AE-May-2013-98-Ind-A1

National Exams May 2013 98-lnd-A1 Operations Research 3 hours duration NOTES: 1 If doubt exists as to the interpretation of any question, the candidate is urged to submit with the answer paper, a clear statement of any assumptions made. 2 Any non-communicating calculator is permitted. This is an Open Book exam. Note to candidates: You must indicate the type of calculator being used, i.e. write the name and model designation of the,calculator, on the first left hand sheet of the exam workbook. 3 There may be more questions than you are able to answer in the allotted time. Although the total value of the questions is 180, any marks achieved will be considered toward the 100 total requirements. Page 1of 4

20 10 10 20 1. An airline flies planes onthe following route: Vancouver - Calgary - Toronto - New York - Vancouver. The distances are asfollows: Vancouver to Calgary, 687 km, Calgary to Toronto, 2690 km,Toronto to New York 571 km, and NewYork to Vancouver 3910 km.At each stop the plane may purchase Vancouver itis 0.88 $11, in Calgary 0.15 $/[, inToronto 0.95 $/[, and in New York 1.05 $/1. The plane's fuel tank holds 12000 I,and for safety, the plane must carry atleast 600 I.upon landing. The amount offuel used per km on each legofthe flight is[1+(average amount offuel intank on legof flighU2000)]. Assume that this average amount offuel is [(1/2) (amount offuel intank atstart of leg + amount offuel intank atend of leg)]. Formulate an LP that can be usedto minimize the fuel cost incurred in completing the schedule. Donot solve. 2. Consider the following LP problem Subject to XI 2: 0, i;; 1, 2. a. Use the Simplex Method tofind the optimal solution. b. Forwhat range ofcoefficient "3~inthe objective function does the solution remain optimal? 3. A catering company must meet itsdaily demand for clean napkins byeither buying new napkins, using regular laundry service which requires afull day turnaround, or special overnight service. Supplied with the demand for each ofthe next 5days, and the costsfor new napkins, regular laundry service, and overnight laundry service you have been asked todevelop a model which will minimize the cost ofprocuring the napkins over the next 5days. The company currently has no napkins in usable condition, clean ordirty, on the premises or at the laundry. Formulate this problem as a Minimum Cost Network Flow model. 98-Ind-A1IMay 2013 Page 2 of4

-------~--- 20 7. Robert Blue is trying to find a parking place near his favorite restaurant. He is approaching the restaurant from the west, and his goal is to park as nearby as possible. The available parking places are pictured in Figure 2. Robert is nearsighted and cannot see ahead: he can only see whether the space he is at now is empty. When Robert arrives at an empty space, he must decide whether to park there or to continue to look for a closer space. Once he passes a space, he cannot return to it. Robert estimates that the probability that space t is empty is pt. If he does not end up with a parking space, he is embarrassed and incurs a cost M (M is a big positive number). If he does park in space t he incurs a cost Itl. Show how Robert can use dynamic programming to develop a parking strategy that minimizes his expected cost. 20 20 FIGURE 2 ~[-TIIJ-TtI2-TI ... 8]E!J[Q][]@] ...0 Location of Parking Places 0 - Restaurant 8. A tool crib has exponential inter-arrival and service times and services a very large group of mechanics. The mean time between arrivals is 4 minutes. It takes 3 minutes on the average for tool-crib attendant to service a mechanic. The attendant is paid $10 per hour and the mechanic is paid $15 per hour. Would it be advisable to have a second tool-crib attendant? 9. As manager of the Western Wheat Cooperative you have one week to arrange delivery of an order for 6000 tons of wheat, stored among 5 silo locations, to arrive at the port of Rotterdam no later than 3 weeks hence. You will be ready to transport the wheat from the silos starting the morning of day 7, first by rail to one of 4 ports on ~heGreat Lakes, and from there by ship. There is only one ship sailing for Rotterdam within the next three weeks at each port. The amount of wheat at each silo is provided, as well as the rail cost/ton and transport time from each silo to each port. Also each ship departure date, expected arrival date in Rotterdam, ship capacity and shipping cost per.ton to Rotterdam are known. To simplify describing the mathematical model, you obtained the following specific travel times. The trains deliver the goods at each port early in the morning and the ships depart late in the day, providing plenty of time to load the ships. There is an incentive for early delivery in $/ton/day, and an overnight storage costs at each port in $/ton/day. Provide an LP model deriving the optimal shipping plan. 98-Ind-A1/May 2013 Port A Port B Port D Port C Days From Silo 1to Days From Silo 2 to Days From Silo 3 to Days From Silo 4 to Days From Silo 5 to Ship Departure Day Ship Arrival Day 2 ·1 4 3 1 8 17 1 6 2 5 3 11 21 4 2 3 1 4 9 17 5 2 1 6 4 10 19 Page 4 of4