Solution to Preventive Maintenance1
We start with a spreadsheet layout, as shown here:
A
B
C
D
1 Simulation of policy 1 (never replace)
2
Week
Beginning state Revenue Replace?
3
1
1
100
0
4
2
1
100
0
5
3
2
80
0
6
4
2
80
0
7
5
2
80
0
8
6
3
50
0
9
7
4
10
0
10
8
4
10
0
11
9
4
10
0
12
10
4
10
0
13
11
4
10
0
14
12
4
10
0
15
13
4
10
0
16
14
4
10
0
17
15
4
10
0
18
16
4
10
0
19
17
4
10
0
20
18
4
10
0
21
19
4
10
0
22
20
4
10
0
E
F
Cost
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Profit
100
100
80
80
80
50
10
10
10
10
10
10
10
10
10
10
10
10
10
10
G
H
I
Random # Ending State
0.678
1
0.852
2
0.469
2
0.156
2
0.936
3
0.689
4
0.989
4
0.549
4
0.867
4
0.403
4
0.648
4
0.449
4
0.699
4
0.163
4
0.469
4
0.982
4
0.928
4
0.095
4
0.926
4
0.425
4
J
K
Average weekly profit
L
M
Inputs
Cost of replacing
$200
Weekly revenue as a function of the state of the machine
State Revenue
1
$100
2
$80
3
$50
4
$10
Initial state
1
Description of machine deterioriation from one week to the next
State at beginning
State at beginning of next week
of this week
1
2
3
4
1
0.7
0.3
0
0
2
0
0.7
0.3
0
3
0
0
0.6
0.4
4
0
0
0
1
The right side of the spreadsheet contains basic input information: the average
profit earned by the machine over 50 weeks (K2), the cost of replacing the
machine (K5), the various revenue values associated with different machine
states (J9:K12), the initial state of the machine (K14), and the probabilities
associated with the changes in the state of the machine from week to week.
(K19:N22).
J
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Average weekly profit
Inputs
Cost of replacing
K
$18.60
L
M
N
=AVERAGE(F3:F52)
$200
Weekly revenue as a function of the state of the machine
State Revenue
1
$100
2
$80
3
$50
4
$10
Initial state
1
Description of machine deterioriation from one week to the next
State at beginning
State at beginning of next week
of this week
1
2
3
4
1
0.7
0.3
0
0
2
0
0.7
0.3
0
3
0
0
0.6
0.4
4
0
0
0
1
Based on 11-25 (p. 612) in Practical Management Science (2 nd ed., Winston and Albright, 2001
Duxbury Press). Solution by David Juran, 2002.
1
N
$18.60
On the left side of the spreadsheet, we have a row for every simulated week. The
state of the machine at the start of the first week is “1”, representing “excellent”.
For all the other weeks, the beginning state is simply the ending state from the
previous week.
The revenue (column C) is a function of the state of the machine, as is the
replacement policy. (For this particular replacement policy, we wouldn’t need
such an elaborate model, but we need a way to model different policies within
the same basic layout.)
The numbers in column D will be binary (zeroes or ones), indicating whether we
do or don’t replace the machine under specific circumstances. The cost of
replacing the machine will be captured in column E, and will affect net profit in
column F.
A
B
C
D
E
F
G
H
1 Simulation of policy 1 (never replace)
=D4*$K$5
=C4-E4
2
Week
Beginning state Revenue Replace?
Cost
Profit
Random # Ending State
=K14
3
1
1
100
0
0
100
0.678
1
4
2
1
100
0
0
100
0.852
2
5
3
2 =H3
80
0
0
80
0.469
2
6
4
2
80
0
0
80
0.156
2
7
5 =VLOOKUP(B4,$J$9:$K$12,2,0)
2
80
0
0
80
0.936
3
8
6
3
50
0
0
50
0.689
4
9 This will
7 contain a different
4
10 for each proposed
0
0
10Depending
0.989
4
decision rule
replacement
policy.
on the
previous week
a one
10 state8of the machine at
4 the end of the10
0 we will have0a zero or a one,
10 where 0.549
4
the decision
machine. Here,
so it's10
always a zero.
11 represents
9
4 to replace the 10
0 we never replace,
0
0.867
4
12
10
4
10
0
0
10
0.403
4
13
11
4
10
0
0
10
0.648
4
=IF(B4=1,(((G4<$K$19)*(1))+((G4>$K$19)*(2))),IF((B4=2),(((G4<$L$20)*(2))
14
12
4 +((G4>$L$20)*(3))),IF((B4=3),(((G4<$M$21)*(3))+((G4>$M$21)*(4))),4)))
10
0
0
10
0.449
4
15
13
4
10
0
0
10
0.699
4
16
14
4
10
0
0
10
0.163
4
The tricky part is to use the probabilities in K19:N22 to model random
deterioration of the machine from week to week. One way to accomplish this is
to generate uniform random variables (column G) and use the complicated “IF”
statement in column H to make changes in the machine state. (We will
eventually get Crystal Ball to generate the random numbers, but we haven’t done
anything about that yet.)
How does this “IF” statement work? First, it is important to understand how a
uniform random number works. These uniform random numbers have a lower
limit of zero and an upper limit of 1, a range in which all possible numbers are
equally likely. The probability that the uniform random variable X is less than 0.5
= 0.5; the probability that X is less than 0.9 is 0.9, and so forth.
Now consider this statement:
=((G5<$K$19)*(1))+((G5>$K$19)*(2))
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In English, this translates as “1 if G5 is less than K19 and 2 if G5 is not less than
K19”. We can have this cell return a 1 or a 2, based on the probability that G5 is
less than K19.
We use nested “IF” statements to expand this idea into four different probability
distributions, each representing a different condition for the machine at the
beginning of the week.
Here is a probability tree for this problem, showing what the “IF” statement is
doing:
Excellent at End of Week
70.0%
Good at End of Week
30.0%
Begin in Excellent Condition
Average at End of Week
0.0%
Bad at End of Week
0.0%
Excellent at End of Week
0.0%
Preventive Maintenance
Good at End of Week
70.0%
Average at End of Week
30.0%
Begin in Good Condition
Bad at End of Week
0.0%
Excellent at End of Week
0.0%
Good at End of Week
0.0%
Begin in Average Condition
Average at End of Week
60.0%
Bad at End of Week
40.0%
Excellent at End of Week
0.0%
Good at End of Week
0.0%
Average at End of Week
0.0%
Begin in Bad Condition
Bad at End of Week
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Prof. Juran
Once we have the ending state for a particular week in column H, it becomes the
beginning state for the following week.
Now for the Crystal Ball part. A Crystal Ball spreadsheet is like any other Excel
file, except that it contains two special kinds of cells. The first is called an
assumption cell (a random variable). The other is called a forecast cell (very
similar to the objective function in an optimization model).
The assumption cells in this model are the uniform random variables in column
G. (Make sure that you have some numbers in these cells before you try to define
them.)
Selecting cell G3, click on the define assumption button
distribution gallery. Select Uniform, and click OK.
. This opens the
We want a uniform distribution from 0 to 1, so type in these values for the two
parameters, and then click OK.
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The cell turns bright green:
F
1
2
3
4
5
Profit
100
100
80
G
H
Random # Ending State
0.678
1
0.852
2
0.469
2
We could use the same procedure to define all of the other assumption cells, but
that would be tedious. Luckily, Crystal Ball has copy and paste buttons,
and
. Select the assumption cell you want to copy (G3), and click the Crystal Ball
copy button (not the regular Excel copy button). Then select the cells you want to
define as assumptions (G4:G52), and click the Crystal Ball paste button. They will
all turn green.
A
B
C
D
1 Simulation of policy 1 (never replace)
2
Week
Beginning state Revenue Replace?
3
1
1
100
0
4
2
1
100
0
5
3
2
80
0
6
4
2
80
0
7
5
2
80
0
8
6
3
50
0
9
7
4
10
0
10
8
4
10
0
11
9
4
10
0
12
10
4
10
0
E
F
Cost
0
0
0
0
0
0
0
0
0
0
Profit
100
100
80
80
80
50
10
10
10
10
G
H
Random # Ending State
0.350
1
0.852
2
0.469
2
0.156
2
0.936
3
0.689
4
0.989
4
0.549
4
0.867
4
0.403
4
Now we need to define a forecast cell, which is a cell that Crystal Ball will track
through the simulation experiment, and for which we will be able to extract
summary statistics, charts, etc.
In our case, we are interested in the long-run average profit of the machine over
50 weeks, which is cell K2. Select cell K2 and click on the Crystal Ball define
forecast button
. The only things Crystal Ball needs to define a forecast are a
name and the units. In this case, call it Average Weekly Profit, and use dollars as
the units.
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Click OK, and the forecast cell will turn blue.
To actually run the simulation, we need to specify run preferences by clicking on
the run preferences button
.
There are a lot of features in the run preferences dilaog box, accessed through the
selections across the right side of the box (Trials, Sampling, Speed, etc.). We will
mostly be concerned with Trials and Sampling.
For now, select 1000 trials and leave the other default settings.
Crystal Ball has buttons for controlling the simulation run, similar to the buttons
on a DVD player:
Play
Stop
Rewind
Single Step
The single step button is useful for performing one iteration at a time to ensure
that the simulation is behaving properly. Once we are satisfied that the
calculations are working properly, we click on the play button and Crystal Ball
runs all 1,000 trials.
During the run, we can watch the forecast window to see what the distribution of
average weekly profit looks like. After the run is over, we can look at the results
in various ways; most often we will use either the frequency chart or the
summary statistics.
To answer this question, we need to make models of the other replacement
policies. We can copy this spreadsheet using Excel’s Edit – Move Or Copy Sheet
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function. The four models are all identical except for the functions in column D.
Here are pictures of the four models, showing the function in cell D6:
A
B
C
D
1 Simulation of policy 1 (never replace)
2
Week
Beginning state Revenue Replace?
3
1
1
100
0
4
2
1
100
0
5
3
1
100
0
6
4
1
100
0
7
5
1
100
0
8
6
1
100
0
A
B
C
D
1 Simulation of policy 2 (replace bad machines)
2
Week
Beginning state Revenue Replace?
3
1
1
100
0
4
2
1
100
0
5
3
2
80
0
6
4
2
80
0
7
5
2
80
0
8
6
2
80
0
E
F
Cost
0
0
0
0
0
0
0
E
G
H
Profit
Random # Ending State
100
0.545
1
100
0.189
1
100
0.383
1
100
0.195
1
100
0.649
1
100
0.620
1
F
G
H
Cost
Profit
Random # Ending State
0
100
0.259
1
0
100
0.964
2
0
80
0.056
2
0
80
0.170
2
0
80
0.203
2
=(H5=4)*(1)+(H5<4)*(0)
0
80
0.119
2
A
B
C
D
E
F
G
H
1 Simulation of policy 3 (replace bad and average machines)
2
Week
Beginning state Revenue Replace?
Cost
Profit
Random # Ending State
3
1
1
100
0
0
100
0.816
2
4
2
2
80
0
0
80
0.384
2
5
3
2
80
0
0
80
0.703
3
6
4
1
100
1
200
-100
0.115
1
7
5
1
100
0
0
100
0.333
1
=(H5>2)*(1)+(H5<3)*(0)
8
6
1
100
0
0
100
0.415
1
A
B
C
D
E
F
G
H
1 Simulation of policy 4 (replace bad, average, and good machines)
2
Week
Beginning state Revenue Replace?
Cost
Profit
Random # Ending State
3
1
1
100
0
0
100
0.786
2
4
2
1
100
1
200
-100
0.880
2
5
3
1
100
1
200
-100
0.014
1
6
4
1
100
0
0
100
0.239
1
7
5
1
100
0 =(H5>1)*(1)+(H5<2)*(0)
0
100
0.080
1
8
6
1
100
0
0
100
0.709
2
Now we can run all four models simultaneously, and compare the results after
the run is over.
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Preventative Maintenance
$70
Policy 3
Average Weekly Profit
$60
Policy 2
$50
$40
$30
Policy 4
Policy 1
$20
$10
$$-
$2
$4
$6
$8
$10
$12
$14
Standard Deviation of Weekly Profit
It looks like policies 2 and 3 are both reasonable, while policies 1 and 4 are clearly
inferior.
Some Crystal Ball notes:
If you lose your forecast windows, you can get them back by clicking the forecast
windows button:
.
Your frequency charts will look slightly different from the ones in this document.
You need to learn how to use the forecast preferences menu to make the charts
and statistics look the way you want.
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