Probabilistic Dynamic Programming

A company want to find out the best solution to assign the spare parts to 4 machines to minimize the probabilities of failure. There are only FOUR spare parts available at all. And each machine can assign not more than THREE Spare parts. The probabilities of failure are summarized in the following table.

	Machine
Spare parts assigned		A	B	C	D
	0	0.30	0.40	0.20	0.40
	1	0.225	0.25	0.15	0.30
	2	0.15	0.10	0.10	0.15
	3	0.055	0.02	0.04	0.10

For example, if no spare part is going to be assigned, then the overall probability of failure of all four locations is 0.30*0.40* 0.20*0.40 = 0.0096

1.What is the recursive relationship between the optimal decision at a stage and a previous optimum decision?

Hint: Define the recursive function and related variables such as: Let i be the current stage, di be the number of spare parts, xi be the decision of spare parts to be assigned to this machine. Pi(xi) be the probability of failure at stage i given that xi spare parts are being assigned,

The recursive function linking up two stages is: f_i(d_i)=min

2.Calculate the optimal solution that has a minimum probability of failure with details procedures.