On Fuzzy Three Level Large Scale Linear Programming Problem

Transcription

1 J. Stat. Appl. Pro. 3, No. 3, (2014) 307 Journal of Statistics Applications & Probability An International Journal On Fuzzy Three Level Large Scale Linear Prograing Proble O. E. Ea 1,, A. M. Abdo 1 and A. A. Abohany 2 1 Departent of Inforation Systes, Faculty of Coputer Science and Inforation, Helwan University, P.O. Box 11795, Egypt 2 Director of Electronic Docuent Manageent Systes, Huan Resources Behalf, Teleco Egypt, Sart Village, B7, Egypt Received: 20 Apr. 2014, Revised: 26 Jun. 2014, Accepted: 28 Jun Published online: 1 Nov Abstract: This paper suggests an algorith to solve a three level large scale linear prograing proble with fuzzy nubers, where all coefficients of the objective functions are syetric trapezoidal fuzzy nubers. A three-level prograing proble can be thought as a static version of the Stackelberg strategy. The suggested algorith uses a linear ranking function at each level to define a crisp odel which is equivalent to the fuzzy nuber, then all decision akers attepts to optiize its proble separately as a large scale prograing proble using Dantzig and Wolfe decoposition ethod. Therefore, we handle the optiization process through a series of sub probles that can be solved independently. Finally, a nuerical exaple is given to clarify the ain results developed in this paper. Keywords: Large scale probles; linear prograing; ranking; three-level prograing; decoposition algorith. MSC 2000: 90C06; 90C05; 90C99. 1 Introduction The basic concept of the three level prograing (TLP) technique is that a first level decision aker (FLDM) sets his goals and/or decisions and then asks each subordinate level of the organization for their optia which are calculated in isolation; the second-level decision aker (SLDM) decision is then subitted and odified by the FLDM; Finally, the third-level decision aker (TLDM) decision is subitted and odified by the FLDM and SLDM with consideration of the overall benefit for the organization; and the process continued until an optial solution is reached [8]. Most studies in ultilevel field are focused on bi-level optiization proble [9, 13, 14, 15, 16]. Saraj and Safaei [16] introduced an approach to solve a fuzzy linear fractional bilevel ulti-objective prograing proble (FLFBLMOP) by using of Taylor series and Kuhn-Tucker conditions approach. The Taylor series is an expansion of a series that represents a function. By using the Kuhn-Tucker conditions, the proble is reduced to a single objective. Coonly, when forulating a large scale prograing odel which closely describes and represents the real world decision situations, various factors of the real syste should be reflected in the description of the objective functions and constraints. Naturally, these objective functions and constraints involve any paraeters and the experts ay assign the different values, which can be represented by fuzzy nubers. In the traditional approaches of large scale systes, paraeters are fixed at soe values in an experiental and/or subjective anner through the experts understanding of the nature of the paraeters. In practice, however, it is natural to consider that the possible values of these paraeters are often only abiguously known to experts understanding of the paraeters as fuzzy nuerical data, which can be represented by eans of fuzzy subsets of the real line known as fuzzy nubers [2,3,4]. Fuzzy linear prograing probles have an essential in fuzzy odeling which can forulate uncertainty in actual environent. Afterwards any authors [1, 6, 10, 11, 17] considered various types of the fuzzy linear prograing and propose several approaches for solving these probles. Recently, notable studies have been done in the area of large scale linear and nonlinear prograing probles with block angular structure based on Dantzig and Wolfe decoposition ethod [3, 12]. Corresponding author e-ail: ea o e@yahoo.co

2 308 O. E. Ea et. al. : On Fuzzy Three Level Large Scale Linear... This paper is organized as follows: we start in Section 2 by forulation the odel of the three level large scale linear prograing proble with fuzzy nubers. In Section 3, presents a decoposition algorith for a three level large scale linear prograing probles with fuzzy nubers based on linear ranking function. An algorith followed by a flowchart for solving a three level large scale linear prograing proble with fuzzy nubers is suggested in Section 4 and Section 5. In addition a nuerical exaple is provided in Section 6 to clarify the results. Finally, conclusion and future works are reported in Section 7. 2 Proble Forulation and Solution Concept The three level large scale linear prograing proble with fuzzy nubers in objective functions (TLLSLPPFN) ay be forulated as follows: [First Level] x 1,x 2 F 1 (x)= x 1,x 2 c 1 j x j, (1) [Second Level] wherex 3,...,x solves x 3,x 4 F 2 (x)= x 3,x 4 c 2 j x j, (2) [Third Level] wherex 5,...,x solves x 5,x 6 F 3 (x)= x 5,x 6 c 3 j x j, (3) x G. wherex 7,...,x solves (4) where G={a 01 x 1 + a 02 x 2 + a 0 x b 0, d 1 x 1 b 1, d 2 x 2 b 2, d x b, x 1,...,x 0}. In the above proble (1)-(4), x j R,( j = 1,2,...,) be a real vector variables, c i j an n-diensional row vector of fuzzy paraeters in the objective functions, G is the large scale linear constraint set where, b=(b 0,...,b ) T is (+1) vector, and a 01,...,a 0,d 1,...,d are constants. Therefore F i : R R,(i=1,2,3) be the first level objective function, the second level objective function, and the third level objective function, respectively. Moreover, FLDM has x 1,x 2 indicating the first decision level choice, SLDM and TLDM have x 3,x 4 and x 5,x 6 indicating the second decision level choice and the third decision level choice, respectively. Definition 1. For any(x 1,x 2 G 1 ={x 1,x 2 (x 1,...,x ) G}) given by FLDM and(x 3,x 4 G 2 ={x 3,x 4 (x 1,...,x ) G}) given by SLDM, if the decision-aking variable (x 5,x 6 G 3 = {x 5,x 6 (x 1,...,x ) G}) is the Pareto optial solution of the TLDM, then(x 1,...,x ) is a feasible solution of TLLSLPPFN. Definition 2. If x R is a feasible solution of the TLLSLPPFN; no other feasible solution x G exists, such that F 1 (x ) F 1 (x); so x is the Pareto optial solution of the TLLSLPPFN.

3 J. Stat. Appl. Pro. 3, No. 3, (2014) / A Decoposition Technique for the TLLSLPPFN Based on Linear Ranking To solve the TLLSLPPFN using a decoposition technique based on a linear ranking technique, the FLDM convert fuzzy nuber for into equivalent crisp for by using linear ranking function [17], then he/she gets the optial solution using the decoposition ethod [7] to break the large scale proble into n-sub probles that can be solved directly. Then by inserting the FLDM decision variable to the SLDM for hi/her to seek the optial solution using linear ranking function [17] cobine with the decoposition ethod [7]. Finally the TLDM do the sae action till he obtains the optial solution of his proble which is the optial solution to TLLSLPPFN. Definition 3.[17] Ã=(a,b,c,d) F(R), then a linear ranking function is defined as R(Ã)=a+b+ 1 2 (d c). Definition 4. [17] Let Ã(a 1,b 1,c 1,d 1 ) and B(a 2,b 2,c 2,d 2 ) be two trapezoidal fuzzy nubers and x G. A convenient ethod for coparing of the fuzzy nubers is by using of ranking functions. A ranking function is a ap fro F(R) into the real line. So, the orders on F(R) as follows: 1. à B if and only if R(Ã) R( B). 2. Ã> B if and only if R(Ã)>R( B). 3. Ã= B if and only if R(Ã)=R( B) where à and B are in F(R). 3.1 The First-Level Decision-Maker (FLDM) Proble The first-level decision-aker proble of the (TLLSLPPFN) is as follows: [First Level] x 1,x 2 F 1 (x)= x 1,x 2 c 1 j x j, (5) x G. The FLDM transfors fuzzy nubers for into equivalent crisp for by using of linear ranking function equation. The FLDM proble can be reduced to the following for [17]: [First Level] F 1 (x)= c 1 j x j, (6) x G. To obtain the optial solution of the FLDM proble, the FLDM solves his proble by the decoposition technique [7]. The decoposition principle is based on representing the TLLSLPP in ters of the extree points of the sets d j x j b j,x j 0, j= 1,2,...,. To do so, the solution space described by each d j x j b j,x j 0, j= 1,2,..., ust be bounded and closed, for ore details see [7]. To obtain the optial solution of the FLDM proble of the TLLSLPPFN is as follows: Suppose that the extree points of d j x j b j,x j 0 are defined as ˆx jk,,2,3, where x j defined by: x j = k j β jk ˆx jk, j = 1,...,. (7) and β jk 0, for all k and k j β jk = 1. Now, the FLDM proble in ters of the extree points to obtain the following aster proble of the FLDM are forulated as stated in [7]: k 1 c 11 ˆx 1k β 1k + k 2 c 12 ˆx 2k β 2k + + k n c 1n ˆx nk β nk, (8)

4 310 O. E. Ea et. al. : On Fuzzy Three Level Large Scale Linear... k 1 a 01 ˆx 1k β 1k + k 2 a 02 ˆx 2k β 2k + + k n k 1 k 2 k n a 0n ˆx nk β nk b 0, β 1k = 1, β 2k = 1, β nk = 1, β jk 0,for all j andk. The new variables in the FLDM proble are β jk which deterined using Balinski s algorith [5]. Once their optial values β jk are obtained, then the optial solution to the original proble can be found by back substitution as follow: x j = k 1 β jk ˆx jk, j = 1,2,3. (9) It ay appear that the solution of the FLDM proble requires prior deterination of all extree points ˆx jk. To solve the FLDM proble by the revised siplex ethod, it ust deterine the entering and leaving variables at each iteration. Let us start first with the entering variables. Given C B and B 1 of the current basis of the FLDM proble, then for non-basic β jk : Where z jk c jk = C B B 1 P jk c jk, (10) a j ˆx jk 0 C jk = C j ˆx jk and P jk = 1 0 Now, to decide which of the variables β jk should enter the solution it ust deterine:. (11) z jk c jk = in{z jk c jk }. (12) Consequently, if z jk c jk 0, then according to the axiization optiality condition, β jk ust enter the solution; otherwise, the optial has been reached. The SLDM and the TLDM solve their aster proble by the decoposition ethod [7] as the FLDM. 3.2 The Second-Level Decision-Maker (SLDM) Proble According to the echanis of the TLLSLPPFN, the FLDM variables x F 1,xF 2 SLDM proble of the (TLLSLPPFN) can be written as follows : [Second Level] ust be given to the SLDM; hence, the x 3,x 4 F 2 (x)= x 3,x 4 c 2 j x j, (13) x G, where G 2 =(x F 1,xF 2,...,c ). The SLDM proble can be reduced to the following for [17] : [Second Level] F 2 (x)= c 2 j x j, (14)

5 J. Stat. Appl. Pro. 3, No. 3, (2014) / x G 2. To obtain the optial solution of the SLDM proble; the SLDM solves his proble by the decoposition technique [7] as the FLDM. 3.3 The Third-Level Decision-Maker (TLDM) Proble According to the echanis of the TLLSLPPFN, the SLDM variables x F 1,xF 2,xS 3,xS 4 ust be given to the TLDM; hence, the TLDM proble can be written as follows: [Third Level] x 5,x 6 F 3 (x)= x 5,x 6 c 3 j x j, (15) x G 3 where G 3 =(x F 1,xF 2,xS 3,xS 4,...,x ). The TLDM proble can be reduced to the following for [17] : [Third Level] F 3 (x)= c 3 j x j, (16) x G 3. To obtain the optial solution of the TLDM proble; the TLDM solves his proble by the decoposition technique [7] as the FLDM. Now the optial solution(x F 1,xF 2,xS 3,xS 4,xT 5,xT 6,...,xT ) of the TLDM is the optial solution of the TLLSLPPFN. 4 An Algorith for Solving a TLLSLPPFN A solution algorith to solve (TLLSLPPFN) is described in a series of steps. This algorith uses linear ranking function to convert the fuzzy nuber into real line to overcoe the coplexity nature of the three level large scale linear prograing proble with fuzzy nubers, and uses the constraint ethod of the three level optiization to facility the large scale nature. Inserting the variables value of every higher level decision aker to his lower level decision aker break the difficulty faces the proble. The suggested algorith can be suarized in the following anner. Step 1. Forulate FLDM proble in for (8) and solve it. Step 2. If FLDM obtain his optial solution set (x 1,x 2 )=(x F 1,xF 2 ) go to Step 9, otherwise go to Step 3. Step 3. Copute R(Ã) for all the cofficients of the proble (1)-(4) where à is trpezoidal Fuzzy nuber, go to Step 4. Step 4. Set. Step 5. Copute z jk c jk = C B B 1 P jk C jk, go to Step 6. Step 6. If z jk c jk 0, then go to Step 7; otherwise, the optial solution has been reached, go to Step 8.

6 312 O. E. Ea et. al. : On Fuzzy Three Level Large Scale Linear... Step 7. Set k=k+1, go to Step 5. Step 8. The decision aker use arithetic on fuzzy nubers xã = (xa 1,xb 1,xc 1,xd 1 ), Ã+ B = (a 1 a 2,b 1 b 2,c 1 + c 2,d 1 + d 2 ) to transfor crisp for into equivalent fuzzy nubers for, go to Step 2. Step 9. If SLDM obtain his optial solution set (x 1,x 2,x 3,x 4 )=(x F 1,xF 2,xS 3,xS 4 ) then go to Step 11, otherwise go to Step 10. Step 10. Forulate SLDM proble in for (8), go to Step 3. Step 11. If the TLDM obtain the optial solution go to Step 13, otherwise go to Step 12. Step 12. Forulate TLDM in for (8), go to Step 3. Step 13. Put (x F 1,xF 2,xS 3,xS 4,xT 5,xT 6,...,xT is an optial solution for three-level large scale linear prograing proble with fuzzy nubers, then stop. 5 A Flowchart for Solving a TLLSLPPFN A flowchart to explain the suggested algorith for solving a three-level large scale linear prograing proble with fuzzy nubers is described as follows. Fig. 1: A three level large scale linear prograing proble with fuzzy nubers in objective functions flowchart.

7 J. Stat. Appl. Pro. 3, No. 3, (2014) / Reark 1. For TLLSLPPFN, the winqsb package is suggested as a basic solution tool. 6 Nuerical Exaple [First Level] x 1,x 2 F 1 (x 1,x 2 )= x 1,x 2 (2,3,1,1)x 1 +(1,3,1,1)x 2 +(1,2,1,1)x 5, where x 3,x 4,x 5,x 6 solves [Second Level] x 3,x 4 F 2 (x)= x 3,x 4 (3,5,1,1)x 3 +(2,3,1,1)x 4 +(1,2,1,1)x 5 +(1,2,1,1)x 6, where x 5,x 6 solves [Third Level] x 5,x 6 F 3 (x 5,x 6 )= x 5,x 6 (1,2,1,1)x 3 +(3,5,1,1)x 5 +(2,4,1,1)x 6. x 1 + x 2 + x 3 + x 4 + x 5 + x 6 50, 2x 1 + x 2 40, 5x 3 + x 4 12, x 5 + x 6 20, x 5 + 5x 6 80, x 1,x 2,x 3,x 4,x 5,x Firstly, the FLDM solves his proble as follow: 1- Applying ranking function R(Ã)=a+b+ 2 1 (d c) where Ã=(a,b,c,d) F(R) to transfor the fuzzy nuber for into equivalent crisp for. So, the proble reduces to F 1 (x)=5x 2 + 3x 5 subject to x G. 2- Set, so the slack variable x 7 convert coon constraint into equation x 8,x 9,x 10 are artificial variables as : Let s identify iteration 0 as: x 1 + x 2 + x 3 + X 4 + x 5 + x 6 + x 7 = 50. X B =(X 7,x 8,x 9,x 10 ) T,X B =(50,1,1,1) T,C B =(0, M, M, M),B=1,B 1 = 1. Now iteration 1 for sub proble 1 where is : Minw 1 = 5x 1 4x 1 M, 2x 1 + x 2 40, x 1,x 2 0, ˆx 11 =(0,40), w 1 = 160 M. For sub proble 2 where j = 2 Minw 2 = M, 5x 3 + x 4 12, x 3,x 4 0, ˆx 21 =(0,0), w 2 = M. For sub proble 3 where j = 3 Minw 3 = x 5 x 6 M, x 5 + x 6 20, x 5 + 5x 6 80, x 5,x 6 0. ˆx 31 =(20,0), w 3 = 60 M. After 4 iterations the FLDM obtain his optial solution (x F 1,xF 2,xF 3,xF 4,xF 5,xF 6 )=(10,20,0,0,20,0).

8 314 O. E. Ea et. al. : On Fuzzy Three Level Large Scale Linear... Now set (x F 1,xF 2 )=(10,20) to the SLDM constraints. Secondly, the SLDM solves his/her proble as follows: Applying ranking function R(Â) = a+b+ 2 1 (d c) where à = (a,b,c,d) F(R) to transfor the fuzzy nuber for into equivalent crisp for. So, the proble reduces to F 2 (X)=8x 3 + 5x 4 + 3x 5 + 3x 6 x G 2. The SLDM do the sae action like FLDM till he obtains the optial solution (x S 3,xS 4,xS 5,xS 6 ) = (0,12,8,0), now set (x S 3,xS 4 )=(0,12) to the TLDM constraints. Finally, the TLDM solves his/her proble as follows: Applying ranking function R(Â) = a+b+ 1 2 (d c) where à = (a,b,c,d) F(R) to transfor the fuzzy nuber for into equivalent crisp for. So, the proble reduces to F 3 (x 5,x 6 )=8x 5 + 6x 6, x G 3. The TLDM do the sae action like FLDM and SLDM till he obtain the optial solution(x T 5,xT 6 )=(8,0). SO (x F 1,xF 2,xS 3,xS 4,xT 5,xT 6 ) = (10,20,0,12,8,0) is the optial solution for three level large scale linear prograing proble with fuzzy nubers. Where F 1 =(48,106,38,38), F 2 =(32,52,20,20) and F 3 =(24,40,8,8). 7 Conclusion This paper suggested an algorith to solve a three level large scale linear prograing proble with fuzzy nubers, where all coefficients of the objective functions are trapezoidal fuzzy nubers. A three-level prograing proble can be thought as a static version of the Stackelberg strategy. The suggested algorith used a linear ranking function at each level to define a crisp odel which is equivalent to the fuzzy nuber, then all decision akers attepts to optiize its proble separately as a large scale prograing proble using Dantzig and Wolfe decoposition ethod. Therefore, we handled the optiization process through a series of sub probles that can be solved independently. The solution algorith has a few features: 1. It cobines both a decoposition algorith and linear ranking function to obtain an optial solution solution for the TLLSLPPFN. 2. The objective functions and constraints involve any paraeters and the descision akers ay assign the different values. 3. It can be efficiently coded. 4. It is found that the decoposition based ethod generally leads better results than the traditional siplex-based ethods. Especially, the efficiency of the decoposition-based ethod increased sharply with the scale of the proble. Finally, a nuerical exaple was given to clarify the ain results developed in this paper. However, there are any other aspects, which should by explored and studied in the area of a large scale ulti-level optiization such as: 1- Large scale ulti-level fractional prograing proble with fuzzy paraeters in the objective functions and in the constraints and with integrality conditions. 2- Large scale ulti-level fractional prograing proble with stochastic paraeters in the objective functions and in the constraints and with integrality conditions. 3- Large scale ulti-level fractional prograing proble with rough paraeters in the objective functions and in the constraints and with integraity conditions. References [1] A. Abas, and P.Karai, Ranking functions and its application to fuzzy DEA, International Matheatical Foru, 30(3) (2008) [2] T. Abou - El- Enin, on the solution of a special type of large scale integer linear vector optiization probles with uncertainty data through TOPSIS approach, International Journal of Conteporary Matheatical Sciences, (6) (2011) [3] T. Abou - El- Enin, on the solution of a special type of large scale linear fractional ultiple objective prograing probles with uncertainty data, Applied Matheatical Sciences, (4) (2010) [4] M.A. Abo-sinna, and Abou - El- Enin, An interactive algorith for large scale ultiple objective prograing probles with fuzzy paraeters through TOPSIS approach, Yugoslav Journal of Operations Research, (21) (2011) [5] M. Balinski, An Algorith For Finding All Vertices of Convex Polyhedral Sets, SIAM Journal. 9 (1961)

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