Linear Programing Model

By Author: Pierce Brosnan
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Introduction to the linear programing model :

A linear programing model helps the business community to maximize the profit by using the available resources or to minimize the cost of expenses. The linear programing model is designed as a model in the following ways:

1. An objective function of linear function is created which is to be maximized or to be minimized.

2. The above objective function depends on certain constraints which will be represented in the form of inequalities. Here the constraints equations will be represented in “≤” for maximization model and for the minimization model it will have “≥”.

3. All the variables involved should have a non-negative values.

Problem on a Linear Programing Model

Ex 1: A furniture dealer deals in only chairs and tables. He can invest only 50,000 dollars. He has a storage capacity of only 100 pieces. His cost price of a chair is 500 dollars and of a table is 1200 dollars. He can earn a profit of 180 dollars on the sale of the table and 75 dollars on the sale of one chair. Assuming that he can sell all the items he buys, ...
... formulate a linear programing model to maximize the profit.

Sol 1: Let us take two variables x and y to represent the number of tables and chairs respectively.

Therefore the cost of x tables = 1200x and the cost of y chairs = 500y.

Here the total investment cannot be more than 50,000, therefore,

The total cost = 1200x + 500y ≤ 50,000. This is the first constraint inequality.

Here, since the storage capacity is for only 100 pieces, we have x + y ≤ 100. This is the second constraint equation. Since the number of chairs and the number of tables non-negative, we have x ≥ 0, y ≥ 0.

Now, the profit on x tables is 180x and profit on y chairs = 75y.

Here, the objective is to maximize the profit, therefore, the objective function is 180x + 75y.

Hence the linear programing model is given by:

Maximize Z = 180x + 75y

Subject to the constraints

1200x + 500y ≤ 50,000

x + y ≤ 100

x ≥ 0, y ≥ 0.

The above problem can be solved by graphical method.

More Problem on a Linear Programing Model

Ex 2: A dietitian wishes to mix two kinds of food , X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. One kg of food X costs 6 dollars and one kg of food Y costs 10 dollars. Formulate the linear programing model to minimize the cost.

Sol : Let the mixture contain x kg of food X and y kg of food Y.

Given, one kg of food X contains 10 units of vitamin A.

Therefore, the mixture of x kg of food X and y kg of food Y will contain x + 2y units of vitamin A. But the mixture must contain 10 units of vitamin A.

Therefore x + 2y ≥ 10 and for vitamin B, it is 2x + 2y ≥ 12 and for vitamin C, it is 3x + y ≥ 8.

The cost will be 6x + 10y.

Therefore the linear programing model is given by:

Minimize Z = 6x + 10y

Subject to the constraints

x + 2y ≥ 10

2x + 2y ≥ 12

3x + y ≥ 8

x ≥ 0, y ≥ 0.

Hence the problem.

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