1. Replacement Problem

2. Introduction
The problem of replacement is felt when the job performing units such as men, machines, equipments, parts etc., become less effective or useless due to either sudden or gradual deterioration in their efficiency, failure or breakdown.
By replacing them with new ones at frequent intervals, maintenance and other overhead costs can be reduced.
However, such replacements would increase the need of capital cost for new ones.

3. Introduction… Example:
A machine becomes more and more expensive to maintain after a no. of years, a railway time table gradually becomes more and more out of date, an electric light bulb fails all of a sudden, pipeline is blocked, or an employee loses his job etc.
In all such situations, there is a need to formulate a most economic replacement policy for replacing faulty units or to take some remedial special action to restore the efficiency of deteriorating units.

4. Introduction…
Following are the situations when the replacement of certain items needs to be done:
An old item has failed and does not work at all, or the old item is expected to fail shortly.
The old item has deteriorated and works badly or requires expensive maintenance.
A better design of equipment has been developed.

5. Introduction…
Replacement problems can be classified into the following two categories:
When the equipment deteriorate with time and the value of he money
1. Does not change with time.
2. Changes with time.
When the units fail completely all of a sudden.

6. Types of Failure Gradual Failure
It is progressive in nature. That is, as the life of an item increases, its operational efficiency also deteriorates resulting in
1. increased running (maintenance & operating) cost
2. decrease in its productivity
3. decrease in the resale or salvage value.
Mechanical items like pistons, rings, bearings etc. and automobile tyres fall under this category.

7. Types of Failure Sudden Failure
This type of failure occurs in items after some period of giving desired service rather than deterioration while in service. The period of desired service is not constant but follows some frequency distribution which may be progressive, retrogressive or random in nature.
1.Progressive failure: If the probability of failure of an item increases with the increase in its life, then such failure is called progressive failure. (light bulb or tubes fails progressively)

8. Types of Failure
2.Retrogressive failure: If the probability of failure in the beginning of the life of an item is more but as time passes the chances of its failure become less, then such failure is said to be retrogressive.
3.Random failure: In this type of failure, the constant probability of failure is associated with items that fail from random causes such as physical shocks, not related to age. For example, vacuum tubes in air born equipment have been found to fail at a rate independent of the age of the tube.

9. Replacement of Equipment that deteriorates gradually
Generally the cost of maintenance & repair of certain items increases with time and a stage may come when these costs become so high that it is more economical to replace the item by a new one.
Replacement policy when value of money does not change with time
The aim here is to determine the optimum replacement age of an equipment/item whose running/maintenance cost increases with time and the value of money remains static during that period. Let

10. Replacement policy when value of money does not change with time…
C : capital cost of equipment
S : Scrap (waste) value of equipment
n : number of years that equipment would be in use
f(t) : maintenance cost function
A(n) : average total annual cost.
Case 1: When t is a continuous variable. If the equipment is used for n years, then the total cost incurred during this period is an under:

12. Replacement policy when value of money does not change with time…
Case 2: When t is a discrete variable. Here period of time is considered as fixed and n, take the values 1,2,3,…. Then

14. Example: When should the machine be replaced?
A firm is considering replacement of a machine, whose cost per year is Rs. 12,200 and the scrap value, Rs The running (maintenance & operating) costs in rupees are found from experience to as follows:
Year 1 2 3 4 5 6 7 8
Running Cost
200
500
800
1200
1800
2500
3200
4000
When should the machine be replaced?

15. We have given:
The running cost, f(n)
The scrap value (s) = 200
Cost of the machine (C)= 12,200
To find out the optimal time n when the machine should be replaced, we need to calculate an avg. Total cost per year during the life of the machine.

16. (1) n (2) f(n) (3) (4) C-S (5) (6) 3167
Year of service n
(2)
Running cost
f(n)
(3)
Cumulative running cost
(4)
Depreciation
cost
C-S
(5)
Total cost
TC=(3)+(4)
(6)
Avg. cost
A(n)=(5)/(1) 1
200
12000
12200 2
500
700
12700
6350 3
800
1500
13500
4500 4
1200
2700
14700
3675 5
1800
16500
3300 6
2500
7000
19000
3167 7
3200
10200
22200
3171 8
4000
14200
26200
3275
3167

17. Replacement policy when value of money changes with time
When the time value of money is taken into consideration, we shall assume that (1) the equipment in question has no salvage value, and (2) the maintenance costs are incurred in the beginning of the different time periods.
Since it is assumed that the maintenance cost increases with time and each cost is to be paid just in the start of the period, Let the money carry a rate of interest r per year.

18. Replacement policy when value …
Thus, a rupee invested now will be worth (1+r) after a year, (1+r)2 after 2 years, and so on.
In this way the Rs. invested today will be worth (1+r)n after n years. Or, if we have to make a payment of 1 Rs. In n years time, it is equivalent to making a payment of (1+r)-n Rs. Today.
The quantity (1+r)-n is called the present worth factor (pwf) of 1 Rs. spent in n years time from now onwards

19. Replacement policy when value …
Thus, a rupee invested now will be worth (1+r) after a year, (1+r)2 after 2 years, and so on.
In this way the Rs. invested today will be worth (1+r)n after n years. Or, if we have to make a payment of 1 Rs. In n years time, it is equivalent to making a payment of (1+r)-n Rs. Today.
The quantity (1+r)-n is called the present worth factor (pwf) of 1 Rs. spent in n years time from now onwards

20. Replacement policy when value…
(1+r)n is known as the payment compound amount factor (Caf) of 1 Rs. spent in n years time.
Let the initial cost of the equipment be C and let Rn be the operating cost in year n.
Let v be the rate of interest in such a way that v= (1+r)-1 is the discount rate (pwf).
Then the present value of all future discounts costs Vn associated with a policy of replacing the equipment at the end of each n years is given by

21. Replacement policy when value…
Then the present value of all future discounts costs Vn associated with a policy of replacing the equipment at the end of each n years is given by

22. Replacement policy when value…
Since v is the value of money, it will always be less than 1, and therefore 1-v will always be +ve. This implies that Vn/(1-Vn+1) will always be +ve.
The expression which lies between Rn-1and Rn is called the weighted avg. cost of all the previous n years with weights 1,v,v2,…..,vn-1 respectively.

23. Replacement policy when value…
Hence the optimal replacement policy of the equipment after n periods is:
(a) Do not replace the equipments if the next period’s operating cost is less than the weighted avg. of previous cost.
(b) Replace the equipments if the next period’s operating cost is greater that the weighted avg. of previous costs.

24. Selection of the best equipment amongst two
Step 1 Considering the case of 2 equipments (A & B), we first find the best replacement age for both equipments by making use of Rn-1<(1-v)Vn<Rn. Let the optimal replacement age for A & B comes out to be n1 and n2 respectively.
Step 2 Compute the fixed annual payment (or weighted avg. cost) for A & B by using the formula

25. Selection of the best equipment amongst two
and substitute n=n1 for A and n=n2 for B
Step 3

26. Example
Let the value of money be assumed to be 10% per year and suppose that machine A is replaced after 3 years whereas machine B is replaced after every 6 years. The yearly costs of both the machines are given below:
Year 1 2 3 4 5 6
Machine A
1,000
200
400
Machine B
1,700
100
300
500
Determine which machine should be purchased (replaced).

27. Solution :

29. Replacement of equipment that fails suddenly