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# Statistics: A Linear Programming

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*10. Consider the production, sales inventory, and capacity limits stated below for the manufacture of small turbine engines used in auxiliary power units, ground support equipment and various other uses.

The company expects to market the unit for \$50,000 each in quarters one and two and to raise the price to \$55,000 for the third quarter.

At that price it is expected that demand for the three periods will be 250, 300, and 300 respectively.

Cost of production is expected to increase by \$1000 each quarter, with period one cost at \$28,000. If overtime production must be used cost of production during overtime will be 20% higher than normal production costs, due to higher wage rates.

Production capacity for the three periods is 250, 300, 300, respectively, and overtime capacity is 100, 100, 125 for each period, respectively.

Inventory costs, including an estimate of opportunity costs of the money and labor invested is \$5,000 for periods one and two, and \$5,500 for period three.

The costs of lost sales, an amount to indicate loss of customer goodwill and loyalty, is estimated to be very high, since there are several new, aggressive competitors entering the market. It is estimated at \$40,000 per lost sale.

At commencement of period 1 there are 100 units in inventory. Due to expected high demand in period four it is desired to have not less than 50 units in inventory.

a. Develop a table containing sales, lost sales, product, overtime production, and inventory at period completion for each period.

b. What is the total profit for the three periods combined?

c. If inventory costs were reduced by 50% would the production schedule change?

d. If demand in period one were to be higher than expected, what would happen to the solution?

e. If demand in period two were 400 vice 300 what would be the effect on schedule and profitability?

f. If demand in period two were 500 vice 300 what would be the effect on schedule and profitability? Compare this result with that found in (e).

g. If overtime capacity could be increased by 20% by changing the overtime differential to 40% vice 20% how would that effect profitability given the conditions of (b), (e), and (f) respectively?

h. What appears to be the overriding factor in the above example and what steps would you recommend to the company?