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Dynamic Programming : Reliability of Components

An electronic device consists of 3 main components arranged in series so that the failure of one component causes the failure of the whole device. The reliability of the device can be improved by having up to, 2 standby units in parallel for each component. The total capital available for the design is £10,000. The reliability Rk and Cost Ck of having ak standby units for each component is given below:

Number of Component 1 Component 2 Component 3
Standbys k Rk Ck Rk Ck Rk Ck
0 0.6 1000 0.7 3000 0.5 2000
1 0.8 2000 0.8 5000 0.7 4000
2 0.9 3000 0.9 6000 0.9 5000

The total reliability of the device is the product of the reliability of its three components. Formulate the problem of maximising the total reliability of the device within the total capital available using dynamic programming. Give a reason why your formulation is valid and solve the problem.

I am struggling with the formulation of this problem. Do I have to multiply the reliabilities and if the standbyunits are there do we use the probability of the device 'failing' ?

Please help me out with this problem.

Solution Summary

A dynamic programming problem is solved. The solution is detailed and well presented.