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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 110, No. 1, pp. 53–74, JULY 2001 Efficient Solution Concepts and Their Relations in Stochastic Multiobjective Programming R. CABALLERO,1 E. CERDÁ,2 M. M. MUÑOZ,3 L. REY,4 5 AND I. M. STANCU-MINASIAN Communicated by Y. C. Ho Abstract. In this work, different concepts of efficient solutions to problems of stochastic multiple-objective programming are analyzed. We center our interest on problems in which some of the objective functions depend on random parameters. The existence of different concepts of efficiency for one single stochastic problem, such as expected-value efficiency, minimum-risk efficiency, etc., raises the question of their quality. Starting from this idea, we establish some relationships between the different concepts. Our study enables us to determine what type of efficient solutions are obtained by each of these concepts. Key Words. Stochastic multiobjective programming, expected-value efficiency, minimum-variance efficiency, minimum-risk efficiency, efficiency in probability. 1. Introduction In many multicriteria decision problems, some parameters take unknown values at the moment of making the decision. This uncertainty can be due to problems of observing the parameters themselves or that their values depend on such factors as nature, decisions of other agents, etc. If 1 Professor, Department of Applied Economics (Mathematics), University of Málaga, Málaga, Spain. 2 Professor, Department of Foundations of Economic Analysis, Universidad Complutense de Madrid, Madrid, Spain. 3 Professor, Department of Applied Economics (Mathematics), University of Málaga, Málaga, Spain. 4 Professor, Department of Applied Economics (Mathematics), University of Málaga, Málaga, Spain. 5 Senior research worker, Center for Mathematical Statistics, Romanian Academy, Bucharest, Romania. 53 0022-3239兾01兾0700-0053$19.50兾0 2001 Plenum Publishing Corporation 54 JOTA: VOL. 110, NO. 1, JULY 2001 these parameters are random variables, the resulting problem is called stochastic multiobjective programming problem. There is much research in the literature dealing with the study of such problems, among which we could mention the books by Goicoechea, Hansen, and Duckstein (Ref. 1), Stancu-Minasian (Ref. 2), Slowinski and Teghem (Ref. 3), and the articles by Teghem, Dufrane, Thauvoye, and Kunsch (Ref. 4), Stancu-Minasian and Tigan (Ref. 5), Stancu-Minasian (Ref. 6), Urli and Nadeau (Ref. 7), Ben Abdelaziz, Lang, and Nadeau (Refs. 8–9). The review of these works shows that the solution of such problems involves always transforming the problem into a deterministic one, which is called the equivalent deterministic problem. This transformation is carried out by using some statistical characteristic of the random variables involved in the problem (expected value, variance, etc.). In the literature, this has led to different definitions of the efficient solution concept for the same stochastic multiobjective programming problem, so that different efficient sets are obtained for the same stochastic problem. In principle, these sets are noncomparable, since they utilize different characteristics of the initial problem. In this work, we consider five such sets: expected value, minimum variance, expected-value minimum standard deviation, minimum risk-efficiency, and efficiency in probability. We raise the following question: do these efficiency sets have some relationship to each other? An initial answer is given in Caballero, Cerdá, Muñoz, and Rey (Ref. 10), in which relations between two of the efficient sets were established. Specifically, it is shown that, given certain conditions, the minimum-risk efficient solution set and the efficiency in probability set are reciprocal. This paper deals with the analysis of other relations between the previously mentioned sets. On the one hand, we relate the concept of expected-value standard-deviation efficiency to that of expected-value efficiency and minimum standard-deviation efficiency; on the other hand, we relate the concept of efficiency in probability to that of expected-value standard-deviation efficiency. 2. Efficient Solution Concepts for a Stochastic Multiobjective Programming Problem Let us consider the stochastic multiobjective programming problem (SMP) Min(z̃1 (x, c̃), z̃2 (x, c̃), . . . , z̃q (x, c̃)), x ∈D where the following notations and assumptions are employed: (i) x∈⺢ n is the vector of decision variables of the problem and c̃ is a random vector whose components are random continuous JOTA: VOL. 110, NO. 1, JULY 2001 55 variables, defined on the set ⺕⊂⺢ s. We assume given the family ⺖ of events (that is, subsets of ⺕) and the distribution of probability P defined on ⺖ so that, for any subset of ⺕, A⊂⺕, A∈⺖, the probability P(A) is known. Also, we assume that the distribution of probability P is independent of the decision variables x 1 , . . . , xn . (ii) The functions z̃1 (x, c̃), z̃2 (x, c̃), . . . , z̃q (x, c̃) are defined on ⺢nB⺕. (iii) The set of feasible solutions D⊂⺢ n is nonempty, compact, and convex. Let z̄k (x) denote the expected value of the kth objective function, and let σ k (x) be its standard deviation, k∈{1, 2, . . . , q}. Let us assume that, for every k∈{1, 2, . . . , q} and for every feasible vector x of the (SMP) problem, the standard deviation σ k (x) is finite. As previously pointed out, in the literature, different concepts of efficient solutions exist for the (SMP) problem. In this section, we present some which share the common feature that the efficient-solution concept is defined from a multiobjective problem which is generated by applying the same criterion to all the stochastic objectives separately. The first concept to look at is the expected-value efficient solution. This concept is obtained from the construction of a multiobjective problem in which the objective functions vector is the expected value of the vector of stochastic objectives of the initial problem. Definition 2.1. Expected-Value Efficient Solution. The point x∈D is an expected-value efficient solution of the (SMP) problem if it is Pareto efficient to the following problem: (E) Min (z̄1 (x), z̄2 (x), . . . , z̄q (x)). x ∈D Let E E be the set of expected-value efficient solutions of the (SMP) problem. The next concept considered is that of the minimum-variance efficient solution. In this case, the concept comes from obtaining the variance of each stochastic objective and outlining the multiobjective problem of minimizing such variances. Definition 2.2. Minimum-Variance Efficient Solution. The point x∈ D is a minimum-variance efficient solution for the (SMP) problem if it is a Pareto efficient solution for the problem: (σ 2) Min(σ 21 (x), σ 22 (x), . . . , σ 2q (x)). x ∈D Let Eσ 2 be the set of efficient solutions of the problem (σ 2 ). 56 JOTA: VOL. 110, NO. 1, JULY 2001 Next, we define the concept of expected-value standard-deviation efficiency. In this case, the concept of efficiency arises from the construction of a problem with 2q objectives involving the expected value and the standard deviation of each stochastic objective. Definition 2.3. Expected-Value Standard-Deviation Efficient Solution or Eσ Efficient Solution. The point x∈D is an expected-value standarddeviation efficient solution for the (SMP) problem if it is a Pareto efficient solution to the problem (Eσ ) Min(z̄1 (x), . . . , z̄q (x), σ 1 (x), . . . , σ q (x)). x ∈D Let E Eσ be the set of expected-value standard-deviation efficient solutions of the (SMP) problem. Finally, we give the concepts of efficiency for two criteria of maximum probability. As we will see next, in order to define these two concepts, the minimum-risk criterion (concept of minimum-risk efficiency) and the Kataoka criterion (efficiency in probability) are applied respectively to each stochastic objective. Definition 2.4. Minimum-Risk Efficient Solution for the Levels u1 , u2 , . . . , uq . See Stancu-Minasian and Tigan (Ref. 5). The point x∈D is a minimum-risk vectorial solution for levels u1 , u2 , . . . , uq if it is a Pareto efficient solution to the problem: (MR(u)) Max(P(z̃1 (x, c̃)⁄u1 ), . . . , P(z̃q (x, c̃)⁄uq )). x ∈D Let E MR(u) be the set of efficient solutions for the problem (MR(u)). Definition 2.5. Efficient Solution with Probabilities β 1 , β 2 , . . . , β q or β -Efficient Solution. The point x∈D is an efficient solution with probabilities β 1 , β 2 , . . . , β q if there exists u∈⺢ q such that (xt, ut )t is a Pareto efficient solution to the problem: (K(β )) Min (u1 , . . . , uq ), x,u s.t. P{z̃k (x, c̃)⁄uk }¤ β k , kG1, 2, . . . , q, x∈D. Let E K ( β )⊂⺢ n denote the set of efficient solutions with probabilities β 1 , β 2 , . . . , β q for the (SMP) problem. JOTA: VOL. 110, NO. 1, JULY 2001 57 It may be noted that these definitions of efficient solutions are obtained by applying the same transformation criterion to each one of the stochastic objectives separately (expected value, minimum variance, etc.), and by building afterward the resulting deterministic multiobjective problem. In this sense, it is necessary to point out that: (a) The concepts of expected value, minimum variance, etc., weak and properly efficient solution can be defined in a natural way. (b) The concepts of minimum-risk efficiency and β -efficiency require setting a priori a vector of aspiration levels u or a probability vector β . This implies that, in both cases, the efficient set obtained depends on the predetermined vectors in such a way that, in general, different level and probability vectors give rise to different efficient sets, u ≠ u′ ⇒ E MR (u) ≠ E MR (u′), β ≠ β ′ ⇒ E K ( β ) ≠ E K ( β ′). (c) The concept of expected-value standard-deviation efficient solution is an extension to the multiobjective case of the concept of the mean-variance efficient solution that Markowitz defines (Ref. 11) for the stochastic mono-objective problem of the portfolio selection. Note that, in the concept that we have just defined, instead of using the variance of each stochastic objective, we take the standard deviation. In this way, we have the two statistical moments corresponding to each stochastic objective in the same measuring units. Since the square root function is strictly increasing, the set of efficient solutions does not vary in the problem if we substitute standard deviation for variance; see White (Ref. 12). (d) The efficiency in probability criterion is a generalization of the one presented by Goicoechea, Hansen, and Duckstein (Ref. 1), who define the same concept taking the same probability β for all the stochastic objectives and with the probabilistic equality constraints taking the form P{z̃k (x, c̃)⁄uk }Gβ . This notion was introduced by Stancu-Minasian (Ref. 6), considering the Kataoka problem in the case of multiple criteria. Starting from the given definitions, we obtain different sets of efficient solutions for the (SMP) problem. This fact can give rise to some confusion. Although one could consider that the concepts previously defined are not 58 JOTA: VOL. 110, NO. 1, JULY 2001 connected, since they utilize different statistical characteristics of the stochastic objectives, it will be shown that they are closely related. We begin by studying the relationship between the expected-value standard-deviation efficient solutions and the efficient solutions of probabilities β 1 , . . . , β q . Next, we analyze the existence of relationships between the minimum expected-value efficient solutions, minimum-variance efficiency, and the expected-value standard-deviation efficient solutions, corresponding to the problem of 2q objectives that includes both the expected value and the standard deviation of the objectives. First, we present some results of efficient sets for deterministic multiobjective programming problems that will be used to establish the relations in stochastic multiple-objective problems. 3. Preliminary Results We present some relations between the efficient sets of several problems of deterministic multiobjective programming. These results will be used later for the analysis of the concepts of efficient solutions for multiple-objective stochastic problems. Let f and g be vectorial functions defined on the same set H ⊂ ⺢ n, with f: H ⊂ ⺢ n → ⺢ q and g: H ⊂ ⺢ n → ⺢ q, and let α , γ be nonnull vectors with q real components, that is, α , γ ∈⺢ q and α , γ ≠ 0. Let us consider the following multiobjective problems: Min( f1 (x), . . . , fq (x), γ 1 g1 (x), . . . , γ q gq (x)), (1) Min( f1 (x), . . . , fq (x)), (2) Min( γ 1 g21 (x), . . . , γ q g2q (x)), (3) x ∈D x ∈D x ∈D with γ ∈⺢q, γ ≠ 0. Let E ( f, γ g), E ( f ), E (γ g2) be the sets of efficient points of problems (1), (2), (3) respectively. The following theorem relates these problems to each other. With the superscripts d and p, we denote weak and proper efficiency, respectively. Theorem 3.1. Let us assume that g(x)H0 for every x∈D. Then: (i) (ii) (iii) E ( f ) ∩ E (γ g2 ) ⊂ E ( f, γ g); E ( f ) ∪ E (γ g2 ) ⊂ E d ( f, γ g); E d ( f ) ∪ E d(γ g2 ) ⊂ E d ( f, γ g). JOTA: VOL. 110, NO. 1, JULY 2001 59 Proof. (i) Let x∈E ( f )∩ E (γ g 2 ). Let us show that x∈E ( f, γ g) by reductio ad absurdum. We assume that x∉E ( f, γ g). Then, there exist an x*∈D such that fk (x*)⁄fk (x) and γ k gk (x*)⁄ γ k gk (x), for every k∈{1, 2, . . . , q}, there being an s∈{1, 2, . . . , q} for which the inequality is strict, fs (x*)Ffs (x) or γ s gs (x*)Fγ s gs (x). Therefore, x∉E ( f ) or x∉E (γ g2 ), due to γ k gk (x*)⁄ γ k gk (x), implies γ k g 2k (x*)⁄ γ k g2k (x), contrary to x∈E ( f ) ∩ E (γ g2 ). (ii) Let x∈E ( f ) ∪ E (γ g2 ). Let us see that x∈E d ( f, γ g) by reductio ad absurdum. We assume that x∉E d ( f, γ g). Then, there exist a vector x*∈ D that weakly dominates x and so verifies fk (x*)Ffk (x) and γ k gk (x*)Fγ k gk (x), for every k∈{1, 2, . . . , q}. Thus, x∉E ( f ) and, due to γ k gk (x*)Fγ k gk (x), implies γ k g2k (x*)Fγ k g2k (x), x∉E (γ g2 ), contrary to x∈E ( f ) ∪ E (γ g2 ). (iii) Let x∈E d ( f ) ∪ E d(γ g2). Let us see that x∈E d ( f, γ g) by reductio ad absurdum. We assume that x∉E d ( f, γ g). Then, there exist a vector x*∈ D that weakly dominates the vector x and therefore verifies that fk (x*)Ffk (x) and γ k gk (x*)Fγ k gk (x), for every k∈{1, 2, . . . , q}. Thus, x∉E d ( f ) and, due to γ k gk (x*)Fγ k gk (x), implies γ k g2k (x*)F γ k g2k (x), x∉E d (γ g2 ), contrary to x∈E d ( f ) ∪ E d (γ g2 ). From (iii), it is obvious that E d ( f ) ∩ E d (γ g2 ) ⊂ E d ( f, γ g) is also verified. Furthermore, as E ( f ) ⊂ E d( f ) and E (γ g2 ) ⊂ E d (γ g2 ), then E ( f ) ∪ E (γ g2 ) ⊂ E d ( f ) ∪ E d (γ g2 ). Thus, (ii) can be deduced from (iii). 䊐 Let us consider again the functions f and g. Let the problem be Min( f1 (x)Cα 1 g1 (x), . . . , fq (x)Cα q gq (x)). x ∈D (4) 60 JOTA: VOL. 110, NO. 1, JULY 2001 Let E (α ) and E p (α ) denote the efficient solutions set and the properly efficient set respectively for problem (4). We will now present some relations between these sets and the sets of efficient solutions and properly efficient solutions for problem (1). Theorem 3.2. For every α , γ ∈⺢ q, with α k , γ k ≠ 0 and sign(α k )Gsign (γ k ), kG1, 2, . . . , q, the following relation holds: E (α ) ⊂ E ( f, γ g). Proof. Let x∈E (α ). Let us assume that x∉E ( f, γ g). In this case, there is a solution x* that dominates the solution x, that is, fk (x*)⁄fk (x) and γ k gk (x*)⁄ γ k gk (x), for every k∈{1, 2, . . . , q}, and there exist at least one s∈{1, 2, . . . , q} for which the inequality is strict, that is, fs (x*)Ffs (x) or γ s gs (x*)Fγ s gs (x). From this point onward, since fk (x*)⁄fk (x), γ k gk (x*)⁄ γ k gk (x), sign(α k )Gsign(γ k ), the following inequalities are verified: fk (x*)Cα k gk (x*)⁄fk (x)Cα k gk (x*), for every k∈{1, 2, . . . , q}, (5) fk (x)Cα k gk (x*)⁄fk (x)Cα k gk (x), (6) for every k∈{1, 2, . . . , q}. From (5) and (6), we obtain fk (x*)Cα k gk (x*)⁄fk (x)Cα k gk (x), for every k∈{1, 2, . . . , q}. In particular, for kGs, we have the results below: (a) if fs (x*)Ffs (x), fs (x*)Cα s gs (x*)Ffs (x)Cα s gs (x*), and the following inequality is obtained from (6): fs (x*)Cα s gs (x*)Ffs (x)Cα s gs (x); (b) if γ s gs (x*)Fγ s gs (x), fs (x*)Cα s gs (x*)Ffs (x*)Cα s gs (x), and since fs (x*)⁄fs (x), we obtain fs (x*)Cα s gs (x*)Ffs (x)Cα s gs (x). JOTA: VOL. 110, NO. 1, JULY 2001 61 Therefore, for every k∈{1, 2, . . . , q}, fk (x*)Cα k gk (x*)⁄fk (x)Cα k gk (x), and there is at least a subscript s∈{1, 2, . . . , q} for which the inequality is strict, fs (x*)Cα s gs (x*)Ffs (x)Cα s gs (x), which implies that the solution x* dominates the solution x; therefore, we reach a contradiction with the hypothesis of x being the efficient solution to problem (4). 䊐 A natural question is whether Theorem 3.1 is true for the set of properly efficient solutions. Next, we prove that, given certain conditions, this relationship is preserved for properly efficient solutions. For this purpose, we define problems Pf,γ g(λ , µ) and Pα (ω ), obtained by applying the weighting method to problems (1) and (4) respectively as follows: q (Pf,γ g (γ , µ)) Min λ tf (x)C ∑ µk γ k gk (x), x∈D (Pα (ω )) k G1 q Min ∑ ω k ( fk (x)Cα k gk (x)). x ∈D k G1 We use the results available in the literature about the relationships between the optimal solutions to the weighting problem and the efficient solutions to the multiobjective problem. Some results [see for example Chankong and Haimes (Ref. 13)], applied to problem (1) and its associated weighted problem Pf,γ g (λ , µ), are as follows: If f and (γ 1 g1 , . . . , γ q gq )t are convex functions, D is convex, and x* is a properly efficient solution for the multiobjective problem (1), there exist some weight vectors λ , µ with strictly positive components such that x* is the optimal solution for the weighted problem Pf,γ g (λ , µ). (b) For each vector of weights with strictly positive components, the optimal solution to the weighted problem Pf,γ g (λ , µ) is properly efficient for the multiobjective problem (1). (a) Proposition 3.1. If f and ( γ 1 g1 , . . . , γ q gq )t are convex functions, D is a convex set, and sign(α k )Gsign(γ k ), for every k∈{1, 2, . . . , q}, then E p(α ) ⊂ E p ( f, γ g). 62 JOTA: VOL. 110, NO. 1, JULY 2001 Proof. If f and (γ 1 g1 , . . . , γ q gq )t are convex functions and if D is a convex set, then the sets of properly efficient solutions to problems (1) and (4) are obtained from the associated weighted problems for strictly positive weight vectors. We will prove that any solution to the optimization problem Pα (ω ), with ω H0, is the solution to problem Pf,γ g (λ , µ) for some vector (λ t, µt )tH0. Let x∈E p (α ). Then, given the established hypotheses, there exists a vector ω H0 for which x is the solution for problem Pα (ω ). Let us assume that, for every k∈{1, 2, . . . , q}, α k , γ k ≠ 0. Then, by making λ k Gω k , µ k G ω k α k 兾γ k , λ k , µkH0, since ω H0, we obtain that x is the optimal solution to problem Pf,γ g (λ , µ). For some i∈{1, 2, . . . , q}, if α i Gγ i G0, then the proof would be the same, since in problem (1) the function gi is not involved and since in problem (4) the ith objective would be fi . 䊐 In general, the inverse inclusion does not hold, as it is shown by the following example. Example 3.1. Let us consider the following problem: Max (x, y), x,y s.t. x2Cy2 ⁄1, x, y¤0, with f (x, y)Gx, g(x, y)Gy, γ G1. The set of efficient points for this problem is {(x, y)t ∈⺢2兾x2Cy2 G1, x, yH0} and is represented in Fig. 1. We outline the solution of the problem Max xCα y, x,y s.t. x2Cy2 ⁄1, x, y¤0, with α H0. For each fixed α H0, the optimal solution of the resulting problem is one of the properly efficient solutions to the original bicriterion problem. JOTA: VOL. 110, NO. 1, JULY 2001 63 Fig. 1. Proposition 3.2. If f and (γ 1 g1 , . . . , γ qgq )t are convex functions, then E p ( f, γ g) ⊂ * α ∈Ω E p (α ), with ΩG{α ∈⺢ q兾sign(α k )Gsign(γ k ), kG1, 2, . . . , q}. Proof. As in the previous case, the proof of the proposition is carried out by demonstrating that any solution to the problem Pf,γ g (λ , µ) is a solution to the problem Pα (ω ) for some vector α ∈⺢n, with sign(α k )G sign(γ k ), k∈{1, 2, . . . , q}, and for some ω H0. Let x∈E p( f, γ g). Then, as f and (γ 1 g1 , . . . , γ q gq )t are convex functions, there exist vectors λ , µH0 for which x is the solution to the problem Pf ,γ g (λ , µ). By making ω k Gλ k , α k G µk γ k 兾 ω k , since ω , µH0, we therefore obtain that x is also the solution to the problem Pα (ω ). 䊐 Note that, from Propositions 3.1 and 3.2, if f and ( γ 1 g1 , . . . , γ q gq )t are convex functions and if sign (α k )Gsign(γ k ), for every k∈{1, 2, . . . , q}, the sets of properly efficient solutions to problems (1) and (4) verify the 64 JOTA: VOL. 110, NO. 1, JULY 2001 following properties: (a) Every properly efficient solution to problem (4) is properly efficient for problem (1) (b) Setting γ ∈⺢ q with nonnull components, the set of properly efficient solutions to problem (1) is a subset of the union in α of the sets of properly efficient solutions for problem (4). By combining both results, the following corollary can be stated. Corollary 3.1. If f and (γ 1 g1 , . . . , γ q gq )t are convex functions, then for every γ ∈⺢ q, E p ( f, γ g)G * E p (α ), α ∈Ω with ΩG{α ∈⺢ q 兩sign(α k )Gsign(γ k ), kG1, 2, . . . , q}. 4. Relations between Expected-Value Efficient Solutions, MinimumVariance Efficient Solutions, and Expected-Value Standard-Deviation Efficient Solutions Let us consider again the (SMP) problem and the sets of efficient solutions expected value (E E ), minimum variance (E σ 2), and expected value standard deviation (E Eσ ) associated with the problem. Let E dE , E σd 2 , E dEσ be the sets of weakly efficient solutions associated with problems (E), (σ 2), (Eσ ) respectively. If we consider fk (x)Gz̄k (x), gk (x)Gσ k (x), and if we choose γ k G1, given that, for every k∈{1, 2, . . . , q}, it is verified that σ k : ⺢n → ⺢ +, then the relations between these efficient sets are deduced directly from Theorem 3.1 in Section 3 as follows: (i) (ii) EE ∩ Eσ ⊂ E Eσ . Every solution which is both expected-value efficient and minimum-variance efficient is also expected-value standard-deviation efficient solution. E E ∪ E σ 2 ⊂ E dEσ . Every expected-value solution or minimumvariance efficient solution is an expected-value standard-deviation weakly efficient solution. 2 JOTA: VOL. 110, NO. 1, JULY 2001 (iii) E dE ∪ E σd 65 ⊂ E dEσ . The set of expected-value standard-deviation weakly efficient solutions includes the union of the set of expected-value weakly efficient solutions and the set of minimumvariance weakly efficient solutions. 2 In Section 6, we present an example which illustrates these results. Now, we move on to study the relations between the expected-value standard-deviation efficient solutions and the efficient solutions with probabilities β 1 , β 2 , . . . , β q . 5. Relations between Expected-Value Standard-Deviation Efficient Solutions and Efficient Solutions with Probabilities β 1 , β 2 , . . . , β q Next, we analyze the existing relationships between the expected-value standard-deviation efficient solutions and efficient solutions with probabilities β 1 , β 2 , . . . , β q . Given the stochastic multiobjective programming problem (SMP), we consider its associated problems (Eσ ) and (K(β )). In order to determine the set of efficient solutions Eσ of the (SMP) problem, we need to know only the expected valued and the standard deviation of each objective function of the stochastic problem. However, obtaining the β -efficient solution set for the (SMP) problem is more complex because the distribution functions of the stochastic objectives are involved in this definition of efficient solution, and so it is necessary to specify additional hypotheses about the stochastic objective functions and the probability distributions of the random parameters involved. The studies carried out up to now have focused mainly on linear objective functions (see for example Ref. 14) and linear fractional objective functions (see Ref. 6). As for the type of probability distribution normally used for the stochastic vectors, the multivariate normal distribution is generally considered, or it is assumed that the vector of random parameters depends linearly on a single random variable (hypothesis of simple randomness). In these cases and others, it is possible to obtain the distribution function of the stochastic objective, although in general this is a complex task. We now analyze the existing relationship between the set of expectedvalue standard-deviation properly efficient solutions and the set of properly efficient solutions with probabilities β 1 , β 2 , . . . , β q when the objective functions are linear and the random parameter vector follows a normal distribution or it verifies the hypothesis of simple randomness. Before going any further, we outline the two cases that we are going to analyze. 66 JOTA: VOL. 110, NO. 1, JULY 2001 Case 1. Normal Linear Case. Let us assume that the kth objective function takes the form z̃k (x, c̃)Gc̃ kt x, where c̃k is a random vector multinormal with expected value c̄k and positive-definite variance and covariance matrix Vk . Let us also assume that 0∉D. Under such hypotheses, the expected value of the random variable z̃k (x, c̃)Gc̃ kt x is z̄k (x)Gc̄ kt x, its standard deviation is σ k (x)G1xtVk x, and the distribution function of the random variable calculated in uk is P(c̃ kt x⁄uk )GΦ((ukAc̄ kt x)兾1xt Vk x), where Φ is the standardized normal distribution function. From now on, the probabilistic constraint P(c̃ kt x⁄uk )¤ β k is equivalent to c̄ kt xCΦ−1 ( β k )1xtVk x⁄uk , an inequality that can be expressed as z̄k (x)Cα k σ k (x)⁄uk , with α k GΦ−1 ( β k ). Since the expected value is linear and the function 1xtVk x is convex [see Stancu-Minasian (Ref. 2)], this inequality defines a convex set if β k ¤0.5, since in this case α k GΦ−1 ( β k )¤0. Case 2. Simple Linear Randomness Case. Let us assume that the kth objective function takes the form z̃k (x, c̃)Gc̃ kt x, where c̃k is a random vector, which depends linearly on the random variable t̃k , in such a way that c̃k Gc1kCt̃k c2k . Let trk be the expected value of t̃k ; let υk be its standard deviation, υkFS; and let Fk be it distribution function, which we assume to be strictly increasing. We assume also that, for every x∈D, it is verified that ck2txH0. JOTA: VOL. 110, NO. 1, JULY 2001 67 If these hypotheses are verified, we obtain the following results: (a) The expected value of c̃ kt x is z̄k (x)Gck1t xCtrk ck2t x, and its standard deviation is σ k (x)Gυk ck2t x. (b) The distribution function of the random variable c̃ kt x valued in uk is P(c̃ kt x⁄uk )GFk ((ukAck1t x)兾ck2t x); therefore, P(c̃ kt x⁄uk )¤ β k is equivalent to 2t c k1t xCF −1 k ( β k )ck x⁄uk , an inequality that defines a convex set for every β k , and which can be expressed also as z̄k (x)Cα k σ k (x)⁄uk , rk )兾υk . with α k G(F −1 k ( β k )At Having analyzed these two cases, we see that the probabilistic constraint P(z̃k (x, c̃)⁄uk )¤ β k is equivalent to z̄k (x)Cα k σ k (x)⁄uk . Therefore, the set of β -efficient solutions for the problem (SMP) coincides in both cases with that of the following multiobjective problem: (Kα ) Min(z̄1 (x)Cα 1 σ 1 (x), . . . , z̄q (x)Cα q σ q (x)); x ∈D that is, the β -efficient set is obtained via a problem of q objective functions in which each objective function takes the expected value of the stochastic problem plus its standard deviation weighted by a coefficient that depends on the predetermined or fixed probability. Naturally emerging from this idea is the comparison between the set of efficient solution of the problem (Kα ) and the set of expected-value standard-deviation efficient solutions of the original stochastic multiobjective 68 JOTA: VOL. 110, NO. 1, JULY 2001 problem. If we make fk (x)Gz̄k (x), gk (x)Gσ k (x), γ k G1, from the results obtained in Section 3, we can relate the efficient solutions sets of both problems. Since γ k is the parameter that weights the standard deviation of the stochastic objective, and since in Section 3 the hypothesis sign(γ k )Gsign(α k ) holds, then if we restrict to the analysis γ k G1, in order to maintain the relations obtained in previous results, it will be necessary that α kH0. Let us see what this involves in each case: Case 1. Normal linear case: α k GΦ−1( β k ), and so α kH(⁄) 0, if β kH(⁄) 0.5. rk )兾υk , and Case 2. Linear simple randomness case: α k G(F −1 k ( β k )At so α kH(⁄) 0, if β kH(⁄) Fk (trk ). Therefore, in both cases, the fact that the parameter α k takes a strictly positive value implies that the fixed probability must be high. Therefore, from Theorem 3.2, Propositions 3.1 and 3.2, and Corollary 3.1, we can assert that, if the stochastic objectives of the (SMP) problem fulfill the hypotheses in Case 1 or Case 2, then it is verified that: Given a fixed vector of probabilities β 1 , . . . , β q , such that the associated α k , kG1, 2, . . . , q, are strictly positive, the set of efficient solutions with probabilities β 1 , . . . , Bq is a subset of the set of expected-value standard-deviation efficient solutions, E Kα ⊂ E Eσ , where E Kα will denote the set of efficient solutions for problem (Kα ). (b) Regarding properly efficient solutions, we point out that, given that the functions z̄k (x) and σ k (x) are convex in both cases, the results from Proposition 3.1, Proposition 3.2, and Corollary 3.1 are verified, with which, for each vector of probabilities β 1 , . . . , β q such that the associated α k are strictly positive, it is verified that: The set of properly efficient solutions with probabilities β 1 , . . . , β q is a subset of the set of expected-value standarddeviation efficient solutions, E pKα ⊂ E pEσ ; The union of sets properly β -efficient, corresponding to probabilities such that the associated α k are strictly positive, gives a set that coincides with the set of expected-value standard-deviation properly efficient solutions, *α H0 E pKα GE pEσ . (a) JOTA: VOL. 110, NO. 1, JULY 2001 69 Therefore, the criteria of expected-value standard-deviation efficiency and efficiency in probability are closely related, at least in the cases analyzed. Now, we analyze the results for the cases studied. In both instances, our study enables us to see how the application of the efficiency in probability criteria gives an entire range of solutions, as a function of the fixed probabilities that include some of the expected-value standard-deviation efficiencies and all those corresponding to low probabilities, efficient solutions that, in general, are impossible to obtain by means of the expected-value standard-deviation efficiency criteria. This fact corroborates the idea that the expected-value standard-deviation efficiency is appropriate when the decision maker is risk averse (the hypothesis which is held, among others, in the models of portfolio selection in order to consider this criterion of efficiency as an appropriate one). All this leads to the following question: Is it possible to obtain β efficient solutions via some other criteria? Note that, when we fix a low probability for some stochastic objective in Cases 1 and 2, its standard deviation is weighted negatively; furthermore, the lower the probability, the smaller the corresponding weight is. In such cases, let us consider the following problem: Min(z̄1 (x), . . . , z̄q (x), γ 1 σ 1 (x), . . . , γ q σ q (x)), x ∈D (7) where γ k G1 if the fixed probability is high and γ k G−1 if the fixed probability is low. Then, from Theorem 3.2, we can assert that the set of β efficient solutions is a subset of the set of efficient solutions for problem (7). Regarding the relations for the properly efficient sets shown in Proposition 3.1 and Corollary 3.1, these hold only if low probabilities correspond to stochastic objectives that verify the hypothesis in Case 2, simple randomness, since in this case the function −σ k (x) is linear and thus convex. Finally, following the paper of Caballero, Cerdá, Muñoz, and Rey (Ref. 10), which analyzes the conditions under which the analysis of efficiency with probabilities β 1 , β 2 , . . . , β q and minimum-risk efficiency with aspiration levels of u1 , u2 , . . . , uq of the (SMP) problem are equivalent, it can be asserted that, given that the necessary conditions for reciprocity between efficient solutions with probabilities β 1 , β 2 , . . . , β q and minimum-risk efficiency with aspiration levels u1 , u2 , . . . , uq of the (SMP) problem are verified, the relationship established between the set of efficient solutions with probabilities β 1 , β 2 , . . . , β q and the set of expected-value standard-deviation properly efficient solutions is verified also between the latter and the minimum-risk efficient solutions with aspiration levels u1 , u2 , . . . , uq . In order to illustrate these results, we present an example. 70 JOTA: VOL. 110, NO. 1, JULY 2001 6. Example Let us consider the following stochastic biobjective problem: Min (c̃ 1t x, c̃ 2t x), x s.t. x1Cx2 ¤1, x1C3x2 ⁄10, A2⁄Ax1Cx2 ⁄2, x1 , x2 ¤0, where c̃i Gc 1i Ct̃i c 2i , iG1, 2, with c 11 G(−7, −12)t, c 21 G(6, 5)t, c 12 G(3, A5)t, c 22 G(4, 8)t ; t̃1 follows the normal distribution of expected value 1 and variance 4, t̃1 ∼ N(1, 4), and t̃2 follows the exponential distribution with parameter λ G2. From the data, the expected-value standard-deviation efficiency problem of the above mentioned stochastic problem is Min (−x1A7x2 , 12x1C10x2 , 5x1Ax2 , 2x1C4x2 ), x s.t. x1Cx2 ¤1, x1C3x2 ⁄10, A2⁄Ax1Cx2 ⁄2, x1 , x2 ¤0. Figure 2 shows the set of expected-value standard-deviation efficient solutions for the problem. Points E1 G(1, 3), E2G(0, 2), S1G(0, 1), S2G (1, 0) are the optimal solutions for the expected-value problem of the first and second objective functions, and for the minimum-variance problem of stochastic objectives 1 and 2, respectively. This implies that the segment E1E2 joining E1 and E2 is the set of expected-value efficient points (E E ≡ E1E2) and that the segment S1S2 is the minimum-variance efficient set (E σ 2 ≡ S1S2). The expected-value standard-deviation efficient solutions are points of the segments S1S2, S1E2, E2E1. Note that E E ≡ E dE , Eσ 2 ≡ E σd 2 , E Eσ ≡ E dEσ . As it can be observed, the expected-value standard-deviation efficient set includes the expected-value and minimum-variance efficient sets, which is shown in the results obtained in Section 4. JOTA: VOL. 110, NO. 1, JULY 2001 71 Fig. 2. On the other hand, the problem of the efficiency with probabilities β 1 , β 2 , is Min (−x1A7x2CΦ−1( β 1 )(12x1C10x2 ), x 3x1A5x2CF −1 ( β 2 )(4x1C8x2 )), s.t. x1Cx2 ¤1, x1C3x2 ⁄10, A2⁄Ax1Cx2 ⁄2, x1 , x2 ¤0. The sets of efficient solutions to this problem for different probability vectors is shown in the Table 1, where the column ‘‘efficient set’’ contains the efficient set that is obtained for the probability vector given in the second column. The results in the table show that the set of efficient solutions with probabilities β 1 and β 2 changes according to the fixed probabilities, and this helps illustrate the theoretical results obtained in Section 5. The probability 72 JOTA: VOL. 110, NO. 1, JULY 2001 Table 1. Efficient sets of the problem for different values of the probabilities. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ( β 1 G0.90, β 2 G0.99) ( β 1 G0.75, β 2 G0.99) ( β 1 G0.60, β 2 G0.99) ( β 1 G0.30, β 2 G0.99) ( β 1 G0.90, β 2 G0.80) ( β 1 G0.75, β 2 G0.80) ( β 1 G0.60, β 2 G0.80) ( β 1 G0.30, β 2 G0.80) ( β 1 G0.90, β 2 G0.40) ( β 1 G0.75, β 2 G0.40) ( β 1 G0.60, β 2 G0.40) ( β 1 G0.30, β 2 G0.40) ( β 1 G0.90, β 2 G0.10) ( β 1 G0.75, β 2 G0.10) ( β 1 G0.60, β 2 G0.10) ( β 1 G0.30, β 2 G0.10) Minimum variance efficient set Minimum variance efficient set and segment S1E2 Eσ efficient set Eσ efficient set and segment E1P Solution (x1 G0, x2 G1) Segment S1E2 Expected value efficient set and segment S1E2 Expected value efficient set and segments S1E2 and E1P Segment S1E2 Solution (x1 G0, x2 G2) Expected value efficient set Expected value efficient set and segment E1P Expected value efficient set and segment S1E2 Expected value efficient set Solution (x1 G1, x2 G3) Expected value efficient set and segment E1P vectors for which the standard deviation has an associate positive weight are such that β 1H0.5 and β 2H0.6321205. From this, the following can be outlined: (a) In general terms, we point out that the dimension of the efficiency in probability set varies according to the disparity existing between the fixed probabilities, as is shown in Lines 4, 8, or 13 in Table 1. (b) For probability vectors such that β 1H0.5 and β 2H0.6321205 [which in problem (Kα ) implies that α H0], the solutions obtained are subsets of the expected-value standard-deviation efficient set. For example, for the probability vector (β 1 G0.75, β 2 G0.8), the efficient set obtained is S1E2; but if probability β 1 is lowered to 0.6, keeping the second value the same (β 2 G0.8), the new set includes the previous one and all of the expected-value efficient solutions. Furthermore, in some cases, e.g. for the probability vector ( β 1 G 0.6, β 2 G0.99), the efficient set that gives us the efficiency in probability criterion coincides with the efficient set Eσ . (c) When the fixed or predetermined probabilities are such that β 1 ⁄0.5 or β 2 ⁄0.6321205 (see Line 4 and Lines 7–16 of Table 1), the resulting efficient set can include points which are not part of the efficient set Eσ . In all instances, these points are those of the segment E1P. JOTA: VOL. 110, NO. 1, JULY 2001 73 7. Conclusions In stochastic programming, four basic areas can be distinguished: theory; approximations兾information and bounds; computations and algorithms; and applications (Ref. 15). Our paper belongs to the first area (theory) in a multiobjective context. A question often raised in stochastic programming is whether the theoretical results obtained are applicable when we deal with real-world problems. We think that, in some cases, it is possible to carry out a statistical estimation of the parameters in the model; therefore, we can apply directly the techniques proposed in this paper (for example, the Markowitz model, to solve real problems in finance). In other cases, some specific algorithm or method has to be developed to relate the theory to the applications (for example, Goicoechea, Hansen, and Duckstein present in Ref. 1 the PROTRADE method to solve a real problem in land reclamation and management). Yet in other cases, it is necessary to use other techniques, such as computer simulation, for analyzing the system behavior and optimizing performances. From the results obtained, we can assert that the concepts of efficient solution considered in this work for a single problem of stochastic multiobjective programming are closely related, under certain conditions. The relations established can help to obtain efficient solutions to a problem with the characteristics described here, since these concepts include different statistical characteristics of the stochastic objectives and apparently do not have to have any relationship to each other. Based on our results, it is possible to deal with attaining efficient solutons from a different perspective since, given a particular problem, we are able to see from the established relationships what concept of efficiency is the most appropriate or the one that best fits the preferences of the decision maker. Therefore, this study helps to choose between different efficiency criteria for solving stochastic multiple-objective problems. In this sense, the richness of the efficiency in probability criterion may be highlighted, since by varying the fixed probability for each stochastic objective, different efficient sets are obtained. Also, the fact that the decision maker has to fix a probability for each stochastic objective is more of an advantage than a hindrance, since in a certain way, it determines the risk that he or she is willing to take in each of the stochastic objectives. 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