Support Vector Machine Approach for Partner Selection of Virtual Enterprises* Jie Wang1,2, Weijun Zhong1, and Jun Zhang2 1
School of Economics & Management, Southeast University, Nanjing, Jiangsu 210096, China 2 Laboratory for High Performance Scientific Computing and Computer Simulation, Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA
[email protected],
[email protected]
Abstract. With the rapidly increasing competitiveness in global market, dynamic alliances and virtual enterprises are becoming essential components of the economy in order to meet the market requirements for quality, responsiveness, and customer satisfaction. Partner selection is a key stage in the formation of a successful virtual enterprise. The process can be considered as a multi-class classification problem. In this paper, The Support Vector Machine (SVM) technique is proposed to perform automated ranking of potential partners. Experimental results indicate that desirable outcome can be obtained by using the SVM method in partner selections. In comparison with other methods in the literatures, the SVM-based method is advantageous in terms of generalization performance and the fitness accuracy with a limited number of training datasets.
1 Introduction An important activity in the formation of a virtual enterprise (VE) is the selection of partners. How to select appropriate partners to form a team is a key problem for successful operation and management of VEs. This problem has attracted much attention recently [1, 2]. Partner selection is an unstructured and multi-criterion decision problem. Qualitative analysis methods are commonly used in many research works [3]. However, quantitative analysis methods for partner selection are still a challenge. Existing quantitative methods in the related literatures can be classified into several categories: mathematical programming models, weighting models, genetic algorithms, dynamic clustering, neural network and fuzzy sets. Talluri and Baker [4] proposed a two-phase mathematical programming approach for partner selection by designing a VE, in which the factors of cost, time and distance were considered. The weighting model includes the linear scoring model and analytic hierarchy process (AHP). The linear scoring model assigns weights and scores arbitrarily. In the AHP model, the *
This work was supported by grant No. 70171025 of National Science Foundation of China and grant No. 02KJB630001 of Research Project Grant of JiangSu, China.
J. Zhang, J.-H. He, and Y. Fu (Eds.): CIS 2004, LNCS 3314, pp. 1247–1253, 2004. © Springer-Verlag Berlin Heidelberg 2004
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priorities are converted into the ratings with regard to each criterion using pairwise comparisons and the consistency ratio [5]. Clustering technology can also be used in the partner selection process, which is based on the rule that multiple potential partners can be classified into one class if no significant difference on evaluation criteria exists in these partners. While artificial neural network (ANN) approach seems to be the best one available that efficiently combines qualitative and quantitative analysis to ensure the objective of selection processes, the prerequisite for the ANN approach is a large number of training data and the method may easily lead to local optimum. This paper proposes a new approach to partner selection process by utilizing the Support Vector Machine technique. Based on well developed machine learning theory, Support Vector Machine (SVM) is a supervised learning technique that has received much attention for superior performances in various applications, such as pattern recognition, regression estimation, time series predication and text classfication. To employ SVM for distinguishing more than two classes, several approaches have been introduced [7]. In this paper, we focus on how to use binary SVM technique in the multiclass problem of partner selection.
2 Design of SVM-Based Partner Selection System Basically partner selection is a process that produces partner rank in the order of overall scores of their performance according to a certain criterion system. Let us consider a pool of potential partners containing k independent organizations, and a criterion system containing d sub-criteria. Define xi to be a feature vector of length d for the i-th potential partner.
xi = ( xi1 , xi 2 ,..., xij ,..., xid ) , ( i=1,..., k; j=1,..., d ) where xij is the value of the j-th criterion for the i-th potential partner. 2.1 Selection Criterion System Partners are selected based on their skills and resources to fulfill the requirements of the VE [4]. The selection process is based on multiple variables such as organizational fit, technological capabilities, relationship development, quality, price, and speed [6]. A three-layer selection criterion is developed as shown in Fig.1. The hierarchical structure includes goal, criteria and sub-criteria. The hierarchy can easily be extended to more detailed levels by breaking down the criteria into sub-criteria. Some sub-criteria are determined by five rating levels: outstanding, above average, average, below average and unsatisfactory. Since the SVM method requires that each data be represented as a vector of real numbers, such sub-criteria should be converted into numerical data simply by using 5 to represent the best level and 1 for the lowest level.
Support Vector Machine Approach for Partner Selection of Virtual Enterprises Goal
Criteria
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Sub-criteria
Quality
C1: Pass rate in quality control C2: Total quality management C3: Quality system certification C4: Remedy for quality problems
Cost
C5: Product Price C6: Delivery cost
Time
C7: Due date
Partner Rank
C8: Attitude
Service
C9: Maintenance quality C10:Cooperation and information exchange Financial status
C11: Balance ratio C12: Investment / revenue
Compatibility
C13: Enterprise culture C14: Management compatibility C15: Information format
Environment
C16: Geographical limitation C17: Political environment C18: Law restriction C19: Economy environment C20: Organization credit standing
Fig. 1. The three-layer criterion structure for supplier partner selection
2.2 Binary Support Vector Machine SVM is basically a binary classifier. For a two-class classification problem, given a set G of training data (xi,yj), for i=1,2,…,n, i.e., the input vectors x i ∈ \ d with corresponding labels yi ∈ {1, −1} , here +1 and –1 indicate two classes, SVM seeks the solution of the following optimization problem:
min w , b ,ξ
s.t.
l 1 T ω ω + C ∑ ξi 2 i =1
yi (ω T φ ( xi ) + b) ≥ 1 − ξi
ξi ≥ 0,
i = 1,..., n
where C is a parameter to be chosen by the user. A larger C corresponds to assigning a larger penalty to errors. ξi is slack variables when the training data is not linearly separable in the feature space. The decision function is
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n
f ( x) = ( wi x) + b = ∑ α i yi ( xi i x) + b i =1
2.3 Key Problems of Design
Using the idea of SVM, the class number is equal to the number of potential partners. We need to find a classifier with the decision function, f(x) such that y = f(x), where y is the class label for x. K is a variable determined by the actual number of training data. Obviously the SVM method cannot be applied directly. The following three problems need to be solved when applying the SVM method to the selection process: 1. decompose the ranking problem into classification problems, 2. construct multi-class classification SVM from the binary SVM, 3. transform results of classification into numerical values.
3 System Implementation 3.1 One-to-One Comparison of Potential Partners
Given two partners denoted by i and j, with the feature vectors xi and xj of dimension d, their ranks are represented by F(xi) and F(xj) respectively. We define the vector of cij of dimension 2d by combining xi and xj
cij = ( xi1 , x j1 ,..., xim , x jm ,..., xid , x jd ) (i,j =1,..., k; i≠j, m=1,...,d) Let yij be the class label, according to the overall score of the partners i and j, we have ⎧+1, ⎪ yij = ⎨ 0, ⎪ −1, ⎩
F ( xi ) > F ( x j ) F ( xi ) = F ( x j )
i, j = 1,..., k and j ≠ i
F ( xi ) < F ( x j )
Therefore the class number for two partners is 3, which does not change with the total number of training samples. The original training sets represented by (xi , yj) is transformed into the format of (cij , yij). In this case, the SVM approach can be applied to classify cij to determine the relative performance of any pair of partners. 3.2 Multi-class Classification SVM
Through the format variation of (cij , yij), the selection process can be initiated by a three-class classification process. The one-against-one method is used to construct all the possible classifiers where each one is generated between two classes chosen out of the k classes from the training data [8]. The decision function for the class pair pq is defined by fpq(x). Since fpq(x)=–fqp(x), there exist k(k–1)/2 different classifiers for a k-class problem. The “max wins” algo-
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rithm is used for class identification of the one-against-one method. In the “max wins” algorithm each classifier casts one vote for its preferred class, and the final result is the class with the most votes. k
the class of X = arg max p
∑
sign( f pq ( x))
q ≠ p , q =1
When more than one class have the same number of votes, each point in the unclassifiable region is assigned to the closest class using the real valued decision functions as: k
the class of X = arg max p
∑
f pq ( x)
q ≠ p , q =1
Based on the above one-against-one algorithm, we developed three binary classifiers between every two classes. 3.3 Transformation of Classification Results
In order to compute the final rank of the partners, we have to decide how to transform the results of classification into the final rank of partners. The idea of Round Robin are utilized here in which each partner is compared with every other partners in the same test dataset. Let n be the total number of the test datasets, xi be the feature vector of the i-th partner, then cij is constructed by combining xi with every other xj ( j = 1,…,n and j≠i). Using 3-class SVM in Section 3.2, the n-1 class labels yij are calculated for each cij. Define
⎧ 2, yij = 1, ⎪ g i ( x j ) = ⎨ 1, yij = 0, i, j = 1,..., n and j ≠ i ⎪ 0, y = −1, ij ⎩ where gi(xj) is the score of the i-th partner with respect to the j-th partner. And define n
f ' ( xi ) = ∑ gi ( x j )
i = 1,..., n
j =1 j ≠i
where f ' ( xi ) is the final score of the i-th partner. Therefore, the order of potential partners in the test dataset can be determined according to the absolute value of each f ' ( xi ) in either ascending or descending order.
4 Experimental Analysis Using the ideas discussed in Section 3, a partner selection system based on the multiclass SVM method was implemented. We point out that in the multiclass SVM method, the kernel function and the parameter adjustment are very important. We used a polynomial function.
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K ( x, xi ) = [( x ⋅ xi ) + 1] q We ran experiments with different kernel parameters and measured accuracy and CPU time for the training time. Table 2 indicates that the values of q and C have direct effect on the accuracy and training time of the system. Table 1. Accuracies and trainning times for kernel parameter q and penalty coefficient C Kernel Function
Kernel Parameters
Polynomial function K ( x , x ) = [( x ⋅ x ) + 1] i
i
q
Accuracy(%)
Training time(second)
q=2, C=0.1 q=2, C=0.2 q=2, C=0.3
96.6 96.6 97.1
31 24 24
q=3, C=0.1
98.1
22
q=3, C=0.2 q=3, C=0.3
98.1 98.1
20 19
q=4, C=0.1
97.5
24
q=4, q=4, q=5, q=5,
C=0.2 C=0.3 C=0.1 C=0.2
97.6 97.6 97.1 97.1
22 22 31 25
q=5, q=8, q=8, q=8,
C=0.3 C=0.1 C=0.2 C=0.3
97.1 96.1 96.1 95.6
23 56 56 43
5 Conclusion By transforming the ranking of partners into multiple binary classification problems, we proposed and implemented a new approach to the solution of partner selection of virtual enterprise. Results of our experiments indicate that in comparison with other methods reported in the literatures, the SVM-based method presented in the paper is advantageous in terms of achieving certain fitness accuracy with a limited number of training datasets. More works need to be done in the selection of the kernel function and parameters. Determining the best kernel parameters would be an interesting topic for future research. Furthermore, the comparison of the SVM-based system with those based on other methods such as PCA and Fisher are desirable in order to demonstrate that better accuracy can be obtained by the adoption of SVM in the ranking system.
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