decision has been made to perform certain repairs on the outlet works of a small

Question

decision has been made to perform certain repairs on the outlet works of a small dam. For a particular - Inch gate valve. There are three available Alternatives: Of the valve is replaced; the probability of a failure of valve seats, over the life of the Project is 30%; the of failure of the valve stem is 20%; and of failure of the valve body is 10%. If the valve is repaired; the probability of a failure of valve seats, over the life of the project is 40%; the of failure of the valve stem is 30%; and of failure of the valve body is 20%. If the valve is left as it is; the probability of a failure of valve seats, over the life of the project is 60 %, probability of failure of the valve stem is 50%; and of failure of the valve body is 40%. The present worth of cost of future repairs and service disruption of failure of the seats is $10,000; the of the cost of a failure of stem is $20,000; and The present worth of cost of a failur of the body $30,00; repairing the valve now is $10,000; and replacing it is $20,000 If the criterion is to minimize expected costs, which alternative is best? Explain in deatil

Explanation / Answer

In this problem, you have to decide on a poject of damaged valve. It can be repaired, replaced or can be left as it is.

Alternative 1: If it is replaced, then 60% chance of failure is there. so there is 40% probability of it sucess. Out of 60% failure, 30% may be due to valve sheet, 20% due to valve stem and 10% due to valve body .

Now consider cost. replacing cost is $20,000. This cost is a must if repairing is made. Further company has to incur $10,000 if valve sheet has failed. Then total cost will be $20,000+$10,000 = $30,000. Probability of incurring this $30,000 cost is 30%.

If valve stem fail ,then it has to bear an extra cost of $20,000 for repairing stem. Thus total cost will be $20,000+$20,000=$40,000. Its probability is 20%

If valve body fail, then extra cost is $30,000. Thus total cost is $20,000+$30,000= $50,000. It has a probability of 10%.

There is 40% chance that everything will be Ok. In that case total cost will be $20,000 (i.e. replacement cost only).

Hence expected cost of repairing here is:

0.3 x $30,000 +0.2 x $40,000 + 0.1 x $50,000 + 0.4 x $20,000 = $30,000

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Now consider second alternative: Here repairing of valve is done. It has a cost of $10,000. There is 40% chance of failure of valve sheet. In that case additional $10,000 cost is required. Thus probability of incurring $10,000+$10,000=$20,000 cost is 40%.

After repairing there is 30% chance that valve stem will fail. In that case extra cost of $20,000 is required. Thus probability of incurring a totalcost of $10,000+$20,000=$30,000 is 30%.

Also after repair body of valve may fail. In that case extra cost for correcting the body is $30,000. Thus total cost of $10,000+$30,000=$40,000 has a probability of 10%.

Above three situation of further failure is covering 80% probability. Balance 20% chance exist when it will operate smoothly after repair. In that case only cost is $10,000. Considering all the situations together, expected cost is
0.4 x $20,000 + 0.3 x $30,000 + 0.1 x $40,000 + 0.2 x $10,000 = $23,000

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Alternative 3: If the position is left alone then there is more than 100% probability that system will fail. There is 60% chance of failure of valve sheet. In that situation it has to incur a cost of $10,000.

There is 50% chance that valve stem will fail. In that case cost will be $20,000.

Finally there is 40% chance that valve body will fail. In that case cost will be $30,000. Thus average cost of correcting them will be:

(0.6 x $10,000) + (0.5 x $20,000) +(0.4 x $30,000) / (0.6+0.5+0.4) = $28,000 / 1.5 = $18,667

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Result: Compare above three alternatives. You will observe that expected cost of third one is lowest. So it is better to left it as it is for minimizing expected cost.

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