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Paper on cost benefit analysis
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Improved Cost Benefit Analysis Model
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved1
• Objective– To build a more robust and generalized model for estimating maintenance
cost. • Work Plan
– Review existing model and addition of new parameters• Net Revenue Formulation
– Study different type of Multi System Models • Markov Renewal Process• Queuing Theory• Monte Carlo Method• Aggregation method
– Parameters for Cost Calculation– Cost Comparison– Observations
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved
Improved Cost Benefit Analysis(CBA) Model
2
Existing model • Estimation of Costs
M S F i i i,j,k i,j,ki,j,k
i=1 j=1 k=1i,j,k i,j,k i,j,k i
LT RGPH IIC ( BDT NMHRNet Revenue
AVCPMH )+ SPC ( BDT RGPH )
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved3
i,j,kBDT : Breakdown time for machine under maintenance
strategy for fault
i,j,kNMHR : Number of Man Hour required for machine under
maintenance strategy for fault and
i,j,kAVCPMH : Average Cost per Man hour in $ / Hour
i,j,kLBC : Labor cost for machine under maintenance strategy for
fault i,j,kSPC : Spare Parts cost for machine under maintenance strategy for
fault
Improved CBA Model…Net Revenue
For Individual Maintenance Strategies we use,
Where,
WT is working time of machine λ(t) is failure rate
RGPH is revenue generated per hour is repair rate of machine
IIC is initial investment cost
FMC is fixed maintenance cost
NMHR is number of man hours
AVCPMH is average cost of man hour
SPC is spare parts cost
BDT is break down time
RGPH is revenue generated per hour
M
n=1
{(WT ×RGPH) -[ IIC+FMC+f NMHRNet Revenue =
×AVCPMH) +SPC+( ( ) BDT×RGPH)]}R
t
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved4
Net Revenue ={(WT ×RGPH) -[ IIC+FMC+(NMHR×AVCPMH)+SPC+(BDT×RGPH)]}
( ) tt e
( ( ))Rf f t
Improved CBA Model…Multi Unit Models
• Markov Renewal Process– In this model state transition diagram is made, conditional Probability distribution of future state depends on present state
• Queuing Theory– Probability distributions are used, inter relating different states
• Monte Carlo Method– In this process by simulation, probability distribution and random variables are assigned
• Aggregation Method– In this method all other states are aggregated to new state which is sum of other state
probabilities.
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved5
Type 1 Markov Renewal Process
• Initial machine stage is defined by 0, makes transitions into states [1] 1, 2 ..... S, and comes back to state 0.
• Circle (o) represents a state with regeneration point and
• Rectangular (□) a state with non-regeneration point.
Fig. 1: Type-1 Markov model
Improved CBA Model…Markov Model
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved6
[1] Toshio.N, Shunji.O, “Markov Renewal Processes with some non regeneration points and their applications to reliability ” Microelectronics and Reliability, Vol. 15, pp. 633 to 636.
• is the first-passage time distribution from the moment the process enters state i at time 0 to state j entered at time t [1]
• is the mean number of visits to state j during (0, t) if the process starts in state i at time 0, which is referred to as the "renewal function.“
• Probability that the process is in state j at time t if the process starts in state i at time 0.
( ) Pr{ ( ) 0 | (0) }i j jS S S iG t N t Z s
( ) { ( ) | (0) }i j jS S S iM t E N t Z s
( ) Pr{ ( ) | (0) }i jS S j iP t Z t s Z s
Improved CBA Model…Markov Model
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved7
Source: [1] Toshio.N, Shunji.O, “Markov Renewal Processes with some non regeneration points and their applications to reliability ” Microelectronics and Reliability, Vol. 15, pp. 633 to 636.
0 1 0 1 0 2 0 0 0 1
(1) (1,2,.., )( ) ( ) ( ) ( )* ( )S S S S S S
nS S S SP t Q t Q t Q t P t
0 0 0 1
(1,2,.., 1) (1,2,.., )( ) ( ) ( )j S S S Sj j
j jS SP t Q t Q t
0 0( )* ( )
j jS S S SG t P t 2,3,..jS n
0 0 1 0 0 0 0( ) 1 ( ) ( )* ( )
jS S S S S S S SP t Q t G t P t
0 1 0 2 0 0 0 1
(1)( ) ( ) ( )* ( )S S S S S S S SQ t Q t G t P t
0 1 0 0
(1,2,.., ) (1,2,.., )( ) ( )S S S Sn
n nQ t Q t
Asterisk (*) denotes the pairwise Stieltjes convolution [1]
• Probabilities from one state to other states
These are the mass functions [1] Q(t) which is the probability that after entering state i,
Process next makes transitions into states and finally enters state j and
Improved CBA Model…Transition Probabilities
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved8
Source: [1] Toshio.N, Shunji.O, “Markov Renewal Processes with some non regeneration points and their applications to reliability ” Microelectronics and Reliability, Vol. 15, pp. 633 to 636.
Improved CBA Model…2 Type MRP
0 0 0 0
( )
0
( ) ( )S S
ni
S Si
g s q s
0 0 0 0( ) ( ) /[1 ( )]
j jS S S S S Sg s q s g s
0 0 0 0 0 0( ) ( ) /[1 ( )]S S S S S Sm s g s g s
0 0 0 0( ) ( ) /[1 ( )]
j jS S S S S Sm s q s g s
0 0 0 0 01
( ) 1 ( ) /[1 ( )]j
n
S S S S S Si
p s q t g s
0 0 0 00
( )( ) [ ( ) ( )] /[1 ( )]j j S S j
jS S S S S Sp s q s q s g s
1,2,3,..js n
Fig2 : 2 Type Markov Renewal Process
After taking Laplace-Stieltjes transforms these are results for mean number of visits,first passage time distribution
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved9
Source: [1] Toshio.N, Shunji.O, “Markov Renewal Processes with some non regeneration points and their applications to reliability ” Microelectronics and Reliability, Vol. 15, pp. 633 to 636.
Improved Cost Benefit Analysis…Spare part system Type 3 Markov Renewal Process
0 0 0 1 0 4 0 0( ) [1 ( ) ( )] /[1 ( )]S S S S S S S Sp s q s q s h s
0 1 0 1 0 00 2 0 3
(1) (1)( ) [ ( ) ( ) ( )] /[1 ( )]S S S SS S S S S Sp s q s q s q s h s
00 1 0 2 0 0 0
(1) (1,2)( ) [ ( ) ( )] /[1 ( )]S S S S S S S Sp s q s q s h s
0 3 0 3 0 3 0 0
(1) (1,3)( ) [ ( ) ( )] /[1 ( )]S S S S S S S Sp s q s q s h s
0 4 0 00 4 0 4( ) ( )[1 ( )] /[1 ( )]
S S S S S S S Sp s q s q s h s
0 0 0 0 0 0 0 4 0 4
(1,2) (1,3)( ) ( ) ( ) ( ) ( )S S S S S S S S S S
where
h s q s q s q s q s
Probabilities into different states
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved10
Improved Cost Benefit Analysis…Spare part system Type3 MRP
0 10
( ) ( ) ( )t
S SQ t F u dA u 0 2
(1)
0
( ) ( ) [ ( )* ( )]t
S SQ t F u d A u B u
0 0
(1,2)
0
( ) [ ( )* ( )] ( )t
S SQ t A u B u dF u 0 3
(1)
0
( ) [ ( )* ( )] ( )t
S SQ t A u B u dF u
0 0
(1,3)
0 0
( ) [ ( ) ( )] ( ) ( )t u
S SQ t F u F v dA v dB u v 0 4
0
( ) ( ) ( )t
S SQ t A u dF u 0 4( ) ( )S SQ t B t
Mass Functions of different states
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved11
Fig 4: State Transition Model of machine into different states
Improved CBA Model…State Transition Diagram of Machine Working Conditions
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved12
• Mean Time to System Failure
μ is mean sojourn time P is probability between different states
• System Availability
where N(s) represents system working timeD(s)represents cumulative distribution of system failure
( )( )
N sAD s
0 1 12 2 4 24
10 12 20
( )1
p pMTSFp p p
Improved CBA Model…Transition Probabilities
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved13
Source: [2] S.K. Singh, Cost-benefit analysis of a 2-unit priority-standby system with patience-time for repair, Reliability, IEEE Transactions april1997, volume 40..
• Probability of n machines are down [2]
Where
• Steady State Availability where
is expected downtime of machine is expected uptime of machine
s
s d
EA
E E
dE
sE
0n
nn CQ N Q
0!!n
n
C n c
nN Qc c
1
00 1
!!n n
nc Nn
C C n cn n c
nQ N Nc c
Fig 5: Queuing Theory [2]
Improved CBA Model…Queuing Theory
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved14
Source: [3] V.N. Achutha Naikan, P.Naga Srinivasa Rao, Maintenance Cost benefit analysis of production shops availability Approach, IEEE Transactions on reliability,2005 Page(s): 404 – 409..
Improved CBA Model…Formulation of Queuing Theory
• Expected number of machines in the upstate [3]
Probability that n machines are working
• Expected number of machine in the failed state [3]
• System Failure [2]
where λ is failure rate (Exponential distribution) [3]
0 0
N N
s n N nn n
E nP nQ
0
N
d nn
E nQ
0
N
eff n sn
P E
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved15
Improved CBA Model…Parameters for Cost Analysis
Parameters Markov Renewal Process(In terms of Probability)
Queuing Theory(In terms of Probability)
Availability 0.67 0.79
System Failure Rate 0.06 0.05
Mean sojourn time 0.5 0.7
Repair Time 0.02 0.025
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved16
N
n=1
{(WT ×RGPH) -[ IIC+FMC+(f(t, ) NMHRNet Revenue =
×AVCPMH) +SPC+( BDT×RGPH)]}
Using these in the net revenue calculation
Improved CBA…Case Study: Condition Based Maintenance
17
Parameters Values
Revenue generated per hour $200
Total number of operating hours available 720 hours
Average number of hours in break down 7.2 hours
Assume number of man hours 6
Spare parts cost $500
Average cost per man hour $100
Initial investment cost (one time cost) $10000
Operation Period in months
Markov Renewal Process ($)
Queuing Process ($)
CBA Model($)
1 132200 133340 133479.8
2 264400 266680 276874
3 396600 402000 420374
6 793200 835040 851000
12 1586400 1680080 1712000
24 3172800 3372160 3434000
Improved CBA Model…Revenue Generation
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved18
Improved CBA Model…Result
1 2 3 6 12 240
400000
800000
1200000
1600000
2000000
2400000
2800000
3200000
Markov Renewal Process Queuing Process
Reve
nue
Gene
rate
d ($
)
Operation Period in months
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved19
Improved CBA Model…Observations
• In Markov Renewal Process complexity increases for multi unit systems• Large number of states and computation accordingly is time consuming• Queuing Process is better compared to Markov renewal process• Queuing Theory produces better revenues compared to Markov renewal
Process
• Enhancing net revenue for multi unit systems with other models• Calculating Revenues by surveying machines uptime, downtime,
availability and other important features
Future Scope
Copyrights © 2013 IIT Kanpur-Boeing Inc. USA. All rights reserved20