Creative Sports Design (CSD) manufactures a standard-size racket and an oversize racket. The firm’s rackets are extremely light due to the use of a magnesium-graphite alloy that was invented by the firm’s founder. Each standard-size racket uses 0.125 kilograms of the alloy and each oversize racket uses 0.4 kilograms; over the next two-week production period only 80 kilograms of the alloy are available. Each standard-size racket uses 10 minutes of manufacturing time and each oversize racket uses 12 minutes. The profit contributions are $10 for each standard-size racket and $15 for each oversize racket, and 40 hours of manufacturing time are available each week. Management specified that at least 20% of the total production must be the standard-size racket. How many rackets of each type should CSD manufacture over the next two weeks to maximize the total profit contribution? Assume that because of the unique nature of their products, CSD can sell as many rackets as they can produce.

165 oversize rackets = 32 machine hours (79.71% of total production)

42 standard size rackets = 7 machine hours (20.29% of total production)

total profit contribution = (165 x $15) + (42 x $10) = $2,895

Explanation:

                                         materials          machine hours      profit

standard size                    0.125 kg              1/6                        $10

oversize                             0.4 kg                 1/5                        $15

constraints 80 kilograms of materials

40 hours of manufacturing

profit per machine hour:

standard size  $10 x 6 = $60 x 40 hours = $2,400 (total possible production = 240 rackets)

oversize  $15 x 5 = $75 x 40 hours = $3,000 (total possible production = 200 rackets)

profit per kilogram of alloy:

standard size  $10 / 0.125 = $80 x 80 kgs = $6,400 (total possible production = 480 rackets)

oversize  $15 / .4  = $37.50 x 80 hours = $3,000 (total possible production = 200 rackets)

since the most important constraint is the manufacturing hours available, the company should try to produce the products that yield the highest contribution margin per machine hour. In this case, at least 20% of total production must be standard size rackets, so the remaining 80% should be oversize rackets that yield a higher profit.

165 oversize rackets = 32 machine hours (79.71% of total production)

42 standard size rackets = 7 machine hours (20.29% of total production)

total manufacturing time = 40 hours

if we produce 166 oversize rackets and 41 standard size rackets, total manufacturing time will exceed 40 hours (40.03 hours exactly).