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Measurement and Control
Transactions of the Institute of
http://tim.sagepub.com/content/34/6/720
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DOI: 10.1177/0142331211402901
2012 34: 720 originally published online 1 July 2011
Transactions of the Institute of Measurement and Control
ZH Che, Tzu-An Chiang and Zhen-Guo Che
supply chain planning considering supplier relationship management
Using analytic network process and turbo particle swarm optimization algorithm for non-balanced
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Using analytic network process and turbo
particle swarm optimization algorithm for
non-balanced supply chain planning
considering supplier relationship
management
ZH Che
1, Tzu-An Chiang
2and Zhen-Guo Che
3Abstract
Supply chain planning has been regarded as a key strategic decision-making activity for the enterprises under the current business circumstances. From the point of supply chain planning, the important issues are to find suitable and quality partners and to decide upon an appropriate production– distribution plan. In this study, hence, we address to develop a decision methodology for supply chain planning in multi-echelon non-balanced supply chain system, taking into account such four criteria as cost, quality, delivery and supplier relationship management and considering quantity discount and capacity constraints. The proposed methodology is based on the analytic network process and turbo particle swarm optimization (TPSO), to evaluate partners and to determine an optimal supply chain network pattern and production–distribution mode. Finally, to demonstrate the performance of the proposed TPSO algorithm, comparative numerical experiments are performed by TPSO, particle swarm optimization (PSO) and genetic algorithm (GA). Empirical analysis results demonstrate that TPSO can outperform PSO and GA in non-balanced supply chain planning problems.
Keywords
Analytic network process, non-balanced supply chain, particle swarm optimization, supplier relationship management, supply chain planning
Introduction
Emerging in the late 1980s, the concepts of supply chain have been widely used in manufacturing management. Some world-renowned companies, such as Dell, IBM and HP, had gained profound benefits from supply chain management (SCM). To supply chain planning, selection of suitable sup-pliers/partners not only means a basis, but also a critical step. Selection of suppliers has been referenced in a lot of research, such as Sha and Che (2005, 2006), Choi and Chang (2006), Huang and Keskar (2007), Wang and Che (2007), Che and Wang (2008), Ha and Krishnan (2008), Kheljani et al. (2009) and Yue et al. (2010), and cost, quality and delivery are always the most frequently used criteria for selecting suppliers (Dahel, 2003; Liao and Rittscher, 2007; Wadhwa and Ravindran, 2007; Wang and Che, 2008); quantity discount should also be taken into account for costs (Dahel, 2003; Darwish, 2009; Li and Liu, 2006; Xia and Wu, 2007; Xie et al., 2010).
According to Herrmann and Hodgson (2001), supplier relationship management (SRM) is a process of managing suppliers, where cost reduction, experience sharing and pro-curement co-ordination are achieved through partnership, to offer an integrated management tool. As suggested by David
and Stuart (2001), in addition, the relationship between two business partners has changed from an antagonistic one to a symbiotic one, where the stress is placed on mutual reliance and technical co-operation so that the partners can reach an agreement. With respect with the trends of SRM and the effects incurred on supply chains, SRM is taken into account in this study, in conjunction with cost, quality and delivery as criteria for partner selection. For this, we have adopted an analytic network process (ANP) approach to evaluate the weight of each criterion for partner performance evaluation in supply chain system.
1
Department of Industrial Engineering and Management, National Taipei University of Technology, Taipei, Taiwan, Republic of China
2Department of Business Administration, National Taipei College of Business, Taipei, Taiwan, Republic of China
3Institute of Information Management, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
Corresponding author:
ZH Che, Department of Industrial Engineering and Management, National Taipei University of Technology, 1, Sec. 3, Chung-Hsiao E. Rd, Taipei 106, Taiwan, Republic of China
E-mail: [email protected]
Transactions of the Institute of Measurement and Control 34(6) 720–735
Ó The Author(s) 2011 Reprints and permissions:
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In real-life production systems, there are many uncer-tainties (Lee and Rosenblatt, 1986; Xu et al., 2006), and hence production characteristics of individual members within a system should be taken into account before plan-ning and construction of supply chain systems to ensure the fulfilment of production goals. In that, supply chains that involve production loss are called non-balanced supply chain system as shown in Figure 1. Assimilation of such system may bring our research closer to real-life production practices.
As in the above statement, in this study, we emphasizes a multi-echelon non-balanced supply chain planning for choosing the suitable partners from a number of potential participators to become involved in the system, and to make the optimal production–distribution planning deci-sions. Gen and Cheng (1997) pointed out that the multi-stage logistic problem is like the combination of the multiple-choice Knapspack problem with the capacitated location–allocation problem as an NP-hard problem. In our study, the capacity, production and transportation losses, and quantity discount are considered, and hence our problem becomes even more difficult. In addition, in the optimization processes of common particle swarm opti-mization (PSO), more infeasible solutions may be created after particle position and velocity updating. A novel PSO approach, hence, called the turbo particle swarm optimiza-tion (TPSO) for supply chain planning is presented for solving the proposed problem in this paper. The TPSO adopts the stepwise approach to check and adjust the solu-tions based on a variable-demand dependency mechanism, which can avoid producing the repeated infeasible solutions in updating processes.
Therefore, there are five major study purposes:
1. Developing the partner relationship assessment procedure for SRM that includes partner relationship (PR) and infor-mation technology (IT) criteria. As we know, so far, there has never been any procedure proposed that performs the PR evaluation in a supply chain system.
2. Building an ANP procedure for appraising the relative weights among criteria for partner evaluation.
3. Proposing a multiple objective optimization model, and its objectives taken into account include: minimum cost, minimum quality (defective rate), maximum delivery
(on-time delivery rate), and maximum PR coefficient (PRC).
4. Applying a variable-demand dependency mechanism to develop a TPSO and applying the TPSO to solve the mul-tiple objective optimization model.
5. Comparing the solving performance of TPSO, PSO and genetic algorithm (GA; Sha and Che, 2006) to verify that TPSO has excellent capabilities for solving the problems defined in this study. The criteria for the performance comparison are the objective value and the convergence time.
The rest of this paper is organized as follows. Section 2 is the literature review about SCM, SRM, ANP and PSO. Section 3 gives a framework to explain this study, and estab-lishes the PR assessment procedure, ANP procedure, optimi-zation mathematical model and TPSO solving model. Section 4 presents a numerical example to illustrate the application of TPSO for obtaining optimal strategies. The results thus obtained are also compared with those of PSO and GA to validate the efficiency of the proposed algorithm. Conclusions are shown in Section 5.
Literature review
Supply chain management and supplier relationship
management
The concept of SCM was first proposed by Houlihan (1984), which was an important advancement in logistics. In the early stages, the concepts and techniques from industry dynamics were utilized to deal with physical distribution and transportation operations. Then, some scholars described it as an integrated activity covering the entire series of logistic operations and categories running from suppliers down to end-users.
Choi and Hartley (1996) held that businesses hoped to quicken distribution, shorten delay, reduce cost and improve quality through managing suppliers in the supply chain. Moreover, the supply chain is dedicated to maximizing the value produced by an entire supply chain, and SCM is intended to gain maximum benefit from a supply chain through the integrated management of logistics, flow of
Production System Production System Production System . . Input Output Loss Supply Chain System
Residual Value
Delivery Delivery
cash, business and information. Some reviews about the con-cept for supply chain planning are available in the literature, such as Che and Wang (2008), Ha and Krishnan (2008), Che (2010), Kheljani et al. (2009), Yue et al. (2010) and Arora et al. (2010).
From the above discussions, we can see that supply chain planning is intended to integrate suppliers, manufacturers, logistic firms and vendors in order to effectively run a supply chain under low cost and high quality, and improve competence for the chain. Corbett et al. (1999) stated that a company allies closely with upstream and downstream sup-pliers/partners to create a highly competitive supply chain; then members of the supply chain can share market, reduce investment, improve transport services and shorten time-to-market. Hence, the relationship between two business part-ners has changed from an antagonistic one to a symbiotic one. For this, SRM seems especially important in supply chain planning.
According to Gulledge (2002), SRM refers to the auto-mated accomplishment of a series of processes between companies and suppliers, including planning, scheduling, delivery and payment, which is an integration and exchange of data and processes between companies and suppliers.
Most applications of SRM for companies focus on
e-Procurement at present, in that it is hoped that more
transparent upstream and downstream requirements
planning can be secured through the functions of
e-Procurement to achieve collaborative supply. Many busi-nesses work on SRM software, including i2 Technologies, Oracle, SAP, PeopleSoft and SAS. Although the SRM applications provided by the developers differ in function, with the advancement in the Internet, the entire SRM IT has been made more convenient, and one can enjoy infor-mation sharing once linked up with the Internet.
From the definitions and development for SRM, we can know that the core values affecting SRM lie in long-term PR and IT. Janz et al. (1997) suggested that partners might come to know each other and reach an agreement through interactive information sharing. Information is the most significant strategic sourcing for companies; it is a fixed trend that organizations learn from partnerships through
strategic alliance and then develop new capabilities
(Simonin, 1999).
Many SCM papers have discussed the importance of PR and IT, such as Prahinski and Benton (2004), Kulp et al. (2004), Benton and Maloni (2005), Kannan and Tan (2005), and Narasimhan and Nair (2005). Hence, the two factors, PR and IT, were taken into account under SRM as impact factors for evaluation and selection of partners.
The concept of analytic network process
ANP was evolved from an analytic hierarchy process. At the beginning of ANP execution, a problem is resolved into different clusters, which incorporate many elements, and a network graph is constructed to present the interplay between clusters and elements according to the definitions and requirements from decision makers. Saaty (1996) thought that the dependent interplay between clusters and elements could be analysed with graphs, where every
element in a graph was connected with each other so that the graph could not be divided into two or more discon-nected graphs. Clusters in the graph were used to represent system structure framework, where the elements referred to impact factors in the system, and interplay might exit within every cluster or between elements.
According to Huang et al. (2005), ANP is applicable in quantitative and qualitative data analysis as well as in solving the dependence and feedback relationships between elements. Also, Ravi et al. (2005), Agarwal et al. (2006), Gencer and Gu¨rpinar (2007), and Tseng et al. (2009) pointed out ANP is one of the most widely used multiple criteria decision-making methods and has been proved to be useful for selection prob-lems such as strategy selection, supplier selection and location selection. Hence, in this study, ANP was used to determine the relative weights of the criteria for the evaluation and selec-tion of partners.
The concept of particle swarm optimization
Proposed by Kennedy and Eberhart (1995), PSO is an opti-mization tool in evolutionary computation inspired by the collective intelligence of bird flocks and the explorative and exploitative features of particle swarms, and improvement has been made to it by many scholars. Sivanandam and Visalakshi (2009) developed a parallel orthogonal PSO to schedule heterogeneous tasks dynamically on to heteroge-neous processors in a distributed setup. Arumugam et al. (2009) presented three PSO algorithms to solve the optimal control problem for the two-stage steel annealing processes and analysed the efficacy, efficacy and validity of each algo-rithm. Zhang and Qiu (2010) proposed a hybrid PSO approach to solve the travelling salesman problem based on cluster analysis on the swarm with the k-centres method. Zhang and Cai (2010) presented a novel hybrid PSO algo-rithm with dynamic inertial weight and chaotic search to solve the economic load dispatch problem, and results show that the proposed PSO algorithm can save considerable cost of economic load dispatch. In addition, Agrawal et al. (2008b) used PSO to solve the highly constrained multi-objec-tive environmental/economic dispatch problem and the results show it able to provide a satisfactory compromise solution in almost all the trials. Agrawal et al. (2008a) used an interactive PSO in multiple objective problems, and the results show that the PSO intends to find a compromise solu-tion along with a set of Pareto optimal solusolu-tions having a higher utility.
Chau (2006) stated that the PSO is based on a set of pre-sumed latent solutions, which are continuously updated to obtain the optimal value. Li et al. (2006) stated that PSO has constructive co-operation between particles, i.e. particles in the swarm share their information. Han and Ling (2008) mentioned that PSO has memory, i.e. the knowledge of good solutions is retained by all particles. Yang et al. (2007) denoted that PSO is rapidly converging towards an optimum, simple to compute, easy to implement and free from the com-plex computation in GA. Also, Zhang et al. (2006) and Zhang and Cai (2010) applied both PSO and GA on limited resource programming, and the results suggested that the performance of PSO is better than that of GA. There are still many other
variants of PSO (Abraham et al., 2010; Deng et al., 2010; Jain et al., 2010; Lu et al., 2010; Singh et al., 2010); for more details please refer to the corresponding references. Since the operating efficiency of PSO is better than that of GA, the proposed solving approach TPSO, in this study, is based on PSO.
Research framework and problem
formulation
Research framework
The numbers of upstream and downstream partners associ-ate closely with each other to form a supply chain. Therefore, in this study, we will gather the information on each partner and assess the relative weight of each criterion by ANP. Then, an optimization mathematical model is con-structed, taking into account the relevant costs (production cost, transportation cost, etc.) and capacity constraints. The TPSO is applied in minimization of the overall supply chain cost to find decision results satisfying the constraints and an optimal production decision-making system is developed for decision makers. The research framework is presented in
Figure 2, and the developments of PR assessment, ANP procedure, mathematical model and TPSO solution model for this study are to be laid out in following sub-sections.
Partner relationship assessment procedure for
supplier relationship management
The following procedure is adopted to derive a PRC for each partner.
. Step 1: Complete PR and IT coefficient assessment tables, as shown in Table 1. Item 1 and Item 2 are factors affecting PR and IT, respectively I and K items in total. The degree
to which item i and k respectively in PRijand ITkjaffect
PR and IT of partner j is measured in high, medium and
low instead of 3, 2 and 1 score. PR relative weight WPRj
and IT relative weight WITj for the partner j is to be
cal-culated with the following formula:
WPRj¼ PI i¼1 PRij 3I , WITj¼ PK k¼1 ITkj 3K Optimal fitness value
Optimum decision framework for non-balance multi-echelon supply chain
Optimal capacity allocation for supplier Supplier information ANP TPSO Calculate optimal fitness
Construct supply chain framework Construct evaluation selection model quantity discount Delivery SRM Cost Quality Import mathematical model Assessment criteria Most on-time delivery Optimum SRM Minimum cost Maximum
quality Supplier database Section 3.2 Section 3.3 Section 3.4 Section 3.5
. Step 2: Calculate the relative weight WPRCj of partner j
with the following formula: WPRCj¼ sw1WPRjþ sw2WITj
swu: Weight, swu.0 and
X2
u¼1
swu¼ 1, for u ¼ 1, 2
. Step 3: Find PRC for each partner according to PRC index table as shown in Table 2.
. Step 4: Complete partner PRC table for each firm in supply chain system as presented in Table 3.
Analytic network process for partner evaluation
This section presents an ANP procedure, proposed by Meade and Sarkis (1998), for evaluating the partners in the supply chain system. Its procedure is shown in Figure 3 and described step-by-step as follows.
. Step 1: Construct the model framework. The first step is to construct a model framework for the problem being solved as shown in Figure 4. We can see clearly the evaluation criteria (dimensions) and subcriteria (performance indices) in the figure, and we should consider whether there is inde-pendence or feedback residing in that.
. Step 2: Calculate the pairwise comparison matrix. Pair up the evaluation criteria (dimension) and the subcriteria
(performance index) to make matrix computations to eval-uate the independence and feedback properties between them, as shown in matrix A,
A¼ aij ¼ 1 a12 a1n a21 1 a2n .. . .. . . . . .. . an1 an2 1 2 6 6 6 6 4 3 7 7 7 7 5, where aij¼ 1 aji ; i, j¼ 1, 2 . . . , n:
The assessment coefficients (aij) are obtained through
expert questionnaire surveys or major decision makers in each departments based on the measurement scales sug-gested by Saaty (1996), where pairwise comparisons were classified into nine levels of importance, as presented in Table 4.
. After all the pairwise comparisons are finished, the two-stage algorithm created by Meade and Sarkis (1998) is used to find the row and column mean of the matrix Table 1 Partner relationship (PR) and information technology (IT) coefficient assessment
PR IT Item 1 2 . . . J 1 2 . . . J Item 1 PR11 PR12 . . . PR1J IT11 IT12 . . . IT1J 1 2 PR21 : . . . : IT21 : . . . : 2 : : : . . . : : : . . . : : I PRI1 : . . . PRIJ ITK1 : . . . ITKJ K WPR WPR1 WPR2 . . . WPRJ WIT1 WIT2 . . . WITJ WIT
Construct problem model framework Calculate the pairwise comparison matrix Calculate the Supermatrix Testing of consistency CR < 0.1 CR > 0.1
Figure 3 Relative weights of criteria for analytic network process partner evaluation. CR, crossover rate.
Table 3 Partner relationship coefficient (PRC) value Partner
1 2 . . . J
WPRC WPRC1 WPRC2 . . . WPRCJ
PRC X1 X2 . . . XJ
Xj2{1,2,3,4,5}, j ¼ 1,2, . . . , J.
Table 2 Partner relationship coefficient (PRC) index
WPRCj 0.0–0.2* 0.2–0.4* 0.4–0.6* 0.6–0.8* 0.8–1.0
PRCj 1 2 3 4 5
with the approximate eigenvector W; the equation is pre-sented as follows: Wi¼ XJ j¼1 aij= XI i¼1 aij ! =J
where Wi is the relative weight of evaluation criterion
(dimension) and subcriterion (performance index) i. J is the total number of columns in matrix. I is the total number of rows in matrix.
. Step 3: Test the consistency. According to Saaty (1996),
CR <0.1 means that in a comparison matrix, the deviation
for factor weight assessment is acceptable, and thus there is consistency, otherwise re-examination must be made and assessments of the pairwise comparisons must be modified; detailed formula and specifications are as follows: CR¼ CI/ RI, where CR (Consistency Ratio) indicates consistency, CI (Consistency Index) indicates consistency index, and there is CI¼lmaxn
n1 , where lmaxdenotes a maximum eigenvector
in a pairwise comparison matrix, and n is the number of comparative factors. RI denotes random index.
. Step 4: Calculate the Supermatrix. As an important feature for ANP, the Supermatrix is used to represent relationship between elements and their strength; according to the weights for each feasible solution and the relative weights between the criteria, the overall weight for each solution can be found.
As suggested by Saaty (1996), the supermatrix is a complete comprehensive matrix consisting of multiple submatrices with
clusters and elements included in clusters respectively distrib-uted on the right and upper side of the matrix, where the submatrices are formed based on the eigenvectors (W in this case) from pairwise element comparisons. A blank or 0 means that the clusters and elements are independent of each other without dependence, yet if the matrix elements are dependent on each other, after a certain number of multiplication oper-ations, the matrix may converge to a fixed extreme value, then the weight can be obtained through lim
i!‘T
2iþ1. Figure 5
pre-sents the matrix pattern for the supermatrix.
Matrix A is created by matching each criteria with the indices, matrix B is created by matching each index with the criteria, and matrices C and D are created by matching the criteria and indices respectively; last of all, they aggregate into a supermatrix.
Optimization mathematical model for solving supply
chain planning problems
Find the quota for various partners to minimize total cost in the supply chain using cost, quality, delivery and SRM as criteria, and taking customer requirements, partner produc-tion capacity limit, producproduc-tion and transportaproduc-tion loss rates into account. It is a multi-echelon, single-product supply chain framework with multi-level partners and multiple cus-tomer requirements. Assumptions of the mathematical model are: 1) all customer demands are known; 2) no shortage is allowable; 3) the order quantity is subject to maximum and minimum production capacity; 4) the quantity discount from supply members in respect to order handling is taken into consideration; 5) transportation costs are calculated on a unit basis, and do not change with quantity; and 6) produc-tion and transportaproduc-tion loss rates are taken into consideraproduc-tion at the same time, which means a non-balanced supply chain system.
The optimization mathematical model is as follows.
Objective function.
The four criteria – cost, quality, delivery and SRM – are taken into consideration, and the relative weight for each criterion was found through ANP; then multi-objective mathematical programming approach isSupplier selection Delivery SRM Quality Cost Criteria (dimensions)
Subcriteria (performance indices)
Quality defect rate Transportation
cost
Production cost Information
technology Partner
relationship On-time
delivery rate
Figure 4 Model framework for partner assessment criteria. SRM, supplier relationship management.
Table 4 Pairwise comparison scale for analytic network process preferences
Numerical rating Verbal judgments of preferences
1 Equal importance
3 Moderate importance
5 Strong importance
7 Very strong or demonstrated importance
9 Extreme importance
incorporated to select the best partners and production quota for them.
1. Cost model Z1 has two parts, production cost C1 and
transportation cost C2, where quantity, production loss
rate and fixed cost are taken into account; to find mini-mum cost, the objective function is as follows:
Min Z1¼ C1þ C2 C1¼ XI1 i¼1 XJ j¼1 XZ z¼1 dZi:jðQÞPCi:j PK k¼1 Qði:j Þðiþ1:kÞ 1 PDRi:j 2 6 6 6 4 3 7 7 7 5þ XI1 i¼1 XJ j¼1 SOi:jFi:j C2¼ XI1 i¼1 XJ j¼1 XK k¼1 TCði:j Þðiþ1:kÞ Qði:j Þðiþ1:kÞ 1 TDRði:j Þðiþ1:kÞ " #
2. As to quality model Z2, it is hoped that partners minimize
their defective rate through continuous improvement on quality; the objective function is as follows:
Min Z2¼ XI1 i¼1 XJ j¼1 QDRi:j XK k¼1 Qði:j Þðiþ1:kÞ
3. For delivery model Z3,it is expected that partners
maxi-mize their on-time delivery rate to ensure sufficient supply and prevent shortage:
Max Z3¼ XI1 i¼1 XJ j¼1 DRi:j XK k¼1 Qði:j Þðiþ1:kÞ
4. In the SRM model Z4, it could be known that PRC is
composed of PR and IT from the above PRC inferences, only a long-term relationship can enrich the overall com-petence for the supply chain under shared information and technologies, and superior IT may provide more transpar-ent upstream and downstream sourcing schemes for col-laborative works. Hence, PRC maximization may provide quick response to supply chain failure; the objective func-tion is as follows: Max Z4¼ XI1 i¼1 XJ j¼1 SRi:j XK k¼1 Qði:j Þðiþ1:kÞ
5. As the four criteria in this case are in different units, stan-dardization (Ng, 2007) and T-score conversion are neces-sary (Wang and Che, 2007); together with ANP relative
weight, a multi-objective mathematical model (ZT) for
total minimization can be found, and the objective func-tion is as follows:
Min ZT¼ Wð 1Z1þW2Z2 W3Z3 W4Z4Þ
Constraints.
Limits on quantity discount, partner produc-tion capacity and demand quantity under producproduc-tion and transportation losses are taken into account; the models are detailed as follows.The discount changes with quantity and is group-based for each partner, and the equations are
Let Q¼ Qði:jÞðiþ1: kÞ, dZi:jðQÞ ¼
d0i:j, d1i:j, .. . dn1 i:j , dni:j, 8 > > > > > > > > < > > > > > > > > : Q\ Y1 Y1 Q \ Y2 .. . Yn1 Q \ Yn Yn Q d0
i:j¼ 1 . d1i:j. d2i:j. . . dni:j.0, for i¼ 1, 2 . . . , I; j ¼ 1, 2, . . . , j
Yn. Yn1. . Y2. Y1.0
Qði:j Þðiþ1:kÞ˜0 and2 Integer for all i, j, k
The total transportation quantity that partner of each echelon and level received should be in compliance with the upper and lower capacity limits with production and transportation costs taken into consideration; the equations are
Li:j XK k¼1 Qði:j Þðiþ1:kÞ 1 PDRi:j Ui:j, for i¼ 1, 2 . . . , I 1; j ¼ 1, 2, . . . , J Li:k XJ j¼1
Qði1:j Þði:kÞ 1 TDRði1:j Þði:kÞ
Ui:k,
for i¼ 2, 3 . . . , I; k ¼ 1, 2, . . . , K
With production and transportation cost taken into consid-eration, the total transportation quantity that partner of each echelon and level received should be in compliance with the quantity demanded, and finally in compliance with customer demand, and the equations are
XJ
j¼1
Qði1:j Þði:kÞ1 TDRði1:j Þði:kÞ1 PDRi:j
¼ Di:k,
for i¼ 2, 3 . . . , I 1; k ¼ 1, 2, . . . , K
XJ
j¼1
Qði1:j Þði:kÞ1 TDRði1:j Þði:kÞ
¼ DI:k,
for i¼ I; k ¼ 1, 2, . . . , K
The binary constrained variables and the relative weights for each criterion add up to 1, and the variable should not be negative, and the equations are
SOi:j2 ð0, 1Þ, for i ¼ 1, 2 . . . , I 1; j ¼ 1, 2, . . . , J
Wt.0 and
X4
t¼1
Wt¼ 1, for t ¼ 1, 2, 3, 4
Turbo particle swarm optimization solving model for the
mathematical model
Many new methods for particle and velocity updating had been proposed, and our study is based on the population criteria criteria indices indices
A
B
C
D
rule proposed by Shi and Eberhart (1998), where a computa-tion acceleracomputa-tion technique was used to improve the compu-tation efficiency for PSO – the approach is termed TPSO. Figure 6 shows the computation processes for TPSO, and following are the solution steps.
. Step 1: Set number of particles and epoch, inertia weight and maximum velocity, where number of particles is
representative of search points, and a larger epoch would mean a larger number of evolutional searches for all par-ticles, as well as longer time spent. The convergence rate for fitness is dependent on the weight, an optimum weight may be acquired through experimental design and the maximum velocity refers to the maximum scope of move-ment for particle.
. Step 2: Randomly generate an initial velocity and position for each particle, where velocity ranges between 0 and 1, but the position variable should be generated under the conditions that multi-period customer demand, capacity limit and integer requirement are met. Consider the pro-posed problem in which the supply chain system with I
echelons and Jimembers of echelon i, and then we can
set up a search space of PI1i¼1ðJi3Jiþ1Þ dimensions.
Every particle is composed of PI1i¼1ðJi3Jiþ1Þ discrete
points. The value of each discrete point is restricted within the upper and lower capacity limits, and it repre-sents that the quantity of product is transported from the corresponding upstream partner to the relevant down-stream partner. The position of each particle indicates the apropos pattern of the products shipped from each upstream partner to the downstream partner.
. Step 3: Begin iteration from 1.
. Step 4: Calculate fitness function value for each particle
according to the specified objective function ZT, and the
resulting fitness (objective solution) value varies with par-ticle position. In the multi-echelon non-balanced supply chain planning problem, the objective is to minimize the integrated criterion, which integrates cost, quality, delivery and PRC. Then the following equation is adopted as fit-ness function in the light of the description in Section 3.4 and the particle with the minimal fitness value should be reserved during the computation processes.
Fitness function¼ ZT¼ Wð 1Z1þW2Z2 W3Z3 W4Z4Þ
. Step 5: Record pbest (the best solution that the particle has obtained at the current iteration) and gbest (the best solution obtained from the population); if pbest is better than gbest, then revise gbest in memory, at the
Echelon 4 Echelon 1 S1.1 Echelon 2 Echelon 3 S4.1 S4.2 S4.3 S4.4 S2.1 S2.2 S2.3 S1.2 S1.3 S1.4 S3.1 S3.2 S3.3
Figure 7 Supply chain framework.
YES
NO Begin
J=J+1 Randomly generate an initial velocity
and position for each particle
Set J=1 Specify relevant parameters
Record pbest and gbest
Termination condition satisfied
Update particle velocity and position
End
Calculate particle fitness
Computation acceleration technique Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8
same time. Each particle modifies the particle velocity for next search according to gbest, which means storage of optimal feasible solutions from particles of various generations.
. Step 6: Update the current position and velocity of each particle by the following formula.
vnewi ¼ wvold
i þ c1rand1 xpi xoldi
þ c2rand2 xg xoldi
xnewi ¼ Int xold i þ v
new i
vnew
i and voldi are the new/old velocities of particle i. xnewi and
xold
i are the new/old positions of particle i. c1and c2
cogni-tion and social learning factors. xpi is the best solution that particle i has obtained at the current iteration. xgiis the best solution obtained from xpiin the population. w is the inertia
weight. rand1and rand2are random numbers within [0,1].
Int xold i þ vnewi
is the integer value of xold i þ vnewi .
. Step 7: Find feasible values of variables which satisfy the constraints in a stepwise way based on variable-demand dependency after updating the particle position and veloc-ity with a computation acceleration technique, and those repetitive updating calculations can be reduced.
The computation procedure can be algorithmically stated as follows.
For each j ∈(1…n)
when i =1
If Xij ≥dj then
Xij = dj, other variables equal 0
If Xij + X(i +1) j ≥dj then
X(i +1)j = dj –Xij, other variables equal 0
:
If Xij + X(i +1)j +…+ X(m –1)j ≥dj then
X(m–1)j = dj –{ Xij +X(i +1)j +…+X(m – 2)j },
other variables equal 0
If Xij + X(i +1)j +…+ X(m–1)j < dj then
Xmj = dj−{ Xij +X(i +1)j +…+X(m – 1)j }
Endfor
Combinations of Xij variables create dj demands
under n limit conditions, where i ∈(1…m) and j ∈(1…n). combinations of Xij variables dj X11+X21+…+Xm1 d1 X12 +X22 +…+Xm2 d2 : : X1n +X2n +…+Xmn dn
. Step 8: Obtain the optimal result of supply chain planning when the termination condition is specified as maximum epoch number. If the condition has not been met, then repeat Step 4.
Table 5 Related data
PC TC QD DR PRC PDR TDR U L Echelon 1 S1.1 10 2 0.15 0.85 5 0.001 0.002 1000 100 S1.2 12 1 0.02 0.99 5 0.003 0.001 1000 100 S1.3 10 2 0.01 0.88 4 0.001 0.002 1000 100 S1.4 8 1 0.18 0.99 4 0.002 0.001 1000 100 Echelon 2 S2.1 10 2 0.02 0.9 4 0.001 0.002 1000 100 S2.2 12 1 0.2 0.95 4 0.002 0.001 1000 100 S2.3 8 2 0.03 0.97 3 0.001 0.003 1000 100 Echelon 3 S3.1 8 2 0.01 0.98 3 0.001 0.002 1000 100 S3.2 10 1 0.01 0.96 4 0.002 0.001 1000 100 S3.3 12 1 0.02 0.95 5 0.003 0.001 1000 100 Echelon 4 Demand Period 1 2 3 S4.1 300 200 100 S4.2 200 300 500 S4.3 300 100 200 S4.4 200 400 200
PC, production cost; TC, transportation cost; QD, quality defect rate; DR, on-time delivery rate; PRC, partner relationship coefficient; PDR, production loss rate; TDR, transportation loss rate; U, maximum production limit; L, minimum production limit.
Table 6 Quantity discount
Section Interval Discount rate
1 0 to under 400 1
2 400 to under 700 0.9
Numerical examples and
comparison results
We would like to use a 4–3–3–4 supply chain network framework to validate the method proposed here, as shown in Figure 7. The numbers 4, 3, 3, 4 respectively represent the number of partners for echelon 1 to 4, and total market demand is divided into three periods; there are 1000 units for each period. Factors, such as costs (produc-tion, transportation) and capacity limits, are different for each member in the supply chain, as shown in Table 5. Assume that all partners use the same conditions for quan-tity discount as shown in Table 6, and the fixed cost is set at 3000 for all of them. The framework considers a single-product, three-period demand, where products are manu-factured under the same framework and distributed to dif-ferent members on the supply chain, thus the total cost for the supply chain can be minimized through differentiated
quantity quota. S1.1 on echelon 1 is used as an example to
derive the PRC value.
Complete the PR and IT coefficient assessment table, as shown in Table 7. High, medium and low are substituted for 3, 2 and 1 score, then calculate PR and IT for the first
partner as WPR1 ¼ ð3 þ 2 þ 3 þ 3Þ=3ð4Þ ¼ 0:917 and WIT1¼
ð2 þ 3 þ 3Þ=3ð3Þ ¼ 0:889. Assuming sw1¼ sw2¼ 0:5, we can
find that WPRC1 ¼ 0:5ð0:917Þ þ 0:5ð0:889Þ ¼ 0:903, and
PRC1¼ 5 can be obtained from Table 1. PRC2, PRC3 and
PRC4are also figured out in the same way as PRC1; then the
partner PRC table would be completed and shown in Table 8. There are four criteria (dimensions) taken into account: cost, quality, delivery and SRM, and there are six subcriteria (performance indices) taken into consideration: production cost, transportation cost, quality loss rate, one-time delivery rate, PR and IT. Figure 2 presents the dependence and feed-back relationship between them. Then the four criteria and six indices are paired for comparison matrix computation, and each criterion is matched with the indices to form a 6 3 4 matrix A as shown in Table 9. Each index is matched with the criteria to form a 4 3 6 matrix B as shown in Table 10. Similarly, a 4 3 4 matrix C and a 6 3 6 matrix D are also created. Assume that there are consistent among all pairwise comparison matrices and there is no significant correlation between them. Hence, matrices C and D are 0, and those plus the previous matrices A and B, and a 10 3 10 superma-trix come into being as presented in Table 11. Then masuperma-trix multiplication operations are performed, and thus a stable
extreme value is obtained at T16 as presented in Table 12.
The results indicate that the weights of cost, quality, delivery
and SRM are respectively W1¼ 0.2978, W2¼ 0.1634,
W3¼ 0.1839 and W4¼ 0.3549.
Optimization for the multi-objective programming model here is solved with TPSO, and the period 1 demand in Table 5 is taken as an example. The optimum parameter design sug-gested by Shi and Eberhart (1998) is taken into account, where 16 (2 3 2 3 2 3 2) combinations were obtained from
the number of particle (P¼ 10 and 20), number of epoch
(E¼ 500 and 1000), weight (W ¼ 0.9 and 1.2) and maximum
velocity (Vmax¼ 10 and 50). Then optimum parameter
com-bination is obtained through experimental design as shown in Table 13 to find the best fitness function. As to the computa-tion of time, a Pentium(R)4 CPU 3.00 GHz, 512 MB is used jointly with Windows XP, and epoch number is mainly used as termination condition for TPSO. The data of T-score and Table 7 Partner relationship (PR) and information technology (IT) coefficient assessment for each partner
PR IT
Item 1
Partner Partner
Item 2
1 2 3 4 1 2 3 4
1. Mutual trust High Middle High Middle Middle High Middle High 1. Information science
2. Profit sharing Middle High High Middle High Middle Middle Low 2. Information sharing
3. Jointly forge long-term competitive advantages
High Middle Middle Low High Middle Low High 3. Information resource
integration capability
4. Risk sharing High High Middle High
WPR 0.92 0.83 0.83 0.67 0.89 0.78 0.56 0.78 WIT
Table 9 Comparison matrix for the importance of criteria and indices
A matrix Cost Quality Delivery SRM
PC 0.3767 0.1623 0.1322 0.0662 TC 0.2993 0.0999 0.0823 0.0511 QD 0.0677 0.184 0.0638 0.1015 DR 0.0525 0.1239 0.441 0.0829 PR 0.1128 0.2546 0.1655 0.3892 IT 0.091 0.1754 0.1151 0.3091
SRM, supplier relationship management; PC, production cost; TC, transportation cost; QD, quality defect rate; DR, on-time delivery rate; PR, partner relationship; IT, information technology.
Table 8 Partner relationship coefficient (PRC) value of each partner Partner
1 2 3 4
WPRC 0.903 0.806 0.694 0.722
time for each experiment is got through 10 tests, where the mean of the tests is used. The results show 28618.62 will be the optimal T-score with a computation time of 1.83 s, and
the parameter combination is: E¼ 1000, P ¼ 20, W ¼ 1.2, and
Vmax¼ 50.
The GA optimum parameter design is as shown in
Table 14 (Wang and Che, 2007), epoch number (E)¼ 500
and 1000, crossover rate (CR)¼ 0.3 and 0.6, and mutation
rate (MR)¼ 0.03 and 0.05. Finally, we have the optimum
combination: E¼ 1000, PO ¼ 20, CR ¼ 0.5 and MR ¼ 0.06.
T-score for the three periods of demand is derived through optimum parameter designs for TPSO, PSO and GA; 20 tests are ran to take the mean, and the results are compared in Table 15. The purpose of this experiment is to validate the optimization of the fitness function found with TPSO. We can gain a clear understanding from Tables 16 and 17 that the TPSO demands for various periods and t-test for both PSO
and GA all reject H0at the 95% confidence level, which
indi-cate that the average T-score and time for TPSO are superior to those for PSO and GA. In other words, the results validate Table 11 Initial supermatrix
Super matrix Cost Quality Delivery SRM TC PC QD DR PR IT
Cost 0 0 0 0 0.5308 0.5902 0.1758 0.1758 0.1758 0.1664 Quality 0 0 0 0 0.148 0.1305 0.6069 0.0888 0.1285 0.0744 Delivery 0 0 0 0 0.1035 0.0908 0.0888 0.6069 0.0888 0.1757 SRM 0 0 0 0 0.2177 0.1884 0.1285 0.1285 0.6069 0.5834 TC 0.3767 0.1623 0.1322 0.0662 0 0 0 0 0 0 PC 0.2993 0.0999 0.0823 0.0511 0 0 0 0 0 0 QD 0.0677 0.184 0.0638 0.1015 0 0 0 0 0 0 DR 0.0525 0.1239 0.441 0.0829 0 0 0 0 0 0 PR 0.1128 0.2546 0.1655 0.3892 0 0 0 0 0 0 IT 0.091 0.1754 0.1151 0.3091 0 0 0 0 0 0
SRM, supplier relationship management; TC, transportation cost; PC, production cost; QD, quality defect rate; DR, on-time delivery rate; PR, partner relationship; IT, information technology.
Table 12 Supermatrix converged at T16
Super matrix Cost Quality Delivery SRM PC TC QD DR PR IT
Cost 0 0 0 0 0.2977 0.2977 0.2977 0.2977 0.2977 0.2977 Quality 0 0 0 0 0.1634 0.1634 0.1634 0.1634 0.1634 0.1634 Delivery 0 0 0 0 0.1839 0.1839 0.1839 0.1839 0.1839 0.1839 SRM 0 0 0 0 0.355 0.355 0.355 0.355 0.355 0.355 PC 0.1865 0.1865 0.1865 0.1865 0 0 0 0 0 0 TC 0.1387 0.1387 0.1387 0.1387 0 0 0 0 0 0 QD 0.098 0.098 0.098 0.098 0 0 0 0 0 0 DR 0.1464 0.1464 0.1464 0.1464 0 0 0 0 0 0 PR 0.2438 0.2438 0.2438 0.2438 0 0 0 0 0 0 IT 0.1866 0.1866 0.1866 0.1866 0 0 0 0 0 0
SRM, supplier relationship management; PC, production cost; TC, transportation cost; QD, quality defect rate; DR, on-time delivery rate;; PR, partner relationship; IT, information technology.
Table 10 Comparison matrix for the importance of indices and criteria
B matrix PC TC QD DR PR IT
Cost 0.5308 0.5902 0.1758 0.1758 0.1758 0.1664
Quality 0.148 0.1305 0.6069 0.0888 0.1285 0.0744
Delivery 0.1035 0.0908 0.0888 0.6069 0.0888 0.1757
SRM 0.2177 0.1884 0.1285 0.1285 0.6069 0.5834
PC, production cost; TC, transportation cost; QD, quality defect rate; DR, on-time delivery rate; PR, partner relationship; IT, information technology; SRM, supplier relationship management.
Table 16 Turbo particle swarm optimization (TPSO) and particle swarm optimization (PSO) comparisons TPSO PSO Period tpm tpm9 tps tps9 pm pm9 ps ps9 1 28517.75 1.73 672.83 0.33 29837.64 79.54 1448.83 34.22 2 28741.27 1.40 1594.91 0.39 29742.32 91.30 1097.48 29.18 3 28119.64 1.45 1797.50 0.41 29095.97 103.21 1495.19 34.38
t-Test (a¼ 0.05) t-Statistic P-value Decision
1 H0: mtpm mpm 3.48 0.001 Reject H0 H0: mtpm9 mpm9 10.17 1E-12 Reject H0 2 H0: mtpm mpm 2.33 0.013 Reject H0 H0: mtpm9 mpm9 13.78 1E-16 Reject H0 3 H0: mtpm mpm 1.99 0.027 Reject H0 H0: mtpm9 mpm9 13.23 4E-16 Reject H0
tpm, mean of T-score of TPSO; tpm9, mean of time of TPSO; tps, Std. of T-score of TPSO; tps9, Std. of time of TPSO; pm, mean of T-score of PSO; pm9, mean of time of PSO;
ps, Std. of T-score of PSO; ps9, Std. of time of PSO.
Table 13 T-score combinations for turbo particle swarm optimization (TPSO) parameter experimental design
E 500 1000 W Vmax p¼ 10 p¼ 20 p¼ 10 p¼ 20 0.9 10 33949.69 (0.39a) 34497.05 (0.65) 34188.41 (0.71) 33005.87 (1.48) 50 31527.40 (0.43) 31230.82 (0.85) 31156.71 (1.04) 30997.55 (1.90) 1.2 10 34524.42 (0.37) 34020.07 (0.72) 34239.18 (0.76) 34076.00 (1.27) 50 30460.81 (0.50) 28929.61 (0.81) 29759.48 (0.84) 28618.62 (1.83) a, time (s).
Table 14 T-score combinations for genetic algorithm (GA) parameter experimental design
E 500 1000
CR MR po¼ 10 po¼ 20 po¼ 10 po¼ 20
0.3 0.03 37308.51 (47.44a) 37169.66 (102.25) 37088.33 (104.45) 36996.91 (213.45)
0.05 37286.27 (60.77) 37063.52 (125.17) 37190.22 (119.27) 36873.39 (256.53)
0.6 0.03 37312.48 (55.39) 37084.11 (111.97) 36958.80 (112.75) 36795.62 (239.99)
0.05 37258.01 (66.88) 36887.92 (136.47) 36802.43 (128.66) 36316.68 (274.86)
a, time (s).
Table 15 Turbo particle swarm optimization (TPSO), particle swarm optimization (PSO) and genetic algorithm (GA) comparisons
TPSO PSO GA
Period tpb tpm tpm9 pm pm9 gm gm9
1 26467.67 28517.75 1.73 29837.64 79.54 36603.79 279.31
2 26482.63 28741.27 1.40 29742.32 91.30 36999.57 265.86
3 26120.61 28119.64 1.45 29095.97 103.21 35839.82 285.85
tpb, best of T-score of TPSO; tpm, mean of T-score of TPSO; tpm9, mean of time of TPSO; pm, mean of T-score of PSO; pm9, mean of time of PSO; gm, mean of T-score of GA;
Table 17 Turbo particle swarm optimization (TPSO) and genetic algorithm (GA) comparisons TPSO GA Period tpm tpm9 tps tps9 gm gm9 gs gs9 1 28517.75 1.73 672.83 0.33 36603.79 279.31 212.84 18.83 2 28741.27 1.40 1594.91 0.39 36999.57 265.86 248.96 4.71 3 28119.64 1.45 1797.50 0.41 35839.82 285.85 477.40 12.33
t-Test (a¼ 0.05) t-Statistic P-value Decision
1 H0: mtpm mgm 25.92 8E-19 Reject H0 H0: mtpm9 mgm9 70.63 1E-28 Reject H0 2 H0: mtpm mgm 11.38 3E-11 Reject H0 H0: mtpm9 mgm9 264.93 7E-42 Reject H0 3 H0: mtpm mgm 9.46 1E-09 Reject H0 H0: mtpm9 mgm9 110.31 4E-33 Reject H0
tpm, mean of T-score of TPSO; tpm9, mean of time of TPSO; tps, Std. of T-score of TPSO; tps9, Std. of time of TPSO; gm, mean of T-score of GA; gm9, mean of time of GA;
gs, Std. of T-score of GA; gs9, Std. of time of GA.
Table 18 Results of non-balanced supply chain planning by turbo particle swarm optimization (TPSO)
From/To S2.1 S2.2 S2.3
Echelon 1 S1.1 [1a, 1b, 1c] 103d, 105e, 105f (1g, 1h, 1i) – –
S1.2 [3, 3, 3] – 913, 910, 910 (0, 1, 1) – S1.3 – – – S1.4 – – – From/To S3.1 S3.2 S3.3 Echelon 2 S2.1 [1, 1, 1] 0, 0, 103 (0, 0, 1) 101, 103, 0 (1, 1, 0) – S2.2 [2, 2, 2] – – 910, 907, 907 (1, 1, 1) S2.3 – – – From/To S4.1 S4.2 S4.3 S4.4 Echelon 3 S3.1 [0, 0, 1] 0, 0, 101 (0, 0, 1) – – – S3.2 [1, 1, 0] – 99, 0, 0 (1, 0, 0) 0, 101, 0 (0, 1, 0) – S3.3 [3, 3, 3] 301, 201, 0 (1, 1, 0) 103, 301, 501 (1, 1, 1) 301, 0, 201 (1, 0, 1) 201, 401, 201 (1, 1, 1)
a, period-1 production loss of s (PL); b, period-2 PL; c, period-3 PL; d, period-1 transportation quantity between s (TQ); e, period-2 TQ; f, period-3 TQ; g, period-1 transportation loss between s (TL); h, period-2 TL; i, period-3 TL; –, out of operation.
20000 25000 30000 35000 40000 45000 1 133 265 397 529 661 793 925 Generation T-score Period-1 demand Period-2 demand Period-3 demand
the stability and reliability of TPSO in this study. In addition, we found the optimum supply chain network and transpor-tation quantity quota between the partners from the experi-mental results in Table 18, where production loss and transportation loss are included. Taking period-1 demand for example, from the fact that S1.3, S1.4, S2.3 and S3.1 are
out of operation, we find the optimum supply chain network
{2–2–2–4}, and S1.1transports 103 units to S2.1, where one
unit of transportation loss is deducted and one unit of
pro-duction loss is also deducted from S2.1. Those add up to 101
units, which are transported to S3.2, then one unit of
trans-portation loss is deducted and one unit of production loss is
deducted from S3.2; those add up to 99 units, which are
trans-ported to S4.2. Similarly, we can find that on echelon 1, S1.1
and S1.2 should input 103þ1þ913þ3 ¼ 1020 units, after
deducting production and transportation loss for various ech-elons. The total demand 1000 on echelon 4 can be finally met,
where the 1000 come from S4.1¼ 300, S4.2¼ 200, S4.3¼ 300
and S4.4¼ 200. TPSO demand convergence for various
peri-ods in this experiment is presented in Figure 8.
Conclusions
In this study, we propose a multi-echelon, single-product and multi-period non-balance supply chain planning decision sup-port methodology. Based on that, optimum supply chain structure and production–distribution decisions are derived, under the condition that customer demand can be met. The total cost for the supply chain is minimized through differen-tiated quantity quota, and based on this system, companies can effectively pick out suitable supply chain partners and optimal production resource allocation, as well as help deci-sion makers find optimal plans and make an assessment.
In the methods proposed in this study, production and transportation losses are taken into account, and the required quantity of production predicted by upstream partners in advance is incorporated in SRM as criteria for partner eval-uation to perfect the evaleval-uation model. ANP technique is also used to handle subjectively the relative weights between cri-teria. Finally, TPSO algorithm is utilized to solve the multi-objective optimization mathematical model, and then the supply chain network structure and production–distribution plan are determined. A case study with a 4–3–3–4 supply chain framework is used to demonstrate the application of this model. The analysis of the results indicate that the model is capable of quickly and effectively constructing a supply chain structure by selecting the suitable supply chain partners and figuring out the optimal production–distribution quantity. Moreover, comparisons between PSO and GA algo-rithms are made to validate the optimization performance of the TPSO model proposed in this study. We find from the comparisons that the approach created in this study is more stable and reliable.
Acknowledgements
We wish to thank the Editor and the anonymous referees for their valuable comments. This research was partially financially supported by the National Science Council under project No. NSC 98-2410-H-027-002-MY2.
Nomenclature
Notations for the optimization mathematical model develop-ment for non-balanced supply chain planning problems in Section 3.4 are as follows.
i Echelon index for supply chain system,
i¼ 1,2,3 . . . I
I Total number of echelons
j,k Partner index, j¼ 1,2,3 . . . J; k ¼ 1,2,3 . . . K
J,K Number of partners
Q(i.j)(iþ1.k) Quantity of product is transported from
partner j of echelon i to partner k of echelon iþ1
Ui.j Maximum production limit for partner j of
echelon i
Li.j Minimum production limit for partner j of
echelon i dz
i:jðQÞ dzi:jis a function of Q, indicating discount
from partner j of echelon i at stage z TCði, j Þðiþ1:kÞ Unit transportation cost for transit from
partner j of echelon i to partner k of echelon iþ1
TDRði:j Þðiþ1:kÞ Transportation loss rate for transit from
partner j of echelon i to partner k of echelon iþ1
PDRi, j Production loss rate for partner j of echelon i
PCi:j Production cost for partner j of echelon i
SOi:j 1 Operation ongoing for partner j of echelon i
0 Otherwise
Fi:j Fixed cost for partner j of echelon i
Di:j Demand quantity for partner j of echelon i
QDRi:j Defective rate for partner j of echelon i
DRi:j On-time delivery rate for partner j of
echelon i
PRCi:j Partner relationship assessment coefficient
for partner j of echelon i
Wt Weight of criterion t
[X] Integer value of X
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