Operations Research: How Mathematics Optimizes Complex Decisions

From Scientific Decision Analysis to Combinatorial Optimization to Modern Decision Intelligence

Feature Article | mAInthink Insights | Science & Methods

Companies, public institutions, and investors face a fundamental challenge: They must make decisions whose economic or societal impact depends not solely on individual measures, but on the interplay of numerous alternatives.

Which investments should be funded? Which projects collectively generate the highest value contribution? How can limited resources be allocated optimally? And how does the best decision change when budgets, priorities, or framework conditions are adjusted?

The traditional approach often involves evaluations, prioritization lists, business cases, and management decisions.

Operations Research takes a different approach:

It does not merely evaluate individual alternatives but mathematically determines the best possible combination of all permissible decisions.

For decades, this discipline has provided a scientific foundation for optimizing complex economic, technical, and organizational systems.

Today, it is gaining new practical significance through powerful optimization algorithms, parallel computing methods, artificial intelligence, and interactive decision-making platforms.

Table of Contents

1. What Is Operations Research?
2. The Scientific Development of Operations Research
3. What Problems Does Operations Research Solve?
4. The Mathematical Foundations
5. Combinatorial Optimization and Exponential Complexity
6. The Most Important Optimization Methods
7. Global Optimality: What Can Be Mathematically Proven
8. Portfolio Optimization Instead of Traditional Prioritization
9. Multi-period planning and resource constraints
10. Multi-Objective Optimization and Competing Objectives
11. Operations Research Under Uncertainty
12. Operations Research, AI, and Decision Intelligence
13. Operations Research in Business and Public Administration
14. Limitations and Prerequisites of Mathematical Optimization
15. From Mathematical Solutions to Real-World Decisions
16. Frequently Asked Questions
17. Conclusion and Scientific Context

1. What is Operations Research?

Operations Research, often referred to as OR or, in the British English-speaking world, as Operational Research, is an interdisciplinary scientific field dedicated to the analysis and optimization of complex decision-making problems.

In particular, it combines:

• Applied Mathematics
• Mathematical optimization
• Probability theory and statistics
• Computer Science and Algorithm Theory
• Economics
• Systems Analysis and Decision Sciences

The central goal is to model decision-making problems in such a way that different courses of action can be systematically examined and optimized based on defined objective functions.

Unlike purely descriptive analytical methods, operations research does not merely answer the question of what happens in a system.

It examines which decisions should be made under the given conditions to best achieve a defined objective.

For example, a company may know which of its investment projects are expected to yield a positive return on investment.

However, this does not necessarily indicate which combination of these projects will generate the highest economic value within a limited overall budget.

This is precisely where a classic operations research problem begins.

2. The Scientific Development of Operations Research

The modern development of operations research began primarily during World War II.

At that time, scientists from various disciplines investigated complex operational problems, such as the efficient use of limited resources, the coordination of logistical processes, and the optimization of military systems.

After the war, the methods developed were increasingly applied to industrial and economic problems.

A major milestone was the development of the Simplex method by George B. Dantzig in 1947.

This made linear optimization an important tool for mathematical decision support.

In the decades that followed, other significant methods emerged:

• Dynamic programming
• Integer and mixed-integer optimization
• Branch-and-bound
• Network optimization
• Stochastic optimization
• Multi-objective optimization
• Metaheuristic Methods
• Robust Optimization

Today, operations research is an internationally established research discipline.

Important scientific organizations include INFORMS, EURO, IFORS, and the German Society for Operations Research (GOR).

Operations Research is therefore not a new manifestation of artificial intelligence, but rather an independent scientific foundation of modern decision optimization.

3. What problems does Operations Research solve?

Operations Research becomes relevant wherever multiple decision alternatives, limited resources, and definable objective functions intersect.

The fields of application are correspondingly broad.

Investment and Capital Allocation

Which investments should be made within a limited budget?

Project Portfolio Optimization

Which combination of projects generates the highest overall economic or strategic contribution?

Production Planning

How can production volumes, machine capacities, and order sequences be optimized?

Supply Chain Optimization

How can inventory levels, transportation routes, delivery capacities, and production sites be efficiently coordinated?

Energy and Infrastructure Planning

Given limited resources, which investments in networks, facilities, or maintenance measures yield the greatest benefits?

Workforce and Resource Planning

How can employees, machinery, time slots, or other scarce resources be optimally allocated?

Public Budgets

Given the financial, legal, and political framework, which combination of public policies yields the greatest defined societal impact?

Healthcare

How can capacities, treatment appointments, medical resources, and care structures be improved?

Despite their different fields of application, many of these problems share a common mathematical structure:

A decision that is as good as possible—or optimal—must be found within a limited, permissible decision space.

4. The Mathematical Foundations

A mathematical optimization problem fundamentally consists of three elements:

decision variables, an objective function, and constraints.

4.1 Decision Variables

Decision variables describe which decisions can be made.

For a binary project decision, for example:

Where:

• _xi=1: Project i is selected.
• _{xi = 0}: Project i is not selected.

Other optimization problems use integer or continuous variables.

4.2 Objective Function

The objective function describes which quantity is to be maximized or minimized.

For a portfolio with additive project values, a simplified maximization function is:

Here, _vi denotes the economic or strategic value contribution of a project.

4.3 Constraints

Real-world decisions are subject to constraints.

A typical budget constraint is:

Where:

• _ci: Capital requirement for project i
• B: Total available budget

Additional constraints may, for example, reflect human resources, technical dependencies, legal requirements, or time constraints.

The complete optimization problem thus consists of finding a feasible combination of decision variables that best satisfies the objective function.

This model structure forms the basis for numerous practical operations research applications.

5. Combinatorial Optimization and Exponential Complexity

A particularly important subfield of operations research is combinatorial optimization.

It examines decision problems in which the best feasible solution is sought from a finite, often very large set of discrete alternatives.

For n independently selectable binary projects, the number of theoretically possible combinations is:

This implies:

These numbers initially describe the total number of possible binary combinations, not the number of actually feasible portfolios.

Budget constraints, project dependencies, and other constraints can significantly reduce the feasible solution space.

Nevertheless, the theoretical number of possible combinations illustrates the magnitude of the problem.

A complete manual evaluation of all alternatives is practically impossible even for relatively small portfolios.

Exponential complexity does not mean that every combination must be checked individually

Modern optimization algorithms can rule out large portions of the decision space without fully evaluating every single combination.

To do this, they use mathematical bounds, relaxations, structural properties of the problem, and intelligent search strategies.

Therefore, under suitable conditions, an optimization method can efficiently solve even very large combinatorial decision problems.

However, the existence of powerful algorithms does not guarantee that any given problem can be solved globally optimally within a short time.

The actual computational complexity depends, among other things, on the model structure, the constraints, the problem size, and the solution method used.

6. The Most Important Optimization Methods

Operations Research encompasses various mathematical solution methods.

Which method is appropriate depends on the structure of the decision problem.

6.1 Linear Programming

Linear optimization addresses problems with linear objective functions and linear constraints.

It is used, for example, in production, transportation, and resource planning.

6.2 Integer Programming

Integer optimization takes into account decision variables that can only take integer values.

This is particularly relevant for discrete decisions, such as when projects are either fully selected or excluded.

6.3 Mixed-Integer Linear Programming

Mixed-Integer Linear Programming (MILP) combines continuous and integer decision variables within a linear model.

This allows, for example, investment decisions to be linked to continuous budget or capacity variables.

6.4 Branch-and-Bound

Branch-and-Bound systematically decomposes the decision space into subproblems.

Using mathematical bounds, it is possible to determine which subspaces cannot contain a better solution.

These subspaces then do not need to be investigated further.

6.5 Branch-and-Cut

Branch-and-Cut extends Branch-and-Bound by introducing additional valid inequalities, known as cutting planes.

These can tighten the mathematical relaxations and speed up the search for optimal solutions.

6.6 Dynamic Programming

Dynamic programming breaks down suitable problems into interrelated subproblems.

It is used, among other things, in sequential decision-making problems, inventory planning, and certain resource allocation problems.

6.7 Heuristics and Metaheuristics

Heuristic methods efficiently search for good, feasible solutions without, in principle, guaranteeing a proof of global optimality.

Well-known examples include:

• Genetic Algorithms
• Simulated Annealing
• Tabu Search
• Local Search

These methods can be particularly useful for large or complex problems.

6.8 Hybrid Optimization Methods

Hybrid approaches combine different methods.

For example, heuristics can initially generate high-quality feasible solutions, while exact methods then attempt to improve these solutions or prove their optimality.

Machine learning can also be used to select search strategies, predict parameters, or accelerate individual solution steps.

The actual benefits of such combinations must be investigated using appropriate benchmarks.

7. Global Optimality: What Can Be Mathematically Proven

A key strength of mathematical optimization is that, under certain conditions, not only can a good solution be found, but its global optimality can also be proven.

A globally optimal solution has the best possible objective function value within the defined feasible solution space.

Formally, for a maximization problem, the following holds:

Where:

• X: the set of all feasible solutions
• x^*: globally optimal solution
• f(x): objective function value of a solution

Global optimality can be proven, for example, using mathematical bounds.

The Optimality Gap

For mixed-integer optimization problems, the difference between the best feasible solution found and a valid bound on the global optimum is often considered.

For a maximization problem with a positive best objective function value, a relative optimality gap can be defined, for example, as follows:

Where:

• LB: value of the best feasible solution found so far
• UB: valid upper bound for the global optimum

A gap of zero, when calculated exactly, means that global optimality has been proven.

Practical solvers take numerical tolerances and defined termination criteria into account.

A small gap can also indicate that the solution found is already very close to the theoretically best possible objective function value.

A crucial scientific limitation

A global optimum is always an optimum of the mathematical model—not necessarily the objectively best decision in the real world.

If the input data is incomplete, utility assumptions are incorrect, or relevant constraints are not taken into account, a mathematically optimal solution may be economically unsuitable.

Therefore, model validation, sensitivity analyses, and the verification of underlying assumptions are essential components of a rigorous optimization process.

8. Portfolio Optimization Instead of Traditional Prioritization

A particularly relevant area of application is the optimization of investment and project portfolios.

Traditional portfolio methods often evaluate projects individually.

This results in a ranking, for example, based on ROI, NPV, strategic importance, or weighted evaluation criteria.

Such prioritization answers the question:

Which individual project appears particularly attractive?

However, it does not necessarily answer:

Which combination of all available projects generates the highest total value given the existing constraints?

A simplified example

A company has an investment budget of 10 million euros.

There are four projects to choose from:

When prioritizing based on absolute NPV, Project A ranks first.

If A is selected, the remaining budget of 4 million euros can be used to fund either Project C or Project D.

The combination of A and C achieves a total NPV of 16 million euros with a budget expenditure of 10 million euros and is the best permissible combination in this example.

The equally valid combination B + C achieves an NPV of 15 million euros with a capital outlay of 9 million euros.

In this example, A + C is indeed optimal.

However, this does not mean that the ranking is necessarily incorrect.

Rather, the example shows that the optimal portfolio decision arises from comparing permissible combinations and does not follow solely from the isolated evaluation of individual projects.

Even small changes in project values, budgets, or dependencies can result in the best combination no longer consisting of the highest-ranked individual projects.

Why Maximizing ROI Requires Special Care

Maximizing total NPV is not the same as maximizing portfolio ROI.

If portfolio ROI is defined as the ratio of aggregate net income to aggregate capital employed, a different objective function emerges:

Mathematically, this type of ratio optimization differs from the maximization of an additive total value.

Furthermore, pure ROI maximization may favor very small portfolios.

Therefore, before any optimization, it must be clarified whether, for example, the total NPV, the ROI, the strategic benefit, or another objective should be maximized.

Defining the objective function is itself a critical management decision.

9. Multi-period Planning and Resource Constraints

Real-world investment decisions often span multiple fiscal years.

For example, a project may require high investments in the first year, tie up additional resources in the second year, and only generate economic returns later on.

An appropriate mathematical model must take this temporal structure into account.

For a simplified multi-period portfolio selection, the following may apply:

subject to:

Where:

• _cit: capital requirement of project i in period t
• _Bt: available budget in period t

In addition, resource constraints can be modeled:

Here, _airt denotes a project’s resource requirement for resource r in period t.

Project Dependencies

Logical relationships between projects can also be expressed mathematically.

If Project B may only be carried out if Project A has been selected:

If two projects cannot be carried out simultaneously:

If a project must be implemented:

Such constraints are essential because the actual quality of an investment portfolio is often determined not only by the budget but also by numerous technical, organizational, and strategic dependencies.

10. Multi-Objective Optimization and Competing Objectives

Many real-world decision-making problems involve more than a single objective.

For example, a company may want to simultaneously:

• Maximize economic value contribution
• Limit risks
• Reduce CO₂ emissions
• Achieve strategic goals
• Avoid resource constraints

These objectives may compete with one another.

Multi-objective optimization therefore examines decisions with multiple objective functions.

Formally, such a problem can be represented as:

subject to:

Weighted Objective Functions

One approach is to combine multiple normalized objective functions into a weighted overall function:

Here, _wj represents the weight of the respective objective function.

The weights must be justified based on the subject matter.

Pareto Optimality

Another approach is to determine Pareto-optimal solutions.

A solution is Pareto-optimal if no other feasible solution is at least as good across all objective measures and strictly better in at least one objective measure.

This often results not in a single, unambiguously best decision, but in a set of efficient alternatives.

Management must then decide which trade-offs are acceptable.

Operations research does not, therefore, replace strategic responsibility.

It makes the consequences of different preferences mathematically analyzable.

11. Operations Research Under Uncertainty

Not every relevant decision variable is known at the time of decision-making.

Future revenues, project costs, resource availability, and market conditions may be uncertain.

The classical deterministic optimization model operates with fixed parameters.

Additional methodological approaches exist for decision-making problems involving uncertainty.

Stochastic Optimization

Stochastic optimization takes random variables and, where applicable, different future scenarios into account.

For example, expected values, risk measures, or multi-stage decisions can be taken into account.

Robust optimization

Robust optimization seeks solutions that remain valid or perform well even under defined adverse parameter deviations.

Scenario Analysis

Scenario analysis examines how optimal decisions change under different assumptions.

Sensitivity Analysis

Sensitivity analysis examines how sensitive the results are to changes in individual input parameters.

These methods are particularly relevant for long-term investment decisions.

A mathematically optimized portfolio should therefore not be evaluated solely on the basis of its performance under a single future scenario.

Equally important is the question of how stable the decision remains under plausible changes.

12. Operations Research, Artificial Intelligence, and Decision Intelligence

Operations Research, Artificial Intelligence, and Decision Intelligence are related but not identical.

Operations Research

Operations Research provides mathematical methods for modeling and optimizing decision-making problems.

Artificial Intelligence

Artificial Intelligence encompasses various techniques, such as machine learning, pattern recognition, natural language processing, and automated decision-making processes.

Decision Intelligence

Decision Intelligence integrates data, decision models, analytical methods, and organizational decision-making processes.

The goal is to improve the quality, transparency, and practical feasibility of decisions.

The Connection

One possible interaction involves machine learning first forecasting future project costs or revenues.

An operations research model can then calculate which combination of projects is optimal under the forecasted conditions.

Finally, a Decision Intelligence application presents the results in a way that allows decision-makers to compare alternatives, adjust assumptions, and understand the consequences.

The three approaches thus fulfill different, complementary functions.

Artificial intelligence can help improve the basis for decision-making. Operations research can mathematically optimize appropriate decisions. Decision intelligence can integrate these insights into real-world decision-making processes.

This division of roles is a possible architectural model, not a universally applicable distinction between all methods.

13. Operations Research in Business and Public Administration

The mathematical foundations of operations research are relevant to both private-sector and public-sector decision-making.

13.1 Industrial Companies

Industrial companies often have to coordinate numerous investment projects across locations, production lines, and business units.

Individual projects compete for capital, personnel, and technical capacity.

Mathematical portfolio optimization can determine which combination generates the highest defined total value under these conditions.

13.2 Energy Companies

Energy companies face complex investment decisions regarding grids, generation facilities, maintenance, and infrastructure.

In doing so, they must take into account technical dependencies, supply reliability, resources, and long-term planning horizons.

13.3 Pharmaceutical Industry

Research and development portfolios contain projects with varying probabilities of success, cost structures, and development times.

Operations research models can assist in the selection and scheduling of such portfolios.

13.4 Real Estate Industry

In large real estate portfolios, investment, renovation, and development initiatives may compete with one another.

The optimal combination depends on capital requirements, value contribution, scheduling, and potential dependencies.

13.5 Public Budgets

Government budget and investment decisions also follow mathematical allocation structures.

However, public decisions differ from private-sector investment decisions in one key respect:

Social benefits cannot always be directly expressed as financial returns.

Instead, factors such as service quality, infrastructure condition, climate impact, social accessibility, or other public policy objectives can be taken into account.

To this end, appropriate benefit criteria must be developed, operationalized, and weighted transparently.

Legal obligations, earmarking, and political decision-making authority remain binding framework conditions.

Mathematical optimization can support public decision-making, but it cannot replace democratic legitimacy and political accountability.

14. Limitations and Prerequisites of Mathematical Optimization

Operations research is powerful, but it is no universal guarantee of correct decisions.

The quality of the results depends on several factors.

Model Quality

The mathematical model must adequately and accurately represent the relationships relevant to decision-making.

Data Quality

Incomplete or erroneous data can lead to inappropriate results.

Objective Function Quality

A mathematical method optimizes the defined objective function—regardless of whether it fully represents an organization’s actual objectives.

Computational Complexity

Certain optimization problems are computationally difficult.

Not every problem can be solved globally optimally within a given time frame.

Uncertainty

The results may be sensitive to future developments or uncertain parameters.

Organizational Feasibility

A mathematically attractive decision can cause practical difficulties if organizational, political, or cultural conditions have not been adequately taken into account.

It follows that:

Mathematical optimality, economic feasibility, and organizational feasibility are interrelated but not identical.

A professional optimization process must take all three dimensions into account.

15. From the Mathematical Solution to the Real-World Decision

The scientific methods of operations research have been known for decades.

However, a major challenge lies in their practical integration into management processes.

Traditionally, optimization models are often developed and calculated by specialized departments and then provided as decision-making templates.

Modern decision intelligence applications take a more interactive approach.

For example, decision-makers can:

1. modify budgets or resource constraints.
2. Select or exclude projects.
3. Adjust strategic weightings.
4. Calculate alternative scenarios.
5. Compare the economic consequences of different portfolio combinations.

This allows a one-time optimization calculation to evolve into an iterative decision-making process.

StratePlan as a specialized application

StratePlan, the decision intelligence technology from mAInthink GmbH, is specifically designed for the mathematical optimization of investment and project portfolios.

The application combines combinatorial optimization with the interactive analysis of budget, resource, and project dependencies.

A key focus is on the ability to immediately recalculate portfolio decisions within management and decision-making bodies.

The underlying problem class belongs to operations research, specifically to discrete portfolio and resource allocation.

StratePlan should therefore be classified as a specialized application of mathematical decision optimization, not automatically as a universal solver for all operations research problems.

The optimality guarantees, computation times, and model classes supported depend on the specific technical implementation and the respective problem formulations.

The economic value lies in the ability to apply mathematical optimization to real-world strategic decision-making processes.

16. Frequently Asked Questions

Is Operations Research a form of artificial intelligence?

Not necessarily. Operations Research is an independent scientific discipline involving mathematical optimization, simulation, and decision-making methods. However, there are numerous overlaps and opportunities for combination with artificial intelligence.

What is the difference between Operations Research and Mathematical Optimization?

Mathematical optimization is a central methodological subfield of operations research. Operations research also encompasses, for example, simulation, queueing theory, stochastic modeling, and other methods of decision analysis.

What is combinatorial optimization?

Combinatorial optimization examines the selection or arrangement of discrete alternatives under defined conditions. Typical examples include project portfolio selection, route planning, assignment problems, and certain scheduling problems.

Can Operations Research compute a global optimum?

Yes, for suitable mathematical models and with appropriate solution methods, global optimality can be proven. However, this is not guaranteed for every problem class or every given computation time.

Why is traditional project prioritization not always sufficient?

A ranking evaluates individual projects. Portfolio optimization, on the other hand, examines the overall impact of feasible project combinations. The best combination does not necessarily consist of the highest-ranked individual projects.

What is the difference between simulation and optimization?

A simulation examines what results may arise under certain assumptions and decisions.

Optimization searches within a defined decision space for solutions that best satisfy an objective function.

Both approaches can be combined.

What does NP-hard mean?

NP-hard refers to a class of computationally difficult problems, which includes numerous combinatorial optimization problems.

No general algorithm is known for these problems that can optimally solve all instances in polynomial time. However, this does not imply that every practical instance must be difficult.

Can companies use operations research without their own mathematics department?

Yes. Specialized software and decision intelligence applications can make mathematical modeling and optimization accessible to business users.

The correct formulation of the problem, reliable input data, and the subject-matter validation of the results remain crucial.

Is StratePlan an operations research application?

Yes. The mathematical optimization of investment and project portfolios is part of operations research.

StratePlan specifically addresses combinatorial selection and allocation problems under defined constraints.

17. Conclusion: Operations Research as the Scientific Foundation for Better Decisions

Operations research demonstrates that complex decisions do not have to rely exclusively on experience, intuition, or isolated individual assessments.

Many economic, technical, and public decision-making problems can be mathematically modeled and systematically optimized.

This approach becomes particularly relevant when numerous alternatives compete with one another, resources are limited, and interdependencies must be taken into account.

The key insight is:

The quality of individual projects does not necessarily determine the quality of the entire portfolio. What matters is the combination that generates the highest defined total value under the actual constraints.

Operations Research provides the scientific methods for investigating such relationships.

Mathematical optimization methods enable the search for high-quality solutions and, under suitable conditions, solutions that are demonstrably globally optimal.

Decision Intelligence can translate these results into transparent, interactive decision-making processes.

This creates a link between decades of mathematical research and the practical requirements of modern business management.

Operations Research makes complex decision-making problems mathematically analyzable. Decision Intelligence makes the results usable for decision-makers.

Further Scientific Resources

Professional societies and foundational resources

• INFORMS – Operations Research & Analytics: https://www.informs.org/Explore/Operations-Research-Analytics
• EURO – Association of European Operational Research Societies: https://www.euro-online.org/
• German Society for Operations Research (GOR): https://www.gor-ev.de/

Standard Academic Texts

• Hillier, F. S. & Lieberman, G. J.: Introduction to Operations Research.
• Bertsimas, D. & Tsitsiklis, J. N.: Introduction to Linear Optimization.
• Nemhauser, G. L. & Wolsey, L. A.: Integer and Combinatorial Optimization.
• Ehrgott, M.: Multicriteria Optimization.
• Birge, J. R. & Louveaux, F.: Introduction to Stochastic Programming.

Technical Deep Dive

• Gurobi – Mixed-Integer Programming: https://www.gurobi.com/resources/blog/mixed-integer-programming-an-introduction-to-the-basics
• Gurobi – Provable Optimality: https://www.gurobi.com/resources/blog/provable-optimality

Experience Operations Research in Practice

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