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A NOVAL APPLICATION OF DUANE PROCESS FOR MODELING TWO GRADED MANPOWER SYSTEM WITH DIRECT RECRUITMENT IN BOTH THE GRADES
Ch. Ganapathi Swamy1, K. Srinivasa Rao 2
department of Community Medicine, GSL Medical College, Rajahmundry, Andhra Pradesh, India. Email: ganesh051981@gmail.com 2Dept.of statistics, Andhra University, Visakhapatnam, Andhra Pradesh, India.
Email: ksraoau@yahoo.co.in
Abstract
Human Resource Management and other companies rely heavily on manpower models. Manpower planning was a prerequisite for effective organization administration. The construction and analysis of two graded manpower models with direct Duane recruiting processes in both graduates is the subject of this paper. Duane's recruitment procedure was capable of identifying time-dependent recruitments. Poisson and non-homogeneous Poisson processes are used in the Duane recruitment process as precise instances for specified parameter values. It is assumed that the organization has two grades and that the recruitment procedure is based on the Duane Process. The processes of leaving and promotion are Poisson processes. The model's transient behavior was investigated by deriving unambiguous expressions for system characteristics such as the mean number of employees in each grade, the mean durational stay of an employee in each grade, and the variance of number of employees in each grade using differential equations. The model's sensitivity analysis of parameters shows that the Duane recruiting process has a substantial impact on system performance indicators. It was also noted that this model incorporates rates of recruitment that are increasing, decreasing, or stable. This model proved helpful in analyzing organizational manpower issues.
Keywords - Two graded Manpower model, Duane process, Time dependent recruitment rate, Sensitivity analysis.
I. Introduction
Planning the organization with the manpower structure in mind was a prerequisite for getting the most out of the resources. Due to their usefulness in creating strategies for Human Resource Development and resource allocation, much work has been reported on manpower models. Seal was a pioneer in the mathematical modelling of labour systems [1]. Silock looked at the observable fact of labour yield, which is related to the study of demography [2]. Bartholomew examined manpower models based on probability distributions of an employee's total service time in the organization [3] [4]. Ugwuowo and Mc Clean, as well as Wang, have examined manpower models and various approaches to their development [5] [6].
The parameters of the manpower model, such as the mean number of employees in each grade, the mean duration of stay an employee in each grade, and the variation of number of employees in each grade, were required for effective analysis and design of manpower systems Srinivasa Rao [7]. Kannan Nilakantan investigated the manpower models staffing rules and their extension to individual outsourcing [8]. Jeeva and Geetha looked at manpower models in a hazy environment [9]. Lalithadevi and Srinivasan used geometric process and shock models to investigate
a single graded manpower system [10]. Osagiede and Ekhosuehi used continuous-time Markov chains and sparse stochastic measures to investigate Manpower models [11].
The graded manpower models with poisson processes have been examined by Srinivasa Rao, K. et al., Kondababu et al., and Govinda Rao et al. They implied that the hiring procedure was timesensitive [12] [13] 14] [15] [16]. Parameswari, K., and Srinivasan [17] used a geometric technique to investigate the reduction in manpower for a two-graded system. Amudha.T and Srinivasan.A, discussed the problem of time to recruitment for a two-graded system, taking into account the loss of personnel in the form of an I.I.D Exponential random variable sequence [18]. Saral, L. et al. established a two-tiered personnel structure and a recruiting policy based on two thresholds [19]. Srividhya, K. et al. investigated manpower loss in a multi-graded organization [20]. Jayanthi et al. (2018) looked at a single graded manpower system and looked at the time to recruitment with a break-even point [21]. Tamas Banyai et al. used Markov chains to study a model for analyzing human resource use [22]. Arokkia Saibe,P et al. investigated two stochastic models based on the assumption that manpower shortages and inter-policy decision delays constitute two distinct sequences of independent and identically distributed random variables with two distinct breakdown thresholds [23]. They assumed that the hiring procedure was time-sensitive.
However, in many real-world circumstances in corporate offices and government agencies, the recruitment process was time-sensitive and did not necessarily follow the Poisson process. As a result, non stationary models must be considered for correct analysis. Srinivasa Rao et al. [24] recently developed two graded manpower models based on non-homogeneous Poisson recruitment. Srinivasa Rao,K et al. [25] have studied on two grade manpower model with Duane recruitment process. They realized that the recruitment rate was linearly proportional to time and that the duration between recruitment was distributed in an exponential manner. However, the recruiting rate in many organizations may increase/decrease/remain steady and time-dependent. The time-dependent non-stationary recruitment process can be fully characterized by the Duane process, which follows a Weibull distribution of inter-recruitment time. Little is reported in the literature on two hierarchical workforce models that use the Duane recruitment process directly at both grades. Therefore, this paper uses the Duane recruitment process of both graduates to develop and analyze a two-step manpower model. Poisson and non-homogeneous processes are two examples of the Duane process. The concept can be applied to a variety of organizations thanks to the recruitment in both grades. The remainder of the paper was laid out as follows:
II. Two graded manpower model with direct recruitment:
Consider a personnel system with two grades, each of which has its own recruitment process. The grade I recruiting process was supposed to be a Duane process, with the mean recruitment rate being a power function of time t and the form A1(t) = a1b1tb1-1. The grade II recruiting process was considered to be a Duane process with a mean recruitment rate of A2(t) = a2b2tb2-1. A Poisson process with parameter is used to promote students from grade I to grade II. Poisson processes with parameters and are used in the grade I and grade II leaving processes, respectively. "Figure 1" depicted a schematic diagram depicting the two-grade manpower concept.
The grade I recruiting process was supposed to be a Duane process, with the mean recruitment rate being a power function of time t and the form A1(t) = a1b1tb1-1. The grade II recruiting process was considered to be a Duane process with a mean recruitment rate of A2(t) = a2b2tb2-1. A Poisson process with parameter is used to promote students from grade I to grade II. Poisson processes with parameters and are used in the grade I and grade II leaving processes, respectively. "Figure 1" depicted a schematic diagram depicting the two-grade manpower concept.
Figure 1: Two grade manpower model with direct Duane recruitment process.
Let Pn,m(t) denote the probability that there are 'n' employees in grade-1 and 'm' employees in grade-2 at time 't' in the organization. Then the difference-differential equations of the model are
¿Pa m(tJ 3t
- UL (t) + Л , Сt) + na + nß + my] Pnim(t} + [lL 0)] Pn _km (t)
+ [Л, GOlP^^ft) + «Ol + l)Pn + Un-i(t} +[3(n + l)Pn+Vn(t) +(m +■ 1)уРп m+i(t)
(1)
ÎPD ;:tJ
э г
= -UL (Ö +Л3 fc> + па + nß] Pn D(t) -h Ui (i)]Pn_i,D(Ö+|3(n + l)Pn+lj0(t> + yPE LW
(2)
¿Pim'tJ
™ = -tiL (Ö + Л2 (t) + +my]PD^(t> + Uï (i}]PDjn_L(Ô + Pi^it) + (m + l>yPD>m+L(lD
Let P(S1,S2; t) be the joint probability generating function then
(3)
(4)
P(SL,S;; t) =InYnSnLS? Pn,m(t-) (5)
Multiplying the equations (1) to (4) with corresponding SJS? and summing over all n=0,1,2,...; m= 0,1,2,...; we get
-1) +I2(£)CÎ2- D]P
After simplification, we get
Solving the equation (7) by Lagrangian's method, the auxiliaiy equations are
1 a (.S ±— V3 II +ji(.S'3 -Li f Oj - LI [¿! (rt Ls"i - lJ +i3 (rf Cj3 - 0]P
(6)
(7)
(8)
With the initial conditions that there are N employees in grade-1 and M employees in grade-2 in the organization at time t=0. i.e., Pv.jh№ = land Pz =0 fort>0.
To solve the equation (8) the functional forms of A1(t) and A2(t) are required.
Since the recruitment processes follow Duane processes we have the mean recruitment rates of grade-1and grade-2 in the system are A1(t) =a1b1t b1-1 and A1(t) = a2b2tb2-1 respectively, where A>0, a1, b1, a2 and b2 are constants Solve the equations in (8) we get A = (S: - 1)*-*
where Sf = 1 - (1 - - ^jJ [H-<*+№ _ ff-rt]j , s? = [ 1 - (1 - S^"*]»
A,B and C are arbitrary constants. (9)
The joint probability generating function of the number of employee in grade-1 and in grade-2 is PCS^S,; t) = C.Exp [aLbL(SL - 1} [e"^ * a dv]
+ azb2 (Sz - 1) [e-r* ¿e*9
(10)
Substituting the value of 'C' from equation (9) in the equation (10), the joint probability generating function of the number of employees in the grade-1 and grade-2 are obtained as
Where X = = 1 - U -
Y = S« = [l-Cl ISj < lJSjl < 1
III. Characteristics of the model:
(11)
The characteristics of the model are obtained by using the equation (11).Expanding PCSL.S;; t)and collecting the constant terms, we get the probability that there is no employee in the
organization as
P0J>($ =
Where, XL
{e*p {OLbL (-1) jf +
a^ (-1) [Bjf - [XL Yl]\
(12)
Taking S; = 1 in equation (11), we get the probability generating function of the number of employees in grade-1 as
P(SL;0 = [[1-&-S^-W]"] [^[Mi&i
SJ<1 (13)
Expanding P(SL;t) and collecting the constant terms, we get the probability that there is no employee in grade-1 as
P0(t) = [[l - [eap [a.b, (-1) [e-i-P^V-PK^dv
1
The mean number of employees in grade-1 is
The probability that there is at least one employee in grade-1 is UL(t) = l-fjjCt)
(14)
(15)
(16)
The average duration of stay of an employee in grade-1 is
H'VCfj
[№ -(« +JD"] +aibl ffte;n[, l- *■ - t^ ^ 1j
(jt+JJ.I 1 - [[l-r^fl*]"] [«cp [a i- iJ [e-t^^V^^JVvfhi - LJdv .
^♦fl J
(17)
The variance of the number of grade-1 is
VL{t) = [jVo-^^JLl - s-^JtJ -Hfl^ [B-^^1 Sle^P* v&i-Vdv -
(c+ji.i
Coefficient of variation of the number of employees in grade-1 is «^(jt) = \ where, Vi(t) and Li(t)
are given in equations (18) and (15) respectively (19)
Similarly, taking Sj^ = 1 in equation (11), then we get the probability generating function of the
number of employees in grade-2 as aI b, a (S- — U
V^'i
Y - Gœ + ß)
+ r-ta+P I Г J r-(ir+JîJ I y Jj
,i.a+0)vvib1-l)dv_e-rt Г
Jn
1 _ gtL-Ла Jr (g+ftlr
r+JJ.I
;; 1 — Cl — ^>8-^]*] where, Is.I < 1 (20) Expanding P(S, ;tj and collecting the constant terms, we get the probability that there is no employee in grade-2 as
The mean number of employees in grade-2 is
Ln (f) = Me~Yt +N
Y-ic+ß. I
(22)
The probability that there is at least one employee in grade-2 is f/2Ct) = l - P0 (t)
The average duration of stay of employees in grade-2 is
The variance of the number of employees in grade-2 is lUt) = Ate-^tl -e-n
(24)
Coefficient of variation of the number of employees in grade-2 is
where,V2(t) and L2(t) are given in equations (25) and (22) respectively.
(25)
(26)
The mean number of employees in the organization is L =L1+ L2 where, L1(t) and L2(t) are given in
equations (15) and (22) respectively.
(27)
IV. Numerical illustration and results
A numerical illustration was used to explain the model's behaviour in this subdivision. For the recruiting, advancement, and leaving rates of the organization, several values of the parameters were explored. Because the manpower model's performance characteristics were particularly timesensitive, the transient behaviour of the model was investigated by computing performance measures with the following set of values for the model parameters: t= 1.5,2,2.5,3and 3.5; a=3,4,5,6 and 7; P=4,5,6,7 and 8; y=5,6,7,8 and 9; ai=5,10,15,20 and 25; bi=3,4,5,6 and 7; a2=5,10,15,20 and 25; b2=3,4,5,6 and 7; N=1000,1100,1200,1300 and 1400 ; M=600,700,800,900 and 1000.
Performance measures such as the mean number of employees in grades I and II, the mean duration of stay of a grade I employee and in grade II, the variance of the number of employees in grades I and II, and the coefficient of variation of the number of employees in grades I and II were computed and presented in Tables 1 and 2. Figures 2a, 2b, 3a and 3b show the link between parameters and performance measures.
Table 1 demonstrated that the performance indicators in grades I and II were extremely time sensitive. The mean number of employees in grade I and grade 2 in the organization increased from 4.0180 to 24.1946 and 8.1050 to 45.6831, respectively, while time (t) varied from 1.5 to 3.5. When all other factors were held constant, the mean period of stay of an employee in grade I and grade II in the company increased from 0.5845 to 3.4564 and 1.7138 to 9.1367, respectively.
When all other parameters were held constant, the mean number of employees in grade I was not influenced and in grade II it increased from 8.4203 to 8.6416.When all other parameters were held constant, the mean duration of stay of an employee in grade I was not influenced and in grade II it increased from 1.7533 to 1.7848.
When all other parameters were held constant, the mean number of employees in grade I was not influenced and in grade II it increased from 8.4203 to 8.6416.When all others parameters were held constant, the mean duration of stay of an employee in grade I was not influenced and in grade II it increased from 1.7533 to 1.7848.
Table 1: Values of L1, L2, W1 and W2 for different values of parameters.
t a 3 Y a1 b1 a2 b2 N M L1 L2 W1 W2
1.5 3 4 5 5 3 5 3 1000 600 4.0180 8.1050 0.5845 1.7138
2 3 4 5 5 3 5 3 1000 600 7.4352 13.5421 1.0628 2.7769
2.5 3 4 5 5 3 5 3 1000 600 11.9497 22.0582 1.7071 4.4219
3 3 4 5 5 3 5 3 1000 600 17.5364 32.7960 2.5052 6.5599
3.5 3 4 5 5 3 5 3 1000 600 24.1946 45.6831 3.4564 9.1367
1.5 3 4 5 5 3 5 3 1000 600 4.0180 8.1050 0.5845 1.7138
1.5 4 4 5 5 3 5 3 1000 600 3.5804 8.4013 0.4604 1.7508
1.5 5 4 5 5 3 5 3 1000 600 3.2370 8.6505 0.3744 1.7862
1.5 6 4 5 5 3 5 3 1000 600 2.9553 8.8632 0.3118 1.8188
1.5 7 4 5 5 3 5 3 1000 600 2.7189 9.0462 0.2646 1.8483
1.5 7 5 5 5 3 5 3 1000 600 2.5174 8.7440 0.2282 1.8003
1.5 7 6 5 5 3 5 3 1000 600 2.3435 8.4931 0.1994 1.7634
1.5 7 7 5 5 3 5 3 1000 600 2.1921 8.2806 0.1763 1.7350
1.5 7 8 5 5 3 5 3 1000 600 2.0589 8.0977 0.1573 1.7130
1.5 7 8 6 5 3 5 3 1000 600 2.0589 6.5928 0.1573 1.2559
1.5 7 8 7 5 3 5 3 1000 600 2.0589 5.7212 0.1573 0.9933
1.5 7 8 8 5 3 5 3 1000 600 2.0589 5.0994 0.1573 0.8147
1.5 7 8 9 5 3 5 3 1000 600 2.0589 4.6108 0.1573 0.6856
1.5 7 8 9 10 3 5 3 1000 600 4.1178 5.9837 0.2791 0.7104
1.5 7 8 9 15 3 5 3 1000 600 6.1767 7.3565 0.4126 0.8309
1.5 7 8 9 20 3 5 3 1000 600 8.2356 8.7293 0.5492 0.9739
1.5 7 8 9 25 3 5 3 1000 600 10.2944 10.1021 0.6863 1.1236
1.5 7 8 9 25 4 5 3 1000 600 19.7548 15.5521 1.3170 1.7280
1.5 7 8 9 25 5 5 3 1000 600 35.6026 24.0877 2.3735 2.6764
1.5 7 8 9 25 6 5 3 1000 600 61.6965 37.3243 4.1131 4.1471
1.5 7 8 9 25 7 5 3 1000 600 104.0989 57.6922 6.9399 6.4102
1.5 7 8 9 25 7 10 3 1000 600 104.0989 60.9278 6.9399 6.7698
1.5 7 8 9 25 7 15 3 1000 600 104.0989 64.1634 6.9399 7.1293
1.5 7 8 9 25 7 20 3 1000 600 104.0989 67.3990 6.9399 7.4888
1.5 7 8 9 25 7 25 3 1000 600 104.0989 70.6346 6.9399 7.8483
1.5 7 8 9 25 7 25 4 1000 600 104.0989 84.7664 6.9399 9.4185
1.5 7 8 9 25 7 25 5 1000 600 104.0989 107.9303 6.9399 11.9923
1.5 7 8 9 25 7 25 6 1000 600 104.0989 145.3699 6.9399 16.1522
1.5 7 8 9 25 7 25 7 1000 600 104.0989 205.2306 6.9399 22.8034
1.5 3 4 5 5 3 10 3 1000 600 4.0180 8.1050 0.5845 1.7138
1.5 3 4 5 5 3 10 3 1100 600 4.0208 8.1838 0.5849 1.7230
1.5 3 4 5 5 3 10 3 1200 600 4.0235 8.2627 0.5853 1.7327
1.5 3 4 5 5 3 10 3 1300 600 4.0263 8.3415 0.5856 1.7428
1.5 3 4 5 5 3 10 3 1400 600 4.0290 8.4203 0.5860 1.7533
1.5 3 4 5 5 3 10 3 1400 700 4.0290 8.4756 0.5860 1.7610
1.5 3 4 5 5 3 10 3 1400 800 4.0290 8.5309 0.5860 1.7688
1.5 3 4 5 5 3 10 3 1400 900 4.0290 8.5863 0.5860 1.7767
1.5 3 4 5 5 3 10 3 1400 1000 4.0290 8.6416 0.5860 1.7848
The performance metrics in grade I and grade II employees in the organization were extremely sensitive to time, as shown in Table 2. When other parameters were held constant, it was discovered that time (t) varies from 1.5 to 3.5, the variance of the number of employees in grade I and grade II increased from 4.0180 to 24.1946 and 8.1042 to 45.6831, respectively, and the coefficient of variation of the number of employees in both grades decreased from 0.4989 to 0.2033 and 0.3512 to 0.1480.
When the promotion rate (a) from grade I to grade II increased from 3 to 7, the variance of the number of employees in grade I decreased from 4.0180 to 2.7189 and increased from 8.1042 to 9.0456, and the coefficient of variation of the number of employees in grade I increased from 0.4989 to 0.6065 and decreased from 0.3512 to 0.3325, when all other parameters remained constant.
Figure 2a: Relation between the parameters and performance measures
Figure 2b: Relation between the parameters and performance measures
When the leaving rate (p) of an employee in grade I increases from 4 to 8, the variance of the number of employees in grade I and grade II decreases from 2.7189 to 2.0589 and 9.0456 to 8.0974, respectively, while the coefficient of variation of the number of employees in both grades I and II increases from 0.6065 to 0.6969 and 0.3325 to 0.3514, respectively, when other parameters remain constant.
When the leaving rate (y) of an employee in grade II increases from 5 to 9, the variance of the number of employees in grade I is unaffected, while in grade II it decreases from 8.0974 to 4.6108. When other parameters are held constant, the coefficient of variation of the number of employees in grade I is unaffected, while in grade II it increases from 0.3514 to 0.4657. When the recruitment rate parameter (a1) of employees in grade I was changed from 5 to 25, the variance of the number of employees in grade I and grade II increased from 2.0589 to 10.2944 and 4.6108 to 10.1021 respectively,
while the coefficient of variation of the number of employees in both grade I and grade II decreased from 0.6969 to 0.3117 and 0.4657 to 0.3146 when the other parameters remained constant.
Table 2: Values of V1,V2, CV1 and CV2 for different values of parameters.
T a [3 Y a1 b1 a2 b2 N M V1 V2 CV1 CV2
1.5 3 4 5 5 3 5 3 1000 600 4.0180 8.1042 0.4989 0.3512
2 3 4 5 5 3 5 3 1000 600 7.4352 13.5421 0.3667 0.2717
2.5 3 4 5 5 3 5 3 1000 600 11.9497 22.0582 0.2893 0.2129
3 3 4 5 5 3 5 3 1000 600 17.5364 32.7960 0.2388 0.1746
3.5 3 4 5 5 3 5 3 1000 600 24.1946 45.6831 0.2033 0.1480
1.5 3 4 5 5 3 5 3 1000 600 4.0180 8.1042 0.4989 0.3512
1.5 4 4 5 5 3 5 3 1000 600 3.5804 8.4006 0.5285 0.3450
1.5 5 4 5 5 3 5 3 1000 600 3.2370 8.6498 0.5558 0.3400
1.5 6 4 5 5 3 5 3 1000 600 2.9553 8.8626 0.5817 0.3359
1.5 7 4 5 5 3 5 3 1000 600 2.7189 9.0456 0.6065 0.3325
1.5 7 5 5 5 3 5 3 1000 600 2.5174 8.7435 0.6303 0.3382
1.5 7 6 5 5 3 5 3 1000 600 2.3435 8.4927 0.6532 0.3431
1.5 7 7 5 5 3 5 3 1000 600 2.1921 8.2802 0.6754 0.3475
1.5 7 8 5 5 3 5 3 1000 600 2.0589 8.0974 0.6969 0.3514
1.5 7 8 6 5 3 5 3 1000 600 2.0589 6.5928 0.6969 0.3895
1.5 7 8 7 5 3 5 3 1000 600 2.0589 5.7212 0.6969 0.4181
1.5 7 8 8 5 3 5 3 1000 600 2.0589 5.0994 0.6969 0.4428
1.5 7 8 9 5 3 5 3 1000 600 2.0589 4.6108 0.6969 0.4657
1.5 7 8 9 10 3 5 3 1000 600 4.1178 5.9837 0.4928 0.4088
1.5 7 8 9 15 3 5 3 1000 600 6.1767 7.3565 0.4024 0.3687
1.5 7 8 9 20 3 5 3 1000 600 8.2356 8.7293 0.3485 0.3385
1.5 7 8 9 25 3 5 3 1000 600 10.2944 10.1021 0.3117 0.3146
1.5 7 8 9 25 4 5 3 1000 600 19.7548 15.5521 0.2250 0.2536
1.5 7 8 9 25 5 5 3 1000 600 35.6026 24.0877 0.1676 0.2038
1.5 7 8 9 25 6 5 3 1000 600 61.6965 37.3243 0.1273 0.1637
1.5 7 8 9 25 7 5 3 1000 600 104.0989 57.6922 0.0980 0.1317
1.5 7 8 9 25 7 10 3 1000 600 104.0989 60.9278 0.0980 0.1281
1.5 7 8 9 25 7 15 3 1000 600 104.0989 64.1634 0.0980 0.1248
1.5 7 8 9 25 7 20 3 1000 600 104.0989 67.3990 0.0980 0.1218
1.5 7 8 9 25 7 25 3 1000 600 104.0989 70.6346 0.0980 0.1190
1.5 7 8 9 25 7 25 4 1000 600 104.0989 84.7664 0.0980 0.1086
1.5 7 8 9 25 7 25 5 1000 600 104.0989 107.9303 0.0980 0.0963
1.5 7 8 9 25 7 25 6 1000 600 104.0989 145.3699 0.0980 0.0829
1.5 7 8 9 25 7 25 7 1000 600 104.0989 205.2306 0.0980 0.0698
1.5 3 4 5 5 3 10 3 1000 600 4.0180 8.1042 0.4989 0.3512
1.5 3 4 5 5 3 10 3 1100 600 4.0208 8.1830 0.4987 0.3495
1.5 3 4 5 5 3 10 3 1200 600 4.0235 8.2617 0.4985 0.3479
1.5 3 4 5 5 3 10 3 1300 600 4.0263 8.3405 0.4984 0.3462
1.5 3 4 5 5 3 10 3 1400 600 4.0290 8.4193 0.4982 0.3446
1.5 3 4 5 5 3 10 3 1400 700 4.0290 8.4746 0.4982 0.3435
1.5 3 4 5 5 3 10 3 1400 800 4.0290 8.5298 0.4982 0.3424
1.5 3 4 5 5 3 10 3 1400 900 4.0290 8.5851 0.4982 0.3412
1.5 3 4 5 5 3 10 3 1400 1000 4.0290 8.6404 0.4982 0.3402
When the recruitment rate parameter (b1) in grade I is changed from 3 to 7, the variance of the number of employees in grade I and grade II increases from 10.2944 to 104.0989 and 10.1021 to
57.6922, respectively, while the coefficient of variation of the number of employees in both grades I and II decreases from 0.3117 to 0.0980 and 0.3146 to 0.1317, respectively, when the other parameters remain constant.
When the recruitment rate parameter (a2) in grade II changes from 5 to 25, the variance of the number of employees in grade I does not change and in grade II it increases from 57.6922 to 70.6346, while the coefficient of variation of the number of employees in grade I does not change and in grade II it decreases from 0.1317 to 0.1190.
Figure 3a: Relation between the parameters and performance measures.
Figure 3b: Relation between the parameters and performance measures.
When the recruitment rate parameter (b2) of employees in grade II varies from 3 to 7, the variance of the number of employees in grade I is unaffected, and in grade II it increases from 70.6346 to 205.2306, while the coefficient of variation of the number of employees in grade I is unaffected, and in grade II it decreases from 0.1190 to 0.0698.
When other parameters were held constant, the initial number of employees in grade I (N) varied from 1000 to 1400, the variance of the number of employees in grade I and grade II increased from 4.0180 to 4.0290 and 8.1042 to 8.4193, respectively, and the coefficient of variation of the number of employees in grade I and grade II decreased from 0.4989 to 0.4982 and 0.3512 to 0.3446.
When other parameters were fixed, the variance of the number of employees in grade I was not influenced and in grade II it was increasing from 8.4193 to 8.6404, coefficient of variation of the
number of employees in grade I was not influenced and in grade II it was decreasing from 0.3446 to 0.3402, when the primary number of employees in grade II (M) varies from 100 to 500.
V. Sensitivity analysis of the model
The model was sensitivity tested with respect to the value of time (t), recruitment rates A1(t) and A2(t), promotion rate parameter (a), and leaving parameters(p) and (y) of both grade I and grade II, as well as all other parameters combined on the mean number of employees in grade I and grade II, mean duration of stay of an employee in grade I and grade II, and variance of the number of employees in grade I and grade II.
For different value of t,a,p,y,a1,b1,a2 and b2 the mean number of employees in grade I and in grade II, mean duration of stay of an employee in grade I and in grade II, the variance of the number of employees in grade I and in grade II were computed and presented in Table 3a and 3b with variation of -15%,-10%,-5%,0%,5%,10% and 15% of the model parameters.
Time had a significant impact on the performance measurements (t). The mean number of employees, mean duration of stay of an employee, and variation of the number of employees in grade I and grade II increased when t increased from -15 % to 15%.
The mean number of employees, mean duration of stay of an employee, and variation of the number of employees in grade I decreased and in grade II it was increasing as the promotion rate parameter (a) increased from -15 % to 15%. The mean number of employees, mean duration of stay of an employee, and variation of the number of employees in grade I and grade II decreased as the leaving rate parameter ( p) in grad-1 increased from -15 % to 15%.When the leaving rate parameter (y) in grad-2 is increased from -15 % to 15%, the mean number of employees, mean duration of stay of
Table 3a:The values of Li(t),L2(t),Wi(t),W2(t),V1(t) and Vi(t) for different Values of t, a, ft, y, ai, bi, ai and bi
Para -meters Performance Measure -15% -10% -5% 0% +5% +10% +15%
t=2 L1 5.2497 5.9333 6.6624 7.4356 8.2524 9.1123 10.0152
L2 16.7277 18.7380 20.9764 23.4109 26.0219 28.7973 31.7300
W1 0.7539 0.8499 0.9530 1.0629 1.1792 1.3019 1.4308
W2 3.5231 3.9175 4.3407 4.7972 5.2901 5.8205 6.3878
VI 5.2497 5.9333 6.6624 7.4356 8.2524 9.1123 10.0152
V2 16.7276 18.7379 20.9763 23.4109 26.0219 28.7973 31.7300
a=3 L1 7.8716 7.7206 7.5754 7.4356 7.3010 7.1711 7.0457
L2 23.0525 23.1764 23.2957 23.4109 23.5220 23.6292 23.7329
W1 1.2022 1.1528 1.1065 1.0629 1.0218 0.9831 0.9466
W2 4.7743 4.7801 4.7879 4.7972 4.8075 4.8187 4.8305
VI 7.8716 7.7206 7.5754 7.4356 7.3010 7.1711 7.0457
V2 23.0525 23.1764 23.2957 23.4109 23.5220 23.6292 23.7329
(3=4 L1 8.0287 7.8206 7.6232 7.4356 7.2571 7.0870 6.9247
L2 23.7293 23.6162 23.5103 23.4109 23.3172 23.2286 23.1447
W1 1.2549 1.1854 1.1216 1.0629 1.0086 0.9585 0.9120
W2 4.8301 4.8173 4.8064 4.7972 4.7895 4.7833 4.7784
VI 8.0287 7.8206 7.6232 7.4356 7.2571 7.0870 6.9247
V2 23.7293 23.6161 23.5103 23.4109 23.3171 23.2286 23.1446
Table 3b:The values of Li(t),L2(t),Wi(t),W2(t),V1(t) and Vi(t) for different Values of t, a, ft, y, ai, bi, ai and b2
Para Performance -15% -10% -5% 0% +5% +10% +15%
-meters Measure
L1 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
L2 26.8975 25.5983 24.4462 23.4109 22.4709 21.6107 20.8184
Y=5 W1 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629
W2 6.3982 5.7752 5.2484 4.7972 4.4064 4.0650 3.7647
VI 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
V2 26.8974 25.5983 24.4461 23.4109 22.4709 21.6107 20.8184
L1 6.3205 6.6922 7.0639 7.4356 7.8074 8.1791 8.5508
L2 22.8696 23.0500 23.2305 23.4109 23.5913 23.7717 23.9521
a1=5 W1 0.9046 0.9572 1.0100 1.0629 1.1158 1.1688 1.2218
W2 4.7704 4.7742 4.7834 4.7972 4.8146 4.8351 4.8582
VI 6.3205 6.6922 7.0639 7.4356 7.8074 8.1791 8.5508
V2 22.8696 23.0500 23.2305 23.4109 23.5913 23.7717 23.9521
L1 4.7683 5.5453 6.4298 7.4356 8.5783 9.8752 11.3459
L2 22.2169 22.5701 22.9665 23.4109 23.9089 24.4666 25.0909
b1=3 W1 0.6870 0.7953 0.9200 1.0629 1.2257 1.4108 1.6209
W2 4.6951 4.7717 4.7796 4.7972 4.8525 4.9345 5.0407
VI 4.7683 5.5453 6.4298 7.4356 8.5783 9.8752 11.3459
V2 22.2169 22.5701 22.9664 23.4109 23.9089 24.4666 25.0909
L1 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
L2 20.4589 21.4429 22.4269 23.4109 24.3948 25.3788 26.3628
a2=10 W1 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629
W2 4.1923 4.3939 4.5955 4.7972 4.9988 5.2004 5.4021
VI 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
V2 20.4589 21.4429 22.4269 23.4109 24.3948 25.3788 26.3628
L1 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
L2 16.4841 18.5091 20.8066 23.4109 26.3602 29.6977 33.4714
b2=3 W1 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629
W2 3.3778 3.7927 4.2635 4.7972 5.4015 6.0854 6.8587
VI 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
V2 16.4841 18.5091 20.8066 23.4109 26.3602 29.6977 33.4714
L1 4.0256 4.0276 4.0297 4.0318 4.0338 4.0359 4.0380
L2 13.4547 13.5138 13.5729 13.6321 13.6912 13.7503 13.8094
N=1500 W1 0.5855 0.5858 0.5861 0.5864 0.5866 0.5869 0.5872
W2 2.8212 2.8257 2.8307 2.8360 2.8418 2.8479 2.8543
VI 4.0256 4.0276 4.0297 4.0318 4.0338 4.0359 4.0380
V2 13.4537 13.5128 13.5719 13.6310 13.6901 13.7491 13.8082
L1 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
L2 23.4075 23.4086 23.4097 23.4109 23.4120 23.4131 23.4143
M=500 W1 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629 1.0629
W2 4.7969 4.7970 4.7971 4.7972 4.7973 4.7974 4.7975
VI 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356 7.4356
V2 23.4075 23.4086 23.4097 23.4109 23.4120 23.4131 23.4143
The mean number of employees, mean duration of stay of an employee, and variation of the number of employees in grade I and grade II increased when the recruitment rate parameter (a1) in grade I increased from -15 % to 15%. The mean number of employees, mean duration of stay of an
employee, and variation of the number of employees in grade I and grade II increased when the recruitment rate parameter (b1) in grade I increased from -15 % to 15%.
When the recruitment rate parameter (a2) of employees in grade II goes from -15 % to 15%, the mean number of employees, the mean duration of stay of an employee, and the variance of the number of employees in grade I are unaffected, while they increase in grade II. When the recruitment rate parameter (b2) of employees in grade II is increased from -15 % to 15%, the mean number of employees, mean duration of stay of an employee, and variance of the number of employees in grade I are unaffected, but they increase in grade II.
V1.Comparative Studies of the models:
In this part, a comparison of the generated model with a model with homogeneous Poisson recruitment was shown. Table 4 shows the performance measures of both models for various values of t=1.6, 1.7, 1.8, 1.9, and 2.
Table 4: Comparative study of models with Homogeneous and Non-Homogeneous
Recruitments.
t Characteristics Measured Non-Homogeneous recruitment Homogeneous recruitment Deference Percentage of Variation
t=1.6 L1 2.1021 0.7348 1.3673 65.0445
L2 4.7680 2.3187 2.4493 51.3695
W1 0.3421 0.2017 0.1404 41.0406
W2 1.1085 0.6329 0.4756 42.9048
VI 2.1021 0.7348 1.3673 65.0445
V2 4.7680 2.3187 2.4493 51.3695
t=1.7 L1 2.2347 0.7245 1.5102 67.5795
L2 4.7067 1.9718 2.7349 58.1065
W1 0.3575 0.2008 0.1567 43.8322
W2 1.1500 0.6342 0.5158 44.8522
VI 2.2347 0.7245 1.5102 67.5795
V2 4.7067 1.9718 2.7349 58.1065
t=1.8 L1 2.3724 0.7193 1.6531 69.6805
L2 4.7803 1.7598 3.0205 63.1864
W1 0.3738 0.2003 0.1735 46.4152
W2 1.2040 0.6611 0.5429 45.0914
VI 2.3724 0.7193 1.6531 69.6805
V2 4.7803 1.7598 3.0205 63.1864
t=1.9 L1 2.5127 0.7168 1.7959 71.4729
L2 4.9365 1.6303 3.3062 66.9746
W1 0.3906 0.2001 0.1905 48.7711
W2 1.2579 0.6972 0.5607 44.5743
VI 2.5127 0.7168 1.7959 71.4729
V2 4.9365 1.6303 3.3062 66.9746
t=2.0 L1 2.6543 0.7155 1.9388 73.0437
L2 5.1432 1.5513 3.5919 69.8378
W1 0.4079 0.2000 0.2079 50.9684
W2 1.3082 0.7320 0.5762 44.0453
VI 2.6543 0.7155 1.9388 73.0437
V2 5.1432 1.5513 3.5919 69.8378
Table 4 shows that the percentage variation of the performance measures between the two models increased as time progressed. The assumption of the Duane recruitment process was found to have a considerable impact on all of the manpower model's performance measures. Time has a substantial impact on system performance, and the proposed model can more correctly forecast system performance.
VII. Conclusion
The purpose of this work is to build and analyze a two-graded manpower model with direct recruitment in both grades for non-stationary recruitment. The Duane recruitment procedure was capable of identifying recruitments that were time-dependent. The model's characteristics were obtained explicitly to assist HR Managers in adopting optimal operating policies, such as the mean number of employees in each grade, the mean duration of stay of an employee in each grade, the variance of the number of employees in each grade, and the coefficient of variation of an employee in each grade in the organization. The model's sensitivity study revealed that the Duane recruitment process has a considerable impact on system performance indicators. A comparison of the suggested model with Poisson recruitment reveals that the proposed model predicts system properties more accurately. When the recruiting was done in a time-dependent manner, the performance measures could be anticipated more correctly and realistically using the evolving model. This model can also be expanded by factoring in cost considerations while determining the best values for the parameters, which will be done elsewhere.
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