## (Solved) Problem 3 (5 marks) A salesperson for Fuller Brush has three

Please note, initial price is very negotiable .

I would like to put the attached non-linear programming problem into an Excel solver format. The objective is to minimize cost, but also find out what wage is paid at each level of sales:

Find wages @ the following sales figures: \$0, \$5000, \$50,000

Objective is to minimized cost.

Problem 3 (5 marks)

A salesperson for Fuller Brush has three options: quit, put forth a low-e?ort level, or put forth a

high-e?ort level. Suppose for simplicity that each salesperson will either sell \$0, \$5000, or \$50000

worth of brushes. The probability of each sales amount depends on the e?ort level in the manner

described in the following table.

E?ort

Low

0.6

0.3

0.1

Size of Sale

\$0

\$5000

\$50000

Level

High

0.3

0.2

0.5

If the salesperson is paid \$w, he or she earns a bene?t w1/2 . Low e?ort costs the salesperson 0

bene?t units while high e?ort costs 50 bene?t units. If the salesperson were to quit Fuller and work

elsewhere he or she could earn a bene?t of 20. Fuller wants all salespeople to put forth a high-e?ort

level. The question is how to minimize the cost of doing it. The company cannot observe the level

of e?ort put forth by a salesperson, but they can observe the size of his or her sale. Thus, the wage

is completely determined by the size of the sale. Fuller must then determine w0 = wage paid for \$0

in sale, w5 = wage paid for \$5000 in sales, and w50 = wage paid for \$50000 in sales. These wages

must be set so that the salespeople value the expected bene?t from high e?ort more than quitting

and more than low e?ort. Formulate an NLP that can be used to ensure that all salespeople put

forth high e?ort. (This problem is an example of agency theory.)

Solution

Decision variables. Let w0 , w5 and w50 represent the wage paid by Fuller Brush to a salesperson

if the size of sales is \$0, \$5000 and \$50000, respectively.

Objective function. The expected cost associated with a salesperson who puts forth a high-e?ort

level is

0.3w0 + 0.2w5 + 0.5w50

Fuller Brush wants to minimize this cost; therefore, the objective function is

min z = 0.3w0 + 0.2w5 + 0.5w50

Constraints. The expected bene?ts for a salesperson who puts forth a high-e?ort level is

1/2

0.3w0

1/2

+ 0.2w5

1/2

+ 0.5w50 ? 50

while low-e?ort level earns expected bene?ts of

1/2

0.6w0

1/2

+ 0.3w5

1/2

+ 0.1w50 ? 0

To ensure that salespeople value the expected bene?t from high e?ort more than quitting and more

than low e?ort, we need, respectively,

1/2

0.3w0

1/2

0.3w0

1/2

+ 0.2w5

1/2

+ 0.2w5

1/2

1/2

+ 0.5w50 ? 50 ? 20

1/2

+ 0.5w50 ? 50 ? 0.6w0

1/2

+ 0.3w5

1/2

+ 0.1w50

To summarize, the NLP is:

min z = 0.3w0 + 0.2w5 + 0.5w50

1/2

1/2

1/2

s.t.

0.3w0 + 0.2w5 + 0.5w50 ? 70

1/2

1/2

1/2

?0.3w0 ? 0.1w5 + 0.4w50 ? 50

w0 , w5 , w50 ? 0

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This question was answered on: Oct 15, 2019

Solution~000.zip (25.37 KB)

STATUS

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Approved

Oct 15, 2019

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