The last three decades ha ve witnessed development of optimization under fuzziness and randomness also called Fuzzy Stochastic Optimization. The main objective of this new field is the need for basing many human decisions on information which is both fuzzily imprecise and probabilistically uncertain. Consistency indexes providing a union nexus between possibilities and probabilities of uncertain events exist in the literature. Nevertheless, there are no reliable transformations between them. This calls for new paradigms for coping with mathematical models involving both fuzziness and randomness. Fuzzy Stochastic Optimization (FSO) is an attempt to fulfill this need. In this paper, we present a panoramic view of Fuzzy Stochastic Optimization emphasizing the methodological aspects. The merits of existing methods are also briefly discussed along with some related theoretical aspects.
Mathematical programming is a very useful tool in the hands of a decision maker. As a matter of fact, many decision problems including allocation of resources, transportation, affectation, production, etc. may be cost into an optimization framework. An interested reader may consult the paper by Dan [
According to Zadeh [
P ( A ) = E ( μ A ) = ∫ Ω μ A ( x ) d x .
For details on probability of a fuzzy event, the interested reader is invited to consult [
Given a set X and a probability space Ω , a probabilistic set A on X is a fuzzy set of X × Ω , whose membership function:
μ A : X × Ω → [ 0 , 1 ] ( x , ω ) ↣ μ A ( x , ω )
is measurable.
Intersections and unions of fuzzy probability sets are defined using T-norms or Conorms respectively. We refer the reader for properties of probabilistic sets to Hirota [
To put things in context, consider a continuous random variable X with density function f ( x , θ ) , where θ is a parameter describing the density function. If θ can be generated as a fuzzy number θ ˜ , then X has density f ( x , θ ˜ ) and the probability of the event: “ X is between c and d” is a fuzzy set whose α -cuts are defined as follows:
[ P ˜ ( c ≤ X ≤ d ) ] α = { ∫ c d f ( x , θ ) d x | θ ∈ θ ˜ α ; ∫ − ∞ ∞ f ( x , θ ) d x = 1 } .
The first two moments of X are defined, through their α -cuts as follows.
m ˜ X α ( θ ) = { ∫ − ∞ ∞ x f ( x , θ ) d x | θ ∈ θ ˜ α ; ∫ − ∞ ∞ f ( x , θ ) d x = 1 } ,
( σ ˜ X 2 ) α ( θ ) = { ∫ − ∞ ∞ ( x − m X ( θ ) ) 2 f ( x , θ ) d x | θ ∈ θ ˜ α ; m X ( θ ) ∈ m ˜ X α ( θ ) ; ∫ − ∞ ∞ f ( x , θ ) d x = 1 } .
Random variables X i , i = 1 , 2 , ⋯ , n with fuzzy parameters and having joint density function f ( x 1 , ⋯ , x n , θ ˜ ) and marginal density function f i ( x i , θ ˜ ) are said to be independent if, for α ∈ ( 0,1 ] and for all θ ∈ θ ˜ α ,
f ( x 1 , ⋯ , x n , θ ) = ∏ i = 1 n f i ( x i , θ ) . (1)
For readers interested in more details on uncertain probabilities, we recommend the authoritative scientific papers by Buckley and Eslami [
A fuzzy random variable (frv) on a probability space ( Ω , F , P ) is a fuzzy-valued function:
X α : Ω → F 0 ( ℝ ) ω ↣ X ω
such that for every Borel set A of ℝ and for every α ∈ ( 0,1 ] , ( X α ) − 1 ( A ) ∈ F . Here F 0 ( ℝ ) and X α stand for the set of fuzzy numbers and the set-valued function:
X α : Ω → 2 ( ℝ ) ω ↣ X ω α = { x ∈ ℝ | X ω ( x ) ≥ α }
respectively.
Zadeh’s decomposition principle extends quite naturally to frvs. Moreover X α ( α ∈ ( 0,1 ] ) are random intervals and frvs possesses the Radon-Nikodym property. A considerable body of literature has grown out of the concept of frv, in a wide range of fields (e.g. [
A random fuzzy variable is a map from a possibilistic set to a probabilistic one, verifying some measurability conditions [
Other approaches to blending epistemic and aleatory forms of uncertainty include imprecise probabilities in the style of Peter Walley [
Fuzzy Stochastic Optimization (FSO) models are numerous and varied (e.g. [
In this paper, we consider 5 categories of FSO models. This classification is far form being exhaustive. It just reflects the main models encountered in the literature. The classification has been done according to the following criteria: the way fuzziness and randomness enter into the problem, the mathematical formalism used to account of involved hybrid uncertainty (see above), the structure of the optimization problem, the number of objective functions, the number of stages and the numbers of levels. The first class (C1) contains mathematical programming problems under fuzziness and randomness. Models on this class have one objective function and a limited number of constraints. Moreover the decision is taken here and now. For examples of models in category C1, we refer the reader to papers [
Examples of models in Categories C2 and C3 may be found in references [
It is also worth mentioning that problems in classes C 1 , ⋯ , C 4 may be extended to the multi objective case, leading to C 1 l , ⋯ , C 4 l (l denoting the number of objectives considered). The reader is referred to [
FSO models have the following common features:
1) Fuzziness and randomness are, in some way, involved in these models.
2) There is no agreed definition of optimum.
3) There are different perspectives about the problem situation.
4) The fuzzy stochastic optimization problem should be approximated by a deterministic one.
5) The closedness of the appropriate deterministic problem to the original one as well as the tractability of the former is desirable traits.
Early attempts to solve decision problems involving fuzziness and randomness dated back to the 1980’s [
As suitable models are central to conceptual formulation of FSO problems, so suitable mathematical methods are central to their quantitative treatment. The presence of both possibilistic and probabilistic information within an Optimization framework is a harbinger of computational nightmares if one were to approach the problem without any simplifications. The game is to come up with approximations which tradeoff faithful representation of reality for computational tractability. Although the trajectory of FSO follows a situation-specific approach that do not lead to a generalization, we introduce below a general scheme from which most of existing FSO methods boil down. This algorithmic framework unifies a variety of seemingly different methods that have been derived from disparate approaches.
Before proceeding further, we need the following notations:
DO : Transformation that converts the original problem into a form suitable for treatment.
If the problem at hand is ready for treatment, then DO = I (identity transformation).
DF : Deffuzification.
That is a process that associates to a given fuzzy system a deterministic counterpart.
DR : Derandomization.
That is a procedure that immunizes from randomness.
DFR : Transformation that deals simultaneously with fuzziness and randomness.
DL(P) : stands for the problem resulting in applying DL to P, where L may be O, F, R or FR.
We are now in a position to depict a general scheme for solving a FSO problem.
Step 0: make a choice of DO, DF, DR, DFR.
Step 1: Apply T 0 = D O to the original problem (P) to obtain T 0 ( P ) .
Step 2: Choose T 1 ∈ { D F , D R , D F R } and apply T 1 to T 0 ( P ) to obtain T 1 ( T 0 ( P ) ) .
Step 3: Choose T 2 as follows:
- If T 1 = D F R then T 2 = I .
- Otherwise take T 2 ∈ { D F , D R } − T 1 .
- Apply T 2 to T 1 ( T 0 ( P ) ) to get T 2 ( T 1 ( T 0 ( P ) ) ) .
Step 4: Solve the resulting deterministic problem T 2 ( T 1 ( T 0 ( P ) ) ) .
The choice in Step 0 can be fulfilled in many ways not all of them being equally satisfactory. This choice poses a double challenge. First, the transformations retained should not lead to a bad caricature of the reality. Secondly, they should be such that the resulting deterministic program could be computationally tractable. A FSO method becomes attractive if these two challenges are successfully overcome.
| Type of constraints | Transformation | Deterministic counter part |
|---|---|---|
| a ˜ x ≤ b ˜ | D F 1 | D F 1 ( a ˜ x ≤ b ˜ ) = { x | a ˜ α x ≤ b ˜ α ; ∀ α ∈ ( 0,1 ] } |
| a ˜ x ≤ b ˜ | D F 2 | D F 2 ( a ˜ x ≤ b ˜ ) = { x | a ˜ α i x ≤ b ˜ α i ; α i ∈ ( 0,1 ] ; i = 1, ⋯ , p } |
| a ˜ x ≤ b ˜ | D F 3 | D F 3 ( a ˜ x ≤ b ˜ ) = { x | T ( a ˜ x ≤ b ˜ ) ≥ β } where T ∈ { P o s s , N e c } and β is a prescribed threshold. Poss and Nec stand for possibility and necessity respectively. |
| a ˜ x ≤ b ˜ | D F 4 | D F 4 ( a ˜ x ≤ b ˜ ) = { x | P o s s ( a ˜ x ≤ b ˜ ) ≥ α and N e c ( a ˜ x ≤ b ˜ ) ≥ δ } α and |
| a ¯ x ≤ b ¯ | D R 1 | D R 1 ( a ¯ x ≤ b ¯ ) = { x | ( E a ¯ ) x ≤ E ( b ¯ ) } where E stands for the expectation operator. |
| a ¯ x ≤ b ¯ | D R 2 | D R 2 ( a ¯ x ≤ b ¯ ) = { x | P r o b ( a ¯ x ≤ b ¯ ) ≥ β } where Prob is the probability measure and β a fixed threshold. |
For the sake of space, we briefly discuss in the section below, section (5.2.2). Only some approaches for models of different categories.
(a) Flexible programming with random data
Consider the following fuzzy stochastic model:
( M 1 ) { m i ˜ n f ( x , r ˜ ) g i ( x , s ˜ i ) ≤ ˜ b i ; i = 1, ⋯ , m x ∈ X ⊂ ℝ n
where r ˜ , s ˜ i are random variables on ( Ω , F , P ) ; f , g i ( i = 1 , ⋯ , m ) are functions of ℝ n × Ω . “ m i ˜ n ” and “ ≤ ˜ ” indicate that the minimization and the inequality are not strict imperatives.
There are two main approaches for solving Problem (M1) in the literature. The symmetrical approach boils down from the general scheme depicted in section (5.2.1) by using the following transformations.
T 0 ( M 1 ) = ( M ′ 1 ) where ( M ′ 1 ) is the following problem. Find x ∈ X such that:
g i ( x , s ¯ i ) ≤ ˜ b ¯ i ; i = 0 , 1 , ⋯ , m .
Here g 0 = f , s ¯ 0 = r ¯ and b ¯ 0 plays the role of a fixed benchmark for the value of the objective function.
T 1 ( M ′ 1 ) = ( M ″ 1 ) , where ( M ″ 1 ) is as follows. Find
arg max x ∈ X ( μ D ( x , s ¯ ) )
where
μ D ( x , s ¯ ) = ∧ i = 0 m μ i ( x , s ¯ i )
and “ ∧ ” stands for an operator used to translate the semantic meaning of the “and” connective.
It is worth noticing that μ i ( i = 0 , 1 , ⋯ , m ) are memberships functions of probabilistic sets representing the constraints:
g i ( x , s ¯ i ) ≤ ˜ b ¯ i ; i = 0 , 1 , ⋯ , m .
T 1 ( M ″ 1 ) = ( M ‴ 1 ) where ( M ‴ 1 ) is obtained from ( M ″ 1 ) by immunizing from randomness through the expectation operator. That is ( M 1 ) ′ ′ ′ is the following deterministic optimization problem.
max x ∈ X E ( μ D ( x , s ¯ ) ) .
Solving this optimization problem yields a satisfying solution of Problem (M1). The term symmetrical refers to the fact that the objective function and the constraints are considered as equivalent concepts.
To solve problem ( M 1 ) ′ ′ ′ , we need an analytical expression of the distribution of μ D ( x , s ¯ ) denoted by F μ D ( x , s ¯ ) . The following result, the proof of which may be found in [
Theorem 1
(1) If μ D ( x , ω ) = min i = 0 , 1 , ⋯ , m μ i ( x , ω ) then
F μ D ( x , ω ) ( z ) = ∑ i = 0 m F μ i ( x , ω ) ( z ) − ∑ j < k F μ j ( x , ω ) μ k ( x , ω ) ( z , z ) + ∑ i < j < k F μ i ( x , ω ) μ j ( x , ω ) μ k ( x , ω ) ( z , z , z ) + ⋯ + ( − 1 ) m − 2 F μ 0 ( x , ω ) μ 1 ( x , ω ) ⋯ μ m ( x , ω ) ( z , ⋯ , z ) .
(2) If μ D ( x , ω ) = γ min i μ i ( x , ω ) + ( 1 − γ ) min ( 1 , ∑ i μ i ( x , ω ) ) , where γ stands for a coefficient of compensation ranging between 0 and 1, then
F μ D ( x , ω ) ( z ) = L − 1 ( g 1 ( s ) ⋅ g 2 ( s ) ) ,
where
g 1 ( s ) = ∫ 0 ∞ e − ( s γ ) Z F min i μ i ( x , ω ) ( z ) d z ,
g 2 ( s ) = ∫ 0 ∞ e − ( s 1 − γ ) z F min ( 1 , ∑ i μ i ( x , ω ) ) ( z ) d z
and L − 1 stands for the inverse of the Laplace transform.
For a perspective on the asymmetrical approaches for solving Problem (M1). We refer the reader to [
(b) Mathematical programming with fuzzy random coefficients
Advances on elucidating properties of fuzzy random variables have triggered considerable work on how to deal with mathematical programs with fuzzy random (fr) or random fuzzy (rf) coefficients. It is worth mentioning that the general scheme described in section (5.2.1) applies here as well. Typical, but by no means unique, transformations are used. To provide a taste of what is done on optimization with fuzzy random data, we consider the following Program (M2). We deliberately restrict ourselves to the linear case so that basic ideas are illustrated in a relatively simple context. For non linear programs, we refer the reader to [
Consider the following fuzzy stochastic program:
( M 2 ) { max c x ∑ a ˜ i j x j ≤ b ˜ i ; i = 1 , ⋯ , m x j ≥ 0
Let
H α ( ω ) = ( A ˜ α ( ω ) , b ˜ α ( ω ) ) , H α + ( ω ) = ( A ¯ α ( ω ) , b ¯ α ( ω ) ) ,
and
H α − ( ω ) = ( A _ α ( ω ) , b _ α ( ω ) ) ,
where
A ˜ α ( ω ) , A ¯ α ( ω ) , A _ α ( ω )
stand for matrices
( a ˜ i j ω α ) i , j , ( a ¯ i j ω α ) i , j , ( a _ i j ω α ) i , j
respectively and a ˜ i j ω α denotes the α -level set of a ˜ i j ω , while a ¯ i j ω α , a _ i j ω α stand for the upper and lower endpoints of a ˜ i j ω α respectively. Similarly, b ˜ α ( ω ) , b ¯ α ( ω ) and b _ α ( ω ) stands for vectors:
( b ˜ 1 ω α ⋮ b ˜ m ω α ) , ( b ¯ 1 ω α ⋮ b ¯ m ω α ) , ( b _ 1 ω α ⋮ b _ m ω α )
respectively. Moreover W denotes the optimum of Problem ( M 2 ) and W α ∗ stands for the optimum value of ( M 2 α ∗ ) , where ∗ ∈ { − , + } and ( M 2 α − ) and ( M 2 α + ) are the following programs respectively:
( M 2 α − ) { max c x A _ α ( ω ) x ≤ b _ α ( ω ) x ≥ 0
( M 2 α + ) { max c x A ˜ α ( ω ) x ≤ b ˜ α ( ω ) x ≥ 0
g i α r is the penalty for unit of discrepancy between ∑ a ¯ i j ω α r and b _ i ω α r . β stands for a prescribed threshold fixed by the Decision Maker. f H α ∗ denotes the density function of H α ∗ ( ∗ ∈ { − , + } ) and ( V α ∗ ) i is the ith decision region of ( M 2 α ∗ ) ( ∗ ∈ { − , + } ) .
All transformations used in this table are well explained in
It is well known that constraints on probability induce non convexities which preclude application of powerful convex optimization techniques. So the viability of approach 1 in
| Transformation | Resulting deterministic problem ( P ′ ) | Approach | references | |
|---|---|---|---|---|
| T 1 | T 2 | |||
| D F 2 | D R 2 | max c x P r o b [ ∑ j = 1 n a ¯ i j ω α r x j ≤ b _ i ω α r ] ≥ β , r = 1 , ⋯ , p ; i = 1 , ⋯ , m ; x j ≥ 0 , j = 1 , ⋯ , n | 1 | [ |
| D F 4 | D R 5 | Find the fuzzy probability distribution of W and its expectation given respectively by: F ˜ W ( x ) = ∪ α ∈ ( 0,1 ] α [ F H α + ( x ) , F H α − ( x ) ] E ˜ W ( x ) = ∪ α ∈ ( 0,1 ] α [ E ( H α − ) , E ( H α + ) ] | 2 | [ |
| D F 2 | D R 3 | max [ c x − E ( ∑ r = 1 p ∑ i = 1 m g i α r y i α r ( ω ) ) ] y i α r ( ω ) = ∑ j = 1 n a ¯ i j ω α r x j + b _ i ω α r | 3 | [ |
| D F 4 | D R 4 | Use the random simplex method to find x α solution of max c x a _ α ( ω ) x ≤ b _ α ( ω ) ; a ¯ α ( ω ) x ≤ b ¯ α α ∈ ( 0,1 ] and construct x ∗ = ∪ α ∈ ( 0 , 1 ] α x α | 4 | [ |
on probability on a feasible convex set. Moreover, approach 2 relies on the following result.
Theorem 2. Consider the Problem (P) and all related notation below.
( P ) { max c x , ∑ j = 1 n a i j ∗ x j ≤ b i ∗ , i = 1 , ⋯ , m , x j ≥ 0 , j = 1 , ⋯ , n ,
where c , x ∈ ℝ n and ∗ means that a i j and b i are either fuzzy quantities or random variables, “ ∗ ” takes “+” if the data is random and it takes “−” if the data is fuzzy; then the distribution of H α ∗ and its expectation are respectively:
F H α ∗ ( x ) = ∑ i = 1 R ∫ { H ∗ ( ω ) | H α ∗ ( ω ) ∈ ( V α ∗ ) i and W α ∗ < x } f H α ∗ ( s ) d s
E ( H α ∗ ) = ∑ i = 1 R ∫ ( V α ∗ ) i W α ∗ f H α ∗ ( s ) d s .
For the proof of this result, we invite the reader to consult [
Important part of research in this realm includes the generalization of existing ideas by using other transformations and the driving up to the performance analysis in the side of effectiveness rather than sticking on computational efficiency. Something that should also be pursued with focused attention is the search for solutions of fuzzy stochastic nature, in a way to reflect the fuzzy random structure of the problem.
The case where one has both fuzzy and random data in a same optimization problem, can be converted into the case one has fuzzy random data, by considering involved fuzzy and random quantities as degenerate fuzzy random variables (e.g. [
In addition to shedding light on analysis of Optimization problems under hybrid uncertainty, the proposed algorithm may be the only one resort for solving optimization problems under randomness and fuzziness, where random fuzzy variable is the only one format available for data. It is our belief that exploring the duality between a mathematical program with fuzzy random data and its counterpart with related random fuzzy parameters, will help fostering a more deep understanding of characteristics of Fuzzy Stochastic Optimization problems.
(c) Mathematical program with random variables having fuzzy parameters
We now move to solution procedures for optimization problems, where another specific tool, namely uncertain probability, is used for modeling hybrid imprecision in the data of the problem. The mathematical program on which ideas will be illustrated is as follows.
( M 3 ) { min c x A i ∗ x ≤ b i ∗ ; i = 1 , ⋯ , m x ∈ X = { x ∈ ℝ n | x ≥ 0 }
where “ ∗ ” means the daturn is random with some fuzzy parameters. This model has many applications in situations where experts who provide data of the optimization problem feel more comfortable in coupling their vague perception with hard statistical data. To this end, they may prefer to represent these data in the form of random variables with vague parameters. As an example, consider a portfolio selection problem where, due to stock expects’ judgments and investors’ different opinions, the security returns are modeled as random variables with fuzzy parameters. Model (M3) has not received much attention in the literature. The art here is to find an approximation scheme simple enough to be computable, but not so simple that useful detail is lost. This is done in [
This means one has to consider the following problem.
( M ′ 3 ) { min c x P ˜ [ ∑ j = 1 n a i j ∗ x ≤ b i ∗ ] ≥ δ ˜ i ; i = 1 , ⋯ , m x ∈ X
where P ˜ stands for uncertain probability and δ ˜ i , i = 1 , ⋯ , m are fuzzy thresholds fixed by the Decision maker. The question that immediately arises is that of converting Problem ( M ′ 3 ) into a standard mathematical program.
Fascinating results in connection with this issue have been obtained in the literature. For the sake of space, we only give below one of these results. To this end, we need the following lemma.
Lemma 1. Assume a i j ∗ ( j = 1 , ⋯ , n ) and b i ∗ ; i = 1 , ⋯ , m are independent and normally distributed random variables with fuzzy parameters. Put
ξ i ∗ ( x ) = ∑ j = 1 n ( a i j ∗ x j − b i ∗ )
Then ξ i ∗ ( x ) is also normally distributed with fuzzy mean m ˜ ξ i ( x ) and fuzzy variance σ ˜ ξ i 2 ( x ) .
Let now the fuzzy numbers δ ˜ i ; i = 1 , ⋯ , m be threshold fixed by the Decision maker. As m ˜ ξ i ( x ) , σ ˜ ξ i 2 ( x ) and δ ˜ i are fuzzy numbers, their α -cuts are intervals denoted respectively as follows.
m ˜ ξ i α ( x ) = [ m ξ i α L ( x ) , m ξ i α U ( x ) ]
σ ˜ ξ i 2 ( x ) = [ σ ξ i 2 α L ( x ) , σ ξ i 2 α U ( x ) ]
δ ˜ = [ δ i α L , δ i α U ] .
Theorem 3. Under assumption of Lemma (1), Problem ( M ′ 3 ) is equivalent to the following mathematical program:
( M ″ 3 ) { min c x ϕ ( − m ξ i α U ( x ) σ ξ i 2 α U ( x ) ) ≥ δ i α U ; i = 1 , ⋯ , m x ∈ X ; α ∈ ( 0 , 1 ]
where ϕ stands for the density function of the standard normal distribution.
It is worth noting that Problem ( M ″ 3 ) has infinitely many constraints and it can be handled by semi-infinite mathematical programming techniques [
One key ingredient for dealing with models in the category C4 is the lattice isomorphism between the set of fuzzy random sets and the family of their α -level sets [
Here is an example of a fuzzy stochastic robust programming model.
( P R ) { max c x A _ 1 x 1 + A _ 2 x 2 + ⋯ + A _ n x n ⊆ B _ x 1 ≥ 0, x 2 ≥ 0, ⋯ , x n ≥ 0
where c , x ∈ ℝ n and A _ i , B _ are fuzzy sets of ℝ n . Components of A _ denoted by a _ i j are fuzzy numbers, while components of B denoted by b _ i are convex fuzzy sets of ℝ having random characteristics. More precisely, we assume like in [
( P ′ R ) { max c x A _ 1 α x 1 + ⋯ + A _ n α x n ⊆ B α , ∀ α ∈ ( 0,1 ] x 1 ≥ 0, x 2 ≥ 0, ⋯ , x n ≥ 0
To tackle randomness surrounding B _ , we resort to chance constrained programming approach. This approach requires that the constraints need to be satisfied with some higher probability. For instance, for a stochastic constraint:
D ( ω ) x ≤ e ( ω )
One may require that
P r [ { ω | D ( ω ) x ≤ e ( ω ) } ] ≤ 1 − l (2)
where l is a fixed level of probability. Constraint (2) can be merely written:
D ( ω ) x ≤ e ( ω ) ( l )
where e ( ω ) ( l ) = F − L ( l ) ⋅ F being the cumulative distribution of e. For details on Chance Constrained programming, we refer the reader to [
( P ″ R ) { max c x ∑ j = 1 n a _ i j α x j ≥ b _ i α ( l ) ; i = 1 , ⋯ , m ; α ∈ ( 0 , 1 ] ∑ j = 1 n a ¯ i j α x j ≤ b ¯ i α ( l ) ; i = 1 , ⋯ , m ; α ∈ ( 0 , 1 ] x j ≥ 0 ; j = 1 , ⋯ , n
This resulting program is a semi-infinite optimization problem and complex optimization techniques are required to solve it. Nevertheless, if the image of the membership function of b _ i denoted by ( I m μ b ¯ i ) is finite for every i, say I m μ b ¯ i = { α 1 i , ⋯ , α k i } then (see [
( P ‴ R ) { max c x ∑ j = 1 n a _ i j α s i x j ≥ b _ i α s i ( l ) ; i = 1 , ⋯ , m ; s i = 1 , ⋯ , k i ∑ j = 1 n a ¯ i j α s i x j ≤ b ¯ i α s i ( l ) ; i = 1 , ⋯ , m ; s i = 1 , ⋯ , k i x j ≥ 0 ; j = 1 , ⋯ , n
Problem ( P ‴ R ) is a standard linear program, that can be solved by existing linear programming software.
To illustrate how a Fuzzy Stochastic dynamic model can be solved, we briefly discuss a multistage decision process, involving randomness and fuzziness. The state space of the process at hand is
X = { x 1 , ⋯ , x k , x k + 1 , ⋯ , x n } .
We assume that the termination set, denoted by T, consists of states x k + 1 , ⋯ , x n and we put T ¯ = X / T . We also assume that the constraints of the process are defined through state-dependent fuzzy sets C ( x k ) over the control space U. The membership function of C ( x k ) is denoted by μ C ( μ t | x t ) . The conditional probabilities of a state transition under a control μ t are given as follows.
p ( x t + 1 | x t , μ t ) = { 1 if x t + 1 = x t ∈ T 0 if x t + 1 = x t ∈ T ¯ 0 if x t ∈ T , x t + 1 ≠ x t
Suppose now that the sequent inputs u 0 , u 1 , ⋯ , μ t , ⋯ are determined by a stationary policy function Π : T ¯ → U . For a policy Π , in a given state x i ∈ T ¯ , the decision D Π ( x t ) with respect to Π is the confluence of the constraint in the transition from x t to x t + 1 , C Π ( x i ) and the next decision D Π ( x t + 1 ) with respect to Π . If x t + 1 ∈ T , we set D Π ( x t + 1 ) = G ( x t + 1 ) for any Π .
Given x t the state x t + 1 is a random variable characterized by the conditional probability function p ( x t + 1 | x t , u t ) . Thus
D Π ( x t ) = C Π ( x t ) ∩ E D Π ( x t + 1 )
or
u D ( x t | Π ) = u C ( Π ( x t ) | x t ) ∩ E u D ( x t + 1 | Π )
where
E u D ( x t + 1 | Π ) = ∑ x t + 1 ∈ T ¯ p ( x t + 1 | x t , Π ( x t ) ) ⋅ u D ( x t + 1 | Π ) + ∑ x t + 1 ∈ T p ( x t + 1 | x t , Π ( x t ) ) ⋅ u G ( x t + 1 )
and hence,
u D ( x t | Π ) = u C ( Π ( x t ) ) ∧ [ ∑ x t + 1 ∈ T ¯ p ( x t + 1 | x t , Π ( x t ) ) ⋅ u D ( x t + 1 | Π ) + ∑ x t + 1 ∈ T p ( x t + 1 | x t , Π ( x t ) ) ⋅ u G ( x t + 1 ) ] . (3)
Given a policy Π = ( Π ( x 1 ) , ⋯ , Π ( x k ) ) we consider the following vectors:
u C ( Π ) = ( u C ( Π ( x 1 ) | x 1 ) , ⋯ , u C ( Π ( x k ) ) x k ) ( constraint vector )
u D ( Π ) = ( u D ( x 1 | Π ) , ⋯ , u D ( x k | Π ) ) ( control vector )
u G = ( u G ( x k + 1 ) , ⋯ , u G ( x n ) ) ( goal vector )
Let’ also introduced the following transition matrices:
P T ¯ ( Π ) = ( p i j ( Π ) ) i , j = 1 , ⋯ , k
P T ( Π ) = ( p i j ( Π ) ) j = k + 1 , ⋯ , n i = 1 , ⋯ , k
with these notations, the system of equations in (3) can be put in a more compact form.
u D ( Π ) = u C ( Π ) ∧ [ P T ¯ ( Π ) ⋅ u D ( Π ) + P T ( u ) ⋅ u G ] (4)
Π is said to be proper, if there is a natural number K such that ‖ p T ¯ K ‖ < 1 , 1 ≤ K ≤ k where k is the number of states in T ¯ . The set of proper policies is denoted by Π p . Π p is a finite set. So we may write Π p = { Π 1 , ⋯ , Π r } , where r is finite. An optimal policy denoted by Π o p t is the one that maximizes u D ( Π ) . It is shown that under mild assumptions, the termination set T is reachable from any initial state in T ¯ . Moreover, an optimal policy Π o p t ∈ Π p can be obtained from the solution of the following fixed point problem:
V ( ω ) = ∨ ν = 1 r { C ν ∧ [ A ν ω + b ν ] }
where ω , A ν , b ν and c ν stand for u D ( Π ) , P T ¯ ( Π ν ) , P T ( Π ν ) . u G and u C ( Π ν ) respectively.
The literature on multiperiod programming under fuzziness and randomness is rather limited. This could be due to the lack of tractable methodologies. The optimality principle, so profusely used in deterministic dynamic programming, does not lend itself better in a fuzzy stochastic environment. Nevertheless attempt to integrate fuzziness and randomness in a dynamic programming framework is found in [
The prevailing research direction in the Fuzzy Stochastic dynamic programming framework is the improvement of this Neural network approach in a way to address drawbacks of existing models like the impossibility of considering simultaneously related fuzzy and stochastic characteristics.
Hierarchical decision making process is extremely practical to such decentralized systems as agriculture, government policy, economic systems, finance, etc. For more realism on their multilevel decision models, researches have investigated ways for incorporating fuzziness into these settings. Related solution concepts like Stackelberg strategy have been elaborated and interesting results have been obtained. Powerful algorithms for singling out desired solutions have also been pushed forward. These approaches are at the root of extensions to Fuzzy stochastic multilevel methodologies. In a method that integrate fuzzy random simulation, neural network training and Genetic algorithm has been developed to find a satisfying solution of a fuzzy random multilevel program. Another interesting approach for the two-level case is found. Here essential ingredients used are fuzzy random versions of the Nash and the Stackelberg-Nash equilibria. The former is used for the lower level problem, while the latter is used for the overall problem. These approaches give rise to a number of challenging issues including effectiveness and convergence of proposed methods. Currently researchers are busy pondering these thorny issues.
In a way to illustrate how a multilevel optimization problem under randomness and fuzziness can be solved, we briefly discuss below an approach proposed by Sakawa, Nishizaki and Katagiri [
To illustrate how algorithms for dealing with Fuzzy stochastic multilevel programs work, consider the following optimization problem:
( P M L ) { Minimize ( C ¯ ˜ 11 x 1 + C ¯ ˜ 12 x 2 ) for D M 1 Minimize ( C ¯ ˜ 21 x 1 + C ¯ ˜ 22 x 2 ) for D M 2 subject to : A x ≤ b ; x ≥ 0,
where D M i stands for Decision maker i, C ¯ ˜ l j , l = 1 , 2 ; j = 1 , 2 are fuzzy random variables defined on a probability space ( Ω , F , P ) and whose realized values C ¯ ˜ l j ( ω ) , l = 1 , 2 ; j = 1 , 2 are fuzzy numbers with membership functions of the form:
u C ¯ ˜ l j ( ω ) ( t ) = { L ( d l j ( ω ) − t β l j ) if t ≤ d l j ( ω ) R ( t − d l j ( ω ) δ l j ) if t > d l j ( ω )
where L and R are respectively non decreasing and non increasing functions of ℝ such that L ( 0 ) = R ( 0 ) = 1 and d l j ( ω ) , β l j ( ω ) and δ l j ( ω ) are respectively the center value, the left spread value and the right spread value of C ¯ ˜ l j ( ω ) . Using Zadeh extension principle; we have that:
u C ¯ ˜ l j ( ω ) x ( ν ) = { L ( d l ( ω ) x − ν β l x ) if ν ≤ d l ( ω ) x R ( ν − d l ( ω ) x δ l x ) if ν > d l ( ω ) x
where d l ( ω ) = ( d l 1 ( ω ) , d l 2 ( ω ) ) , β l = ( β l 1 , β l 2 ) and δ l = ( δ l 1 , δ l 2 ) ; l = 1 , 2 . Suppose that the decision makers are happy in minimizing their objective functions under the condition that all coefficients C ¯ ˜ l j ( ω ) , l = 1 , 2 ; j = 1 , 2 belongs to α -level sets of corresponding fuzzy random variables, for a fixed α . Then Problem ( P M L ) reads:
( P ′ M L ) { Minimize C ¯ 1 x for D M 1 Minimize C ¯ 2 x for D M 2 subject to : x ∈ X = { x ∈ ℝ 2 | A x ≤ b ; x ≥ 0 } C ¯ 1 j ∈ C ¯ ˜ 1 j α , C ¯ ˜ 2 j ∈ C ¯ ˜ 2 j α ; j = 1 , 2.
To handle imprecision inherent to objective functions, we consider fuzzy goals, say the objective value should be substantially less than some given value. These goals are elicited by D M 1 and D M 2 . Assume the membership functions of these goals are μ 1 and μ 2 for objective 1 and objective 2 respectively. It is worth mentioning that μ l ( C ¯ ˜ l ( ω ) x ) , l = 1 , 2 vary randomly. A deterministic counterpart of Problem ( P ′ M L ) may be obtained by adopting the following strategy, closely related to chance constrained programming. The strategy is to require that the probability that the degrees of satisfaction of fuzzy goals μ l ( C ¯ ˜ l ( ω ) x ) , l = 1 , 2 be greater to some optimal values ( max h 1 , max h 2 ) be not less than some fixed permissible levels ( θ 1 , θ 2 ). With these requirements, Problem ( P ′ M L ) can be written:
( P ″ M L ) { Maximize h 1 for D M 1 Maximize h 2 for D M 2 subject to : P ( { ω | μ 1 ( C ¯ ˜ 1 ( ω ) x ) ≥ h 1 } ) ≥ θ 1 P ( { ω | μ 2 ( C ¯ ˜ 2 ( ω ) x ) ≥ h 2 } ) ≥ θ 2 x ∈ X ; C l j ∈ C ¯ ˜ l j α l ; l = 1 , 2 ; j = 1 , 2.
where h l , l = 1 , 2 are target values and θ l , l = 1 , 2 are permissible probability levels, specified by the decision makers. Before discussing how to find a satisfying solution of the original multilevel program using the auxiliary problem ( P ″ M L ), we need the following notations. We assume that the random parameter associated to the fuzzy random variable C ¯ ˜ l is denoted by d ¯ l and is expressed as d ¯ l = d l 1 + t ¯ l d l 2 , where t ¯ l is a random variable with probability distribution T l . Moreover, we denote by F ∗ the pseudo inverse of a real-valued function F of ℝ . That is for θ ∈ ℝ , F ∗ ( θ ) = inf { r ∈ ℝ | F ( r ) ≥ θ } . It is shown that Problem ( P ″ M L ) is equivalent to the following program:
( P ‴ M L ) { Maximize μ 1 ( Z 1 α F ( x ) ) for D M 1 Maximize μ 2 ( Z 2 α F ( x ) ) for D M 2 subject to x ∈ X
where Z l α F ( x ) = ( T l ∗ ( θ l ) d l 2 + d l 1 − L ∗ ( α ) β l ) x . To find a satisfying solution of Problem ( P ‴ M L ), we first solve the following problem:
( P M L i v ) { Maximize min { μ 1 ( Z 1 α F ( x ) ) , μ 2 ( Z 2 α F ( x ) ) } subject to x ∈ X
It is an easy matter to see that the above program is equivalent to:
( P M L v ) { Maximize v subject to ( T 1 ∗ ( θ ) d 1 2 + d 1 1 − L ∗ ( α ) β 1 ) x ≤ μ 1 ∗ ( v ) ( T 2 ∗ ( θ ) d 2 2 + d 2 1 − L ∗ ( α ) β 2 ) x ≤ μ 2 ∗ ( v ) x ∈ X
Let x ∗ the solution of Problem ( P M L v ), if the decision maker 1 is satisfied with the membership function values μ l ( Z l α F ( x ∗ ) ) , l = 1 , 2 , then x is regarded as a satisfactory solution of Problem ( P M L ). Otherwise, the decision maker 1 specifies a minimal satisfactory level δ ∈ ( 0,1 ] and one solves the mathematical program:
Minimize ( T 2 ∗ ( θ 2 ) d 2 2 + d 2 1 − L ∗ ( α ) β 2 ) x subject to ( T 1 ∗ ( θ 1 ) d 1 2 + d 1 1 − L ∗ ( α ) β 1 ) x ≤ μ 1 ∗ ( δ ) x ∈ X
A part from this level-set based fractile approach for a multilevel optimization problem under fuzziness and randomness, there is also a possibility-based probability approach.
Significant progress has been made on ways for dealing with situations in which randomness and fuzziness are inextricably involved in a multi objective programming framework. These advances have generated more interest for this class of optimization problems. In a theoretical ground several concepts of generalized Pareto optimality have been introduced and properly characterized (see [
· methods for generating Pareto optimal solutions,
· interactive methods,
· goal programming methods.
As an example of a method for solving a fuzzy stochastic multi objective program. Consider the following multi objective programming problem:
( P P O ) { Minimize ( C ¯ ˜ 1 x , ⋯ , C ¯ ˜ h x ) subject to x ∈ X = { x ∈ ℝ n | A x ≤ b , x ≥ 0 }
where x is an n-dimensional decision vector, A an m × n matrix, b an m-dimensional vector and C ¯ ˜ l = ( C ¯ ˜ l 1 , ⋯ , C ¯ ˜ l n ) where C ¯ ˜ l j , l = 1 , ⋯ , k and j = 1 , ⋯ , n are discrete fuzzy random variables. That is value of C ¯ ˜ l j are triangular fuzzy numbers C ˜ l j s j , depending on a scenario s l ∈ { 1, ⋯ , S l } which occurs with probability p l s l . The membership function of C ˜ l j s j is:
μ C ˜ l j s l ( t ) = { max { 1 − d l j s l − t β l j , 0 } if t ≤ d l j s l max { 1 − t − d l j s l δ l j , 0 } if t > d l j s l
where d l j s l , β l j , δ l j are respectively the center value, the left spread value and the right spread value of C ¯ l j ( ω ) . Following this pattern, we can define the membership function of the fuzzy random variable C ¯ ˜ l j as:
μ C ¯ ˜ l j ( ω ) ( t ) = { max { 1 − d ¯ i j ( ω ) − t β i j , 0 } if t ≤ d ¯ i j ( ω ) max { 1 − t − d ¯ i j ( ω ) δ i j , 0 } if t > d ¯ i j ( ω )
Making use of the extension principle, we have:
μ C ¯ l s l ( x ) ( ν ) = { max { 1 − d l s l x − ν β l x , 0 } if ν ≤ d l s l x max { 1 − ν − d l s l x δ l x , 0 } if ν > d l s l x
and
μ C ¯ ˜ l ( ω ) x ( ν ) = { max { 1 − d ¯ l ( ω ) x − ν β l x , 0 } if ν ≤ d ¯ l ( ω ) x max { 1 − ν − d ¯ l ( ω ) x δ l x , 0 } if ν > d ¯ l ( ω ) x .
Assume that the decision maker has a fuzzy goal for each of the objective function, say “the objective value should be substantially less or equal than some given value”. Denote the membership of this fuzzy goal by μ G ˜ . Piecewise linear functions are generally used. Now the degree of possibility that the objective function C ¯ ˜ l x attains the fuzzy goal G ˜ l is given by:
Π C ¯ ˜ l x ( G ˜ l ) = sup y min { μ C ¯ ˜ l x ( y ) , μ G ¯ ˜ l ( y ) } ;
Or in terms of scenario s l ∈ { 1, ⋯ , S l } .
Π C ¯ ˜ l s l x ( G ˜ l ) = sup y min { μ C ¯ l s φ x ( y ) , μ G ¯ φ ( y ) } ; l = 1 , ⋯ , k .
One may use the expectation operator to convert the original problem into deterministic terms. In this case, one maximizes the expected degree of possibility that each of the original functions involving fuzzy random coefficients attains the fuzzy goals. In this case, the resulting deterministic problem is:
( P ′ M O ) { Maximize E [ Π C ¯ ˜ 1 x ( G ˜ 1 ) ] ⋮ Maximize E [ Π C ¯ ˜ k x ( G ˜ k ) ] subject to x ∈ X
It is proved that Problem ( P ′ M O ) is equivalent to the following multi objective program:
( P ″ M O ) { Maximize Z 1 Π , E ( x ) ⋮ Maximize Z k Π , E ( x ) subject to x ∈ X
where
Z l Π , E = ∑ j = 1 n ( β l j − ∑ s l = 1 S l p l s l d l j s l ) x j + Z l 0 ∑ j = 1 n β l j x j − Z l 1 + Z l 0 ,
and
Z l 0 = max s j ∈ { 1 , ⋯ , S l } max x ∈ X ∑ j = 1 n d l j s l x j ; l = 1 , ⋯ , k
Z l 1 = max s j ∈ { 1 , ⋯ , S l } min x ∈ X ∑ j = 1 l d l j s l x j ; l = 1 , ⋯ , k .
A Pareto optimal solution of Problem ( P ″ M O ) may be obtained by using well-known results of deterministic multi objective programming.
Fuzzy stochastic optimization is currently applied in a wide variety of domains spanning engineering Economy, Finance, Ecology, Energy, etc. The relevance of FSO was recognized once it became clear that it can be used as a unifying framework for handling situations where fuzziness and randomness co-occur in an optimization setting.
To assert that it is more useful to conceive imprecision in an optimization setting as a variegated whole is not to minimize important research works that have done in Fuzzy Optimization and Stochastic Programming. It is instead to assert that new perspectives for coping with complex real life problems may be gained by integration of both approaches than exclusion. It is virtually impossible to encompass the full spectrum of FSO applications within the scope of a paper. In
Fuzzy stochastic system is a fascinating area that allows generating satisfying outputs from fuzzy random inputs subject to a given system restriction. Investigating such systems represents an important research avenue in the discipline of mathematical modeling under uncertainty. Apart from being helpful in several applications, fuzzy stochastic systems can help shed some light on intricate
| Domain | Example of applications |
|---|---|
| Engineering | Planning of solid waste management [ |
| Economy | Marketing [ |
| Fiance | Bank hedging decision problems [ |
| Ecology | Environmental system planning [ |
| Energy | Electric power planning [ |
| Water resources | Reservoir strategies [ |
| Reliability | Time-dependent reliability [ |
| Network | Shortest path problem [ |
| Renewal processes | Queuing applications [ |
theoretical issues like impacts of infinitesimal changes on inputs of a mathematical model under uncertainty. When systems under scrutiny boil down to mathematical programs, we fall into the realm of fuzzy stochastic optimization, the subject matter of this paper. Fuzzy Stochastic Optimization has triggered, during the last two decades, an explosion of research worldwide both on theoretical issues, algorithmic aspects and concrete applications. It is therefore worth-while to pause once a while and ask ourselves, what we have done, what is done currently and what are the perspectives in this field. This paper has been written in this spirit. Its coverage has been made with a broad target audience in view.
We have highlighted the modeling power of FSO, its broad applicability as well as significant advances made in terms of both the quantity and the quality of results obtained. The variety of related subjects and the current profusion of scientific publications in the field, reflect its vitality and diversity. Some lines for further developments on this fascinating field are briefly outlined below:
1) Despite the fact that most existing approaches for solving FSO problems bow to epistemologically based prescriptive procedures for dealing with problems involving uncertainty, the model validation has been treated mostly from the perspective of efficiency (doing things right). Efforts should be devoted to strike a good balance between efficiency and effectiveness of designed FSO methods. Some indexes, including poss-prod and prod-poss that serve as windows through which one can appraise the desirability of a given action under considered transformations, have been introduced, but more is needed.
2) Decision makers would perhaps be best served if from a deep comparative analysis of good FSO methods, user-friendly Decision Support Systems (DSSs) could be pushed forward. Such (DSSs) would be of great help advise Decision makers with the choice of the most suitable methods for their problems.
3) In some sensitive cases, it would be preferable to have as a solution to a FSO problem, a fuzzy stochastic quantity in a way to reflect the vagueness and the randomness surrounding the problem. So an interesting direction for future research is that of defining and characterizing fuzzy random solutions for a Fuzzy Stochastic Optimization problem. Algorithms for singling out such solutions should also be designed and implemented.
4) Description of high quality case studies is also needed, in order to have more solid arguments in favour of the underniable importance of FSO methods.
We very much hope to see FSO advances so far, so fast that its influence transforms both the theory and practice of mathematical programming under uncertainty.
The authors declare no conflicts of interest regarding the publication of this paper.
Mangongo, Y.T., Kampempe, J.D.B. and Luhandjula, M.K. (2021) A Kaleodoscopic View of Fuzzy Stochastic Optimization. American Journal of Operations Research, 11, 283-308. https://doi.org/10.4236/ajor.2021.116018