A Guide to Mathematical Optimization: What, Why, and How

Every day, businesses face thousands of decisions: where to allocate resources, how to schedule operations, which routes to take, how to price their products, etc. Making these decisions at scale and under real-world constraints is one of the hardest challenges organizations face. That's where mathematical optimization comes in.

At its core, mathematical optimization is about finding the best possible outcome from a set of available options, whether that means minimizing costs, maximizing revenue, or making the most of limited resources. In this article we will review what mathematical optimization is, the different types available, and how organizations across industries are using it to make smarter, faster, and more confident decisions.

Key Takeaways

• Mathematical optimization is about making better decisions. Optimization works by defining decision variables (what you can control) and constraints (your limitations), then finding the mathematically best outcome, whether that's minimizing costs, maximizing revenue, or allocating resources efficiently.
• There are many types of optimization. The main types include Linear Programming (continuous decision variables, with linear equations for both the objective and the constraints), Mixed Integer Programming (handles whole-number and binary decisions like scheduling), and Nonlinear Optimization (choosing from a vast set of possibilities). The right type depends on the problem.
• The business benefits of optimization go beyond just efficiency. Optimization can benchmark your current performance against the theoretical best, reveal the hidden cost of arbitrary rules or policies, and uncover process improvements that aren't obvious on the surface.
• Optimization spans every industry. Whatever the sector, the common thread is the same: turning complex, constraint-laden decisions into fast, data-driven recommendations.
• Optimization is an ongoing need, not a one-time fix. Because real-world data changes constantly, optimization needs to be applied continuously rather than treated as a one-off analysis.

What Is Mathematical Optimization

Mathematical optimization is the discipline of selecting the best possible outcome from a set of available alternatives according to a defined objective, while adhering to a set of constraints or rules. In practice, it involves formulating problems (such as maximizing profits, minimizing costs, or achieving optimal resource allocation) into mathematical models where decision variables, objectives, and constraints are precisely articulated. 

These mathematical models are solved using specialized algorithms within optimization software and return data-driven recommendations that improve decision-making. At its core, mathematical optimization uses prescriptive analytics to provide a rigorous framework to guide choices in complex, real-world environments, such as supply chain network design, workforce scheduling, and inventory management. 

To understand mathematical optimization, we also need to understand the components that comprise an optimization model.

• Sets: Sets define the groups of things the optimization model is working with, such as products, customers, machines, locations, or time periods. They give the model its basic structure by answering: “What are we making decisions about?” For example, consider the 50 states in the United States. Each state within the set is referred to as an element. California is one element in the states set. Mississippi is another element in the states set. The total number of elements in the states set, or the cardinality, is 50. 
• Parameters: Parameters are the known inputs to the model, such as costs, prices, demand, capacity, distance, or available resources. They describe the facts of the problem before any decisions are made. Given a set of states defined above, a parameter could be the total population in each state, the median home price in each state, and/or the unemployment rate in each state. Parameters serve as necessary input data the optimization model needs in order to find optimal solutions. 
• Decision variables: Decision variables represent the choices the model is allowed to make. Decision variables can be continuous, integer, or binary variables. For example, how much of each product to produce, which routes to use, which customers to serve, or how many workers to schedule. 
• Objective Function: The objective function tells the model what “best” means, and takes the form of either a minimization objective or a maximization objective. Common examples are minimizing total cost, maximizing total profit, minimizing risk or volatility, maximizing customer reach, minimizing the absolute value of the difference between two sets of decision variables, and so on. The objective function is tailored to the specific problem being solved for. 
• Constraints. Constraints are the rules and limits the solution must follow. They capture real-world restrictions such as budgets, capacity limits, staffing requirements, delivery deadlines, regulations, or minimum service levels.

Types of Mathematical Optimization

There are many different types of mathematical optimizations. The most common ones include:

Variable-Based Optimization

• Continuous Optimization: the decision variables used in the model are continuous. This means that the variables can take any value from a set of real numbers, without any gaps between them. Because of the continuity assumption, continuous optimization allows the use of calculus techniques.
• Discrete Optimization: in contrast to continuous optimization, variables in discrete optimization are restricted to separate values. Discrete optimization is often used in scheduling and logistics.

Equation-Based Optimization

• Linear Programming (LP): in linear programming, both the objective function (for example, costs) and constraints (such as resources limitations) are linear functions. It is often used for resource allocation or transportation problems.
• Nonlinear Programming (NLP): unlike linear programming, in non-linear programming the constraints and/or objective function are not linear equations, or the objective function is not a linear function. It is often used for trajectory planning for robotics or portfolio optimization in finance.
• Mixed Integer Programming (MIP): it involves both continuous (decimal) and integer (whole number or binary) variables, subject to specific constraints. It is commonly used for complex decision-making, such as resource allocation, production planning, and scheduling.

Other Types of Optimization

• Combinatorial Optimization: combinatorial optimization aims to find the optimal solution from a finite, but usually vast set of possibilities. It is often used for vehicle routing, airline crew scheduling, or telecommunications network configuration. 
• Multi-Objective Optimization: the process of simultaneously optimizing two or more competing objectives, such as staffing a call center that balances total operational cost with average caller wait time. 
• Dynamic Optimization: dynamic optimization determines the best decision path over time and is commonly used to plan growth trajectories or manage inventories

The Benefits of Mathematical Optimization in Practice

Why Use Optimization?

Mathematical optimization is crucial for decision-making, for a few reasons:

• It uses a rigorous approach to decision-making and takes into account all of the factors that influence decisions 
• It translates real-world situations into quantitative models and uses a structured, data-driven approach to improve the objectivity of decisions as much as possible 
• It allows organizations to objectively analyze the trade-offs of resource allocation, scheduling, and logistics, while ensuring that operational constraints are rigorously respected
• It often uncovers hidden insights about cost drivers or potential process improvements that go beyond surface-level efficiency gains
• It enables measurable improvements in performance
• It supports robust decision-making under uncertainty and bolsters confidence in critical business choices

Some problems will not require formal optimization. For example, the problem may be simple, or the answer may be obvious, or there may be no decision variables, so we have no choices. But most problems are complex, require making decisions, and so benefit from mathematical optimization. In addition, as many problems change over time, particularly data, optimization is required on an ongoing basis.

Key Benefits of Mathematical Optimization

There are many benefits to using optimization in a business setting:

• Operational efficiency: We can make decisions that mean we better utilize our resources. For example, we can use optimization to maximize production with existing resources within existing facilities. We can reduce energy consumption; we can reduce transportation manpower, commissions or overhead costs. In finance, we can minimize the number of shares held to track an index.
• Revenue maximization and cost minimization: Optimization can be used to increase business revenue or earnings, or reduce costs. For example, we can minimize production costs by making more product from existing facilities, or make a product of certain characteristics from cheaper materials, all without violating the constraints we are under.
• Understanding and performance assessment: Optimization gives a unique insight into situations where decisions are involved. It can be used to benchmark performance, for example current performance against the best possible performance. Optimization provides information about the costs of limitations, for example: what additional profits could be made if a limit were moved or removed? In the same way it can give insight into the implied costs of policy decisions or arbitrary rules.

Applications of Mathematical Optimization

Financial Services

• Portfolio optimization: Allocate capital across a set of assets (such as stocks, bonds, and commodities) to maximize expected returns for a given level of risk tolerance, while respecting constraints such as liquidity requirements or regulatory guidelines.
• Credit risk and loan portfolio management: Determine the optimal mix of loans and credit products to issue across different borrower segments, balancing expected revenue against default risk, capital reserve requirements, and regulatory constraints.
• Algorithmic trading and order execution: Optimize the timing, sizing, and routing of trade orders to minimize market impact and transaction costs, while achieving target execution prices and managing exposure to price volatility across trading venues and liquidity conditions.

Read the FICO case study with HSBC

Manufacturing

• Production line sequencing: Determine the most efficient order in which different products, batches, or operations should be processed on shared equipment, such as refineries, LNG plants, or battery manufacturing lines, to minimize setup times, changeover costs, and downtime while meeting demand and quality requirements
• Raw material procurement and inventory optimization: Decide how much of each input material to order, from which suppliers, and at what frequency to balance holding costs against stockout risks, while respecting budget constraints, supplier lead times, and minimum order quantities.
• Facility layout and machine placement: Determine the optimal physical arrangement of machines, workstations, and storage areas on the factory floor to minimize material handling distances, reduce bottlenecks, improve workflow between production stages, and make the best use of available floor space.

Read the FICO case study with Pepsico

Transportation & Logistics

• Vehicle routing and delivery schedule: determine the most efficient set of routes for a fleet of vehicles to service with a known set of customers or locations, while minimizing costs such as total distance, or travel time, and respecting constraints like vehicle capacity, delivery time windows, and driver regulations.
• Driver and maintenance scheduling: Assign drivers to vehicles, and vehicles to routes, while simultaneously planning maintenance activities to maximize fleet availability, minimize downtime, and comply with labor laws, safety regulations, and service requirements.
• Load consolidation and freight cost optimization: Combine multiple smaller shipments, orders, or loads into larger, more efficient ones, so that trucks, containers, or other transportation assets are better utilized while minimizing transportation costs and reducing empty space.

Read the FICO case study with Avis Europe

Telecommunications

• Network design and bandwidth allocation​: Determine the optimal placement and configuration of network infrastructure including cellular towers, fiber routes, and so on, ensuring maximum coverage and network robustness to quickly recover from failures, congestion, or outages.​
• Call center staffing​: Forecast call volumes and optimize workforce staffing to ensure customer demand is met at the lowest possible cost while maintaining service-level agreements such as average wait time or call abandonment rates.
• Equipment placement: Determine the optimal placement for network assets such as antennas, base stations, routers, switches, and so on to maximize coverage, capacity, and service quality while minimizing deployment and overall operational costs.
• Pricing and plan structure: Structure broadband, mobile, and bundled service plans in a way that maximizes revenue and market share while meeting customer demand, competitive pressures, and regulatory constraints. 

Read the FICO case study with Eolo

Health & Life Sciences

• Medical resources allocation: Distribute limited resources, such as hospital beds, ventilators, staff, medical supplies, and so on across patients, departments, or regions in a way that maximizes patient outcomes and system efficiency while minimizing inequities.
• Organ transplant matching: Maximize life-saving transplants and survival outcomes by matching donors to patients based on medical compatibility, urgency, availability, geographic proximity, and fairness criteria.
• Clinical trials: Design, plan, and execute studies more efficiently by selecting the best trial sites, allocating patients across locations, coordinating dosing schedules, and sequencing trial phases to minimize time and cost while maximizing statistical power and patient safety​.

Read the FICO case study with Boeing and Karolinska University Hospital

Energy & Utilities

• Load balancing: Continuously match electricity supply with demand across the grid, ensuring that generation is dispatched and flows are routed in the most efficient way to minimize costs, reduce losses, and maintain system reliability.​
• Reservoir management: Determine how water, oil, or gas reservoirs are utilized over time by determining the optimal rates of extraction, storage, and release in order to maximize energy production, economic returns, and resource longevity while respecting environmental, safety, and regulatory constraints.
• Unit commitment: Determine which power generation units (e.g., thermal plants, hydro, renewables) should be turned on or off, and when, over a given planning horizon to meet forecasted electricity demand at the lowest cost while respecting operational constraints such as minimum up/down times, ramp rates, and reserve requirements.

A great real-life use case of optimization in the energy field is the partnership between FICO and Shell. Using FICO^® Xpress Optimization, Shell developed the Platform for Advanced Control and Estimation (PACE) and used it to solve complex problems across diverse asset types (such as Liquefied Natural Gas or Chemical). As a result, Shell saw notable improvements in asset value, including:

• Improved product quality: enhanced control led to higher standards in product output
• Increased margins: optimization of economic functions helped boost financial performance
• Reduced greenhouse gas emissions: more precise control contributed to sustainability goals
• Lower operational costs: Greater efficiency decreased overall costs

Read the full case study on the Shell and FICO Xpress partnership.

Communication & Retail

• Ad spend optimization: Allocate budgets across channels, campaigns, or audience segments in such a way that maximizes key performance indicators such as impressions, clicks, conversions, or return on ad spend (ROAS), while staying within budget, targeting, and timing constraints
• Price optimization: Determine the best prices for products across categories, stores, or customer segments in order to maximize revenue, profit, or market share while respecting constraints such as price and cross-price elasticities, competitor pricing, aggregate price level changes, inventory levels, and promotional strategies
• Inventory management: Determine the right levels of inventory to hold, replenish, and distribute across stores, warehouses, and channels in order to meet customer demand while minimizing holding costs, stockouts, and waste. Robust optimization practices are commonly used to control for uncertainty in forecasts and predictions used in the models

Read the FICO case study with Procter & Gamble

FICO's Solutions for Optimization

FICO^® Xpress Optimization products and services allow businesses to quickly and easily apply optimization techniques to solve their business problems, faster. Our deep portfolio of optimization options enables users to easily build, deploy and use optimization solutions that meet their needs. FICO^® Platform also provides advanced simulation and optimization capabilities.

• Test FICO Xpress for free for 60 days by requesting a free license
• Learn how Pepsico used Xpress to execute hundreds of large-scale optimization models daily
• Learn more about Xpress's continuous improvement in mixed integer programming speed
• Explore the enterprise optimization capabilities in FICO Platform