The Suez Canal Problem In the Suez Canal Problem we consider a multi-million dollar question of traffic flow through the Suez Canal. The figure below illustrates the major features of the canal. The Suez Canal is 163 kilometers long, 300-365 meters wide, and 21 meters deep. This short canal can dramatically reduce the distance and cost of shipping goods around the coast of Africa. For Sea Port Port Said anean NORTH Mediterr example, the distance between Jeddah (Saudi Arabia) and the port of Constanza (Black Sea) is 11771 miles via the Cape of Good Hope, while it is only 1698 mile via the Suez canal, thus a saving of 86% in distance is achieved. A saving of 23% in distance is also achieved by using the Suez Canal for the trip from Rotterdam in Holland to Tokyo in Japan if compared with the route round the African coast. Almost 7% of sea transported world trade passes through the Suez canal each year. Ballah Bypass The region of the canal available for ship travel is Capacity: 17 ships approximately 169 meters in width. This is not enough to allow passage in both directions. As a result, the ships must travel, in convoys either North-South or South-North. Generally, the demand for passage is the same in both directions. The only positions along the canal which allow for ships to pass are at the Ballah Bypass and the Bitter Lakes. 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 16 kilometers, from 100 to 116 kilometers south of Port Bitter Lakes Capacity: 36 ships Said. In order for ships to pass, one convoy must pull to the side and dock at floating moorings. The movement of ships in the opposite direction makes unmoored ships too unstable and likely to run aground. Unfortunately, the Ballah Bypass has room for only 17 ships to moor. The Bitter Lakes can accommodate up to 36 ships at a time. Traditionally, the convoy which moves South to North (SN) never stops. The Convoys which move North to South (NS) pull aside at the Ballah Bypass and the Bitter Lakes to allow the convoy moving SN to pass. The timing is important, so that the convoys will meet each other at the proper places. Since the SN convoy never docks, its size is not limited by the Sea Red Port Tewfik SOUTH size of the passing zones. However, the first NS convoy can be at most 36 ships and the second at most 17 ships. This limits the number of ships moving NS to 53. Under the assumption that there must be the same number of ships moving SN as NS, the total shipping each day is limited to 106. But this is much more than is actually moved through the canal. The size of the canal is not the limiting factor. In the first model of traffic in the canal, we will assume that all ships move through the canal at 14 km/hr. Moving more rapidly produces too much wash on the shore, damaging the shoreline and endangering the population which lives on the shore. At this speed, the ships must maintain a minimum separation of 10 minutes. Since there is little maneuvering room in the canal, it is essential that the ships have enough room to stop if the ship preceding them through the canal has difficulty. The Egyptian Canal Authority is interested in moving as many ships through the canal as possible under the speed and separation conditions stated. At present, they average 67 ships each day. If they are able to increase the number of ships by only two each day, at $100,000 a ship, this would mean an annual increase of $73,000,000 for the Egyptian treasury. Just why is it impossible to move 106 ships through the canal? Once we know the answer to that, we can consider how to increase the number of ships which actually are passed through the canal and, in fact, determine the largest number possible. What is needed is a method of representing the movement of a convoy through the canal. We need a mathematical model of traffic through the Suez Canal. The model we choose for this problem is a simple graph. The vertical axis represents the physical canal and the units are kilometers, while the horizontal axis represents time, and is measured in hours. We can trace the path of a ship as it travels through the canal by drawing lines on this graph. Notice that the Ballah Bypass and Bitter Lakes are marked. We can describe the passage of a ship moving SN by a line moving up the page with a slope of 14 and ship moving NS by a line moving down the page with a slope of -14. On the graph below, plot a path that describes a ship that leaves Port Said at 2:00, arrived at the Ballah Bypass at 5:30, traveled into the Bypass for another 30 minutes and docked at 6:00. After 1 hour it began its journey again, arriving at the Bitter Lakes at approximately 10:00. It traveled into the Lakes for 30 minutes, docking at 10:30. After 90 minutes, it again began its passage, arriving at Port Tewfik at 16:00 hours. If we can describe the path of the first and last ships in each of the convoys by using the graph described above, we can solve the problem of the Suez Canal. For the sake of simplicity, let’s assume that the last ship in a NS convoy must reach the beginning of the passing zone when the first ship of a SN convoy reaches the same end. Question 1: If a convoy of ten ships leaves Port Said at time 0, when must the last ship in a 15 ship SN convoy leave so that the convoys pass at Bitter Lakes? Question 2: Why can’t 106 ships go through the canal in a day? Question 3: If we choose to send 36 ships NS in Convoy A and 25 ships SN in Convoy B, is it possible to get the maximum number of ships (17) through in Convoy C? When should the first ship in Convoy C leave and when will the last pass Port Tewfik. Would this system work in the real canal? Explain your answer. Question 4: Does the number of ships in Convoy B affect the time which the 17 ships in Convoy C have to have completed their trip through the canal? Question 5: What is the largest number of ships that can transit the canal in 24 hours? Show your work. Question 6: Suppose the Port Authority decides to make traffic one-way each day. For example, on Monday traffic would be North-South only, while on Tuesday it would be South-North only. How many ships could move through the canal in a 24-hour period? Question 7: How many ships could pass through the canal if the Port Authority used a 48-hour schedule? Question 8: Suppose the Egyptian government is considering enlarging the canal at Ismalia to create another area for ships to pass. This area would be approximately 5 kilometers in length, from 80 to 85 kilometers south of Port Said, and would accommodate 10 ships. How many additional ships would this improvement allow to pass through the canal in a 24-hour period? Question 9: How can we generalize this problem? What is the largest number of ships that can pass through the canal in 24 hours if the ships travel at speed S with a separation time of T minutes? Reference: Griffiths, Jeff, From an address at the Third International Conference on the Teaching of Mathematical Modeling and Applications, University of Kassel, West Germany, September, 1988. .