IEOR 3609: Advanced Optimization
Homework 1
Advanced Optimization代写 1.Suppose that you are interested in choosing a set of investments 1, . . . , 7 using binary (0-1) Model the following constraints
- The completed homework must be submitted online on gradescope (latexed, or scan/photo of hand-written solutions).
- Please be mindful of the submission deadline. The submission link will be open for 24 hours after the deadline. However, 1 point will be deducted for every 2 hours of delay after the deadline.
- You are allowed to discuss the assignment with others but the write-up must be individual work. Please clearly mention in your write-up if you have discussed the solution with someone.
1.(20points) Advanced Optimization代写
Suppose that you are interested in choosing a set of investments 1, . . . , 7 using binary (0-1) Model the following constraints:
(a)You cannot invest in all of them.
(b)You must choose atleast one of them.
(c)Investment 1 cannot be chosen if investment 3 is chosen.
(d)Investment 4 can be chosen only if investment 2 is also chosen.
(e)You must choose either both investment 1 and 5 or neither.
(f)You must choose either at least one the investments 1,2,3 or at least two investments from 2,4,5,6.
2.(20 points) Advanced Optimization代写
At Blair General Hospital, six types of surgical operations are performed. The types of operations each surgeon is qualifified to perform (indicated by an X) are given in the table below. Suppose that surgeon 1 and surgeon 2 dislike each other and cannot be on duty at the same time. Formulate an IP whose solution will determine the minimum number of surgeons required so that the hospital can perform all types of surgery.
3.(20 points)
A company must complete three jobs. The amounts of processing time (in minutes) required are shown in the table below. A job cannot be processed on machine j unless for all i < j the job has completed its processing on machine i. Once a job begins its processing on machine j, the job cannot be preempted on machine j. The flflow time for a job is the difffference between its completion time and the time at which the job begins its fifirst stage of processing. Formulate an IP whose solution can be used to minimize the average flflow time of the three jobs.
(Hint: Here, the decision variables are the starting times for each job on each machine. Add linear constraints to ensure that a job cannot begin to be processed on a machine until all earlier portions of the job are
completed. You also need to ensure that only one job will occupy a machine at any given time. For example, on machine 1 either job l is completed before job 2 begins, or job 2 is completed before job 1 begins. You will need a binary variable (say y1) to decide which of these two situations should occur for machine 1 in the solution. A pair of constraints should be formulated for the two situations in such a way that for y1 = 0, one of them becomes void and the other is imposed, and vice-versa for y1 = 1. Similarly, for machine 2,3,4.)
4.(20points) Advanced Optimization代写
PSI believes it will need the amounts of generating capacity shown in the table below during the next fifive years. The company has a choice of building (and then operating) power plants with the specififications shown in the table below. Formulate an IP to minimize the total costs of meeting the generating capacity requirements of the next fifive years. (In this problem, once a power plant is built, it cannot be closed, and the annual operating cost must be paid for all the remaining years, including the year in which it is built).
5.(20 points)
Reconsider the previous problem. Suppose that at the beginning of year 1, power plants 1-4have been constructed and are in operation. At the beginning of each year, PSI may shut down a plant that is operating or reopen a shut down plant. The cost associated with reopening or shutting down a plant are shown in the table below. Formulate IP to minimize the total cost of meeting the demands of the next five years.
(Hint: Let
Xit = 1if the plant i is operated during year t.
Yit = 1 if the plant i is shut down in the beginning of year t.
Zit = 1 if the plant i is reopened at beginning of year t.
You must ensure that if Xi,t−1 = 1 and Xit = 0, then Yit = 1. You must also ensure that if Xi,t−1 = 0 and Xit = 1, then Zit = 1)
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