Problem 1

Fox Enterprises is considering six projects for possible construction over the next 4 years. Fox can undertake any of the projects partially or completely. A partial undertaking of a project will prorate both the return and cash outlays proportionately. The expected (present value) returns and cash outlays for the projects are given in the following table.

1. Formulate the problem as a linear program, and determine the optimal project mix that maximizes the total return using AMPL and MINOS or CPLEX. Ignore the time value of money.
2. Suppose that if a portion of project 2 is undertaken, then at least an equal portion of project 6 must be undertaken. Modify the formulation of the model, and find the new optimal solution.
3. In the original model, suppose that any funds left at the end of the year are used in the next year. Find the new optimal solution, and determine how much each year "borrows" from the preceding year. For simplicity, ignore the time value of money.
4. Suppose that in the original model the yearly funds available for any year can be exceeded, if necessary, by borrowing from other financial activities within in the company. Ignoring the time value of money, reformulate the LP model, and find the optimal solution.

Problem 2

A large department store operates 7 days a week. The manager estimates that the minimum number of salespersons required to provide prompt service is 12 for Monday, 18 for Tuesday, 20 for Wednesday, 28 for Thursday, 32 for Friday, and 40 for each of Saturday and Sunday. Each salesperson works 5 days a week, with two consecutive off-days staggered throughout the week. For example, if 10 salespersons start on Monday, 2 can take their off-days on Tuesday and Wednesday, 5 on Wednesday and Thursday, and 3 on Saturday and Sunday. How many salespersons should be contracted, and how should their off-days be allocated? Use AMPL and MINOS or CPLEX to find an optimal solution.

Problem 3

Beerco manufactures ale and beer from corn, hops, and malt. At present, 40 lbs of corn, 30 lbs of hops, and 40 lbs of malt are available. A barrel of ale sells for $40 and requires 1 lb of corn, 1 lb of hops, and 2 lbs of malt. A barrel of beer sells for $50 and requires 2 lbs of corn, 1 lb of hops and 1 lb of malt. Beerco can sell all ale and beer that is produced. To optimize profits, Beerco runs the following AMPL script:

# beerco.txt

var Ale >= 0; # Barrels of ale produced

var Beer >= 0; # Barrels of beer produced

maximize Revenue: 40 * Ale + 50 * Beer;

subject to Corn: Ale + 2*Beer <= 40;

subject to Hops: Ale + Beer <= 30;

subject to Malt: 2*Ale + Beer <= 40;

option solver cplex;

option cplex_option ‘sensitivity’;

solve;

display Ale, Beer;

display Ale.down, Ale.current, Ale.up, Ale.rc;

display Beer.down, Beer.current, Beer.up, Beer.rc;

display Corn.down, Corn.current, Corn.up, Corn.dual;

display Hops.down, Hops.current, Hops.up, Hops.dual;

display Malt.down, Malt.current, Malt.up, Malt.dual;

Use the output from the script shown on the next page to answer the following questions:

1. How many barrels of ale and beer should be produced to maximize Beerco's profit?
2. For what range of values of the price of ale does the solution you indicated in part (a) remain optimal?
3. Suppose that Beerco has the opportunity to purchase an additional 5 lbs of one of the ingredients (corn, hops, or malt) for $50. Which ingredient, if any, should it purchase? Explain your answer in terms of the output shown on the next page.

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CPLEX 9.1.3: sensitivity

ILOG CPLEX, licensed to "AMPL Student Edition for Taha Textbook".

CPLEX 9.1.3: optimal solution; objective 1200

2 dual simplex iterations (1 in phase I)

suffix up OUT;

suffix down OUT;

suffix current OUT;

Ale = 13.3333

Beer = 13.3333

Ale.down = 25

Ale.current = 40

Ale.up = 100

Ale.rc = 0

Beer.down = 20

Beer.current = 50

Beer.up = 80

Beer.rc = 0

Corn.down = 20

Corn.current = 40

Corn.up = 50

Corn.dual = 20

Hops.down = 26.6667

Hops.current = 30

Hops.up = 1e+20

Hops.dual = 0

Malt.down = 20

Malt.current = 40

Malt.up = 50

Malt.dual = 10

Problem 4

A farmer has just purchased an unstocked farm with 1,040 acres of pasture. He has a budget of $10,400 to spend on stocking the farm. He can buy sheep, pigs, or cattle. The current price, estimate of annual revenue, and the number of acres required per animal are given the table below.

To optimize profits, the farmer runs the following AMPL script:

# farm.txt

var sheep >= 0; # Number of Sheep

var pigs >= 0; # Number of Pigs

var cows >= 0; # Number of Cows

maximize profit: 12*sheep + 7*pigs + 40*cows;

subject to budget: 7*sheep + 10*pigs + 100*cows <= 10400;

subject to land: sheep + 0.5*pigs + 3*cows <= 1040;

option solver cplex;

option cplex_options 'sensitivity';

solve;

display sheep, pigs, cows;

display sheep.down, sheep.current, sheep.up, sheep.rc;

display pigs.down, pigs.current, pigs.up, pigs.rc;

display cows.down, cows.current, cows.up, cows.rc;

display budget.down, budget.current, budget.up, budget.dual;

display land.down, land.current, land.up, land.dual;

Use the output from the script (shown on the next page) to answer the following questions:

1. How should the farmer stock the farm in order to maximize his profit in the first year?
2. By how much must the revenue per head of cattle increase before the answer to part (a) changes? Explain your answer.
3. Suppose that the owner of the neighboring farm has offered to sell the farmer 104 acres of land at a price of $10 per acre. Would you recommend that the farmer purchase this additional land? Explain your answer.

CPLEX 9.1.3: sensitivity

ILOG CPLEX, licensed to "AMPL Student Edition for Taha Textbook".

CPLEX 9.1.3: optimal solution; objective 12960

2 dual simplex iterations (1 in phase I)

suffix up OUT;

suffix down OUT;

suffix current OUT;

sheep = 800

pigs = 480

cows = 0

sheep.down = 4.9

sheep.current = 12

sheep.up = 14

sheep.rc = 0

pigs.down = 6.32911

pigs.current = 7

pigs.up = 17.1429

pigs.rc = 0

cows.down = -1e+20

cows.current = 40

cows.up = 48.1538

cows.rc = -8.15385

budget.down = 7280

budget.current = 10400

budget.up = 20800

budget.dual = 0.153846

land.down = 520

land.current = 1040

land.up = 1485.71

land.dual = 10.9231

Price: $29.2

Solution: The downloadable solution consists of 18 pages, 1120 words and 11 charts.
Deliverable: Word Document