Abstract: This study presents a multivariate approach for solving supplier selection problem. The approach is based on Principal Component Analysis (PCA) that uses information obtained from eigen values to combined different ratio measures defined by every input and every output.
INTRODUCTION
Supplier selection is one of the most critical activities of purchasing management in a supply chain, because of the key role of supplier`s performance on cost, quality, delivery and service in achieving the objectives of a supply chain. Supplier selection is a Multiple Criteria Decision-Making (MCDM) problem which is affected by several conflicting factors. Consequently, a purchasing manager must analyze the trade off among the several criteria. MCDM techniques support the decision-makers DMs in evaluating a set of alternatives. Depending upon the purchasing situations, criteria have varying importance and there is a need to weight criteria (Dulmin and Mininno, 2003). The criticality of supplier selection is evident from its impact on firm performance and more specifically on final product attributes such as cost, design, manufacturability, quality and so forth.
Banker and Khosla (1995) classified the supplier selection process as an important Operations Management (OM) decision area. They suggest that OM research should attempt to identify the supply chain management practices that provide competitive advantage.
Yahya and Kingsman (1999) used Saaty`s Analytic Hierarchy Process (AHP) method to determine priority in selecting suppliers. The AHP has found widespread application in decision-making problems, involving multiple criterion systems of many levels. The strongest features of the AHP are that it generates numerical priorities from the subjective knowledge expressed in the estimates of paired comparison matrices. The method is surely useful in evaluating suppliers` weights in marketing, or in ranking order, for instance. It is, however, difficult to determine suitable weight and order of each alternative. Weber (1996) used data envelopment analysis in supplier selection problems especially when multiple conflicting criteria have to be considered. DEA identifies an `efficient frontier` from the inputs and outputs to be evaluated creating Decision Making Units (DMU`s) and then the efficiency of each of these DMUs are compared to the efficient frontier by using identifying the most efficient DMU.
In the present study, an alternative methodology is proposed to aid purchasing managers in identifying and selection suppliers. This approach attempts to address the need for flexible models that are highly customized to meet an individual firm`s particular needs. This is a multi-objective approach to supplier selection that attempts to provide a useful decision support system for a purchasing manager faced with multiple suppliers and tradeoffs such as price, delivery reliability and product quality.
PRINCIPLE COMPONENT ANALYSIS
Principal Component Analysis (PCA) is widely used in multivariate statistics such as factor analysis. It is used to reduce the number of variables under study and consequently ranking and analysis of Decision-Making Units (DMUs), such as industries, universities, hospitals, cities, etc (Azadeh et al., 2002, 2003). These DMUs utilize a variety of sources as inputs to produce several outputs. The purpose of another study was to describe and demonstrate the applicability of two multivariate statistical techniques, namely Principal Component Analysis (PCA) and corresponding analysis, as analysis tools for quality professional. Using a principal component factor analysis with a rotation technique identifies seven hotels out of 33 hotel attributes and determines the level of satisfaction and service quality of Hong Kong hotels among Asian and Western travelers (Tat and Raymond, 2000). A multivariate analysis proposed a framework for measuring the efficiency of investment in IT that addresses some shortcomings include measurement errors, lags between investment and benefits, redistribution of profits and mismanagement of IT resources. This study suggests the utilization of multivariate technique as an objective optimization approach for the comparative efficiency evaluation of the maintenance department (Kamal et al., 2000). Furthermore, PCA captured the measurement correlations and reconstructed each variable to define associated residuals and sensor validity index. The beverage data was analyzed using PCA and cluster analysis (Rossi and Thomas, 2001). A multivariate analysis was used to test whether there is any relationship between airline flight delays and the financial situation of an airline (Bhat, 1995).
The objective of PCA is to identify a new set of variables such that
each new variable, called a principal component, is a linear combination
of original variables. Second, the first new variable y1 accounts
for the maximum variance in the sample data and so on. Third, the new
variables (principal components) are uncorrelated. PCA is performed by
identifying Eigen structure of the covariance or singular value decomposition
of the original data. Here, the former approach will be discussed. It
is assumed there are p variables (indexes) and k DMUs and suppose X =
(x1...xp)kxp is a kxp matrix composed
by xij`s defined as the value of jth index for ith DMU and
therefore X = (x1m...xkm)T (m = 1,...,
p). Furthermore, suppose
lmj is the coefficient of m-th variable for the jth principal component. The lmj`s are estimated such that the following conditions (1, 2 and 3) are met:
| • | Y1 accounts for the maximum variance in the data, Y2 accounts for the maximum variance that have not been accounted by y1 and so on |
(1) |
(2) |
For obtaining the lij`s and consequently p vectors (y1j...ykj) (j = 1...p) and PCA scores the following steps are performed:
Step 1:Calculate the sample mean vector
(3) |
(4) |
(5) |
Step 2:Calculate the sample correlation matrix.
p diagonal matrix whose jth diagonal element is
Step 3:Solve the following equation
where, Ip is a pxp identity matrix. We obtain the ordered
p characteristic roots (eigenvalues) λ1≥λ≥2...≥λp
with
Those characteristic vectors compose the principal components Yi. The components in eigenvectors are respectively the coefficients in each corresponding Yi:
(6) |
RESEARCH DESIGN
In this research we use a case study which was illustrated by Narasimhan et al. (2001). Five inputs and four outputs are used for analysis in the following approach.
For the purpose of the PCA evaluation, items on the supplier capability Questionnaire were grouped into the following categories, constituting the input variables:
| • | Quality Management Practices and systems (QMP) |
| • | Documentation and Self-Audit (SA) |
| • | Process/Manufacturing Capability (PMC) |
| • | Management of the firm (MF) |
| • | Cost Reduction capability(CR) |
These five categories were measured with a composite score between 0 and 1. the score was computed as the proportion of yes answers to individual questionnaire items in the category. Blank and not applicable responses were not considered in the calculation of the proportion of the yes responses.
Items on the supplier performance Assessment Questionnaire were grouped into the following categories, constituting the output variables:
| • | Quality |
| • | Price |
| • | Delivery |
| • | Cost Reduction Performance (CRP) |
These categories were also measured with a composite score between 0 and 1. To compute the score, the proportion of yes answer was evaluated in each category to provide an objective measure of the variables in the category.
Table 1 shows the scaled composite scores for the input and output variables for the 23 suppliers in order to maintain confidentiality the data were have scaled by dividing each factor by its factor mean score.
Twenty output/input ratios for the supplier`s attributes were defined:
| d1 | = | Quality/QMP |
| d2 | = | Quality/SA |
| d3 | = | Quality/PMC |
| d4 | = | Quality/Mgt. |
| d5 | = | Quality/CR |
| d6 | = | Price/QMP |
| d7 | = | Price/SA |
| d8 | = | Price/PMC |
| d9 | = | Price/Mgt. |
| d10 | = | Price/CR |
| …. | ||
| d19 | = | CRP/Mgt. |
| d20 | = | CRP/CR |
The decision maker would like to select the supplier that provides the best combination of the performance parameters. In statistical terms, these supplier s are extreme observations that lie away from the data.
| Table 1: | Data matrix for inputs and outputs |
| Table 2: | Correlations between output/input ratios |
| **Correlation is significant at the 0.01 level (1-trailed), *Correlation is significant at the 0.05 level (1-trailed) | |
Thus, a procedure that identifies outlying suppliers regardless of the importance the purchasing manager attaches to each performance parameter of the supplier. Most of the variability in the data set is contained in the first few linear combinations of variables, i.e., the principal components. In other words, PCA is employed to identify the principal components that are respectively, different linear combinations of the performance variables so that the principal components can be multiplied by their Eigen values to obtain a weighted measure of the variables.
The first step in PCA consists of testing whether the variables show a sufficient level of correlation. To this extern; both the correlation matrix (Table 2) and the Bartlett`s test of sphericity have been analyzed. The coefficients are the usual Pearson correlations (one-tailed) and confirm that significant correlations can be found.
Bartlett`s test of sphericity is a useful instrument to test the null hypothesis that the correlation matrix is an identity matrix. If the null hypothesis cannot be rejected and the sample size is reasonably large the decision maker should reconsider the use of multivariate analysis since the dependent variables are not correlated. In this case, the null hypothesis is rejected at the 0.001 level.
A rule-of-thumb for determining the number of components to extract is to consider the Eigen value greater than one criterion. The so-called Eigen value screen plot is often useful in graphically determining the number of factors extracted (Fig. 1). In the present analysis, four component are a workable solution (since the residual components have all Eigen value less than 1).
Component 1, 2, 3 and 4 account for approximately 0.93 % of the total variance of the variable (Table 3).
The percentage of variance explained by each component represents its relative importance.
| Fig. 1: | Eigen values versus component |
| Table 3: | Total variance explained by the components weights of the rotated component |
The term rotated component in the heading in Table 4 is due to the fact that an orthogonal rotation is usually carried out on the component that are initially extracted. Usually the initial component extraction does not give interpretable results. One of the purposes of rotation is to obtain components that can be named and interpreted more clearly. in other words rotation makes the larger loadings of the different variables on each component larger than before and smaller loadings smaller than before. There are a number of different methods of extraction-here the varimax method has been chosen. The rotated loadings of each variable on each component are reported in Table 4.
The interpretation of the Table 4 leads the decision maker to conclude that component 1 loads on matters concerning cost reduction performance, component 2 loads on matters concerning quality, component 3 loads on matters concerning delivery and component 4 loads on matters concerning price.
| Table 4: | Matrix of components (loading smaller than 0.1 are omitted) |
| Table 5: | Final ranking of suppliers |
For each variable considered (d1, d2, d3, ..., d20), a coefficient Wi (i = 1 to 20) is obtained by multiplying the loadings on each component by the percentage of variance explained by the component For instance, Wi is obtained as follows:
Wi = 0x0.463477+0.907x0.316012+0.295 |
x0.090295+0x0.061295 = 0.31326 |
The coefficient is then multiplied by the value of the corresponding variable (d1 to d20) for each supplier to get final supplier score (Table 5). Based on these score the final ranking is obtained.
Summing up, supplier number 19 ends up as the supplier that provides the best performances with respect to the three component identified.
CONCLUSION
This study describes a multiple-attribute approach based on the use of principal component analysis, aimed at helping purchasing managers to formulate viable sourcing strategies in the changing marketplace. An application of the methodology using actual data retrieved from a firm operating in the bottling industry is illustrated. PCA has proved to be capable of handling multiple conflicting attributes inherent in supplier selection criteria.
PCA`s strength is in simultaneously considering multiple inputs and multiple outputs without any need for a priori assignment of weights. PCA also has distinct advantage over the traditional methods of selection suppliers `performances in that it is not necessary to state the performance measures in the same units. The relevant attributes of suppliers can in fact be measured in any unit such as money, percentages and qualitative subjective judgments. Furthermore since PCA does not force the decision maker into a standardized rating for the attributes this technique is less subjective than the traditional methods.