On Parametric Multi-level Multi-objective Fractional Programming Problems with Fuzziness in the Constraints

In this paper, a new concept of the fuzzy stability set of the first kind for multi-level multi-objective fractional programming (ML-MOFP) problems having a single-scalar parameter in the objective functions and fuzziness in the right-hand side of the constraints has been introduced. Firstly, A parametric ML-MOFP model with crisp set of constraints is established based on the -cut approach. Secondly, a fuzzy goal programming (FGP) approach is used to find an -Pareto optimal solution of the parametric ML-MOFP problem. Thus, the FGP approach is used to achieve the highest degree of each membership goal by minimizing the sum of the negative deviational variables. Finally, the fuzzy stability set of the first kind corresponding to the obtained -Pareto optimal solution is developed here, by extending the Karush-Kuhn-Tucker optimality conditions of multi-objective programming problems. An algorithm to clarify the developed fuzzy stability set of the first for parametric ML-MOFP problem as well as Illustrative numerical example are presented. Original Research Article Osman et al.; BJMCS, 18(5): 1-19, 2016; Article no.BJMCS.28531 2


Introduction
Hierarchical optimization or multi-level mathematical programming (ML-MP) techniques are extensions of Stackleberg games for solving decentralized planning problems with multiple decision makers (DMs) in a hierarchical organization [1]. The basic concept of multi-level programming technique is that the first-level decision maker (FLDM) sets his/her goal and/or decision, and then asks each subordinate level of the organization for their optima, that calculated in isolation. The lower level decision makers' decisions are then submitted and modified by the FLDM in consideration of the overall benefit for the organization the process continues until a satisfactory solution is reached [2,3]. ML-MP are common in government policies, competitive economic systems, supply chains, vehicle path planning problems, and so on [4]. During the past few decades, ML-MP [1,2,5] have been deeply studied and many methodologies have been developed for solving such problems. Baky [3] studied FGP algorithm for solving a decentralized bi-level multiobjective programming problem. The solution of bi-level large scale quadratic programming problem with stochastic parameters in the constraints has been studied by Emam et al. [6]. Saad et al. [7] presented a method for solving a three-level quadratic programming problem where some or all of its coefficients in the objective function are rough intervals. Pramanik and Roy [1] adopted fuzzy goals to specify the decision variables of higher level DMs and proposed weighted/ unweighted FGP models for solving ML-MP to obtain a satisfactory solution. Emam applied an interactive approach on bi-level integer multi-objective fractional programming problem [8]. Multi-level decision-making problems were recently studied by Chen and Chen [9].
Fractional optimization problem is one of the most difficult problems in the field of optimization. Optimization of the ratio of two functions is called fractional programming (ratio optimization) problem [10]. Indeed, in such situations, it is often a question of optimizing a ratio of output/employee, profit/cost, inventory/sales, student/cost, doctor/patient, and so on subject to some constraints [11]. Such type of problems in large hierarchical organizations of complex and conflicting multi-objectives formulate ML-MOFP problems. Omran et al. [12] extended the fuzzy approach to solve a three-level fractional programming problem with rough coefficient in the constraints.
In real world decision-making situations, mathematical programming models involving fuzzy parameters were viewed to be more realistic versions than the conventional one [9]. Therefore, the parameters involved in the right-hand side of the constraints of the parametric ML-MOFP problem are assumed to be characterized by fuzzy numbers.
Osman [13] introduced the notions of the solvability set, stability set of the first kind (the set of all parameters corresponding to the efficient solution) and stability set of the second kind and analyzed these concepts for parametric convex programming. Stability of multi-objective non-linear programming problems with fuzzy parameters in the constraints was studied by Kassem and Ammar [14]. Saad [15] presented stability of proper efficient solutions in multi-objective fractional programming problems under fuzziness. Saad and Hughes [16], considered bicriterion integer linear fractional programs with single-scalar parameter in the objective functions. Recently, a parametric study on multi-objective integer quadratic programming problems under uncertainty has been presented by Emam [17].
Parametric programming investigates the effect of predetermined continuous variations in the objective function coefficients and the right-hand side of the constraints on the optimum solution [18]. In parametric analysis the objective function and the right-hand side vectors are replaced with parameterized function ሺߠሻ and ሺߙሻ, where ߠ and ߙ are the parameter of variation. The general idea of parametric analysis is to start with the ߙ-Pareto optimal solution at ߠ = ߠ ܽ݊݀ ߙ = ߙ * . Then by utilizing the Karush-Kuhn-Tucker (KKT) optimality conditions the fuzzy stability set of the first kind (the set of parameters for which the solution at at ߠ = ߠ ܽ݊݀ ߙ = ߙ * remain optimal and feasible) is determined [19].
Different basic notions like solvability set and stability set of the first kind for parametric multi-objective programming have been studied in several papers. In the present research, these notions have been extended to introduce the fuzzy stability set of the first kind for parametric fuzzy ML-MOFP problems. The proposed parametric fuzzy ML-MOFP problem involves a single-scalar parameter in the objective functions and fuzzy parameters in the right-hand side of the constraints. Firstly, a numerical parametric ML-MOFP model is established based on a confidence level (ߙ-level) then, a FGP approach is considered for finding an ߙ-Pareto optimal solution for such problem. In FGP approach, the membership functions for the defined fuzzy goals are developed. Also, in the proposed approach, membership goals of the objective functions are linearized. Then, the highest degree of each membership goals is achieved by minimizing the sum of the negative deviational variables. Secondly, after obtaining an ߙ-Pareto optimal solution, the parametric FGP model is set up. Thus, based on the Kuhn-Tucker optimality conditions for multi-objective programming problems, we apply KKT conditions on the parametric FGP model of the parametric fuzzy ML-MOFP problem to formulate a system of equations. Then, the fuzzy stability set of the first kind, obtained from the reduced system of equations.
The rest of this paper is organized as follows. Section 2 presents the parametric fuzzy ML-MOFP problems formulation and introduces its solution concepts. Section 3 explains the developed FGP approach for solving such problems. Section 4 proposes the fuzzy stability set of the first kind for parametric fuzzy ML-MOFP problems. An algorithm for obtaining the fuzzy stability set of the first kind for parametric fuzzy ML-MOFP problems is introduced in section 5. Illustrative example is given in section 6. This paper ends with some concluding remarks in section 7.
Definition 1 [19]. Let ܾ ෨ be a fuzzy subset of ܴ with membership function ߤ ෩ . It is said that ܾ ෨ is a fuzzy number if the following conditions are satisfied: • ܾ ෨ is normal, i.e., there exists an ‫ݔ‬ ∈ ܴ such that ߤ ෨ ሺ‫ݔ‬ሻ = 1, • ߤ ෨ is upper semicontinuous, i.e., ሼ‫:ݔ‬ ߤ ෨ ሺ‫ݔ‬ሻ ≥ ߙሽ is a closed subset of ܷ for all ߙ ∈ [0, 1], • The 0-level set ܾ ෨ is a compact subset of ܷ.
Definition 2 [14]. The ߙ-level set of the vector of fuzzy parameters ܾ ෨ , is defined as an ordinary set ‫ܮ‬ ఈ ൫ܾ ෨ ൯ for which the degree of its membership function exceeds the level set ߙ ∈ [0, 1], where: Based on the parametric fuzzy ML-MOFP model ሺ2ሻ − ሺ5ሻ, with single-scalar parameter ߠ ∈ ܴ in the objective functions and fuzzy parameters in the right hand side of the constraints. Let ߤ ෩ , be the membership functions which represents the fuzziness in the corresponding vector ෩ . Thus, for a specified value of ߙ = ߙ * ∈ [0,1], estimated by all DM, the parametric ߙ-(ML-MOFP) problem reformulated as follow: where the crisp system of constraints, in equation (10), at an ߙ-level denoted by ‫ܩ‬ ఈ which form a compact set.
Assuming that the parametric ߙ-(ML-MOFP) problem has an ߙ-Pareto optimal solution * at ߠ .

Fuzzy Goal Programming Approach of Parametric Fuzzy ML-MOFP Problems
In the proposed FGP approach for parametric ߙ -(ML-MOFP) in order to obtain the compromise (satisfactory) solution that is an ߙ-Pareto optimal solution. The vector of objective functions for each DM is formulated as a fuzzy goal characterized by its' membership function ߤ ቀ ೕ ൫௫,ఏ బ ൯ቁ ሺ݅ = 1,2, … , ‫ݐ‬ሻ, ሺ݆ = 1,2, … , ݇ ሻ [1,2]. The model formulation and solution process are carried out at ߠ = ߠ .

Characterization of membership functions
To define the membership functions of the fuzzy goals [3], each objective function's individual maximum is taken as the corresponding aspiration level, as follows: where ‫ݑ‬ , give the upper tolerance limit or aspired level of achievement for the membership function of ݆݅ ௧ objective function at ߠ = ߠ . Similarly, each objective function's individual minimum is taken as the corresponding aspiration level, as follows: where ݃ , give the lower tolerance limit or lowest acceptable level of achievement for the membership function of ݆݅ ௧ objective function. Assuming that the values of ݂ ሺ, ߠ ሻ ≥ ‫ݑ‬ , ሺ݅ = 1,2, … , ‫ݐ‬ሻ, ሺ ݆ = 1,2, … , ݇ ሻ, are acceptable and all values ݂ ሺ, ߠ ሻ ≤ ݃ , are absolutely unacceptable. And all values ݃ ≤ ݂ ሺ, ߠ ሻ ≤ ‫ݑ‬ described by the membership function ߤ ቀ ೕ ൫௫,ఏ బ ൯ቁ = ߤ ଵ , as shown in Fig. (1), for the ݆݅ ௧ fuzzy goal [1]:

Fig. 1. Membership functions of maximization type for ሺ, ሻ
Following the basic concept of multi-level programming problems the first level decision maker sets his/her goals and/or decisions and then asks subordinate level for their optima [1][2][3]. Therefore, to study the fuzzy stability set of the first kind the vector of decision variables , ሺ݅ = 1,2, … , ‫ݐ‬ − 1ሻ, ሺ݇ = 1,2, … , ݊ ሻ for the top levels are taken as binding constraints for the ‫ݐ‬ ௧ -level problem as follows:

Fuzzy goal programming methodology
In the decision-making context, each decision maker is interested in maximizing his or her own objective function; the optimal solution of each DM, when calculated in isolation, would be considered as the best solution and the associated value of the objective function can be considered as the aspiration level of the corresponding fuzzy goal [2]. In fuzzy programming approach, the highest degree of membership is one. So, for the defined membership function in equation (13), the flexible membership goals having the aspired level unity can be represented as follows [20]: or equivalently as: where ݀ ି , ݀ ା ≥ 0 with ݀ ି × ݀ ା = 0, represent the under-and over-deviations, respectively, from the aspired levels [1,3,4].
In the classical methodology of goal programming, the under-and over-deviational variables are included in the achievement function for minimizing them depends upon the type of the objective functions to be optimized [1,3]. Thus, considering the goal achievement problem at the same priority level, the equivalent proposed FGP model of the parametric ߙ-(ML-MOFP) problem can be formulated as follows: where ܼ represents the achievement function consisting of the weighted under-deviational variables of the fuzzy goals. The numerical weights ‫ݓ‬ ି represent the relative importance of achieving the aspired levels of the respective fuzzy goals these values are determined as [20]: -݃ , ሺ݅ = 1,2, … , ‫ݐ‬ሻ, ሺ ݆ = 1,2, … , ݇ ሻ, ሺ23ሻ

Linearization of parametric membership goals
It can be easily realized that the parametric membership goals in equation (13) are inherently non-linear in nature and this may create computational difficulties in the solution process. To avoid such problems, a linearization procedure is presented in this section. Following Pal et al. [11] the parametric ݆݅ ௧ membership goals with single-scalar parameter ߠ can be presented as: Considering the expression of ݂ ሺ, ߠ ሻ , the above goal in equation (25) can be represented as: Considering the method of variable change presented by Pal et al. [11], the goal expression in equation (28)

The Fuzzy Stability Set of the First Kind for Parametric Fuzzy ML-MOFP Problem
Now, the main question is: Having solved the parametric ߙ-(ML-MOFP) problem to what extents can its data with respect to ߙ and ߠ be changed without invalidating the efficiency of its ߙ-Pareto optimal solution (compromise solution)?
Thus the definition of the set of feasible parameters, solvability set and the fuzzy stability set of the first kind for parametric ߙ-(ML-MOFP) problem is given as follows: The fuzzy stability set of the first kind of the parametric ߙ-(ML-MOFP) problem is the set of all parameters corresponding to one ߙ-Pareto optimal solution [14,15]. It is easy to see that the fuzzy stability of the parametric ߙ -(ML-MOFP) model (7)-(10) implies the stability of the parametric FGP model which is defined as follows:

Utilization of the Karush-Kuhn-Tucker optimality conditions corresponding to parametric FGP model
The Lagrangean function of parametric FGP model (39)-(46) follows as [16,19]: where ߣ, ߦ, ߰, ߤ, ‫,ݒ‬ ߟ, ߶, ߛ and ߜ are the Lagrange multipliers. Then the Karush-Kuhn-Tucker necessary optimality conditions [16,19] corresponding to the parametric FGP model (39)-(45), which has the above Lagrangean function, will have the following form:  67), the fuzzy stability set of the first kind ሺ , ߙሻ for parametric fuzzy multi-level multi-objective fractional programming problem with single-scalar parameter in the objective functions and fuzziness in constraints will be obtained.

Algorithm for Determination of the Fuzzy Stability Set of the First Kind ሺ , ߙሻ
Following the above discussion, an algorithm will be developed for obtaining the fuzzy stability set of the first kind ሺ , ߙሻ for parametric fuzzy ML-MOFP problem as follows: : ‫݊݅ܽݐܾ‬ ܽ ‫݁ݏ݅݉ݎ݉ܿ‬ solution of ‫ݐ‬ℎ݁ ‫݈ܾ݉݁ݎ‬ Step 1.
Set the value of ߙ, acceptable for all decision makers.

Step 2.
Postulate that ߠ = ߠ at the first.

Step 3.
Compute the individual maximum and minimum values for each objective function.

Step 4.
Set the goals and the upper tolerance limits for each objective function in all levels, according to equations (11)-(12).

Step 8.
Do the linearization procedures for each parametric membership goal according to equations (28)-(30) at ߠ = ߠ .

Step 9.
Solve the ℓ ௧ level FGP model to get the decision variables ℓ௩ = ℓ௩ * .

Illustrative Example
To demonstrate the proposed algorithm for finding the fuzzy stability set of the first kind, consider the following parametric fuzzy ML-MOFP problem with single-scalar parameter in the objective functions and fuzziness in the right hand side of the constraints.
Stage I: finding the α-compromise solution of the parametric fuzzy ML-MOFP problem.
: determination of the fuzzy stability set of the ϐirst kind ሺ , ࢻሻ Step 12.

Step 13.
Obtain the Lagrangian function, for the final FGP model, as in equation (47).
Apply the Kuhn-Tucker optimality conditions to find, the fuzzy stability set of the first kind, equations (48)-(67).
Reduce the system of equations (48)-(67), to obtain the fuzzy stability set of the first kind ሺ , ߙሻ and Stop.
[ ] The individual maximum and minimum values are summarized in Table 1. The decided aspiration levels, upper tolerance limits and the weights ‫ݓ‬ are also given. The coefficient of the linearized membership goals are presented in Table 2.
Stage II: determination of the fuzzy stability set of the first kind ሺ , ߙሻ To determine the fuzzy stability set of the first kind ሺ , ߙሻ of the parametric fuzzy ML-MOFP problem, the coefficients of the linearized membership goals in the parametric form are recalculated and summarized in Table 3. and Table 4 respectively.