Question

AlamoPrint is a small printing company who currently have several contracts from manufacturers of Italian foods. For these customers AlamoPrint prints two products – boxes for pizzas and wrappers which are wrapped around foil pasta dishes. Demand is strong for these two products and AlamoPrint can sell all the boxes and wrappers it can print.

AlamoPrint delivers the printed goods to its customers and has a maximum delivery capacity per working day of 60000 wrappers or 30000 boxes or an equivalent combination of both.

Two people are employed who set up the printing machines, load and unload work on to the machines, and pack finished material. Each person is employed for 8 hours per day, the length of the working day at AlamoPrint, and can complete the tasks require for a batch of 500 wrappers in 5 minutes and for a batch of 100 boxes in 6 minutes. Both people can switch seamlessly between tasks.

AlamoPrint has two machines each of which can print either boxes or wrappers during the 8 - hour working day at the rate of 200 wrappers per minute or 10 boxes per minute.

It is hot work producing boxes, so for health and safety reasons production of boxes, by all processes, is limited to a maximum of 20% of total production of boxes and wrappers.

If AlamoPrint charges $5 for 100 boxes and $2 for 100 wrappers, how many should it plan to print each 8 - hour working day to maximize income? In deriving your answer, exact values for decision variables (i.e. results accurate to the nearest two decimal places) are required. You may also assume that fractional values for decision variables are acceptable, because production is continuous. You must develop an LP model for this problem and solve it using graphical method.

Answer 1

LP model is following:

Let W and B be the number of wrappers and boxes to be produced.

Max 2W/100 + 5B/100

s.t.

W/60000 + B/30000 <= 1 or, W + 2B <= 60000   (whole eqn multiplied by 60000)

5W/500 + 6B/100 <= 2*8*60 or, W + 6B <= 96000   (whole eqn multiplied by 100)

1W/200 + 1B/10 <= 2*8*60 or, W + 20B <= 192000 (whole eqn multiplied by 200)

B <= 20%*(W + B) or,  -.2W+.8B <= 0

W, B >= 0

Solution using graphical method is following:

Optimal production plan:

Number of Wrappers to be produced = 45,333.33

Number of Boxes to be produced = 7,333.333

total profit = $ 1,273.33