8. A manufacturer of wire cloth products can produce four basic product lines: (1) industrial wire cloth; (2) insect screen; (3) roofing mesh; and (4) snow fence. He can sell all that he can produce of each product line for the next year.

The production process for each product line is essentially the same. Aluminum wire is purchased in large

diameters of approximately 0.375 inches and drawn down to finer diameters of 0.009 to 0.018 inches. Then the fine wire is woven on looms, in much the same manner as textiles. Various types of different wire meshes are produced, depending on product line. For example, industrial wire cloth consists of meshes as fine as 30 wires per inch, whilesnow fence has approximately 6 wires per inch. The production process is limited by both wire-drawing capacity and weaving capacity, as well as the availability of the basic raw material, large-diameter aluminum wire.

For the next planning period, there are 600 hours of wire-drawing machine capacity, 1000 hours of loom capacity, and 15 cwt (hundred weight) of large-diameter aluminum wire. The four product lines require the following inputsto make a thousand square feet of output:

Aluminum Wire drawing Weaving

wire(cwt) (100's of hrs) (100's of hrs)

Industrial cloth 1 1 2

Insect screen 3 1 1

Roofing mesh 3 2 1.5

Snow fence 2.5 1.5 2

The contributions from industrial cloth, insect screen, roofing mesh, and snow fence are 2.0, 3.0, 4.2, and 4.0,

respectively, in 100's of dollars per thousand square feet.

a) Formulate a linear program to maximize the firm's contribution from the four product lines.

b) Since roofing mesh and snow fence are really new product lines, analyze the problem graphically, considering onlyindustrial wire cloth and insect screen.

i) What is the optimal solution using only these two product lines?

ii) Over what range of contribution of screen cloth will this solution remain optimal?

iii) Determine the shadow prices on the resources for this solution. Give an economic interpretation of these

shadow prices.

iv) Suppose that the manufacturer is considering adding wire-drawing capacity. Over what range of wiredrawing

capacity will the shadow price on this constraint remain unchanged? In this range, what happens to the other shadow prices?

c) Now, considering the fullline of four products, answer the following, making use of the information developed

in (b).

i) Will it pay to transfer resources into the production of roofing mesh or snow fence?

ii) Without performing the calculations, what will be the form of the optimal solution of the full product line?

What is the largest number of product lines that will be produced?

iii) Suppose that in the future some new product line is developed. Is it possible that one of the product lines not

presently included in the optimal solution will then be included? Explain.