22] proposed a dynamic neighbourhood PSO in a two-objective optimization problem. Coello and Lechunga [12] proposed a multi-objective PSO by dividing the objective space into small grids. Subsequently, the ideal solution can be selected using roulette wheel selection based on grid segmentation, and the optimum solution in the storage space becomes the reference for guidance in the next generation Fieldsend and Singh [16] calculated the dominated trees and identified new particles by dividing the space based on the dominated trees. Because the non-dominated solution closest to the storage space is the global solution for the search in the next generation, the next position of the particles can be calculated. Mostaghim and Teich [28] calculated a sigma (σ) value and selected a solution closest to the σ of the storage space as the guidance for selecting the global optimum. Based on these multi-objective PSO algorithms, the majority of studies have focused on selecting the global optimum to improve the algorithm performance and the variation 9

among solutions.

3. Problem Formulation 3.1 Simple example for the proposed problem Fig. 1 shows a simple example to demonstrate the function of the proposed optimization model in solving the supplier selection problem by considering ASP and ALB. In the figure, the product comprises seven parts (A, B, C, D, E, F, and G). Each part can be provided by three potential suppliers, and the entire connector relationship of the parts can also be constructed. Fig. 1 shows numerous combinations of connector relationships, part suppliers, and assembly sequences, although not every combination is a quality plan that meets the demands of customers. The proposed multi-objective optimization model is applied to this combination planning problem to identify the optimal quality plan with a superior assembly sequence, suitable supply chain partner combination, and proper assembly assignment. [Insert Fig. 1 about here] 3.2 Assumptions and notations The assumptions considered for supplier evaluation that integrate ASP and ALB in this study are detailed as follows: 1) Each part or component is purchased from only one supplier, and no supply shortages occur during the purchasing process. 2) The assembly production does not commence until all parts and components have been delivered to the central assembly plant. 3) The number of workstations and the sequence of the workstations are known, and any workstation can address any assigned assembly activity. 4) The ASP is based on the connector concept. Information on the parts for each

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connector and the precedence relationship are already known. 5) No assembly type, direction, or tool is considered for the assembly activities. The following notations are used for the multi-objective optimization model: i, j

Part index, i =1, 2, 3,…, I , j =1, 2, 3,…, I

I

Total number of parts

s, t

Supplier index for part i , s =1,2,3,…, S i , t =1,2,3,…, S j

Si , S j

Total number of suppliers for parts i and j

m ,n

Connector index, m =1, 2, 3,…, M , n =1, 2, 3,…, M

M

Total number of connectors

p

Assembly sequence index of a connector, p =1, 2, 3,…, P

P

Total number of assembly sequences for a connector, M ＝ P

q

Assembly sequence index of a workstation, q =1, 2, 3,…, Q

Q

Total number of assembly sequences for a workstation

c

Designated cycle time

D

Demand volume

Gm

Precedence connector sets of connector m

MTis

Manufacturing time for the production of part i by supplier s

DTis

Transportation time for the distribution of part i from supplier s Respective part assembly times for parts i and j by suppliers s and

ATmisjt

t in connector m OTmn

Processing time from connector m to connector n

QLT

Lead time for suppliers to deliver parts to the central assembly plant

QATq

Assembly time for all connected parts at workstation q

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QOTq

Processing time of the connectors at workstation q

Tq

Total processing time at workstation q

Tsum

Sum of processing times for finishing a product

T

Time from order acceptance by the assembly plant to delivery

cmax

Maximum workstation processing time in the assembly line

EOL

Assembly efficiency of the assembly line

Apmn

1　From connector m to connector n in assembly sequence p  0 　Otherwise

Bqm

1 Connector m is assigned to workstation q  0 Otherwise

C qpm

1　 Connector m is assigned to workstation q in assembly sequence p  0 　 Otherwise

Dmij

1　In connector m, part i and part j are assembled  0 　Otherwise

H mis

1　Supplier s of part i in connector m  0 　Otherwise

3.3 Multi-objective optimization model for supplier selection with ASP and ALB This section presents the proposed multi-objective optimization model for supplier selection with ASP and ALB. The objective function and constraints are detailed as follows. Objective function: Objective 1 – Minimize the assembly line cycle time to increase the total output.

Min cmax

(3)

The cycle time of the assembly line is the longest workstation processing time, which is expressed as: 12

cmax  max { Tq q  1,2,3,..., Q }.

(4)

The processing time at each workstation is the sum of the assembly time and processing time at the workstation, which can be obtained using: Tq  QATq  QOTq 　　q ,

QATq 

QOTq 

M

I

I

Si S j



m 1 i 1 j 1 s 1 t 1

M M

(5)

ATmisjt Dmij H mis H mjt Bqm　 　　q ,

(6)

P

  OTmn ApmnCqpn　　q ,

(7)

m 1 n 1 p 1

Objective 2 –Maximize the assembly line efficiency to ensure smooth flow. (8)

Max EOL

The production efficiency of the assembly line is the ratio of the total workstation processing time to the cycle time and the number of workstations (indicated as a percentage), expressed as:

EOL 

Tsum  100% . Q  c max

(9)

The workstation processing times are the sum of the processing times of each workstation: Q

Tsum   Tq 　

(10)

q 1

Objective 3 – Minimize the product delivery time to ensure a rapid response to market demands. Min T

(11)

The product delivery time is the sum of the lead time of the part/component assembly and the manufacturing time of assembly activities finished on the assembly line.

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T  Tsum D  QLT

(12)

The lead time accounts for the supplier’s manufacturing time for parts and components and the transportation time. In this study, the longest lead time is adopted because the assembly activity does not commence until all parts and components have been delivered. QLT  max

D  MTis  DTis

{(

H mis | i  1,2,3,..., I ; s  1,2,3,..., S i ; m  1,2,3,..., M }

) (13)

Constraints: Each part/component is only assigned to one connector and is provided by only one supplier. Si M



s 1 m 1

H mis  1　　　i

(14)

Each connector is assigned to only one assembly sequence and one workstation. Q

P

  Cqpm  1　　　m

(15)

q 1 p 1

This ensures that the assigned connector processing does not commence until the immediate predecessor’s processing assignment has been completed. Q

Q

P

P

　　　m, n  Gm   qpCqpm    qpCqpn　 q 1 p 1

q 1 p 1

(16)

This ensures the number of workstations to which the part or component assembly is assigned does not exceed the designated number. Q

 qCqpm  Q　　　p  P, m

(17)

q 1

This ensures the cycle time of a workstation does not exceed the designated cycle time.

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cmax  c　

(18)

Stressing the 0–1 integer restriction for decision-making variables Apmn , Bqm , Cqpm , Dmij , and H mis . Apmn ={0,1} Bqm ={0,1}

p, m, n

(19)

q, m

(20)

Cqpm ={0,1}

q, p, m

(21)

Dmij ={0,1}

m, i, j

(22)

H mis ={0,1}

m, i, s

(23)

4. Proposed MMPSO approach for solving the optimization model This study proposes an MMPSO to solve a multi-objective optimization problem for supplier selection by integrating ASP with ALB techniques. The MMPSO combines a guided search algorithm and MPSO. In this algorithm, the assembly sequence is indirectly generated by a priority-based coding approach which is coordinated with guidance information such as precedence constraints. Moreover, the supplier information is updated through mutation operations. In the subsequent processes, the global optimum is selected by employing the generation compared weight method to guide the updated position. The MMPSO algorithm process shown in Fig. 2 is described in detail as follows: [Insert Fig. 2 about here] 4.1 Solution coding Each particle is a string of bits depicting the scheme of a feasible solution. Each string comprises the following three parts: (a) assembly sequence; (b) part suppliers; 15

and (c) work assignment (Fig. 3), and the length of a particle is 2M+I. The assembly sequence, work assignment, and supplier information are coded with symbols. In previous works solving ALBPs, a string is indicated as station-oriented and/or take-oriented [4]. This study employs the take-oriented approach to indicate the assembly sequence with connectors, and applies the station-oriented approach to indicate a work assignment with a connector. Fig. 4 shows that under ASP, the assembly sequence is (7, 9, 3, 5, 4, 2, 1, 8, 6). Following the assignment of parts, Connector 1 is assigned to Station 3, Connector 2 is assigned to Station 2, and so on. Similarly, for the indication of the supplier string, Part A is supplied by Supplier 2, Part B is supplied by Supplier 1, and so on. [Insert Figs. 3 and 4 about here] 4.2 Generation of initial population The number of particles of the initial population is twice that of the populations, Po , with the primary purpose of increasing the number of non-dominated solutions

for the initial search. In the assembly processes, a set of operations is restricted by precedence relationships according to the technological characteristics of products. The precedence constraints represent a typical operational mode constrained by the connector order. In this paper, each precedence constraint requires that Connector i has to be assembled before Connector j. The assembly sequence and supplier information are generated randomly, and the assembly sequence string employs the guided generation method based on a precedence graph to confirm the constraints on assembly. This is a priority-based coding method [27, 29], and the initial assembly sequence is determined based on the assembly precedence constraints on each connector. Moreover, a feasible assembly sequence string is generated also based on the precedence constraints. For example, Fig. 4 shows a randomly generated assembly

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sequence (7, 9, 5, 3, 4, 2, 1, 8, 6). Considering the precedence constraints, Connector 3 should precede Connectors 4 and 5, and the assembly sequence with Connector 5 must follow Connector 3; therefore, the guided approach yields the initial assembly sequence (7, 9, 3, 5, 4, 2, 1, 8, 6). Next, the assembly assignment is based on the initial assembly sequence and supplier information, and it satisfies the constraints shown in Eqs. (14)–(18), thereby yielding the initial assignment result. The assembly is assigned to a workstation with a specific sequence based on the assembly sequence of the connector. For the initial cycle time and work assignment, a calculation mechanism is proposed based on the initial cycle time and work assignment modified from Kim et al. [25] and Nearchou [29], the process of which is detailed as follows. Step 1. Set the initial cycle time. The initial cycle time is a ratio of the sum of the manufacturing time of product Tsum to the number of workstations Q . Step 2. Assign the connector to the previously listed Q -1 workstations in accordance with the assembly sequence. During the assignment, the workstation processing time assigned to the connector does not exceed the designated initial cycle time. Finally, the remaining connectors are assigned to the final workstation Q . Step 3. Calculate the processing time of each workstation Tq and the processing time of the potential workstation PTq , which is the sum of Tq and the manufacturing time of the first connector assigned to the subsequent workstation q  1 , where q  1,2,3,..., Q  1 .

Step 4. Set c  max { Tq q  1,2,3,..., Q } and pc  min { PTq q  1,2,3,..., Q  1}. If c > pc or constraint Eqs. (14)–(18) are not satisfied, reassign the assembly sequence

and return to Step 1; otherwise, c is the cycle time.

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Step 5. Set the largest cycle time of the population as the designated upper limit of the cycle time during the follow-up evolution. 4.3 Calculation of the objective functions Substitute the data/figure of the assembly sequence, supplier combination, and assembly assignment in the population of the assessment model and calculate the assembly line cycle time (Eq. (3)), assembly line efficiency (Eq. (8)), and delivery time (Eq. (11)). 4.4 Population sorting Sort the population by level into non-dominated solution sets based on the multi-objective function value of each particle. First, select any non-dominated solution individuals from the population (Level 1), and then select any non-dominated solution individuals from the remaining individuals (Level 2). Screen the individuals and increase the level value until the entire population is sorted. Select the first 50% of particles in the population for subsequent evolution procedures. 4.5 Calculation of fitness value In previous studies, multi-objective fitness values have been calculated primarily by using the fixed-weight, random-weight, or adaptive weight methods. Guo et al. [20] developed an improved adaptive weight approach, the generation compared weight method, for damper distribution of transmission towers. Their results indicated that compared with the adaptive weight approach, the generation compared weight method has stronger wide-area search capabilities for addressing the local optimum solution. Therefore, this study applies the generation compared weight method to calculate the fitness function values of the particles, and the particles with the largest fitness function value are selected to guide the evolution of the subsequent sequence. The calculation procedure for the N-objective generation compared weight method and

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fitness values is detailed as follows. Step 1. Assume the maximization of the Nth objective function and the minimization of the remaining objective functions are optimal; first, define the intermediate variable t k of the k th objective function as follows:

tk  1 / N　 ;G  1,

(24)

1 tk  Z kG,min / Z kG,min 　 ; G  1,

(25)

1 t N  Z NG,max / Z NG,max 　 ; G  1,

(26)

where Z kG,min is the minimum of the k th objective function in generation G, and Z NG, max is the maximum of the N-objective function in G.

Step 2. Calculate the generation compared weight. The weight in the calculation equation shows the comparison between the extent of evolution of the objective functions in two neighbouring generations to facilitate equilibrium development among the objective functions, as follows. N

wk  tk /  tk

(27)

k 1

Step 3. Calculate the fitness function values for each particle. The maximization of the fitness function value is optimal, where  is a non-zero fractional value (because the denominator cannot be zero). In a fitness function, a partial factor of each objective function for the particles comprises the following two parts: (a) the generation compared weight wk among the multiple objectives, and (b) the parameter 1 /( Z kmax  Z kmin ) of the internal comparison. The search pressure for the population to evolve to the negative ideal point can be guaranteed, with which, the pressure can be distributed evenly among the objectives. The fitness function value is calculated as follows: 19

N 1

f   wk k 1

( Z kmax  Z k )   ( Z N  Z kmin )    w N ( Z kmax  Z kmin )   ( Z kmax  Z kmin )  

(28)

4.6 Identification of pBest and gBest The first 50% of the selected particles are set as the initial pBest . The objective function values of these particles are calculated and compared with the current generation population and the preceding generation population; subsequently, the better particles are selected to update pBest . For the selection of gBest in the particle swarm, the best fitness function values obtained in the previous step are applied to guide the next assembly sequence evolution. 4.7 Update of particle velocity and position Based on the pBest for each particle, and through the gBest particle retrieved from the generation compared weight approach, update the assembly sequence velocity and position for each particle in the population. To update the assembly sequence velocity, we employ the velocity function (Eq. (1)) of the constriction factor method introduced by Maurice (1999), and then update the position based on Eq. (2). The integer function Int[ xidj 1 ] is introduced for obtaining the integer value of xidj 1 by eliminating its decimal. The assembly sequence string of the new particle should be adjusted using the guided generation method with priority-based coding under the precedence constraints. Conversely, the string of suppliers is updated using multi-point mutation (Fig. 5). First, we randomly select a string and the number of bits to be mutated. Next, a mutation position is generated randomly from the chromosome, and the bit code of the position is changed to a random number within the total number of suppliers. Repeat the process until the desirable number of bits to be mutated is obtained. Subsequently, based on the updated assembly sequence string and supplier string, 20

proceed with the assignment of assembly activities to obtain a new workstation string while satisfying the constraints in Eqs. (14)–(18). Within the designated workstation cycle time, the assembly activities are assigned to Q -1 workstations, based on the assembly sequence of the connector, and the final assembly activity is assigned to the Q th workstation. Substitute the data figures with the updated assembly sequence (Eqs. (4)–(7)), supplier combination (Eqs. (9)–(10)), and assembly assignment (Eqs. (12)–(13)) of the assessment model to obtain the new objective function for each particle. [Insert Fig. 5 about here] 4.8 Elite-preservation mechanism To retain optimal solutions during the evolution, the original and new parents form a new population which is then divided by level into non-dominated solution sets. The first 50% of the population is reserved for the next evolution, and the other 50% is eliminated. The reserved individuals become a new population, including a new assembly sequence, new suppliers, and new workstations. 4.9 Search to minimize the cycle time All assembly activities are conducted within a specific cycle time. During each evolution, the largest cycle time obtained after the initial assembly assignment serves as the upper limit for a given cycle time during the subsequent evolution. The cycle time decreases by a specific unit in each evolution until the current shortest cycle time is reached, and a generation is then obtained, after which the next stage of evolution ensues. 4.10 Determination of termination criteria If a specific number is not reached in a generation, Subsection 4.5 is repeated.

21

5. Illustrative Example and Results Analysis 5.1 Case description This section presents a case study of a computer manufacturer (Company A) as an example. Each computer has 22 parts and components, and each part/component has 5 potential suppliers. Fig. 6 shows the computer explosion diagram. Company A receives a request from a customer for 1,000 computers. [Insert Fig. 6 about here] Because a computer has 22 parts/components and there are 23 connectors, there are 23 ASP steps for each computer (Table 1). We employ a matrix to indicate the limit on the assembly sequence for each step. In Table 2, the value “1” indicates a precedent sequence limit between the two connectors. For example, ( M 1 , M 5 )=1 implies that connector M 1 should come prior to M 5 in assembly. According to the computer connector data sheet and the assembly sequence precedence matrix, a connector-based precedence graph can be plotted (Fig. 7), and we can observe the connector and sequence limitations. [Insert Tables 1 and 2 and Fig. 7 about here] Based on the foregoing data and precedence relation, we can create part/component data, a precedence relation matrix, and a part or connector relation matrix database. Corresponding with suppliers’ lead time (including manufacturing time and transportation time), the assembly plant’s connector time, and the assembly time for various suppliers, we can solve the assembly assignment and the optimization cycle time, and ensure assembly line efficiency and product delivery time through the decision-making module. In this study, the computer assembly activities in the assembly plant involve four workstations with a given assembly sequence. 5.2 Experimental results 22

To address the multi-objective optimization problems of supplier evaluation for connector-based ASP and Type-2 ALB techniques, this study employs a MMPSO to solve the problem. The parameters of the MMPSO algorithm include the number of populations, Po , the number of generations, G , the cognitive parameter, 1 , social parameter  2 , and the largest velocity, V max . For the population size for setting the PSO parameters, previous studies have typically selected 30 particles [48]. Clerc and Kennedy [11] argued that the sum of 1 and  2 should be greater than 4 to ensure that the particle swarm to which the constriction factor is added is optimized under the acceleration function. However, these studies did not indicate an optimal combination of these two parameters; only the setting of the two parameters 1  2  2.05 is presented. Zhang et al. [48] further argued when 1  2.8 and 2  1.3 , the convergence speed of the population can be accelerated to improve the algorithm’s capability. To determine an appropriate setting for V max , Clerc and Kennedy [11] argued that a favourable convergence can be obtained without limiting V max . Conversely, Eberhart and Shi [15] argued that when executing the PSO algorithm to which constriction factors are added, the execution performance can be improved if the speed limitation is considered. Mohemmed et al. [27], who employed a particle swarm to solve a shortest-path problem, directly set the V max of all search dimensions to  3000 . Based on this discussion, we set Po as 30, 1 and  2 as two levels ( (1  2.05, 2  2.05) and (1  2.8, 2  1.3) ), and V max as two levels (  3000 or no limit). Setting G as 50 may result in a convergence effect during the evolution. To optimize the performance of the decision-making system, it is necessary to conduct a study that is designed for the parameters that can obtain the optimal

23

parameter combination for comparing different algorithms. In this study, the mean number of non-dominated solutions and the mean ratio of Pareto-optimal solutions serve as measurement indices for the algorithm performance [2]. The formula of the ratio of Pareto-optimal solutions for the i th algorithm (Eq. (29)) indicates that the Pareto-optimal solutions in Pi are not dominated by any other solutions in P, Pi is the set of Pareto-optimal solutions obtained from the i th algorithm, and P is the union of the sets of Pareto-optimal solutions for all I

algorithms （ P   Pi 　 ）. Y  X implies that solution X is dominated by solution i 1

Y , and Pi  { X  Pi | Y  P : Y  X } implies that the dominated solutions, X, in P are removed from the solution set Pi. The algorithm was coded in Visual Basic, the database was built using Microsoft Access 2003, and the algorithm was executed using a personal computer (CPU, Intel Core 2 Duo T7100 1.8 GHz; RAM, 2039 MB). Each experiment was repeated 30 times to obtain the Pareto optimization results for the comparison (   0.05 ).

RPOS ( Pi ) 

Pi  { X  Pi | Y  P : Y  X } Pi

(29)

Table 3 shows the experimental results of 1 ,  2 , and the V max parameter settings. According to the results, a superior mean ratio for the Pareto-optimal solutions is achieved when ( 1 ,  2 , V max )=(2.8, 1.3, no limit), indicating that the quality of the Pareto-optimal solutions obtained using the parameter setting is superior to that obtained using other parameter settings. When ( 1 ,  2 , V max )=(2.8, 1.3, no limit) and G  50 , Po is set as (30, 50, 70) and 100, respectively. Table 4 shows the experimental results, which indicate that the quality of the obtained Pareto-optimal solutions are optimal when the population is higher.

24

[Insert Tables 3 and 4 about here] Finally, when ( 1 ,  2 , V max )=(2.8, 1.3, no limit), and Po  100 , then G is set at (50, 100, 150) and 200, respectively. Table 5 shows that the quality of the Pareto-optimal solutions is superior for higher generations. Finally, this study sets the parameters of MMPSO algorithm as ( 1 ,  2 , V max ) = (2.8, 1.3, no limit), Po  100 , and G  200 .

[Insert Table 5 about here] The results obtained from the MMPSO experiment are compared with NSGA-II and gNSGA-II to solve the performance comparison (the number of iterations n  30 ,

  0.05 ). Table 6 shows the analysis results. This study compares the variance between the number of non-dominated solutions and the ratio of the Pareto-optimal solutions for NSGA-II and gNSGA-II with two independent normal population variances of the F-test. A statistically significant variance exists between the number of non-dominated solutions (p=0.0281) and the ratio of Pareto-optimal solutions (p=9.04E-18) for NSGA-II and gNSGA-II; thus, the unequal variance t test is employed to compare the performance of these two methods in solving the problem. The results of the unequal variance t test indicate that the variance between NSGA-II and gNSGA-II is non-significant (p=0.4094), although significant variance was observed between NSGA-II and NSGA-II (p=8.83E-07) for the ratio of Pareto-optimal solutions. This implies that the quality of the Pareto-optimal solutions obtained from gNSGA-II is superior to that obtained from NSGA-II. Moreover, an identical approach is employed to compare the performance of MMPSO and gNSGA-II. The results show that MMPSO not only yields more non-dominated solutions than gNSGA-II does, but the mean ratio of the Pareto-optimal solutions indicates that it also has superior Pareto-optimal solutions. Fig. 8 shows the 25

non-dominated solutions obtained by the MMPSO, gNSGA-II, and NSGA-II algorithms. The figure clearly shows the majority of non-dominated solutions obtained using gNSGA-II and NSGA-II are dominated by those obtained using MMPSO. [Insert Table 6 and Fig. 8 about here] Finally, this study selected three Pareto-optimal solutions obtained using MMPSO and summarizes the retrieved objective values, as well as the optimum strategies (Table 7), in which the optimum strategy includes the optimal assembly sequence, supply chain partner combination, and assembly assignment. The objective values of cycle time, delivery time, and assembly line efficiency are 153, 613905, and 0.9967 for Solution 1. In this solution, Connectors M1, M2, M5, M6, M8, M12, M15, M16, M18, M20, and M22 are assigned to Station 1 and the assembly sequence of these connectors is M20 M8 M22 M15 M12 M2 M16 M1 M6 M5 M18. Components G, K, S, A, M, I, U, N, H, J, B, L, P, F, E, and Q are introduced in Station 1 to complete the assembly task, depending on the assembly sequence of the connectors. In addition, the second supplier of G, the third supplier of K, and the fifth supplier of S are selected to provide each specific component. [Insert Table 7 about here]

6. Conclusion and Suggestions From the perceptive of a central assembly plant, this study combined the supplier selection problem with ASP and Type-2 ALB techniques, allowing the assembly plant to consider the production status and efficiency of the assembly line while conducting supplier selection to facilitate the production of parts or components and minimize the time-to-market or delivery time. For this problem, a multi-objective optimization

26

model was developed by considering the following objective conditions: (a) the minimum cycle time of the assembly line; (b) the minimum delivery time, and (c) the maximum assembly efficiency. In addition, this study compared MMPSO, gNSGA-II, and NSGA-II, and showed that MMPSO is significantly superior to NSGA-II and gNSGA-II in two assessment indices for multi-objective algorithms (the mean number of non-dominated solutions and the mean ratio of Pareto-optimal solutions), indicating that we can obtain more and higher-quality Pareto-optimal solution sets by using MMPSO. This study also shows that gNSGA-II is significantly superior to NSGA-II in the mean ratio of Pareto-optimal solutions, indicating that the quality of Pareto-optimal solutions obtained using the guided method is superior to those obtained using a random search method. Throughout the study, we also identified certain areas that require further research: 1) Different weights should be applied to various objectives by considering customer demands or manufacturer assessments. 2) We should further consider part-related engineering information such as combination, assembly directions, and assembly tools for ASP. 3) Any limitations on the assembly assignment (e.g., the type of workstation, the assembly tool available), and assign parts/components to suitable workstations should be taken into account. 4) Future works can apply heuristic methods to the assignment of ASP and ALB to improve the execution efficiency of the algorithm and the quality of the obtained Pareto-optimal solution sets.

Acknowledgements The author thanks the Ministry of Science and Technology, Taiwan, ROC, for partial financially supporting under grants MOST 104-2221-E-027-044 and MOST 105-2221-E-027-059. The author also thanks Miss Y.Y. Lin for writing and

27

performing the programs and acknowledges the editors and anonymous reviewers for their helpful comments and suggestions, which greatly improved the presentation of this paper.

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31

Captions for Figures Fig. 1 A simple illustrative example Fig. 2 MMPSO algorithm procedure Fig. 3 Particle scheme Fig. 4 Coding method Fig. 5 Multi-point mutations of supplier chromosome Fig. 6 Computer explosion diagram [38] Fig. 7 Connector-based precedence graph Fig. 8 Non-dominated solutions of the three algorithms

Captions for Tables Table 1 Computer connector data sheet Table 2 Computer assembly sequence precedence matrix Table 3 Experiment results of MMPSO for parameters 1 ,  2 , and V max Table 4 Experiment results of MMPSO for parameter Po Table 5 Experiment results of MMPSO for parameter G Table 6 Test results of the three algorithms Table 7 Optimal strategy combination

32

Connector Assembly relation M1 A-B M2 B-C M3 D-E M4 C-D M5 F-G M6 D-G

Connector-based precedence M1 Customer Demands

S

M2 M4

M3

M6

F

M5

Part

Suppliers

A B C D E F G

A1, A2, A3 B1, B2, B3 C1, C2, C3 D1, D2, D3 E1, E2, E3 F1, F2, F3 G1,G2,G3

Possible solutions Station 1 Assembly sequence a. M1 → M3 → M2 Supplier A1,B2,C1,D3,E2 b. M3 → M1 A3,B2,D1,E2 c. M5 → M1 A2,B1,F1,G2

Station 2 M4

Station 3 M5 → M6 F2,G3 M2 → M4 M5 → M6 C3 F3,G3 M2 → M3 → M4 M6 C3,D1,E3, G1

... Proposed optimization model Objectives:

Min cmax Max EOL Min T cmax  max Tq q  1,2,3,..., Q T EOL  sum  100% Q  cmax





T  Tsum  D  QLT

...

Si M

Constraints:

H mis  1　　　i



C qpm  1　　　m

s 1 m 1 Q P

q 1 p 1

Station 1 Station 2 Assembly sequence a. M5 → M3 → M1 M2 → M4 Supplier A1,B2,D3,E2,F1,G2 C1 b. M3 → M1 M2 → M4 A3,B2,D1,E2 C3 c. M1 → M3 M2 → M4 A2,B2,D3,E1 C1

Station 3 M6 M5 → M6 F3,G3 M5 → M6 F1,G1

...

Fig. 1 A simple illustrative example

33

...

Optimal solutions for assembly sequence planning and supplier selection



Chromosome coding

Guide generation method

Initial assembly sequence population

Calculation mechanism of initial cycle time and work assignment

4.1 Initial supplier population

Initial workstation population

4.2

Calculation of objective function

4.3

Population sorting

4.4

Calculation of fitness value

4.5

pBest and gBest update

4.6

Update of particle velocity and position

4.7

Elite-preservation strategy

4.8 NO

Minimization of cycle time

4.9 Yes Determination of termination criteria

NO

4.10

YES Optimal strategies

Fig. 2 MMPSO algorithm procedure

Connector index

7

…

Supplier index

6

Assembly sequence

2

…

Supplier

4

Station index

3

Assignment

Fig. 3 Particle scheme 34

…

2

7

Assembly sequence (Before)

9

5

3

4

2

1

8

6

1

8

6

Adjust by precedence constraints converson Assembly sequence (After) Supplier Assignment

7

9

3

5

4

2

2

1

4

3

5

2

5

1

4

3

2

1

2

2

3

1

4

1

Fig. 4 Coding method

1

3

4

2

1

5

4

3

5

4

2

5

After conversion 1

5

3

2

1

2

Fig. 5 Multi-point mutations of supplier string

Fig. 6 Computer explosion diagram [38]

35

M5 Pre ce d en c e :M 1

M1 P re ce d en c e: －

M6 Pre ce d en c e :M 1

M3 P re ce d en c e: － M2

M18 P re c e d e n ce : M1、M 5 M19 P re c e d e n ce : M1、M 6

M4 Pre ce d en c e :M 3

P re ce d en c e: － M7 P re ce d en c e: － M8 P re ce d en c e: － M9 P re ce d en c e: － M 10 P re ce d en c e: － M 11 P re ce d en c e: － M 12 P re ce d en c e: －

S

F

M 13 P re ce d en c e: － M 14 P re ce d en c e: － M 15 P re ce d en c e: － M 16 P re ce d en c e: － M 17 P re ce d en c e: － M 20 P re ce d en c e: － M 21 P re ce d en c e: － M 22 P re ce d en c e: － M 23 P re ce d en c e: －

Fig. 7 Connector-based precedence graph

36

Assembly line efficiency

MMPSO gNSGA-II NSGA-II

cycle time

delivery time

Fig. 8 Non-dominated solutions of the three algorithms

Table 1 Computer connector data sheet connector No. M1 M2 M3

M4 M5 M6 M7 M8 M9 M 10

M 11 M 12

assembly relation of parts/components

connector No.

A, K, P A, B A, C A, D A, E A, F A, L A, M A, N K, O A, J H, J

M 13

M 14 M 15 M 16 M 17 M 18 M 19 M 20

M 21 M 22 M 23

37

assembly relation of parts/components I, J G, J G, N H, L I, M E, K, Q F, K, R G, K, S H, K, T I, K, U J, K, V

Table 2 Computer assembly sequence precedence matrix M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16 M17 M18 M19 M20 M21 M22 M23

( 1 ,

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

M13

M14

M15

M16

M17

M18

M19

M20

M21

M22

M23

0

0 0

0 0 0

0 0 1 0

1 0 0 0 0

1 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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

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

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table 3 Experiment results of MMPSO for parameters 1 ,  2 , and V max No limit  3000 V max Average number of Average ratio of Average number of Average ratio of

2 )

(2.05, 2.05) (2.8, 1.3 )

non-dominated solutions 12.23 13.73

Pareto-optimal solutions 0.1992 0.2386

non-dominated solutions 13.67 13.90

Pareto-optimal solutions 0.2561 0.5228

Table 4 Experiment results of MMPSO for parameter Po Po 30 50 70 100

Average number of non-dominated solutions 13.90 15.83 16.20 17.86

Average ratio of Pareto-optimal solutions 0.2586 0.2128 0.2987 0.4881

Table 5 Experiment results of MMPSO for parameter G G 50 100 150 200

Average number of non-dominated solutions 17.87 17.62 21.10 20.00

38

Average ratio of Pareto-optimal solutions 0.0947 0.3174 0.4402 0.4948

Table 6 Test results of the three algorithms Number of

Ratio of Pareto-optimal solutions

non-dominated solutions gNSGA-II v.s NSGA-II gNSGA-II average NSGA-II average F-value P-value(F<=f) T-Statistic P-value(T<=t) MMPSO v.s gNSGA-II MMPSO average gNSGA-II average F-value P-value(F<=f) T-Statistic P-value(T<=t) Performance ranking

16.4667 16.7333 0.4854 0.0281 -0.2302 0.4094

0.3002 0.0130 48.1507 9.04E-18 5.9154 8.83E-07

20.0000 16.4667 2.4589 0.0091 2.8691 0.0030

0.8396 0.3002 0.5090 0.0370 9.1393 1.06E-12

1. MMPSO; 2. gNSGA-II; 2. NSGA-II

1. MMPSO; 2. gNSGA-II; 3. NSGA-II

Table 7 Optimal strategy combination Objective value

Solutions

(

f1 , f 2 , f 3 )

Optimal strategies

Assignment Station 1

Station 2

Station 3

Station 4

M20 M8 M22

1

153, 613905, 0.9967

Assembly M15 M12 M2 sequence M16 M1 M6

2

Assembly sequence Supplier

3

153.5, 613655, 0.9931

M9 M14 M19 M13

M3 M4 M7 M17

T3, V4, O1

R5

C4, D3

M6 M5 M18 M10 M11

M9 M14 M19 M13

M3 M4 M7 M17

F4, E4, Q1, O4

R1

C4, D1

M23 M11 M9 M21

M10 M3 M4 M19

M7 M16 M2 M14

V5, T1

O2, C2, D4,R1

L4, B4

M5 M18

Supplier

153.25, 616650, 1.0

M21 M23 M10 M11

Assembly sequence Supplier

G2*, K3, S5, A5, M4, I3, U2, N3, H1, J3, B1, L1, P5, F3,E4, Q4 M20 M22 M8 M2 M15 M12 M16 M21 M23 M1 G2, K1, S4, I3, U3, A5, M2, B1, N3, H1, J3, L1, T4, V5, P1 M17 M1 M12 M13 M15 M20 M8 M22 M6 M5 M18 I4, M1, A4, K3, P3, H4, J3, G3, N3, R1, U3, F4, E4, Q5

* Supplier No.

39

Highlights

   

Propose a multi-objective optimization mathematical model for supplier selection. ASP and ALB are introduced into the mathematical model. A MMPSO method based on particle swarm optimization is proposed to solve the model. Provide a case study of a computer assembly plant to illustrate the MMPSO method.

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