4'nInternational

1. 4'nInternational Confcrcnce oflranian Opcrations Research Society May 18-19,201I Faculty Mathematical of Sciences, University Guilan of A new Nefwork DEA model Roza Azizi., lslamic Azad University, Karaj Branch - Departmcnt of Mathematics- aziziroza@gmail.com Reza Kazemi Matin, Islamic Azad University, Karaj Branch - Depaftment of Mathematics- rkmatin@kiau.ac.ir Abstract: In recent years, there have been various researches which handle network systemsin DEA literafure. Network systems are the systems which consume some inputs to produce some outputs which some of them in some processesare used as the inputs of some of the other processes (intermediateproducts) to produce the final outputs. Most of the current approachesdid not have any specified model foi all the casesof network systems.In this paper a brief review of Kao approach is given and a new model for network systemswill be presentedwhich computesefficiency measurefor all the casesof nefwork systems. Ke).words: Network systems,Specifiedmodel, Efficiency measure. 1. INTRODUCTION K^o (2009) APPROACH 2. REVTEWoN DEA is a mathematical programming approach for Based on the most common notations in DEA literature, measuring the efficiency and the performance of any let f\flJ 0 : 1 , . . . , n ) c o n s u m e i n p u t s x U ( i : 1 , . . . , m ) to organizations, such as, commercial, educational, serving and etc. produceoutputsYo (r:1,...,s). The conventional DEA models, consider each DMU as Modcl (1) shows the multiple form of CCR rnodel a black box without any look to the subsystemsactivities. which is firstly presented Chameset al. (i978). by In rcccnt years, a number ofDEA studieshave focused on Zu't '" systems with network structure. For example, Fare and 0. - Mtu +- Grosskopf (2000) divided network systemsto four cases, LV ,T,. Static non-network model, Fixed but allocative Inputs, Static nctwork model, Dynamic network model, Lu'Y, j=t.....rt (l) s.1 <l explanation of these cases and examples of them are rn +* thcir paper. Sexton and Lewis (2003) study the Major /,''^ t i League Baseball performance in a two-stage process. In r - 1 , . . . , s , i =\ , . . . , n t v,,u,Z0 their study money is used in the form of player salaries as The CCR model is based on the constant returns to inputs to acquire talent as outputs of first stage and in scale (CRS) assumption for the production set. Model (l) second stage talent is used as inputs to win games as final is in its input-oriented version. The efficiency score of outputs. And in 2004 they study the Major League this model is between zero and one.p\,fll.is called Baseball performance in network structure. Another study "efficient" respect to this model if and only if its is Kao (2009) which divided network systems to series, computed efficiency scorebe equal to one. parallel, and systems which do not have thcse two Model (1) cannot show logical efficiency score for structures. network systcmsbecauseit does not consider the products In the systems which do not have series or parallel which are produced and consumed between processes struchrres, most of thc authors like Kao and Fare and (intermediate products). So, it is necessary to introduce Grosskopf (2000), did not introduce any specified model the model which can use intermediateproducts. for this case and just wrote the restrictions of each Kao (2009) has been divided network systems to three processesin their model and this way for the time when caseswhich will be presentedas the follows: there are a few processesor a few inputs or outputs need 1. Seriessystem much lime and thc possibility of error is high. Figure I shows a system consisting of p (p: I ,. . .,q) In this paper, we introduce the model which is processes connectedin senes. appiicable for series and parallel systems, and also for network systems which do not have series or parallel structures. Based on the above discussion and to introduce the method to find the efficiency measure of all the casesof Figure Series l. system sys:ems with network structures, the rest of this paper will zlis the dth intermediate product which is consumed unfold as following. Section 2 is devoted to a brief revrew of Kao (2009) paper. A new Nehvork DEA model will in pth process andwlis the weight of dth intermediate introduced in section 3, which have the ability of finding product in pth process.Kao (2009) have been introduced the efficiency measureof all the network systems, seriesmodel as follows: 663