Replacement theory is concerned with the problem of replacement of machines electricity bulbs, men etc. due to their deteriorating efficiency, failure or breakdown. Replacement is usually carried out under the following situations:

(i)                 When existing items have outlived their effective lives and it may not be economical to continue with them anymore.

(ii)               Items which might have been destroyed either by accident or otherwise.

The above replacement situations may be categorized into the following four categories.

(a)    Replacement of items that deteriorates with time e.g., machine tools, vehicles equipment building etc.

(b)   Replacement of items which do not deteriorate but fail completely after certain amount of use eg electric bulbs T.V. parts etc.

(c)    Replacement of an equipment (or item) that becomes out of date due to new developments, e.g., ordinary weaving looms by automatic looms mechanized accounting system by computer system etc.

(d)   The existing working staff in an organization gradually diminishes due to death retirement retrenchment and other reasons. The replacements are thus needed.

O.R. METHODOLOGY OF SOLVING REPLACEMENT PROBLEMS

O.R. provides a methodology for tackling replacement problem. The steps adopted for tacking such replacement problem in O.R. have been discussed as below.

(i) Identify the items to be replaced and also their mechanism. There can be two types of failures viz. gradual and sudden. Item such as machines, equipment etc, follow gradual failure mechanism and they deteriorate with time. Such type of failures account for increased expenditure in the form of operating costs, decrease in the productivity of the equipment and decrease in the value of the equipment i.e., the resale or salvage value Items which follow sudden failure mechanism may fail anytime, thus precipitating cost of failure. The cost of failure in some cases may be quite high as compared to the value of the item itself. Sometime sudden failure of an item may cause loss of production and may also account for damaged or faulty products. In some cases failures may involve safety risks to personnel as well. To avoid the cost of sudden failure, the concern should try to predict when such failures are likely to occur and try to replace the item before it actually fails.

(ii) Collect the data relating to the depreciation cost and the maintenance cost over a time period from the available sources for the items which follow gradual failure mechanism. In the case items following sudden failure mechanism collect the data for failure rates cost of replacement for failed items and cost of preventive replacement.

(iii)on using the above data suitable model in OR (as discussed in the following sections) may be evolved for determining the exact time of replacing the involved items.

REPLACEMETN OF ITEMS THAT DETERIORATES WITH TIME (WITHOUT CHANGE IN MONEY VALUE). For finding the optimum replacement period of the items pertaining to this class we will consider basically two categories of costs. In one category, we have maintenance and operating cost which tend to increase as the equipment ages. In the other category, we have depreciation cost which diminishes with the age of the equipment. Further we disregard the time value of money. The optimum replacement of the equipment is calculated according to the following rules.

1. if the scrap value of the equipment is zero i.e., the depreciation cost is not given. Then replace the equipment when the maintenance cost becomes greater than the current average cost.

2. if we are given the resale value or the depreciation cost, the maintenance cost and the cost of the equipment then the optimum replacement period is determined by the minimum value of the average cost to date.

Let us first consider a simple situation which consists of minimizing the average annual cost of equipment whose maintenance cost is a function increasing with time and whose scrap value is constant. As the time value of money is not to be considered the interest rate is zero and the calculation can be based on average annual cost.

Let

C = the capital cost of a certain item say a machine.

S (t) = the salvage or scrap value of the item after t years operations.

F (t) = operating (or maintenance) cost of the item at time t.

N = optimum replacement period of the item.

Now the annual cost of the machine at any time t

= capital cost-scrap value + maintenance cost at time t = c-s(t) + f(t)

And since the total maintenance cost incurred on the machine during n year is

                   n

?n f(t) dt or ? f (t) the total cost. T incurred on the item during n years is

                   t = 0

given by: T=C – s(t) ?n f(t) dt

               n

or C-S(t) ? f(t). Thus the average annual total cost incurred on the item per year during n years is given by:

It can be easily shown that this solution TA = f(n) is minimum for T, provided that f(t) is non-decreasing and f(O) = 0. Hence an item should be replaced when th average cost to date becomes equal to the current maintenance cost.

Remark 1. If TA goes on diminishing upto rth year and again increases from (r+1) th year the rth is considered as the most economic year for replacement.

2. under the assumption that maintenance or operating costs would be increasing monotonously we can depict the average cost curve and the period corresponding to minimum point of the curve represents the optimum replacement period.

3. the two categories of costs required to determine the optimum period for replacement of an equipment which has a gradual failure mechanism are:

(i) Depreciation cost                          (ii) Operating cost.

(i) Computation of depreciation cost.

There are several methods available for computing depreciation. Some of the methods which are widely used are described below:

(a)    Straight line method. This method provides for depreciation by means of equal periodic charges over the assumed life of the asset. Depreciation under this method may be calculated by using the following formula:

                                            cost of the asset-scrap value

Cost of depreciation =

                                     No of years of useful life of the asset

(b)   Residual value method. Depreciation here is calculated on the basis of he written down value of an asset. The written down value means the net depreciated value of an asset every year. For example if the original cost of the asset is Rs. 10,000 and the depreciation @ 20% the depreciation in the first year will be Rs.2,000 and in the next year , 20% of (Rs. 10,000 – 2,000) = Rs. 8,000 i.e., Rs. 1,600 and so on.

(c)    The depreciation cost may also be determined by estimating the resale value of the asset at the end of each year. The cost in the beginning of the year less the resale value at the end of the year will give the depreciation cost.

3. Computation of operating cost:

This includes cost incurred under the following heads.

(a)    Cost of new spares used

(b)   Cost of repairs

(c)    Cost of maintenance staff of the equipment

(d)   Preventive maintenance cost

(e)    Cost of loss due to breaskdown

(f)    Cost incurred on safety measures for life of human beings.

This cost can only be estimated on the basis of past experience.

Example 2. A machine shop has a press which is to be replaced as it wears out. A new press is to be installed now and an optimum replacement plan is to for next 7 years after which the press is no longer required. Following data are available.

Year                                        cost of new machine         salvage value                operating cost

1                                              500                                    250                                150                      

2                                              525                                    125                                200

3                                              550                                    75                                  250

4                                              600                                    50                                  300

5                                              650                                    40                                  375

6                                              725                                    25                                  450

7                                              800                                    0                                    575

Find an optimum replacement policy.

Solution. using the given information the minimum annual cost of the press is computed in the following table.

1                                 150                 150                     250            500              400                 400

2                                 200                 350                     125            525              750                 375

3                                 250                 600                     75              550              1075               358

4                                 300                 900                     50              600              1450               363

5                                 375                 1275                   40              650              1885               377

6                                 450                 1725                   25              725              2425               404

7                                 575                 2300                   0                800              3100               443

Since the annual average value Ta is minimum at the end of third year the press should be replaced after every third year.

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