Consider the same scenario as in the previous exercise. Assume F is an EME distribution (as defined in Section 9.2) with the parameter p very small.

Consider a single machine and three jobs. The distribution of job j, j = 1, 2, 3, is discrete uniform mover the set {10 − j, 10 − j + 1,…, 10 + j}. Find the schedule(s) that minimize and compute the value of the objective function under the optimal schedule.

Consider the same setting as in the previous exercise. Find now the schedule that minimizes where the function h(Cj ) is defined as follows. Is the Largest Variance first (LV) rule or the Smallest Variance first (SV) rule optimal?

Consider two jobs with discrete processing time distributions: The two jobs have deterministic due dates. The due date of the first job is D1 = 2 and the due date of the second job is D2 = 4. Compute E(max(L1, L2)) and max(E(L1), E(L2)) under EDD.

Consider the framework of Section 10.2. There are 3 jobs, all having a discrete uniform distribution. The processing time of job j is uniformly distributed over the set {5−j, 5−j + 1,…, 5 +j −1, 5 +j}. The discount factor β is equal to 0.5. The weight of job 1 is 30, the weight of …

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Redo the instance in Exercise 10.7 with the discount factor β = 1. Determine all optimal policies. Give an explanation for the results obtained and compare the results with the results obtained in Exercise 10.7. Exercise 7: Consider the framework of Section 10.2. There are 3 jobs, all having a discrete uniform distribution. The processing …

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Consider Example 10.3.5 with the linear deterioration function a(t) = 1 + t. Instead of the two jobs with exponentially distributed processing times, consider two jobs with geometrically distributed processing times with parameters q1 and q2. Compute the expected makespan under the two sequences. Example 10.3.5: Consider two jobs with exponential processing times. The rates …

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Construct a counterexample for the stochastic problem with exponential processing times and deterministic due dates showing that if λjwj ≥ λkwk ⇔ dj ≤ dk the λw rule does not necessarily minimize

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 distributed) with mean 1/µ. Show that the expected time job j spends on the machine is equal to where E(Xj ) is the expected processing time of job j. Give an explanation why …

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Consider the same model as in Exercise 10.11 but assume now that the processing time of job j is exponentially distributed with rate λj . Assume that the repair time is exponentially distributed with rate µ. (a) Show that the number of times the machine breaks down during the processing of job j is geometrically …

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