Management Science - Linear Programming
The Silver Star Bicycle Company must determine how many new sport model bicycles to produce each of the next four quarters. The demand for each of the next four quarters is as follows: first quarter, 60 bicycles; second quarter, 55 bicycles; third quarter, 75 bicycles, fourth quarter, 65 bicycles. Sliver Star must meet demand on time. At the beginning of the first quarter, Silver Star has an inventory of 35 bicycles. At the beginning of each quarter, Silver Star must decide how many bicycles should be produced during that quarter. For simplicity, we assume that bicycles manufactured during a quarter can be used to meet demand for that quarter.
During each quarter, Silver Star produces up to 35 bicycles with regular-time labor at a total cost of $400 per bicycle. By having employees work overtime during a quarter, Silver Star can produce additional bicycle with overtime labor at a cost of $450 per bicycle. At the end of each quarter (after production has occurred and the current quarter’s demand has been satisfied), a carrying or holding cost of $20 per bicycle is incurred. Formulate the linear programming to determine a production schedule to minimize the sum of production and inventory costs during the next four quarters. Do not solve.
So what I've done for variables is Pa,1 (for bikes @400 in quarter 1) and Pb,1 (for overtime bikes @450 in quarter 1), I1 for inventory at end of quarter 1 and so on.
So minimize equation 400Pa1 + 450Pb1 + 400Pa2 + 450Pb2 + 400Pa3 + 450Pb3 + 400Pa4 + 450Pb4 + 20I1 + 20I2 + 20I3 + 20I4
Constraints:
Pa1 + Pb1 - I1 = 25
Pa2 + Pb2 - I2 = 55
Pa3 + Pb3 - I3 = 75
Pa4 + Pb4 - I4 = 65
Pai < 35 where all values of i = 1,2,3,4
Pai >/ 0 where all values of i = 1,2,3,4
Pbi >/ 0 where all values of i = 1,2,3,4
Ii >/0 where all values of i = 1,2,3,4
So am I on the right track here? I'm having trouble on how to show that the bikes at $450 (overtime) can't be produced until 35 bikes have been produced at $400.
Was it the right approach to use two different variables for the two prices?
Thanks in advance.