markov processes manualcomputerbased homework solution data mining and forecast management mgmt e -...
TRANSCRIPT
![Page 1: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/1.jpg)
Markov Processes
ManualManual
ComputerComputerBasedBased
Homework SolutionHomework Solution
Data Mining and Forecast Management
MGMT E - 5070
![Page 2: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/2.jpg)
Machine Operation ProblemA manufacturing firm has developed a transition matrix containingA manufacturing firm has developed a transition matrix containing
the probabilities that a particular machine will operate or break down the probabilities that a particular machine will operate or break down in the following week, given its operating condition in the present week.in the following week, given its operating condition in the present week.
REQUIREMENT:REQUIREMENT:
Assuming that the machine is operating in week 1, that is, the initial state is ( .4 , .6 ) :Assuming that the machine is operating in week 1, that is, the initial state is ( .4 , .6 ) :
1.1. Determine the probabilities that the machine will operate or break down in weeksDetermine the probabilities that the machine will operate or break down in weeks 2, 3, 4, 5, and 6.2, 3, 4, 5, and 6.2.2. Determine the steady-state probabilities for this transition matrix algebraically and Determine the steady-state probabilities for this transition matrix algebraically and indicate the percentage of future weeks in which the machine will break down.indicate the percentage of future weeks in which the machine will break down.
Problem 1
![Page 3: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/3.jpg)
Machine Operation Problem
.16 .24.16 .24
.4 .6.4 .6
.8 .2.8 .2
.48 .12.48 .12
.64 .36 Week No. 2
( .4 , .6 )( .4 , .6 )
![Page 4: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/4.jpg)
Machine Operation Problem
.256 .384.256 .384
.4 .6.4 .6
.8 .2.8 .2
.288 .072.288 .072
.544 .456 Week No. 3
( .64 , .36 )( .64 , .36 )
![Page 5: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/5.jpg)
Machine Operation Problem
.2176 .3264.2176 .3264
.4 .6.4 .6
.8 .2.8 .2
.3648 .0912.3648 .0912
.5824 .4176 Week No. 4
( .544 , .456 )( .544 , .456 )
![Page 6: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/6.jpg)
Machine Operation Problem
.23296 .34944.23296 .34944
.4 .6.4 .6
.8 .2.8 .2
.33408 .08352.33408 .08352
.56704 .43296 Week No. 5
( .5824 , .4176 )( .5824 , .4176 )
![Page 7: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/7.jpg)
Machine Operation Problem
.226816 .340224.226816 .340224
.4 .6.4 .6
.8 .2.8 .2
.346368 .086592.346368 .086592
.57384 .426816 Week No. 6
( .56704 , .43296 )( .56704 , .43296 )
![Page 8: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/8.jpg)
Machine Operation Problem
.4X.4X11 .6X .6X11
.8X.8X22 .2X .2X22
P(O) = 1XP(O) = 1X11 P(B) = 1X P(B) = 1X22
OPERATE BREAKDOWN
P (O) = .4XP (O) = .4X11 + .8X + .8X22 = 1X = 1X11 ( (dependent equation))
P (B) = .6XP (B) = .6X11 + .2X + .2X22 = 1X = 1X2 2 ((dependent equation))
1X1X11 + 1X + 1X22 = 1 ( = 1 (independent equation))
![Page 9: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/9.jpg)
Machine Operation Problem
.6X.6X11 + .2X + .2X22 – 1.0X – 1.0X22 = 0 = 0
becomes……becomes……
.6X.6X11 - .8X - .8X22 = 0 = 0
.4X.4X11 + .8X + .8X22 – 1.0X – 1.0X11 = 0 = 0
becomes……becomes……
- .6X- .6X11 + .8X + .8X22 = 0 = 0
Setdependentequationsequal to
zero
![Page 10: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/10.jpg)
Machine Operation Problem
.6X.6X11 - .8X - .8X22 = 0 = 0.6 ( 1X ( 1X11 + 1X + 1X22 = 1 ) = 1 ) .6X.6X11 + .6X + .6X22 = .6 = .6 -1.4X-1.4X22 = -.6 = -.6 XX22 = = .4285 = P ( = P ( BREAKDOWN BREAKDOWN ))
Since XSince X11 + X + X22 = 1, then: = 1, then:
1 – X1 – X22 = X = X11
1 - .4285 = 1 - .4285 = .5715 = P ( = P ( OPERATION OPERATION ))
STEADY-STATE PROBABILITIES
![Page 11: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/11.jpg)
![Page 12: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/12.jpg)
![Page 13: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/13.jpg)
![Page 14: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/14.jpg)
![Page 15: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/15.jpg)
Newspaper ProblemNewspaper Problem
A city is served by two newspapers – The Tribune and the Daily News. Each Sunday, readers purchase one of the newspapers at a stand. The following transition matrix contains the probabilities of a customer’s buying a particular newspaper in a week, given the newspaper purchased the previous Sunday.
Problem 2
![Page 16: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/16.jpg)
Newspaper ProblemNewspaper Problem
REQUIREMENT:
1. Determine the steady-state probabilities for the transition matrix algebraically, and explain what they mean.
![Page 17: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/17.jpg)
Newspaper Problem
.65 X.65 X11 .35 X .35 X11
.45 X.45 X22 .55 X .55 X22
P(T) = XP(T) = X11 P(DN) = X P(DN) = X22
Tribune Daily News
TribuneDaily News
![Page 18: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/18.jpg)
Newspaper Problem
P ( T ) = .65XP ( T ) = .65X11 + .45X + .45X22 = 1X = 1X11 ( ( dependent equation ))
P ( DN ) = .35XP ( DN ) = .35X11 + .55X + .55X22 = 1X = 1X22 ( ( dependent equation ))
1X1X11 + 1X + 1X22 = 1 ( = 1 ( independent equation ))
![Page 19: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/19.jpg)
Newspaper ProblemNewspaper Problem
.65X1 + .45X2 = 1X1
.65X1 + .45X2 – 1X1 = 0
- .35X1 + .45X2 = 0
.35X1 + .55X2 = 1X2
.35X1 + .55X2 – 1X2 = 0
.35X1 - .45X2 = 0
![Page 20: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/20.jpg)
Newspaper ProblemNewspaper ProblemSTEADY - STATE PROBABILITIES
.35X.35X11 - .45X - .45X22 = 0 = 0
.35 ( 1X( 1X11 + 1X + 1X22 = 1 ) = 1 )
.35X.35X11 + .35X + .35X22 = .35 = .35 - .80X- .80X22 = - .35 = - .35 XX22 = = .4375 = P ( = P ( Daily NewsDaily News ) )
Since XSince X11 + X + X22 = 1, then: = 1, then:
1 – X1 – X22 = X = X11
1 - .4375 = 1 - .4375 = .5625 = P ( = P ( TribuneTribune ) )
![Page 21: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/21.jpg)
![Page 22: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/22.jpg)
![Page 23: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/23.jpg)
![Page 24: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/24.jpg)
![Page 25: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/25.jpg)
Fertilizer ProblemFertilizer Problem
Problem 3
![Page 26: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/26.jpg)
Fertilizer ProblemFertilizer ProblemPROBABILITY TRANSITION MATRIXPROBABILITY TRANSITION MATRIX
This Spring
Next Spring
![Page 27: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/27.jpg)
Fertilizer ProblemFertilizer ProblemThe number of
customers presently
using each brand
of fertilizer is shown below:
![Page 28: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/28.jpg)
Fertilizer ProblemFertilizer Problem
![Page 29: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/29.jpg)
Fertilizer ProblemTransition Matrix
Plant Plus Crop Extra Gro Fast
.4 .3 .3.4 .3 .3 .5 .1 .4.5 .1 .4 .4 .2 .4.4 .2 .4
![Page 30: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/30.jpg)
Fertilizer ProblemTransition Matrix
Plant Plus Crop Extra Gro Fast
.4X.4X11 .3X .3X11 .3X .3X11
.5X.5X2 2 .1X .1X22 .4X .4X22
.4X.4X33 .2X .2X33 .4X .4X33
P (PP) = 1XP (PP) = 1X11 P(CE) = 1X P(CE) = 1X22 P(GF) = 1X P(GF) = 1X33
![Page 31: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/31.jpg)
Fertilizer ProblemTHE EQUATIONS
P (PP) = .4XP (PP) = .4X11 + .5X + .5X22 + .4X + .4X33 = 1X = 1X1 ( 1 ( DEPENDENT ) )
P (CE) = .3XP (CE) = .3X11 + .1X + .1X22 + .2X + .2X33 = 1X = 1X2 ( 2 ( DEPENDENT ))
P (GF) = .3XP (GF) = .3X11 + .4X + .4X22 + .4X + .4X33 = 1X = 1X3 ( 3 ( DEPENDENT ) )
1X1X11 + 1X + 1X22 + 1X + 1X33 = 1 = 1 ( ( INDEPENDENT ) )
![Page 32: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/32.jpg)
Fertilizer Problem
P (PP) = .4X1 + .5X2 + .4X3 - 1.0X1 = 0
P (CE) = .3X1 + .1X2 + .2X3 - 1.0X2 = 0
P (GF) = .3X1 + .4X2 + .4X3 – 1.0X3 = 0
1X1 + 1X2 + 1X3 = 1 ( INDEPENDENT )
Setdependentequationsequal to
zero
![Page 33: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/33.jpg)
Fertilizer ProblemFertilizer Problem
P (PP) = - .6XP (PP) = - .6X11 + .5X + .5X22 + .4X + .4X33 = 0 = 0
P (CE) = .3XP (CE) = .3X11 - .9X - .9X22 + .2X + .2X33 = 0 = 0
P (GF) = .3XP (GF) = .3X11 + .4X + .4X22 - .6X - .6X33 = 0 = 0
Setdependentequationsequal to
zero
![Page 34: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/34.jpg)
Fertilizer Problem
.3X1 - .9X2 + .2X3 = 0
.3X1 + .4X2 - .6X3 = 0
- 1.3X2 + .8X3 = 0
.3 ( 1X1 + 1X2 + 1X3 = 1.0 ) .3X1 + .3X2 + .3X3 = .3 .3X1 + .4X2 - .6X3 = 0
- .1X2 + .9X3 = .3
CANCEL OUT VARIABLE X1
![Page 35: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/35.jpg)
Fertilizer ProblemFertilizer ProblemCANCEL OUT VARIABLE X2
- 1.3X2 + .8X3 = 0 -13 ( .1X2 + .9X3 = .3 ) - 1.3X2 + 11.7X3 = - 3.9
- 10.9X3 = - 3.9 X3 = .357798
![Page 36: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/36.jpg)
Fertilizer Problem
-1.3X2 + .8 ( .358 ) = 0 - 1.3 X2 = - .286 X2 = .220
X1 + .220 + .358 = 1.0 X1 = 1.0 - .578 X1 = .422
SOLVING FOR THE REMAINING VARIABLES
![Page 37: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/37.jpg)
Fertilizer ProblemFertilizer Problem
ΣΣ = 9,000 1.00 = 9,000 1.00 9,0009,000
![Page 38: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/38.jpg)
![Page 39: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/39.jpg)
![Page 40: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/40.jpg)
![Page 41: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/41.jpg)
![Page 42: Markov Processes ManualComputerBased Homework Solution Data Mining and Forecast Management MGMT E - 5070](https://reader034.vdocuments.mx/reader034/viewer/2022051401/56649d305503460f94a08148/html5/thumbnails/42.jpg)
Markov Processes
ManualManual
ComputerComputerBasedBased
Homework SolutionHomework Solution