case study, results & discussion -...

159

Chapter 7

CASE STUDY, RESULTS & DISCUSSION

160

Chapter 7CASE STUDY, RESULTS & DISCUSSION

7.0. INTRODUCTION

The developed model can be used in the block replacement

decisions for a bock of Computers, a block of LCD televisions in hotel

industry, a block of air conditioners, a block of pressure gauges in

filling plants etc.

In this work, a case study is done with reference to a block of

Computer and computer based system. The results of the case study

are presented in the following sections.

7.1 CASE STUDY: Computer And Computer Based System

With the advent of Computers and the widespread of the

internet and fibre optics network across the world, the huge

population comprising good number of unemployed Indian youth

provided a platform for MNCs to open up their ICT branches in India.

Also Banking, Insurance, Railways, Corporate houses etc. are

becoming IT savvy and using computers in large scale for their

operations. Consequently the decisions on capital investment on

Computer and Computer based system became important and a need

for scientific approach for the replacement decisions is felt.

In this study a block of 1000 computers is considered. Though

it is difficult to identify the specific repairable intermediate states, to

make the model simplistic the repairs are grouped as follows:

161

Minor Repairs: Non-working of USB ports, keyboard, mouse, color

flickering, network card connections, LAN card, etc.,

Major repairs: Non-working of mother-board, monitor,

XGA/SVGA/VGA cards, SMPS, memory modules, processor

overheating, etc.,

State of the item: Status of the computer system at a particular

time period is considered as the state of the system.

As two intermediary states are considered between working and

complete failure states, the states can be defined as follows:

State 1: The item (Computer) is in working condition

State 2: The item (Computer) is in minor repairable condition

State 3: The item (Computer) is in major repairable condition

State 4: The item (Computer) is in complete failure

This leads to four-state discrete-time Markov chain with the

following state transition or state space diagram (Fig. 7.1)

Fig. 7.1: State transition or space diagram

1

3 4

2

162

7.1.1 Block Replacement Decision Using First Order Markov

Chain:

N=1000 computers

1C =Individual replacement cost = Rs.40000

2C = Minor repair cost=Rs.1000

3C = Major repair cost= Rs.8000

4C =each item cost under group replacement=Rs.30000/computer

nr =Nominal rate of interest=20%=0.2

020.0045.0085.0850.0XXXXX IV0

III0

II0

I00

Generator of Markov Process, TPM,

10004000.02222.02000.01778.00824.01059.01294.06824.00294.00388.00494.08824.0

P

At the end of the first period, the state probabilities can be calculated

from

PXX 01 )PXX( n0n

44434241

34333231

24232221

14131211

IV0

III0

II0

I0

IV1

III1

II1

I1

PPPPPPPPPPPPPPPP

XXXXXXXX

(where I,II, III & IV represents functional, minor repairable, major

repairable and complete failure states respectively; and Pij = Probability

of items switching over from ith state to jth state in a period)

10004000.02222.02000.01778.00824.01059.01294.06824.00294.00388.00494.08824.0

020.0045.0085.0850.0

163

The transition probabilities for the future time periods are calculated

based on the first order Markov process.

The state probabilities of items at any time period ‘n’ can be computed

as = (probability of items in different states in initial period) X nTPM

i.e. n0n PXX , n=1,2,3,…

OCTAVE software is used to calculate the probabilities of items falling

in different states for future periods (Appendix I ).

The transition and state probabilities for future time periods

and the respective number of failures (n), minor repairs (n) and

major repairs (n) are shown in Appendix II.

The calculations for the replacement decision (without the

influence of Inflation) using first order Markov Chain are shown on

Table 7.1.

Period

(n)

IndividualReplacement cost=n*C1

MinorRepaircost =n*C2

Majorrepair cost

= n*C3

Maintenance

cost(R)R

(lacs)

Groupreplacement cost= N * C4

Total cost(lacs)

AnnualAverage

cost(lacs)

1 27.96 0.61 4.15 32.73 32.73 300.00 332.73 332.732 49.87 0.62 4.19 54.69 87.42 300.00 387.42 193.713 73.75 0.63 4.19 78.58 166.00 300.00 466.00 155.334 100.68 0.63 4.19 105.52 271.53 300.00 571.53 142.885 132.03 0.64 4.19 136.86 408.39 300.00 708.39 141.676 169.33 0.64 4.18 174.17 582.56 300.00 882.56 147.097 214.30 0.65 4.17 219.13 801.70 300.00 1101.70 157.388 269.13 0.65 4.17 273.96 1075.66 300.00 1375.66 171.95

Table 7.1: Computations for block replacement model without theinfluence of Inflation using FOMC

As the average annual cost over a period of 5 years is minimum

and started increasing from 6th year onwards, it can be inferred that

economics can be achieved if the block replacement is done at the end

of 5th year.

164

7.1.2 Influence Of Forecasted Inflation On Block Replacement

Decision Using First Order Markov Chain

Price Inflation for Computer and Computer Based System in

India is studied and forecasted. A regression forecasting model with

trigonometric function is developed to accommodate the cyclical

fluctuations of real time inflation pattern. For this purpose Price

Inflation (based on WPI) for Computer and Computer based system

over a period of time is studied, forecasted and compared with actual

values for the known periods (Sec. 6.5.1).

The developed regression equation is:

π/4)π.5802sin(t59.0369t2.512Ft

Further to this, the Price Inflation for Computer and Computer

based system for the forthcoming time periods is predicted by the

developed Regression model with trigonometric function, which

yielded relatively minimal errors and are tabulated in the Table 6.7

and plotted in the Fig . 6.2.

The forecasted inflation ( tφ ) data thus obtained is used to

calculate the real value of money (or real rate of interest) by

Fisherman’s relation. This real rate of interest is used to calculate

Present worth Factor (PWF).

Real interest rate= tr = t

tn

φ1φr

, from Fisherman’s relation

And for the same data considered in the above section i.e. in

Sec. 7.1.1, the calculation of various costs for replacement decision

165

under the influence of inflation and time value of money using first

order Markov chain is shown in Table 7.2.

1 2 3 4 5 6 7

Period (n) Inflation, (%)

Real rate ofinterest, rt =

(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )

Discountfactor (n-1) n-1 Ind. Replacement

cost= n*C1 * n-1

1 -18.56 0.473 0.679 1 1 27.962 -17.41 0.453 0.688 0.6882 1.6882 34.323 -0.49 0.206 0.829 0.6876 2.3759 50.714 0.65 0.192 0.839 0.5900 2.9659 59.4125 17.58 0.021 0.980 0.9217 3.8877 121.696 18.72 0.011 0.989 0.9477 4.8355 160.497 35.65 -1.115 1.130 2.086 6.9220 447.158 36.8 -0.123 1.140 2.502 9.4243 673.43

8 9 10 =(7+8+9) 11 12 13 = (11+12) 14 = (13/6)

MinorRepair cost

=n*C2 * n-1

Majorrepair cost

=n*C3*n-1

Maintenancecost(R)(in lacs)

R(in lacs)

Groupreplacement

cost =N* C4* n-1

Total cost(TC)

(in lacs)

Weightedaverage annual

cost(Rs. in lacs)

0.61 4.15 32.73 32.73 300.00 332.73 332.730.43 2.88 37.64 70.37 206.47 276.84 163.980.43 2.88 54.04 124.41 206.29 330.70 139.190.37 2.47 62.26 186.67 177.01 363.69 122.620.59 3.86 126.15 312.83 276.52 589.35 151.590.61 3.96 165.07 477.91 284.33 762.24 157.631.36 8.71 457.23 935.14 625.96 1561.11 225.521.64 10.4 685.53 1620.68 750.68 2371.36 251.62

Table 7.2: Computations for block replacement model with theinfluence of Inflation using FOMC


and started increasing from 5th year onwards, it can be inferred that

economics can be achieved if the block replacement is done at the end

of 4th year.

i.e. when the inflation with up-trend is considered, the optimal

block replacement period is advanced to 4th year from 5th year.

166

7.1.2.1 Influence of Forecasted Inflation on Block replacement

decision using Second Order Markov chain

N=1000 computers

1C =Individual replacement cost = Rs.40000

2C = Minor repair cost=Rs.1000

3C = Major repair cost= Rs.8000

4C =each item cost under group replacement=Rs.30000/computer

nr =Nominal rate of interest=20%=0.2

Real interest rate= tr = t

tn

φ1φr

, from Fisherman’s relation

020.0045.0085.0850.0XXXXX IV0

III0

II0

I00

To generate/estimate state transition probabilities for future periods,

Weighted Moving Transition Probability Method (WTPM), a

parsimonious model that approximates second order Markov chain is

employed.

Generators of Markov process (TPMs) are

10004000.02222.02000.01778.00824.01059.01294.06824.00294.00388.00494.08824.0

P 2-n 0.2δ, 2n

10005105.00774.00791.03328.01554.00637.00716.07092.00749.00480.00577.08192.0

P 1-n 0.8δ, 1n

167

The state probabilities of items in different states can be computed as

= (probability of items in different states in initial period) X nTPM

i.e. nij0n PXX , where n=1,2,3,…

The transition and state probabilities for future time periods and the

respective number of failures(n), minor repairs (n) and major repairs

(n) are shown in Appendix IV.

The forecasted inflation data (refer Sec. 6.5.1, Table 6.7) is used

to calculate the real value of money (or real rate of interest) by

Fisherman’s relation. This real rate of interest is used to calculate

Present Worth Factor (PWF).

And the calculation of various costs for block replacement

decision under the influence of inflation and time value of money

using second order Markov chain is shown in Table 7.3.

As the weighted average annual cost over a period of 4 years is

minimum and started increasing from 5th year onwards, it can be

inferred that economics can be achieved if the block replacement is

done at the end of 4th year.

i.e. when the inflation with up-trend is considered, the optimal

block replacement period (using SOMC also) is advanced to 4th year

from 5th year.

168

1 2 3 4 5 6 7

Period(n)

Inflation, (%)


(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )


cost= n*C1 * n-1

1 -18.56 0.473 0.679 1 1 27.962 -17.41 0.453 0.688 0.6882 1.6882 34.323 -0.49 0.206 0.829 0.6876 2.3759 34.924 0.65 0.192 0.839 0.5900 2.9659 33.835 17.58 0.021 0.980 0.9217 3.8877 58.336 18.72 0.011 0.989 0.9477 4.8355 66.527 35.65 -0.115 1.130 2.0865 6.9220 162.258 36.8 -0.123 1.140 2.5022 9.4243 215.62

8 9 10 =(7+8+9) 11 12 13 =

(11+12) 14 = (13/6)

Minor Repaircost =n*C2 * n-1

Major repaircost =n*C3*n-1

Maintenance cost(R)

(lacs)R

(lacs)

Groupreplacement

cost =N* C4* n-1

Total cost(TC)

(in lacs)


cost(Rs. in lacs)

0.61 4.15 32.73 32.73 300.00 332.73 332.730.43 2.88 37.64 70.37 206.47 276.84 163.980.45 3.05 38.43 108.80 206.29 315.10 132.620.41 2.74 36.99 145.80 177.01 322.82 108.840.69 4.50 63.53 209.33 276.52 485.85 124.970.75 4.86 72.14 281.47 284.33 565.81 117.011.75 11.24 175.25 456.72 625.96 1082.69 156.412.22 14.16 232.01 688.74 750.68 1439.42 152.73

Table 7.3: Computations for block replacement model with theinfluence of Inflation using SOMC

When the influence of predicted inflation and net worth of the money

is considered, both ways using FOMC and SOMC resulted in the early

replacement of block of computers at the age of 4 years.

Note:A detailed discusion on the results is made in section 7.1.4.

169

7.1.3 Reengineering Treatment On Block Replacements

Table 7.3 shows that the block of computers should be replaced

at the end of 4th year. However this section looks into the possibility of

extending the optimal block replacement age by reengineering the

computer hardware in lieu of replacement option using thin client

technology. Thin client is a generic term for a group of emerging

technologies that reduces costs of hardware, maintenance and

support. Thin client solution can be very well used in IT intensive

banking, insurance, railways etc.

Thin client card, is a PC networking solution to computer

maintenance problems. It will make the maintenance department free

from regular upgradations and maintenance related issues. Also it

makes possible to run latest software on old PCs (P-1/P-II/P-III/P-

IV/celeron/AMD). Thin client solution shows dramatic experience in

the “cost saving” and “Performance enhancement”.

With the following given data, Table 7.4 and Table 7.5 gives the

average annual cost of block replacement without and with

reengineering using second order Markov chain respectively.

Withoutreengineering

Withreengineering

Total number of items in the system N=1000 N=1000Individual replacement cost per unit C1 = Rs.40000 C1r = Rs.30000Minor repair cost C2 = Rs. 1000 C2r = Rs.1000Major repair cost C3 =Rs. 8000 C3r = Rs.8000Group replacement cost (per computer) C4 = Rs.30000 C4r = Rs.30000Reengineering cost*** (Table 7.3B) ---- Cr =Rs.2200000Nominal rate of interest

nr 28% nr 28%

Table 7.3A: Data for reengineering treatment calculations

170

*** Note 1: Break-up for reengineering cost:

Item descriptionQuantity required for

1000 computers netwrok

Each item

cost (Rs.)

Total cost

(Rs.)

Thin client cards(1 card for every clientcomputer)

1000 Nos. 1300 1300000

1 GB RAM at the server side forevery 10 client computer systems

100 Nos. 4000 400000

Server capacity enhancement &installation charges(1 server for every 100 clientcomputers)

10 Nos. 50000 500000

Total reengineering cost = Rs. 22.00 lacs 2200000Table 7.3B: Break-up for reengineering cost

Note 2: As reengineering is aimed at the extension of optimal block

replacement age beyond 4th year (Table 7.3), reengineering will be

done at the end of 4th year. Therefore forecasted inflation from 5th year

onwards (Table 6.7) is employed.

The calculations for the optimal decision without reengineerng

(but replacement at the end of 4th year) are made in following section

Sec.7.1.3.1 and the calculations with reengneering are made in

Sec.7.1.3.2.

171

7.1.3.1 Forecasted Inflation Based Block Replacement Decision

Without Reengineering Using Second Order Markov Chain

020.0045.0085.0850.0XXXXX IV0

III0

II0

I00

To generate/estimate state transition probabilities for future

periods, weighted Moving Transition Probability Method (WTPM), a


employed.

In case of replacement without reengineering, it is considered to

be the regeneration point of whole life where the operating cost

function initially starts. Therefore the transition probabilities

(Appendix IV) that are considered for the new computer systems are

taken into account here also.


10004000.02222.02000.01778.00824.01059.01294.06824.00294.00388.00494.08824.0

P 2-n 0.2δ, 2n

10005105.00774.00791.03328.01554.00637.00716.07092.00749.00480.00577.08192.0

P 1-n 0.8δ, 1n



major repairs (n) are shown in Appendix IV.

172

The average annual block replacement cost is computed in

Table 7.4.

1 2 3 4 5 6 7



(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )


cost= n*C1 * n-1

1 17.58 0.089 0.919 1 1 27.962 18.72 0.078 0.928 0.9275 1.9275 46.253 35.65 -0.056 1.060 1.1231 3.0506 57.034 36.8 -0.064 1.069 1.2207 4.2713 70.005 53.73 -0.167 1.201 2.0806 6.3519 131.67

8 9 10 =(7+8+9) 11 12 13 =

(11+12) 14 = (13)/(6)


Major repair cost= n*C3*n-1

Maintenancecost(R)(lacs)

R(lacs)

Groupreplacement

cost = N* C4*n-1

Total cost(TC)

(in lacs)


cost(Rs. in lacs)

0.61 4.15 32.73 32.73 300.00 332.73 332.730.57 3.88 50.72 83.45 278.25 361.70 187.650.75 4.98 62.76 146.22 336.93 483.15 158.380.86 5.67 76.54 222.77 366.22 588.99 137.891.55 10.16 143.40 366.17 624.18 990.36 155.91

Table 7.4: Computations for block replacement modelwithout reengineering


and started increasing from 5th year, it can be inferred that economics

can be achieved if the block replacement is done at the end of 4th year.

173

7.1.3.2 Forecasted Inflation Based Block Replacement Decision

WITH REENGINEERING Using Second Order Markov Chain

To generate/estimate state transition probabilities for future

periods, weighted Moving Transition Probability Method (WTPM), a


employed

The initial state transition probabilities (X0 ) for the

reengineering model are considered as

I II III IV0 0 0 0 0X X X X X 0.780 0.087 0.073 0.060

In case of replacement with reengineering, as the maintenance

related issues are relatively less, the following Markov process

generators (TPMs) are assumed.


n-2

0.9718 0.0205 0.0038 0.00380.8966 0.0575 0.0345 0.0115

P0.8767 0.0685 0.0274 0.02740.0000 0.0000 0.0000 1.0000

0.2δ, 2n

n-1

0.9526 0.0308 0.0115 0.00510.8621 0.0805 0.0345 0.0230

P0.8493 0.0822 0.0411 0.02740.0000 0.0000 0.0000 1.0000

0.8δ, 1n



major repairs (n) are shown in Appendix VI.

174

The average annual optimal block replacement cost is computed

in Table 7.5.

1 2 3 4 5 6 7 8

Period(n)

Inflation, (%)


(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )

Discountfactor (n-1) n-1

Initial cost ofre-

engineering(lacs)

Ind.Replacement

cost= n*C1*n-1

1 17.58 0.089 0.919 1 1 22.00 20.272 18.72 0.078 0.928 0.9275 1.9275 --- 20.093 35.65 -0.056 1.060 1.1231 3.0506 --- 25.974 36.8 -0.064 1.069 1.2207 4.2713 --- 30.145 53.73 -0.167 1.201 2.0806 6.3519 --- 54.85

9 10 11 =(7+8+9+10) 12 13 14 =

(12+13) 15 = (14)/(6)


Major repaircost =n*C3*n-1

Maintenancecost(R)(lacs)

R(lacs)

Groupreplacementcost = N*C4*n-1

Total cost(TC)

(in lacs)

Weighted averageannual cost(Rs. in lacs)

0.34 1.08 43.70 43.70 300.00 343.70 343.700.33 1.04 21.47 65.18 278.25 343.43 178.170.42 1.27 27.67 92.85 336.93 429.78 140.880.47 1.40 32.02 124.87 366.22 491.10 114.970.84 2.42 58.11 182.99 624.18 807.18 127.07

Table 7.5: Computations for block replacement model withreengineering

The average annual block replacement cost (Rs.93.86 lacs) is

minimum during 4th year. This is less than the average annual block

replacement cost (Rs.137.89 lacs, Table 7.4) without reengineering.

Therefore, it is suggestible to go for reengineering the computer

network with Thin client technology at the end of 4th year and replace

the block of computers at the end of 8th year (4 years from Table 7.3

before reengineering + 4 years from Table 7.5 after reengineering).

175

7.1.4 RESULTS AND ANALYSIS:Table 7.6 and Fig. 7.2 furnish the summary of numerical results

obtained (Table 7.1, Table 7.2 and Table 7.3) for block replacement

decision calculated using - first order Markov chain with and without

inflation, and Weighted Moving Transition Probability (WMTP)

methods.

Timeperiod (n

years)

Weighted Av. Annual Cost (lacs)

Using First Order Markov ChainUsing WMTP for

second orderMarkov chain

without inflation with inflation with inflation1 332.73 332.73 332.732 193.71 163.98 163.983 155.33 139.19 132.624 142.88 122.62 108.845 141.67 151.59 124.976 147.09 157.63 117.017 157.38 225.52 156.418 171.95 251.62 152.73

Table 7.6: Numerical results for weighted average annual costs (withconstant maintenance cost)

Fig. 7.2 Weighted Average Cost per Year (withconstant maintenance cost)

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8Time period (years)

Ave

rage

cos

t ( in

lacs

) per

yea

r

FOMC withoutInflationFOMC with Inflation

SOMC with Inflation

176

When the influence of inflation is not considered, First

Order Markov Chain (FOMC) model resulted in the replacement age

as 5 years.

When the influence of predicted inflation and net worth

of the money is considered, both ways using FOMC and SOMC

resulted in the early replacement of block of computers at the age of

4 years i.e. when the inflation with up-trend is considered, the

optimal block replacement period is advanced to 4th year from 5th

year.

If the decision would have been taken without

considering the influence of inflation and money value, it might be

required to allocate and incur huge amounts on maintenance.

In modern economy with the increased business activity,

Inflation, dominantly, is playing a macro economic role influencing the

replacement decisions. If the macro economic variables such as

inflation are not considered in decision-making, service systems

particularly in the field of computers will face pathetic conditions and

incur heavy losses.

When the influence of inflation is considered, WMTP technique

resulted in optimal replacement age as 4 years with less annual

average cost of (Rs.108 lacs) comparative with first order Markov

chain model.

Therefore, for the block of the computers considered for study in

this work, 4 years is the optimal replacement period.

177

However the reengineering at the end of 4th year with thin client

technology in lieu of replacement resulted in reduction in average

annual block replacement cost with optimal replacement period

extended by four years (Sec. 7.1.3.2, Table 7.5). In other words,

reengineering resulted in extending the optimal replacement period to

8 years.

Table 7.7 shows the summary of the numerical results obtained

(Table 7.4 and table 7.5) for annual average block replacement costs

from 5th year onwards without and with reengineering.

Timeperiod (n

years)

Using WMTP method(Second order Markov Chain)

Without reengineering With reengineering

Total cost (lacs)Weighted

Av. AnnualCost(lacs)

Total cost (lacs)Weighted

Av. AnnualCost(lacs)

5 332.73 332.73 343.70 343.706 361.70 187.65 343.43 178.177 483.15 158.38 429.78 140.888 588.99 137.89 491.10 114.979 990.36 155.91 807.18 127.07

Table 7.7 Numerical results for weighted average annual blockreplacement cost without and with reengineering

The average annual block replacement cost (Rs.114.97 lacs) is

minimum during 8th year. This is less than the average annual block

replacement cost (Rs.137 lacs) without reengineering.

Therefore, it is suggestible to go for reengineering the computer

network with Thin client technology at the end of 4th year and replace

the block of computers at the end of 8th year (4 years from Table 7.3

before reengineering + 4 years from Table 7.5 after reengineering).

178

The overall savings are shown in the Table 7.8.

Cost Comparison:

Time period (Years)

Total Cost (maintenance + replacement)(in lacs)

withoutreengineering

(Computers arereplaced at the end

of 4th year)

withreengineering

(Computers arereengineered at

the end of 4th year)1-4 years (From Table 7.3) 322.82 145.80*5-8 years (From Table 7.7) 588.99 491.10Total 911.81 636.90

Savings 274.91% of savings 30.14%

Table 7.8 Cost comparison for block replacement without and withreengineering

*As the reengineering is done at the end of 4th year, only maintenancecost is to be taken; group replacement cost need not be considered.

The cost savings are 30.14% by the way of reengineering.

Therefore for the block of computers considered for study in this work,

8 years is the optimal replacement period with reengineering at the

end of 4th year.

179

7.2 MODEL BEHAVIOUR - FEW MORE OBSERVATIONS

The replacement decision, obtained through the sections 7.1.1,

and 7.1.2, is with constant maintenance cost and forecasted inflation

data. To understand the impact of the variable maintenance cost and

different inflation trends, block replacement strategy is evaluated in

the following cases:

a) Variable maintenance cost – low initial maintenance cost with big

increments during later periods – in the same range of

maintenance cost.

b) Variable maintenance cost - high initial maintenance cost with

small increments during later periods – in the same range of

maintenance cost.

c) Rapid up-trend in inflation (assumed inflation data)

d) Sluggish up-trend in inflation (assumed inflation data)

180

7.2(a) Replacement decision with low initial maintenance cost

and big increments during the later time periods

To understand the influence of variable maintenance cost, a

pattern with low initial maintenance cost (minor and major repair

costs) and big increments at later periods is considered as given in

Table 7.9.

Timeperiod(years)

1 2 3 4 5 6 7 8

Minorrepair cost(Rs.)

200 500 700 900 1100 1300 1300 1300

Majorrepair cost(Rs.)

2000 5000 8000 11000 13000 13000 13000 13000

Table 7.9: Variable maintenance cost - with low initialmaintenance cost and big increments during the later time periods

Note: Variable maintenance cost (minor and major repair costs) is

taken with in the range of fixed maintenance cost.

The remaining data is (taken) same as in the case of fixed

maintenance cost (Sec. 7.1.1).

181

(i) The computations for block replacement decision (without the

influence of inflation) using FOMC with low initial maintenance cost

and big increments during the later time periods are shown in the

Table 7.10.

1 2 3 4 5 6 7

Period

(n)

Individual

Replacement =n*C1

EachMinorrepaircost(C2 )

TotalMinorRepaircost =n*C2

EachMajor

repair cost(C3 )

TotalMajor

repair cost= n*C3

Maintenance

cost(R)

1 27.96 0.002 0.12 0.02 1.03 29.122 49.87 0.005 0.31 0.05 2.61 52.803 73.75 0.007 0.44 0.08 4.19 78.394 100.60 0.009 0.57 0.11 5.77 107.035 132.03 0.011 0.70 0.13 6.81 139.556 169.33 0.013 0.84 0.13 6.80 176.987 214.30 0.013 0.84 0.13 6.79 221.948 269.13 .01300 0.85 0.13 6.78 276.77

8 9 10 11

R(lacs)

Groupreplacementcost = N * C4

Total cost (lacs) Annual Averagecost (lacs)

29.12 300.00 329.12 329.1281.92 300.00 381.92 190.96

160.32 300.00 460.32 153.44267.36 300.00 567.36 141.84406.91 300.00 706.91 141.38583.89 300.00 883.89 147.31805.83 300.00 1105.83 157.97

1082.60 300.00 1382.60 172.82Table 7.10: Computations for block replacement model without the influence of

inflation using FOMC with low initial maintenance cost and big increments duringlater periods

From Table 7.10, as the average annual cost is minimum over a

period of 5 years and started increasing from 6th year onwards, it can

be inferred that economics can be achieved if the block replacement is


It is observed that the replacement decision remains same as in

the case of fixed maintenance cost (Sec. 7.1.1, Table 7.1)

182

(ii) Table 7.11 shows the computations for block replacement decision

(with the influence of inflation) using FOMC with low initial

maintenance cost and big increments during the later time periods.

The data (except the minor and major repair costs) is same as in

the case of fixed maintenance cost (Sec. 7.1.1).

1 2 3 4 5 6 7 8

Period(n)

Inflation, (%)


(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )


Ind.Replacementcost= n*C1

* n-1

EachMinorrepair

cost (C2 )

1 -18.54 0.473 0.679 1 1 27.96 0.0022 -17.41 0.453 0.688 0.6882 1.6882 34.32 0.0053 -0.49 0.206 0.829 0.6876 2.3759 50.71 0.0074 0.65 0.192 0.839 0.5900 2.9659 59.41 0.0095 17.58 0.021 0.980 0.9217 3.8877 121.69 0.0116 18.72 0.011 0.989 0.9477 4.8355 160.49 0.0137 35.65 -0.115 1.130 2.0865 6.9220 447.15 0.0138 36.8 -0.123 1.140 2.5022 9.4243 673.43 0.013

9 10 11 12 =(7+9+11) 13 14 15 =

(13+14) 16 = (15/6)

TotalMinorRepaircost =n*C2 *n-1

EachMajorrepaircost(C3 )

TotalMajorrepaircost =n*C3*n-

1

Maintenance cost(R) R

Groupreplacement

cost =N* C4* n-1

Total cost(TC)

Weightedaverage

annual cost(Rs. in lacs)

0.12 0.02 1.03 29.12 29.12 300.00 329.12 329.120.21 0.05 1.80 36.34 65.46 206.47 271.94 161.070.30 0.08 2.88 53.91 119.37 206.29 325.67 137.070.33 0.11 3.40 63.15 182.53 177.01 359.55 121.220.65 0.13 6.28 12.86 311.16 276.52 587.68 151.160.79 0.13 6.44 16.77 478.90 284.33 763.24 157.840.17 0.13 1.41 46.30 941.99 625.96 1567.96 226.510.21 0.13 1.69 69.25 1634.5

6750.68 2385.24 253.09

Table 7.11: Computations for block replacement model with the influence ofinflation using FOMC with low initial maintenance cost and big increments duringlater periods



183




the case of fixed maintenance cost (Sec. 7.1.2, Table 7.2). It is also to

be noted that the increase in inflation resulted in the advancement of

replacement period.

(iii) Table 7.12 shows the computations for block replacement

decision (with the influence of inflation) using SOMC with low initial

maintenance cost and big increments during the later time periods.


the case of fixed maintenance cost (Sec. 7.1.2.1).






the case of fixed maintenance cost (Sec. 7.1.2.1, Table 7.3). It is also

to be noted that the increase in inflation resulted in the advancement

of replacement period.

184

1 2 3 4 5 6 7 8

Period(n)

Inflation, (%)


(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )



* n-1

EachMinorrepair

cost (C2 )

1 -18.54 0.473 0.679 1 1 27.96 0.0022 -17.41 0.453 0.688 0.6882 1.6882 34.32 0.0053 -0.49 0.206 0.829 0.6876 2.3759 34.92 0.0074 0.65 0.192 0.839 0.5900 2.9659 33.83 0.0095 17.58 0.021 0.980 0.9217 3.8877 58.33 0.0116 18.72 0.011 0.989 0.9477 4.8355 66.52 0.0137 35.65 -0.115 1.130 2.0865 6.9220 162.25 0.0138 36.8 -0.123 1.140 2.5022 9.4243 215.62 .01300

9 10 11 12 =(7+9+11) 13 14 15 =

(13+14) 16 = (15/6)

Total MinorRepair cost

=n*C2 * n-1


Total Majorrepair cost

=n*C3*n-1

Maintenance cost(R) R

Groupreplacemen

t cost =N* C4* n-1

Total cost(TC)

Weightedaverage


0.12 0.02 1.03 29.12 29.12 300.00 329.12 329.120.21 0.05 1.80 36.34 65.46 206.47 271.94 161.070.32 0.08 3.05 38.29 103.76 206.29 310.05 130.500.37 0.11 3.77 37.98 141.74 177.01 318.76 107.470.75 0.13 7.31 66.41 208.16 276.52 484.68 124.670.97 0.13 7.90 75.40 283.56 284.33 567.90 117.442.28 0.13 18.27 182.80 466.37 625.96 1092.34 157.802.89 0.13 23.01 241.53 707.90 750.68 1458.58 154.76

Table 7.12: Computations for block replacement model with the influence ofinflation using SOMC with low initial maintenance cost and big increments duringlater periods.

185

Table 7.13 and Fig. 7.3 furnish the summary of the numerical

results obtained (Table 7.10, Table 7.11 and Table 7.12), for block

replacement decision with low initial maintenance cost and big

increments during the later periods, using - first order Markov chain

with and without inflation, and Weighted Moving Transition

Probability (WMTP) methods.

Timeperiod (n

years)

Using First Order Markov Chain Using SecondOrder Markov chain

without inflation with inflation with inflationAnnual Av.cost(lacs)

Weighted Av.Annual Cost(lacs)

Weighted Av. AnnualCost(lacs)

1 329.12 329.12 329.122 190.96 161.07 161.073 153.44 137.07 130.54 141.84 121.22 107.475 141.38 151.16 124.676 147.31 157.84 117.447 157.97 226.51 157.88 172.82 253.09 154.76

Table 7.13: Numerical results for annual average costs with low initialmaintenance cost and big increments during later periods

fig. 7.3 Average cost per year under low initialmaintenace cost with big increments

0

50

100

150

200

250

300

350

1 2 3 4 5 6 7 8

Time period (years)

Ave

rage

cos

t per

yea

r in

lacs

Without inflation

With inflation uinsgFOMCWith inflation usingSOMC

186

With the low initial maintenance cost and big increments during later

periods:

- When the influence of inflation is not considered, First Order

Markov Chain (FOMC) model resulted in the replacement age as 5

years, and

- When the influence of inflation is considered, FOMC and SOMC

calculations resulted in the early replacement of block of

computers at the age of 4 years.

It can be observed that the optimal replacement decision (block

replacement at the end of 4th year) remains same as in the case of

fixed maintenance cost (Sec. 7.1.4).

187

7.2(b) Replacement Decision With High Initial Maintenance Cost

And Small Increments

To understand the influence of variable maintenance cost,

another pattern with high initial maintenance cost (minor and major

repair costs) and small increments at later periods is considered as

given in Table 7.14.

Note: Variable maintenance cost (minor and major repair costs) is

taken in the range of fixed maintenance cost.

Timeperiod(years)

1 2 3 4 5 6 7 8

Minorrepair cost(Rs.)

1000 1100 1200 1300 1400 1400 1400 1400

Majorrepair cost(Rs.)

8000 9000 10000 11000 12000 13000 13000 13000

Table 7.14: Variable maintenance cost - with high initial maintenancecost and small increments during the later time periods

The remaining data is (taken) same as in the case of fixed

maintenance cost (Sec. 7.1.1).

188

(i) The computations for block replacement decision (without the

influence of inflation) using FOMC with low initial maintenance cost

and big increments during the later time periods are shown in the

Table 7.15.

1 2 3 4 5 6 7

Period

(n)

Individual

Replacement =n*C1

EachMinorrepaircost(C2 )

TotalMinorRepaircost =n*C2

EachMajor

repair cost(C3 )

TotalMajor

repair cost= n*C3

Maintenance

cost(R)

1 27.96 0.010 0.61 0.08 4.15 32.732 49.87 0.011 0.68 0.09 4.71 55.273 73.75 0.012 0.75 0.10 5.24 79.764 100.60 0.013 0.82 0.11 5.77 107.285 132.03 0.014 0.89 0.12 6.29 139.226 169.33 0.014 0.90 0.13 6.80 177.047 214.30 0.014 0.91 0.13 6.79 222.008 269.13 0.014 0.92 0.13 6.78 276.83

8 9 10 11

R(lacs)

Groupreplacementcost = N *

C4

Total cost (lacs) Annual Averagecost (lacs)

32.73 300.00 332.73 332.7388.00 300.00 388.00 194.00

167.77 300.00 467.77 155.92275.06 300.00 575.06 143.76414.28 300.00 714.28 142.85591.32 300.00 891.32 148.55813.33 300.00 1113.33 159.04

1090.17 300.00 1390.17 173.77Table 7.15: Computations for block replacement decision without the influence of

inflation using FOMC with high initial maintenance cost and small incrementsduring later periods





It is observed that the replacement decision remains same as in the

case of fixed maintenance cost (Sec. 7.1.1, Table 7.1)

189

(ii) Table 7.16 shows the computations for block replacement decision

(with the influence of inflation) using FOMC with high initial

maintenance cost and small increments during the later time periods.


the case of fixed maintenance cost (Sec. 7.1.1).

1 2 3 4 5 6 7 8

Period(n)

Inflation, (%)


(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )



* n-1

EachMinorrepair

cost (C2 )

1 -18.54 0.473 0.679 1 1 27.96 0.0102 -17.41 0.453 0.688 0.6882 1.6882 34.32 0.0113 -0.49 0.206 0.829 0.6876 2.3759 50.71 0.0124 0.65 0.192 0.839 0.5900 2.9659 59.41 0.0135 17.58 0.021 0.980 0.9217 3.8877 121.69 0.0146 18.72 0.011 0.989 0.9477 4.8355 160.49 0.0147 35.65 -0.115 1.130 2.0865 6.9220 447.15 0.0148 36.8 -0.123 1.140 2.5022 9.4243 673.43 0.014

910

1112 =

(7+9+11)

13 14 15 =(13+14) 16 = (15/6)

Total MinorRepair cost

=n*C2 * n-1


TotalMajorrepaircost =n*C3*n-1

Maintenance

cost(R)R

Groupreplacement

cost =N* C4* n-1

Total cost(TC)


cost(Rs. in lacs)

0.61 0.08 4.15 32.73 32.73 300.00 332.73 332.730.47 0.09 3.24 38.04 70.77 206.47 277.25 164.220.52 0.10 3.61 54.84 125.62 206.29 331.92 139.700.48 0.11 3.40 63.30 188.93 177.01 365.95 123.380.82 0.12 5.79 128.32 317.25 276.52 593.78 152.730.85 0.13 6.44 167.80 485.06 284.33 769.39 159.111.90 0.13 14.16 463.22 948.29 625.96 1574.25 227.422.30 0.13 16.98 692.72 1641.0

1750.68 2391.69 253.77

Table 7.16: Computations for block replacement model with the influence ofinflation using FOMC with high initial maintenance cost and small incrementsduring later time periods



190




the case of fixed maintenance cost (Sec. 7.1.2, Table 7.2). It is also to

be noted that the increase in inflation resulted in the advancement of

replacement period.

(iii) Table 7.17 shows the computations for block replacement

decision (with the influence of inflation) using SOMC with high initial

maintenance cost and small increments during the later time periods.


the case of fixed maintenance cost (Sec. 7.1.2.1).






the case of fixed maintenance cost (Sec. 7.1.2.1, Table 7.3). It is also

to be noted that the increase in inflation resulted in the advancement

of replacement period.

191

1 2 3 4 5 6 7 8



(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )



* n-1

EachMinorrepair

cost (C2 )

1 -18.54 0.473 0.679 1 1 27.96 0.0102 -17.41 0.453 0.688 0.6882 1.6882 34.32 0.0113 -0.49 0.206 0.829 0.6876 2.3759 34.92 0.0124 0.65 0.192 0.839 0.5900 2.9659 33.83 0.0135 17.58 0.021 0.980 0.9217 3.8877 58.33 0.0146 18.72 0.011 0.989 0.9477 4.8355 66.52 0.0147 35.65 -0.115 1.130 2.0865 6.9220 162.25 0.0148 36.8 -0.123 1.140 2.5022 9.4243 215.62 0.014

9 10 11 12 =(7+9+11) 13 14 15 =

(13+14) 16 = (15/6)

TotalMinorRepaircost =n*C2 *n-1

EachMajorrepair

cost (C3 )

Total Majorrepair cost

=n*C3*n-1

Maintenance cost(R)

(Rs. inlacs)

R(Rs. inlacs)

Groupreplacemen

t cost =N* C4* n-1

Total cost(TC)

(Rs. inlacs)

Weightedaverage


0.61 0.08 4.15 32.73 32.73 300.00 332.73 332.730.47 0.09 3.24 38.04 70.77 206.47 277.25 164.220.55 0.10 3.81 39.28 110.06 206.29 316.35 133.150.54 0.11 3.77 38.15 148.21 177.01 325.23 109.650.96 0.12 6.75 66.05 214.27 276.52 490.79 126.24

10.05 0.13 7.90 75.48 289.75 284.33 574.09 118.7220.45 0.13 18.27 182.98 472.73 625.96 1098.70 158.7230.12 0.13 23.01 241.75 714.49 750.68 1465.17 155.46

Table 7.17: Computations for block replacement model with the influence of inflationusing SOMC with high initial maintenance cost and small increments during laterperiods

192


results obtained (Table 7.15, Table 7.16, Table 7.17), for block

replacement decision with high initial maintenance cost and small

increments during the later periods, using - first order Markov chain

with and without inflation, and Weighted Moving Transition

Probability (WMTP) methods.

Timeperiod (n

years)

Using First Order Markov Chain Using SecondOrder Markov chain

without inflation with inflation with inflationAnnual Av.cost(lacs)

Weighted Av.Annual Cost(lacs)

Weighted Av. AnnualCost(lacs)

1 332.73 332.73 332.732 194.00 164.22 164.223 155.92 139.7 133.154 143.76 123.38 109.655 142.85 152.73 126.246 148.55 159.11 118.727 159.04 227.42 158.728 173.77 253.77 155.46

Table 7.18: Numerical results for annual average costs with highinitial maintenance cost and small increments during later periods

Fig. 7.4 Average cost per year under high initialmaintenanace cost with small increments

0

50

100

150

200

250

300

350


Ave

rage

cos

t per

yea

r

Without inflationusing FOMCWith inflation usingFOMCWith inflation usingSOMC

193

With the high initial maintenance cost and small increments

during later periods:

- When the influence of inflation is not considered, First Order

Markov Chain (FOMC) model resulted in the replacement age as 5

years, and

- When the influence of inflation is considered, FOMC and SOMC

calculations resulted in the early replacement of block of

computers at the age of 4 years.

It can be observed that the optimal replacement decision (block

replacement at the end of 4th year) remains same as in the case of

fixed maintenance cost (Sec. 7.1.4).

Therefore, when the maintenance cost is varied with different

trends – high initial maintenance cost and small increments; and with

low initial maintenance cost and big increments; – in the same range of

fixed maintenance cost, it is observed that the replacement decision

remains the same as in the case of fixed maintenance cost.

194

7.2(c) Influence Of Changing Inflation TrendsIn the Section 7.1.2, to know the influence of inflation, the

forecasted inflation is used. However to evaluate behaviour of the

developed Block Replacement model under different inflation patterns,

two trends – rapid up-trend and sluggish up-trend – in inflation are

considered.

The data for these rapid and sluggish (inflation) up-trends with

respect to the forecasted inflation (refer Sec. 6.5.1, Table 6.7) is shown

in Table 7.19 and plotted in Fig. 7.5.

Timeperiod(year)

Forecastedinflation (%)

Rapidincrease

in inflation(%)

Sluggishincrease ininflation (%)

1 -18.54 -18.54 -18.542 -17.41 -5 -153 -0.49 5 -124 0.65 20 -95 17.58 30 -66 18.72 40 -37 35.65 50 18 36.8 60 4

Table 7.19: Inflation trends: Rapid and Sluggish up-trends in inflationwith respect to the forecasted inflation.

Fig. 7.5 Inflation trends:Rapid and Sluggish up-trends with respect to forecasted inflation

-30-20-10

010203040506070

1 2 3 4 5 6 7 8

Time period (years)

Infla

tion(

%) Forecasted

inflationRapid increase ininflationSluggish increasein inflation

195

(i) Table 7.20 shows the computations for influence of the rapid

increase in inflation on block replacement decision using FOMC with

fixed maintenance cost.

The remaining data (except the inflation values) is same as in the case

of forecasted inflation (Sec 7.1.1).

1 2 3 4 5 6 7



(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )


cost= n*C1 * n-1

1 -18.54 0.473 0.679 1 1 27.962 -5 0.263 0.792 0.7916 1.7916 39.483 5 0.143 0.875 0.7656 2.5572 56.464 20 0.000 1.000 1 3.5572 100.685 30 -0.077 1.083 1.3773 4.9346 181.856 40 -0.143 1.167 2.1613 7.0960 366.007 50 -0.200 1.250 3.8146 10.910 817.498 60 -0.250 1.333 7.4915 18.402 2016.19

8 9 10 =(7+8+9) 11 12 13 = (11+12) 14 = (13/6)

MinorRepair cost

=n*C2 * n-1

Majorrepair cost

=n*C3*n-1

Maintenance

cost(R)R

(Rs. inlacs)

Groupreplacemen

t cost =N* C4* n-1

Total cost(TC)

(Rs. in lacs)


cost(Rs. in lacs)

0.61 4.15 32.73 32.73 300.00 332.73 332.730.49 3.31 43.29 76.02 237.50 313.52 174.990.48 3.21 60.16 136.19 229.68 365.88 143.070.63 4.19 105.52 241.71 300.00 541.71 152.280.88 5.77 188.51 430.23 413.20 843.44 170.921.39 9.05 376.45 806.68 648.41 1455.10 205.052.48 15.94 835.92 1642.6

11144.40 2787.02 255.43

4.93 31.29 2052.42 3695.03

2247.46 5942.49 322.92Table 7.20: Computations for the influence of rapid up-trend in

Inflation on block replacement decision using FOMC



196


done at the end of 3rd year.

It is observed that when there is a rapid up-trend in inflation,

the optimal replacement period is advanced by one year with respect

to block replacement decision with forecasted inflation (Sec. 7.1.2,

Table 7.2).

(ii) Table 7.21 shows the computations for influence of the sluggish

up-trend in inflation on block replacement decision using FOMC with

fixed maintenance cost.

The remaining data (except the inflation values) is same as in

the case of forecasted inflation (Sec 7.1.1).





It is observed that when there is a sluggish up-trend in

inflation, the optimal replacement period is delayed by one year with

respect to block replacement decision with forecasted inflation (Sec.

7.1.2, Table 7.2).

197

1 2 3 4 5 6 7



(rn - t ) /(1 + t )

PWF, =

1/(1+ rt )


cost= n*C1 * n-1

1 -18.54 0.473 0.679 1 1 27.962 -15 0.412 0.708 0.7083 1.7083 35.323 -12 0.364 0.733 0.5377 2.2461 39.664 -9 0.319 0.758 0.4360 2.6822 43.905 -6 0.277 0.783 0.3765 3.0587 49.716 -3 0.237 0.808 0.3451 3.4038 58.437 1 0.188 0.842 0.3555 3.7593 76.188 4 0.154 0.867 0.3672 4.1265 98.83

8 9 10 =(7+8+9) 11 12 13 =

(11+12) 14 = (13/6)


Majorrepair cost

=n*C3*n-1

Maintenancecost(R)

(Rs. in lacs)R

(Rs. in lacs)

Groupreplacement

cost =N* C4* n-1

Totalcost(TC)

( in lacs)

Weightedaverage


0.61 4.15 32.73 32.73 300.00 332.73 332.730.44 2.96 38.73 71.47 212.50 283.97 166.220.33 2.25 42.26 113.73 161.33 275.06 122.460.27 1.83 46.01 159.75 130.82 290.57 108.330.24 1.57 51.53 211.28 112.95 324.23 106.000.22 1.44 60.10 271.39 103.53 374.92 110.140.23 1.48 77.90 349.29 106.65 455.94 121.280.24 1.53 100.61 449.90 110.17 560.08 135.72

Table 7.21: Computations for the influence of sluggish up-trend inInflation on block replacement decision using FOMC


results obtained (Table 7.2, Table 7.20 and Table 7.21), for the

influence of rapid and sluggish up-trends in inflation (with respect to

the forecasted inflation) on block replacement decision, using first

order Markov chain.

198

Timeperiod( years)

Weighted Average annual cost per year(Rs. in lacs)

forForecastedinflation

(from Table 7.2)

for Rapidincrease ininflation

(from Table 7.20)

for Sluggishincrease ininflation

(from Table 7.21)1 332.73 332.73 332.732 163.98 174.99 166.223 139.19 143.07 122.464 122.62 152.28 108.335 151.59 170.92 106.006 157.63 205.05 110.147 225.52 255.43 121.288 251.62 322.92 135.72

Table 7.22: Average cost per year for rapid and sluggish up-trends in

inflation

Fig. 7.6 Weighted Average cost per year fordifferent inflation trends

0

50

100

150

200

250

300

350


Ave

rage

cos

t per

yea

r

for Forecastedinflationfor Rapid increase ininflationfor Sluggish increasein inflation

The optimal replacement age for the block items in the case of

forecasted inflation is 4 years. Where as, in case of rapid up-trend in

inflation, the optimal replacement age is advanced to 3rd year and

instead, in case of sluggish up-trend in inflation, the optimal

replacement age for the block of items is delayed to 5th year.

199

Summary: This chapter deals with the case study in which the

developed replacement model using Markov chains approach is

applied to evaluate the replacement strategies and determine the age

at which block replacement can be done that is economical, for a

block of computers and computer based system. To understand the

performance of the developed model, the replacement age obtained

under various conditions is tabulated in Table 7.23 and Table 7.24.

Replacement age in years

Without reengineering Withreengineering

Using First OrderMarkov Chain

Using Second OrderMarkov chain

Using Second OrderMarkov chain

withoutinflation

withinflation with inflation with inflation

With constantmaintenance cost 5 years 4 years 4 years 8 yearsVariable maintenancecost: low initialmaintenance cost withbig increments duringlater periods

5 years 4 years 4 years --

Variable maintenancecost: High initialmaintenance cost withsmall incrementsduring later periods

5 years 4 years 4 years --

Table 7.23: Replacement ages with the developed models

Replacement age in yearsfor

Forecastedinflation

for Rapidincrease ininflation

forSluggish

increase ininflation

4 years 3 years 5 years

Table 7.24: Replacement ages with variable trends in inflation

case study, results & discussion -...

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