Goal Programming (Lindo)

First West Chemical

First West Chemical Company produces two chemical ingredients for pharmaceutical firms; formula X and formula Y.

Production of each ingredient requires two processes.

A unit of Formula X requires 4 hours in process 1 and 3 hours in process 2.

A unit in formula Y requires 2 hours in process 1 and 5 hours in process 2.

The normal operating production times for the two processes are as follows: process 1, 70 hours and process 2, 60 hours.

The production process of formula X results in one unit of a by-product called XZ. Four (4) units of X produce 1 unit of XZ.

The production process for formula Y yields 5 units of a by-product, YK for each unit of formula Y.

The unit profits for formulas X and Yare $10,000 and $15,000 respectively.

By products XZ yields $6000 unit profit.

By product YK yields a $3000 unit profit for up to 15 units.

Because of the limited market and the danger involved in handling the material, any by product YK in excess of 15 units must be destroyed at a unit cost of $4,000.

Also due to storage regulations, no more than 15 units of Formula X can be produced.

The management at this chemical company has established the following goals in order of importance:

1. Avoid any underutilization of normal operation hours of each of the two processes. Each of equal importance.

2. Meet the outstanding orders for 8 units of formula X and 7 units of Formula Y. Each of equal importance.

3. Limit any overtime operation of each of the two production process to 10 hours. Each of equal importance.

4. Achieve a profit goal of $220,000

5. Limit the production of by-product YK to 15 units.

6. Minimize the overtime operation of the production processes. Each of equal importance.

Formulate this problem into a mathematical model and using LINDO determine the solution that best satisfies these goals.

this is what I have so far.....

GOAL #1: Avoid any underutilization of normal operation hours of each of the two processes. Each of equal importance.

Process #1

4FX + 2FY + S1M - S1P = 70

where

FX = # of units of formulea X
FY = # of units of formulea Y
S1M = UNDERutilization of process 1 at 70 hours
S1P = OVERutilizaztion of process 1 at 70 hours

Process #2

3FX + 5FY + S2M - S2P = 60

where

FX = # of units of formulea X
FY = # of units of formulea Y
S2M = UNDERutilization of process 2 at 60 hours
S2P = OVERutilizaztion of process 2 at 60 hours

GOAL 1 CONSTRAINT

4FX + 2FY + S1M - S1P = 70
3FX + 5FY + S2M - S2P = 60

GOAL 1 OBJECTIVE FUNCTION

MIN Z = S1M + S2M

GOAL #2: Meet the outstanding orders for 8 units of formula X and 7 units of Formula Y. Each of equal importance.

GOAL 2 CONSTRAINT

FX + S3M - S3P = 8
FY + S4M - S4P = 7

GOAL 2 OBJECTIVE FUNCTION

MIN S3M + S4M

GOAL 3. Limit any overtime operation of each of the two production process to 10 hours. Each of equal importance.

The normal operating production times for the two processes are as follows: process 1, 70 hours and process 2, 60 hours.

Anything greather than 70hrs or 60hrs is considered OT.

GOAL 3 CONSTRAINT

S1P + S12M - S12P = 10
S2P + S22M - S22P = 10

GOAL 3 OBJECTIVE FUNCTION

MIN S12P + S22P

where

S12M = negative deviation of OT from 10 hours in process 1
S12P = OT beyond 10 hours in process 1
S22M = negative deviation of OT from 10 hours in process 2
S22P = OT beyond 10 hours in process 2

GOAL 4. Achieve a profit goal of $220,000

10000FX + 15000FY + 6000XZ + 3000YK - YM4000 - YN4000 = 220000

YM + YN = 1

YM = LESS THAN 15 UNITS OF XXXXXXX
YN = GREATER THAN 15 UNITS OF XXXXXX

GIN YM
GIN YN

GOAL 5. Limit the production of by-product YK to 15 units.

15 UNITS OF YK = 5 * 15 = 75 UNITS OF FY

THEREFORE,

FY <= 75

GOAL 6. Minimize the overtime operation of the production processes. Each of equal importance.

MIN S12P + S22P