Programacion Lineal
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UNIVERSIDAD DE ORIENTE NÚCLEO DE ANZOÁTEGUI EXTENSIÓN REGIÓN CENTRO SUR ANACO- EDO- ANZOÁTEGUI
Programación lineal BACHILLERES: ANDREINA SOLANO INDIRA PÉREZ JOSÉ GARCÍA ANDREINA MARTÍNEZ MARBELLA LÓPEZ
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Programación lineal FRANCIS GELVIZ MARZO, 2010
ÍNDICE INTRODUCCIÓN……………………………………………...…..02 DESARROLLO 1. Modelo Modelo de transport transportee ……………… ……………………… ………………… ………………03 ……03 1.1.
Método de la esquina noroeste…………… no roeste………………………09 …………09
1.2. 1.2. Méto Método do del del costo costo míni mínimo mo…… ………… ………… ………… ………… ………. ….14 14 1.3. 1.3. Méto Método do de aprox aproxim imac ació ión n Vogel… Vogel……… ………… ………… ………. …..1 .177 1.4. 1.4. Método Método de los mul multip tiplic licado adores res…… …………… …………… ………… …….21 .21 2. Modelo Modelo de asignación… asignación………… ………………… ………………… ………………… …………..30 ..30 2.1 2.1
Méto Método do de hú húng ngar aro… o……… ………… ………… ………… ………… ………. ….....…3 …311
3. Modelo Modelo de transbord transbordo……… o……………… ……………… ………………… ……………….4 …….400 CONCLUSIÓN……………………………………………………..48 BIBLIOGRAFIA………………………………………………...…49
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Programación lineal INTRODUCCIÓN La Programación Lineal es una pequeña parte de una teoría matemática que se ha consolidado en el siglo XX con el nombre de Opti Optimi miza zaci ción ón.. En gene genera ral, l, se trat trataa de un conj conjun unto to de técn técnic icas as matem matemát ática icass qu quee inten intentan tan obten obtener er el mayo mayorr prove provech cho o posibl posiblee de sistemas económicos, sociales, tecnológicos,... cuyo funcionamiento se puede describir matemáticamente de modo adecuado. En la actualidad es una herramienta común que ha ahorrado miles o millo millone ness de dó dóla lare ress a mucha muchass comp compañ añías ías y nego negoci cios os,, incluy incluyen endo do industrias medianas en distintos países del mundo. Segú Según n las cara caracte cterís rístic ticas as de las func funcion iones es del del proble problema ma y de las las variables se tienen diferentes tipos de problemas de Programación Mate Matemá máti tica ca.. Si toda todass las las func funcio ione ness del del prob proble lema ma,, ob obje jeti tivo vo y restr restric iccio cione ness son son linea lineales les,, se tiene tiene un probl problem emaa de Prog Program ramac ació ión n Lineal. Así, la PL trata la planeación de las actividades para obtener un resultado óptimo, esto es, el resultado que mejor alcance la meta especificada (según el modelo) entre todas las opciones de solución. Aunque la asignación de recursos a las actividades es la aplicación más más frecu frecuen ente, te, la PL tiene tiene much muchas as otra otrass po posi sibil bilid idad ades es.. De hech hecho, o, cualqui cualquier er problema problema cuyo cuyo modelo modelo matemáti matemático co se ajuste ajuste al formato formato general del modelo de PL es un problema de PL. En este trabajo describiremos algunos métodos como lo son el método de transporte, asignación y transbordo con el propósito de conocer
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Programación lineal 4. MODELO MODELOS S DE TRANSP TRANSPORT ORTE: E: Es una clase especial de programación lineal que tiene que ver con transportar un articulo desde sus fuentes (es decir, fábricas) hasta sus destinos (es decir, bodegas). El objetivo es determinar el programa de transporte y que al mismo tiempo satisfaga los límites de la oferta y dema demand nda. a. En el mode modelo lo se supo supone ne qu quee el cost costo o de tran transp spor orte te es proporcional a la cantidad de unidades transportadas en determinada ruta. En general, se pude ampliar el problema de transporte a otras áreas de operación, entre otras el control de inventarios, programación de empleos y asignación de personal.
Descripción general de un modelo de transporte: 1) Un conjunto de m puntos de suministro a partir de los cuales se
envía un bien. El punto i de suministro abastece a lo sumo S i unidades. 2) Un conjunto de n puntos de demanda a los que se envía el bien.
El punto de demanda j debe de recibir por lo menos d j unidades del bien enviado. 3) Cada unidad producida en el punto de suministro i y enviada al
punto de demanda j incurre en el costo variable c ij. Sea X ijij= número de unidades enviadas desde el punto de suministro i y
envi enviad adaa al pu punt nto o de dema demand ndaa j ento entonc nces es la form formul ulac ació ión n de un
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Programación lineal min s.a: (Restricciones de suministro) (Restricciones por demanda) xij
. •
Si el problema es un transporte equilibrado entonces se dice:
Entonces el suministro total es igual a la demanda, siendo así se podría escribir como: min s.a (Restricciones de suministro) (Restricciones de demanda) xij
. •
Si el problema es un transporte no equilibrado debido al que el suministro total excede a la demanda total, entonces se tiene lo
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Programación lineal a. Se debe debe de crear crear un punto punto de demand demandaa fictic ficticio io que tiene tiene una demanda igual a la cantidad de suministro en exceso. b. A los puntos puntos de demanda demanda ficticios ficticios se les asigna asigna un costo costo cero, cero, ya que, no son envíos reales.
•
Si el problema es un transporte no equilibrado debido al que el suministro total es menor que la demanda total, entonces se tiene lo siguiente:
a. Si es estricta estrictament mentee menor no tiene tiene soluci solución ón factibl factible. e. b. b. Se perm permit itee la po posi sibi bili lida dad d de deja dejarr sin sin sati satisf sfac acer er part partee de la demanda. c. (En (En este este tipo tipo de situ situac ació ión n se asoc asocia ia un unaa pena penali liza zaci ción ón con con la demanda no cumplida). La estructura especial de un problema de transporte permite asegurar que haya una solución básica no artificial de inicio, obtenida con uno de los métodos siguientes: 1. Método Método de de la esquina esquina noroes noroeste te 2. Métod Método o del del costo costo mínimo mínimo 3. Método Método de de aproxi aproximaci mación ón de de Vogel Vogel 4. Método Método de multipli multiplicad cadore oress
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Programación lineal Corpower tiene tres centrales eléctricas que cubren las necesidades de cuatro ciudades. Cada central suministra los números siguientes de kilowatts-hora (kwh) de electricidad: Planta 1= 35 millones Planta 2= 50 millones Planta 3= 40 millones Las deman demanda dass de po pote tenc ncia ia pico pico en esta estass ciuda ciudade dess que ocurr ocurren en al mismo tiempo (2p.m), son como sigue (en kwh): Ciudad 1= 45 millones Ciudad 2= 20 millones Ciudad 3= 30 millones Ciudad 4= 30 millones. Los costos por enviar un millón de kwh de electricidad de la planta a la ciudad dependen de la distancia que debe viajar la electricidad. Formule un PL para minimizar el costo de satisfacer la demanda de potencia pico de esta ciudad.
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Programación lineal Demanda
45
20
30
30
(millones de (kwh)
Tabla1: Costos de envió y demanda para Corpower
Solución: Paso 1: Se comienza por definir una variable para cada decisión que
debe tomar Corpower. Debido a que Corpower debe de determinar cuanta potencia se envía desde cada planta a cada ciudad, se define (para i= 1, 2,3 y j=1, 2, 3,4) xij = numero de (millones) de kwh producidos en la planta i y enviados
a la ciudad j. Escrib ibir ir en térm términ inos os de las las vari variab able les, s, el cost costo o tota totall de Pas Paso o 2: Escr suministrar las demanda de potencia pico a las ciudades de 1 a 4, de la siguiente manera: 8x11+6x12+10x13+9x14 (Costo de enviar potencia desde la planta 1) +9x21+12x22+13x23+7x24 (Costo de enviar potencia desde la planta 2)
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Programación lineal para satisfacer su demanda pico, esta restricción se conocerá como restricción por demanda. Lo cual se escribe de la siguiente manera:
Restricciones por suministro.
x11+x12+x13+x14
(Restricción de suministro de la planta 1)
x21+x22+x23+x24
(Restricción de suministro de la planta 2)
x31+x32+x33+x34
(Restricción de suministro de la planta 3)
Restricciones por demanda.
x11+x21+x31
(Restricción de demanda de la ciudad 1)
x12+x22+x32
(Restricción de demanda de la ciudad 2)
x13+x23+x33
(Restricción de demanda de la ciudad 3)
x14+x24+x34
(Restricción de demanda de la ciudad 4)
Paso 4: Colocar la restricción de no negatividad, puesto que, las x
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Programación lineal x11+x12+x13+x14 x21+x22+x23+x24
Restricciones de suministro
x31+x32+x33+x34
x11+x21+x31 x12+x22+x32
Restricciones de demanda
x13+x23+x33 x14+x24+x34 xij≥0∀ Paso 6 : Hallar la solución optima a través de alguno de los métodos
utilizados para resolver problemas de este tipo, ya sea, el método de la esquina ina
NOROE OROES STE, TE,
costo
meno norr,
apro proxima imación
Vogel
o
multiplicadores. (Los cuales serán descritos más adelante). De esta manera se encuentran planteados los problemas de transporte y las características vistas anteriormente son las que se presentan en
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Programación lineal Paso 1: Asignar todo lo mas que se pueda a la celda seleccionada y
ajus ajustar tar las las canti cantida dade dess asoci asociad adas as de ofert ofertaa y deman demanda da resta restand ndo o la cantidad asignada. Paso 2: Salir del renglón o la columna cuando se alcance la oferta o
demanda cero, y tacharlo para indicar que no se puede hacer mas asignaciones a ese renglón o columna. Si un renglón o una columna dan cero al mismo tiempo, tachar solo uno el renglón o la columna y dejar una oferta (demanda) cero en el renglón (columna) que no se tachó. qued edaa exac exactam tamen ente te un reng renglón lón o colum columna na sin tacha tachar, r, Pas Paso o 3: Si qu detenerse. En caso contrario, avanzar a la celda de la derecha si se acaba de tachar una columna, o a la de abajo si se tachó un renglón. Seguir con el paso 1.
Ejercicio nº1: (Método de la esquina noroeste) Siguiendo con el ejemplo dado anteriormente de Corpower tenemos la sigu siguie ient ntee form formul ulac ació ión, n, aplic aplicar arem emos os el mé méto todo do de la esqu esquin ina a noroeste para hallar una solución óptima. ó ptima.
Corpower tiene tres centrales eléctricas que cubren las necesidades de
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Programación lineal Demanda
45
20
30
30
(millones de (kwh)
Los costos por enviar un millón de kwh de electricidad de la planta a la ciudad dependen de la distancia que debe viajar la electricidad. Formule un PL para minimizar el costo de satisfacer la demanda de potencia pico de esta ciudad.
Min
z
=8x11+6x12+10x13+9x14+9x21+12x22+13x23+7x24+14x31+9x32
+16x33+5x34 s.a x11+x12+x13+x14 x21+x22+x23+x24
Restricciones de suministro
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Programación lineal Planta 1
35
8
//////////
6
///////////10
Planta 2
10
9
20
12
20
9
10
Planta 3
////////// 45
14
/////////// 20
30
/////////// 9
35
13
/////////// 7
50
16
30
5
40
30
Luego hacemos el cálculo tomando los datos de la tabla optima. El costo para satisfacer la demanda de electricidad es: Z= 35x8+10x9+20x12+20x13+10x16+30x5= 1180.
Ejercicio nº2: (Método de la esquina noroeste) Una compañía tiene 3 almacenes con 15, 25 y 5 artículos disponibles respectivamente. Con estos productos disponibles desea satisfacer la dema demand ndaa de 4 clie client ntes es qu quee requ requie iere ren n 5, 15 15,, 15 y 10 un unid idad ades es respectivamente. Los costos asociados con el envío de mercancía del
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Programación lineal
Demanda
5
15
15
10
Luego hacemos el cálculo tomando los datos de la tabla optima. El costo para satisfacer la demanda es: Z= (5x10) + (10x0) + (5x7) + (15x9) + (5x20) + (5x18) Z= 410
Ejercicio nº3: (Método de la esquina noroeste) Una Una compa compañía ñía de renta renta de autos autos tien tienee prob problem lemas as de distri distribu buci ción ón debido a que los acuerdos de renta permiten que los autos se entreguen en lugares diferentes diferentes a aquellos en que originalmente fueron rentados. Por el momento hay 2 lugares (fuentes) con 15 y 13 autos en exceso respectivamente y 4 lugares (destinos) en los que se requieren 9; 6; 7 y
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Programación lineal Origen
Lugares 1
1
9
2
//////////
3
//////////
Demanda:
9
Lugares 2
45
6
17
14
//////////
18
0
/////////// 0 6
Lugares 3
Lugares 4
/////////// 21 /////////// 30 19
6
////////// 0
3
7
7
31 0
15 13 3
9
Z= (9x45) + (6x17)+ (7x19)+ (6x31)+ (3x0) Z= 826$
4.2 Método Método del costo mínim mínimo. o. Este método determina una mejor solución de inicio, porque se concentra en las rutas menos costosas. Paso 1: Se inicia asignando todo lo posible a la celda que tenga el
mínimo costo unitario (los empates se rompe en forma arbitraria).
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Programación lineal
Ejercicio nº1: (Método del costo mínimo) Siguiendo con el ejemplo dado anteriormente de Corpower tenemos la siguiente formulación, aplicaremos el método del costo mínimo para hallar una solución óptima.
Corpower tiene tres centrales eléctricas que cubren las necesidades de cuatro ciudades. A De
Ciudad 1
Ciudad 2
Ciudad 3
Ciudad 4
Suministros (millones de kwh)
Planta1 Planta 2 Planta 3 Demanda
$8 $9 $14 45
$6 $12 $9 20
$10 $13 $16 30
$9 $7 $5 30
35 50 40
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Programación lineal Luego hacemos el cálculo tomando los datos de la tabla optima. El costo para satisfacer la demanda de electricidad es: Z= (15x8) + (20x6) + (30x9) + (20x13) + (10x16) + (30x5)= 1080
Ejercicio nº2: (Método del costo mínimo) Una compañía tiene 3 almacenes con 15, 25 y 5 artículos disponibles respectivamente. Con estos productos disponibles desea satisfacer la dema demand ndaa de 4 clie client ntes es qu quee requ requie iere ren n 5, 15 15,, 15 y 10 un unid idad ades es respectivamente. Los costos asociados con el envío de mercancía del
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Programación lineal 4.3 Método de Aproximación Vogel.
Es una versión mejorada del método del costo mínimo, que en general produce mejores soluciones de inicio. Determin minar ar para para cada cada reng renglón lón (colu (columna mna)) un unaa medid medidaa de Pas Paso o 1: Deter penalización restando el elemento del costo unitario mínimo en el rengl renglón ón (colu (columna mna)) del del eleme elemento nto con con costo costo un unita itario rio sigu siguien iente te al mínimo del mismo renglón (columna). Paso 2: Identificar el renglón o columna con mayor penalización.
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Programación lineal c) Si todos los renglones y columnas que no se tacharon tienen cero oferta y demanda (restante), determinar las variables básicas cero por el método del costo mínimo. Detenerse. d) En cualquier otro caso, seguir en el paso 1.
Ejercicio nº1: (Método de Aproximación Vogel) Siguiendo con el ejemplo dado anteriormente de Corpower tenemos la siguiente siguiente formulación, formulación, aplicarem aplicaremos os el método método de Aproximac Aproximación ión Vogel para hallar una solución óptima.
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Programación lineal
Planta 2
9
12
13
7
50
Planta 3
14
9
16
5
40
45
20
30
30
Procedemos a obtener las penalizaciones. Ciudad 1 Planta 1 Planta 2
//////
8
40 9
Ciudad2
10
6
////// 12
Ciudad 3
Ciudad 4
Suministro
Penalización de renglón
25
10
///////
9
35
P1 P2 P3 P4
5
13
///////
7
50
2
2
2
2
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Programación lineal dema demand ndaa de 4 clie client ntes es qu quee requ requie iere ren n 5, 15 15,, 15 y 10 un unid idad ades es respectivamente. Los costos asociados con el envío de mercancía del almacén al cliente por unidad se dan en la siguiente tabla: Cliente
Almacén 1 2 3
1
2
3
4
10 12 0
0 7 14
20 9 16
11 20 18
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Programación lineal
Demanda
5
15
15
10
Penalización por columna P1
10-12=2
7-0=7
16-9=7
18-11=7
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Programación lineal Paso 4: La información obtenida en el paso 4 equivaldrá a calcular el
renglón de z de la tabla simplex. Paso 5: Debe de determinarse quien entra y quién sale.
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Programación lineal A De
Ciudad 1
Ciudad 2
Ciudad 3
Ciudad 4
Suministros (millones de kwh)
Planta1 Planta 2
$8 $
$6 $
$10 $
$9 $
35
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Programación lineal u3 + v1 =14(ecuación 4) u2 + v2 =12 (ecuación 5) u2 + v3 =13 (ecuación 6)
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Programación lineal Planta 3
//////////
14
45
Ciudad 1
9
10
16
30
5
20
30
30
Ciudad 2
Ciudad 3
Ciudad 4
40
Suministro
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Programación lineal
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Programación lineal
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