Impact of Operational and Restoration Interdependencies on Cost and Disruptive Effect in Multilayered Infrastructure Networks

Interdependency relationships that are of interest to this study can be classified as operational interdependencies (affecting the operations of infrastructure networks) and restoration interdependencies (affecting the restoration of disrupted infrastructure networks). Rinaldi et al. (2001) expressed a comprehensive set of operational interdependencies subtypes, including physical, cyber, logical, and geospatial interdependencies. Physical relates to the flow of commodities and asset functionality, cyber relates to information flow through the telecommunications network, geospatial is based on proximity, and logical is any other type of relationship. Using these definitions, Ouyang (2014) categorized 10 critical infrastructure interdependencies based on historical disaster scenarios. During this same analysis, no other set of operational interdependency subtype definitions could categorize all 10 historical examples. Rinaldi et al. (2001)’s four interdependency subtypes largely affect the operations of infrastructure networks and constitute the operational interdependency types used in the present work.

Sharkey et al. (2016) identified five different restoration interdependencies subtypes that only influence the recovery of disrupted networks and deal with recovery task scheduling and resource management. These include traditional precedence, effectiveness precedence, options precedence, time-sensitive options, and competition for resources. Traditional precedence requires Task A in network one to be accomplished before Task B in network two can be started (e.g., de-energize power lines before tree cleanup). Effective precedence means if Task A in network one has not been completed, work on Task B in network two can continue at a slower rate or an extended processing time (e.g., restoring power to pump house speeds flooded road recovery versus pumping by truck). Options precedence means at least one task of two or more in a network(s) must be completed before Task B in a different network is allowed to start (e.g., either power is restored or a generator is brought before a water pump can be used to clear floodwater). Time-sensitive options must be done by a certain deadline, or an additional recovery task will be generated (e.g., restore power to lift station by a certain time or a cleanup task will be needed). Competition for resources can affect restoration activities (e.g., one generator needed at two geographically separated locations). The restoration interdependency subtype of competition for resources is not considered in this work based on the assumption of sufficient resources due to the minimal damage event simulated; however, an example of this type of relationship is expressed in the work of González et al. (2016).

Additionally, Gonzalez et al. (2016) also identified a way in which the geospatial interdependency can be construed as a restoration interdependency by taking into consideration cost savings from scheduling adjacent work and only expending resources once for site preparation (e.g., excavation for the repair of colocated utilities that were both damaged in an earthquake). Four of the five restoration interdependency subtypes (excluding competition for resources) identified by Sharkey et al. (2016), plus the geospatial repair subtype identified by Gonzalez et al. (2016), affect the restoration of interdependent infrastructure networks and comprise the restoration interdependencies in this work.

Although interdependent infrastructure recovery modeling is still an emerging science, some significant progress has been achieved. Lee et al. (2007) developed the interdependent layer network (ILN) model, which sought to find the least cost recovery strategy by minimizing the cost of flow, the unmet weighted demand, and service disruption caused by interdependencies. Using a MIP, this model generated optimal recovery strategies while considering physical, logical, and geospatial operational interdependencies; however, it did not include any restoration interdependencies. The ILN model has been influential and other authors have used and modified it for various interdependent infrastructure recovery applications (Cavdaroglu et al. 2013; Loggins and Wallace 2015).

Sharkey et al. (2015) built upon the model proposed by Cavdaroglu et al. (2013) and Nurre et al. (2012) to develop the interdependent integrated network design and scheduling (IINDS) problem. This model’s objective was to understand the timing of recovery, scheduling repairs with parallel workgroups, and maximizing the network’s cumulative performance over a finite period of time. This work added several unique elements, the most important to the present work is the identification and addition of restoration interdependencies, of which only traditional precedence and time-sensitive options were modeled. This model was limited by the absence of three restoration interdependency subtypes (effective precedence, options precedence, and geospatial repair) and two operational interdependency subtypes (logical and cyber).

González et al. (2016) developed what they called the interdependent network design problem (INDP) and other variations to include the consideration of time dependency, iterative heuristics, and stochastics based on parameter uncertainty (González 2017). These models seek to find the least cost recovery strategies. These models can handle multiple operational interdependency types based on certain coupling strategies of interdependent layers, although only physical and geospatial operational interdependency subtypes were used in the problem instances. This model’s limitations include the lack of cyber and logical operational interdependency subtypes, the lack of explicit formulation to handle various interdependency subtypes simultaneously, and the exclusion of most restoration interdependency subtypes.

Almoghathawi et al. (2019) developed a multiobjective restoration model seeking to maximize resilience while finding the least-cost recovery strategy. They analyzed power and water systems by using a fictional dataset generated using algorithms. The primary advantage of this model was the explicit inclusion of a resilience measure in the objective function that allowed the future exploration of the balance between “withstanding a disruption” and “recovering from a disruption.” This model’s limitations consist of considering only physical operational interdependencies, no restoration interdependencies, and using a fictitious dataset.

None of the aforementioned models address the multiple objectives and listed operational and restoration interdependencies (Table 1). Every model examined included one or more of the three primary objective functions. The most likely reasons that not all models have included all three primary objectives are due to the fact that (1) models are typically purpose-built for some stakeholder-specific objectives, and (2) there has not been a formalization of these primary objectives found in restoration literature. The most common objective between these models was least-cost recovery, followed by minimal recovery time and minimal disruptive effect. Lee et al. (2007) included cost, unmet demand, and weighted time in one objective function, which essentially combined elements of all three objectives. Almoghathawi et al. (2019) used two objectives but included an element of weighted time, which is considered as combining elements of all three objectives.

Least cost, repair time, and disruptive effect are not the only objectives that are possible in recovery operations, they have a striking similarity to the construction or project management trilemma of cost, quality, and time. While there are critics of this approach (Atkinson 1999), it has guided project management for over 70 years and perhaps has influenced these restoration objectives. There is validity in seeking least cost recovery strategies given that financial resources are finite. Optimizing repair time is an objective that is critical for time-sensitive operations, such as defense sector infrastructure with mandated uptimes and limited uninterrupted power supply (Theony 2020). Minimizing disruptive effect of critical assets ensures that infrastructure is supporting life-saving functions (O’Rourke 2007; White et al. 2016). Therefore, these common objective functions are assumed to have utility and value to stakeholders involved in recovery operations.

The models listed in Table 1 do not fully address all interdependency subtypes for four reasons. First, the models have focused on some of the operational interdependency subtypes and not considered the restoration-specific interdependency subtypes given that they were formalized only within the last five years (González et al. 2016; Sharkey et al. 2016) and have not been integrated into all restoration models. Second, exploration of operational interdependency subtypes has been limited by data accessibility issues, which continues to be a problem within interdependent infrastructure restoration modeling (Buldyrev et al. 2010; NIAC 2018; Ouyang 2014; Peretti 2014; Rinaldi et al. 2001). Third, different model purposes and goals have limited the need to include all the various types of interdependency subtypes. Some of the models mentioned could have possibly been adapted or expanded to include additional interdependency subtypes but how they were presented in the literature was insufficient to incorporate all the various subtypes. The limitation of inherently incorporating all known interdependency subtypes is exacerbated by inconsistent nomenclature and categorization. For example, Lee et al. (2007) used a different set of operational interdependency subtypes, which can be considered as both physical and logical from the subtypes used in this study. These problems have led to no other model making the integration of all known interdependency subtypes an inherent part of the model.

The present work seeks to address the absence of a model which considers all the various interdependency types identified within the context of the most common objective functions. The proposed model is designated as the base interdependent infrastructure recovery model (BIIRM). The BIIRM provides a starting point for future models that seek to incorporate interdependent infrastructure analysis in modeling and simulation efforts.

To describe the overall system, let $\mathcal{G}(\mathcal{N},\mathcal{A})$ be a digraph consisting of a set of nodes, $\mathcal{N}$, and a set of arcs, $\mathcal{A}$, indexed as $i$ and $(i,j)$, respectively. To further define this digraph, sets must be defined regarding infrastructure layers, commodities, node and arc subsets, work crews, spaces, operational interdependency types, and time periods.

Let $\mathcal{K}$ be a set of infrastructure layers constructed in a multiplex fashion and let ${\mathcal{L}}^k$ be a subset of commodities that are restricted to flow only within the infrastructure layer $k\in \mathcal{K}$, where ${\cup }_{k\in \mathcal{K}}{\mathcal{L}}^k=\mathcal{L}$. Similarly, let ${\mathcal{N}}^k$ and ${\mathcal{A}}^k$ be subsets of nodes and arcs, respectively, that play an active role in the flow of commodities within a given infrastructure layer $k\in \mathcal{K}$, meaning ${\cup }_{k\in \mathcal{K}}{\mathcal{N}}^k=\mathcal{N}$ and ${\cup }_{k\in \mathcal{K}}{\mathcal{A}}^k=\mathcal{A}$. Also, let ${\mathcal{N}}^{\prime k}$ and ${\mathcal{A}}^{\prime k}$ be the damaged subset of nodes and arcs respectively, where ${\mathcal{N}}^{\prime k}\subseteq {\mathcal{N}}^k$ and ${\mathcal{A}}^{\prime k}\subseteq {\mathcal{A}}^k$. Let there be a work crew $w\in {\mathcal{W}}^k$ who work within a given infrastructure layer $k\in \mathcal{K}$, where ${\cup }_{k\in \mathcal{K}}{\mathcal{W}}^k=\mathcal{W}$. Let there be a collection of spaces $s\in \mathcal{S}$ that are mutually exclusive and comprehensive of the region of interest, where every node is in one and only one space, and every arc is in at least one space. Therefore, the set of spaces $\mathcal{S}$ helps define the geospatial operational interdependencies. Let $\mathrm{\Psi }$ be a set of other operational interdependency types, including physical, cyber, and logical. This additional indexing based on operational interdependency subtype is what allows for layered relationships to exist between node pairs to handle complex interdependent operations. Also, $\mathcal{T}$ is the set of time periods used in the evaluation of the model.

The preceding sets deal with the model at large, but specific interdependency sets are also required to describe the various relationships. In all restoration interdependency relationships included in this model, there is assumed to be a parent-to-child relationship, where the child task in infrastructure layer $\tilde{k}\in \mathcal{K}$ depends on the parent task(s) in infrastructure layer $k\in \mathcal{K}$. Either a node or arc may play the role of parent or child, thus creating node-to-node, node-to-arc, arc-to-node, and arc-to-arc relationships, indexed as $(i,\tilde{i})$, $(i,(\tilde{i},\tilde{j}))$, $((i,j),\tilde{i}))$, and $((i,j),(\tilde{i},\tilde{j}))$, respectively. These parent-to-child relationships are defined as node-based or arc-based depending on the parent asset type being a node or arc, respectively. Therefore, for the four different restoration interdependency subtypes defined by Sharkey et al. (2015) that are used in this model, we have node-based traditional precedence $(NTP)$, effectiveness precedence $(NEP)$, options precedence $(NOP)$, and time-sensitive options $(NTS)$. There are equivalent sets for the arc-based relationships designated as sets $ATP,AEP,AOP$, and $ATS$. These eight different sets provide a comprehensive manner in which to describe four of the five restoration interdependency subtypes used. These special sets are similar to those described by Sharkey et al. (2015), even though only two restoration-specific subtypes were fully used. The geospatial repair subtype is described based on the repair of an arc or node. The presentation of the mathematical formulation for the restoration interdependency constraints is abbreviated by only explaining the relationships used in the scenario described later.

There are decision variables within the model responsible for the flow of material, assigning recovery tasks, completing recovery tasks, operability, and recovery task location. The flow of materials is designated by $x_{ijlt}^k$, which is the flow of commodity $l\in {\mathcal{L}}^k$ across arc $(i,j)\in {\mathcal{A}}^k$ within infrastructure layer $k\in \mathcal{K}$ at time period $t\in \mathcal{T}$. Assignment of recovery tasks is designated by a binary variable ${\alpha }_{\mathit{iwt}}^k$ or ${\alpha }_{ijwt}^k$ (Greek alpha), which is equal to 1 if work crew $w\in {\mathcal{W}}^k$ is assigned to start work at time period $t\in \mathcal{T}$ and continue working until finished repairing node $i\in {\mathcal{N}}^{\prime k}$ or arc $(i,j)\in {\mathcal{A}}^{\prime k}$, respectively, within infrastructure $k\in \mathcal{K}$, and 0 otherwise. In an effectiveness precedence relationship, there is an additional binary assignment variable denoted as ${\alpha }_{i_{e}wt}^k$ or ${\alpha }_{i{j}_{e}wt}^k$ (Greek alpha), which employs a subscript $e$ on the node or arc index to denote an assignment with an extended processing time. The completion of a recovery task is denoted by the binary variable ${\beta }_{iwt}^k$ or ${\beta }_{ijwt}^k$, which is equal to 1 if node $i\in {\mathcal{N}}^{\prime k}$ or arc $(i,j)\in {\mathcal{A}}^{\prime k}$ in infrastructure layer $k\in \mathcal{K}$ is completed by work crew $w\in {\mathcal{W}}^k$ at the start of time period $t\in \mathcal{T}$, and 0 otherwise. The binary variable $y_{it}^k$ or $y_{ijt}^k$ denotes the operability of a node or arc, which is equal to 1 if node $i\in {\mathcal{N}}^k$ or arc $(i,j)\in {\mathcal{A}}^k$ in infrastructure layer $k\in \mathcal{K}$ is operable by the start of time period $t\in \mathcal{T}$, and 0 otherwise. Operability is controlled by whether the node or arc is damaged, the repair is completed, and any operational interdependencies with other networks. The location of recovery activities is controlled by binary variable $z_{st}$, which is equal to 1 if a recovery task (node- or arc-based) is started in space $s\in \mathcal{S}$ in time period $t\in \mathcal{T}$.

Parameters within the model can be divided into those that affect the cost, flow, scheduling, operational interdependencies, and restoration interdependencies. Cost parameters can be further delineated into site preparation, repair, assignment, and flow costs. The site preparation cost is defined as $g_{st}$, which represents the average cost of preparing a site $s\in \mathcal{S}$ at time period $t\in \mathcal{T}$. The repair costs are defined for all $k\in \mathcal{K}$ and $t\in \mathcal{T}$ as $q_{it}^k$ and $q_{ijt}^k$ for any node $i\in {\mathcal{N}}^{\prime k}$ and arc $(i,j)\in {\mathcal{A}}^{\prime k}$, respectively, which is generated from a unit cost table based on the type of facility and an assumed reference size (DoD 2020). The assignment cost represents the national average for a general laborer working on that type of infrastructure layer $k\in \mathcal{K}$ at time period $t\in \mathcal{T}$ and is defined as $a_{wt}^k$ (Latin a) for every work crew $w\in {\mathcal{W}}^k$. The flow cost, $c_{ijlt}^k$, is based on the infrastructure owner’s cost for operations and maintenance of flowing commodity $l\in {\mathcal{L}}^k$ along arc $(i,j)\in {\mathcal{A}}^k$ of infrastructure $k\in \mathcal{K}$ at time period $t\in \mathcal{T}$.

The flow and scheduling parameters are defined for supply and demand, flow capacity, normal processing time, and extended processing time. For all $k\in \mathcal{K}$ and $t\in \mathcal{T}$ the supply or demand of commodity $l\in {\mathcal{L}}^k$ of a particular node $i\in {\mathcal{N}}^k$ is defined by $b_{ilt}^k$, where if $b_{ilt}^k<0$ it is a demand node, if $b_{ilt}^k=0$ it is a transshipment node, and if $b_{ilt}^k>0$ it is a supply node. For ease of notation, subscripts are added to ${\mathcal{N}}^k$ to denote a further subset indicating demand, transshipment, and supply by ${\mathcal{N}}_D^k,{\mathcal{N}}_T^k$, and ${\mathcal{N}}_S^k$, respectively when necessary. Flow is capacitated through an arc $(i,j)\in {\mathcal{A}}^k$ by $u_{ijt}^k$ for all shared commodities $l\in {\mathcal{L}}^k$ within a given infrastructure layer $k\in \mathcal{K}$ at time period $t\in \mathcal{T}$. For all $k\in \mathcal{K}$ each damaged node $i\in {\mathcal{N}}^{\prime k}$ or arc $(i,j)\in {\mathcal{A}}^{\prime k}$ has an associated normal processing time, $p_i^k$ of $p_{ij}^k$, respectively. Similarly, there is an extended processing time for those nodes and arcs that are included in an effectiveness precedence relationship defined as $e_i^k$ or $e_{ij}^k$, respectively. These sets, variables, and parameters provide the background to discuss the formulation and development of the BIIRM.

The literature focuses on minimizing cost, disruptive effect, and repair time. Costs associated with recovery of a disrupted system include repair costs, assignment costs, site preparation costs, and costs of flowing commodities. The equation associated with the cost objective is as follows:

The cost objective has 10 terms, as shown in Eq. (1). The first term is the cost of site preparation. The second and third terms are the arc-based repair costs associated with either normal or extended recovery assignments, respectively. The fourth and fifth terms are the assignment costs for arc-based work, depending on whether a normal or extended processing time is used. The sixth and seventh terms are the node-based repair costs, and the eighth and ninth terms are the node-based assignment costs similar to the arc-based ones. The 10th term is the flow cost of commodities throughout the entire network.

The second primary objective is minimizing disruptive effect and is shown in Eq. (2). Various forms of this objective are presented in literature that seek to ensure demand is met at critical nodes or that critical nodes and arcs are operational. In contrast to using only unmet demand, which restricts applicability to a subset of nodes, the inclusion of all nodes and arcs based on operability allows the model to target critical assets that are not strictly listed as a demand node. Therefore, the surrogate used for minimizing disruptive effect is to maximize the operability at the critical nodes and arcs based on the nodal weight, ${\mu }_{it}^k$, and arc weight, ${\mu }_{ijt}^k$. Weights are assigned by a collaboration of stakeholders to reflect the value infrastructure or infrastructure services provided

The third primary objective is reducing the time required to recover critical assets. Time is integrated into nearly all the variables and parameters, which is a similar integration of this objective, as shown in the works of Lee et al. (2007) and Almoghathawi et al. (2019). The time index allows for capturing the importance of time and ensuring rapid recovery of critical assets. Of note, the nodal and arc weight parameters that signify an asset’s criticality are also indexed by time, thus allowing a user to define when certain critical assets are most needed or relevant in the recovery process.

The two explicitly defined objectives $A$ and $B$, along with the implicit time objective, are weighted in a combined overall objective function. This combination enables recovery personnel to tailor recovery to emphasize cost, operability, or speed. Having described the notation and objective functions, the BIIRM can be presented. This will be done by introducing the overall objective, the network flow and scheduling constraints, the operational interdependency constraints, and the restoration interdependency constraints.

The following is the summarized version of the BIIRM based mainly on node-based constraints for the network flow and scheduling portion. Any additional arc-based constraints are noted where applicable but are not shown. Restoration interdependency constraints use the applicable asset-to-asset relationship, which is defined in each subsection

Subject to

The combined objective function balances minimizing cost and disruptive effect while addressing time by using a time index within the two objective functions [Eq. (3)]. A general flow balance equation for all nodes is presented in Eq. (4). Two slack variables are used to capture unmet demand of a specific commodity $(x_{ilt}^{-,k})$ and surplus of a specific commodity $(x_{ilt}^{+,k})$. Flow is restricted based on starting node, ending node, and arc operability as shown in Eqs. (56)–(7), respectively. A damaged node may become operable once repairs are complete [Eq. (8)]. A damaged node can be repaired only once, as shown in Eq. (9). Only one work crew can be assigned to repair a node, limiting any compounding positive or negative effect that could be possible with multiple crews being assigned [Eq. (10)]. A damaged node cannot be completed until it has been assigned and the normal processing time has elapsed [Eq. (11)]. A work crew can only be assigned to one restoration activity at a given time until the work task is completed [Eq. (12)]. Eqs. (8910)–(11) have corresponding arc-based equivalents not shown in the aforementioned equations, which substitute the arc indices for the node index.

These flow and scheduling constraints provide the base recovery model similar to other integrated network design and scheduling problems used in infrastructure recovery (González et al. 2016; Nurre et al. 2012; Sharkey et al. 2015). Specifically, Eqs. (456)–(7) were adapted from González et al. (2016) and Eqs. (1011)–(12) were inspired by Sharkey et al. (2015). However, both operational and restoration interdependencies must be integrated to address interdependencies.

Operational interdependencies affect the operations of the infrastructure networks by the propagation of failure. The controlling parameter, ${\gamma }_{i\tilde{i}\psi t}^{k\tilde{k}}$, is a time-indexed parent-child node pairing between infrastructure layers. A new set ${\mathcal{N}}_{\tilde{i}\psi }^{k\tilde{k}}$ is used as a subset of parent nodes $i\in {\mathcal{N}}^k$, which have an operational interdependency relationship with a given child node $\tilde{i}\in {\mathcal{N}}^{\tilde{k}}$ based on some operational interdependency type $\psi$. This parameter takes on values of 1 or a fractional amount based on the number of parent nodes in the pairing when an operational interdependency exists between the node pairs consisting of parent node(s) $i\in {\mathcal{N}}^k$ and child node $\tilde{i}\in {\mathcal{N}}^{\tilde{k}}$. This means if two parent nodes are required for a child node to operate, then the interdependency parameter would be equal to one half

The operability of a child node depends on the parent node’s operability and the operational interdependency relationship parameter, which is shown in Eq. (13) and adapted from González et al. (2016). For example, in a simple physical interdependent relationship between $i\in {\mathcal{N}}^k$ and $\tilde{i}\in {\mathcal{N}}^{\tilde{k}}$, the operational interdependency parameter ${\gamma }_{i\tilde{i}\psi t}^{k\tilde{k}}$ would be equal to 1, and therefore, the child node $\tilde{i}\in {\mathcal{N}}^{\tilde{k}}$ depends on the operability of the parent node $i\in {\mathcal{N}}^k$. If the parent node is a demand node, it is essential to ensure that the demand must be met in order for the parent node to be operable, as shown in Eq. (14).

Restoration interdependencies include traditional precedence, effectiveness precedence, options precedence, time-sensitive options, and geospatial repair constraints. The first four subtypes exhibit various asset-to-asset relationships as follows: traditional precedence utilizes arc-to-arc relationships, effective precedence utilizes node-to-arc relationships, options precedence utilizes arc-to-node relationships, and time-sensitive options utilize node-to-node relationships. Each asset-to-asset type of relationship is possible for the first four restoration interdependency subtypes with slight variations to the subsequent constraints. Geospatial repair is handled differently and is addressed following the presentation of the first four restoration interdependency subtypes.

A realistic dataset was used based on a modified version of the CLARC County dataset including the social infrastructure systems (Little et al. 2020; Sharkey et al. 2018). The CLARC dataset represents a county or regional-scale database; however, a municipal size dataset was desired to parallel the size of a military installation. Therefore, a 10% sampling size was taken based on asset type within the CLARC database while ensuring that at least one of each asset type was represented to preserve the diversity of operations by assets. This resulted in a network approximately the same size and scope as a military installation. This dataset will be referenced as the BIIRM dataset.

The data was then reconfigured into a multiplex construct, a one-to-one mapping of a given node as it is reflected in any layer in which that node functions as a supply, transshipment, or demand node (Bianconi 2018). Each arc is assumed to operate and exist only within a given layer. Reflecting nodes based on demand across multiple layers increased the overall node count (if counting reflected nodes separately) well beyond the original 10% sampling. While the reflection of nodes, increases the number of nodes used for a given instance the multiplex structure is revealing of whether or not operational interdependencies exist. The original CLARC database notes 2,631 instances where one infrastructure depends on another (Sharkey et al. 2018). The construction of the database into a multiplex structure maintained all unidirectional dependencies, but highlighted the 703 interdependencies within that number based on nodes having different functions (i.e., demand, transshipment, and supply) within different reflected infrastructure layers. This insight was critical in setting up the interdependency constraints correctly.

Additional significant changes to the dataset included integrating cost information from DoD cost tables (DoD 2020), additional communications infrastructure information to support cyber interdependencies, and addition of another transportation and emergency response commodity of people, which are considered the workforce for the various assets within the networks. Table 2 summarizes the nodes, arcs, and additions made to the dataset.

A critical part of the current research is incorporating various types of operational interdependencies simultaneously. The original dataset included only physical and geographic interdependencies; however, with the addition of communication infrastructure and the commodity of people, cyber and logical interdependency types were established. The cyber interdependencies represent infrastructure systems that depend on communication to provide the service from that infrastructure layer (e.g., emergency responders) or systems controlled by Supervisory Control and Data Acquisition (SCADA) systems. The logical interdependencies are based on certain assets or facilities that require workers to be present to provide the infrastructure service from those infrastructure assets (e.g., power plant, water treatment plant). Table 3 summarizes the number of operational interdependency subtype relationships across the associated infrastructure layers. Geographical interdependency relationships were employed during the damage event due to a simulated flood event to specific portions of the network.

The damage scenario represents a major flood event, which significantly inundates the lower-lying areas of the network. This causes damage to all different types of networks. The assets damaged include some that have operational and restoration interdependencies and some that do not. Table 4 summarizes the damage simulated to nodes and arcs across the five infrastructure layers within the BIIRM dataset.

Based on the damage scenario, several of the recovery tasks exhibit restoration interdependencies or precedence recovery. These recovery tasks range from downed power lines due to trees that have fallen to pumping flooded streets and refueling generators if necessary. Table 5 summarizes the various restoration interdependencies, the coupling method employed in the BIIRM formulation, and a description of the scenarios.

The damage scenario was first analyzed using all nine of the interdependent relationships across varying weights among the two explicit objective functions to provide an overview of the solution landscape. These solutions resulted in a Pareto optimal front, which highlighted the intuitive low expenditure yield of minimal operability improvement. The Pareto front also showed the diminishing returns on increased spending over a particular weighted operability. The Pareto front could be used to determine a “sweet spot” for temporary or expedient recovery operations. For example, the initial weighted operability value following disruption was 35,538 and after $2.6 million the weighted operability value increased to 45,563, which correlates to a 10,025 value increase. However, over the next $7 million the highest increase is only 1,534, which exemplifies diminishing returns. This correlates to infrastructure assets and services that have high cost, but minimal impact to the weighted operability. This understanding can help focus resources to achieve the greatest amount of recovery using temporary and expendable assets. Additionally, some nonessential functions might need to wait until follow-on efforts are made.

Fig. 1 illustrates the balance between cost and operability. The model’s input parameters remained constant throughout the evaluated time periods for this scenario (e.g., costs, node-, and arc-priority weights did not fluctuate over time). Although the operability objective and the combined objective values did not increase and decrease monotonically, respectively, when compared to the cost, the overall objective did decrease consistently at every time period, thus illustrating the tradeoff between cost and operability within the overall convex combination.

The ability to analyze a recovery scenario with and without interdependencies shows the necessity of acknowledging both operational and restoration interdependencies to create the most accurate site picture. Most current modeling efforts incorporate physical and geospatial operational interdependencies subtypes, if any are included. Therefore, this was used as a base and compared against a simulation that added all the operational interdependency subtypes. These results are tabulated in Table 6.

The exclusion of cyber and logical operational interdependency subtypes in this damage scenario meant overestimating the operability shortly after disruption by 0.3%, whereas by the end of 12 8-h time periods, the operability was overestimated by 2.3%. While these are the figures for this instance, other scenarios may show greater or lesser disparities depending on the operational interdependency relationships. The ability to include various, multiple, and sometimes compounding interdependent relationships allows for a more accurate estimate of timelines and achievable operability.

A series of simulations were conducted to understand the effect of restoration interdependencies on cost and operability. The simulation that included traditional precedence (TP), effective precedence (EP), options precedence (OP), and time-sensitive options (TS) was assumed to be the closest reflection to reality from the simulations. Multiple simulations were done by removing one or more restoration interdependency subtypes. In terms of cost, the simulation of TP, EP, and OP was the most closely matched simulation of all the others, effectively showing that in this instance, TS did not play a significant role when combined with the other restoration interdependency types. All simulations except TP and EP overestimated the cost initially, which can be understood as assigning and repairing more work initially that might not be possible due to precedence requirements. This means the model suggested fixing more than can be fixed due to neglecting certain interdependencies. At Time Period 4, all the simulations except for the one with no restoration interdependencies started to underestimate the cost for the remainder of the recovery efforts, which can be interpreted as giving a low estimate of the actual cost. Only the simulation of no restoration interdependencies consistently overestimated the cost when considered against the assumed picture of reality. Fig. 2 illustrates the overestimation and underestimation against the simulation with all the restoration interdependency subtypes in terms of cost.

The damage resulted in all the recovery simulations starting with 65.9% of the network operable. The networks were then restored, with every simulation experiencing a significant increase in operability around Time Period 5 due to a restoration of a critical node. The closest approximation to reality on percent operable is assumed to be the simulation, including all the restoration interdependency types. In terms of operability, all of the simulations followed the same general trend. The TP and EP simulation improved rapidly along with all others except the TP, EP, OP, and TS simulation and then had no improvement to operability over the later Time Periods 5 to 12. The TP, EP, OP, and TS simulation lagged behind every other simulation for Time Periods 1 to 4, but then achieved a greater percent operable than TP-only and TP and EP simulations. The performance of the TP, EP, OP, and TS simulation over the other two seems to indicate that options precedence appreciably affects the system’s operability or recovery time in this scenario by creating desirable recovery strategies leading to higher operability. However, the inclusion of time-sensitive options restricted the TP, EP, OP, and TS simulation so it was not able to achieve as high operability. In the instance of the simulation with TP, EP, and OP interdependency subtypes and no restoration interdependencies, the percent operable remained consistently over the assumed reality (i.e., TP, EP, OP, and TS simulation). Fig. 3 illustrates the percent operability throughout recovery operations for the various simulations, with enlarged windows for Time Periods 1 to 4 and 5 to 12. The TP-only and no restoration interdependencies simulations both experienced a decrease in the operability, which is a manifestation overall objective function balancing cost and disruptive effect as well as Eq. (14)’s limitation of operability being based on demand being met. Therefore, in both of these simulations the model restricted flow for one or two time periods causing some assets to be classified as nonoperable.

In summary, the inclusion and exclusion of restoration interdependency subtypes made the estimations of overall network operability either high or low. In terms of operability, effective precedence and options precedence provide alternate recovery strategies and increase the solution space. Traditional precedence and time-sensitive options restrict the solution space by eliminating certain recovery pathways.