Replacement theory in optimization techniques

replacement theory

The replacement theory addresses situations wherein various items, such as machinery, electric light bulbs, and computers, necessitate replacement due to diminishing efficiency or complete breakdown. This deterioration can occur gradually over time or suddenly due to unforeseen events. Several reasons contribute to the replacement problem:

1. Poor Performance or Costly Maintenance: The old item may function poorly or require expensive maintenance.

2. Complete Failure: The old item might have failed due to accidents or other reasons, rendering it inoperable, or it's expected to fail imminently.

3. Availability of Better Designs: A more efficient or improved design of machinery or equipment may become available in the market.

Failure can manifest in two primary types:

1. Gradual Failure: This is a progressive decline in operational efficiency as the item's lifespan increases, leading to increased running costs, decreased productivity, and reduced resale value. Mechanical components like pistons, bearings, and automobile tires typically experience gradual failure.

2. Sudden Failure: This type of failure occurs after a certain period of service rather than through gradual deterioration. Sudden failure can be categorized as progressive, retrogressive, or random:
a. Progressive Failure: The probability of failure increases with the item's lifespan. Light bulbs and tubes are examples of items prone to progressive failure.
b. Retrogressive Failure: The probability of failure is higher at the beginning of the item's life but decreases over time.
c. Random Failure: In this case, failure occurs due to random causes unrelated to the item's age. For instance, vacuum tubes in airborne equipment may fail at a constant rate independent of their age.

Replacement scenarios typically fall into two main categories:

1. Replacement of Capital Equipment: This involves replacing machinery and equipment that deteriorates over time, such as machine tools, buses, and aircraft in transportation organizations.

2. Individual or Group Replacement: This entails replacing items that fail completely, such as light bulbs or tubes.
Regarding replacement decisions based on increasing maintenance costs over time:
If maintenance costs increase with time and the scrap value of the machine remains constant:

- In continuous time measurement, the average annual cost is minimized by replacing the machine when the cumulative cost equals the current maintenance cost.
- In discrete time measurement, the optimal replacement time is when the next period's maintenance cost exceeds the current average cost.

: The cost of a machine is Rs. 6100 and its scrap value is only Rs. 100.
The maintenance costs are found from experience to be:

year :	1	2	3	4	5	6	7	8
Maintenance cost in Rs :	100	250	400	600	900	1250	1600	2000

When should machine be replaced?

Solution: First, we find an average cost per year during the life of the machine as follows:
Total cost in first year = maintenance cost in the year + loss in purchase price = 100 + (6100 − 100) = Rs. 6100.
Therefore, the average cost in the first year = Rs. 6100.
Total cost up to two years = maintenance cost up to two year + loss in purchase price = (100 + 250) + 6000 = Rs. 6350.
Therefore, average cost in first two years = Rs. 3175.
In a similar fashion, average cost per year during first three years = 6750/3 = Rs. 2250.
Average cost per year during first four years 7350/4 = Rs.1837.50.
Average cost per year during first five years 8250/5 = Rs. 1650.
Average cost per year during first six years 9500/6 = Rs. 1583.33
Average cost per year during first seven years 11100/7 = Rs. 1585.71.
The computations are summarized in the following table:

year (n)	Maintenance cost (R_n)	Total maintenance cost(∑R_n)	Depreciation cost(C - S)	Total cost (P_n)	Average cost P_n/n
(1)	(2)	(3)	(4)	(5)=(3)+(4)	(6)
1	100	100	6000	6100	6100
2	250	350	6000	6350	3175
3	400	750	6000	6750	2250
4	600	1350	6000	7350	1837
5	900	2250	6000	8250	1650
→6	1250	3500	6000	9500	1583
7	1600	5100	6000	111000	1586
8	2000	7100	6000	131000	1638

Here it is observed that the maintenance cost in the 7th year becomes greater than the average cost for 6 years [i.e. R7 > P (6)/6]. Hence the machine should be replaced at the end of the 6th year.
Alternatively, last column of the above table shows that the average cost starts increasing in the 7th year. So the machine should be replaced before the beginning of the 7th year, i.e., at the end of the 6th year.