⚡ electrical engineering

Embedding Economic Input-Output Models in Systems of Systems: An MBSE and Hetero-functional Graph Theory Approach

This paper introduces a novel methodology that integrates Economic Input-Output models into a Model-Based Systems Engineering framework using Hetero-functional Graph Theory, thereby creating a scalable, ontologically unified approach to characterize and analyze the interdependent dynamics of complex Anthropocene systems of systems.

Original authors: Mohammad Mhadi Naderi, Megan S. Harris, John C. Little, Amro M. Farid

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohammad Mhadi Naderi, Megan S. Harris, John C. Little, Amro M. Farid

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Connecting the Dots in a Chaotic World

Imagine the world as a giant, bustling city where everything is connected. The economy, the weather, the water supply, and our daily lives are all part of one massive, tangled web. Scientists call this the Anthropocene (the age where humans are the main drivers of change).

The problem is that we usually study these things separately. Economists look at money, hydrologists look at water, and ecologists look at nature. But in reality, if the economy booms, it might drain the water supply. If the water dries up, the economy crashes.

This paper proposes a new way to study these connections. It wants to build a universal translator that lets economists, engineers, and environmental scientists speak the same language and see the whole picture at once.

The Three Tools in the Toolbox

To build this translator, the authors use three specific tools. Think of them as a Blueprint, a Dictionary, and a Calculator.

1. The Blueprint: MBSE (Model-Based Systems Engineering)

The Analogy: Imagine you are building a complex LEGO castle. Before you start, you draw a detailed picture of how every brick fits together. You don't just write a list of bricks; you draw the structure.
In the Paper: This is called MBSE. It uses a visual language (SysML) to draw "blueprints" of how systems work. Instead of just writing equations, the authors draw diagrams showing how money flows, how factories produce goods, and how resources are used. It makes the invisible structure of the economy visible.

2. The Dictionary: HFGT (Hetero-functional Graph Theory)

The Analogy: Imagine you have a blueprint in English, but your construction crew only speaks "Math." You need a dictionary to translate the drawing into a language they understand.
In the Paper: This is HFGT. It's a mathematical framework that acts as a universal dictionary. It takes the visual blueprints (from MBSE) and translates them into a rigorous mathematical structure. It treats everything—water, money, electricity, and time—as "operands" (things being moved or changed) and "processes" (actions being taken). This ensures that the math is perfectly consistent with the drawing.

3. The Calculator: The Economic Input-Output (EIO) Model

The Analogy: Think of the economy as a giant recipe book. If you want to bake a cake (Final Demand), you need flour, eggs, and sugar (Inputs). But to get the flour, you need to mill wheat, which needs electricity, which needs coal. The EIO model tracks all these hidden steps.
In the Paper: This is the EIO model. It's a classic way economists calculate how much of everything is needed to produce a final product. The paper asks: Can we take this classic recipe book and fit it inside our new Blueprint and Dictionary system?

The Experiment: The "Technology Choice" Game

To prove their idea works, the authors created a fake, simplified economy (a "synthetic" example).

The Scenario: Imagine a small world with three industries: Manufacturing, Construction, and Agriculture.
The Twist: Each industry has a choice of technology.
- Manufacturing has only one way to work.
- Construction can use old-school tools or modern robots.
- Farming can use manual labor, hybrid machines, or fully automated drones.
The Goal: The system needs to produce a specific amount of goods (Final Demand) while using the least amount of resources (Water and Capital) possible.

The Challenge: The authors took this "Technology Choice" problem and tried to solve it in two ways:

The Old Way: Using standard economic math equations.
The New Way: Drawing it as a Blueprint (MBSE), translating it to the Dictionary (HFGT), and running it through the Calculator.

The Result: A Perfect Match

The paper found that both methods gave the exact same answer.

The "New Way" correctly calculated that the economy should use modern robots for construction and a mix of manual/hybrid methods for farming to save money and water.
Why this matters: It proves that the new system (MBSE-HFGT) can handle complex economic math without losing any precision.

Why Should You Care? (The "So What?")

Think of the old way of studying the world as looking at a puzzle with the pieces separated into different boxes. One box has the water pieces, one has the money pieces, and one has the nature pieces. You can't see the full picture.

This paper builds a single table where you can dump all the pieces from all the boxes and see how they fit together.

Better Decisions: If a city wants to build a new factory, they can now instantly see how it will affect the local water supply, the job market, and the carbon footprint all at once.
No More Silos: It stops economists from talking to themselves and engineers from talking to themselves. They can finally collaborate using the same "language."
Future Proofing: Because the system is built on a flexible "dictionary" (HFGT), it can easily add new pieces later. If we discover a new type of energy or a new economic crisis, we can just add a new "block" to the blueprint without breaking the whole system.

In a Nutshell

This paper is about building a super-powerful simulation engine. It takes the complex, messy reality of our economy and nature, draws it clearly (MBSE), translates it into perfect math (HFGT), and proves that it works just as well as the old, isolated methods. It's a step toward solving the world's biggest problems by finally understanding how everything is connected.

1. Problem Statement

The paper addresses the critical challenge of modeling Anthropocene Systems of Systems (SoS), where human, natural, and engineered systems are deeply interdependent. Specifically, it highlights a gap in current modeling practices regarding Economic Input-Output (EIO) models:

Isolation: Economic models are often treated as isolated entities or coupled with other systems (hydrology, energy, ecology) using exogenous drivers, failing to capture bidirectional feedback loops and complex interdependencies.
Lack of Unified Ontology: Existing coupling strategies often rely on specialized interfaces, require simplifying assumptions, or lack a shared language to represent heterogeneous domains (e.g., money vs. physical units) consistently.
Analytical vs. Structural Disconnect: Traditional EIO models provide strong analytical rigor but lack the graphical clarity and structural extensibility needed to integrate them into broader, multi-domain System-of-Systems frameworks.

2. Methodology

The authors propose a novel workflow that integrates Model-Based Systems Engineering (MBSE) and Hetero-functional Graph Theory (HFGT) to internalize EIO models within a unified SoS framework.

A. The MBSE Component (SysML)

Goal: To graphically define the form (structure) and function (behavior) of the economic system using a common ontology.
Tools:
- Block Definition Diagram (BDD): Represents the system's structural elements (sectors, resources, buffers) and their relationships (aggregation, decomposition).
- Activity Diagram (ACT): Represents the flow of processes, transforming inputs (factors of production) into outputs (products), capturing inter-sectoral transactions and final demand.
Outcome: Creates a domain-independent Reference Architecture (RA) that maps economic concepts to system engineering concepts (Resources, Processes, Operands).

B. The HFGT Component

Goal: To translate the graphical SysML models into a rigorous, computable mathematical framework capable of large-scale quantitative analysis.
Core Concepts:
- Operands ( $L$ ): Objects being processed (e.g., water, capital, goods).
- Processes ( $P$ ): Activities transforming operands.
- Resources ( $R$ ): Entities enabling processes (e.g., the economy itself).
- Capabilities ($ES$): The specific execution of a process by a resource (Subject + Verb + Object).
Mathematical Formulation:
- The system is modeled as an Engineering System Net (a type of Petri net).
- Incidence Tensors: Positive ( $M^+$ ) and Negative ( $M^-$ ) hetero-functional incidence tensors are generated to track the flow of operands through buffers and capabilities.
- Optimization: The system is solved using the Hetero-functional Network Minimum Cost Flow (HFNMCF) problem, a convex optimization program that minimizes total factor costs subject to flow balance and capacity constraints.

C. The Integration Workflow

Modeling: An EIO model (specifically a Rectangular Choice of Technology - RCOT model) is first defined mathematically.
Graphical Abstraction: The RCOT model is mapped to SysML diagrams (BDD and ACT) to define its form and function.
Computational Translation: The SysML model is processed by an HFGT toolbox to generate incidence matrices and tensors.
Simulation: The resulting mathematical structure is solved as an HFNMCF problem to reproduce the economic dynamics.

3. Key Contributions

First Application to Economics: This is the first study to apply MBSE and HFGT specifically to Economic Input-Output models, establishing a bridge between systems engineering and economic theory.
Unified Ontology: The paper demonstrates that economic sectors, technologies, and factors of production can be represented using the same ontological structure (Resources, Processes, Operands) used for physical systems like water or energy networks.
Internalization of EIO: The framework successfully "internalizes" EIO models, treating them as endogenous components of a larger SoS rather than external drivers, allowing for seamless coupling with other domains without custom interfaces.
Generalization of RCOT: The study proves that the Rectangular Choice of Technology (RCOT) model is a special case of the broader MBSE-HFGT framework. This suggests that more complex EIO variants (e.g., Multi-Region IO, Environmentally Extended IO) can also be embedded.
Knowledge Co-production: By using a shared graphical language (SysML) and mathematical framework (HFGT), the methodology facilitates collaboration between economists, ecologists, and engineers.

4. Results

The authors validated the methodology using a synthetic Rectangular Choice of Technology (RCOT) example involving:

3 Sectors: Manufacturing, Construction, Agriculture.
6 Technologies: Different technological alternatives for each sector.
2 Factors: Water and Capital.

Key Findings:

Exact Equivalence: The MBSE-HFGT framework reproduced the exact results of the traditional RCOT optimization.
- Objective Function ( $Z$ ): 805.724 (Total factor cost).
- Final Output Vector ( $x$ ): Matched perfectly across all technology options.
- Factor Use ( $\phi$ ): Capital (498.92) and Water (342.00) usage matched the traditional model.
Structural Insight: The Petri net representation (Fig. 4) successfully visualized the flow of operands (tokens) through the economic system, making the "black box" of economic transactions transparent and traceable.
Scalability: The generation of incidence matrices ( $M = M^+ - M^-$ ) from the SysML model confirmed that the framework can automatically handle the complexity of multi-technology choices without manual matrix re-derivation.

5. Significance and Future Implications

Addressing Anthropocene Challenges: The framework provides a scalable tool to tackle "wicked problems" (e.g., climate change, resource scarcity) where economic, social, and environmental systems are inextricably linked.
Decision Support: By preserving analytical precision while adding graphical clarity, the approach supports better-informed policy decisions that account for trade-offs and synergies across sectors.
Interdisciplinary Convergence: The work aligns with initiatives like the NSF's Growing Convergence Research (GCR), offering a practical pathway to integrate diverse disciplines (hydrology, land use, economics) into a single, consistent modeling environment.
Future Work: The authors are currently applying this framework to the Chesapeake Bay Watershed to model the interdependencies between eutrophication, agricultural impacts, and economic growth, validating the approach in a real-world, multi-stakeholder context.

In summary, this paper establishes a foundational methodology for treating economic systems not as isolated mathematical abstractions, but as integral, heterogeneous components of a global System of Systems, enabling more robust, transparent, and integrated sustainability modeling.