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Solving a possibilistic linear program through compromise programming

Jiménez López, Mariano,Rodríguez Uría, Ma. Victoria,Arenas Parra, Mar,Bilbao Terol, Amelia

Abstract

In this paper we propose a method to solve a linear programming problem involving fuzzy parameters whose possibility distributions are given by fuzzy numbers. To address the above problem we have used a preference relationship of fuzzy numbers that leads us to a solving method that produces the so-called $\alpha$-degree feasible solutions. It must be pointed out that the final solution of the problem depends critically on this degree of feasibility, which is in conflict with the optimal value of the objective function. Then DM faces a bi-objective problem that we will solve through a Compromise Programming approach, whose solution lets the Decision-Maker express his own preferences about feasibility versus optimality. Our proposed method will be illustrated by a numerical example

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RZA`{ B ? ; w K \, K K D ) + ? ; {p z b K \?  K K,?  (  w d D bY K  ) w K  @ , K   ,333, K  z D K \ )>K y 7i H E?z K K ) w K K @ , K K  ,333, K K E D  }Rt PZ99k }ssT 8R bYBR }BRR$A$j$k I$R$AZ$BtR s $%t Ak PZ99k tZ8AR 7Bj%$t s _BRR$A$j$R${ N$ts _Bs8 qYBZY ]B8}B8$R _Bs88$t d !Z s}}Bs{Y $t Y }%$BZR }s} sjjBbI ZR B BAs$t s PZ99k B}$8sj RBjZ$Bt $t Y BA`{$% R}s{ PB }BAj8 w d D  yt Y }Rt bB, b }B}BR BAs$t$t s tBt PZ99k B}$8sj RBjZ$Bt $t Y I{$R$Bt %s$sAjR R}s{ B Y Rs8 }BAj8 qB IB Ys Y PBjjBb$t JZR$BtR >H s$RU d D <Bb {st b Iht Y PsR$A$j$k BP s I{$R$Bt %{B - bYt Y }ss8R BP {BtRs$tR s PZ99k tZ8AR2 1 D <Bb {st b Iht Y B}$8sj$k PB st BA`{$% PZt{$Bt b$Y PZ99k }ssT 8R2 7%sj sZYBR Ys% RZI$I Y$R }BAj8 7B8 BP Y8 Ak {BtR$I$t tB Btjk PZ99k }ss8R AZ sjRB Y PZ99k BsjR BP Y I{$R$Bt 8s, $ qsts,s stI VRsk >d(H 7s,sbs >H !YR Ak ZR$t RB8 js$BtRY$} BP {B8}s$RBt BP PZ99k tZ8AR $ ]sItsR stI zIsk >nH 7s,sbs >H !Z s}}Bs{Y AjBtR B Y R{BtI BZ} G ZR s PZ99k js$BtRY$} B {B8}s PZ99k tZ8AR I%jB}I Ak F$8t9 >H Ys %$Pk BBI }B}$R RZ{YU I$R$tZ$RYsA$j$k s$Btsj$k PZ99k }Rts$Bt stI BAZRtRR qY$R 8YBI $R {B8}Zs$Btsjjk ^{$t B RBj% st N_ }BAj8 A{sZR $ }R%R Y j$ts$k BP Y }BAj8 PB }Rt$t Y }Pt{ js$BtRY$} BP PZ99k tZ8AR Ys b b$jj ZR b RYBZjI {sjj Y tB$BtR BP Y -}{I $t%sj stI Y -}{I %sjZ BP s PZ99k tZ8A <$j}t >xH IhtR Y -}{I $t%sj BP s PZ99k tZ8A K y  bY${Y b$jj A tBI o{ w K y D  sRU o{ w K y D )> o y @ ,o y  H) }8 @  W  d y w _ D Q_, 8 d ( /  d y w _ D Q_ ] w 1 D bY W y w _ D stI / y w _ D s Y jP stI Y $Y YstI R$I BP Y 88ARY$} PZt{$Bt BP Y PZ99k tZ8A K y  VtI Y -}{I %sjZ BP s PZ99k tZ8A K y  bY${Y b$jj A tBI o~ w K y D sRU o~ w K y D ) o y d L o y 1 1 w n D yP K y ) w y d ,y 1 ,y n ,y ; D $R s s}9B$Isj PZ99k tZ8A $R -}{I $t%sj stI $R -}{I %sjZ {st A {sj{ZjsI %k sR$jkU o{ w K y D ) } y d L y 1 1 ,y n L y ; 1 ]  o~ w K y D ) y d L y 1 L y n L y ; ; w ; D *ht$$Bt d 3B stk }s$ BP PZ99k tZ8AR K y kK K b Iht Y js$BtRY$} BP PZ99k }Pt{ - w K y, K K D  YBZY Y PBjjBb$t 88ARY$} PZt{$Bt >HU - w K y, K K D ) T  z  Z (R$ o y 1  o K d c ( o y !  o K A o y !  o K A  w o y A  o K ! D R$ ( ; J o y d  o K 1 ,o y 1  o K d o dR$ o y d  o K 1 k ( w x D bY - p K y, K K Q $R Y I BP }Pt{ BP K y B% K K  yP - p K y, K K Q    b ItB $ Ak K y   K K stI b Rsk Ys K y $R s Yst K K s jsR $t s I   GYt - p K y, K K Q ) d 1 bRskYsK y stI K K s $tI$&t dM  F$8?t9 N?B}9 z iBI?Z9 =?s  VtsR  m V $jAsB *ht$$Bt 1 V I{$R$Bt %{B ? ; {p z $RRs$IBA  TPsR$Aj B }BAj8 w d D $PU 8$t i @ ,333,E \ - w y i ?,K i D i )  w E D Ys $R B Rsk $P K y i ?   K K i ,  i )d ,333,E w  D qYPB sP w x D b b$ Y PBjjBb$t -}RR$BtU J w d   D o y E  L o y E @ o ?  o K E  L w d   D o K E @ w M D qY R BP sjj  TPsR$Aj I{$R$Bt %{BR $R ItBI Ak  w  D  3B8 Iht$$Bt 1 Y PBjjBb$t }B}k $88I$sjk PBjjBbRU  @ c  p w  @ D V w   D w  D k 8stR BP Iht$$Bt 1 b Ys% stRbI Y hR JZR$BtU YBb B Iht PsR$A$j$k bYt Y {BtRs$tR $t%Bj% PZ99k tZ8AR 7$t{  $tI${sR Y PsR$A$j$k I BP s{Y {BtRs$t d   {st A $tT }I sR Y I BP ZtPsR$A$j$k $ d   $%R ZR s 8sRZ BP $R, BP %$Bjs$Bt BP RZ{Y {BtRs$t qB stRb Y R{BtI JZR$BtU YBb B Iht Y B}$8sj$k PB st BA`{$% PZt{$Bt b$Y PZ99k }ss8R b {BtR$I Y PBjjBb$t }BAj8U $t$8$9 K F )K ? RZA`{ B U ? ; w \,K D ) \ ? ; {p z b \?  K,?  ( i w d( D bY Y PZ99k }ss8R s Btjk $t Y BA`{$% PZt{$Bt V{{BI$t B Y st,$t 8YBI $%t $t Iht$$Bt d b {st Rsk YsU *ht$$Bt n ? L ; w \,K D $R Rs$I B A st s{{}sAj B}$8sj RBjZ$Bt B }BAj8 w d( D $P $ $R %$hI YsU K ?  @ b  K ? L ,  ? ; w \,K D w dd D 3B8 Iht$$Bt d Y sAB% -}RR$Bt {st A b$t sR PBjjBbRU o   ?   o  0 ? L @ o  0 ?   o  0 ? @ L o  0 ? L   o  0 ? L @  d 1 w d1 D RB Ys o  0 ? @ L o  0 ?  1  o  0 ? L @ L o  0 ? L  1 w dn D qYPB PB8 JZs$Bt w n D stI Iht$$Bt n Y PBjjBb$t }B}BR$$Bt {st A RsIU _B}BR$$Bt d ? L ; {p z $R st s{{}sAj B}$8sj RBjZ$Bt B }BAj8 w d( D $P stI Btjk $P $ $R st B}$8sj RBjZ$Bt B Y PBjjBb$t {$R} j$ts }Bs8U $t$8$9 o~ w K  D ? RZA`{ B U ? ; w \,K D w d; D 7Bj%$t s _BRR$A$j$R${ N$ts _Bs8 qYBZY ]B8}B8$R _Bs88$t d bY o~ w K  D ) w o~ w K  @ D ,o~ w K   D ,333,o~ w K  z D D  V{{BI$t B Y$R RZj b 8sk RsAj$RY Y PBjjBb$t Iht$$Bt jsI B Y $t$$sj PZ99k 8BIj w d D  *ht$$Bt ; ? L ; {p z $R Rs$I B A st  TB}$8sj RBjZ$Bt B }BAj8 w d D $P $ $R st B}$8sj RBjZ$Bt B Y PBjjBb$t }BAj8U $t$8$9 o~ w K  D ? 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