Session ‘Emerging Applications of OR: Stochastic Models in Manpower Planning’,
24th European Conference on Operational Research, EURO XXIV, Lisbon, July 2010.
The limiting behavior of
the mixed push-pull manpower model
Tim De Feyter * and Marie-Anne Guerry
* Center
for Business Management Research, HogeschoolUniversiteit Brussel, K.U.Leuven Association, Stormstraat 2, B-1000
Brussels, Belgium
VLEKHO-HONIM
Introduction
Manpower Planning is concerned with predicting and
controlling the (internal) personnel structure of an
organization.
A manpower system is classified into k exclusive
subgroups, resulting in the states of the system
S1,…,Sk.
To model the dynamics in the manpower system
(transitions between the states), traditionally two
approaches are used in Manpower Planning:
 Push-models (based on Markov theory)
 Pull-models (based on Renewal theory)
VLEKHO-HONIM
Introduction
Push-models
Every employee in a state i has the same probability wi
to leave the organization and the same probability
pij to make a transition to state j.
 A certain expected number of transitions in each time period
Pull-models
Every employee in a state i has the same probability wi
to leave the organization and the same conditional
probability sji to make a transition to fill a vacancy
in state j.
 Transitions only if there are vacancies in other states
Bartholomew D.J., Forbes A.F. and McClean S.I. (1991). Statistical techniques for
manpower planning. Chichester: Wiley publishers.
VLEKHO-HONIM
Introduction
Push-model
R.r1
n1(t)
n1(t).p12
R.r2
n1(t).w1
n2(t).p21
n2(t)
n2(t).w2
Pull-model
(1-s11-s12).V1
n1(t)
V2.s21
(1-s21-s22).V2
n1(t).w1
V1.s12
n2(t)
n2(t).w2
The stocks at time t are denoted by the row vector n(t) = [n1(t) n2(t) … nk(t)] with
k = number of states in the manpower system. In time-discrete models, the
dynamics in the manpower system can be expressed as systems of difference
equations.
VLEKHO-HONIM
Introduction
Push-model
n(t )  n(t  1).P  R.r
Pull-model
V(t)=[n*-n(t-1).(1-W)]
n(t)=n*-V(t).S
Notations
k
number of states in the system
ni(t)
number of staff in state i on time t
n(t)
(1 × k) row vector with entries ni(t)
pij
transition rate from state i to state j
P
(k × k) transition matrix with entries pij
R
total number of recruitments
ri
proportion of R(t) recruited in state i
r
(1 × k) recruitment vector with entries ri
VLEKHO-HONIM
Notations
k
number of states in the system
ni(t)
number of staff in state i on time t
n(t)
(1 × k) row vector with entries ni(t)
ni*
desired number of staff in state i
n*
(1 × k) stock vector with entries ni*
sij
rate of vacancies state i filled from state j
S
(k × k) transition matrix with entries sij
wi
rate of vacancies state i filled from state j
W
(k × k) diagonal matrix with entries wi
Mixed push-pull model (De Feyter, 2007)
In HRM-literature, the consensus has grown that firms
seldom apply one unique personnel strategy, but
mix several strategies to enable success on several
separate markets. Consequently, a mix of push and
pull transitions might occur in the same personnel
systems at the same time. The mixed push-pull
model allows studying the dynamics in a manpower
systems if push as well as pull transitions occur.
De Feyter T. (2007). Modeling mixed push and pull promotion flows in Manpower
Planning, Annals of Operations Research, 155(1), 25-39.
VLEKHO-HONIM
Mixed push-pull model (De Feyter, 2007)
(1-s11-s12).V1
n1(t).w1
n1(t)
R.r1
n1(t). (1-w1-V2.s21).q12
V2.s21
V1.s12
n2(t).w2
(1-s21-s22).V2
n2(t)
R.r2
VLEKHO-HONIM
n2(t). (1-w2-V1.s12).q21
Mixed push-pull model (De Feyter, 2007)
In the mixed push-pull approach, the dynamics in the
system is modeled by the following set of difference
equations:
n(t )  V (t )   n(t  1)  I  W   V (t ) S  Q  Rr
V (t )  Max 0; n*  n(t  1)  I  W 
Q={qij} is a row stochastic matrix. Its elements represent the transition
probabilities for employees in group i at time t to be in group j at time t+1,
under the condition that they do not leave the system nor make a pull
transition during time interval.
_____
The Max-operator is hereby defined as Max { A, B}  {Max ( Ai , Bi )} .
VLEKHO-HONIM
Asymptotic behavior of the MPPM
MP research involves studying the limiting behavior
of the expected number of employees n(t ) as t  
In the mixed push-pull model, this is a complex
problem, since:
1. In each time interval [t,t+1) the transitions
may be determined by both push and pull
probabilities or only by push probabilities;
2. This system may be different for the different
states.
VLEKHO-HONIM
Illustration
Evolution group 1
n*   43 71
70
n(0)  19 98 
50
n(t)
60
 0,14 0,86 
Q

 0,37 0, 63 
Rr   7 31
groep 1
ne(0,0)
30
n*/(1-w)
20
10
0
1
2
3
4
5
6
7
8
9
Evolution group 2
140
120
100
n(t)
 0, 09 0 
W 

0
0,
26


40
80
groep 2
60
ne(0,0)
40
n*/(1-w)
20
0
1
VLEKHO-HONIM
2
3
4
5
6
7
8
9
Asymptotic behavior of the MPPM
Theorem (De Feyter, 2007)
Let A = I–S.Q and M= [I-W][Q-A]. If in every time
period both pull and push transitions occur in all
the states, the row sums of M are strictly less than
one and M is such that:
(i) M is a power-nonnegative matrix (1) or
(ii) M+I is a totally nonnegative matrix (2)
then the mixed push-pull models converges to
_____
1.
2.
ne  (n *  A  R  r )( I  M ) 1
k
A matrix A is called power-nonnegative of degree k (with k a positive integer) if  k : A  0
k is the smallest integer for which this condition holds.
A matrix is called totally nonnegative if all his minors are nonnegative.
VLEKHO-HONIM
and
Asymptotic behavior of the MPPM
However, these are not necessary conditions.
We studied this further for k = 2 by
reformulating the difference equations as:
n(t )  n(t  1)  M (t )  n* (t ) 
.
V (t )   n*  n(t  1)  I  W   
 I
i
ii
 I
i
ii
(t )  A  R  r

(t )

A : I  S  Q


M (t ) :  I  W    Q   I ii (t )  A 
i 


 (t ) :  i Vi (t )  0 
VLEKHO-HONIM
Illustration
 0 0, 45 
 0,14 0,86 
 0, 09 0 
S 
 Q
 W 

0, 26 
 0, 09 0 
 0,37 0, 63 
0
M 
M 
M 
M 
 0,66 1,01 
 ( I  W )[Q  ( I  SQ)]  

0,
29

0,14


0,78 
 0,13
 (I  W )  Q  

0,
27
0,
47


1 0 
 0,66 1,01 
 ( I  W )[Q  
 ( I  SQ)]  

0
0
0,
27
0,
47




0,78 
0 0
 0,13
 ( I  W )[Q  
 ( I  SQ)]  

0
1
0,
29

0,14




Asymptotic behavior of the MPPM
Theorem
In case the system is stable from a particular value of
t on and for all values i, the evolution of the stock
vector converges to:
ne  (n*   Iii  A  R r ) ( I  M )1
i 
Theorem
If the evolution according to M++ is convergent, than
the same is true for the evolution according to M-+ ,
M+- and M--.
VLEKHO-HONIM
Evolution group 1
Illustration
120
100
n(0)   33 32 
ne(1,1)
n(t)
n*   52 69 
ne(0,1)
ne(0,0)
n*/(1-w)
20
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
Evolution group 2
100
90
80
n(t)
0 

0, 2 
ne(1,0)
60
40
 0,84 0,16 
Q

 0, 60 0, 40 
Rr  19 17 
 0,3
W 
0
groep 1
80
70
groep 2
60
ne(1,1)
50
ne(1,0)
40
ne(0,1)
30
ne(0,0)
20
n*/(1-w)
10
0
1
3
5
7
9
11 13 15 17 19 21 23 25 27 29
Asymptotic behavior of the MPPM
Properties of matrix M
1) Row sums are between 0 and 1
2) The diagonal elements are between -1 and 1; the
other elements between -1 and 1.
3) Therefore, we can show that the largest eigenvalue
is positive and smaller than 1.
4) And the smallest eigenvalue is smaller than 1 (but
can only smaller than -1 if Det M++ is positive and
Trace M++ is negative)
VLEKHO-HONIM
Illustration
n*   33 29  n(0)   0 0  Rr   7 1
 0 0, 45 
 0, 04 0,96 
 0,39 0 
S 
 Q
 W 

0,
09
0
0,95
0,
05
0
0,
04






Conclusion
• We extended the results on the limiting
behavior of the mixed push-pull model in
previous work by relaxing the conditions for
which we know that the stocks converge.
• Further research is necessary to find out
under which conditions the system becomes
stable (so far, all the examples we know, the
system converges or flips between ‘two
converging stocks’)
VLEKHO-HONIM
Conclusion
Therefore, the properties of the products of two
different matrices M  should be investigated, since:
– If the model keeps switching between the same two
systems M1 and M2:
ne  n*   I1  A  I 2  A  M1   R r   I  M1    I  M 2  M1 
1
– The model would become stable if from a particular value
*
of t on and for all values i: ni (t )  ni (t  1) 1  wi  stays
positive or negative. To evaluate this, we need to
investigate the evolution n(t) by relaxing the assumption of
stable systems which are determined by only one fixed M 

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