Modeling
Professor Mahbob wants to assign tasks of grading 6 Operations Research (2 sections total) homework to his 2 graders. Based on the content of the tasks, Mahbob believes that one of the graders would perform better than the other one.
So, he prepared the following table indicating the potentials of successfully grading. The higher the score, the better the grader is in performing the task. Mahbob will assign 3 tasks to each grader.
Moreover, tasks 5 and 6 are related and will be assigned to the same grader. Formulate an ILP to maximize the total potential of achieving tasks distributing the tasks among graders.
|
Task
|
|
Grader
|
1
|
2
|
3
|
4
|
5
|
6
|
|
1
|
100
|
85
|
40
|
45
|
70
|
82
|
|
2
|
80
|
70
|
90
|
85
|
80
|
65
|
Define the decision variables clearly (you need to use one index not two).
Formulate the objective function implicitly (compactly).
Formulate the objective function explicitly.
Formulate a constraint to ensure that each grader will be assigned 3 tasks.
Formulate a constraint to ensure that the tasks 5 and 6 go to the same grader.
Complete the model formulating variable domain constraints.
Write an AMPL code to solve the model.
Solve the model using the code in (g) report your solution.
Write each of the following using "summation" and "for all" notation using index as compact as possible.
We want to decide installation of a facility on 8 different sites. Using the decision variables formulate the following constraints:
Installation will be on 3 sites out of 8.
At least 2 of the sites 1,2,4,5 must be installed.
A facility should not be installed on both of the sites 3 and 8.
A facility can be installed at site 1 only if one is installed at site 4.
Regarding single machine multiple jobs sequencing problem formulate constraints to satisfy: if job i precedes job j then job k precedes job l. Explain how the constraints work in your own words (the formulation is provided in your text book).
AMPL
Please stop by the announcement board facing the entrance door of Systems Engineering Department. On the board, the Future magazine is pinned. On the magazine, you will see an unsolved SUDOKU puzzle. Yes...Please solve it using AMPL.
Paste your AMPL code.
Paste the solution of the puzzle.
B&B Algorithm
Write the LP relaxation of the above MILP
Solve the model constructing the B&B tree. Clearly number the nodes of the tree in the order of your computations. Use AMPL to solve your sub-problems. Stop branching if you cannot obtain optimal solution after 10 computations (the last node number should be 10).
Report your upper and lower bounds for the optimal MILP solution.