Consider the model in Exercise 10.11. Assume that the jobs are subject to precedence constraints that take the formof chains. Show that Algorithm 3.1.4 minimizes the total expected weighted completion time. Exercise 11: Consider the model in Theorem 10.1.1 with breakdowns. The up-times are exponentially distributed with rate ν and the down-times are i.i.d. (arbitrarily …

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Consider the discrete time stochastic model described in Section 10.2. The continuous time version is a stochastic counterpart of the problem 1 |

Consider the stochastic counterpart of with the processing time of job j arbitrarily distributed according to Fj . All jobs have a common random due date that is exponentially distributed with rate r. Show that this problemis equivalent to the stochastic counterpart of the problem (that is, a problem without a due date but with …

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Show that if in the model of Section 10.3 the deterioration function is linear, i.e., a(t) = c1 + c2t with both c1 and c2 constant, the distribution of the makespan is sequence independent.

Show, through a counterexample, that LEPT does not necessarily minimize the makespan in the model of Section 10.3 when the distributions are merely ordered in expectation and not in the likelihood ratio sense. Find a counterexample with distributions that are stochastically ordered but not ordered in the likelihood ratio sense.

Consider the two processing time distributions of the jobs in Example 10.3.6. Assume the deterioration function a(t) = 1 for 0 ≤ t ≤ 1 and a(t) = t for t ≥ 1 (i.e., the deterioration function is increasing convex). Show that SEPT minimizes the makespan. Example 10.3.6: Consider two jobs with discrete processing time …

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Consider the discrete time counterparts of Theorems 10.4.3 and 10.4.4 with geometric processing time distributions. State the results and prove the optimality of the WSEPT rule.

Generalize the result presented in Theorem10.4.6 to the case where the machine is subject to an arbitrary breakdown process.

Generalize Theorem10.4.6 to include jobs which are released at different points in time.

Consider the following discrete time stochastic counterpart of the deterministic mode The n jobs have a common random due date D. When a job is completed before the due date, a discounted reward is obtained. When the due date occurs before its completion, no reward is obtained and it does not pay to continue processing …

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