5. The Pinewood Furniture Company produces chairs and tables from two resources—labor and wood. The company has 80 hours of labor and 36 board-ft. of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood, whereas a table requires 10 hours of labor and 6 board-ft. of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day to maximize profit. a. Formulate a linear programming model for this problem. b. Solve this model by using graphical analysis.
9. Solve the following linear programming model graphically: maximize Z = 3×1 + 6×2 subject to 3×1 + 2×2 … 18 x1 + x2 Ú 5 x1 … 4
x1, x2 Ú 0
29. Solve the following linear programming model graphically: minimize Z = 8×1 + 2×2 subject to 2×1 – 6×2 … 12 5×1 + 4×2 Ú 40 x1 + 2×2 Ú 12 x2 … 6
x1, x2 Ú 0
35. Assume that the objective function in Problem 34 has been changed from Z = 30×1 + 70×2 to Z = 90×1 + 70×2. Determine the slope of each objective function, and discuss what effect these slopes have on the optimal solution.
Chapter 3: problem 22, 23, 46, 47
22. Solve the linear programming model formulated in Problem 21 for Rucklehouse Public Relations graphically. a. Determine the sensitivity ranges for the daily cost of a telephone interviewer and the number of personal interviews required.
b. Does the firm conduct any more telephone and personal interviews than are required, and if so, how many more?
c. What would be the effect on the optimal solution if the firm was required by the client to increase the number of personal interviews conducted from 800 to a total of 1,200?
23. Solve the linear programming model formulated in Problem 21 for Rucklehouse Public Relations by using the computer. a. If the firm could reduce the minimum interview requirement for either telephone or personal interviews, which should the firm select? How much would a reduction of one interview in the requirement you selected reduce total cost? Solve the model again, using the computer, with the reduction of this one interview in the constraint requirement to verify your answer.
b. Identify the sensitivity ranges for the cost of a personal interview and the number of total interviews required.
46. John Hoke owns Hoke’s Spokes, a bicycle shop. Most of John’s bicycle sales are customer orders; however, he also stocks bicycles for walk-in customers. He stocks three types of bicycles—road-racing, cross-country, and mountain. A road-racing bike costs road-racing bike costs $1,200, a cross-country bike costs $1,700, and a mountain bike costs $900. He sells road-racing bikes for $1,800, cross-country bikes for $2,100, and mountain bikes for $1,200. He has $12,000 available this month to purchase bikes. Each bike must be assembled; a road-racing bike requires 8 hours to assemble, a cross-country bike requires 12 hours, and a mountain bike requires 16 hours. He estimates that he and his employees have 120 hours available to assemble bikes. He has enough space in his store to order 20 bikes this month. Based on past sales, John wants to stock at least twice as many mountain bikes as the other two combined because mountain bikes sell better.
Formulate a linear programming model for this problem.
47. Solve the linear programming model formulated in Problem 46 for Hoke’s Spokes by using the computer. a. Should John Hoke try to increase his budget for purchasing bikes, increase space to stock bikes, or increase labor hours to assemble bikes? Why?
b. If John hired an additional worker for 30 hours at $10 per hour, how much additional profit would he make, if any?
c. If John purchased a cheaper cross-country bike for $1,200 and sold it for $1,900, would this affect the original solution?
Chapter 4: problem 12, 17,18, 27, 50, 65,66,67
12. The Hickory Cabinet and Furniture Company produces sofas, tables, and chairs at its plant in Greensboro, North Carolina. The plant uses three main resources to make furniture—wood,upholstery, and labor. The resource requirements for each piece of furniture and the total resources available weekly are as follows:
The furniture is produced on a weekly basis and stored in a warehouse until the end of the week, when it is shipped out. The warehouse has a total capacity of 650 pieces of furniture. Each sofa earns $400 in profit, each table, $275, and each chair, $190. The company wants to know how many pieces of each type of furniture to make per week to maximize profit. a. Formulate a linear programming model for this problem. b. Solve the model by using the computer.
17. The Midland Tool Shop has four heavy presses it uses to stamp out prefabricated metal covers and housings for electronic consumer products. All four presses operate differently and are of different sizes. Currently the firm has a contract to produce three products. The contract calls for 400 units of product 1, 570 units of product 2, and 320 units of product 3. The time (in minutes) required for each product to be produced on each machine is as follows:
Machine 1 is available for 150 hours, machine 2 for 240 hours, machine 3 for 200 hours, and machine 4 for 250 hours. The products also result in different profits, according to the machine they are produced on, because of time, waste, and operating cost. The profit per unit per machine for each product is summarized as follows:
The company wants to know how many units of each product to produce on each machine to maximize profit. a. Formulate this problem as a linear programming model. b. Solve the model by using the computer.
18. The Cabin Creek Coal (CCC) Company operates three mines in Kentucky and West Virginia, and it supplies coal to four utility power plants along the East Coast. The cost of shipping coal from each mine to each plant, the capacity at each of the three mines, and the demand at each plant are shown in the following table
The cost of mining and processing coal is $62 per ton at mine 1, $67 per ton at mine 2, and $75 per ton at mine 3. The percentage of ash and sulfur content per ton of coal at each mine is as follows:
Each plant has different cleaning equipment. Plant 1 requires that the coal it receives have no more than 6% ash and 5% sulfur; plant 2 coal can have no more than 5% ash and sulfur combined; plant 3 can have no more than 5% ash and 7% sulfur; and plant 4 can have no more than 6% ash and sulfur combined. CCC wants to determine the amount of coal to produce at each mine and ship to its customers that will minimize its total cost. a. Formulate a linear programming model for this problem. b. Solve this model by using the computer.
50. In Problem 49, the optimal solution results in a fractional (i.e., noninteger) number of lawyers being hired. Explain how you would go about logically determining a new solution with a whole (integer) number of lawyers being hired, and discuss the difference in results between this new solution and the optimal noninteger solution obtained in Problem 49.
65. Valley Fruit Products Company has contracted with apple growers in Ohio, Pennsylvania, and New York to purchase apples that the company then ships to its plants in Indiana and Georgia, where they are processed into apple juice. Each bushel of apples produces 2 gallons of apple juice. The juice is canned and bottled at the plants and shipped by rail and truck to warehouses/ distribution centers in Virginia, Kentucky, and Louisiana. The shipping costs per bushel from the farms to the plants and the shipping costs per gallon from the plants to the distribution centers are summarized in the following tables:
Formulate and solve a linear programming model to determine the optimal shipments from the farms to the plants and from the plants to the distribution centers in order to minimize total ship-ping costs.
66. In the event of a disaster situation at Tech from weather, an accident, or terrorism, victims will be transported by emergency vehicles to three area hospitals: Montgomery Regional, Radford Memorial, and Lewis Galt. Montgomery Regional is (on average) 10 minutes away from Tech, Radford Memorial is 20 minutes away, and Lewis Galt is 35 minutes away. Tech wants to analyze a hypothetical disaster situation in which there are 15 victims with different types of injuries. The emergency facilities at Montgomery Regional can accommodate, at most, 8 victims; Radford Memorial can handle 10 victims; and Lewis Galt can admit 7 victims. A priority has been assigned for each victim according to the hospital that would best treat that victim’s type of injury, as shown in the following table (where 1 reflects the best treatment).
For example, for victim 1’s type of injury, the best hospital is Montgomery Regional, the next best is Radford Memorial, and Lewis Galt is the third best. a. Formulate and solve a linear programming model that will send the victims to the hospital best suited to administer to their specific injuries while keeping the average transport time to 22 minutes or less.
b. Formulate and solve a linear programming model that will minimize the average transport time for victims while achieving an average hospital priority of at least 1.50 or better.
67. Two coffee house chains, Startrak and Mill Mountain, dominate the Columbus market. Startrak has developed three new marketing strategies, 1, 2, and 3, for the fall season, which include various combinations of media and print advertising, and promotions. Mill Mountain knows Startrak plans a marketing campaign, and it has developed three strategies of its own, A, B and C, to offset those of Startrak. The following table shows the percentage market share gains (or losses) expected by Startrak for each of its three strategies given the possible strategies of Mill Mountain. Startrak obviously wants to maximize its gains, whereas Mill Mountain wants to minimize Startrak’s gains.
Startrak wants to select a mix of its strategies such that its gains are the same regardless of the strategies of Mill Mountain; Startrak’s objective is to select a “mixed strategy” (i.e., the probabili-ties of selecting each of its three strategies) in such a way as to maximize its minimum expected gains. This is a problem in the field known as “Game Theory,” and this decision scenario can be formulated and solved as a linear programming problem. If the value of the game that Startrak wants to maximize is denoted by “V,” and the probabilities that Startrak will select strategies 1, 2 and 3, are respectively ;P1,< ;P2,< and ;P3,< formulate and solve a linear programming model for this game situation.