CASOS ESPECIALES DEL MÉTODO GRÁFICO
DANIELA LEE PERNETT KATHERINA MARMOLEJO MEJÍA MARY QUINTERO PARRA GLORIA TORREGROZA
TRABAJO DE INVESTIGACIÓN DE OPERACIONES
NELSON ZUÑIGA
UNIVERSIDAD DEL ATLANTICO FACULTAD DE INGENIERÍA PROGRAMA DE INGENIERÍA QUÍMICA PUERTO COLOMBIA, ATÁNTICO !"#
CONTENIDO
INTRODUCCIÓN " "$ PROGRAMACIÓN LINEAL % "$"$ SUPUESTOS DE LA PROGRAMACIÓN LINEAL $ CLASES DE SOLUCIÓN
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'$ CASOS ESPECIALES % '$"$ REDUNDANTES O SOBRANTES '$$ SOLUCIÓN ÓPTIMA MÍLTIPLE & '$'$ SOLUCIÓN ÓPTIMA NO ACOTADA '$%$ SOLUCIÓN NO FACTIBLE & %$ CONCLUSIONES BIBLIOGRAFÍA
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PROGRAMACIÓN LINEAL L( )*+*(-(./01 2/13(2 34 51 (2+*/6-+ 31 32 .5(2 43 *34532731 4/65(./+134 *3(234 8 2+ 953 43 :54.( 34 ;(*23 51( 4+25./01 +)6/-/<(;( )(*( (4= (5-316(* 2( )*+;5.6/7/;(; *34)3.6+ ( 2+4 *3.5*4+4 8 (4= (5-316(* 2+4 :313>/./+4 )*/1./)(2-3163 3.+10-/.+4$ C5(1;+ 43 ?(:2( ;3 +)6/-/<(./01, 32 +:@36/7+ 34 -(/-/<(* + -/1/-/<(* >51./+134 2/13(234 31 7(*/(4 7(*/(:234 *3(234 .+1 *346*/../+134 2/13(234, +)6/-/<(1;+ 51( >51./01 +:@36/7+ 6(-:/1 2/13(2$ E46( 6.1/.( -(63-6/.( 34 *32(6/7(-3163 1537( 953 45*3 31 32 4/2+ 8 6/313 -5.?( ()2/.(./01 31 2( *34+25./01 ;3 )*+:23-(4 31 32 -:/6+, 4+:*3 6+;+, ;3 2(4 C/31./(4 S+./(234$ B*373-3163, 2( ()2/.(./01 -4 .+-1 ;3 2( )*+*(-(./01 2/13(2 (:(*.( 32 )*+:23-( 313*(2 ;3 (4/1(* *3.5*4+4 2/-/6(;+4 316*3 (.6/7/;(;34 .+-)36/6/7(4 ;3 2( -3@+* -(13*( )+4/:23 31 >+*-( 0)6/-($ E463 )*+:23-( ;3 (4/1(./01 )53;3 45*/* .5(1;+ ;3:( 323/*43 32 1/732 ;3 ./3*6(4 (.6/7/;(;34 953 .+-)/631 )+* *3.5*4+4 34.(4+4 )(*( *3(2/<(*2(4$ L( 7(*/3;(; ;3 4/65(./+134 ( 2(4 953 43 )53;3 ()2/.(* 346( ;34.*/)./01 34 4/1 ;5;( -58 *(1;3, 8 7( ;34;3 2( (4/1(./01 ;3 /146(2(./+134 )*+;5.6/7(4 ( 2+4 )*+;5.6+4, ?(46( 2( (4/1(./01 ;3 2+4 *3.5*4+4 1(./+1(234 ( 2(4 13.34/;(;34 ;3 51 )(=4 ;34;3 2( )2(13(./01 (*=.+2(, ?(46( 32 ;/43+ ;3 51( 63*()/( ;3 *(;/(./01 36.$ N+ +:46(163, 32 /1*3;/3163 .+-1 ;3 6+;(4 346(4 4/65(./+134 34 2( 13.34/;(; ;3 (4/1(* *3.5*4+4 ( 2(4 (.6/7/;(;34$ E4.+3* 51( (263*1(6/7( )53;3 43* -4 .+-)23@( ;3 2+ 953 )(*3.3 8( 953 43 ;3:31 4(6/4>(.3* 7(*/(4 )(56(4 8 2( )*+*(-(./01 2/13(2 )5;+ /* 51 )(4+ (;32(163 8 ;/7/;/* 346+4 .*/63*/+4 31 2(4 .(63+*=(4 *346*/../+134 8 +:@36/7+$ L(4 *346*/../+134 4+1 (95322(4 .+1;/./+134 953 ;3:3 .5-)2/* 51( 4+25./01 953 346 :(@+ +:43*7(./01$ S/ -4 ;3 51( (263*1(6/7( 4(6/4>(.3 6+;(4 2(4 *346*/../+134, 32 +:@36/7+ 43 54( )(*( 34.+3* 316*3 6+;(4 2(4 (263*1(6/7(4 2( 953 -4 +)6/-/.3 2+4 *3.5*4+4$
1. SUPUESTOS DE LA PROGRAMACIÓN LINEAL P(*( .(;( -+;32+ 3/4631 7(*/(4 45)+4/./+134 8 2( 56/2/;(; ;3 51 -+;32+ 43 *32(./+1( ;/*3.6(-3163 .+1 2( *3(2/;(; ;3 2+4 45)5346+4$ L( )*/-3*( 34 51( Suposición de Proporción, 2( .5(2 6/313 953 73* .+1 2( >+*-( 2/13(2 ;3 2(4 >51./+134$ S5 +:@36/7+ 34 2/13(2, 346+ 95/3*3 ;3./* 953, 2( .+16*/:5./01 (2 +:@36/7+ ;3 .5(295/3* ;3./4/01 34 ;/*3.6(-3163 )*+)+*./+1(2 (2 7(2+* ;3 2( 7(*/(:23 ;3 ;3./4/01$ L( 4351;( 34 2( Suposición de Adición, 346( 34 .5(1;+ 2( .+16*/:5./01 ;3 51( 7(*/(:23 ( 2( >51./01 +:@36/7+ 34 /1;3)31;/3163 ;3 2+4 7(2+*34 ;3 2(4 +6*(4 7(*/(:234$ T(-:/1, 346 2( Suposición de ser Divisible, 34 ;3./*, 2( )+4/:/2/;(; ;3 6+-(* 51( >*(../01 ;3 .5(295/3* 7(*/(:23$ S/ 2( 45)+4/./01 ;3 43* ;/7/4/:23 1+ 34 72/;(, 316+1.34 43 54(* 2( 6.1/.( ;3 )*+*(-(./01 2/13(2 3163*($
F/1(2-3163, 2( 26/-( 45)+4/./01 34 32 Supuesto de Certeza, 8 346( 34 953 2( )*+*(-(./01 2/13(2 1+ )3*-/63 /1.3*6/;5-:*3 31 2+4 7(2+*34$ E/4631 2/-/6(./+134 )*.6/.(4 )(*( 32 54+ ;3 2( )*+*(-(./01 2/13(2$ U1( 43 *32(./+1( .+1 2+4 .2.52+4$ E1 313*(2 43 13.34/6( 51( .+-)56(;+*($ D34(>+*651(;(-3163, 2(4 .(2.52(;+*(4, (51 2(4 )*+*(-(:234, 4+1 )+.+ 6/234, )5346+ 953 2( )*+*(-(./01 2/13(2 6/313 13.34/;(; ;3 *(1 .(16/;(; ;3 -3-+*/( + (2-(.31(-/316+$ S/ 1+ 43 6/313 (..34+ ( 51( .+-)56(;+*(, 43 346(* 2/-/6(;+ ( )*+:23-(4 -58 431./22+4$ L( +6*( 2/-/6(./01 43 *3>/3*3 (2 .+46+ ;3 >+*-52(* 51 )*+:23-( ;3 )*+*(-(./01 2/13(2$ A1634 ;3 3-)*31;3* 51( ()2/.(./01 ;3 )*+*(-(./01 2/13(2, ;3:3 .+14/;3*(*43 2( ;/4)+1/:/2/;(; 8 32 .+46+ ;3 2+4 ;(6+4 13.34(*/+4$ CLASES DE SOLUCIÓN$ CASOS ESPECIALES
1. REDUNDANTES O SOBRANTES E1 2+4 -+;32+4 ;3 )*+*(-(./01 2/13(2 3/463 51 6/)+ ;3 *346*/../+134 953 1+ ;343-)3(1 1/11 *+2 31 2( ;363*-/1(./01 ;32 .+1@516+ 4+25./01 ;3 /5(2 -(13*( 31 2( 4+25./01 0)6/-(, 2+ 953 .+12237( ( .+1.25/* 953 346(4 4+1 *3;51;(1634 + /113.34(*/(4$
Ejemplo: L( -(**+95/13*=( P5*+ C53*+ )*3631;3 /11+7(* 31 32 -3*.(;+ .+1 ;+4 6/)+4 ;3 .(*63*(4 ;31+-/1(;+4 A 8 B$ C(;( 51+ ;3 322(4 ;3:3 )(4(* )+* 6*34 +)3*(./+134 (1634 ;3 45 .+-3*./(2/<(./01 32 .+*63 ;3 2+4 -+2;34 31 32 -(63*/(2, 51/01 ;3 2+4 -+2;34 8 .+16*+2 ;3 .(2/;(;$ L(4 .(*63*(4 6/)+ A *395/3*31 " ?+*(4 ;3 .+*63, ?+*(4 31 2( 51/01 ;3 454 -+2;34 8 " ?+*( ;3 .+16*+2 ;3 .(2/;(; 2(4 .(*63*(4 6/)+ B *395/3*31 , 8 " ?+*(4 *34)3.6/7(-3163$ E2 -(*31 .+16*/:56/7+ )+* .(;( .(*63*( 6/)+ A 8 B 34 ;3 "!!!!! 8 &!!! *34)3.6/7(-3163$ L( -(**+95/13*=( ;/4)+13 .+-+ -/-+ 43-(1(2-3163 ' ?+*(4 31 .+*63 ;3 -+2;34, % ?+*(4 .+4/31;+ 346+4 -+2;34 8 "& ?+*(4 ;3 .+16*+2 ;3 .(2/;(;, ;3:/;+ (2 2/-/6(;+ )3*4+1(2 .+16*(6(;+$ C+1 :(43 31 2( /1>+*-(./01 45-/1/46*(;( ;363*-/13 2(4 51/;(;34 ( )*+;5./* 43-(1(2-3163 ;3 .(;( *3>3*31./( )(*( -(/-/<(* 2(4 56/2/;(;34$ Las variables :
C(16/;(; ;3 .(*63*(4 6/)+ A ( )*+;5./* 43-(1(2-3163 Y C(16/;(; ;3 .(*63*(4 6/)+ B ( )*+;5./* 43-(1(2-3163
Las restricciones : X + 2 Y ≤ 36
6 X + 8 Y
≤ 48
X + Y ≤ 15
Función Objetivo: Z máx =100000 X + 95000 Y
No negatividad : X , Y ≥ 0
L( *>/.( *34526(163 34 2( 4/5/3163
L( 4+25./01 0)6/-( .+**34)+1;3 ( ! 8 Y 8 2( >51./01 +:@36/7+ 953;(*=($ Z máx =100000 ( 0 )+ 95000 ( 6 )
Z máx = $ 570000
C2(*(-3163 43 )53;3 +:43*7(* 953 2(4 *346*/../+134 *(>/.(;(4 .+1 2( *3.6( (<52 8 73*;3 1+ ;363*-/1(1 32 .+1@516+ 4+25./01, )+* 31;3 43 ;31+-/1(1 *346*/../+134 *3;51;(1634 + 4+:*(1634$ $ SOLUCIÓN ÓPTIMA MLTIPLE
S3 ?( 7/46+ .+1 (163*/+*/;(; 953 )(*( 2( 4+25./01 ;3 51 3@3*././+ ;3 )*+*(-(./01 2/13(2, 31 32 .(4+ -4 431./22+, .+14/463 31 *3)*34316(* .(;( 51( ;3 2(4 *346*/../+134 8 31.+16*(* 32 )+2=+1+ >(.6/:23 .+1@516+ 4+25./01 + *3/01 >(.6/:23 31 32 .5(2 )+* *(<+134 6*/+1+-6*/.(4 31 51+ ;3 454 7*6/.34 43 31.5316*( 2( -3@+* *34)5346(, 34 ;3./*, 2( 4+25./01 0)6/-($ S/1 3-:(*+, 51( ;3 2(4 7(*/(1634 953 )53;3 )*34316(* 51 3@3*././+ ;3 )*+*(-(./01 2/13(2 .+14/463 31 6313* -4 ;3 51( 4+25./01 0)6/-($ E46+ +.5**3 .5(1;+ 2( >51./01 +:@36/7+ 34 395/7(23163 ( 51( ;3 2(4 >51./+134 ;3 *346*/../01$
E46+4 )*+:23-(4 ;3:31 ;3 *34+273*43 ;3 >+*-( 953 )*3;+-/13 32 (12/4/4 ;3 4314/:/2/;(;, 34 ;3./* 51( 73< 31.+16*(;(4 2(4 -26/)234 4+25./+134 /5(234 43 ;3:3 )*+.3;3* (2 .+-)+*6(-/316+ ;32 .+145-+ ;3 2+4 *3.5*4+4 8 *346*/../+134, 37/;3163-3163 )*37(23./31;+ 32 .+1.3)6+ ;3 )*+;5.6/7/;(; ;3 2+4 *3.5*4+4 -4 2/-/6(;+4 8 .+46+4+4$
Ejemplo:
L( 3:(1/463*=( M53:234 J(-4 ?( *3./:/;+ 51( *(1 .(16/;(; ;3 )(*634 )*3>(:*/.(;(4 )(*( 2( 32(:+*(./01 ;3 -34(4$ L(4 -34(4 953 )53;31 32(:+*(*43 ;3 2(4 )(*634 )*3>(:*/.(;(4 4+1 ;3 ;+4 -+;32+4, -+;32+ A 8 B, 8 346(4 1+ *395/3*31 -4 953 43* 314(-:2(;(4 8 )/16(;(4$ E46( 43-(1( 43 ?( ;363*-/1(;+ ;3;/.(* ?+*(4 ;3 314(-:23 8 "& ;3 )/165*( )(*( 32(:+*(* 2( -(8+* .(16/;(; ;3 -34(4 )+4/:234 631/31;+ 31 .5316( 953 .(;( -34( -+;32+ A *395/3*3 ;3 # ?+*(4 ;3 314(-:23 8 ;3 )/165*( *34)3.6/7(-3163, 8 953 .(;( -34( -+;32+ B *395/3*3 ;3 "% ?+*( ;3 314(-:23 8 " ;3 )/165*( *34)3.6/7(-3163$ S/ 32 -(*31 ;3 56/2/;(; 34 ;3 !! )+* .(;( -34( -+;32+ A 8 "!! )+* .(;( -34( -+;32+ B$ D363*-/13 32 -+;32+ (;3.5(;+ ;3 )*+;5../01 )(*( 346( 43-(1($
Variables:
C(16/;(; ;3 -34(4 -+;32+ A ( >(:*/.(* 346( 43-(1( Y C(16/;(; ;3 -34(4 -+;32+ B ( >(:*/.(* 346( 43-(1(
Restricciones :
7 X + 14 Y
2 X + Y
≤ 98
≤ 15
Función objetivo:
Z máx =200 X + 100 Y
No negatividad : X , Y ≥ 0
L( *>/.( *34526(163 43*=(
C+-+ )*/-3*( (1+6(./01, 43 ;3:3 6313* 31 .5316( 953 2( *3.6( (<52 8 -+*(;( 4+1 2(4 *346*/../+134, 8 2( 73*;3 34 2( >51./01 +:@36/7+$ C+-+ 43 )53;3 +:43*7(*, -3;/(163 2( 3+-36*=( 2( *3.6( 953 *3)*34316( 2( >51./01 +:@36/7+ 34 )(*(232( ( 2( *3.6( 953 *3)*34316( 2( *346*/../01 N$
C+-+ 1+4 )+;3-+4 ;(* .5316( 31 ;+4 )516+4 43 )*34316(1 4+25./+134 0)6/-(4, 953 4+1 2+4 )516+4 C 8 D$
O:43*73-+4 2( 4+25./01 0)6/-( -26/)23
Z máx ( A )=200 ( 0 ) + 100 ( 0 )= 0 Z máx ( B )= 200 ( 0 )+ 100 ( 7 )=700
Z máx ( C )=200 ( 7,5 ) + 100 ( 0 )=1500 Z máx ( D )=200 ( 16 / 3 )+ 100 ( 13 / 3 )= 1500
E/4631 316+1.34 ;+4 4+25./+134 0)6/-(4
S+25./01 0)6/-( N" #$& 8 Y !
S+25./01 0)6/-( N "' 8 Y"''
P(*( *34+273* 32 )*+:23-( ;3:3 ?(:3* 51 31>+953 :(4(;+ 31 >(.6+*34 6(234 .+-+ 51 (12/4/4 ;3 4314/:/2/;(; ;+1;3 43 631( 31 .5316( 32 .+145-+ ;/46/16+ ;3 ;363*-/1(;+4 *3.5*4+4 ?+*(4 314(-:23 74$ ?+*(4 )/165*( 8 >(.6+*34 36*(4 (2 -+;32+ .+-+ 2+ )53;3 223(* ( 43* 31 3463 .(4+ 51( 13.34/;(; ;3 34)(./+ ;3 (2-(.31(-/316+$ E1 .+1.254/01, 2( 4+25./01 ;3)31;3 ;3 95/31 6/313 2( .()(./;(; ;3 6+-(* ;3./4/+134 8( 953 3463 ;3:3 ?(.3* 51 *(./+./1/+ (;/./+1(2 (2 -3*+ ;34(**+22+ ;32 -+;32+ ;3 )*+*(-(./01 2/13(2$
'$ SOLUCIÓN ÓPTIMA NO ACOTADA
U1( 7(*/(163 -4 953 )*34316(1 2+4 -+;32+4 ;3 )*+*(-(./01 2/13(2 .+**34)+1;3 ( 2+4 -+;32+4 ;3 4+25./01 0)6/-( 1+ (.+6(;(, 34 ;3./* )*+:23-(4 .+1 /1>/1/6(4 4+25./+134 0)6/-(4$ E1 2( 7/;( *3(2 *(1 )(*63 ;3 346+4 )*+:23-(4 43 ;3:31 ( 51 -(2 )2(163(-/316+ ;3 2(4 *346*/../+134, 4/1 3-:(*+ 34 .+-1 953 3463 6/)+ ;3 )*+:23-(4 43(1 37(25(;+4 31 2( 7/;( (.(;-/.($
Ejemplo: L( .+-)(=( .+-3*./(2/<(;+*( ;3 .+-)56(;+*34 )+*66/234 346 )*+-+./+1(1;+ ;+4 1537(4 )*+;5.6+4, 51 .+-)56(;+* )+*66/2 6/)+ A 8 32 6/)+ B, ;(;+ 953 43 31.5316*(1 31 )*+-+./01 43 )53;3 (435*(* 32 .5:*/-/316+ ;3 .5(295/3* .(16/;(; ;3 ;3-(1;(, 4/1 3-:(*+ 3/4631 ;+4 )+2=6/.(4 953 2( 3-)*34( ;3:3 6313* 31 .5316($ U1( ;3 322(4 34 953 2( .(16/;(; ;3 .+-)56(;+*34 )+*66/234 6/)+ A 953 43 731;(1 1+ )53;3 43* -31+* 953 2+4 ;3 6/)+ B, 8 2( 4351;( 34 953 43 ;3:31 ;3 731;3* )+* 2+ -31+4 !!! .+-)56(;+*34 )+*66/234 ;3 .5(295/3* 6/)+$
D(;+ 953 43 31.5316*(1 31 )*+-+./01 32 )*3./+ ;3 7316( ;3 (-:+4 .+-)56(;+*34 )+*66/234 395/7(23 ( ""&! ;02(*34$
D363*-/13 2( .(16/;(; ;3 51/;(;34 953 ;3:31 731;3*43$
Variables:
C(16/;(; ;3 :3:/;(4 6/)+ A ( 731;3* Y C(16/;(; ;3 :3:/;(4 6/)+ B ( 731;3*
Restricciones :
X ≥Y
X + Y ≥ 2000
Función Objetivo:
Z máx =1150 X + 1150 Y
No negatividad : X , Y ≥ 0
L( *>/.( *34526(163 43*=(
E4 .2(*+ 953 31 3463 3@3*././+ 2(4 7(*/(:234 )53;31 (5-316(* -3@+*(1;+ /1;3>/1/;(-3163 2( >51./01 +:@36/7+, 31 346+4 .(4+4 43 ;/.3 953 2( 4+25./01 0)6/-( 1+ 34 (.+6(;(, )+* 2+ .5(2 2(4 )+4/:234 4+25./+134 4+1 /1>/1/6(4$
%$ SOLUCIÓN INFACTIBLE
E2 .(4+ ;3 2( 4+25./01 /1>(.6/:23 34 -4 6=)/.+ ;3 2+ )314(;+, 8 .+**34)+1;3 ( 2+4 .(4+4 31 2+4 .5(234 1+ 3/4631 4+25./+134 953 .5-)231 .+1 6+;(4 2(4 *346*/../+134$ E4 -58 .+-1 73* 3463 >310-31+ )*+;5.6+ ;3 /17/(:234 )*+)+*./+134 ;3 +>3*6( 8 ;3-(1;($
Ejemplo:
L( *3)+463*=( S336 C(3 ;343( )2(1/>/.(* 2( )*+;5../01 ;3 ;+4 )(463234 953 631;* 953 316*3(* ( 45 .2/3163 31 51 -34, 32 .+16*(6+ /1;/.( 953 2( *3)+463*=( 43 .+-)*+-363 ( 316*3(* )+* 2+ -31+4 '!! )(463234 .5(295/3*( 43( 45 6/)+ 6/)+ A, 6/)+ B + .5(295/3*( ;3 2+4 ;+4$ L+4 )(463234 6/)+ A 6/3131 51 6/3-)+ ;3 32(:+*(./01 ;3 ?+*(4, 8 51 6/3-)+ ;3 ;3.+*(./01 ;3 ' ?+*(4, -/316*(4 32 6/)+ B, 6/313 51 6/3-)+ ;3 32(:+*(./01 ;3 ' ?+*(4 8 51 6/3-)+ ;3 ;3.+*(./01 ;3 " ?+*($ L( .+-)(=( .5316( 346(4 ;+4 43-(1(4 .+1 &&! ?+*(4 )(*( 32(:+*(./01 8 .+1 %! ?+*(4 ;3 ;3.+*(./01 ;3:/;+ ( 45 2/-/6(;+ )3*4+1(2$
T31/31;+ 31 .5316( 953 32 -(*31 ;3 56/2/;(; ;3 .(;( 6/)+ ;3 )(4632 34 ;3 &!! )(*( 32 6/)+ A 8 "!! )(*( 32 6/)+ B$ D363*-/13 -3;/(163 51 -+;32+ ;3 )*+*(-(./01 2/13(2 32 )2(1 ;3 )*+;5../01 953 -(/-/.3 2(4 56/2/;(;34$
Variables:
C(16/;(; ;3 )(463234 6/)+ A ( )*+;5./* 31 51 -34$ Y C(16/;(; ;3 )(463234 6/)+ B ( )*+;5./* 31 51 -34$
Restricciones:
2 X + 3 Y ≤ 550
3 X + Y
≤ 480
X + Y ≥ 300
Función Objetivo:
Z máx =8500 X + 8100 Y
No negatividad : X , Y ≥ 0
L( *>/.( *34526(163 34 2( 4/5/3163
E7/;3163-3163 1+ 3/463 >+*-( (251( ;3 4(6/4>(.3* 6+;(4 2(4 *346*/../+134, )+* 31;3 43 .+1.2583 953 1+ 3/463 4+25./01 >(.6/:23$
BIBLIOGRAFIA IZAR LANDETA, J5(1$ F51;(-316+4 ;3 I17346/(./01 ;3 +)3*(./+134 )(*( (;-/1/46*(./01$ "*( 3;$ S(1 L5/4 P+6+4= U1/73*4/;(; A5601+-( SLP, " .
