Modeling Fuzzy Data Envelopment Analysis

with Expectation Criterion

Xiaodong Dai, Ying Liu, and Rui Qin�

College of Mathematics & Computer Science, Hebei UniversityBaoding 071002, Hebei, China

daixiaodong@cmc.hbu.edu.cn, yingliu@hbu.edu.cn, qinrui07@163.com

Abstract. This paper presents a new class of fuzzy expectation data en-velopment analysis (FEDEA) models with credibility constraints. Sincethe proposed model contains the credibility of fuzzy events in the con-straints and the expected value of a fuzzy variable in the objective, thesolution process is very complex. Thus, in the case when the inputs andoutputs are mutually independent trapezoidal fuzzy variables, we dis-cuss the equivalent nonlinear forms of the programming model, whichcan be solved by standard optimization software. At the end of this pa-per, one numerical example is also provided to illustrate the efficiency ofdecision-making unites (DMUs) in the proposed model.

Keywords: Data envelopment analysis, Credibility constraint, Fuzzyvariable, Expected value, Efficiency.

1 Introduction

Data envelopment analysis (DEA) was initially proposed by Charnes, Cooperand Rhodes [1]. It is an evaluation method for measuring the relative efficiencyof a set of homogeneous DMUs with multiple inputs and multiple outputs. Sincethe first DEA model CCR [1], DEA has been studied by a number of researchersin many fields [2,3,4].

The advantage of the DEA method is that it does not require either a prioriweights or the explicit specification of functional relations between the multipleinputs and outputs. However, when evaluating the efficiency, the data in tradi-tional DEA models must be crisp, and the efficiency is very sensitive to datavariations. To deal with stochastic data variations, some researchers proposedseveral DEA models. For example, Cooper, Huang and Li [5] developed a sat-isficing DEA model with chance constrained programming; Olesen and Peterso[6] developed a probabilistic constrained DEA model. For more stochastic DEAapproaches, the interested readers may refer to [7,8,9].

On the other hand, to deal with fuzziness in the real world problems, Zadeh[10] proposed the concept of fuzzy set. Recently, the credibility theory [11], meanchance theory and fuzzy possibility theory [12] have also been proposed to treat

� Corresponding author.

Y. Tan, Y. Shi, and K.C. Tan (Eds.): ICSI 2010, Part II, LNCS 6146, pp. 9–16, 2010.c© Springer-Verlag Berlin Heidelberg 2010

10 X. Dai, Y. Liu, and R. Qin

fuzzy phenomena existing in real-life problems. For more theories and applica-tions of credibility theory and mean chance theory, the interested readers mayrefer to [13,14,15,16]. In fuzzy environments, some researchers extended the tra-ditional DEA and proposed several fuzzy DEA models. For example, Entani,Maeda and Tanaka [17] developed a new pair of interval DEA models; Saen [18]proposed a new pair of assurance region-nondiscretionary factors-imprecise DEAmodels, and Triantis and Girod [19] proposed a mathematical programming ap-proach to transforming fuzzy input and output data into crisp data.

This paper attempts to establish a new class of fuzzy DEA models basedon credibility theory [11], and discuss its equivalent nonlinear forms when theinputs and outputs are mutually independent trapezoidal fuzzy variables. Therest of this paper is organized as follows. In Section 2, we present the fuzzyexpectation DEA models with fuzzy inputs and fuzzy outputs. Section 3 discussesthe equivalents of credibility constraints and the expectation objective in somespecial cases. In Section 4, we provide a numerical example to illustrate therelative efficiency in the proposed model and the effectiveness of our solutionmethod. Section 5 draws our conclusions.

2 Fuzzy DEA Formulation

The traditional DEA model, which was proposed by Charnes, Cooper and Rhodes(CCR) [1], is built as

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

maxu,v

vT y0

uT x0

subject to vT yi

uT xi≤ 1, i = 1, · · · , n

u ≥ 0, u �= 0v ≥ 0, v �= 0,

(1)

where xi represent the input column vector of DMUi, x0 represents the inputcolumn vector of DMU0; yi represent the output column vector of DMUi, y0

represents the output column vector of DMU0; u ∈ �m and v ∈ �s are theweights of the input and output column vectors.

Model (1) is used to evaluate the relative efficiency of DMUs with crisp in-puts and outputs. However, in many cases, we can only obtain the possibilitydistributions of the inputs and outputs. Thus in this paper, we assume that theinputs and outputs are characterized by fuzzy variables with known possibilitydistributions. Based on fuzzy expected value operator and credibility measure[20], we can establish the following fuzzy expectation DEA model

maxu,v

VEDEA = E[

vT η0uT ξ0

]

subject to Cr{uT ξi − vT ηi ≥ 0} ≥ αi, i = 1, 2, · · · , nu ≥ 0, u �= 0v ≥ 0, v �= 0,

(2)

where the notations are illustrated in Table 1.

Modeling Fuzzy Data Envelopment Analysis with Expectation Criterion 11

Table 1. List of Notations for Model (2)

Notations Definitions

ξ0 the fuzzy input column vector consumed by DMU0

ξi the fuzzy input column vector consumed by DMUi, i = 1, · · · , nη0 the fuzzy output column vector produced by DMU0

ηi the fuzzy output column vector produced by DMUi, i = 1, · · · , nu ∈ �m the weights of the fuzzy input column vector

v∈ �s the weights of the fuzzy output column vector

αi ∈ (0, 1] the predetermined credibility level corresponding to the ith constraint

In model (2), our purpose is to seek a decision (u, v) with the maximum valueof E

[vT η0/uT ξ0

], while the fuzzy event {uT ξi − vT ηi ≥ 0} is satisfied at least

with credibility level αi. Thus, we adopt the concept of expectation efficientvalue to illustrate the efficiency of DMU0. The optimal value of model (2) isreferred to as the expectation efficient value of DMU0, and the bigger the valueis, the more efficient it is.

Model (2) is very difficult to solve. Therefore, in the next section, we willdiscuss the equivalent forms of model (2) in some special cases.

3 Deterministic Equivalent Programming of Model (2)

In the following, we first handle the constraint functions.

3.1 Handing Credibility Constraints

When the inputs and outputs are mutually independent trapezoidal fuzzy vec-tors, the constraints of model (2) can be transformed to their equivalent linearforms according to the following theorem.

Theorem 1. Let ξi = (Xi−ai, Xi, Xi+bi, Xi+ci), ηi = (Yi−ai, Yi, Yi+bi, Yi+ci)be independent trapezoidal fuzzy vectors with ai, bi, ci, ai, bi, ci positive numbers.Then Cr{uT ξi − vT ηi ≥ 0} ≥ αi in the model (2) is equivalent to

gi(u, v) ≥ 0, (3)

where gi(u, v) = uT (Xi − (2αi − 1)ai) − vT (Yi + (2αi − 1)ci + 2(1 − αi)bi).

Proof. It is obvious that uT ξi − vT ηi = (uT (Xi − ai) − vT (Yi + ci), uT Xi −vT (Yi + bi), uT (Xi + bi) − vT Yi, u

T (Xi + ci) − vT (Yi − ai)). When 0.5 < αi < 1(i = 1, · · · , n), according to the distributions of uT ξi − vT ηi and the definitionof the credibility measure, we have

uT (Xi − ai) − vT (Yi + ci) < 0 < uT Xi − vT (Yi + bi).

12 X. Dai, Y. Liu, and R. Qin

Thus, Cr{uT ξi − vT ηi ≥ 0} ≥ αi is equivalent to

uT (Xi − (2αi − 1)ai) − vT (Yi + (2αi − 1)ci + 2(1 − αi)bi) ≥ 0.

The proof of the theorem is complete.

By the transformation process proposed above, we have turned the constraintfunctions of model (2) to their equivalent linear forms. In the following, we willdiscuss the equivalent form of the objective.

3.2 Equivalent Representation of the Expectation Objective

In this section, we first deduce some formulas for the expected value of thequotient of two independent fuzzy variables.

Theorem 2. Suppose ξ = (X−a, X, X+b, X+c) and η = (Y −a, Y, Y +b, Y +c)are two mutually independent trapezoidal fuzzy variables, where a, b, c, a, b, c arepositive numbers, b < c, b < c and X > a or X < −c. Then we have

E[

ηξ

]= − a

2(c−b) + b−c2a + 1

2a

(Y + b + c−b

a X)

ln XX−a

+ 12(c−b)

(Y + a

c−b(X + b))

lnX+cX+b .

(4)

Proof. We only prove the case X > a, and when X < −c, the proof is similar.When X > a, ξ is a positive fuzzy variable. Thus, we have

E[

ηξ

]=

∫ +∞0

Cr{

ηξ ≥ r

}dr − ∫ 0

−∞ Cr{

ηξ ≤ r

}dr

=∫ +∞0 Cr{η − rξ ≥ 0}dr − ∫ 0

−∞ Cr{η − rξ ≤ 0}dr.(5)

Since η − rξ = (Y − rX − (rc + a), Y − rX − rb, Y − rX + b, Y − rx + (c + ra)),according to credibility measure of a fuzzy event, we have

Cr{η − rξ ≥ 0} =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

1, if r < Y −aX+c

12 + Y −r(X+b)

2(a+r(c−b)) , if Y −aX+c ≤ r < Y

X+b12 , if Y

X+b ≤ r < Y +bX

12 + Y −rX+b

2(ra+c−b), if Y +b

X ≤ r < Y +cX−a

0, if r ≥ Y +cX−a ,

Cr{η − rξ ≤ 0} =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

0, if r < Y −aX+c

12 − Y −r(X+b)

2(a+r(c−b)) , if Y −aX+c ≤ r < Y

X+b12 , if Y

X+b ≤ r < Y +bX

12 − Y −rX+b

2(ra+c−b), if Y +b

X ≤ r < Y +cX−a

1, if r ≥ Y +cX−a .

In the following, we calculate E[η/ξ] according to five cases.(i) If (Y − a)/(X + c) > 0, i.e. Y > a, then (5) becomes to

E[

ηξ

]=

∫ Y −aX+c

0 1dr +∫ Y

X+bY −aX+c

(12 + Y −r(X+b)

2(a+r(c−b))

)dr +

∫ Y +bXY

X+b

12dr

+∫ Y +c

X−a

Y +bX

(12 + Y −rX+b

2(ra+c−b)

)dr = Y −a

X+c + M1 + 12 (Y +b

X − YX+b ) + M2,

Modeling Fuzzy Data Envelopment Analysis with Expectation Criterion 13

where

M1 = 12

(Y

X+b − Y −aX+c

)+

∫ YX+b

Y −aX+c

(− X+b

2(c−b) + 12(c−b)

ac−b (X+b)+Y

ac−b +r

)dr

=(

12 − X+b

2(c−b)

)(Y

X+b − Y −aX+c

)+ 1

2(c−b)

(a

c−b (X + b) + Y)

ln X+cX+b ,

M2 =(

12 − X

2a

) (Y +cX−a − Y +b

X

)+ 1

2a

(c−ba X + Y + b

)ln X

X−a .

Therefore, formula (4) is valid for this case.(ii) If (Y − a)/(X + c) ≤ 0 < Y/(X + b), i.e. 0 < Y ≤ a, then (5) becomes to

E[

ηξ

]=

∫ YX+b

0

(12 + Y −r(X+b)

2(a+r(c−b))

)dr +

∫ Y +bXY

X+b

12dr +

∫ Y +cX−a

Y +bX

(12 + Y −rX+b

2(ra+c−b)

)dr

− ∫ 0Y −aX+c

(12 − Y −r(X+b)

2(a+r(c−b))

)dr = M1 + 1

2 (Y +bX − Y

X+b ) + M2 − M3,

where M2 is the same with the M2 in Case (i), and

M1 =(

12 − X+b

2(c−b)

)Y

X+b + 12(c−b)

(a

c−b (X + b) + Y)

ln a(X+b)+Y (c−b)a(X+b) ,

M3 =(

X+b2(c−b)− 1

2

)(Y

X+b− Y −aX+c

)− 1

2(c−b)

(a

c−b (X + b) + Y)

ln a(X+c)a(X+b)+Y (c−b) ,

Therefore, formula (4) is valid for this case.(iii) If Y/(X + b) ≤ 0 < (Y + b)/X , i.e. −b < Y ≤ 0, then (5) becomes to

E[

ηξ

]=

∫ Y +bX

012dr +

∫ Y +cX−a

Y +bX

(12 + Y −rX+b

2(ra+c−b)

)dr − ∫ Y

X+bY −aX+c

(12 − Y −r(X+b)

2(a+r(c−b))

)dr

− ∫ 0Y

X+b

12dr = Y +b

2X + M2 − M3 + Y2(X+b) ,

where M2 is the same with the M2 in Case (i), and

M3 = X+c2(c−b)

(Y

X+b − Y −aX+c

)− 1

2(c−b)

(a

c−b (X + b) + Y)

ln X+cX+b ,

Therefore, formula (4) is valid for this case.(iv) If (Y + b)/X ≤ 0 < (Y + c)/(X − a), i.e., −c < Y ≤ −b, then (5) becomesto

E[

ηξ

]=

∫ Y +cX−a

0

(12 + Y −rX+b

2(ra+c−b)

)dr − ∫ Y

X+bY −aX+c

(12 − Y −r(X+b)

2(a+r(c−b))

)dr − ∫ Y +b

XY

X+b

12dr

− ∫ 0Y +b

X

(12 − Y −rX+b

2(ra+c−b)

)dr = M2 − M3 − 1

2

(Y +bX − Y

2(X+b)

)− M4,

where M3 is the same with the M3 in Case (iii), and

M2 =(

12 − X

2a

)Y +cX−a + 1

2a

(c−ba X + Y + b

)ln X(c−b)+a(Y +a)

X−a ,

M4 = − (12 + X

2a

)Y +bX − 1

2a

(c−ba X + Y + b

)ln X

X(c−b)+a(Y +a).

Therefore, formula (4) is valid for this case.(v) If (Y + c)/(X − a) ≤ 0, i.e. Y ≤ −c, then (5) becomes to

E[

ηξ

]= − ∫ Y

X+bY −aX+c

(12 − Y −r(X+b)

2(a+r(c−b))

)dr − ∫ Y +b

XY

X+b

12dr − ∫ Y +c

X−a

Y +bX

(12 − Y −rX+b

2(ra+c−b)

)dr

− ∫ 0Y +cX−a

1dr = −M2 − 12

(Y +bX − Y

2(X+b)

)− M3 + Y +c

X−a ,

14 X. Dai, Y. Liu, and R. Qin

where M3 is the same with the M3 in Case (iii), and

M2 = X+c2(c−b)

(Y

X+b − Y −aX+c

)− 1

2(c−b)

(a

c−b (X + b) + Y)

ln X+cX+b .

Therefore, formula (4) is valid for this case. The proof of the theorem is complete.

3.3 Deterministic Equivalent Programming

Denote the inputs and outputs of DMU0 as ξ0 = (ξ1,0, · · · , ξm,0)T and η0 =(η1,0, · · · , ηs,0)T . Suppose that ξj,0 = (Xj,0 − aj,0, Xj,0, Xj,0 + bj,0, Xj,0 + cj,0)and ηk,0 = (Yk,0 − ak,0, Yk,0, Yk,0 + bk,0, Yk,0 + ck,0) are mutually independenttrapezoidal fuzzy variables, where aj,0, bj,0, ck,0, aj,0, bj,0, ck,0 are positive num-bers, and Xj,0 > aj,0, Yk,0 > ak,0 for j = 1, · · · , m, k = 1, · · · , s. Then accordingto Theorem 2, we have

f0(u, v) = E[

vT η0uT ξ0

]= − a

2(c−b) + b−c2a + 1

2a

(Y + b + c−b

a X)

ln XX−a

+ 12(c−b)

(Y + a

c−b (X + b))

lnX+cX+b ,

(6)

where

a =∑m

j=1 ujaj,0, b =∑m

j=1 ujbj,0, c =∑s

k=1 vkck,0, a =∑s

k=1 vk ak,0,

b =∑s

k=1 vk bk,0, c =∑s

k=1 vk ck,0, X =∑m

j=1 ujXj,0, Y =∑s

k=1 vkYk,0.

As a consequence, when the inputs and outputs are mutually independent trape-zoidal fuzzy variables, the model (2) can be transformed into the following equiv-alent nonlinear programming

maxu,v

f0(u, v)

subject to gi(u, v) ≥ 0, i = 1, 2, · · · , nu ≥ 0, u �= 0v ≥ 0, v �= 0,

(7)

where f0(u, v) and gi(u, v) are defined by (6) and (3), respectively.The model (7) is a nonlinear problem with linear constraints, which can be

solved by standard optimization solvers.

4 Numerical Example

In order to illustrate the solution method for the proposed FDEA, we providea numerical example with five DMUs, and each DMU has four fuzzy inputsand four fuzzy outputs. In addition, for each DMU, the inputs and outputs arecharacterized by mutually independent trapezoidal fuzzy variables, as shown inTable 2. For simplicity, we assume that α1 = α2 = · · · = α5 = α.

With model (7), we obtain the results of evaluating all the DMUs with cred-ibility level α = 0.95 with Lingo software [21], as shown in Table 3. From the

Modeling Fuzzy Data Envelopment Analysis with Expectation Criterion 15

Table 2. Four Fuzzy Inputs and Outputs for Five DMUs

DMUi Input 1 Input 2 Input 3 Input 4

i=1 (2.8, 3.0, 3.1, 3.4) (2.0, 2.1, 2.3, 2.4) (2.5, 2.7, 2.9, 3.0) (3.6, 3.9, 4.1, 4.2)i=2 (1.8, 1.9, 2.0, 2.2) (1.4, 1.6, 1,7, 1.8) (2.2, 2.3, 2.4, 2.5) (3.1, 3.2, 3.4, 3.7)i=3 (2.5, 2.6, 2.8, 2.9) (1.8, 2.0, 2.4, 2.5) (2.0, 2.5, 2.8, 3.0) (4.1, 4.2, 4.4, 4.5)i=4 (3.0, 3.1, 3.3, 3.5) (4.1, 4.3, 4.5, 4.6) (3.8, 3.9, 4.0, 4.1) (4.6, 4.8, 4.9, 5.0)i=5 (4.8, 4.9, 5.0, 5.3) (6.1, 6.2, 6.4, 6.6) (4.4, 4.5, 4.8, 5.0) (5.2, 5.5, 5.6, 5.8)

DMUi Output 1 Output 2 Output 3 Output 4

i=1 (4.0, 4.1, 4.2, 4.4) (3.0, 3.2, 3.4, 3.5) (3.6, 3.8, 4.1, 4.2) (4.8, 4.9, 5.0, 5.1)i=2 (3.4, 3.8, 4.0, 4.2) (4.0, 4.3, 4.5, 4.6) (3.5, 3.6, 3.7, 3.9) (4.0, 4.1, 4.3, 4.4)i=3 (4.5, 4.8, 5.0, 5.5) (3.8, 3.9, 4.0, 4.1) (3.0, 3.1, 3.3, 3.4) (4.3, 4.5, 4.6, 4.7)i=4 (4.8, 5.0, 5.1, 5.4) (4.3, 4.4, 4.5, 4.7) (5.2, 5.3, 5.4, 5.5) (6.0, 6.2, 6.4, 6.8)i=5 (5.8, 6.0, 6.3, 6.4) (6.5, 6.7, 6.8, 6.9) (4.9, 5.0, 5.2, 5.4) (5.9, 6.3, 6.5, 6.8)

Table 3. Results of evaluation with α=0.95 in model (1)

DMUs Optimal solution (u,v) Efficiency value

DMU1 (0.0000,0.0000,0.5685,1.0000,0.0000,0.0000,0.0000,0.9946) 0.8946490DMU2 (0.5397,0.0000,1.0000,0.0000,0.0000,0.0000,0.0000,0.7259) 0.8965842DMU3 (0.0000,0.0000,0.0663,1.0000,0.7791,0.0000,0.0000,0.0000) 0.8660948DMU4 (0.0000,0.0000,0.0000,1.0000,0.0000,0.0000,0.3486,0.4003) 0.9149018DMU5 (0.0000,0.0000,0.0000,1.0000,0.7440,0.0000,0.0000,0.0000) 0.8268326

results, we know that DMU4 is the most efficient with expectation efficient value0.9149018, followed by DMU2 and DMU1, which implies that DMU4 has thebest position in competition. If the DMUs with less expectation efficient valueswant to improve their position in competition, they should decrease their inputs.Therefore, with the expectation efficient values, the decision makers can obtainmore information and thus make better decisions in competition.

5 Conclusions

This paper proposed a new class of fuzzy DEA models with credibility constraintsand expectation objective. In order to solve the proposed model, for trapezoidalfuzzy inputs and outputs, we discussed the equivalent representation for theconstraints and the objective. With such transformations, the proposed DEAmodel can be turned into its equivalent nonlinear programming, which can besolved by standard optimization softwares. At last, a numerical example wasprovided to illustrate the efficiency of DMUs in the proposed DEA model.

Acknowledgments. This work was supported by the National Nature ScienceFoundation of China (NSFC) under Grant No. 60974134.

16 X. Dai, Y. Liu, and R. Qin

References

1. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the Efficiency of DecisionMaking Units. European Journal of Operational Research 2, 429–444 (1978)

2. Cook, W.D., Seiford, L.M.: Data Envelopment Analysis (DEA)-Thirty Years on.European Journal of Operational Research 192, 1–17 (2009)

3. Cooper, W.W., Seiford, L.M., Tone, K.: Data Envelopment Analysis. SpringerScience and Business Media, New York (2007)

4. Emrouznejad, A., Parker, B.R., Tavares, G.: Evaluation of Research in Efficiencyand Productivity: A Survey and Analysis of the First 30 Years of Scholarly Liter-ature in DEA. Socio-Economic Planning Sciences 42, 151–157 (2008)

5. Cooper, W.W., Huang, Z.M., Li, S.X.: Satisficing DEA Models under Chance Con-straints. Annals of Operations Research 66, 279–295 (1996)

6. Olesen, O.B., Peterso, N.C.: Chance Constrained Efficiency Evaluation. Manage-ment Science 41, 442–457 (1995)

7. Desai, A., Ratick, S.J., Schinnar, A.P.: Data Envelopment Analysis with StochasticVariations in Data. Socio-Economic Planning Sciences 3, 147–164 (2005)

8. Gong, L., Sun, B.: Efficiency Measurement of Production Operations under Un-certainty. International Journal of Production Economics 39, 55–66 (1995)

9. Retzlaff-Roberts, D.L., Morey, R.C.: A Goal Programming Method of Stochas-tic Allocative Data Envelopment Analysis. European Journal of OperationalResearch 71, 379–397 (1993)

10. Zadeh, L.A.: Fuzzy Sets. Information and Control 8, 338–353 (1965)11. Liu, B.: Uncertainty Theory: An Introduction to its Axiomatic Foundations.

Springer, Berlin (2004)12. Liu, Z.Q., Liu, Y.K.: Type-2 Fuzzy Variables and their Arithmetic. Soft

Computing 14(7), 729–747 (2010)13. Lan, Y., Liu, Y.K., Sun, G.: Modeling Fuzzy Multi-period Production Planning

and Sourcing Problem with Credibility Service Levels. Journal of Computationaland Applied Mathematics 231, 208–221 (2009)

14. Qin, R., Hao, F.F.: Computing the Mean Chance Distributions of Fuzzy RandomVariables. Journal of Uncertain Systems 2, 299–312 (2008)

15. Qin, R., Liu, Y.K.: A New Data Envelopment Analysis Model with Fuzzy Ran-dom Inputs and Outputs. Journal of Applied Mathematics and Computing (2009),doi:10.1007/s12190-009-0289-7

16. Qin, R., Liu, Y.K.: Modeling Data Envelopment Analysis by Chance Method inHybrid Uncertain Environments. Mathematics and Computers in Simulation 80,922–950 (2010)

17. Entani, T., Maeda, Y., Tanaka, H.: Dual Models of Interval DEA and its Extensionto Interval Data. European Journal of Operational Research 136, 32–45 (2002)

18. Saen, R.F.: Technology Selection in the Presence of Imprecise Data, Weight Re-strictions, and Nondiscretionary Factors. The International Journal of AdvancedManufacturing Technology 41, 827–838 (2009)

19. Triantis, K., Girod, O.: A Mathematical Programming Approach for MeasuringTechnical Efficiency in a Fuzzy Environment. Journal of Productivity Analysis 10,85–102 (1998)

20. Liu, B., Liu, Y.K.: Expected Value of Fuzzy Variable and Fuzzy Expected ValueModels. IEEE Transaction on Fuzzy Systems 10, 445–450 (2002)

21. Mahmoud, M.E.: Appendix II: Lingo Software. Process Systems Engineering 7,389–394 (2006)