Download Efficient Solution Concepts and Their Relations in Stochastic

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
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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
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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 ).
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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.
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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
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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).
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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)
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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).
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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).
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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.
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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
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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.
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(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.
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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.
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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
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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)
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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.
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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.
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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
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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.
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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.
Finally, we note that the preliminary results obtained in Section 3 are
applicable to any deterministic multiobjective problem.
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