Linear Programming - Model Formulation & Graphical Method

Diet Problem

Example 3

Three nutrient components namely, thiamin, phosphorus and iron are found in a diet of two food items - A and B. The amount of each nutrient (in milligrams per ounce, i.e., mg/oz) is given below:

The cost of food items A and B is Rs. 2 per oz and Rs. 1.70 per oz respectively. The minimum daily requirements of these nutrients are atleast 1.00 mg of thiamin, 7.50 mg of phosphorus, and 10.00 mg of iron. Formulate this problem in the linear programming form.

Solution.

Let x₁ and x₂ be the number of units (ounces) of A and B respectively. The objective here is to minimize the total cost of the food items, which is given by the linear function

Minimize z = 2x₁ + 1.7x₂

0.12x₁ + 0.10x₂ ³ 1.0
0.75x₁ + 1.70x₂ ³ 7.5
1.20x₁ + 1.10x₂ ³ 10.0

x₁ ³ 0, x₂ ³ 0

Blending Problem

Example 4

The manger of Deep Sea Oil Refinery must decide on the optimal mix of two possible blending processes of which the inputs and outputs per production run are given in the following table:

Formulate the blending problem as a linear programming problem.

Solution.

Let x₁ and x₂ be the number of production runs of process A & B respectively. The objective here is to maximize the profit. The decision problem can be formulated as

Maximize z = 400x₁ + 500x₂

subject to

6x₁ + 3x₂ £ 300
2x₁ + 6x₂ £ 250
6x₁ + 4x₂ ³ 120
8x₁ + 5x₂ ³ 100

x₁ ³ 0, x₂ ³ 0

Although all the above illustrations were highly simplified, the decision problems shown have close counterparts in actual companies. The effort required to formulate a real situation depends on the complexity of the problem.

By studying the above examples and answering the formulation exercises given at the end of this chapter, you will acquire considerable experience of formulating a model.

"Experience is a dim lamp, which only lights the one who bears it." -Louis Ferdinand