MARKOV CHAIN EXAMPLEPersonnel Modeling

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

STATE SPACE

•S = {N1, N2, N3, N4, V}

• V for Vacancy

• Every time period, the employee moves according to a probability

MODELED AS A MARKOV CHAIN

• Discrete time periods

• Stationarity– transitions stay constant

over time– transitions do not depend

on time in grade

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

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

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

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

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

COMPUTATION STRATEGY

PN1N2 N3 N4

V

• Substitute stochastic equation for first component of P

• Solve Linear System via Gaussian Elimination

...more COMPUTATION STRATEGY

• Start with arbitrary a0

• calculate a1, a2, a3, ...

• will converge to

CONVERGENCE TO

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15

ITERATION

N1

N2

N3

N4

V

CONVERGENCE IS QUICK

0

0.5

1

1.5

iteration

sqrt

(sum

sqr

err

or)

FOR GRINS

• Changed PN4,V to 0.0

= [0.09, 0.16, 0.23, 0.46, 0.06]

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

LABOR COSTS

• Salaries– CN1 = $12,000

– CN2 = $21,000

– CN3 = $25,000

– CN4 = $31,000

• Total Workforce = 180,000

• Cost = 180K * (C * ) = $3.7B

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?

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

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

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

BUILDING AN N4 FROM AN N1CUMULATIVE

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20

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