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Engineering Optimization

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Robust solutions using fuzzy chance constraints

F. Alberto Camposa; J. Villara; M. Jiménezb

a_{Instituto de Investigación Tecnológica, Universidad Pontificia Comillas de Madrid.} c/Santa Cruz de Marcenado, 26, Madrid, 28015. Spain

b_{Departamento de Estadística y Econometría, Universidad de Málaga. Spain} To link to this article: DOI: 10.1080/03052150600603165

URL:http://dx.doi.org/10.1080/03052150600603165

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Vol. 38, No. 6, September 2006, 627–645

Robust solutions using fuzzy chance constraints

F. ALBERTO CAMPOS*†, J. VILLAR† and M. JIMÉNEZ‡

†Instituto de Investigación Tecnológica, Universidad Pontificia Comillas de Madrid, c/Santa Cruz de Marcenado, 26, 28015 Madrid, Spain

‡Departamento de Estadística y Econometría, Universidad de Málaga, Spain

(Received 9 February 2005; in final form 4 August 2005)

It is well known that optimization problems for the decision-making process in real environments should consider uncertainty to attain robust solutions. Although this uncertainty has been usually modelled using probability theory, assuming a random origin, possibility theory has emerged as an alternative uncertainty model when statistical information is not available, or when imprecision and vagueness have to be considered. This article proposes two different criteria to obtain robust solutions for linear optimization problems when the objective function coefficients are modelled with possibility distributions. To do so, chance constrained programming is used, leading to equivalent crisp optimiza-tion problems, which can be solved by commercial optimizaoptimiza-tion software. A simple case example is presented to illustrate the use of the proposed methodology.

Keywords: Robustness; Possibility theory; Fuzzy linear programming; Chance constraints

1. Introduction

In real decision-making problems, where input data are not precisely known, it becomes nec-essary to obtain solutions which are in some sense insensitive to potential input modifications. Otherwise, seemingly good decisions could perform very badly when similar but not identical input data values are presented. For a long time, these types of solutions, popularly called

robust solutions, have been considered one of the major challenges for operational research,

leading to several similar interpretations ofrobustness (see, for example, Paraskevopoulos

et al.1991, Mulveyet al.1995, Kouvelis and Yu 1997, Ben-Tal and Nemirovski 1998, 2000,

Averbakh 2001). Most approaches to robustness are based on uncertainty modelling to opti-mize against not only the set of most expected inputs, but also against other less favourable ones that could reasonably occur.

Traditionally, uncertain knowledge has been modelled usingprobability theory, which leads to two types of optimization problems. The first type, calledexpected value models, is con-structed under the attitude of risk indifference, optimizing the expected objective function subject to some expected constraints. However, sometimes it is interesting to consider that some unfavourable event could occur, leading to the second type, calledchance constrained

*Corresponding author. Email: [email protected]

Engineering Optimization

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programming (CCP), developed by Charnes and Cooper (see Charnes and Cooper 1959).

This second approach consists in optimizing the decision when the unfavourable success probability is bounded by a minimum fixed probability threshold (see Mayer 1998 for an interpretation of this threshold), corresponding to the low risk decisions usually adopted under the risk management discipline (see Jorion 1997). Probabilistic CCP models can be converted into deterministic (crisp) equivalent ones for some particular cases (see Wischmeier and Smith 1978, Segarraet al.1985), allowing them to be solved by traditional deterministic mathematical programming.

Although probability theory is widely understood and accepted by the engineering community, it generally requires sufficient statistical information (not always available) to compute theprobability distributionsof the uncertain inputs, usually entails complex calcu-lations, and might not provide the correct interpretation when the major source of imperfect information is not only the uncertainty but also the imprecision and vagueness, so often in subjective expert knowledge. While uncertainty deals with the relation between a piece of information and the available knowledge, imprecision deals with the information content and vagueness refers to ill-defined information boundaries (see Smets 1997). Among the different alternatives to knowledge modelling,possibility theoryintroduced by Zadeh in 1978 (Zadeh 1978) allows uncertainty, imprecision and vagueness to be modelled (see Villar 1997) by using

fuzzy sets(Zadeh 1965) aspossibility distributions, and can therefore be more flexible than

probability theory in many realistic situations (see Dubois and Prade 1993). Additionally fuzzy problems are usually less complex to solve than their probabilistic counterparts, although the results obtained are less informative and have a more controversial interpretation (see Dubois and Prade 1989).

Different approaches have been adopted to solve linear optimization models with fuzzy coefficients. Many of them select the cheapest solution with an acceptable possibility degree to occur (see, for example, Luhandjula 1987). Nevertheless, minor changes in the coeffi-cient assumptions can lead to completely worse objective values, which mean that these methodologies tend to be too optimistic and therefore too risky. For this reason, in this article robust solutions have been computed using CCP models, similar to those suggested in Tanaka (1984), Tanaka and Asai (1984), and Buckley (1988), but where CCP models are not explicitly employed.

Following the idea of probabilistic CCP models, in a fuzzy environment where uncertain data are modelled withpossibility distributions, the decision can be optimized for a set of scenarios with possibility to occur greater than or equal to a minimum possibility threshold. As is remarked in Dubois (1987), one of the advantages of using fuzzy CCP models (see, for example, Luhandjula 1983, 1996) is that they can be more often transformed into equivalent linear crisp (deterministic) models than probabilistic CCP models, usually based on normally distributed coefficients. In particular, in Dubois (1987) an approach based on fuzzy CCP models is used to obtain feasible solutions, although the objective function optimization is not studied under the CCP point of view. For a wider spectrum of fuzzy CCP models where constraints and objective function are studied, refer, for example, to Liu (1999, 2001). In these works the pessimistic (or low risk) solution, defined from the maximin and minimax CCP formulations, is obtained solving the problem with hybrid intelligent algorithms.

The approach proposed in this article computes the pessimistic solution by transforming the fuzzy linear decision-making problem into a linear crisp optimization model (LP) (see Fuller and Zimmermann 1993 for fuzzy non-linear programming problems). In order to formulate the LP with a moderate amount of effort, modern modelling languages such as AMPL (see Foureret al.1993), GAMS (see Brookeet al.1998) and OPL (see Hentenryck 1999) can be used. Solvers such as CPLEX (see Cplex 1997) can be called from the modelling languages to compute the numerical solutions.

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Additionally, this work proposes to look for a complementary pessimistic solution that can be obtained from a minimin CCP formulation, when a desirable objective function value, or objective threshold, is previously selected, and the possibility of any worst objective function value is minimized. With this approach, an auxiliary LP to look for the trivial solution must previously be solved, followed by an auxiliary linear–fractional crisp optimization problem (LFP) to obtain the non-trivial solution. Specific algorithms for LFP exist, some of them are shown, for example, in Schaible and Ziemba (1981), Craven (1988) and Shi (2001). The only software found in the literature that solves the LFP is WINGULF (see Bajalinov and Pannell 1993). It is based on the well-known simplex algorithm (see Gass 1969) that uses the simple steepest ascent and the highest step pivot selection (see Heesterman 1983). It should be noted that a variety of methods exist to approximate LFP by means of LP. For example, a method that directly transforms a LFP into a LP by using a variable transformation is proposed in Charnes and Cooper (1962). Dubois (1987) suggests a method based on the bisection method of Bolzano, finding a feasible solution of a complex linear region instead of looking for a root (see also procedures shown in Ibaraki 1981 and Schaible and Ziemba 1981). Duttaet al.(1992) propose an approximate algorithm based on Zimmermann’s method to find compromise solutions (see Zimmermann 1978).

Inuiguchi’s approach (see Inuiguchiet al.1993, Inuiguchi and Ramik 2000, Inuiguchi 2004) also obtains the two proposed pessimistic solutions by means of the so-calledfractile model (leading to the LP model) and the so-calledmodality model(leading to the LFP model). How-ever, in these works the equivalency between the fractile and modality models, and the LP and LFP models, is not proved in detail, missing some significant considerations. For example, in the modality model the transformation into a LFP model is only possible if a realistic objec-tive threshold is selected. The approach provides a methodology to select realistic objecobjec-tive thresholds when too ambitious ones are initially selected. Additionally, the main relationships between the fractile and modality models are not explicitly described when possibility distri-butions are used to represent the uncertainty. Finally, the robustness of the obtained solutions has not been analysed and described with a real case example. In Camposet al.(2005) the proposed pessimistic solutions have been successfully applied to compute the Cournot equilib-rium in an electricity market where the uncertainty of the demand curve has been modelled by using possibility distributions. GAMS modelling language, CPLEX solver and the proposed algorithm in Dubois (1987) have been used to compute the corresponding equilibriums in the Spanish electricity market, where large-scale optimization problems must be tackled.

The rest of this article is organized as follows. The next section introduces some basic concepts of fuzzy sets and possibility theory. Section 3 proposes a comparison criterion to determine the more robust of two candidate solutions of an optimization problem under uncer-tainty. Then, by means of simplified crisp optimization problems, it proposes two different methods based on fuzzy CCP models to obtain robust solutions for linear decision-making problems when the coefficients of the objective function are fuzzy numbers. The relations between the two methods are also described. The proposed methodology is applied, in section 4, to a case example. Finally, concluding remarks are given in section 5.

2. Fuzzy sets and possibility theory

Fuzzy sets (see Zadeh 1965) are a generalization of classic sets where gradual memberships are allowed. A fuzzy setX˜is characterized by a membership functionµ_X_˜(x)∈ [0,1]relating each elementx ∈to its compatibility degree withX˜.is called the universal or referential set (figure 1).

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Figure 1. Classic setvs. fuzzy set.

Anα-cut of a fuzzy setX˜ is a classic setXαobtained fromX˜ for eachα∈ [0,1]such that

Xα =

x∈ X

µ_X_˜(x)≥α

. (1)

A fuzzy set is said to be a fuzzy number when its membership function is continuous, normal and convex (see figure 2). Normality means that the maximum value of the membership function is one, that is

∃x ∈ X

µ_X_˜(x) =1. (2)

Convexity means that all theα-cuts are nested closed intervals decreasing withα, that is

Xα =[aα, bα]∀α∈[0,1];Xα1⊆Xα2, α1> α2, α1, α2∈[0,1]. (3)

Algebraic real number operations can be generalized to fuzzy numbers applying the so-called

extension principle(see Zadeh 1975, Yager 1986). In particular it is interesting to note that the

addition of fuzzy numbers and the multiplication of a fuzzy number by a scalar are closure operations for trapezoidal or triangular membership functions (see figure 3). Thus, when these particular fuzzy numbers are represented by their four characteristic points, it holds

xa, xb, xc, xd+ya, yb, yc, yd=xa+ya, xb+yb, xc+yc, xd+yd γ·xa, xb, xc, xd=γ·xa, γ·xb, γ ·xc, γ ·xd γ >0 γ·xa, xb, xc, xd=γ·xd, γ ·xc, γ ·xb, γ·xa γ <0

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One of the most important applications of fuzzy set theory is possibility theory, introduced by Zadeh in 1978 (see Zadeh 1978), where fuzzy sets are used to model what it is known

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Figure 3. Trapezoidal and triangular fuzzy numbers.

about the value of a variable. In this setting given a fuzzy setX˜ defined overrepresenting the knowledge of a variableX, then, for eachx ∈,µ_X_˜(x)is interpreted as the degree of possibility that X takes the valuex (see Dubois and Prade 1988, 1991). The membership functionµ_X_˜(x)is then called a possibility distribution forX, which is denoted byπ(x):

π(x)=µ_X_˜(x)∀x∈. (5)

Given a possibility distributionπfor a variableX, possibility theory states that the possibility thatXtakes its value in a classic setF is given by the possibility measure(F )induced byπ

(F )=max

x∈F π(x). (6)

The fundamental axiom of the possibility measureinduced by a possibility distributionπ is given by

(F∪G)=max((F ), (G)). (7)

Given a possibility measure(F )a dual necessity measureN (F )can be defined by

N (F )=1−(Fc) (8)

whereFc_{is the complement of}_F_{. Note that this definition is coherent with the intuition that}

an eventF becomes more necessary when its complementFc_{becomes more impossible.}

A dual fundamental axiom for the necessity measureN is given by

N (F∩G)=min(N (F ), N (G)). (9)

3. Robust solutions using fuzzy chance constraints

3.1 A robustness concept in uncertain optimization problems

Real decision-making processes often lead to optimization problems under uncertainty where an objective function, representing the net cost of a decision, must be minimized. In these cases a unique optimal decision for all the likely scenarios is not possible, being necessary to attain some kind of robust decisions, although the definition of robustness is still a matter of discussion. This section proposes a robustness concept based on the values of those net costs with enough certainty to occur, leading to the following formal definition.

Given the optimization problem‘Min’D∈S(ω)˜ Z(D, ω)˜ , where its objective functionZ(D, ω)˜

and feasible region S(ω)˜ depend on the n-dimensional uncertain coefficient vector ˜

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necessity measure, see, for example, Prade 1985, Klir and Folger 1988), then it will be said that a decisionD1is more robust than a decisionD2 for a given certaintyβ ∈ [0,1]if and

only if the costz∗₁such that the certainty thatZ(D, ω)˜ is higher thanz∗₁ is at most 1−β, is less than the correspondingz₂∗forZ(D2,ω)˜ , that is

z₁∗< z₂∗ where z∗_i =Max

z:C

{Z(Di,ω)˜ ≤z} ∩ { Di ∈S(ω)˜ }

≤β

.

For example, whenCis a probability measure, this definition leads to compare the Value at Risk z∗₁with confidence levelβ(usually chosen in [0.9, 0.99]), obtained with the cost distribution Z(D1,ω)˜ , and the correspondingz∗2withZ(D2,ω)˜ , that is, to compare the percentiles 100·β%

calculated for each cost distribution (see Jorion 1997). In this article the selected uncertainty measureCis the necessity measureN described in the above section.

This definition of robustness is consistent with the common understanding that decision making should consider not only the set of most expected coefficients, but also other less favourable ones that could reasonably occur.

3.2 The fuzzy linear optimization problem considered

To be able to reach crisp robust solutions for a linear optimization model with badly known unit costs, in the sense described above, uncertainty can be modelled using possibility distributions. The obtained fuzzy linear optimization model can be formulated as follows:†

F P ≡‘Min’

D∈S

˜ Z(D) =

n

j=1

˜

cj ·Dj (10)

whereS= { D=(D1, . . . , Dn)/A· D≤b,D ≥0}is a polyhedron,c˜j is a fuzzy number

with membership functionµc_˜j(cj), and according to the extension principleZ(˜ D) is a fuzzy

number with a known membership functionµ_Z(_˜ _D) (z)for eachD.

Since the objective function takes fuzzy values, traditional optimization techniques cannot directly be applied to solve this type of problem. Additionally, due to the fact that fuzzy right-hand sides in LP constraints are equivalent in the dual programming to fuzzy cost coefficients (see Rodder and Zimmermann 1977), in the sequel not only approaches to solve fuzzy objec-tives but also to solve fuzzy constraints are briefly reviewed (see Inuiguchiet al.1990 for an extended review).

For example, Verdegay (1982) finds the same fuzzy solution defined in Rodder and Zimmermann (1977) by optimizing a crisp parametric linear problem, obtained by considering the unit costs of eachα-cut instead of the fuzzy unit costs. In practice this solution can be difficult to obtain because the optimization must be realized for eachα∈ [0,1].

When triangular unit costs are considered, Tanakaet al.(1984) obtain a crisp solution with an auxiliary crisp problem based on the weighted average of the upper and lower characteristic points of the triangular membership functions. However, an incorrect configuration of the selected weights can provide too risky solutions and the robustness notion is not clarified.

Using some results of Bitran (1980) related to convex sets, Delgadoet al.(1987) define a realistic crisp solution by solving a multiple objective linear programming problem (MLP) with 2n_{objectives, for a fixed}_α_{-cut of the fuzzy unit costs. However, computational efficiency}

can be questioned, especially whennis large. Something similar occurs with the approach described in Luhandjula (1987).

†_{In this formulation the minimum operator appears within quotation marks since the objective function is a fuzzy}

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In Buckley (1989) a crisp parametric linear problem to calculate a fuzzy solution by using the lower bounds of eachα-cut is proposed, and Sakawa and Yano (see their works between 1985 and 1989 in Sakawa and Yano 1990) do the same with a fixedα-cut. The same comments suggested in Verdegay (1982) should be considered.

In Rommelfangeret al.(1989), the procedure suggested in Zimmermann (1978) is used to obtain a compromise solution of the MLP obtained considering the lower and upper bounds ofr α-cuts of the fuzzy unit costs. As in Delgadoet al.(1987), computational efficiency can be questioned especially whenris large.

When triangular unit costs are considered, Lai and Hwang (1992) formulate an auxiliary LP with three objectives. Firstly, the most possible value of the net cost is minimized. Secondly, to maximize the possibility of having lower costs, the difference between the most possible value and the smaller one is maximized. Finally, to minimize the risk of having higher costs, the dif-ference between the greater and the most possible net cost is minimized. As in Rommelfanger

et al.(1989) the procedure suggested in Zimmermann (1978) is used to obtain a compromise

solution of this MLP. However, a lot of compromise solutions can be obtained.

In Ching-Lai et al.(1993) triangular unit costs are also used, and the most possibleα -efficient solution is defined by optimizing the weighted average of the three characteristic points. The same comments suggested in Tanakaet al.(1984) can be applied here.

In Jamison and Lodwick (2001) the fuzzy objective and constraints are reformed into an unconstrained fuzzy function by penalizing the objective for possible constraint violations, the aim being to optimize the expected midpoint of the image of this fuzzy function. However, a special algorithm is needed to find the optimum, since no commercial software can be used.

Since the above approaches tend to find too optimistic solutions, and since in real problems the net cost could be too sensitive to unit costs uncertainty, the objective of this work has been to compute robust solutions using fuzzy CCP models. These solutions should judge positive in most of the possible scenarios, doing not too badly in any of them.

To be able to deal with real optimization problems, a computationally efficient search-ing procedure must be designed. In the next subsections two robust criteria for decisions in the fuzzy linear problem FLP are proposed. It is shown that both criteria lead to dif-ferent crisp optimization problems that can easily be solved by commercial optimization software.

3.3 The robust primal CCP

In order to establish a robustness criterion, the decision-maker can fix a parameterα∈ [0,1] to quantify the risk he is ready to assume. From a pessimistic point of view, for every possible decisionD, the maximum (net) costZmax(D, α) that results for the prefixed riskαis obtained

by solving the following possibility constraint (figure 4):

˜

Z(D) ≥Zmax(D, α)

=α. (11)

Note that in the case of continuous membership functions the above equality holds true. Therefore, any cost higher thanZmax(D, α) has a possibility degree to occur less than the

prefixed riskα. Whenα=0, the constraintZ(˜ D) < Z max(D, α) must hold true, which means

that any cost with non-null possibility to occur is less thanZmax(D, α) .

According to the definition given in section 3.1, when the uncertainty measureCselected is a necessity measureNandβ=1−α, then a robust solutionD∗is the decision that provides

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Figure 4. Objective value with riskα.

the smallest costZmax(D, α) , and is obtained by solving the following problem:

Min

D_∈S

Zmax(D, α)

s.t.

˜

Z(D) ≥Z_max(D, α)

=α.

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SinceZmax(D, α) has no analytical expression, it is convenient to transform this problem into

a maximin CCP formulation, using an intermediate variableZmax:

Min

D_∈S Max

Zmax

s.t.

˜

Z(D) ≥Zmax

=α.

(13)

According to (7) the possibility constraint defined in (13) can be transformed using the maximum operator:

˜

Z(D) ≥Zmax

= Max z≥Zmax

µ_Z(_˜ _D) (z). (14)

Substituting (14) into (13) the crisp problem to solve now is

Min

D∈S Max

Zmax

s.t. Max z≥Zmax

µ_Z(_˜ _D)(z)=α.

(15)

Since the resolution of (15) can sometimes be too difficult, the following proposition has been used to find a crisp robust solution.

PROPOSITION1 If(D∗, Z_max∗ )is an optimal solution of(15)then there does not exist a greater

value thanZ_max∗ whose membership to the fuzzy objective isα,that is,the maximum defined

in(14)is achieved forz=Z∗_max:

z≥ Z

∗

max

µ_Z(_˜ _D∗) (z)=α

=Z∗_max. (16)

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Suppose that there is a greater value than the optimal Z_max∗ with fuzzy objective membershipα:

∃z0>

Z_max∗ µ_Z(_˜ _D∗)

(z0)=α. (17)

In this situation(D∗, z0)is a feasible solution of (15) since

α=µ_Z(_˜ _D∗)(z0)≤Max

z≥z0

µ_Z(_˜ _D∗)(z)

≤ Max z≥Z∗max

µ_Z(_˜ _D∗)(z)

=α. (18)

Furthermore, its objective valuez₀ is greater thanZ∗_maxwhich contradicts the optimality of

(D∗, Z_max∗ ).

COROLLARY1 Problem(15)can be transformed into an easier crisp one formulated as

Min

D∈S

Max Zmax∈µ−_Z(_˜1_D)(α)

Zmax (19)

whereµ−_˜1

Z(D) (α)is the inverse image atαof the fuzzy objectiveZ(˜ D) (figure5).

Remark 1

(a) Problem (13) can also be formulated in terms of the dual necessity measureN:

Min

D_∈S Max

Zmax

s.t. N

˜

Z(D) < Z _max

=1−α.

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For an introduction to the use of possibility and necessity in optimization problems see Buckley (1988).

(b) When the objective coefficients are trapezoidal fuzzy numbers, that is,c=(ca_,_cb_,_cc_,_cd₎_,

then

µ−_˜1 Z(D)(α)=

(1−α)· ca· D+α· cb· D, α· cc· D+(1−α)· cd· D

. (21)

Hence, problem (19) is the result of applying a variant of the Hurwicz decision rule, that is, the result of averaging one of the most possible valuescc_·_D_{and one of the least possible}

valuescd_·_D_{of the fuzzy objective}_Z(˜ _D) _: Min

D∈S

α· cc· D+(1−α)· cd· D

. (22)

Clearly this problem is linear, and can easily be solved by commercial optimization software.

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(c) Solution of (19) is given by the decisionD∗∈Sand the cost vector:

c∗_α=

max

c1/µc˜1(c1)≥α

c₁, . . . , max

cn/µcn˜ (cn)≥α

cn

. (23)

Additionally the following conditions hold true:

c· D∗≤ c∗_α· D∗≤ c∗_α· D∀ D∈S,∀c/µc_˜j(cj)≥α. (24)

Note that decision D∗∈S coincides with the compromise decision that minimizes the maximum cost function in the MLP described in Delgadoet al.(1987), which is

Min

D_∈S

(c1· D, . . . ,c2n· D)

cj ∈E(α) j =1, . . . ,2n (25)

whereE(α)is constituted by vectors whosejth component is equal to either the upper or the lower bound of theα-level set ofc˜j.

3.4 The robust dual CCP

An alternative way to look for a robust solution consists of fixing a maximum (net) costZmax

in order to minimize the possibilityα(D, Z max)that the fuzzy objectiveZ(˜ D) is greater than

Zmax(figure 6):

min

D_∈S

α(D, Z max)=min

D_∈S

{ ˜Z(D) ≥Zmax}. (26)

In this case, the resulting optimal solution D∗ is robust in the sense of the definition in section 3.1, when the uncertainty measureCis a necessity measureN, and when the certainty degreeβis equal to the optimal necessity degree 1−α(D∗, Zmax). BesidesD∗is such that the

constraintZ(˜ D∗)≥Z_maxholds true with the smaller possibility and the variableα(D, Z _max) equals the possibility value for which the constraint holds.

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Using the fuzzy CCP approach, the problem to be solved has now a minimin CCP formulation, which is

Min

D∈S Min

0≤α≤1α

s.t. { ˜Z(D) ≥Zmax} =α. (27)

According to (14) this problem can now be formulated as follows:

Min

D∈S Min

0≤α_≤1α

s.t. Max z≥Zmax

µ_Z(_˜ _D)(Z)α. (28)

PROPOSITION2 If(D∗, α∗)is an optimal solution of(28)verifying

∃z∗≥Zmax/µZ(˜D∗)(z∗)=1, (29)

then:

(a) The possibility thatZis greater than or equal to the fixed costZmaxis one,that is, α∗ =1

(possiblyZmaxhas been underestimated and the problem has no informative solutions).

(b) Any decision D ∈S has α(D, Z max)=1 (all alternatives are equally robust, and

therefore,another robust criterion should be used,or a higher costZmaxselected).

Proof Byreductio ad absurdum.

(a) Ifα∗<1, as(D∗, α∗)is a feasible solution then Max

z_≥Zmax

µ_Z(_˜ _D∗)(z)

=µ_Z(_˜ _D∗)(z∗)=1=α∗<1, (30)

which is a contradiction.

(b) IfD ∈Sverifiesα(D, Z max) <1, since(D∗, α∗)is the optimal solution then(D, α) must

be worse, that is

1> α(D, Z max)=Max

z_≥Zmax

µ_Z(_˜ _D)(z)≥ Max z_≥Zmax

µ_Z(_˜ _D∗)(z)

=α∗, (31)

which is a contradiction with the previous proof.

PROPOSITION3 If(D∗, α∗)is an optimal solution of (28)verifying

∀z≥ZmaxisµZ(˜ D∗)(z) <1, (32)

then the maximum defined in constraint(28)is reached at least forz=Zmax:

{Zmax} ⊂

z≥Zmax/µZ(˜ D∗)(z)=α∗

. (33)

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Suppose that the maximum defined in constraint (28) is not reached forZmax, then

∃z∗> Zmax/α∗ =µZ(˜ D∗)(z∗) > αmax=µZ(˜ D∗)(Zmax). (34)

SinceZ(D∗)is a fuzzy number it is normal and thus

∃z1/α1=µZ(˜ D∗)(z1)=1. (35)

According to the hypothesis, α∗<1 and z1< Zmax must both hold true. Consider now

¯

α∈(α_max, α∗)and denote theα-cut ofZ(˜ D∗)as Zα =

z/µ_Z(_˜ _D∗)(z)≥α

. (36)

The convexity ofZ(˜ D∗)implies that theα-cuts are intervals for eachα∈ [0,1]and since α_max <α < α¯ ∗< α₁=1 the following conditions are satisfied:

Zα1⊆Zα∗⊆Zα¯ ⊆Zαmax. (37)

Sincez∗ ∈Zα∗,z1∈Zα1then (37) implies thatz∗,z1∈Zα¯. Due toz1< Zmax< z∗and since

Zα_¯ is an interval thenZmaxbelongs toZα¯ too, which is in contradiction with the fact thatZmax

belongs toZαmax, since

αmax=µZ(˜ D∗)(Zmax) <α.¯ (38)

PROPOSITION4 The optimal objectiveZxof the problem

Min

D∈S Max z∈µ−1

˜

Z(D)(1)

Z (39)

is less thanZmaxif and only if(32)holds true,that is

Zx < Zmax⇐⇒ ∀z≥ZmaxisµZ(˜ D∗)(z) <1. (40)

Proof (⇒)Suppose thatZx =Max_z_∈_µ−1

˜

Z(Dx ) (1)

Z < Zmax. Then it is

Max z≥Zmax

µ_Z(_˜ _Dx₎(z) <1. (41)

If(D∗, α∗)is an optimal solution of (28), then α∗ =Max

z≥Zmax

µ_Z(_˜ _D∗)(z)≤ Max z≥Zmax

µ_Z(_˜ _Dx₎(z) <1. (42)

Therefore

∀z≥ZmaxisµZ(˜ D∗)(z) <1. (43) (⇐)Suppose(D∗, α∗)is an optimal solution of (28) verifying

∀z≥Z_maxisµ_Z(_˜ _D∗)(z) <1. (44)

AsZx is the optimal objective of (39) then

Zx ≤Z∗ = Max z∈µ−1

˜

Z(D∗)(1)

Z. (45)

Sinceµ_Z(_˜ _D∗)(Z∗)=1 and (44) holds true, the next conditions must also hold true:

Zx≤Z∗< Zmax. (46)

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COROLLARY2 According to the previous proposition:

(c) If the optimal solution Zx _{of problem} ₍₃₉₎ _{is greater than or equal to} _Z

max, then

(Proposition2)all decisions are equally optimum,and the possibility thatZ˜ is greater

than or equal toZmaxis one.

(d) If the optimal solutionZx _{of problem}₍₃₉₎_{is less than} _Z

max, then(Proposition3)the

optimum can be obtained from the following crisp problem(note that ifZx _≥_Z

maxthen

α∗is greater than1):

α∗=Min

¯

D∈S

µ_Z(_˜ _D)(Zmax) <1. (47)

Remark 2

(a) The reader can check thatα∗ >0 if and only ifZmax< Zxwhere

Zx =Min

D∈S Sup z/µ_Z(_˜_D)(z)>0

Z. (48)

(b) The choice made for the maximum costZmaxis not a trivial issue. An easy solution is to

model the aspiration levelZmaxby a symmetrical triangular fuzzy number:

˜

Zmax=(zmax−σ zmax, zmax, zmax+σ zmax)σ zmax>0. (49)

In order to set the pair {zmax, σ zmax}, the decision-maker must fixσ zmax, that can be

interpreted as a measure of the fuzzy dispersion, and the centrezmax, that can be estimated

as the most possible maximum cost obtained solving the following LP:

z_max=max

D∈S n

j₌1

¯

cj ·Dj (50)

wherec¯j is the middle point of the 1-cut ofc˜j (see figure 7).

In this case a similar procedure can be adopted to solve the new fuzzy CCP model described in (27), except that now, instead of comparingZ˜ withZmax, the fuzzy number

˜

Z− ˜Zmaxmust be compared with zero.

(c) Problem (27) can be formulated in terms of the dual necessity measureN:

Min

D∈S Min

0≤α≤1α

s.t. N{ ˜Z(D) < Zmax} =1−α. (51)

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(d) When the objective coefficients are trapezoidal fuzzy numbers, that is,c=(ca,cb,cc,cd), then

µ−_˜1

Z(D)(1)= [c

b_·_D,_cc_·_D_]_.

(52)

Thus, problem (39) becomes

Zx =Min

D∈S{

cc· D}. (53)

IfZx _{< Z}

max< Zxis satisfied, according to the definition of the membership function of

a trapezoidal fuzzy number, problem (47) results in the following LFP:

α∗ =Min

D∈S

µ_Z(_˜ _D) (Zmax)∈(0,1)⇐⇒α∗ =Min

D∈S

cd· D−Zmax

cd· _D− _cc· _D. (54)

Graphically (figure 8) it consists in minimizing the distance betweencd_·_D_and_Z

max(to

minimize the risk of having cost higher thanZmax), and conversely maximizing the distance

betweencd_·_D_and_cc_·_D_{(to maximize the possibility of having cost less than}_Z

max).

3.5 Relations between primal and dual CCP

There exists reciprocity between the two criteria presented in the previous subsections:

1. If (D∗, Z_max∗ ) is an optimal solution of problem (13) for a fixed risk parameter α, then (D∗, α) is an optimal solution of problem (27) for a fixed cost Z∗_max such that α=(Z(˜ D∗)≥Z∗_max).

2. If(D∗, α∗)is an optimal solution of problem (27) for a fixed costZmax, then(D∗, Zmax)is an

optimal solution of problem (13) for a fixed risk parameterα∗such thatα∗=(Z(˜ D∗)≥ Zmax).

In addition, the following considerations between the formulations of the two dual models (13) and (27) could be stated:

• The dual CCP model with a maximin CCP formulation (or a minimax CCP formulation, if (10) is defined with a maximization criterion) is a CCP model with a minimin CCP formulation.

• The requirement fixed in one model, being in this case a parameter, is the objective in the other one, being now a variable.

• Possibility constraints are the same in both models.

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4. Case example

To show the robustness of the proposed solutions, a simple case study is considered in this subsection (see Camposet al.2005 for a more complex case study), based on a single-period inventory problem (see Chen and Hsieh 1999, Hsieh 2002).

Suppose that a merchant produces two productsP1andP2with production costsc1andc2

equal to 3 and 4, respectively, and that productsP1 andP2 can be stored in two storagesA

andB. WhileAis from his ownership, with a unit store costcAequal to 0,Bmust be rented,

with an uncertain unit storec˜Brepresented by the following triangular fuzzy number:

µ_C_˜

B(cB)=

    

2·(cB−(1/2)) (1/2)≤cB ≤1 (3−cB)/2 1≤cB ≤3

0 elsewhere

. (55)

Assume that the merchant can sell the products with the following unit sale prices:

• ProductP1is sold at an uncertain unit sale priceb˜1, reflecting the presence of an elastic

demand of productP1. In this case the uncertainty has been modelled with the following

triangular fuzzy number:

µ_b_˜₁(b1)=

    

2·(b1−5) 5≤b1≤5.5

2·(6−b1) 5.5≤b1≤6

0 elsewhere

. (56)

• ProductP2is sold at a crisp unit priceb2equal to 7.

Additionally, suppose that storageAhas a capacity limit equal to 100 and storageBequal to 10, and that the unit sizes for productsP1andP2are 2 and 6, respectively. The objective is to

find the amount of each product that the merchant should produce in order to maximize his profit.

Obviously, the merchant has to maximize the function defined as the sale income minus the total cost (including the total production cost and the total storage cost), or equivalently, to minimize the opposite function, which is equivalent to optimizing the following fuzzy linear optimization model:

_Min

(D1A,D1B,D2A,D2B)∈S

(c1− ˜b1)·(D1A+D1B)+(c2−b2)·(D2A+D2B)+ ˜cB·(D1B+D2B),

(57) whereDij is the stored quantity in storagej of productPi forj =A, B andi=1,2. The

feasible regionSof the model is obtained from the limits of the storage capacity:

S= {(D1A, D2A, D1B, D2B)≥0:2·D1A+6·D2A≤100,2·D1B+6·D2B ≤10}.

(58) The aim of this case study is to compare the robustness of the proposed solutions with the solution obtained from the traditional crisp model when solved for the most possible values of c˜B andb˜1 (in the sequel, the MPapproach), which can be considered a risk indifferent

approach. Similar conclusions can be obtained when comparing with other approaches such as those shown in section 3.1.

The solution with theMPapproach(c˜B =1,b˜1=2.5)is

MP= {D₁∗_A=50, D₁∗_B =5, D∗₂_A=0, D₂∗_B =0}. (59)

Therefore, the merchant should produce 55 and 0 of productsP1andP2, respectively, and in

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When the merchant fixes a risk parameterα, the robust primal CCP model proposed in this article in (13) becomes

Min (D1A,D1B,D2A,D2B)∈S

Max Zmax

Zmax

s.t. {(c1− ˜b1)·(D1A+D1B)+(c2−b2)·(D2A+D2B)

+ ˜cB·(D1B+D2B)≥Zmax} =αα∈ [0,1].

(60)

Applying the results from remark b of section 3.3, the equivalent LP to solve is

Min D1A,D1B,D2A,D2B

α·(−2.5·(D1A+D1B)+(D1B+D2B))+(1−α)·(−2·(D1A+D1B)

+3·(D1B+D2B))−3·(D2A+D2B). (61)

Solving it with a risk parameterαequal to 0.5, not as optimistic as the MP approach, the obtained solution is

Primal=D₁∗_A=50, D₁∗_B=0, D₂∗_A=0, D∗₂_B =0. (62)

And, therefore, in this case the merchant should produce 50 and 0 of productsP1 andP2,

respectively, and storageBis no longer necessary.

Additionally, the robust dual CCP model proposed in this article in (27) when, for example, the merchant fixes a maximum target costZmax= −105 is

Min (D1A,D1B,D2A,D2B)∈S

Min

0≤α≤1α

s.t. ((c1− ˜b1)·(D1A+D1B)+(c2−b2)·(D2A+D2B)

+ ˜cB·(D1B+D2B)≥ −105)=α.

(63)

To apply the results presented in remark d of section 3.4, the optimal objectivesZx_and_Z xof

the following LPs must be found:

Zx= Min

−2.5·(D1A+D1B)+(D1B+D2B)−3·(D2A+D2B)

Zx= Min

−2·(D1A+D1B)+3·(D1B+D2B)−3·(D2A+D2B).

(64)

The reader can easily check thatZx _{= −127}_._{5 and}_Z

x = −100. SinceZx < Zmax< Zx is

satisfied, then the robust dual CCP model is equivalent to the following LFP:

α∗= Min

−2·(D1A+D1B)+3·(D1B+D2B)−3·(D2A+D2B)+105

0.5·(D1A+D1B)+2·(D1B+D2B)

.

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Solving it using WINGULF software, the solution is the same as that with the primal approach. Therefore, the merchant should not necessarily use storageB. The dual approach provides the minimum possibility degreeα∗of having higher cost thanZmax= −105, which in this case

is a low possibility degree equal to 0.2.

In order to compare the robustness of solution ‘use storage B’, obtained with the MP approach, and the solution ‘don’t use storageB’, obtained with the proposed approaches, the original fuzzy objective function (defined as the sale income minus the total cost) has been computed for both solutions, and shown in figure 9.

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Figure 9. Fuzzy costs for the obtained solutions.

From a pessimistic point of view, that is, for higher objective costs with lower possibility to occur than the most possible ones (corresponding to unit store costc˜Bhigh and unit sale price

˜

b1low), the proposed solution ‘don’t use storageB’ results in higher profits (less costs) than

the solution ‘use storageB’. Otherwise, both solutions provide more similar results, although the proposed solution ‘don’t use storageB’ is logically slightly worse. This reflects the fact that the proposed approaches only protect the merchant against the one side event ‘objective cost unexpectedly high’, as can be deduced from the original fuzzy constraint used in both primal and dual approaches (equations (60) and (63)):

(c₁− ˜b₁)·(D₁A+D1B)+(c2−b2)·(D2A+D2B)+ ˜cB·(D1B+D2B)≥Zmax

=α. (66)

Summarizing, the proposed solutions imply a profit loss under optimistic scenarios, but sig-nificantly reduce losses when more pessimistic but possible situations occur, showing the robustness of these new approaches.

5. Conclusions

While uncertainty has traditionally been modelled with probability distributions, it is well known that this approach requires sufficient statistical information to compute them. Since possibility theory does not presuppose the existence of statistical data (although weaker sta-tistical interpretations do exist), it can be used as an alternative uncertainty model, especially when vagueness of perception or subjectivity comes out.

This article uses possibility distributions to model the uncertainty of objective coefficients (as unit cost) when linear optimization problems are considered, and computes robust solu-tions by using two criteria borrowed from traditional stochastic risk management. In the first criterion, a possibility threshold of having higher net costs must be decided. The proposed robust solution is computed by minimizing the maximum net cost with possibility equal to the selected possibility threshold. In the second one, an estimate of a maximum realistic desired net cost is necessary. In this case, the proposed robust solution is obtained by minimizing the possibility of having net costs higher than the maximum desired cost. It is shown that both

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resolution criteria can be considered dual in the sense that: (a) the requirement fixed in one of them is the objective of the other one; (b) the resolution of one of them allows solving the other one.

In order to compute the two proposed solutions, chance constrained programming (CCP) has been used. The two CCP models obtained (one of them with either a maximin or a mini-max formulation and the other one with a minimin formulation) have been transformed into different crisp optimization problems. In particular, when trapezoidal or triangular possibility distributions are considered, these crisp optimization problems are linear or fractional, and can therefore be easily solved by commercial optimization software.

The proposed methodology can be especially interesting for real linear optimization prob-lems under uncertainty because (a) no special assumptions are needed (such as those required by probability theory), and (b) when trapezoidal possibility distributions are considered, they can be solved with reasonable computational efficiency using standard linear optimization techniques. Finally, although less informative than probability theory, this methodology can be very helpful in decision-making processes. The authors are currently applying the proposed approach to solve Cournot equilibrium in electricity markets (see Camposet al.2005 for a real case study solved in 10 min with approximately 84,000 constraints and 120,000 variables).

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