CHAPTER 7 STOCHASTIC ANALYSIS OF MANPOWER LEVELS AFFECTING BUSINESS 7.1 Introduction

Transcription

1 CHAPTER 7 STOCHASTIC ANALYSIS OF MANPOWER LEVELS AFFECTING BUSINESS 7.1 Introduction Consider in this chapter a business organization under fluctuating conditions of availability of manpower and business with a special emphasis given to a new and prevailing idea of business to go off with the manpower leading to crisis state. The different states have been discussed under the assumption that changes from availability to shortage and shortage to availability occur in exponential times with different parameters.an expression for Rate of Crisis under steady state (C ) is derived and steady state costs have also been worked by assuming different costs for the parameters under different conditions. The content of this chapter has been published as a paper with the title Stochastic Analysis of Manpower Levels Affecting Business in International Mathematical Forum, Volume 7, No. 44, 2012,

2 After Operations Research becoming very popular and found its application in any field whether it is Business or Economics or Science or otherwise the approach to business has completely changed and that companies run on commercial basis, wish to keep only the optimum level of any resources needed to meet company s requirement at any time during the course of the business and manpower is not an exception. This is spelt in the sense that a company does not want to keep manpower more than what is required. Hence the measures taken are retrenchment and recruitment. Recruitment is done when the business is busy and shed manpower when the business is lean. Equally true with the labor, has the option to switch over to other jobs because of better working condition, better emolument, proximity to their living place or other reasons. Under such situations the company may face crisis because the manpower shortage will affect its business, may even loose business and profit will get affected. If skilled laborers and technically qualified persons leave the business the company will be constrained to hire paying heavy price or pay overtime to employees, this will affect the profitability of the company. In this chapter are considered two characteristics namely manpower and business. Formulas for the steady state rate of crisis and the steady state probabilities are derived. The situations may be that the manpower may be fully available or hardly available and business may fluctuate between full availability to nil availability. It goes off when the manpower becomes nil. This is so because the experts may take the business along with them or those who have brought good will to the concern may carry the clients off the concern. The business depends fully on the availability of manpower. The steady state probabilities of the continuous Markov chain describing the transitions in various states are derived

3 and critical states are identified for presenting the cost analysis. illustrations are provided. Numerical 7.2 Assumptions 1. There are two levels of Manpower namely Manpower is full and Manpower is nil. 2. There are two levels of business namely (1) business is fully available (2) business is lean or nil. 3. The time T during which the Manpower remains continuously full and becomes nil has exponential distribution with parameter λ 10 and the time R required to complete full recruitment from nil level is exponentially distributed with parameter μ The busy and nil periods of the business are exponentially distributed with parameters a and b respectively. 5. T and R are independently distributed random variables. 6. When manpower becomes nil, the business is lost and becomes NIL. 7.3 System Analysis The Stochastic Process X ( t ) describing the state of the system is a continuous time Markov chain with 3 points state space as given below in the order of Manpower, and Busness

4 S = { (0 0 ), ( 1 0 ), (1 1 ), } (7.2.1) Where 1-Refers to full availability of manpower and busy period in the case of business. 0-Refers to nil level of manpower and business. The system is in state ( i j ) when the manpower is in state i and business is in state j for ( i j ) = ( 0 0 ) or ( 1 0 ) or ( 1 1 ) and there is not the state ( 0 1 ) as the business is lost when the manpower goes off. The infinitesimal generator Q of the continuous time Markov chain of the state space is given below which is a matrix of order 6. MP / B (0 0) ( 1 0 ) ( 1 1 ) (0 0) - μ 01 μ 01 0 Q = ( 1 0 ) 10 (b + 10 ) b ( 1 1 ) 10 a (a + 10 ) Let π = [ π 00 π 10 π 11 ] be the steady state probability vector of the matrix Q., then π Q = 0, π e = 1 ( ) Using (7.2.2 ) the steady state probabilities are :

5 Π 00 =, Π 10 =, Π 11 = (7.2.3) The only crisis state in this model is ( 1 1 ) since if manpower becomes nil the business goes off with manpower. So the crisis rate of the steady state is given by C = Using the steady state probabilities C = 10 x ( ) 7.4 Numerical Illustration The steady state probabilities and the rate of crises are found using the formulas ( ) and ( ) respectively. Taking,, a =10, b = = 5, 01 = 8, the following are obtained: 00 = , 10 = and 11 = The formula for finding the steady state probability cost is Steady state probability cost = I j [ + ] ( ) Now assigning the values 10 = 5, 7, 9, 11 and 15 the corresponding rate of crises are determined as below in the table:

6 λ 10 C The graph is shown below to show the trend of the curve. The crisis rate increases when λ 10 increases when all other values remain the same λ 10 - C graph The rate of crisis increases as the value 10 increases when values of a, b and µ 01 remain fixed. The steady state costs in different situations are determined taking the values:

7 = 50, = 40,, = 65, = 48.where = 50, refers to cost of, funds at state zero, = 40 refers to cost of funds at state 1, = 65 cost of business at state zero and = 48 refers to cost of business at state 1. S.No Steady state probability Cost Total When 10 =5, µ 01 = 8; b = 12, a = 4,8,10, 14 and 18, equation ( ) gives A C Obseve that when busy period goes up the rate of crisis comes down when other values b, 10 and µ 01 remaining constant. The graph is shown below:

8 a - C graph When 10 =5, µ 01 = 8; a =10 and b = 2, 4, 6, 12, 14, and 20, equation gives, b C

9 Observe that when the lean period goes up the rate of crisis also goes up the other values a, 10 and µ 01 remaining constant. The graph is given below to show the effect of b increasing. b - C graph When a =10, b = 12 ; 10 =5 µ 01 = 1, 3, 5, 7 and 10, equation ( ) gives the following C values. µ 01 C

10 µ 01 - C graph It is observed that the rate of crisis increases as time for recruitment takes longer time when all other values a, b and 10 above graph. remain fixed.this effect is shown in the λ 10 and µ 01 C ( λ 10) C ( µ 01)

11 λ 10 and µ 01 - (C ( λ 10), C ( µ 01 ) graph The combined graphs with respect to parameters 10 and µ 01 is given above observe that when the time for which the manpower remaining the same goes up the rate of crisis also goes up and when the time for recruitment of staff to meet the shortage goes up the rate of crisis also goes up. These are shown using bar graphs, the effect of 10 and µ 01 for the same values are shown by the bars drawn adjacent to each to other. The bar lengths increase in both the cases. a&b C ( a ) C ( b )

12 a&b - (C ( a ), C ( b ) ) graph The combined graphs with respect to parameters a and b is given above and observe that when the busy period goes up the rate of crisis comes down whereas when the lean period goes up the rate of crisis also gos up. These are shown using bar graphs, the effect of a and b for the same values are the bars drawn adjacent to each to other. The bar lengths decrease in the case of parameter a increasing and the opposite is the effect in the case of parameter b increasing. 7.5 Observation It is found that as the value of parameter 10 ( time for which the manpower remains the same increases) increases the crisis rate also increases. If 01 increases ( the time for recruitment increases ) the rate of crisis will also imcrease.if a the busy period of the business increases rate of crisis decreases and if b the lean period increases rate of crisis increases. A comparitive study is done

13 using the combined bar graphs. Also observe that the cost of doing business is very high if the manpower is lean and business is also lean but the cost is higher when there is manpower but no business, the business has to be got by paying premium and finally the cost of business is least when the manpower is full and there is full business and so the business can do very well. But, it has to be noted that, there is always a threat of business getting into crisis state if manpower leaves especially experts and experienced people leave the concern. 7.6 Model II In the previous model is derived an expression for crisis state under steady state conditions and also the steady state probabilities considering availability and shortage in manpower and business with the new and prevailing concept of business to go off with manpower. In this model instead of two states for manpower to fluctuate, considerd are three states namely manpower is fully available, moderately available and not all available. The business to fluctuate between two states full availability and zero availability. The business becomes nil when manpower becomes nil. 7.7 Assumptions 1. The time T during which the manpower remains continuously moderate and becomes nil has exponential distribution with parameter λ 10. The time R required to complete recruitment for filling up of vacancies from level nil to moderate level is exponentially distributed with parameter μ The time T during which the Manpower remains continuously full and becomes nil has exponential distribution with parameter λ 20 and the time R required to complete full recruitment from nil level is exponentially distributed with parameter μ 02

14 3. The period of time T during which the Manpower is continuously moderate becomes full has exponentially distribution with parameter λ 21 and the period of time R required for recruitment from insufficient level to full is exponentially distributed with parameter μ 12.The random variables T and R ; T and R :T and R are all independent. 4. The busy and lean periods of the business are exponentially distributed with parameters a and b respectively. The business becomes nil when the manpower goes off. 5. The state of the system is ( I j ) when manpower is in state i and the business is in state j. The state ( 0 1 ) is not considered as our model is based on the assumption that once manpower becomes zero the business goes off. This is because the experts of the concern may carry the business along with them or good will of the concern built over a period is taken away by them. 7.8 System Analysis The Stochastic Process X ( t ) describing the state of the system is a continuous time Markov chain with 5 points state space as given below in the order of Manpower, and Busness S = { (0, 0 ), ( 1 0 ), (1, 1 ),, ( 2 0 ), ( 2 1 ) } (7.8.1) here 2- Refers to full availability in the case of manpower 1-Refers to semi availability or insufficiently available manpower and it refers to busy period in the case of business. 0-Refers to nil level of manpower or business.

15 The infinitesimal generator Q of the continuous time Markov chain of the state space is given below which is a matrix of order 5. MP / B (0 0) ( 1 0 ) ( 1 1 ) ( 2 0 ) ( 2 1 ) (0 0) 1 μ 01 0 μ 02 0 ( 1 0 ) λ 10 2 b μ 12 0 Q = ( 1 1 ) λ 10 a 3 0 μ 12 ( 2 0 ) λ 20 λ b ( 2 1 ) λ a 5 1 = -( μ 01 + μ 02 ), 2 = -( λ 10 + μ 12 +b ), 3 = -( λ 10 + μ 12 +a ), 4 = -( λ 20 +λ 21 +b ), 5 = -( λ 20 + λ 21 +a ) (7.8.2) Q matrix is of order 5. The steady state probability vector satisfies the following equations of the matrix Q Q = 0 and e = 1 ( )

16 = [ ] and e = ( ) t are vectors of order 1 x 5 and 5 x 1 respectively. If Q is taken as MP / B (0 0) ( 1 0 ) ( 1 1 ) ( 2 0 ) (0 0) 1 μ 01 0 μ 02 Q = ( 1 0 ) λ 10 2 b µ 12 ( 1 1 ) λ 10 a 3 0 ( 2 0 ) λ 20 λ Then Q c Q ( 7.8.5) r (λ λ a) where r = (λ 20 0.λ 21 a ) and c = ( 0 0 µ 12 b ) t using (7.8.3 ), the steady state probabilities explicitly are found Π = ( ( ) The crises states are ( 1 1 ) and ( 2 1 ) because as per assumption manpower becoming moderate to zero or full to zero will lead to absolute business loss

17 hence they are critical states. Therefore the rate of crisis for the steady state is given by : = λ λ (7.8.7 ) Steady state costs and numerical illustration The steady state costs are determined by using the formula C ij = ij [ + ], ( ) where c im stands for cost of money at states i = o or 1, c JB stands for cost of business at state j = o or1. Assumed are the following values for different parameters to find the steady costs and the rate of crises:.λ 10 = 3,.λ 20 = 2.λ 21 = 1,.μ 01 = 2.μ 02 = 1.μ 12 = 2 a = 1 b = 2 Using the above values in ( ), gives.r ( - Q ) -1 = ( ).r ( - Q ) -1 e = Now using ( ), the steady state probabilities are as given below 00 = , 10 = , 11 = , 20 = , 21 = Using ( ) the rate of crisis is: = Assuming the following costs and arrive at steady state costs and the expected total cost = 15, = 11,, = 7, = 16.where = 20,

18 S. No Steady state probability Cost of state 1 Π Π Π Π Π Total expected cost (7.8.9) 7.9 Observation It is observed that the steady state cost is very high when both business and manpower are nil where as it is very less when business is full and cases when manpower is full or medium because business has to be obtained by paying premium and the labor has to be paid high. REASONS FOR SELECTION OF MODELS AND THEIR APPLICATION TO INDUSTRIES / ORGANIZATIONS The new concept but most prevalent almost in all industries is introduced namely the business to go off with manpower. The reasons are Experts or silled persons in

19 that business may carry the business along with them or the person who brings good will to the company because of his business acumen or liasion capacity will take the busines along with him. So there is no state here as manpower nil and business full. The paper has two models of two characteristics only. Various operating characteristics are determined when business and manpower fluctuate between two levels, in the second model business fluctuates between two levels but manpower amoung three levels. Numerical examples and graphs explain various truths about the business.nowdays people get good education, equip themselves well academically and technically and job opportunities are more with attractive salaries that they switch over jobs when opportunities come, they are also likely to carry the present business because of their unique skill or good will created over a period of time. This is common in any manufacturing company and software companies.