Question

6 6 RS Question 2 (Marks:20) The accountant of Royal Products provided the following information for purposes of determining the optimal product mix Product King (K) Product Queen (Q) Marginal income per unit R30 R36 Machine hours required per unit Marginal income per machine hour R6 Ranking in terms of machine hours 2nd Labour hours required per unit 6 Marginal income per labour hour R6 Ranking in terms of labour hours Annual demand for product in units 12 000 15 000 Labour hours available per annum 65 000 Machine hours available per annum 82 000 3 R10 14 Required: Determine the optimal product mix by means of the application of linear programming techniques. (Hint: Start with your objective function and all necessary constraints.)

Answer 1

Let x and y denote the number of units produced for the product King and Queen respectively.

The linear programming model for the given problem is:

Maximise Z = 30x + 36y

Subject to the Constraints:

6x + 6y < 82,000 ( Machine hours )

i.e, x+ y < 13,666.67

3x + 6y < 65,000 ( Labour hours )

i.e, x + 2y < 21,666.67

x,y> 0

x <12,000

y<15,000

Intersection point:

x + y = 13,666.67

x + 2y = 21,666.67

i.e, y = 8,000

x = 5,666

X(Units)	Y(Units)	MARGINAL INCOME(R)
12,000	1,666	419,976 (12,000*30 + 1,666*36)
0	10,833	389,988 (0*30 + 10,833*36)
5,666	8,000	457,980 (5,666*30 + 8,000*36)

Therefore, maximum marginal income can be earned by producing 5,666 units of product "KING" and 8,000 units of product "QUEEN"

As the demand is restricted to 12,000 units for product king so we are not considering option where production of more than 12,000 units is possible. We are ignoring those options.

Also, there is demand of product queen upto 15,000 units but constraints does not allow that much production of the product.

1400of (0,13627) Hortio, 1833) (60, 15000) Itooop 13600 1200 focool Goos $6+ (5666,8000) Pero 6000 (12000,4833) 4000 3000 2600 tooot (12.001,0) (13667,0 3000 4000 1000 2000 Coon 6000 7000 8000 1oooo 11000 12000 13000 14000 900 n-areo: Product King yr aris : Product Queen.