Page 212 - DCOM303_DMGT504_OPERATION_RESEARCH

Unit 10: Sequencing Problems and Replacement Theory

1st Case: Theorem: Statement Notes

The cost of maintenance of a machine is given as a function increasing with time and its scrap
value is constant.
1. If the time is measured continuously, then the average annual cost will be minimized by
replacing the machine when the average cost to that date becomes equal to the current
maintenance cost.

2. If the time is measured in discrete units, then the average annual cost will be minimized
by replacing the machine when the next period’s maintenance cost becomes greater than
the current average cost.

Example: The cost of a machine is ` 6, 100 and its scrap value is ` 100 only. The maintenance
costs are found from experience to be:

Year 1 2 3 4 5 6 7 8
Maintenance Cost (`) 100 250 400 600 900 1,250 1,600 2,000

Solution:
Table showing optimum period of replacement of the machine

Replacement at the Running cost Total Rn C – S Total cost Average cost
end of the year Rn (`) Rn (`) (`) Pn (`) (`)
1 100 100 6,000 6,100 6,100
2 250 350 6,000 6,350 3,175
3 400 750 6,000 6,750 2,250
4 600 1,350 6,000 7,350 1,837.05
5 900 2,250 6,000 8,250 1,650
6 1,250 3,500 6,000 9,500 1,583.33
7 1,600 5,100 6,000 11,100 1,585.07
8 2,000 7,100 6,000 13,100 1,637.05

Inference

The machine has to be replaced at the end of the 6th year or at the beginning of the 7th year as the
maintenance cost of the 7th year becomes higher than the average cost of maintaining the
machine at the end of the 6th year.

Example: A machine owner finds from his past records that the costs per year of
maintaining a machine different and which are given below:

Year 1 2 3 4 5 6 7 8
Maintenance cost in (`) 1,000 1,200 1,400 1,800 2,300 2,800 3,400 4,000
Resale price (`) 3,000 1,500 750 375 200 200 200 200

If the purchase price of that machine is ` 6,000, then at what age the replacement is due?

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