Exam 4: Linear Programming: an Algebraic Approach

Use the technique developed in this section to solve the minimization problem.

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

A nutritionist at the Medical Center has been asked to prepare a special diet for certain patients. He has decided that the meals should contain a minimum of 700 mg of calcium, 16 mg of iron, and 56 mg of vitamin C. He has further decided that the meals are to be prepared from foods A and B. Each ounce of food A contains 40 mg of calcium, 1 mg of iron, 2 mg of vitamin C, and 2 mg of cholesterol. Each ounce of food B contains 35 mg of calcium, 0.5 mg of iron, 7 mg of vitamin C, and 7 mg of cholesterol. ​ Determine how many ounces of each type of food the nutritionist should use in a meal so that the cholesterol content is minimized and the minimum requirements of calcium, iron, and vitamin C are met. ​

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows: For the first year's production, a sum of $7,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed 198,000 hr; and the amount of labor for on-site work is to be less than or equal to 207,000 labor-hours. Determine how many houses of each type Boise should produce (market research has confirmed that there should be no problems with sales) to maximize its profit from this new venture. __________ houses of Standard Model. __________ houses of Deluxe Model. __________ houses of Luxury Model.

Find the pivot element to be used in the next iteration of the simplex method.

Solve the linear programming problem by the simplex method.

A furniture company is introducing a new line of executive desks made from a specially selected grade of walnut. Initially, three models - A, B, and C - are to be marketed. Each model A desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model B desk requires hr for fabrication, 1 hr for assembly, and 1 hr for finishing; each model C desk requires hr, hr, and hr for fabrication, assembly, and finishing, respectively. The profit on each model A desk is $34, the profit on each model B desk is $37, and the profit on each model C desk is $28. The total time available in the fabrication department, the assembly department, and the finishing department in the first month of production is 279 hr, 181 hr, and 166 hr, respectively. To maximize company's profit, how many desks of each model should be made in the month? What is the largest profit the company can realize? Are there any resources left over?

Use the technique developed in this section to solve the minimization problem. __________; __________; __________; __________ __________; __________; __________

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

A farmer has 150 acres of land suitable for cultivating crops A and B. The cost of cultivating crop A is $40/acre, whereas that of crop B is $60/acre. The farmer has a maximum of $7,400 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 3,300 labor-hours available. If he expects to make a profit of $170/acre on crop A and $180/acre on crop B, how many acres of each crop should he plant in order to maximize his profit? __________ acres of crop A, __________ acres of crop B What is the largest profit the farmer can realize? $__________ Are there any resources left over? $__________ left over, __________ acres of land left over, __________ labor-hours left over

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

The optimal value attained by the objective function of the primal problem may be different from that attained by the objective function of the dual problem.

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

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 = __________ C = __________

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

Solve the linear programming problem by the simplex method. ​

Find the pivot element to be used in the next iteration of the simplex method.

Natsano has at most $60,000 to invest in the common stocks of two companies. He estimates that an investment in company A will yield a return of 30%, whereas an investment in company B, which he feels is a riskier investment, will yield a return of 40%. If he decides that his investment in the stocks of company A is to exceed his investment in the stocks of company B by at least $30,000, determine how much he should invest in the stocks of each company in order to maximize the returns on his investment. Investment in company A $ __________. Investment in company B $ __________.

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows: For the first year's production, a sum of $7,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed 198,000 hr; and the amount of labor for on-site work is to be less than or equal to 207,000 labor-hours. Determine how many houses of each type Boise should produce (market research has confirmed that there should be no problems with sales) to maximize its profit from this new venture.