Replacement Theory

Replacement Theory in Operations Research – An Algorithm using MS-Excel

Replacement Theory in Operations Research

The Replacement Theory in Operations Research is used 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. The ‘Replacement Theory’ is used in the cases like; existing items have out-lived, or it may not be economical anymore to continue with them, or the items might have been destroyed either by accident or otherwise. The above discussed situations can be solved mathematically and categorised on some basis like:

  • Items that deteriorate with time e.g. machine tools, vehicles, equipment  buildings etc,
  • Items becoming out-of-date due to new developments like ordinary weaving looms by automatic, manual accounting by tally etc.
  • Items which do not deteriorate but fail completely after certain amount of use like electronic parts, street lights etc (Group Replacement) and
  • The existing working staff in an organization gradually diminishing due to death, retirement, retrenchment & otherwise (Staff Replacement).

Replacement Policy for Equipments which Deteriorate Gradually

Let us see the first case of gradual failure of items with time. Consider the example of a Motor Vehicle; the pattern of failure here is progressive in nature i.e. as the life of vehicle increases; its efficiency decreases. This results in additional expenditure in running or maintaining this vehicle and at the same time its resale value (also called as scrap value) also keeps on decreasing. The above case makes this situation a typical case for applying ‘Replacement Theory’.


A transport company purchased a motor vehicle for rupees 80000/-. The resale value of the vehicle keeps on decreasing from Rs 70000/- in the first year to Rs 5000/- in the eighth year while, the running cost in maintaining the vehicle keeps on increasing with Rs. 3000/- in the first year till it goes to Rs. 20000/- in the eighth year as shown in the below table. Determine the optimum replacement policy?

Example on Replacement Theory

The MS-Excel Files of this Algorithm can be downloaded from the links provided further in this post. The cost of the equipment is to be entered in the cell B1 (as shown by the green cell with 80000). Now, enter the scrap values and the running costs as entered in the green columns C5 to C12 and D5 to D12.  The algorithm will now automatically calculate the solution which is as shown in the below figure.

Algorithm for Replacement Theory in Operations Research

The answer can be fetched from the last column. See the pattern; the average total cost (ATC) at first starts dipping from Rs. 13000/- till it reaches Rs. 11850/- in the cell H8. From H9 it again starts increasing. This cost at which the ATC is lowest in a particular year (after which it starts increasing again) gives the optimum replacement period and cost of the vehicle.

Solution: The vehicle needs to be replaced after four years of its purchase wherein the cost of maintaining that vehicle would be lowest at an average of Rs 11850/- per year.

Clarification on the Methodology

There are two considerations here. First, the running cost (Rn) is increasing every year at the same time the vehicle is depreciating in its value. This depreciation is ‘(C-S)’ i.e. in the first year the scrap value of the vehicle is Rs. 70000/- which was purchased for Rs. 80000/- . So, the vehicle is depreciated by Rs. 10000/- in year one and so on (see column F).

Thus the total cost in keeping this vehicle is this depreciation and its maintenance. The maintenance is made cumulative by adding previous years running cost to it every successive year. Let’s make this simple!

The depreciation is Rs. 10000/- in the first, 19000/- in the second, 25000/- in the third and so on. See here, the vehicle is depreciated by Rs. 25000/- “by” the third year and not “in” the third year. Note that the non-cumulative cost of depreciation “in” the third year would be Rs. 6000/- [Rs. 25000/ minus Rs. 19000/, see the cells F6 and F7]

As, the depreciation in itself is a cumulative function here, we make the running cost cumulative also. That means the cost of maintaining the vehicle “by” the particular years. So, the cost of maintaining the vehicle “by” the third year is Rs. 11400/- (D5+D6+ D7 or 3000+3600+4800).

Hence the total cost incurred by the third year would be Rs. 25000 + Rs. 11400 = Rs. 36400 (see cell G7). Finally, the “average cost” of keeping this vehicle for three years would be 36400 divided by 3 years i.e. Rs. 12133.33 as can be seen from cell H7 and so on.

Notations Used:

  • C –  (Capital) Cost of Equipment
  • S –  Scrap (or Resale) Value
  • Rn –  Running (or Maintenance) Cost
  • E Rn –  Cumulative Running Cost
  • (C-S) –  Depreciation
  • TC –  Total Cost
  • ATC –  Average Total Cost

How to Download?

Instruction on – How to use the Algorithm Files?

  1. Download the Algorithm file and save it to a location like ‘My Documents’ (This is your Master File)
  2. Create a copy for yourself by using “Save As” command and save the file with a new name maybe into a new location ‘Desktop”
  3. Do not delete the ‘cell values’ at any point of time during the use; rather; over-write the values. The reason being, once you delete or clear the excel sheet, all the formulae used are lost.

Try This Out !

  • Playing with existing Numerical: Start changing the values in the “Rn” and “S” value columns or better if you change the cost of the vehicle only. For example, in the cell B1; use values lesser than or greater than Rs. 80000/- and observe how the solution changes for different purchase costs.
  • To solve a new Numerical: Start overwriting new values for C, S and Rn in the green cells and get the solutions. If say scrap value is not given then you can over-write all the existing scrap values with ‘zeroes’. If the numerical has data gathered for lesser years / periods (in our case the data gathered was for 8 years); then overwrite the remaining values with ‘zeroes’. Do not save the file or if you want to, then use ‘Save As’.

PowerPoint Presentation 

Download this PPT – Replacement Theory Models in Operations Research by Dr. Rajesh Timane

Other Replacement Models


When time-value of money is considered

Replacement of Items that fail suddenly


  1. Prof Sachin Kadu says

    This would definitely help the students sir. I can ask them now to check the answers of solved and unsolved numericals.

    • safaa says

      sir i am research scholar my topic a study of usaing quantitative method to rationalty replacement decision .
      we are using exponential function .
      did you have any information about that pleace help my if you can

  2. Sheetal says

    can we create similar excel sheets for direct answers for sequencing and simulation problems?

  3. Vineet says

    Its good to understand and will surely help .. sir would be looking for some more material on Manufacturing Economics. It will be of great help if u can share some topics.

    • says

      @ Vineet

      Specify any topic in Manufacturing Economics in perticular about which you want theory and or numericals. Pradeep Suryavanshi, a PD student persuing Operations will get in touch with you. You guys can exchange the study material.

  4. Vineet says

    The topics I am looking for are as Follows:
    Unit 1: Introduction to manufacturing economics
    Unit 4: Econometrics
    Unit 10 : Modelling and Simulation..

    NUMERICALS on Expected topics

    In case you get any of the material please upload it…I have spoken to Pradip also regarding the material..

    Keep Posting Sir.

  5. Moirangthem Rakesh singh says

    I’m needing to understand more on this theory, so please help me as much as you can…..

    Thanks and regards

    Rakesh Singh

  6. Sachin.P.M says

    Please visit my blog and look for replacement ppt. I have prepared form your data only. Thank you for sharing the information.

  7. Ravi Patil says

    Rajesh Sir,
    We want rules for sequencing along with assumptions in replacement theory and sequencing theory with some illustrations if possible.

  8. Tschoep says

    Great topic. I needs to spend some time studying much more or understanding more. Thanks for magnificent information.

  9. Damion Kleinsmith says

    I am happy to search out numerous helpful info right here in the publish, we need develop more strategies in this regard, thanks for sharing . . .

  10. Rajesh Mahajan says

    It helped me a lot. Subject allotted to me is applied operations research, accounts and Quantitative decision making

  11. D Wright says

    Hi my friend! I wish to say that this article is amazing, nice written and include almost all vital infos. I’d like to see more posts like this .

  12. Keshala says

    in your example, let’s say when that machine is 2 years old, another machine which is a new model, is available. the optimal period of replacement of this new machine is 4 years with an average cost of Rs.10,500. should the machine be replaced with the new one and if so, what will be the best period of replacement?

    if you can please explain how to work out this problem.
    thank you.

    • rajesh says

      Keshala, Hope the mail answered your query.

      It becomes easy by using those algorithms in excel sheets. Once you get those individual replacement years and costs, you then just need to interpret them.

  13. Chris Brandau says

    I rarely comment, however i did a few searching and wound up here Replacement Theory Algorithm | Rajesh

  14. David says

    Hello, my name is David C.

    I just found this post and already love it.

    See you soon on others forum topics 😉

    • rajesh says

      It is the function of scrap value and maintenance cost. Also, as an equipment get used and gets older, it looses its value year on year, sometimes decreasing monotonically to such an extend that the equipment may have to be sold free of cost.

  15. salimath says

    The vehicle needs to be replaced after four years of its purchase wherein the cost of maintaining that vehicle would be lowest at an average of Rs 11850/- per year. I have copied and pasted your solution. But the optimal replacement policy says that replacement is economical only when the running cost is more than Rs.11850/- That is not the case here. It is Rs.8000/- Could you explain the fallacy?

    • rajesh says

      You got the numbers wrong. First of all, the running cost in the replacement year (4th year) is Rs. 5000. But we are not interested in absolute cost, rather, in cumulative cost. So, we need the running cost incurred during all these 4 years rather than in 4th year; which is, Rs. 16,400..

      Secondly, the vehicle now can be sold in Rs. 49,000 which means it is depreciated by Rs. 31,000 in these 4 years.

      Thirdly, adding these two (running and scrap) and not the running cost alone decides the policy.So, the total cost in maintenance and depreciation is = 16,400 + 31,000 = 47,400.

      This cumulative cost in 4 years gives an average cost of replacement = 47,400 / 4 = Rs. 11,850 per year.

      Hope I made it clear.

  16. Marc says

    I’ve got some recommendations for your blog you might be interested in hearing. Either way, great website Dr. Rajesh Timane and I look forward too seeing it improve over time.

    – Marc

  17. Dreggie says

    You really make it seem so easy with your presentation but I find this topic to be actually something that I think I would never understand. It seems too complicated and very broad for me. I’m looking forward for your next post, I’ll try to get the hang of it!

  18. ankit ardeshana says

    i am not get(understand) the second table in solution which is given in N D Vohra book.(same problem)
    how they calculate the Depreciation cost in 2nd table.
    The data on the operating cost per year and resale price of equipment A whose
    purchase price is Rs.10000 are given below:
    Year 1 2 3 4 5 6 7
    Cost (Rs) 1500 1990 2300 2900 3600 4500 5500
    (Rs) 5000 2500 1250 600 400 400 400
    (i) What is the optimum time for replacement?
    (ii) When equipment A is two years old, equipment B, which is a new model
    for the same usage, is available. The optimum period for replacement is
    four years with an average cost of Rs. 3600. Should we change equipment
    A with equipment B? If so, When?

    • rajesh says

      @ Ankit,

      Optimum time for replacement is 6 years at a cost of Rs. 4398.33/-. For the rest follow these steps –

      1) You can download the Excel Sheet from following link Excel Algorithm on Replacement
      2) Select File Menu – Download as – Microsoft Excel (.xlsx)
      3) Over-write the new values from your problem
      4) Compare these readymade answers and write your inferences.

      If you still face any problem, let me know.

    • rajesh says


      It needs to be seen; what you are talking about and the nature of problem. Can you type-in the example/problem/numerical you came across?

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