(4.) 10 Points – As the head of the training department in your company, you wan
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Question
(4.) 10 Points – As the head of the training department in your company, you want to estimate the “optimum” amount of training to give your employees. Your dilemma: If you don’t train them enough, the profits they produce for your company will probably be small. However, if you train them too much, the increase in profits due to the additional training may be smaller than the cost to train the employees, i.e. a diminishing returns situation. The gross profitability of an employee is the profit generated by the employee before taking into consideration the cost of training the employee. The average annual gross profitability per employee for different amounts of training has been determined on the basis of the following sample information:
Gross
Profit / Year
($K)
Training Received (weeks)
59.2
2
66.0
4
68.0
7
70.0
9
71.2
11
71.4
13
72.0
15
70.0
17
67.0
20
(a) Develop a quadratic model for estimating the net annual profit (i.e., gross profit - cost of training) generated by an employee as a function of the number of weeks of training that the employee receives. Assume that it costs $1,144 to provide an employee with one week of training. – (7 Points)
(b) Calculate the maximum number of weeks of training that is economically justified to give each employee. – (3 Points)
Gross
Profit / Year
($K)
Training Received (weeks)
59.2
2
66.0
4
68.0
7
70.0
9
71.2
11
71.4
13
72.0
15
70.0
17
67.0
20
Explanation / Answer
it costs $1,144=$1.444K to provide an employee with one week of training.
hence total cost for training=$(1.444*number of weeks of training)K
now a table is made to calculate the net profit
gross profit($K) training received(weeks) cost training($K)=weeks*1.444 net profit=gross-cost
59.2 2 2.888 56.312
66.0 4 5.776 60.224
68.0 7 10.108 57.892
70.0 9 12.996 57.004
71.2 11 15.884 55.316
71.4 13 18.772 52.628
72.0 15 21.660 50.340
70.0 17 24.548 45.452
67.0 20 28.880 38.120
a) now a quadratic regression of netprofit(y) on $K on number of weeks of training received(x)
the quadratic equation is as obtained from MINITAB is
Y= 55.73 + 1.038 x - 0.09601 x**2 where y is measured in $K [answer]
b) the maximum number of weeks of training that is economically justified to give each employee is that value of x for which Y is maximum.
now Y=55.73 + 1.038 x - 0.09601 x**2
differentiating Y with respect to x and equating it to 0 we get
dY/dx=0
or, d[55.73 + 1.038 x - 0.09601 x**2]/dx=0
or, 1.038-2*0.09601x=0
or, x=1.038/0.19202=5.40568=5 weeks [aprrox] [answer]
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