IE 417
Farmer Jones
Page 772 Problem 3
Presented by:
Geoffrey Cheung
Chris Mui
Tamara Vail
Presented to:
Dr. Sima Parisay
Tuesday, January 18, 2011
Industrial and Manufacturing Engineering Department
College of Engineering
California State Polytechnic University, Pomona
Table of Contents
Problem Statement .......................................................................................................................... 2
Solving the Problem ........................................................................................................................ 3
Decision Tree .............................................................................................................................. 3
Given Information ....................................................................................................................... 4
Probabilities ............................................................................................................................. 4
Costs and Profits ...................................................................................................................... 4
Calculated Information (using Bayes)......................................................................................... 5
WinQSB Input ............................................................................................................................. 6
WinQSB Output .......................................................................................................................... 7
WinQSB Decision Tree ............................................................................................................... 8
Best Decision............................................................................................................................... 9
Sensitivity Analysis ...................................................................................................................... 10
Summary Table ......................................................................................................................... 10
Sensitivity Analysis Graph ........................................................................................................ 11
Turning Point Calculation ......................................................................................................... 11
Best Decision............................................................................................................................. 12
Utility Function ............................................................................................................................. 13
Utility Function Examples ........................................................................................................ 14
Utility WinQSB Input ............................................................................................................... 15
Utility WinQSB Output ............................................................................................................. 16
Utility WinQSB Decision Tree ................................................................................................. 17
Report to Manager ........................................................................................................................ 18
Appendix ....................................................................................................................................... 19
A. Sensitivity Analysis – WinQSB Input/Output...................................................................... 19
1
Problem Statement
Farmer Jones must determine whether to plant corn or wheat. If he plants corn and the weather is
warm, he earns $8,000; if he plants corn and the weather is cold, he earns $5,000. If he plants
wheat and the weather is warm, he earns $7,000; if he plants wheat and the weather is cold, he
earns $6,500. In the past, 40% of all years have been cold and 60% have been warm. Before
planting, Jones can pay $600 for an expert weather forecast. If the year is actually
cold, there is a 90% chance that the forecaster will predict a cold year. If the year is actually
warm, there is an 80% chance that the forecaster will predict a warm year. How can Jones
maximize his expected profits? Also find EVSI and EVPI.
2
Solving the Problem
Decision Tree
3
Given Information
Probabilities

P(Warm)=0.6
In the past 60% of all years have been warm.

P(Cold)=0.4
In the past 40% of all years have been warm.

P(PredictCold|Cold)=0.9
There is a 90% chance that the forecaster will predict it to be cold given the year was
actually cold.

P(PredictWarm|Warm)=0.8
There is an 80% chance that the forecaster will predict it to be warm given the year was
actually warm.
Costs and Profits



Cost of Forecaster: $600
Plants Corn (profit)
o Cold Year: $5000
o Warm Year: $8000
Plants Wheat (profit)
o Cold Year: $6500
o Warm Year: $7000
4
Calculated Information (using Bayes)
C=Cold
W=Warm
PC=PredictCold
PW=PredictWarm
Bayes' Theorem
P(Si)
Prior Prob
P(C) =
P(W) =
Highlighted means given information
P(Oj|Si)
P(Si and Oj)
Likelihood
Joint Prob
P(Oj)
Probability of outcome
0.4 P(PC|C) =
0.9 P(PC and C) = P(C)P(PC|C)
P(E) = P(PC and C)+P(PC and W)
=(0.45)(0.6) =
0.36
= 0.27+0.165 =
0.48
0.6 P(PW|C) = 0.1 P(PW and C) = P(C)P(PW|C)
P(D) = P(PW and C)+P(PW and W)
= (0.45)(0.4) =
0.04
= 0.385+0.18 =
0.52
P(PW|W) = 0.8 P(PC and W) = P(W)P(PC|W)
= (0.55)(0.3) =
0.12
P(PC|W) = 0.2 P(PW and W) = P(W)P(PW|W)
= (0.55)(0.7) =
0.48
P(Si|Oj)
Posterior Prob.
P(C|E) = P(C and PC)/P(PC)
= 0.27/0.435 =
0.750
P(W|E) = P(W and PC)/P(PC)
= 0.165/0.435 =
0.250
P(C|PW) = P(C and PW)/P(PW)
= 0.18/0.565 =
0.077
P(W|PW) = P(W and PW)/P(PW)
= 0.385/0.565 =
0.923

P(Predict Cold)=0.48
There is a 48% chance that the forecaster will predict that it will be a cold year.

P(Predict Warm)=0.52
There is a 52% chance that the forecaster will predict that it will be a warm year.

P(Cold|Predict Cold)=0.75
There is a 75% chance that it will be a cold year given that the forecaster predicted the
year would be cold.

P(Warm|Predict Cold)=0.25
There is a 25% chance that it will be a warm year given that the forecaster predicted the
year would be cold.

P(Cold|Predict Warm)=0.077
There is a 7.7% chance that it will be a cold year given that the forecaster predicted the
year would be warm.

P(Warm|Predict Warm)=0.923
There is a 92.3% chance that it will be a warm year given that the forecaster predicted the
year would be warm.
5
WinQSB Input
6
WinQSB Output
7
WinQSB Decision Tree
8
Best Decision
The best decision then would be to not hire the forecaster and just plant corn. This gives an expected
value of $6800. Actual profit can range from $5000 on a cold year to $8000 on a warm year.
The best decision path has been highlighted in green on the WinQSB decision tree below.
9
Sensitivity Analysis
Summary Table
Vary Cost of Expert Weather Forecast
Cost
Expected
Outcome
-$700
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$600
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$500
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$475
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$450
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$425
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$420
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
HireForecaster -> Plant
Hire forecaster. If cold weather predicted,
plant wheat. If warm weather predicted,
plant corn.
HireForecaster -> Plant
Hire forecaster. If cold weather predicted,
plant wheat. If warm weather predicted,
plant corn.
HireForecaster -> Plant
Hire forecaster. If cold weather predicted,
plant wheat. If warm weather predicted,
plant corn.
HireForecaster -> Plant
Hire forecaster. If cold weather predicted,
plant wheat. If warm weather predicted,
plant corn.
-$419
-$410
-$400
-$300
$6,800.88
$6,809.88
$6,819.88
$6,919.88
WinQSB Output
Decision to Make
10
Sensitivity Analysis Graph
WinQSB inputs and outputs for the sensitivity analysis can be found in Appendix A.
Turning Point Calculation
𝑃𝑙𝑎𝑛𝑡 (𝑤𝑖𝑡ℎ𝑜𝑢𝑡 ℎ𝑖𝑟𝑖𝑛𝑔 𝑓𝑜𝑟𝑒𝑐𝑎𝑠𝑡𝑒𝑟) = 6800
𝐻𝑖𝑟𝑒 𝑓𝑜𝑟𝑒𝑐𝑎𝑠𝑡𝑒𝑟 = 7219.88 − 𝑥
7219.88 − 𝑥 = 6800
X = 419.88
Turning point is at $419.88
11
Best Decision
After doing the sensitivity analysis the best decision to make changed. If the cost of an expert
weather forecast is more than $419.88, then the best decision is to still not hire a forecaster and
just plant corn. However, if the cost of an expert weather forecast is less than $419.88, then the
best decision is to hire a forecaster. If the forecaster predicts cold weather, our best decision
would be to plant wheat. If the forecaster predicts warm weather, our best decision is to plant
corn.
12
Utility Function
Utility Function
1.2
y = 0.0003x - 1.2222
Utility Value
1
0.8
0.6
Utility Function
0.4
0.2
0
0
2000
4000
6000
8000
10000
Terminal Value ($)
Utility
0.00
0.17
0.42
0.56
0.58
0.72
0.83
1.00
Terminal Values (Profits)
4400
5000
5900
6400
6500
7000
7400
8000
The utility values are simple to calculate once you find the formula for the function. In this case
the formula is y = 0.0003x - 1.2222. The closer to 0 the smaller the net value of return and the
closer to 1 the larger the net value return.
13
Utility Function Examples

If one wants to know the utility at a terminal profit of $6500 you plug into the formula
x=6500 and then solve for y which in this case equals 0.58.

Another way to solve is if one knows the utility and wants to find the net value of return
associated with it. If y=0.17 then you rearrange the equation to sole for x:
𝑦 = 0.0003𝑥 − 1.2222
0.0003𝑥 = 𝑦 + 1.2222
𝑦
𝑥= ⁄
0.0003 + 1.2222⁄0.0003
𝑥 = 0.17⁄0.0003 + 1.2222⁄0.0003 = 5000
Therefore the net value of return at a utility of 0.17 is $5000.
14
Utility WinQSB Input
15
Utility WinQSB Output
16
Utility WinQSB Decision Tree
17
Report to Manager
Dear Manager,
After careful consideration and analysis, our best option is not to hire a forecaster and
to just plant corn. This option gives us an expected profit of $6,800.00. A key factor to this
decision is the cost of $600.00 for the forecaster. Please keep in mind that expected profit is not
the exact amount of profit you will receive; it is the profit that we expect to see over time if this
process was repeated many times. The profit can range from $5000 to $8000. If we are able to
procure a forecaster at a smaller rate then we would have another option. We performed a
sensitivity analysis by changing the cost of hiring the forecaster. A condensed summary table is
included below. We calculated the cost of the forecaster at the turning point to be $419.88. If we
are able to find a forecaster at that rate or less, our second option would be to hire a forecaster. If
cold weather is predicted we will plant wheat while if warm weather is predicted we would plant
corn. Using a forecaster with these options gives us an expected value of $6800.88 and the
actual profit can range from $4400 to $7400.
Vary Cost of Expert Weather Forecast
Cost
Expected
Outcome
-$600
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$420
$6,800.00
Plant -> PlantCorn
Don't hire forecaster, just plant corn.
-$419
$6,800.88
HireForecaster -> Plant
-$300
$6,919.88
HireForecaster -> Plant
WinQSB Output
Decision to Make
18
Hire forecaster. If cold weather predicted,
plant wheat. If warm weather predicted,
plant corn.
Hire forecaster. If cold weather predicted,
plant wheat. If warm weather predicted,
plant corn.
Appendix
A. Sensitivity Analysis – WinQSB Input/Output
-$700 WinQSB Input
19
-$700 WinQSB Output
20
-$600 WinQSB Input
21
-$600 WinQSB Output
22
-$500 WinQSB Input
23
-$500 WinQSB Output
24
-$475 WinQSB Input
25
-$475 WinQSB Output
26
-$450 WinQSB Input
27
-$450 WinQSB Output
28
-$425 WinQSB Input
29
-$425 WinQSB Output
30
-$420 WinQSB Input
31
-$420 WinQSB Output
32
-$419 WinQSB Input
33
-$419 WinQSB Output
34
-$410 WinQSB Input
35
-$410 WinQSB Output
36
-$400 WinQSB Input
37
-$400 WinQSB Output
38
-$300 WinQSB Input
39
-$300 WinQSB Output
40