Understanding the math behind StratePlan: Why better decisions need a different math logic
Many investment and prioritization decisions look like a "list of projects". Mathematically, they are something else: a combinatorial decision space that grows exponentially with each additional option. If you don't model this space, you can't optimize it.
Who is this page for?
- C-Level & supervisory bodies: to understand why "good individual projects" do not automatically result in the best portfolio.
- CFO/Controlling: to formally summarize opportunity costs and restrictions (budget, risk, ESG, capacities).
- Public sector: to understand why funding logic, departmental thinking and election cycles structurally lead to suboptimal portfolios.
The core problem: decision-making spaces are growing exponentially
Each additional project does not create "one more item" on a list, but a new dimension in the solution space. The number of possible portfolio combinations follows the logic 2^N:
- 10 projects → 2^10 = 1,024 combinations
- 20 projects → 2^20 = 1,048,576 combinations
- 50 projects → 2^50 ≈ 1.125 quadrillion combinations
This is the point at which classic committee processes, Excel logic and heuristics reach a mathematical limit.
Local vs. global optimum
Local optimum means: a solution that works better than obvious alternatives.
Global optimum means: the best solution in the entire decision space.
Many organizations improve local decisions (better scores, better business cases) without calculating the overall decision space. As a result, the best combinations often remain invisible.
Why heuristics are structurally incomplete
Typical rules and restrictions from municipal and corporate decision-making processes such as "top 5 according to NPV", "IRR > WACC", "payback < 3 years" or "strategic beacons first" are operationally comprehensible. Mathematically, they have a weakness: they evaluate projects in isolation, not as an interdependent portfolio.
A project with a low individual value can generate the highest overall impact in combination with other projects. A project with a high individual value can displace better combinations if restrictions apply.
The solution: Formal modeling instead of gut feeling
Decision mathematics begins where a portfolio is formulated as a model:
- Decision variables: xi ∈ {0,1} (project is chosen or not)
- Objective function: e.g. maximize total value, impact, NPV, utility index
- Secondary conditions: Budget, capacities, risk, CO₂, minimum quotas, dependencies and many more...
A Simple Model (Simplified)
Maximize:
∑ (Valuei × xi)
subject to:
∑ (Costi × xi) ≤ Budget
∑ (Emissioni × xi) ≤ CO₂ limit
xi ∈ {0,1}
This basic principle corresponds to a (multi-)restrictive Knapsack problem and forms the basis for real portfolio models with multiple dimensions and interdependencies.
What you will learn on this platform
- Why 2^N is the real "invisible space" behind portfolio decisions
- How constraints dominate decisions (budget, capacity, ESG, risk)
- Why "prioritizing" is not the same as "optimizing"
- How opportunity costs can be made visible ex ante
- How to turn data into a decision-capable model
Basics: decision spaces & optimization
The mathematical basis: 2^N local vs. global optima, restrictions, objective functions and model logic.
Mathematics Deep Dive: The 5 Building Blocks That Truly Matter
- Decision Variables: What choices exist (xi)?
- Objective Function: What is being maximized (value, impact, NPV, benefit)?
- Constraints: What limits the decision space (budget, CO₂, capacity, risk, quotas)?
- Interdependencies: Which projects depend on or exclude others?
- Optimization: How is the best combination found within the entire decision space?
Projects do not disappear - they are better positioned and optimally planned over several years
In a mathematically optimized investment system, projects are not discarded. Instead, they are reprioritized, postponed or strategically repositioned, so that they make the maximum economic contribution to the overall portfolio at the optimum time under given budget, capacity and risk restrictions the maximum economic contribution to the overall portfolio.
The decisive factor here is the multi-year perspective. Investment decisions are not made in isolation for a single year, but are optimized in the context of 2-, 3-, 5- or 10-year plans.
Liquidity generated by optimization in the start year is systematically carried over to the following year year. This increases the available investment budget for the next period. This subsequent year is then also optimized again.
The effect: projects can be added as soon as they fit into the globally optimized portfolio under the new budget, capacity and return conditions, Capacity and return conditions fit into the globally optimized portfolio. This creates a dynamic multi-year optimization in which each optimization period Optimization period structurally improves the investment opportunities of the following years.
Final thought
Those who do not calculate the decision space manage complexity - and do not optimize. Understanding mathematics here does not mean "learning formulas", but rather modeling the structure of decisions in such a way that the global optimum can become visible at all.
Visualization of a 2^50 decision space:
The visualization shows the 2^50 decision space of a large global corporation using the example of 50 projects under limited budgets. The underlying decision space is domain-independent and can be applied identically to municipal projects, budget decisions and infrastructure portfolios.
2^50 possible combinations correspond to an order of magnitude greater than the number of stars in over 2,800 Milky Ways.
This dimension makes it clear: without algorithmic optimization, the selection is in fact based on heuristic approximations - not on a complete calculation of the global optimum.
A size comparison:
our Milky Way and a corporate decision space with "only" 50 projects
of 1.125 quadrillion possible project combinations
