Markov Chain: A Predictive Model for Manpower Planning


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1 JASEM ISSN All rights reserved Fulltext Available Olie at wwwaolifoad wwwbiolieorgbr/a J Al Sci Eviro Maage May 27 Vol 2 (3) * EZUGWU, VO; 2 OLOGUN, S,2 Deartmet of Mathematics ad statistics Uiversity of Uyo, AwaIbom State, Nigeria ABSTRACT: The use of Mathematical models for maower laig has icreased i recet times for better maower laig quatitatively I resect of orgaizatioal maagemet, umerous revious studies have alied Marov chai models i describig title or level romotios, demotios, recruitmets, withdrawals, or chages of differet career develomet aths to cofirm the actual maower eeds of a orgaizatio or redict the future maower eeds The movemets of staff called trasitios are usually the cosequeces of romotios, trasfer betwee segmets or wastage ad recruitmet ito the system The obective of the study is to determie the roortios of staff recruited, romoted ad withdraw from the various grades ad to forecast the academic staff structure of the uiversity i the ext five years I this aer, we studied the academic staff structure of uiversity of Uyo, Nigeria usig Marov chai models The results showed that there is a steady icrease i the umber of Graduate Assistats, Seior Lecturer ad Associate rofessors, while, there is a steady decrease i the umber of Assistat Lecturer, Lecturer II, Lecturer I, ad Professor i the ext five years The model so develoed ca oly be alied whe there is o cotrol o recruitmet but the research ca be exteded to iclude cotrol o recruitmet The model ca also be alied i school erollmet roectio JASEM htts://dxdoiorg/434/asemv2i37 Keywords: Marov Chai, Trasitio Probability Matrix, Maower Plaig, Recruitmet, Promotio, Wastage A Marov chai (Discrete Time Marov Chai, DTMC), amed after a Russia Mathematicia, Adrey Marov i 97, is a radom rocess that udergoes trasitio from oe state to aother o a state sace Marov rocess is a sythesis of movemets betwee states to describe the relocatios of members of the trasfer robability matrix to differet states, o the basis of the mobility tred of historical data, MegChua et al, (24) I the resect of orgaizatioal maagemet, umerous revious studies have alied the Marov chai i describig title or level romotios, demotios, or chages of differet career develomet aths to cofirm the actual maower eeds of a orgaizatio or redict the future maower eeds Otimizatio of maower ad forecastig maower eeds i moder coglomerates are essetial art of the future strategic laig ad a very imortat differet ature of busiess imeratives, (Adisa, 25) The Marov chai model allows us to aswer questios from olicy maers For examle, it allows easy comutatio of various statistics at both idividual ad aggregate levels At the idividual level, it ca be used to describe the robabilistic rogressio for a staff at a give career stage At aggregate level, it ca be used to derive iformatio o overall cotiuatio rates ad searatio behavior which are critical iuts i develoig retetio rograms Marov chai model has bee widely used i differet fields icludig Educatio to study studets erolmet roectio both i secodary schools ad tertiary istitutios Educatio system is comarable to a hierarchal orgaizatio i which after a academic sessio, three ossibilities arise i the ew status of the studets; the studets may move to the ext higher class, may reeat the same class, or leave the system successfully as graduates or dro out of the system before attaiig the maximum qualificatio (Nyadwai ad Keedy, 26) Modelig the maower maagemet maily cocers the redictio of future behavior of emloyees, (Rachid ad Mohamed, 23) Tred researchers used Marov chai model associated or itegrated to describe the chage of the rocess i light of its historical evolutios, (Bartholomew, 99) Marov chai is oe of the techiques used i oeratios research with ossibilities view that maagers i orgaizatioal decisio maig bodies use, Hamedet at, (23) Maower laig is the rocess by which the maagemet determies how a *Corresodig author
2 558 orgaizatio should move from its curret maower ositio to its desired maower ositio Through laig, maagemet strives to have the right umber ad right ids of eole, at the right laces at the right time, doig thigs which result i both the orgaizatio ad idividual receivig maximum logru beefits Colligs ad Wood (29) defied maower laig as the rage of hilosohies, tools ad techiques that ay orgaizatio should deloy to moitor ad maage the movemet of staff both i terms of umbers ad rofiles These movemets of staff called trasitios are usually the cosequeces of romotios, trasfer betwee segmets or wastage ad recruitmet ito the system The aroach to maower olicy i most Nigeria uiversities aears to be guided by the traditioal method of uttig the right umber of eole i the right lace at the right time or arragig the suitable umber of eole to be allocated to various obs usually i a hierarchal structure, (Igboaugo ad Oifade, 2) Maower laig ivolves two stages The first stage is cocered with the detailed laig of maower requiremets for all tyes ad levels of emloyees throughout the laig eriod ad the secod stage is cocered with the laig of maower sulies to rovide the orgaizatio with the right tyes of eole from all sources to meet the laig requiremets A adage says, he who fails to la, las to fail The laig rocess is oe of the most crucial, comlex ad cotiuig maagerial fuctios which embraces orgaizatioal develomet, maagerial develomet, career laig ad successio laig The rocess of maower laig may rightly be regarded as a multiste rocess icludig various issues such as ; Decidig goals or obectives, Auditig of iteral resources, Formulatio of recruitmet la, Estimatig future orgaizatioal structure ad maower requiremets ad develoig a huma resource la Effective maower laig is very crucial which orgaizatios, lie large comaies, academic system, federal ad state admiistratio must carryout, sice huma resources are cosidered as the most crucial, volatile ad otetially uredicted resource which a orgaizatio utilizes The redictio of maower is subect to how curret suly of emloyees will chage iterally These chages are observed by aalyzig what haeed i the ast, i terms of staff retetio or movemet, extraolatig ito the future to see what haes with the same tred of the ast Marov chai is a useful tool i redictio ad has bee used extesively i may areas of huma edeavors Rachid ad Mohamed (23) reseted a redictive model of umbers of emloyees i a hierarchal deedet time system of huma resources, icororatig subsystems that each cotais grades of the same family The roosed model was motivated by the reality of staff develomet which cofirms that the ath evolutio of each emloyee is usually i its family of grades Kwo et at, (26) used Maro chai model ad ob coefficiet to ivestigate the differece of maower status betwee US ad Korea uclear idustry ad to redict the future maower requiremets i Korea The worforce laig, o the basis of established rocess, requires a good owledge of those deloyed i the establishmet, as well as etry, droout at romotio of emloyees i order to reach a future la fit ad desired admiistratio i determiig the future olicies of the worforce system, Touama, (25) alied Marovia models ad trasitio robability matrix to aalyze the movemet of the worforce i Jorda roductivity comaies To achieve his aim, he collected secodary data related to worforce movemet selected from aual reorts of Jordaia roductivity comaies (otash, hoshate ad harmaceutical) for year 24 The urose of maower laig is to get a better matchig betwee maower requiremets ad maower availability Parma et al, (23) cosidered a otimizatio model for maower system where vacacies are filled u by romotio ad recruitmet i automatio system egieerig rivate limited They roosed a method for the determiatio of trasitio robability of romotio ad recruitmet vector by usig Marovia theory with certai assumtios The aroach to maower laig i Nigeria uiversities are guided by the traditioal method of uttig the right umber of eole i the right lace at the right time or arragig a suitable umber of eole to be allocated to various obs usually i a hierarchal structure However, this method is difficult i that it does ot offer comutatioal tools that will eable admiistrators to determie ossible lie of actio to be tae or rovides tools to geerate alterative olicies ad strategies Rahela, (25) aalyzed maower data of higher learig istitutio usig Marov chai His obective was to desig a laig model for roectig uiversity faculty emloymet uder alterative olicies suggested by the govermet Wayi ad Shou, (25) focused o the imroved gray Marov model i huma resource iteral suly forecastig, so that eterrises ca reasoably redict their iteral huma resource suly through Marov model ad rovide imortat guaratee for eterrises to develo huma resources strategic laig Ehosuehi, (23) examied the assage of staff i a faculty usig oe state absorbig Marov
3 559 chai He cosidered two cases ivolvig regardless of staff leavig itesio ad staff uwilligess to leave Babu ad Rao, (23) carried out studies o two graded maower model with bul recruitmet i both grades They assumed that the orgaizatio is havig two grades ad recruitmet is doe i both grades i bul Osagiede ad Ehosuehi, (26) reseted the use of Marovia model to roect the future erolmet of studets i a uiversity where they removed the assumtio of certai costat values i the rate of ew itae by some revious authors ad rovided a better method for calculatig the costat value of icremet i the ew itae The obective of this study is to determie the roortios of staff recruited, romoted ad withdraw from the various grades ad to forecast the academic staff structure of the uiversity i the ext five years MATERIALS AND METHODS Descritio of Study Area: I this aer, the academic staff maower system of uiversity of Uyo is modeled usig Marov chai redictive model The obective of the study is to determie the roortio of staff romoted, withdraw, ad recruited ito the various grades, ad to redict the maower structure of the Uiversity of Uyo for several years to come if the olicy of recruitmet ad romotio remai uchaged I the uiversity, vacacies are filled by romotio from the servig staff or by recruitmet from outside the system Promotio is doe every year ad each staff is romoted every three years after meetig the requiremets for romotio, otherwise, o romotio for that articular staff I the uiversity, we have categories of academic staff They are; Graduate Assistat (GA), Assistat Lecturer (AL), Lecture II (LII), Lecturer I (LI), Seior Lecturer (SL), Associate Professor (AP), Professor (P) These grades form the states of the system These states are mutually exclusive ad collectively exhaustive Each member of staff trasitios from oe grade to the ext higher grade Here, we use first order timeideedet homogeeous Marov chai A Marov chai is a sequece of radom variables X,,, X 2 X with the Marov roerty, amely that the robability of movig to ext 3 state deeds etirely o the reset state, ot o the revious states ie [ X = / X, X,, X ] = P[ X = / X = i], i, 2 + ( ) + The basic maower model is of the form ( t ) = +, i, =,2 + i,, i i= Equatio ( 2) ca be exressed as ( t + ) = + + R ( 3) i i ii This gives the umber of staff i grade at time t + R i ( 2) t + is ( t + ) = ( t + ) = ( t ) ( t ) + i = = i= ( 4) i The the total umber of staff i the system at time NOTATIONS AND THEIR MEANINGS t =,2,,T ; Plaig eriods, with T beig the horizo, the value of t reresets a sessio i =,2,, ; The states of the system which rereset the various grades of academic staff i the uiversity These states are mutually exclusive ad collectively exhaustive t ; The umber of academic staff (romotio flow) who trasitio from grade i to at the th sessio ( ) i ; The umber of staff i grade i at the begiig of t th sessio i R W i Pi ; The umber of recruits ito grade at the begiig of t th sessio ; Wastage from grade i ; The robability of a staff movig from grade i to grade at the begiig of t th sessio = R
4 56 i ; The robability of staff i grade i leavig the system at the t th sessio ; The robability of recruitmet ito grade at the begiig of t th sessio Where is a state outside the system ASSUMPTIONS OF THE MODEL These are the assumtios of the model; There are seve grades of academic staff i the system 2 Promotio is doe every year ad every staff is romoted every three years after obtaiig the required umber of ublicatios, otherwise there is o romotio for that articular staff 3 There is o double romotio ad o demotio, ie =, f i + ad i i 4 The robability of withdrawal of staff ad robability of recruitmet of ew staff are ideedet with robabilities ad i i,, =,2,, resectively That is the robability that staff who leave the system i grade i will be relaced by recruit who eter the system i grade, ie = ( )( ) i i 5 There is recruitmet i all the grades =,2,, such that = 6 The trasitio robability matrix (TPM) is statioary overtime The trasitio robability matrix is gives as 2 P = ( 5) = i W This meas that P is a submatrix i Such that i= ESTIMATION OF TRANSITION PROBABILITY MATRIX t Let ( ) i = be the flow level from grade i to durig the t th eriod The, the distributio is i,, multiomial with robabilities, i, =,2! i = ( ) [ ( ) ( ) ( )] = i t i t, i t,, i t i ( 6) 2 i i=! i= ( ) The lielihood fuctio give by (lidgree, 993) is i = i i, i, =,2,, i ad t =,2,, T ( 7) (Uche ad Ezeue, 99) The distributio is give as Sice we assume that the trasitio robability matrix is statioary over time, the maximum lielihood estimates of ( t i ) is calculated by oolig equatio ( ) 7, that is
5 56 i T i ( 8) t= = T i t= PREDICTIVE MODEL FOR FUTURE ACADEMIC STAFF STRUCTURE Let =,,, 2 t t be a row vector of grade sizes at the begiig of the t th sessio, that is The ( + ) = ( )Q ( 9) where ad = ( t + ) = ( t + ), ( t + ), ( t ) 2 3 Q= [ + ] 2 where, the ca ad bottom bar idicate the exected ad vector resectively sice = ( )( ) i i I order to obtai the arameters of the model, secodary data for academic staff flow (romotios) i the uiversity of uyo, Nigeria for the eriods betwee 2/22 ad 24/25 were collected The data are reseted i the tables below DATA; the tables below are the data for academic staff flow based o recruitmet, romotio ad wastage for each sessio uder study
6 562 Table ; Academic staff flow based o recruitmet, romotio ad wastage for 2/22 sessio W i i R Table 2; Academic staff flow based o recruitmet, romotio ad wastage for 22/ W i i R Table 3; Academic staff flow based o recruitmet, romotio ad wastage for 23/ W i i R Table 4; Academic staff flow based o recruitmet, romotio ad wastage for 24/ W i i R
7 563 Table 5: Pooled academic staff flow based o recruitmet, romotio ad wastage for all the sessios W i i R RESULTS AND DICUSSIONS From the ooled academic staff flow based o recruitmet, romotio ad withdrawal, the trasitio robability matrix P, the wastage robabilities, ad the recruitmet robabilities i are obtaied usig equatio ( 8) as follows, P = = 57 ad i = [ ] Usig the trasitio robability matrix P, the wastage robability ad the recruitmet robability i the matrix Q is obtaied as
8 Q = Usig equatio ( ) , the redicted academic staff structure for the ext five years if the olicy for romotio ad recruitmet remaied uchaged is give i the table below Table 6: The redicted academic staff structure for uiversity of Uyo i the ext five sessios SESSION ( t ) 25/ / / / / The results idicate that grade 7 (rofessor) has the highest wastage robability This could be attributed to their age ad closeess to retiremet, while grades 2 ad 3(Assistat Lecturer ad Lecturer II) have the highest recruitmet robability I matrix Q, the etries i colums 6 of each trasitio matrix arise from relacemet matrix ad recruitmet olicy of the uiversity as ew etrats are recruited ito all the grades That is, i the colums, the etry is the robability that a staff, either reeated gradei, is romoted from grade i to or the staff is recruited ito to relace the leavers i grade i The zero etries idicate that there is o trasitio betwee the corresodig years of romotio The maor diagoal elemets (reetitio robabilities) of matrices P ad Q are relatively large, while, the uer offdiagoal elemets (romotio robabilities) are relatively small This meas that o average, a isigificat umber of staff i each grade is beig romoted every year The reaso could be that most of the academic staff are ot actively ivolved i research to meet with the coditios for romotio or the maagemet of the uiversity is ot much iterested i romotio of staff ad this coditio will ot augur well with the staff i articular ad the system i geeral Coclusio: The redictio made i this research has show that, there is a steady icrease i the umber of Graduate Assistats, Seior Lecturer ad Associate rofessors, while, there is a steady decrease i the umber of Assistat Lecturer, Lecturer II, Lecturer I, ad Professor i the ext five years However, the system may grow out of roortio if there is o cotrol o recruitmet This research ca be exteded to iclude recruitmet cotrol by lacig cotrol o recruitmet A model for this ca be develoed REFERENCES Adisa, C (25) The Develomet of Maower Modelig ad Otimizatio: A Case Study of Asia Leadig Eergy Coglomerates Global Joural of Huma Resource Maagemet 3(2): Babu, PK;Rao, KS (23)Studies o Two Graded Maower Model with Bul Recruitmet i Both Grades Iteratioal Joural o Huma Resource Maagemet Research, (2); 5 Bartholomew, DJ; Forbes, AF; McClea, SI (99) Statistical Techiques for Maower Plaig (2 d ed) Joh Willy & Sos, Chichester Colligs, DG; Wood, G (29) Huma Resources Maagemet; A Critical Aroach Routledge, Lodo Ehosuehi, VU (23) Evolutio of Career Patters of Academic Staff i a Faculty i the Uiversity of Bei, Nigeria Ife Joural of Sciece, 5(): 8 9
9 565 Hamed, AT; Kiyoumars, J; Arash, H; Afshi, AK; Ami, T; Seyed, AA (23) Alicatio of Marov Chai i Forecastig Demad of Tradig Comay Iterdisciliary Joural of Cotemorary Research i Busiess, 5(): 7 74 Igboaugo, AC; Oifade, MK (2)Marov Chai Aalysis of Maower Data of Nigeria Uiversity Joural of Iovative Research i Egieerig ad Sciece, 2(2): 7 23 Kwo, H; Byug, J; EuiJi, L; ByugHoo, Y (26) Predictio of Huma Resource Suly/Demad i Nuclear Idustry Usig Marov Chai Model ad Job Coefficiet Trasactios of Korea Nuclear Society Autum Meetig, Gyeogi, Korea, November 23, 26 Lidgree, BW (993) Statistical Theory (4 th editio) Chama & Hall, New Yor MegChua, T; YaRig, L; PiHeg, C (24)A Study o Predictig the Turover of Nursig Staff of Differet Educatio Bacgrouds; Usig the Absorbig Marov Chai Method Joural of Quality, 2(6): Nyadwai, MJ; Keedy, J (26) Statistical Modelig of Keya Secodary School Studets Erollmet : A Alicatio of Marov Chai Model IOSR Joural of Mathematics, 2(2): 8 Parma, D; Raisighai, S; Mawaa, P (23) Alicatio of Marovia Theory i Maower Plaig: A Case Study Global Research Aalysis, 2(2): Rachid, B; Mohamed, T (23) A Marov Model For Huma Resources Suly Forecast Dividig the HR System ito Subgrous Joural of Service Sciece ad Maagemet, 6: 2 27 Rahela, AR (25) Aalyzig Maower Data of Higher Learig Istitutios; A Marov Chai Aroach Iteratioal Joural of Huma Resource Studies, 5(3): WaYi, D; Shou, L (25)Alicatio of Marov Chai Model i Huma Resource Suly Forecastig i Eterrises Iteratioal Coferece o Comutatioal Sciece ad Egieerig (ICCSE, 25) Touama, HY (25) Alicatio of Marovia Models ad Trasitio Probabilities Matrix to Aalyze the Worforce Movemet i Jordia Productivity Comaies Idia Joural of Research, 4(6): Uche, PI; Ezeue, PO (99)The Marovia Model of academic Staff Flow i Tertiary Istitutio Joural of Nigeria Statistical Associatio, 6 (): Osagiede, AA; Ehosuehi, VU (26)Marovia Aroach to School Erolmet Proectio Process Global Joural of Mathematical Sciece, 5(): 7