Algebra: Math solves a UK supply chain crisis | mathematics

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Nando put it succinctly on Twitter last month: “The UK supply chain has a bit of a mare at the moment.” Getting things on supermarket shelves, through your mailbox or into a restaurant kitchen, has become a problem lately. It’s hard to know exactly where to blame, although Covid and Brexit certainly played a role. What we can do is be thankful for algebra, because things would be so much worse without it.

You probably have mixed feelings about algebra. Even if you could handle it and manage it at school, you’ve probably wondered why it’s important to solve an equation involving x to the power of 2 or why you want to find a and b when a + b = 3 and 2a – b = 12. You may feel that your doubts are justified : You’ve probably never done algebra in your life after school. But that doesn’t mean that the jumble of letters, numbers, and missing things we call algebra is useless. Whether it’s groceries at the supermarket, a new TV, or a package from Aunt Emily, they all make it to your home with some attempts to solve the equation and find the missing number. Algebra is the mathematics that comes to us.

Algebra has been around for thousands of years. The word comes from the Arabic word algebra in the title of a 9th-century book on arithmetic, but the ancient peoples of Babylon, India, China and Africa were solving algebraic equations long before that. It is, basically, the art of finding unknown numbers, given some other numbers. The hidden factor required in Latin was usually referred to as “squash” or “thing”, and so algebra was often known as “the art of the quaz”: the art of the thing.

The early adopters of algebra did not have the luxury of solving equations: until the sixteenth century, everything was written in words. An early student of the art of the iris might find themselves face to face with something like this: two men leading oxen along the road and one saying to the other, “Give me two oxen and I will get as many oxen as possible.” Then the other said, “Now give me two oxen and I will get double.” the number you have.” How many bulls and how many were each?

Delivery is a daunting algebraic challenge, with more variables than I’ve encountered in any test

This poser comes from a compendium of riddles, published around AD 800, called Problems of Young Aggravation. It’s not much different from the questions we all encountered in math lessons at school and the main use of algebra today is still to count the numbers of bulls on the road – to stock your butcher’s counter in your supermarket.

“Warehouse storage is a complex issue,” says Anna Moss, principal data scientist at Ocado Technology. Moss’ role includes ensuring that the quantities of inventory required from suppliers are sufficient to meet customer demand, but do not exceed the warehouse’s storage capacity, and most importantly, reduce food waste.

You might not be surprised to learn that Moss is an expert in mathematics. In the past, she worked for Intel and frequently publishes her mathematics research in academic journals. Applying such expertise to grocery delivery may seem overkill, but the logistical puzzles involved are as challenging as anything I’ve encountered elsewhere.

Logistics calculations begin with algebra — linear algebra, to be exact. This is algebra where variables (data about warehouse stock, for example) tend to be processed in ways that do not depend on the square, cube, or other power. So y = 4x would be practical in linear algebra; y = 4×2 will not.

Linear algebra explores solutions to sets of equations that together contain everything you need to discover relationships between variables. Their equations are, essentially, mathematical spreadsheets where a single operation can process a huge set of data, revealing the relationships between all of them and allowing a mathematician to refine one particular selected result. The same trick lies in Google searches, flight scheduling, and parcel delivery; Even the way a virtual shopping cart is delivered to your computer screen includes linear algebra in the logistics of online routing information.

The UK supply chain crisis has left holes in the shelves. Photo: Justin Thales/AFP/Getty Images

However, Logistics did not stand idly by with linear algebra. It has been developed into algorithms for “linear programming”, “mixed integer programming” and many other odd-sounding mathematical procedures, such as “harmonic optimization”, “greedy heuristics” and “simulated annealing”.

“You can think of this as computational algebra,” says Keith Moore of US logistics software company Autoscheduler.AI. And all of this is done for one purpose only: Delivery to every customer, on time and in full – OTIF as it is known in the trade. And as anyone working in supermarkets after Brexit knows, this is never possible. “In every distribution center I’ve been around, practical limitations prevent OTIF from being optimally optimized,” Moore says.

Moore’s mission is to maximize what’s possible for a wide range of clients, including Unilever and Procter & Gamble. Does not use paper, pen or calculator. “Even in a single distribution center, they collect gigabytes of data every minute and that data is constantly changing. It is not impractical to have analysts and people sitting in a room doing the math to make decisions, it is completely impractical.”

Instead, the necessary algebra is programmed into software. Of course, the exact nature of the algorithm in action is a trade secret. That’s why many companies refused to talk to me for this article: they were worried that their mathematicians might say too much. Sainsbury, DPD and Hermes all declined an interview request on the grounds that the sports tricks used to improve their services were, as DPD’s PR said, “not something they want their competitors to know.”

The movements of Ocado’s robot command collectors are controlled by algorithms. Photograph: Peter Nichols/Reuters

What we do know is that delivery is a frighteningly algebraic challenge, with far more variables than you’ve encountered in any exam question. For example, Ocado’s optimization algorithms take into account how ordered items are packed into as few bags as possible, as well as the best route a robot would take in a warehouse or by a personal shopper in a store picking products off the shelves. But they also have to take into account the time period you selected, the capacity of the truck and a myriad of other factors such as achieving a minimum environmental impact. “All of these criteria are given weights based on their relative importance, and this weighted set acts as one value that must be optimized,” Moss says. “In addition, our problem changes all the time, as customers place new orders and modify existing ones. Our algorithms have to adapt to these rapid changes.”

Then there are the optimal delivery methods, given the location of warehouses and stores relative to your home address. “We know the travel distances between all pairs of these sites,” Moss says. “The problem is finding the best truck routes or how to assign orders to pickup trucks and sequencing each truck to deliver their assigned orders.”

Assuming you can find drivers and fuel, the mathematics of efficient cargo delivery is actually an example of a long-standing problem for mathematicians: the traveling salesman problem. Cut to the bone, that is: How do you find the shortest route that allows you to visit a number of sites only once?

You don’t have to treat this as an algebra problem, strictly speaking, although linear algebra does provide one angle of attack. Others come through disciplines derived from algebra such as graph theory. However, the exact nature of the math is somewhat moot, as there is no way to solve the traveling salesman problem just by dealing with a realistic number of destinations.

While there are six options for travel from the distribution center to three destinations, there are 479 million possible ways to deliver to just 12 destinations. A single parcel delivery driver might deliver 60 or 70 parcels a day and there are trillions of possible routes for that relatively small number of deliveries.

No one even expects a computer to get through them all, so software, like UPS’s Orion, guesses the best methods, scans for its problems and then improves them. This is an approach known as heuristics. It is another relative who is highly capable in school algebra and tries to get as close as possible to optimal solutions. Although it begins with guesswork, it is still validated through mathematics. “The math gives us confidence that we are close enough to the best option,” says Ravi Ahuja, founder and CEO of logistics optimization companies Optym and Axele.

Nowhere is this more important than in the aviation industry. Here, the variables include what flight routes are, how many flights per route and at what time, and of course, how best to schedule each aircraft, crew members and passenger for maximum profit. “Timing is everything for airlines, as this allows inbound planes to connect with outbound planes, crews to move from one flight to the next, and passengers to fly when they want to and make flight connections,” Ahuja says. “For a large passenger airline that flies thousands of flights a day, this is a huge, very complex mathematical problem.”

Solve it with algebra. “We use many approaches, including greedy heuristics—one decision at a time but the best—or mixed integer programming, which relies on linear algebra,” Ahuja says. He’s proud to say that his algebra skills once allowed him to find the perfect solution to a hideous set of equations on behalf of an airline, and the company has seen its profitability skyrocket by millions of dollars annually.

So learning algebra wasn’t a waste of time – for Ahuja and the others keeping our shipments coming in these tough times, at least. To be fair Moss never felt like that, math has always been her favorite subject and she’s built her career on “a solid math foundation she learned in school,” as she puts it. “I consider myself lucky,” she says. “I still enjoy mathematics and have the opportunity to do the things that really matter to me and make a difference in society.”

Not that all these mathematicians keep things professional. “I sleep well every night knowing we’re helping companies reduce costs and run greener operations,” Moore says. “But from a personal perspective, building an algorithmic fantasy football team that completely ruined the 2019 NFL season was pretty cool too.”

Michael Brooks is the author of The Art of More: How Mathematics Created Civilization, published by Scribe UK (£18.99) this month. To support Guardian and Observer, order your copy at guardianbookshop.com. Delivery charges may apply

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