Question

A firm wants to decide on its production mix for two of its expensive products X and Y. The profit per unit of x is estimated to be LE 30,000, while the profit for unit of y is estimated to be LE10,000. Each unit of x requires 100 production hours while each unit of y requires 50 production hours and the firm has a total of 500 production hours available per week.

1. Develop a model for this problem and find the optimal solution.
2. Given the same production requirements, if the firm must produce at least 4 units of y, what is the optimal solution in this case?
3. Given the same production requirements, if the max to be produced from x is 2 units, what is the optimal solution?
4. Consider the constraints in b and c together, what is the optimal solution?

Answer 1

Q1) LP Model

Objective Function Maximize Profit Z = 30000X+10000Y

s.t. Constraints - 100X+50Y<=500 --> Production Hours constraint

X,Y>=0 --> Non-negativity constraint

Excel solution-

Maximum Profit = 150000 $ which produces 5 units of X and 0 units of Y

Solution	X	Y
Decision variable	5	0	Maximize
Z	30000	10000	150000
Production Hours	100	50	500	500

Solver with constraints

Q2) Additional constraint which produces atleast 4 units of Y

Objective Function Maximize Profit Z = 30000X+10000Y

s.t. Constraints - 100X+50Y<=500 --> Production Hours constraint

Y>=4 --> Units constraint for Y

X,Y>=0 --> Non-negativity constraint

Excel solution-

Maximum Profit = 130000 $ which produces 3 units of X and 4 units of Y

Solution	X	Y
Decision variable	3	4	Maximize
Z	30000	10000	130000
Production Hours	100	50	500	500
Y units	0	1	4	4

Solver with constraints -

Q3) Additional constraint which produces atmost 2 units of X

Excel Solution-

Maximum Profit = 120000 $ which produces 2 units of X and 6 units of Y

Solution	X	Y
Decision variable	2	6	Maximize
Z	30000	10000	120000
Production Hours	100	50	500	500
X units	1	0	2	2

Solver with constraints -

Q4) Additional constraints where X<=2 and Y>=4

Maximum Profit = 120000 $ which produces 2 units of X and 6 units of Y

Excel Solution -

Solution	X	Y
Decision variable	2	6	Maximize
Z	30000	10000	120000
Production Hours	100	50	500	500
X units	1	0	2	2
Y units	0	1	6	4

Solver with constraints -