OPSEARCHDOI 10.1007/s12597-013-0162-1

THEORETICAL ARTICLE

A manpower planning model based on lengthof service under varying class sizes

Arindam Gupta ·Anindita Ghosal

Accepted: 14 September 2013© Operational Research Society of India 2013

Abstract Study of different types of manpower planning models in different orga-nizations are very much useful for proper planning and implementation of differentobjectives. These types of studies are generally solved by deterministic way in pre-viously. Some scientists have addressed these problems in stochastically and basedon age of employee. The present work is an attempt to develop a probabilisticsmanpower planning model under the setup where the classes are varying sizes andpromotion occurs only on the basis of length of service of employee not age.

Keywords Manpower planning · Stochastic · Length of service

1 Introduction

The study related to manpower planning is necessary for appropriate management ofan organization. There is a continuous movement along various grades of an organi-zation due to need with the organization or causes may will be dictated by outside ofthe system. So, it is necessary to study the movement of the employees along variousgrades in an organization.

The length of service has a great role in the life of an employee. In many organi-zations all benefits of an employee depend on his/her length of service. So, when wewant to study manpower planning of an organization in properly it is necessary totake a key component “length of service”.

A. Gupta (�)Department of Statistics, University of Burdwan, Burdwan, Indiae-mail: guptaarin@gmail.com

A. GhosalDepartment of Statistics, Midnapore College, Midnapore, India

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The main goal of an employee in an organization is when he/she will get a pro-motion in the next grade. The time of occurrence of promotion in an employee’s lifemay be defined in terms of his/her age or length of service. It is possible to enter inemployment in different age. So, it is better to consider length of service rather thanage of an employe for promotion criterion.

The size of the different grades of an organization may be fixed due to differentcauses like financial and amount of work at each grade. Due to the vacancies arisethe recruitment and promotion may be needed. Internal movement comes out on thissituation. Abodunde and McClean [1, 6, 7] have considered several manpower plan-ning situations and developed models on Markov and Semi-markov processes. Khanand Chattopadhyay [5] and Chattopadhyay and Khan [2] have developed differentoccupational mobility on the basis of salary and number of job offers. Mukherjee andChattopadhyay [8] developed stochastic solution to the problem related to recruit-ment and promotion of staff members in case of an airline. Chattopadhyay and Gupta[3] has extended this work in the case when grade varies with sizes. These solutionsare based on age of employee. But the time point when an employee be promoted tonext higher grade it will be depends on the length of service not only age of employee.This work will try to solve this problem based on length of service of the employeeunder varying class in stochastically.

2 Staffing problem

It is necessary to study the internal movements of the employees over differ-ent grades for staffing problem. For the proper planning of the carrier of theemployees the problems related to the recruitment and promotion within the orga-nization are needed to analyze. The length of service of an employee in a par-ticular grade is the key random variable in studying the internal movement in anorganization.

2.1 Study parameters

r = Number of the graded job categories in an organization.m(t) = Total size of the organization at a particular point t.ah = Minimum recruitment age.Mi(t) = Number of staffs at the ith category as decided by the managementpreviously corresponding to the time point t.xw = Retirement age which is same and fixed for all employees.λ

′0h(t) = Number of persons recruited each year after attaining age ah at a

particular time point t at the first category.λx(t) = Number of persons whose length of service is x at a particular point t.px(t) = The proportion of population whose length of service is x that continuesto be in service at length of service is (x + 1) at a particular point t. (SurvivalRate)Zx(t) = The number of persons whose length of service x who survive upto thelength of service (x + 1) at a particular point t.

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xi(t) = The minimum possible length of service at which ith promotion takesplace at a particular point t.p0x(t) = Probability of survival of a person of length of service x in the systemat a particular point t.

Here px(t) can be interpreted as the conditional probability and p0x(t) beunconditional probability.

2.2 Organizational setup

In different organizations there are different recruitment and promotion policy. In thiswork we assume a very common setup which prevails in many organization. The var-ious grades constituting the system are categorized on the basis of seniority i.e. lengthof service not employee’s age. Here any other job efficiency is ignored in consider-ation of promotion. The total size of the system (m(t)) is time dependent. The ageof retirement is fixed which is same for all employees. The minimum age at recruit-ment is also fixed. The number of staff members at x length of service (λx(t)) will bedetermined by voluntary and involuntary wastage. Involuntary wastage include lossesdue to death, ill-death and retirement. Voluntary wastage arises out of withdrawalsby individual on their own. There is no demotion in the system in assumption. Thesurvival rate from length of service x to (x + 1) are expected to depend on bothlength of service and grade. Under the present setup these are depended on lengthof service.

The exact length of service at which promotions to different categories can befind out on the basis of given the λx(t) values and the staff strength require-ment. Solution of the above problem based on age under stochastic setup havebeen developed by [3, 8]. A more general form of the above problem is the situ-ation where the distribution of length of service of the staff members at differentcategories are known at a particular time point and the number of workers to beincreased in the next few years can also be estimated. The present work attemptsto develop a stochastic solution of such a problem on the basis of length ofservice.

3 Probabilistic solution

The survival rates px(t) and the number of persons at different length of service λx(t)

should be treated as random due to “withdrawls” as a random event which is dependson unpredictable human behaviour. The complete length of service (CLS) is unpre-dictable and should be treated as random variable. For any organization λx(t) shouldbe treated as random samples from the corresponding CLS distribution dependingupon the px(t) values.

Here, the staff strength requirements be random. Promotions should be given atlength of service which may have to be random. The numbers remaining in the ser-vice at different length of service are also unpredictable. All this implies that the lifetable functions should be treated as random variables.

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Suppose, Zx(t) be the number of persons at length of service x who survive up tolength of service (x +1), then Zx(t) = [λx(t)px(t)] = Rounded value of the productλx(t)px(t).

So, the actual number of persons whose length of service x in category i is givenby ∫ xi(t)

xi−1(t)

Zx(t)dx (1)

where xi−1(t) is the minimum length of service at which (i − 1)th promotion takesplace and the range of x values corresponds to only those persons who have got theirpromotions at length of service xi−1(t).

Since the stuff strength in category i should not exceed Mi(t) we have thecondition ∫ xi(t)

xi−1(t)

< Mi(t) (2)

Here, xi(t) determines the minimum possible length at which ith promotion takesplace. In case the number of persons to be promoted is larger than the requirement,the management may promote on the basis of other criterion. Of course this solutionis quite ad hoc and crude but it is difficult to find a better solution. Under the prob-abilistic set up, the above condition only may be satisfied with a high probability. Inorder to get a solution, we have to make some assumptions regarding the underlyingpopulation distributions of the different random variables involved in the model. Wemay assume that

px(t) ∼ Beta(a, b).

Since the data set is available only for a particular time point we have to assumethat the distribution of the different random variables involved are time homogeneousotherwise the related parameters can not be estimated from data.

Under this assumptionpx ∼ Beta(a, b)

as px takes values between 0 and 1 and px(t) values generally increase up to a valueof x and then start decreasing.

Given λ′0,

λx(t) ∼ Binomial(λ′

0, p0x

)(according to [4]).

Here it is quite reasonable to assume that

Mi(t) ∼ Poission(li)

since for large number of categories the probability that a particular person belongsto the ith category is very small. The problem is to find xi(t) in such a way that theprobability of the random event given in Eq. 1 is sufficient large.

The lengths of service at which successive promotion take place cannot be exactlypredicted. Thus x1(t) (i.e. that length of service at which the first promotion takesplace) can be treated as random variable distributed over (0, xw − xh) with a rea-sonable value of the average. Without loosing much significance, we may reasonableassume x1(t) to have an exponential distribution with probability density function

f (x1(t)) = λ exp(−λx1(t)), x1(t) > 0.

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In general, x1(t) may be similarly assumed to have a Exponential distribution over(xi−1(t), ∞). The average age at promotion may of course vary from one stage toanother.

Our problem now involves four sets of random variables viz λx(t), px(t), xi(t)

and mi(t). For first promotion, we are required to find the value of x1(t), which willmake the probability of the event in Eq. 2 sufficiently large.

Considering xi(t) as random here, it is difficult to derive the distribution of theabove integrate generally, since Zx(t)s have varying parameters

P

[∫ xi(t)

xi−1(t)

Zx(t)dx ≤ Mi(t)

]

=∞∑

mi=1

P

[∫ xi (t)

xi−1(t)

Zx(t)dx ≤ mi

]P [Mi(t) = mi ]

=∞∑

mi=1

∫ ∞

j=0P

[∫ j

x=0Zx(t)dx ≤ mi |xi(t) − xi−1(t) = j

]f (j)djP [Mi(t) = mi ]

(3)

where, f (j) = P [xi(t) − xi−1(t) = j ].Here, xi−1(t) is known, since in the first promotion case it is zero, and in all other

cases it may be assumed known from the previous step.Using the relation, f (j) = λ exp(−λl), j > 0, Eq. 3 becomes

∞∑mi=1

∫ ∞

j=0P

[∫ j

x=0Zx(t)dx ≤ mi |xi(t)−xi−1(t) = j

]λ exp(−λj)djP [Mi(t)=mi ]

(4)It is easy to show that this series is convergent.

Under the conditional set up one can find the value of j from the following relation:

P

[∫ j

x=0Zx(t)dx ≤ mi

]≥ (1 − α) (5)

where α may be sufficiently small so that the system requirement is satisfied.

4 An example

In [9] an real life example is given which is stated below. An airline requires 200assistant flight attendants, 300 flight attendants and 50 supervisors. Assistant flightattendants are recruited at age 21 and, if still in service, retire at age 60. The survivalrates and age distribution are given in Tables 1 and 2 respectively. If persons are to berecruited age 21, at what ages will promotion take place when for the next year staffrequirements will increase by 10 %.

The above example is on age of staff. In this, all staff are to be recruited at age 21not at other ages. So, we can think that the length of service of any staff at age 21 bezero. The survival rates and age distribution which are given in the above example

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Table 1 Survival rates atdifferent ages x px x px

21 0.600 41 0.95222 0.800 42 0.94723 0.800 43 0.94224 0.800 44 0.93625 0.800 45 0.93026 0.850 46 0.92327 0.875 47 0.91528 0.900 48 0.90629 0.925 49 0.89630 0.950 50 0.88531 0.958 51 0.87332 0.962 52 0.86033 0.965 53 0.84634 0.967 54 0.83135 0.968 55 0.81536 0.968 56 0.79837 0.968 57 0.78038 0.965 58 0.76139 0.961 59 0.74140 0.957

Table 2 Staff age distribution at a particular time point

Assistant flight attendants Flight attendants Supervisors

Age Number Age Number Age Number

21 90 26 40 42 522 50 27 35 43 423 30 28 35 44 524 20 29 30 45 325 10 30 28 46 3Total 200 31 26 47 3

32 20 48 633 18 49 234 16 50 035 12 51 036 10 52 437 8 53 338 0 54 539 8 55 040 8 56 341 6 57 2Total 300 58 0

59 2Total 50

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Table 3 Survival rates atdifferent length of service x px x px

0 0.600 20 0.9521 0.800 21 0.9472 0.800 22 0.9423 0.800 23 0.9364 0.800 24 0.9305 0.850 25 0.9236 0.875 26 0.9157 0.900 27 0.9068 0.925 28 0.8969 0.950 29 0.88510 0.958 30 0.87311 0.962 31 0.86012 0.965 32 0.84613 0.967 33 0.83114 0.968 34 0.81515 0.968 35 0.79816 0.968 36 0.78017 0.965 37 0.76118 0.961 38 0.74119 0.957

Table 4 Staff length of service distribution at a particular time point

Assistant flight attendants Flight attendants Supervisors

Age Number Age Number Age Number

0 90 5 40 21 51 50 6 35 22 42 30 7 35 23 53 20 8 30 24 34 10 9 28 25 3Total 200 10 26 26 3

11 20 27 612 18 28 213 16 29 014 12 30 015 10 31 416 8 32 317 0 33 518 8 34 019 8 35 320 6 36 2Total 300 37 0

38 2Total 50

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Fig. 1 The nature of the E(Zx)

0 10 20 30

020

4060

80

Length of Service

EZ

x

may be converted on the length of service in place “age of staff”. The converted tableof survival distributions are given in Tables 3 and 4 respectively. After conversionnow we can use the above example based on length of service in this work. Thisexample can be treated as a special case of this working problem.

Using the data of Table 3, it has been found that Beta(14.352, 1.893) fits the dataon px well under the assumption of time homogeneity. By estimating the p0x valuesfrom Table 4, λx values have been generated from Binomial distribution using com-puter simulation (after adjusting for the total size of the system). Hence, it is easy togenerate the Zx values. Figure 1 shows the nature of the E(Zx) values.

Since here it is difficult to estimate the Poission parameter li’s corresponding tothe different job categories, the mi values are assumed to be given. But it is easyto extend the analysis to the case where mi’s are random variables when we havesufficient number of observations at each length of service corresponding to differentorganizations.

On the basis of the simulated values of Zx (at a particular time point), we haveestimated the length of service at first promotion from the probability statement givenin Eq. 5 by α as 0.05. From our simulation study most probable length of service is 4for the 1st promotion. Similarly by starting at length of service 4 the most probablelength of service at 2nd promotion has been found to be 28.

5 Concluding remarks

This work is helpful to getting the time of promotion in an organization stochastically.This work can be extended by considering the other type of promotion criterion likework ability, qualification of an employe etc.

References

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3. Chattopadhyay, A.K., Gupta, A.: A stochastic manpower planning model under varying class sizes.Ann. Oper. Res. 155(1), 41–49 (2007)

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