## Linear Programming

Linear ProgrammingINDIVIDUAL ASSIGNMENT

Linear Programming Problem

A paper recycling company converts newspaper, mixed paper, white office paper,and cardboard into pulp for newsprint, packaging paper, and print stock qualitypaper. The following table summarizes the yield for each kind of pulp recoveredfrom each ton of recycled material.

Recycling Yield

Newsprint Packaging Print Stock

Newspaper 85% 80% —

Mixed Paper 90% 80% 70%

White Office Paper 90% 85% 80%

Cardboard 80% 70% —

For instance, a ton of newspaper can be recycled using a technique that yields 0.85tons of newsprint pulp. Alternatively, a ton of newspaper can be recycled using atechnique that yields 0.80 tons of packaging paper. Similarly, a ton of cardboard canbe recycled to yield 0.80 tons of newsprint or 0.70 tons of packaging paper pulp.

Note that newspaper and cardboard cannot be converted to print stock pulp usingthe techniques available to the recycler.

The cost of processing each ton of raw material into the various types of pulp issummarized in the following table, along with the amount of each of the four rawmaterials that can be purchased and their costs.

Newsprint Packaging Print Stock Purchase Cost Tons Available

Per Ton

Newspaper $6.50 $11.00 — $15 600

Mixed Paper $9.75 $12.25 $9.50 $16 500

White Office Paper $4.75 $7.75 $8.50 $19 300

Cardboard 7.50 $8.50 — $17 400

The recycler wants to determine the least costly way of producing 500 tonsof newsprint pulp, 600 tons of packaging paper pulp, and 300 tons of print stockquality pulp.

a. Formulate a LP Model for the problem

b. Create a spreadsheet model for this problem and solve it.

c. What is the optimal solution?

Use Solver to create a Sensitivity Report for the problem andanswer the following questions.

d. Is the solution degenerate?

e. Is the solution unique?

c. How much should the recycler be willing to pay to acquire more cardboard?

f. If the recycler could buy 50 more tons of newspaper at a cost of $18 per ton,should they do it? Why or why not?

g. What is the recycler’s marginal cost of producing each of the three differenttypes of pulp?

h. By how much would the cost of converting white office paper into newsprinthave to drop before it would become economical to use white office paper forthis purpose?

i. By how much would the yield of newsprint pulp per ton of cardboard have to increase

before it would become economical to use cardboard for this purpose?

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