Mathematical Formulation of Linear Programming Problems

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There are mainly four steps in the mathematical formulation of linear programming problem as a mathematical model. We will discuss formulation of those problems which involve only two variables.

• Identify the decision variables and assign symbols x and y to them. These decision variables are those quantities whose values we wish to determine.

• Identify the set of constraints and express them as linear equations/inequations in terms of the decision variables. These constraints are the given conditions.

• Identify the objective function and express it as a linear function of decision variables. It might take the form of maximizing profit or production or minimizing cost.

• Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation.

There are many real life situations where an LPP may be formulated. The following examples will help to explain the mathematical formulation of an LPP.

01. A diet is to contain at least 4000 units of carbohydrates, 500 units of fat and 300 units of protein. Two foods A and B are available. Food A costs 2 dollars per unit and food B costs 4 dollars per unit. A unit of food A contains 10 units of carbohydrates, 20 units of fat and 15 units of protein. A unit of food B contains 25 units of carbohydrates, 10 units of fat and 20 units of protein. Formulate the problem as an LPP so as to find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimum requirements.

Suggested answer:

The above information can be represented as

Let the diet contain x units of A and y units of B.

\ Total cost = 2x + 4y

The LPP formulated for the given diet problem is

Minimize Z = 2x + 4y

subject to the constraints

02. In the production of 2 types of toys, a factory uses 3 machines A, B and C. The time required to produce the first type of toy is 6 hours, 8 hours and 12 hours in machines A, B and C respectively. The time required to make the second type of toy is 8 hours, 4 hours and 4 hours in machines A, B and C respectively. The maximum available time (in hours) for the machines A, B, C are 380, 300 and 404 respectively. The profit on the first type of toy is 5 dollars while that on the second type of toy is 3 dollars. Find the number of toys of each type that should be produced to get maximum profit.

Suggested answer:

Mathematical Formulation

The data given in the problem can be represented in a table as follows.

Let x = number of toys of type-I to be produced

y = number of toys of the type - II to be produced

\ Total profit = 5x + 3y

The LPP formulated for the given problem is:

Maximise Z = 5x + 3y subject to the constraints