decision trees november 18, 2013. example: we have a machine which may break down at any time over...
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Decision Trees
November 18, 2013
Example: we have a machine which may break down at any timeover the next three years. We can replace it now, at a cost of $40,000, or we can keep it in service till it breaks. That willcost us $10,000 in lost production, and we will have to pay tohave it replaced. Every year, there is a 30% chance that the cost of a replacement will go up by $5,000, though we don’texpect there to be more than one such increase in the next threeyears. Our MARR is 20%.
Example: we have a machine which may break down at any timeover the next three years. We can replace it now, at a cost of $40,000, or we can keep it in service till it breaks. That willcost us $10,000 in lost production, and we will have to pay tohave it replaced. Every year, there is a 30% chance that the cost of a replacement will go up by $5,000, though we don’texpect there to be more than one such increase in the next threeyears. Our MARR is 20%.
Your company has developed a new design of mobile phone.You need to decide whether to manufacture and market it inChina or North America.
If you manufacture in China, the fixed costs will be $4,000,000, andthe variable costs per phone will be $15.
If you manufacture in North America, the fixed costs may be anywherefrom $3,000,000 to $4,000,000, and the variable costs per phone willbe between $15 and $20. Shipping across the Pacific will add $10 perphone.
If you sell at $30 in China, you expect to sell 300,000-400,000 units.If you sell for $40, you can expect these figures to be halved. In NorthAmerica, if you sell at $40 you expect to sell 100,000 to 300,000, andthese figures will be halved if you sell at $50.
North America
China
Start with a decision node…
Make further decisions…
Make in China
Make in NA
Sell in China
Sell in China
Sell in NA
Sell in NA
Make further decisions…
$30
$30
$40
$40
$40
$50
$40
$50
Sell China
Sell NA
Sell China
Sell NA
Label leaf nodes…
$30
$30
$40
$40
$40
$50
$40
$50
Sell China
Sell NA
Sell China
Sell NA
NN40NN40
NN50NN50
CC30CC30
Construct `chance’ trees from leaf nodes…
CC30CC30
Sell 300,000
Sell 400,000
Calculate PW (omit (P/F) factors, since they’rethe same on all branches; figures in $000’s)
CC30CC30
Sell 300,000
Sell 400,000
PW = -4,000+300(30-15)= -4,000+4,500= 500
PW = -4,000+400(30-15)= -4,000+6,000= 2,000
Calculate average PW
CC30CC30
Sell 300,000
Sell 400,000
PW = 500
PW = 2,000
E(x)=1,250
Calculate variance and standard deviation
CC30CC30
Sell 300,000
Sell 400,000
PW = 500
PW = 2,000
E(x)=1,250
Var(x) = 0.5 × (1,250-500)2 + 0.5 × (1,250-2,000)2 = 562,500
s.d. = 750
For some leaf nodes, the process is more complicated…
NN40NN40
Fixed cost 300,000
Fixed cost 400,000
VC 15
VC 15
VC 20
VC 20
Sell 100,000
Sell 100,000
Sell 100,000
Sell 100,000
Sell 300,000
Sell 300,000
Sell 300,000
Sell 300,000
NN40
NN40
VC 15
VC 20
Sell 100K
Sell 300KFC 300K
PW=-3,000+100(40-15) = -500
PW=-3,000+300(40-15) = 5,500
NN40
NN40
VC 15
VC 20
Sell 100K
Sell 300KFC 300K
PW=-3,000+100(40-15) = -500
PW=-3,000+300(40-15) = 5,500
FC 400K
PW= -1,000
PW= 3,000
PW= -1,500
PW= 4,500
PW= -2,000
PW= 2,000
NN30
NN30
VC 15
VC 20
Sell 100K
Sell 300KFC 300K
PW= -500
PW= 5,500
FC 400K
PW= -1,000
PW= 3,000
PW= -1,500
PW= 4,500
PW= -2,000
PW= 2,000
E(x) = 1,250Var = 4,570,000s.d. = 2,137
Mark these results on the decision tree
$30
$30
$40
$40
$40
$50
$40
$50
Sell China
Sell NA
Sell China
Sell NA
Mean = 1,250, sd = 750
Mean = 375, sd = 625
Mean = -1,000, sd = 1,500
Mean = -1,500, sd = 1,250
Mean = 1,250, sd = 2,137
Mean = -250, sd = 1,639
Make in China
Make in NA
Mark these results on the decision tree
$30
$30
$40
$40
$40
$50
$40
$50
Sell China
Sell NA
Sell China
Sell NA
Mean = 1,250, sd = 750
Mean = 375, sd = 625
Mean = -1,000, sd = 1,500
Mean = -1,500, sd = 1,250
Mean = 1,250, sd = 2,137
Mean = -250, sd = 1,639
Make in China
Make in NA
The China/China option is mean-variance dominant
$30
$30
$40
$40
$40
$50
$40
$50
Sell China
Sell NA
Sell China
Sell NA
Mean = 1,250, sd = 750
Mean = 375, sd = 625
Mean = -1,000, sd = 1,500
Mean = -1,500, sd = 1,250
Mean = 1,250, sd = 2,137
Mean = -250, sd = 1,639
Make in China
Make in NA
The Monte Carlo Method
Illustrative example:
How to calculate π using a shotgun
Step One: Draw a circle in a square
Note that, if the side of the square is 2, then the area
of the square is 4 and the area of the circle is π
2
Blast the target with your shotgun
Count the number of holes inside the circle and the
number inside the square. Their ratio approximates π/4
In this case, we find that π is about 4 × 10 ⁄ 13 = 3.08
To get an improved estimate, we take more shots
j = 0;
for (i=1; i <= 1000000; i++){
x = rnd();y = rnd();
if ((x2 + y2) < 1) j++;
pi_Estimate = 4 * real(i)/real(j)
}
j = 0;
for (i=1; i <= 1000000; i++){
x = rnd();y = rnd();
if ((x2 + y2) < 1) j++;
pi_Estimate = 4 * real(i)/real(j)
}
In the square
j = 0;
for (i=1; i <= 1000000; i++){
x = rnd();y = rnd();
if ((x2 + y2) < 1) j++;
pi_Estimate = 4 * real(i)/real(j)
}
Is it in the circle?
π
π
π
π
Applying Monte Carlo analysis to economic problems:
Your company has developed a new design of mobile phone.You need to decide whether to manufacture and market it inChina or North America.
If you manufacture in China, the fixed costs will be $4,000,000, andthe variable costs per phone will be $15.
If you manufacture in North America, the fixed costs may be anywherefrom $3,000,000 to $4,000,000, and the variable costs per phone willbe between $15 and $20. Shipping across the Pacific will add $10 perphone.
If you sell at $30 in China, you expect to sell 300,000-400,000 units.If you sell for $40, you can expect these figures to be halved. In NorthAmerica, if you sell at $40 you expect to sell 100,000 to 300,000, andthese figures will be halved if you sell at $50. It will take 2 to 4 yearsto begin manufacture, and your MARR is between 10 and 15%.
Step 1: Build the deterministic model
(this may be a computer program or set of spreadsheetformulae)
PW(NN30) = (Fixed_Cost – Sales(30 -Variable_Cost))(P/F, i, N)
PW(NN40) = (Fixed_Cost – Sales(40 -Variable_Cost))(P/F, i, N)
Do a sample calculation to make sure it works.
Step 2: Decide what the probability distribution is for each random variable:
Fixed Cost
p(F
ixed
Cos
t)
$3M $3.5M $4M
Step 3: The random numbers you are going to generate will be uniformly distributed between 0 and 1.
This is fine if the variable you are modelling happens to be uniformly distributed between 0 and 1, but what if it’s normally distributed between $3M and $4M?
First form the cumulative distribution function
Fixed Cost$3M $4M
p(F
ixed
Cos
t < x
)
Then discretize it
x2$3M $4Mx1 x3
p(x<x1)
p(x<x2)
p(x<x3)
Pick z = rnd()
x2$3M $4Mx1 x3
p(x<x1)
p(x<x2)
p(x<x3)
z
Now if p(x<x2) < z < p(x<x3), X =(x2+x3)/2
x2$3M $4Mx1 x3
p(x<x1)
p(x<x2)
p(x<x3)
z
X
Step 3a: The random numbers you are going to generate will be uniformly distributed between 0 and 1.
This is fine if the variable you are modelling happens to be uniformly distributed between 0 and 1, but what if it’s a discrete random variable?
For example:
``There is a 30% probability that the government will levy a surcharge on phones imported from China.’’
Let z = rnd(); if z > 0.3, no surcharge
In this way, we generate random estimates for eachparameter in the problem. Then we plug these parameters into the deterministic model, and getone possible value of the present worth.
PW(NN30)1 = (Fixed_Cost1 – Sales1(30 -Variable_Cost1))(P/F, i1, N1)
Now do this a thousand times.
PW(NN30)1000 = (Fixed_Cost1000 – Sales1000(30 -Variable_Cost1000))(P/F, i1000, N1000)
PW(NN30)2 = (Fixed_Cost2 – Sales2(30 -Variable_Cost2))(P/F, i2, N2)
.
.
.
What do we do with the 1,000 PW’s?
We create a histogram:
Find highest and lowest PW’s.
Divide the interval between them into N binsof equal size. (N could be 10 or thereabouts)
Count the number of cases in each bin.
Then repeat for other possible strategies
Cases inBin
Present worth ($000’s)
-500 5500
NN30
Cases inBin
Present worth ($000’s)
-500 5500
NN30
CC30
Making a decision:
The histogram provides somewhatmore information than mean + variance.
We may want to choose the strategy thathas no chance of making a loss.
Other applications of the Monte Carlo method:
First used in the Manhattan project
N. Metropolis and S. Ulam, "The Monte Carlo Method", Journal of the American Statistical Association, volume 44, number 247, pp. 335–341 (1949)
Used to simulate combustion in diesel engines:
Start with N1 packets of air andN2 packets of fuel.
Each packet of fuel has anN1/(N1+N2) chance of mixingwith a packet of air.
After mixing has taken place, there is a delay for vaporization, thena second delay before combustion starts.
After combustion starts, each packetof fuel has a certain chance of mixingwith air, and a certain chance of mixingwith an already-combusted packet.
The outputs of the process are histograms of various output parameters:
Soot emissions Nox emissions