This section lays out the MIP development, which is denoted as the base interdependent infrastructure recovery model (BIIRM). The section starts by describing the general BIIRM notation to include sets, variables, and parameters. The section then describes the three BIIRM objectives followed by three sets of constraints. The first main section of constraints is focused on network flow of commodities and scheduling damage repair. The next section of constraints incorporates operational interdependencies. The final section of constraints incorporates restoration interdependencies.The multilayered nature of the BIIRM employs both multiplex structuring (i.e., one-to-one nodal reflections in various layers) and multislice structuring (i.e., adds element of time) (Bianconi 2018). The combination of these multilayered structures allowed for the analysis of operational and restoration interdependencies. These multilayered structures will be employed for a network comprised of 150 key infrastructure assets and the associated linear assets to establish connectivity across five infrastructure layers. This network is described in detail following the formulation of the BIIRM.General NotationTo describe the overall system, let G(N,A) be a digraph consisting of a set of nodes, N, and a set of arcs, 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 K be a set of infrastructure layers constructed in a multiplex fashion and let Lk be a subset of commodities that are restricted to flow only within the infrastructure layer k∈K, where ∪k∈KLk=L. Similarly, let Nk and Ak be subsets of nodes and arcs, respectively, that play an active role in the flow of commodities within a given infrastructure layer k∈K, meaning ∪k∈KNk=N and ∪k∈KAk=A. Also, let N′k and A′k be the damaged subset of nodes and arcs respectively, where N′k⊆Nk and A′k⊆Ak. Let there be a work crew w∈Wk who work within a given infrastructure layer k∈K, where ∪k∈KWk=W. Let there be a collection of spaces s∈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 S helps define the geospatial operational interdependencies. Let Ψ 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, 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 k˜∈K depends on the parent task(s) in infrastructure layer k∈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,i˜), (i,(i˜,j˜)), ((i,j),i˜)), and ((i,j),(i˜,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 xijltk, which is the flow of commodity l∈Lk across arc (i,j)∈Ak within infrastructure layer k∈K at time period t∈T. Assignment of recovery tasks is designated by a binary variable αiwtk or αijwtk (Greek alpha), which is equal to 1 if work crew w∈Wk is assigned to start work at time period t∈T and continue working until finished repairing node i∈N′k or arc (i,j)∈A′k, respectively, within infrastructure k∈K, and 0 otherwise. In an effectiveness precedence relationship, there is an additional binary assignment variable denoted as αiewtk or αijewtk (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 βiwtk or βijwtk, which is equal to 1 if node i∈N′k or arc (i,j)∈A′k in infrastructure layer k∈K is completed by work crew w∈Wk at the start of time period t∈T, and 0 otherwise. The binary variable yitk or yijtk denotes the operability of a node or arc, which is equal to 1 if node i∈Nk or arc (i,j)∈Ak in infrastructure layer k∈K is operable by the start of time period t∈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 zst, which is equal to 1 if a recovery task (node- or arc-based) is started in space s∈S in time period t∈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 gst, which represents the average cost of preparing a site s∈S at time period t∈T. The repair costs are defined for all k∈K and t∈T as qitk and qijtk for any node i∈N′k and arc (i,j)∈A′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∈K at time period t∈T and is defined as awtk (Latin a) for every work crew w∈Wk. The flow cost, cijltk, is based on the infrastructure owner’s cost for operations and maintenance of flowing commodity l∈Lk along arc (i,j)∈Ak of infrastructure k∈K at time period t∈T.The flow and scheduling parameters are defined for supply and demand, flow capacity, normal processing time, and extended processing time. For all k∈K and t∈T the supply or demand of commodity l∈Lk of a particular node i∈Nk is defined by biltk, where if biltk<0 it is a demand node, if biltk=0 it is a transshipment node, and if biltk>0 it is a supply node. For ease of notation, subscripts are added to Nk to denote a further subset indicating demand, transshipment, and supply by NDk,NTk, and NSk, respectively when necessary. Flow is capacitated through an arc (i,j)∈Ak by uijtk for all shared commodities l∈Lk within a given infrastructure layer k∈K at time period t∈T. For all k∈K each damaged node i∈N′k or arc (i,j)∈A′k has an associated normal processing time, pik of pijk, respectively. Similarly, there is an extended processing time for those nodes and arcs that are included in an effectiveness precedence relationship defined as eik or eijk, respectively. These sets, variables, and parameters provide the background to discuss the formulation and development of the BIIRM.Infrastructure Recovery ObjectivesThe 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: (1) Cost objective: A=∑t∈T(∑s∈Sgstzst+∑k∈K(∑w∈Wk(∑(i,j)∈A′k(qijtk(αijwtk+αijewtk)+awtk(pijkαijwtk+eijkαijewtk))+∑i∈N′k(qitk(αiwtk+αiewtk)+awtk(pikαiwtk+eikαiewtk)))+∑l∈Lk∑(i,j)∈Akcijltkxijltk))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, μitk, and arc weight, μijtk. Weights are assigned by a collaboration of stakeholders to reflect the value infrastructure or infrastructure services provided (2) Disruption objective: B=∑t∈T∑k∈K(∑i∈Nkμitkyitk+∑(i,j)∈Akμijtkyijtk)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.Integrating Restoration InterdependenciesRestoration 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.Traditional PrecedenceTraditional precedence is when a parent recovery task at arc (i,j)∈A′k must be accomplished before a child recovery task at arc (i˜,j˜)∈A′k˜ can be started, which is the arc-to-arc or ((i,j),(i˜,j˜)) relationship in the arc-based traditional precedence (ATP) set (15) ∑τ=1t∑w∈Wkβijwτk≥∑w∈Wk˜αi˜j˜wtk˜,∀ ((i,j),(i˜,j˜))∈ATP,t∈TBased on the definition of traditional precedence, the parent arc must be completed before the child arc can be started, as shown in Eq. (15). When the parent asset is a demand node, demand must be met to start the child restoration task and maintain the total demand throughout the restoration activity. While these are not shown due to the arc-to-arc relationship, similar constraints are shown in the effective precedence relationship.Effectiveness PrecedenceEffective precedence is when a parent recovery task at node i∈N′k must be accomplished for a child recovery task at arc (i˜,j˜)∈A′k˜ to proceed at a normal processing time; however, if the parent node is not completed, then the child recovery task at arc (i˜,j˜)∈A′k˜ can still proceed at an extended processing time. It should be noted that when programming these relationships, it is as if there is a traditional precedence relationship for the normal processing time and an extended processing time if the traditional precedence conditions are not met (16) ∑τ=1t∑w∈Wkβiwτk≤∑τ=1t∑w∈Wk˜αi˜j˜ewτk˜+∑τ=1min[T,t−pik]∑w∈Wkαiwτk,∀ (i,(i˜,j˜))∈NEP,t∈T(17) 1−xilt−,k−biltk≥∑w∈Wk˜αi˜j˜wtk˜,∀ (i,(i˜,j˜))∈NTP|biltk<0,l∈Lk,t∈TThe difference between traditional and effective precedence is the child node’s ability to be completed before the parent node, so long as the child task is processed at the extended processing time [Eq. (16)]. The traditional precedence restriction of meeting demand at the parent node before starting on the child arc is still effective for the assignment variable associated with normal processing time, as shown in Eq. (17).Effective precedence relationships adjust several equations already previously presented (18) ∑w∈Wk(αiwtk+αiewtk)≤1,∀ i∈N′k,k∈K,t∈T(19) βiwtk≤∑τ=1min[T,t−pik]αiwτk+∑τ=1min[T,t−eik]αiewτk,∀ i∈N′k,(i˜,i)∈NEP,((i˜,j˜),i)∈AEP,w∈Wk,k∈K,t∈T(20) ∑τ=1min[T,t+pik−1]∑i∈N′kαiwτk+∑τ=1min[T,t+pijk−1]∑(i,j)∈A′kαijwτk+∑τ=1min[T,t+eik−1]∑((i˜,i)∈NEP,((i˜,j˜),i)∈AEP)αiewτk+∑τ=1min[T,t+eijk−1]∑((i˜,(i,j))∈NEP,((i˜,j˜),(i,j))∈AEP)αijewτk≤1+∑τ=pik+1t∑i∈N′kβiwτk+∑τ=pijk+1t∑(i,j)∈A′kβijwτk,∀ w∈Wk,k∈K,t∈TThese modified constraints describe how only one work crew can be assigned to repair a node either at a normal or extended processing time [Eq. (18); compare Eq. (10)]. A damaged node cannot be completed until it has been assigned and the normal or extended processing time has elapsed [Eq. (19); compare Eq. (11)]. For example, a damaged node i∈N′k (e.g., Fire Station) with a normal processing time of two time periods and an extended processing time of three time periods at Time Period 4 could be repaired (i.e., βiw4k=1) so long as the repair was assigned in Time Periods 1 or 2 at a normal processing time or in Time Period 1 at an extended processing time. A work crew can only be assigned to one restoration activity at a given time until it is completed, regardless of whether the work crew is working at a normal processing time or at an extended processing time [Eq. (20); compare Eq. (12)]. Therefore, returning to the Fire Station example, there were three options to assign a work crew in order to make sure the Fire Station was operable by Time Period 4, but only one of the three options can be picked based on Eq. (20). Additionally, because the Fire Station was in an effectiveness precedence relationship there is one other task that has to be complete prior to normal processing time, therefore the options are trimmed down to at most two options: (1) normal processing assignment at Time Period 2 (based on mandatory task for normal processing time equal to one time period); or (2) extended processing time assignment at Time Period 1. Eqs. (18) and (19) have corresponding arc-based equivalents.Options PrecedenceOptions precedence is when at least one parent arc must be completed before a child recovery task can begin. This precedence relationship is achieved by summing over the parent-child pairs similar to the traditional precedence, as shown in Eq. (21). Similar to traditional precedence, node-based relationships must ensure demand is met at parent nodes and remains throughout the child recovery task’s duration (21) ∑τ=1t∑w∈Wk∑((i,j),i˜)∈AOPβijwτk≥∑w∈Wk˜αi˜wtk˜,∀ ((i,j),i˜)∈AOP,t∈TMathematically traditional precedence completion [Eq. (15)] can be thought of as a special case of options precedence [Eq. (21)]. However, in describing restoration activities they are used differently. Traditional precedence relationships are often used in a chain of events (e.g., Task A before B, Task B before C, and so on). Options precedence are almost exclusively used as a single event where there are two or more tasks that could satisfy the precedence relationship. Therefore, both restoration interdependencies are used separately.Time-Sensitive OptionsTime-sensitive options are those in which a parent node i∈N′k must be operable or child recovery task at node i˜∈N′k˜ must be accomplished by a certain deadline, θii˜kk˜(22) yitk+∑τ=1θii˜kk˜∑w∈Wk˜βi˜wτk˜≥1,t=θii˜kk˜,…,T,∀ (i,i˜)∈NTS(23) ∑w∈Wkαi˜wtk˜=0,t=1,…,θii˜kk˜−pi˜k˜−1,∀ (i,i˜)∈NTSThe child recovery task must be completed by the deadline or the parent node must be operable [Eq. (22)]. By definition, the child recovery task cannot be assigned until the normal processing time before the deadline so that one task is completed by the deadline [Eq. (23)].Geospatial RepairNodes and arcs are also geospatially located within at least one space s∈S. Each space is mutually exclusive and comprehensive. This restoration interdependency subtype allows for cost savings during recovery operations by selecting tasks within a geographical region, where recurring costs for mobilization and site preparation can be avoided. This selection process assumes the crews work in a collaborative environment and are managed by a central authority (Lee et al. 2007). (24) ∑w∈Wkgisk(αiwtk+αiewtk)≤zst,∀ i∈N′k,s∈S,k∈K,t∈TWhen a recovery task at node i∈N′k is assigned at either a normal or extended processing time, then a variable indicating work in that region is used to indicate some site preparation costs will be necessary [Eq. (24)]. Eq. (24) has a corresponding arc equivalent similar to others used in this model. This concludes the abbreviated formulation of the BIIRM.