Algebraic Modeling of Social Systems Evolution: Application to Sustainable Development Strategy

Abstract

This paper presents ALMODES, a discrete-time modeling approach for social systems that uses matrix algebra and directed graphs. The method bridges the gap between static network analysis and continuous System Dynamics, offering a transparent framework that reduces data requirements. The method enables clear causal mapping, rapid simulation, straightforward sensitivity analysis, and natural hybridization with agent-based or discrete-event models. Two case studies illustrate its utility for sustainable-development strategy: in an urban public-health setting, modernization and sanitation policies drive sustained declines in disease despite growth, whereas reversing the population-to-modernization link triggers a morbidity rebound that can be prevented by strengthening the modernization-to-sanitation pathway; in a high-tech services Balanced Scorecard model, a baseline backlog spike depresses customer satisfaction, aggressive hiring shortens the spike but erodes income, and coordinated boosts to training and incentives (about twelve percent productivity gain) remove the backlog early, stabilize customers, and improve income, highlighting human-capital policy as a robust lever. ALMODES thus supports pragmatic policy design under limited, expert-elicited parameters. Future research will address uncertainty quantification, time-varying structures and shocks, automated calibration and empirical validation at scale, optimization and control design, richer integration with hybrid simulation, participatory interfaces for stakeholders, and standardized benchmarks across domains.

1. Introduction

This paper introduces the Algebraic Modeling of Evolution of System (ALMODES) method, which employs algebraic techniques, specifically matrix algebra and its connection with directed graphs (digraphs), to represent and analyze the dynamic behavior of social systems. ALMODES addresses several limitations of Forrester’s System Dynamics, as it is grounded in a simple algebraic mechanism with discrete time and a clear connection to directed graphs. This approach eliminates the need for approximate computational methods, reduces data requirements, and enhances accessibility for non-experts. Notably, the use of a discrete time scale aligns naturally with the characteristics of many social phenomena, often spanning intervals from days to years. This feature allows for a more natural hybridization of ALMODES with other approaches, such as agent-based modeling (ABM) [1] or discrete event-simulation (DES) [2,3].

The simplicity of the method—its foundation in matrix and vector algebra—is likewise advantageous when the model must account for uncertainty. One should not anticipate difficulties in incorporating uncertainty in assessments represented by interval or fuzzy numbers. For a small number of uncertain parameters, a straightforward approach is to rerun the simulation for different parameter values (the relatively low algorithmic complexity prevents substantial computational costs). When the statistical distributions are known, an appropriate modification of the algorithm that introduces random values into successive iterations should not pose a problem.

The approach proposed in the article is illustrated with two models related to sustainable development strategies. Since the method is completely new, relatively simple examples were chosen for its application. These are better suited to demonstrate the mechanism of the method than more complex real-world cases.

The first example is a long-term public health model in the urban area. This example presents the foundational principles of applying the ALMODES method. The second example, based on a real model of a high-technology service company, relates to simulating the company’s development using the strategic tool Balanced Scorecard (BSC). The simulation using method ALMODES highlights the importance of stable management in human resources, particularly the implementation of an appropriate incentive policy for employees.

Existing research on the evolution of social systems leaves a clear gap between (i) static digraph/matrix methods (e.g., ISM, DEMATEL, FCM), which primarily characterize equilibrium structure, and (ii) continuous-time System Dynamics, whose reliance on ODE solvers, extensive data, and complex numerics can impede transparency and adoption. This paper addresses that gap by introducing ALMODES—a discrete-time, matrix—digraph framework that reduces data requirements, clarifies model structure, and remains compatible with hybrid approaches. The research hypothesis is that ALMODES can reproduce salient dynamic patterns of social systems and support robust policy design using limited, expert-elicited parameters; specifically, in the public-health and BSC case studies, targeted feedback adjustments and human-capital policies will stabilize trajectories and improve outcomes.

The remainder of this article is organized as follows. Literature review is presented in Section 2. Section 3 outlines the core assumptions of the ALMODES method, its framework, and its application to sustainable strategy problems. Section 4 and Section 5 present two case studies demonstrating the method’s potential for analyzing sustainable development of social systems: a public health model (Section 4) and a Balanced Scorecard (BSC) model (Section 5). Section 6 discusses the results and their broader implications, along with potential avenues for further research.

2. Literature Review

2.1. Systemic Approach in Social Sciences

The systemic approach in social sciences has yielded diverse solutions, often characterized by a static perspective. These solutions involve achieving a stable state of the system through a series of operations, potentially infinite in number. Implicitly, these methods assume that internal processes within the system evolve rapidly compared to the observation period. The primary interest of the researcher lies in the final, typically stable state, which serves as a valuable source of information about the system under investigation. Two pioneering research projects at Battelle were among the first attempts to model complex social systems. These efforts led to the development of two graphical representation methods for problem analysis: Interpretive Structural Modeling (ISM) and the Decision Making Trial and Evaluation Laboratory (DEMATEL).

The ISM was introduced by Warfield in the Batelle reports in the early 1970s [4,5]. ISM is a consistently active area of research, as indicated by contemporary publications. This is evidenced by, among others, two recently published extensive literature reviews devoted to ISM. In a systematic literature review, Kumar and Goel analyzed 1480 publications related to the application of Interpretive Structural Modeling (ISM), sourced from the Scopus database and published between 2000 and 2020 [6]. Their study found a rapid exponential increase in the number of annual publications using the ISM technique since 2000. The paper of Sreenavasan et al. which presents a bibliometric overview of ISM research, with a focus on its linkages to the Sustainable Development Goals (SDGs) and the impact of COVID-19; it also identified several emerging topics for future ISM research, such as blockchain and IoT, environmental management systems, climate change adaptation, smart cities, and humanitarian logistics and their potential linkages to the SDGs [7].

The Decision Making Trial and Evaluation Laboratory (DEMATEL) method, first proposed by Gabus and Fontela [8,9], is a widely applied analytical tool. Its prominence and evolution are highlighted in a systematic review by Si et al. [10], which analyzed 346 papers published between 2006 and 2016. The review categorized these studies into five groups based on the approach used: classical, fuzzy, grey, hybrid ANP-DEMATEL, and other variants.

Recent research continues to demonstrate the method’s adaptability. For instance, Wang et al. [11] developed an Input-Output (IO) DEMATEL model to evaluate industrial development, clarifying the systemic impact of input-output linkages on cleaner production and sustainable consumption. Similarly, Li et al. [12] applied an integrated gray DEMATEL and Analytical Network Process (ANP) technique to weigh the criteria for evaluating green mining performance (GMP) in underground gold mines.

Both ISM and DEMATEL utilize digraphs and matrices to represent relationships. ISM employs Boolean matrices to identify relationships and system structure [13]. In contrast, DEMATEL measures direct impacts on a ratio scale, typically limited to a few points (e.g., 1, 2, 3, 4). Through appropriate scaling, DEMATEL can assess total interactions within the digraph, considering paths of any length. The elements of the output matrix can be combined to compare the significance of system components and their relative positions as ‘causal’ or ‘affected.’

The Weighted Influence Non-linear Gauge System (WINGS) method [14] evolved as a natural extension of DEMATEL. Empirical observations reveal that system components are not homogeneous. Consequently, assessing the system’s state requires not only considering the components’ mutual influences but also their internal strength, which reflects their significance within the system. This internal strength can vary depending on the context, encompassing organizational or political power, size, and other relevant features.

Taken together, these studies show how the WINGS family turns messy, interdependent factors into actionable causal maps across domains. In place-branding, Adamus-Matuszyńska et al. build a systemic model of Katowice and, via an extended WINGS procedure, surface policy levers for image change [15]; in supply chains, Kaviani et al. integrate Delphi and BWM with WINGS to identify, weight, and link reverse-logistics barriers in automotive—pinpointing economic barriers as most critical [16]; and in built-heritage, Radziszewska-Zielina & Śladowski fuse fuzzy WINGS with structural analysis to compare adaptive-reuse variants under imprecise, interdisciplinary criteria [17].

Methodologically, Michnik & Grabowski extend WINGS with interval arithmetic to admit uncertain or divergent expert judgments while retaining transparent, mathematically sound aggregation for group decisions [18]; later advances include fuzzy D-WINGS (D numbers + triangular fuzzy numbers) with MICMAC to handle ambiguity [19], and a rough–fuzzy WINGS-ISM pipeline that exposes hierarchical/causal structure in agricultural sustainable-supply-chain indicators (e.g., the driving role of regulation and subsidies) [20]. Most recently, Zhou et al. go beyond DEMATEL/WINGS-style uniform attenuation with SIDA, which incorporates structural constraints and unequal importance diffusion to reveal “hidden” critical elements in a pharmaceutical ecosystem [21].

The Fuzzy Cognitive Map (FCM), introduced by Kosko [22], is a knowledge representation framework rooted in cognitive mapping and fuzzy logic. FCMs have found widespread application in the fields of social sciences and engineering (see e.g., a monograph [23]). Examples of recent literature reviews include the following: Jiya et al. trace the evolution of fuzzy cognitive maps (FCMs) from the original, conventional model to recent extensions, highlighting learning algorithms, applications, and the distinctive features of each variant [24]. Karatzinis & Boutalis complements this with a broad survey of 80 studies across 15 engineering sub-domains, organized by task type and learning family, and provides clear tables, figures, and future research directions [25].

MICMAC (Matrice d’Impacts Croisés-Multiplication Appliquée à un Classement) a structural analysis tool, employs matrix multiplication to categorize system variables into clusters and ultimately identify key variables [26]. This method exclusively considers the relative order of variables. The primary role of MICMAC analysis, as consistently highlighted in recent studies, is to analyze and categorize variables based on their driving and dependence power. This is most frequently performed as a validation and enrichment step for models developed using Interpretive Structural Modeling (ISM). The synergy between ISM, which establishes the hierarchical relationships between variables, and MICMAC, which classifies their influence, provides a more robust understanding of complex systems.

Recent literature showcases the versatility of the ISM-MICMAC framework across multiple domains. For instance, Ali applied it to identify enablers for green manufacturing [27], and Primadasa et al. (2025) used it to model indicators for Circular Supply Chain Management (CSCM) [28]. In strategic management, Goel et al. identified barriers to Industry 4.0 implementation in SMEs [29], while Quynh classified key strategies for advancing Net-Zero Carbon Procurement (NZCP) [30].

The above-mentioned methods are of a static nature, as evidenced by the fact that their outcome is typically a certain, generally stable final state of the system. However, observing changes over time provides a better understanding of the system and is often indispensable. The study of system evolution is central to numerous fields within natural science. In disciplines such as physics (e.g., classical and quantum mechanics), chemistry, and engineering (e.g., system dynamics and signal processing), mathematical models are widely employed to describe temporal changes. In these fields, time is treated as a continuous variable, and evolution is modeled using differential equations. J. Forrester’s extension of this technique to social systems is known as System Dynamics (SD) [31,32].

Forrester’s System Dynamics (SD) approach has gained significant popularity across various disciplines. In the field of management, its utility is exemplified by the influential monographs of Sterman [33] and Morecroft [34]. The approach’s versatility is further demonstrated by its application in specialized fields, such as water supply and demand management. A recent literature review by Naeem confirms that in this domain, SD is frequently integrated with other methods [35]. The other example of a hybridization of SD with other methods is the paper which reviews how system dynamics (SD) and agent-based modeling (ABM) are used to analyze the complex, multi-stakeholder system of construction waste management, contrasting SD’s strength in macro-level feedback/policy analysis with ABM’s ability to capture heterogeneous micro-level behaviors [36].

The use of continuous time and ordinary differential equations (ODEs) within SD presents certain challenges. This approach demands high levels of data accuracy and completeness, which can be difficult to achieve in the context of social sciences. Furthermore, the complex mathematical techniques involved can hinder communication with users, potentially leading to reduced trust and acceptance of the results. Additionally, the accuracy of the outcomes can be influenced by the specific algorithm used for approximating solutions to systems of ODEs. These limitations represent the primary weaknesses of Forrester’s System Dynamics.

2.2. Sustainable Development Strategy

A sustainable development strategy [37] is a dynamic, coordinated framework designed to meet present human needs while preserving the environment and resources for future generations. The concept, deeply rooted in the Brundtland Report’s definition, emphasizes the balance of economic growth, environmental protection, and social equity, respecting the limitations imposed by current technology, societal organization, and environmental capacity [38]. Such strategies integrate the objectives of various sectors—economic, social, and environmental—into mainstream planning processes and policy-making to maximize human well-being while ensuring the enduring health of the planet [39]. Across two reviews, the SDGs have so far had their strongest effects on discourse—with only scattered normative/institutional change and little evidence of major budget shifts or strong policy coherence [40]. Meanwhile, SDG research itself has surged (12,176 Web of Science papers, 2015–2022), clustering around climate, circular economy, health, and governance; output is concentrated in the USA, China, and the UK (15 citations/article), and the authors flag gaps and priorities for future work [41]).

The following three papers examine the localization of the Sustainable Development Goals (SDGs) at the municipal (city-government) level. Across these complementary studies, the literature converges on what cities need to meaningfully localize the 2030 Agenda. Krellenberg et al. compare sustainability strategies in Hamburg, Magdeburg, St. Petersburg, and Milwaukee using a five-level strategic framework plus SDSN guidance, finding persistent gaps in integrated visions, clear targets, and indicators—while arguing SDGs can structure more coherent, monitorable urban plans [42]. Salvador and Sancho add a management lens, proposing four institutional capacities—strategic leadership, analytical/data capability (and governance), organizational/managerial structures, and collaborative network capacity—and show via a Barcelona case that robust, combined capacity is a precondition for consistent climate and sustainability policy [43]. Extending to the Global South, Teixeira et al.’s case of Birigui (Brazil) documents how municipal adoption of SDGs is propelled by partnerships and institutional pressures (e.g., the National Confederation of Municipalities’ “SDGs Mandala”), and catalogues concrete, locally tailored practices for planning, monitoring, and implementation. Taken together, these works suggest effective SDG localization hinges on marrying integrated, target-and-indicator-driven strategies with the organizational capacities and cross-sector collaborations that turn goals into executable, trackable municipal programs [44].

Highly cited research, such as the bibliometric analysis by Mishra et al. [41], shows that successful cities align sectoral policies—spanning transportation, housing, green infrastructure, public health, and inclusive services—under unified strategic frameworks, involving both local stakeholders and cross-sector partnerships to advance social equity, economic vitality, and ecological resilience. Public health has emerged as a central issue within municipal sustainable development strategies, reinforcing the need for clean environments, access to healthcare, urban greening, and resilience to climate-related health challenges. These approaches emphasize robust monitoring systems, transparency, and participatory mechanisms to ensure adaptability and responsiveness to evolving local realities [45,46].

Public health management, which is the focus of the first application model in this paper, is a field extensively analyzed from a complex systems perspective. The literature offers several frameworks for classifying relevant methodologies, as highlighted in a comprehensive review by McGill et al. The authors categorize the approaches into four primary domains: (i) mapping approaches (e.g., Causal Loop Diagrams, System Mapping); (ii) simulation modeling (e.g., System Dynamics, Agent-Based Modeling, Discrete Event Simulation); (iii) network analysis (e.g., Social Network Analysis); and (iv) ‘system framing’, which involves the qualitative use of systems thinking to guide evaluation design and interpretation [47]. Complementing this, a review by Mansouri et al. centers on quantitative simulation techniques, identifying three core methods: System Dynamics (SD), Agent-Based Modeling (ABM), and Hybrid Models that integrate multiple approaches, such as combining SD with ABM [48].

Beyond these classification frameworks, a notable trend in recent research is the integration of optimization techniques with simulation. For instance, Rehman developed a hybrid simulation-optimization model to enhance resource allocation in a hospital’s Emergency Department by combining Discrete Event Simulation (DES) with goal programming [49]. In another application, Zhang employed dynamic integer linear programming to optimize emergency medical logistics, uniquely minimizing costs by factoring in demand urgency via time-penalty functions [50].

The systemic model proposed by Maruyama [51], particularly after its analysis with the Fuzzy Cognitive Mapping (FCM) approach by Tsaridas and Margaritis [52], has served as a foundational framework for extensive research into health issues. A recent review by Sarmiento et al. [53] underscores this influence, analyzing 25 participatory FCM projects across nine countries that tackle various health challenges. The application of such participatory systems thinking extends beyond health. For instance, Currie et al. [54] presented a system-dynamics model, co-developed with residents and municipal authorities, to evaluate sanitation technology portfolios. Similarly, Falcone and De Rosa [55] utilized participatory FCM to derive policy strategies for optimizing municipal solid-waste systems. In a more general follow-up to Maruyama’s original idea, Parreño and Pablo-Marti [56] conducted extensive interviews with experts in Ecuador to construct an FCM that identifies key elements influencing the short- and long-term well-being of citizens.

Employment fluctuations—in the form of high employee turnover [57,58] or unstable work arrangements—may significantly impede progress toward organizational sustainability. Research demonstrates that sustainable business practices are strongly associated with improved employee retention, job satisfaction, and organizational commitment, while high turnover increases training costs, disrupts knowledge transfer, and can lower fidelity to evidence-based or sustainable business practices [59]. Furthermore, people-oriented initiatives such as shortened working weeks, flexible scheduling, and work-life balance programs have been shown to reduce absenteeism and burnout while promoting social well-being and economic resilience, key pillars of sustainable human resource management [60]. Addressing the problem of customer and employment fluctuations through the lens of sustainable development not only enhances organizational competitiveness but is also central to fostering inclusive, resilient, and equitable economies.

3. The ALMODES Method

This section presents a detailed description of the ALMODES method. In earlier work [61], an incorrect assumption was made that the final state of the system at period n was calculated as the sum of all previous states from 1 to $n-1$. This approach was inspired by methods such as DEMATEL [8,9] and WINGS [14]. However, upon closer examination, it became apparent that a more appropriate approach, consistent with common practices in natural sciences, involves calculating the next system state as a function of the previous state. The iterative nature of these models, where successive system states are derived from preceding ones, implies that the entire historical trajectory of the system from its initial state exerts an indirect influence on its current condition. Consequently, the current state encapsulates the cumulative impact of the system’s entire history.

The ALMODES method is based on the assumption that the behavior and evolution of a system are driven by causal relationships between its components. The model acknowledges limited interactions with the environment, such as inputs received from and outputs exerted upon the external world, in accordance with established principles of system analysis. ALMODES shares this core assumption with Forrester’s System Dynamics.

Another assumption is related to the discrete time measurement. This assumption posits that within each specified time interval, a causal component triggers a state change in its directly related component. It is backed up by observation that in social science, the majority of phenomena unfold over significantly longer time scales compared to those observed in natural science. Furthermore, data collection often occurs at intervals of at least a day, with exceptions in specialized financial domains. This justifies the use of discrete-time models to analyze social phenomena.

3.1. Problem Structuring

As mentioned in the Introduction, ALMODES is related to other methods that can be collectively referred to as the systems approach. Therefore, the preliminary stage of the method, which involves building a model—i.e., problem structuring—can proceed similarly to other systems methods. The user—an individual or, as is most commonly the case, a group interested in solving the problem—takes steps to identify the most important components of the system and their interrelationships. The next step is to quantitatively estimate the states of these components and their mutual influences.

The state of system components can be quantified using either natural units (for measurable quantities) or conventional measures (for complex, indivisible components). While natural units are straightforward, conventional measures require careful definition to accurately represent the state of abstract or complex elements. The units and values of the interactions between the components should also be chosen accordingly. This is particularly common in decision-making models involving intangible concepts.

An example of the ALMODES procedure is described in the public health model (Section 4). It serves as the introductory model that illustrates the way of working with ALMODES before we can proceed to more complex cases. As the public health is a crucial component of the human well-being in their living space, this example is inextricably linked to sustainable city development.

The next example refers to the sustainable development of a high-tech service company. It is based on the popular Balanced Scorecard (BSC) model. It is more detailed than the first example and can serve as a tool for strategic management in a company. Both models address the issue of sustainable development and demonstrate the possibilities of the ALMODES method.

3.2. The ALMODES Method Framework

The proposed method adheres to the principle that the studied system can be effectively modeled as a network of causal relationships. A directed graph serves as a perfect representation for this purpose.

Figure 1 illustrates an example of a system model graph. In this figure, and all subsequent ones, we follow the convention that an absent arc between initial vertex $Ci$ and final vertex $Cj$ (where i and j are natural numbers) indicates no coupling. This includes self-coupling, meaning the arc from $Ci$ to itself is omitted (note that the component’s index uniquely identifies it, so we may use $Ci$ or simply i interchangeably).

We associate a matrix $\mathbf{A}={\left[a_{ij}\right]}_{i,j=1,\dots ,N}$ with the digraph, where N is the number of system components. This matrix, referred to as the interaction matrix or dependency matrix, is analogous to the coincidence (or adjacency) matrix, but its elements can be any real number, representing the strength of interaction between system components. The $(i,j)$ element of successive powers of the matrix $\mathbf{A}$ (i.e., ${\mathbf{A}}^2$, ${\mathbf{A}}^3$, etc.) represents the algebraic sum of the path values between vertices $Ci$ and $Cj$. The path value is calculated as the product of the values along the path’s arcs. For instance, a two-arc path from C3 to C1 via C2 in Figure 1 has a value of $a_{32}{a}_{21}$. This value contributes to the (3, 1) element of ${\mathbf{A}}^2$.

We will assume that a direct interaction, represented by a single arc, takes a unit of time. Indirect interactions, involving multiple arcs, will require a corresponding number of time units. This establishes a discrete time framework, where time is measured in consecutive natural numbers.

To describe the evolution of a system with n components, we employ vector calculus, a fundamental tool in many natural sciences. We define the system’s state at time step t as an n-dimensional real-valued vector $\mathbf{s}\left(t\right)$, where $t=1,2,\dots$. Given the interaction matrix $\mathbf{A}$, the influence of other components on a specific component is captured in the corresponding column. To determine the system’s state at the next time step, we multiply the current state vector $\mathbf{s}\left(t\right)$ by the interaction matrix $\mathbf{A}$ from the right: $\mathbf{s}\left(t\right)=\mathbf{s}(t-1)\mathbf{A}$. For convenience, we will represent the state vector $\mathbf{s}\left(t\right)$ as a row vector in subsequent discussions. The successive powers of the interaction matrix $\mathbf{A}$ capture the cumulative effects of interactions within the system over time. In general:

$$\mathbf{s}\left(t\right)=\mathbf{s}(t-1)\mathbf{A}=\mathbf{s}\left(0\right){\mathbf{A}}^t.$$

(1)

In summary. Concepts that are elements of the system are represented by the nodes in the digraph. They are variables whose values are given by the $\mathbf{s}\left(t\right)$—system’s state vector. Interrelations between concepts (including self-relations) are the elements of the matrix $\mathbf{A}$. They play the roles of parameters modifying the state of the system and are represented by arcs in the digraph.

In the following subsection we present the properties of simple systems, composed of just a few elements. These serve as the building blocks for more complex structures.

3.3. Evolution of a Two-Component System–Causal Loop

The two-nodes system with three links is presented in Figure 2. Depending on the values of the interactions it can represent reinforcing or balancing loop as the special cases.

A simple reinforcing or balancing loop is generated by the matrix

$$\left[\begin{array}{cc}1& a_{12}\\ a_{21}& 0\end{array}\right].$$

(2)

The dynamics of such a two-nodes system are described by the following iterative equation

$$\mathbf{s}\left(t\right)=[s_1\left(t\right),s_2\left(t\right)]=[s_1(t-1)+a_{21}{s}_2(t-1),a_{12}{s}_1(t-1)],$$

(3)

where $a_{12}$ is a rate of change.

With $a_{21}=1$ it will reveal a behavior of the reinforcing loop, while with $a_{21}=-1$–the balancing loop. Both these loops co-create the population model described in the next subsection. On the right hand side of Figure 2, there is simulation of the balancing loop with $a_{12}=0.05$ and $\mathbf{s}\left(0\right)=[1.05,0.05]$.

3.4. Evolution of a Three-Component System

We will explore two intriguing three-node system configurations. The first configuration, depicted on the left of Figure 3, is governed by the following interaction matrix:

$$\left[\begin{array}{ccc}1& 0& 0\\ a_{21}& 1& 0\\ a_{31}& 0& 1\end{array}\right].$$

(4)

The evolution of the state of this system is given by:

$$\mathbf{s}\left(t\right)=[s_1\left(t\right),s_2\left(t\right),s_3\left(t\right)]=[s_1(t-1)+a_{21}{s}_2(t-1)+a_{31}{s}_3(t-1),s_2(t-1),s_3(t-1)]$$

(5)

When $a_{21}>0$, it becomes the inflow rate and $s_2$ represents inflow. For $a_{31}<0$, it becomes outflow rate and $s_3$ represent outflow. $s_1$ plays a role of a container, similarly to the case of System Dynamics. This configuration can model many various scenarios like the number of customers with inflow and outflow from/to a market. Example of its simulation is on the right on Figure 3 with $\mathbf{s}\left(0\right)=[1,1,1]$, $a_{21}=0.05$ and $a_{31}=-0.03$. According to Equation (5) it reveals the linear growth with the net rate $0.02$.

The evolution of the second structure is shown on the left in Figure 4 with its simulation example on the right. It is governed by the following interaction matrix

$$\left[\begin{array}{ccc}1& a_{12}& a_{13}\\ 1& 0& 0\\ -1& 0& 0\end{array}\right].$$

(6)

The dynamics of this system are described by the following equation

$$\mathbf{s}\left(t\right)=[s_1\left(t\right),s_2\left(t\right),s_3\left(t\right)]=[s_1(t-1)+s_2(t-1)-s_3(t-1),a_{12}{s}_1(t-1),a_{13}{s}_1(t-1)]$$

(7)

This structure models a population $s_1\left(t\right)$ that is modified by two factors: $s_2\left(t\right)$ represents population growth at a rate of $a_{12}$ ($a_{21}=1$), while $s_3\left(t\right)$ represents population decline at a rate of $a_{13}$ ($a_{31}=-1$). Again, this is a one of the most useful substructures of many systems. E.g. in biological terms, $a_{12}$ corresponds to the birth rate, and $a_{13}$ to the death rate. The example of the model simulation with $\mathbf{s}\left(0\right)=[1,0,0]$, $a_{12}=0.1$ and $a_{13}=0.14$ is presented on the right of Figure 4. According to Equation (7) it reveals non-linear behavior. In both ALMODES and System Dynamics, nonlinear behavior can emerge from the recursive application of linear operations. Even when the underlying models consist of linear matrix-vector calculations (ALMODES) or linear differential equations (SD), the presence of feedback loops introduces non-linearity as the system evolves over time.

3.5. Scalability

As models increase in size and complexity, their scalability—encompassing computational performance, structural manageability, and governance—becomes a primary determinant of their practical utility. Like System Dynamics (SD), the ALMODES framework faces certain scalability challenges common to large-scale modeling. These shared limitations include structural bloat, which complicates model visualization and maintenance; the high computational cost of propagating uncertainty; and the nontrivial governance tasks of versioning and testing complex models.

Despite these similarities, the simpler algebraic foundation of ALMODES allows it to circumvent several critical problems inherent in SD. Specifically, ALMODES is not constrained by the computational demands of solving stiff or nonlinear ODEs, which force tiny timesteps in SD simulations. It also avoids the numerical instability and boundary mismatches that arise in SD when coupling systems with vastly different time scales.

Furthermore, ALMODES offers distinct advantages over Fuzzy Cognitive Maps (FCM). It is inherently free from the convergence and stability problems that can plague FCMs, where iterative calculations may oscillate or diverge, especially in large models. Additionally, ALMODES does not rely on arbitrary threshold functions, which in FCMs have a strong influence on outcomes despite lacking a direct interpretation within the system being modeled.

4. Modeling Public Health Dynamics with ALMODES

Since the method is being introduced in its present form for the first time, a simple illustration of its procedure is necessary. The foundational model of public health, which originated in the 1960s, will serve this purpose. It was presented by Maruyama in a highly influential publication [51]. This model was subsequently analyzed as a FCM by Tsadiras and Margaritis [52]. It is a good example of the problems that can arise in the sustainable development of a city.

Table 1. The concepts in the public health model.

Concept	Unit	Initial Value
C1 Population of a City	[100,000]	0.5
C2 Migration into a City	[10,000]	0.3
C3 Modernization	[conventional]	1
C4 Garbage per Area	[25,000 t]	1
C5 Sanitation Facilities	[conventional]	1
C6 Number of Diseases per 1000 Residents	[100]	1
C7 Bacteria per Area	[conventional]	1

provides a list of concepts, while Figure 5 illustrates the corresponding map. For our purposes, the model has undergone minor modifications; however, its main structure remains the same. Among others, we have incorporated an additional negative impact of Diseases on Migration, reflecting the awareness of potential migrants regarding the city’s health conditions and their subsequent decision-making. For measurable concepts, we have selected units that align with initial values within the approximate range of [0, 1]. Furthermore, the values of impacts have been calibrated to accurately represent the relationships between these concepts. Similarly, relationships involving abstract concepts have been adjusted to appropriate quantities.

4.1. Starting Point Model

In accordance with FCM principles, conclusions are drawn from the relative positions of concepts at a stable equilibrium. In this instance, the authors observed a correlation between increased urban migration, population growth, and associated waste generation. However, concurrent modernization and sanitation efforts have stabilized bacterial levels and led to a decline in per capita disease rates [52].

In ALMODES, we can employ natural units without restrictions on the range of model parameters (unlike FCM, which is confined to the interval $[0,1]$ or $[-1,1]$). This flexibility allows us to monitor the system’s evolution at every stage from its inception. The choice of time unit, ranging from 3 to 12 months, is determined by data availability and expert assessments of the system’s evolutionary pace (in our case, data aligns with a 12-month interval). Despite these differences, ALMODES and FCM have yielded comparable results.

4.1.1. Estimation of Model Parameters

This illustrates some model-building steps. All units and values refer to a one-year time period. The initial population is 50,000 (measurable quantity). Using a unit of 100,000, the initial value is 0.5. We simplify by assuming a stable population (births and deaths are roughly equal), represented by a self-coupling value of 1. (This parameter can easily model population growth or decline due to birth/death imbalances). Initial net migration is 3000 per year (0.3 in units of 10,000), affecting population size. Since the population unit (100,000) is 10 times larger than the migration unit (10,000), the migration’s impact on population is set to 0.1.

Garbage is also a measurable quantity, with a unit of 25,000 tons. The initial value is 1 (current value). Each population unit increase (100,000) increases garbage by 50,000 tons (two units), so the population’s impact on waste is 2 (more precisely, 2 tons per person).

Population affects city modernization—a complex concept with no natural unit. It is convenient to use 1 as the initial state, remaining constant over time (self-coupling = 1). However, modernization can increase each period depending on population size (expenditure proportional to local taxes). Experts estimated that for each population unit (100,000), modernization investment increases its level by 20% (impact value 0.2). Other model quantities are estimated similarly.

4.1.2. ALMODES Results

As shown in Figure 6, over a 20-period horizon, the city exhibits sustainable development. As illustrated in the left panel of Figure 6, driven by the Population-Modernization-Migration feedback loop, these quantities exhibit exponential growth proportional to one another. While Garbage generation and Bacteria increases proportionally, adequate levels of Modernization and Sanitation effectively reduce Diseases levels to a minimum from around the 10th period (right panel on Figure 6). Although Diseases level eventually reaches zero, this may be an artifact of the model, as garbage-generated bacteria are not the sole source of diseases in society. This effect can be understood as minimizing sanitary risks originating from garbage.

4.2. Overcrowded City Model

In a densely populated city, population growth can impede modernization. Reversing the Population-Modernization impact within the FCM model led to cyclical behavior [52]. To address this, an intervention was implemented, adjusting the activation levels of certain concepts based on external factors. In the context of public health, to represent local authority intervention, the Sanitation Facilities concept was assigned its maximum value of +1. Following this adjustment, the system reached a stable equilibrium [52].

ALMODES enables a far more precise observation of this process than is achievable with the FCM model. A reduction in the Population’s impact on Modernization from 0.2 to $-0.1$ leads to a crisis in the latter and its virtual collapse starting from period 13. In turn, the influence of Modernization on Migration causes its significant reduction and subsequent stabilization within the range of approximately 0.33–0.36. The Population, however, exhibits approximately linear growth (left panel of Figure 7). In the right panel of Figure 7, an increase in Garbage and Bacteria, proportional to the Population, can be observed. Nevertheless, the Sanitation variable ceases to grow in period 13 and stabilizes thereafter. As a result, after an initial decline, morbidity (Diseases) begins to rise around period 17. This signals a more drastic effect that becomes apparent when the simulation is extended to 80 periods, revealing a sharp increase in morbidity in the 60th period. This, coupled with negative Migration, ultimately led to the city’s complete depopulation.

Within the ALMODES framework, we can propose a similar intervention. To counter the challenges of urban overcrowding, a potential strategy is to substantially enhance the role of Sanitation in Modernization. This can be achieved by increasing the impact of Modernization on Sanitation from 0.1 to 0.15. The effects of this intervention are illustrated in Figure 8. Compared with the preceding situation (Figure 7), the Population grows similarly, and the decline in Migration is less pronounced. Due to the effect of overpopulation, Modernization still falls steeply (Figure 8, left panel). In this instance, however, Sanitation levels off at a higher value, and unlike the previous case, morbidity does not start to rise (Figure 8, right panel).

5. Applying ALMODES to Sustainable Strategy in a High-Tech Service Sector

In this section, we introduce a more sophisticated model designed to support sustainable strategies within commercial organizations. This model is based on the Balanced Scorecard (BSC) approach [62,63], which is widely cited in academic literature and enjoys considerable popularity among practitioners (see, for example, recent reviews [64,65]). Our example extends a System Dynamics model originally developed for a high-tech services company [66]. In the original research, the BSC framework was utilized to capture the company’s structure and processes, while System Dynamics was employed to conduct simulations.

Figure 9 presents a comprehensive overview of the system, which is divided into four main components corresponding to the four perspectives of the BSC. For clarity, each perspective is marked with a different color:

• Customer perspective (blue): Number of customers, customer growth, and customer decline.
• Internal Processes Perspective (violet): Service backlog, service demand (increase), and service delivery (decrease).
• Learning and Growth Perspective (red): Number of service employees, employee growth and employee decline. Parameters determine the employees productivity, taking into account spending on salaries, training and incentives.
• Financial Perspective (green): Revenue, expenses, personnel costs, training costs, marketing and sales costs, and net income.

In the following subsections, we will delve into the specifics of each perspective. We adhere to the convention that the influence parameter (depicted as an arrow in the network) corresponds to the perspective of the influencing concept. One month is selected as the fundamental time unit. Initial values for variables and parameters are derived from both internal and external data gathered within the company. For each concept, the initial value represents its state at time zero. Parameters are assumed to remain constant throughout the simulation, except in cases where scenario testing involves changes to specific parameters.

5.1. Customer Perspective

• Concepts

• Number of customers ($cu\ge 0$).The core concept, influenced by customer growth and decline ($cu\left(0\right)=100$).
• Customer Growth ($cui$). Represents the rate at which new customers are acquired through sales and marketing efforts ($cui\left(0\right)=0.5$). This growth rate is influenced by marketing and sales spending ($cosm$) and the potential customer growth rate ($cuir$).
• Customer Decline ($cud$). Represents the rate at which customers leave due to various factors, primarily long wait times ($cud\left(0\right)=0.5$). This decline rate is influenced by the service backlog ($seb$) and the customer decline rate factor ($cudr$).

• Parameters

• Average order size ($seoc$). The average number of service units ordered by a customer per time period ($seoc\left(0\right)=3$).

5.2. Internal Processes Perspective

• Concepts

• Service Backlog ($seb\ge 0$). This key metric represents the accumulation of unfulfilled service requests ($seb\left(0\right)=75$). It increases with the total service orders ($seo$). The backlog decreases as services are delivered ($sed$). A high service backlog triggers an increase in the number of service employees ($emsei$)—more hiring. The rate of this increase is influenced by the backlog level and a parameter $emseir$.
• Total Service Orders ($seo$). It is a product of the number of customers ($cu$) and the average order size ($seoc$) ($seo\left(0\right)=300$).
• Service Delivery ($sed$). The quantity of services delivered is directly proportional to the number of service employees ($emse$) and their average productivity ($supe$) ($sed\left(0\right)=280$).

• Parameters

• Customer decline rate factor ($cudr$). It measures the influence of service backlog on customer decline rate ($cudr\left(0\right)=0.01$).
• Rate of service employment increase ($emseir$). It is a rate that increases service employee number, proportionally to the service backlog ($emseir\left(0\right)=0.03$).
• Unit price ($pr$): Together with service delivery it determines the revenue ($pr\left(0\right)=1.5$).

5.3. Learning and Growth Perspective

• Concepts

• Number of Service Employees ($emse\ge 0$): The number of service employees is influenced by both hiring ($emsei$) and attrition ($emsed$) ($emse\left(0\right)=170$).
• The Increase in Service Employment ($emsei$): It is driven by the service backlog ($seb$) with a parameter hiring rate ($emseir$) ($emsei\left(0\right)=1$).
• The Decrease in Service Employment ($emsed$): It is proportional to the current number of employees with a parameter attrition rate ($emsedr$) ($emsed\left(0\right)=2$).

• Parameters

• Average productivity ($supe$). The core parameter measuring the performance of service employees ($supe\left(0\right)=1.75$)
• Attrition rate ($emsedr$). It measures the rate of employees leaving the service ($emsedr\left(0\right)=0.013$).
• Average salary ($apc$). It is a unit personal cost per service employee. ($apc\left(0\right)=2$).
• Average training cost ($cot$). It is a unit cost of professional training per service employee ($cot\left(0\right)=0.1$).
• Average incentives cost ($coi$). It covers cost of all kinds of incentives per service employee ($coi\left(0\right)=0.1$).

5.4. Financial Perspective

• Concepts

• Revenue ($rv$). It is generated by multiplying the quantity of delivered services ($sed$) by the average price per service unit ($pr$) ($rv\left(0\right)=440$).
• Marketing and Sales Costs ($cosm$). The spending dedicated to marketing and sales that influence the growth of customers through the rate ($cuir$) ($cosm\left(0\right)=11$).
• Salary Costs ($cop$). It is calculated by multiplying the number of service employees ($emse$) by the average salary per employee ($apc$) ($cop\left(0\right)=340$).
• Training Costs ($cot$). It is calculated by multiplying the number of service employees ($emse$) by the average training spending per employee ($atc$) ($cot\left(0\right)=17$)
• incentives cost ($coi$) It is calculated by multiplying the number of service employees ($emse$) by the average spending for incentives per employee $aic$ ($coi\left(0\right)=17$).
• Income ($in$) is the difference between revenue ($rv$) and expenses ($ex$) ($in\left(0\right)=60$).

• Parameters

• Customer growth rate ($cuir$). It measures rate of customers growth ($cui$) per unit of marketing and sale spending ($cuir\left(0\right)=0.05$)

In Figure 9, three dashed lines are depicted. They represent the assumed links between the average costs per employee (salary, training, and incentives) on one side, and the rate of service employment increase ($emseir$) and average productivity ($supe$) on the other. These connections provide the foundation for managerial interventions in the strategic scenarios discussed below.

For simplicity, we assume that factors such as overhead costs remain constant during the analysis period, and thus do not significantly impact the system’s dynamics. Additionally, we assume a linear relationship between taxes and income, which simplifies the model without compromising its core insights.

5.5. Simulation and Analysis

To identify key drivers of system behavior and inform sustainable strategic planning, we conducted simulations over a 30-month period, using a monthly time step. While this example serves as an illustration, it highlights the potential of ALMODES to uncover critical insights.

A preliminary review of the simulation results indicates strong nonlinear effects and significant fluctuations in the values of the variables. Given that the uncertainty of external and internal data increases over the longer term, we will focus our attention on the first 18 months of the simulation. No later than approximately 10–12 months later, the simulations should be repeated, taking into account the changed situation.

5.5.1. Initial System State

Figure 10 and Figure 11 illustrate the initial state of key system variables. A significant concern arises from the substantial increase in the service backlog between months 3 and 16, peaking almost 91 units in 7th month and drops to zero in 20th month (see Figure 11, left). This backlog surge is correlated with a 4% decline in customer numbers, from 100 to about 96 in 15th month, likely due to increased wait times Figure 10, left). The inadequate number of service employees (Figure 10, right) is identified as the root cause of the service backlog and subsequent customer loss. Income fluctuates within the range of 53-62 units, reaching the value of about 61 units in the 20th month (see Figure 11, right).

5.5.2. Scenario 1: Increased Hiring

To address the service backlog and customer loss, a potential intervention involves more aggressive hiring. By increasing the parameter $emseir$ from 0.03 to 0.039 (a 30% increase), coupled with raising the average salary to 2.2, we observe the following effects:

• The maximum backlog decreased to 85, reaching its peak earlier (at 5th month) and returning to zero sooner, in 15th month (Figure 12, left panel).
• The decline in average customer numbers is slightly mitigated, reducing by about 2.5%, from a maximum of 100 to a minimum of 97.5 (Figure 13, left panel).
• Employment rises to a higher level compared to the initial scenario, which contributes to the reduction in backlog (Figure 13, right panel).
• Income drops significantly, falling from an initial value of 60 to 28 units by the 10th month (Figure 12, right panel)

5.5.3. Scenario 2: Increased Training and Incentives Budget

An alternative scenario involves increasing the training and incentives budget. Company analysis indicated that doubling both training and incentives spending per employee in a coordinated manner is the optimal strategy. Smaller increases in these budgets produce less significant effects, while larger increases yield diminishing returns. This approach is expected to raise average productivity by approximately 12%. The resulting outcomes are as follows:

• The backlog begins to decrease immediately, reaching zero as early as the 4th month (Figure 14, left panel).
• The number of customers remains virtually stable, peaking at over 105 in the 17th month (Figure 15, left panel).
• Improved efficiency allows for a reduced workforce, with employee numbers declining to around 145 during the first 17 months (Figure 15, right panel).
• Income rises to about 78 units by the 6th month, then gradually declines to a range of 65–67 units between the 16th and 20th months Figure 14, right panel).

5.5.4. Validation and Sensitivity Analysis

Because this paper introduces a novel method, it is illustrated using simplified models. This prevents a comprehensive validation of these models (see e.g., [67]). Although the model described here refers to the same organization and its processes, and is also based on the Balanced Scorecard (BSC) like the one in [66], it is not identical, making a direct comparison of the results impossible. Nevertheless, qualitative similarities stemming from the conceptual resemblance of both models can be observed. In both models, fluctuations in the service backlog and the number of customers, among other things, can be seen. These become apparent at certain parameter values. Due to the different structure of the ‘Growth and Learning’ submodel, the fluctuations in the number of employees that occur in this work are not visible in [66].

Initial validity tests, including extreme condition tests and degenerate tests consistent with methodology described by Sargent [67], confirm the robustness of the presented model. For instance, an increase in the customer growth rate ($cuir$) parameter leads to a proportional increase in the number of customers relative to the baseline. Similarly, a rise in the attrition rate ($emsedr$) parameter results in reduced employment and an increase in the backlog. Conversely, when the customer growth rate ($cuir$) is set to zero, the model shows a continuous decline in the number of customers, accompanied by a drop in employment and revenue.

For the purpose of a sensitivity analysis, we will investigate the effect of changes to the key parameter, average productivity ($supe$), within a range of $\pm 10\%$ to $\pm 20\%$. The resulting impact on the primary model variables is shown in Figure 16 and Figure 17. A qualitative analysis of the differences presented in these graphs, even without referring to the exact values, allows conclusions to be drawn.

At first glance, all variables behave rationally and in line with intuition. When productivity declines, the service backlog increases approximately in proportion to the magnitude of the drop. Conversely, higher productivity accelerates the reduction of the backlog and extends the period during which it remains at zero (Figure 17, left panel).

A similar pattern holds for the number of customers: under a decline in productivity, the customer count contracts more rapidly with a 20% reduction than with a 10% reduction, and the trough is approximately proportional to the size of the decrease. When productivity rises, however, the increase in customer numbers is smaller than the decline observed under reduced productivity, and moving from +10% to +20% no longer yields a proportional effect (Figure 16, left panel). This suggests that productivity and backlog effects alone are insufficient to expand the customer base; additional measures, for example in marketing and sales, may be required.

Employment exhibits the expected inverse relationship with productivity: it rises in rough proportion to a fall in productivity but decreases non-proportionally when productivity increases. This asymmetry runs contrary to the pattern observed for the number of customers (Figure 16, right panel). Revenues, by contrast, vary approximately in proportion to changes in productivity in both directions; a 20% decline in productivity quickly leads to losses (Figure 17, right panel).

As noted above, improving productivity and reducing the backlog were, by themselves, insufficient to increase the number of customers. Therefore, to extend the sensitivity analysis, a simulation was performed by varying the effectiveness of marketing and sales activities. In the model, this effectiveness is measured by the customer growth rate ($cuir$) parameter from the financial perspective. The results are illustrated in Figure 18 and Figure 19.

Similar to the previous analysis, the changes induced by modifying the customer growth rate exhibit logical behavior. The number of customers increases as the $cuir$ parameter rises and decreases as it falls. In this case, however, the changes are weaker, and there is no asymmetry between upward and downward modifications, unlike the dependency on productivity changes (Figure 18, left panel). The changes in employment follow a similar pattern, exhibiting a significantly smaller amplitude than the changes caused by variations in productivity (Figure 18, right panel).

The backlog also shows considerably less variability than in the previous analysis and, as expected, grows with an increase in the $cuir$ parameter (Figure 19, left panel). Financial performance is also proportional to the changes in $cuir$ and is substantially more stable (Figure 19, right panel).

In summary, this sensitivity analysis indicates that the model responds correctly to changes in the customer growth rate and that this parameter has a weaker overall impact on the simulation results. In practice, this suggests that a significant increase in the number of customers would likely also require increased spending on marketing and sales.

5.5.5. Simulations Summary

The results suggest that initiatives aimed at boosting productivity offer a more effective solution than aggressive employment policies. Naturally, such improvements cannot be pursued indefinitely; after several months, alternative approaches may become more advantageous. Additionally, other parameters can be adjusted and tested within the model. Overall, the complex interactions among factors underscore the potential for diverse intervention strategies, often requiring simultaneous adjustments across multiple parameters.

6. Conclusions and Future Research Directions

This paper introduced the Algebraic Modeling of Evolution of System (ALMODES), a novel method designed to analyze the dynamics of complex social systems. By employing a discrete-time, matrix-based framework, ALMODES offers a practical and accessible alternative to traditional System Dynamics. Its effectiveness was demonstrated through two distinct case studies—a public health model and a corporate strategy model—where it successfully identified key intervention points to stabilize system behavior and support sustainable outcomes. The primary contribution of ALMODES is that it bridges the gap between static, structural analysis methods (like DEMATEL, WINGS and FCM) and the complexities of continuous-time differential equation models. In doing so, it presents several distinct advantages:

• Accessibility and Transparency: Its foundation in matrix algebra and directed graphs provides an intuitive and transparent approach to model building, making it accessible to practitioners and stakeholders who may not be experts in advanced mathematics.
• Reduced Data Requirements: The method is well-suited for social systems where quantitative data is often scarce, as it can effectively operate using limited, expert-elicited parameters.
• Computational Efficiency: The algebraic mechanism is computationally straightforward, allowing for rapid simulation and sensitivity analysis without the significant processing costs associated with solving complex systems of differential equations.
• Ease of Hybridization: Its discrete-time nature aligns well with other modeling techniques, creating clear opportunities for hybridization with methods like Agent-Based Modeling (ABM) or Discrete-Event Simulation (DES).

At the same time, as a novel method, ALMODES has several current limitations that must be acknowledged:

• Model Simplification and Validation: The case studies presented in this paper were intentionally simplified to clearly illustrate the method’s mechanics. Consequently, they have not undergone the comprehensive validation required to confirm their accuracy against complex, real-world data, and a direct quantitative comparison with existing models was not possible.
• Reliance on Expert Judgment: The model’s reliance on expert-elicited parameters, while practical, introduces a degree of subjectivity. The framework does not yet include a formal methodology for parameter estimation or validation to mitigate potential bias.
• Assumption of Linearity: While the recursive application of the model can produce nonlinear emergent behavior, the core relationships between system components are defined by linear parameters in the interaction matrix. This may not adequately capture systems governed by inherently strong, nonlinear dynamics.

These limitations pave the way for valuable future research. The immediate priority is to apply ALMODES to more complex, data-rich case studies to rigorously validate its performance against empirical data and established models. Beyond this, several key development paths are envisioned:

• Uncertainty and Robustness: To move beyond deterministic analysis, future work should incorporate interval or fuzzy parameters and introduce stochastic draws in the iteration process. This will allow for reporting distributions of outcomes and defining robustness envelopes for policy recommendations.
• Time-Varying Structures: The model can be extended to allow the interaction matrix, $A\left(t\right)$, to evolve over time in response to policy phases or external shocks. This would enable the study of system resilience and adaptation under structural breaks.
• Control and Optimization: The framework can be enhanced by embedding multi-objective policy search (e.g., balancing service, social, environmental, and financial goals) and implementing simple model-predictive control to provide rolling decision support.
• Methodological Refinements: Further research should focus on developing structured methodologies for parameter elicitation and exploring extensions that can more explicitly incorporate non-linear relationships into the model’s algebraic core.

In summary, ALMODES proves to be a versatile and promising tool for decision support in sustainable development and strategic management. While further validation is needed, its unique combination of analytical rigor and user accessibility makes it a valuable addition to the systems modeling toolkit.