Determining Efficient Solutions to Multiple. Objective Linear Programming Problems

Transcription

1 Applied Mathematical Sciences, Vl. 7, 2013, n. 26, HIKARI Ltd, Determining Efficient Slutins t Multiple Objective Linear Prgramming Prblems P. Pandian and M. Jayalakshmi Department f Mathematics, Schl f Advanced Sciences VIT University, Vellre-14, India pandian61@rediffmail.cm, m.jayalakshmi@vit.ac.in Cpyright 2013 P. Pandian and M. Jayalakshmi. This is an pen access article distributed under the Creative Cmmns Attributin License, which permits unrestricted use, distributin, and reprductin in any medium, prvided the riginal wrk is prperly cited. Abstract A new psteriri methd namely, mving ptimal methd is prpsed fr slving multiple bjective linear prgramming prblems. It prvides efficient slutin line segments t the prblem with the percentage level f satisfactin f each f the bjective at each pint n it which is very much useful t decisin makers fr chsing an efficient slutin accrding t their level f satisfactin n the bjective functins. Illustrative example is presented t clarify the idea f the prpsed apprach. Mathematics Subject Classificatin : 90B06, 9050, 90C05, 90C29 Keywrds: Multiple bjective linear prgramming prblem, Efficient slutin, Mving ptimal methd, Efficient line segment, Level f satisfactin 1. Intrductin In the classical linear prgramming mdel, a single linear bjective functin with linear cnstraints is nly cnsidered. In practice, many cnstrained linear mdels may invlve mre than ne bjective with cnflicting nature. Such prblems are termed as multiple bjective linear prgramming (MOLP) prblems. The MOLP mdels can arise in the fields f Science, Engineering and Management Science. The main aim f a MOLP prblem is t ptimize k different linear bjective functins, subject t a set f linear cnstraints where k 2. In the MOLP prblem,

2 1276 P. Pandian and M. Jayalakshmi ptimizing all bjective functins at the same time is nt pssible because f the cnflicting nature f the bjectives. The cncept f ptimality in the MOLP prblem is replaced with that f efficiency [1,10]. In the literature, a variety f algrithms [ 6, 2, 15, 14, 7, 11, 13, 4, 9, 8, 3, 12] fr finding efficient slutins t multi-bjective ptimizatin prblems have been prpsed. Fr getting gd slutin frm a set f efficient slutins, a decisin maker is needed t prvide additinal preference infrmatin and t identify the mst satisfactry slutin. Hwang and Masud [6] classified the methds fr slving multi-bjective ptimizatin prblems int three categries namely, priri methds, interactive methds and psteriri methds. The psteriri methds prvide the whle picture f slutins space f the prblem. Hence, these methds are mst preferred ne fr decisin makers. cnsidered the fuzzy apprach t slve the MOLP prblem. In this paper, we prpse a new psteriri methd fr finding efficient slutins and efficient slutin line segments t the MOLP prblem, called. mving ptimal methd. In the prpsed methd, slving the jth bjective related prblem using the ptimal slutin f kth bjective related prblem as an initial slutin by the simplex methd and cllecting all feasible slutins btained by each f iteratin f the simplex methd; this prcess is repeated fr all pssible pairs f the bjective related prblems ; cllecting all efficient slutins frm the set f the slutins btained and the percentage level f satisfactin f each f the bjective functin fr each efficient slutin is prvided fr the benefit f decisin makers. Hence, the mving ptimal methd is based n the simplex methd nly which differs ttally frm utility functin methd, gal prgramming apprach, min-max methd, tw phase methd, fuzzy prgramming technique, genetic apprach and evlutinary apprach. Numerical example is given fr better understanding the prcedures f the prpsed methd. Efficient slutin line segments t MOLP mdels prvided by the mving ptimal methd are very much helpful fr the decisin makers t evaluate the ecnmical activities and make self satisfied managerial decisins when they are handling MOLP mdels. 2. Preliminaries Cnsider the fllwing MOLP prblem : (P) Maximize Z( X ) = (Z1( X ),Z2( X ),..., Zk ( X )) subject t AX B, X 0 n where Zi : R R ( i = 1,2,..., k ) is linear, A is an m n real matrix, m n B = ( b 1,..., b m ) R and X R, an n-dimensinal Euclidian space.

3 Determining efficient slutins 1277 n Let P = { U R ; AU B andu 0} be the set f all feasible slutin t the prblem (P). Nw, we need the fllwing definitins which can be fund in [1,2,3] Definitin 2.1 A feasible pint X P is said t be an efficient slutin fr (P) if there exists n ther feasible pint X f the prblem (P) such that Z i ( X ) Zi( X ), i = 1,2,..., k and r ( Z X ) > Zr( X ), fr sme r { 1,2, K,k}. That is, an efficient slutin is the slutin that cannt be imprved in ne bjective functin withut deterirating their perfrmance in at least ne f the rest. Fr simplicity, ( Z1( X ), Z2( X ),..., Zk ( X )) is called a slutin t the prblem (P) if X is a slutin t the prblem (P). Nw, the prblem (P) can be separated int k number f single bjective LP prblems as fllws: ( P t ) Maximize f t (X ) subject t AX B, X 0, where t = 1,2,..., k. t Definitin 2.2 : Let X be the ptimal slutin t the prblem ( P t ), t = 1,2,..., k. Then, the value f the bjective functin (Z1( X 1 ),Z2( X2),..., Z k ( X k )) is called the ideal slutin t the MOLP prblem (P). Definitin 2.3 : An efficient slutin X P is said t be the best cmprmise slutin t the prblem (P) if the distance between the ideal slutin and the bjective value at X, (Z1( X ), Z2( X ),..., Zk ( X )) is minimum amng the efficient slutins t the prblem (P). 3. Mving Optimal Methd Nw, we define the fllwing new terms namely, efficient slutin line segment and the percentage level f satisfactin f the kth bjective f the MOLP prblem fr the given slutin t the prblem (P) which are used in the prpsed methd.

4 1278 P. Pandian and M. Jayalakshmi Definitin 3.1: Let (Z1( X ), Z2( X ),..., Zk ( X )) and (Z1( U ), Z2( U ),..., Zk ( U )) be the bjective value f the prblem (P) at the efficient slutins X and U respectively. The line segment jining the pints (Z1( X ), Z2( X ),..., Zk ( X )) and (Z1( U ), Z2( U ),..., Zk ( U )) is said t be an efficient slutin line segment if each pint n the line segment is an efficient slutin t the prblem (P). Definitin 3.2: The percentage level f satisfactin f the bjective f the prblem P ) fr the slutin U t the prblem (P), LS( Zt ; U ) is defined as fllws: ( t Zt ( U ) 100 Zt ( X ) t if the prblem LS( Zt; U ) = 2Z ( ) ( ) if the prblem t Xt Zt U 100 Zt ( Xt ) ( P ) t ( P ) t is maximizatin type is minimizatin type. Nw, we intrduce a new methd namely, mving ptimal methd fr finding efficient slutins and efficient slutin line segment with their percentage level f satisfactin t the MOLP prblem. The prpsed methd prceeds as fllws: Step 1: Cnstruct k number f single bjective LP prblems frm the given prblem (P) as fllws: ( P t ) Maximize Z t (X ) subject t AX B, X 0, where t = 1,2,..., k. Step 2: Cmpute the ptimal slutin t the prblem P ) using the simplex methd fr t = 1,2,..., k. Let the ptimal slutin be the ideal slutin t the given prblem (P). ( t X t, t = 1,2,..., k. Then, btain Step 3: (a) Take t { 1,2,..., k}. (b) Cmpute all feasible slutins t the prblem ( P s ), s = 1,2,..., k, u t up t its ptimal slutin cnsidering the ptimal slutin t the prblem ( P t ), X t as an initial basic feasible slutin t the prblem ( P s ) using the simplex methd.

5 Determining efficient slutins 1279 Step 4: Repeat the Step 3. fr each value f t in {1,2,..., k} and cllect all slutins t the given prblem (P) with its bjective values. Step 5: Cllect all efficient slutins t the given prblem (P) frm the Step 4. and Cmpute the distance between the ideal slutin t the given prblem (P) and the bjective value f the given prblem at each f the efficient slutin, that is, each efficient slutin t the given prblem (P). Step 6. Draw a plygn cnnecting the efficient slutins t the given prblem (P) such that atleast ne f the c-rdinates is in ascending rder and mark the efficient slutin line segments which are the sides f the plygn. Step 7: Find the best cmprmise slutin t given prblem (P) and cnstruct the satisfactin level table fr the efficient slutins f the prblem (P). Step 8: The values f the decisin variables f the prblem (P) fr a pint n the efficient slutin line segment can be btained using the bjective values at the pint and the matrix methd. Remark 3.1: In general, the prpsed methd prvides an infinite number f efficient slutins t the given MOLP prblem which are very much useful t decisin makers fr selecting a slutin accrding t their satisfactin. The prpsed methd fr slving the MOLP prblem is illustrated by the fllwing example. Example 3.1: Cnsider the fllwing MOLP prblem: (P) Maximize Z ( X ) = ( Z1( X ), Z2( X )) = ( x1 + 2x2,2x1 + x2) subject t x 1 + 3x 2 21; x 1 + 3x 2 27 ; 4x 1 + 3x2 45 ; 3x 1 + x2 30 ; x 1, x2 0. Nw, by the Step 1. t Step 6. f the prpsed methd, the ideal slutin I = ( 14,21 ) and the efficient slutins X 1 = (0,7 ), X 2 = (3,8), X 3 = (6,7) and X 4 = (9,3) and their crrespnding bjective values are A = ( 14, 7 ), B = ( 13, 14), C = ( 8, 19) and D = ( 3, 21) respectively, are btained. By the Step 7, the line segments DC, CB and BA, that is, the sides f the plygn ABCD are efficient slutin segments. Nw, by the Step 8., M = ( 10,17) is the pint n CB such that the distance between M and I is minimum. S, M = ( 10,17) is the best cmprmise slutin t the given prblem.

6 1280 P. Pandian and M. Jayalakshmi Nw, by the Step 9, the pint crrespnding t M = ( 10,17) is ( 4.8, 7.4) and the level f satisfactin table is given belw: S.N. Efficient slutin (X) Z (X ) LS ( Z 1 ; X ) LS ( Z 2 ; X ) 1 X 1 = (0,7) A = ( 14, 7) G 1 = ( 3t1,7 + t1) ; t 1 [0,1] Z ( G 1 ) LS ( Z 1 ; G1 ) LS ( Z 2 ; G1 ) 3 X 2 = (3,8) B = ( 13, 14) G2 = ( 3 + 3t2,8 t2) ; t 2 [0,1] Z ( G2) LS ( Z 1 ; G2) LS ( Z 2 ; G2) 5 X 3 = (6,7) C = ( 8,19) G3 = ( 6 + 3t3,7 4t3) ; t 3 [0,1] Z ( G3) LS ( Z 1 ; G3) LS ( Z 2 ; G3) 7 X 4 = (9,3) D = ( 3, 21) X 5 = (4.8,7.4) M = (10,17) where Z ( G1 ) = (14 t1,7 + 7t1) ; Z ( G2) = (13 5t2,14 + 5t2) ; Z ( G3) = (8 11t3,19 + 2t3) ; 14 t ( ; ) t LS Z1 G1 = 100; ( ; ) 1 LS Z2 G1 = 100 ; t ( ; ) t LS Z1 G2 = 100 ; ( ; ) 2 LS Z2 G2 = 100 ; t ( ; ) t LS Z1 G3 = 100 and ( ; ) 3 LS Z2 G3 = Remark 3.2: Fr the Example 4.1., the best cmprmise slutin given by De and Bharti Yadav [3] is (8,19), but by the prpsed methd, we prvide three efficient

7 Determining efficient slutins 1281 slutin line segments, Z ( G 1 ), Z ( G2) and Z ( G3) including the best cmprmise slutin (10,17) which is better than (8,19). 4. Cnclusin In this paper, the mving ptimal methd fr finding efficient slutins t the MOLP prblem is prpsed. The efficient slutin line segments with level f satisfactin prvided by the mving ptimal methd enables the decisin makers t chse an apprpriate slutin depending n their financial psitin and their level f satisfactin f bjectives. In near future, we extend the mving ptimal methd t multiple bjective integer linear prgramming prblems and multiple bjective fuzzy linear prgramming prblems. REFERENCES [1] V. Chankng and Y.Y. Haimes, Multi-bjective Decisin Making Thery and Methdlgy, Nrth-Hlland, New Yrk, [2] H.K.Chen and H.W.Chu, Slving multibjective linear prgramming prblem A generic apprach, Fuzzy sets and systems, 82(1996) [3] P. K. De and Bharti Yadav, An algrithm fr btaining ptimal cmprmise slutin f a multi bjective fuzzy linear prgramming prblem, Internatinal Jurnal f Cmputer Applicatins, 17(2011), [4] S.I. Gass and P. G. Ry, The cmprmise hypersphere fr multibjective linear prgramming, Eurpean Jurnal f Operatinal Research, 144(2003), [5] A.M. Geffrin, Prper efficiency and the thery f vectr maximizatin, Jurnal f Mathematical Analysis and Applicatins, 22(1968), [6] C.L. Hwang and A.S.M. Masud, Multiple bjective decisin making : Methds and Applicatins: A sate f the art survey, Lecture ntes in Ecnmics and Mathematical Systems, Vl.164, Spring-Verglag, Berlin, [7] M. Ida, Efficient slutin generatin fr multiple bjective linear prgramming based n extreme ray generatin methd, Eurpean Jurnal f Operatinal Research, 160(2005), [8] Ivan P. Stanimirvic, Milan Lj. Zlatanvic and Mark D. Petkvic, On the linear weighted sum methd fr multi-bjective ptimizatin, Facta Acta Universitatis, 26 (2011), !

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