Linear Programming 2.13 The Apple Company has made a contract with the governmen
ID: 430236 • Letter: L
Question
Linear Programming
2.13 The Apple Company has made a contract with the government to supply 1200 microcomputers this year and 2500 next year. The company has the production capacity to make 1400 microcomputers each year. and it has already committed its production line for this level. Labor and management have agreed that the production line can be used for at most 80 overtime shifts each year, each shift costing the company an additional $20,000. In each overtime shift, 50 microcomputers can be manufactured. Units made this year but used to meet next year's demand must be stored at a cost of $100 per unit. (a) Formulate a linear programming model for finding a production schedule that minimizes cost. (b) Explain how an optimal production schedule can be obtained by just thinking carefully about the problems without even considering a linear programming for mutation.Explanation / Answer
a. let the number of overtime shifts be "x" , "y" for both the years (year 1 and year 2 respectively). total computers made in overtime = 50(x+y). total regular schedule = 1400 computers per year or 2800 computers for the 2 years. Total demand = 1200+2500 = 3700
cost of overtime shift = (x+y)*20,000 = $20,000(x+y). number of computers made in 1st year = regular schedule+overtime schedule production = 1400+50x
units to be carried forward next year = total production in 1st year - demand for 1st year = 1400+50x - 1200 = 200+50x
cost of storage = 100*(200+50x) = 20,000+5,000x
totak cost = cost of overtime+cost of storage = $20,000(x+y)+ 20,000+5,000x = 25,000x+20,000y+20,000. This is the objective function and we have to minimize it subject to the constraints:
(1) 50*(x+y) = 900 (total demand for 2 years - production capacity for 2 years)
(2)x,y>=0
b. the first year does not require any overtime as the company has the capacity to make 1400 microcomputers. Requirement for the 1st year = 1200. stock at the end of 1st year = 1400. stock that can be carried forward for the 2nd year = 1400 - 1200 = 200.
Thus production required for 2nd year = 2500 - 200 = 2300 microcomputers. Normal production = 1400. Thus computers that will be made in the overtime shift = 2300 - 1400 = 900.
50 microcomputers can be made in each shift. so number of shifts required = 900/50 = 18.
total cost of storing 200 computers from 1st year = 200*100 = $20,000.
total cost of overtime = 18 shifts *20,000 = $360,000. Thus minimum total cost = 20,000+360,000 = $380,000
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