BUS 323 MIDTERM EXAMINATION I                                                                                                                                                                  

Nov. 4, 2003

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FALL 2003-2004

  1. Cem Yılmaz is in charge of loading cargo ships for International Cargo Company (ICC) at the port in Aliağa, İzmir. Cem is preparing a loading plan for an ICC freighter destined for Ghana. An agricultural commodities dealer wants to transport the following products aboard this ship.

Paul can elect to load any and/or all of the available commodities. However, the ship has three cargo holds with the following capacity restrictions.

More than one type of commodity can be placed in the same cargo hold. However, because of balance considerations, the weight in the forward cargo hold must be within 10% of the weight in the rear cargo hold and the center cargo hold must be between 40% to 60% of the total weight on board.

1. Formulate LP model for this problem.

2. Create a spreadsheet model and define Solver parameters on the Excel and Solver figures below.

SOLUTION:

X_ij = tons of commodity i stored in hold j

Maks.         70(X₁₁+X₁₂+X₁₃) + 50(X₂₁+X₂₂+X₂₃) + 60(X₃₁+X₃₂+X₃₃) + 80(X₄₁+X₄₂+X₄₃)

Kısıtlar

X₁₁ + X₁₂ + X₁₃ < 4800

X₂₁ + X₂₂ + X₂₃ < 2500

X₃₁ + X₃₂ + X₃₃ < 1200

X₄₁ + X₄₂ + X₄₃ < 1700

X₁₁ + X₂₁ + X₃₁ + X₄₁ < 3000

X₁₂ + X₂₂ + X₃₂ + X₄₂ < 6000

X₁₃ + X₂₃ + X₃₃ + X₄₃ < 4000

40 X₁₁ + 25 X₂₁ + 60 X₃₁ + 55 X₄₁ < 145000

40 X₁₂ + 25 X₂₂ + 60 X₃₂ + 55 X₄₂ < 180000

40 X₁₃ + 25 X₂₃ + 60 X₃₃ + 55 X₄₃ < 155000

0.9*(X₁₃ + X₂₃ + X₃₃ + X₄₃ ) < X₁₁ + X₂₁ + X₃₁ + X₄₁

X₁₁ + X₂₁ + X₃₁ + X₄₁ <  1.1*(X₁₃ + X₂₃ +   X₃₃ + X₄₃)

0.4 * Toplam < X₁₂ + X₂₂ + X₃₂ + X₄₂

X₁₂ + X₂₂ + X₃₂ + X₄₂ < 0.6 * Toplam

X_ij > 0

(Not: Toplam = X₁₁ + X₁₂ + X₁₃ +X₂₁ + X₂₂ + X₂₃ + X₃₁ + X₃₂ + X₃₃ + X₄₁ + X₄₂ + X₄₃)

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

For instance, a ton of newspaper can be recycled using a technique that yields 0.85 tons of newsprint pulp. Alternatively, a ton of newspaper can be recycled using a technique that yields 0.80 tons of packaging paper.

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

a.        a. Write the mathematical model of the problem.

b.       b. Show the formulas in the cells, F10, C14, C24

F10: =SUM(C10:E10)

C14: =SUMPRODUCT(C4:C7;C10:C13)

C24: =SUMPRODUCT(C10:E13;C19:E22)+SUMPRODUCT(F10:F13;F19:F22)

c.        c.  Is the solution unique?

Yes

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

$0.  There is a surplus of cardboard already

e.        e. 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?

Yes. The recycler currently pays $15 per ton.  The shadow price is $3.14.  So the recycler should be             willing to pay up to $18.14 to acquire up to 348 additional tons.

f.         f. What is the recycler’s marginal cost of producing each of the three different types of pulp?

Newspaper $28.99, Packaging Paper $36.43, Print Stock Paper $37.70.

g.       g. By how much would the cost of converting white office paper into news print have to drop before it would become economical to use white office paper for this purpose?

$1.87.

h.       h. By how much would the yield of news print pulp per ton of cardboard have to increase before it would become economical to use cardboard for this purpose?

24.499999/28.9915 = 0.845072