13-7. Klein Industries manufactures three types of portable air compressors: small, medium, and large, which have unit profits of $20.50, $34.00, and $42.00, respectively.

The projected monthly sales are:

Small Medium Large

Minimum 14,000 6,200 2,600

Maximum 21,000 12,500 4,200

The production process consists of three primary activities: bending and forming, welding, and painting.

The amount of time in minutes needed to process each product in each department is shown below:

Small Medium Large Available Time

Bending/forming 0.4 0.7 0.8 23,400

Welding 0.6 1.0 1.2 23,400

Painting 1.4 2.6 3.1 46,800

How many of each type of air compressor should the company produce to maximize profit?

a. Formulate and solve a linear optimization model using the auxiliary variable cells method and write a short memo to the production manager explaining the sensitivity information.

b. Solve the model without the auxiliary variables and explain the relationship between the reduced costs and the shadow prices found in part a.