Mathematics of shipping
The Mathematics of shipping encompasses the analytical techniques and algorithms employed in the global shipping and delivery industry, which is vital for transporting packages, mail, and cargo worldwide. This industry generates substantial revenue, with express delivery companies alone earning $130 billion in 2009. The mathematical approach is crucial in three main areas: container packing, vehicle routing, and package tracking.
For container packing, efficient algorithms, such as the First-fit algorithm, are utilized to determine the optimal arrangement of items within shipping containers, though achieving absolute optimality remains challenging. Vehicle routing involves calculating the shortest and most cost-effective delivery routes, influenced by various factors including fuel costs and traffic, exemplified by the well-known Traveling Salesman Problem.
Finally, package tracking uses technologies like barcodes and Radio-frequency identification (RFID) to monitor shipments, enhancing the efficiency of package sorting and delivery. While RFID holds promise for streamlined tracking, it raises privacy concerns. Overall, the integration of mathematics in shipping not only improves operational efficiency but also contributes to the industry's financial viability.
Mathematics of shipping
Summary: A variety of mathematical concepts, including packing, routing, and tracking, are necessary to make the process of shipping goods more efficient.
The shipping and delivery industry is a vast global business that is responsible for delivering packages, postal mail, and commercial cargo all over the world. In 2009 alone, express delivery companies made $130 billion in revenue worldwide, the U.S. Postal Service delivered 177 billion pieces of mail, and ocean liners transported more than $4.6 trillion worth of goods between nations. With so many items being delivered to so many different places, there is a need for mathematics to help manage the complex delivery network and ensure that deliveries are made correctly, safely, cheaply, and quickly. Mathematics has had a significant impact in three key areas of the shipping industry: container packing, vehicle routing, and package tracking.
Container Packing
To minimize transportation costs and maximize profit, a shipper would naturally prefer to pack cargo into as few shipping containers as possible. Determining the optimal way to arrange items in a container is a deceptively difficult problem. Given a set of differently sized objects, the Bin-Packing Problem is to find the order in which to place the objects so that they fill the minimum number of bins. Testing every permutation of packing the objects would be too time-consuming, so an efficient and simple algorithm is required.
A common packing procedure is the First-fit algorithm, where the objects are ordered from largest to smallest, and each object is placed in the first available bin that will hold it. It can be proven mathematically that this algorithm is not guaranteed to produce the optimal packing. In the worst case, the result can be far from optimal and require the use of more bins than a more sophisticated packing. The First-fit algorithm is an example of an approximation algorithm, which means it produces a good approximate answer but not necessarily the optimal arrangement of objects. Other more sophisticated bin-packing algorithms have been developed, but as of 2010, no efficient algorithm was known that always produced the optimal packing.
In practice, there are more considerations to packing shipping containers. Some packages will be irregularly shaped and do not stack well. Some cargo is fragile and must be secured separately. Sometimes, a delivery vehicle will make several stops, so the packages that are delivered first should be packed into a container last to make them easily accessible.
Through World War II, most cargo was shipped in wooden crates of various sizes. A big step forward came in 1956, when trucker Malcolm McLean patented the modern shipping container made of corrugated steel. This sturdy container was easier to move between truck, rail, and ocean liner. More importantly, having a standard-size container meant that packing procedures could be standardized. Prior to 1956, it was estimated that loose cargo cost $5.86 per ton to load. After the standardized container was introduced, it was estimated the loading cost dropped to 16 cents per ton, a 3600% improvement.
Vehicle Routing
Cargo travels by a variety of transportation modes, including truck, rail, air freight, and ocean liners. The goal of routing is to determine a vehicle for each piece of cargo to be delivered and then find the shortest delivery route for each of the vehicles. The Traveling Salesman Problem is a simple mathematical example of a routing problem. In practice, the value of a route is not determined by just the distance. The problem is complicated by considerations such as personnel, fuel costs, traffic, tolls, and tariffs.
Mathematical analysis of delivery routes can lead to huge improvements in shipping efficiency. As the first Postmaster General of the United States, Benjamin Franklin ordered careful surveying of delivery routes, refined the post office accounting practices, and increased public access to mail. Under this new system, the U.S. Postal Service became profitable for the first time, and it is estimated that the mail delivery time between major cities was cut in half.
The routing problem is an example of a problem studied in operations research, the branch of mathematics that studies the cost-effectiveness of decisions made by corporate management such as scheduling and personnel assignments. The field of operations research has its origins in World War II, when the Allied Forces were interested in coordinating the manufacturing and organization needed to mobilize the military. One of the early researchers in operations research was Tjalling Koopmans, who proposed a mathematical model for the routing problem for shippers.
Package Tracking
It is important for a shipper to carefully track a package until it reaches its destination. A common system for identifying a package is the barcode. By encoding the destination as a sequence of black and white bars, the packages can be sorted quickly by automated sorting machines equipped with laser scanners. The U.S. Postal Service has developed a special barcode that encodes the address as a sequence of short and tall black bars. The mail is first read by an Optical Character Recognition (OCR) program, which translates the handwritten address into a barcode. The barcode is stamped onto the package and then automatically sorted to be sent to the next distribution center.
Radio-frequency identification (RFID) is a tracking technology that could potentially have a large impact on the shipping industry. A small electronic tag that emits a radio signal would be placed on each item to be shipped. Generally, this tag is a microchip just a few millimeters on a side. Potentially, this microchip would allow a shipper to determine the entire contents of a shipping container without ever opening the container. However, the technology still needs to be refined to make RFID a cheaper alternative to the barcode. Furthermore, since an item could theoretically still be tracked after the delivery is made, RFID technology is somewhat controversial because of privacy concerns.
Bibliography
Hillier, Frederick. Introduction to Operations Research. New York: McGraw-Hill, 2009.
Lodi, Andrea, Silvano Martello, and Daniele Vigo. “Recent Advances on Two-Dimensional Bin Packing Problems.” Discrete Applied Mathematics 123 (2002).
Palmer, Roger. The Bar Code Book. Peterborough, NH: Helmers, 2007.
Roberti, Mark. “The History of RFID Technology.” RFID Journal (2005).