A Multi-Level Multi-Objective Quadratic Programming Problem with Fuzzy Parameters on Objective Functions

This paper proposes analgorithm to solve multi-level multi-objective quadratic programming problemwith fuzzy parameters in the objective functions, This algorithm uses the tolerance membership function conceptsand multi-objective optimization at each level to develop a fuzzy Max-Min decision model for generating satisfactory solution after applying linear rankingmethod ontrapezoidal fuzzy numbers in the objective functions, An illustrative example is included to explain the results.


INTRODUCTION
Multi-level programming (MLP) techniques are developed to solve decentralized problems that contain multiple decision-makers in hierarchical organization, where each unit or department independently seeks its own interest, but is affected by the actions of other unit through externalities.Three level programming is a class of Multi-level programming problem in which there are three independent decision-makers.The field of multi-level programming which defines the art and science of making such decisions studied in [2, 3, and 4].
Fuzzy approach uses the concept of tolerance membership to develop a fuzzy max-min decisionModel for generating Pareto optimal (satisfactory) solution [1 and 3] In [3] Osman et al. presented the concepts of tolerance membership function and multi-objective optimization at each level to develop a fuzzy max-min decision model for generating (satisfactory) solution for three level multi-objective programming problem.
Recently several linear or nonlinear programming problems with fuzzy parameters on objective functions and their solution method have been presented, such as, in [4, 5, 6,7, 8, 9, 10and11]. In [4] Under the rules of simplex technique and the operations on trapezoidal fuzzy numbersEmamet al. ,suggested a new solution method to solve bi-level linear fractional integer programming problem with trapezoidal fuzzy numbers in the objective functions of the two levels .In [5]Dashetal. aimed to present a method in which a fuzzy multi objective nonlinear programming problem is reduced to crisp using ranking function and then the crisp problem is solved by fuzzy programming technique. In [6]Nasseri defined a quadratic programming problem with trapezoidal and/or triangularfuzzy numbers in the cost coefficients,constraint coefficients, and right-hand sides then used linear ranking method to solve the problem.

PROBLEM FORMULATION AND SOLUTION CONCEPT
Let ∈ , ( = 1, 2, 3),be a vector of variables which indicates the first decision level's choice, thesecond decision level's choice and the third decision level's choice andF i : → ( = 1, 2, 3),be the first level objective function, the second level objective function and the third level objective function, respectively. Assume thatthe first level decision maker is (FLDM),the secondlevel decision maker is(SLDM) and the third level decision maker is(TLDM). 1 , 2 and 3 ≥ 2 ,the FLDM, SLDM, and TLDM have 1 , 2 and 3 objective functions, respectively.LetGbe the set of feasible choices {( 1 , 2, 3 )}. Therefore a(MLMOQPP) with fuzzy parameters in the objective functionsmay be formulated as follows: Subject to: Where F i (x , q , C ), is a multi-level multi-objective quadratic programming problemwith fuzzy parameters in the objective functions.

Definition 1 [3]:
Let 1 , 2 , 3 be the feasible regions of FLDM, SLDM and TLDM, respectively. For any is the optimal solution of the TLDM, then ( 1 , 2, 3 ) is a feasible solution of the(MLMOQPP) with fuzzy parameters in the objective functions.

RANKING METHOD
To solve (MLMOQPP) with fuzzy parameters in the objective functions a linear ranking method technique is used to convert fuzzy number form into equivalent crisp form.

Definition 3.2[10]:
A (a 1 , b 1 , c 1 , d 1 ),B (a 2 , b 2 , c 2 , d 2 )are two trapezoidal fuzzy numbers and x∈R. Ranking function is a convenient method for comparing the fuzzy numbers which is a map from F(R) into the real line. So, the orders on F(R) as follow:

A = B if and only if ℜ A = ℜ(B ).
Where are in F(R). Nowafter applying linear ranking methodthe problem will be formulated as follow: Subject to:

FUZZY APPROACH FOR SOLVING (MLMOQPP)
To solve the (MLMOQPP) by using fuzzy approach, first the satisfactory solution that is acceptable for the FLDM is obtained, then the FLDM's decision variables and goals with some leeway are given to the SLDM to seek the satisfactory solution according to him/her, then the SLDM's decision variables and goals with some leeway are given to the TLDM to seek the satisfactory solution according to him/her and finally to arrive at the solution which is closest to the optimal solution of the FLDM. [3] 4.1.The problem ofthe FLDM.
The problem of theFLDM is solved individually;FLDM'soptimal solution is achieved by applying the following steps, firstby obtaining the best and the worst solutions oftheProblem Second Usingthe value of ( 1 * ,  k f 1 )to build membership functions as follows: Third by solving the Tchebycheff problem as follow: Subject to: Whose solution is assumed to be: [ Now the solution of the three level decision makers is disclosed. However, three solutions are usually different because of nature between three levels objective functions.
The FLDM knows that using the optimal decisions F x 1 as a control factors for the SLDM are not practical. It is more reasonable to have some tolerance that gives the SLDM an extent feasible region to search for his/her optimal solution, and reduce searching time or interactions, also the SLDM do the same action with the TLDM. In this way, the range of decision variables 1 , 2 should be around The FLDM goal may be reasonably consider all k f 1 ≥ F k f 1 = 1,2, … 1 are absolutely acceptable and Second, the SLDM goal may reasonably consider all  I n t e r n a t i o n a l J o u r n a l o f C o m p u t e r s a n d T e c h n o l o Third, the TLDM may be willing to build a membership function for his/her objective functions, so that he/she can rate the satisfaction of each potential solution. In this way, the TLDM has the following membership functions for his/her goals: Finally, in order to generate the satisfactory solution, which is also a Pareto optimal solution with overall satisfaction for all DMs, theTchebycheff problem can be solved as follow:

AN ALGORITHM
In this section an algorithm is presented to solve (MLMOQPP) with fuzzy parameters in the objective functions, the algorithm is illustrated in the following series steps: Step 1: Computeℜ for all the coefficients ofthe (MLMOQPP) with fuzzy number in the objective functions (1) -(3), where is a trapezoidal fuzzy number.
Step 2: Convert the (MLMOQPP) with fuzzy number in the objective functions (1) -(3)from the fuzzy form to the crisp form.
Step 5: The FLDM finds the individual best solutions * 1k f and individual worst  k f 1 for each objective of the FLDM.
Step 6: State membership functions of the FLDM, Step 7: Solve the Tchebycheff problem of the FLDM.

Step 8:Obtain satisfactory level of FLDM
Step 11: Solve the Tchebycheff problem of the SLDM.
Step 14: State membership functions of the TLDM Step 15: Solve the Tchebycheff problem of the TLDM.
Step 20: If 2 t ,then go to step 17, otherwise go to step 21.
Step 21: A compromise solution 0 X of the (MLMOQP) problem is obtained and  is overall satisfaction for all decision -makers.

5.1.A flowchart:
A flowchart to explain the suggested algorithm is described as follow: