Problems Integer Programming Models
 Problems for Integer Programming - Product Mix with Variations

Three products can be produced at two machining centers. The products may be produced in fractional amounts. The linear relationships describing this situation are listed below. The variables are:

A, B and C are the amounts of the three products in units.

R1 and R2 are the amounts of raw materials used in kilograms.

T1 and T2 are the times used in the two machining centers.

Linear expressions describing important quantities associated with the products have been derived and are shown below.

 Profit: P = 2A + 3B + 2.55C - 0.6R1 - 0.8R2 Time required on machine 1: T1 = 0.5A + 0.8B + 1C (hours) Time required on machine 2: T2 = 0.8A + 0.6B + 0.2C (hours) Raw material 1 used: R1 = 0.1A + 0.2B + 0.075C (kilograms) Raw material 2 used: R2 = 0.05A + 0.1B + 0.05C (kilograms) Market Limits: A <= 100, B <= 200, C <= 100.

The linear programming model when objective is to maximize profit is:

Max Z = 2A + 3B + 2.5C - 0.6R1 - 0.8R2

subject to:

 Machine 1: 0.5A + 0.8B + 1C <= 100 (hours) Machine 2: 0.8A + 0.6B + 0.2C <= 100 (hours) RM 1 0.1A + 0.2B + 0.075C - R1 = 0 RM 2: 0.05A + 0.1B + 0.05C - R2 = 0 Market: A <= 100, B <= 200, C <= 100. Nonnegativity: A >= 0, B >= 0, C >= 0, R1 >= 0, R2 >= 0.

The following paragraphs describe modifications of the situation. The modifications are not cumulative. Show the changes in the model necessary to describe the new situation. Some changes require the introduction of integer variables while others require the incorporation of nonlinear functions. In either case add as few variables as possible.

When the model uses integer variables it must have the linear form.

a. The revenue for each product is reduced by an advertising cost as illustrated in the figure for product A. In order to sell any A, \$40 must be expended. If no A is sold, the cost is not expended. Product B and C have similar advertising costs, \$5 for B and \$60 for C.

b. The raw material costs are nonlinear in the following fashion.

• Raw Material R1: The unit cost for the first 50 kilograms is \$0.60 per kg.. The unit cost for amounts greater than 50 but less than 100 is \$0.70 per kg. The unit cost for purchases above 100 kg. is \$0.75 per kg..
• Raw Material R2: The unit cost for the first 50 kilograms is \$0.80 per kg.. The unit cost for amounts greater than 50 but less than 100 is \$0.75 per kg. The unit cost for purchases above 100 kg. is \$0.70 per kg..

c. Add the logical restriction that if you produce more than 20 of product A, then you must produce at least 5 of product B.

d. We specify that you can produce at most two of the three products.

Operations Research Models and Methods
Internet
by Paul A. Jensen