REPLACEMENT MODELS
Dr. Devendra Choudhary Department of Mechanical Engineering
Govt. Engineering College Ajmer
Replacement theory
Replacement theory deal in the decision making process of replacing a used equipment with a substitute; mostly a new equipment of better usage.
The replacement might be necessary due to the deteriorating property or failure or breakdown of particular equipment.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Replacement models
‘When to replace’……….an item, machine, automobile etc.
Situations are analyzed mathematically.
Hazard functionh(t)
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Types of replacement models
Model-I: Aging of machines
Items that deteriorate with time e.g. machine tools, vehicles, equipment buildings …
Model-II: Availability of similar new machines with better usages
Items becoming out-of-date due to new developments like ordinary weaving looms by automatic, manual accounting by tally, computers, cars ...
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Types of replacement models
Model-III: Aging of machines with time value of money
In Model-I, the depreciation of money (or present worth factor) is considered in calculations.
Model-IV: Group replacement
Replacement of Items that fail suddenly
Items which do not deteriorate but fail completely after certain amount of use like electronic parts, street lights...
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-I: Aging of machines
To Find The Best Replacement Age (Time) of a Machine
The total cost of machine in nyears
= Cost of machine – Scrap value of machine + Total maintenance cost in nyears.
Replace when average total cost is lowest
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-I: Aging of machines
Do not replace if the next years maintenance cost is less than the previous years average total cost.
Replace if the next years maintenance cost is greater than the previous years average total cost.
Model-I: Aging of machines
The maintenance cost and resale value per year of a machine whose purchase price is Rs. 7000 is given below:
When should the machine be replaced?
Year 1 2 3 4 5 6 7 8
Maint.
cost
900 1200 1600 2100 2800 3700 4700 5900
Resale value
4000 2000 1200 600 500 400 400 400
Model-I: Aging of machines
Year Purchase price
Maintenan ce cost
Cum.
Maint.
Cost
Resale
value Total cost Ave. total cost
1 7000 900 900 4000 3900 3900.0
2 7000 1200 2100 2000 7100 3550.0
3 7000 1600 3700 1200 9500 3166.7
4 7000 2100 5800 600 12200 3050.0
5 7000 2800 8600 500 15100 3020.0
6 7000 3700 12300 400 18900 3150.0
7 7000 4700 17000 400 23600 3371.4
8 Dr. Devendra Choudhary, Govt. Engineering College, Ajmer7000 5900 22900 400 29500 3687.5
Model-I: Aging of machines
Year Purchase price
Maintenan ce cost
Cum.
Maint.
Cost
Resale
value Total cost Ave. total cost
1 7000 900 900 4000 3900 3900.0
2 7000 1200 2100 2000 7100 3550.0
3 7000 1600 3700 1200 9500 3166.7
4 7000 2100 5800 600 12200 3050.0
5 7000 2800 8600 500 15100 3020.0
6 7000 3700 12300 400 18900 3150.0
7 7000 4700 17000 400 23600 3371.4
8 Dr. Devendra Choudhary, Govt. Engineering College, Ajmer7000 5900 22900 400 29500 3687.5
Model-I: Aging of machines
Year Purchase price
Maintenan ce cost
Cum.
Maint.
Cost
Resale
value Total cost Ave. total cost
1 7000 900 900 4000 3900 3900.0
2 7000 1200 2100 2000 7100 3550.0
3 7000 1600 3700 1200 9500 3166.7
4 7000 2100 5800 600 12200 3050.0
5 7000 2800 8600 500 15100 3020.0
6 7000 3700 12300 400 18900 3150.0
7 7000 4700 17000 400 23600 3371.4
8 7000 5900 22900 400 29500 3687.5
0 1000 2000 3000 4000 5000 6000 7000
1 2 3 4 5 6 7 8
Maintenance cost Resale value Ave. total cost
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
Machine A costs Rs 9000. Annual operating cost are Rs 200 for the 1st year, and then increase by Rs 2000 every year.
Determine the best age at which to replace the machine. If the optimum replacement policy is followed, what will be the average yearly cost of owing and operating the machine ? (Assume that the machine has no resale value when replaced, and that future cost are not discounted).
Machine B costs Rs 10,000. Annual operating costs are Rs 400 for the first year, and then increase by Rs 800 every year.
You have now a machine of type A which is one year old. Should you replace it with B, and if so when ?
Suppose you are, just ready to replace machine A with another machine of the same type, when you hear that machine B will become available in a year. What you should do ?Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
Year Purchase price
Maintena nce cost
Cum.
Maint.
Cost
Resale
value Total cost Ave.
total cost
1 9000 200 200 0 9200 9200.0
2 9000 2200 2400 0 11400 5700.0
3 9000 4200 6600 0 15600 5200.0
4 9000 6200 12800 0 21800 5450.0
5 9000 8200 21000 0 30000 6000.0
6 9000 10200 31200 0 40200 6700.0
Machine A should be replaced at the end of the 3rdyear. The average yearly cost in this situation will be Rs 5200.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
Year Purchase price
Maintena nce cost
Cum.
Maint.
Cost
Resale
value Total cost Ave.
total cost
1 10000 400 400 0 10400 10400.0
2 10000 1200 1600 0 11600 5800.0
3 10000 2000 3600 0 13600 4533.3
4 10000 2800 6400 0 16400 4100.0
5 10000 3600 10000 0 20000 4000.0
6 10000 4400 14400 0 24400 4066.7
7 10000 5200 19600 0 29600 4228.6
Machine B should be replaced at the end of the 5thyear. The average yearly cost in this situation will be Rs 4000.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
Should we replace Machine A with Machine B?
Model-II: Availability of similar new machines with better usages
Should we replace Machine A with Machine B?
Replace existing machine with new one if:
Lowest average cost of new machine < Lowest average cost of existing machine
4000 (M/C B) < 5200 (M/C A)
Model-II: Availability of similar new machines with better usages
Should we replace Machine A with Machine B?
Replace existing machine with new one if:
Lowest average cost of new machine < Lowest average cost of existing machine
4000 (M/C B) < 5200 (M/C A)
Since the lowest average cost Rs 4000 for machine B is less than the lowest average cost Rs 5200 for machine A, the Machine A should be replaced by Machine B.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
When should we replace Machine A with Machine B?
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
When should we replace Machine A with Machine B?
The machine A is replaced by machine B at the time (age) when its running cost of the next year exceeds the lowest average yearly cost Rs 4000 of machine B.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-II: Availability of similar new machines with better usages
When should we replace Machine A with Machine B?
The machine A is replaced by machine B at the time (age) when its running cost of the next year exceeds the lowest average yearly cost Rs 4000 of machine B.
Year 1 2 3 4 5 6
Maintenan ce cost of m/c A
200 2200 4200 6200 8200 10200
Ave. Total cost ob
m/c B
4000 4000 4000 4000 4000 4000
DifferenceDr. Devendra Choudhary, Govt. Engineering College, Ajmer-3800 -1800 200 2200 4200 6200
Model-II: Availability of similar new machines with better usages
When should we replace Machine A with Machine B?
The running cost of third year of machine A is Rs 4200 which is more than the lowest average yearly cost Rs 4000 of machine B.
Therefore, the machine A should be replaced by machine B when its age is 2 years.
Since the machine A is one year old now, therefore it should be replaced after one year from now.
Year 1 2 3 4 5 6
Maintenance cost of
m/c A 200 2200 4200 6200 8200 10200
Lowest Ave. Total
cost of m/c B 4000 4000 4000 4000 4000 4000 Difference -3800 -1800 200 2200 4200 6200
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with time value of money
The value of money changes with time.
If we borrow Rs 100 at interest of 10% per year then after one year we have to return Rs 110.
Thus Rs 110 after one year from now are equivalent to Rs 100 today.
If the interest on Rs 1 is Rs ‘i’ per year, then the present value of Rs 1 to be spent after nyears from now is Rs 1/(1+ i)n
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with time value of money
Present Value or Present Worth of ‘P’ rupees spent after ‘n’ years from now.
P*(1 + i)-n
Present worth factor or discounted factor = (1 + i)-n
Model-III: Aging of machines with time value of money
When to replace
Replace if the operating cost of the next period (year) is greater than the weighted average of the previous cost.
Model-III: Aging of machines with time value of money
When to replace
Replace if the operating cost of the next period (year) is greater than the weighted average of the previous cost.
Or
When weighted average annual cost is minimum
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with time value of money
A firm pays Rs. 10000 for its equipment. Its
operating and maintenance costs is about Rs. 2500 per year for the first two years and then go up by Rs. 1500 per year. When such equipment be replaced? The interest rate is 10%.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with time value of money
Year Price Maint . Cost PWF
PW of maint.
Cost
Comm.
PW of maint.
Cost Resal
e value
PW of resale value
PW of total cost
Commul ative PWF
Weighted average total
cost 1 10000 2500 1.0000 2500 2500 0 0 12500 1.0000 12500 2 10000 2500 0.9091 2272.7 4772.72 0 0 14772.7 1.9091 7738.1 3 10000 4000 0.8264 3305.7 8078.51 0 0 18078.5 2.7355 6608.7 4 10000 5500 0.7513 4132.2 12210.7 0 0 22210.7 3.4869 6369.8 5 10000 7000 0.6830 4781.1 16991.8 0 0 26991.8 4.1699 6473.1 6 10000 8500 0.6209 5277.8 22269.6 0 0 32269.6 4.7908 6735.7 7 10000 10000 0.5645 5644.7 27914.4 0 0 37914.4 5.3553 7079.8
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with time value of money
Year Price Maint . Cost PWF
PW of maint.
Cost Comm.
PW of maint.
Cost Resal
e value
PW of resale value
PW of total cost
Commul ative PWF
Weighted average total
cost 1 10000 2500 1.0000 2500 2500 0 0 12500 1.0000 12500 2 10000 2500 0.9091 2272.7 4772.72 0 0 14772.7 1.9091 7738.1 3 10000 4000 0.8264 3305.7 8078.51 0 0 18078.5 2.7355 6608.7 4 10000 5500 0.7513 4132.2 12210.7 0 0 22210.7 3.4869 6369.8 5 10000 7000 0.6830 4781.1 16991.8 0 0 26991.8 4.1699 6473.1 6 10000 8500 0.6209 5277.8 22269.6 0 0 32269.6 4.7908 6735.7 7 10000 10000 0.5645 5644.7 27914.4 0 0 37914.4 5.3553 7079.8
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with time value of money
A manufacturer offered two machines A and B. A is priced at Rs 5000 and running cost are estimated at Rs 800 for each of the first 5 years, increasing by Rs 200 per year in the 6th and subsequent years.
Machine B, which has the same capacity as A, costs Rs 2500 but with running cost of Rs 1200 per year for the first six years, increasing by Rs 200 per year there after.
If money is worth 10% per year, which machine should be purchased?
(Assume that the machine will eventually be sold for scrap at negligible price.)
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-III: Aging of machines with PWF
Year Price Maint.
Cost PWF
PW of maint.
Cost
Comm.
PW of maint.
Cost
PW of total cost
Commulat ive PWF
Weighted average total cost 1 5000 800 1.0000 800.0 800.0 5800.0 1.0 5800.0 2 5000 800 0.9091 727.3 1527.3 6527.3 1.9 3419.0 3 5000 800 0.8264 661.2 2188.4 7188.4 2.7 2627.8 4 5000 800 0.7513 601.1 2789.5 7789.5 3.5 2234.0 5 5000 800 0.6830 546.4 3335.9 8335.9 4.2 1999.1 6 5000 1000 0.6209 620.9 3956.8 8956.8 4.8 1869.6 7 5000 1200 0.5645 677.4 4634.2 9634.2 5.4 1799.0 8 5000 1400 0.5132 718.4 5352.6 10352.6 5.9 1764.1 9* 5000 1600 0.4665 746.4 6099.0 11099.0 6.3 1752.0 10 5000 1800 0.4241 763.4 6862.4 11862.4 6.8 1755.0 11 5000Dr. Devendra Choudhary, Govt. Engineering College, Ajmer2000 0.3855 771.1 7633.5 12633.5 7.1 1768.3
Model-III: Aging of machines with PWF
Year Price Maint.
Cost PWF
PW of maint.
Cost
Comm.
PW of maint.
Cost
PW of total cost
Commulat ive PWF
Weighted average total cost 1 2500 1200 1.0000 1200.0 1200.0 3700.0 1.0 3700.0 2 2500 1200 0.9091 1090.9 2290.9 4790.9 1.9 2509.5 3 2500 1200 0.8264 991.7 3282.6 5782.6 2.7 2113.9 4 2500 1200 0.7513 901.6 4184.2 6684.2 3.5 1917.0 5 2500 1200 0.6830 819.6 5003.8 7503.8 4.2 1799.5 6 2500 1200 0.6209 745.1 5748.9 8248.9 4.8 1721.8 7 2500 1400 0.5645 790.3 6539.2 9039.2 5.4 1687.9 8* 2500 1600 0.5132 821.1 7360.3 9860.3 5.9 1680.2 9 2500 1800 0.4665 839.7 8200.0 10700.0 6.3 1689.0
Model-III: Aging of machines with PWF
Time to replace Lowest ave.
weighted cost Machine A
9 yr 1752.0
Machine B
8yr 1680.2
It would be better to purchase machine B instead of A.
Model-IV: Group replacement
There are certain items which do not deteriorate but fail completely after certain amount of use. These kinds of failures are analysed by the method called as group replacement theory.
It should be noted that, group replacement does involve periodic simultaneous replacements along with individual replacements in between.
Few examples are fluorescent tubes, light bulbs, electronic chips, fuse etc.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
A long period between group replacements results in increase in cost of individual replacements, while frequent group replacements are definitely costly.
It is found that replacing these random failing items simultaneously at specific intervals is economical as compared to replacing them only when an item fails.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
The following morality rates have been observed for a certain type of light bulb:
There are 1000 bulbs in use and it costs Rs 10 to replace an individual bulb which has burnt out. If all bulbs were replaced simultaneously it would cost Rs 2.5 per bulb. It is proposed to replace all bulbs at fixed intervals, whether or not they have burnt out and to continue replacing burnt out bulbs as they fail. What would be the best policy to adopt?
Week 1 2 3 4 5
% failing by
week end 10 25 50 80 100
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
Week (xi)
Cumulative % failure till the end
of the week
% failure during the
week
ProbabilityPithat a new bulb fails during the week
1 10 10 0.10
2 25 15 0.15
3 50 25 0.25
4 80 30 0.30
5 100 20 0.20
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
Cost of individual replacement:
Expected life = ∑ xi*Pi
Expected life = 1*.10+2*.15+3*.25+4*.30+5*.20
= 3.35 weeks
Ave. cost of individual replacement per week
= (# of units in operation/Expected life)*Cost of individual replacement per unit
Ave. cost of individual replacement per week = (1000/3.35)*10 = 2985
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
Cost of group replacement:
Let Nibe the number of replacements at the end of ithweek, while all 1000 bulbs were new initially, then we have
N0= 1000
N1= N0*P1 = 1000*0.10=100
N2= N0*P2+N1*P1 = 1000*0.15+100*.10=160
N3= N0*P3+N1*P2 +N2*P1 =
1000*0.25+100*.15+160*.10=281
N4= N0*P4+N1*P3 +N2*P2 +N3*P1 =377.1
N5= N0*P5+N1*P4 +N2*P3 +N3*P2 +N4*P1
=349.89
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
Average cost of group replacement:
Week Failure during week
Cumm.
Failure during week
Cost of individual replacement
Cost of group replacement
Total cost
Average cost of group replacement
1 100 100 1000 2500 3500 3500
2 160 260 2600 2500 5100 2550
3 281 541 5410 2500 7910 2636.67
4 377.1 918.1 9181 2500 11681 2920.25
Model-IV: Group replacement
Average cost of group replacement:
Week Failure during week
Cumm.
Failure during week
Cost of individual replacement
Cost of group replacement
Total cost
Average cost of group replacement
1 100 100 1000 2500 3500 3500
2* 160 260 2600 2500 5100 2550
3 281 541 5410 2500 7910 2636.67
4 377.1 918.1 9181 2500 11681 2920.25
Model-IV: Group replacement
What would be the best policy:
Week Failure during week
Cumm.
Failure during week
Cost of individual replacement
Cost of group replacement
Total cost
Average cost of group replacement
1 100 100 1000 2500 3500 3500
2* 160 260 2600 2500 5100 2550
Ave. cost of individual replacement per week = 2985
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
Model-IV: Group replacement
What would be the best policy:
Week Failure during week
Cumm.
Failure during week
Cost of individual replacement
Cost of group replacement
Total cost
Average cost of group replacement
1 100 100 1000 2500 3500 3500
2* 160 260 2600 2500 5100 2550
Ave. cost of individual replacement per week = 2985 Group replacement is more economical. So, replace all
the bulbs at the end of second week and also replace those fail during the first and second weeks.
Dr. Devendra Choudhary, Govt. Engineering College, Ajmer
