An Interactive Fuzzy Satis(cid:12)cing Method for Multiobjective Linear Programming Problems with Random Fuzzy Variables Using Possibility-based Probability Model

This paper formulates multiobjective linear programming problems where each coeﬃcient of the objective functions is expressed by a random fuzzy variable. Assuming that the decision maker concerns about the probability that each of the objective function values is smaller than or equal to a certain target value, the fuzzy goals of the decision maker for the probabilities are introduced. Then, the possibility-based probability model to maximize the degrees of possibility with respect to the attained probability is considered. For solving transformed deterministic problems eﬃciently, particle swarm optimization for nonlinear programming problems is introduced. An interactive fuzzy satisﬁcing method is presented for deriving a satisﬁcing solution for a decision maker eﬃciently by updating the reference probability levels. An illustrative numerical example is provided to demonstrate the feasibility and eﬃciency of the proposed method.


Introduction
In actual decision making situations, we must often make a decision on the basis of vague information or uncertain data. For such decision making problems involving uncertainty, there exist two typical approaches: probability-theoretic approach [1][2][3][4] and fuzzy-theoretic one [5][6][7]. Stochastic programming, as an optimization method based on the probability theory, have been developing in various ways including two stage problems considered by Dantzig [8] and chance constrained programming proposed by Charnes et al. [9].
In most practical situations, however, it is natural to consider that the uncertainty in real world decision making problems is often expressed by a fusion of fuzziness and randomness rather than either fuzziness or random-ness. For handling not only the decision maker's vague judgments in multiobjective problems but also the randomness of the parameters involved in the objectives and/or constraints, Sakawa and his colleagues incorporated their interactive fuzzy satisficing methods for deterministic problems [5,10] into multiobjective stochastic programming problems, through the introduction of several stochastic programming models such as expectation optimization [11,12], variance minimization [11], probability maximization [11,13,14] and fractile criterion optimization [11], to derive a satisficing solution for a decision maker from Pareto optimal solution sets.
In multiobjective stochastic programming problems, it is implicitly assumed that uncertain parameters or coefficients can be expressed as random variables in probability theory. This means that the realized values of random parameters under the occurrence of some event are assumed to be definitely represented with real values.
However, it is natural to consider that the possible realized values of these random parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the experts' ambiguous understanding of the realized values of random parameters under the occurrence of events as fuzzy numbers. From such a point of view, a fuzzy random variable was first introduced by Kwakernaak [15], and its mathematical basis was constructed by Puri and Ralescu [16]. Studies on linear programming problems with fuzzy random variable coefficients, called fuzzy random linear programming problems, were initiated by Wang and Qiao [17] and Qiao, Zhang and Wang [18] as seeking the probability distribution of the optimal solution and optimal value.
On the other hand, from a viewpoint of ambiguity and randomness different from fuzzy random variables [15][16][17], by considering the experts' ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables was proposed, and mathematical programming problems with random fuzzy variables were formulated together with the development of a simulation-based approximate solution 6 An Interactive Fuzzy Satisficing Method for Multiobjective Linear Programming Problems with Random Fuzzy Variables Using Possibility-based Probability Model method [19]. A recently published book of Sakawa et al. [20] is devoted to introducing the latest advances in the field of multiobjective optimization under both fuzziness and randomness on the basis of authors' continuing research works. Special stress is placed on interactive decision making aspects of fuzzy stochastic multiobjective programming for human-centered systems under uncertainty in most realistic situations when dealing with both fuzziness and randomness.
Under these circumstances, in this paper, we consider multiobjective linear programming problems involving random fuzzy variables. To deal with the formulated random fuzzy multiobjective linear programming problems, we assume that the decision maker concerns about the probabilities that each of the objective function values is smaller than or equal to a certain target value. By considering the imprecise nature of the human judgments, we introduce the fuzzy goals of the decision maker for the probabilities. Assuming that the decision maker is willing to maximize the degrees of possibility with respect to the attained probability, we consider the possibility-based probability model for random fuzzy multiobjective linear programming problems. For solving transformed deterministic problems efficiently, particle swarm optimization for nonlinear programming problems [22] is introduced. Then, we present an interactive fuzzy satisficing method to derive a satisficing solution for the decision maker by updating the reference possibility levels. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

Random fuzzy variables
In the framework of stochastic programming, it is implicitly assumed that the uncertain parameter which well represents the stochastic factor of real systems can be definitely expressed as a single random variable. This means that the realized values of random parameters under the occurrence of some event are assumed to be definitely represented with real values.
Depending on the situations, however, it is natural to consider that the possible realized values of these random parameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the experts' ambiguous understanding of the realized values of random parameters as fuzzy numbers. From such a point of view, a fuzzy random variable was first introduced by Kwakernaak [15], and its mathematical basis was constructed by Puri and Ralescu [16].
From the expert's experimental point of view, however, the experts may think of a collection of random variables to be appropriate to express stochastic factors rather than only a single random variables. In this case, reflecting the expert's conviction degree that each of random variables properly represents the stochastic factor, it would be quite reasonable to assign the different degrees of possibility to each of random variables. For handling such an uncertain parameter, a random fuzzy variable was defined by Liu [19] as a function from a possibility space to a collection of random variables, which is considered to be an extended concept of fuzzy variable [21]. It should be noted here that the fuzzy variables can be viewed as another way of dealing with the imprecision which was originally represented by fuzzy sets. Although we can employ Liu's definition, for consistently discussing various concepts in relation to the fuzzy sets, we define the random fuzzy variables by extending not the fuzzy variables but the fuzzy sets.

Definition 1 (Random fuzzy variable) Let Γ be a collection of random variables. Then, a random fuzzy variableC is defined by its membership function
In Definition 1, the membership function µC assigns each random variableγ ∈ Γ to a real number µC(γ). It should be noted here that if Γ is defined as R, then (1) becomes equivalent to the membership function of an ordinary fuzzy set. In this sense, a random fuzzy variable can be regarded as an extended concept of fuzzy sets. On the other hand, if Γ is defined as a singleton Γ = {γ} and µC(γ) = 1, then the corresponding random fuzzy variableC can be viewed as an ordinary random variable.
When taking account of the imprecise nature of the realized values of random variables, it would be appropriate to employ the concept of fuzzy random variables. However, it should be emphasized here that if mean and/or variance of random variables are specified by the expert as a set of real values or fuzzy sets, such uncertain parameters can be represented by not fuzzy random variables but random fuzzy variables.
As a simple example of random fuzzy variables, we consider a Gaussian random variable whose mean value is not definitely specified as a constant. For example, when some random parameterγ is represented by the Gaussian random variable N (s i , 10 2 ) where the expert identifies a set {s 1 , s 2 , s 3 } of possible mean values as (s 1 , s 2 , s 3 ) = (90, 100, 110), if the membership function µC is defined by thenC is a random fuzzy variable. More generally, when the mean values are expressed as fuzzy sets or fuzzy numbers, the corresponding random variable with the fuzzy mean is represented by a random fuzzy variable.

Problem formulation
In the remainder of this paper, considering the possible realized values of random parameters are often only ambiguously known to the experts, we focus on multiobjective linear programming problems with random fuzzy variables. For that purpose, we formulate the random fuzzy multiobjective linear programming problems Computational Research 2(1): 5-11, 2014 7 expresses by . . ,C ln ) is a random fuzzy variable coefficient vector. Here, assume thatC lj is a Gaussian random variable whose mean value is a fuzzy numberM lj characterized by the membership function where the shape functions L and R are nonincreasing continuous functions from [0, ∞) to [0, 1], m lj is the mean value, and α lj and β lj are positive numbers which represent left and right spreads. Fig. 1 illustrates an example of the membership function µM lj (τ ). Let Γ be a collection of all possible Gaussian random variables N (s, σ 2 ) where s ∈ (−∞, ∞) and σ 2 ∈ (0, ∞). Then, the membership function ofC lj is expressed as Using the Zadeh's extension principle, each objective functionC l x is expressed as a random fuzzy variable characterized by the membership function whereγ l = (γ 1 , . . . ,γ n ).
By substituting (4) into (5), the membership function of a random fuzzy variable corresponding to the objective functionC l x in (2) is rewritten as where s l = (s l1 , . . . , s ln ).
ObservingC l x is expressed as a random fuzzy variable with the membership function µC x defined by (6), it is significant to realize that the fuzzy random programming models cannot be applied.

Possibility-based probability model
Assuming that the decision maker (DM) concerns about the probability that each of the objective function valuesC l x is smaller than or equal to a certain target values f l , we introduce the probability P which is expressed as a fuzzy set P l with the membership function where f l , l = 1, . . . , k are target values specified by the DM as constants.
Considering the imprecise nature of the DM's judgments for the probabilitiesP l with respect to the random fuzzy objective function valuesC l x, l = 1, . . . , k, we introduce the fuzzy goalsG l , l = 1, . . . , k such as "P l should be greater than or equal to a certain value." Such fuzzy goalsG l , l = 1, . . . , k can be quantified by eliciting corresponding membership functions probability, we consider the possibility-based probability model for multiobjective random fuzzy integer programming problems formulated as or equivalently

Interactive fuzzy satisficing method
Observing that (11) can be regarded as a multiobjective programming problem, a complete optimal solution that simultaneously minimizes all of the objective functions does not always exist when the objective functions conflict with each other. Thus, instead of a complete optimal solution, a solution concept of Pareto optimality plays an important role in multiobjective programming [5].
To be more specific, in the general form of multiobjective programming problem where X denotes the set of all feasible solutions, since there does not always exist a solution minimizing all of the objective functions simultaneously, the solution concept of Pareto optimality plays an important role and it is defined as follows.

Definition 2 (Pareto optimal solution)
A point x * ∈ X is said to be a Pareto optimal solution if and only if there does not exist another x ∈ X such that z i (x) ≤ z i (x * ) for all i ∈ {1, . . . , k} and z j (x) ̸ = z j (x * ) for at least one j ∈ {1, . . . , k}.
As can be seen from Definition 2, in general there exist an infinite number of Pareto optimal solutions if the feasible region X is not empty. In real-world decision making problems, to make a reasonable decision or implement a desirable scheme, the decision maker should select one point from among the set of Pareto optimal solutions [5].
For the multiobjective programming problem (11) under consideration, in order to generate a candidate for the satisficing solution which is also Pareto optimal, the DM is asked to specify the reference levels of h l , l = 1, . . . , k, called reference possibility levels. For the reference levelsĥ l , l = 1, . . . , k specified by the DM, the corresponding Pareto optimal solution, which is, in the minimax sense, nearest ito the requirement or better than that if the reference possibility levels are attainable, is obtained by solving the augmented minimax problem [5] minimize max l=1,...,k where ρ is a sufficiently small positive number.
It should be noted here that (13) involves the possibility constraints ΠP l (G l ) ≥ h l , l = 1, . . . , k, solution methods for ordinary mathematical programming problems cannot be directly applied. Realizing such difficulty, we deal with the possibility constraints in the following way.
From (9), the possibility constraints ΠP l (G l ) ≥ h l in (11) or (13) are equivalently replaced by the conditions that there exists a p l such that µP l (p l ) ≥ h l and µG l (p) ≥ h l , namely, and This implies that there exists a vector (p l , s l ,ū l ) such that which can be equivalently transformed into the condition that there exists a vector (s l ,ū l ) such that µM lj (s lj ) ≥ h l , j = 1, . . . , n, In view of (3), it follows that Computational Research 2(1): 5-11, 2014 9 where L ⋆ (h l ) and R ⋆ (h l ) are pseudo inverse functions defined as L ⋆ (h l ) = sup{t | L(t) ≥ h l } and R ⋆ (h l ) = sup{t | R(t) ≥ h l }. Hence, (15) is rewritten as the equivalent condition that there exists aū l such that is equivalently transformed as where Φ is a probability distribution function of the standard Gaussian random variable N (0, 1). From the monotone increasingness of Φ, (17) is rewritten as In this way, the augmented minimax problem (13) is equivalently transformed into minimize max l=1,...,k Observing that (19) is an ordinary nonconvex nonlinear programming problem, it is possible to employ particle swarm optimization for nonlinear programming (PSONLP) [22] for obtaining an approximate solution.
Following the above discussions, we can now construct an interactive algorithm for deriving the satisficing solution for the DM from among the Pareto optimal solution set.
Step 4: The DM is supplied with the corresponding Pareto optimal solution x * . If the DM is satisfied with the current objective function values h * l , l = 1, . . . , k, then stop. Otherwise, ask the DM to update the reference possibility levelsĥ l , l = 1, . . . , k by considering the current values of objective functions h l , l = 1, . . . , k, and return to step 3.
It should be noted for the DM that any improvement of one objective function value can be achieved only at the expense of at least one of the other objective function values for the fixed target values f l , l = 1, . . . , k.
Suppose that the DM sets the target values f l , l = 1, 2 as f 1 = −300, f 2 = 100, and determines the linear membership functions µG l , l = 1, 2. For the initial reference possibility levelsĥ l = 1, l = 1, 2, the corresponding augmented minimax problem is solved, and the obtained result is shown at the column labeled "1st" in Table 3. DM is not satisfied with these values of objective functions, and the DM updates the reference probability levels as (1.0, 0.7) for improving the values of objective functions h 1 at the expense of h 2 . For the updated reference fractile levels, the corresponding augmented minimax problem is solved, and the obtained result is shown at the column labeled "2nd" in Table 3.
A similar procedure continues until the DM is satisfied with the values of objective functions. In this example, we assume that the satisficing solution for the DM is derived in the third interaction.

Conclusions
In this paper, random fuzzy multiobjective linear programming problems have been considered. For tackling the formulated problems, it has been assumed that the decision maker concerns about the probabilities that each of the objective function values is smaller than or equal to a certain target value. By introducing the fuzzy goals of the decision maker for the probabilities and assuming that the decision maker is willing to maximize the degrees of possibility with respect to the attained probability, an interactive fuzzy satisficing method has been presented for deriving a satisficing solution for the decision maker by updating the reference possibility levels. It was shown that all of the problems to be solved in the proposed interactive fuzzy satisficing method can be solved through particle swarm optimization for nonlinear programming (PSONLP). An illustrative numerical example was provided to demonstrate the feasibility of the proposed method. However, further computational experiences should be carried out for several types of numerical examples. From such experiences the proposed computational method must be revised. Applications of the proposed method to the real world decision making situations will be required in the near future. Extensions to other stochastic programming models will be considered elsewhere. Also extensions to integer programming problems involving random fuzzy variable coefficients will be required in the near future.