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pg+214 - IU wants to determine a formula for grading each...

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Unformatted text preview: IU wants to determine a formula for grading each job. To do this, it will assign a point value to the score for each factor that a job requires. For example, suppose level 2 of factor 1 yields a point total of 10, level 3 of factor 2 yields a point total of 20, and level 3 of factor 3 yields a point value of 30. Then a job with these requirements would have a point total of 10 + 20 + 30. A job’s hourly salary equals half its point total. IU has two goals (listed in order of priority) in setting up the points given to each level of each job factor. Goal 1 When increasing the level of a factor by one, the points should increase by at least 10. For example, level 2 of factor 1 should earn at least 10 more points than level 1 of factor 1. Goal 1 is to minimize the sum of deviations from this requirement. Goal 2 For the benchmark jobs in Table 71, the actual point total for each job should come as close as possible to the point total listed in the table. Goal 2 is to minimize the sum of the absolute deviations of the point totals from the desired scores. Use preemptive goal programming to come up with appro- priate point totals. What salary should a job with skill lev- els of 3 for each factor be paid? 14 A hospital outpatient clinic performs four types of operations. The profit per operation, as well as the minutes of X—ray time and laboratory time used are given in Table 72. The clinic has 500 private rooms and 500 intensive care rooms. Type 1 and Type 2 operations require a patient to stay in an intensive care room for one day while Type 3 and Type 4 operations require a patient to stay in a private room for one day. Each day the hospital is required to perform at least 100 operations of each type. The hospital has set the following goals: Goal 1 Earn a daily profit of at least $100,000. Goal 2 Use at most 50 hours daily of X-ray time. Goal 3 Use at most 40 hours daily of laboratory time. The cost per unit deviation from each goal is as follows: Goal 1 Cost of $1 for each dollar by which profit goal is unmet Goal 2 Cost of $10 for each hour by which X-ray goal is unmet Goal 3 Cost of $8 for each hour by which laboratory goal is unmet Formulate a goal programming model to minimize the daily cost incurred due to failing to meet the hospital’s goals. Group B 15 Consider a maximization problem with the optimal tableau in Table 73. The optimal solution to this LP is z = 10,x3 = 3, x4 = 5, x1 = x2 = 0. Determine the second-best bfs to this LP. (Hint: Show that the second-best solution must be a bfs that is one pivot away from the optimal solution.) 16 A camper is considering taking two types of items on a camping trip. Item 1 weighs a1 lb, and item 2 weighs (12 lb. Each type 1 item earns the camper a benefit of c, units, and each type 2 item earns the camper c2 units. The knapsack can hold items weighing at most b lb. a Assuming that the camper can carry a fractional number of items along on the trip, formulate an LP to maximize benefit. I) Show that if Q 2 g 02 a1 then the camper can maximize benefit by filling a knap- sack with 5—; type 2 items. 0 Which of the linear programming assumptions are violated by this formulation of the camper’s problem? 17 You are given the tableau shown in Table 74 for a maximization problem. Give conditions on the unknowns a1, a2, a3, b, and c that make the following statements true: a The current solution is optimal. II The current solution is optimal, and there are alter- native optimal solutions. I: The LP is unbounded (in this part, assume that b 2 0). 18 Suppose we have obtained the tableau in Table 75 for a maximization problem. State conditions on a1, a2, a3, b, C], and 02 that are required to make the following statements true: a The current solution is optimal, and there are alter- native optimal solutions. b The current basic solution is not a basic feasible solution. TABLE 73 z x‘ 12 . la 14 . ms 1 2 l 0 0 10 0 3 2 1 0 3 0 4 3 0 1 5 TA B L E 71 Factor Level ‘ Jul! 1 '2 3 fleshed Score 1 4 4 4 105 2 3 3 2 93 3 2 2 2 75 4 1 1 2 68 TA B L E 72 Type at flporatlun 1 2 3 4 Profit ($) 200 150 100 80 X-ray time (minutes) 6 5 4 3 Laboratory time (minutes) 5 4 3 2 21 4 mmn 4 The Simplex Algorithm and Goal Pmuramminu ...
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