Mathematical Optimization & Operations Research – From Science to Investment Decisions

Scientific foundations, mathematical modeling, algorithms, and optimization methods for business, research, and public institutions.

Complex investment and resource decisions are mathematical problems. Whether companies are optimizing their CAPEX portfolios, public institutions are allocating limited budgetary resources, or scientists are examining combinatorial decision spaces, the focus is always on the same fundamental question.

Which combination of decisions achieves the best possible outcome given specific objectives and constraints?

The answer requires more than data analysis, project prioritization, or the calculation of individual metrics. It requires mathematical modeling, systematic optimization, and a transparent evaluation of viable alternatives.

This page connects operations research, decision science, applied mathematics, and computational science with concrete application areas in business management and public investment planning.

From mathematical theory to a calculable decision.

Discover the scientific foundations →

01 | Management, Finance & Investments

CAPEX Managers & Investment Directors

Capital Allocation, Capital Budgeting, and Mathematical CAPEX Optimization

CAPEX managers face the challenge of allocating limited investment budgets across numerous competing projects. In doing so, they must take into account financial returns, strategic goals, resource constraints, and time dependencies.

Traditional project prioritization first answers the question of which individual projects appear particularly attractive. Mathematical portfolio optimization answers a more advanced question:

Which combination of all available projects maximizes economic benefit within the given CAPEX budget?

Relevant Fields:

• Capital Allocation Optimization
• Capital Budgeting
• Investment Portfolio Optimization
• Mixed-Integer Programming (MIP/MILP)
• Multi-Period Investment Planning
• Resource-Constrained Optimization
• Scenario Optimization
• Capital Expenditure Management

A mathematical optimization model can combine projects based on their investment costs, expected benefits, and interdependencies. Mandatory measures can be taken into account, as can budget limits or capacity constraints.

This shifts the focus from evaluating individual investments to optimizing the entire investment portfolio.

CFOs & Finance Executives

Corporate Finance, Financial Modeling, and Optimal Capital Allocation

For CFOs, the allocation of scarce financial resources is a key management task. Investment decisions influence return on investment, liquidity, risk, strategic flexibility, and long-term competitive position.

Mathematical optimization makes it possible to examine different uses of capital within a single decision-making model.

Relevant fields:

• Corporate Finance
• Financial Engineering
• Strategic Financial Planning
• Financial Modeling
• Risk-Adjusted Optimization
• Investment Appraisal
• Real Options Analysis
• Sensitivity Analysis

Economic metrics such as net present value (NPV), expected cash flow, or risk-adjusted benefit can be included in the objective function. Multi-period models also take into account different cash flows, maturities, and capital commitments.

The result is a systematic framework for determining how existing capital can be deployed as effectively as possible under defined assumptions.

Portfolio & Program Managers

Project Portfolio Optimization Instead of Isolated Project Prioritization

In traditional project portfolio management, projects are evaluated, categorized, and prioritized. However, the ranking of individual projects is not necessarily identical to the optimal overall portfolio.

For example, a high-priority project may consume so many resources that several other projects—which, taken together, would be more economically attractive—cannot be implemented.

Relevant Fields:

• Project Portfolio Management (PPM)
• Project Portfolio Optimization (PPO)
• Project Selection Problem (PSP)
• Resource-Constrained Project Scheduling (RCPSP)
• Multi-Project Scheduling
• Strategic Portfolio Management (SPM)
• Dependency Management
• Portfolio Balancing

Mathematical optimization makes it possible to analyze combinations of projects under shared constraints. This approach not only manages the project portfolio but also treats it as a cohesive optimization problem.

Strategic Business Planners

Strategic Planning, Scenario Analysis, and Multi-Period Optimization

Strategic business planning requires decisions spanning multiple years. Investments, capacities, locations, technologies, and organizational resources are all interrelated.

A decision made in the current planning period can either enable or preclude future options.

Relevant fields:

• Strategic Resource Allocation
• Multi-Period Optimization
• Dynamic Programming
• Scenario Analysis
• Strategic Portfolio Optimization
• Long-Term Investment Planning
• Robust Optimization

Mathematical models can capture temporal relationships and compare different strategic development paths. The combination of scenario analysis and optimization is particularly relevant: Not only are the consequences of a given strategy calculated, but a suitable decision alternative can be identified for each scenario.

Asset & Resource Managers

Resource Allocation, Capacity Planning, and Asset Optimization

Companies have limited resources: capital, personnel, machinery, equipment, time, and technical capacity. These resources often need to be allocated simultaneously to multiple projects and business units.

Relevant Fields:

• Resource Allocation Optimization
• Capacity Planning
• Asset Management Optimization
• Maintenance Optimization
• Workforce Optimization
• Network Optimization
• Scheduling Optimization

A mathematical optimization model takes into account available capacities and competing demands. This allows resource conflicts to be systematically identified and permissible allocations to be calculated.

02 | Public Sector & Infrastructure

Political and Public Decision-Makers

Public Resource Allocation and Optimization of Social Impact

Public institutions make investment decisions under specific conditions. In addition to budget and resource constraints, legal obligations, political priorities, earmarked funds, and social impact goals must be taken into account.

Unlike private-sector investments, the focus is often not on monetary returns. Instead, public measures can be evaluated based on defined utility criteria.

Relevant Fields:

• Public Decision Making
• Welfare Economics
• Multi-Criteria Decision Analysis (MCDA)
• Public Investment Management
• Social Welfare Optimization
• Public Resource Allocation

Mathematical optimization can help determine the combination of public policies that achieves the highest modeled total utility under given criteria.

The political decision regarding goals and priorities remains with the democratically elected decision-maker. Mathematics optimizes within these parameters.

Budget and Finance Officials

Public Budget Optimization and Multi-Year Budget Planning

Public budgets are multi-period allocation systems with competing demands. Municipalities, states, and federal institutions must allocate financial resources among mandatory tasks, investments, infrastructure, and policy programs.

Relevant Fields:

• Public Finance
• Budget Allocation Optimization
• Multi-Period Budget Planning
• Cost-Effectiveness Analysis
• Constraint-Based Budgeting
• Fiscal Planning
• Multi-Objective Optimization

A mathematical model can take into account available funds, legal obligations, project dependencies, and defined impact criteria. This makes it possible to mathematically compare different budget combinations.

Infrastructure & Investment Planners

Infrastructure Optimization and Long-Term Resource Allocation

Infrastructure investments are characterized by high capital commitment, long useful lives, and complex dependencies. Roads, bridges, schools, administrative buildings, energy facilities, and utility networks compete for limited public funds.

Relevant Fields:

• Infrastructure Asset Management
• Infrastructure Portfolio Optimization
• Network Optimization
• Reliability Engineering
• Maintenance Scheduling
• Lifecycle Optimization
• Multi-Period Capital Planning

Mathematical optimization can help combine investment and maintenance measures while taking into account budget, urgency, dependencies, and impact.

Administrative Modernization & Controlling

Analytical Decision Support for Public Institutions

The digitization of public administration generates large volumes of data. However, the mere availability of data does not in itself determine which measures should be implemented.

Relevant Fields:

• Public Sector Analytics
• Decision Support Systems
• Performance Management
• Multi-Criteria Decision Making
• Mathematical Modeling
• Prescriptive Analytics
• Scenario Planning

A scientifically grounded optimization approach can integrate administrative data, evaluation criteria, and institutional constraints into a unified decision-making model.

03 | Mathematics & Operations Research

Mathematicians & Applied Mathematicians

Applied Mathematics, Mathematical Programming, and Discrete Optimization

Mathematical optimization examines the determination of optimal solutions within defined decision spaces. An optimization problem fundamentally consists of decision variables, an objective function, and constraints.

A general formulation is:

Maximize:

max f(x)

Subject to the constraints:

g_i(x) ≤ 0, i = 1, ..., m

h_j(x) = 0, j = 1, ..., p

Depending on the problem, the decision variables can be continuous, integer, or binary.

Binary variables are frequently used when selecting investment projects:

x_i ∈ {0,1}

• x_i = 1: Project is selected.
• x_i = 0: The project is not selected.

A simple portfolio optimization problem can be formulated as a binary optimization problem:

Objective function:

max Σ v_i · x_i

Budget constraint:

Σ c_i · x_i ≤ B

Here, v_i denotes the utility, c_i the investment costs, and B the available budget.

This basic model is known as the 0-1 knapsack problem. Real-world investment portfolios may also involve dependencies, multiple resources, time constraints, and competing objectives.

Relevant fields:

• Applied Mathematics
• Discrete Mathematics
• Integer Programming
• Linear Programming
• Nonlinear Optimization
• Combinatorial Optimization
• Global Optimization
• Mathematical Economics

Operations Research Specialists

Operations Research as the Scientific Foundation for Complex Allocation Decisions

Operations Research deals with the mathematical analysis and optimization of complex decision-making problems. The discipline combines mathematical modeling, algorithmic solution methods, statistics, and systematic decision analysis.

Typical problem classes:

• Knapsack Problems
• Generalized Assignment Problems
• Set covering problems
• Facility Location Problems
• Scheduling Problems
• Network Flow Problems
• Resource Allocation Problems
• Portfolio Selection Problems

In combinatorial problems, the number of possible solutions often grows very rapidly.

For n independently selectable binary projects, there are initially 2ⁿpossible selection combinations.

For 100 projects, this corresponds to:

²,¹⁰⁰ ≈ 1.27 ×^10³⁰ theoretical combinations.

However, modern optimization methods do not need to examine all combinations individually. Methods such as branch-and-bound, branch-and-cut, presolve, and mathematical relaxations can eliminate large portions of the search space.

What matters is not exhaustive enumeration, but the mathematically grounded search for optimal or demonstrably sufficiently good solutions.

The actual computational complexity depends on the model structure, constraints, data, and the algorithms used.

Optimization Engineers

Translating Mathematical Models into Powerful Optimization Systems

Optimization engineering combines mathematical optimization with software development and practical problem-solving. Real-world decision problems are translated into formally defined models and solved using appropriate methods.

Relevant fields:

• Mathematical Programming
• Constraint Programming
• Solver Engineering
• Model Formulation
• Algorithm Selection
• Decomposition Methods
• Optimization Software Architecture
• Computational Performance Analysis

Key tasks include formulating objective functions, modeling constraints, selecting suitable algorithms, and evaluating solution quality.

Mathematical modeling is just as important as computational power. A powerful algorithm can only optimize the problem that has actually been formulated in the model.

Numerical Analysts

Numerical Analysis, Stability, and Mathematical Computation Methods

Numerical analysis examines the computational solution of mathematical problems, taking into account accuracy, stability, and efficiency.

Relevant fields:

• Numerical Stability
• Floating-Point Arithmetic
• Numerical Conditioning
• Convergence Analysis
• Approximation Methods
• Error Analysis
• Sparse Matrix Computation
• Numerical Linear Algebra

Numerical methods are particularly important for continuous and nonlinear optimization problems. Even in mixed-integer optimization problems, numerical properties, scaling, and tolerances can influence the reliability of the computation.

It is therefore essential to distinguish between mathematical optimality, numerical accuracy, and practical solution quality.

04 | Data Science, Systems Analysis & Technology

Data Scientists & Decision Scientists

From Predictive Analytics to Prescriptive Analytics

Data science examines data, identifies patterns, and develops statistical or algorithmic models. Decision science focuses on the systematic improvement of decision-making.

The distinction between three analytical levels is particularly relevant:

• Descriptive Analytics: What happened?
• Predictive Analytics: What is likely to happen?
• Prescriptive Analytics: What decision should be made given specific goals and constraints?

Prescriptive analytics combines forecasts, mathematical models, and optimization methods.

For example, a forecasting model can estimate the expected economic benefits of various investment projects. An optimization model then calculates which combination of projects should be selected given budget and resource constraints.

Relevant fields:

• Data Science
• Decision Science
• Machine Learning
• Predictive Analytics
• Prescriptive Analytics
• Bayesian Decision Theory
• Multi-Criteria Decision Analysis
• Decision Support Systems

Systems Analysts & Systems Engineers

Mathematically Modeling and Optimizing Complex Systems

Systems analysis examines the structure, relationships, and interactions of complex systems. Systems engineering deals with the systematic development and integration of technical and organizational systems.

Numerous interactions can occur in investment portfolios:

• Projects require shared resources.
• Measures are temporally dependent on one another.
• Individual investments are absolutely necessary.
• Projects are mutually exclusive.
• Investment decisions influence future capacities.

Such relationships can be modeled mathematically as constraints.

Relevant fields:

• Systems Engineering
• Systems Analysis
• Systems Optimization
• Constraint Modeling
• Model-Based Systems Engineering
• Complex Systems
• Simulation Modeling
• Requirements Engineering

The quality of an optimization result depends largely on whether these system relationships have been modeled completely and correctly.

Computational Scientists

Scientific Computing and High-Performance Computing

Computational science combines mathematical models with computer-aided computational methods. For large-scale optimization problems, algorithmic efficiency, memory requirements, and computing architecture become key factors.

Relevant Fields:

• Scientific Computing
• Computational Mathematics
• High-Performance Computing
• Parallel Computing
• Numerical Optimization
• Computational Complexity
• Performance Engineering

Parallel computing can distribute computations across multiple processor cores. Whether and to what extent this results in a speed advantage depends on the algorithm, the problem structure, and the portion of the computation that can be parallelized.

For interactive optimization systems, the ability to quickly re-evaluate changed constraints or budgets is particularly important.

Algorithm Engineers

Algorithm Design, Complexity Theory, and Combinatorial Search Methods

Algorithm engineering focuses on the development, implementation, and experimental evaluation of high-performance algorithms.

For mathematical optimization problems, the following questions are central, among others:

• How is the search space structured?
• Which subspaces can be mathematically ruled out?
• What bounds can be computed?
• When can a solution be proven to be optimal?
• Which approximation methods are appropriate for large problems?
• How does the runtime change as the problem size increases?

Relevant fields:

• Algorithm Design
• Computational Complexity Theory
• Branch-and-Bound
• Branch-and-Cut
• Dynamic Programming
• Heuristic Search
• Metaheuristics
• Approximation Algorithms
• Parallel Algorithms

A key quality criterion for mathematical optimization methods is the distinction between a found feasible solution and a solution that is proven to be optimal.

For difficult problems, the so-called optimality gap can also be used to assess the quality of a found solution based on mathematical bounds.

05 | Common Scientific Foundations

From the Problem Statement to the Mathematical Decision Model

Regardless of whether an investment manager, a mathematician, or a public budget official is examining a problem, mathematical optimization follows a common basic structure.

Step 1: Define decision variables

What decisions can be made? For example, selecting, postponing, or rejecting individual investment projects.

Step 2: Formulate the objective function

Which outcome should be maximized or minimized?

• Net Present Value
• Economic Return
• Social benefit
• Security of supply
• Resource efficiency
• Risk Minimization

Step 3: Model Constraints

What conditions must be strictly adhered to? These include budget constraints, resource availability, technical dependencies, legal obligations, and time constraints.

Step 4: Select a Solution Method

Depending on the mathematical structure, linear and integer optimization, constraint programming, dynamic programming, or heuristic methods may be appropriate.

Step 5: Validate Results

The calculated solution must be checked for feasibility, model assumptions, sensitivity, and economic or institutional plausibility.

Step 6: Simulate Decisions

Changes in budgets, priorities, or constraints can lead to a new optimal solution. This makes mathematical optimization a tool for interactive decision support.

06 | Scientific Optimization in Practice: StratePlan

StratePlan is the technology developed by mAInthink for the mathematical optimization of complex investment and project portfolios.

The approach combines combinatorial optimization with the systematic consideration of budget, resource, and project dependencies.

The focus is not merely on evaluating individual projects, but on calculating economically or strategically suitable combinations of projects.

The underlying scientific disciplines include, in particular:

• Operations Research
• Combinatorial Optimization
• Mathematical Programming
• Decision Science
• Prescriptive Analytics
• Resource Allocation Optimization
• Computational Science

For businesses, economic metrics such as NPV or other defined targets can be used. For public institutions, weighted benefit and impact criteria can be modeled instead.

The objective function and constraints are tailored to the specific use case.

Same projects. Same budget. Better combinations. Better results.

Mathematical optimization does not replace management’s responsibility or political decision-making authority. It expands the basis for decision-making by providing systematically calculated alternatives.

Learn more about StratePlan →

07 | Frequently Asked Questions

What is the difference between Operations Research and Data Science?

Operations Research focuses on the mathematical optimization of decisions subject to constraints. Data Science deals more broadly with deriving insights from data, including statistical analysis and forecasting. Both disciplines complement each other, particularly in the field of prescriptive analytics.

Is mathematical optimization the same as artificial intelligence?

No. Mathematical optimization is a distinct field of research with a long scientific tradition. It can be combined with artificial intelligence methods, but it is not the same as machine learning.

Can the global optimum always be computed?

Not every real-world optimization problem can be proven to have a globally optimal solution within practical computation times. Computability depends on the problem structure, model size, mathematical properties, and available computational resources. With suitable models and solution methods, global optimality can be mathematically proven.

Why is traditional project prioritization often insufficient?

A ranking evaluates projects individually. Portfolio optimization additionally takes into account the interactions between projects as well as shared budget and resource constraints. As a result, a different combination of projects can achieve a higher overall benefit.

What does combinatorial optimization mean?

Combinatorial optimization examines the selection or arrangement of discrete decision alternatives. Typical applications include project portfolio selection, resource planning, production control, and network optimization.

What role does numerical analysis play?

Numerical analysis examines the accuracy, stability, and efficiency of computational mathematical methods. It is particularly relevant for the reliable implementation and evaluation of numerical optimization algorithms.

Can public budgets be mathematically optimized?

Yes, provided that objectives, evaluation criteria, and constraints can be appropriately formalized. Legal obligations, earmarked funds, and political priorities must be explicitly taken into account. Mathematical optimization calculates solutions within these parameters but does not replace democratic decision-making.

What is the difference between simulation and optimization?

A simulation examines the effects of a given decision or scenario. Optimization searches within a defined decision space for the best permissible solution according to an objective function. Both methods can be combined.

08 | From Science to Better Decision-Making

Operations research, mathematical modeling, decision science, and computational science form the scientific foundations of modern optimization systems.

Their common goal is to formally describe complex decision-making problems, systematically examine feasible alternatives, and transparently improve the quality of decisions.

• For mathematicians, the focus is on model and solution quality.
• For technical analysts, algorithms, computing power, and validation are crucial.
• For CAPEX managers and CFOs, the economic impact is what counts.
• For public institutions, the priority is the most effective use of limited resources.

Different perspectives. Shared mathematical foundations. Quantifiable decision alternatives.

StratePlan applies these principles to complex investment and project portfolios.

Understand mathematical optimization. Calculate economic potential.

Discover StratePlan now →