6.1The 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 |
Time | Number of Waiters and Busboys Required |

1 | 3am-7am | 3 |

2 | 7am-11am | 12 |

3 | 11am-3pm | 17 |

4 | 3pm-7pm | 11 |

5 | 7pm-11pm | 12 |

6 | 11pm-3am | 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 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:

| Distance to School | |||

Sector | School in Sector B | School in Sector C | School in Sector E | Number of Students |

A | 5 | 9 | 6 | 700 |

B | 0 | 5 | 12 | 500 |

C | 4 | 0 | 7 | 100 |

D | 7 | 3 | 5 | 800 |

E | 12 | 8 | 0 |
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 | Minimum Fuel Required (1,000 gal.) | Maximum Fuel Allowed (1,000 gal.) | Regular Fuel Consumption (1,000 gal.) |
Fuel Price per Gallon |

Atlanta-Los Angeles | 25 | 36 | 13 | 4.25 |

Los Angeles-Houston | 16 | 23 | 7 | 4.55 |

Houston-New Orleans | 8 | 18 | 3 | 4.15 |

New Orleans-Atlanta | 12 | 20 | 5 | 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.

Formulate this as a LP problem to minimize the cost.

- How many gallons should be purchased in each city?
- 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:

| To | ||

From | Phoenix | Cleveland | Chicago |

Reno | 10 | 16 | 19 |

Denver | 12 | 14 | 13 |

Pittsburgh | 18 | 8 | 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.

- How many units should be shipped from each manufacturing facility to each of the retail stores if cost is to be minimized?
- 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?

| Network | |||

Viewing Hours | A | B | C | Independent |

1-2pm | 21.2 | 18.1 | 11.3 | 9.5 |

2-3pm | 18.9 | 15.5 | 17.1 | 12.3 |

3-4pm | 19.2 | 18.5 | 10.9 | 9.7 |

4-5pm | 11.5 | 19.4 | 16.8 | 12.8 |

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