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?…show more content… 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 | 4002,500 |
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.
Total student miles traveled by bus = 6200
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.