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 Xray 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 Xray goal is
unmet Goal 3 Cost of $8 for each hour by which laboratory goal
is unmet Formulate a goal programming model to minimize the daily
cost incurred due to failing to meet the hospital’s goals. Group B 15 Consider a maximization problem with the optimal
tableau in Table 73. The optimal solution to this LP is z =
10,x3 = 3, x4 = 5, x1 = x2 = 0. Determine the secondbest
bfs to this LP. (Hint: Show that the secondbest solution
must be a bfs that is one pivot away from the optimal
solution.) 16 A camper is considering taking two types of items on
a camping trip. Item 1 weighs a1 lb, and item 2 weighs (12
lb. Each type 1 item earns the camper a beneﬁt of c, units,
and each type 2 item earns the camper c2 units. The knapsack
can hold items weighing at most b lb. a Assuming that the camper can carry a fractional
number of items along on the trip, formulate an LP to
maximize beneﬁt. I) Show that if Q 2 g
02 a1
then the camper can maximize beneﬁt by ﬁlling a knap sack with 5—; type 2 items. 0 Which of the linear programming assumptions are
violated by this formulation of the camper’s problem? 17 You are given the tableau shown in Table 74 for a
maximization problem. Give conditions on the unknowns
a1, a2, a3, b, and c that make the following statements true: a The current solution is optimal. II The current solution is optimal, and there are alter
native optimal solutions. I: The LP is unbounded (in this part, assume that b 2 0). 18 Suppose we have obtained the tableau in Table 75 for
a maximization problem. State conditions on a1, a2, a3, b,
C], and 02 that are required to make the following statements
true:
a The current solution is optimal, and there are alter
native optimal solutions. b The current basic solution is not a basic feasible
solution. TABLE 73 z x‘ 12 . la 14 . ms
1 2 l 0 0 10
0 3 2 1 0 3
0 4 3 0 1 5 TA B L E 71
Factor Level ‘
Jul! 1 '2 3 ﬂeshed Score
1 4 4 4 105
2 3 3 2 93
3 2 2 2 75
4 1 1 2 68
TA B L E 72
Type at ﬂporatlun 1 2 3 4
Proﬁt ($) 200 150 100 80
Xray 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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 Spring '09
 VLADIMIRLBOGINSKI

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