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(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: 

Please note the answer must be in Excel format.

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

 


 

 


Solution details:

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