Exam 4: Linear Programming: an Algebraic Approach

National Business Machines Corporation manufactures two models of fax machines: A and B. Each model A costs $150 to make, and each model B costs $105. The profits are $45 for each model A and $30 for each model B fax machine. If the total number of fax machines demanded each month does not exceed 2,500 and the company has earmarked no more than $600,000/month for manufacturing costs, find how many units of each model National should make each month in order to maximize its monthly profits. What is the largest monthly profit the company can make?

Solve the linear programming problem by the simplex method.

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

A farmer has 150 acres of land suitable for cultivating crops A and B. The cost of cultivating crop A is $40/acre and that of crop B is $60/acre. The farmer has a maximum of $7,800 available for land cultivation. Each acre of crop A requires 20 labor-hours, and each acre of crop B requires 25 labor-hours. The farmer has a maximum of 3,200 labor-hours available. He has also decided that he will cultivate at least 60 acres of crop A. If he expects to make a profit of $170/acre on crop A and $220/acre on crop B, how many acres of each crop should he plant in order to maximize his profit?

Use the simplex method for solving nonstandard problems to solve the given linear programming problem. x = __________ y = __________ z = __________ C = __________

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 50 deluxe cabins and 190 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 140 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 340 deluxe and 800 standard cabins for their 15-day cruise in May. It costs $44,000 to operate a type-A vessel and $54,000 to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?

An oil company operates two refineries in a certain city. Refinery I has an output of 200, 100, and 100 barrels of low-, medium-, and high-grade oil per day, respectively. Refinery II has an output of 100, 200, and 600 barrels of low-, medium-, and high-grade oil per day, respectively. The company wishes to produce at least 1,000, 1,400, and 3,000 barrels of low-, medium-, and high-grade oil to fill an order. If it costs $500/day to operate refinery I and $700/day to operate refinery II, determine how many days each refinery should be operated to meet the requirements of the order at minimum cost to the company. What is the minimum cost?

Solve the linear programming problem by the simplex method.

Use the simplex method for solving nonstandard problems to solve the given linear programming problem. x = __________ y = __________ P = __________

Rewrite the linear programming problem as a maximization problem with constraints involving inequalities of the form (with the exception of the inequalities , , and ). Minimize subject to

Use the simplex method for solving nonstandard problems to solve the given linear programming problem. x = __________ y = __________ P = __________

Ashley has earmarked at most $350,000 for investment in three mutual funds: a money market fund, an international equity fund, and a growth-and-income fund. The money market fund has a rate of return of 7%/year, the international equity fund has a rate of return of 12%/year, and the growth-and-income fund has a rate of return of 16%/year. Ashley has stipulated that no more than 25% of her total portfolio should be in the growth-and-income fund and that no more than 50% of her total portfolio should be in the international equity fund. To maximize the return on her investment, how much should Ashley invest in each type of fund? What is the maximum return?

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

Solve the linear programming problem by the simplex method. Minimize subject to

Use the simplex method for solving nonstandard problems to solve the given linear programming problem.

Solve the linear programming problem by the simplex method.

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

Consider the linear programming problem. Sketch the feasible set for the linear programming problem.

Use the simplex method for solving nonstandard problems to solve the given linear programming problem. x = __________ y = __________ C = __________

A company manufactures two products, A and B, on two machines, I and II. It has been determined that the company will realize a profit of $4/unit of product A and a profit of $5/unit of product B. To manufacture 1 unit of product A requires 7 min on machine I and 9 min on machine II. To manufacture 1 unit of product B requires 9 min on machine I and 3 min on machine II. There are 6 hr of machine time available on machine I and 4 hr of machine time available on machine II in each work shift. What is the largest profit the company can realize?