INSIGHTS | OPERATIONS RESEARCH & BUSINESS MANAGEMENT
Operations Research in Management: Mathematical Optimization Directly in the Boardroom
How operations research, mathematical portfolio optimization, and live simulation are transforming strategic investment decisions—from CAPEX planning to the boardroom.
Companies invest significant resources in evaluating, prioritizing, and planning their projects. Yet one key question often remains unanswered: Is the selected combination of investment projects truly the best given the existing budget, resource, and dependency constraints?
This is precisely where Operations Research (OR) comes in. This scientific discipline examines how complex decision-making problems can be mathematically modeled and optimized.
While Operations Research has been used for decades in production, logistics, transportation planning, and resource allocation, its integration into strategic management processes opens up further applications: mathematical optimization right where investment decisions are made—in the boardroom.
This transforms what is often an upstream analysis into an interactive tool for corporate management.
1. What is Operations Research in Management?
Operations research is the scientific discipline concerned with the mathematical analysis and optimization of complex decision-making problems. It employs, among other things, mathematical programming, combinatorial optimization, simulation, stochastic models, and algorithmic methods to determine the best possible or demonstrably optimal solutions under defined conditions.
In business management, this means that strategic and operational decisions should not be based solely on empirical data, rankings, or isolated profitability analyses, but should instead systematically take into account the interactions between available courses of action.
Typical decision-making questions include:
- Which investment projects should be implemented given a limited CAPEX budget?
- How can resources be optimally allocated across multiple projects and time periods?
- Which combination of projects yields the highest modeled economic benefit?
- How do additional constraints affect the optimal portfolio?
- What are the effects of budget cuts or changes in strategic priorities?
Operations research thus differs from purely descriptive analyses: It examines not only what has happened or could happen, but also which permissible course of action should be preferred under a defined objective function.
2. Operations Research and Business Management: Two Disciplines Converging
Traditional business management deals with strategy, corporate objectives, budgeting, investments, organization, and governance.
Operations Research provides methods for mathematically structuring and optimizing selected management decisions.
The two disciplines complement each other particularly well in situations where many interrelated decisions must be made simultaneously.
2.1 The Traditional Separation Between Analysis and Decision-Making
In many companies, investment proposals are initially developed at the decentralized level. Controlling and Finance consolidate the data, assess profitability and risks, and prepare decision-making documents.
The executive board then receives a prioritized list of projects or several pre-calculated scenarios.
If a key assumption changes during the meeting, a new analysis may be necessary. This is particularly true when budget changes affect numerous project dependencies and resource constraints.
2.2 From Static Reports to Interactive Decision-Making Models
A mathematical optimization model can partially bridge this gap.
Strategic targets (revenue, expenses, and weightings/dependencies) and other constraints are integrated into a single model. This allows for the consistent analysis of different management objectives.
Company management continues to determine objectives, assumptions, and priorities. The optimization process then calculates suitable project combinations.
Operations research does not, therefore, replace corporate leadership. It expands the analytical basis for decision-making.
3. Why Project Prioritization Is Not Portfolio Optimization
Project prioritization and portfolio optimization follow different methodological approaches.
3.1 Project prioritization evaluates individual projects
In prioritization, projects are evaluated—for example, based on net present value, return on investment, strategic importance, or a weighted scoring system—and then ranked.
Such a ranking can be valuable for decision-making. However, it does not automatically answer the question of which combination of projects is best within a given budget.
3.2 Portfolio optimization considers the overall combination
Mathematical portfolio optimization simultaneously takes into account:
- Investment costs and available budgets
- Economic or strategic target contributions
- Project dependencies and exclusion criteria
- Resource availability and capacity limits
- Multi-period investment and resource planning
- Mandatory projects and regulatory requirements
A highly rated individual project is therefore not necessarily part of the optimal overall portfolio. Several smaller projects can collectively achieve a higher contribution to objectives.
The crucial management question is not merely: Which projects are good? But rather: Which combination is the best under the given conditions?
4. The Mathematical Foundations of Portfolio Optimization
A typical CAPEX portfolio can be formulated as a binary optimization problem.
4.1 Decision Variables
For each project i, a binary variable is defined:
x_i ∈ {0,1}
Where:
x_i = 1: Project i is selected.x_i = 0: Project i is not selected.
4.2 Objective Function
If each project is assigned an additive objective contribution v_i, a simplified objective function is:
Maximize Z = Σ(i=1 to n) v_i · x_i
The objective contribution can, for example, be a net present value, a monetized benefit, or a consistently defined strategic utility.
In the case of non-additive effects, risks, or interactions between projects, correspondingly extended models are required.
4.3 Budget Constraint
For a given investment budget B and project costs c_i, the following applies:
Σ(i=1 to n) c_i · x_i ≤ B
The selected projects must not exceed the defined budget.
4.4 Dependencies and Additional Constraints
If Project A depends on the completion of Project B, the condition might be, for example:
x_A ≤ x_B
Additional constraints may reflect personnel capacities, technical dependencies, project durations, or annual budget limits.
The mathematical model thus describes a feasible decision space within which an optimization algorithm searches for the best solution.
4.5 Combinatorial Complexity
For n independent binary project decisions, there are theoretically 2 to the power of n possible combinations.
For 30 projects, this results in:
2³⁰ = 1,073,741,824 possible selection combinations.
This number describes the theoretical selection space before taking constraints into account. It does not mean that an optimization algorithm must compute every combination individually. Modern methods can systematically exclude large portions of the search space.
5. Operations Research in the Boardroom: Optimization as Part of the Board Meeting
The integration of operations research into the boardroom primarily changes the timeline of decision support.
Instead of discussing only pre-prepared project rankings, corporate leadership can draw on a mathematical model that calculates alternative decisions based on defined objectives and constraints.
5.1 From Report to Interactive Analysis
A possible approach consists of five steps:
- Data Set: Investment projects, costs (expenditures), benefit contributions (revenues), and constraints are prepared.
- Initial portfolio: The model calculates a permissible and, as far as possible, optimal combination of projects.
- Management Discussion: The executive board and finance team review budget limits, priorities, and assumptions.
- Recalculation: Revised parameters are incorporated into the model and optimized again.
- Decision: Company leadership evaluates the results and makes a documented decision.
In this way, mathematical optimization can become an interactive component of strategic decision-making processes.
5.2 The Role of Human Decision-Making
Even a portfolio calculated to be mathematically optimal remains dependent on the quality of the model.
An optimization algorithm can only take into account the objectives, assumptions, and constraints that are included in the model.
Strategic responsibility, business judgment, and governance therefore remain with management.
6. Live Simulation: What Happens When the Board Changes Its Parameters?
The practical value of live optimization is particularly evident when key conditions change during a meeting.
6.1 Scenario: The CAPEX budget is reduced
A company is planning an investment portfolio with a budget of 20 million euros. During the executive board meeting, it is decided to limit the available funds to 17 million euros.
Simply cutting the most recently prioritized projects may not result in the best remaining combination.
An optimization model, on the other hand, can completely recalculate the portfolio within the new budget limit.
6.2 Scenario: A project becomes mandatory
The executive board decides that a project required by regulation must be implemented regardless of its economic evaluation.
The new constraint is, for example:
x_MandatoryProject = 1
The model takes this requirement into account and searches for the best combination of projects within the remaining decision space.
6.3 Scenario: Resources Become a Bottleneck
The budget is not always the only decisive constraint. Technical specialists, production capacity, or project management resources may also be limited.
By applying additional resource constraints, you can examine how the optimal project composition changes.
The live simulation allows for an immediate comparison of the consequences of changed management specifications, provided that the model and computation times permit it.
7. Real-World Example: 30 CAPEX Projects and Over a Billion Theoretical Combinations
Let’s consider a company with 30 planned investment projects.
Each project has an investment requirement, an economic target contribution, and, where applicable, dependencies on other projects.
With binary project selection, there are theoretically more than one billion possible combinations.
7.1 Why a spreadsheet ranking isn’t enough
A spreadsheet can document project costs, net present values, and priorities very well.
However, a project list sorted by ROI or net present value is not a general algorithm for solving a combinatorial portfolio optimization problem.
The problem does not fundamentally lie with Excel as software. With appropriate Solver methods, optimization models can also be solved using spreadsheets. The decisive factor is whether actual mathematical portfolio optimization is performed or whether merely a ranking is processed.
7.2 Sample Calculation with Six Projects
A simplified portfolio contains the following hypothetical projects:
| Project | Investment | Net Present Value |
|---|---|---|
| A | €6 million | €9 million |
| B | €5 million | 7 million € |
| C | €4 million | €6 million |
| D | 3 million € | €4 million |
| E | €2 million | 3 million € |
| F | 2 million € | 2 million euros |
The available budget is 10 million euros.
A selection based solely on the highest absolute net present value would initially favor Project A. Together with Project C, this results in a net present value of 15 million euros.
The combination of B, C, and E, on the other hand, requires 11 million euros and is therefore not permitted.
The combination of B, D, and E requires exactly 10 million euros and achieves a net present value of 14 million euros.
The combination of A and C also requires exactly 10 million euros and yields 15 million euros. It is therefore optimal in this simplified example.
The example shows how budget constraints determine the selection. For larger portfolios and additional dependencies, a systematic mathematical calculation becomes particularly relevant.
8. What is the significance of operations research for the CEO, CFO, and board of directors?
8.1 CEO: Translating Strategic Priorities into Action
For CEOs, the benefit lies in linking strategic objectives to concrete resource and investment decisions.
Growth, transformation, efficiency, and resilience often compete for the same financial resources.
An optimization model can clarify which combination of projects best supports the defined goals given the available resources.
8.2 CFO: Systematically Improving Capital Allocation
For CFOs, the connection between capital allocation, CAPEX planning, and portfolio optimization is particularly relevant.
Mathematical optimization can help consistently compare investment alternatives and highlight the consequences of different budget decisions.
It does not replace financial evaluation or risk management, but rather complements them with a systematic selection process.
8.3 Supervisory Board: Understanding the Basis for Decisions
For supervisory boards and other oversight bodies, documented optimization models can improve the transparency of investment decisions.
Of particular relevance are the objectives, constraints, scenarios, and differences between proposed alternatives.
Mathematical optimization thus creates an additional analytical foundation for governance and oversight without altering the responsibilities of the governing bodies.
9. Which optimization methods does operations research enable in management?
The mathematical and algorithmic implementation depends on the structure of the respective decision-making problem.
9.1 Linear and Mixed-Integer Optimization
Linear programming and mixed-integer linear programming (MILP) are among the established methods for modeling allocation and selection problems.
Binary project decisions can be combined with linear objective functions and constraints.
9.2 Combinatorial Optimization
Combinatorial optimization examines the selection, arrangement, and combination of discrete alternatives.
It is particularly relevant when decisions are interdependent and the number of possible combinations grows rapidly with the number of variables.
9.3 Exact Algorithms and Heuristic Methods
Under suitable conditions, exact optimization methods can prove the global optimality of a solution or specify a quantified optimality gap.
Heuristic and metaheuristic methods, on the other hand, search for good solutions without generally providing proof of global optimality.
Which approach is appropriate depends on the model structure, problem size, desired solution quality, and available computation time.
9.4 Parallel Computing and Computing Power
Parallel computing can accelerate certain optimization methods by distributing appropriate computational steps across multiple processor cores.
The actual speedup depends on the algorithm, the problem structure, and the degree of parallelization. Therefore, a short computation time or a proof of global optimality cannot be inferred solely from the number of processing cores.
10. Operations Research, Artificial Intelligence, and Decision Intelligence
Operations Research, Artificial Intelligence, and Decision Intelligence are often discussed together, but they serve different functions.
10.1 Predictive Analytics and Prescriptive Analytics
Predictive analytics deals with forecasting future developments.
Prescriptive Analytics examines which actions are appropriate under certain assumptions and objectives.
Operations Research provides key mathematical methods for this purpose.
10.2 The Connection Between Forecasting and Optimization
For example, a machine learning model can estimate future demand, costs, or failure probabilities.
An optimization model can then use these estimates to allocate resources, investments, or actions subject to constraints.
The quality of the decision depends on both the forecast and the modeling of the optimization problem.
10.3 Decision Intelligence as an Organizational Framework
Decision Intelligence integrates data, models, decision logic, and organizational processes with the goal of systematically supporting decision-making.
Within this framework, operations research can provide the mathematical basis for calculating alternative courses of action.
The combination of mathematical optimization and management processes makes scientific methods applicable to specific business decisions.
11. How can Operations Research be integrated into existing business processes?
A successful implementation requires not only suitable algorithms but also robust organizational and technical integration.
11.1 Data from Finance, Controlling, and Project Management
Relevant information can come from ERP systems, project management solutions, financial planning, and spreadsheets.
It is important to have a consistent definition of project costs (expenses), value contributions (revenue), time periods, and constraints.
11.2 Governance and Model Ownership
The company must determine who defines objectives and constraints, approves data, and validates optimization results.
Changes to model assumptions should be documented and versioned as needed.
11.3 Integration into the Decision-Making Cycle
Mathematical portfolio optimization can be integrated into annual CAPEX planning processes, multi-year investment programs, and intra-year reallocations.
For live sessions, it is advisable to validate data and the model in advance and to interactively modify only clearly defined parameters.
12. Prerequisites and Limitations of Mathematical Management Optimization
Operations research does not automatically guarantee economically successful decisions.
12.1 Data Quality and Model Validation
Incorrect cost assumptions, incomplete dependencies, or inconsistent target contributions can lead to inappropriate results.
Therefore, data verification, domain-expert model validation, and plausibility checks are essential components of a sound optimization process.
12.2 Uncertainty and Risk
Future investment returns and project costs are often uncertain.
Depending on the application, scenario analyses, sensitivity analyses, robust optimization, or stochastic methods may be required.
12.3 Global optimality is always relative to the model
A mathematically global optimum is the best permissible solution within the defined objective function and constraints.
It is not automatically synonymous with the best decision from all real-world economic perspectives.
12.4 Real-time computation depends on the problem structure
Many optimization problems can be solved quickly. Others require significant computation time or the use of approximate methods.
For a robust real-time application, model size, solver behavior, computation times, and result quality must be tested under realistic conditions.
13. Applications of Operations Research in Business Management
13.1 CAPEX and Investment Portfolio Optimization
Companies can use mathematical optimization to allocate investment budgets across projects, locations, or business units.
13.2 Strategic Resource Allocation
The allocation of limited human, technical, or financial resources to competing projects is a classic optimization problem.
13.3 Multi-Year Investment Planning
Multi-period models can account for annual budget constraints, project durations, dependencies, and resource availability.
13.4 Infrastructure and Public Budgets
Public institutions also face complex allocation decisions. Rather than maximizing monetary returns exclusively, appropriate utility models can take into account social impact and political objectives.
Legal obligations, earmarked funds, and democratically legitimized priorities remain binding framework conditions in this context.
13.5 Research, Development, and Transformation
Research and development portfolios, transformation programs, and strategic initiatives can also be supported by mathematical selection and allocation models.
14. StratePlan: Operations Research as an Online Decision Service
With StratePlan, mAInthink GmbH is developing an application that makes mathematical portfolio optimization accessible for economic and public decision-making processes.
The approach combines operations research methods with an application-oriented interface for analyzing project and investment portfolios.
14.1 From Project Data to an Optimized Combination
The starting point is structured project data such as investment requirements, revenues, and relevant constraints.
The optimization model examines which combination of projects achieves the highest modeled target contribution under the defined conditions.
14.2 Scenarios for Management and Finance
Changes to budgets or other model parameters can be used to recalculate the portfolio.
This allows for the comparison of different strategic objectives without having to manually rebuild the entire project evaluation.
14.3 CAPEX Live Boardroom Simulation
The CAPEX Live Boardroom Simulation brings the optimization approach into the management meeting.
Company leadership can discuss alternative conditions and examine their effects on the calculated project combination.
The practical speed depends on the specific model structure and the scale of the problem.
14.4 Decision Intelligence 4 All
The Decision Intelligence 4 All concept aims to make mathematical optimization accessible even outside of specialized operations research departments.
The Online Decision Service is designed to simplify the application of scientific optimization methods without requiring users to program mathematical models or algorithms themselves.
Same Budget. Better Combinations. Greater Impact.
Optimize your own projects mathematically now →
15. Frequently Asked Questions About Operations Research in Management
What does operations research mean in business management?
Operations research in business management refers to the application of mathematical models and optimization techniques to business decisions, such as capital allocation, project portfolio selection, resource planning, and investment management.
Can operations research be used directly in an executive board meeting?
Yes. Once the data, model, and technical infrastructure are in place, alternative budget and resource decisions can be recalculated and compared during a meeting. The required computation time depends on the complexity of the model.
What is the difference between project prioritization and mathematical portfolio optimization?
Project prioritization ranks individual projects according to evaluation criteria. Portfolio optimization, on the other hand, calculates the best possible or demonstrably optimal combination of multiple projects under shared constraints.
Is operations research the same as artificial intelligence?
No. Operations Research is a distinct scientific discipline. Both fields use mathematical methods that are partly related and can be combined, for example, in forecast-based optimization.
Can mathematical optimization increase ROI?
Under the right conditions, mathematical optimization can identify better project combinations and thereby unlock economic potential. However, a concrete increase in ROI depends on project data, model assumptions, and available alternatives and is not generally guaranteed.
What data is required for CAPEX portfolio optimization?
Typically, project identifiers, investment costs, revenues, and relevant constraints are required. For more complex models, additional factors such as project dependencies, time periods, and resource requirements are also needed.
How many project combinations are there for 30 projects?
With 30 binary selection decisions, there are theoretically 2 to the power of 30, or 1,073,741,824 combinations. Constraints limit the permissible combinations; optimization algorithms do not necessarily have to examine all theoretical possibilities individually.
Does mathematical optimization replace the executive board’s decision?
No. Management sets goals, priorities, and framework conditions and continues to bear responsibility for the decision. Optimization provides mathematically grounded decision support.
16. Conclusion: Operations Research Becomes a Strategic Management Competency
Operations research is not a new field. What is new is the ability to integrate its methods more closely into interactive strategic decision-making processes.
When mathematical models, available data, and appropriate software come together, the optimization of investment and project portfolios can be directly incorporated into management discussions.
The executive board then does not have to rely exclusively on static rankings and pre-selected scenarios. It can examine how alternative parameters affect the best possible combination of projects.
This changes the role of operations research: from a specialized analytical function to a potentially recurring capability in strategic corporate management.
Mathematics does not provide business acumen. However, it can significantly improve the quality of the basis for decision-making.
Decision Intelligence 4 All. Powered by Operations Research & Mathematical Optimization.
