The Uniqueness of the Overall Assurance Interval for Epsilon in DEA Models by the Direction Method

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1 Available online at Vol., No., Summer 5 Journal of New Researches in Mathematics Science an Research Branch (IAU) he Uniqueness of the Overall Assurance Interval for Epsilon in DEA Moels by the Direction Metho S.Mehrabian Department of Mathematics, Faculty of Mathematical Science & Computer, Kharazmi University, Kara, Iran Receive Spring 5, Accepte autumn 5 Abstract he role of non-archimeean in the DEA moels has been clarifie, so that the associate linear programs can be infeasible (for the multiplier sie) an unboune (for the envelopment sie) with an unsuitable choice of. his paper shows that the overall assurance interval for in DEA moels is unique by the concept of extreme irections. Also, it presents an assurance value for using only simple computations on inputs an outputs of DMUs. Keywors: Data Envelopment Analysis (DEA), Non-Archimeean Infinitesimal, Extreme Directions. aress: saei_mehrabian@yahoo.com

2 S.Mehrabian /JNRM Vol., No., Summer 5. Introuction DEA is a mathematical metho for etermining the relative efficiency of ecision maing units (DMUs). he ata are input-output observations for a number of DMUs using varying amounts of the same inputs to prouce varying amounts of the same outputs. Charnes et al. (978) propose a linear programming for etermining the relative efficiency of DMUs. In recent years, DEA has enoye both rapi growth an wiesprea acceptance. A new bibliography in website contains almost 9 stuies employing the methoology of DEA. In these stuies, the two most frequently use moels are the Charnes, Cooper an Rhoes (CCR) moel an Baner, Charnes an Cooper (BCC) moel, both of which involve the non- Archimeean. Even though some researchers prefer to apply the Archimeean DEA moels for their researches, the DEA literature shows that the non-archimeean DEA moels are still wiely accepte an applie to a large number of practical problems. Mehrabian et al. () efine the overall assurance interval of the non-archimeean for all of DMUs in CCR an BCC moels. hey have shown that an assurance value for using a single LP is enough for fining non-archimeean. In this paper, the concept of extreme irections in mathematical programming is use to provie strong support for the valiity an uniqueness of the overall assurance interval. Moreover, it is shown that an assurance value for can be etermine using only simple computations on inputs an outputs of DMUs. -Non-Archimeean DEA Moels an the Overall Assurance Interval for ε Consier n DMUs, each consuming varying amounts of m inputs in the prouction of s outputs. he m n matrix of inputs is enote by X an the s n matrix of outputs by Y. Furthermore, x i enotes the amount consume of the i th input by the th ecision maing unit, an y r enotes the amount prouction of its r th output. Finally, X an Y enote, respectively, the vector of inputs an outputs for the th DMU. he input-oriente linear programming problem formulation for the CCR an CC

3 he Uniqueness of the Overall Assurance Interval for Epsilon in DEA Moels 5 moels (both the envelopment an the multiplier sies) is as follows: CCR p : Envelopment Sie min ( S S ) s.t. Y S Y, X X S,, S, S. CCR : Multiplier Sie maxuy s.t. VX, BCC p UY VX, U, V. : Envelopment Sie min ( S S ) s.t. Y S Y, X X S,,, S, S. BCC : Multiplier Sie maxuy s.t. VX, u UY VX u, U, V, u free where is a row vector of units. () () Mehrabian et al. () introuce the overall assurance interval for as where min,, n [, ], such that is the optimal value of the following problem: max s.t. VX, UY VX, U, V. () Each element of the overall assurance interval [, ] is efine as an assurance value of the non-archimeean for feasibility/bouneness of the multiplier/envelopment sie in the CCR moel for all DMUs.. he Directions Metho he concept of extreme irections plays an important role in the theory of mathematical programming (see Bazaraa et al. (6) an Murty (99)). his concept is use to evelop a new metho which we call the Directions Metho, for calculating an overall assurance interval of [, ]. Definition : Let S be a nonempty convex set in nonzero vector in n n. A is calle a irection of S if for each x S, x S for all. wo irections an of S are calle istinct if for any. A irection of S is calle an extreme irection if it can not be written as a positive combination of two istinct irections, that is, if

4 S.Mehrabian /JNRM Vol., No., Summer 5 6 for for some., then Lemma: Given that S x : Ax b, x is a nonempty set where A is an m n. hen, is a irection of S if an only if, an A. Proof: See Bazaraa et al. (6). heorem : Suppose that the set of S x : Ax b, x is not empty an let,, be the extreme irections of the set S. hen, there is a finite optimal solution to LP of min cx : x S only c for,,. if an where A = X c(ε) = Y I X ε ε, x = I, b = Y θ S, S Now, suppose that,, are extreme irections for CCR (,, n ), where (,,, ),,,. In orer to guarantee the bouneness of CCR, we show that there is a positive such that c( ) for,, which is equivalent to the following inequality:,,, (5) Lemma : he set H :,,, is nonempty, for,, n. Proof: See Bazaraa et al. (6).. he Uniqueness of the Overall Assurance Interval Let CCR be the CCR p moel for the evaluating DMU. Without losing generality, we suppose in (). herefore, the matrix form of CCR is as follows: min c( ) x s.t. Ax b, x, () Proof: For a given, we only nee to show that there is a {,, } such that an. By contraiction, suppose that for,, t an for t,,. herefore, c( ) for,, t an c( ) for t,, imply that CCR is boune for all, which is a contraiction.

5 he Uniqueness of the Overall Assurance Interval for Epsilon in DEA Moels 7 heorem : he problems CCR (,, n) are boune for where min : H,, (6) min :,, n. (7) Proof: he proof is obvious from Lemma. By the following theorem, we are now able to show that the overall assurance interval introuce in Definition is equal to the one obtaine by the irections metho. heorem : Proof:.Suppose outcomes: Case.. We consier two. Let. Since [, ] is the largest overall assurance interval (see Mehrabian et al. (6)), there is at least one for which the CCR problem is unboune an this is impossible by heorem. Case. Let. Hence, there is at least one such that the CCR problem is unboune. From (7), there is a ( n) such that.now, (6).herefore, implies that the ( ) c. his means that CCR problem is unboune, which contraicts heorem. From the above two cases, we can euce that. 5. An Example In the following example, we will obtain a unique upper boun for non-archimeean in the CCR moel. Consier the following ata omain consisting of three DMUs each consuming one input to prouce one output (able). Input able : hree DMUs with one input an one output For calculating, we nee to obtain all the extreme irections (,,,, ) for each CCR problem (,,) as reporte in able. DMU DMU DMU Output implies that there exists a H that

6 S.Mehrabian /JNRM Vol., No., Summer 5 8 CCR CCR CCR,,,,,,,,,, 5 5 5,,,,, 5 5 5,,,,, 5 6,,,,, 9 9 9,,,,,,,,,,,,,,,,,,,,,,,,, 5 5 5,,,,, 9 9 9,,,,, able : he extreme irections for the problems CCR (,,) We have, an herefore, min,,.. So, heorem implies that CCR problems (,,) are boune for each. Also, it is obtaine that. hus, 6. An approach for obtaining an assurance value It is prove that for etermining an assurance value of the non-archimeean in the CCR moel, solving the following LP is enough (see []). he following theorem presents an assurance value for using only simple computations on inputs an outputs of DMUs. heorem : min{ p, pq} Is an assurance value of the non-archimeean in the CCR moel where, p / max X an q min X / Y Proof: It is sufficient that, prove, V, U is belongs to the feasible region of the problem P, where V For this means, we have p an U pq. P:max s.t. VX,, n UY VX,, n U, V. (8) V X = px, U Y V X = pqy px = py (q X Y ), =, n Also, pq U an p V. In the given example, p / max { X } / max{,,}

7 he Uniqueness of the Overall Assurance Interval for Epsilon in DEA Moels 9 an q min { X / Y } min{,, } herfore, / min{ p, pq}. 7. Conclusion In this paper, it is shown that the overall assurance interval for in DEA moels is unique by the concept of extreme irections. Also, it is provie an assurance value for non-archimeean epsilon, using only arithmetic operations on the inputs an outputs of DMUs.

8 S.Mehrabian /JNRM Vol., No., Summer 5 References [] Bazaraa, M. S., John J. Jarvis, an H. D. Sherali (6), Linear Programming an Networ Flows, John Wiley an Sons, hir Eition, New Yor. [] Baner, R. D., A. Charenes, an W. W. Cooper (98), Some Moels for Estimating echnical an Scale Inefficiencies in Data Envelopment Analysis, Management Science, Vol., No. 9, pp [8] Charnes, A., W. W. Cooper an E. L. Rhoes (978), Measuring the Efficiency of Decision Maing Units, European Journal of Operational Research, Vol., No. 6, pp. 9-. [] Mehrabian, Saei, Gholam R. Jahanshahloo, Mohamma R. Alirezaee an Gholam R. Amin, (), An Assurance Interval for the Non- Archimeean Epsilon in DEA moels, Operations Research, 8(), pp.-7. [5] Murty, K. G. (985), Linear Programming, John Wiley an Sons, New Yor.