[35 points] Jobco Shop has 10 outstanding jobs to be processed on a single machi

Question

[35 points] Jobco Shop has 10 outstanding jobs to be processed on a single machine. The following table provides processing times is measured from time 0: and due dates. All times are in days and due time ProcessingDue Time Time Job 20 98 100 50 32 60 80 150 30 10 If job 4 precedes job 3, then job 9 must precede job 3. The objective is to process all 10 jobs with the minimal total delay. Formulate the model as an ILP. Use Excel or GAMS to solve the problem, and print out your Excel or GAMS codes and solution.

Explanation / Answer

In fact the whole procedure can be viewed as a Markov Chain over the set of all feasible solutions of the problem to be solved. The fine tuning of the program is in the selection of the random neighbour, i.e., selecting the probabilities qxy, and in the definition of the probability to accept a worse neighbour. This is usually taken to be Pr{accept y | f(y) > f(x)} = e (f(y)f(x))/T , with T a large enough constant. If the neighbourhood is symmetric and the probabilities q have been chosen in such a way that the Markov Chain is ergodic, then it converges to its stationary distribution. If T is chosen large enough then the stationary distribution has positive probability mass only on the set of global optima. It is an extremely interesting and important open research question how fast the convergence is, even for specific optimization problems. So far, the only information we have is that simulated annealing appears to work well on some problems and it works much less on some others, without having any insight why. There is quite a lot of research on fast convergence of Markov Chains in the counting literature, or equivalently in the literature on sampling implicitly de- fined objects randomly

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