Network investments with a limited budget - how combinatorial optimization maximizes impact

Why classic expansion decisions fail - and how optimization unleashes real impact

Classification

Network investments are among the most capital-intensive and strategically sensitive decisions in companies, Municipalities and infrastructure sectors. Whether fiber optics, energy, logistics, branch networks, service outlets or distribution routes: The budget is almost always limited, while the number of potential investment points is large.

This is precisely where a classic mathematical problem meets real-life decision-making reality: the Traveling Salesman Problem (TSP) - extended by budget, Priority and impact constraints.

The central question is no longer:
Where do we invest everywhere?
But rather:
What sequence, selection and combination of investments maximizes impact under scarce resources?

1. The TSP as a model for network investments

The classic traveling salesman problem describes the task of visiting a set of nodes (locations) in such a way that:

• every relevant point is considered
• Costs (e.g. distance, time, effort) are minimized
• the overall route is optimal

Applied to network investments, this means

• Nodes: Investment points (locations, regions, network nodes)
• Edges: Costs, dependencies, implementation effort
• Goal: maximum impact with minimum use of resources

In practice, however, the problem is significantly more complex than the classic TSP.

2. Why network investments are not a linear problem

Typical investment decisions are often made on a linear basis:

• Ranking by ROI
• Prioritization according to political or regional pressure
• successive expansion "from the outside in" or vice versa

However, these approaches ignore systematic effects:

• Network effects (value is only created through connection)
• Dependencies between investment points
• Economies of scale and thresholds
• temporal sequences

The result: high investment costs with a disproportionately low impact.

3. The real problem: TSP under constraints

Real network investments involve additional restrictions:

• limited budget
• Priorities (critical regions, key customers, regulatory requirements)
• Dependencies (node A makes node B useful)
• Partial benefits (not every node delivers value in isolation)

Mathematically, it is a combination of:

• Traveling Salesman Problem
• Knapsack problem
• Portfolio optimization

This combination cannot be solved by human intuition.

4. The most common mistake: completeness instead of impact

A classic mistake in network investments is:
"If we're going to invest, let's do it as comprehensively as possible."

This leads to

• too many half-finished networks
• low capacity utilization
• high capital commitment
• politically "beautiful" but economically weak solutions

Optimal solutions are often not complete, but targeted and combined.

5. Sequence is more important than area

With the TSP, it is not only relevant which points are visited, but in which order. Applied to investments, this means

• incorrectly set initial investments block the budget
• correctly set starting nodes multiply later effects
• some investments are only worthwhile with an existing basis

6. Why experience and Excel are not enough

Above a certain network size, the number of possible variants explodes:

• 10 investment points → millions of combinations
• 15 investment points → billions of variants
• including sequence → exponential explosion

Excel, workshops and priority lists artificially reduce this complexity - and thus and thus create a systematic loss of efficiency.

Proof (formal): Why experience and Excel are not structurally sufficient

The structural limit of classic decision-making approaches for network investments is mathematically justified. Even with moderate network sizes the solution space does not grow linearly, but facultatively or exponentially. This effect is independent of experience, organization or tool selection.

6.1st selection problem: Subsets with a limited budget

Let n be the number of potential investment points. Due to a limited budget Budget, only a subset of these points can be realized. The number of all possible subsets is given by :

|\u1d4f(n)| = 2n

Examples:

• n = 10:²¹⁰ = 1,024 combinations
• n = 15:²¹⁵ = 32,768 combinations

This number only describes the selection - not yet in sequence. The actual complexity only arises in the next step.

6.2nd sequence problem: classic symmetric TSP

In the symmetric traveling salesman problem (TSP) with a fixed starting point and identical evaluation of outward and return directions, the number of possible round trips is Round trips:

|\u1d4fTSP(n)| = (n - 1)! / 2

Examples:

• n = 10: 9! / 2 = 181,440 tours
• n = 15: 14! / 2 = 43,589,145,600 tours

Even without a budget restriction, with 15 points there are over 43 billion possible routes.

6.3. real investment problem: selection and sequence

In real network investments, not all points are expanded. Instead, a subset of size k is selected and an optimal sequence is an optimal sequence is determined.

There is a fixed subset of the size k:

(k - 1)! / 2

possible round trips. The number of subsets of this size is:

n over k = n! / (k! - (n - k)!)

The complete search space thus results as:

Σ (k = 2 to n) [ (n over k) - (k - 1)! / 2 ]
  

6.4. result: Order of magnitude of the search space

Number of points (n)	Selection only (2ⁿ)	Sequence only ((n-1)!/2)	Selection + sequence (Σ)
10	1.024	181.440	≈ 556,036 (≈ 1.11 million without directional reduction)
15	32.768	43.589.145.600	≈ 127,661,752,459 (≈ 255 billion without directional reduction)

6.5. consequence

From about 10-15 investment points, the decision space moves far beyond far beyond what Excel can enumerate or what human human experience can reliably survey.

Excel inevitably reduces this space through pre-selection, Heuristics or linear assumptions. Experience replaces calculation by intuition. Neither leads to optimal solutions, but to structurally suboptimal decisions.

The limiting factor is therefore not competence, but Combinatorics. Network investments of this kind are not a problem of experience, but a a pure optimization problem.

7. Network investments as an optimization problem

Network investments are a combinatorial optimization problem:

• Target value: maximum overall effect
• Variables: Selection and sequence of investments
• Constraints: Budget, time, dependencies, risks

This is the only way to see where budget has real leverage.

8. The strategic added value

Systemically optimized network investments lead to

• greater impact with the same budget
• less political and operational friction
• transparent, justifiable decisions
• better scalability

9. Governance and liability perspective

Calculated, comprehensible decision-making logic reduces liability risks, Reputational damage and political attack surfaces. Transparency itself thus becomes a strategic asset.

Conclusion

Network investments with a limited budget are not a distribution problem, but an optimization problem.

Those who continue to invest linearly distribute the budget - but do not maximize the effect. Those who understand networks as a combinatorial system achieve more results with fewer resources.

The crucial question is not:
How much can we invest?
But rather:
Which investment route generates the maximum overall benefit under real constraints?

Network investments with a limited budget - Calculate combinatorial optimization and impact now