Formulation of Linear Programming-Maximization Case

Definition: Linear programming refers to choosing the best alternative from the available alternatives, whose objective function and constraint function can be expressed as linear mathematical functions.

Maximization Case: Let’s understand the maximization case with the help of a problem. Suppose a firm produces two products A and B. For producing the each unit of product A, 4 Kg of Raw material and 6 labor hours are required. While, for the production of each unit of product B, 4 kg of raw material and 5 labor hours is required. The total availability of raw material and labor hours is 60 Kg and 90 Hours respectively (per week). The unit price of Product A is Rs 35 and of product, B is Rs 40.

This problem can be converted into linear programming problem to determine how many units of each product should be produced per week to have the maximum profit. Firstly, the objective function is to be formulated. Suppose x₁ and x₂ are units produced per week of product A and B respectively. The sale of product A and product B yields Rs 35 and Rs 40 respectively. The total profit will be equal to

Z = 35x₁+ 40x₂ (objective function)

Since the raw material and labor is in limited supply the mathematical relationship that explains this limitation is called inequality. Therefore, the inequality equations will be as follows:

Product A requires 4 kg of raw material and product B also requires 4 Kg of Raw material; thus, total consumption is 4x₁+4x₂, which cannot exceed the total availability of 60 kg. Thus, this constraint can be expressed as:

4x₁ + 4x₂ ≤ 60

Similarly, the second constraint equation will be:

6x₁ + 5x₂ ≤ 90

Where 6 hours and 5hours of labor is required for the production of each unit of product A and B respectively, but cannot exceed the total availability of 90 hours.

Thus, the linear programming problem will be:

Maximize Z = 35x₁+ 40x₂ (profit)

Subject to:

4x₁ + 4x₂ ≤ 60 (raw material constraint)
6x₁ + 5x₂ ≤ 90 (labor hours constraint)
x₁, x₂ ≥ 0 (Non-negativity restriction)

Note: It is to be noted that “≤” (less than equal to) sign is used as the profit maximizing output may not fully utilize all the resources, and some may be left unused. And the non-negativity condition is used since the x₁ and x₂ are a number of units produced and cannot have negative values.