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南台科技大學 專題討論報告 指導老師：黃振勝 學 生：陳智凱 學 號：Ｍ９８Ｕ０１１０ 中華民國98年12月16日 Source： Journal of the Chinese Institute of Industrial Engineers, Vol. 22, No. 3, pp.243-251 (2005) 結合層級分析法與資料包絡法分 析法之績效評估模式 A PERFORMANCE EVALUATION MODEL BASED ON AHP AND DEA 劉志明、許漢昇、王信智、李海光 國立清華大學工業工程與工程管理學系 Abstract • This research uses the concept of the Analytical Hierarchy Process (AHP) method and fuzzy set theory to modify the CCR model of the Data Envelopment Analysis (DEA) method to develop a more effective performance evaluation method which can be used to evaluate the performance of a smaller number of business units, which is usually prohibited in the traditional DEA method. • In order to prove the applicability of the proposed method, a case study for the selection of suppliers for the key component of a LCD manufacturing company is given. • The result shows that it can be used to choose among a small number of suppliers in a very efficient way and also can provide improvement suggestions for each supplier. Program 1. Introduction 2. Literature Review 3. A performance evaluation model based on AHP and DEA 4. An implementation case 5. Conclusion 1.1 Background • In the present competitive business environment, an enterprise must have an evaluation mechanism to measure the performance of every business unit in order to direct the entire organization or system toward a common goal. In other words, it needs a performance evaluation system. The goal of performance evaluation can be classified into two major parts: efficiency and effectiveness. • A high-performance system must consider both efficiency and effectiveness, i.e., pursue the highest effectiveness by using the most efficient method. 1.2 Purpose of research 1. Investigate the fitness problem and the limitations of DEA, and try to propose a method to increase its application domain. 2. Apply DEA to the evaluation and selection of suppliers in the business environment. 1.3 Structure of sections The structure of this paper is as follows. In the second section of this paper, some related literature on DEA will be given. In the third section, a performance evaluation model, which is based on AHP and DEA, is developed. In the fourth section, the proposed model is applied to the supplier evaluation and selection problem of a LCD key component manufacturing company. Finally, some conclusions are drawn. 2. Literature Review 2.1 DEA models In research , data envelopment analysis of the CCR model with Analytical Hierarchy Process Performance Assessment Model 2.2 Investigation of the weighting limitations of DEA 1. The absolute range The values of the weighting variables in the DEA model are confined by the upper and lower bounds, however, those bounds are obtained through historical data or experts’ opinion as shown in formula (7). 2.2 Investigation of the weighting limitations of DEA 2. The assurance region(AR) In the application of DEA, one situation may occur such that the ratio of two factors has some kind of relationship, and under this condition it is required to confine the corresponding ratio for the weights of these two factors to a range as shown in formula[8]. The determination of the upper and lower bounds for an AR model is usually based on the subjective opinion of experts. 2.2 Investigation of the weighting limitations of DEA 3. The polyhedral cone-ratio(polyhedral cone-ratio) In the DEA model, it requires the weighting variables to satisfy the restriction of a multipleface cone, as shown in formula (9). 3. A performance evaluation model based on AHP and DEA 3.1. Establish the range for each evaluation criterion The establishment of the range for each evaluation criterion has three phases, as shown in Figure 2. Table 1. The relative importance rating scale of AHP Table 2. The fuzzy scale table 1. Establish the pair-wised comparison matrix with range values. First, Table 1 is referred to establish the rating of relative importance for each pair of factors. Then, Table 2 is referred to get the scale of the fuzzy value between different factors. Finally, by adding and subtracting the fuzzy value from the relative importance rating value, one can obtain the range of the relative importance rating value, which can be expressed as ， After having obtained all the and , the pair-wise comparison matrix in range values can be given as shown in Table 3.The relationship between and its bounds can be expressed in formula (10). 2. Find the range of weight for each measure. • Calculate the range for each column vector: • After having calculated all the row vector ,the range of the weight of each factor can be obtained through a normalization process as shown in formula (12). 3. Consistence test The consistence test can be done by calculating the consistence ratio (C.R.) as shown in formulas (13) and (14). Phase 3: Integration of weights from multiple decision-makers. If there are several people involved in the decisionmaking, each person may have different range for the weight of a measure. So it’s necessary to integrate the rating of all the decision-makers to form a single range of the weight of a measure. The integration method adopted here is to average the weights of all the evaluators. The formulas for calculating the mean values are given in formulas (15) and (16). represents the final evaluation of the lower bound of the ith index, represents the lower bound provided by the kth decision-maker, and N represents the total number of decisionmakers. represents the final evaluation of the upper bound for the ith index, and represents the upper bound provided by the kth decision-maker. 3.2. Homogenize data After having assigned the range for the weight of each measure, the next step is to homogenize the rating scores for all the measures into a common numerical scale. The homogenization method applied here is to assign the highest value of a measure for a DMU to be 1, and the values for other DMUs are rescaled accordingly as shown in formulas (17) and (18). Homogenize the scores of output variables (thelarger-the-better index), as shown in formulas(17): represents the original score of the kth DMU of the rth the-small-the-better index and represents its homogenized value. Homogenize the scores of input variables (thesmaller-the-better index), as shown in formula (18): Where represents the original score for the ith measure of the kth DMU and represents the its normalized value. 3.3. A performance evaluation model by combining AHP and DEA Through the establishment of the range for the weight and the normalization of the score, the modified DEA model is shown in formula (19). 3.3. A performance evaluation model by combining AHP and DEA and ， represent the upper and lower bounds for the input and output variables respectively. In the same way, the super-efficiency model can be shown in formula (20): 3.4. The improvement model for DMUs with low efficiency In order to investigate the improvement model for DMUs with low efficiency, its dual model needs to be studied first. Since the objective function of the model in this research is nonlinear, it must be first transformed into a linear model. 3.4. The improvement model for DMUs with low efficiency We can find the optimal solution by using the LINGO software, and then use the found values of those variables to rewrite the objective function. Assume denotes the optimal solution of the-smaller-the-better variables, then the constant can then be used to rewrite the original model into a linear programming model. 3.4. The improvement model for DMUs with low efficiency Because is the optimal solution of the-smaller-the-better variables, if both the numerator and denominator of the objective function are divided by the constant , it will not change the solution of the original model. So in the situation the denominator of the objective Function equals to one. Suppose , then the linear programming model can be shown in formula (21) : 3.4. The improvement model for DMUs with low efficiency After having transformed the original model into its dual model, the relationship cx∗ = y∗b must exist if x∗ is the optimal solution of the primal problem And y∗ is the optimal solution of the dual problem according to the strong dual theorem. Then, when the optimal solution is found, the values of the objective function of the primal problem and the dual problem should be the same. So the calculating process of the improvement alternatives can be simplified by using the performance value (the value of the primal problem) found for DMUs. 3.4. The improvement model for DMUs with low efficiency The improvement alternatives of the DMUs with low efficiency are given below: The improvement goal for the an input item can be expressed as formula (22). 3.4. The improvement model for DMUs with low efficiency Where is the calculated performance value of that DMU. The improvement goal for the output item can be expressed as formula (23). 4. An implementation case The case study is a LCD backlight manufacturing company (C company) which needs to select suppliers for its aluminum part. There are four related persons to get involved in the decision process. Five measures with the-higher-thebetter values and four measures with the-lowerthe-better values are chosen. The rating information for suppliers is given in Table 4. 4.1 Establish ranges for the weights of evaluation measures of the C company’s suppliers Using the relative importance values for the thelarger-the-better measures of the first decisionmaker as an example, Table 5 gives the pairwise comparison matrix for those measures expressed in range values. After some calculation, the ranges of weights can be obtained in Table 6. Since the value of C.R. = 0.043 ≤ 0.1, the consistence test is passed. Table 5. The pair-wise comparison matrix for range values of the relative importance of the -higher- the- better measures of the first decision-maker 4.1 Establish ranges for the weights of evaluation measures of the C company’s suppliers The same method can be applied to obtain the range values of the relative weights of measures for other three decision-makers. By averaging the range values of the four decision-makers, the final range values of weights can be obtained as shown in Table 7 and Table 8. 4.2 Homogenize data The homogenize data are shown in Table 9. Table 9. The normalized data for the rating of suppliers for the C company 4.3 The performance evaluation model for the case study Using supplier A as an example, the objective function can be used to find the maximum performance value of supplier A, while the restrict functions will set the maximum performance value of all the suppliers to be 1, and also provide a bound for the weight of each factor. The mathematical programming model for the performance evaluation of supplier A is illustrated in Table 10. From the output of LINDO, the performance value of supplier A is only 0.9828, which is below the established range values of the weighting factor(Tables 6~8) in section 4.1. So supplier A belongs to the low-performance group, and should be given a lower priority in the selection process. The same method can be applied to other suppliers to obtain their performance values as shown in Table 11. Table 11. The performance values of suppliers The result shows that the performance values of suppliers C and D are 1, which is within the range of the weight established by evaluators. Since these two suppliers have the highest performance value, they should be the most favorable candidates of choice. The performance values of suppliers A and E are very close to 1, so they can be considered as the second group to choose from. The super-efficiency model can be applied to the two suppliers with performance values equal to 1, then the ranking between them can be obtained. Taking supplier C as an example, the superefficiency model can be constructed as shown in Table 12. From the output of LINDO, the performance value of supplier C is 1.053, and supplier D is 1.1277. So supplier D has a higher possibility to be the best supplier. 4.4 Improvement models for each supplier We can find the optimal solution by using the LINDO software, and then use the found values of variables and of improvement models for each supplier. Alternatives for the improvement of the input and the output items for each supplier is shown in Tables 13 and 14. Table 13 Alternatives for the improvement of the output items for each supplier Table 14. Alternatives for improvement of input items for suppliers 5. Conclusion By combining the methodologies of DEA, AHP and fuzzy set theory, a modified DEA method is proposed in this research. It has the following substantial results: 1.It can be used to evaluate the performance of each business unit, and also can suggest the directions for improvement of the lowperformance units. 2. It has a wider application scope, especially when there are only a small number of business units under evaluation. 5. Conclusion 3. Through the establishment of the range values for weights, the evaluation result is more acceptable than the original DEA method which may rate several DMUs with the same performance value of 1. From the case study results, it has been shown that the proposed method can be used to many practical problems like the selection of suppliers for a component, which usually involves a small number of candidates. Q&A THANKS! REFERENCES 1. Andersen, P. and N. C. Petersen. “A Procedure for Ranking Efficient Units in Data Envelopment Analysis,” Management Science, 39, 1261-1264 (1993). 2. Charnes, A., W. W. Cooper and E. Rhodes, “Measuring the Efficiency of Decision Making Units,” European Journal of Operational Research, 2(6), pp. 429-444 (1978). 3. Hung H. L., “Use DEA to Analyze the Operation Performance of Manufacturing Industry ,” Thesis, National Chemg Kung University, (2002). 4. Kao C., S. N. Huang, and Toshiyuki Sueyoshi, Performance Evaluation of Management-DEA , Hua-Tai Publisher,Taiwan (2003). 5. Lo F. Y., C. F. Chien, and J. T. Lin, “A DEA Study to Evaluate theRrelative Efficiency and investigate the district reorganization of the Taiwan power company,” IEEE Trans. Power Syst.,16, pp. 170–178 Feb. (2001). 6. Narasimhan. R., S. Talluri and D. Mendez. “Supplier Evaluation and Rationalization via Data Envelopment Analysis: An Empirical Examination,” The Journal of Supply Chain Management; Summer (2001). 7. Sun S., Data Envelopment Analysis-Theory and Applications, Yang-Chih publisher, Taipei (2004). 心得 讀完此篇DEA理論論文，讓我知道原本的ＤＥＡ權重是沒有 設限的，但利用層級分析法規範出權重範圍，這樣就不會 讓投入及產出項無效率的項變的很有效率；並且也可以針 對每項投入與產出項提出需要改善的項目。而且加入了層 級分析法計算出來的績效值更具客觀性，同時也比原始Ｄ ＥＡ模式更符合實際情況，而且也可以運用在投入及產出 項少的評選上。