Original URL: https://www.theregister.com/2014/06/18/feature_logistics_travelling_salesman_problem/
Finding the formula for the travelling salesman problem
Inside the mathematics of supply chain logistics
Posted in On-Prem, 18th June 2014 08:31 GMT
What do heuristics, graph theory and doughnuts have in common? Each of them, in its own way, underpins one of the most challenging parts of the logistics process: planning delivery routes.
Every day, millions of products find their way from manufacturers to distributors, retailers and finally to homes and businesses everywhere.
Making sure those products get there on time is a gargantuan task which has its roots in a mathematical conundrum from the 1800s: the travelling salesman problem.
In its simplest form, the problem takes a hypothetical salesperson tasked with visiting a certain number of places in one day. How can the salesperson visit all of the places via the shortest route?
This problem is a product of graph theory – the study of nodes and lines – and it is the perfect kind of mathematics for logistics networks. After all, these are effectively collections of the same thing.
“If you think of what the supply chain looks like, you draw circles on paper and label them,” says Jeff Karrenbauer, president and founding director of Insight, which specialises in planning solutions for logistics and supply-chain operations.
“You draw lines between them that represent transportation links.”
The travelling salesman problem may be a fun puzzle to solve at home with about six nodes. But for logistics firms dealing with thousands of circles and lines, it is far more challenging.
The problem lies in the fact that as you add places to visit, the number of possible combinations increases exponentially. And the reason for that lies with the logistics route planner’s nemesis: the factorial.
Dizzying numbers
The factorial of a number, n, is the product of all the positive integers equal to or less than itself, and is denoted as n! . The factorial of 4 (4!), for example, is 4 x 3 x 2 x 1 (24).
That is how many combinations you would have for a travelling salesman with four places to visit. 5! Is 120. 6! Is 720.
Now, think about a single driver making 25 stops in a day. That provides 1.551121 x 1025 different combinations (25!).
Given that there are about 1025 stars in the visible universe, the average delivery driver would earn a lot of overtime doing that on paper – and would probably have to skip breakfast.
“It's a very special situation, because once you have decided that every node will be visited once, every node has exactly one predecessor and every node has one successor. It's what is called a combinatorial optimisation problem,” says Jim Bookbinder, director of the Management of Integrated Manufacturing Systems research group at Canada’s University of Waterloo.
In real life, the whole thing becomes intractable long before you reach 25 stops, explains Danny Bausch, director of analytics and product management at Insight.
“That problem doesn't yield well to optimisation. Once you get above about nine stopping locations, the number of combinations explodes exponentially and optimisation routines melt down,” he says.
To ease the load, logistics experts must try to bring in another kind of mathematics: heuristics. This is a way of finding approximate solutions to mathematical problems when there are simply too many potential inputs to handle.
“At some point the brain says there are too many things going on"
We have found them in popular computing applications before, such as anti-virus software, for example, where many files must be checked quickly, often in real time, to look for malicious bugs.
“At some point the brain says there are too many things going on so I will follow some rules of thumb,” says Bausch.
For example, 25 different stops may be too difficult to analyse using the computing technology available. But you may be able simply to look at the map where the drops are to be made and divide them into more easily manageable chunks, for example north, south, east and west quadrants. Each of those could then be subject to its own travelling salesman analysis.
After all, it is unlikely that a driver would ever want to go directly from the southernmost point in a daily route to the northernmost point, so that is one of many combinations it wouldn’t make sense to compute.
“That's a manual heuristic. Those rules can be put together like that,” says Bausch. “They are a series of steps that the computer understands well.”
There are a handful of commonly used heuristics when it comes to mathematical problems of this sort. One of them is the sweep algorithm, which uses a line that sweeps across the map to find potential intersections between the different points and come up with an optimal order for deliveries.
Another is the exchange heuristic. “If an order turns out not to be a good sequence, then you operate on it with exchanges, saying ‘trade stop one for stop two, and switch nine for six’,” says Bausch. “You can do that for a long time, too.”
Yet another, far more complex algorithm deals with quadratic assignments. It considers the sweep and exchanges together and the expected value.
Back in the real world
One difficulty with the travelling salesman problem is that it plays out on paper in a spatial system unsullied by worldly constraints.
Back in the unforgiving world of logistics, things are far more complex, according to Bookbinder.
“There are so many other issues that the optimisation on the routing part isn't the be all and end all,” he says.
Capacity is one such constraint. A van may be able to take only so much product, based on its size. A manufacturing or distribution plant may have only so much stock.
Road quality is another. Which are the good, fast roads to use and how well do they connect the stops on the route?
“I may start out with a full truck making deliveries to customers but I may end up back in the factory,” says Bookbinder.
“If there are suppliers of raw materials nearby, where I can deviate a little from the delivered route, then I can mix pickup and deliveries. Then I can really have an efficient route. That has become a standard variation on the vehicle routing problem.”
Your goals may also be different. Delivery in as short a time as possible may be the primary objective in some cases. In others, the lowest cost may top the list.
Some organisations may have a need to factor carbon emissions into their supply-chain management operation to meet a corporate sustainability requirement.
“These things are expressed as thousands or millions of additional equations. All of them describe a network which is a series of locations and the transportation links between them, such as ports, suppliers, cross docks and customers,” says Bookbinder.
Thinking inside the box
Those calculations address the big picture, ranging from 30-mile delivery routes to global logistics systems, but sometimes the mathematical problems simply explore the inside of a 40ft van.
Even the mathematics of loading and unloading a truck creates software challenges, says Jim Lodwick, transport and logistics consultant at Rocket Consulting, which makes warehouse management and supply-chain visibility software for clients.
“Mathematics will be used to sequence things such as the right way to load the vehicles so that the product comes off in the right order,” he says.
Other things to consider include product sizes, the weight and dimensions of the vehicle and how much weight you can put over its various axles.
UPS: what goes where
Road sense
This kind of complexity is why UPS devoted so much attention to its On-Road Integrated Optimization and Navigation (Orion) system, designed to optimise deliveries across its vast global network of drop points.
The system began its first rollouts last year and will be deployed to all 55,000 north American routes by 2017, the firm says. Currently it is in operation only in the US.
Remember that a 25-stop route has the same number of combinations as there are stars in the visible universe, give or take a few trillion?
The average UPS truck makes between 125 and 175 stops a day, creating more possible combinations than there are nanoseconds in the earth’s history, the firm says.
The system not only uses advanced heuristics to carve up delivery routes while taking into account many of the constraints mentioned, but it also optimises the placement of packages inside the vehicle.
This enables drivers to avoid moving more than 30 inches to select the next package in the truck. Every second shaved off a driver’s time could represent hundreds of thousands of dollars when replicated across the entire UPS network.
So, where do the doughnuts come in?
Mathematics doesn’t solve everything. Bausch recalls one delivery problem in which a contact at a particular location insisted on taking deliveries outside lunch hours.
Unfortunately, this threw a mathematical spanner in the works for the planning software, which needed to schedule a delivery at noon to make everything workable. The alternative would be to send a dedicated truck to that address just for a single delivery.
Instead, Bausch says, the delivery dispatcher called the person in question, who it turned out just really wanted to take lunch during those hours.
“What about if we bring you a sandwich?” the dispatcher asked.
“Make it doughnuts and you have a deal.”
Problem solved, and not an equation in sight. Sometimes, even mathematics needs a little sugar coating. ®