Download - MARKOV CHAIN EXAMPLE
![Page 1: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/1.jpg)
MARKOV CHAIN EXAMPLEPersonnel Modeling
![Page 2: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/2.jpg)
DYNAMICS
• Grades N1..N4
• Personnel exhibit one of the following behaviors:– get promoted– quit, causing a vacancy
that is filled during the next promotion period
– remain in grade– get demoted
![Page 3: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/3.jpg)
STATE SPACE
•S = {N1, N2, N3, N4, V}
• V for Vacancy
• Every time period, the employee moves according to a probability
![Page 4: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/4.jpg)
MODELED AS A MARKOV CHAIN
• Discrete time periods
• Stationarity– transitions stay constant
over time– transitions do not depend
on time in grade
![Page 5: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/5.jpg)
TRANSITION DIAGRAM
1
2
3
4V
0.1
0.1
0.1
0.10.1
0.20.1
0.1
0.60.5
0.31.0
0.30.6
0.8
![Page 6: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/6.jpg)
PROBABILITY TRANSITION MATRIX
1 2 3 4 V
1 0.1 0.6 0.3
2 0.1 0.5 0.3 0.1
3 0.1 0.6 0.2 0.1
4 0.1 0.8 0.1
V 1
= P
![Page 7: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/7.jpg)
MEASURES OF INTEREST
• Proportion of the workforce at each level
• Expected labor costs per year
• Expected annual cost of Entry-level training
• PDF of passage from N1 to N4
![Page 8: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/8.jpg)
TRANSITION PROBABILITY CALCULATION
• Start with employee in N1• a0 = [1, 0, 0, 0, 0]• a1 = a0 * P• a1 = [0.1, 0.6, 0, 0, 0.3]• a2 = a1 * P
37.0
1
1
1
2
1,
1,22
1,11
1
VV
N
N
N
Pa
Pa
Pa
a
![Page 9: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/9.jpg)
STEADY STATE PROBABILITIES
• a0 * P * P * P * P * ....
• P is singular (rank 4)
• P is stochastic– rows sum to 1
is the stationary probability distribution
N1 is the proportion of the time spent in state N1
![Page 10: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/10.jpg)
COMPUTATION STRATEGY
PN1N2 N3 N4
V
• Substitute stochastic equation for first component of P
• Solve Linear System via Gaussian Elimination
![Page 11: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/11.jpg)
...more COMPUTATION STRATEGY
• Start with arbitrary a0
• calculate a1, a2, a3, ...
• will converge to
![Page 12: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/12.jpg)
CONVERGENCE TO
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15
ITERATION
N1
N2
N3
N4
V
![Page 13: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/13.jpg)
CONVERGENCE IS QUICK
0
0.5
1
1.5
iteration
sqrt
(sum
sqr
err
or)
![Page 14: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/14.jpg)
FOR GRINS
• Changed PN4,V to 0.0
= [0.09, 0.16, 0.23, 0.46, 0.06]
![Page 15: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/15.jpg)
ENTRY-LEVEL TRAINING
• 12% of the time we are in state V
• Cost of ELT = – 12%– times the Workforce size– times the cost of training
![Page 16: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/16.jpg)
LABOR COSTS
• Salaries– CN1 = $12,000
– CN2 = $21,000
– CN3 = $25,000
– CN4 = $31,000
• Total Workforce = 180,000
• Cost = 180K * (C * ) = $3.7B
![Page 17: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/17.jpg)
EXCURSION
• Promotion probabilities unchanged
• Allow attrition to reduce workforce– PV,N1 = 0.6 results in
workforce of 108,000
• How much $ saved?
• How fast does it happen?
![Page 18: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/18.jpg)
LABOR COSTS
3.35
3.4
3.45
3.5
3.55
3.6
3.65
3.7
100000 120000 140000 160000 180000 200000
WORKFORCE
$B
![Page 19: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/19.jpg)
CONVERGENCE TO 75% WORKFORCE (135K)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10
ITERATION
N1
N2
N3
N4
V
![Page 20: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/20.jpg)
CONVERGENCE TO 60% WORKFORCE (108K)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10
ITERATION
N1
N2
N3
N4
V
![Page 21: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/21.jpg)
BUILDING AN N4 FROM AN N1CUMULATIVE
0
0.05
0.1
0.15
0.2
0.25
0 5 10 15 20
![Page 22: MARKOV CHAIN EXAMPLE](https://reader030.vdocument.in/reader030/viewer/2022033019/56813650550346895d9dd2f0/html5/thumbnails/22.jpg)
BUILDING AN N4 FROM AN N1MARGINAL
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 5 10 15 20