Generalized tolerance sensitivity and DEA metric sensitivity

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1 Croatia Operatioal Research Review 69 CRORR 6(205), Geeralized tolerace sesitivity ad DEA metric sesitivity Luka Neralić, ad Richard E. Wedell 2 Faculty of Ecoomics ad Busiess, Uiversity of Zagreb Trg J. F. Keedy 6, 0000 Zagreb, Croatia leralic@efzg.hr 2 Katz Graduate School of Busiess, Uiversity of Pittsburgh Pittsburgh, PA 5260, USA wedell@pitt.edu Abstract. This paper cosiders the relatioship betwee Tolerace sesitivity aalysis i optimizatio ad metric sesitivity aalysis i Data Evelopmet Aalysis (DEA). Herei, we exted the results o the geeralized Tolerace framework proposed by Wedell ad Che ad show how this framework icludes DEA metric sesitivity as a special case. Further, we ote how recet results i Tolerace sesitivity suggest some possible extesios of the results i DEA metric sesitivity. Keywords: sesitivity, tolerace, DEA, metric, stability, Chebyshev Received: September 22, 204; accepted: March 3, 205; available olie: March 30, 205 DOI: /crorr Itroductio Tolerace sesitivity aalysis i liear programmig was proposed by Wedell [9, 20, 2]. This work was origially focused o the Chebyshev orm ad showed how to fid a maximum tolerace, typically iterpreted as the maximum percetage withi which selected coefficiets or terms (e.g., objective fuctio coefficiets) could vary while maitaiig the same optimal basis. As oted by Ward ad Wedell [8], Wedell [22], ad Wedell ad Che [24], a umber of subsequet papers proposed various theoretical extesios as well as applicatios icludig sesitivity of matrix coefficiets, facility locatio problems, ad multiple objective problems. Shortly before Tolerace sesitivity was proposed, Chares et al. [2] wrote the semial paper o Data Evelopmet Aalysis (DEA), proposig a methodology for performig a relative efficiecy evaluatio of etities called Decicsio Makig Uits (DMUs) which use the same iputs ad produce the Correspodig author Croatia Operatioal Research Society

2 70 Luka Neralić ad Richard E. Wedell same outputs. Sice the, DEA has become a fast growig area with, accordig to the bibliography of Emrouzejad [9], more tha four thousad refereces of works o DEA. For a excellet itroductio to the theory ad use of DEA see, for example, [7] ad [6]. The importace of variatios i the data, or what we will call sesitivity aalysis, was apparet i the early days begiig with Chares et al [] cosiderig sesitivity with respect to a sigle output. This was followed by Chares ad Neralić [4], Neralić [3], Neralić ad Wedell [4] ad others. Oe of the approaches to DEA sesitivity, proposed by Chares et al [3, 5], was to apply metric cocepts i a way to make it possible to determie allowable variatios i all iputs ad outputs for oe DMU i a DEA model. I particular, usig a l p orm to characterize distace, these papers propose optimizatio models to determie a maximum radius, called a radius of stability, withi which variatios of the iputs ad outputs i a selected DMU would remai efficiet, if the selected DMU was origially efficiet, ad would remai iefficiet, if it was origially iefficiet. Sice the, research o DEA metric sesitivity has primarily focused o the Chebyshev orm. For a further discussio see, for example, [8] as well as [7] ad [6]. Both Tolerace sesitivity ad DEA metric sesitivity have remarkable similarities, but iterestig differeces. Herei, we exted the results o a geeralized Tolerace framework proposed i [24] ad we show how the geeralized framework icludes DEA metric sesitivity as a special case. Further we ote how recet results i Tolerace sesitivity suggest extesios of the results for DEA metric sesitivity. The paper proceeds as follows. Sectio 2 gives a statemet of the geeral Tolerace framework from Wedell ad Che together with some theoretical extesios. Sectio 3 applies this geeral framework to DEA, yieldig the results of Chares et al [3], Cooper et al [8], ad Cooper et al [7] o metric-based DEA sesitivity aalysis. Sectio 4 gives some cocludig observatios ad discusses potetial future research, icludig how recet results i Tolerace sesitivity ca be applied to yield extesios of the results i DEA metric sesitivity. 2. A geeralized tolerace sesitivity aalysis framework Let c deote a vector of parameters i R t, ad let f(x,c) ad g(x,c) deote vectors of fuctios o X with parameters c where XX RR is a give costrait set. Cosider the optimizatio problem: MMMMMM {ff(xx, cc): gg(xx, cc) 0,xx XX } () I this paper we follow ad exted the otatio give i Wedell ad Che [24]. Ay overlap betwee this otatio ad otatio used i DEA will be clear from the cotext.

3 Geeralized tolerace sesitivity ad DEA metric sesitivity 7 Observe that whe the vector f has multiple compoets, () deotes a multiple objective problem. Let x* deote a feasible solutio to () whe c = cc. Cosider a perturbatio structure of c i which c = cc + γc ' where C ' is a give s x t matrix ad where the parameters γ ca take o values i a give set Γ, called a a priori parametric regio, deotig a regio withi which γ may vary. The a priori set Г, which always icludes 0 ad which we assume is closed, eables us to exploit kow a priori limits o variatios of the parameters γ to obtai stroger results. Herei we cosider the case whe s = t ad C ' is a oegative, diagoal matrix. The diagoal elemets C ' ii are sometimes called perturbatio rates ad deoted as a vector c '. Observe that whe γ = 0 we have c = cˆ. Suppose that whe γ = 0 the solutio x* satisfies some specified optimality coditio, ofte characterized as a set of equatios ad/or iequalities i γ. For example, i a liear programmig problem the optimality coditio could be that the same basis is optimal, that the same solutio is optimal, that the same variables are basic, etc. Let P(x*), called a critical regio, deote the set of γ for which the specified optimality coditio holds. For problems with a liear structure, P(x*) is typically a covex polyhedro. A basic questio i sesitivity aalysis is: How much ca γ deviate from 0 withi Γ ad still remai i the critical regio P(x*)? To elimiate the trivial case, we assume that P(x*) is a proper subset of R s. Tolerace sesitivity aalysis is a approach to aswer this questio i the case whe "how much" is determied by the size of a hyperbox withi which the parameters γ ca vary simultaeously ad idepedetly over Γ. For a further discussio see, for example, [8, 22, 24, ]. The Tolerace sesitivity problem ca be characterized as: sup ρ(uu) s.t. γγ Γ { γγ: γγ BB(uu)} γγ PP(xx ) (2) where uu is a oegative vector i R 2t, BB(uu) is a hyperbox, also called a tolerace box, characterized as B( u) = { γ : ui γ i ut+ i, for i =, 2,, t}, ρρ(uu) deotes a fuctio that characterizes the size of the hyperbox " " deotes a if-the implicatio. If uu deotes a feasible solutio to (2), observe that the parameters γ may vary simultaeously ad idepedetly over the regio B(uu) Γ while preservig the specified optimality coditio for xx*. Traditioal Tolerace sesitivity ca be viewed as settig all compoets of u equal to oe umber, which we call a tolerace ad deote as τ. I this case, the hyperbox is a hypercube correspodig to the set { γ : γ τ }, where is

4 72 Luka Neralić ad Richard E. Wedell the Chebyshev orm ad the tolerace τ deotes the size ρρ(uu) of the hypercube. Problem (2) correspods to fidig the maximum tolerace τ, deoted as τ*, where the optimality coditio γ P(x*) holds for all γ Γ whe γ τ. Thus, the traditioal Tolerace problem for computig τ* ca be stated as: sup τ s.t. γγ ϵ ΓΓ γγ: γ τ γγ ϵ PP(xx ) (3) Observe that the maximum tolerace τ* is itself a tolerace iff the sup i (3) is achieved. Also ote that the maximum tolerace correspods to a hypercube * B( u ) where ui =τ for all i. Whe c' = cc the τ*x00%, called the maximum Tolerace percetage, deotes the maximum percetage withi which the parameters c may vary simultaeously ad idepedetly from their origial values ĉ while maitaiig the optimality property. I additio, observe that settig a compoet c i ' = 0 is a coveiet way to specify that o perturbatios o ĉ i will be cosidered, which would be the case if we kew that ĉ i was the kow value of c i. A key advatage of usig traditioal Tolerace sesitivity is the ease with which a maager ca ituitively iterpret ad readily use the maximum tolerace percetage i sesitivity aalysis. For example, i cosiderig sesitivity aalysis of the objective fuctio coefficiets i liear programmig, the maximum tolerace percetage ca deote the maximum percetage by which the objective fuctio coefficiets may vary from their origial values while maitaiig the same optimal basic feasible solutio. From (3) we ca give a simple characterizatio of the case whe τ* =. Theorem. The maximum tolerace τ* = iff γγ ΓΓ γγ P(x*) (4) Observe that, for the special case whe there is o a priori iformatio o the allowable rage of perturbatios (amely, whe Г = R t ), τ* is fiite iff P(x*) is a proper subset of R t. Lettig P(x*) c deote the relative complemet of P(x*), we ow give a equivalet characterizatio of (3). Theorem 2. A equivalet represetatio of (3) is if γ γ Γ { P x* c } (5) s.t. ( ) Proof. The equivalece whe τ* = follows from Theorem. Thus, cosider the case whe τ* is fiite, lettig if deote the ifimum i (5). We first show γ ' Γ { P x* c } with that τ* if. If τ* > if, the by (5) there exists a ( )

5 Geeralized tolerace sesitivity ad DEA metric sesitivity 73 γ ' < τ*. But lettig τ = γ ' i (3) gives a cotradictio. We ow show that if τ*. Cosider a positive sequece {ε k } covergig to 0. For each k, let τ k = τ* + ε k. For each k there is by (3) some γ k k k Г with γ τ ad k γ P x* c k k. Sice for each k, if γ τ, it follows that if τ*. QED ( ) The set P( x* ) c i (5) is ofte ot closed. I the followig importat special case, whe there is o a priori iformatio o the variatios of the parameters, this presets o problem. Theorem 3. If Г = R t the τ* equals the ifimum if γ s.t. γ Px ( *) c (6) The proof is immediate. Note that, sice problems (3) ad (6) are equivalet whe Г = R t, oe ca choose whichever problem is easier to solve for a particular situatio. Ofte (e.g., for Tolerace sesitivity i liear programmig) problem (6) is used. However i DEA, as we will see, problem (3) is easiest whe evaluatig the Tolerace sesitivity of a iefficiet DMU. Whe Г does ot equal R t dealig with the closure of P(x*) c is more delicate. This is illustrated i the example below where the ifimum i (5) is ifiite but where substitutig the closure of P(x*) c for P(x*) c i (5) gives a fiite ifimum. Example. Cosider the simple liear programmig problem: Max {(2x + x 2 ): x + x 2 =, x, x 2 0} for which x* = (, 0). Lettig c' = (, ), the perturbed problem is Max {(2+γ )x + ( + γ 2 )x 2 : x + x 2 =, x, x 2 0}. Thus, we have P(x*) = {(γ, γ 2 ): γ 2 - γ }. If we have the a priori iformatio that Г = {(γ, γ 2 ): γ -.5, γ 2.5}, the the ifimum i (5) is ifiity but the ifimum i (5) with the closure of P(x*) c substituted for P(x*) c is.5. Buildig o the work of Wedell [20] ad others, the solutio of (5) whe Г does ot equal R t is explored i a subsequet paper. Typically, the critical regio P(x*) i applicatios of Tolerace sesitivity correspods to a covex polyhedro. I this case, as oted by Hladik [], the basic idea of Tolerace sesitivity i liear systems as give by (2) is to fid a maximal box lyig iside a covex polyhedro. Here a box BB(uu) beig called maximal meas that there is o other box lyig i P(x*) ad beig a strict superset of BB(uu). Note that the box give by traditioal Tolerace sesitivity is

6 74 Luka Neralić ad Richard E. Wedell maximal amog the set of boxes that are hypercubes. Further, the work o Tolerace sesitivity has focused o the case whe P(x*) has a exteral represetatio 2 such as P(x*) = {γ: Dγ d}. As we will see, DEA metric sesitivity is oe applicatio for which the tolerace framework usig a iteral represetatio of a polyhedro is used. 3. DEA metric sesitivity as Tolerace sesitivity We ow cosider a applicatio to DEA. As usual i DEA, we assume that there are DMUs to be evaluated with >, where each cosumes give amouts of m differet iputs to produce give amouts of s differet outputs. Specifically, let X j R m deote the vector of iputs used by DMU j ad let Y j R s deote the vector of correspodig outputs. We assume that X j 0 ad Y j 0. Let X ij ad Y rj deote compoets i ad r of the respective vectors ad, as commo i DEA, assume that each DMU has at least oe positive iput ad oe positive output compoet. Followig [3] ad [7, 8], we focus for simplicity o the Additive Model which has the productio possibility set S = {(x,y): x λ jx, y j λ jy, j λ =, λ 0}. For ay two give poits (x',y') ad j j= j= j= (x'',y'') i S we say that (x',y') domiates (x'',y'') if x x'' ad y' y'' with at least oe iequality beig strict. A give poit (x*,y*) S is said to be efficiet if there is o poit (x, y) S that domiates it. Otherwise it is said to be iefficiet. As show by Chares et al [3], testig whether or ot a poit (x, y) is efficiet ca be easily determied by solvig a auxiliary liear program. Let DMU 0 deote a selected DMU with correspodig iputs ad outputs X 0 ad Y 0. As i [3] we cosider a radius of stability with respect to a give orm withi which the iputs ad outputs of DMU 0 may vary while preservig a DMU s curret classificatio efficiet or iefficiet. Followig [3] ad [7, 8], we assume that the iput ad output values of all other DMUs remai uchaged, ad we focus o the Chebyshev (l ) orm. I particular, for the Chebyshev orm observe that the radius of stability is a umber τ* 0 satisfyig the followig property where d + ad r d are positive umbers: i 0 τ < τ*, x i [X i0 τ d i, X i0 + τ d i ], y r [Y r0 - τ d + r, Y r0 + τ d + r ], (7) ad DMU 0 is efficiet (iefficiet) implies that (x,y) is also efficiet (iefficiet); ad τ* is the largest umber for which this property holds. 2 For a discussio of a exteral ad iteral represetatios of a polyhedro see, for example, page 07 of Schrijver [5].

7 Geeralized tolerace sesitivity ad DEA metric sesitivity 75 Chares et al [3] ad Cooper et al [7, 8] give the followig results. Theorem 4. If DMU 0 is iefficiet the s.t. τ* = Max θ Yrjλj θ d + r Y for r =,,s (8) r 0 j= λ, θ 0 Theorem 5. If DMU 0 is efficiet the Xijλj + θd i X for i =,,m i0 j= λ = j j= s.t. τ* = Mi θ Yrjλj + θ d + r Y for r =,,s (9) r 0 j=, j 0 Xijλj θd i X for i =,,m i0 j=, j 0 λ = j j=, j 0 λ, θ 0 where j 0 refers to the efficiet DMU 0 that is beig aalyzed. I particular, ote that if all compoets of d ad d + equal, the Theorems 4 ad 5 above respectively correspod to Theorems 3.4 ad 3.2 i [3]. The followig multiple optimizatio problem for the Additive DEA model correspods to problem () i Tolerace sesitivity: Mi x Max y s.t. (x, y) S (0) Let the coordiates X 0 ad Y 0 of DMU 0 correspod to the feasible solutio x* i (). With (X 0, Y 0 ) deotig a efficiet (iefficiet) DMU, we ow show how Theorems 4 ad 5 above ca be viewed as a applicatio of Tolerace sesitivity to problem (0).

8 76 Luka Neralić ad Richard E. Wedell Lettig C ' = (C' X, C ' Y ) whose oegative diagoal elemets are d i ad d + r, we ca represet poits x ad y resultig from perturbatios from X 0 ad Y 0 as (X 0 + γ X C ' X ) ad (Y 0 + γ Y C ' Y ) respectively. Observe that if d = X 0 ad d + = Y 0 the γ X ad γ Y represet % variatios of x ad y from X 0 ad Y 0 respectively as i (7). This correspods to a observatio i [8] that τ* i Theorems 4 ad 5 ca be iterpreted as a ratio relative to the positive weights used for each costrait. Also, i a similar aalysis for the Ratio (CCR) model, Chares et al [5] observe that τ* deotes the miimum percetage chage ecessary i every iput ad output to brig the test uit to its earest ustable poit. Note that the iterpretatios of the perturbatios are idetical to those i Tolerace sesitivity. Below we idividually cosider the cases whe DMU 0 is iefficiet ad efficiet. 3.. Whe DMU 0 is iefficiet First we assume that DMU 0 is iefficiet. I this case the critical regio P(x*) is the set for which a perturbatio gives a iefficiet DMU. Thus, whe DMU 0 is iefficiet, ote that P(x*) equals the set of perturbatios (γ X, γ Y ) where (X 0 + γ X C ' X,Y 0 + γ Y C ' Y ) is i the covex hull of {(X j,y j ): j =,, } plus the polyhedral coe, excludig the poit 0, geerated by {(e i,0): i =,, m} ad {(0,-e r ): r =,, s}. Cosider the traditioal Tolerace problem (3). Observe that problem (3) is equivalet to problem (8) whe Γ has o limits. Sice the supremum i (3) is idetical whether or ot we exclude 0 from the polyhedral coe, we ca iclude the 0 poit i the coe, so that P(x*) correspods to a iterior represetatio of a polyhedro. Iterestigly, i cotrast to Wedell [2] ad Hladik [] who give a closed form solutio for a exteral represetatio of a polyhedro P(x*) whe Γ = R t (based o computig the miimum distaces to each face of the polyhedro), Chares et al [3] ad Cooper et al [7, 8] deal directly with a iteral represetatio that exploits the structure of P(x*) by showig that a optimal solutio to (3) ca be foud by movig from (X 0, Y 0 ) i the directio θ(- d i, d + r ). I doig this, they obtai (8) i Theorem 4 as a equivalet optimizatio model to the traditioal Tolerace problem (3). Thus, for a iefficiet DMU we coclude that DEA metric sesitivity is a special case of the geeralized tolerace framework.

9 Geeralized tolerace sesitivity ad DEA metric sesitivity Whe DMU 0 is efficiet Now suppose that DMU 0 is efficiet. I this case, the critical regio P(x*) is the set for which a perturbatio correspods to a efficiet DMU; ad therefore P(x*) c correspods to iefficiet DMUs. Thus, P(x*) c equals the set of perturbatios (γ X, γ Y ) where (X 0 + γ X C ' X,Y 0 + γ Y C ' Y ) is i the covex hull of {(Xj,Yj): j=,, } plus the polyhedral coe, excludig the poit 0, geerated by {(e i,0): i =,, m} ad {(0,-e r ): r =,, s}. I particular, observe that P(x*) is a proper subset of R m+s. Sice the case above correspods to havig Γ = R t, it follows from Theorem 3 that (6) is a equivalet characterizatio of (3). Chares et al [3] ad Cooper et al [7, 8] essetially observe that problem (6) is the liear programmig problem (9) give i Theorem 5. Thus, for a efficiet DMU we coclude agai that DEA metric sesitivity is a special case of the geeralized tolerace framework. 4. Coclusios ad observatios This paper cosiders the relatioship betwee Tolerace sesitivity aalysis i optimizatio ad metric sesitivity aalysis i DEA. Specifically, it exteds the results o the geeralized Tolerace framework proposed by Wedell ad Che ad shows how this framework icludes DEA metric sesitivity as a special case. Further, as oted below, the paper shows how recet results i Tolerace sesitivity suggest some possible extesios of the results i DEA metric sesitivity. I a related paper Sigh [6] recetly proposed a volume-maximizig tolerace approach for DEA metric sesitivity. This approach, while ot directly related to the results of Chares et al [3] ad Cooper et al [7, 8], offers a alterative method of costructig a stability regio by maximizig the volume of the regio (buildig o the work of Wag ad Huag [7]). Critiques of the volume-maximizig approach iclude: a lack of a ituitive ratioale for the maximizig volume objective (see [22]); ad the eed to solve a oliear optimizatio problem to perform sesitivity aalysis o a optimal basis of a liear program (see [0]). Also, while ot directly relevat herei, we observe that Jahashahloo et al [2] has recetly show how models proposed by Chares et al [3] ad Cooper et al [7, 8] ca be combied ito oe model with the same results, thereby savig some computatioal work. Oe coclusio from the results herei is that DEA metric sesitivity uses a similar framework to traditioal Tolerace sesitivity. However, the relatioship betwee them is oly clear whe cosiderig the geeral framework give herei. Furthermore, there are sigificat differeces i DEA metric sesitivity from previous work i Tolerace sesitivity. Oe is that i cotrast

10 78 Luka Neralić ad Richard E. Wedell to cosiderig sesitivity with respect to just optimal solutios, DEA metric sesitivity cosiders variatios of a iefficiet (i.e., a ooptimal ) solutio. Aother is that DEA cosiders both iteral ad exteral represetatios of a polyhedro, whereas traditioal Tolerace sesitivity cosiders just exteral represetatios. Give the relatioship betwee DEA metric sesitivity ad Tolerace sesitivity, it is iterestig to cosider how recet work i Tolerace sesitivity ca yield extesios of DEA metric sesitivity results. I particular, buildig o the results of Filippi [0], Hladik [], ad Wedell [20], a follow-up paper by the authors herei show how to obtai larger maximum toleraces by exploitig a priori parametric iformatio ad how to expad the maximum tolerace regio. Iterestigly, the DEA Tolerace relatioship also suggests other possible research topics such as buildig o the work of Wedell [23] to cosider trade-offs betwee the degree of efficiecy of a DMU ad its sesitivity to perturbatios. We hope that this paper motivates research o such trade-offs. Ackowledgemet This research was partially supported by Grat No of the Miistry of Sciece, Educatio ad Sports of the Republic of Croatia. We would like to ackowledge a suggestio from W. W. Cooper to the secod author to examie the relatioship betwee Tolerace sesitivity ad sesitivity i DEA. Refereces [] Chares A, Cooper W. W., Golay, L., Seiford, L. M. ad Stutz, J. (985). Foudatios of data evelopmet aalysis for Pareto-Koopmas efficiet empirical productio fuctios. Joural of Ecoometrics, 30, doi:0.06/ (85) [2] Chares A, Cooper W. W. ad Rhodes E. (978). Measurig the efficiecy of Decisio Makig Uits. Europea Joural of Operatioal Research, 2(6), doi:0.06/ (78) [3] Chares, A., Haag, S., Jaska, P. ad Semple, J. (992). Sesitivity of efficiecy calculatios i the additive model of data evelopmet aalysis. Joural of Systems Scieces, 23(5), doi:0.080/ [4] Chares, A., ad Neralić L. (990). Sesitivity aalysis of the additive model i data evelopmet aalysis. Europea Joural of Operatioal Research, 48, doi:0.06/ (90) [5] Chares, A., Rousseau, J. J., ad Semple, J. (996). Sesitivity ad stability of efficiecy classificatios i data evelopmet aalysis. The Joural of Productivity Aalysis, 7(), 5 8. doi:0.007/bf

11 Geeralized tolerace sesitivity ad DEA metric sesitivity 79 [6] Cook, W. D. ad Sieford, L. M. (2009). Data Evelopmet Aalysis (DEA) Thirty years o. Europea Joural of Operatioal Research, 92(), 7. doi:0.06/j.ejor [7] Cooper, W. W., Seiford, L. M. ad Toe, K. (2007). Data Evelopmet Aalysis, 2 d Editio. New York: Spriger. [8] Cooper, W. W., Li, S., Seiford, L. M., Toe, K., Thrall, R. M. ad Zhu, J. (200). Sesitivity ad stability aalysis i DEA: Some recet developmets. Joural of Productivity Aalysis, 5, [9] Emrouzejad A. (203) Data Evelopmet Aalysis homepages. [Accessed o 28 July 20]. [0] Filippi, C. (2005). A fresh view o the tolerace approach to sesitivity aalysis i liear programmig. Europea Joural of Operatioal Research, 67(), 9. doi:0.06/j.ejor [] Hladik, M. (20). Tolerace aalysis i liear systems ad liear programmig. Optimizatio Methods ad Software. 26(3), doi:0.080/ [2] Jahashahloo, G. R., Lotfi, F. H., Shoja, N., Tohidi, G. ad Razavya, S. (2005). A oe-model approach to classificatio ad sesitivity aalysis i DEA. Applied Mathematics ad Computatio, 69(2), doi:0.06/j.amc [3] Neralić, L. (997). Sesitivity i Data Evelopmet Aalysis for arbitrary perturbatios of data. Glasik Matematicki, 32, [4] Neralić, L. ad Wedell, R. E. (2004). Sesitivity i Data Evelopmet Aalysis usig a approximate iverse matrix. Joural of the Operatioal Research Society, 55(), doi:0.057/palgrave.jors [5] Schrijver, A. (986). Theory of Liear ad Iteger Programmig. New York: Joh Wiley & Sos. [6] Sigh, S. (200). Multiparametric sesitivity aalysis of the additive model i Data Evelopmet Aalysis. Iteratioal Trasactios i Operatioal Research, 7(3), doi:0./j x. [7] Wag, H. F. ad Huag, C. S. (993). Multi-parametric aalysis of the maximum tolerace i a liear programmig problem. Europea Joural of Operatioal Research, 67(), doi:0.06/ (93)90323-f. [8] Ward, J. E. ad Wedell, R. E. (990). Approaches to sesitivity aalysis i liear programmig. Aals of Operatios Research, 27(), doi:0.007/bf [9] Wedell, R. E. (982). A preview of a tolerace approach to sesitivity aalysis i liear programmig. Discrete Mathematics, 38(), doi:0.06/ x(82) [20] Wedell, R. E. (984). Usig bouds o the data i liear programmig: The tolerace approach to sesitivity aalysis. Mathematical Programmig, 29(3), doi:0.007/bf [2] Wedell, R. E. (985). The tolerace approach to sesitivity aalysis i liear programmig. Maagemet Sciece, 3(5), doi:0.287/msc [22] Wedell, R. E. (997). Liear Programmig 3: The Tolerace Approach. I: Gal, T. ad Greeberg, H. J. (Eds.), Advaces i Sesitivity Aalysis ad Parametric Programmig (pp ). Bosto/Dordrecht/Lodo: Kluwer Academic Press. [23] Wedell, R. E. (2004). Tolerace sesitivity ad optimality bouds i liear programmig. Maagemet Sciece, 50(6), doi:0.287/msc

12 80 Luka Neralić ad Richard E. Wedell [24] Wedell, R. E. ad Che, W. (200). Tolerace sesitivity aalysis: Thirty years later. Croatia Operatioal Research Review,, 2 2.