What is (Mathematical) Optimization?

Document created by neillcrossley@fico.com on Aug 17, 2017Last modified by neillcrossley@fico.com on Sep 12, 2017
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At FICO we consider Optimization as the mathematical process of finding the best decision (usually highest profit, or lowest cost) for a given business problem within a defined set of constraints.


We take known & unknown (forecasts or predictions) input variables, including defined constraints and objectives, and use Optimization solvers and algorithms to find the best solutions (that maximize or minimize) the objective while honoring the constraints). Often there a many possible solutions and comparative scenario analysis is undertaken to identify which of the possible solutions provides the best overall results.

Opt Graphic.PNG


Alternatively we can consider the Wikipedia definition for Mathematical Optimization:

" In mathematics, computer science and operations research, mathematical optimization is the selection of a best element (with regard to some criterion) from some set of available alternatives. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains."


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. Standard capabilities include scalable high-performance solvers and algorithms, flexible modeling environments, rapid application development, comparative scenario analysis and reporting capabilities, for on-premises and cloud installations.


Why use optimization?

Optimization is a rigorous approach which takes into account all of the factors that influence decisions in business. Optimization implies careful modeling of the business, a process which itself invariably gives valuable insights. Benefits include operational efficiency, revenue maximization, cost minimization, performance assessment and understanding the effects of changes in input data.


Key optimization considerations include:

  • Decision variables. These are the things we can change, the things we need to decide upon. For example, how much product will we make? Where? How should we make it? How should we transport it?
  • Constraints. These state the limitations on our decisions. For example, in a logistics problem each mode of transport has a maximum speed and maximum payload. Operations may be limited to so many hours in a day.

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 need optimization. In addition, as many problems change over time, particularly data, optimization is required on an ongoing basis.



Optimization has a track record across a wide range of industries and commerce. Organizations use optimization to improve the quality of their decision making. Optimization functionality is a logical extension to many software products, making them more valuable to their users. In addition, optimization is a must-have tool for consultants engaged in any aspect of business performance or business process improvement. Why not take a look at our Solutions pages to see the range of industries and business problems that we regular use optimization to improve.



These are the main business benefits of optimization:

  • 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 making a product of certain characteristics from the cheapest materials, all without violating the constraints we are under. We can use optimization to increase revenue, i.e., produce the maximum possible from a particular operation, carry as much as possible from one point to another or provide the highest level of insurance cover. We can construct a portfolio with characteristics as close as possible to target characteristics; we can maximize the number of transactions between a given set of sell and buy bids.
  • 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. 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. Further insight is gained when an optimization model of a process or situation is created; making the model is instructive as is performing what-if analyses. Once an optimization model is built, it can be used for what-if analysis, for instance, by modeling the impact of a new opportunity, plant, ingredient, or process.
  • Sensitivities: How does the overall profit change if various data items change? For example, in a production situation, how much does the unit cost of item I at Factory A have to change before we switch production to Factory G?


FICO Xpress Optimization History & Facts

  • 36 years of experience in modeling and optimization
  • 33 years of experience in mixed integer optimization
  • 20 years of experience in nonlinear optimization
  • 17 years Xpress-Mosel, modeling and solving environment
  • 17 years of FICO Decision Optimization - applying optimization to improve decision strategies


FICO® Xpress Optimization Innovations


1983: LP solver running on PCs

1992: Parallel MIP (1997 on distributed PC/Linux networks)

1995/1996: Commercial branch and cut algorithm

1998: Bound switching in dual simplex

2003: Lift-and-project cuts

2009: Parallel MIP heuristics

2010: LP/MIP solver crosses 64-bit coefficient indexing threshold

2013: Tight integration of different NLP technologies

2013:  Automatic solver selection for NLP

2014:  Parallel simplex

2016: Task based parallel MIP

2017:  barrier warm start and parallel crossover parallel black box optimization


1983: General purpose algebraic modeling language (mp-model)

2001: Algebraic modeling language combining modeling, solving, and programming (Mosel)

2005: Profiler and debugger for a modeling language

2005: User-controlled parallelism at the model level

2010: Algebraic modeling language supporting distributed computing

2012: Mosel remote launcher

2014: Parallel profiler and debugger; robust   optimization

2017: Cloud based optimization model and solution development