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MARKOV CHAIN EXAMPLE

MARKOV CHAIN EXAMPLE. Personnel 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

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MARKOV CHAIN EXAMPLE

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  1. MARKOV CHAIN EXAMPLE Personnel Modeling

  2. 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

  3. STATE SPACE • S = {N1, N2, N3, N4, V} • V for Vacancy • Every time period, the employee moves according to a probability

  4. MODELED AS A MARKOV CHAIN • Discrete time periods • Stationarity • transitions stay constant over time • transitions do not depend on time in grade

  5. TRANSITION DIAGRAM 0.1 1 0.6 0.5 2 0.3 1.0 0.1 0.6 0.3 3 0.2 0.1 0.1 0.8 0.1 V 4 0.1 0.1

  6. PROBABILITY TRANSITION MATRIX = P

  7. 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

  8. 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

  9. STEADY STATE PROBABILITIES • a0 * P * P * P * P * .... • P is singular (rank 4) • P is stochastic • rows sum to 1 • p is the stationary probability distribution • pN1 is the proportion of the time spent in state N1

  10. COMPUTATION STRATEGY • p = p * P • 1 = pN1 + pN2 +pN3 +pN4 +pV • Substitute stochastic equation for first component of p = p * P • Solve Linear System via Gaussian Elimination

  11. ...more COMPUTATION STRATEGY • Start with arbitrary a0 • calculate a1, a2, a3, ... • will converge to p • p = [0.16, 0.24, 0.24, 0.24, 0.12] • SUPRIZED?

  12. CONVERGENCE TO p

  13. CONVERGENCE IS QUICK

  14. FOR GRINS • Changed PN4,V to 0.0 • p = [0.09, 0.16, 0.23, 0.46, 0.06]

  15. 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

  16. LABOR COSTS • Salaries • CN1 = $12,000 • CN2 = $21,000 • CN3 = $25,000 • CN4 = $31,000 • Total Workforce = 180,000 • Cost = 180K * (C * p) = $3.7B

  17. 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?

  18. LABOR COSTS

  19. CONVERGENCE TO 75% WORKFORCE (135K)

  20. CONVERGENCE TO 60% WORKFORCE (108K)

  21. BUILDING AN N4 FROM AN N1CUMULATIVE

  22. BUILDING AN N4 FROM AN N1MARGINAL

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