MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 363718 10.1155/2013/363718 363718 Research Article An Integrated Model of Material Supplier Selection and Order Allocation Using Fuzzy Extended AHP and Multiobjective Programming Li Zhi 1 Wong W. K. 1 Kwong C. K. 2 Tang Yang 1 Institute of Textiles and Clothing The Hong Kong Polytechnic University Hunghom Kowloon Hong Kong polyu.edu.hk 2 Department of Industrial and Systems Engineering The Hong Kong Polytechnic University Hunghom Kowloon Hong Kong polyu.edu.hk 2013 26 2 2013 2013 15 10 2012 14 12 2012 2013 Copyright © 2013 Zhi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a supplier selection and order allocation (SSOA) model to solve the problem of a multiperiod supplier selection and then order allocation in the environment of short product life cycle and frequent material purchasing, for example, fast fashion environment in apparel industry. At the first stage, with consideration of multiple decision criteria and the fuzziness of the data involved in deciding the preferences of multiple decision variables in supplier selection, the fuzzy extent analytic hierarchy process (FEAHP) is adopted. In the second stage, supplier ranks are inputted into an order allocation model that aims at minimizing the risk of material purchasing and minimizing the total material purchasing costs using a dynamic programming approach, subject to constraints on deterministic customer demand and deterministic supplier capacity. Numerical examples are presented, and computational results are reported.

1. Introduction

Most manufacturers nowadays face cutthroat competition in the ever-changing market, which leads to establishment of network organizations. Supply chain management offers an integrated decision-making framework to administer such organizations. One of the key functions of supply chain management is the purchasing strategy. For a general producer, purchased items (e.g., raw materials) can account for 60% of total sales; purchasing shares normally account for 50% to 90% of total turnover in an industrial company . Therefore, it is important to manage the process of supplier selection and the strategy of order allocation in order to construct a competitive and effective purchasing function.

Suitable suppliers can give a company a competitive edge and are instrumental to cost reduction and improvement in product quality. Various techniques have been presented to effectively evaluate and select suppliers. For order allocation problem, material purchasing managers firstly determine the optimal materials quantities purchased from each supplier during the purchasing period. An integrated mathematical programming model has then been established to solve supplier selection and order allocation problems based on various assumptions adapting to real-world production.

1.1. Supplier Selection

To deal with supplier selection, many methodologies have been proposed, including categorical methods, case-based reasoning systems , statistical models , total cost of ownership models , mathematical programming, techniques for order preference by similarity to an ideal solution (TOPSISs) , and analytic hierarchy processes (AHP) . Among these methods, the analytic hierarchy process (AHPs), first proposed by Saaty (1977), is a popular multiple criteria decision-making technique  combining qualitative with quantitative criteria. It is used to rank potential suppliers in a hierarchy system [7, 8]. However, the AHP frequently fails to adequately accommodate the inherent uncertainty and imprecision associated with mapping decision-makers’ perceptions on extracted numbers. It is difficult to respond to preferences of decision-makers by assigning precise numerical values. As a result, the fuzzy analytic hierarchy process (FAHP) is proposed [9, 10] which incorporates both the fuzzy nets theory and the AHP. To prioritize decision variables, Chan and Kumar  proposed the fuzzy extent analytic hierarchy process (FEAHP) which is used in different types of problem [12, 13]. In addition, order allocation is another important topic, especially in a multiple-supplier environment.

1.2. Order Allocation

For material purchasing process, after choosing suitable suppliers, order allocation is the next important stage to determine the optimal materials quantities purchased from each supplier, especially in a multiple-supplier environment. Various techniques have been developed to solve the optimal order allocation problem, including linear programming , nonlinear programming [15, 16], mixed-integer programming , and artificial intelligence technique . These methods mainly focused on single objective optimization, that is, minimize cost. However, in a real-world supply chain environment, the decision-maker must consider uncertain factors along the supply chain. To reduce the risk, many scholars  proposed multiobjective optimization models to identify appealing tradeoffs between two or more conflicting objectives involved in the order allocation process. Furthermore, to deal with uncertainty, Xu and Nozick  proposed a two-stage mixed-integer stochastic programming model, which quantified the tradeoff between the risk and cost on the basis of ordering, thus determining optimal supplier sourcing decisions for varying levels of risk tolerance. To solve the multiperiod order allocation problem, dynamic programming has been utilized. Wagner and Whitin  employed a dynamic programming solution algorithm to solve a dynamic lot-sizing problem with the objective of minimizing the total cost, under time-varying demands for a single item, inventory holding charges, and setup costs. Basnet and Leung  extended the model, which has been proposed by Wagner and Whitin, to a multi-item order allocation problem, with multiple suppliers during a multiperiod planning horizon. Alidaee and Kochenberger  solved the single-sink, fixed charge transportation problem by using a dynamic programming method which is able to determine optimal order quantities from a set of potential suppliers to achieve the minimization of cost based on the total materials demands. Li et al.  compared periodically purchasing from the spot market with signing a long-term contract with a single supplier with consideration of fluctuant stochastic demand and price. Sawik  investigated the problem of multiperiod supplier selection and order allocation in make-to-order environment and proposed a mixed-integer programming approach to incorporate risk that uses conditional value-at-risk via scenario analysis, which is capable of optimizing the dynamic supply portfolio by calculating value-at-risk of cost per part and minimizing expected worst-case cost per part simultaneously.

Based on the aforementioned discussion, few studies [33, 34] have investigated material purchasing problems by integrating supplier evaluation and order allocation together. Specifically, research on material purchasing problems with consideration of the features of the fast fashion environment such as imprecise supplier evaluation measure and multiperiod and multiobjective order allocation, has not been reported so far.

The main purpose of this paper is to develop a supplier selection and order allocation (SSOA) model, which is an effective multicriteria decision-making model, to handle material purchasing. Various features in fast fashion environment will be considered, including imprecise supplier evaluation measure, multiple order allocation objectives, varying purchasing prices, supplier capacities, and customer demands in different periods. The SSOA model will combine FEAHP with multiobjective dynamic linear programming technique to generate effective material purchasing solutions.

The rest of this paper is organized as follows. Section 2 presents the mathematical model of optimal order allocation based on supplier rankings. In Section 3, a supply selection (ranking) and order allocation model is developed. In Section 4, experimental results to validate the performance of the proposed model are presented. Conclusions are drawn and future research is suggested in Section 5.

2. Problem Description

To meet customers’ demands and make a healthy profit, a manufacturer must make an effective sourcing plan based on customers’ orders. In the textile industry, manufacturers always need to source common materials (e.g., white fabric) from suppliers over a planning horizon of different periods in order to encourage competition among suppliers and ensure access to a wide variety of goods or services. Therefore, selection of suitable suppliers and an optimal order allocation plan become crucial. This study proposes a model to handle optimal order allocation based on supplier ranking.

The assumptions of this study are as follows.

Each supplier can provide materials for manufacturers, and suppliers have different production capacities.

Manufacturers can get information on each supplier in terms of production capacity and price at the beginning of each planning horizon.

There is no inventory of materials, and manufacturers need to purchase all materials for production.

Let I={1,,N} represent the set of N suppliers, J={1,,M} the set of M customer orders, and T={1,,H} the set of T planning periods. xit denotes the order quantity from supplier i. cit denotes the capacity of supplier i (iI) in period t. pit denotes the unit price of the material purchased from supplier i (iI) in period tT. Dt represents manufacturers’ demands for materials based on customers’ orders, known ahead of material purchasing. ri represents the relative risk index of supplier i, which indicates that a higher value of ri can generate a higher real purchasing risk. Supplier ranking and order allocation investigated in this research can be formulated as follows: (1)minE(xit)=min(t=1Ti=1npit·xit),(2)minF(xit)=min(t=1Ti=1nri·xit),(3)s.t.,(0xitcit),i,t,(4)i=1nxit=Dt,i.

Formula (1) minimizes the total purchasing costs of materials for all customers’ orders. Formula (2) minimizes the total purchasing risks (e.g., delay risk and defect risk). Formula (3) represents that the order quantities are not more than the supplier’s maximum capacity in any purchasing periods. Formula (4) requires that the supply must satisfy the demands of manufacturers.

3. Methodologies for Supplier Selection and Order Allocation

This paper proposes an effective supplier selection (ranking) and order allocation (SSOA) model based on the FEAHP and dynamic programming (DP). This model comprises an FEAHP-based supplier/criteria ranking process and a DP-based order allocation process (Figure 1).

The processes of SSOA.

The details of the SSOA model are described as follows.

3.1. The FEAHP Method

The AHP has been widely used to address multicriteria decision-making problems. It only requires a discrete scale from one to nine. However, human judgement is uncertain of criteria’s preferences. The linguistic assessment of human feelings and judgement is vague and cannot be represented reasonably in precise numbers. Hence, triangular fuzzy numbers are used to decide the priority of decision variables. Synthetic extent analysis is used to decide the final priority weights based on triangular fuzzy numbers.

3.1.1. Triangular Fuzzy Numbers and Representation of Preferences

A fuzzy set [35, 36] is characterized by a membership function, which assigns to each object a grade of membership ranging from 0 to 1. The general terms “large”, “medium” and “small” are used in fuzzy set to capture a range of numerical values. If l, m, and u, respectively, denote the smallest possible value, the most promising value, and the largest possible value that describe a fuzzy event, the triangular fuzzy number (TFN) can be denoted as a vector (l,m,u), where lmu. When l=m=u, it is a nonfuzzy number by convention. The membership function can be defined as (5)μ(xM)={(x-1)(m-1),1xm,(u-x)(u-m),mxu,0otherwise.

TFNs M1, M3, M5, M7, and M9 are used to represent the pairwise comparison of decision variables from “equal” to “absolutely preferred”, and TFNs M2, M4, M6, and M8 represent the middle preference values among them. Figure 2 shows the membership functions of the TFNs, Mi=(li,mi,ui), where i=1,2,,9, and li,mi,ui are the lower, middle, and upper values of the fuzzy number Mi, respectively.

The membership functions of triangular fuzzy numbers.

3.1.2. Fuzzy Extent Analytic Hierarchy Process (FEAHP)

The FEAHP was originally introduced by Chang (1996). Some calculation steps are essential and explained as follows.

Let X={x1,x2,x3,,xn} be an object set and G={g1,g2,g3,,gn} a goal set. According to Chang’s method, each object is taken, and extent analysis of each goal is performed. Therefore, m extent analysis values for each object can be obtained with the following signs: Mgi1,Mgi2,,Mgim, i=1,2,,n, where Mgij (j=1,2,,m) are the triangular fuzzy numbers (TFNs).

Step 1.

Constructing a hierarchical structure with decision elements, decision-makers are required to make pairwise comparisons between decision alternatives and criteria using a nine-point scale (Table 1). All matrices are developed, and all pairwise comparisons are obtained from each decision-maker.

Triangular fuzzy numbers.

Linguistic variables Positive triangular fuzzy number Positive reciprocal triangular fuzzy number
Extremely strong ( 8,9 , 9 ) ( 1 / 9,1 / 9,1 / 8 )
Intermediate ( 7,8 , 9 ) ( 1 / 9,1 / 8 ,1/7)
Very strong ( 6,7 , 8 ) ( 1 / 8,1 / 7,1 / 6 )
Intermediate ( 5,6 , 7 ) ( 1 / 7,1 / 6,1 / 5 )
Strong ( 4,5 , 6 ) ( 1 / 6,1 / 5,1 / 4 )
Intermediate ( 3,4 , 5 ) ( 1 / 5,1 / 4,1 / 3 )
Moderately strong ( 2,3 , 4 ) ( 1 / 4,1 / 3,1 / 2 )
Intermediate ( 1,2 , 3 ) ( 1 / 3,1 / 2,1 )
Equally strong ( 1,1 , 2 ) ( 1 / 2,1 , 1 )
Step 2.

The fuzzy synthetic extent value with respect to the ith object is defined as: (6)Sj=j=1mMgij×[i=1nj=1mMgij]-1.

To obtain j=1mMgij, the fuzzy addition operation of m extent analysis values for a particular matrix is performed as (7)j=1mMgij=(j=1mlj,j=1mmj,j=1muj).

To obtain [i=1nj=1mMgij], the fuzzy addition operation of Mgij(j=1,2,,m) values is performed as (8)i=1nj=1mMgij=(i=1nli,i=1nmi,i=1nui).

And the inverse of the previous vector is computed as (9)[i=1nj=1mMgij]-1=(1i=1nui,1i=1nmi,1i=1nli).

Step 3.

As M1=(l1,m1,u1) and M2=(l2,m2,u2) are two triangular fuzzy numbers, the degree of M2=(l2,m2,u2)M1=(l1,m1,u1) possibility of is defined as (10)V(M2M1)=supyx{min(μM1(x),μM2(y))} can be expressed as follows: (11)V(M2M1)=hgt(M2M1)=μM2(d)={1,ifm2m1,|l1-u2(m2-u2)-(m1-l1)|,otherwise.

Equation (11) (Figure 3) indicates that d is the ordinate of the highest intersection point D between μM1 and μM2. To compare M1 and M2, the values of V(M2M1) and V(M1M2) are needed.

The intersection between M1 and M2.

Step 4.

The degree of possibility that the convex fuzzy number is greater than k convex fuzzy Mi (i=1,2,,k) numbers can be defined by (12)V(MM1,M2,,Mk)=V[(MM1)and(MM2),,and(MMk)]=minV(MMi),(i=1,2,3,,k).

Assume that d(Ai)=minV(SiSk) for k=1,2,,n; k1. Then, the weight vector is given by (13)W=(d(A1),d(A2),,d(An))τ, where Ai  (i=1,2,3,,n) are n elements.

Step 5.

Via normalization, the normalized weight vectors are (14)W=(d(A1),d(A2),,d(An))τ, where W is a nonfuzzy number.

The upward composition of these weights (from the lowest to the top level) generates the ranking scores (weights) of elements at the lowest level (i.e., suppliers) in fulfilling the topmost objective (i.e., suppliers ranking).

3.1.3. FEAHP-Based Supplier Ranking

As discussed in the introduction, supplier ranking gives decision-makers an effective technique to choose suitable suppliers. In this research, supplier ranking is implemented by the FEAHP method. The procedure is detailed as follows.

Step 1 (define criteria for supplier selection).

To define effective criteria for supplier selection, this research collects promising candidate criteria based on existing research results (Dickson 1966; Chan and Kumar  2007), and any additional criteria deemed important for manufacturers are included.

On the basis of the candidate criteria selected, structured interviews are used to evaluate these criteria by three senior specialists, including a senior designer and two purchasing managers denoted by (R1), (R2), and (R3), respectively. To evaluate candidate criteria, the respondents are requested to use the linguistic assessment of human feelings (Table 1). Upon receiving the inputs of the respondents, the criteria are identified and averaged. If there are too many criteria, the pairwise comparison can become a difficult and time-consuming process. To overcome these problems, the criteria’s average value in top 5 is selected. In this paper, the 5 final criteria are (1) overall cost of products (C1); (2) quality of product (C2); (3) risk factors (C3); (4) a supplier’s profile (C4); (5) service performance of a supplier (C5).

Step 2 (define subcriteria for supplier selection).

To evaluate suppliers more precisely, each selected criterion in Step 1 needs to be further represented by several subcriteria. The identification and selection of these subcriteria can be implemented as described in Step 1. If the subcriteria are still obscure, they can be re-represented by sub-subcriteria using the same process.

Step 3 (structure the hierarchical model and each criterion’s weight).

In this step, the FEAHP hierarchy model is built, and the weight of each supplier selection model is calculated. The developed FEAHP model, based on the identified criteria, subcriteria and subcriteria’s subcriteria, has five levels: goals, criteria, subcriteria, subcriteria’s subcriteria, and candidates. Figure 4 shows the 5-level hierarchy for supplier selection. The goal of supplier selection for manufacturers is identified on the first level. The second level (criteria) contains 5 criteria mentioned in Step 1. The third and fourth levels consist of subcriteria and subcriteria’s subcriteria (subcriteria’s subcriteria are not considered in this paper’s numerical experimentation). The lowest level of the hierarchy contains alternatives. That is, different suppliers are evaluated in order to pick the best ones. As shown in Figure 4, different suppliers are used to represent the arbitrary ones which manufacturers wish to evaluate.

General hierarchy for supplier selection.

The FEAHP model (Figure 4) is generally applicable to any type of supplier selection by manufacturers as it covers many important factors and their related criteria, subcriteria, and subcriteria’s subcriteria.

In order to obtain the priority weight of each criterion on each level, a second structure is done in a similar manner as Step 1. The interview consisting of factors on each level of the FEAHP model is used to collect the judgments of pairwise comparisons from all evaluation team members. This judgments is performed using pairwise comparisons, which are elaborated in Section 3.1.1. An example of the pairwise comparison matrix is shown in Table 2.

The fuzzy evaluation of criteria of the overall objective.

C 1 C 2 C 3 C 4 C 5 Weights
C 1 ( 1,1 , 1 ) ( 2,3 , 4 ) ( 3,4 , 5 ) ( 5,6 , 7 ) ( 2,3 , 4 ) 0.43
C 2 ( 0.25,0.33,0.5 ) ( 1,1 , 1 ) ( 3,4 , 5 ) ( 2,3 , 4 ) ( 4,5 , 6 ) 0.33
C 3 ( 0.2,0.25,0.33 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) ( 3,4 , 5 ) ( 2,3 , 4 ) 0.13
C 4 ( 0.14,0.17,0.2 ) ( 0.25,0.33,0.5 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) ( 2,3 , 4 ) 0.02
C 5 ( 0.25,0.33,0.5 ) ( 0.17,0.2,0.25 ) ( 0.25,0.33,0.5 ) ( 0.25,0.33,0.5 ) ( 1,1 , 1 ) 0.09
Step 4 (measure supplier performance and identify supplier priority).

After obtaining the priority weight of each criterion and subcriterion, the third structured interview is designed and modified. This interview collects the weights of alternatives to identify the best suppliers.

The priority weight is determined for alternatives in this step. The competitive rivals that are supposed to be suppliers for manufacturers are compared by each subcriteria standard. After finding the local weight of each alternative in subcriteria, the global weight of each alternative in each criterion can be calculated. The evaluation of the global weight of each alternative can be obtained by multiplying the global weights of subcriteria and the local weight of each alternative. Based on the global priority, the weight of each alternative can be evaluated and summarized. An example of FEAHP-based supplier ranking is described in Section 4.1.2.

3.2. Dynamic Programming

As purchasing price is time-varying in the model, the cost objective is judiciously captured by the following dynamic value function: (15)V1,t=min0xitcit{i=1npitxit+V1,t+1(Dt-i=1nxit)}, where stage is decision dates in time periods, t=1,2,,T. The decision variable is the quantities ordered from supplier i, xit=0,1,,cit.

To account for both objectives, a distance-to-ideal framework is employed to integrate the risk and cost objective functions, using the optimal values of individual objectives obtained earlier.

To incorporate the ideal values of risk and cost, the sum (weights) of deviations from such ideal values is minimized. Hence, a dynamic value function is derived as follows: (16)V2,t=min0xitcit{wc1i=1npitxit+wc3i=1nxitwi+V2,t+1(Dt-i=1nxit)}, where V2,t is the minimum total weighted deviation. wc1 is the cost weight defined by decision-makers using the FEAHP. wc3 is the risk weight defined by decision-makers using the FEAHP.

4. Numerical Experiments

To validate the effectiveness of the proposed SSOA model, a series of experiments are conducted to obtain industrial data from an apparel manufacturer. The manufacturer needs to purchase a specified amount of raw fabric from 3 appropriate material suppliers for the production of its customers’ orders. The 3 suppliers have been selected from its N collaborative suppliers. The manufacturer seeks to determine how much should be purchased from the 3 key suppliers in order to minimize its overall cost and maximize its utility over a multiperiod planning horizon.

4.1. Experiment for FEAHP-Based Supplier Ranking

The FEAHP starts from the pairwise comparison matrices of five criteria (Table 2). Based on these matrices, the weights of suppliers and criteria are calculated and presented in Table 3. The supplier’s information and manufacturer’s demands are shown in Tables 4 and 5, respectively. The solutions to single objectives are shown in Table 6, where cost is minimum. Table 7 shows the order allocation based on minimum risk.

The fuzzy evaluation of the attributes of criterion C1.

A 1 A 2 A 3 A 4 Weights
A 1 ( 1,1 , 1 ) ( 2,3 , 4 ) ( 3,4 , 5 ) ( 3,4 , 5 ) 0.49
A 2 ( 0.25,0.33,0.5 ) ( 1,1 , 1 ) ( 3,4 , 5 ) ( 2,3 , 4 ) 0.31
A 3 ( 0.2,0.25,0.33 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) ( 3,4 , 5 ) 0.09
A 4 ( 0.2,0.25,0.33 ) ( 0.25,0.33,0.5 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) 0.11

The fuzzy evaluation of the attributes of criterion C2.

A 5 A 6 A 7 Weights
A 5 ( 1,1 , 1 ) ( 4,5 , 6 ) ( 2,3 , 4 ) 0.55
A 6 ( 0.17,0.2,0.25 ) ( 1,1 , 1 ) ( 3,4 , 5 ) 0.19
A 7 ( 0.25,0.33,0.5 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) 0.26

The fuzzy evaluation of the attributes of criterion C3.

A 8 A 9 A 10 A 11 Weights
A 8 ( 1,1 , 1 ) ( 4,5 , 6 ) ( 4,5 , 6 ) ( 1,1 , 2 ) 0.59
A 9 ( 0.17,0.2,0.25 ) ( 1,1 , 1 ) ( 4,5 , 6 ) ( 2,3 , 4 ) 0.39
A 10 ( 0.17,0.2,0.25 ) ( 0.17,0.2,0.25 ) ( 1,1 , 1 ) ( 3,4 , 5 ) 0.01
A 11 ( 0.5,1 , 1 ) ( 0.25,0.33,0.5 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) 0.01

The fuzzy evaluation of the attributes of criterion C4.

A 12 A 13 A 14 Weights
A 12 ( 1,1 , 1 ) ( 3,4 , 5 ) ( 3,4 , 5 ) 0.51
A 13 ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) ( 3,4 , 5 ) 0.18
A 14 ( 0.2,0.25,0.33 ) ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) 0.31

The fuzzy evaluation of the attributes of criterion C5.

A 15 A 16 Weights
A 15 ( 1,1 , 1 ) ( 3,4 , 5 ) 0.52
A 16 ( 0.2,0.25,0.33 ) ( 1,1 , 1 ) 0.48

This paper presents the results of 6 experiments.

The criteria for selection of global suppliers are as follows:

overall cost of products (C1): product price (A1), freight cost (A2), penalty for delayed payment (A3), and tariff and custom duties (A4);

product quality (C2): rejection rate (A5), response to changes (A6), and rate of warranty claims (A7);

risk factors (C3): lead time (A8), political stability (A9), geographical location (A10), and inability to meet further requirements (A11);

supplier’s profile (C4): financial status (A12), performance history (A13), and production capacity (A14);

service performance of suppliers (C5): remedy for quality problems (A15) and delivery schedule (A16).

These criteria can be found in the hierarchical structure shown in Figure 5.

Hierarchy for supplier selection.

4.1.1. Determination of Criteria and Subcriteria Weights

The example of the pairwise comparison matrices shows that the fifth row and column attach importance to the row’s criterion relative to the column’s criterion (Table 2).

Due to a good cost performance, the criterion for the first row is slightly preferred to the one on product quality, risk factors, and service performance of suppliers (the fuzzy values of (2,3,4), (3,4,5), and (2,3,4), resp.), which is strongly preferred to the supplier’s profile (the value of (5,6,7)). Due to a good quality performance, the criterion for the second row and column is moderately more important than the service performance of suppliers (the value of (4,5,6)). Having fewer risk factors, the third row’s criterion is slightly preferred to a good profile (value of (3,4,5)). Decision makers only need to fill in the upper half of the comparison matrix by assuming that the pairwise comparison of cost and service performance is (2,3,4), following the pairwise comparison of service performance and cost (0.25,0.33,0.5). The value of (1,1,1) is assigned to diagonal elements.

Calculate various decision alternatives of fuzzy numbers based on Section 3 as follows: (17)Sc1=(13,17,21)×(156.95,145.78,135.16)=(0.23,0.37,0.60),Sc2=(10.25,13.33,16.5)×(156.95,145.78,135.16)=(0.18,0.29,0.47),Sc3=(6.40,8.50,10.67)×(156.95,145.78,135.16)=(0.11,0.19,0.30),Sc4=(3.59,4.75,6.03)×(156.95,145.78,135.16)=(0.06,0.10,0.17),Sc5=(1.92,2.20,2.75)×(156.95,145.78,135.16)=(0.03,0.05,0.08). Compare the following decision alternatives: (18)V(Sc1Sc2)=1,V(Sc1Sc3)=1,V(Sc1Sc4)=1,V(Sc1Sc5)=1,V(Sc2Sc1)=0.75,V(Sc2Sc3)=1,V(Sc2Sc4)=1,V(Sc2Sc5)=1,V(Sc3Sc1)=0.29,V(Sc3Sc2)=0.54,V(Sc3Sc4)=1,V(Sc3Sc5)=1,V(Sc4Sc1)=0.27,V(Sc4Sc2)=0.05,V(Sc4Sc3)=0.42,V(Sc4Sc5)=1,V(Sc5Sc1)=0.87,V(Sc5Sc2)=0.72,V(Sc5Sc3)=0.33,V(Sc5Sc4)=0.21. Calculate the following decision alternatives’ weights: (19)d(c1)=min(1,1,1,1)=1,d(c2)=min(0.75,1,1,1)=0.75,d(c3)=min(0.29,0.54,1,1)=0.29,d(c4)=min(0.27,0.05,0.42,1)=0.05,d(c5)=min(0.87,0.72,0.33,0.21)=0.21. Priority weights form W=(1,0.75,0.29,0.05,0.21) vector. Normalizing the W vector: (20)w1=11+0.75+0.29+0.05+0.21=0.43,w2=0.751+0.75+0.29+0.05+0.21=0.33,w3=0.291+0.75+0.29+0.05+0.21=0.13,w4=0.051+0.75+0.29+0.05+0.21=0.02,w5=0.211+0.75+0.29+0.05+0.21=0.09.

After normalization of the values, priority weights of the main goal are calculated as (0.43,0.33,0.13,0.02,0.09). The results (principal vectors) show that the criteria have the following approximate priority weights: cost (0.43), quality (0.33), risk (0.13), supplier’s profile (0.02) and service performance of suppliers (0.09).

Different attributes are compared by each criterion separately with the same procedure as discussed above. The fuzzy evaluation matrices of attributes and the weight vectors of subcriteria are shown in Tables 3, 4, 5, 6, and 7.

4.1.2. Calculate the Suppliers’ Weights

Similarly, the fuzzy evaluation matrices of decision alternatives and the corresponding weight vector of each alternative with respect to the corresponding attributes are determined. The priority weights of suppliers with respect to each criterion are given by adding each supplier’s weight to each corresponding attribute’s weight. The results are shown in Tables 8, 9, 10, 11, and 12.

The fuzzy evaluation of the subcriteria of criterion C1.

A 1 A 2 A 3 A 4 Alternative priority
Weight 0.49 0.31 0.09 0.11 weight
Alternatives
S 1 0.51 0.51 0.69 0.87 0.57
S 2 0.23 0.23 0.08 0.01 0.19
S 3 0.26 0.26 0.23 0.12 0.24

The fuzzy evaluation of the subcriteria of criterion C2.

A 5 A 6 A 7 Alternative priority
Weight 0.55 0.19 0.26 weight
Alternatives
S 1 0.42 0.49 0.53 0.46
S 2 0.28 0.23 0.15 0.24
S 3 0.30 0.28 0.32 0.30

The fuzzy evaluation of the subcriteria of criterion C3.

A 8 A 9 A 10 A 11 Alternative priority
Weight 0.59 0.39 0.01 0.01 weight
Alternatives
S 1 0.51 0.53 0.69 0.68 0.52
S 2 0.21 0.23 0.08 0.11 0.22
S 3 0.28 0.24 0.23 0.21 0.26

The fuzzy evaluation of the subcriteria of criterion C4.

A 12 A 13 A 14 Alternative priority
Weight 0.51 0.18 0.31 weight
Alternatives
S 1 0.39 0.49 0.51 0.45
S 2 0.28 0.21 0.17 0.23
S 3 0.33 0.30 0.32 0.32

The fuzzy evaluation of the subcriteria of criterion C5.

A 15 A 16 Alternative priority
Weight 0.52 0.48 weight
Alternatives
S 1 0.39 0.35 0.37
S 2 0.33 0.41 0.37
S 3 0.28 0.24 0.26

Finally, the priority weight of each supplier can be calculated by multiplying the weight of each corresponding criterion. The results are shown in Table 13. The summary of the overall attributes is shown in Table 13. It should be noted that among the three given suppliers, “S1” has the highest weight and therefore is selected as the best supplier to satisfy the goals and objectives of the manufacturing company. Table 13 also shows the final score of each supplier s’ results and rankings. As can be seen, S1 (0.5) scores higher than S2 (0.23) and S3 (0.23). The important results are shown in Figures 6 and 7.

Demand information.

Period 1 2 3
Demand 6 6 6

The suppliers based on the criteria.

Final priority weights of the suppliers.

4.2. Experiment for Dynamic Order Allocation

A real apparel manufacturer purchasing environment usually has the following four scenarios. (1) A manufacturer’s demand for common material is the same in all planning periods; in order to obtain orders steadily, suppliers reserve a certain capacity and offer a reasonable price. (2) Demands for common material are steady, but suppliers do not reserve a certain capacity; so, price and capacity fluctuate in different planning periods. (3) Suppliers’ prices are different throughout the planning period. (4) Suppliers’ capacities and prices are different throughout the planning periods.

4.2.1. Scenario 1: Suppliers Can Reserve a Certain Capacity and Offer a Reasonable Price in All Planning Periods

After getting the weight score of each supplier and criterion in the first stage, wc1=0.43 and wc2=0.13. In addition to having the capacity and price information on each supplier, the dynamic approach mentioned in Section 3.2 can be rewritten as follows: (21)V2,t=min0xitcit{0.43i=1npitxit+0.13i=1nxitwimin0xitcit2+V2,t+1(Dt-i=1nxit)}.

If the manufacturer entirely just focuses on minimizing cost alone, the risk will increase substantially by 215% than entirely focusing on minimizing risk. On the opposite extreme, if the manufacturer just entirely focuses on minimizing risk, this would raise its cost by 20% than just entirely focuses on minimizing cost only.

By applying the bi-objective dynamic function, the trade-off solution incurs at 13.9% and 94% higher than the objective of minimizing cost and minimizing risk respectively. These experimental results show that the proposed dynamic programming approach can generate a better trade-off solution than entirely focusing on minimizing cost and minimizing risk alone. (Tables 13, 14, 15, 16, and 17).

Price and capacity information.

Supplier Ordering price (per unit) Capacity
Period 1 Period 2 Period 3
S 1 12 12 12 6
S 2 10 10 10 6
S 3 11 11 11 6

Optimal order quantities with respect to minimum risk.

Period 1 2 3 Total risk Total cost
S 1 6 6 6 30 216
S 2
S 3

Optimal order quantities with respect to minimum cost.

Period 1 2 3 Total risk Total cost
S 1 94.7 180
S 2 6 6 6
S 3

Optimal order quantity with respect to minimum cost and risk.

Period 1 2 3 Total risk Total cost
S 1 4 3 2 58 205
S 2 1 1
S 3 1 3 3
4.2.2. Scenario 2: Suppliers Have Different Capacities and Offer Different Prices in Different Planning Periods

After getting the weight score of each supplier and criterion in the first stage, wc1=0.43 and wc2=0.13. In addition to having the capacity and price information on each supplier, the dynamic approach mentioned in Section 3.2 can be rewritten as follows: (22)V2,t=min0xitcit{0.43i=1npitxit+0.13i=1nxitwimin0xitcit2+V2,t+1(Dt-i=1nxit)}.

If the manufacturer entirely just focuses on minimizing cost alone, the risk will increase substantially by 141% than entirely focusing on minimizing risk. On the opposite extreme, if the manufacturer just entirely focuses on minimizing risk, this would raise its cost by 20.7% than just entirely focuses on minimizing cost only.

By applying the bi-objective dynamic function, the trade-off solution incurs at 9.2% and 77% higher than the objective of minimizing cost and minimizing risk respectively. These experimental results show that the proposed dynamic programming approach can generate a better trade-off solution than entirely focusing on minimizing cost and minimizing risk alone. (Tables 18, 19, 20, 21, and 22).

Demand information.

Period 1 2 3
Demand 6 6 6

Price and capacity information.

Supplier Ordering price (per unit) Capacity
Period 1 Period 2 Period 3
S 1 12 11 14 5
S 2 11 12 10 6
S 3 9 11 10 4

Optimal order quantities with respect to minimum risk.

Period 1 2 3 Total risk Total cost
S 1 6 6 6 30 222
S 2
S 3

Optimal order quantities with respect to minimum cost.

Period 1 2 3 Total risk Total cost
S 1 5 72.24 184
S 2 2 2
S 3 4 1 4

Optimal order quantity with respect to minimum cost and risk.

Period 1 2 3 Total risk Total cost
S 1 3 5 3 53.1 201
S 2 3
S 3 3 1

If the manufacturer entirely just focuses on minimizing cost alone, the risk will increase substantially by 61.1% than entirely focusing on minimizing risk. On the opposite extreme, if the manufacturer just entirely focuses on minimizing risk, this would raise its cost by 12.2% than just entirely focuses on minimizing cost only.

By applying the bi-objective dynamic function, the trade-off solution incurs at 4.1% and 17.8% higher than the objective of minimizing cost and minimizing risk respectively. These experimental results show that the proposed dynamic programming approach can generate a better trade-off solution than entirely focusing on minimizing cost and minimizing risk alone. (Tables 23, 24, 25, 26, and 27).

Demand information.

Period 1 2 3 4
Demand 6 6 6 6

Price and capacity information.

Supplier Ordering price (per unit) Capacity
period 1 2 3 4
S 1 12 11 14 10 5
S 2 11 12 10 11 6
S 3 9 11 10 11 4

Optimal order quantities with respect to minimum risk.

Period 1 2 3 4 Total risk Total cost
S 1 5 5 5 5 54.76 276
S 2
S 3 1 1 1 1

Optimal order quantities with respect to minimum cost.

Period 1 2 3 4 Total risk Total cost
S 1 5 88.21 246
S 2 6 6
S 3 6 1

Optimal order quantity with respect to minimum cost and risk.

Period 1 2 3 4 Total risk Total cost
S 1 5 5 5 64.48 256
S 2 2
S 3 1 1 4 1
4.2.3. Scenario 3: The Manufacturer’s Demands and the Supplier’s Prices Are Different throughout the Planning Periods

After getting the weight score of each supplier and criterion in the first stage, wc1=0.43 and wc2=0.13. In addition to having the capacity and price information on each supplier, the dynamic approach mentioned in Section 3.2 can be rewritten as follows: (23)V2,t=min0xitcit{0.43i=1npitxit+0.13i=1nxitwimin0xitcit2+V2,t+1(Dt-i=1nxit)}.

If the manufacturer entirely just focuses on minimizing cost alone, the risk will increase substantially by 28% than entirely focusing on minimizing risk. On the opposite extreme, if the manufacturer just entirely focuses on minimizing risk, this would raise its cost by 4% than just entirely focuses on minimizing cost only.

By applying the bi-objective dynamic function, the trade-off solution incurs at 1% and 16% higher than the objective of minimizing cost and minimizing risk respectively. These experimental results show that the proposed dynamic programming approach can generate a better trade-off solution than entirely focusing on minimizing cost and minimizing risk alone. (Tables 28, 29, 30, 31, and 32).

Demand information.

Period 1 2 3
Demand 11 9 10

Price and capacity information.

Supplier Ordering price (per unit) Capacity
Period 1 Period 2 Period 3
S 1 95 95 99 4
S 2 87 87 89 4
S 3 93 91 91 6

Optimal order quantities with respect to minimum risk.

Period 1 2 3 Total risk Total cost
S 1 4 4 4 90.69 2802
S 2 1
S 3 6 5 6

Optimal order quantities with respect to minimum cost.

Period 1 2 3 Total risk Total cost
S 1 1 116.8 2706
S 2 4 4 4
S 3 6 5 6

Optimal order quantity with respect to minimum cost and risk.

Period 1 2 3 Total risk Total cost
S 1 4 4 0 105.21 2728
S 2 4 4 4
S 3 3 1 6

If the manufacturer entirely just focuses on minimizing cost alone, the risk will increase substantially by 32% than entirely focusing on minimizing risk. On the opposite extreme, if the manufacturer just entirely focuses on minimizing risk, this would raise its cost by 4% than just entirely focuses on minimizing cost only.

By applying the bi-objective dynamic function, the trade-off solution incurs at 1% and 15.7% higher than the objective of minimizing cost and minimizing risk respectively. These experimental results show that the proposed dynamic programming approach can generate a better trade-off solution than entirely focusing on minimizing cost and minimizing risk alone. (Tables 33, 34, 35, 36, and 37).

Demand information.

Period 1 2 3 4
Demand 11 9 10 12

Price and capacity information.

Supplier Ordering price (per unit) Capacity
period 1 2 3 4
S 1 95 95 99 96 5
S 2 87 87 89 90 4
S 3 93 91 91 93 6

Optimal order quantities with respect to minimum risk.

Period 1 2 3 4 Total risk Total cost
S 1 5 5 5 5 121.38 3950
S 2 1
S 3 6 4 5 6

Optimal order quantities with respect to minimum cost.

Period 1 2 3 4 Total risk Total cost
S 1 1 2 160.29 3816
S 2 4 4 4 4
S 3 6 5 6 6

Optimal order quantity with respect to minimum cost and risk.

Period 1 2 3 4 Total risk Total cost
S 1 5 5 5 140.43 3853
S 2 4 4 4 4
S 3 2 6 3
4.2.4. Scenario 4: Demands, Price, and Capacity Are Different throughout the Planning Periods

After getting the weight score of each supplier and criterion in the first stage, wc1=0.43 and wc2=0.13. In addition to having the capacity and price information on each supplier, the dynamic approach mentioned in Section 3.2 can be rewritten as follows: (24)V2,t=min0xitcit{0.43i=1npitxit+0.13i=1nxitwimin0xitcit2+V2,t+1(Dt-i=1nxit)}.

If the manufacturer entirely just focuses on minimizing cost alone, the risk will increase substantially by 20% than entirely focusing on minimizing risk. On the opposite extreme, if the manufacturer just entirely focuses on minimizing risk, this would raise its cost by 5% than just entirely focuses on minimizing cost only.

By applying the bi-objective dynamic function, the trade-off solution incurs at 1% and 9% higher than the objective of minimizing cost and minimizing risk respectively. These experimental results show that the proposed dynamic programming approach can generate a better trade-off solution than entirely focusing on minimizing cost and minimizing risk alone. (Tables 38, 39, 40, 41, and 42).

Demand information.

Period 1 2 3
Demand 11 9 10

Price and capacity information.

Supplier Ordering price (per unit) Capacity
period 1 2 3 1 2 3
S 1 95 95 99 4 6 5
S 2 87 87 89 4 7 7
S 3 93 91 91 6 5 6

Optimal order quantities with respect to minimum risk.

Period 1 2 3 Total risk Total cost
S 1 4 6 5 85.7 2818
S 2 1
S 3 6 3 5

Optimal order quantities with respect to minimum cost.

Period 1 2 3 Total risk Total cost
S 1 1 103.5 2688
S 2 4 7 7
S 3 6 2 3

Optimal order quantity with respect to minimum cost and risk.

Period 1 2 3 Total risk Total cost
S 1 4 2 93.2 2708
S 2 4 7 4
S 3 3 6
5. Conclusions

This paper investigates the topic of multiobjective order allocation based on supplier selection in the purchasing stage with single material and multiple suppliers taken into consideration. A mathematical model for investigation is established, which considers minimizing the total cost and risk in all purchasing processes. These objectives are particularly useful for manufacturing companies to survive in a make-to-order environment and improve the performance of supply chain management.

The SSOA model comprises two processes, namely, an FEAHP-based supplier/criteria ranking process and a DP-based order allocation process. In the FEAHP process, the weight of each criterion and supplier is obtained. Based on their weights, the optimal order allocation solution is obtained using the DP technique.

The effectiveness of the proposed optimization model is validated by using real data from a manufacturing company. The experimental results show that the proposed model can handle order allocation effectively.

The proposed optimization model can handle order allocation based on supplier selection. Further research will consider the effects of various uncertainties on supply chain management, such as uncertain customers’ orders and possible material shortages.

Acknowledgment

The authors would like to acknowledge the financial support of The Hong Kong Polytechnic University under the RPUG project.

de Boer L. Labro E. Morlacchi P. A review of methods supporting supplier selection European Journal of Purchasing and Supply Management 2001 7 2 75 89 2-s2.0-0012074924 10.1016/S0969-7012(00)00028-9 Aamodt A. Plaza E. Case-based reasoning: foundational issues, methodological variations, and system approaches AI Communications 1994 7 1 39 59 2-s2.0-0028401306 Degraeve Z. Labro E. Roodhooft F. Evaluation of vendor selection models from a total cost of ownership perspective European Journal of Operational Research 2000 125 1 34 58 2-s2.0-0033704252 10.1016/S0377-2217(99)00199-X ZBL0959.90027 Deng H. Yeh C. H. Willis R. J. Inter-company comparison using modified TOPSIS with objective weights Computers and Operations Research 2000 27 10 963 973 2-s2.0-0034074074 10.1016/S0305-0548(99)00069-6 ZBL0970.90038 Saaty T. L. How to make a decision: the analytic hierarchy process European Journal of Operational Research 1990 48 1 9 26 10.1016/0377-2217(90)90057-I Steuer R. E. Na P. Multiple criteria decision making combined with finance: a categorized bibliographic study European Journal of Operational Research 2003 150 3 496 515 2-s2.0-1642405236 10.1016/S0377-2217(02)00774-9 ZBL1044.90043 Srdjevic B. Srdjevic Z. Bi-criteria evolution strategy in estimating weights from the ahp ratio-scale matrices Applied Mathematics and Computation 2011 218 4 1254 1266 10.1016/j.amc.2011.06.006 ZBL1229.65070 Ishizaka A. Balkenborg D. Kaplan T. Does AHP help us make a choice? An experimental evaluation Journal of the Operational Research Society 2011 62 10 1801 1812 10.1057/jors.2010.158 van Laarhoven P. J. M. Pedrycz W. A fuzzy extension of Saaty's priority theory Fuzzy Sets and Systems 1983 11 1–3 229 241 10.1016/S0165-0114(83)80083-9 MR727206 ZBL0528.90054 Buckley J. J. Fuzzy hierarchical analysis Fuzzy Sets and Systems 1985 17 3 233 247 10.1016/0165-0114(85)90090-9 MR819361 ZBL0602.90002 Chan F. T. S. Kumar N. Global supplier development considering risk factors using fuzzy extended AHP-based approach Omega 2007 35 4 417 431 2-s2.0-33748889528 10.1016/j.omega.2005.08.004 Wang Y. M. Luo Y. Hua Z. On the extent analysis method for fuzzy AHP and its applications European Journal of Operational Research 2008 186 2 735 747 2-s2.0-35348973437 10.1016/j.ejor.2007.01.050 ZBL1144.90011 Dagdeviren M. Yuksel I. Developing a fuzzy analytic hierarchy process (ahp) model for behavior-based safety management Information Sciences 2008 178 6 1717 1733 10.1016/j.ins.2007.10.016 Ghodsypour S. H. O'Brien C. A decision support system for supplier selection using an integrated analytic hierarchy process and linear programming International Journal of Production Economics 1998 56-57 199 212 2-s2.0-0032156562 10.1016/S0925-5273(97)00009-1 Fazlollahtabar H. Mahdavi I. Ashoori M. T. Kaviani S. Mahdavi-Amiri N. A multi-objective decision-making process of supplier selection and order allocation for multi-period scheduling in an electronic market International Journal of Advanced Manufacturing Technology 2011 52 9–12 1039 1052 2-s2.0-79951886441 10.1007/s00170-010-2800-6 Tang Y. Wang Z. Fang J. A. Controller design for synchronization of an array of delayed neural networks using a controllable probabilistic PSO Information Sciences 2011 181 20 4715 4732 2-s2.0-79960561565 10.1016/j.ins.2010.09.025 Ghodsypour S. H. O'Brien C. The total cost of logistics in supplier selection, under conditions of multiple sourcing, multiple criteria and capacity constraint International Journal of Production Economics 2001 73 1 15 27 2-s2.0-0035979629 10.1016/S0925-5273(01)00093-7 Guo Z. Wong W. Leung S. Li M. Applications of artificial intelligence in the apparel industry: a review Textile Research Journal 2011 81 18 1871 1892 10.1177/0040517511411968 Tang Y. Gao H. Kurths J. Fang J. Evolutionary pinning control and its application in uav coordination IEEE Transactions on Industrial Informatics 2012 8 4 828 838 10.1109/TII.2012.2187911 Tang Y. Wang Z. Wong W. Kurths J. Fang J. Feedback learning particle swarm optimization Applied Soft Computing 2011 11 8 4713 4725 10.1016/j.asoc.2011.07.012 Zhu W. Tang Y. Fang J. Zhang W. Adaptive population tuning scheme for differential evolution Information Sciences 2013 223 164 191 10.1016/j.ins.2012.09.019 Zhu W. Fang J. Tang Y. Zhang W. Xu Y. Identification of fractional-order systems via a switching differential evolution subject to noise perturbations Physics Letters A 2012 376 45 3113 33120 10.1016/j.physleta.2012.09.042 Zhu W. Fang J. Tang Y. Zhang W. Du W. Digital IIR filters design using differential evolution algorithm with a controllable probabilistic population size PLoS One 2012 7 e40549 Demirtas E. A. Üstün Ö. An integrated multiobjective decision making process for supplier selection and order allocation Omega 2008 36 1 76 90 2-s2.0-34247187936 10.1016/j.omega.2005.11.003 Soylu B. Kapan Ulusoy S. A preference ordered classification for a multi-objective maxmin redundancy allocation problem Computers and Operations Research 2011 38 12 1855 1866 2-s2.0-79955068159 10.1016/j.cor.2011.02.024 Tang Y. Wang Z. Wong W. K. Kurths J. Fang J. Multiobjective synchronization of coupled systems Chaos 2011 21 2, article 025114 10.1063/1.3595701 MR2849986 Xu N. Nozick L. Modeling supplier selection and the use of option contracts for global supply chain design Computers and Operations Research 2009 36 10 2786 2800 2-s2.0-62549098117 10.1016/j.cor.2008.12.013 ZBL1160.90325 Wagner H. M. Whitin T. M. Dynamic version of the economic lot size model Management Science 1958 5 89 96 MR0102442 10.1287/mnsc.5.1.89 ZBL0977.90500 Basnet C. Leung J. M. Y. Inventory lot-sizing with supplier selection Computers & Operations Research 2005 32 1 1 14 10.1016/S0305-0548(03)00199-0 MR2091734 ZBL1076.90002 Alidaee B. Kochenberger G. A. A note on a simple dynamic programming approach to the single-sink, fixed-charge transportation problem Transportation Science 2005 39 1 140 143 2-s2.0-16244412317 10.1287/trsc.1030.0055 Li S. Murat A. Huang W. Selection of contract suppliers under price and demand uncertainty in a dynamic market European Journal of Operational Research 2009 198 3 830 847 10.1016/j.ejor.2008.09.038 MR2517286 ZBL1176.90444 Sawik T. Selection of a dynamic supply portfolio in make-to-order environment with risks Computers & Operations Research 2011 38 4 782 796 10.1016/j.cor.2010.09.011 MR2735266 ZBL1202.90166 Lin R. H. An integrated FANP-MOLP for supplier evaluation and order allocation Applied Mathematical Modelling 2009 33 6 2730 2736 2-s2.0-60549117496 10.1016/j.apm.2008.08.021 ZBL1205.90051 Mafakheri F. Breton M. Ghoniem A. Supplier selection-order allocation: a two-stage multiple criteria dynamic programming approach International Journal of Production Economics 2011 132 1 52 57 2-s2.0-79955480928 10.1016/j.ijpe.2011.03.005 Zadeh L. A. Fuzzy sets Information and Computation 1965 8 338 353 MR0219427 ZBL0139.24606 Kasabov N. K. Song Q. Denfis: dynamic evolving neural-fuzzy inference system and its application for time-series prediction IEEE Transactions on Fuzzy Systems 2002 10 2 144 154 2-s2.0-0036530967 10.1109/91.995117