Two component mixture of Laplace and Laplace type distributions with applications to manpower planning models

Introduction

In two parameters Laplace type distributions and three parameter Laplace type distributions. Distributions it was observed that the point at 𝑋 = 𝜇 is zero, which is seemingly not useful in some practical situations. So to lift the drift at the mean point 𝜇 one way of consideration is assuming that the random variable follows Laplace distribution with probability 𝑝 and two parameter Laplace type distribution with probability1 -𝑝( 0 ≤ 𝑝 ≤ 1). This idea is imbedded in generating the mixture distributions. Hence in this paper it was developed the probability density function of two component mixture of Laplace and Laplace type distributions. The distributional properties of this distribution are also studied. This was applied to a practical manpower situation. This contains two parameters𝜇 𝑎𝑛𝑑 𝛽. To maintain the parsimonious property of the modal in two components of the mixture population, the same parameters are considered. If𝑝 = 1, this distribution becomes Laplace distribution with parameters𝜇 𝑎𝑛𝑑 𝛽. And if𝑝 = 0, the distribution becomes two parameter Laplace type distributions.

Properties of the Distribution

The distribution function of the random variable X is For different values of 𝜇, 𝛽𝑎𝑛𝑑 𝑝, the frequency curves of the distribution are plotted and shown in figure: 1

Mean, Mode and median of the distribution

The mean of this random variable X is

This distribution is a bimodal distribution for the value of 0 ≤ 𝑝 ≤

The distance between two modes is 2

The median of this distribution can be obtained by solving the equation

That is for this distribution the Median = Mean and hence the distribution is said to be symmetric at 𝑥 = 𝜇

Moment generating function

The moment generating function of the two-component mixture of Laplace and Laplace type variate is

Using the transformations and on simplification we get

Characteristic function

The characteristic function of the variate X is

Using the transformation, y= 𝑥 -𝜇 and on simplification one can get

Central Moments of the Distribution

The central moments of this distribution can be obtained as

Using the transformation and on simplification, one can get

In particular the first four central moments are 𝜇 1 = 0,𝜇 2 = 2𝛽 2 , 𝜇 3 = 0, 𝜇 4 = 𝛽 4 (360 -366𝑝) 8 Hence the variance of the distribution is2𝛽 2 (6 -5𝑝). Since the distribution is symmetric, its skewness is zero and the kurtosis of the distribution is

9

Hazard rate of the distribution

The hazard rate of the distribution is ℎ(𝑥) = 𝑓(𝑥)

1-𝐹(𝑥)

Where 𝑓(𝑥) and 𝐹(𝑥) are as given in equations ( 1) and ( 2) respectively. 10

Additive property of the distribution

For verify the additive property of the distribution, consider two independent variables 𝑋 1 and 𝑋 2 with density function

The characteristic functions are

Then the characteristic function of the sum of the random variables, 𝑋 1 + 𝑋 2 is

The lack of closed form of the characteristic function does not permit to have a single way of identifying the distribution of the sum of the random variables. That is additive property was not possessing by this distribution. However, from Cramer 1955 one can invent the characteristic function by using the inversion formula

Estimation of the Parameters

This section is on the the estimation of the parameters arising from the mixture of Laplace and Laplace type distribution. The drift parameter 𝑝 (mixing parameter) can be estimated by using the method of moments along with the location and scale parameters.

Since the estimation of the location and scale parameter involves the drift parameter, estimate 𝑝 can be obtained by equating the sample kurtosis with the population kurtosis.

We have 𝜇 4 = 𝛽 4 (360 -366𝑝) and 𝜇 2 = 2𝛽 2 (6 -5𝑝)

Therefore 𝑝 can be estimated by solving the following quadratic equation and picking up the value which lie between 0 and 1 as an estimate of the parameter

11

Where the RHS is the sample Kurtosis. The moment estimators 𝜇 and 𝛽 2 𝑝 as 𝜇̂= 𝑥̅ and

Where, 𝑠 2 is the sample variance.

The variance of these estimators is given by 𝑉

13

Where 𝜇 𝑖 is the ith central moment Therefore,

As these variances tends to zero as𝑛 → ∞, these estimators 𝑥̅ and 𝑠 2 are consistent for 𝜇 and 𝜎 2 .

The maximum likelihood estimation

The maximum likelihood estimation of 𝑝 and 𝛽 2 can be obtained after estimating the value of 𝜇 through median similar to the location parameter estimated in two parameter Laplace type distributions. After substituting the estimate of 𝜇 the likely hood equation of the sample is

Differentiating with respect to 𝑝 and 𝛽 2 the likelihood equations are

Solving these equations iteratively for 𝑝 and 𝛽 2 using the Numerical techniques like Newton Raphson's Method, we can obtain the maximum likely hood estimators.

Application of mixture of Laplace and two Parameters Laplace type bimodal distribution to Manpower Planning

In this section we consider the application of the developed distribution to the manpower planning models. The complete length of service of an employee in an organization is the total duration of time that employee spends in an organization. In general, the complete length of service can be viewed as a random variable as it is determined by a number of factors in nature. This is one of the most important features characterizing the manpower situation and is important in manpower models, which describe the manpower structure of the organization and useful in obtaining optimal man power policies. Several researchers have approximated the complete length of service distributions by different probability distributions. Silcock (1954) [1] have considered that the complete length of service of an employee fallows and exponential distribution. However exponential distribution is unimodal having long right tail with constant hazard rate. But in many private and public-sector organizations the frequency curve of the complete length of service of an employee in an organization gradually increase up to a point and decline up to some other point and again increase and decrease, that is the curve is bimodal. This phenomenon may be due to the imposition of probation period in many organizations. After an employee joining the organization, because of social and behavioral factors he is leaving and his complete length of service forms bimodal. Since the complete length of service of an employee cannot be negative, and cannot be greater than a specified value A, we can truncate the theoretical distribution given in section -2 at '0' in left and at A in right and impose the condition that the location parameter 𝜇 > 0. Then we can approximate the complete length of service of an employee fallows left truncated mixture of Laplace and Laplace type distribution.

Then the probability density function of the complete length of service is And 𝐹(0) = [

Then 𝑔(𝑥) = The probability that an employee served in an organization for the length of time 't' is 𝐺(𝑡), which is survival function of the employee in the organization is defined as the compliment of distribution function. We have distribution function of complete length of service as )}] 𝑓𝑜𝑟 𝑡 ≥ 𝜇 18

The force of separation, which is known as rate of labour wastage can be obtained as

Therefore, for this manpower model the rate labour wastage is

19

The average time that an employee spends in the organization is

Using the transformation and on simplification, one can get

Using the transformation and on simplification, one can get

That is the variability of the complete length of service of an employee is

The renewal density of this model is

24

Where 𝑔(𝑡) and 𝐺(𝑡)are as defined in (18

Using the values of 𝑔(𝑡) and 𝐺(𝑡) in the (19) one can get the ) (t h

Recruitment Polices For Manpower Model

In this section, we study the different recruitment policies of an organization, which play a vital role in Manpower Planning strategies. An organization can be viewed as having two grades. The first grade is training grade and the second grade is organization grade. The organization itself in general possess an hierarchical structure. The transfer between training grade (grade I) and organization grade (grade II) is considered as promotion. The whole organization is considered as single unit. Let 𝑁 1 and 𝑁 2 are the sizes of training grade and organization grade respectively. Then the total number of employees in the organization is 𝑁 1 + 𝑁 2 = 𝑁. Here 𝑁 2 is assumed to be known as fixed constant. In any organization the promotion policies are of two types namely (i) promotion by seniority (i.e.) the most senior member of the grade I is promoted to grade II and (ii) promotion by random. This is usually done based on merit. Let 𝑤 1 be the individual loss rate in training grade and 𝑤 2 is the individual loss rate in organization grade and p is the promotion rate. Since the input and output of each grade must balance, for grade II, it can be written as

In equilibrium, the expected input to the organization per unit time is

In general, 𝑤 1 , 𝑤 2 𝑎𝑛𝑑 𝑝 are function of time. The value of 𝑝, 𝑤 1 𝑎𝑛𝑑 𝑤 2 are independent of time, when the system is equilibrium.

Let the 𝜃 1 be the average time spent in grade I. Then under study state the expected number of vacancies arriving per unit time in this grade is

28

From this equation the throughput for the whole system is equal to the number of vacancies in the grade I.

Promotion by Seniority: Let a ( 𝑡 𝑇 ) denote the age distribution of the system at time '𝑡' give that the organization is established at time𝑡 = 0. Thus, the probability that an employee chosen at random is having a length of service as 𝑇 in (𝑡, 𝑡 + 𝛿𝑡) is ( Which gives the study state age distribution. In promotion by seniority, the loss form grade II is replaced by the seniority of member of grade I. It follows that at any time every individual in grade II has length of service at least as long as any individual in grade I and hence there is some thresh hold value 𝑡 1 such that all individuals with length of service less than 𝑡 1 are in grade I. 𝑡 1 is a random variable, but for large values of grade sizes the approximate formula for its expected value is given by Bartholomew, D.J. (1973) [2] as

31

Therefore, under study state the expected number of proportion per unit time will be the promotion of new recruits whose service is at least at the threshold length𝑡 1 . i.e.

36

By solving the (36), the explicit values can be obtained. For different values of 𝜇 𝑝, 𝛽 𝑎𝑛𝑑 𝑝 we complete the value of 𝑅 and presented in Table -

References

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