Optimal manpower recruitment by stochastic ... - Semantic Scholar
Int J Syst Assur Eng Manag (June 2010) 1(2) DOI 10.1007/s13198-010-0016-7
RESEARCH ARTICLE
Optimal manpower recruitment by stochastic programming in graded manpower systems P. Tirupathi Rao • P. R. S. Reddy • A. V. S. Suhasini
Received: 10 February 2010 / Revised: 11 March 2010 The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2010
Abstract Human resource is one of the most effective resources in the development of any organization or a Nation. Optimal utilization of manpower is a pivotal challenge to a manager of a dynamic management system. Achieving the objectives by meeting the constraints on various feasibilities is the core problem before statisticians, model developers and OR scientists. In this paper we develop a model with Stochastic Programming Problem of manpower recruitment for an organization where the graded manpower system is observed. The averages, variance of number of employees in initial and final grades before leaving/resigning organization are computed by a suitable bivariate probability mass function. The costs like setup cost, recruitment cost, salaries, Cost of Training, Promotion Costs etc., are considered while developing a model. Sensitivity analysis was also carried out and observed the model behaviour. This model is a suitable tool to serve the decision maker for making optimal decisions on Recruitment policies for determining the optimal recruitment cost, to decide number of employees to be recruited through various grades, to estimate the number of employees for Promotion and to Retirement etc. Keywords Manpower planning Stochastic programming Optimal recruitment policy
P. T. Rao P. R. S. Reddy (&) A. V. S. Suhasini Department of Statistics, S.V. University, Tirupati 517 502, India e-mail: [email protected] P. T. Rao e-mail: [email protected] A. V. S. Suhasini e-mail: [email protected]
1 Introduction In the competitive world of today, manpower planning draws a significant attention of researchers. Determining manpower planning policies is one of the most critical aspects of an organization. The management of any organization not only optimizes the expertise and skills of its human resources, but may also select the optimal number and suitable type of employees available at the right time. Manpower forecasts are important for assisting the Human resource agencies in the policy-making process. Forecasts based on an inaccurate analysis can cause either undersupply or oversupply of manpower. Management should not only be mindful of the outcome of the performance reward systems but also the process of how to implement those systems. Manpower planning depends on the highly unpredictable human behaviour and the uncertain social environment in which the system functions. The study of probabilistic or stochastic models of Manpower systems have been proposed and studied extensively in the past by many researchers. Optimization models are developed to obtain the business strategy of the organization, and in turn the manpower management goal is to find appropriate size of manpower supply. Markov Analysis (MA) is a very suitable technique mostly used in manpower planning. Markov chains make it possible to predict the size of manpower per category (per state) as well as transitions occurring within a given time period in the future (Resignation, Dismissal, Retirement, Death, etc.). Markovian models have been applied in examining the structure of manpower systems in terms of the proportion of staff in each grade or age profile of staff under a variety of conditions and evaluating policies for controlling manpower
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systems. The movements of individuals are characterized by replacements (renewals) according to probabilistic laws, and such models of manpower systems are referred as renewal models. An important element to consider in designing a manpower supply–demand model is the time delays occurring within the system. Delays can increase the instability of a system, leading to undesirable behaviors. Any organizational structure is generally built on a graded manpower system in which a member of the organization can belong to only one of the several mutually exclusive grades. There are several grades in a large number of manpower organizations each grade is further subdivided into several categories for administration reasons. These categories may be several departments or sections within grades or divisions consisting of persons who have completed 0 years of service, 1 year of service, 2 years of service, etc. The proportion of promotion will be different for each category and hence dependent not only on the grade size but also on the category size. Graded Manpower systems have been studied from different points of view by several researchers. An employee in a lower grade is eligible for promotion to the next higher grade and the probability of promotion is dependent on the grade and category of the employee. The flow of people in manpower systems, moving employees in various states can be subdivided into recruitment stream, the transition between the state and the outflow from the system. The concept of linear programming is used to develop a graded population structure where both the recruitment and transfer rates are in between various grades, which are controlled by management. Recruitment control refers to an effective control of recruitment policies to obtain an optimal supply of recruits for a system at any time. There is much evidence in the literature on modeling of manpower. Various kinds of approaches in manpower planning through mathematical models using Rigid and flexible hierarchy and computer simulation models were reviewed by Dill et al. (1966). A minimum risk manpower scheduling technique is developed by Roger FJ (1967) by assuming possible work load can be described by a finite set of mutually exclusive manpower requirements. A stochastic model of migration, occupational and vertical mobility based on theory of semi—markov process is derived by Ginsberg (1971). An optimization model to determine recruitment plans was developed by Gary et al. (1975). A discrete renewal model for graded population using linear programming techniques is developed by Vajda (1985). Total Number of vacancies available in an organization was studied through a semi-markov model of a manpower system by Yadavalli and Natarajan (2001). stochastic models of manpower systems of pressure impact
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on promotion with time dependent behaviour is studied by Yadavalli et al. (2002). Stochastic programming in cluster based optimum allocation of recruitment was discussed by Jeeva et al. (2004). Two graded manpower System using Stochastic modeling developed by Rao et al. (2006). A queuing approach to determine the optimal promotion policy and total optimal cost of promotion in a manpower planning system is developed by Yadavalli et al. (2006). LPP model based on integer programming to determine effective/optimal manpower size is applied by Akinyele et al. (2007). A stochastic manpower planning model under the set up of varying class sizes and promotion was developed by Chattopadhyay et al. (2007). A linear programming based effective maintenance and manpower planning strategy is described by Kareem and Aderoba (2008). Most of the organizations employed the people on graded system. Usually an employee is recruited for initial grade and later his/her services will be used by updating their work skills or by upgrading than to higher grades. The phenomena of wastage and promotion will be observed at each and every stage of initial grades. The mobility of human resources from the current grade to the next grade with in organization may be due to promotion. The optimal requirement of manpower sizes in each grade under study are probabilistic rather Deterministic. Stochastic modeling is more rational than any other conventional models. In this paper we develop a stochastic programming problem for optimization of recruitment and other costs by considering the constraints like salaries and investments on various heads. The objective function and influencing constraints are modeled by assuming linear programming approach. We have considered an organization with two grades of manpower requirements such as direct recruitment to grade-I and also for grade-II, promotion from grade I to grade II and retrenchment with and without allowing promotions are there in our study. Let N(t), M(t) are the Number of employees in grade I, grade II at time t, respectively. Grade I is an initial stage of recruitment and Grade II is a promotion stage and leaving grade is a retirement stage. Let k, d, a, b, c be the rates of recruitment to grade I; direct recruitment to grade II; retrenchments from grade I; promotion from grade I to grade II; retirements from grade II respectively. All are assumed as parameters of Poisson process. The duration of stay at grade I and grade II are assumed as exponential with parameters b, aand c respectively, further they are independent to each other. Let Pn;m ðtÞ be the probability that there are ‘n’ employees in grade I and ‘m’ employees in grade II at time t. The Joint Probability mass function at steady state is
Int J Syst Assur Eng Manag (June 2010) 1(2)
Pn;m ¼ e
h
kðc2 þudÞ k2 ðeþbÞ
i n m k d kb þ ðn! m!Þ1 aþb c cða þ bÞ
for m, n = 0, 1, 2, … (Rao et al. 2006) The Average number k of employees in Grade I is EðNÞ ¼ aþb and grade II is Eð M Þ ¼
d kb þ c cða þ bÞ
2 Stochastic programming problem The input/output rates of recruitment/promotion/retirements depend on various mentioned parameters namelya, b, c, d, k. An organization’s human resources are to be optimally utilized by meeting various constraints. A stochastic non-linear programming problem by developing the objective function of cost minimization, influenced by the constraints of various costs like recruitment, salaries, Training, maintenance and overheads during shortage of staff is required. The size of the staff in each grade and optimal size of employees are also considered for developing suitable constraints. Let ‘A’ be a setup cost for an organization in recruitment procedure. Let r1 be the cost of recruitment per individual belong to grade I and Let r2 be the cost of recruitment per individual belong to grade II. Let p1, p2 be the penalty costs per individual due to shortage of staff in grade I and grade II, respectively. Let s1, s2 be the salaries paid to each individual in grade I, grade II, respectively. Let N0 ; M0 be the initial required sizes of Manpower (no. of employees) in grade I and grade II, respectively. Let M0 EðMÞ [ 0 be the shortage of Manpower in grade II and let N0 EðNÞ [ 0 be the shortage Manpower in grade I. The process of recruitment will be carried out when the shortage in the size of the employees in grade I and grade II. The phenomena of penalty cost (or) cost paid to employees who recruited to deal overtime works (or) out sourcing manpower to handle the work due to employee scarcity. The cost of payment for salaries is applicable to the section of Manpower, who is currently working with organization. Therefore the work load on which the recruitment costs and storage costs are to be bared in grade-I is N0 - E(N). and in grade-II is M0 - E(M). Considering the above work loads of an organization, The cost function is defined as KðTÞ ¼ A þ r1 N0 þ r2 M0 þ p1 N0 þ p2 M0 þ ðs1 r1 p1 Þ EðNÞ þ ðs2 r2 p2 Þ EðMÞ This implies KðTÞ ¼ C þ C1 EðNÞ þ C2 EðMÞ considering the values of EðNÞ and EðMÞ This implies
by
KðTÞ ¼ C þ C1
k d b k þ C2 þ aþb c c aþb
ð3:1Þ
where C ¼ A þ ðr1 þ p1 ÞN0 þ ðr2 þ p2 ÞM0 ; C1 ¼ s1 r1 p1 ; C2 ¼ s2 r2 p2 Usually in any two graded recruitment systems, the initial grade size is more than that of next grade. i.e., N0 C M0. Similarly the salary in initial graded is less than grade II i.e., s1 B s2. The Same phenomena may be observed in holding costs, recruitment costs and penalty costs also. Therefore we may observe the relationships between grade I and grade II regarding the above mentioned costs as r1 B r2, p1 B p2 and so on. Let s1 ; s2 be the salaries per individual in grade I and grade II, respectively, Let S be the maximum salary (or allowable budget) that a company can pay to its employees. Then the constraint with salary limit is k d kb s1 þ þ s2 S ð3:2Þ aþb c cða þ bÞ Let p1, p2 be the penalties/shortage costs per individual to meet the requirements of grade I and grade II respectively and let P be the maximum allowable staff shortage cost. Then the constraint with Shortage of staff is k d kb p1 N 0 þ p2 M 0 P ð3:3Þ aþb c cða þ bÞ Let r1 , r2 be the recruitment costs per one individual of grade I, grade II, respectively and let R be the maximum allowable cost of an organization for recruitment purpose. Then the constraint with recruitment cost is k d kb r1 N0 þ r2 M0 R: ð3:4Þ aþb c cða þ bÞ Let Z1 be the number of man hours per one individual to work in grade I., then the total expected Man hours at grade k I isZ1 aþb . If total available Man hours as ‘N0 ’ People are recruited isZ1 N0 . Then the Constraint with man hours for k Z1 N 0 . grade I, is Z1 aþb k This implies aþb N0 . The Shortage phenomena is k observed in grade I when N0 aþb Then the constraint with shortage cost in grade I Manpower is k N0 aþb
ð3:5Þ
Let Z2 be the number of man hours per one individual to work in grade-II then the total expected man hours at kb grade-II is Z2 dc þ cðaþbÞ . If the total available Man hours as M0 people are recruited is Z2 :M0 , then the constraint kb with Manpower for grade – II isZ2 dc þ cðaþbÞ Z2 M 0 .
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Therefore the constraint with shortage cost in grade II Manpower is d kb þ M0 ð3:6Þ c cða þ bÞ Let a, b, c are non-influenced factors on recruitment (or) promotion scheme then a, b, c are considered to be constants. Assuming a,b and c are non controllable parameters of manpower planning models. The objective function can be written as Min K(T) = C ? d1k ? d2d; where C¼ A þ ðr1 þ p1 Þ N0 þðr2 þ p2 Þ M0 ; d1 C1 C2 :b C2 þ ¼ ð3:7Þ d2 ¼ ; a þ b c:ða þ bÞ c C1 ¼ s1 r1 p1 ; C2 ¼ s2 r2 p2 Assuming k, d are the decision parameters, a, b, c are known constants, then the programming problem is influenced on set of constraints. The constraint on salaries is k a11 þ d a12 S; where s1 s2 b 1 s2 :b ¼ þ a11 ¼ s1 þ ; aþb c a þ b c ða þ bÞ s2 ð3:8Þ a12 ¼ c The constraint on penalties due to shortage of staff is k a21 þ d a22 p1 N0 þ p2 M0 P; where p1 p2 b 1 p2 b ¼ þ p1 þ ; a21 ¼ aþb c a þ b c ða þ bÞ p2 ð3:9Þ a22 ¼ c The constraint on recruitment cost due to the shortage of staff is k a31 þ d a32 r1 N0 þ r2 M0 R; where r1 r2 b 1 r2 b ¼ r1 þ ; a31 ¼ þ aþb c a þ b c ða þ bÞ r2 a32 ¼ ð3:10Þ c The constraint with the required/necessary size of staff in grade-I is 1 k a41 N0 ; where a41 ¼ ð3:11Þ aþb The constraint with the required (or) necessary size of staff in grade-II is b 1 k a51 þ d a52 M0 ; where a51 ¼ ; a52 ¼ c:ða þ bÞ c ð3:12Þ All the decision parameters under study are non-negative. All other variables under study are also considered as
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non-negative. Therefore N0, M0 C 0, S, R, P, C 0, s1, s2, r1, r2, p1, p2 C 0 and more over a, b, c, d, k C 0.
3 Numerical illustration and analysis In order to study the behaviour and to extract various decision parameters. The inputs are considered from the observations of a local Battery manufacturing industry. Data on the employee structure, graded systems and salaries structures of organization are obtained from the secondary data sources. We have considered the information of initial salary structures. The increments of salaries are simulated by assuming the 10, 20, 30%, Increments to the total salaries. Most of the times the existing staff is attending the work load of shortage. We have considered the summarized information and approximate salaries, over time charges. Regarding recruitment cost also the organization is having their own human resource management department. HR department is looking after the recruitment formalities. We considered the estimated work load in the mentioned grades through our developed model. The mentioned recruitment costs are considered through the experience and the past data. The study has concentrated on the influencing factors of decision parameters. The rate of recruitment to grade I (k) and the direct rate of recruitment to grade II (d) are estimated through various inputs like, salaries (s1, s2); Shortage costs (p1, p2); recruitment costs (r1, r2); The required sizes of work force at grade I and grade II (N0, M0); The total budget allocations for salaries, for shortage maintenance, for running the HR department (S, P, R). Further the sensitivity analysis is carried out through the rate of retirement from grade I (a); the rate of promotion from grade I (b) and the rate of recruitment from grade II (c).The employee’s requirements into grade I and grade II (k,d) are obtained by using MS excel solver and TORA softwares. The cost function values are calculated for different sets of variables, includes s1, s2, S; p1, p2, P; r1, r2, R; N0, M0; a, b, c, k, d and A. At a glance table for outputs k, d and K (T) is presented for varying values of a, b, c and other input variables namely salaries, shortage costs, recruitment costs and staff size requirements. This model is applied for exploring the dependable parameters k, d for various varying values of A, s1, s2, S, p1, p2, P, r1, r2, R etc., usually recruitment cost is under control of Human Resource Development Department of the organization. The organization has its own set up for procedure of recruitment through the department. The independent parameters such as a, b, c, N0, M0 are considered as inputs along with the above mentioned factors. From Fig. 1, it is observed that k is an increasing function and d is a decreasing function of N0 as the remaining parameters are constant. From Fig. 2, d is an increasing function of M0.
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Figure 3 reveals that lambda (k) is an increasing function of alpha (a) and delta (d) is non sensitive to alpha (a). Figure 4 indicates that lambda (k) is an increasing of Beta (b) and delta (d) remain zero and non sensitive to Beta (b). Figure 5 reveals that delta is an increasing function of gamma where as lambda is non sensitive to gamma. It is observed that there is no direct recruitment in grade II (d = 0) and it is non-sensitive to promotion grade of employee from grade I to grade II (b remain same) where as the recruitment rate of grade I (k) is an increasing function of the promotion rate from grade I to grade II (b). From Figs. 6 and 7 it is observed that cost function is an increasing function of both N0 and M0. The influence of decision parameters on the cost function is reviewed from the Figs. 8, 9, and 10, it is observed that the cost function is non sensitive to a, c and increasing function of b.
grade II and a range of 27.89 (%28) to 32.08 (%32) recruitments to grade I per 100 to the existing manpower for A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000, N0 = 150; M0 = 110; a = 5; b = 10. The rate of retrenchments/retirements from grade II are ranging from 30 to 75 suggests 22.5 (%23) recruitment to grade I and range from 9 to 45 direct recruitments to grade II per 100 of existing manpower. For A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000; N0 = 150; M0 = 110; a = 5; b = 10. The rate of recruitment in grade I will be increased by increasing the requirement to grade I and also the rate of direct recruitment in grade II is decreased with increase size in grade I.The rate of direct recruitment to grade II is an increase function of the optimal requirement of manpower M0. The rate of recruitment to grade I is positively influenced by its retrenchments. The recruitment rate to initial grade is highly influenced by the promotion rate from grade I to grade II (b). The direct recruitment to grade II is highly influenced by the rate of vacancies to grade II or rate of retirement from grade II. The impact of retirements effects recruitments of grade II where as it has no impact on the recruitment to grade I. Attaining optimality in cost reduction has inverse impact on the size of manpower requirement in grade I and grade II. The total cost of an organization can be optimal by decreasing the values of required size in grade I and grade II. The optimal objective cost is not influenced by either rate of retrenchments or the rate of retirements where the objective function is increasing with the increase of rate of promotion from grade I to grade II. Influence of decision parameters on total cost of an organization can be minimized by decreasing the promotion from grade I to grade II. The rate of recruitments in grade I and direct recruitment rate to grade II, influenced by total layout of budgets on salary, staff shortage cost and recruitment expenditure.
4 Findings and conclusions The optimal size of manpower in grade I ranging from 113 to 180 suggests the recruitments of range 16.875 (%17) to 27. in grade I and 2 to 9 in the direct recruitment of grade II per 100 size of existing manpower. For A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000, M0 = 110, a = 5; b = 10; c = 25. The optimal size of manpower in grade II ranging from 83 to 138 suggests the recruitment rate of range 26.25–30 to grade I; the direct recruitment rate ‘0’ in between the range 116 to 138 for M0 for A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000; N0 = 150, a = 5; b = 10; c = 25. Important factors of influence on recruitment rates are lies with a, b, c. If the rate of leaving/retrenchment from grade I is ranging from 5 to 23, it suggests the recruitments to grade I are in the range from 22.5 to 49.5 per hundred of existing manpower of grade I and further suggests only 5 direct recruitments to grade II. For A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000; N0 = 150; M0 = 110; b = 10; c = 25. The rate of promotions from grade I to grade II are ranging from 14 to 30 suggests no direct recruitment to
Appendix 1 See Table 1.
Table 1 The decision parameters Lambda and Delta; the optimum cost for various values of alpha, beta, gamma, N0, M0 N0
M0
113
110
a 5
b
c
10
25
k
d
K(T)
1687.5
875
1979750
128
1912.5
725
2092250
143
2137.5
575
2204750
123
Int J Syst Assur Eng Manag (June 2010) 1(2) Table 1 continued N0
M0
a
b
c
k
d
K(T)
158
2362.5
425
173
2587.5
275
2429750
2625
0
2306500
150
150
150
150
83
5
10
25
2317250
94
2775
0
2438300
105
2925
0
2570100
121
3000
275
2792200
132
3000
550
2948400
2250
500
2261000
110
110
110
5
10
25
9
2850
500
2261000
13
3450
500
2261000
17
4050
500
2261000
21
4650
500
2261000
2789.683 2996.479
0 0
2268746 2309113
20
3083.333
0
2326067
24
3231.928
0
2355072
28
3241.071
0
2298607
5
14 18
5
25
10
30
2250
900
2261000
40
2250
1700
2261000
50
2250
2500
2261000
60
2250
3300
2261000
70
2250
4100
2261000
Appendix 2 Inputs of manpower requirements See Figs. 1 and 2.
Required size for grade I with parameters
N0
lambda
delta
113 120 128 135 143 150
1687.5 1800 1912.5 2025 2137.5 2250
875 800 725 650 575 500
2500
parameters
Fig. 1 For the values of s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; M0 = 110, a = 5; b = 10; c = 25
lamda delta
2000 1500 1000 500 0 0
50
100
150
200
Required size in grade I
Required Size In grade II with parameters
M0
lambda
delta
105 116 121 127 132 138
2925 3000 3000 3000 3000 3000
0 137.5 275 412.5 550 687.5
4000
parameters
Fig. 2 For the Values of s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; b = 10; c = 25
lamda delta
3000 2000 1000 0 0
50
100
required Size in gradeII
123
150
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Inputs of decision parameters See Figs. 3, 4 and 5.
Fig. 4 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; M0 = 110, a = 5; c = 25
beta
14 16 18 20 22 24
lambda delta 2789.683 0 2899.254 0 2996.479 0 3083.333 0 3161.392 0 3231.928 0
objective function with alpha
objective function
alpha lambda delta 13 3450 500 15 3750 500 17 4050 500 19 4350 500 21 4650 500 23 4950 500
6000 5000 4000 3000 2000 1000 0
lamda delta
0
5
10
15
20
25
Alpha
Objective function
Fig. 3 For the Values of s1 = 15000; s2 = 25000; S =7000000; p1 = 7000; p2 =10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; M0 = 110, b = 10; c = 25
objective function with beta 4000 lamda delta
3000 2000 1000 0 0
5
10
15
20
25
30
Fig. 5 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; M0 = 110, a = 5; b = 10
objective function
Beta
gamma lambda delta 30 2250 900 35 2250 1300 40 2250 1700 45 2250 2100 50 2250 2500 55 2250 2900
objective function with gamma 4000
lamda delta
3000 2000 1000 0 0
10
20
30
40
50
60
Gamma
Inputs of manpower requirements See Figs. 6 and 7.
Fig. 7 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; b = 10; c = 25
113 120 128 135 143 150
Objective function 1979750 2036000 2092250 2148500 2204750 2261000
Objective function with Required size in grade I objective function
N0
2300000 2200000 2100000
Objective function
2000000 1900000 0
50
100
150
200
Required size in grade I Objective function with required size in grade II
M0
105 116 121 127 132 138
Objective function 2570100 2714100 2792200 2870300 2948400 3026500
3100000
objective function
Fig. 6 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; M0 = 110, a = 5; b = 10; c = 25
3000000 2900000 2800000 Objective function
2700000 2600000 2500000 0
50
100
150
Required size in grade II
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Inputs of decision parameters
Fig. 9 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; c = 25
alpha 13 15 17 19 21 23
objective function 2261000 2261000 2261000 2261000 2261000 2261000
objective functionith alpha
2262000
objective function
2261000
2260000 0
10
20
30
alpha
objective function with beta
beta
14 16 18 20 22 24
objective function 2268746.03 2290134.33 2309112.68 2326066.67 2341303.8 2355072.29
objective function
Fig. 8 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; b = 10; c = 25
objective function
See Figs. 8, 9 and 10.
2360000 2340000 2320000
objective function
2300000 2280000 2260000 0
10
20
30
Fig. 10 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; b = 10
objective gamma function 30 2261000 35 2261000 40 2261000 45 2261000 50 2261000 55 2261000
References Chattopadhyay et al (2007) A stochastic manpower planning model under varying class sizes. Ann Oper Res 155:41–49 Dill WR, Gawer DP, Weber WL (1966) Models and modeling for manpower planning. Manag Sci 13(4):B142–B167 Gary L et al (1975) A model for manpower management. Manag Sci 21(12) Ginsberg RB (1971) Semi-Markov process and mobility. J Math Sociol 1:233–262 Glen JJ (1977) Length of service distributions in markov manpower models. Oper Res Q 28:975–982 Jeeva M et al (2004) An application of stochastic programming with Weibull distribution–cluster based optimum allocation of recruitment in manpower planning. Stoch Anal Appl 22(3):801–812 Jewelt RJ (1967) A minimum risk manpower schedule technique. Manag Sci 13(10):B578–B592 Kareem B, Aderoba AA (2008) Linear programming based effective maintenance and manpower planning strategy: a case study. J Comput Internet Manag 16(2):26–34
123
objective function
beta
objective function with gamma 2262000
objective function
2261000
2260000 0
20
40
60
gamma
Srinivasa Rao K, Srinivasa Rao V, Vivenkanada Murty M (2006) On two graded manpower planning model. Opsearch 43(3): 225–237 Taiwo AS (2007) Determination of the optimal manpower size using linear programming model. Res J Bus Manag 1(1):30–36 Vajda S (1985) The use of linear programming in manpower planning; Homenaje Al Profesor Sixto Rios Trabajos De Estadistica De Investigacion Operativa 36(3):301–311 Yadavalli VSS, Natarajan R (2001) A semi-Markov modelo f manpower system. Stoch Anal Appl 19(6):1077–1086 Yadavalli VSS, Natarajan R, Udayabhaskaran S (2002) Time dependent behaviour of Stochastic models of manpower system—impact of pressure on promotion. Stoch Anal Appl 20(4): 863–882 Yadavalli VSS, Setlhare K, Mishra SS, Rajalakshmi R (2006) An analysis of an optimal promotion policy for a manpower system using a queuing approach. Manag Dyn. http://findarticles.com/p/ articles/mi_qa5465/is_200604/ai_n21402995/
RESEARCH ARTICLE
Optimal manpower recruitment by stochastic programming in graded manpower systems P. Tirupathi Rao • P. R. S. Reddy • A. V. S. Suhasini
Received: 10 February 2010 / Revised: 11 March 2010 The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2010
Abstract Human resource is one of the most effective resources in the development of any organization or a Nation. Optimal utilization of manpower is a pivotal challenge to a manager of a dynamic management system. Achieving the objectives by meeting the constraints on various feasibilities is the core problem before statisticians, model developers and OR scientists. In this paper we develop a model with Stochastic Programming Problem of manpower recruitment for an organization where the graded manpower system is observed. The averages, variance of number of employees in initial and final grades before leaving/resigning organization are computed by a suitable bivariate probability mass function. The costs like setup cost, recruitment cost, salaries, Cost of Training, Promotion Costs etc., are considered while developing a model. Sensitivity analysis was also carried out and observed the model behaviour. This model is a suitable tool to serve the decision maker for making optimal decisions on Recruitment policies for determining the optimal recruitment cost, to decide number of employees to be recruited through various grades, to estimate the number of employees for Promotion and to Retirement etc. Keywords Manpower planning Stochastic programming Optimal recruitment policy
P. T. Rao P. R. S. Reddy (&) A. V. S. Suhasini Department of Statistics, S.V. University, Tirupati 517 502, India e-mail: [email protected] P. T. Rao e-mail: [email protected] A. V. S. Suhasini e-mail: [email protected]
1 Introduction In the competitive world of today, manpower planning draws a significant attention of researchers. Determining manpower planning policies is one of the most critical aspects of an organization. The management of any organization not only optimizes the expertise and skills of its human resources, but may also select the optimal number and suitable type of employees available at the right time. Manpower forecasts are important for assisting the Human resource agencies in the policy-making process. Forecasts based on an inaccurate analysis can cause either undersupply or oversupply of manpower. Management should not only be mindful of the outcome of the performance reward systems but also the process of how to implement those systems. Manpower planning depends on the highly unpredictable human behaviour and the uncertain social environment in which the system functions. The study of probabilistic or stochastic models of Manpower systems have been proposed and studied extensively in the past by many researchers. Optimization models are developed to obtain the business strategy of the organization, and in turn the manpower management goal is to find appropriate size of manpower supply. Markov Analysis (MA) is a very suitable technique mostly used in manpower planning. Markov chains make it possible to predict the size of manpower per category (per state) as well as transitions occurring within a given time period in the future (Resignation, Dismissal, Retirement, Death, etc.). Markovian models have been applied in examining the structure of manpower systems in terms of the proportion of staff in each grade or age profile of staff under a variety of conditions and evaluating policies for controlling manpower
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systems. The movements of individuals are characterized by replacements (renewals) according to probabilistic laws, and such models of manpower systems are referred as renewal models. An important element to consider in designing a manpower supply–demand model is the time delays occurring within the system. Delays can increase the instability of a system, leading to undesirable behaviors. Any organizational structure is generally built on a graded manpower system in which a member of the organization can belong to only one of the several mutually exclusive grades. There are several grades in a large number of manpower organizations each grade is further subdivided into several categories for administration reasons. These categories may be several departments or sections within grades or divisions consisting of persons who have completed 0 years of service, 1 year of service, 2 years of service, etc. The proportion of promotion will be different for each category and hence dependent not only on the grade size but also on the category size. Graded Manpower systems have been studied from different points of view by several researchers. An employee in a lower grade is eligible for promotion to the next higher grade and the probability of promotion is dependent on the grade and category of the employee. The flow of people in manpower systems, moving employees in various states can be subdivided into recruitment stream, the transition between the state and the outflow from the system. The concept of linear programming is used to develop a graded population structure where both the recruitment and transfer rates are in between various grades, which are controlled by management. Recruitment control refers to an effective control of recruitment policies to obtain an optimal supply of recruits for a system at any time. There is much evidence in the literature on modeling of manpower. Various kinds of approaches in manpower planning through mathematical models using Rigid and flexible hierarchy and computer simulation models were reviewed by Dill et al. (1966). A minimum risk manpower scheduling technique is developed by Roger FJ (1967) by assuming possible work load can be described by a finite set of mutually exclusive manpower requirements. A stochastic model of migration, occupational and vertical mobility based on theory of semi—markov process is derived by Ginsberg (1971). An optimization model to determine recruitment plans was developed by Gary et al. (1975). A discrete renewal model for graded population using linear programming techniques is developed by Vajda (1985). Total Number of vacancies available in an organization was studied through a semi-markov model of a manpower system by Yadavalli and Natarajan (2001). stochastic models of manpower systems of pressure impact
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on promotion with time dependent behaviour is studied by Yadavalli et al. (2002). Stochastic programming in cluster based optimum allocation of recruitment was discussed by Jeeva et al. (2004). Two graded manpower System using Stochastic modeling developed by Rao et al. (2006). A queuing approach to determine the optimal promotion policy and total optimal cost of promotion in a manpower planning system is developed by Yadavalli et al. (2006). LPP model based on integer programming to determine effective/optimal manpower size is applied by Akinyele et al. (2007). A stochastic manpower planning model under the set up of varying class sizes and promotion was developed by Chattopadhyay et al. (2007). A linear programming based effective maintenance and manpower planning strategy is described by Kareem and Aderoba (2008). Most of the organizations employed the people on graded system. Usually an employee is recruited for initial grade and later his/her services will be used by updating their work skills or by upgrading than to higher grades. The phenomena of wastage and promotion will be observed at each and every stage of initial grades. The mobility of human resources from the current grade to the next grade with in organization may be due to promotion. The optimal requirement of manpower sizes in each grade under study are probabilistic rather Deterministic. Stochastic modeling is more rational than any other conventional models. In this paper we develop a stochastic programming problem for optimization of recruitment and other costs by considering the constraints like salaries and investments on various heads. The objective function and influencing constraints are modeled by assuming linear programming approach. We have considered an organization with two grades of manpower requirements such as direct recruitment to grade-I and also for grade-II, promotion from grade I to grade II and retrenchment with and without allowing promotions are there in our study. Let N(t), M(t) are the Number of employees in grade I, grade II at time t, respectively. Grade I is an initial stage of recruitment and Grade II is a promotion stage and leaving grade is a retirement stage. Let k, d, a, b, c be the rates of recruitment to grade I; direct recruitment to grade II; retrenchments from grade I; promotion from grade I to grade II; retirements from grade II respectively. All are assumed as parameters of Poisson process. The duration of stay at grade I and grade II are assumed as exponential with parameters b, aand c respectively, further they are independent to each other. Let Pn;m ðtÞ be the probability that there are ‘n’ employees in grade I and ‘m’ employees in grade II at time t. The Joint Probability mass function at steady state is
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Pn;m ¼ e
h
kðc2 þudÞ k2 ðeþbÞ
i n m k d kb þ ðn! m!Þ1 aþb c cða þ bÞ
for m, n = 0, 1, 2, … (Rao et al. 2006) The Average number k of employees in Grade I is EðNÞ ¼ aþb and grade II is Eð M Þ ¼
d kb þ c cða þ bÞ
2 Stochastic programming problem The input/output rates of recruitment/promotion/retirements depend on various mentioned parameters namelya, b, c, d, k. An organization’s human resources are to be optimally utilized by meeting various constraints. A stochastic non-linear programming problem by developing the objective function of cost minimization, influenced by the constraints of various costs like recruitment, salaries, Training, maintenance and overheads during shortage of staff is required. The size of the staff in each grade and optimal size of employees are also considered for developing suitable constraints. Let ‘A’ be a setup cost for an organization in recruitment procedure. Let r1 be the cost of recruitment per individual belong to grade I and Let r2 be the cost of recruitment per individual belong to grade II. Let p1, p2 be the penalty costs per individual due to shortage of staff in grade I and grade II, respectively. Let s1, s2 be the salaries paid to each individual in grade I, grade II, respectively. Let N0 ; M0 be the initial required sizes of Manpower (no. of employees) in grade I and grade II, respectively. Let M0 EðMÞ [ 0 be the shortage of Manpower in grade II and let N0 EðNÞ [ 0 be the shortage Manpower in grade I. The process of recruitment will be carried out when the shortage in the size of the employees in grade I and grade II. The phenomena of penalty cost (or) cost paid to employees who recruited to deal overtime works (or) out sourcing manpower to handle the work due to employee scarcity. The cost of payment for salaries is applicable to the section of Manpower, who is currently working with organization. Therefore the work load on which the recruitment costs and storage costs are to be bared in grade-I is N0 - E(N). and in grade-II is M0 - E(M). Considering the above work loads of an organization, The cost function is defined as KðTÞ ¼ A þ r1 N0 þ r2 M0 þ p1 N0 þ p2 M0 þ ðs1 r1 p1 Þ EðNÞ þ ðs2 r2 p2 Þ EðMÞ This implies KðTÞ ¼ C þ C1 EðNÞ þ C2 EðMÞ considering the values of EðNÞ and EðMÞ This implies
by
KðTÞ ¼ C þ C1
k d b k þ C2 þ aþb c c aþb
ð3:1Þ
where C ¼ A þ ðr1 þ p1 ÞN0 þ ðr2 þ p2 ÞM0 ; C1 ¼ s1 r1 p1 ; C2 ¼ s2 r2 p2 Usually in any two graded recruitment systems, the initial grade size is more than that of next grade. i.e., N0 C M0. Similarly the salary in initial graded is less than grade II i.e., s1 B s2. The Same phenomena may be observed in holding costs, recruitment costs and penalty costs also. Therefore we may observe the relationships between grade I and grade II regarding the above mentioned costs as r1 B r2, p1 B p2 and so on. Let s1 ; s2 be the salaries per individual in grade I and grade II, respectively, Let S be the maximum salary (or allowable budget) that a company can pay to its employees. Then the constraint with salary limit is k d kb s1 þ þ s2 S ð3:2Þ aþb c cða þ bÞ Let p1, p2 be the penalties/shortage costs per individual to meet the requirements of grade I and grade II respectively and let P be the maximum allowable staff shortage cost. Then the constraint with Shortage of staff is k d kb p1 N 0 þ p2 M 0 P ð3:3Þ aþb c cða þ bÞ Let r1 , r2 be the recruitment costs per one individual of grade I, grade II, respectively and let R be the maximum allowable cost of an organization for recruitment purpose. Then the constraint with recruitment cost is k d kb r1 N0 þ r2 M0 R: ð3:4Þ aþb c cða þ bÞ Let Z1 be the number of man hours per one individual to work in grade I., then the total expected Man hours at grade k I isZ1 aþb . If total available Man hours as ‘N0 ’ People are recruited isZ1 N0 . Then the Constraint with man hours for k Z1 N 0 . grade I, is Z1 aþb k This implies aþb N0 . The Shortage phenomena is k observed in grade I when N0 aþb Then the constraint with shortage cost in grade I Manpower is k N0 aþb
ð3:5Þ
Let Z2 be the number of man hours per one individual to work in grade-II then the total expected man hours at kb grade-II is Z2 dc þ cðaþbÞ . If the total available Man hours as M0 people are recruited is Z2 :M0 , then the constraint kb with Manpower for grade – II isZ2 dc þ cðaþbÞ Z2 M 0 .
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Therefore the constraint with shortage cost in grade II Manpower is d kb þ M0 ð3:6Þ c cða þ bÞ Let a, b, c are non-influenced factors on recruitment (or) promotion scheme then a, b, c are considered to be constants. Assuming a,b and c are non controllable parameters of manpower planning models. The objective function can be written as Min K(T) = C ? d1k ? d2d; where C¼ A þ ðr1 þ p1 Þ N0 þðr2 þ p2 Þ M0 ; d1 C1 C2 :b C2 þ ¼ ð3:7Þ d2 ¼ ; a þ b c:ða þ bÞ c C1 ¼ s1 r1 p1 ; C2 ¼ s2 r2 p2 Assuming k, d are the decision parameters, a, b, c are known constants, then the programming problem is influenced on set of constraints. The constraint on salaries is k a11 þ d a12 S; where s1 s2 b 1 s2 :b ¼ þ a11 ¼ s1 þ ; aþb c a þ b c ða þ bÞ s2 ð3:8Þ a12 ¼ c The constraint on penalties due to shortage of staff is k a21 þ d a22 p1 N0 þ p2 M0 P; where p1 p2 b 1 p2 b ¼ þ p1 þ ; a21 ¼ aþb c a þ b c ða þ bÞ p2 ð3:9Þ a22 ¼ c The constraint on recruitment cost due to the shortage of staff is k a31 þ d a32 r1 N0 þ r2 M0 R; where r1 r2 b 1 r2 b ¼ r1 þ ; a31 ¼ þ aþb c a þ b c ða þ bÞ r2 a32 ¼ ð3:10Þ c The constraint with the required/necessary size of staff in grade-I is 1 k a41 N0 ; where a41 ¼ ð3:11Þ aþb The constraint with the required (or) necessary size of staff in grade-II is b 1 k a51 þ d a52 M0 ; where a51 ¼ ; a52 ¼ c:ða þ bÞ c ð3:12Þ All the decision parameters under study are non-negative. All other variables under study are also considered as
123
non-negative. Therefore N0, M0 C 0, S, R, P, C 0, s1, s2, r1, r2, p1, p2 C 0 and more over a, b, c, d, k C 0.
3 Numerical illustration and analysis In order to study the behaviour and to extract various decision parameters. The inputs are considered from the observations of a local Battery manufacturing industry. Data on the employee structure, graded systems and salaries structures of organization are obtained from the secondary data sources. We have considered the information of initial salary structures. The increments of salaries are simulated by assuming the 10, 20, 30%, Increments to the total salaries. Most of the times the existing staff is attending the work load of shortage. We have considered the summarized information and approximate salaries, over time charges. Regarding recruitment cost also the organization is having their own human resource management department. HR department is looking after the recruitment formalities. We considered the estimated work load in the mentioned grades through our developed model. The mentioned recruitment costs are considered through the experience and the past data. The study has concentrated on the influencing factors of decision parameters. The rate of recruitment to grade I (k) and the direct rate of recruitment to grade II (d) are estimated through various inputs like, salaries (s1, s2); Shortage costs (p1, p2); recruitment costs (r1, r2); The required sizes of work force at grade I and grade II (N0, M0); The total budget allocations for salaries, for shortage maintenance, for running the HR department (S, P, R). Further the sensitivity analysis is carried out through the rate of retirement from grade I (a); the rate of promotion from grade I (b) and the rate of recruitment from grade II (c).The employee’s requirements into grade I and grade II (k,d) are obtained by using MS excel solver and TORA softwares. The cost function values are calculated for different sets of variables, includes s1, s2, S; p1, p2, P; r1, r2, R; N0, M0; a, b, c, k, d and A. At a glance table for outputs k, d and K (T) is presented for varying values of a, b, c and other input variables namely salaries, shortage costs, recruitment costs and staff size requirements. This model is applied for exploring the dependable parameters k, d for various varying values of A, s1, s2, S, p1, p2, P, r1, r2, R etc., usually recruitment cost is under control of Human Resource Development Department of the organization. The organization has its own set up for procedure of recruitment through the department. The independent parameters such as a, b, c, N0, M0 are considered as inputs along with the above mentioned factors. From Fig. 1, it is observed that k is an increasing function and d is a decreasing function of N0 as the remaining parameters are constant. From Fig. 2, d is an increasing function of M0.
Int J Syst Assur Eng Manag (June 2010) 1(2)
Figure 3 reveals that lambda (k) is an increasing function of alpha (a) and delta (d) is non sensitive to alpha (a). Figure 4 indicates that lambda (k) is an increasing of Beta (b) and delta (d) remain zero and non sensitive to Beta (b). Figure 5 reveals that delta is an increasing function of gamma where as lambda is non sensitive to gamma. It is observed that there is no direct recruitment in grade II (d = 0) and it is non-sensitive to promotion grade of employee from grade I to grade II (b remain same) where as the recruitment rate of grade I (k) is an increasing function of the promotion rate from grade I to grade II (b). From Figs. 6 and 7 it is observed that cost function is an increasing function of both N0 and M0. The influence of decision parameters on the cost function is reviewed from the Figs. 8, 9, and 10, it is observed that the cost function is non sensitive to a, c and increasing function of b.
grade II and a range of 27.89 (%28) to 32.08 (%32) recruitments to grade I per 100 to the existing manpower for A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000, N0 = 150; M0 = 110; a = 5; b = 10. The rate of retrenchments/retirements from grade II are ranging from 30 to 75 suggests 22.5 (%23) recruitment to grade I and range from 9 to 45 direct recruitments to grade II per 100 of existing manpower. For A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000; N0 = 150; M0 = 110; a = 5; b = 10. The rate of recruitment in grade I will be increased by increasing the requirement to grade I and also the rate of direct recruitment in grade II is decreased with increase size in grade I.The rate of direct recruitment to grade II is an increase function of the optimal requirement of manpower M0. The rate of recruitment to grade I is positively influenced by its retrenchments. The recruitment rate to initial grade is highly influenced by the promotion rate from grade I to grade II (b). The direct recruitment to grade II is highly influenced by the rate of vacancies to grade II or rate of retirement from grade II. The impact of retirements effects recruitments of grade II where as it has no impact on the recruitment to grade I. Attaining optimality in cost reduction has inverse impact on the size of manpower requirement in grade I and grade II. The total cost of an organization can be optimal by decreasing the values of required size in grade I and grade II. The optimal objective cost is not influenced by either rate of retrenchments or the rate of retirements where the objective function is increasing with the increase of rate of promotion from grade I to grade II. Influence of decision parameters on total cost of an organization can be minimized by decreasing the promotion from grade I to grade II. The rate of recruitments in grade I and direct recruitment rate to grade II, influenced by total layout of budgets on salary, staff shortage cost and recruitment expenditure.
4 Findings and conclusions The optimal size of manpower in grade I ranging from 113 to 180 suggests the recruitments of range 16.875 (%17) to 27. in grade I and 2 to 9 in the direct recruitment of grade II per 100 size of existing manpower. For A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000, M0 = 110, a = 5; b = 10; c = 25. The optimal size of manpower in grade II ranging from 83 to 138 suggests the recruitment rate of range 26.25–30 to grade I; the direct recruitment rate ‘0’ in between the range 116 to 138 for M0 for A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000; N0 = 150, a = 5; b = 10; c = 25. Important factors of influence on recruitment rates are lies with a, b, c. If the rate of leaving/retrenchment from grade I is ranging from 5 to 23, it suggests the recruitments to grade I are in the range from 22.5 to 49.5 per hundred of existing manpower of grade I and further suggests only 5 direct recruitments to grade II. For A = 100000; s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000, r1 = 500; r2 = 800, R = 100000; N0 = 150; M0 = 110; b = 10; c = 25. The rate of promotions from grade I to grade II are ranging from 14 to 30 suggests no direct recruitment to
Appendix 1 See Table 1.
Table 1 The decision parameters Lambda and Delta; the optimum cost for various values of alpha, beta, gamma, N0, M0 N0
M0
113
110
a 5
b
c
10
25
k
d
K(T)
1687.5
875
1979750
128
1912.5
725
2092250
143
2137.5
575
2204750
123
Int J Syst Assur Eng Manag (June 2010) 1(2) Table 1 continued N0
M0
a
b
c
k
d
K(T)
158
2362.5
425
173
2587.5
275
2429750
2625
0
2306500
150
150
150
150
83
5
10
25
2317250
94
2775
0
2438300
105
2925
0
2570100
121
3000
275
2792200
132
3000
550
2948400
2250
500
2261000
110
110
110
5
10
25
9
2850
500
2261000
13
3450
500
2261000
17
4050
500
2261000
21
4650
500
2261000
2789.683 2996.479
0 0
2268746 2309113
20
3083.333
0
2326067
24
3231.928
0
2355072
28
3241.071
0
2298607
5
14 18
5
25
10
30
2250
900
2261000
40
2250
1700
2261000
50
2250
2500
2261000
60
2250
3300
2261000
70
2250
4100
2261000
Appendix 2 Inputs of manpower requirements See Figs. 1 and 2.
Required size for grade I with parameters
N0
lambda
delta
113 120 128 135 143 150
1687.5 1800 1912.5 2025 2137.5 2250
875 800 725 650 575 500
2500
parameters
Fig. 1 For the values of s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; M0 = 110, a = 5; b = 10; c = 25
lamda delta
2000 1500 1000 500 0 0
50
100
150
200
Required size in grade I
Required Size In grade II with parameters
M0
lambda
delta
105 116 121 127 132 138
2925 3000 3000 3000 3000 3000
0 137.5 275 412.5 550 687.5
4000
parameters
Fig. 2 For the Values of s1 = 15000; s2 = 25000; S = 7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; b = 10; c = 25
lamda delta
3000 2000 1000 0 0
50
100
required Size in gradeII
123
150
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Inputs of decision parameters See Figs. 3, 4 and 5.
Fig. 4 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; M0 = 110, a = 5; c = 25
beta
14 16 18 20 22 24
lambda delta 2789.683 0 2899.254 0 2996.479 0 3083.333 0 3161.392 0 3231.928 0
objective function with alpha
objective function
alpha lambda delta 13 3450 500 15 3750 500 17 4050 500 19 4350 500 21 4650 500 23 4950 500
6000 5000 4000 3000 2000 1000 0
lamda delta
0
5
10
15
20
25
Alpha
Objective function
Fig. 3 For the Values of s1 = 15000; s2 = 25000; S =7000000; p1 = 7000; p2 =10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; M0 = 110, b = 10; c = 25
objective function with beta 4000 lamda delta
3000 2000 1000 0 0
5
10
15
20
25
30
Fig. 5 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; M0 = 110, a = 5; b = 10
objective function
Beta
gamma lambda delta 30 2250 900 35 2250 1300 40 2250 1700 45 2250 2100 50 2250 2500 55 2250 2900
objective function with gamma 4000
lamda delta
3000 2000 1000 0 0
10
20
30
40
50
60
Gamma
Inputs of manpower requirements See Figs. 6 and 7.
Fig. 7 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; b = 10; c = 25
113 120 128 135 143 150
Objective function 1979750 2036000 2092250 2148500 2204750 2261000
Objective function with Required size in grade I objective function
N0
2300000 2200000 2100000
Objective function
2000000 1900000 0
50
100
150
200
Required size in grade I Objective function with required size in grade II
M0
105 116 121 127 132 138
Objective function 2570100 2714100 2792200 2870300 2948400 3026500
3100000
objective function
Fig. 6 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; M0 = 110, a = 5; b = 10; c = 25
3000000 2900000 2800000 Objective function
2700000 2600000 2500000 0
50
100
150
Required size in grade II
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Inputs of decision parameters
Fig. 9 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; c = 25
alpha 13 15 17 19 21 23
objective function 2261000 2261000 2261000 2261000 2261000 2261000
objective functionith alpha
2262000
objective function
2261000
2260000 0
10
20
30
alpha
objective function with beta
beta
14 16 18 20 22 24
objective function 2268746.03 2290134.33 2309112.68 2326066.67 2341303.8 2355072.29
objective function
Fig. 8 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; b = 10; c = 25
objective function
See Figs. 8, 9 and 10.
2360000 2340000 2320000
objective function
2300000 2280000 2260000 0
10
20
30
Fig. 10 For the Values of s1 = 15000; s2 =25000; S =7000000; p1 = 7000; p2 = 10000; P = 300000; r1 = 500; r2 = 800; R = 100000; N0 = 150; a = 5; b = 10
objective gamma function 30 2261000 35 2261000 40 2261000 45 2261000 50 2261000 55 2261000
References Chattopadhyay et al (2007) A stochastic manpower planning model under varying class sizes. Ann Oper Res 155:41–49 Dill WR, Gawer DP, Weber WL (1966) Models and modeling for manpower planning. Manag Sci 13(4):B142–B167 Gary L et al (1975) A model for manpower management. Manag Sci 21(12) Ginsberg RB (1971) Semi-Markov process and mobility. J Math Sociol 1:233–262 Glen JJ (1977) Length of service distributions in markov manpower models. Oper Res Q 28:975–982 Jeeva M et al (2004) An application of stochastic programming with Weibull distribution–cluster based optimum allocation of recruitment in manpower planning. Stoch Anal Appl 22(3):801–812 Jewelt RJ (1967) A minimum risk manpower schedule technique. Manag Sci 13(10):B578–B592 Kareem B, Aderoba AA (2008) Linear programming based effective maintenance and manpower planning strategy: a case study. J Comput Internet Manag 16(2):26–34
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Srinivasa Rao K, Srinivasa Rao V, Vivenkanada Murty M (2006) On two graded manpower planning model. Opsearch 43(3): 225–237 Taiwo AS (2007) Determination of the optimal manpower size using linear programming model. Res J Bus Manag 1(1):30–36 Vajda S (1985) The use of linear programming in manpower planning; Homenaje Al Profesor Sixto Rios Trabajos De Estadistica De Investigacion Operativa 36(3):301–311 Yadavalli VSS, Natarajan R (2001) A semi-Markov modelo f manpower system. Stoch Anal Appl 19(6):1077–1086 Yadavalli VSS, Natarajan R, Udayabhaskaran S (2002) Time dependent behaviour of Stochastic models of manpower system—impact of pressure on promotion. Stoch Anal Appl 20(4): 863–882 Yadavalli VSS, Setlhare K, Mishra SS, Rajalakshmi R (2006) An analysis of an optimal promotion policy for a manpower system using a queuing approach. Manag Dyn. http://findarticles.com/p/ articles/mi_qa5465/is_200604/ai_n21402995/
