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max z = 3x + 2x2 12 s1 82 rhs 2x1 + 5x2 < 8 3x1 + 7x2 < 10 *1, 12 2 1 0 0 0 1/3 1 -2/3 4/3 0 1 7/3 0 s.t. 1 10 1/3 10/3 (a) The range of values of the right hand side bị of the first constraint for which the current basis remains optimal is from to (b) The range of values of the right hand side b2 of the second constraint for which the current basis remains optimal is from to (c) If the right hand side of the first constraint is changed to bị = 7, the new optimal solution will be #1= 22=O (d) If the right hand side of the second constraint is changed to b=5, the new optimal solution will be x1=, r2:

Practical Management Science

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter7: Nonlinear Optimization Models

Section: Chapter Questions

Problem 50P

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2

rhs
max z = 3x1 + 2x2
2x1 + 5x2 <
3x1 + 7x2 < 10
S1
82
s.t.
8
1
1
10
0 1/3 1 -2/3 4/3
7/3 0
1/3
X1, X2 > 0
0 1
10/3
(a) The range of values of the right hand side b1 of the first constraint for which
the current basis remains optimal is from to
(b) The range of values of the right hand side b2 of the second constraint for which
the current basis remains optimal is from
to
(c) If the right hand side of the first constraint is changed to bi
optimal solution will be r1=O, x2
7, the new
(d) If the right hand side of the second constraint is changed to b2=5, the new
optimal solution will be x1:

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Based on Grossman and Hart (1983). A salesperson for Fuller Brush has three options: (1) quit, (2) put forth a low level of effort, or (3) put forth a high level of effort. Suppose for simplicity that each salesperson will sell 0, 5000, or 50,000 worth of brushes. The probability of each sales amount depends on the effort level as described in the file P07_71.xlsx. If a salesperson is paid w dollars, he or she regards this as a benefit of w1/2 units. In addition, low effort costs the salesperson 0 benefit units, whereas high effort costs 50 benefit units. If a salesperson were to quit Fuller and work elsewhere, he or she could earn a benefit of 20 units. Fuller wants all salespeople to put forth a high level of effort. The question is how to minimize the cost of encouraging them to do so. The company cannot observe the level of effort put forth by a salesperson, but it can observe the size of his or her sales. Thus, the wage paid to the salesperson is completely determined by the size of the sale. This means that Fuller must determine w0, the wage paid for sales of 0; w5000, the wage paid for sales of 5000; and w50,000, the wage paid for sales of 50,000. These wages must be set so that the salespeople value the expected benefit from high effort more than quitting and more than low effort. Determine how to minimize the expected cost of ensuring that all salespeople put forth high effort. (This problem is an example of agency theory.)

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