Thirty-nine million, nine hundred sixteen thousand, eight hundred. That is how many routes home Odysseus could have taken.
The Greek hero wandered the Mediterranean for 10 years after the Trojan War, stopping at 11 places along the way. Had he been bound by fate to visit all 11 but free to choose the order, the number of possible routes would be 11 factorial, or 39,916,800. If measuring the distance of a single route took him 10 seconds, comparing every combination would require 12 years and eight months.
Odysseus' Dilemma Is the Traveling Salesman Problem
Mathematicians call the puzzle Odysseus faced the traveling salesman problem, which asks for the shortest route that visits each of several cities once. With few destinations, it poses little difficulty — three stops yield just six combinations. But as stops multiply, the number of possibilities grows exponentially, as the Odysseus case shows, far beyond what a person or a calculator can handle.
Mathematical optimization was devised to slash the time and cost of finding the best solution. Its core is building equations that quickly identify the best combination of variables. The field grew out of the problem of extracting maximum results from limited resources under real-world constraints, as researchers sought ways to dramatically shorten the calculation of optimal solutions while honoring those constraints.
Navigation systems, which must rapidly compute optimal routes much like the traveling salesman problem, are the most familiar everyday example. Mathematical optimization is also used to allocate volumes in logistics warehouses, schedule factory production equipment, arrange merchandise in retail stores and deploy corporate workforces.
Finding Answers to Real Problems
"Mathematical optimization does not stay confined to academic papers," said Lee Kyung-sik, a professor of industrial engineering at Seoul National University. "Defining real-world problems and finding answers is the essence of mathematical optimization."
Mathematical optimization rests on four basic components: decision variables, external factors, an objective function and constraints. Decision variables are elements the decision-maker can control and choose. External factors, by contrast, are conditions beyond the decision-maker's control. When a company draws up a production plan, the products and volumes are decision variables, while exchange rates and raw material prices are external factors.
The objective function is the ultimate goal of the decision-making process — minimizing costs or maximizing profits, the very reasons companies adopt mathematical optimization. Constraints are the practical conditions that must be observed, such as laws, internal rules and terms of business with clients.
"If you define the problem well, you have all but solved half of it," Lee said. "When you clearly specify these four elements within a real decision-making problem, a vague concern turns into a problem you can solve."
Once the basic elements are defined, the next step is to build a mathematical model by plugging them in. The same problem can yield different optimal solutions depending on the model used. Models are divided by form into techniques including linear programming, integer programming and convex optimization. Convex optimization has drawn renewed attention in the information technology industry after it emerged that SpaceX uses it to calculate the ground landing trajectory of its Falcon 9 rocket.
Solving Complex Corporate Problems
Applying mathematical optimization to a company's complex problems inevitably produces sprawling mathematical models. The software tools used to enter and compute long, complicated equations are known as solvers. Solvers can be run through programming languages and calculate optimal solutions from mathematical models and data. They also handle multiple objective functions at once and evaluate various scenarios efficiently.
Among the most widely used solvers worldwide is Gurobi. Developed by the U.S. company of the same name, the software is used in management decisions by Air France and the German energy systems firm Encoord, among others.
"There are still many attempts to solve problems simply by using Excel," said Oliver Bastert, Gurobi's chief technology officer. "But the larger and more complex the problem a company wants to solve, the more such attempts run into limits." Bastert said Gurobi's speed and accuracy in deriving optimal solutions allow users to evaluate many scenarios quickly.
Few companies, however, can handle solvers themselves and build mathematical optimization capabilities in-house. Consulting firms have moved into that gap. Notably, consulting work in this field is no longer just a contest of numbers. In the field, engagements begin with pinpointing exactly what is troubling the client, because even companies that seek out a provider after feeling the need for mathematical optimization often cannot state their own problems and goals precisely. Getting a project off to a proper start requires reading the blind spots hidden behind the documents.
Optimizing Inventory, Staffing and Store Displays
LG CNS, which runs a mathematical optimization consulting business, makes thorough preparations before defining a client's problem, examining the client's work processes, related organizations, business data and existing operational know-how.
"Mathematical optimization consulting requires basic knowledge of the client's industry, but we derive the client's problems and solutions only after comprehensively understanding its business model, core competencies, competitor trends and recent changes in the business environment," said Lee Joo-han, an optimization consulting specialist at LG CNS.
Since starting its mathematical optimization business in 2017, LG CNS has won more than 200 projects through the first half of this year. A signature result was a project that cut a domestic air conditioner maker's warehouse inventory by about half. The client had struggled because its indoor and outdoor units were produced at different rates. The two must ship as matched pairs, and unmatched units were moved to a separate warehouse and piled up. Seventy percent of the products made sat in warehouses at any given time, and the inventory management costs fell squarely on the client.
Using process data, LG CNS improved the production sequence and volumes of the indoor and outdoor units through mathematical optimization. Without adding production lines or new equipment, it fixed the mismatch in production cycles by changing how production plans were drawn up. Warehouse inventory fell 57% from the previous level, and inventory management costs dropped by hundreds of millions of won a year.
Sharpened by AI
Experts agree that industry interest in mathematical optimization will keep growing. Companies increasingly feel the need for swift decisions that keep pace with a changing business environment, and mathematical optimization helps by reducing unnecessary risk and quickly presenting better options. Companies and research institutes studying the field are responding to that demand, refining the technology by combining it with generative artificial intelligence, agentic AI and quantum computing.
"The theme across every industry right now is adopting intelligent automation as AI spreads," Lee of LG CNS said. "Because faster and more accurate decisions are needed, the importance of mathematical optimization will grow along with it."
