Linear Programming Model Formulation
Model formulation is the process of transforming a real word problem into linear programming model.
A company manufactures two products A and B.
The resources are the capacities Machine-1, Machine-2, and Machine-3.
The available capacities are 50,25,and 15 hours respectively.
Product A requires 1 hour of Machine-2 and 1 hour of Machine-3.
Product B requires 2 hours of Machine-1, 2 hours of Machine-2 and 1 hour of Machine-3.
The profit contribution of products A and B are Rs 5 and Rs 4 respectively.
Formulate the linear programming model
Solution:
The contents of the statement of the problem can be summarized as follows
| Machines | A | B | Availability |
| Machine-1 | 0 | 2 | 50 |
| Machine-2 | 1 | 2 | 25 |
| Machine-3 | 1 | 1 | 15 |
| Profit | 5 | 4 | - |
Decision variables
Here Products A and B are competing variables and Machine-1, Machine-2, and Machine-3 are available resources.
Let x1, x2 denotes units of A, B respectively
The LP Model
As the profit contributions of A and B are Rs 5 and Rs 4 respectively. The objective of the problem is to maximize the profit Z,
Hence objective function is
Maximize Z = 5x1 + 4x2
The utilization of Machine hours by products x1 and x2 should not exceed the available capacity.
This can be shown as follows
For Machine-1 : 0x1 + 2x2 ≤ 50
For Machine-2 : 1x1 + 2x2 ≤ 25
For Machine-3 : 1x1 + 1x2 ≤ 15
There is no negative production. Hence we can write,
x1, x2 ≥ 0
Thus, the linear programming model is
Maximize Z = 5x1 + 4x2
Subject to
2x2 ≤ 50
x1 + 2x2 ≤ 25
x1 + x2 ≤ 15
and x1, x2 ≥ 0
This material is intended as a summary. Use your textbook for detail explanation.
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