Ecological Principles and Metrics for Improving Material Cycling Structures in Manufacturing Networks

Substantial progress has been made in the past two decades in the field of sustainable manufacturing. Many are looking for technologies that will solve the environmental problems related to manufacturing, see, e.g., Refs. [1] and [2]. In 2001, a comprehensive study on environmentally benign manufacturing, however, found that there was no evidence that the environmental problems from our production systems are solvable by a “silver bullet” technology [3,4]. Rather, the need for systems-based solutions was noted. Achieving true sustainable manufacturing requires adoption of a systems view that goes beyond a single factory and company. Close collaboration between multiple companies, stakeholders, and even public–private partnerships around internal and external value chains and resource networks have to be pursued. Although all manufacturers strive for efficient and effective resource utilization, we still have a long way to go in order to move from the current “take-make-waste” society to a truly cyclical production paradigm.

Nature is an excellent source of inspiration for sustainable manufacturing in the guise of ecological systems, which are canonical examples of sustainability. Industrial symbiosis, named for the analogous close and often long-term interactions between two or more biological species, occurs when multiple firms or facilities achieve higher system efficiency through the exchange of “waste” energy and materials. An EIP is characteristic of a type of industrial symbiosis that occurs among firms collocated in a bounded geographic area, typically an industrial park. The exchanges characteristic of EIPs has proven environmental benefits, most often reducing resource consumption and emissions. For example, the mutualistic relationships within the well-known Kalundborg EIP have resulted in the EIPs yearly CO₂ emission being reduced by 240,000 tons, water savings of 3 × 10⁶ m³ through recycling and reuse, 30,000 tons of straw being converted to 5.4 × 10⁶l of ethanol, 150,000 tons of yeast replacing 70% of the soy protein in traditional feed mix for more than 800,000 pigs, and the recycling of 150,000 tons of gypsum from desulfurization of flue gas (SO₂), replacing the import of natural gypsum (CaSO₄) [5].

Unfortunately, symbiotic relationships do not guarantee success, as the world of production and development is constantly fluctuating and can be difficult to predict in the long term. Monetary problems halt the implementation of many exciting EIP plans, companies that must fill such plans may remain unconvinced that moving locations would be financially beneficial, and things may fall apart for a number of reasons between an EIP's early development and its maturation. Furthermore, there is also no real design guidance for establishing sustainable symbiotic industrial resource networks beyond typical ad-hoc approaches. The solution to these difficulties may be found within the biological systems that were the inspirations for EIPs. Biological systems, specifically ecosystems, have evolved to maintain their function in nonideal and fluctuating conditions. A better understanding of biological ecosystems is needed in order to identify and apply key components from ecosystems that support the development and formation of sustainable industrial manufacturing systems.

The goal of biologically inspired design of industrial manufacturing networks is to reproduce the sustainable cycling and recycling that characterize ecosystems, ideally achieving a highly efficient closed-loop flow of materials. This paper investigates the cycling of material/food through ecosystems and industrial systems using ecological metrics. The goals of this article are: (a) to review relevant principles and findings from ecology related to FWs and their functioning; (b) to compare industrial case studies against ecological system using key metrics, specifically cyclicity; (c) to identify key differentiating components between current industrial systems and ecologies; and (d) to provide suggestions on how to improve industrial symbiosis based on the findings.

This paper demonstrates how insight and improvements to cyclical networks can be made based on structural information only. That is, only the presence and direction of flows between two actors in the network is required and information on flow magnitude is not necessary to provide insight into how to improve material cycling. Although detailed flow knowledge allows for more precise network representations, the trade-off is the effort associated with collecting this quantitative flow information. The cost of information is less with structural metrics that only require knowing a link exists between two actors than with flow metrics that require knowing amounts of material or energy transfer.

The importance of characterizing the anatomy of ecological networks is given by Strogatz: “structure always affects function” [6]. By reproducing the structure of these biological networks, the hope is that the functioning of human systems will replicate the inherently sustainable natural world as well. The appropriate application of ecological principles and analyses depends on building models that specify how principles from biology are translated to industry, and back again. One biological model for ecosystems is an FW. Somewhere during the process of translating this model to industry, the defining characteristic of an ecosystem (the weblike structure) is discarded and industry is left with a unidirectional, top–down “food chain” [7]. Companies within an industrial park or components in an industrial cycle are cast as species, and the material and energy exchanges between them are analogous to the transfer of caloric energy that supports the species (metabolism). At first glance, the comparison may seem a complete one; however, the transfer of ecological properties and principles to industry is highly complex and much is missing. Definitions have led to the sustainable design slogan “waste equals food,” a slogan that although consistent with systems in nature does not fully capture the important workings of ecological systems. The lack of a well-translated framework has led to many discrepancies in the implementation and interpretation of ecological principles and how they advise the organization of industrial system [8]. A framework built on real and complete ecological knowledge is of the utmost importance to accomplish this goal. Extensive literature exists to aid in the successful translation of many desirable properties found in nature to industry, e.g., Refs. [9–12].

Models and structural metrics have been developed to analyze and explain specific properties of ecosystems, such as the system's ability as a whole to withstand environmental fluctuations and support diverse species, which could be immensely beneficial to industry [13,14]. Findings indicate that FWs are composed of multiple pathways through which material and energy flow, strongly connected groups of interacting species compartments with weak interactions between the compartments, and a modular structure that is hypothesized to increase the overall systems stability by localizing interactions and disruptions [15–17]. This hypothesis, however, has been difficult to fully resolve [18,19]. Replicating ecosystem robustness and stability in human systems could ease the damage caused by supply chain disruptions, which reduce the share price of the affected companies so significantly that 80% of companies worldwide consider better protection of supply chains top priority [20].

In 1969, Odum recognized that ecological systems, particularly mature ones, are associated with a high degree of internal recycling of energy and materials such that the amount of new inputs into the system is small compared to what is transformed among the system components [9]. Human systems in contrast (e.g., agricultural ones) are geared for production rather than efficiency, resembling young rather than mature natural systems. Odum has suggested mimicking mature systems would help shift the focus of human systems from production to efficiency [9]. One desirable property of mature systems is a complex FW structure, a proliferation of connections between species that exchange material and energy [21]. The centripetal nature of FW structure also is a selling point for industry. When a species becomes more efficient in use or acquisition of a resource, its population increases. Centripetality results in this singularly focused positive change being cascaded through the system such that all the populations of species involved are benefited [17,22]. Translated to industry, this would mean that a change which benefits one company within an EIP translates into a park-wide positive net effect.

A hypothesis within industrial ecology is that diversity, in the sense that a wide range of species types are contained within any system, could contribute to a more stable system: when one firm departs, the system may adapt or recover by another actor(s) stepping in to fulfill the supplying role [23].

An analysis of 40 FWs by Briand indicates that connectance, which is a measure of the number of direct to the total possible interactions in a web and an important parameter in the previous ecosystem analyses, declines as variability of the environment increases [24]. Following a line of reasoning strongly influenced by May's theoretical analysis [25], Briand argues that differences in connectance values for ecosystems in stable and unstable environments are the result of limitations in feeding periods caused by environmental fluctuations, which can lead organisms to depend upon intermittent, intense feedings. This suggests a structure which is dependent on the stability of resources, a property of interest for industries.

Ecologists use simple unweighted digraphs to quantify the characteristics of FW, where every link indicates direction, in this case from resource (prey) to consumer (predator) [17]. Species or functional groups are represented in the digraphs such that any species with identical predators and prey are grouped as trophic species, which has been found to reduce methodological bias in the data [13,17,26–28].

Ecologists use structural measures and metrics for the analysis of ecological FWs [14,24,29–31]. The meaning and calculation of each ecological measure/metric is best understood within the context of an organizational matrix. Organizational matrices are used by ecologists to collect and document the exchanges between species or functional groups within the community at hand. These matrices can document a variety of interactions from predator–prey exchanges to all interactions in a community, including any competitive interactions. Three types of matrices are commonly used by ecologists: a FW matrix [F], a community matrix [C], and an adjacency matrix [A].

We use an FW matrix for the representation of the exchanges in industrial systems and ecological FWs. Analogous to a connectivity matrix [32], an FW matrix is concerned only with the structural information (links and nodes) of a network and defines the pathways that exist by which material and energy flows from one compartment to another. It is blind to information such as flow rate, quality, and the type of flow. A link exists as long as some physical quantity directly joins two nodes. Only flow existence and direction are captured. An FW matrix [F] captures the observed consumer–producer interactions that include both plant and animal species (Ecologists commonly use different terms to describe these interactions such as plant–herbivore versus Predator–prey but for simplicity we use predator–prey to describe any interaction where one species is consumed by another.). The left half of Fig. 1 depicts a hypothetical FW represented as a directional digraph; the right half represents the web as an FW matrix [F]. Since a species (N) can be both predator and prey, the result is a square matrix. A value of 1 indicates the existence of a flow while 0 indicates the absence of a flow. In Fig. 1-right, a value of 1 for L₁₃ indicates that species 1 (S₁) contributes resources to species 3 (S₃). Alternatively, one can interpret that element of the matrix to mean S₃ “consumes” S₁. Each row in an FW matrix captures the flow of resources from one species to all species in a web and each column captures the input of resources to a particular species from all species in the web. In other words, if predator j feeds on prey-i, then f_ij = 1; the interaction (or link, L) is accounted for exactly once in the FW matrix. The maximum number of links L scales as (N)*(N − 1) if a given species does not eat itself, and as (N₂) if cannibalism is allowed (noted as a 1 on the diagonal). A structural adjacency matrix [A] is the transpose of the FW matrix.

A wide variety of metrics have been developed to understand the link between structure and behavior of ecological systems [32,33]. The structural measures and metrics used most frequently by ecologists can be calculated using the N × N structural FW matrix (such as given in Fig. 1); that is, all calculations are simply based on binary information on whether or not a link exists between two actors in the matrix. The following are key metrics typically used by ecologists to perform structural assessments of FWs:

• Number of species or actors (N)—The total number of actors in a network, sometimes termed “species richness.” Represented by the size (number of rows or, as the two are equal, columns) of the FW matrix [F] [24].

• Number of links (L)—The number of direct links or interactions between actor in a network. Represented by the total number of nonzero interactions in the FW matrix [F] [24]

• Linkage density (L_d)—The ratio of the total number of links to the total number of actors in a network.

• Number of prey (n_prey)—Actors which are eaten by at least one other. Represented by the number of nonzero rows in the FW matrix [F]. “Prey” in an industrial network transfer material or energy to be utilized by others; that is, they are producers [14].

• Number of predator (n_predator)—Actors that eat at least one other. Represented by the number of nonzero columns in an FW matrix [F]. “Predators” in an industrial network receive material or energy from others; that is, they are consumers [14].

• Prey to predator ratio (P_r)—The ratio of the number of actors eaten by another to the number of actors that eat another. That is, the ratio of prey:predators or producers:consumers. This is the number of nonzero rows in an FW matrix [F] divided by the number of nonzero columns.

• Specialized predator fraction (P_s)—The number of predators (consumers) eating only one actor divided by the total number of consumers in the network. This is the sum of the number of columns with only one nonzero element in the FW matrix [F] divided by the total number of columns with nonzero elements.

• Generalization (G)—The average number of prey eaten per predator in a web, which corresponds to the average number of producers a consumer interacts with in an industrial network. Generated by summing the columns in an FW matrix [F] and dividing this figure by the number of columns with nonzero elements (the number of predators/consumers).

• Vulnerability (V)—The average number of predators per prey in a web, which corresponds to the average number of consumers a producer interacts with in an industrial network. Generated by summing the rows in an FW matrix [F] and dividing by the total number of rows with nonzero elements (the number of prey/producers).

• Cyclicity (λ_max)—A measure of the strength and presence of cyclic pathways present in the system [32,34]. Obtained by finding the maximum real eigenvalue of the transpose of the FW matrix [F].

Cyclicity is a key metric that addresses resource cycling in ecosystems and will be discussed in more detail in the following section.

Cyclicity (λ_max) is a measure of the number and proliferation of cyclic pathways present in the system [32] as the number of steps in the path approaches infinity. Cyclicity is obtained by finding the maximum real eigenvalue of a web's structural adjacency matrix [A] that is the transpose of the FW matrix [F]. The maximum real eigenvalue (λ_max) is a measure of the proliferation of pathways that originate and terminate at the same node. There is a greater potential for flows to remain within the system as pathways proliferate, and so λ_max is indicative of the resulting internal cycling [17].

The use of eigenvalues to determine cyclicity (also known as “pathway proliferation rate”) of a system combines results from graph theory and linear algebra [17]. The proof presented by Borrett et al. uses the Perron–Frobenius theorem, which guarantees that there is only one real eigenvalue that is greater than or equal to all other eigenvalues (λ₁ ≥ λ_i for i = 2,…, n) in adjacency matrices associated with a strongly connected network [17]. In networks where it is possible to reach every node from every other node, only the maximum (dominant) eigenvalue is left to represent the pathway proliferation rate of the system as the limit of the number of indirect links (pathways between two nodes which consist of more than one link) goes to infinity.

Cyclicity can be 0, 1, or greater than 1.This is illustrated in Fig. 2, which is based on the similar figure by Fath and Halnes [32,35]. Zero cyclicity indicates that no internal cycles are present, Fig. 2(a). In these networks, energy traveling through the system never passes through a component twice. A value of one is representative of a network where only simple closed-loop pathways exist, Fig. 2(b). Those networks that have cycles made up of one link (self-loops) or have cycling only if link direction is ignored may have a maximum eigenvalue of either 1 or 0 [17]. A network with a maximum eigenvalue greater than 1 indicates that the network is made up of complex looped pathways, as described in Fig. 2(c). The larger the cyclicity, the more complex and numerous the paths are between components, creating a system that is more interconnected. Most FWs are composed of networks where large subsets of “nodes” are strongly connected such that the maximum eigenvalue is greater than 1, indicating the existence of multiple cyclic pathways.

With respect to cyclicity, the dynamics and stability of FWs are significantly influenced by nutrient recycling and decomposition [36]. In ecosystems, the detritivores (earthworms, fungi, and bacteria, for example) are responsible for the decomposition of dead organic matter and the distribution of nutrients to the system, often known as the “recyclers of the biosphere.” This decomposition and redistribution create a fixed cyclic structure in the system as measured by cyclicity [37].

Given the importance of cyclicity in FWs, we now look at how ecological metrics can be applied to industrial manufacturing systems, specifically, EIPs. In Ref. [38], the success of proposed versus existing EIPs was examined and compared to the performance of ecological networks using structural ecosystem metrics. The goal of mimicking median values of the FW metrics stems from the belief that form follows function. EIPs that match the form of FWs will function more like the FWs. For this comparison, a comprehensive dataset of 48 EIPs was compiled and analyzed. The EIPs in this dataset can be classified based on the current (or as current as possible) status of the EIP: existing, proposed, or failed. EIPs currently (or as current as possible) active/in operation/existing are often termed “successful” in the literature as they have been fully or mostly implemented and are still running. The locations of the EIPs listed span the globe, ranging from various locations in the U.S. to China to Denmark and France. The proposed EIPs are often based on an existing industrial park where additional linkages between existing and new companies to increase the symbiotic relationships have been suggested. Some of these EIPs exist entirely on paper. Three EIPs that were fully or mostly implemented are no longer in operation for unknown reasons.

Note that this analysis defined the system boundary of the EIP as not including customers that may consume products (including energy) created by the EIP actors. Although including this actor might change some of the properties examined, it would not change the analysis of cycling since it is not possible in most cases to account for returns to the EIP from this type of actor. Moreover, although defining system boundaries is always a challenge, the goal of a successful EIP is to reduce raw material inputs and waste outputs by increasing the cycling of material between actors in the EIP. Material and energy outputs to customers outside of the EIP boundary do not fall into this category. Thus, the system boundary definition used here is the most relevant to the industrial goals that EIPs are designed to achieve.

Cannibalism is excluded from this analysis both for FWs and EIPs to permit accurate comparisons. Cannibalism in an EIP would correspond to a particular actor directly recycling or reusing parts of its own waste stream. Although admittedly important, these self-loops do not contribute to the cyclicity measured in a system (see below), and information on this potential link is not easily obtained.

The 48 EIPs are ranked in four groups based upon the existence and complexity of the internal cycling measured by the ecological metric cyclicity. As mentioned, the characteristic cycling seen in FWs is an especially desirable function for developing sustainable manufacturing as it embodies the major goal for eco-industrial networks: closed-loop manufacturing. The four groups are as follows:

• Class A: The EIPs with a designation of class A are representative of highly complex internal cycling. This is defined as those EIPs with a cyclicity value greater than or equal to 3 (λ_max ≥ 3). These EIPs represent the top tier of collected systems.

• Class B: The EIPs with a designation of class B are representative of complex internal cycling. This is defined as those EIPs with a cyclicity value greater than 1 (3 > λ_max > 1).

• Class C: The EIPs with a designation of class C contain simple internal cycling. This is defined as those EIPs with a cyclicity value equal to 1 (λ_max = 1).

• Class D: The EIPs with a designation of class D have no internal cycling present. This is defined as those EIPs with a cyclicity value equal to 0 (λ_max = 0). All of the EIPs in this grouping pass along a by-product to another industry for use rather than disposal; however, they do not have the more complex cycling that results from the reintroduction of that by-product into the system.

Table 1 shows the resulting rank ordering of the 48 EIPs with respect to cyclicity (λ_max) as well as the corresponding values for linkage density (L_D), the prey–predator ratio (P_R), generalization (G), and vulnerability (V). The median values for these metrics from a post 1993 data set consisting of 50 natural FWs also have been included.

When compared to median values for FWs, the values for EIPs highlight that EIPs do not match the values characteristic of ecosystems. All ecological metric values for EIPs markedly differ with the average values calculated for ecosystems. In Table 2, a summary of the differences is given in terms of best and worst values.

The largest and most consistent differences between EIPs and biological ecosystems occur for cyclicity and linkage density, which are both metrics that characterize the type and presence of connections within the system. None of the EIPs reach a cyclicity or density of links close to that which is found for biological ecosystems. As seen in Table 2, the closest EIP has a cyclicity of 3.87 and a linkage density of 3.13 as compared to the respective median FW values of 4.24 and 5.04. The other structural metrics calculated (ratio of prey to predators, generalization, and vulnerability) also fall short of FW median values but to a lesser degree.

For all metrics, those EIPs which have high cyclicity values (three or greater) show the smallest difference from FW averages. Those EIPs that have no internal cycling (cyclicity of zero) or basic internal cycling (cyclicity of one) show the largest difference from FW averages.

EIPs in all groupings come closest on average to matching the median values for the FW metric vulnerability (V) and prey to predator ratio (P_R), with both metrics coming in with an average value for all EIP types 19% lower than FWs. However, the EIPs with the highest cyclicity (λ_max ≥ 3) come closest to average FW values for all metrics. This supports the notion that EIP designers and decision makers should be aiming for structural designs that converge on FW metrics in order to obtain the highest cyclicity.

The results of the existing and failed EIPs are due partially to a response to external stimuli. Industrial networks from which EIPs are built are “complex adaptive systems” where the system does not adapt with any coordination but rather, it is the components that change in their own best interest in response to external conditions [39]. EIPs experience a certain amount of purposeful coordination between the participating companies, but still experience to different degrees the complex adaptive system response. The best scenarios would be expected to be those created on paper as these have been ideally designed with the total purpose of optimizing a coordinated network behavior. The proposed EIPs show a tendency to come slightly closer to FW averages than the existing or failed EIPs. This is most likely due to the fact that the proposed EIPs have not had to deal with the realities of operation yet; on paper, one may make a very beautiful design; however, in actual operation, the design may not be possible. These “proposed” EIPs, however, still possess cycling well below that found in biological systems.

An EIP with no internal cycling seems contrary to what one expects of a bio-inspired industrial network because one of the most influential and identifying characteristics of biological networks is the prevalence and importance of materials and energy cycling within the system. What is observed in the EIP results represents the difference between a simple “waste = food” analogy and a truly biologically inspired FW, the two concepts are illustrated by Fig. 3. The industrial networks of class D, all of which have a cyclicity of zero, follow the linear structure of the food chain in Fig. 3-left. Even though many of these networks exchange and reuse by-products, the system is still made up of a linear chain of relationships, characterized by the food chain in Fig. 3-left. The industrial networks having cyclicity greater than zero, those in classes A, B, and C, begin to show some of the ecological benefits characteristic of strong internal cycling, characterized by the FW in Fig. 3-right. The EIPs in these higher classes exhibit median values for all the metrics used here closer to medians for the FWs.

The characteristic cycling seen in FWs is an especially desirable function for developing sustainable manufacturing. This cycling of materials and energy in FWs brings with is a host of other industry desirable properties and functions. The successful establishment of cycling in EIPs can also be understood through the presence and relative strength of indirect effects. Indirect effects have been shown to be very important to the workings of ecosystems [40,41] and investigation of their influence in industrial networks has been done as well [42,43]. Key aspects on how to achieve greater cyclicity will be discussed.

As mentioned, cyclicity is a measure of internal cycling in networks. As energy and materials savings in EIPs are highly dependent on the successful cycling of waste and by-products, cyclicity is an important metric. The lower cyclicity values highlight the less complex internal cycling present in the structure of EIPs as compared to FWs. The median value of cyclicity for FWs is more than one and a half times larger than EIPs, indicating that FWs generally have a much more complex set of pathways. Although many EIPs fall into the category of having at least one single cyclic loop that all connected components participate in (cyclicity = 1), a number of the EIPs show a cyclicity of zero, meaning no cyclic structure is present. This is essentially a failure on the part of the EIP designers to reproduce the structure and function of FWs.

High cyclicity values (≫1) relate strongly to the overall proportion of the energy retained or used within the system versus that which is lost or discarded by the system. This relationship is also reflected in an analysis done on thermodynamic power systems [45]. The results of that analysis suggest that designing EIPs with a high cyclicity structure may lead to more efficient closed-loop industrial networks. Despite consumer and financial support, recycling in industrial systems still only accounts for a small fraction of mobilized matter. Most recycling is in the form of metals collected and shipped to an offsite recycling facility [46], which from an ecosystem perspective corresponds to the export of material to other systems. The potential for onsite reuse of water and other by-products is immense and much better reflects the role of the detritivores (decomposers) in an ecosystem. These organisms consume waste or dead animals (detritus), which recycles matter and energy back into the living components as opposed to being exported.

The internal cycling in FWs is very strongly influenced by the presence of detritivores. Over half of all the material in an FW is connected to a decomposer-type species such as fungi, which recycles unused material or dead matter (detritus) and returns it back to the system. Decomposers ensure the presence of FW pathways that include all other species in the system because the connections due to this consumption pattern contribute to many other existing cycles. Even limited connections to an actor that functions similarly in an EIP would dramatically increase connectivity, and thereby efficiency.

A detritus-type actor for an EIP is an actor that functions in waste treatment (i.e., composting), recovery and recycling (i.e., repair, remanufacture, reuse, resale), or agriculture (i.e., farm, zoo, landscaping, green house, golf course). Additionally, to qualify as a detritus-type actor, there must be at least one link entering and leaving said actor. This last criterion is based on the fundamental functional description of a detritus/decomposer in an FW and ensures that the detritus-type actor is an active participant of the EIP.

The types of interactions present clearly influence the magnitude of the differences between EIP and FW performance (Table 1); EIPs fall closer to FWs without cannibalism and detrital interactions [38], suggesting that the failure to include such roles in EIPs is at least partially responsible for their lower cyclicity relative to FWs. The top six EIPs listed in Table 1 have one or more detritus-type actors, which can comprise from one-third to one-half of the total actors in the group. Four of the top EIPs have some form of composting or agriculture-type actor. The EIPs in this top group tended to have a larger than average linkage density as well.

The presence of detrital actors in EIP results in complex cycling even when fewer connections exist relative to FWs (i.e., linkage density is lower). The lowest EIP in the top group, Kitakyushu Resource Recovery Park in Japan, has a low linkage density and prey–predator ratio in comparison to the rest of the group, while still having a high cyclicity. The explanation is that all the interactions in the system are to and from only 1 of the 11 actors: the resource recovery facility, which is the acting detritus. Thus, the Kitakyushu Resource Recovery Park has 100% of the total links in the system passing through its detritus actor. Clark Special Economic Zone also has a lower linkage density as compared to a majority of the top EIPs. Of the 51 links between the 20 actors in the Clark Special Economic Zone, those actors that saw the most connections were the five composting/processing/recovery facilities and 84% of the total links in this system passed through these detrital-type actors.

There are six EIPs listed in Table 1 that ranked as D class, exhibiting zero internal cycling. These EIPs are characteristic of cyclicity equal to zero, and low linkage density. Connecticut Newsprint ranks the lowest out of all the EIPs in comparison to FWs. Interesting is it does in fact have a composting and a recycling component, but these actors fail to provide any benefits with regard to structure; they each only have one connection with the rest of the system. Triangle J located in NC, another EIP in this bottom group, has a wastewater treatment plant which interacts with three other actors; however, similar to Connecticut Newsprint, it too fails to be an “active-enough” participant to have an impact on the internal cycling. So, we see it is not enough to simply have a “detrital” component in an EIP; it must be an active participant in the system in order to create cycles of materials and energy.

This role of the detritus actor is a key differentiator between EIPs. For example, EIP Kalundborg ranks in the bottom half of the C class EIPs, those exhibiting only basic internal cycling. Comparing Kalundborg to Pomacle–Bazancourt, the top ranking EIP, Fig. 4, highlights the level of participation of the detritus actors in each system. All except one of the 15+ cycles in Pomacle–Bazancourt involve the two detritus actors. Kalundborg also has two detritus actors. The difference is that only one of the two detritus actors participates in only two of the three existing cycles. So, Kalundborg has far fewer cycles because the detritus actors are disengaged from a majority of the system, while those EIPs in the top-performing group have a majority of their total links involved in a cycle and highly involved detritus actors.

Therefore, it is not enough to simply have a detrital component in an EIP, but it must be an active participant in the system in order to create cycles of materials and energy. An EIP with no internal cycling seems contrary to what one expects of a bio-inspired industrial network because one of the most influential and identifying characteristics of biological networks is the prevalence and importance of materials and energy cycling within the system.

Most of the detrital/decomposer actors in the systems examined here constitute agricultural users. However, industrial recyclers, wastewater treatment plants, or other actors that reuse and remanufacture products from inputs derived from waste streams of other EIP actors would fulfill this same role. The important role of these actors for cycling should encourage industrial networks that do not currently have recycling-type facilities to invest in them, and motivate the search for ways to reduce barriers that discourage this role with respect to other types of material flows.

Due to the ecological importance of the decomposer/detritus actor, we also quantified the frequency of their occurrence in each EIP and plotted this frequency against EIP cyclicity (Fig. 5). The plot area between zero and one on the x-axis is shaded to indicate that the value of cyclicity cannot fall between these two limits.

The two data points circled in Fig. 5 have no active detrital actors but still have a greater than zero cyclicity. These EIPs are Harjavalta with a cyclicity of 2.0 and the Lower Mississippi Corridor with a cyclicity of 1.0. The Harjavalta industrial area in Finland is a full industrial park that includes a wastewater treatment plant and an industrial cleaning facility, but these two companies are not included in the material and energy exchange diagram provided in the literature [47]. The existence of a detrital-type actor cannot be discounted as being a contributor to the success of this EIP as the wastewater treatment plant and cleaning facility may contribute behind the scenes to the overall structure. The material and energy exchanges between firms in the industrial network as documented in the literature are shown in Fig. 6. Although none of the companies within the network fall into the functional categories defined above for a detritus-type actor, they do all meet the active participant requirement, with five of the six actors having at least one connection entering and leaving. This is why the cyclicity is so high.

The outlier in Fig. 5 with a cyclicity of 1.0 but no active detritus is the Lower Mississippi Corridor. Figure 7 gives a visual description of the material and energy exchanges between firms showing that the cyclicity of the Lower Mississippi Corridor results from three bidirectional links between three different pairs of actors. Technically, a bidirectional link (or two actors linked in both directions) does create a cycle; however, it is not the complex cycling of ecosystems that EIPs strive to reproduce.

Ecology distinguishes between direct and indirect relationships and flows. A direct relationship is one that is formed between actors that are linked by a material or energy flow. An indirect relationship is formed if two actors interact through at least one intermediary so that material or energy flows through at least one other actor [48]. This can be, but does not require spatial or temporal separation. Per these definitions, indirect relationships are also prevalent in manufacturing systems. For example, the relationship between original equipment manufacturer and a tier 2 supplier is indirect because it goes through a tier 1 supplier first.

Although many EIPs focus on direct flows from one actor to another, ecologists have noted the significance of indirect flows, i.e., flows between actors that have indirect relationships. The ecologists Salas and Borrett found in a set of 50 FWs that when significant cycling was present indirect flows nearly always dominated direct flows [49]. This and other literature over the last 20 years has established the dominance of indirect effects in ecosystems [48,50]. The characteristic cycling of materials and energy in FWs is one of the most desirable properties to sustainably minded industry networks. The apparent relationship between cycling in the system and indirect effects should also be a design property of interest for industrial networks and can be examined to enhance understanding and increase material cycling in manufacturing networks.

Indirect flow effects can be determined by looking at paths of length greater than one. A path is the route traced by following some quantity of material or energy and is made up of either chains or cycles. A path with a length greater than one indicates that the material or energy being followed interacts with more than two actors in the system. The two methods for path formation are chains and cycles. Both methods limit flows through transfer efficiencies relating to dissipation and export and chains apply an additional limitation by way of their length [17,51].

Graph theory enables the calculation of the number of paths of different lengths in the system by raising the adjacency matrix [A] to a power that represents the path length being investigated [5,52]. Thus, [A]⁴ provides all the paths in the network represented by [A] that have a length of four. This can also be done for what is known as the flow intensity matrix [G]. The flow intensity matrix highlights the amount of flow (kg, kJ, units, etc.) that is indirectly circulated through the system (circulated using paths of length greater than one).

A number of distinct patterns have arisen from the investigation of indirect effects in ecosystems. The tendency for the number of paths to increase geometrically without bound as path length increases, known as pathway proliferation, was first applied to the study of ecosystems in the early 1980s [52–54] and has been studied more recently in FWs [17,32]. Pathway proliferation only occurs if there is more than one cycle in the network. As noted previously, the rate of increase in the number of paths with path length is measured by cyclicity.

The relative magnitude of cyclicity can be used as a descriptor of indirect flows in the network. Pathway proliferation has a strong influence on the development and significance of indirect flows [17]. A faster rate of pathway proliferation, or a higher cyclicity, signifies that indirect pathways are more numerous. Because indirect pathways that involve multiple actors will process larger flows, a higher cyclicity increases the possibility that indirect flows will dominate direct flows [17].

Paths of specific lengths can be found by raising the adjacency matrix to a power that represents the path length being investigated [5,52]. Thus, to find paths of length two or greater, the matrix [A] is raised to the powers 2, 3, 4, and so on. Figure 8 shows path lengths of 1–100 for the 48 EIPs investigated. These were calculated by raising each of the adjacency matrices of the 48 EIPs to the powers 1–100. Each line in Fig. 8 represents an EIP. The pathway proliferation rate for FWs has been shown to increase with a power-law degree distribution [17,32,53,54]. This relationship can be seen in Fig. 8 inset (a) [52]. The curves for those EIPs with the highest cyclicity, those curves on top in Fig. 8, most strongly resemble the curve found by Patten for ecosystem behavior shown Fig. 8 inset (a) [52].

Figures 9 through 11 show the pathway proliferation rate of the 48 EIPs plotted in Fig. 8, broken down by EIP rank using cyclicity. The goal of this break down is to show that the relationship between indirect path lengths and the presence and strength of cycles as seen for FWs occurs in industry.

Figure 9 plots the top six EIPs, designated as group A, which have a cyclicity of three or greater. The figure clearly shows that the EIPs with a relatively high cyclicity display a rapid rate of increase in the number of paths with path length. The power-law degree distribution seen for these EIPs in group A closely matches the FW behavior. This topological similarity means that the network robustness to random node deletion that has been related to this structure [55,56] may be translated for those EIPs that closest match the cyclicity seen in FWs. The metric robustness is a parameter that requires knowledge of the quantities of materials and energy flowing between actors, information that is not currently available for EIPs.

Figure 10 shows the class B EIPs, those with cyclicity greater than one, but note that the rate of increase is not as rapid as that for the top-tier EPS. This is another supporting factor for EIP designers to strive for higher cyclicity values in their networks; EIPs with higher cyclicity will have a structure that supports a level of dominance of indirect flows that is on par with what is seen in FWs.

Figure 11 shows the class C and class D EIPs that have a cyclicity of one or zero. Figure 11-left shows that for those EIPs with some form of basic cycling present in their structure, there is no guarantee that the number of paths in the system will be able to increase with path length. Figure 11-right shows that with no cycling there is no pathway proliferation. Figures 8 through 11 confirm that pathway proliferation can only occur when there is more than one cycle in the network, and that this can be confirmed by measuring a cyclicity that is greater than 1. Thus, cyclicity is a key metric for assessing and improving direct as well as indirect material flows in manufacturing networks.

Ecological principles used for understanding ecological FW structures are relevant to improving manufacturing networks. Groupings of EIPs were made in terms of both their economic status (proposed, existing, or failed) and the level of internal cycling in the network structure (high, medium, basic, and none) based on the metric cyclicity. When examined using selected structural FW metrics commonly used by ecologists for FW analysis, the analyzed groupings create a more complete perspective between each other and biological FWs in terms of their success in being “bio-inspired.” None of the systems, despite their status, successfully match the average values found for biological ecosystems. Based upon these results, it is clear that the biological ecosystem, in the sense of the aforementioned structural metrics, has yet to be fully mimicked by industrial networks. For example, it is unlikely that high cyclicity values can be achieved in EIPs that lack actors fulfilling the role of detritus/decomposers. This suggests that EIP designers must incorporate analogous interactions in their industrial networks to achieve the strong cycling characteristic of FWs.

Many approaches to sustainability emphasize this as a values-driven act that provides for individual and societal well-being. This is undoubtedly true, but values and motivation themselves do not constitute a framework to determine how well a circular economy functions, although they may encourage the search for appropriate solutions. This paper demonstrates that ecological networks composed of predator prey linkages can be used to interpret and guide the construction of sustainable industrial organizations through analysis of structure–function relationships. Currently, none of the EIPs identified come close to matching median amounts of cycling seen in FWs. Cyclicity is also a measure of indirect cycling and pathway proliferation rate, i.e., the rate that the number of paths increases as path length increases. Higher cyclicity values correspond to the greater amount of indirect cycling and pathway proliferation rates. This is important because pathway proliferation rate is representative of indirect links in the system, and it has been shown that in FWs when significant cycling was present, indirect flows were nearly always found to dominate direct flows. The analysis of pathway proliferation and indirect links is novel in the context of closed-loop manufacturing systems.

This material is based upon work supported by the National Science Foundation under Grant Nos. CMMI-0600243, CBET-0967536, and CBET-1510531. 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.

EIP =

eco-industrial park

FW =

food web

G =

generalization

L =

number of links

L_d =

linkage density

P_r =

predator to prey ratio

S =

number of species

V =

vulnerability

λ_max =

cyclicity