A Chance Constraint Approach to Multi Response Optimization Based on a Network Data Envelopment Analysis

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1 Journal of Otimization in Industrial Engineering 1 (013) A Chance Constraint Aroach to Multi Resonse Otimization Based on a Network ata Enveloment Analysis Mahdi Bashiri a* Hamid Reza Rezaei b a Phd candidate eartment of Industrial Engineering hahed University Tehran Iran b Mc eartment of Industrial Engineering hahed University Tehran Iran Received 6 October 011; Revised 19 August 01; Acceted 7 ecember 01 Abstract In this aer a novel aroach to multi resonse otimization is resented. In the roosed aroach resonse variables in treatments combination occur with a certain robability. Moreover we assume that each treatment has a network style. ue to the robabilistic nature of the treatment combination the suggested aroach can comute the efficiency of each treatment under the desirable reliability. The aroach has been constructed based on a network data enveloment analysis and the chance constraint method. Finally a numerical examle shows alicability of the roosed method to a multi resonse roblem. The results indicate that the roosed aroach is a caable method for analyzing the best treatment and comared to other existing methods it erforms better in efficient treatment determination. Keywords: Multi Resonse Otimization; Chance Constraint; Network ata Enveloment Analysis; Knasack Aroach. 1. Introduction Multi resonse otimization (MRO) is a useful and wellstudied method in industrial roblems. In multile resonse roblems resonse variables must be considered generally in comlex roduct/rocess designs. In this regard some revious studies were done based on the MAM (multi attribute decision-making) aroaches such as data enveloment analysis (EA) and TOPI. In most of them the aim was to aggregate several criteria into a single criterion. Yet there are some aroaches which identify the non-dominated solution by areto solution search methods. On the other hand in multi resonse roblems there are treatments with controllable factors and resonse variables in a single stage; it means that we have many controllable factors as inuts and many resonses as exerimental results. In this structure the desirability of each treatment is checked to find otimum factors levels among all of the treatments. Nevertheless the exeriment can be more comlex than one stage and even with network structure for exeriments. In other words in addition to relations between inut/s and outut/s of each stage it is ossible for the outut/s of intermediate stage to be consumed as inut for the * Corresonding author bashiri.m@gmail.com neighborhood stage or to lay the role of final resonse in the exeriment. o in this situation the exerimental style is more comlex. This condition is illustrated in Figure 1. Inut 1 Inut n tage 1 Inut... tage Resonse 1 Fig. 1. The network structre of the multile resonse roblem The deterministic asect of exeriment design is a secial case for the robabilistic view with 100% confidence level. Moreover in some roblems we accet a considered risk to forecast future events and we can disregard the scenarios according to the risk interval. Therefore we are interested in determining the most efficient and reliable treatment with a considered confidence level. In this study to find the most efficient treatment an aroach is roosed based on the Network EA (Lewis and exton 004) and chance constraint method (Charnes and Cooer 1959). The first one hels to solve the comlexity of structure and the second one is.. tage K Resonse 49

2 Mahdi Bashiri et al./ A Chance Constraint Aroach to... considered the desired confidence level for the MRO roblem. The rest of this aer is organized as follows: the related literature on multi resonse otimization is reviewed in the next section. ection 3 involves the statement of roblem and its alication. ection 4 focuses on the roosed aroach for determining the most efficient treatment/s with a secific reliability level. In ection 5 a numerical examle is given and the comarison results between the roosed method and other existing aroaches are reorted. Finally some concluding remarks are resented in the last section.. Literature Review As an interesting subject multile resonse roblems have attracted the attention of many researchers. Existing methods in this area include both simle weighted-sum aroaches (Gauri and Pal 010) and more comlex regression and mathematical rogramming aroaches (Al-Refaei et al. 009) Princial Comonents Analysis(Fung and Kang 005) Fuzzy Logic (Tarng 000) Grey Rational Analysis (Manivannan et al.011) and multi criteria decision making (MCM) aroaches (Refaie 010; Lan 009). Caoraletti et al. (1999) roosed an inut EA model for NTB characteristics using y ij y i and ij as inuts. The authors only considered their exeriment to judge about controllable factors status efficiency. Liao and Chen (00) roosed an inut-oriented model (CCR EA model) that uses the normalized mean of resonses as inuts (if TB or NTB) or oututs (if LTB). Like the aforementioned study only the data of the actual exeriments were used. Liao (004) also used a Neural Network to estimate the mean resonses of all factor level combinations and then used a CCR EA model to select the best level based on Normalized /N ratios (as oututs) (Charnes et al 1978). In another study Gutiérrez and Lozano (010) used an Artificial Neural Network to train the relation between the resonses and all factor level combinations. ince then ata Enveloment Analysis (EA) is used to determine the efficient factor level combinations and then to select among factors levels mean square deviations of the quality characteristics are used as EA inuts. Tsai et al. (008) resented an otimization rocedure which uses EA to analyze multile inuts/oututs and costs for exeriments with mixtures. The authors assessed the method in a real case. The survey on EA extensions shows that Lewis and exton (004) suggested a novel model that alies to MUs and consists of a network of ub-mus. They considered a method which yields the efficiency scores of both MU and sub-mu. Camanho and yson (005) resented an alication of stochastic EA with chance rogramming. Also Bruni et al. (009) roosed a stochastic model for data enveloment analysis (EA) based on the theory of joint robabilistic constraints. The authors evaluated their roosed method in a sulier selection case study. ome studies have also been done on MRO. For examle Chiao and Hamada (001) considered exeriments with correlated multile resonses. The means variances and correlations deend on exerimental factors. iaz Garcia et al. (005) studied the alication of some stochastic rogramming aroaches to resonse surface otimization. They assumed a normal distribution for resonse surface coefficients (resonse surface methodology RM) and comared their aroach with the results of some methods such as E- Model V- Model and Lexicograhy. The RM is used to find a suitable aroximation of the true relationshi between resonse variables and controllable factors. In another study Hejazi et al. (011) roosed goal rogramming for multi resonse roblems with stochastic arameters ( ). In the study the mean and variance were considered as goals and goal rogramming led to tend the characteristics to the desirable values. ometimes the main roblem which occurs in multi resonse otimization roblems is that when the mean square error (ME) of the regression model is high the ability of the model to describe the relationshi of the resonse variable and the controllable factors is oor (Kim et al. 001). To overcome this roblem the ANN can be used as a roer substitute method for resonse estimation. ome authors comared the resonse surface and regression models with ANN in model building and the reciseness of ANN was verified in their results (Namvar-Asl et al. 008; Tsao 008). Table 1 shows a summary of existing aroaches classified according to the EA technique and its alications. In addition deterministic and uncertainty asects of MRO are another imortant asect considered to survey the characteristics of ublished works and the novelty of roosed methods. It is obvious that because of the stochastic nature of exerimentation the robabilistic view of treatment is an essential and attractive issue. Therefore the stochastic view of the MRO roblem leads to the identification of more reliable levels of controllable factors. Moreover the network structure of some exeriments and its comlexity structure have not been studied in revious studies. 50

3 Journal of Otimization in Industrial Engineering 1 (013) Table 1 The summary of related ublished works in the literature based on EA EA Problem tatus Alication Published works MRO imle Network eterministic uncertainty imle tructure Network structure Other Ref. [1] * * * Ref.[7] * * * Ref.[ ] * * * Ref.[13] * * * Ref[1617] * * * Ref[18190] * * The roosed method * * * 3. tatement of the Problem In this section more details of multi resonse roblem which is discussed in this aer are exlained. First the network structure of exeriments is evaluated. Most of multile resonse roblems existing in the literature have a single stage structure so that in each treatment controllable factors have the role of inuts and resonse variables are considered as oututs while structure of exeriments may be more comlex. According to Figure 1 and its network structure formal aroaches designed for simle exeriments cannot determine the otimum treatment accurately because of the structure of the exeriment which involves more than one stage. For more exlanation suose that we have a refining rocess. The core refining rocess is simle distillation. Because crude oil is made u of a mixture of hydrocarbons this first basic refining rocess is aimed at searating the crude oil into its fractions the broad categories of its comonent hydrocarbons. Crude oil is heated and ut into a still (a distillation column) and different roducts boil off and can be recovered at different temeratures. After that some roducts of the revious stage are added to the unfinished roduct and new inuts are used as inuts to a vacuum distillation unit. It is worth noting that consideration of a single stage for this roblem is wrong because the inuts of the second stage do not have effect on the first stage and entering the second stage inuts to the first stage leads to mistakes in the analysis of exeriment. In this real case and many other similar rocesses some of inut/s outut/s and intermediate roducts (unfinished roducts) exist in a multi stage exerimental design. Therefore it seems that the network structure of Figure 1 is more suitable for the mentioned roblems. In this aer in addition to the network style of exeriments uncertainty situation increases the roblem comlexity. Paying attention to the robabilistic asect of MRO in this study lets us to consider the deterministic situation as a secial case of robabilistic aroach with 100% confidence level. For defining the robabilistic situation of the roblem suose that according to the collected exerimental data there are lots of different results during a secial eriod of time and the analysis of them should be done considering the stochastic nature. One of the oular aroaches to stochastic analysis is the scenario-based aroach and we have used the clustered data according to the scenarios. Therefore there are some treatments with their occurrence robability in every scenario. The roblem in the form of robabilistic scenarios is resented in Table. This table shows the simle kind of exeriment (one stage) in a redefined scenario. Let us define the roblem arameters for each treatment j j=1 n: i=1 m controllable factor index. r=1 q resonse index. =1 P scenario index. y 1 Y j = ( j th scenario. y qj ) T resonses vector in the jth Trt and the Table Characteristics of the treatments in each scenario Probability of scenario cenario Trt occurrence P th 1 Pr Factors x 11 x m1 x 1 Resonses n x m x 1 n x mn y 11 y 1 y 1 n y q 1 yq y qn 51

4 Mahdi Bashiri et al./ A Chance Constraint Aroach to... For this roblem identification of the best treatment with the minimum risk and best ranking is the main objective. 4. The Proosed Aroach In this section to determine the efficiency of each treatment ata Enveloment Analysis and its extension are reresented for the deterministic situation. Then by adding the robabilistic aroach to the efficiency model the roosed stochastic efficiency model for the network MRO roblem is resented. Finally the stes of the roosed method are described ata Enveloment Analysis EA is one of the useful MAM techniques. EA measures the efficiency of organizations with multile inuts and oututs. These organizations are called decision-making units (MUs). Also by alying the EA to the related roblems the rank of each MU is secified based on its efficiency score. In this aer EA is used for multi resonse roblems so that after exerimental designs each treatment is considered as a MU. Then the efficiency score of each treatment hels us in the otimization stage. In this study the MU and ub-mu are the equivalent of treatment (trt) and sub-treatment (sub-trt) resectively in the design of exeriment resectively. We assume that each trt is comrised of a set of sub-trts. Each inut to a ub-trt is either exogenous to the trt or is the outut of another ub-trt. imilarly each outut from a ub-trt either leaves the trt as a final resonse variable or is an inut to another ub-trt. The network EA model (outut-oriented) can be illustrated in this way: Let k be the index of the trt in the exeriments. For d=1 and s t=1 s t X dsi = The level of inut i consumed by ub-trt s in trt d for i = 1 I Y dsto = The level of intermediate roduct o roduced by ub-trt s and consumed by ub-trt t in trt d for o= 1 O Z dsr = The value of resonse r roduced by ub-trt s in trt d for r = 1 R = The weight laced on ub-trt s in trt d by ub-trt s in trt k sk = The inverse efficiency of ub-trt s in trt k. For ub-trt s at trt k we let Max (1).t. d 1 sk X dsi X ksi i 1... I () Ydtso Yktso o 1... O (3) d 1 t1 t1 Ydsto sk Yksto o 1... O (4) d 1 t1 t1 d 1 Zdsr skz ksr r 1... R (5) 0; d 1... sk 0 sk is the inverse efficiency value of sub-trt s at trt k. Also Inequalities () and (3) ensure that the reference ub-trt consumes no more of each inut and each intermediate roduct as does ub-trt s at Trt k. Inequalities (4) and (5) guarantee that the reference ubtrt roduces at least as much as each intermediate roduct and each resonse as does ub-trt s at Trt k. * For each sub-trt by calculating the value of sk we can comute the efficient value of outut (Z*) or intermediate roducts (Y*) according to Equations (6) and (7). * * Z ksr Zdsr r 1... R (6) d 1 Y * * s ksr Ydstr r 1... R (7) d 1 t 1 where sub-trt and * is the otimal value of for the reference * Z ksr is the level of resonse variable r roduced by sub-trt s at trt k if it would be efficient. uose that we have a structure with two stages and we have comuted all of efficiency scores in the first stage of first Trt. To calculate the overall efficiency score we need to relace the otimal value of oututs or intermediate roduct (according to Equations (6) and (7)) in the next sub-trt. Hence for the second sub-trt the inverse efficiency score is comuted as follows: Max (8) 1.t. (9) d 1 X dsi X ksi i 1... I Y * dtso Yktso o 1... O (10) 1 t1 t1 d (11) d 1 Zdsr sk Zksr r 1... R 5

5 Journal of Otimization in Industrial Engineering 1 (013) ; d 1... (1) sk 0 (13) To obtain the overall inverse efficiency for each treatment we have: ** Zksr s1 k min (14) r1... R Z ksr s1 ** where Z ksr is the level of outut r that ub-trt s in trt k would roduce under these conditions if it were efficient. Notice that 1 because variables are returns to d 1 the scale. By calculating the inverse efficiency scores the treatment rank can be determined (for more exlanation see [1]). 4.. The robabilistic aroach In this section the robabilistic asect of the roblem is added to the roosed model. We have some scenarios with certain occurrence robability and in each scenario a set of treatments network structure are tested. Knasack aroach is used to comute (1- )% confidence level to determine the efficiency of each trt. efinition 1: (cenario efficiency). Trt k is efficient with resect to scenario if it is imossible to find a feasible solution for the following roblem: d 1 d 1 X Y t d 1 1 Z dsi dsr X dtso Z ksi ksr Y t 1 ktso i 1... I o 1... O r 1... R (15) (16) (17) We have 1- % reliability for selecting the best treatment. In the stochastic framework for each trt uncertainty arameters are reresented by the random variables. These variables can be considered as the aroximation of the known distribution function or the set of scenarios. efinition : 1- % confidence level is erformed if: Max sk (18).t. ~ ~ Xdsi Xksi i 1... I d1 ~ ~ Ydtso Y ktso o 1... O d1 t1 t1 1 (19) ~ ~ Ydtso 1... sk Yktso o O d1 t1 t1 ~ ~ Zdsr sk Zksr r 1... R d1 where inut/s outut/s and intermediate roduct are reresented by the random variable defined on a given Z ~ ksr robability sace. For examle is the rth resonse of sub-trt s at trt k under the stochastic situation. efinition 3: (robabilistic knasack roblem) Knasack aroach is characterized by the allocation of limited resources to cometing items. The items are associated with resource requirements as well as rewards. In this aer we have a knasack with 1- % volume. Each scenario has a certain volume and an individual effect on the maximization of the inverse efficiency score of sub-trt and treatment. Therefore this aroach adds big M and 0-1 decision variable to decide which constraints can be removed from the knasack based on disregarding % (as the maximum risk-value) of the overall knasack volume The roosed multile resonse otimization aroach In this section the multi resonse otimization method is roosed. The roosed aroach is illustrated in Figure and its exlanation is rovided below: 53

6 Mahdi Bashiri et al./ A Chance Constraint Aroach to... Normalization of the inuts & oututs values esigning and erforming the exeriment Efficiency calculation for each stage trt by Equations (0)-(7) o the inverse efficiency of sub-trts have significant differences? etermination of the total Confidence level 1- (Family confidence level) T No: ecrease1- T Yes Calculation of overall efficiency for each trt by Eq. (0) etermination of each stage confidence level 1- (Individual confidence level) election of the best treatment with 1- T % reliability Fig.. The roosed multile resonse otimization aroach te1. Exeriment information etermine the controllable factors resonse variables and quality characteristics (i.e. larger-the-better (LTB) nominal-the-best (NTB) or smaller-the-better (TB)) of the resonse variables and collect the exeriment data. te. Normalization of the gathered data Normalize all of the results to reduce the effects of various measuring units of the resonse variables on efficiency analysis in the Network EA. Note that in most of MRO aroaches the analyzer does not attend to cost considerations of factors levels and the controllable factors' values are not considered. However in the EA aroach we assume that the analyzer is interested in decreasing the inut variable value. We can use one of the available normalization formulas (according to the TB LTB and NTB). te3. Efficiency calculation for each sub-trt Comute the inverse efficiency score for each sub-trt at each trt according to following model: Max (0).t. d 1 sk X dsi X dsi i 1... I 1... P Ydtso Y d 1 t1 t1 o 1... O; 1... P ktso M (1) () Y d 1 t P d 1 Z dsr r 1... R dst M M sk Z Y sk kst t 1 ksr (3) (4) P Pr. (5) 1 0; d 1... (6) sk 0 =01 (7) where 1 = 1 t.in this relation is the number of stages (sub-trt) in the exeriment structure and is the comlement of the imosed reliability level (riskvalue) for each stage and t has the same concet for the overall inverse efficiency calculation. Equation (0) is the objective function for determining the efficiency score of the sth sub-trt at the kth treatment and Equation (5) defines a binary knasack constraint which guarantees the violation of the constraints for a subset of scenarios whose cumulative robability is less than the comlement of the imosed confidence level. Also is a binary variable for eliminating the th scenario in the comlement of the imosed confidence level interval. 54

7 Journal of Otimization in Industrial Engineering 1 (013) Other equations have the same concets mentioned in section 4.1 with a scenario treatment. te 4. Treatment selection based on efficiency scores etermine the best treatment among all based on the overall efficiency score of all treatments. To this end get the overall efficiency in accordance with Equation (8). Notice that k is the inverse of efficiency in k th treatment. 5. A Numerical Examle In this section a hyothetical examle is resented. The structure of exeriments at each treatment is given in Figure 3. Controllable Factor 3 Controllable Factor 1 Resonse 1 Controllable Factor tage 1 Intermediate Product tage Resonse As it is shown in the figure each treatment consists of two stages. The first stage has two controllable factors each at three levels and two oututs. Other controllable factors are an inut for the second stage. The detailed information about the numerical examle is given in Table 3. In this regard suose that we exeriment with 9 treatments. The aggregated data show that we have 4 scenarios with the robabilities of and 0.05 resectively. In this examle 80% overall confidence level (1- ) is defined by the decision maker. Thus giving attention to Table 3 Information about the numerical examle cenario (occurrence robability Pr) 1 (Pr=0.1) (Pr=0.35) 3 (Pr=0.5) 4 (Pr=0.05) Fig. 3. The network structre of the examle T the individual confidence level for each sub-trt shows that 90% confidence level is suitable for running the roosed MILP model and comuting each efficiency score. We used a classic otimization software (Lingo 8) for calculating the efficiency of each treatment. The results summarized in Table 4 show that the 4 th treatment has the highest efficiency score among the others ( 4 1 ). To further clarify the roosed aroach the comuted variables for the 7th treatment are reorted in Table 5. Controllable Factors Resonse variables Intermediate roduct Trt Factor 1 Factor Factor 3 Resonse1 Resonse

8 Mahdi Bashiri et al./ A Chance Constraint Aroach to... Table 4 Efficiency results of the roblem for each treatment Treatments efficiency score Rank Table 5. Efficiency calculation for the 7th treatment considering 81% confidence level Treatment 17 * Inverse Efficiency score Efficiency core Table 5 reveals that the 7 th treatment has got 0.37 efficiency score with 0.81% confidence level (two stages each with 90% confidence level) Method analysis The roosed aroach emhasizes cost consideration. Most of multi resonse otimization aroaches overlook controllable factors values but in some cases the amount of controllable factors (with different costs) are not equal in different treatments. In such cases using the roosed aroach is therefore more aroriate. uose two treatments with two factors each at two levels according to Table 6. Most of existing aroaches select the first treatment; however it seems that the second one is aroriate too. Table 6 Regarding or disregarding of costs in MRO Factors Resonses Trial A B Resonse 1 Resonse No To check the good erformance of the roosed aroach we generated random data for all of the treatments. By increasing the LTB resonses in a certain trt and scenarios the reasonable behavior existed for its efficiency value. This concet is resented in Table 7. Table 7 The effect of resonse changes on the inverse efficiency score (the first resonse of the 7th treatment) cenario number Current value of the first resonse Changed values of the first resonse Current value of the inverse efficiency score Modified value of the inverse efficiency score To evaluate the efficiency of the roosed method we imlement two existing aroaches and comare the results. We solve the mentioned multi resonse roblem by means of the deterministic EA and Network EA. The results are reorted in Table 8. All of the surveyed aroaches can be used for determining the efficient trt but the main roblem is which method works with more accuracy and caability. As it clear in Table 8 the mean of inverse efficiency score for Network EA (1.197) is greater than EA (1.079); therefore the Network EA is more caable of identifying sources of inefficiency in comlex models thereby otentially yielding greater analysis insights into exerimental design imrovements. Moreover the comarison of EA and Network EA shows that the number of efficient trt is less for Network EA and ranges and standard deviations of inverse efficiencies for Network EA are more sread. The defined criteria show that the Network EA can roose better solutions for analyzer so that the analyzer can secify the source of inefficiency by considering the subtrt efficiencies in the network structure. By decreasing the confidence level from 100% the model can determine the resources of inefficiencies better. For examle considering 81% reliability level for the numerical examle leads to a greater mean for the robabilistic kind of Network EA comared with the deterministic EA and Network EA. Table 8 shows that the robabilistic Network EA works better than the others so that it has a high mean of inverse efficiency scores (1.445) and rang of differences (1.73). Moreover the roosed method suggests only one trt as the otimal solution whereas the others have more than one solution. In other words the roosed aroach hels analyzer to identify the areto otimal solution by acceting the risk value in a network structure of exeriments. Controllable Factor 1 Controllable Factor Controllable Factor 3 tage 1 Resonse Resonse 3 Fig. 4. The simlified structre of the numerical examle for using in the simle EA 56

9 Journal of Otimization in Industrial Engineering 1 (013) In this section behavior of inefficiencies with resect to the risk value is evaluated. It is worth noting that the invers efficiency score increases or (in the worst case) remains stable with growing the risk values. This illustrates the imortance of choosing aroriate threshold robability levels to avoid incorrect classifications of treatments. Table 9 resents the risk value and its effect on the inverse efficiency scores. This table is related to the inverse efficiency score of the first stage in the numerical examle. Table 8 Comarison results obtained for the EA Network EA and the roosed aroach Trt No. Inverse Efficiency core EA Network EA (the roosed aroach) No Mean Rang Table 9 Effects of confidence level changes on the inverse efficiency score in the examle Confidence level % or 100-(ercent of risk-value) Trt We can analyze the stability of efficiency scores and ranks according to risk intervals. For examle a risk value is suggested by the analyzer but the results would be stable for a higher confidence level. o we can check the neighborhood interval of the threshold robability level to reort the most reliable level. In Figure 5 the behavior of the inverse efficiency score with resect to the risk-values is evaluated. In this figure the first stage of numerical examle is used to interret the results of Table 8. The figure demonstrates that increasing the risk leads to cutting the overla scenarios or constraints in the roosed model. To evaluate the validity of the roosed aroach we generated five random samles according to the mentioned structure. The results show the caability of the roosed method through aroriate scores in comarison to the other aroaches related to EA. All of the scores are reorted in Table 10. Inverse efficiency score ۵٠ ٧٠ ٩٠ ١٠٠ Confidence level trt 1 trt trt 3 trt 4 trt 5 trt 6 trt 7 trt 8 Fig. 5. Behavior of the inverse efficiency score for the first stage with resect to the confidence level 6. Conclusion In this aer an attemt was made to roose an efficient aroach to find the best controllable factors levels in a joint robabilistic situation and network form of 57

10 Mahdi Bashiri et al./ A Chance Constraint Aroach to... exeriments. First an MILP model based on the network EA and knasack aroach was resented. Then the efficiency score of each sub-trt was obtained by the roosed model according to the individual desirable confidence level. After that the overall efficiency score for each treatment was calculated. Finally the mentioned aroach was discussed in numerical examles. The results show that the suggested aroach not only rooses a unique treatment to ot but it also can rank the treatments according to a wide range of efficiency scores. As a future research correlated scenarios for resonse variables can be studied so that each resonse variable influences the others. Moreover during designing the exeriments the designer needs to evaluate the treatments considering a structure that flaws can be refined for Rerocessing; therefore other network structures (with a recursive loo) can be considered in future research. Table. 10 efined criteria for comaring the methods caability based on inverse efficiency scores in five samles amles No.1 Mean Rang 1 * ** 3 *** *EA **Network EA *** The roosed aroach 7. References [1] Al-Refaei A. Al-urgham L. and Bata N. (009). Otimal Parameter esign by Regression Technique and Grey Relational Analysis. Proceedings of World Congress on Engineering. London U.K [] Al-Refaie A. (010). Grey-EA aroach for solving the multi-resonse roblem in Taguchi method. Proceeding of the Institution of Mechanical Engineering-Part B. Journal of Engineering Manufacture [3] Bruni M.E. Conforti. Beraldi P. and Tundis E. (009). Probabilistically constrained models for efficiency and dominance in EA. International Journal of Production Economics [4] Camanho A.. and yson R.G. (005). Cost efficiency measurement with rice uncertainty: a EA alication to bank branch assessments. Euroean Journal of Oerational Research [5] Caoraletti L.E. ul J.H. and Womer N.K. (1999). Performance evaluation based on multile attributes with nonarametric frontiers. Omega [6] Charnes A. and Cooer W.W. (1959). Chanceconstrained rogramming. Management cience [7] Charnes A. Cooer W.W. and Rhodes E. (1978). Measuring the efficiency of decision making units. Eur. J. Oerat. Res [8] Chiao C. and Hamada M. (001). Analyzing exeriments with correlated multile resonses. Journal of Quality Technology [9] iaz-garcia J. A. Ramos-Quiroga R. and Cabrera- Vicencio E. (005). tochastic rogramming methods in the resonse surface methodology" Comutational tatistics & ata Analysis [10] Fung C-P. and Kang P-C. (005). Multi-resonse otimization in friction roerties of PBT comosites using Taguchi method and rincile comonent analysis. Journal of Materials Processing Technology [11] Gauri.K. and Pal. (010). Comarison of erformances of five rosective aroaches for the multiresonse otimization International Journal of Advanced Manufacturing Technology [1] Gutiérrez E. and Lozano. (010). ata Enveloment Analysis of multile resonse exeriments. Alied Mathematical Modeling [13] Hejazi T.H. Bashiri M. iaz-garcia J. A. and Noghondarian K. (011). Otimization of Probabilistic Multile Resonse urfaces Alied Mathematical Modeling [14] Kim K.J. Byun J.H. Min. I.J. Jeong. (001) Multiresonse surface otimization: concet methods and future directions Tutorial; Korea ociety for Quality Management. [15] Lan T.. (009). Taguchi otimization of multi-objective CNC Machining using TOPI. Information Technology Journal [16] Lewis H.F. and exton T.R. (004). Network EA: efficiency analysis of organizations with comlex internal structure. Comuters & Oerations Research [17] Liao H.C. and Chen Y.K. (00). Otimizing multiresonse roblem in the Taguchi method by EA based ranking method. International Journal of Quality & Reliability Management [18] Liao H.C. (004). A data enveloment analysis method for otimizing multi-resonse roblem with censored data in the Taguchi method. Comuters and Industrial Engineering [19] Manivannan. evi.p. Arumugam R. and udharsan N.M. (011). Multi-objective otimization of flat late heat sink using Taguchi-based Grey relational analysis. International Journal of Advanced Manufacturing Technology [0] Tarng Y.-. Yang W.-H. and Juang.C. (000). The use of fuzzy logic in the Taguchi method for the otimization of the submerged arc welding rocess. Int. J. Adv. Manuf. Technol [1] Tsai Chih-Wei. Tong Lee-Ing. and Wang Chung-Ho. (008). Using EA to otimize a mixture design with multile resonse incororating cost considerations. Journal of cientific & Industrial Research

11 Journal of Otimization in Industrial Engineering 1 (013) [] Tsao C. (008) Comarison between resonse surface methodology and radial basis function network for corecenter drill in drilling comosite materials; The International Journal of Advanced Manufacturing Technology

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