Métodos de optimización para la toma de decisiones
Instrucciones. Resolver los siguientes problemas. Problema 1
Alden Enterprises produces two products. Each product can be produced on one of two machines. The length of time needed to produce each product (in hours) on
each machine is as shown in Table 57. Each month, 500 hours of time are available on each machine. Each month, customers are willing to buy up to the quantities of each product at the prices given in Table 58. The company’s goal is to maximize the revenue obtained from selling units during the next two months. Formulate an LP to help meet this goal.
TABLE 57 Product
Machine 1 (Hrs)
Machine 2 (Hrs)
4 7
3 4
1 2
TABLE 58 DEMANDS Product
PRICES
Month 1
Month 2
Month 1
Month 2
100 140
190 130
$ 55 $ 65
$ 12 $ 32
1 2
Empresa: Alden Enterprises 2 Products / 2 machines Time available per month on each machine = 500 hrs Goal = Maximize the revenue during next 2 months. Restrictions: Time available per month for each machine is 500 hrs. X1 < 500 X2 < 500 2 Products / 2 machines Time available per month on each machine = 500 hrs Goal = Maximize the revenue during next 2 months. Formulate LP.
1. Variables de decision: Xa11 = Producto 1 realizado en maquina 1 durante el mes 1
Xa21 = Producto 1 realizado en maquina 1 durante el mes 2 Xa12 = Producto 1 realizado en maquina 2 durante el mes 1 Xa22 = Producto 1 realizado en maquina 2 durante el mes 2 Xb11 = Producto 2 realizado en maquina 1 durante el mes 1 Xb21 = Producto 2 realizado en maquina 1 durante el mes 2 Xb12 = Producto 2 realizado en maquina 2 durante el mes 1 Xb22 = Producto 2 realizado en maquina 2 durante el mes 2
2. Función Objetivo: Max Z= 44 Xa111 + 18,24Xa122 + 33 Xa211 +13,68 Xa222 + 127,4 Xb111 + 58,24 Xb122 + 72,8 Xb211 + 33,28 Xa222
Problema 2
Gotham City Hospital serves cases from four diagnostic-related groups (DRGs). The profit contribution, diagnostic service use (in hours), bed-day use (in days), nursing care use (in hours), and drug use (in dollars) are given in Table 60. The
hospital now has available each week 570 hours of diagnostic services, 1,000 beddays, 50,000 nursing hours, and $50,000 worth of drugs. To meet the community’s minimum health care demands at least 10 DRG1, 15 DRG2, 40 DRG3, and 160 DRG4 cases must be handled each week. Use LP to determine the hospital’s optimal mix of DRGs.
Datos del problema: Availability per week = 570 hrs of diagnostic services Bed days = 1,000 Nursing hours = 50,000 Worth of drugs = $ 50,000 Community minimum health care demands at least: 10 DRG 1 15 DRG 2 40 DRG 3 160 DRG 4
1. Variables de decision:
X1 = Servicio tipo DRG 1 X2 = Servicio tipo DRG 2 X3 = Servicio tipo DRG 3 X4 = Servicio tipo DRG 4 3. Función Objetivo:
Max Z = 2000X1 + 1500X2 + 500X3 + 300X4 Diagnostic serv. (hrs) 7X1 + 4X2 + 2X3 + X4 ≤ 570 Bed day (days) 5X1 + 2X2 + X3 ≤ 1,000 Nursing-hours (hrs) 30X1 + 10X2 + 5X3 + X4 ≤ 50,000 Worth of Drugs ($) 800X1 + 500X2 + 150X3 + 50X4 ≤ 50,000
Community demands: DRG 1 DRG 2 DRG 3 DRG 4
X1 X2 X3
≤ 10 ≤ 15 ≤ 40 X4 ≤ 160
Utilizando el Solver:
Grafica de servicios y su demanda 240 220 200 180 160 140 120 100 80 60 40 20 0 Servicios tipo DR1 Servicios tipo DR2 Servicios tipo DR3 Servicios tipo DR4 Servicios
Demanda Minima
Problema 3 (extraído del libro de texto, problema 9, página 68)
Graphically determine two optimal solutions to the following LP. min z 3 x1 5 x2 1
s.t .
3 x1 2 x2 36 3 x1 5 x2 45 x
1
, x2
0
2 3
De la Ecuación 1
De la Ecuación 2
Igualando las ecuaciones se obtiene la intersección de las rectas:
Al sustituir los valores obtenidos en la ecuación se obtiene: min z 3 x1 5 x2
() ()
Solucion con solver: Productos
X1
X2
RESULTADOS
Funcion Objetivo
3
5
45
Restriccion 1
3
2
36
≥
36
A (minutos)
Restriccion 2 INICIO
3
5
45
≥
45
B (minutos)
10
3
X1
10
X2
3
1
2
La solución que satisface la gráfica es x 1=10, x2=3, z=45
