Question

A firm manufactures pills in two sizes A and B. Size A contains 2 mgs of aspirin, 5 mgs of bicarbonate and 1 mg of codeine. Size B contains 1 mg. of aspirin, 8 mgs. of bicarbonate and 6 mgs. of codeine. It is found by users that it requires at least 12 mgs. of aspirin, 74 mgs. of bicarbonate and 24 mgs. of codeine for providing immediate relief. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LLP.

Answer 1

(i) Variables: Let x₁ and x₂ represents the pills in two sizes A and B.

	A	B	Requirement (at least)
Aspirin	2 mg	1 mg	12 mg
Bicarbonate	5 mg	8 mg	74 mg
Codeine	1 mg	6 mg	24 mg

Requirement of Aspirin 2x₁ + x₂ ≥ 12

Requirement of Bicarbonate 5x₁ + 8x₂ ≥ 74

Requirement of Codeine x₁ + 6x₂ ≥ 24

(ii) Objective function:

Number of pills required for a patient = x₁ + x₂

let Z = x₁ + x₂

∴ Minimize Z = x₁ + x₂ is the objective function.

(iii) Non-negative restrictions:

Since the number of pills of size A and B cannot be negative,

We have x₁, x₂ ≥ 0

Hence, the mathematical formulation of the LLP is minimize Z = x₁ + x₂

Subject to the constraints

2x₁ + x₂ ≥ 12

5x₁ + 8x₂ ≥ 74

x₁ + 6x₂ ≥ 24

x₁, x₂ ≥ 0