Mathematical Optimization: Calculating the Global Optimum for CAPEX, Capital Allocation, and Project Portfolios

Mathematical Optimization uses mathematical models and algorithms to determine, from among many possible decisions, a solution that optimizes a defined objective under given constraints.

In the context of CAPEX, capital allocation, and project portfolio management, this means, for example:

Which combination of projects maximizes our NPV, ROI, or strategic portfolio value within a limited budget?

The challenge lies not primarily in evaluating individual projects.

It lies in the number of possible combinations.

With 20 independent yes/no projects, there are theoretically up to:

2^20 = 1,048,576 combinations.

With 50 projects:

2^50 ≈ 1.13 × 10^15 combinations.

For 100 projects:

2^100 ≈ 1.27 × 10^30 combinations.

This is exactly where combinatorial optimization, operations research, mixed-integer programming, constraint optimization, and specialized portfolio optimization algorithms come into play.

Don’t just rank projects. Calculate the portfolio.

Table of Contents

• What Is Mathematical Optimization?
• What Is Combinatorial Optimization?
• What Is Combinatorial Optimization Software?
• What Is Mathematical Portfolio Optimization?
• What Is a Portfolio Optimization Algorithm?
• What is an optimization solver for business?
• Operations Research for Capital Allocation
• Mixed-Integer Programming for CAPEX
• The Knapsack Problem as a Business Application
• NPV Portfolio Optimization
• ROI Portfolio Optimization
• Maximize NPV Under Budget Constraints
• What Is Constraint Optimization?
• What Is Budget Constraint Optimization?
• What Is Global Optimization?
• What Is a Global Optimum?
• What Is the Optimal Project Combination?
• Project Ranking vs. Mathematical Optimization
• The Combinatorial Decision Space
• The Mathematical Portfolio Model
• Mathematically Modeling Constraints
• Mathematically Modeling Project Dependencies
• Mathematically Modeling Mandatory Projects
• Example: Why the Combination Is Crucial
• Example: Portfolio Optimization with 150 Projects
• Multi-Objective Optimization
• Multi-Year Portfolio Optimization
• Scenario Optimization
• Mathematical Optimization in the Boardroom
• Mathematical Portfolio Optimization with StratePlan
• Frequently Asked Questions

What Is Mathematical Optimization?

Mathematical optimization refers to the systematic search for the best possible solution to a mathematically defined decision-making problem.

An optimization problem typically consists of three basic components:

1. Decision Variables

What decisions can be made?

In the case of a project portfolio, for example:

Select a project or do not select it.

2. Objective Function

What should be maximized or minimized?

Examples:

• Maximize NPV
• Maximize portfolio value
• Maximize strategic utility
• Minimize costs
• Reduce risk

3. Constraints

What conditions must be met?

Examples:

• CAPEX Budget
• Resources
• Project Dependencies
• Mandatory Projects
• Business Unit Rules
• Capacity Limits
• Time Constraints

Mathematical optimization then searches within the permissible solution space for a solution that best satisfies the defined objective function.

What is Combinatorial Optimization?

Combinatorial Optimization deals with optimization problems in which the best feasible combination is sought from a large set of discrete possibilities.

Project portfolios are a typical example.

For each project, there is initially a binary decision:

Yes or No.

Invest or do not invest.

Select the project or do not select it.

With N independent binary decisions, there are theoretically up to:

2^N possible combinations.

This makes it clear:

The problem isn’t just evaluating projects. The problem is finding the right combination.

What is combinatorial optimization software?

Combinatorial optimization software supports the solution of discrete decision-making problems with a very large number of possible combinations.

Typical business applications include:

• CAPEX Allocation
• Project Portfolio Selection
• Investment Planning
• Resource Allocation
• Production Planning
• Scheduling
• Logistics
• Network Optimization
• Maintenance Planning

Unlike a simple spreadsheet or ranking system, the focus is not merely on the presentation of data.

The software models a mathematical decision space and searches within it for a solution to a defined objective function.

Combinatorial optimization software doesn’t just answer “How good is Project A?”, but “Which combination of A, B, C, D, and all other projects best meets our objectives under the given conditions?”

What Is Mathematical Portfolio Optimization?

Mathematical Portfolio Optimization applies mathematical optimization to the selection and composition of a portfolio.

In a business context, a portfolio might consist of, for example:

• CAPEX Projects
• Strategic Initiatives
• R&D Projects
• Maintenance Projects
• Digitalization Projects
• Transformation Programs
• Infrastructure Investments

Every project requires capital and possibly other resources.

At the same time, every project generates expected financial or strategic value.

Mathematical portfolio optimization determines which projects should be selected together.

The objective function could, for example, be:

Maximize the total NPV of the portfolio.

Or:

Maximize a combined financial and strategic portfolio value.

Always subject to the defined constraints.

What is a portfolio optimization algorithm?

A portfolio optimization algorithm is a mathematical method for determining a suitable or optimal portfolio configuration within a defined model.

Depending on the problem structure, different optimization methods can be used.

These include, for example:

• Linear Programming
• Integer Programming
• Mixed-Integer Programming
• Dynamic Programming
• Constraint Programming
• Branch-and-Bound Method
• Heuristic Methods
• Metaheuristic Methods
• Hybrid Optimization Methods

Which algorithm is suitable depends on the variables, the objective function, the constraints, and the size of the decision space.

The algorithm is not the business objective. It is the mathematical tool for solving the business problem.

What is an Optimization Solver for Business?

An Optimization Solver for Business is a mathematical computational component that solves decision-making problems under defined objective functions and constraints.

For example, a business user might specify:

Budget: maximum €500 million

Engineering capacity: maximum 25,000 hours

Project 17: mandatory

Project 22 requires Project 9

Objective: Maximize total NPV

The solver uses this information to calculate a feasible project combination that optimizes the defined objective function.

The key value for management therefore does not lie in the solver itself.

Its value lies in translating real-world business rules into a computable decision-making model.

Operations Research for Capital Allocation

Operations research uses mathematical models, optimization, and quantitative methods to support complex decision-making.

Capital allocation is a classic application area.

A company has:

• limited capital,
• several investment opportunities,
• varying returns,
• limited resources,
• strategic requirements,
• interdependencies between projects.

Operations research formulates these into a mathematical decision-making problem.

Instead of simply asking:

“Which project has the highest ROI?”

the question is:

“Which combination of all available investments yields the best result under the defined conditions?”

This turns capital allocation into an optimization problem.

Mixed-Integer Programming for CAPEX

Mixed-Integer Programming, or MIP for short, is a class of mathematical optimization models in which some of the decision variables must be integer or binary.

This is particularly well-suited for CAPEX portfolios.

For example, a project can be represented by a binary variable:

xᵢ = 1 → Project is selected

xᵢ = 0 → Project is not selected

Other variables can be continuous.

For example:

• Production volume
• Resource quantities
• Capacities
• Financing shares

Mixed-integer programming can thus combine discrete project decisions with continuous business variables within a single model.

The Knapsack Problem as a Business Application

The knapsack problem is one of the best-known combinatorial optimization problems.

The basic idea:

There are several objects.

Each item has a value and requires a certain capacity.

The total available capacity is limited.

The goal is to find the combination of objects that yields the highest total value within this limit.

Applied to CAPEX:

The business question is:

Which projects should be selected to generate the highest total value within the budget?

Real-world corporate portfolios are often more complex than the classic knapsack problem because additional resources, dependencies, mandatory projects, and multi-year conditions must be taken into account.

NPV Portfolio Optimization

NPV Portfolio Optimization maximizes the total net present value of an investment portfolio under defined constraints.

A simplified objective function is:

Maximize Σ NPVᵢ × xᵢ

subject to:

Σ Investmentᵢ × xᵢ ≤ Budget

The key distinction:

NPV first evaluates the economic value of an individual project.

NPV portfolio optimization then determines which combination of these projects should be financed together.

A positive NPV does not automatically mean that a project is part of the optimal portfolio.

ROI Portfolio Optimization

ROI Portfolio Optimization examines the impact on returns of an entire investment portfolio.

Caution is required when using simple ROI rankings.

A project with a very high ROI may be small.

Another project may have a lower relative ROI but a significantly higher absolute value contribution.

Furthermore, project sizes, budget constraints, and synergy effects can mean that simply sorting by ROI does not result in the optimal portfolio composition.

Therefore, it should first be clearly defined which target metric is actually to be optimized.

For example:

• Total NPV
• Total Value
• Portfolio ROI
• Strategic Value
• Combination of Multiple Objectives

The mathematical objective function must align with the company’s economic objectives.

Maximize NPV Under Budget Constraint

“Maximize NPV Under Budget Constraint” is a classic capital allocation problem.

Assume:

A company has 100 investment projects.

Total requested CAPEX:

€1.2 billion

Available budget:

€750 million

Each project has an expected NPV.

The task is:

Select the combination of projects whose total investment does not exceed 750 million euros and whose total NPV is maximized within the defined model.

Formally simplified:

Maximize Σ NPVᵢ × xᵢ

subject to:

Σ CAPEXᵢ × xᵢ ≤ 750 million €

and:

xᵢ ∈ {0,1}

Additional business constraints can be added later.

What is constraint optimization?

Constraint optimization combines an optimization problem with conditions that every feasible solution must satisfy.

For project portfolios, such conditions might include, for example:

• CAPEX must not exceed €500 million.
• Engineering may take a maximum of 20,000 hours.
• At least €100 million must be invested in Business Unit A.
• Project 17 must be implemented.
• Project 22 may only be implemented if Project 9 is selected.
• Project 31 and Project 32 must not be implemented simultaneously.

The optimization then searches only within the permissible decision space.

Constraints transform theoretical optimization into a model of real-world business decisions.

What is Budget Constraint Optimization?

Budget Constraint Optimization finds the best solution within a fixed budget limit.

For CAPEX, the basic condition is:

Total selected CAPEX ≤ available CAPEX budget

The budget limit forces the company to make selection decisions.

If all projects could be financed, no portfolio selection would be necessary due to the budget.

Scarcity therefore gives rise to the optimization problem.

An important management question is:

“What is the maximum value we can achieve with exactly this budget?”

A second question is:

“How does the achievable value change if we increase or decrease the budget?”

What is global optimization?

Global optimization refers to the search for the best solution across the relevant feasible solution space of a mathematical model, rather than merely considering a locally better solution in the vicinity of an initial solution.

This is relevant for portfolio decisions because small changes to an existing project list do not necessarily lead to the best portfolio configuration.

Sometimes, a seemingly attractive selection must be fundamentally altered to achieve a better combination.

For example, it may be necessary to remove a large project and include several smaller ones.

Global Optimization considers the portfolio decision as a whole.

What is a global optimum?

A global optimum is the best solution within the defined feasible solution space for the specified objective function.

It is important to understand the exact meaning:

“Globally optimal” does not automatically mean “the best real-world business decision in every respect.”

The result is optimal relative to:

• the data used,
• the defined objective function,
• the modeled constraints,
• the assumptions used.

If these assumptions change, the global optimum may also change.

That is why the quality of the decision model is just as important as the optimization algorithm.

Management defines the problem. Mathematics optimizes within that problem.

What is the optimal project combination?

The optimal project combination is the combination of projects that best satisfies the objective function within a defined model while simultaneously adhering to all constraints.

It can differ significantly from a traditional project ranking.

For example, a project might be ranked third and still not be part of the optimal combination.

A project ranked 8th, on the other hand, may be part of the optimal portfolio due to its costs, its value, and its interactions with other projects.

A project’s position in the ranking and its value for the optimal combination are two distinct pieces of information.

Project Ranking vs. Mathematical Optimization

Ranking and optimization thus serve different purposes.

A ranking can provide information about the relative attractiveness of projects.

Optimization answers the portfolio question.

Don’t just prioritize projects. Optimize the portfolio.

The Combinatorial Decision Space

The decision space encompasses all possible decision alternatives in a model.

With 100 binary project decisions, there are theoretically up to 2^100 combinations.

Many of these are not permissible in reality.

For example, a portfolio might:

• exceed the budget,
• require too many engineering resources,
• violate dependencies,
• exclude mandatory projects,
• violate business unit rules.

Constraints remove such invalid solutions from the decision space.

What remains is the feasible decision space.

Within this permissible space, the optimization process searches for the best possible solution for the defined objective function.

The Mathematical Portfolio Model

A simplified CAPEX portfolio model can be formulated as follows.

Decision Variable

For each project i:

xᵢ ∈ {0,1}

xᵢ = 1 means: The project is selected.

xᵢ = 0 means: The project is not selected.

Objective Function

For example:

Maximize Σ Valueᵢ × xᵢ

Budget Constraint

Σ Costᵢ × xᵢ ≤ Budget

Resource Constraint

Σ Resourceᵢ × xᵢ ≤ Available Resources

Additional conditions can be built upon these.

The actual model can be significantly more complex, depending on the business problem.

Mathematically Modeling Constraints

Business rules can be translated into mathematical conditions.

For example:

Total CAPEX ≤ 800 million €

Engineering ≤ 25,000 hours

Business Unit A Investment ≥ 100 million €

Business Unit B Investment ≤ 250 million €

Maximum of 30 projects at the same time

This transforms a verbal management rule into a mathematically verifiable condition.

Every permissible portfolio configuration must satisfy these conditions.

Mathematically Modeling Project Dependencies

Project dependencies can also be formally modeled.

Project B requires Project A

xB ≤ xA

If B is selected, then A must also be selected.

Projects A and B must be implemented together

xA = xB

Projects A and B are mutually exclusive

xA + xB ≤ 1

At least one of A or B must be implemented

xA + xB ≥ 1

This makes technical or organizational dependencies a direct part of portfolio optimization.

Mathematically Modeling Mandatory Projects

A mandatory project can be modeled as a fixed decision.

For mandatory project M, the following applies:

xM = 1

This means that every valid portfolio configuration must include this project.

This is relevant, for example, for:

• Compliance
• Safety
• Regulatory requirements
• Cybersecurity
• Contractual obligations
• Required Maintenance

Optimization no longer determines whether the Mandatory Project will be implemented.

Instead, it optimizes the remaining portfolio around this project.

Example: Why the Combination Is Crucial

A company has a budget of 100 million euros.

Project A has the highest individual value.

If A is selected:

Portfolio Value = 150 million €

If B and C are combined:

Investment = 60 + 40 = 100 million euros

Portfolio Value = 100 + 80 = 180 million €

With the same budget, B + C thus generates 30 million euros more in expected value.

The best individual option is not automatically the best combination.

Example: Portfolio Optimization with 150 Projects

A company has 150 potential CAPEX projects.

Proposed CAPEX:

€1.8 billion

Available budget:

€1.1 billion

In addition:

• €250 million for mandatory projects
• 30,000 engineering hours
• Minimum Business Unit Budgets
• Business Unit Maximum Budgets
• Project Dependencies
• Mutually Exclusive Projects
• Strategic Criteria

A ranking can sort the 150 projects.

However, it does not automatically determine which combination best fulfills the defined objective function under all conditions.

Mathematical Portfolio Optimization therefore models:

Decision Variables + Objective Function + Constraints.

The result is a portfolio configuration calculated within the defined model.

Multi-Objective Optimization

Companies often pursue more than a single objective.

In addition to financial value, the following may be relevant, for example:

• Growth
• Strategic Fit
• Innovation
• Risk Reduction
• Resilience
• Sustainability

Multi-objective optimization takes into account multiple objectives within a defined decision-making framework.

For example, the following can be used:

• objectives can be weighted,
• minimum requirements defined,
• multiple scenarios be calculated,
• trade-offs analyzed

.

It is crucial that the weighting not be arbitrarily determined by mathematics.

Management defines the priorities. The model calculates their consequences.

Multi-Year Portfolio Optimization

Many CAPEX projects span several years.

This results in multiple budget and resource constraints occurring simultaneously.

For example:

CAPEX 2027 ≤ 300 million €

CAPEX 2028 ≤ €350 million

CAPEX 2029 ≤ €400 million

In addition, project start dates, durations, and resource requirements can be taken into account.

The optimization question is then no longer just:

“Which projects should we choose?”

But rather:

“Which projects should we start and finance in which period to best achieve the defined target over the entire planning horizon?”

Scenario Optimization

A mathematical portfolio model can be recalculated under different assumptions.

For example:

Scenario A: Budget = €500 million

Scenario B: Budget = €450 million

Scenario C: Budget = 550 million €

Scenario D: Engineering Capacity -20%

Scenario E: New Mandatory Project

A new feasible decision space is created for each scenario.

As a result, the optimal project combination may also change.

Management can then compare:

• Selected Projects
• Portfolio Value
• Capital Allocation
• Resource Utilization
• Strategic Impact
• Trade-offs

Scenario Planning reveals possible futures. Optimization calculates the best decision within each scenario.

Mathematical Optimization in the Boardroom

Mathematical optimization becomes particularly relevant when assumptions change during a management meeting.

The CFO asks:

“What happens if we reduce CAPEX by €100 million?”

The CEO asks:

“What happens if growth becomes more important?”

The COO asks:

“What happens if engineering capacity falls by 15 percent?”

The Investment Committee asks:

“What happens if Project 27 becomes mandatory?”

Each of these changes alters the mathematical decision problem.

With a prepared portfolio model, the portfolio can be recalculated under the new conditions.

This changes the decision-making process:

Question. Change Constraint. Calculate. Compare. Decide.

Mathematical Portfolio Optimization with StratePlan

StratePlan applies mathematical optimization to real-world CAPEX, investment, and project portfolio decisions.

The decision-making model may include, among other things:

• Project ID
• Investment
• Expected Value or NPV
• Strategic Criteria
• Budgets
• Resources
• Mandatory Projects
• Dependencies
• Business Rules
• Multi-Year Conditions

Based on this, various portfolio questions can be calculated.

For example:

• Which combination of projects maximizes the NPV?
• Which combination maximizes portfolio value?
• How much value can be achieved with a fixed budget?
• Which projects should be selected if CAPEX is reduced?
• Which additional projects become possible with a higher budget?
• How do mandatory projects affect the portfolio?
• How do project dependencies affect the portfolio?
• How do resource constraints influence the selection?
• How does the optimal combination change over several years?
• How do strategic priorities affect capital allocation?

StratePlan thus combines combinatorial optimization, portfolio selection, and management scenario analysis.

The goal is not to replace management decisions with mathematics.

The goal is to make the decision space predictable.

Management defines the objective.

Constraints define the feasible space.

Mathematics calculates the portfolio.

Management makes the decision.

Don't take our word for it. Calculate it.

Frequently Asked Questions

What is Mathematical Optimization?

Mathematical optimization uses mathematical models and algorithms to determine a solution within a defined decision space that optimizes an objective function under given constraints.

What is combinatorial optimization?

Combinatorial optimization searches for the best possible feasible combination from a large set of discrete alternatives. Project portfolios involving yes/no decisions are a typical application area.

What is combinatorial optimization software?

Combinatorial optimization software solves discrete decision problems with many possible combinations. Business applications include, among others, portfolio selection, capital allocation, resource allocation, and scheduling.

What is Mathematical Portfolio Optimization?

Mathematical portfolio optimization calculates the composition of a portfolio based on a defined objective function and real-world constraints such as budget, resources, dependencies, and mandatory projects.

What is a Portfolio Optimization Algorithm?

A portfolio optimization algorithm is a mathematical method for finding a suitable or optimal portfolio configuration. The specific method depends on the structure of the optimization problem.

What is an Optimization Solver for Business?

An optimization solver calculates solutions for mathematically formulated business problems. It processes decision variables, objective functions, and constraints to determine a feasible, optimized solution.

How is operations research used for capital allocation?

Operations research translates capital allocation into a quantitative decision-making problem. Capital, project values, resources, and other conditions are mathematically modeled and optimized together.

What is Mixed-Integer Programming for CAPEX?

Mixed-integer programming allows for the combination of discrete and continuous decision variables. For example, CAPEX projects can be modeled as binary selection decisions and combined with continuous resource or capacity variables.

What is the knapsack problem in business?

The knapsack problem describes the selection of valuable options within a limited capacity. In the context of CAPEX, the options correspond to projects, the capacity corresponds to the budget, and the value corresponds, for example, to the NPV.

What is NPV portfolio optimization?

NPV portfolio optimization seeks a combination of projects that maximizes the total net present value of the portfolio under defined constraints.

What is ROI Portfolio Optimization?

ROI portfolio optimization examines the impact of a combination of projects on the return at the portfolio level. The specific mathematical formulation should align with the desired definition of portfolio ROI and the economic objective.

How can NPV be maximized under a budget constraint?

Each project is modeled using its investment and NPV. A combination of projects is then sought whose total investment does not exceed the budget limit and whose total NPV is maximized within the defined model.

What is Constraint Optimization?

Constraint optimization seeks the best possible solution that simultaneously satisfies defined conditions. Examples include budget limits, resource constraints, dependencies, and mandatory projects.

What is budget constraint optimization?

Budget constraint optimization optimizes a defined objective within a fixed budget. For CAPEX, for example, the total NPV can be optimized within a maximum available investment budget.

What is Global Optimization?

Global optimization aims to determine the best solution across the relevant feasible solution space of a mathematical model, rather than limiting itself to local improvements to an initial solution.

What is a global optimum?

A global optimum is the best solution for the defined objective function within the modeled feasible decision space. This statement applies relative to the model’s data, assumptions, objective function, and constraints.

What is an Optimal Project Combination?

An optimal project combination is the combination of projects that, within a defined model, satisfies all constraints and best achieves the selected objective function.

Why is a project ranking not sufficient?

A ranking evaluates projects individually and produces an order. It does not automatically examine all relevant project combinations under shared budget, resource, and dependency conditions.

How many combinations are there with 100 projects?

With 100 independent binary project decisions, there are theoretically up to 2^100 combinations. This corresponds to approximately 1.27 × 10^30 possible portfolios.

Can mathematical optimization account for project dependencies?

Yes. Dependencies can be formulated as mathematical constraints. For example, it can be specified that Project B may only be selected if Project A is also selected.

Can Mathematical Optimization account for mandatory projects?

Yes. A mandatory project can be modeled as a fixed selection condition and must therefore be part of every valid portfolio configuration.

Can Mathematical Optimization account for multiple resources simultaneously?

Yes. In addition to budget, resources such as engineering hours, IT capacity, FTE, production capacity, or other limited resources can be integrated as additional constraints.

Can Mathematical Optimization plan across multiple years?

Yes. Multi-Year Optimization can account for period-specific budgets, resources, project start dates, durations, and dependencies within a single decision-making model.

What is the difference between scenario planning and optimization?

Scenario Planning defines alternative assumptions about the future or different management conditions. Optimization calculates a solution within each scenario for the defined objective function and the constraints applicable to that scenario.

Does Mathematical Optimization replace management decision-making?

No. Management defines objectives, assumptions, criteria, and constraints and makes the final decision. Mathematical optimization calculates the consequences of these specifications and supports the comparison of possible portfolio configurations.