Profit evaluation of three units compressor standby system by IJARTET

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International Journal of Advanced Research Trends in Engineering and Technology (IJARTET) Vol. 3, Issue 5, May 2016

Profit Evaluation of Three Units Compressor Standby System Upasana Sharma 1, Jaswinder kaur2 Associate Professor, Department of Statistics, Punjabi University, Patiala, India 1 Lecturer, Department of Community Medicine, Gian Sagar Medical College, Rajpura, India 2 Abstract: The paper presents profit analysis of three units compressor standby system working in Refrigeration system of a milk plant. Initially the system consist of two operating and one standby compressor unit. For the functioning of the system at least two compressor units should be in working state .There can be three types of failure i.e serviceable repairable and replaceable type in any unit. The priority of service or repair or replacement has been given to failed unit on FCFS basis. Various measures of reliability such as MTSF, Availability and Profit Analysis has been calculated by using Semi Markov process and regenerative point techniques. For practical utility of our proposed model previous data of different years from Verka milk plant has been gathered and is used for graphical analysis. Keywords: Compressor Unit, Regenerative point technique, Refrigeration System, Semi-Markov process.

I. INTRODUCTION Standby systems are commonly used in many industries and therefore, researchers [1 -3] have spent a great deal of efforts in analyzing such systems to get the optimized reliability results which are useful for effective equipment/plant maintenance. For graphical study, they have taken assumed values for failure and repair rates, and not used the observed values. However, some researchers including [4-7] studied some reliability models collecting real data on failure and repair rates of the units used in such systems. A potential application of the reliability concepts has been recently explored in terms of developing a specific probabilistic model for three units compressor standby system where recently failed unit has been be given priority of service ,repair and replacement by Sharma U. and Kaur J. [8].in the present paper same three units compressor standby system has been studied wherein the priority of service, repair or replacement is given to failed unit on FCFS basis. Initially the system consist of two operating and one standby compressor unit. For the functioning of the system at least two compressor units should be in working state various measures of system effectiveness has been calculated by using semiMarkov process and regenerative point techniques. For practical utility of our proposed model previous data of different years from Verka milk plant has been gathered and is used for graphical analysis.

Notations OI,OII First , Second Compressor are in Operative State SIII Third Compressor is in Stand by State (s) Stieltjes Convolution © Laplace Convolution λi1, λi2, λi3 Failure rates when failure is of serviceable, repairable and replaceable for first, second &third compressor respectively (i= 1,2,3and i symbol used for compressor unit ) αi1, αi2, α13 Repair rates when failure is of serviceable, repairable and replaceable for first, second &third compressor respectively (i= 1,2,3and i symbol used for compressor unit ) FsI , FsII , FsIII Failure category of serviceable type for first, Second and third compressor FrI, FrII, FrIII Failure category of repairable type for first second and third compressor FrepI , FrepII , FrepIII Failure category of replaceable type for First ,Second and third compressor

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FwrI, FwsI, FwrepI First compressor is waiting for Repair, Service, Replacement respectively FwrII, FwsII , FwrepII Second compressor is waiting for Repair, Service, Replacement respectively FwrIII, FwsIII , FwrepIII Third compressor is waiting for Repair,Service,Replacement respectively Gi1(t) , gi1(t) c.d.f and p. d.f of time for service when failure is of serviceable type for first, second and third compressor respectively Gi2(t) , gi2(t) c.d.f and p.d.f of time for repair when failure is of repairable type for first, second and third compressor respectively Gi3(t) , gi3(t) c.d.f and p.d.f of time for replacement when failure is of replaceable type for first , second and third compressor respectively. Qij , qij c.d.f and p.d.f of first passage time from a regenerative state i to j or to a failed state j in (0, t]. øi(t) c.d.f of the first passage time from regenerative state i to a failed state pij , pijk probability of transition from regenerative state i to regenerative state j without visiting any other state in (0,t],visiting state k once in(0,t] qijk p.d.f of first passage from regenerative state i to regenerative state j or to failed state j visting k once in (0,t] Model Description and Assumptions 1) The unit is initially operative at state 0 and its transition depends upon the type of failure category to any of the three states1to3 with different failure rates. 2) All failure times are assumed to have exponential distribution. 3) After each servicing/ repair/replacement at states the unit works as good as new. 4) Priority given to failed unit for service, repair and replacement. Transition Probabilities and Mean Sojourn Times A state transition diagram showing the various states of transition of the system is shown in Table 1. The epochs of entry into states 0, 1, 2, 3, 22, 23, 24, 25, 26 and 27 are regenerative states. States 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 and 21 are down states. The non zero elements pij are given below

 11  , p  12 , p  13 where *  11  12  13 * 02 * 03 * * * * p10  g11 (  ), p20  g12 (  ), p30  g13 ( ) where  21  22  23  31  32  33   * * p14 , p14,22  21 ( 1  g11 (  )); p15 , p15,23  22 ( 1  g11 (  ))     * * p16 , p16,24  23 ( 1  g11 (  )); p17 , p17,25  31 ( 1  g11 (  ))     * * p18 , p18,26  32 ( 1  g11 (  )); p19 , p19,27  33 ( 1  g11 (  ))   21 22 * * p2 ,10 , p10 ( 1  g12 (  )); p2 ,11 , p11 ( 1  g12 (  )) 2 ,22  2 ,23    23 31 * * p2 ,12 , p12 ( 1  g12 (  )); p2 ,13 , p13 ( 1  g12 (  )) 2 ,24  2 ,25    32 33 * * p2 ,14 , p14 ( 1  g12 (  )); p2 ,15 , p15 ( 1  g12 (  )) 2 ,26  2 ,27      * * p3 ,16 , p316,22  21 ( 1  g13 (  )); p3 ,17 , p317,23  21 ( 1  g13 (  ))     * * p3 ,18 , p318,24  21 ( 1  g13 (  )); p3 ,19 , p319,25  31 ( 1  g13 (  ))     * * p3 ,20 , p320,26  31 ( 1  g13 (  )); p3 ,21 , p321,27  31 ( 1  g13 (  ))   p01 

p01  p02  p03  1 p10  p14  p15  p16  p17  p18  p19  1, p10  p 14,22  p15,23  p16,24  p 17,25  p18,26  p19,27  1 p20  p2 ,10  p2 ,11  p2 ,12  p2 ,13  p2 ,14  p2 ,15  1 11 12 13 14 15 p20  p10 2 ,22  p2 ,23  p2 ,24  p2 ,25  p2 ,26  p2 ,27  1

p30  p3 ,16  p3 ,17  p3 ,18  p3 ,19  p3 ,20  p3 ,21  1 p30  p316,22  p317,23  p318,24  p319,25  p320,26  p321,27  1

The mean sojourn time (µi) in the regenerative state ‘i’ is defined as time of stay in that state before transition to any other state: 0   22 

*

, 1 



G



, 2 



, 3 

( t )dt  K1 ,  23 

 24 

( t )dt  K 3 ,  25 

 26 

( t )dt  K 2



G

( t )dt  K 4



G



G 0



G



( t )dt  K 5 ,  27 



G

( t )dt  K 6

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International Journal of Advanced Research Trends in Engineering and Technology (IJARTET) Vol. 3, Issue 5, May 2016

State No.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

OI ,OII ,SIII FsI ,OII ,OIII FrI , OII ,OIII FrepI ,OII ,OIII FusI , FwsII ,OIII FusI , FwrII ,OIII FusI, FwrepII ,OIII FusI , OII , FwsIII FusI , OII , FwrIII FusI,OII , FwrepIII FurI, FwsII , OIII FurI, FwrII , OIII FurI,FwrepII , OIII FurI, OII , FwsIII TABLE 1

T0  N / D

State No.

Status

14 15 16 17 18 19 20 21 22 23 24 25 26 27

Status FurI, OII , FwrIII FurI, OII , FwrepIII FurepI , FwsII ,OIII FurepI , FwrII, OIII FurepI ,FwrepII ,OIII FurepI ,OII , FwsIII FurepI ,OII , FwrIII FurepI,OII, FwrepIII OI , FusII , OIII OI , FurII , OIII OI , FurepII ,OIII OI , OII, FusIII OI ,OII, FurIII OI ,OII, FurepIII

POSSIBLE STATES WITH STATUS

The unconditional mean time taken by the system to transit for any regenerative state ‘j’ when it (time) is counted from the epoch of entrance into state ‘i’ is mathematically state as:

where   m01 p10  m10 p01  m02 p20  m20 p02  m03 p30  m30 p03  m01  m02  m03  m01 p10  m02 p20  m03 p30  m30 p03 * *  m20 p02  m10 p01  p03 3 ( 1  g13 (  ))  p02 2 ( 1  g12 (  )) *  p01 1 ( 1  g11 (  ))

D  1  p01 p10  p02 p20  p03 p30

Availability Analysis Using the arguments of the theory of regenerative processes, In steady state availability of the system is A0  lim( sA0* ( s ))  1 / D1 s0

where N1  0  M 1 p01  M 2 p02  M 3 p03  M 22 p14,22 p01  M 23 p01 p15,23  M 24 p01 p16,24  M 25 p17,25 p01  M 26 p01 p18,26  M 27 p01 p19,27 11 12 13  M 22 p10 2 ,22 p02  M 23 p02 p2 ,23  M 24 p02 p2 ,24  M 25 p2 ,25 p02 15 16 17  M 26 p02 p14 2 ,26  M 27 p02 p2 ,27  M 22 p3 ,22 p03  M 23 p03 p3 ,23

 M 24 p03 p318,24  M 25 p319,25 p03  M 26 p03 p320,26  M 27 p03 p321,27 D1   0  p01 1 ( 1  g11* (  ))  p02 ( 2 ( 1  g12* (  )) ( p03 3 ( 1  g13* (  ))  p01 ( p14,22 m22 ,0  p15,23 m23 ,0  p16,24 m24 ,0



mij   tdQij ( t )   q ij '* ( 0 )

 p17,25 m25 ,0  p18,26 m26 ,0  p19,27 m27 ,0 )  p02 ( p110,22 m22 ,0  p211,23 m23 ,0

m01  m02  m03 

1 ( * )

13 14 15 16  p12 2 ,24 m24 ,0  p2 ,25 m25 ,0  p2 ,26 m26 ,0  p2 ,27 m27 ,0 )  p03 ( p3 ,22 m22 ,0

 0

 p317,23 m23 ,0  p318,24 m24 ,0  p319,25 m25 ,0  p320,26 m26 ,0  p321,27 m27 ,0 )

* m10  m14  m15  m16  m17  m18  m19  1 ( 1  g11 ( )

m10  m

4 1,22

m

5 1,23

m

6 1,24

m

7 1,25

m

8 1,26

m

9 1,27

 1 ( 1  g (  ) * 11

* m20  m2 ,10  m2 ,11  m2 ,12  m2 ,13  m2 ,14  m2 ,15  2 ( 1  g12 ( ) 11 12 13 14 15 * m20  m10 2 ,22  m2 ,23  m2 ,24  m2 ,25  m2 ,26  m2 ,27   2 ( 1  g12 (  )

Proceeding in the similar fashion as above following measures in steady state have also been obtained Busy Period Analysis for Service Time Only

m30  m3 ,16  m3 ,17  m3 ,18  m3 ,19  m3 ,20  m3 ,21  3 ( 1  g (  )

B0  N 2 / D1

* m30  m316,22  m317,23  m318,24  m319,25  m320,26  m321,27  3 ( 1  g13 ( )

where

* 13

Mean Time to System Failure: To determine the mean time to system failure (MTSF) of the system, we regard the failed states of the system as absorbing states and the mean time to system failure (MTSF) when the system starts from state S0 is 1  Ø0 * *( s ) S O s

MTSF  T0  lim

using L’ Hospital’s Rule and putting the value of ø0**(s) , we have

16 N 2  p01 1  p01 p14,22 K1  p02 p10 2 ,22 K1  p03 p3 ,22 K1 19  p01 p17,25 K 4  p02 p13 2 ,25 K 4  p03 p3 ,25 K 4

Busy Period Analysis for Repair Time Only B1  N 3 / D1 where N 3  p02 2  p01 p15,23 K 2  p02 p211,23 K 2  p03 p317,23 K 2 20  p01 p18,26 K 5  p02 p14 2 ,26 K 4  p03 p3 ,27 K 4

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Busy Period Analysis for Replacement Time Only using the above particular case, the following values are estimated as

B2  N 4 / D1

S E  N 5 / D1

11  0.006896,12  0.000586,13  0.04166,  21  0.0000983, 22  0.0001347 , 23  0.00015873  31  0.000345209, 32  0.0010162602 , 33  0.0006648936 11  0.00003868, 12  0.00003879, 13  0.00003865 21  0.0007359,22  0.0007367 , 23  0.0007352 31  0.0000456079,32  0.0000456089,33  0.0000456071

where

C1  3000 ,C2  500,C3  550,C4  800,C5  27700,

N 5  p01  p01 ( p1,0  p14,22  p15,23  p16,24  p17,25  p18,26

C6  7600,C7  7975,N  8887.14331,N 2  2637.208740

 p19,27 )  p01 p14,22 p22 ,0  p02 p210,22 p22 ,0  p01 p17,25 p25 ,0

N 3  2234.277588,N 4  1553.829590,N 5  1.062829

16 19  p02 p13 2 ,25 p25 ,0  p03 p22 ,0 p3 ,22  p03 p25 ,0 p3 ,25

N 6  1.062856,N 7  1.062870,N8  1.0000000

where 18 N 4  p03 3  p01 p16,24 K 3  p02 p12 2 ,24 K 3  p03 p3 ,24 K 2

 p01 p19,27 K 6  p02 p215,27 K 6  p03 p321,27 K 6

Expected Number of Service

Conclusion The measures of system effectiveness are obtained as:

Expected Number of Repairs RE  N 6 / D1 where 11 12 13 14 N 6  p02  p02 ( p2 ,0  p10 2 ,22  p2 ,23  p2 ,24  p2 ,25  p2 ,26 11 8 14  p15 2 ,27 )  p02 p2 ,23 p23 ,0  p01 p1,26 p26 ,0  p02 p2 ,26 p26 ,0

 p03 p317,23 p23 ,0  p03 p26 ,0 p320,26  p03 p25 ,0 p319,25

Expected Number of Replacements R  N 7 / D1 where N 7  p03  p03 ( p3,0  p316,22  p317,23  p318,24  p319,25  p320,26 9  p321,27 )  p01 p16,24 p24 ,0  p02 p12 2 ,24 p24 ,0  p01 p1,27 p27 ,0 18 21  p02 p15 2 ,27 p27 ,0  p03 p24 ,0 p3 ,24  p03 p27 ,0 p3 ,27

Mean time to unit/compressor MTSF =14081.8271 hrs. Availability =.9999999 Busy period for service=.179557 Busy period for repair=0.152123 Busy period for replacement=.105794 Expected number of visits=0.000068 Expected number of services=0.000074 Expected number of repairs=0.000075 Expected number of replacements=0.000072 Profit Analysis The expected total profit incurred to the system in steady state is given by

Expected Number of Visits by Repairman

V0  N8 / D1 where N8 ( s )  p01  p02  p03 Particular Cases For graphical representation, let us suppose that g11 ( t )  11e 11t ,g12 ( t )  12 e 12t ,g13 ( t )  13e 13t g 21 ( t )   21e  21t ,g 22 ( t )   22 e 22 t ,g 23 ( t )   23e  23t g11 ( t )   31e 31t ,g12 ( t )   32 e 32t ,g13 ( t )   33e 33t

P =C0A0-C1B0-C2B1-C3B2-C4V0-C5SE-C6RE-C7R , Where C0= Revenue per unit up time C1=Cost per unit time for which repairman is busy for service C2= Cost per unit time for which repairman is busy for repair C3= Cost per unit time for which repairman is busy for replacement C4=Cost per visit of Repairman C5=Cost per visit of Service C6=Cost per visit of Repair C7= Cost per visit of Replacement

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International Journal of Advanced Research Trends in Engineering and Technology (IJARTET) Vol. 3, Issue 5, May 2016

Graph between Profit vs Revenue per unit time (C0) for different values of cost per unit for which repairman is busy for service (C1)(fig 2)

be noticed that if C2=500, then P>or=or<0 according as or =or<676.2. So for C2=500, revenue per unit up time should be fixed greater than 676.2.Similarly for C2=800 and 1100, the revenue per unit up time should be greater than 721.8 and 767.4 respectively. REFERENCES

It can be interpreted from graph that profit increases with increase in values of revenue per unit up time (C0).It can also be noticed that if C1=3000 , then P>or=or<0 according as C0 >or =or<676.2. So for C1=3000, revenue per unit up time should be fixed greater than 676.2.Similarly for C 1=3300 and 3500, the revenue per unit up time should be greater than 730 and 765.9 respectively. Graph between Profit vs Revenue per unit time (C0) for different values of cost per unit for which repairman is busy for repair (C2) (fig3)

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It can be interpreted from graph that profit increases with increase in values of revenue per unit up time (C0).It can also