AI-supported combination solver for investment decisions

Why most investment decisions are systematically suboptimal - and how combinatorial optimization makes the complete decision space computable for the first time.

In almost every organization, strategic investment decisions are made on the basis of extensive analyses. Business cases are created, projects evaluated, scenarios calculated and budgets allocated. Despite this high level of analysis, a fundamental mathematical problem remains unsolved in most decision-making processes: the complete decision space is not calculated.

When organizations plan investment portfolios, they are faced with a classic problem of combinatorial optimization. In the case of several investment projects, there are not only individual decisions, but a large number of possible combinations of projects, which together form a portfolio.

These combinations grow exponentially. Even with just a few projects, a decision space is created that can no longer be fully analyzed by people, Excel models or traditional project portfolio management tools can no longer fully analyze.

The result is a structural decision-making problem: organizations prioritize projects, analyse scenarios and allocate budgets, without knowing the mathematically optimal combination of all investments.

This is precisely where a new generation of algorithmic systems comes in: AI-supported combination solvers for investment decisions. These systems do not calculate individual projects, but rather the entire decision space of possible project portfolios and identify the global optimum under real restrictions.

The fundamental mathematical problem of investment planning

The planning of an investment portfolio can be formulated mathematically as a combinatorial decision problem. Suppose a company or public organization evaluates a list of potential investment projects.

Each project can either be implemented or not implemented. This results in all possible combinations of these projects from a list of N projects.

The number of possible portfolios results from the function:

2^N

This means that even a relatively small number of projects generates an extremely large decision space.

With ten projects, there are already over a thousand possible project portfolios. With twenty projects, over a million. With fifty projects, more than a quadrillion possible combinations.

This exponential structure is a classic feature of Combinatorial Optimization, a central field of research in operations research and computer science.

In theory, these problems have been known for decades and are described in the scientific literature as variants of the Knapsack problem, the Project Portfolio Selection or mixed Integer Optimization.

In practice, however, the complete decision space is rarely calculated.

Why classic decision-making processes systematically fail

In most organizations, the selection of investment projects follows a relatively similar pattern.

First, projects are analyzed individually. Business cases are drawn up, expected returns are estimated, risks are assessed and strategic priorities are set.

Projects are then prioritized using an evaluation system. This can take the form of scorecards, rankings or strategic weightings.

Finally, projects are included in the portfolio until budget or capacity limits are reached.

From a mathematical point of view, this process corresponds to a so-called greedy procedure.

Greedy algorithms make decisions step by step and select the best option at any given time. They are easy to implement and often intuitively comprehensible.

However, the decisive disadvantage is that they only take local improvements into account. The global optimum of the entire system often remains unrecognized.

In complex investment portfolios, this can lead to projects with a high individual value being selected, although a different combination of projects would generate a significantly higher overall value.

The problem of local optima

A helpful image to explain this problem is a landscape with many hills and a single highest mountain.

Local decision-making processes often move to the nearest hill. As soon as a seemingly good point is reached, it is considered optimal.

However, the actual global optimum - the highest hill in the landscape - remains hidden, because the complete decision space is not systematically examined.

In investment portfolios, this means that companies select good projects, but not necessarily the best combination of all projects.

The difference between a local optimum and a global optimum can lead to significant economic deviations in large investment portfolios lead to considerable economic deviations.

Typical errors in the portfolio decision

The lack of consideration of the complete decision space leads to several systematic errors in investment planning in investment planning.

One common problem is the isolated evaluation of individual projects. If projects are only considered individually, interactions between projects are not taken into account.

Another problem is budget fragmentation. Budgets are allocated to several projects without considering the overall impact of the combination.

Time interdependencies are also often underestimated. Many projects only develop their economic benefits in combination with other initiatives or over several years over several years.

Without mathematical modeling of these interdependencies, portfolios are created, that appear plausible but are not optimal.

The role of combinatorial optimization

Combinatorial optimization deals with precisely this type of problem. The aim is to identify from a large number of possible combinations the one that maximizes or minimizes a certain objective function, that maximizes or minimizes a certain objective function.

In investment portfolios, this objective function typically consists of economic indicators such as Capital value, return, risk or strategic contribution.

There are also secondary conditions such as budget restrictions, capacity limits, Dependencies between projects or regulatory requirements.

Mathematically, this results in an optimization problem with discrete decision variables, which is often formulated as mixed integer programming.

An example of the explosion of the decision space

The following table shows how quickly the decision space grows as the number of projects increases.

This exponential structure explains why classic decision tools are not able to to examine the entire decision space.

AI-supported combination solvers

An AI-supported combination solver addresses precisely this problem.

Instead of analyzing individual projects, the solver models the entire investment portfolio as a mathematical optimization problem.

The decision variables represent the selection of individual projects. Constraints model real restrictions such as budget, capacity or risk.

The solver then systematically searches the decision space and identifies the combination of projects the combination of projects that maximizes the target function.

Modern systems combine methods from several fields of research:

• Operations Research
• Combinatorial optimization
• Mixed integer programming
• Branch-and-bound methods
• Heuristic search algorithms
• Machine Learning

This combination results in powerful decision support, that goes far beyond traditional analysis systems.

The difference between analysis and optimization

Many existing systems in project portfolio management focus on analysis functions.

They answer questions such as:

• How profitable is a project?
• How high is the risk?
• How does the business case change with certain assumptions?

This information is important, but it is not enough to determine the optimal combination of projects.

Optimization systems ask a different question:

Which combination of all projects maximizes the total value of the portfolio under given constraints?

Only from this perspective does the complete decision space become visible.

Practical effects on investment decisions

The difference between heuristic prioritization and mathematical portfolio optimization can have a significant economic impact.

Real-life applications often show that the optimal combination of projects generates generates significantly higher total returns than a classically prioritized portfolio.

The reason for this lies in the interdependencies between projects.

A project with moderate individual value can generate considerable added value in combination with other projects generate considerable added value.

Conversely, several highly valued projects together can form an inefficient portfolio, if they compete for the same resources or have similar risks.

Strategic importance for companies

For companies with large investment budgets, the quality of portfolio decisions is becoming a decisive a decisive competitive factor.

Capital allocation determines which technologies are developed, which markets are opened up and which innovation paths are pursued.

If the decision space is not fully analyzed, resources are often invested in suboptimal projects.

A mathematically optimized portfolio approach can therefore have a significant influence on long-term company performance.

Strategic importance for public budgets

The optimization of investment portfolios is also playing an increasingly important role in the public sector.

Cities and states are faced with the challenge of allocating limited budgets to a large number of infrastructure projects, Educational initiatives and social programs.

The number of possible combinations of these projects is enormous.

Without systematic optimization, there is a risk that investments will not not achieve the maximum possible social impact.

The future of decision intelligence

With increasing computing power and improved optimization algorithms, the calculation of the calculation of complex decision spaces is becoming increasingly practicable.

AI-supported combination solvers open up the possibility of Investment decisions on the basis of the complete mathematical decision space for the first time.

This marks a fundamental change in the way organizations make strategic decisions, how organizations make strategic decisions.

Instead of merely managing complexity, it can now be systematically optimized.

FAQ

What is a combination solver?

A combination solver is an algorithmic system that identifies from a large number of possible combinations the one that maximizes or minimizes a certain objective function.

Why are investment decisions combinatorial problems?

Because every project can either be implemented or not implemented. This results in all possible combinations of these projects from N projects.

Why can't classic tools solve this problem?

The number of possible combinations grows exponentially. Even with just a few projects, the decision space exceeds the possibilities of of classic analysis tools.

Which mathematical methods are used?

Typical methods are mixed integer programming, branch-and-bound, heuristic search methods and various combinatorial optimization techniques.

What are the benefits for companies?

Companies can identify investment portfolios that generate the maximum economic value under real restrictions.

What role does artificial intelligence play?

AI can be used to structure search spaces efficiently, Improve models and support decision-making processes.