A Multi-Level Multi-Objective Quadratic Programming Problem with Fuzzy Parameters in Constraints

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1 International Journal of Engineering Research and Development e-in: X p-in: X Volume Issue 4 (April 06) PP A Multi-Level Multi-Objective Quadratic Programming Problem with uzzy Parameters in Constraints O. E. Emam A. M.Abdo N. M. Bekhit 3 Information ystems Department aculty of Computers and Information Helwan University Cairo Egypt. Information ystems Department aculty of Computers and Information Helwan University Cairo Egypt. 3 Information ystems Department aculty of Computers and Information Helwan University Cairo Egypt. Abstract:- This paper proposes an algorithm to solve multi-level multi-objective quadratic programming (MLMOQP) problem involving trapezoidal fuzzy numbers in the right hand side of the constraint the suggested algorithm uses the linear Ranking Methods to convert the mentioned problem to its equivalent crisp form then uses the interactive approach to obtain the satisfactory solution (preferred solution) in view of the satisfactoriness concept andε -constraint method with considerations of overall satisfactory balance among all of the three levels an illustrative example is included. Keywords:- Multi-Level Programming Trapezoidal uzzy Numbers linear Ranking Methods Interactive Approachε -constraint method. I. INTRODUCTION Multi-level programming (MLP) is indicated as mathematical programming which solves the coordinating problem of the decision-making process in decentralized systems by enhancing the objective of a hierarchical organization while dealing with the tendency of the lower level of the hierarchy to enhance their own objectives. Three level programming (TLP) is a class of Multi-level programming problem in which there are three independent decision-makers. Multi-level programming (MLP) has been applied in many different kinds of the problems ([] [] [3] [5] [6] [8]and [9]). When taking into account some cooperation among the decision makers in different levels with more than one objective function in every level it is appropriate to use an approach or algorithm for obtaining a satisfactory solution to such model many researches utilized the Interactive approach as Theoretical and methodological base for solving such kind of the problems ([5] [7] [9] and [0]). In [5] Emam proposed a bilevel multi-objective integer non-linear programming fractional problem with linear or non-linear constraints at the first phase the convex hull of its original set of constraints was found then the equivalent problem was simplified by transforming it into a separate multi-objective decision-making problem and finally solving the resulted problem by using the e constraint method. In [0] Mishra et al proposed an interactive fuzzy programming method for obtaining a satisfactory solution to a bi-level quadratic fractional programming problem. The main idea behind the fuzzy number is that of gradual membership to a set without sharp boundary in concord with fuzziness of human judgment ([3] [6] and [8]). In [6] Youness et al suggested algorithm to solve a bi-level multi-objective fractional integer.programming problem involving fuzzy numbers in the righthand side of the constraints. In [8] akawa et al. Presented interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. II. PROBLEM ORMULATION AND OLUTION CONCEPT[] Let x i R n (i = 3) be a vector of variables which indicates the first decision level s choice the second decision level s choice and the third decision level s choice and i : R n R N t(i = 3) be the first level objective function the second level objective function and the third level objective function respectively. Assume that the first level decision maker is (LDM) the second level decision maker is (LDM) and the third level decision maker is (TLDM).N N and N 3 the LDM LDM and TLDM haven N and N 3 objective functions respectively. Let G be the set of feasible choices {( )}. Therefor the multi-level multi-objective quadratic programming problem (MLMOQPP) with fuzzy numbers in the right hand side of constrains may be formulated as follows: max x f x f x f N (x ) () 50

2 [3 rd level] max x max 3 x where solve f x f x f N (x ) () where solves f 3 x f 3 x f 3N3 x (3) G(x b) = x R n Ax b x 0 (4) where i (x ) (i =.. n)can be suggested in the following form n n j = i= q ij x i x j + n j = C j x j x =( ) R n +n +n 3 G is a linear constraint functions set and bis a trapezoidal fuzzy number denoted by (A B α β) where A B α β are real numbers and its membership function is given blow: μ b (x) = A x A α if A α x < A if A x < B x B B + β if b < x b + β 0 otherwise (5) III. RANKING METHOD To solve (MLMOQPP) with fuzzy parameters at the right hand side of the constraint the linear ranking method technique is used to convert fuzzy number form into equivalent deterministic crisp form. Definition []: If (E) = (A B α β) (R) then the linear ranking function is definedas R(E)= A + B 4 5 α + 3 β which is known by Yager Ranking Method. (6) f x f x f N (x) () Definition []: E(A B α β )H(A B α β )are two Trapezoidal fuzzy numbers and x R. a Ranking function is a convenient method for comparing the fuzzy numbers which is a map from (R) into the real line. o the orders on (R) asfollow:. E H if and only if R E R(H).. E > H if and only if R E > R(H ). 3. E = H if and only if R E = R(H ). WhereE and Hare in (R). Now after applying linear ranking method the problem will be formulated as follow: max x where solve [3 rd level] max x f x f x f N (x ) () where solves 5

3 max 3 x f 3 x f 3 x f 3N3 x (3) G (x b) = x R n Ax b x 0 (7) Where G is a linear constrain functions set and bis a deterministicnumber. IV. INTERACTIVE APPROACH OR (MLMOQP) PROBLEM To obtain the satisfactory solution (preferred solution) for (MLMOQP) problem the interactive approach through the satisfactoriness conceptε -constraint method and The Kth best method is used. The LDM gives the preferred or satisfactory solutions that are acceptable for the LDM in rank order to the LDM Then LDM will seek for the preferred solution which is close for the LDM and then LDM gives the preferred solution in rank order to the TLDM then the TLDM do exactly like the LDM. The LDM decide the satisfactory solution (preferred solution) of the (MLMOQP) problem according to the minimal satisfactoriness constant and solution test [5]. Definition [4]: To define the range of the objective function the best and the worst solutions of the function is defined as follow: * b Max f ( x) w Min f ( x) j.. N n l = L (8) x G x G Definition [4]: Let s=s 0 at the beginning and let s=s s s n. respectively where s 0 [0] is the satisfactoriness of the decision maker and ᵟ is calculated as follow: ᵟ =( b - w )s + w.l =... L j = n (9) * Definition 3[4]: Themain concept of the ε -constraint method is to optimize one of the objective functions using the other objective functions as constrainstheε -constraint problemincluding satisfactoriness is defined as follow: Max f l ( x ) (0) x G f ( x ) ᵟ l = ( L)j =. N The ε -constraint problem has no solution if f l l ( x )<ᵟl l=...l. so the satisfactoriness should be adjusted to s=s lt + < s lt l =... L t = n till the solution of the problem is obtained. Definition4 [8]: The Kth best method starts from a point which is a non-inferior solution to the problem of the upper level and checks whether it is also non-inferior solution to the problem of the lower level or not. If the first point is not non-inferior solution the Kth best continues to examine the second best solution to the problem of the upper level and so on. The solution of the (MLMOQP) problem will be obtained by solving LDM LDM TLDM problems each one separately. A. The problem of LDM: The LDM problem may be formulated as follow: max f ( ) f ( ).. f N ( ) () 5

4 G () irst the best and the worst of the objective function is calculated (8) Then first level multi-objective decision making problem is transformed into the single- objective decision making problem using the concept of the ε- constraint method including satisfactoriness as follows: max f () G f j ᵟj j =. N ᵟj = b * j w j s t + w j j =. n.wheres t t = n is given by the LDM indicted to LDM satisfaction. (3) The LDM ε-constraint problem has no solution if f ᵟ so the satisfactoriness should be adjusted to s t+ < s t t = n till the satisfactory solution of the problem x x x3 is obtained. econdly according to the interactive mechanism of the MLMOQP problem the LDM variables should be given to the LDM to seek the LDM satisfactory solution so the LDM puts the preferred solutions in order in the format as follow: k k k ).. k +p k +p k (4) Preferred solution( ( +p) preferred ranking k k k ) k k k k +p k +p k ( + + +) ( +p) Where k =0 P= (..n ). B. The problem of LDM: The LDM problem can be formulated as follows: max f x f G f N ( ) (5) (6) The best and the worst solution of LDM problem is calculated (8) then the ε-constraint problem of the LDM including satisfactoriness formulated as follows: max f x (7) 3 G (8) f j ( ) ᵟj j = (. N ) ᵟj = b * j w j s t + w j wheres t t = n is given by the LDM indicted to LDM satisfaction. 53

5 The ε-constraint problem of the LDM including satisfactoriness has no solution; if f ᵟ so the satisfactoriness should be adjusted to s t+ < s t t = n till the satisfactory solution of the LDM problem x x3 is obtained. The satisfactory solution of the LDM problem x x3 is tested to examine the closeness to the LDM satisfactory solution or it may be changed by the following test rule: ( x x x3 ) ( x x x3 ) (9) ( x x x ) 3 and should be given to the TLDM to seek the TLDM non-inferior solution so the LDM puts the preferred solution in order in the format as follow: k k k ).. k +p k +p k Preferred solution( ( +p) preferred ranking (0) k k k ) k k +p k +p k ( ( +) ( +p) Where k =0 P= (..n ). k + k + C. The problem of the TLDM: Thirdly variables and x should be given to the TLDM; hence the TLDM do exactly like the LDM till the satisfactory solution of the TLDM problem T x x3 54 is obtained. The satisfactory solution of the TLDM problem x x3 is tested to examine the closeness to the LDM preferred solution or it may be changed by the following test rule: T ( x x x3 ) ( x x x3 ) () ( x x x ) 3 where is a fairly small positive number given also by LDM. o T x x3 is the satisfactory solution (preferred solution) for (MLMOQP) problem. V. AN ALGORITHM In this section an algorithm is presented to solve (MLMOQPP) with fuzzy parameters in the right hand side of constrains the algorithm is illustrated in the following series steps: tep : use Yager ranking method and Compute R A for all the coefficients of the (MLMOQPP) with fuzzy number in the right hand side of constrains where A a trapezoidal fuzzy number. tep : Convert the (MLMOQPP) with fuzzy number in the right hand side of constrains from the fuzzy form to the crisp form. tep 3: ormulate the (MLMOQPP) tep 4: Use interactive approach to solve the (MLMOQPP). tep 5: solve the problem of the LDM. tep 6: Calculate the best and the worst solution of the LDM problem. tep 7: LDM sets s t t = n tep 8: formulate and solve the ε-constraint problem of the LDM including satisfactoriness. tep 9: If 0. f T ᵟ adjust satisfactoriness s t+ < s t t = n and go to step 8 otherwise go to step tep 0: obtain the satisfactory solution (preferred solution)of the LDM tep : etk = 0. tep : The LDM orders the satisfactory solutions x x x3 tep 3: Given x x to the LDM problem tep 4: solve the problem of the LDM. tep 5: Calculate the best and the worst solution of the LDM problem. tep 6: LDM sets s t t = n. x x x3

6 tep 7: formulate and solve the ε-constraint problem of the LDM. tep 8: I f 7 otherwise go to step 9. ᵟ so the satisfactoriness should be adjusted to s t+ < s t t = n and go to step tep 9: obtain the satisfactory solution of the LDM problem tep 0: test the satisfactory solution of the LDM problem tep : If x x3 then goes to step. tep : etk = 0. x x3 x x3 is close solution to LDM satisfactory solution go to step otherwise set k = k + tep 3: the LDM orders the satisfactory solutions x x3 tep 4: Given x =x s and x to the TLDM problem tep 5: solve the problem of the TLDM. tep 6: Calculate the best and the worst solution of the TLDM problem. tep 7: TLDM sets s 3t t = n. tep 8: formulate and solve the ε-constraint problem of the TLDM. tep 9: If T f 3 8 otherwise go to step 30. ᵟ3 so the satisfactoriness should be adjusted to s 3t+ < s 3t t = n and go to step tep 30: obtain the satisfactory solution of the TLDM problem tep 3: test the non-inferior solution of the TLDM problem tep3: If T x x3 k + then goes to step 3. tep 33: T x x3 T x x3 is a close solution to LDM satisfactory solution go to step 33 otherwise set k = T x x3 is the satisfactory solution (preferred solution) to the (MLMOQP) problem. VI. EXAMPLE To demonstrate the solution for MLMOQP problem with fuzzy parameter at the right hand side let us consider the following example: max [ ] Where solves max x [ ] Where solves [3 rd level] max 3 x3 x3 [ ] G = { (46 5) (3 708) + + (065) 0} By using equation (5) the given problem is converted from the fuzzy form to the crisp form then the problem is formulated as follow: max [ ] Where solves 55

7 [3 rd level] G = { } max x [ ] Where solves max 3 x3 x3 [ ] irst the LDM solves his/her problem separately as follows: Calculate the best and the worst solution of the LDM problem by using equation (8) we get (b b ) = (9.7 54) ( w w ) = (0 0). By using equations () and (3) the ε-constraint problem of the LDM including satisfactoriness s =0.9 which is given by the LDM is formulated as follow: max f x = G = { } The LDM solution is x x x3 = econd = x is given in order by LDM to LDM then the LDM solves his/her problem as follows: Calculate the best and the worst solution of the LDM problem by using equation (8) we get (b b ) = ( ) ( w w ) = (8 4) By using equations (7) and (8) the ε-constraint problem of the LDM including satisfactoriness s =0.6 which is given by the LDM is formulated as follow: Max f = G = { x =.36 0} The LDM solution is By using (9) we will find that x x3 = ( ) where ᵟ= 0. is given by the LDM. ( ) ( ) ( ) x x3 is a satisfactory solution for the LDM from the following test 0. s Third x and x is given in order by LDM to TLDM then the TLDM solves his/her problem as follows: Calculate the best and the worst solution of the TLDM problem by using equation (8) we get (b 3 b 3 ) = (6496) ( w 3 w 3 ) = ( ). 56

8 TLDM do exactly like the LDM the ε-constraint problem of the TLDM including satisfactoriness s 3 =0.3 which is given by the TLDM is formulated as follow: 3 Max f 3 x x = x G = { x x =.36 =.54 0} o the T x x3 ) = ( ) is the solution of the TLDM. inally by using () we will find that T x x3 test where ᵟ= 0. which is given by the LDM. (( ) ) ( ) 0. (( ) ) o T x x3 is a satisfactory solution for the LDM from the following = ( ) is the satisfactory solution (preferred solution) to (MLMOQP) problem. VII. CONCLUION This paper proposed an algorithm to solve multi-level multi-objective quadratic programming (MLMOQP) problem involving trapezoidal fuzzy numbers in the right hand side of the constraint the suggested algorithm used the linear Ranking Methods to convert the mentioned problem to its equivalent crisp form then used the interactive approach to obtain the preferred satisfactory solution (preferredsolution) in view of the satisfactoriness conceptε -constraint method with considerations of overall satisfactory balance among all of the three levels. This algorithm can be applied to problems involving fuzzy number in the objective function or in both the objective function and the constraint. REERENCE []. O. E. Emam A fuzzy approach for bi-level integer nonlinear programming problem Applied Mathematics and Computations 7 (006) 6 7. []. M.. Osman M. A. Abo-inna A. H. Amer and O. E. Emam A multi-level non-linear multi-objective under fuzziness Applied Mathematics and Computations 53 (004) [3]. E. A. Youness O. E. Emam and M.. Hafez implex method for solving bi-level linear fractional integer programming problems with fuzzy numbers International Journal of Mathematical ciences and Engineering Applications 7 (03) [4]. O.E. Emam Interactive Bi-Level Multi-Objective Integer Non-linear Programming Problem Applied Mathematical ciences 5 (65)(0) [5]. O.E. Emam Interactive approach to bi-level integer multi-objective fractional programming problem applied mathematics and computation 3 (03)7-4. [6]. E. A. Youness O. E. Emamand M.. Hafez uzzy Bi-Level Multi-Objective ractional Integer Programming Applied Mathematics 8(6) (04) [7]. Z. A. Kanaya An Interactive Method for uzzy Multi-objective Nonlinear Programming Problems JKAU () (00 ) 03-. [8]. M. akawa I. Nishizaki and Y. Uemura Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters uzzy ets and ystems 09 (000) 3-9. [9]. M. akawa and Ichiro Nishizaki Interactive fuzzy programming for decentralized two-level linear programming problems. uzzy ets and ystems 5 (00) [0].. Mishra and A. Ghosh Interactive fuzzy programming approach to Bi-level Quadratic fractional programming problems Ann Oper Res (43) (006) 5 63 []..A. Adnan and I. H AIkanani Ranking unction Methods or olving uzzy Linear Programming Problems Mathematical Theory and Modeling4(4) (04)