A simple Production and Transportation Problem solved in Lingo 19.0.
Problem Description: This is a generic problem that comes accross in Integer Programming courses. Given a problem description and the associated data, a model and a solution should be derived. A company called "ABC S.A." produces 3 products in 3 factories and supplys them to 3 different markets. Given the indices k = 1, 2, 3, i = 1, 2, 3 and j = 1, 2, 3, derive an Integer Programming problem and derive a solution using Lingo software so that the overall production and transportation costs are minimized. Please consider the following data tables:
Cost of production for a single unit of product k at factory i:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 3 | 5 | 5 |
| i = 2 | 5 | 4 | 2 |
| i = 3 | 4 | 6 | 5 |
Cost of transportation of product k from factory i to market j = 1:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 2 | 4 | 6 |
| i = 2 | 4 | 5 | 2 |
| i = 3 | 5 | 3 | 4 |
Cost of transportation of product k from factory i to market j = 2:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 5 | 3 | 3 |
| i = 2 | 2 | 2 | 5 |
| i = 3 | 4 | 6 | 4 |
Cost of transportation of product k from factory i to market j = 3:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 2 | 5 | 4 |
| i = 2 | 3 | 5 | 2 |
| i = 3 | 6 | 4 | 3 |
Set-up cost for production of product k at factory i:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 20 | 35 | 50 |
| i = 2 | 30 | 40 | 25 |
| i = 3 | 40 | 45 | 35 |
Maximum output production for product k at factory i:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 500 | 950 | 900 |
| i = 2 | 400 | 900 | 850 |
| i = 3 | 900 | 850 | 950 |
Minimum output production for product k at factory i (in the case that we have non-zero production):
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 10 | 5 | 8 |
| i = 2 | 5 | 10 | 5 |
| i = 3 | 4 | 5 | 4 |
Capacity required (in man-hours) for the production of product k at factory i:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| i = 1 | 2 | 2 | 3 |
| i = 2 | 3 | 1 | 2 |
| i = 3 | 4 | 2 | 3 |
Total capacity (in man-hours) for factory i:
| i = 1 | i = 2 | i = 3 |
|---|---|---|
| 2000 | 3500 | 5000 |
Demand for product k in market j:
| k = 1 | k = 2 | k = 3 | |
|---|---|---|---|
| j = 1 | 200 | 350 | 500 |
| j = 2 | 300 | 400 | 250 |
| j = 3 | 400 | 450 | 350 |
Additionally, the next two constraints should be considered: a) Any single factory can not produce more than 2 products. b) Any single product can not be produced in more than 2 factories.