The Kazak Film company needs to cut 15 long rolls and 10 short rolls of film from stock pieces. Each stock piece can be cut in one of two patterns. The first produces 5 long and 2 short rolls; the second yields 3 long and 5 short. Once any part of a piece of stock is cut, anything that remains is scrap. Also, neither pattern should be used more than 4 times because the jig used to cut it will become too inaccurate. Kazak wants to find the allowable combination of patterns that will minimize the number of stock pieces required.

(a) Formulate a mathematical model to decide what patterns to use. Use decision variables x1! number of times pattern 1 is used and x2! number of times pattern 2 is used.

(b) Both variables in your model should be restricted to take on only integer (whole number) values. Explain why.

(c) Enter and solve your model with the class optimization software.

(d) Using a 2-dimensional plot, solve your model graphically for an optimal cutting plan. Remember to consider only integer points.

(e) Explain how your plot shows that the model has alternative optimal solutions.