Answer:
Explanation:
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.
[ where a is a constant]