For this dynamic programming problem and the next one, be sure to (a) define and describe the subproblem; (b) write the recurrence relation for the subproblem, including the base case(s); (e) write pseudo codes showing how a table for the subproblems is filled (i.e., by the bottom up" approach). A mission-critical production system has n stages that have to be performed sequentially; stage i is performed by machine M;. Each machine M; has a probability Ti of functioning reliably and a probability 1 - Ti of failing (and the failures are independent). Therefore, if we implement cach stage with a single machine, the probability that the whole system works is r.12.-In. To improve this probability we add redundancy by having m; copies of the machine M that performs stage i. The probability that all mi copies fail simultaneously is only (1 - r;)", so the probability that stage i is completed correctly is 1-(1-r)" and the probability that the whole system works is l_ [1 (1 r;);). Each machine M; has a cost C, and there is a total budget B to buy machines. (Assume that B and the Gi are positive integers.) Given the probabilities r1, 72, ..., In, the costs C1,C2, ..., Cr, and the budget B, find the maximum reliability that can be achieved within budget B. (10 points) For this dynamic programming problem and the next one, be sure to (a) define and describe the subproblem; (b) write the recurrence relation for the subproblem, including the base case(s); (e) write pseudo codes showing how a table for the subproblems is filled (i.e., by the bottom up" approach). A mission-critical production system has n stages that have to be performed sequentially; stage i is performed by machine M;. Each machine M; has a probability Ti of functioning reliably and a probability 1 - Ti of failing (and the failures are independent). Therefore, if we implement cach stage with a single machine, the probability that the whole system works is r.12.-In. To improve this probability we add redundancy by having m; copies of the machine M that performs stage i. The probability that all mi copies fail simultaneously is only (1 - r;)", so the probability that stage i is completed correctly is 1-(1-r)" and the probability that the whole system works is l_ [1 (1 r;);). Each machine M; has a cost C, and there is a total budget B to buy machines. (Assume that B and the Gi are positive integers.) Given the probabilities r1, 72, ..., In, the costs C1,C2, ..., Cr, and the budget B, find the maximum reliability that can be achieved within budget B. (10 points)