Andy Tyre manages Tyre's Wheels, Inc. Andy has received an order for 1000 standard wheels and 1200 deluxe wheels for next month, and for 750 standard wheels and 1000 deluxe wheels the following months. He must fill all the orders. The cost of regular time production for standard wheels is $25 and for deluxe wheels, $40. Overtime production costs 50% more. For each of the next two months there are 1000 hours of regular time production and 500 hours of overtime production available. A standard wheel requires .5 hour of production time and a deluxe wheel, .6 hour. The cost of carrying a wheel from one month to the next is $2. a) Define the decision variables and objective function for this problem.
b) Write the constraints for this problem.

A) S1R = The number of standard wheels that are produced on regular time production in the first month

S1O = The number of standard wheels that are produced on overtime production in the first month

S2R = The number of standard wheels that are produced on regular time production in the second month

S2O = The number of standard wheels that are produced on overtime production in the second month

D1R = The number of deluxe wheels that are produced on regular time in the first month

D1O = The number of deluxe wheels that are produced on overtime time in the first month

D2R = The number of deluxe wheels that are produced on regular time in the second month

D2O = The number of deluxe wheels that are produced on overtime time in the second month

Y1 = The number of standard wheels that are stored from the first month to the second month.

Y2 = The number of deluxe wheels that are stored from the first month to the second month.

B) S1R + S1O – Y1 = 1000

D1R + D1O – Y2 = 1200

S2R + S2O + Y1 = 750

D2R + D2O + Y2 = 1000

0.5 S1R + 0.6 D1R ≤ 1000

0.5 S2R + 0.6 D2R ≤ 1000

0.5 S1O + 0.6 D1O ≤ 500

0.5 S2O + 0.6D2O ≤ 500

Explanation:

A) Let's first define the decision variables and objective functions;

S1R = The number of standard wheels that are produced on regular time production in the first month

S1O = The number of standard wheels that are produced on overtime production in the first month

S2R = The number of standard wheels that are produced on regular time production in the second month

S2O = The number of standard wheels that are produced on overtime production in the second month

D1R = The number of deluxe wheels that are produced on regular time in the first month

D1O = The number of deluxe wheels that are produced on overtime time in the first month

D2R = The number of deluxe wheels that are produced on regular time in the second month

D2O = The number of deluxe wheels that are produced on overtime time in the second month

Y1 = The number of standard wheels that are stored from the first month to the second month.

Y2 = The number of deluxe wheels that are stored from the first month to the second month.

B) They received an order for 1000 standard wheels and 1200 deluxe wheels. Thus;

S1R + S1O – Y1 = 1000

D1R + D1O – Y2 = 1200

For the second month they received 750 standard wheels and 1000 deluxe wheels. Thus;

S2R + S2O + Y1 = 750

D2R + D2O + Y2 = 1000

For each of the next two months, we are told that there are 1000 hours of regular time production and 500 hours of overtime production. And that standard wheel requires 0.5 hour of production time & deluxe wheel requires 0.6 hour.

Thus;

0.5 S1R + 0.6 D1R ≤ 1000

0.5 S2R + 0.6 D2R ≤ 1000

0.5 S1O + 0.6 D1O ≤ 500

0.5 S2O + 0.6D2O ≤ 500