## Abstract

Networking complex sociotechnical systems into larger Systems of Systems (SoS) typically results in improved performance characteristics including sustainability, efficiency, and productivity. The response, or lack thereof, of many SoS to unexpected constituent system failures undermines their effectiveness in many cases. SoS performance after faults can be improved by improving the SoS’s hard (physical design) or soft (human intervention) resilience. The current approaches to increase resilience are limited due to the cost and necessary of human response increasing non-linearly with SoS scale. The limitations of current approaches require a novel design approach to improve SoS network resilience. We hypothesize that biologically inspired network design can improve SoS resilience. To illustrate this, a systems dynamics model of a Forestry Industry is presented and an optimization search over potential hard and soft resilience approaches is compared to a biologically inspired network improvement. SoS network resilience is measured through the newly developed System of System Resilience Measurement (SoSRM). Our first result provides evidence that biologically inspired network design provides an approach to increase SoS resilience beyond hard and soft resilience improvements alone. Second, this work provides evidence that having a SoS constituent fulfill the ecosystem role of detrital actor increases resilience. Third, this paper documents the first case study using the new SoSRM metric to justify a design decision. Finally, this case study provides a counter-example to the theory that increased sustainability always results in increased resilience. By comparing biologically inspired network redesign and optimized traditional resilience improvements, this paper provides evidence that biologically inspired intervention may be the needed strategy to increase sociotechnical SoS network resilience, improve SoS performance, and overcome the limitations of traditional resilience improvement approaches.

## 1 Introduction

Systems of systems (SoS) combine multiple complex sociotechnical systems through a network architecture to achieve additional functionality [15] (Fig. 1). Modern SoS combine constituent systems such as financial, utility, information, agriculture, industrial, or transportation networks. SoS design is a difficult undertaking due to the nature of SoS as complex multi-level network. Within sociotechnical systems, a population of independent agents (often people) interacts with each other, the technical artifacts in each system, and the environment [6]. As a result, sociotechnical systems are complex with many attributes that hamper design intervention. These attributes include emergence (system behavior that is not reducible to agent behavior), self-organization (including spatial-temporal agent distributions), and non-linear response to system stimuli [710]. Further hindering SoS design efforts, these complex sociotechnical systems (which themselves are difficult to design) are then networked together into larger SoS. One significant challenge in attempting to utilize a design methodology on SoS is that many desirable SoS characteristics (e.g., resilience, safety, sustainability, robustness) are themselves emergent that result from the interactions of the constituent complex sociotechnical systems [1,6,7,1114].

Fig. 1
Fig. 1
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Not surprisingly, these challenges have resulted in SoS design approaches that require significant improvement. Previous efforts in SoS design focused on overcoming the logistical and interface challenges of combining systems, but not on predicting or improving SoS operation [15,16]. As a result, engineers have successfully created SoS, but have difficulty controlling or predicting SoS dynamics [17,18]. Specifically, work is needed in improving the response of SoS to unexpected constituent system failures. Researchers recognize the importance of limiting the impact of failures within complex systems [19], but SoS vulnerabilities still include cascading faults, difficulties in anticipating the scope of failures, and identifying critical infrastructure nodes.

### 3.2 Case Study: Kreuzung Schweizer Mittlelland Forestry Region.

The case study examined was of the forestry industry in (KSM, Switzerland. The original model was built on data collected in a study whose goal was to ensure the KSM continued to be self-sufficient [77]. This case study consists of 12 SoS Links and six systems. Flows shown in Fig. 3 are in kilograms dry matter per capita per week. Stocks are recorded in kilograms dry matter per capita. This SoS has three input flows: Z1: forestry growth, Z2: lumber imports, and Z3: pulp paper imports.

Fig. 3
Fig. 3
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KSM was selected because its sociotechnical architecture reflects both conservation of mass flow between the constituents and the interaction of the human choices. This interaction of human choices is defined in some works as a behavioral network [19]. Previous research has considered either conservation based flows (without behavioral interactions) or behavioral interactions within an artifact (without conservation based flows) [19,40]. Although the links in Fig. 3 represent physical connections between the constituent systems, the material flows also impact constituent decisions. Thus, this case study also incorporates the effects of the KSM “behavior networks” an aspect of SoS that was identified as crucial in previous efforts to explore topography changes to system performance [19]. This paper expands upon Refs. [19,40] because these case studies were technical artifacts (i.e., drivetrains and cooling systems), not sociotechnical systems.

Of note, KSM is not at steady-state. The stocks in systems X1 and X5 increase at a rate of 0.2 kg per capita (0.1%) and 1.0 kg per capita (1%), respectively. Second, only systems X1, X4, and X5 are designed to have a stock. X2, X3, and X6 are designed to have no standing stock (i.e., a mean residence time of zero). These three constituents (X2, X3, and X6) have a direct input/output process. Of course, the processes in these three constituent systems (X2, X3, and X6) take time, but for KSM model formulation, we assume stock accumulation during conversion to be negligible compared to the model time-step of h = 1 week. Time step (h) is chosen to be short enough to ensure that model responses accurately reflect system dynamics, short enough to ensure the sum of the rows of the one-step transition matrix are less than one (a step in calculating TT [76]), and long enough to minimize computational expense.

When estimating the value of SoSRM in dollars, the following assumptions were utilized to provide a rough estimate. KSM gross domestic product for the constituent systems is not available, but the Swiss Central Plain (where KSM is located) 2012 lumber profits was approximately $204,249.94 [78]. To provide a conservative estimate, we will assume that X1’s MOP ($204,249.94/year) is the value of the MOP of the entire KSM used to calculate E[$SoSRM]. Of course, X2-6 MOPs have financial value as well, but as an extremely conservative estimate of the financial value of the resilience improvement approaches in this paper, they will be omitted. ### 3.3 Case Study: System Dynamic Model Formulation. The first key modeling decision is defining the MOP for each constituent system. In ecological systems, SoSRM calculations have been conducted with MOPs defined as standing Biomass [76]. This approach is not a realistic option for man-made SoS for two reasons. Some of the systems are designed to have no standing stock (X2, X3, X6), and standing stock is not a reasonable approximation for constituent success. As shown in Eq. (1), MOP definition plays a critical role in SoSRM calculation. MOPs must be carefully defined to avoid distorting simulation results. MOP formulation assumed that constituents desired to maximize their own utility and would always act in their own self-interest. Utility was defined based on the characteristics of each constituent system. Utilities considered were profit, sustainability, and services provided. Not all constituents considered all possible measures of utility (reflecting competing goals in sociotechnical systems). Although the overall goal of KSM is self-sufficiency, most constituents do not consider sustainability in their MOP. For example, the Forestry system (X1) attempts to maximize profit without engaging in unstainable activities while Consumption of Timber (X4) MOP is driven by ensuring sufficient processed lumber was available within KSM. This provides a model that reflects the disparity in real-world motivations and does not require a utopic SoS where every constituent desires sustainability for sustainable SoS behavior to still be a goal. Appendix A presents our specific MOP definitions and justifications. Additionally, the differential equations for KSM must incorporate the actions of the intelligent agents (people) that control each constituent systems. Intelligent agents can make choices to respond to SoS link failures, such as increasing Forestry (X1) production (e.g., L21) when a recycling link (e.g., L24) is lost. Each constituent system will act to maximize their MOP as defined in Appendix A. The KSM differential equations are presented in Appendix B. Examination of the equations in Appendix B reveal that SoS response depends on model parameters that impact both soft and hard resilience. These parameters are summarized in Table 1. Table 1 KSM model parameters and optimization search space ParameterModel valueOptimization searchSystemResilience type MinMax FLAG1 week052X1Soft FTPP.1.11X2Soft FPPP.1.11X3Soft X6nopenalty.205X6Hard X6maxstorage105X6Hard TRR0.10501X2X4Hard PRR0.6201X3X5Hard ForestMin14,000014,000X1Hard ParameterModel valueOptimization searchSystemResilience type MinMax FLAG1 week052X1Soft FTPP.1.11X2Soft FPPP.1.11X3Soft X6nopenalty.205X6Hard X6maxstorage105X6Hard TRR0.10501X2X4Hard PRR0.6201X3X5Hard ForestMin14,000014,000X1Hard Human response following a constituent fault impacts the soft resilience of a SoS. Human response was incorporated into the model three ways. First, Forestry (X1) responds to supply shortcomings downstream by increasing timber production. However, this response from the Forestry (X1) is subject to a delay time (the variable FLAG) as the additional lumber harvesting resources activate of 1 week. The Production of Paper and Lumber systems (X2 and X3) seeks to maintain zero standing stocks. A simple proportional controller throttles the incoming flows from the Forestry Industry with a proportional constant of 0.1. The higher the proportional constant (FTPP and FPPP), the more responsive is X2 and X3. Constituent design decisions impact the hard resilience of the SoS. These design parameters were implemented into the model in the following ways. First, the Incinerator (X6) has two design parameters that impact resilience. The Incinerator (X6) MOP is driven by the constituent system’s desire to maintain standing stock at zero (efficiently incinerate all incoming waste). There is an amount of waste the Incinerator (X6) can store without incurring a financial penalty (X6nopenalty). Additionally, there is an amount of storage where the Incinerator (X6) has so much stored waste that the cost of storage outweighs the profits made as a waste disposal company (X6maxstorage). The recycling rate implemented by X4 and X5 of timber and paper (Timber Recycle Rate (TRR) and Paper Recycle Rate (PRR)) also impact SoS resilience. Finally, the Forestry System (X1) has a minimum level; it will allow the standing stock of trees to reach (ForestMin). This could be highly conservation motivated where the standing stock in Forestry (X1) is not allowed below the initial level of 14,000 kg per Capita. Conversely, the SoS could have no conservation consideration and allow Forestry (X1) level to reach zero kilograms per Capita. While the first scenario would never allow logging below initial Forestry (X1) population, the second approach would allow for complete deforestation in response to a SoS fault. Model parameter settings during SoSRM evaluation are per Table 1. Verification of the modeling included line-by-line code checks and exercising the KSM model over a variety of initial conditions and faults to ensure system performance matched anticipated response. All optimal results were re-simulated and closely monitored to ensure no unexpected model artifacts drove design solutions. The model was executed within Anylogic 8.4 University Edition. Model unit time was weeks. Simulation runs were conducted on a personal laptop with an Intel® Core i5-7000U CPU operating at 2.50 GHz and 16.0 GB of RAM. ### 3.4 Optimizing Status-Quo Resilience Improvements. An optimization search of the KSM model evaluated the possible resilience improvement by utilizing the traditional approaches of improving soft and hard resilience. Two independent searches were conducted. The first objective function was to maximize SoSRM, while the second objective function was to minimize SoSRM. These independent searches provided two insights. First, they revealed if SoSRM was bounded despite resilience improvement efforts. Second, the maximization search resulted in the highest SoSRM achievable by traditional resilience improvement approaches. Table 1 lists the model’s parameter range search space. The search evaluated 500 samples. These results provide a benchmark to compare traditional approaches to improve resilience against biologically inspired network design. The built-in Optimization Experiment within Anylogic was used to conduct the optimization searches. Anylogic utilizes the OptQuest optimization engine, a proprietary population-based metaheuristic search engine [79]. OptQuest utilizes methods such as scatter search, evolutionary approaches aided by a multivariate linear regression module, neural network to identify new trials, satisfiability data mining approach, and Markov Blankets [80]. ### 3.5 Biologically Inspired Network Improvement. Contrasting traditional resilience improvement approaches, we next implemented a biologically inspired network change. Within SoS, network topography changes often occur slowly and require constituent coordination, making them susceptible to design heuristic intervention. The heuristic tested in this paper is: Ensure the SoS has a constituent system that fulfills the decomposer functional role (detrital actor) found in natural ecosystems. Following this heuristic, we added a link from the Incinerator (X6) to the Forestry (X1) constituent system (Fig. 4). Of course, adding a network link is also potentially a form of adding hard resilience to the SoS. Fig. 4 Fig. 4 Close modal In the original configuration, the Incinerator (X6) took the role of apex predator. All other systems sent flows to X6. No constituent system, however, fulfilled the decomposer functional role. This deficit is unsurprising as man-made systems often do not incorporate detrital actors [74]. As shown in Fig. 4, L16 supplements the original SoS by connecting X6 to X1. This link simulates adding a mulching approach to the KSM waste stream, which returns nutrients and energy to the SoS similar to the decomposer role in natural systems. Mulching improves tree seedling survival, tree growth, soil moisture retention, and tree size in forests [81]. For this SoS, we assume that half of the received waste is eligible for mulching, and the remainder disposed by incineration. The only additional impact of L16 to the SoS constituent dynamics is that the flow of L16 caused X1 stock to increase more rapidly. X1 stock increases at a rate of 3.6 kg/capita per week rather than 0.2 kg/capita per week as in the unmodified SoS. The effectiveness of this biologically inspired network rearrangement was tested by both measuring SoSRM as well as conducting an optimization search per Table 1 to determine the maximum and minimum achievable SoSRM for the biologically inspired KSM. The optimization search provides the maximum possible SoSRM when combining traditional approaches with biologically inspired network re-design. ## 4 Results and Discussion Using the biologically inspired design heuristic resulted in an improved SoSRM over traditional hard and soft resilience improvements. These results are summarized in Table 2. The Best Trial row reports which search iteration resulted in the optimal result, while the <2% from best row records when the search results were within 2% of the optimal result. The <2% value provides an indication of how quickly the search converged to the optimal value. Table 2 KSM SoSRM results summary ParameterStatus-quoBio-inspired redesign Status quo optimization Bio-inspired optimization MinMaxMinMax SoSRM0.9090.9260.7640.9220.7430.931 Best TRIAL384363467434 <2% from Best2304730 FLAG1151.90520 FTPP0.10.110.110.1 FPPP0.10.10.880.110.1 X6nopenalty0.20.20.2251.2575 X6maxstorage11151.0075 TRR0.1050.1050.900.8990 PRR0.620.620.010.8070.0030.803 ForestMin14,00014,00013,995592.332540 ParameterStatus-quoBio-inspired redesign Status quo optimization Bio-inspired optimization MinMaxMinMax SoSRM0.9090.9260.7640.9220.7430.931 Best TRIAL384363467434 <2% from Best2304730 FLAG1151.90520 FTPP0.10.110.110.1 FPPP0.10.10.880.110.1 X6nopenalty0.20.20.2251.2575 X6maxstorage11151.0075 TRR0.1050.1050.900.8990 PRR0.620.620.010.8070.0030.803 ForestMin14,00014,00013,995592.332540 ### 4.1 Kreuzung Schweizer Mittlelland Status-Quo SoSRM. SoSRM calculation for KSM Status-Quo is shown in Table 3. Of note, the KSM SoS is quite resilient initially, with a SoSRM of 0.909 (E[$SoSRM] approximately 2 million dollars). One reason is that the SoS inherently has a large amount of hard resilience in the form of unused reserve capacity in the Forestry (X1) and Incinerator (X6) constituent systems. If inflows for the Forestry (X1) system were to cease, it would take 43.4 years for complete deforestation to occur. Similarly, it would take 50.6 years for Timber Consumption (X4) to be depleted to zero standing stocks. The reserve capacity designed into the KSM also resulted in large values for T* and TT. T* was 111.8 years and TT was 55.9 years. These values show that the KSM is relatively insensitive to short duration faults.

Table 3

KSM status-quo SoSRM calculation

### 4.2 Kreuzung Schweizer Mittlelland Traditional Resilience Improvements.

Traditional methods successfully increased SoSRM from .909 to .922. This results in an improved E[$SoSRM] savings of approximately$300,000. The specific parameter results provide some interesting insights into the KSM SoS. First, the TRR that results in the highest SoSRM was 0 (i.e., no recycling from Consumption of Timber to Production of Timber). This counter-intuitive result may be explained by a closer examination of the SoS response when L24 is broken in Table 3. This is one of the two scenarios in Table 3 when the overall performance of the SoS improves due to a broken link. SoS performance improves because the 0.2 inflow to X2 from L24 is replaced by increasing L21 by 0.2. This 0.2 is the flow that normally increases the Forestry (X1) standing stocks. Thus, the performance of X2-6 MOP is not impacted by severing L24. X1 MOP improves because the total wood exported by X1 increases. X1 MOP does not incur a penalty for reducing the standing stock below the minimum allowable forestry level because Forestry level remains at the initial level (14,000 kg per capita). Although the X1 MOP could be redefined, the current MOP formulation was based on the community’s goal to maintain current forest size. Community decision makers felt that unsustainable activity should be avoided, but they also desired to avoid forest management issues from too large of a forest [77]. Thus, we can see counterintuitively that removing a recycling flow results in increased resilience. This scenario implies that although recycling flows may increase resilience, the implementation approach is also important. The simple integration of a recycling flow into a SoS will not guarantee an improvement in resilience (although one would still expect an increase in sustainability). A final interesting observation is that this case study provides a counterexample to the common idea that increased sustainability (by adding a recycling loop between X4 and X2) always leads to increased resilience.

We must also note that these results find the maximum possible improvement to SoSRM, regardless of cost or technological feasibility. For example, it may not be technically feasible to increase the paper recycling rate to 80%. The optimization search is not meant to provide actionable guidance to the KSM operators, rather this study seeks to provide a baseline allowing us to compare possible improvements from traditional resilience improvement approaches to biologically inspired network redesign.

### 4.3 Kreuzung Schweizer Mittlelland Biologically inspired Improvement SoSRM.

Using Biologically inspired design alone (without optimization) increased SoSRM from 0.909 to 0.926 (Table 4). This results in an improved E[$SoSRM] savings of approximately$390,000. One cause of this increase was that the biologically inspired SoS had an additional link that the Status-Quo SoS did not have. Additionally, the added link (L16) had no impact on constituent MOPs when removed. This outcome is logical because we would expect the SoS to have a SoSRM of 1.0 if there were no broken links. When the newly implemented link (L16) breaks, the system reverts back to Status-Quo but without any links broken. SoSRM was recalculated to ensure that the improvement in SoSRM was not an artifact of adding a link that did not have a negative impact on constituent MOP when removed. This recalculation only incorporated the 12 links in the Status-Quo Model. The recalculated SoSRM results still demonstrated improvement from the Status-Quo SoSRM (0.909 to 0.920).

Table 4

KSM biologically inspired design SoSRM calculation

KSM SoSRM increased more from this biologically inspired redesign than from the maximum possible upgrade through traditional resilience improvement approaches (0.926 versus 0.922). This results in an improved E[$SoSRM] savings of approximately$90,000. Table 4 records the difference in link performance between the Status-Quo and biologically inspired intervention in the Delta column. The increased resilience from the broken links L35, L62, and L61 drives this improved KSM performance. The reason for resilience improving due to these specific broken links is unclear and is the focus of ongoing investigations.

### 4.4 Kreuzung Schweizer Mittlelland Biologically Inspired Improvement Combined With Traditional Approaches.

As a final test for this case study, the same optimization search conducted in Sec. 4.2 on the Status-Quo KSM was repeated for the biologically inspired redesigned KSM. Although traditional resilience improvement approaches were successful in increasing biologically inspired KSM resilience from 0.926 to 0.931, the gain seen in the biologically inspired case due to implementing traditional approaches was smaller than the gain seen for Status-Quo. Traditional approaches were successful in increasing Status-Quo SoSRM by 0.013, while the biologically inspired KSM SoSRM only increased by 0.005. The optimization search to find the lowest SoSRM did result in a lower value than for the Status-Quo scenario (SoSRM 0.743 versus 0.764). The Biologically Inspired Optimization search resulted in a slightly higher possible SoSRM range (difference of 0.188 versus 0.158).

Interestingly, as shown in Table 2, the optimization search for maximum SoSRM found nearly the same parameter results for the both the Status-Quo optimization and the biologically inspired optimization. We observed small differences in the PRR and minimum allowable Forestry level. This same trend is observed for the minimum SoSRM search result parameter values. Both Status-Quo and biologically inspired KSM minimum SoSRM configurations remove the Paper Recycling stream and maximizing the Forestry response lag (FLAG). This agreement indicates that although architecture changes were successful in increasing SoSRM, implementing these independently of traditional hard and soft resilience improvement approaches may be insufficient to guarantee desired SoS performance.

## 5 Conclusion

The examination of the KSM Case Study allowed us to examine our approach to sociotechnical system design, utilizing biologically inspired network design heuristics to increase SoS network resilience. Specifically, our investigation yielded four contributions:

First, we provided evidence that biologically inspired design provides an approach to increase SoS network resilience beyond current approaches. Traditional approaches struggle to increase resilience due to the emergence that manifests because of the complex interactions of humans and technology within the SoS. Optimizing traditional hard and soft resilience improvement approaches only managed to increase KSM SoSRM to 0.922, while shifting the Incinerator (X6) to the decomposer functional role resulted in a higher SoSRM of 0.926. This improvement is especially significant because the optimization search for traditional resilience improvement strategies considers technological changes and investments that may not be possible (e.g., increasing PRR to over 80%).

Second, this work provides evidence that incorporating detrital actors increases SoS network resilience. Although previous studies have emphasized the importance of detrital actors within SoS, Industrial Eco-Parks, and ecosystems, the lack of a resilience metric prevented researchers from providing evidence that adding detrital actors could increase SoS network resilience. Simply by incorporating a link that transformed the Incinerator (X6) from apex predator to a detrital actor, the SoSRM increased from 0.909 to 0.926. This is especially impressive because the traditional SoSRM improvements do not consider technical feasibility or cost. Our estimate of traditional resilience improvements is optimistic, expected gains when cost and technical feasibility are incorporated are expected to be much greater than the 0.027 SoSRM (E[$SoSRM] savings of approximately$90,000) observed in this study.

Third, this paper documents the first case study using the new SoSRM metric to justify a design decision. We hypothesized that transforming the Incinerator (X6) subsystem from apex predator into a detrital actor would increase SoS network resilience. This hypothesis was verified, and the architecture design decision justified when SoSRM increased from 0.909 to 0.926.

Finally, this study provides additional evidence concerning the link between sustainability and resilience. Although some suggest that sustainability and resilience trend together [12,34,82], the optimization to maximize resilience recommended removing recycling from Timber Consumption to Timber Production (L24). This balance between efficiency and resilience in this study strengthens what some ecologists theorize in sustainable natural ecosystems dynamics [83]. Removing L24 caused a net positive increase in resilience but a negative impact on environmental sustainability. Contrasting this example, adding L16 resulted in both an increase in sustainability and resilience. This case study provides a counterexample to the idea that increased sustainability always results in increased resilience. Further work is needed to identify the types of scenarios where adding recycling flows increases resilience.

Although this paper provides a valuable first step by illustrating the application of biologically inspired design to improve sociotechnical SoS network resilience, future studies will focus on identifying other design heuristics for implementation. Sociotechnical SoS network design is conducive to heuristics as a strategy for design, allowing the designer to apply simple guidance to improve SoS performance. Many of these may come from the field of Ecological Network Analysis, an application of graph and network theory. Additional research is focused on attempting to provide insights into the causal mechanisms behind the improvements noted when L16 was implemented into the KSM study. These results will be replicated on other sociotechnical SoS to validate the effectiveness of the design heuristic tested in this paper. Finally, this paper examined the impact of network architecture on network resilience (following link failure), but there are other aspects of SoS performance still requiring investigation (i.e., how to ensure resilient SoS performance following rapid MOP redefinition or node removal). This work provides one area for designer consideration, but more research is needed to create a complete resilient SoS design methodology.

By analyzing KSM, this paper has presented a SoS combined of multiple complex sociotechnical systems through a network architecture. An argument was made that the constant evolution of SoS network structure coupled with the need for constituent concurrence for changes made heuristics a potentially powerful tool to improve sociotechnical SoS network resilience. Traditional design approaches are limited due to the emergence from the interacting population of independent agents, the technical artifacts in each system, and the environment. This paper presents a design methodology applied to a SoS case study that was successful in increasing the manifestation of the emergent property of resilience. This was accomplished by testing the heuristic: Seek to ensure the SoS have a constituent system that fulfills the decomposer functional role (detrital actor).

## Acknowledgment

The authors gratefully acknowledge the support of the Georgia Institute of Technology and the comments of two anonymous reviewers. In addition, we would like to thank the Brook Byers Institute for Sustainable Systems for their unwavering support advancing multiple fronts of sustainability research. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or their respective host institutions.

## Funding Data

• This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1,650,044. In addition, this material is based upon work supported by the National Science Foundation under Grant Nos. CBET-1510531and EFMA-1441208.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

n =

total number of subsystems within the SoS

k =

number of SoS links in the SoS

T* =

“suitably long interval” to determine lost functionality

TT =

amount of time required for 62.3% of energy to flow through the SoS

FLAG =

time delay in forestry response

ForestMin =

minimum allowable forestry level allowable

FPPP =

production proportional controller constant

FTPP =

timber production proportional controller constant

Q(t) =

measure of performance at time t

X6NoPenalty =

incinerator (X6) storage that does not incur a financial penalty

X6MaxStorage =

incinerator (X6) Storage that negates profits from waste management

### Appendix A: Kreuzung Schweizer Mittlelland Measure of Performance Equations

Table 0005
SystemMOP equationVerbal MOP description and justification
Forestry$min{∫02*t*L21+L31+L61dt(L21o+L31o+L61o)*2*t*X1(2*t*)−X1(0).2*2*t*$Forestry (X1) MOP selects the minimum from two conditions. Condition 1 is driven by business interests, while condition 2 is driven by sustainability concerns. 1. Forestry (X1) MOP is a function of the amount of raw lumber sold to the other systems. The amount of lumber sold drives the profit the company can generate. A MOP of 1.0 corresponds to no drop in sales from the initial flow condition. 2. Forestry (X1) MOP is the ratio of actual growth in X1 stock to no disruption growth in X1 stock. This metric is justified by the desire in the initial study for the KSM to maintain self-sufficiency.
Timber Production$∫02*t*L42dtL42o*2*t*$The Profit of the Timber Production (X2) system is driven by the outflow of finished lumber to the Timber Consumption system (X4).
Paper Production$∫02*t*L53dtL53o*2*t*$The Profit of the Paper Production (X3) is driven by the outflow of finished paper products to Paper Consumption (X5).
Timber Consumption$X4(2*t*)X4(0)+1.0*2*t*$Timber Consumption (X4) MOP is derived from the stock of timber maintained within the system. The MOP is the stock of timber after the fault recovery as compared to the no-fault scenario. When Timber MOP falls below X4(0), Timber Consumption MOP is squared to reflect the escalating impact of increased shortages in construction supplies.
Paper Consumption$∫02*t*X5(t)dtX5(0)*2*t*$Paper Consumption (X5) MOP is derived from the stock of paper maintained within the system. The MOP is the stock of paper after the fault recovery as compared to the no-fault scenario.
Incinerator (X6)$∫02*t*1−X6(t)dt1*2*t*$The Incinerator (X6) profit is driven by maximizing the amount of material processed without having a backlog of waste waiting for disposal. Backlog of material (i.e., standing stock within X6) indicates that the Incinerator (X6) is not operating at maximum capacity, thus profit is being lost. This backlog occurs when material accumulates above X6nopenalty. There comes a point where the storage costs of material waiting to be incinerated negates the profit generated by incinerating (X6MaxStorage). Once the Incinerator (X6) has a backlog greater X6MaxStorage, it is no longer making profit. The MOP compares the no-fault Incinerator (X6) flow scenario to the fault scenario by monitoring the stock in X6.
SystemMOP equationVerbal MOP description and justification
Forestry$min{∫02*t*L21+L31+L61dt(L21o+L31o+L61o)*2*t*X1(2*t*)−X1(0).2*2*t*$Forestry (X1) MOP selects the minimum from two conditions. Condition 1 is driven by business interests, while condition 2 is driven by sustainability concerns. 1. Forestry (X1) MOP is a function of the amount of raw lumber sold to the other systems. The amount of lumber sold drives the profit the company can generate. A MOP of 1.0 corresponds to no drop in sales from the initial flow condition. 2. Forestry (X1) MOP is the ratio of actual growth in X1 stock to no disruption growth in X1 stock. This metric is justified by the desire in the initial study for the KSM to maintain self-sufficiency.
Timber Production$∫02*t*L42dtL42o*2*t*$The Profit of the Timber Production (X2) system is driven by the outflow of finished lumber to the Timber Consumption system (X4).
Paper Production$∫02*t*L53dtL53o*2*t*$The Profit of the Paper Production (X3) is driven by the outflow of finished paper products to Paper Consumption (X5).
Timber Consumption$X4(2*t*)X4(0)+1.0*2*t*$Timber Consumption (X4) MOP is derived from the stock of timber maintained within the system. The MOP is the stock of timber after the fault recovery as compared to the no-fault scenario. When Timber MOP falls below X4(0), Timber Consumption MOP is squared to reflect the escalating impact of increased shortages in construction supplies.
Paper Consumption$∫02*t*X5(t)dtX5(0)*2*t*$Paper Consumption (X5) MOP is derived from the stock of paper maintained within the system. The MOP is the stock of paper after the fault recovery as compared to the no-fault scenario.
Incinerator (X6)$∫02*t*1−X6(t)dt1*2*t*$The Incinerator (X6) profit is driven by maximizing the amount of material processed without having a backlog of waste waiting for disposal. Backlog of material (i.e., standing stock within X6) indicates that the Incinerator (X6) is not operating at maximum capacity, thus profit is being lost. This backlog occurs when material accumulates above X6nopenalty. There comes a point where the storage costs of material waiting to be incinerated negates the profit generated by incinerating (X6MaxStorage). Once the Incinerator (X6) has a backlog greater X6MaxStorage, it is no longer making profit. The MOP compares the no-fault Incinerator (X6) flow scenario to the fault scenario by monitoring the stock in X6.

### Appendix B: KSM Flow Equations

Table 0006
LinkDifferential equations (note: only positive flows)Initial status-quo flow(kg/capita/week)Description and reasoning
L21(t)$=4.3−L24(t−FLAG)$
$−Z2(t−FLAG)$
−X2(t − FLAG)*FTPP
3.9Flow from Forestry (X1) to Timber Production is driven by the requirements of Timber Production. Timber production ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FTPP = 0.1 is used to reduce the flow from Forestry (X1) to Timber. L21 flow is a function of the imports from Z2 and recycling flow from L24 The information from Z2 and X2 used to calculate L21 have a time delay of FLAG = 1 week, meaning that it takes 1 week for L21 to update to the needs of the Timber Production System.
L24(t)$=(TRR*L64(t))1−TRR$1.7The recycling flow from Timber Consumption to Timber production is a function of the waste stream from Timber Consumption to the Incinerator (X6). For this SoS, a timber recycling rate (TRR) of 10.5% is used.
L31(t)= 1.2 + L53(tFLAG) + 0.6 − L32(tFLAG) − 4.2 − X3(tFLAG)*FPPP1.2Flow from Forestry (X1) to Paper Production is driven by the requirements of Paper Production. Paper production ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FPPP = 0.1 is used to reduce the flow from Forestry (X1) to Paper Production. L31 is also a function of the imports from Z3 and L32. The information from Z3, L32, and X3 used to calculate L31 have a time delay of FLAG = 1 week, meaning that it takes 1 week for L31 to update to the needs of the Timber Production System.
L32(t)$=0.6*L42(t)2.9$.6Flow of paper pulp from Timber Production to Paper production is a function of timber produced (L42).
L35(t)$=PRR*L65(t)1−PRR$2.6The recycling flow from Paper Consumption to Paper Production is a function of the waste stream from Paper Consumption to the Incinerator (X6). For this SoS, a recycling rate (PRR) of 62% is used.
L42(t)= 1 + L24(t) + L64(t)2.9The timber demand is driven by the Consumption of Timber. The goal of the Timber Production is to replace losses while increasing the stock of X4 at a rate of 1.0 kg per capita per week.
L53(t)$=[L35(t)+L65(t)]*100X5(t)$4.2The paper demand is driven by the Consumption of Paper Products. The goal of the Paper Production is to replace losses while maintaining the stock of paper products.
L61(t)= 1.6 + .8 − L62(tFLAG)1.6Flow from Forestry (X1) to the Incinerator (X6) is used to compensate when there is insufficient flow from Production of Timber to the Incinerator (X6) to run the Incinerator (X6).
L62(t)$=0.8*L42(t)2.9$.8Flow from Timber Production to the Incinerator (X6) is a function of the finished lumber produced. We also assume that if a fault removes the ability of Timber Production to send pulp paper to Paper Production, then this material is sent to the Incinerator (X6) instead (not shown in equation to the left to maintain readability).
L63(t)$=1.1*L53(t)4.2$1.1Flow from Paper Production to the Incinerator (X6) is a function of the amount of paper produced.
L64(t)$=1.7*X4(t)5000$1.7Flow from Timber Consumption to the Incinerator (X6) is a function of the standing stock of timber in X4. We also assume that if a fault removes the ability of Timber consumption to send recyclable material to X2 through L24, then this material is sent to the Incinerator (X6) instead (not shown in equation to the left to maintain readability).
L65(t)$=1.6*X5(t)100$1.6Flow from Paper Consumption to the Incinerator (X6) is a function of the standing stock of Paper in X5. We also assume that if a fault removes the ability of Paper Consumption to send recyclable material to X3 through L35, then this material is sent to the Incinerator (X6) instead (not shown in equation to the left to maintain readability).
Z1(t)= 6.96.9The forest grows at a constant rate.
Z2(t)$=0.2−FTPP*X2(t)$.2Timber Production (X2) ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FTPP = 0.1 is used to reduce the flow from Z2 to Timber Production (X2). Unlike Forestry (X1), there is no lag (FLAG) assumed with this process.
Z3(t)$=0.9−FPPP*X3(t)$.9Paper Production ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FTPP = 0.1 is used to reduce the flow from Z3 to Paper Production. Unlike Forestry (X1), there is no lag (FLAG) assumed with this process.
Y6(t)$(1+X6(t))*6.8*L62(t)+L61(t)2.4$6.8The Incinerator (X6) outflow is driven by two factors. First, the system is designed with zero standing stocks, so if X6 accumulates, Y6 increases. Second, L62 and L61 provide the fuel for the Incinerator (X6), thus outflow is also limited by the available fuel.
LinkDifferential equations (note: only positive flows)Initial status-quo flow(kg/capita/week)Description and reasoning
L21(t)$=4.3−L24(t−FLAG)$
$−Z2(t−FLAG)$
−X2(t − FLAG)*FTPP
3.9Flow from Forestry (X1) to Timber Production is driven by the requirements of Timber Production. Timber production ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FTPP = 0.1 is used to reduce the flow from Forestry (X1) to Timber. L21 flow is a function of the imports from Z2 and recycling flow from L24 The information from Z2 and X2 used to calculate L21 have a time delay of FLAG = 1 week, meaning that it takes 1 week for L21 to update to the needs of the Timber Production System.
L24(t)$=(TRR*L64(t))1−TRR$1.7The recycling flow from Timber Consumption to Timber production is a function of the waste stream from Timber Consumption to the Incinerator (X6). For this SoS, a timber recycling rate (TRR) of 10.5% is used.
L31(t)= 1.2 + L53(tFLAG) + 0.6 − L32(tFLAG) − 4.2 − X3(tFLAG)*FPPP1.2Flow from Forestry (X1) to Paper Production is driven by the requirements of Paper Production. Paper production ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FPPP = 0.1 is used to reduce the flow from Forestry (X1) to Paper Production. L31 is also a function of the imports from Z3 and L32. The information from Z3, L32, and X3 used to calculate L31 have a time delay of FLAG = 1 week, meaning that it takes 1 week for L31 to update to the needs of the Timber Production System.
L32(t)$=0.6*L42(t)2.9$.6Flow of paper pulp from Timber Production to Paper production is a function of timber produced (L42).
L35(t)$=PRR*L65(t)1−PRR$2.6The recycling flow from Paper Consumption to Paper Production is a function of the waste stream from Paper Consumption to the Incinerator (X6). For this SoS, a recycling rate (PRR) of 62% is used.
L42(t)= 1 + L24(t) + L64(t)2.9The timber demand is driven by the Consumption of Timber. The goal of the Timber Production is to replace losses while increasing the stock of X4 at a rate of 1.0 kg per capita per week.
L53(t)$=[L35(t)+L65(t)]*100X5(t)$4.2The paper demand is driven by the Consumption of Paper Products. The goal of the Paper Production is to replace losses while maintaining the stock of paper products.
L61(t)= 1.6 + .8 − L62(tFLAG)1.6Flow from Forestry (X1) to the Incinerator (X6) is used to compensate when there is insufficient flow from Production of Timber to the Incinerator (X6) to run the Incinerator (X6).
L62(t)$=0.8*L42(t)2.9$.8Flow from Timber Production to the Incinerator (X6) is a function of the finished lumber produced. We also assume that if a fault removes the ability of Timber Production to send pulp paper to Paper Production, then this material is sent to the Incinerator (X6) instead (not shown in equation to the left to maintain readability).
L63(t)$=1.1*L53(t)4.2$1.1Flow from Paper Production to the Incinerator (X6) is a function of the amount of paper produced.
L64(t)$=1.7*X4(t)5000$1.7Flow from Timber Consumption to the Incinerator (X6) is a function of the standing stock of timber in X4. We also assume that if a fault removes the ability of Timber consumption to send recyclable material to X2 through L24, then this material is sent to the Incinerator (X6) instead (not shown in equation to the left to maintain readability).
L65(t)$=1.6*X5(t)100$1.6Flow from Paper Consumption to the Incinerator (X6) is a function of the standing stock of Paper in X5. We also assume that if a fault removes the ability of Paper Consumption to send recyclable material to X3 through L35, then this material is sent to the Incinerator (X6) instead (not shown in equation to the left to maintain readability).
Z1(t)= 6.96.9The forest grows at a constant rate.
Z2(t)$=0.2−FTPP*X2(t)$.2Timber Production (X2) ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FTPP = 0.1 is used to reduce the flow from Z2 to Timber Production (X2). Unlike Forestry (X1), there is no lag (FLAG) assumed with this process.
Z3(t)$=0.9−FPPP*X3(t)$.9Paper Production ideally has zero standing stocks, so when stocks accumulate a simple proportional controller with constant FTPP = 0.1 is used to reduce the flow from Z3 to Paper Production. Unlike Forestry (X1), there is no lag (FLAG) assumed with this process.
Y6(t)$(1+X6(t))*6.8*L62(t)+L61(t)2.4$6.8The Incinerator (X6) outflow is driven by two factors. First, the system is designed with zero standing stocks, so if X6 accumulates, Y6 increases. Second, L62 and L61 provide the fuel for the Incinerator (X6), thus outflow is also limited by the available fuel.

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