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[Question 9 Sensitivity Analysis ] Fran Farnsworth travels throughout the Maritime selling a line of three health care Products. Each box of Flex costs \$3 from her supplier, takes Fran 10 minutes (on average) to sell, costs \$0.50 to deliver, and contributes \$2 to Fran’s commission. Each box of Feelgood costs \$5, takes 20 minutes to sell, is left With the customer at the time of sale, and contributes \$4 to Fran’s commission.

Each box of Firmup Costs \$4, takes 12 minutes to sell, costs \$1 to deliver, and contributes \$3 to Fran’s commission. Every week Fran is allowed to draw up to \$1500 worth of Products, and has a weekly delivery expense allowance of \$85. Due to travel Time and other commitments, Fran is limited to 40 hours, or 2400 minutes, per Week for sales.

Suppose x, y, & z represent the number of boxes To sell per week of Flex, Feelgood and Firmup respectively; and C is Fran’s Weekly commissions. Then the LP model is:

 Max C =2x +4 y +3z s.t .3x + 5 y + 4z £ 1500 (\$ - Worth) 10x + 20 y +12z £ 2400 (m - sell time) 0.5x + z £85 (\$ - Delivery) x ³0, y ³0, z ³0

a)How many boxes of each health care product should Fran Sell per week, and what is her maximum weekly commission?

Optimal selling Strategy:

x = 0sell no Flex;

y = 69sell 69 boxes of Feelgood; And

z = 85sell 85 Boxes of Firmup.

max P = 531This will yield the max Commission earning of \$531 per week.

b)Given the optimal solution, how much time does Fran Spend each week to sell the health care products? How much does she spend in Her delivery allowance?

She will spend 40 Hours selling the products, and \$85 of delivery allowance.

c)Should Fran ask her boss for more than \$1500 worth of Products she is allowed to sell per week, assuming no other conditions change? Explain.

NO, because she Only sells \$685 worth products, and still has \$815 unspent.

d)If Fran can hire a baby sitter for Friday afternoon at \$8 per hour, this would allow her to increase her weekly selling time to 45 Hours instead of 40 hours. Should she do it? If so, how specifically would her Selling decisions and commission change?

Ø For Selling Time: shadow Price = \$0.2/min = \$12/hour

allowable Increase = 3260 min = 54.33 hours

ØExtra Selling time? Yes.

·Cost of babysitter = \$8 per hour;

·Benefit of extra selling time = \$12 per hour; and

·Five hours of extra selling is within the limit,

YES, Fran should hire the Babysitter and use the five extra hours for her sales.

e)How would Fran’s selling decision and commission change If her delivery expense allowance were reduced to \$50?

ØDelivery Expense is an active constraint: shadow price = \$0.60 & allowable decrease = \$85

ØBecause Fran’s delivery allowance was reduced by \$35, Her commission earning would drop by \$21 per week (\$0.6×35).

ØShe would have To reduce her sales on Firmup’s. It would affect less on the sales of

Feelgood, Which does not require delivery.

f)A recent sales promotion by her supplier has pressed Fran to sell at least 10 boxes of Flex per week. If she does that, how much Will her weekly commission decrease?

Her weekly Commission would be reduced by \$3 (because the reduced cost is \$0.3 per box for Flex, and \$0.3 × 10 = \$3).

g)For each product, if their profits (commission contribution) were Higher, Fran might wish to sell more of them. How high would the unit profits Have to be to make Fran change her decision to sell? Lower profits would make Products less attractive. How low would they have to be before Fran would Decrease the number of boxes to sell? Fill in the table with your answer.

Profit Range (per box)

 Lower Limit Current Value Upper Limit Flex 0 \$2 \$2.3 Feelgood \$2.50 \$4 \$5.00 Firmup \$2.40 \$3 ∞