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Asia Pacific Journal of Research Vol: I Issue XX, December 2014

ISSN: 2320-5504, E-ISSN-2347-4793

www.apjor.com Page 76

DETERMINATION OF OPTIMAL MANPOWER RESERVE INVENTORY USING SKEW

-LOGISTIC DISTRIBUTION

Dr. N. Vijayasankar

Assistant Professor, Department of Statistics

Annamalai University,Annamalai Nagar – 608002

S.Vijayamirtharaj

Research Scholar, Department of Statistics

Annamalai University

R.Sathiyamoorthi

Retd. Professor, Department of Statistics

Annamalai University

ABSTRACT

In any production or administrative organisation, the manpower is an important factor among the other

factors. There are many organisations where there are two stages such as recruitment stage and work spot

stage. If the manpower availability is not sufficient, then the shortage in the same world affect the

functioning of the organisation and it would lead to loss of potential profits. Hence a reserve inventory of

manpower is maintained. In this paper the optimal level of reserve inventory of manpower is determined

under the assumption that the demand for manpower is a random variable which follows skew logistic

distribution. The optimal size is decided when the random variable follows i) first order statistic ii) nth

order statistic. Numerical illustration is also provided.

Key words: Manpower Planning, Order Statistics, Skew Logistic Distribution


ISSN: 2320-5504, E-ISSN-2347-4793


Introduction

In any organization there are many factors which contribute to the smooth functioning of the

organization and its progress. These factors include the capital investment, marketing strategies and

industrial maintenance. Even in software organizations it is identified that the manpower availability is

very vital. Hence it becomes necessary to maintain proper manpower availability in the organizations.

It is a known fact that the leaving of personnel including labourers is unavoidable. If enough of

manpower is not available it may affect both the production and other activities of the organization, such as

marketing and administrative. So suitable precautionary steps should be taken to keep the manpower

strength at the sufficient level. For this purpose in many organizations especially in software organizations

recruitment is made and sufficient training is given in handling the software technology. Sometimes the

number of recruited persons which in other words the availability manpower in terms of man hours may be

more than requirement and the manpower available above the requirement will lead to loss of profit due to

the excess inventory of manpower. This is called holding cost. If there is shortage of manpower it well lead

to loss of profit. Frequent recruitments is also not desirable. The concept of optimal Reserve between two

machines in series has been discussed in Hanssmann (1961). An extension of this model has been by

Rajagopal and Sathiyamoorthi (2003). Determination of optimal manpower stock has been studied by

G.Arivazhagan et.al (2010). Muthaiyan et.al (2009) have discussed about the estimation of expected time

to recruitment using order statistics.

In this paper it is assumed that the first stage or system-I which is the recruitment and training may

have a breakdown. This affects the functioning of the industry due to want of manpower. This is very much

high and pronounced in software industry. So a reserve inventory of manpower maintained in between the

two nodes namely requirement node, work spot node. The optimal reserve of manpower in terms of man-

hours it is determined under the following assumptions.

Assumptions:

There are two nodes in the maintenance of manpower in the organization, and are such as

recruitment and training node and the work spot node.

If there is a break down at the first node there it will lead to shortage of manpower. It will lead to

the breakdown of the work spot node. The cost of the same is high and prohibitive.

A reserve inventory of trained personnel is maintained in between the two nodes.

If the reserve is more than the requirement then it leads to holding cost. If there isinsufficient stock

then it may lead to breakdown of the second node and the cost of which is highly prohibitive due to

delayed work schedule.

Notations:

h : The cost of holding excess manpower per man-hour.

d :The Shortage cost arising due to shortage of manpower per hour.

s :The stock level of manpower reserve in man-hours.

τ : a random variable denoting the demand for manpower in terms of man-hours, with p.d.f g(.),

which follows skew logistic distribution and c.d.f G(τ).

Results:

Case – I :

It is assumed that the demand for manpower is a random variable denoted as „τ‟. Which follows

firstorder statistics and it has probability density function𝑔 1 𝜏 , with cumulative distribution function

𝐺 1 𝜏 .


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The expected cost due to excess manpower or shortage of manpower is given by

E C = h s − τ

s

0

g 1 τ dτ + d τ − s

∞

s

g 1 τ dτ

Where, g 1 τ = n 1 − G(τ) n−1 g τ

= h s − τ

s

0

n 1 − G(τ) n−1 g τ dτ + d τ − s

∞

s

n 1 − G(τ) n−1 g τ dτ

= 𝑛h s − τ

s

0

1 − G(τ) n−1g τ dτ + nd τ − s

∞

s

1 − G(τ) n−1g τ dτ

nh (A) + nd (B) ……………………….(1)

Where (A) => s − τ s

0 1 − G(τ) n−1 g τ dτ

(B) => τ − s ∞

s 1 − G(τ) n−1 g τ dτ

Now dE (C)

ds= 0 gives the optimum reserve of manpower

𝑑𝐴

𝑑𝑠 = 0 =>

𝑑

𝑑𝑠 s − τ

s

0

1 − G(τ) n−1 g τ dτ

∅ 𝑠 = 0 , ∅′ 𝑠 = 0, 𝜓 𝑠 = 𝑠, 𝜓 𝑠 = 1 By Leibnitz rule

dA

ds= (s − s) 1 − G τ n−1 g s + 0 +

d

ds s − τ

s

0

1 − G τ n−1 g τ dτ

= s

0 1 − G τ n−1 g τ dτ

Now the p.d.f of Skew logistic distribution is given by

g τ = αe−τ

1 + e−τ α+1

𝐺 𝜏 = 1 + 𝑒−𝜏 −𝛼

1 − 𝐺 𝜏 = 1 − 1 + 𝑒−𝜏 −𝛼

= 1 − 1 + 𝑒−𝜏 −𝛼 𝑛−1

𝑠

0

𝛼𝑒−𝜏

1 + 𝑒−𝜏 𝛼+1 𝑑𝜏

𝑙𝑒𝑡𝑡 = 1 − 1 + 𝑒−𝜏 −𝛼

𝑑𝑡 = 0 − −𝛼 1 + 𝑒−𝜏 −𝛼−1 −𝑒−𝜏 𝑑𝜏

𝑑𝑡 = −𝛼𝑒−𝜏(1 + 𝑒−𝜏)−(𝛼+1)

𝑑𝑡 = −𝛼𝑒−𝜏

1 + 𝑒−𝜏 𝛼+1

τ 0 S

t 1−2−𝛼 1 − 1 + 𝑒−𝑠 −𝛼

= tn−1(−dt)

1− 1+e−s −α

1−2−α


ISSN: 2320-5504, E-ISSN-2347-4793


dA

ds = −

1

n 1 − 1 + e−s −α n − 1 − 2−α n ……………. (2)

(B) => 1 − G(τ) n−1 g τ dτ

∞

s

= − 1 − 1 + 𝑒−𝜏 −𝛼 𝑛−1

∞

𝑠

𝛼𝑒−𝜏

1 + 𝑒−𝜏 𝛼+1 𝑑𝜏

𝑡 = 1 − 1 + 𝑒−𝜏 −𝛼

𝑑𝑡 = 0 − −𝛼 1 + 𝑒−𝜏 −𝛼−1 −𝑒−𝜏 𝑑𝜏

𝑑𝑡 = −𝛼𝑒−𝜏(1 + 𝑒−𝜏)−(𝛼+1)𝑑𝜏

𝑑𝑡 = −𝛼𝑒−𝜏

1 + 𝑒−𝜏 𝛼+1 𝑑𝜏

= − tn−1(−dt)

0

1− 1+e−s −α

dB

ds = −

1

𝑛 1 − 1 + 𝑒−𝑠 −𝛼 𝑛 …………………… (3)

𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 2 & 3 𝑖𝑛 (1)

nh −1

n 1 − 1 + e−s −α n − 1 − 2−α n + nd −

1

n 1 − 1 + e−s −α n = 0

−h 1 − 1 + e−s −α n + h 1 − 2−α n − d 1 − 1 + e−s −α n = 0

1 − 1 + e−s −α n −h − d = −h 1 − 2−α n 1 − 1 + e−s −α n h + d = h 1 − 2−α n

1 − 1 + e−s −α n =h 1 − 2−α n

h + d

1 − 1 + e−s −α = h 1 − 2−α n

h + d

1

n

1 + e−s −α = h 1 − 2−α n

h + d

1

n

− 1

1 + e−s = h 1 − 2−α n

h + d

1

n

− 1

−1

α

e−s =

h 1 − 2−α n

h + d

1

n

− 1

−1

α

− 1

τ S ∞

t 1 − 1 + 𝑒−𝑠 −𝛼 0


ISSN: 2320-5504, E-ISSN-2347-4793


−s = log

h 1 − 2−α n

h + d

1

n

− 1

−1

α

− 1

s = log h 1−2−α n

h+d

1

n− 1

−1

α

− 1

−1

………………… (4)

The optimal value of S namely 𝑠 can be obtained from (4) given the values of h, d, n, α.

Case-II :

Te demand „τ‟ is a random variable which has the distribution of nth

order statistic with p.d.fof g n τ . The p.d.f of the n

th order statistic is given as

g n τ = n G τ n−1 g τ Using the p.d.f of the skew logistic distribution we have

E C = h s − τ n

s

0

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ + d τ − s n

∞

s

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ

= nh s − τ

s

0

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ + nd τ − s

∞

s

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ

nh (C) + nd(D) ……………………….. (5)

Where C => s − τ s

0 1 + e−τ −α n−1 αe−τ

1+e−τ α+1 dτ

Now dC

ds=

d

ds s − τ

s

0 1 + e−τ −α n−1 αe−τ

1+e−τ α+1 dτ

∅ s = 0 , ∅ ′ s = 0, ψ s = s, ψ s = 1

dC

ds= s − s 1 + e−τ −α n−1 − 0 +

d

ds s − τ

s

0

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ

= 1 + e−τ −α n−1

s

0

αe−τ

1 + e−τ α+1 dτ

t = 1 + e−τ −α

dt = −α 1 + e−τ −α−1 −e−τ dτ

= αe−τ 1 + e−τ −(α+1) dτ

dt = αe−τ

1 + e−τ α+1 dτ

= tn−1 dt

1+e−s −α

2−α

τ 0 S

t 2−𝛼 1 + 𝑒−𝑠 −𝛼


ISSN: 2320-5504, E-ISSN-2347-4793


dC

ds=

1

n 1 + e−s −α n − 2−α n …………………….. (6)

Also D => τ − s ∞

s 1 + e−τ −α n−1 αe−τ

1+e−τ α+1 dτ

dD

ds=

d

ds τ − s

∞

s

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ

∅ s = 0 , ∅ ′ s = 0, ψ s = s, ψ s = 1

dD

ds= s − s 1 + e−τ −α n−1 − 0 +

d

ds τ − s

∞

s

1 + e−τ −α n−1αe−τ

1 + e−τ α+1 dτ

= − 1 + e−τ −α n−1

∞

s

αe−τ

1 + e−τ α+1 dτ

t = 1 + e−τ −α

dt = −α 1 + e−τ −α−1 0 − e−τ dτ

= αe−τ 1 + e−τ −(α+1) dτ

dt = αe−τ

1+e−τ α+1 dτ

= − tn−1(dt)

1

1+e−s −α

dD

ds= −

1

n+

1

n 1 + e−s −α n …………………… (7)

𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 6 & 7 𝑖𝑛 (5)

nh 1

n 1 + e−s −α n − 2−α n + nd −

1

n +

1

n 1 + e−s −α n = 0

h 1 + e−s −α n − h 2−α n + nd −1

n +nd

1

n 1 + e−s −α n = 0

h 1 + e−s −α n + d 1 + e−s −α n = h 2−α n + d 1 + e−s −α n h + d = d + h 2−α n

1 + e−s −α n =d+h 2−α n

h + d

1 + e−s −α = d + h 2−α n

h + d

1

n

1 + e−s = d+h 2−α n

h + d

−1

nα

e−s = d+h 2−α n

h + d

−1

nα

− 1

τ s ∞

t 1 − 𝑒−𝑠 −𝛼 1


ISSN: 2320-5504, E-ISSN-2347-4793


−s = log d+h 2−α n

h + d

−1

nα

− 1

s = log d+h 2−α n

h + d

−1

nα

− 1

−1

Numerical illustrations:

Giving certain values for the parameters of the distribution and the values for h and d, the following

illustrations provided.

Table-1

Graph-1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

10 20 30 40 50h

first order statistic

n th order statistic

S

first order statistic nth

order statistic

d= 300 α=0.1 n=10 d= 300 α=0.1 n=10

h S h S

10 0.467 10 4.110

20 0.380 20 3.433

30 0.332 30 3.067

40 0.297 40 2.772

50 0.269 50 2.565


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Table-2

Graph-2

Conclusions:

i. It could be observed from the numerical illustration that in the case of both first order and nth

order

statistic as cost of holding of the stock of manpower increases, it is appropriate to have a smaller

reserve inventory of manpower, This is indicated table (1) and graph (1) whereas the value of „h‟

increases the optimal reserve size namely s shows a decline.

ii. If the shortage is more than the loss due to shortage will be higher. Hence an appropriate level of

manpower inventory is maintained. If the shortage cost ‟S‟ is higher than it would be appropriate

the keep a higher level of manpower inventory. So it is observed that as the value of „d‟ increases,

then a corresponding increase in s is seen in both cases namely the first order statistic and the

nth

order statistic as observed in table(2) and the graph(2).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

50 100 150 200 250d

first order statistic

n th order statistic

S

1st order statistic n

th order statistic

h= 100 α=0.1 n=10 h= 100 α=0.1 n=10

d S d S

50 0.058 50 0.693

100 0.099 100 1.098

150 0.129 150 1.386

200 0.154 200 1.610

250 0.175 250 1.814


ISSN: 2320-5504, E-ISSN-2347-4793


Reference:

1. Arivazhagan, G., R. Elangovan and R. Sathiyamoorthi., “Determination of Optimal Manpower

reserve when demand for manpower has fluctuations”, Bulletin of Pure and Applied Sciences.

Vol.29 E, No.2, pp-243-248, 2010.

2. Fred Hanssmann., “Operation Research in production and inventory Control”, Jon willey&

sons.inc, 1961.

3. Muthaiyan, A., A.Sulaiman and R. Sathiyamoorthi.(2009) “A Stochastic model based on order

statistics for estimation of expected time to recruitment”, International Journal of Agriculture

Statistical Science. Vol.5 No: 2, pp -501-508, 2009.

4. Rajagopal, k., R. Sathiyamoorthi., “An extension of the Optimal Reserve Inventory level Between

machines model”, second National Conference on Mathematical & Computational methods-PSG

college of Arts & Science, Coimbatore, 2003

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