Applying the fuzzy ART algorithm to distribution network design

Article history: Received July 15, 2011 Received in Revised form September, 12, 2011 Accepted 28 September 2011 Available online 3 October 2011 Distribution network design is an important issue in supply chain management and plays an important role in making new market development. Because of JIT philosophy, most of managers now have focused on designing appropriate distribution networks. Thus, categorizing distributors and selecting the best ones are crucial for companies. This paper provides a new method to categorize and select distributors. The fuzzy Adaptive Resonance Theory (ART) algorithm is utilized to categorize distributors according to their similarity. To improve the performance of the algorithm, we train the algorithm using the past data. Finally, a numerical example is illustrated to examine the validity of the proposed algorithm. © 2012 Growing Science Ltd. All rights reserved.


Introduction
Supply chain management is the coordination of production, inventory, location and transportation among the participants in a supply chain to achieve the best combination of responsiveness and efficiency for the market being served (Hugos, 2006).One of the important areas in supply chain management (SCM) is partner selection.The competitive advantage of SCM is not only to focus on its core business, but also to establish long-term cooperative partnership with partners.Selection problems are very important in many real-life decision situations.Mousavi et al. (2011a) proposed a fuzzy stochastic approach for multi-attribute group decision making in uncertain situations.They illustrated the effectiveness of their model by applying it on a risk selection problem.Mousavi et al. (2011b) also proposed an integrated DELPHI-AHP-PROMETHEE methodology for a plant location selection problem.In the area of supplier selection, numerous studies are found.Many researchers have investigated the importance of supplier selection problems and their key roles in achieving SCM goals (Bhattacharya et al., 2010;Punniyamoorthy et al., 2011;Vanteddu et al., 2010).A number of methodologies are applied in practice, such as linear and non-linear programing, mixed-integer programming, multi-objective linear programming (Fassnacht & Koese, 2006;Narasimhan et al., 2006;Talluri & Narasimhan, 2005).Along these traditional methods, a number of studies have applied the fuzzy theory to supplier selection problems (Razmi et al., 2009;Wang et al., 2009).Although numerous studies have been done in the context of supplier selection problems, there are a few studies in the context of distributor selection problems.Zou et al. (2011) introduced a rough setbased approach to distributor selection in a supply chain.They proposed a methodology, which is able to perform rule induction for distributors.Lin and Chen (2008) stated that there is little empirical research investigating manufacturers' selection of distributors and then tried to move researchers toward this area by proposing important factors when selecting distributors.Wang and Kess (2006) investigated the distributor selection problem by a case study.They mentioned that task and partnerrelated dimensions in partner selection of international joint ventures that were useful in the distributor relationship.Sharma et al. (2004) proposed a composite distributor performance index (DPI) to evaluate the distributors' performance.

Overview of adaptive resonance theory (ART)
The ART network is a neural algorithm in order to cluster arbitrary data into groups with similar features (Pacella et al., 2004).This network consists of input and output layers.The input layer takes a set of input vectors and gives clusters as output.Input vectors, which are close to each other according to a specific similarity measure, are mapped to the same cluster.If the input does not match any of the stored patterns, the new category can be existed.The ART has the orienting and attention subsystems.These subsystems are responsible for categorization and whether to accept it or not, respectively.Table 1 shows the classical ART clustering algorithms.

Fuzzy ART
The fuzzy ART neural network was first introduced by Carpenter et al. in 1991(Aydın Keskin et al., 2006;Lopes et al., 2005).The fuzzy ART is an unsupervised learning algorithm, which is capable of learning in both off-line and on-line training modes.It is the most recent adaptive resonance framework, which provides a unified architecture for both binary and continuous value inputs.The generalization of learning both analog and binary input patterns is achieved by replacing the appearance of the logical AND intersection operator ( ) in ART1 by the MIN operator (∧) of the fuzzy set theory (Pacella et al., 2004).According to Aydın Keskin et al. (2006), the fuzzy ART involves three main differences in comparison with ART1: • There is a single weight vector connection.
• Non-binary inputs can be processed.
• In addition to vigilance threshold ( ), choice parameter ( ) and learning rate ( ) should be determined.Reviewing the literature of the fuzzy ART shows that in addition to its simplicity, this algorithm has been used frequently by researchers in recent years (Aydın Keskin et al., 2006;Aydın Keskin & Ozkan, 2009;Pandian & Mahapatra, 2009;Pacella & Semeraro, 2011).Fig. 1 shows the basic fuzzy ART architecture.This paper applies the fuzzy ART's classification ability to the distributor categorization and selection area.The fuzzy ART methodology is able to categorize the candidate distributors according to the similarity between input values.Furthermore, some modifications are applied to enhance the classification ability of the algorithm.First, complement coding is used for the normalized data, and then the neural network is trained.The remainder of this paper is organized as follows.The basic concepts, definitions and notations of the proposed algorithm for distributor categorization and selection are introduced in Section 2. In Section 3, an illustrative numerical example is presented, after which this study discusses and shows how the proposed method is effective.Finally, conclusions are presented in Section 4.

Fuzzy ART for distribution network design
A distributor is a firm, which takes ownership of important inventories of products, in which distributor buy from producers and sell to consumers.For the customer, distributors fulfill the 'Time and Place' function, in which they deliver products when and where the customer wants them (Kuo & Liang, 2011).The distributor in a supply chain is not only an important link connecting manufacturing and final customers to transfer products and value, but also the first line listening to customers' voice to directly grasp the pulse of demand.In this paper, the fuzzy ART-based algorithm is proposed for distributor categorization and selection.Fig. 2 shows the phases of the proposed algorithm and the applied fuzzy ART model for distributor selection, respectively.In the following, the stepwise explanation of the proposed method is discussed.

Phase 1. Determining team members and evaluation prerequisites
Step 1) Constituting the team of the decision makers (DMs): The team is developed to identify criteria to evaluate distributors.A brainstorming session or meeting can be held in order to determine the required criteria for distribution according to the product and supply chain of the manufacturing company.
Step 2) Evaluation of distributors: The team determines the grading scale to rate each distributor according to the defined criteria.Phase 3. Categorizing distributors using the fuzzy ART In this phase, the steps of the fuzzy ART algorithm are described below.
Step 1) Initialize the network: In this step, the initial parameters should be determined by the team of the DMs.Parameters for the fuzzy ART algorithm are vigilance threshold ( ) and choice ( ). isresponsible for the number of categories, where 0,1 .If is small, the result is inclined to a rough categorization.On the other hand, if is chosen to be close to 1, many finely divided categories are formed and similarity in each category is much higher and choice parameter is effective in category selection.These parameters are determined based on the type of the problem.
The initial weights for all i and j are taken from the trained network.Also, the number of category is set to the categories in the trained neural network.where i (i=1,2,…,m) is the selected distributor and j (j=1,2,…,n) is the criteria number.
Step 2. Normalization of inputs: Using Eq. 1, each input is first normalized.
Step 3. Complement coding: Complement coding transforms an M-dimensional feature vector I into a 2M-dimensional system input vector.A complement-coded system input represents both the degree to which a feature i is present (a i ) and the degree to which that feature is absent Step 4. Presentation of the input vector NI to the network Step 5. Computation of choice function: Compute the choice function for each existed output node.
The choice function is defined by: where ∧ is fuzzy 'AND' operator and work as minimum function (i.e., x∧y = min(x,y)).
Step 6. Selection of maximum choice function value: The maximum choice function value is selected by: max , ; 1, 2, . . .

Conclusions
In this paper, distributor selection and categorization have been conducted through the fuzzy Adaptive Resonance Theory (ART) algorithm.In the first step, criteria have been defined by the decision makers (DMs), and then these DMs have used a grading scale to rate each distributor regarding these criteria.Furthermore, the fuzzy ART algorithm has been utilized to cluster the distributors with similar features.The proposed approach has enhanced the clustering algorithm proposed by Aydın Keskin et al. (2006) for supplier selection.Then, the numerical example has been conducted to show the effectiveness of our proposed approach.Partner selection problems have been solved by numerous methods.In the context of supplier selection and evaluation, MODM techniques (e.g., goal programming) and MCDM techniques (e.g., AHP, ANP and TOPSIS methods) have been used widely along with mathematical programming methods.On the other hand, in the context of distributor selection and evaluation, few studies could be found in the literature.These studies have utilized MCDM and MODM techniques, rough set theory and artificial intelligence (AI) to deal with this problem.
In reality, when the complexity and ambiguity of information is high, AI methods are better than traditional methods, because they are designed to act like human judgment.In addition, they can learn from the past data.Therefore, the decision maker (DM) should only provide the information needed for the system.The most important contribution of the proposed method was the ability of its clustering for the distributor selection and evaluation problems.The distributors are clustered according to their similarity degrees between them.The fuzzy ART not only determines the best distributors, but also clusters all distributors.This procedure has been very effective in partner selection and evaluation problems.In addition, the drawbacks of the algorithm have been mitigated by training the neural network.The algorithm has been adaptive and has easily applied to firms and companies.This method has been very flexible, and a number of categories have been different by changing the vigilance parameter.The algorithm has been especially good for the large-sized data and its simplicity made it applicable.

Fig. 1 .
Fig. 1.Topological structure of the fuzzy ART Fig. 2. Conceptual model of a supply chain with the fuzzy ART model for distributor selection to the fuzzy ART architecture.The training list is presented as many times as it is necessary for the fuzzy ART to cluster the input patterns.The clustering task is considered accomplished, if the weights in the fuzzy ART architecture do not change during list presentation.The above training scenario is called off-line training.The step-by-step implementation of training is given in the Appendix of this paper.
similar features, when it is confronted by a new input it produces a response that indicates which category the pattern belongs to.The training phase of the fuzzy ART works as follows.Given a list of input patterns, designated as , , … , , we want to train the fuzzy ART to categorize these input patterns into different categories.Obviously, patterns that are similar to each other are expected to be clustered in the same category by the fuzzy ART.In order to achieve the mentioned goal, the training list is presented 4)Step 7. Resonance test: The resonance test defines the appropriate category for the input.Computation of matching function is computed by: Set the choice function value as , 1, and then go back to Step 5. Control the next highest , value.In this way, the matching test continues for all of the , values.If none of , passes the test, a new category is created for the existing input.Thus, the i-th distributor is added to the new category C s+1 .Then, go to Step 4 and compute , for the next input.

Table 3
Alternative distributors and their grade