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Modeling Traffic Flow in the Suez Canal Dan Teague NC School of Science and Mathematics [email protected] After the Six-Days War in 1967 • The Suez Canal was heavily mined and was closed to all traffic for 10 years. • The British eventually cleared the canal of all mines and the Egyptian government and the Suez Canal Authority hired mathematician Jeff Griffiths and his group from Cardiff University to explore the optimal organization for traffic in the canal. http://courses.ncssm.edu/math/talks/conferences/ Crucial to the Egyptian Economy • At the time of its closure, the canal contributed more than 50% of the funds to the treasury of Egypt. • Each ship paid an average of $100,000 for transit through the canal (now $250,000) . • At the time, on average, 67 ships transited the canal each day taking 10-14 hours for passage. http://courses.ncssm.edu/math/talks/conferences/ The Canal The Suez Canal was: • 193 kilometers long • 169 meters wide (for ship travel) • 21 meters deep • The distance between Jeddah (Saudi Arabia) and the port of Constanza (Black Sea) is 11771 miles via the Cape of Good Hope, but only 1698 mile via the Suez canal, a saving of 86% in distance. • Almost 7% of sea transported world trade passes through the Suez canal each year. http://courses.ncssm.edu/math/talks/conferences/ Convoy System • The width of 169 meters in width is not enough to allow passage in both directions. • As a result, the ships must travel, in convoys either North-South or South-North and pass each other at anchorage. • Generally, the demand for passage is the same in both directions. http://courses.ncssm.edu/math/talks/conferences/ Two Convoys going NorthSouth One Convoy going SouthNorth http://courses.ncssm.edu/math/talks/conferences/ No Passing in the Canal The N-S Convoy and the S-N Convoy can pass each other only in the Bitter Lakes and the Ballah Bypass. The N-S Convoy must be anchored to buoys in the canal while the S-N Convoy passes. http://courses.ncssm.edu/math/talks/conferences/ Restrictions The Ballah Bypass has 17 buoys. The Bitter Lakes has 36 buoys. Conditions for the Model 1. At most 36 in 1st N-S Convoy (Convoy A). 2. At most 17 in 2nd N-S Convoy (Convoy B). 3. Same number of ships N-S and S-N each day. 4. Schedule must repeat every 24 hours. http://courses.ncssm.edu/math/talks/conferences/ Develop a Mathematical Model Build a model to capture the current convoy process. Determine the maximum number of ships that can transit the canal each day. Compare the cost and effectiveness of modifications to this optimal value. http://courses.ncssm.edu/math/talks/conferences/ Standard Ship Model Assumptions Speed of all ships is constant at 14 km/hr. Separation for all ships is 10 minutes. Ignore (initially) acceleration and deceleration for docking. Zero-time docking and undocking for simplicity. http://courses.ncssm.edu/math/talks/conferences/ Modeling a Ship’s Transit The Ballah Bypass is 10 kilometers in length, stretching from the 50 to 60 kilometer mark south of Port Said. The Bitter Lakes extend for 20 kilometers, from 100 to 120 kilometers south of Port Said. http://courses.ncssm.edu/math/talks/conferences/ A 10-ship Convoy A NO! http://courses.ncssm.edu/math/talks/conferences/ YES! http://courses.ncssm.edu/math/talks/conferences/ 1. At most 36 in 1st N-S Convoy (Convoy A). 2. At most 17 in 2nd N-S Convoy (Convoy B). 3. Same number of ships N-S and S-N each day. 4. Schedule must repeat every 24 hours. 5. 14 km/hr & 10 minute separation http://courses.ncssm.edu/math/talks/conferences/ What is the First Natural Question? 1. At most 36 in 1st N-S Convoy (Convoy A). 2. At most 17 in 2nd N-S Convoy (Convoy B). 3. Same number of ships N-S and S-N each day. 4. Schedule must repeat every 24 hours. 5. 14 km/hr & 10 minute separation http://courses.ncssm.edu/math/talks/conferences/ Why Not 106 Ships? http://courses.ncssm.edu/math/talks/conferences/ http://courses.ncssm.edu/math/talks/conferences/ First, create a solution: then optimize. What are the Invariants? 100/14 = 7.14 24 – 2(7.14) = 9.72 9.72(60)/10 = 58.3 59 Ships in Convoys A and C. http://courses.ncssm.edu/math/talks/conferences/ A Little Algebra Helps If we let A, B, and C represent the number of ships in Convoy A, Convoy B, and Convoy C, respectively, we require that A+B=C and A + C = 59 with B = 17. http://courses.ncssm.edu/math/talks/conferences/ Analytic solution to the $1,000,000 system of equations. A + B = C, B = 17, A + C = 59 yields A = 21, B = 17, and C = 38 Total of 76 Standard ships per day. http://courses.ncssm.edu/math/talks/conferences/ A = 21 B = 17 C = 38 http://courses.ncssm.edu/math/talks/conferences/ Change Speed • (1) Traveling at 16 km/hr, the first ship in Convoy A must arrive at the 100 km mark at 6.25 hours. • (2) The symmetry of the problem requires the last ship in Convoy C to be at the 100 km mark at 24 – 6.25 = 17.75 hours. • So, we have 11.5 hours of transit time to share between Convoys A and C. If the ships are 12 minutes apart, we have room for 58 ships. Then 58 + 17 = 75 ships can make the transit in 24 hours. http://courses.ncssm.edu/math/talks/conferences/ The Rest of the Story • This is a real problem solved by real people. • Mathematics isn’t the final arbiter of what is “best”. People have to accept the results. http://courses.ncssm.edu/math/talks/conferences/ The Rest of the Story Jeff Griffiths presented his results to the head of the Suez Canal Authority, Meseur Meseur, comparing a variety of alternative to the standard model. Completing his work, he returned to Wales. But Jeff kept thinking that he had missed something simple. http://courses.ncssm.edu/math/talks/conferences/ While taking a bath back home, it hit him. He knew what he had missed. He immediately returned to Egypt and made an appointment with Meseur Meseur. http://courses.ncssm.edu/math/talks/conferences/ A 48 Hour Schedule The Rest of the Story • Jeff presents his results to Meseur Meseur. • Meseur Meseur listened carefully and at the end of Jeff’s presentation, said politely… http://courses.ncssm.edu/math/talks/conferences/ The Rest of the Story Absolutely Not! The Suez Canal has run on a 24 hour schedule every day since 1869. It is impossible to even think of it. Professor Griffiths returned to Wales in despair. http://courses.ncssm.edu/math/talks/conferences/ Misfortune Strikes the Canal How to Clear the Backlog? Meseur Meseur (Head of Suez Canal Authority) “We have this terrible problem. Ships are stacked up at both ends of the canal and it will take weeks to move them all through. What can we do? What can we do?” “…and then I had an idea.” http://courses.ncssm.edu/math/talks/conferences/ “If we ran the canal on a 48 hour schedule, we could significantly increase the number of ships passing through the canal.” “So, I ordered that we use the 48 hour schedule until the backlog was clear, then return to the 24 hour schedule.” http://courses.ncssm.edu/math/talks/conferences/ Dan Teague at [email protected] http://courses.ncssm.edu/math/talks/conferences/