10 mini exercises Please use an excel sheet and show work 5-1 The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 board feet of good-quality redwood. Each bench that Outdoor Furniture produces requires 4 labor hours and 10 board feet of redwood; each picnic table takes 6 labor hours and 35 board feet of redwood. Completed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. a. How many benches and tables should Outdoor Furniture produce to obtain the largest possible profit? Use the graphical LP approach. (Submit both the QM and the graphical solution.) 5-2 MSA Computer Corporation manufactures two models of minicomputers, the Alpha 4 and the Beta 5. The firm employs six technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (i.e., all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor hours to assemble each Alpha 4 computer and 25 labor hours to assemble each Beta 5 model. MSA wants to see at least 10 Alpha 4s and at least 15 Beta 5s produced during the production period. Alpha 4s generate $1,200 profit per unit, and Beta 5s yield $1,800 each. a. Determine the most profitable number of each model of minicomputer to produce during the coming month. b. What would the profit be if they decided to make at least 15 of each model? 5-3 Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $1.50, and each pound of grain costs $0.70. A pound of the dog food must contain at least 9 units of Vitamin 1, 10 units of Vitamin 2, and 6 units of Vitamin 3. A pound of beef contains 6 units of Vitamin 1, 12 units of Vitamin 2, and 4 units of Vitamin 3. A pound of grain contains 12 units of Vitamin 1, 9 units of Vitamin 2, and 10 units of Vitamin 3. Formulate this as a LP problem to minimize the cost of the dog food. a. How many pounds of beef and grain should be included in each pound of dog food? b. What is the cost and vitamin content of the final product? c. What would the distribution of beef and grain be if the cost of beef increased to $3.50 per pound? 5-4 The Feed ‘N Ship Ranch fattens cattle for local farmers and ships them to meat markets in Kansas City and Omaha. The owners of the ranch seek to determine the amounts of cattle feed to buy so that minimum nutritional standards are satisfied, and at the same time total feed Page 1 of 6 20180605174858study_pool costs are minimized. The feed mix can be made up of the three grains that contain the following ingredients per pound of feed: INGREDIENT A B C D STOCK X 3 2 1 6 FEED (OZ) STOCK Y 2 3 0 8 STOCK Z 4 1 2 4 The cost per pound of stocks X, Y, and Z are $3.50, $4.75, and $4.50, respectively. The minimum requirement per cow per month is 4 pounds of ingredient A, 5 pounds of ingredient B, 1 pound of ingredient C, and 8 pounds of ingredient D. The ranch faces one additional restriction: it can only obtain 500 pounds of stock Z per month from the feed supplier regardless of its need. Because there are usually 100 cows at the Feed ‘N Ship Ranch at any given time, this means that no more than 5 pounds of stock Z can be counted on for use in the feed of each cow per month. a. Formulate this as an LP problem. b. Solve using LP software. 6-1 The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report for duty at 3am, 7am, 11am, 3pm, 7pm, or 11pm, and each works an 8-hour shift. The following table shows the minimum number of workers needed during the six periods into which the day is divided. Chang’s scheduling problem is to determine how many waiters and busboys should report for work at the start of each time period to minimize the total staff required for one day’s operation. (Hint: Let Xi equal the number of waiters and busboys beginning work in time period i, where i = 1, 2, 3, 4, 5, 6.) If Chang’s were able to reduce the number of required waiters and busboys by 1 during some period, during which period should they make the reduction? Period 1 2 3 4 5 6 Time 3am-7am 7am-11am 11am-3pm 3pm-7pm 7pm-11pm 11pm-3am Number of Waiters and Busboys Required 3 12 17 11 12 4 6-2 The Arden County, Maryland, superintendent of education is responsible for assigning students to the three high schools in his county. He recognizes the need to bus a certain number of students, for several sectors of the county are beyond walking distance to a school. The superintendent partitions the county into five geographic sectors as he attempts to establish a plan that will minimize the total number of student miles traveled by bus. He also recognizes that if a student happens to live in a certain sector and is assigned to the high school in that Page 2 of 6 20180605174858study_pool sector, there is no need to bus that student because he or she can walk to school. The three schools are located in sectors B, C, and E. The following table reflects the number of high school-age students living in each sector and the busing distance in miles from each sector to each school: Page 3 of 6 20180605174858study_pool Sector A B C D E School in Sector B 5 0 4 7 12 Distance to School School in School in Sector C Sector E 9 6 5 12 0 7 3 5 8 0 Number of Students 700 500 100 800 400 2,500 Each high school has a capacity of 900 students, but must have at least 700. Set up the objective function and constraints of this problem using LP so that the total number of student miles traveled by bus is minimized. Then solve the problem. 6-3 Coast-to-Coast Airlines is investigating the possibility of reducing the cost of fuel purchases by taking advantage of lower fuel costs in certain cities. Since fuel purchases represent a substantial portion of operating expenses for an airline, it is important that these costs be carefully monitored. However, fuel adds weight to an airplane, and consequently, excess fuel raises the cost of getting from one city to another. In evaluating one particular flight rotation, a plane begins in Atlanta, flies from Atlanta to Los Angeles, from Los Angeles to Houston, from Houston to New Orleans, and from New Orleans to Atlanta. When the plane arrives in Atlanta, the flight rotation is said to have been completed, and then it starts again. Thus, the fuel on board when the flight arrived in Atlanta must be taken into consideration when the flight begins. Leg Atlanta-Los Angeles Los Angeles-Houston Houston-New Orleans New Orleans-Atlanta Minimum Fuel Required (1,000 gal.) 25 16 8 12 Maximum Fuel Allowed (1,000 gal.) 36 23 18 20 Regular Fuel Consumption (1,000 gal.) 13 7 3 5 Fuel Price per Gallon 4.25 4.55 4.15 4.22 Along each leg of this route, there is a minimum and a maximum amount of fuel that may be carried. This and additional information is provided in the table above. The regular fuel consumption is based on the plane carrying the minimum amount of fuel. If more than this is carried, the amount of fuel consumed is higher. Specifically, for each 1,000 gallons of fuel above the minimum, 6% (or 60 gallons per 1,000 gallons of extra fuel) is lost due to excess fuel consumption. For example, if 26,000 gallons of fuel were on board when the plane takes off from Atlanta, the fuel consumed on this route would be 13 + 0.06 = 13.06 thousand gallons. If 27 thousand gallons were on board, the fuel consumed would be increased by another 0.06 thousand, for a total of 13.12 thousand gallons. Page 4 of 6 20180605174858study_pool Formulate this as a LP problem to minimize the cost. a. How many gallons should be purchased in each city? b. What is the total cost of his? 6-4 An air conditioning manufacturer produces room air conditioners at plants in Houston, Phoenix, and Memphis. These are sent to regional distributors in Dallas, Atlanta, and Denver. The shipping costs vary, and the company would like to find the least-cost way to meet the demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per month, Atlanta needs 700, and Denver needs 250. Houston has 850 air conditioners available each month, Phoenix has 800, and Memphis has 350. The shipping cost per unit from Houston to Dallas is $8, to Atlanta is $12, and to Denver is $13. The cost per unit from Phoenix to Dallas is $10, to Atlanta is $14, and to Denver is $8. The cost per unit from Memphis to Dallas is $11, to Atlanta is $8, and to Denver is $14. How many units should be shipped from each plant to each regional distribution center? What is the total cost for this? 6-5 Finnish Furniture manufactures tables in facilities located in three cities--Reno, Denver, and Pittsburgh. The tables are then shipped to three retail stores located in Phoenix, Cleveland, and Chicago. Management wishes to develop a distribution schedule that will meet the demands at the lowest possible cost. The shipping cost per unit from each of the sources to each of the destinations is shown in the following table: From Reno Denver Pittsburgh Phoenix 10 12 18 To Cleveland 16 14 8 Chicago 19 13 12 The available supplies are 130 units from Reno, 200 from Denver, and 160 from Pittsburgh. Phoenix has a demand of 140 units, Cleveland has a demand of 160 units, and Chicago has a demand of 200 units. a. How many units should be shipped from each manufacturing facility to each of the retail stores if cost is to be minimized? b. What is the total cost? 6-6 The Gleaming Company has just developed a new dishwashing liquid and is preparing for a national television promotional campaign. The firm has decided to schedule a series of 1-minute commercials during the peak homemaker audience viewing hours of 1pm to 5pm. To reach the widest possible audience, Gleaming wants to schedule one commercial on each of four networks and to have one commercial appear during each of the four 1-hour time blocks. The exposure ratings for each hour, which represent the number of viewers per $1,000 spent, are presented in the following table. Which network should be scheduled each hour to provide the maximum audience exposure? Page 5 of 6 20180605174858study_pool Viewing Hours 1-2pm 2-3pm 3-4pm 4-5pm A 21.2 18.9 19.2 11.5 B 18.1 15.5 18.5 19.4 Network C 11.3 17.1 10.9 16.8 Page 6 of 6 20180605174858study_pool Independent 9.5 12.3 9.7 12.8
Purchase answer to see full attachment