Click to learn more about author Keith D. Foote.

Mathematical optimization has become a problem-solving technology that can be used to automatically generate solutions to business problems. It helps in selecting the best business choices possible. One basic feature of mathematical optimization is the recognition that decisions, and their resulting actions, have far reaching implications which can impact an organization’s operations, and influence future decisions. (It should be noted that mathematical optimization cannot resolve “all” business problems.)

The process of mathematical optimization (MO) is normally used to solve broad, complex business issues, such as shipping routes, supply chain planning, and energy distribution. These issues involve a huge number of options and variables, and can make quick and efficient decision making difficult. Mathematical optimization streamlines the process by rapidly combing through trillions of possible solutions, and finding the best option.

When asked about MO, Dr. Ed Rothberg, Co-Founder and CEO of Gurobi, responded, writing:

“Mathematical optimization helps you make smarter choices about how to use limited resources to produce the best possible business outcome. It starts with a mathematical model of some aspect of your business you can think of it as a “digital twin.” The model captures the resources that are required for your business process (people, trucks, raw materials), the constraints that must be satisfied (parts on hand, capacity of machines, physical constraints), and your objective (minimizing cost, minimizing late deliveries, maximizing profit). This model captures the set of all feasible solutions and the quality of each. A mathematical optimization solver then employs a range of sophisticated algorithms to comb through all possible solutions to find the best.”

A business having resource constraints can use MO to find the most efficient way of using and deploying those resources. The goal becomes maximizing operational efficiency and unlocking new business opportunities.

Examples of Mathematical Optimization

Examples of organizations using mathematical optimization include Uberm Walmart, Microsoft,  FedEx, the National Football League, and Air France. They have used mathematical optimization to achieve tremendous financial and operational benefits. The process has saved billions of dollars and supported revenue growth for businesses around the world.

Regardless of an organization’s goals, MO will automatically consider and analyze all possible solutions and select the best solution (including balances the tradeoffs) to achieve those goals. If a business has resource constraints, the process of mathematical optimization includes those restraints when deciding how to deploy the scarce resources. How successfully a business achieves its goals depends largely on using its resources efficiently. According to Dr. Rothberg:

“Mathematical optimization is a very general technology. It is used in over 40 different industries, ranging from supply chain planning to financial modeling to sports scheduling. The main feature that each of these application areas have in common is a complex set of interconnected decisions that make it impractical for a human to think through the set of all possible outcomes. Mathematical optimization utilizes 70 years of advanced mathematics to focus attention on the most interesting parts of the solution space, allowing it to exhaustively consider a set of possible solutions of truly astronomical size. One big advantage of mathematical optimization over other AI technologies, like machine learning, is that an optimization model captures the behavior of the system, in all possible scenarios. Machine learning learns from historical data, which limits its ability to react to a shifting business landscape.”

The following examples are ways organizations are currently using mathematical optimization:

• Sports Scheduling: Mathematical optimization is used by the NFL to predict the best possible league schedules.

• Government: The FCC has used mathematical optimization to create a two-sided spectrum auction (selling the rights for transmitting signals over specific bands).

• Finance: Betterment uses mathematical optimization to choose the best mix of assets to maximize after-tax returns, and minimize risks.

• Logistics: FedEx saves money by mathematically optimizing the routes of packages moving through their shipping network.

• Electrical Power Distribution: The New York ISO uses mathematical optimization to select the most cost-effective methods to provide electricity.

• Manufacturing: SAP uses mathematical optimization to schedule the manufacture of goods in their factories, with minimal waste.

MIP and Mathematical Optimization

Mixed Integer Programming (MIP) problems have some variables that are continuous, and some that are discrete. The techniques used with MIP were developed many years ago. However, recent advances in algorithms, computing power, and data availability support processing complex business problems at much greater speeds. As a consequence, MIP has had a significant impact on a variety of businesses.

The development of strong algorithms for MIP problems has become a popular theme for research.

Machine Learning and MIP

The primary difference between machine learning and MIP is that ML makes predictions, and MIP makes decisions. For example, when your issue deals with trillions of possible solutions and complicated tradeoffs between competing activities, MIP is designed to find the best solution. Machine learning relies on historical data when make predictions. Mathematical optimization, however, relies on “all available information” of the current business conditions, and include real-time and historical data.

MIP can be combined with machine learning. For example, rather than using machine learning to choose the best offers to present to web customers, machine learning can be merged with MIP to select an array of offers to increase profits. In another scenario, the combination of ML and MIP can be used for predictive maintenance. If an elevator develops a problem, ML can predict the most probable failures, and then MIP can allocate and schedule the needed resources for performing the necessary repairs at minimum cost.

The Popularity of Mathematical Optimization

George B. Dantzig published his Simplex Algorithm in 1947. It described linear programming (also known as linear optimization), which became the foundation of mathematical optimization. A recent survey has shown 85% of the Fortune 500 companies use MO. Curiously, in spite of its extended history, and popularity with large organizations, this type of programming is not terribly well known.

One theory suggests the human mind has problems grasping the large scope of details used in mathematical optimization. Because these problems aren’t tangible or visible to most people, they decide they don’t need it. (Maybe, but people can turn on a light switch without understanding electricity.)

Dr. Rothberg has a different theory. He wrote:

“In the past, there was the common perception in the business world that mathematical optimization applications were difficult to build and maintain. But now – with the latest advancements in mathematical optimization technologies and ease-of-use improvements – it is possible for just about any company (with the right technological tools and people with the right skill sets) to develop and deploy mathematical optimization applications.”

Artificial Intelligence and Mathematical Optimization

There is no single technological solution that will resolve all the challenges businesses encounter. While mathematical optimization offers significant support, when merged with Artificial Intelligence, it offers even more support in assisting with the decision making process, and business outcomes.  AI is promoting huge changes in the way decisions are made, even for smaller companies.

On December 22, 2020, Gurobi Optimization, LLC and o9 Solutions, Inc. – a premier AI-powered cloud-based platform – announced they had formed a partnership. Their shared goal is to deliver state-of-the-art mathematical optimization solutions to businesses across a broad range of industries. o9 will be responsible for making the “Gurobi Optimizer” available as a component of their AI-powered platform, The Gurobi Optimizer will provide businesses with the ability to solve complex business problems, while minimizing operating costs and maximizing productivity.