P. Arokkia Saibe, B. Esther Clara Department of Mathematics, Bishop Heber College, Trichy , Tamil Nadu, India.

Transcription

1 Intenational Jounal o Computational and Applied Mathematics. ISSN olume, Numbe 3 (07), pp Reseach India Publications Stochastic Models o time to Recuitment in a Single Gade Manpowe Sstem with Coelated Wastages due to Exits and Dieent Modiied Renewal Pocesses o Beaking Decisions and Involunta Exits P. Aokkia Saibe, B. Esthe Claa Depatment o Mathematics, Bishop Hebe College, Tich-6007, Tamil Nadu, India. Abstact: Conside an single gaded maketing oganization in which depletion o manpowe takes place due to decisions, exit o pesonnel etc. These depletion causes thee independent dieent souces o loss o manpowe, which is classiied into volunta exit, involunta exit and equent beaks taken b the existing wokes in the oganization. It is assumed that the loss o manpowe due to volunta exit ae coelated. B assuming that the inte-involunta exit times, inte-beaking decision times oms dieent modiied enewal pocesses, mean and vaiance o time to ecuitment ae detemined o two dieent cases o inte-volunta exit times using a univaiate CUM polic o ecuitment. The esults ae numeicall illustated b assuming speciic distibutions and elevant conclusions ae pesented. Kewods: Single gade manpowe sstem, Coelated loss o manpowe, Inte-volunta exit times, Inte-involunta exit times, Inte-beaking decision times, Dieent enewal pocess, Modiied Renewal pocess, Univaiate CUM Polic. AMS Subject Classiication (00): Pima: 90B70, Seconda: 60H30, 60K05. INTRODUCTION: Attition, which leads to depletion o manpowe, is a common phenomenon in an maketing oganization. Wheneve this oganization announces decisions egading sales taget, evision o wages, incentives and pequisites, these decision causes exit

2 944 P. Aokkia Saibe and B. Esthe Claa (volunta, involunta) o pesonnel om the oganization. Anothe wa o depletion ma also be due to the existing pesonnel in the oganization when the takes beak. Thus the thee souces o depletion ae due to volunta exit decisions (e.g. quitting the job, volunta tanse etc.), involunta exit decisions (e.g. dismissal, peiodic tanse etc.) and beaking decisions (e.g. health illness, celebations etc.). Hence it would not be ealistic b combining the depletion poduced b volunta, involunta exit decisions and beaking decisions to om a single souce o depletion. The loss o manpowe due to these decisions will advesel aects the sales tunove o the oganization. Fequent ecuitment is not advisable as it will be expensive due to the cost o ecuitment. As the loss o manpowe is unpedictable, a suitable ecuitment polic has to be designed to ovecome this loss. A univaiate CUM polic o ecuitment [0] based on, the eplacement polic associated with the shock model appoach in eliabilit theo is stated as ollows: Recuitment is made wheneve the cumulative loss o man powe exceeds the beakdown theshold. Seveal models o manpowe sstem have been poposed and studied b man authos [4], [5] and [6] extensivel in the past. Moe speciicall [0] have initiated the stud on inding the expected time to ecuitment o a single gade manpowe sstem using shock model appoach in eliabilit theo b consideing that attition is geneated b polic decisions. [7] have deived the vaiance o time to ecuitment when the beakdown theshold has two components. Then [] have studied the poblem o time to ecuitment with coelated inte decision times and [8] have studied the same poblem when inte decision times oms geometic pocess. Then [] have deived the vaiance o time to ecuitment o coelated wastages. Late [] have studied the poblem o time to ecuitment, when the depletion o manpowe is classiied into exit o pesonnel om the oganisation and equent beaks taken b the existing wokes in the oganisation b consideing coelated wastages due to exits. Depletion o manpowe due to exits can be involunta and volunta exit. It is not so ealistic to assume this exit o pesonnel as a single souce o depletion. Hence ecentl [3] have studied the poblem o time to ecuitment b classiing the exit o pesonnel into volunta and involunta exit and the beaking decision oms modiied enewal pocess in the sense that beoe thee is an loss in manpowe due to beaks, eve beaking decision (decisions such as postponing beaking decisions due to an issues in the oganisation) is associated with the pobabilit 0<p< to poduce loss o manpowe. Ate the ist occuence o loss due to beaks p changes to. Consideing the assumption o modiied enewal pocess onl to the beaking decisions will not be suicient. Since involunta exit decisions (decisions such as iing a pesonnel duing wokload peiod, tanseing the pesons duing the poject peiod etc.) has been associated to a pobabilit 0<q <, beoe the occuence o loss o manpowe. Ate the ist occuence o loss o manpowe pobabilit q equals to. Hence in this pape it is assumed inte-involunta exit decisions and inte beaking decisions oms a modiied enewal pocess. The poblem o time to ecuitment in this pape is

3 Stochastic Models o time to Recuitment in a Single Gade constucted accoding as the inte-volunta exit times ae in the ollowing cases: (Case-I) as a sequence o exchangeable and constantl coelated exponential andom vaiables and (Case-II) as a geometic pocess. Mean and vaiance o time to ecuitment ae obtained using an univaiate CUM polic o ecuitment b assuming speciic distibution o the loss o manpowe and thesholds. The esults ae numeicall illustated and speciic conclusions ae made. MODEL DESCRIPTION: Conside an oganization with single gade in which exit o pesonnel takes place eithe b volunta and involunta exit o pesonnel om the oganization o due to beaks taken b the existing wokes in the oganization. Let Iq, q =,,3, be a sequence o andom vaiables denoting the loss o manpowe due to the q th involunta exit o pesonnel om the oganization with Laplace tansom I (. ) with paamete α, α > 0. Let, =,,3, be a sequence o exponential andom vaiables denoting the loss o manpowe due to the th volunta exit o pesonnel om the oganization with Laplace tansom (. ) with paamete α, α > 0, coelation ρ and the elation v = α ( ρ ). Let Y l, l =,,3, be a sequence o exponential andom vaiables denoting the loss o manpowe due to the l th beak taken b the existing pesonnel om the oganization with Laplace tansom g Yl (. ) with paamete γ, γ > 0. The loss o manpowe ae assumed to be linea and cumulative. Let Iq be the cumulative loss o manpowe in the ist q involunta exits o pesonnel om the oganization. Let be the cumulative loss o manpowe in the ist volunta exits o pesonnel om the oganization. Let Y l be the cumulative loss o manpowe in the ist l beaks. Let U Iq q =,,3 be independent and identicall distibuted exponential andom vaiables epesenting the time between q- th and q th involunta exit o pesonnel om the oganization with mean, λ λ > 0 and oms a modiied enewal pocess with paamete λ, (λ = q λ ), λ > 0 and 0<q < in the sense that, beoe thee is an loss in manpowe, eve involunta exit decision has a ixed pobabilit q o causing loss in manpowe in the oganization. Ate the ist occuence o loss in manpowe, q changes to. Let U =,,3, be independent exponential andom vaiables epesenting the time between - th and th volunta exit o pesonnel om the oganization with mean λ, λ > 0. Let l l =,,3, be independent and identicall distibuted exponential andom vaiables epesenting the time between l- th and l th beak with mean, β β > 0 and oms modiied enewal pocess with paamete β, (β = pβ ), β > 0 and 0<p < in the sense that, beoe thee is an loss in manpowe, eve beaking decision has a ixed pobabilit p o causing loss in manpowe in the oganization. Ate the ist occuence o loss in manpowe, p changes

4 946 P. Aokkia Saibe and B. Esthe Claa to. Let N(t) be the numbe o involunta exit decisions in (0,t], N(t) be the numbe o volunta exit decisions in (0,t] and N(t) be the numbe o beaking decisions in (0,t]. Let Z be the exponential theshold o the cumulative loss o manpowe due to involunta exits o pesonnel om the oganization with paamete θ > 0, Z be the exponential theshold o the cumulative loss o manpowe due to volunta exits o pesonnel om the oganization with paamete θ > 0 and Z be the exponential theshold o the cumulative loss o manpowe due to beaks with paamete θ > 0. Let Z+Z+Z be the beakdown theshold o the oganization. Let W be the time to ecuitment o the oganization with the distibution unction L(.), densit unction l(.) and the Laplace tansom l (.). It is assumed that the loss o man hous due to involunta exits, volunta exits and beaks, inte-involunta and volunta exit times, inte-beaking decision times and beakdown theshold ae stochasticall independent. ANALYTICAL RESULTS: Let the event {W > t} be the time to ecuitment which occus beond the time t, and { IN (t) + N (t) + Y N (t) < Z + Z + Z} epesent the event that the cumulative loss o man hous due to the thee tpes o decisions does not coss the beakdown theshold upto the time t. It is poved that the occuence o these two events ae equal. Hence {W > t} { IN (t) + N (t) + Y N (t) < Z + Z + Z} Now conditioning upon N (t), N (t) and N (t) b using the esult o enewal theo [9], the distibution unction and densit unction o time to ecuitment ae deived. Hence the th moment o the time to ecuitment is detemined b taking the th deivative with espect to s o the Laplace tansom o the time to ecuitment at s = 0. THEOREM- Let Z i, i =,,3, k be a sequence o exchangeable and constantl coelated exponential andom vaiables with the coelation ρ, ρ [,], mean u and the elation v = u(- ρ). I the pobabilit densit unction o Zi i =,,3, k is t u e, u 0,0 t and the k-old convolution o the distibution o these andom u

5 Stochastic Models o time to Recuitment in a Single Gade t u e z dz k i0 k j0 ( k i )! vaiables is z ki i ki t u 0 ZK ( t) e then the Laplace Stieltje s tansom o is Z K k ( )( vs ) (s) ( )( vs ) kvs Poo: B taking the Laplace tansom and using the elation o Laplace tansom and the Laplace Stieltje's tansom, Z v ( s v ( k i j )! i ki s( ) ( k i j )! K (s) k i j k i0 k s j0 ki j Simpliing the above expession, then the Laplace Stieltje's tansom o Zk(t) is educed i to ( ) ( ki) ZK (s) ( vs) k i0 k Z () K t Doing uthe simpliications the Laplace Stieltje's tansom is poved. Henceoth the moments ae deived b taking the th deivative with espect to s o the Laplace Stieltje's tansom o Zk(t) at s=0. Case-I: In case-i inte-volunta exit times ae assumed to om a sequence o exchangeable and constantl coelated exponential andom vaiables with mean λ, coelation ρ with the elation v = λ (- ρ) Using the Theoem- taking deivative with espect to s o the Laplace tansom o W at s=0 gives the mean time to ecuitment o the pesent case. E(W) = C D CD + C3D3 () whee

6 948 P. Aokkia Saibe and B. Esthe Claa D ( ) ( ' (0) ' (0)) ( ) (( ( )) ( ( ))) ( ) U U U U (( ( )) ( ( ))) ( ) (( ( )) ( ( ))) ( ) U U U U 6 (( U ( )) ( ( U ))) ( ) (( (( ))) ( (( )))) ( ) U U ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) ( ) ( ) 7 8 U U U U ( ) 9 ( ( U (( ))) ( (( )))) ( ) U D ( ) ( ' (0) ' (0)) ( ) (( ( )) ( ( ))) ( ) U U U U (( ( )) ( ( ))) ( ) (( ( )) ( ( ))) ( ) U U U U (( ( )) ( U U 6 ( ) U U 7 8 U U U U ( ))) ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) ( ) ( ) 9 ( (( (( ))) ( (( )))) ( ) U U ) D ( ) ( ' (0) ' (0)) ( ) (( ( )) ( ( ))) ( ) 3 U U U U (( ( )) ( ( ))) ( ) (( ( )) ( ( ))) ( ) U U U U 6 (( U ( )) ( ( U ))) ( ) (( (( ))) ( (( )))) ( ) U U ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) ( ) ( ) 7 8 U U U U 9 ( ) (( (( ))) ( (( )))) ( ) U U Now dieentiating twice the Laplace tansom o W with espect to s and at s = 0, the second moment o W is detemined. Fom these esults the vaiance o time to ecuitment o the pesent case is detemined. Theoem-: Let Z k be a sequence o non-negative andom vaiables and a be a positive constant. I a nomalized stochastic pocess { k } k=, whee k =a n Z k, k=,,3, is a geometic pocess with a paamete a, then the Laplace tansom o k is k s K (s) a

7 Stochastic Models o time to Recuitment in a Single Gade Poo: The distibution unction k(t) = F(a k- t) k =,,3, is detemined om the deinition o geometic pocess and b dieentiating the distibution unction with espect to t, the densit unction o vk(t) = a k- (a k- t) k =,, 3, is deived. Now using the popet, Laplace tansom o the convolution o andom vaiables is the poduct o thei Laplace tansoms, the Laplace tansom o vk(t) is deived. Hence the theoem is poved. Coolla: I a =, then the sequence o andom vaiables k, k =,, 3 oms an odina enewal pocess. Remak: I a >, { k } k= is stochasticall deceasing and when 0 < a <, { k } k= oms a stochasticall inceasing sequence. Case-II: In this case inte-volunta exit times ae assumed to om a geometic pocess with a paamete a > 0. Using Theoem-, the mean time to ecuitment o Case-III is detemined b taking ist deivative with espect to s o the Laplace tansom o W at s=0. E(W) = C D4 CD5 + C3D6 () D ( ) ( ' (0) ' (0)) ( ) (( ( )) ( ( ))) ( ) 4 U U U U (( ( )) ( ( ))) ( ) (( ( )) ( ( ))) ( ) U U U U 6 (( U ( )) ( ( U ))) ( ) (( (( ))) ( (( )))) ( ) U U ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) ( ) ( ) 7 8 U U U U ( ) 9 ( ( U (( ))) ( U (( )))) ( ) D ( ) ( ' (0) ' (0)) ( ) (( ( )) ( ( ))) ( ) 5 U U U U (( ( )) ( ( ))) ( ) (( ( )) ( ( ))) ( ) U U U U (( ( )) ( U U 6 ( ) U U 7 8 U U U U ( ))) ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) ( ) ( ) 9 ( (( (( ))) ( (( )))) ( ) U U )

8 950 P. Aokkia Saibe and B. Esthe Claa D ( ) ( ' (0) ' (0)) ( ) (( ( )) ( ( ))) ( ) 6 U U U U (( ( )) ( ( ))) ( ) (( ( )) ( ( ))) ( ) U U U U 6 (( U ( )) ( ( U ))) ( ) (( (( ))) ( (( )))) ( ) U U ( ) (( (( ))) ( (( )))) ( ) (( (( ))) ( (( )))) ( ) ( ) ( ) 7 8 U U U U 9 ( ) (( (( ))) ( (( )))) ( ) U U Taking second deivative o the Laplace tansom o W with espect to s and at s = 0, the second moment o W is deived. Fom these two esults the vaiance o time to ecuitment is detemined o the pesent case. Note: Geometic pocess assumed o the sequence o andom vaiables in the intevolunta exit times educed to an odina enewal pocess when a=. Then the esults o an odina enewal pocess can be educed om Eqn. (). The expessions o the notations used in the equations () and () ae given below. A, A, A, A, A, A, A, A, A h h s s h A h A s A s A s3 s3 4s4 s3 s3 4s4 h3 h3 4h4 h3 h3 4h4 ( A ) h ( A ) s ( A )( A )s h ) ( A )( A )s ( h h ) ( A )( h h ) ( ) ( A )( A )s ( h h ) ( A )( h h ) ( A )(s s )( A ) h ( A )(s s ) ( ) ( ) ( A )( s s )( A ) h ( A )( s s ) ( A )( A )( h h )(s s ) ( ) ( )( ) ( A )( A )( h h )( s s ) ( A )( A )( h h )(s s ) ( )( ) ( )( ) ( A )( A )( h h )( s s ) ( A ) h ( A ) s ( A )( A )s h ) ( )( ) ( A )( A )s ( h h ) ( A )( h h ) ( A )( A )s ( h h ) ( A6 )( h h ) ( ) ( ) ( A )(s s )( A ) h ( A )(s s ) ( A )( s s )( A ) h ( A )( s s ) ( ) ( ) ( A )( A )( h h )(s s ) ( A )( A )( h h )( s s ) ( )( ) ( )( ) ( A )( A )( h h )(s s ) ( A )( A )( h h )( s s ) ( )( ) ( )( )

9 Stochastic Models o time to Recuitment in a Single Gade.. 95 ( A ) h ( A ) s ( A )( A )s h ) ( A )( A )s ( h h ) ( A )( h h ) ( ) ( A )( A )s ( h h ) ( A )( h h ) ( A )(s s )( A9 ) h ( A7 )(s s ) ( ) ( ) ( A )( s s )( A ) h ( A )( s s ) ( A )( A )( h h )(s s ) ( ) ( )( ) 7 ( A9 )( A7 )( h h )( s s ) ( A9 )( A7 )( h h )(s s ) 8 ( )( ) ( )( ) ( A )( A )( h h )( s s ) C C C ( )( ) ( )( ) ( )( ) ( )( ) , A, A, A3, A4, A5, A6, A7, A8, A9,,,, NUMERICAL ILLUSTRATION: The inluence o nodal paametes on the peomance measues namel mean and vaiance o the time to ecuitment is studied numeicall using MATLAB. The peomance measues ae calculated o all the thee cases. Case-I: Eect o ρ, ρ o the mean and vaiance o time to ecuitment is studied b ixing the value o the paametes q=0.9 p = 0.9 θ = 0.00 θ = 0.0, θ = 0.03, α=0.7, α=0.9, γ=0.08, β=0.9, λ=0.8, λ=0.. ρ ρ E(W) (W) x x x x x x x x x x x x0 5 Case-II: Eect o a, ρ o the mean and vaiance o time to ecuitment is studied b ixing the value o the paametes q=0.9 p = 0.6 θ = θ = 0.05, θ = 0.07, α=0.7, α=0.9, γ=0.08, β=0.9, λ=0.8, λ=0..

10 95 P. Aokkia Saibe and B. Esthe Claa ρ a E(W) (W) x x x x x x x x x x x x x x0 3 CONCLUSION: In Case-(I), i the negative value o coelation (loss due to volunta exit) inceases b ixing the othe paametes the mean time to ecuitment and vaiance o time to ecuitment inceases. When the positive value o coelation (loss due to volunta exit) inceases b ixing the othe paametes the mean time to ecuitment vaiance o time to ecuitment deceases. I the negative value o coelation (inte-volunta exit times) inceases b ixing the othe paametes the mean time to ecuitment and vaiance o time to ecuitment inceases. When the positive value o coelation (intevolunta exit times) inceases b ixing the othe paametes the mean time to ecuitment vaiance o time to ecuitment deceases. In Case-(II), o (a<, a>) inceasing the paamete o geometic pocess b ixing the othe paametes the mean time to ecuitment inceases and vaiance o time to ecuitment inceases. I the negative value o coelation (loss due to volunta exit) inceases b ixing the othe paametes the mean time to ecuitment and vaiance o time to ecuitment inceases. When the positive value o coelation (loss due to volunta exit) inceases b ixing the othe paametes the mean time to ecuitment vaiance o time to ecuitment deceases. REFERENCES [] Amala Nanc A., Devi A. and Sinivasan A.,(04), A Stochastic model on the time to ecuitment o a single gade manpowe sstem having coelated inte-

11 Stochastic Models o time to Recuitment in a Single Gade decision times Intenational Jounal o Phsics and mathematical Sciences, 4(), pp [] Aokkia Saibe, P. and Esthe Claa, B. (06), vaiance o time to ecuitment Fo a single gade manpowe Sstem with coelated wastages And the beakdown theshold has Two components, Aabhatta Jounal o Mathematics and Inomatics, 8(), pp [3] Aokkia Saibe, P. and Esthe Claa, B. (07), Time to ecuitment with coelated loss o manpowe unde dieent enewal pocess o exit and beaking decisions oms s modiied enewal pocess, Intenational Jounal o Reseach in Advent Technolog, 5(8), pp [4] Batholomew, D.J. (969), Renewal theo models o Manpowe Sstems, in Manpowe eseach, edito N. A. B., Wilson, pp [5] Clough, D.J. Lewis, C. G. Olive, A.L.(974), Manpowe planning models, English Univesit Pess, London. [6] Ginold, R. C. Mashall, K. T. (977), Manpowe planning models, Noth- Holland, New Yok. [7] Jaalakshmi G. and Sinivasan A. (05), Pobabilistic analsis on time to ecuitment o a single gade manpowe sstem when the beakdown theshold has two components using a dieent polic o ecuitment, Indian Jounal o Applied Reseach, 5(7), pp [8] Lalitha R., Devi A. and Sinivasan A., (04), A Stochastic model on the time to ecuitment o a single gade manpowe sstem with attition geneated b geometic pocess o inte decision times, Jounal o Engineeing Compute and Applied Sciences, 3(7), pp. -5. [9] Medhi, J. (009): Stochastic Pocess, Thid edition, New age Intenational Publishes. [0] Sathamoothi, R. Pathasaath, S. (003), On some stochastic models o manpowe planning using SCBZ popet, Ph. D., Thesis at Depatment o Statistics, Annamalai Univesit. [] Sidhaan J., Elangovan K. and Sinivasan A. (05), Expected time to ecuitment in a single gade manpowe sstem unde coelated wastage, Intenational Jounal o Innovative Science, Engineeing and Technolog, (7), pp

12 954 P. Aokkia Saibe and B. Esthe Claa