A PROCEDURE FOR DISTRIBUTING RECRUITS IN MANPOWER SYSTEMS

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1 Yugoslav Joural of Operatios Research 25 (205) Number DOI: /YJOR32903E A PROCEDURE FOR DISTRIBUTING RECRUITS IN MANPOWER SYSTEMS Virtue U. EKHOSUEHI Departmet of Mathematics Uiversity of Bei P.M.B. 54 Bei City Nigeria virtue.ekhosuehi@uibe.edu Augustie A. OSAGIEDE Departmet of Mathematics Uiversity of Bei P.M.B. 54 Bei City Nigeria augustie.osagiede@uibe.edu Wilfred A. IGUODALA Academic Plaig Divisio Uiversity of Bei P.M.B. 54 Bei City Nigeria wilfred.iguodala@uibe.edu Received: December 203 / Accepted: September 204 Abstract: I this paper we treat the followig problem: Give a stable Gai-type persoflow model ad assumig o egative recruitmet what recruitmet distributio at the step is capable of geeratig a staff-mix that closely follows the desired structure? We relate this problem to the challege of uiversities i Nigeria towards attaiig the desired academic staff-mix by rak specified by the Natioal Uiversities Commissio (NUC). We formulate a populatio-dyamic model cosistig of aggregate-fractioal flow balace equatios withi a discrete-time Markov chai framework for the system. We use MATLAB as a coveiet platform to solve the system of equatios. The utility of the model is illustrated by meas of academic staff flows i a uiversity-faculty settig i Nigeria. Keywords: Gai-type Perso-flow Model; Mapower System; Markov Chai; Natioal Uiversities Commissio; Recruitmet Distributio. MSC: 60J20 9D35.

2 446 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig. INTRODUCTION The settig we cosider is a mapower system stratified ito various categories (states) where egative recruitmet is ot allowed ad a desired staff-mix (structure) is to be attaied via a recruitmet policy implemeted at the step. We formulate a populatio-dyamic model cosistig of aggregate-fractioal flow balace equatios withi a discrete-time Markov chai framework for the system. We adopt the covetio that: t {2 } is the step period startig from the iitial period t. Z The period is a discrete-time scale {0 2 T }. We assume that the umber of recruits ito the multi-grade mapower system i period t is decided by the admiistrative authority of the system. We use the term grade to mea the status of a idividual i a mapower system ad a category to mea the aggregatio of grades. For example i the uiversity system the grades deoted as i are i for Graduate Assistat i 2 for Assistat Lecturer i 3 for Lecturer II i 4 for Lecturer I i 5 for Seior Lecturer i 6 for Reader/Associate Professor ad i 7 for Professor. Our applicatio is focused o the departmets of a faculty i the Uiversity of Bei Nigeria. The uiversity system i Nigeria is regulated by the Natioal Uiversities Commissio (NUC). The commissio provides guidelies for program evaluatio i the uiversity system. Amog the guidelies is the academic staff-mix by rak. The staff-mix by rak is that the existig staff structure for academic staff should closely follow the structure 20:35:45 for Professors/Readers: Seior Lecturers: Lecturer I ad below (excludig the positio of Graduate Assistat) respectively [4]. We perform all computatios i the MATLAB eviromet. The MATLAB program is our choice for this study because it is very flexible ad well-suited for matrix-vector algebra ad it cotais a library of predefied fuctios [2 0]. The Markov chai formulatio is a commo ad attractive approach to modelig a k graded mapower system [ ]. The evolutio of the mapower system is examied by studyig either the structure ( t) [ ( t) 2( t) k ( t)] ' or the relative structure k x( t) i ( t) ( t) i [8 9 22]. Amog the Markov chai formulatios the Gai-type model [7] has received cosiderable attetio [ 6 7]. Specifically the Gai-type perso-flow model where ( t) ( t) for all t 02 ad beig the stable growth factor has bee aalyzed [6 7]. New etrats ito the system are allocated to a grade i accordig to a recruitmet distributio { r i } with r 0 ad i k ri. Several aspects of the Markov system have bee studied i the literature. These iclude: the behavior of the expectatios variaces ad covariaces of the state sizes [9 i

3 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig ] the attaiability ad maitaiability of structures [5 8 2] ad the size order of the state vector [2]. Our study may be see as a aspect of cotrol i mapower systems. Davies [3] ad Bartholomew et al. [] had earlier provided cotrol strategies for mapower systems. However the possibility of obtaiig egative etries i the recruitmet distributio {} r i by the use of the strategies was ot resolved. Istead the egative etries were iterpreted as either retrechmet or redudacy. I this study we circumvet the possibility of obtaiig egative etries i the recruitmet distributio by redefiig the distributio i such fashio that egative recruitmet is ot allowed. Thus the trauma associated with egative recruitmet is evaded. 2. METHODOLOGY Let S be a set with labels correspodig to ay categorizatio of iterest. For istace i the NUC categorizatio we have S { 23} i a ascedig order of categories as: Lecturer I ad below (excludig the positio of Graduate Assistat) Seior Lecturers Professors/Readers. A staff i the system ca belog to oly oe of the categories i S. We use the otatio D to represet a #( S) vector of the desired staff-mix where #( S ) is the cardiality of S. Data for graded mapower systems are ormally available as stocks for each grade or flows betwee grades. We assume that the flows follow a atural order i.e. promotio is from oe grade to the ext higher grade o demotio ad o double promotio. Let 0 S deote the eviromet outside the mapower system. Suppose the grades i to are aggregated ito a sigle category S. The we obtai the time ivariat aggregate-fractioal flow rates betwee the categories deoted as ˆp for the flows: ad S as ad pˆ T ( ) ( ) i i ( t) i i( t) t0 i i ( t) ( t) ( t) ( t) T ( ) ( ) ( ) ( ) i i i i i 0 t0 i i i () T ( ) () t t 0 T ( ) ( ) ( ) ( ) i i i i i0 t0 i i i pˆ ( t) ( t) ( t) ( t) ( ) provided ( ( t) exists where ) () t is the umber of staff members i the grades to aggregated ito a category ad () t is the umber of idividuals movig from grade i to grade j i period t. The hat deotes a estimate. The movemets ij (2)

4 448 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig betwee the categories are govered by the sub-stochastic matrix P ( ˆ ). The use of time ivariat aggregate-fractioal flow rates is ecessary to ease the computatioal agoy of obtaiig the multi-step trasitio matrices. The total stock of the system at period t is #( S ) ( ) ( ) ( ) ( ) N( t) i i ( t) i i ( t) ( t) i 0 ( t) (3) i i i We cosider a stable Gai-type perso-flow model wherei the total stock Nt ( ) satisfies N( t ) N( t) We determie the stable growth rate from historical data usig the method proposed i Ekhosuehi ad Osagiede [5]. Let the curret staff-mix for the system be x ( t) [ x ( t) x#( S) ( t)] ' with x ( t) 0 ad # S x ( t) so that x () t is the proportio of staff curretly i category. The evolutio of the structure x () t at the ext period is computed as P'x () t ad the step evolutio is () t P' x. However ( t) ep'x ad ( t) p (4) e P' x where e is a #( S) vector with each compoet equal to oe. This is because P is sub-stochastic ad ep' e. The shortfall e ep' is due to wastage i the system. So e( I P' ) is called the wastage vector where I is a #( S) #( S) idetity matrix. The total wastage rate at the step is: e( P' ) x ( t). The theoretical uderpiig of our method is based o the followig formula [ 7]: ( t ) P'( t) R( t ) ρ (5) where Rt ( ) is the total umber of recruits i period t+ ad ρ ( ) is a #( S) vector of the recruitmet policy with beig the recruitmet policy for the th category S. The vector ρ is such that eρ. We assume that recruitmet is doe to replace wastage ad to achieve the stable growth rate at the step [5] so that for : Thus we obtai ( t ) ( t) ( ) ( t) ( ) N( t) P' e I P' ρ (6) N( t ) N( t) N( t) x( t ) [( P' ) x( t) ( e( P' ) x( t)) ρ ] (7) The aggregate-fractioal flow balace equatios are obtaied by settig x( t ) D so that

5 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig 449 [( P' ) x( t) ( e( P' ) x( t)) ρ] D (8) It is importat to metio here that whe x() t D i Eq. (8) the the model is aki to the system i []. However we solve the problem for the case whe x() t D. At the step Eq. (8) becomes [( P' ) x( t) ( e( P' ) x( t)) ρ] D (9) We use the costrait eρ to simplify Eq. (9) ad get [( ) ( ) ( ( ) ( )) ] P' x t e e P' x t I ρ D (0) It is worth otig that oe way towards attaiig the desired structure is for the admiistrative authority of the system to retrech all idividuals ad the recruit afresh such that ρ D. This sceario is exhibited by the limitig solutio to Eq. (0) i.e. lim[( P' ) x( t) e ( e( P' ) x( t)) I] ρ lim D. Sice the matrix P is sub-stochastic the 0 lim P lim a 0 where a sup P. This implies that lim P' 0. Therefore ρ D. For we itroduce a #( S) vector system of equatios T ρ ad solve the followig [( P' ) x( t) e ( e( P' ) x( t)) I] T D () et. Eq. () is the populatio-dyamic model. The solutio to Eq. () is computed i the MATLAB eviromet by the followig commad: where T A \ B (2) [( P' ) x( t) e ( e( P' ) x( t)) I] A e D B. The solutio say T ( ) cotais etries which are urestricted i sig. We iterpret the value as follows: 0 idicates the ecessity to recruit ew etrats 0 ito category at the step ad idicates o recruitmet ito category at the step. The distributio of recruits at the step r ( t ) is defied to be #( S ) r( t) T (3)

6 450 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig where T ( ) is a #( S ) vector with if 0 accordig to our 0 if 0 iterpretatio for. The symbol meas is defied to be. The vector r ( t ) is a proxy for ρ. The elemets of r ( t ) deoted as r ( t ) satisfy the relatios r ( t ) 0 ad #( S ) r ( t ) for all t. r ( t ) 0 meas o recruitmet should be made ito category S ad r ( t ) 0 meas that a proportio of r ( t ) out of the total umber of recruits as decided by the admiistrative authority should be recruited ito category i period t. The relative structure of the system after recruitmet at the step is obtaied as: ( ) [( ) x t P' x( t) e ( e( P' ) x( t)) I] r( t ) (4) We use the Euclidea orm t to measure the closeess betwee the desired structure ad the augmeted structure at period t ad the compare it to that of the iitial structure. The orm is give as t t x ( t ) D (5) We provide a guided tour of the computatioal method by writig a computer program implemetable i the MATLAB eviromet (see Appedix). 3. APPLICATION We implemet our procedure o a faculty academic staff structure [4]. The faculty cosists of five departmets amely: Chemistry (CHM) Computer Sciece (CSC) Geology (GLY) Mathematics (MTH) ad Physics (PHY). From the records we obtai the followig for each departmet: CHM Total academic staff stock: [ CHM N ] (0) 25 [ CHM N ] () 25 [ CHM ] [ CHM N (3) 23 ] [ CHM N (4) 22 ] [ CHM N (5) 2 N ] (6) 25. The sub-stochastic matrix ad the curret staff-mix by rak [ CHM ] [ CHM ] 4 P 4 4 x (6) 25 ; [ CHM N ] (2) 25

7 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig 45 CSC [ CSC Total academic staff stock: ] [ CSC N (0) 6 ] [ CSC N () 23 N ] (2) 24 [ CSC ] [ CSC N (3) 22 ] [ CSC N (4) 22 ] [ CSC N (5) 25 N ] (6) 27. The sub-stochastic matrix ad the curret staff-mix by rak GLY [ CSC ] [ CSC ] P x (6) ; Total academic staff stock: [ GLY N ] (0) 4 [ GLY N ] () 2 [ GLY ] [ GLY N (3) 2 ] [ GLY N (4) 3 ] [ GLY N (5) 2 N ] (6) 3. The sub-stochastic matrix ad the curret staff-mix by rak MTH [ GLY ] [ GLY ] P x (6) ; [ GLY N ] (2) 2 Total academic staff stock: [ MTH N ] (0) 24 [ MTH N ] () 26 [ MTH ] [ MTH N (3) 28 ] [ MTH N (4) 28 ] [ MTH N (5) 29 N ] (6) 32. The sub-stochastic matrix ad the curret staff-mix by rak [ MTH ] P [MTH ] 6 x (6) 2 32 ; 32 [ MTH N ] (2) 28

8 452 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig PHY Total academic staff stock: [ PHY N ] (0) 4 [ PHY N ] () 5 [ PHY ] [ PHY N (3) 3 ] [ PHY N (4) 3 ] [ PHY N (5) 4 N ] (6) 5. The sub-stochastic matrix ad the curret staff-mix by rak [ PHY ] [ PHY ] P x (6) [ PHY N ] (2) The desired staff-mix by rak i lie with the NUC specificatio is give by the vector D [ ] '. A ispectio of the trasitio matrices shows a high possibility of stayig o i a category ad cosequetly a low progressio rate betwee a category ad the ext higher category. Noe of the departmetal staff-mix by rak is exact as the NUC specificatio. The mootoe decreasig patter from a category to the ext higher category i the NUC staff-mix is ot a feature of the staff-mix of CHM GLY ad MTH. The staff-mix of CHM is top-heavy while that of CSC is bottom-heavy. It is almost uiform for GLY. PHY satisfies the NUC staff-mix i oe category. We obtai the followig results for each departmet for a 3 step period CHM CHM CHM A B 0.20 r (9) ; CSC CSC CSC A B 0.20 r (9) 0 ; GLY GLY GLY A B 0.20 r (9) ;

9 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig MTH A B MTH r (9) ; 0 MTH PHY PHY A B r (9) PHY The results show a cotractig academic staff stock for CHM GLY ad PHY with CHM GLY PHY ad respectively while the stock for CSC MTH CSC ad MTH is expadig with.059 ad.0340 respectively. At the 3 step period the results for CHM idicate that ew etrats should be recruited ito categories ad 2 but ot ito category 3. However more recruits would be eeded i category 2. I CSC ew etrats should be recruited ito categories ad 3 with greater umber of the recruits ito category 3. There should ot be recruitmet ito category 2. No recruitmet should be made ito category 3 i GLY while ew etrats are required i categories ad 2 with more recruits ito category. A similar result for GLY holds for MTH. New etrats are required i all categories i PHY with the greatest umber of the recruits ito category. The 3 step augmeted structure ad the Euclidea orm 9 as well as that of the iitial structure 6 for each departmet are obtaied usig equatios (4) ad (5) as: [ CHM ] x (9) [ CSC] x (9) [ GLY ] x (9) ; ; ;

10 454 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig [ MTH ] x (9) [ PHY ] x (9) ; ad The Euclidea orm idicates that the augmeted structure improves o the iitial structure i four departmets (CHM CSC GLY ad PHY) but ot i MTH. I particular the augmeted structure for PHY equates the desired structure. 4. CONCLUSION I this study a attempt has bee made to fid a recruitmet distributio that is capable of geeratig a desired structure after oe or more steps i a mapower system where egative recruitmet is ot allowed. The task was to formulate a system of aggregate-fractioal flow balace equatios withi a discrete-time Markov chai framework. Our model complemets the existig model i the literature. Oe of the accomplishmets of the study is the kack to figure out a way to avoid the possibility of obtaiig egative etries i the recruitmet distributio. We have illustrated the usefuless of our approach for a faculty i the Uiversity of Bei Nigeria. The practical challeges of implemetig the model i the uiversity system may iclude bottleecks such as the iadequacy of resources the possibility of overstaffig dearth of applicats with the requisite qualificatios ad cogate experiece etc. These challeges are grey areas for future research. Ackowledgemet: We thak the Editor ad the reviewers for helpful commets which greatly improved the earlier mauscript. REFERENCES [] Bartholomew D. J. Forbes A. F. ad McClea S. I. Statistical techiques for mapower plaig (2 d ed.) Joh Wiley & Sos Chichester 99. [2] Chapma S. J. Essetials of MATLAB programmig (2 d ed.) Cegage Learig USA [3] Davies G. S. Cotrol of grade sizes i a partially stochastic Markov mapower model Joural of Applied Probability 9 (2) (982) [4] Ekhosuehi V. U. Evaluatio of career patters of academic staff i a faculty i the Uiversity of Bei Nigeria Ife Joural of Sciece 5 () (203) 8-9. [5] Ekhosuehi V. U. ad Osagiede A. A. A proposed Markov model for predictig the structure of a multi-echelo educatioal system i Nigeria Moografias Matematicas Garcia de Galdeao 38 (203) [6] Feichtiger G. O the geeralizatio of stable age distributios to Gai-type perso-flow models Advaces i Applied Probability 8 (3) (976)

11 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig 455 [7] Gai J. Formulae for projectig erolmets ad degrees awarded i uiversities Joural of the Royal Statistical Society Series A (Geeral) 26 (3) (963) [8] Gerotidis I. I. O certai aspects of o-homogeeous Markov systems i cotiuous time Joural of Applied Probability 27 (3) (990) [9] Guerry M. A. Properties of calculated predictios of graded sizes ad the associated iteger valued vectors Joural of Applied Probability 34 () (997) [0] Hah B. ad Valetie D. Essetial MATLAB for egieers ad scietists (4 th ed.) Elsevier Ltd. UK 200. [] Haigh J. Stability of mapower systems. The Joural of the Operatioal Research Society 43 (8) (992) [2] Kipouridis I. ad Tsaklidis G. The size order of the state vector of discrete-time homogeeous Markov systems Joural of Applied Probability 38 (2) (200) [3] Nicholls M. G. The use of Markov models as a aid to the evaluatio plaig ad bechmarkig of doctoral programs Joural of the Operatioal Research Society 60 (2009) [4] Osasoa O. Tools for academic plaig I: Uvah I. I. (Editor) Practical Guide o Academic Plaig i Nigeria Uiversities: A Compedium of Academic Plaig Tools (202) [5] Tsaklidis G. M. The evolutio of the attaiable structures of a homogeeous Markov system with fixed size Joural of Applied Probability 3 (2) (994) [6] Tsatas N. ad Vassiliou P.-C. G. The o-homogeeous Markov system i a stochastic eviromet Joural of Applied Probability 30 (2) (993) [7] Vassiliou P. C. G. A ote o stability i Gai-type models i mapower systems Joural of the Operatioal Research Society 3 (980) [8] Vassiliou P. C. G. ad Georgiou A. C. Asymptotically attaiable structures i ohomogeeous Markov systems Operatios Research 38 (3) (990) [9] Vassiliou P. C. G. ad Gerotidis I. Variaces ad Covariaces of the grade sizes i mapower systems Joural of Applied Probability 22 (3) (985) [20] Vassiliou P. C. G. ad Tsaklidis G. The rate of covergece of the vector of variaces ad covariaces i o-homogeeous Markov systems Joural of Applied Probability 26 (4) (989) [2] Vassiliou P. C. G. ad Tsatas N. Maitaiability of structures i ohomogeeous Markov systems uder cyclic behaviour ad iput cotrol SIAM Joural o Applied Mathematics 44 (5) (984) [22] Vassiliou P. C. G. Georgiou A. C. ad Tsatas N. Cotrol of asymptotic variability i o-homogeeous Markov systems Joural of Applied Probability 27 (4) (990) APPENDIX The MATLAB source codes for the computatios clc disp('the Expasio/Cotractio factor.') R=[:]; %The vector of total stock over the years. T=legth(R); t=[:t]; g=exp((2*t*(log(r))'-(6*(t+)*sum(log(r))))/(t*(t^2-))) =3;

12 456 V.U.Ekhoshuehi A.A.Osagiede W.A.Iguodala/ A Procedure for Distributig P=[:]; %Iitial trasitio matrix. P=P^ x=[:]' %Curret structure. disp('matrix of policy coefficiets.') e=oes(3); A=[[P'*x*e+(g-e*P'*x)*eye(3)]; e] D=[ g^-]'; disp('recruitmet policies.') T=A\D if T()>0 r=t(); else r=0; ed if T(2)>0 r2=t(2); else r2=0; ed if T(3)>0 r3=t(3); else r3=0; ed rho=[r r2 r3]'/sum([r r2 r3]) xcal=[p'*x*e+(g-e*p'*x)*eye(3)]*rho k=sum(xcal) xpt=xcal/k EuD0=sqrt(sum((x-[ ]').^2)) EuD=sqrt(sum((xpt-[ ]').^2))