Frustrations, agony and tales of woes that greeted the aftermath of any concluded accreditation exercise informed our interest in addressing this ugly trend that has bedeviled our higher educational system. The dearth or otherwise absence of an appropriate model(s) that will satisfy both the starting matrix and the NUCs staff mix by rank matrix explains this disconnect. A typical challenge here is to reach a desired structure by a certain time in a changing environment or with the smallest possible cost in other to meet up with NUC accreditation minimum bench mark requirement for any higher educational institution. The main objective was to reach a desired structure by a certain time in a changing environment or with the smallest possible cost. Therefore a certain degree of control is sensible at various points in time to the attainment of the desired academic staff structure of any higher institution to monitor the academic staffmix by rank of Academic staff structure of universities not to fall short of NUC requirements for accreditation. In our work, the concept of time as an optimality performance criterion was used to obtain an optimal recruitment control vector for a manpower system modelled by a stochastic differential equation through the necessary condition of Pontryagin theorem. Desired transition matrix P was obtained that is not stochastic but could be further developed into a stochastic matrix as required.

Quality is the ability or degree with a product or service or phenomenon conforms, to an established standard, and which makes it to be relatively superior to others. With respect to education this implies the ability or degree with which an educational system (fadipe, 1999). Quality in education therefore means the relevance and appropriateness of the education programme to the needs of the community for which it is provided. Quality assurance on the other hand, is about consistently meeting product specification or getting things right the first time and every time. Quality assurance in the university system implies the ability of the outputs (

The control models on the other hand have two aspects: maintainability (maintaining a given structure) and attainability (feasibility of attaining a desired structure): for example Nwaigwe (

It is evident that ‘forces’ acting on a manpower system can be divided into two main groups. These are those forces which can be controlled at will by a manpower planner, for example, promotion and recruitment and those which cannot be controlled fully, for example wastage. By judiciously and continually adjusting the controlled variables, we can often get the system to perform in a way consistent with a specific objective. A typical objective may be to reach a desired structure by a certain time in a changing environment or with the smallest possible cost.

The quality of an academic programme becomes a universal concern because the product of one university invariably becomes an employee in another university or other cultures' industrial setting. Also, degree obtained at the end of training in a university is intended to ascertain the level of competency (

As opined by (

The National University Commission (

2.1 Time Optimality

2.1.1 Minimum time Optimality Performance criterion

Here the control strategy is to be chosen in such a way as to transfer the system from an initial state n to a desired state n* in the shortest possible time. This is equivalent to minimizing the performance index,

C = ? t 0 t 1 dt

where t1 is the first instant of time at which the desired state n* is reached.

Pontryagin Maximum Principle Theorem

Letu(t) be an admissible control with corresponding trajectory n* that transfers a controlled system from n0 at time t0 to n1 at some unspecified time t1. Then in order that u* and n* be optimal (that is minimize some performance index) it is necessary that there exist a non-trivial vector ?=(?0,?1,?2)satisfying the Hamiltonian H and co-state equations such that for every t in t0?t?t1, H attains its maximum with respect to, u at u=u(t)

H(?,x,u)=0 and ?0?0at t =t1

where ?is the solution of the co-state equation for u=u(t).

Furthermore, it can be shown that H(?(t),x(t),u[t]) and ?0(t)are constants so that the Hamiltonian H=0 and ?0?0 at each point on an optimal trajectory.

The subject of optimal control has attracted the attention of several authors in the mathematical sciences, for example, Washburn (

In most manpower control problems, there may be more than one way of reaching a desired structure or maintaining a given structure. In situations like this, one is faced with the problem of selecting the control strategy that is best in some sense. This is an aspect of optimal manpower control in which, interest is on the problem of compelling a system in this case, a manpower system to behave in some best possible way. Definitely, the exact control strategy depends on the criterion used to decide what is meant by best. This work will examine the condition under which a manpower system modeled by a differential equation is controllable and uses time as an optimality performance criterion in controlling the manpower system with quadratic index in both state and control spaces.

2.1.1a Markov and Renewal Manpower Models

Ogbogbo, Ebuh and Aronu (

2.1.2 Estimating a Markov Transition Matrix from Observed Data

Markov chains play a central role in Operational Research, frequently being used to describe how a system changes over time. If a system can be adequately modelled as a Markov chain, then numerous theoretical consequences can be applied to the analysis of the system. The behavior of a Markov Chain depends on the values used in the transition matrix which specifies probabilities that the system moves from one state to another in unit time. Standard texts assume that the values of such transition matrices are known. However, in most practical studies, this is not the case and the transition matrix needs to be estimated. One way of doing such estimation is to use data concerning the observed state of the system at successive time ‘point’. If succeed, observations are all the same interval apart then estimating the transition matrix is straightforward. Unfortunately, however, the practitioner is often faced with problems in which, a system has been observed infrequently, where times between successive observations vary. With a large amount of such variation, estimating the transition matrix becomes more complex. Horn (

2.1.3 Markov Chain Analysis of Manpower Data of A Nigerian University

According to Igboanugo and Onifade (

Observably, seven states are required as follows:

Recruitment pool for Lecturer 1 & below as state 1 ( L1')

Recruitment pool for Senior Lecturer as state 2 (SL')

Recruitment pool for Professorial cadre as state 3 ( Pr')

Lecturer 1 & below ? Lecturer 1 and below?Senior Lecturer?Professor?Exit as state 4 (L1?)

Senior Lecturer ?Senior Lecturer?Professor?Exit as state 5 (SL)

Professor ?Professor?Exit as state 6 (Pr)

Exit from the department as state 7 (E)

Let P be the required Transition Probability Matrix given as:

P = a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 71 a 72 a 73 a 74 a 75 a 76 a 77

Where aijis the probability of movement from state i to j. e.g. a15is the probability of being recruited from the pool of L1' to the position of SL and a57 is the probability of exiting from the department. For each i,

If XNis the NUC staff mix and XS is the present staff-mix available in the department, for a case of denial accreditation status that requires one year for the department to readjust

X F = P 1 X S

For cases that requires n years to ensure NUC staff-mix where n stands for interim and full accreditation. e.g 2,3, and 4years

X F = P n X S

In P above, some transitions are impossible e.g. A professor moving to the state of L1?

Starting from state 1, L1'the recruitment can only be to L1< or remain in state 1 unrecruited.

1 2 3 4 5 6 7

State 1 a11 0 0 a14 0 0 0

Staring from state 2, SL' recruitment pool to SL 1 2 3 4 5 6 7

State 2 0 a22 0 0 a25 0 0

Similarly for state 3, Pr' recruitment pool for Pr 1 2 3 4 5 6 7

State 3 0 0 a33 0 0 a36 0

State 4- present staffing of L1<. Transition can only be to state 5 i.e. SL and Pr. State 5 and 6 and exit

1 2 3 4 5 6 7

State 4 0 0 0 a44 a45 a46 a47

Similarly, for state 5 – present staffing of SL, where transition is to Pr or exit

1 2 3 4 5 6 7

State 5 0 0 0 0 a55 a56 a57

For Pr state transition is to exit

1 2 3 4 5 6 7

State 6 0 0 0 0 0 a66 a67

State 7- The Exit state, here no transition

1 2 3 4 5 6 7

State 7 0 0 0 0 0 0 1

Putting all these together,

a ij = a 11 0 0 a 14 0 0 0 0 a 22 0 0 a 25 0 0 0 0 a 33 0 0 a 36 0 0 0 0 a 44 a 45 a 46 a 47 0 0 0 0 a 55 a 56 a 57 0 0 0 0 0 a 66 a 67 0 0 0 0 0 0 1

?j=17aij=1 for all i

Interim - One Year Transition

X F = P 1 X S

X 1 F X 2 F X 3 F X 4 F X 5 F X 6 F X 7 F = a ij X 1 S X 2 S X 3 S X 4 S X 5 S X 6 S X 7 S

Equation becomes,

a11X1S+a14X4S=X1F (1)

a22X2S+a25X5S=X2F (2)

a33X3S+a36X6S=X3F (3)

a44X4S+a45X5S+a46X6S+a47X7S=X4F (4)

a55X5S+a56X6S+a57X7S=X5F (5)

a66X6S+a67X7S=X6F (6)

a67X7S=X7F (7)

Giving rise to 16 unknowns in seven equations. The following additional equations are deduced as follows (available stock of staff in the department):

Total number of L1<

a14+a44 (8)

Representing the sum of those recruited and not promoted

Total number of SL

a25+a45+a55(9)

Typifies, sum of recruited, promoted and not promoted

Total number of Pr

a36+a46+a56+a66 (10)

Equation (8) is as X4S

Equation (9) is as X5S

Equation (10) is as X6S

? a 14 + a 44 = X 4 F

a 25 + a 45 + a 55 = X 5 F

a 36 + a 46 + a 56 + a 66 = X 6 F

Additional equations are (taking the row sum)

a11+a14=1 (11)

a22+a25=1 (12)

a33+a36=1 (13)

a44+a45+a46+a47=1 (14)

a55+a56+a57=1 (15)

a66+a67=1 (16)

Giving rise to 16 equations and 16 unknowns

From equation (10), a67=(1-a66)

Applied to equation ( 6)

1 − a 66 ) X 7 S + a 66 X 6 S = X 6 F (1-a66)X7S+a66X6S=X6F

X S = 0 0 0 X 4 S X 5 S X 6 S X E S , since X 7 S = 0 XS=000X4SX5SX6SXES, since X7S=0

Hence, a66=X6FX5S (17)

From equation (15),a57=(1-a55-a56)

Substituting into equation (5)

a 55 X 5 S + a 56 X 6 S + a 57 X 7 S = X 5 F

If X 7 S = 0If X7S=0

a55X5S+a56X6S=X5F (18)

Assumption 1

From available data, promotion exercise for the last four years; the probability of promotion from SL to Pr given as a56which giving a prior probability relationship between a55 and a56.

For example, for every 9 that remains in SL one is promoted to Pr giving a ratio of a55:a56as 9:1.

Then equation (

a 55 X 5 S + a 55 9 X 6 S = X 5 F

a 55 ( X 5 S + 1 9 X 6 S ) = X 5 F

a55=X5F(X5s+19X6S), then a56and a57

From equation (14)

a 44 + a 45 + a 46 + a 47 = 1

a47=1-a44-a45-a46 (19)

Substitute into equation ( 4)

a 44 X 4 S + a 45 X 5 S + a 46 X 6 S + ( 1 - a 44 - a 45 - a 46 ) X 7 S = X 4 F

a44(X4S-X7S)+a45(X5S-X7S)+a46(X6S-X7S)=X4F-X7S(20)

Since X7S=0

a 44 X 4 S + a 45 X 5 S + a 46 X 6 S = X 4 F

Assumption 2

a46 is the transition from L1< to Pr in the first year which is close to zero

Then equation (20) becomes

a44X4S+a45X5S=X4F (21)

If a 44 : a 45 as 8:2, then equation (21) becomesIf a44:a45 as 8:2, then equation (21) becomes

a 44 X 4 S + 2 10 a 44 X 5 S = X 4 F

a 44 ( X 4 S - 1 5 X 5 S ) = X 4 F

a44=X4FX4S-15X5S (22)

Then follows a45and a46

From equation (8) and since a44is known, then

a14=X4F-a44 (23)

Substituting into equation (11), we get a11

From equation (9),

a25=X5F-a45-a55, (24)

a45and a55 are already known

From equation ( 12)

a22=1-a25 (25)

In equation (10) substitute for a 66 , a 56 and a 46 then we have a 36In equation (10) substitute for a66,a56and a46then we have a36

Then from equation ( 13) we obtain

a33=1-a36 (26)

Additional Equations (Based on Current + Recruited=Final Stock)

a 55 + a 56 + a 57 = X 5 S

a 66 + a 67 = X 6 S

a 44 + a 45 + a 46 + a 47 = X 4 S

a 36 + a 46 + a 56 + a 66 = X 6 F

a 25 + a 41 + a 55 = X 4 F

a 14 + a 44 = X 4 F

a 11 + a 14 = X 1 S

a 22 + a 23 = X 2 S

a 33 + a 36 = X 3 S

P=a1100a140000a2200a250000a3300a360000a44a45a46a470000a55a56a5700000a66a670000001,

X S = X 1 S X 2 S X 3 S X 4 S X 5 S X 6 S X 7 S = 0 . 01 0 . 02 0 . 03 5 8 2 8 1 8 0 , X F = X 1 F X 2 F X 3 F X 4 F X 5 F X 6 F X 7 F = 0 0 0 0 . 45 0 . 35 0 . 20 0 XS=X1SX2SX3SX4SX5SX6SX7S=0.010.020.035828180, XF=X1FX2FX3FX4FX5FX6FX7F=0000.450.350.200

Note: X1S=0.01, X2S=0.02, X3S=0.03are all assigned values as X1S,X2S,X3S?0. This explains the fact that the recruitment pool can never be empty due to prevalent unemployment.

a 11 = 1 - a 14

a 14 = X 4 F - ( X 4 F X 4 S - 1 5 X 5 S )

a 22 = 1 - a 25

a 25 = X 5 F - a 45 - a 55

a 33 = 1 - a 36

a 36 = X 6 F - a 46 - a 56 - a 66

a 44 = X 4 F X 4 S - 1 5 X 5 S

a 45 = X 4 F - X 4 F X 4 S - 1 5 X 5 S X 4 S X 5 S

a 46 = X 4 F - a 44 X 4 S - a 45 X 5 S X 6 S

a 47 = 1 - a 44 - a 45 - a 46

a 55 = X 5 F ( X 5 S + 1 9 X 6 S )

a 56 = X 5 F - a 55 X 5 S X 6 S

a 57 = X 5 F - a 55 X 5 S - a 56 X 6 S X 7 S

a 66 = X 6 F X 5 S

Note: XS is based on the available staff stock in a particular department but in our case study i.e. L1<=5, SL=2 and Pr=1.

P = 1 . 3326 0 0 - 0 . 3326 0 0 0 0 0 . 1796 0 0 - 0 . 8204 0 0 0 0 1 . 7464 0 0 - 0 . 7464 0 0 0 0 0 . 7826 - 0 . 1564 0 0 . 3738 0 0 0 0 1 . 3268 0 . 1464 ? 0 0 0 0 0 0 . 8 0 . 2 0 0 0 0 0 0 1

The matrix P obtained for this test case is not stochastic as the row sum for Rows 2 and 5 does not sum up to 1. Therefore, there is the need to make it stochastic as the matrix is representative of the solution to required staff-mix needed to meet up NUC requirement. Obtained transition matrix P in its non-stochastic state can still be used to obtain the required staff-mix by rank for the accreditation purpose and later made stochastic. This is left for further study (making the obtained matrix become stochastic).

A manpower system is modelled as stochastic differential equation. The concept of time as an optimality performance criterion is proposed to be adapted to obtain optimal recruitment control vector for the manpower system through the necessary condition of Pontryagin theorem. It will be shown that the optimal recruitment control vector to minimised the control time globally.