Operations Management - Linear Programming Problem - Formulation

 

Operations Management

Linear Programming – Formulation

 

Let Q and T are two products to be produced. Product Q requires M₁ kg of raw materials per unit, whereas Product T requires M₂ kg of raw materials per unit. Similarly, Product Q requires L₁ labour hours per unit, whereas Product T requires L₂ labours hours per unit. Profit per unit that can be earned from Product Q is P₁ and profit per unit that can be earned from Product T is P₂.

 

We have to find out how many units of Product Q and how many units of Product T should be produced in order to maximise the total profit. Given that, maximum quantity of raw materials available is K kg and maximum labour hours available is H hours.

 

Formulation (For Maximisation of Profit):

Let X₁ units of Product Q and X₂ units of Product T should be produced in order to maximise the total profit. (Here, X₁ and X₂ are known as decision variables.)

Now we can write the Objective Function as follows:

Maximise Z = P₁X₁ + P₂X₂

 

Subject to Constraints –

M₁X₁ + M₂X₂ ≤ K   and

L₁X₁ + L₂X₂ ≤ H

 

Where, Non-negative Restrictions are –

X₁ ≥ 0, X₂ ≥ 0

Operations Management

Linear Programming – Formulation

Selected Problems

 

Problem: 1

Consider the production planning of The Super Fast Manufacturing Company which makes items P and V. The steel requirement for P is 400 gm per piece and that for V is 350 gm per piece. Both, P and V, are machined on lathe which takes 85 and 50 minutes respectively, and are processed on grinder which requires 55 and 30 minutes respectively. Each unit of P consumes 20 minutes of polishing time. The resource availability is:

 

Total Machine Time	1,450 hours
Total Steel	250 kg

 

30% of total machine time is that of lathe, 50% of grinder and the remaining of polishing. Unit contribution to profits for P and V is Rs 40 and Rs 30, respectively.

 

Formulate this as a linear programming model for determining the number of units of P and V to be produced which would maximise the profit. Given also is the constraint that the company cannot sell more units of item P than item V.

 

Problem: 2

A farmer has a 100 acre farm. He can sell all the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Re 1 per kg for tomatoes, Re 0.75 a head for lettuce, and Rs 2 per kg for radishes. The average yield per acre is 2000 kg of tomatoes, 3000 heads of lettuce and 1000 kg of radishes. Fertilizer is available at Re 0.50 per kg and the amount required per acre is 100 kg each for tomatoes and lettuce and 50 kg for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 men days for tomatoes and radishes and 6 men days for lettuce. A total of 400 men days of labour are available at Rs 20 per man day.

 

Formulate the problem as a linear programming model.