Calculus 220 section 2 6 Further Optimization Problems More Applications notes by Tim Pilachowski To pick up where we left off last time finding maxima or minima is often an objective in the real world Today we take a look at some business related applications Example A In this example it will be important to know which amount is being optimized An efficiency study of the 8 am to noon shift at a factory shows that the number of units N produced by an average worker t hours after 8 am is modeled by the function N t t 3 9t 2 12t Note that the domain is 0 t 4 Question 1 At what time is the number of units produced at its peak Answer noon Question 2 At what time is the rate of production at its peak Answer 11 am stone wall One prime goal in business is to keep costs to a minimum rent or mortgage insurance salaries utility bills expense for purchasing material advertising etc In each example below there will be a function which we want to minimize or maximize the objective function The limitations of reality become constraint functions y length x width Example B Farmer Al needs to fence in 800 yd 2 with one wall being made of stone which costs 24 per yard and the other three sides being wire mesh which costs 8 per yard What dimensions will minimize cost Answer 40 yd long by 20 yd wide Our next example involves minimizing inventory costs ordering costs the money paid for shipping purchased items and carrying costs the money spent on insurance maintenance and taxes to keep items in storage and available for when we might need them Example C For each delivery a company pays a 50 shipping cost Carrying costs based on the average number of units in stock are 4 per unit per year If we assume sales are uniform over the course of a year and we expect sales of 2500 units how often should we place an order so as to minimize costs Scenario 1 If we order all 2500 units at one time the ordering cost will be 50 If we sell them at a steady rate during the year the average number in stock will be 2500 2 1250 so the annual carrying cost will be 1250 4 5000 giving a total cost of 5050 Scenario 2 If we make four equally sized orders of 625 the ordering costs will be 4 50 200 Average inventory will be 625 2 312 5 so annual carrying cost will be 312 5 4 1250 Total cost is 1450 Scenario 3 We ll derive an equation and use calculus to find the means of ordering which will minimize cost First we define variables Let C total cost Let r the number of orders placed during the year ordering costs will be 50r x x Let x the number of units ordered at a time average number in stock will be carrying costs are 4 2 x 2 2 We want to minimize our objective function We are also working under a constraint we want total number of items ordered to equal out projected sales Final answer Order 250 units 10 times per year Example D A company manufactures 6000 units per year We will assume sales are uniformly distributed throughout the year Set up costs are 60 for each production run Carrying costs based on the average number of units in stock are 8 per unit annually Find the number of units to manufacture in each production run which will minimize costs Answer 300 units each in 20 production runs Note This example is actually the same as the previous one The only difference is that it is seen from the pointof view of a manufacturer rather than a distributor