An appliance dealer must decide how many (if any) new microwave ovens to order f

Question

An appliance dealer must decide how many (if any) new microwave ovens to order for next month. The ovens cost $220 and sell for $300. Because the oven company is coming out with a new product line in two months, any ovens not sold next month will have to be sold at the dealer's half price clearance sale. Additionally, the appliance dealer feels he suffers a loss of $25 for every oven demanded when he is out of stock. On the basis of past months' sales data, the dealer estimates the probabilities of monthly demand (D) for 0, 1, 2, or 3 ovens to be .3, .4, .2, and .1, respectively.
The dealer is considering conducting a telephone survey on the customers' attitudes towards microwave ovens. The results of the survey will either be favorable (F), unfavorable (U) or no opinion (N). The dealer's probability estimates for the survey results based on the number of units demanded are:
P(F | D = 0) = .1
P(F | D = 2) .3
P(U | D = 0) = .8
P(U | D = 2) = .1
P(F | D = 1) = .2
P(F | D = 3) .9
P(U | D = 1) = .3
P(U | D = 3) = .1
a.
What is the dealer's optimal decision without conducting the survey?
b.
What is the EVPI?
c.
Based on the survey results what is the optimal decision strategy for the dealer?
d.
What is the maximum amount he should pay for this survey?

Explanation / Answer

Order 0: we have unsold items for which the return is -25

return is -25*(.4*1+.2*2+.1*3) = -25*1.1  = $-27.50

Order 1: we have to sell at a discount if no orders, otherwise sell 1, and unsold items if demand 2 or 3

return is .3*(1/2*300-220) + (1-.3)*(300-220) + -.25*(.2*1+.1*2) = .3*-70+.7*80+-25*(.4) =

-21 + 56 - 10 = $25

Order 2: we have to sell at a discount if 0 or 1 orders, sell 1 or 2, and unsold items if demand 3

return is (.3*2+.4*1)*(1/2*300-220)+(.4*1+(.2+.1)*2)*(300-220)+-25*.1 =1*-70+1*80-25*.1 =

-70 + 80 - 2.5 = $7.50

Order 3:

return is (.3*3+.4*2+.2*1)*(1/2*300-220)+(.4*1+.2*2+.1*3)*(300-220) = 1.9*-70 + 1.1*80 =

-133 + 88 = -$45

Order 1, with a return of $25, as this is the highest return.

b) If we had a perfect information, we would never pay a penalty for underordering or suffer a discounted return from over-ordering

(.4*1+.2*2+.1*3)*(300-220) = 1.1*80 = $88

Then, the value of perfect information is $88 - $25 = $63

c) P(D=0|F) = P(F|D=0)*P(D=0)/(P(F|D=0)*P(D=0)+P(F|D=1)*P(D=1)+P(F|D=2)*P(D=2)+P(F|D=3)*P(D=3))=

.1*.3/(.1*.3+.2*.4+.3*.2+.9*.1)=.03/.26 = 3/26

P(D=1|F) = P(F|D=1)*P(D=1)/(P(F|D=0)*P(D=0)+P(F|D=1)*P(D=1)+P(F|D=2)*P(D=2)+P(F|D=3)*P(D=3))=

.2.4/(.1*.3+.2*.4+.3*.2+.9*.1)=.08/.26 = 4/13

P(D=2|F) = P(F|D=2)*P(D=2)/(P(F|D=0)*P(D=0)+P(F|D=1)*P(D=1)+P(F|D=2)*P(D=2)+P(F|D=3)*P(D=3))=

.3*.2/(.1*.3+.2*.4+.3*.2+.9*.1)=.06/.26 = 3/13

P(D=3|F) = P(F|D=3)*P(D=3)/(P(F|D=0)*P(D=0)+P(F|D=1)*P(D=1)+P(F|D=2)*P(D=2)+P(F|D=3)*P(D=3))=

.9*.1/(.1*.3+.2*.4+.3*.2+.9*.1)=.09/.26 = 9/26

P(D=0|U) = P(U|D=0)*P(0)/(P(U|D=0)*P(D=0)+P(U|D=1)*P(D=1)+P(U|D=2)*P(D=2)+P(U|D=3)*P(D=3))=

.8*.3/(.8*.3+.3*.4+.1*.2+.1*.1)=.24/.39 = 8/13

P(D=1|U) = P(U|D=1)*P(1)/(P(U|D=0)*P(D=0)+P(U|D=1)*P(D=1)+P(U|D=2)*P(D=2)+P(U|D=3)*P(D=3))=

.3*.4/(.8*.3+.3*.4+.1*.2+.1*.1)=.12/.39 = 4/13

P(D=2|U) = P(U|D=`)*P(`)/(P(U|D=0)*P(D=0)+P(U|D=1)*P(D=1)+P(U|D=2)*P(D=2)+P(U|D=3)*P(D=3))=

.1*.2/(.8*.3+.3*.4+.1*.2+.1*.1)=.02/.39 = 2/39

P(D=3|U) = P(U|D=3)*P(3)/(P(U|D=0)*P(D=0)+P(U|D=1)*P(D=1)+P(U|D=2)*P(D=2)+P(U|D=3)*P(D=3))=

.1*.1/(.8*.3+.3*.4+.1*.2+.1*.1)=.01/.39 = 1/39

P(N|D=0 = 1-.1-.8 = .1

P(N|D=1) = 1 - .2 - .3 = .5

P(N|D=2) = 1 - .3 - .1 = .6

P(N|D=3) = 1 - .9 - .1 = 0

P(D=0|N) = P(N|D=0)*P(D=0)/(P(N|D=0)*P(D=0)+P(N|D=1)*P(D=1)+P(N|D=2)*P(D=2)+P(N|D=3)*P(D=3))=.1*.3/(.1*.3+.5*.4+.6*.2+.0*.1)= .03/.35 = 3/35

P(D=1|N) = P(N|D=1)*P(D=0)/(P(N|D=0)*P(D=0)+P(N|D=1)*P(D=1)+P(N|D=2)*P(D=2)+P(N|D=3)*P(D=3))= .5*.4/(.1*.3+.5*.4+.6*.2+.0*.1)= .20/.35 = 4/7

P(D=2|N) = P(N|D=2)*P(D=2)/(P(N|D=0)*P(D=0)+P(N|D=1)*P(D=1)+P(N|D=2)*P(D=2)+P(N|D=3)*P(D=3))= .6*.2/(.1*.3+.5*.4+.6*.2+.0*.1)= .12/.35 = 12/35

P(D=3|N) = 0

If the result of the survey is an F, we have

P(D=0|F) = 3/26

P(D=1|F) = 4/13

P(D=2|F) = 3/13

P(D=3|F) = 9/26

If the order is 0, the return is -25*(1*4/13+2*3/13+3*9/26) = -25*47/26 = -1175/26 = -$45.19

If the order is 1, the return is 3/26*-70+(1-3/26)*80+-25*(1*3/13+2*9/26) = 515/13 = $39.62

If the order is 2, the return is (3/26*2+4/13)*-70+(1*4/13+2*(3/13+9/26))*80 + -25*9/26 =

1835/26 = $70.58

If the order is 3, the return is (3/26*3+4/13*2+3/13)*-70+(1*4/13+2*3/13+3*9/26)*80 =

795/13 = $61.15

We should order 2.

P(D=0|U) = 8/13

P(D=1|U) = 4/13

P(D=2|U) = 2/39

P(D=3|U) = 1/39

If we order 0, the return is (4/13*1+2/39*2+1/39*3)*-25 = -475/39 = -$12.18

If the order is 1, the return is 8/13*-70+(1-8/13)*80+-25*(1*2/39+2*1/39) =-580/39= -14.87

If the order is 2, the return is (8/13*2+4/13)*-70+(1*4/13+2*(2/39+1/39))*80 + -25*1/39 =

-2785/39= -$71.41

If the order is 3, the return is (8/13*3+4/13*2+2/39*1)*-70+(1*4/13+2*2/39+3*1/39)*80 =

-1780/13 = -$136.92

Order 0

P(D=0|N) = 3/35

P(D=1|N) = 4/7

P(D=2|N) = 12/35

P(D=3|N) = 0

If we order 0, the return is (4/7*1+12/35*2)*-25 = -220/7 = -$31.43

If the order is 1, the return is 3/35*-70+(1-3/35)*80+-25*(1*12/35) = 410/7 = $58.57

If the order is 2, the return is (3/35*2+4/7)*-70+(1*4/7+2*12/35)*80 = 340/7 = $48.57

We don't order 3, as the probability of 3 is 0

we order 1

We order 2 if there is an F, 0 if there is an N, and 1 if there is a U.

d) P(F) = .26

P(N) = .39

P(U) = .35

Then, the expected return is .26*1835/26 +-475/39*.39 + 410/7*.35 = $34.10

Since we make $25 if we just take 1, we should pay up to $34.10-$25 = $9.10 for the survey.

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