undergraduate student with a paper in Management Science (Linear programming knowledge is essential) due in mid-May. I would like to know if you could provide any help with the question set below.
Your responses DO NOT need to reach the 4,000 word limit, as I will be attempting the paper on my own. I merely require worked answers as a means to better understand the subject material. Thus, the main issue here would be 1) the accuracy of the responses and 2) worked explanations of your solutions.
Do have a look at the questions below and let me know if you could be of assistance in this, and if so your job specifications (time frame, price, etc.). I will provide the original .pdf of the question paper and any other relevant academic materials as you require.
Looking forward to your reply!
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Our question is motivated by a study of a Queensland coal supply chain. That work considered issues of mine, rail and stockyard management as well as shiploading. We shall restrict our attention to the ship loading issue, and shall further simplify and abstract the problem to, hopefully, make it, simultaneously, manageable by and of sufficient interest to the long sufferig student.
The movement of ships in and out of ports is restricted by the tides. This leads to certain limits being imposed on the timing of the loading process. A similar problem is that of scheduling activities requiring some resource, such as a supervisor at the start of the process, where furthermore the supervisor is only available during certain time periods.
We assume we have a single machine and a set of jobs to be processed by the machine. The machine can process at most one job at a time. The jobs are of different durations, all in the range (0,1] (after suitable scaling). A job, once started must be completed without interruption. All jobs are available at time t = 0. The time (half-)line ft : t ¸ 0g is composed of intervals of length 2: [0,2), [2,4), [4,6), . . . .
We recall that [4,6), for example, is the set ft : 4 · t < 6g. Each interval is divided into an active region when jobs may be started and a passive region when jobs, already started may be completed but no new jobs may be commenced. The active regions are: [0,1], [2,3], [4,5], . . . . The passive regions are: (1,2), (3,4), (5,6), . . . . Thus, for example, if a job of duration 0.6, say, starts at t = 2:7 and thus ¯nishes at t = 3:3, the machine must remain idle until time 4. That is, the next job can commence at t = 4 but not for any t such that 3:3 · t < 4. The general problem is, given a set of jobs how do we arrange them so as to complete all of them using the smallest number of intervals. a. Suppose that we are given the following eight jobs with durations {0:45; 0:58; 0:8; 0:2; 0:15; 0:1; 0:5; 0:7}. Is it possible to complete these jobs using two intervals only? Use a first principles argument to justify your answer. b. Suppose that for some other set of jobs there is a solution using ten intervals. Verify that, possibly after some job interchanges, we can process all these jobs in no more than ten intervals with the ten longest jobs processed in the ten passive regions. c. Using the simplification introduced in b. model the problem in a. as an integer linear program with six binary variables. Note that in order to determine if there is a feasible solution, that is, if the feasible region is non-empty, we may use any objective function we like. Use Solver to either, find a solution,or to determine that there is no feasible solution. d. (Hard) Now consider the general case of our problem. Suppose we have 2n jobs with durations p1, ..., p2n, where pi is greater or equal to 1. At worst, all jobs will be completed using n intervals, by starting the jobs at times 0, 1, 2, ..., 2n - 1. What is the smallest number of intervals required to complete all the jobs? Formulate this as an integer linear program with 4n^2 + n binary variables.
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