Improved Cost Benefit Analysis Model

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 )


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


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.


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


[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


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



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



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


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


Fig 4: State Transition Model of machine into different states

Improved CBA Model…State Transition Diagram of Machine Working Conditions


• 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


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


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


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


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


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


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


Improved Cost Benefit Analysis Model

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