Question

A small manufacturer has employed 5 skilled men and 10 semi - skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2 hrs. work by a skilled man and 2 hrs. work by a semi - skilled man. The ordinary model requires 1 hr by a skilled man and 3 hrs. by a semi - skilled man By union rules no man may work more than 8 hrs per day. The manufacturers clear profit on deluxe model is Rs 15 and on an ordinary model is Rs 10. How many of each type should be made in order to maximize his total daily profit.

Answer 1

Let x articles of deluxe model and y articles of an ordinary model be made.

Numbers cannot be negative.

Therefore,

x, y ≥ 0

According to the question, the profit on each model of deluxe and ordinary type model are Rs 15 and Rs 10 respectively.

So, profits on x deluxe model and y ordinary models are 15x and 10y.

Let Z be total profit, then,

Z = 15x + 10y

Since, the making of a deluxe and ordinary model requires 2 hrs. and 1 hr work by skilled men, so, x deluxe and y ordinary models require 2x and y hours of skilled men but time available by skilled men is 5 × 8 = 40 hours.

So,

2x + y ≤ 40 { First Constraint}

Since, the making of a deluxe and ordinary model requires 2 hrs. and 3 hrs work by semi skilled men, so, x deluxe and y ordinary models require 2x and 3y hours of skilled men but time available by skilled men is 10 × 8 = 80 hours.

So,

2x + 3y ≤ 80 {Second constraint}

Hence the mathematical formulation of LPP is,

Max Z = 15x + 10y

subject to constraints,

2x + y ≤ 40

2x + 3y ≤ 80

x, y ≥ 0

Region 2x + y ≤ 40: line 2x + 4y = 40 meets axes at A₁(20, 0), B₁(0, 40) respectively. Region containing origin represents 2x + 3y ≤ 40 as (0, 0) satisfies 2x + y ≤ 40

Region 2x + 3y ≤ 80: line 2x + 3y = 80 meets axes at A₂(40, 0), (0, \(\frac{80}{3}\)) respectively. Region containing origin represents 2x + 3y ≤ 80.

The corner points are A₁(20, 0), P(10, 20), B₂(0, \(\frac{80}{3}\)).

The value of Z = 15x + 10y at these corner points are

The maximum value of Z is 300 which is attained at P(10, 20).

Thus, maximum profit is obtained when 10 units of deluxe model and 20 units of ordinary model is produced.