Integrating systems thinking and flexibility in infrastructure management

Infrastructure systems are subjected to demands imposed by extreme natural events and increasing socioeconomic pressures. The latter are derived, for example, from a rise in the export of goods and services, and population growth, which is also associated with problems such as access to housing and basic sanitation. According to the World Risk Report 2016, inadequate infrastructure increases the frequency of losses to life and property, as well as the risk that an extreme natural event might become a disaster. Thus, frequently, traditional infrastructure systems design and management have fail to meet the stakeholders’ expectations. The main problems emerge from decisions based on the assumption that systems are static, instead of changing continuously, and on uncertainty management. This paper presents, first, a conceptual proposal that combines systems thinking and flexibility as the main pillars to facilitate systems to respond and change over time. Systems thinking recognizes the dynamic evolution of systems and helps understand how processes and subprocesses interact at different levels (i.e., hierarchical arrangement of processes). Flexibility, which is the ability of a system to respond or change, is crucial to managing better risks and varying demands, thus fostering resilience to disasters and climate change. Incorporating flexibility in the development of infrastructure contributes to enhancing the management of planned or unplanned events, taking advantage of investment/business opportunities, reducing possible cost overruns, and handling the perceptions and interests of stakeholders. Based on the conceptual framework, the second part of the paper presents a proposal to integrate these concepts into a physical and financial infrastructure model. The outcome of this model is a set of management policies that render the actions that best reflect the interest of stakeholders. The conceptual ideas and the proposed model are illustrated with two examples showing the value of integrating systems thinking and flexibility in infrastructure management over time.


Introduction
In 2016, the World Bank estimated that disasters cost the global economy $520 billion annually and force 26 million people into poverty [13]. In addition, it is estimated that since the adoption of the Sendai Framework for Disaster Risk Reduction (2015-2030), some 60 million people in over 100 countries have been displaced by disaster events, mainly floods, storms, and droughts [28]. According to the World Risk Report 2016 [20], inadequate infrastructure and weak logistic chains increase the risk that an extreme natural event might become a disaster. This makes evident the need for developing sustainable infrastructure systems capable of reducing inequality, improving welfare, and adapting to climate change. This problem is significant in emerging economies and developing countries where infrastructure is frequently insufficient. Although infrastructure is critical in enabling long-term development, it is also responsible for more than 70% of greenhouse gas emissions worldwide. The UN report on Infrastructure for Climate Action [47] states that the increased frequency and intensity of events such as wildfires, floods, and droughts cost lives, disrupt economies, and set back development progress that has taken years to establish. Hence, there is a need to balance infrastructure needs on the one hand and its impact on society and economy on the other.
Developing sustainable infrastructure necessitates a set of decisions and actions aimed at maintaining economic, social, and ecological balance, all while preserving the society's quality of life and competitiveness. Building infrastructure is expensive and takes a significant share of public resources. Moreover, infrastructure is constantly changing due to external pressures such as population growth, industrialization, economic evolution, technological needs, and extreme natural events. However, modifications are often expensive, their implementation is difficult to reverse, and it takes time. Besides, damage to infrastructure has critical long-term consequences due to the high replacement cost and the long reconstruction times. As a result, budget assignments for existing and new infrastructure should be carefully evaluated from a financial, technical, environmental (including greenhouse gas (GHG) emissions), and ethical perspective. Furthermore, it is necessary to review the conception, design, implementation, and operation for it to be resilient, robust, and adaptable [38].
In an uncertain environment with the pressures imposed by the consequences of climate change, resilience and sustainability are central to infrastructure development. Resiliency is the system's capacity to recover from or bounce back from some undesirable state to a new condition, e.g., see [1,19]. On the other hand, sustainable infrastructure systems are those that are planned, designed, constructed, operated, and decommissioned to ensure economic, financial, social, and environmental sustainability over the entire infrastructure life cycle [32]. Building resilient and sustainable infrastructure requires a new approach in which the dynamics and continuous change of all constitutive processes are at the center of any decision. Accepting that change plays a crucial role in developing more efficient systems should be integrated into all project life cycle phases. This paper presents a proposal to rethink the problem based on a combination of two fundamental lines of thought: systems thinking and flexibility. This way of addressing the problem is an alternative to rigid management strategies, where improvising or deviating from the original plan is discouraged. Systems thinking incorporates the idea that infrastructure should be considered as a process recognizing its dynamic nature. Flexibility, on the other hand, facilitates change and improves adaptation and decision-making under uncertainty. Therefore, systems thinking and flexibility are necessary to enhance the preparation and response of infrastructure to failures and disasters, thus impacting the long-term welfare of communities by making them more resilient.
Overall, current infrastructure investment needs are exacerbated by the significant growth in demand due to socioeconomic factors (e.g., rapid urbanization, increasing trade), as well as the need to respond to the increasing effects of climate change (e.g., changes in the intensity and frequency of hurricanes). As a result, the current situation has shown a need for a shift in future infrastructure design and management (e.g., water supply, energy distribution, and road networks). This requires revisiting traditional design and operation strategies and developing new conceptual and practical approaches capable of integrating the following essential aspects: (i) phasing development to deferring investments over time, (ii) keeping future commitments open to better managing a continuously and uncertain growing demand, and (iii) being prepared for and responding effectively to the increasing probability of damage caused by extreme events.
The paper's main objective is to present a strategy to overcome the gap between the current practice and the needs for future infrastructure development, as mentioned above. To achieve this goal, three main objectives were defined. The first is to present a conceptual framework that provides a better understanding of how systems evolve. The two main pillars of this framework are systems thinking and flexibility. These two concepts encompass the change needed in an evolving infrastructure and the mechanisms that facilitate it. The second objective of the paper is to present a proposal to integrate these concepts into a physical and financial infrastructure model. The outcome of this model is a set of management policies that render the actions that best reflect the interest of stakeholders. Finally, the paper discusses the importance of this new approach to address the problems of resilient infrastructure and disaster management.
The paper is structured as follows. Section 2 presents a critical discussion on system dynamics, i.e., how systems change and evolve, and focus on the two conceptual pillars of the paper: systems thinking and flexibility. Section 3 discusses the mechanisms of change behind flexible infrastructure management. Because infrastructure design and management are strongly related to decision-making, Sect. 4 points out the different types of decisions involved in managing dynamic systems and emphasizes the problem of sequential decisions. This section also discusses the concept of policy, which is a strategy to manage operation and recovery after a failure. How the system can evolve is presented in Sect. 5. The mathematical model that describes the performance and the evaluation of costs of the three approaches is presented in Sects. 6 and 7. A discussion on resilience and flexibility is also included in Sect. 8. Finally, Sects. 9 and 10 present two illustrative examples that make evident the importance of the proposed approach in comparison with traditional designs.

Evolution of infrastructure over time
Understanding infrastructure: a systems thinking approach Infrastructure could be described and modeled as a system-as described in systems theory [10,27]. Systems thinking is a philosophy for solving many practical problems [5]. During the last decades, systems thinking has emerged as a strategy to understand the dynamics and complexities of the interactions among system components. This includes physical components and the system's "soft" side, such as the behavior and response of all actors, operational decisions, and the interactions among stakeholders. Systems thinking helps understand how subsystems and individual components interact at different levels paying particular attention to emergent properties, which leads to the widely known concept of the holistic view of projects. Furthermore, it pays special attention to emergent properties derived from the existence of holons, which leads to the widely known concept of the "holistic view" of projects. Within the context of project management, Blockley [3] describes systems thinking as: "getting the right information to the right people at the right time for the right purpose in the right form, and in the right way." Systems thinking is supported by three main ideas: i) structuring information hierarchically (i.e., in layers), ii) identifying and recognizing the importance of loops, and iii) understanding the system as a collection of processes [5].
The third pillar is vital in understanding the performance of a dynamic system. A dynamic system is a system whose future states are connected to its current state by some rules usually called evolution rules (also known as the principle of change). By mapping these changes in the state space, it is possible to construct a trajectory of the system in time. Evolution rules are anything that causes the system to move from its current to a new state. In the case of the built environment, this could be, for example, a political decision, a physical intervention, or a financial market-driven force. A vital element of the proposed approach is a shift in how infrastructures are perceived and developed. Overall, it is argued that they should be described and modeled as "dynamic systems" whose structure and performance evolve and may change over time in response to internal and external factors.
Within this context, every system can be described as a hierarchical arrangement of processes, where every process consists of smaller internal processes (i.e., the hierarchical nature of systems thinking) activated throughout the project's lifetime. Embedded in the idea of process is the concept of elasticity and change reflected in continuous adjustments directed toward maintaining functionality. Besides, the concept of process brings into the analysis the idea of systems' continuous evolution (dynamics) through time. This is, the evolution of the relationships and the interactions among its components and the surrounding environment, which leads to continuous changes in the system's processes and their emerging properties. In the case of infrastructure, emergent properties might be, for example, life quality, safety, functionality, etc. Overall, understanding infrastructure as a collection of processes provides a more Fig. 1 Hierarchical structure of some processes that describe infrastructure dynamics dependable insight into its performance and how it should be designed and managed over time. Figure 1 shows a hierarchy of the processes describing the performance of infrastructure systems and some processes relevant to the discussion presented in this paper (i.e., infrastructure management and resilience) are named. This figure also emphasizes the intersections among some processes to strengthen the importance of considering the relationships between components in systems thinking.
Note that there is a link between the concept of processes and life cycle in that they describe all activities or changes in the system during a given timeframe, usually called the project's time mission. Therefore, life cycle analysis is a way to approach engineering projects from a systems thinking perspective. The idea of process has also been recognized in many areas where systems dynamics play an essential role. A case in point is the so-called business process management (BPM), which has attracted attention recently as a critical concept underpinning organizational activities and processes that need managing in the same way as other more tangible corporate assets [41].

Flexibility and adaptability
The dynamics of infrastructure systems are determined by the system's response (planned or unplanned) to changes in the surrounding environment or events that materialize during its time mission. Thus, decision-makers face the challenge of managing systems so that they perform satisfactorily under a wide variety of unforeseen futures, most of which are highly uncertain. Within this context, strategies such as real options, decision trees, road maps, and several policy planning approaches are used to support planning under deep uncertainty [12,16]. More recently, new strategies that focus on incorporating flexibility and adaptability in design, planning, and operation are gaining attention [8,9,14,17,18,36,37,39,45,46].
Flexibility is "the ability of a system to respond or change some of its design or operational parameters easily to keep or add value to the system when subjected to either internal or external demands" [38,46]. This definition has three fundamental elements: (a) the ability to respond/change, (b) how this change occurs (i.e., easiness), and (c) the purpose of change-that is, keeping or adding value to the system. In the development of engineering systems, flexibility contributes to three fundamental issues. First, it makes better use of financial resources. Secondly, changing the design and management paradigms contributes to environmental sustainability by, for example, reducing CO2 emissions and thus climate change effects. Thirdly, it is the best tool for managing uncertainty, which is essential to managing risks and increasing resilience. Flexibility is an attribute of a process; in that sense, implementing flexibility requires describing the problem from the perspective of "systems thinking" [5].
Flexible systems are equipped with specific features that facilitate future changes, such as expansions or contractions of the system capacity [8,12,39,45], build more sustainable infrastructure [21], or changes in demand and technology. Flexibility exists as long as the system is equipped with the potential to change, which can be interpreted as the "potential energy" stored in the system. This potential to change is added to systems at some point before the change is required. Incorporating flexibility within the design and operation of infrastructure or responding to disasters is a crucial source of individual, organizational, and societal resilience, contributing to make better decisions and building more sustainable systems. Conversely, systems that are not designed to be flexible frequently become expensive and inefficient since they may incur additional investments derived, for example, from over-estimating future demands.
It is important to notice that sometimes the terms flexibility and adaptability are used interchangeably. An adaptable system can modify its structure or functionality when required, e.g., respond to perturbations to maintain its functionality over time [11]. Then, flexibility could be interpreted as a necessary feature of adaptable systems. Introducing flexibility and adaptability as system properties is a new concept that contributes to better management of planned/ unplanned events, taking advantage of investment/business opportunities, reducing possible cost overruns, and handling the perceptions and interests of stakeholders [37].

Mechanisms and the nature of change
The mechanisms of change are the processes by which a system modifies its state at a given time. Introducing change into a system can be done at different levels, then, there is no unique interpretation of flexibility. Flexibility is strongly linked with the nature and type of process and its place within the hierarchy. For example, in the area of information systems, [41] identified several types of flexibility. In the context of infrastructure systems, a classification of the main types of flexibility is proposed in Table 1 [34,45].
Infrastructure evolution is linked, to a large extent, with economic growth [30]. Thus, it should continuously accommodate a growth in demand that impact not only the system capacity but also its safety. Note that the problem with safety is not only related to damage and potential losses but also the impact on business continuity. Changing capabilities is not only important to respond to possible threats; it is a mechanism that guarantees the possibility of exploring new opportunities to increase value, for instance, through expansions or technological improvements. Although it might be possible to anticipate possible system changes at the design stage, some frequently result from the need to respond to unknown demands or externalities, which occur at uncertain times, for instance, changes in stakeholders' preferences, new technologies, or regulatory issues.

General aspects
Incorporating flexibility into infrastructure management is in essence a decision-making problem. This means that the system's state and attributes are modified continuously to accommodate changes in demand or capacity (e.g., deterioration). Consider a system whose state at time t is defined by a vector parameter X(t). Then, the mechanisms of change are the processes that define the evolution of the system when it moves from time t to time t + Δt . This may imply that the system state changes from X(t + Δt) with X(t) ≠ X(t + Δt) or remain as it was at time t. Moving to a new state at time t implies making a decision a t ∈ , with A the decision space. This decision clearly depends on the system state at time t and some uncertain parameter , i.e., a t (X(t), ) . This decision materializes in a reward (or loss) for the owner, r t ; and the transition to a new system state. The history of decisions describes the system's evolution, and the order in which they are made constitutes a sequential decision problem [35].

Modeling sequential decisions
There are different strategies to model sequential decisions. An approach extensively used for solving this kind of problem is Markov chains, which consists of making decisions at specific times based on the system state. However, this approach is rather limited since it does not allow modifying possible future system states and transition probabilities. A second approach consists of gathering evidence of possible future scenarios that stem from each available course of action (also known as tree search). In other words, the decision is made at time t based on some measure (e.g., expected value) of all future scenarios from t + 1 onwards. This methodology is useful in problems with limited action spaces and short time windows since it is computationally intensive and may easily become intractable [7,31]. A third alternative consists of observing the system performance continuously until there is "enough" information to decide. The decision as to when to stop observations and make a decision is called the stopping rule. Finally, for more complex problems, there are other options to model sequential decisions such as sparse sampling, which limits the number of possible outcomes compared to the tree search, or the roll-out heuristics that approximate the value of future scenarios. A detailed mathematical description of these strategies is beyond the scope of this paper and can be found, for example, in [2,23,35].

Managing decisions via policies
Managing sequential decision problems might be handled by defining a finite set of decision rules. Decision rules can be expressed simply as "if-then" conditionals that instruct the decision maker as to when to trigger an adaptation process [50]. Given the information available, a structured set of decision rules is called a policy [31]. More precisely, a policy, , is a mapping from the possible system states to the set of actions, A , where the action a t ∈ A is a decision at time t under the policy [15]. By implementing decision rules (policies), stochastic programs can be formulated and solved to determine the optimal sequence of decisions [35,43].
The definition of a policy should be looked at from different perspectives, but overall, it consists of a set of decision rules. These rules define, based on the system's state at a given time, whether an intervention is required or not Description of the type of system modification Flexibility in design Equip the system with the features that facilitate the possibility of implementing future changes throughout its lifetime. Adding new design and operational variables, or new functions is often called modifiability Flexibility in system state Capacity to accommodate deviations from a prescribed evolution path without altering the project's scope and objective. This is, carried out by increasing or decreasing the value of one or a set of state variables Flexibility to evolve Ability to make significant changes to the system, or some of its main processes (e.g., in response to changes in external demands or stakeholders' decisions, e.g., business opportunities) Flexibility to escalate or deescalate Capacity of the system to evolve by adding or removing processes and components to adjust the scope of its functionality. Some authors denominate this form of change simply as scalability. This process can also occur by designing the system as a collection of modules, i.e., subsystems, which may or may not inherit all the functions from the primary system, that are added or removed when needed Flexibility of Management Ability to admit changes in the system management to adjust to the company's policies, changes in demand, or to improve efficiency and their extent. As mentioned before, a policy may impact the different processes that define the system. Sometimes they may be related to physical properties (e.g., materials, geometry), the response capacity (e.g., ductility and inelastic behavior), the operation (maintenance program), or the stakeholders' risk appetite and incentives. Overall, policies should be carefully tailored for every project. Incorporating flexibility does not mean that unexpected events (e.g., increments in demand) are managed randomly. Flexibility should be accompanied by a set of rules (policy), which include an assessment of the situation and a decision about the time and the characteristics of interventions (e.g., size, type). Policies cannot interpreted as prescribed protocols but as guides to evaluate the situation.

Strategies to value a flexible policy
A particular course of action should be accompanied by some metric, i.e., a set of quantifiable indicators. For example, the feasibility of private infrastructure projects is frequently evaluated in terms of the financial gain for stakeholders; e.g., cost-benefit analysis (CBA). CBA requires computing the expected value of the discounted cash flow function Z = B − C with B the benefits and C the investments. In most cases, the CBA for a particular policy, , is computed using simulation. This is, the system is subjected to a large set of feasible demands and the respond according to the policy, i , is recorded to make some statistical inference on the set of outputs. For example, this information can be used to build possible cash flows, from which it is possible to calculate the distribution function for the net present value (NPV) of each policy, or the expected discounted NPV, [Z] , and its standard deviation. The financial evaluation is combined or replaced by an economic assessment in projects with a strong social component. The reason to move to an economic evaluation is that the development of a project has a "value" that cannot be measured only in terms of some monetary unit. This has led to the use of "utility" functions. Utility functions are used extensively to evaluate problems in terms of stakeholders' preferences. Recently, [4] brought up the importance of going beyond utility to consider the idea of obligation. This idea emerges as a result of the "growing urgency of the challenges such as those posed by climate change, possible pandemics, pollution, poverty and loss of biodiversity has brought sharply into focus the need to consider and account for our wider values and worth which is not necessarily financial" [4]. This discussion should be at the center of large projects within the context of climate change and must be a matter of further debate, which is beyond the scope of this paper.

Relationship between actions and decisions
Managing the dynamics of evolutive systems requires a combination of short-and long-term decisions to cope with the immediate needs while keeping open the possibility of modifying its structure or operation in response to how the future unfolds [6,16,24,38]. The main challenge of flexible systems is to balance the time at which an intervention (change) is executed and its extent.
In most cases, performance requirements control decisions about intervention times; for example, interventions might be executed every time the system reaches a safety threshold, i.e., if safety is compromised. Despite the importance of mechanically-driven decisions of interventions, in some cases, they cannot be implemented when required because there are other managerial factors, budget restrictions, or stakeholders' interests that play an essential role in the decision. For example, in practice, there is a natural attitude toward delaying investments [39]. This problem is frequently framed as a stopping decision problem, in which the information is gathered until a certain level of confidence (or urgency) forces the intervention. In addition, in privately operated infrastructure, it is frequent that contracts have requirements on the frequency and extent of interventions; in these cases, decisions are bounded to some legal requirements. In summary, planning for these interventions requires continuous monitoring of the system state, frequent short-term estimations of the system's expected performance (capacity and demand), and an evaluation of future preferences and outcomes [26].

Extent of interventions
Regarding the extent of interventions, there are two ways to examine the problem. The first is assuming that their size may take values in a continuous or discrete space. Then, the system is modified so that the predicted failure probability in a given time window (e.g., a horizon of t p years) is smaller than some predefined target value. As this time window becomes larger, so does the size of the intervention required. This approach is usually known as flexible design and operation of engineering systems [9,38,46]. The second approach consists of redefining the system to improve its performance (e.g., capacity). This option does not necessarily correspond to direct modifications of the system structure. These changes may result from adding new components (expand the state of variables X ) or redefining the system as, for example, a collection of several complementary systems. In this case, possible actions (i.e., changes in the system state) are defined by the stakeholders, and occasionally with the important participation of the government. Adaptation pathways fall within this category [49]. These two alternatives are discussed in Sects. 5.4 and 5.5.
In infrastructure, actions may be divided into i) operational, ii) physical, and iii) a combination of physical and operational strategies. These three types of actions are illustrated in Fig. 2. The first case is associated with changes in how the system is managed, and the second corresponds with a physical intervention that increases capacity "instantly." The third case is a physical intervention that provides a new maximum capacity that is available but not added to the system all at once but progressively. This is carried out using some operational strategies until a maximum capacity, defined by the new physical state, is reached. The latter only occurs on certain actions. For example, if the current system is a road connecting two cities, an alternative action might be to build a railway. Then, building the train infrastructure corresponds to the physical intervention. However, railway management (e.g., trains' frequency and capacity) defines the extra capacity added. It could be adjusted as demand evolves up to the point when it reaches its maximum, as defined by the physical infrastructure.

Formalization of the intervention process
The process and the decisions discussed in Sect. 5.1 can be formalized as follows. Let us consider that the system state at time t i is X(t i ) and that it has some emergent properties (i.e., capabilities) V(t i ) , for example, capacity, resistance or safety. In most systems, V(t) is a vector of system performance indicators; however, for simplicity and without loss of generality, in what follows the problem will be formulated in terms of a single parameter. Furthermore, consider that the expected demand on the system at a time > t i is D( ) . In most cases, it is reasonable to assume that V(t) will decrease, or at most keep its value (e.g., uncontrollable system deterioration), unless there is a proper system maintenance program. Besides, D( ) will maintain its value or grow with time (e.g., traffic growth, climate change effects). At some time , the probability that the system complies with some functional requirements (i.e., fulfill its purpose) will reach a critical value evaluated for instance in terms of failure probability (i.e., P(V( ) ≤ D( )) ). Before reaching the critical condition, an intervention is required to improve the system capabilities V( ) . Then, if A is the set of feasible actions available to modify (mostly update) the system performance, the system capacity after an action a ∈ A has been implemented, can be written as: where p a ( ) is the capacity added at time as a result of the physical intervention only, while q a (t) is the operational component of that action at t > if it exists (Fig. 2).

Flexible response in a continuous action space
In many systems, capacity adjustments over time occur in a continuous space. This means that a change in capacity at time t can take any value V(t) > V(t 0 ) . As mentioned, adding capacity to the system results from combining physical and operational decisions. Figure 3a shows two realizations of the evolution of the system capacity over time, C 1 (t) and C 2 (t) , in response to a specific demand d(t). Clearly, these two policies respond to different management strategies. In a fully flexible system, its performance (e.g., capacity) constantly changes to meet the demand while keeping adaptation costs to the minimum, i.e., C 1 (t) . On the other hand, the response illustrated by C 2 (t) , describes a system where changes occur with less frequency but have larger sizes. In both these cases, decisions regarding the extent of interventions are defined by making successive short-term predictions of the system's expected performance. Short term is used to describe reasonable time intervals that do not compromise the system's value for the users. Finally, Fig. 3a also shows the case where the system is designed for an initial capacity that is kept constant throughout its lifetime; this scenario characterizes an "inflexible" system.

Response based on feasible pathways
In large infrastructure, possible actions are often limited to a small set of feasible options (Fig. 3b), which in many cases depend on a large variety of factors, most of which are not technical. For example, possible actions are limited by access to technology, logistic restrictions, stakeholders' interests, and political issues. The adaptation pathways approach is a strategy to integrate a sequence of possible actions, which can be implemented progressively, depending on the system's evolution (dynamics) over time (Fig. 4). The notation and specific terminology used to represent adaptation pathways can be found elsewhere [16,17]. Adaptation pathways may support decision-making under uncertainty in different ways. Most importantly, it provides a framework to manage uncertainty by taking a sequential decision-making approach to managing the system's evolution. In addition, adaptation pathways help to define not only what decisions but also when decisions are needed. Thus, adaptation pathways can be used to navigate a decision tree as the project's future unfolds; the project evolution becomes a route through the decision tree [17,49]. An adaptation pathways map is drawn based on expert judgment. Figure 3b shows an example of the trajectory of the system capacity over time of a system whose option of change is defined for a set of pathways. This figure shows five possible actions (paths) that are available to enhance the system capacity and also two possible realizations of the decision trajectories that yield changes in system capacity over time, i.e., C 1 (t) and C 2 (t) . The criteria to choose a particular path when needed is a decision that involves many factors. Haasnoot et al. [16] proposed a strategy to support this decision-making process.
Werners et al., [49] present a list of strategies that incorporate adaptation pathways in different contexts and several case examples taken from the area of water management and climate change. They grouped the use of adaptation pathways into three, not necessarily independent groups. First, performance threshold-oriented pathways support infrastructure planning by providing alternative sequences of discrete adaptation measures. Secondly, multi-stakeholderoriented pathways acknowledge that adaptation occurs in a multi-stakeholder setting. It considers multiple drivers and multiple stakeholders with conflicting goals, interests, and contested values. Finally, transformation-oriented pathways view pathways as broad change directions representing different strategic aims.

Definition of the system state
The evolution of a system over time, in response to some deterioration (wearing process) or changes in demand, could be written as an ordered set of  interventions-actions-(including the initial construction) executed throughout its time mission, t m : A = {a t } t≥0 . In practice, the set A cannot be defined at the outset because there might be many possible future paths. Then, the model focuses on the quality of a policy (see Sect. 4.4), i.e., decision strategy. The best policy is the one that maximizes some value function ; e.g., max { (A)} ; where (⋅) is measured in terms of, for instance, some financial metrics (e.g., net present value), or the environmental impact (e.g., CO 2 emissions) throughout the project history.
After the system is put in operation, actions are executed to increase the system's capabilities (capacity) to cope with the growth of demand. Suppose the system does not deteriorate and is subjected to a series of interventions (actions) at different times to respond to changes in demand. Then, the system's capacity at time t, V(t) (emergent property of the system) can be computed as: where V n (t n ) is the capacity added to the system by implementing the nth action at time t n . N(t) is the number of actions implemented in time t, which is unknown. The value V 0 (t 0 ) corresponds to the initial capacity (at construction). The size of interventions, V n (t n ) , may be defined within a continuous system space or chosen from a set of possible actions (e.g., adaptation pathways).
As mentioned in Sect. 5.1, actions might combine both a physical, p, and an operational, q, part (see Sect. 5). The physical part of the nth investment will add a maximum capacity p n to the system. However, not necessarily the total capacity added will be put into service immediately. In some cases, the system will start using a fraction of the installed capacity, i.e., p n, ≤ p n and progressively add capacity (through some management strategy) until it reaches its maximum value p n . In these cases, the contribution to the capacity of the nth intervention at time t > t n would be: with t n the time of the nth intervention. Note that there is an upper limit to the capacity added to the system, i.e., p n . In the case of actions where the operational component does not exist (e.g., increasing the capacity of a bridge column), the term q n (t) = 0 and V n (t n , t) = p n (t n ) . Similarly, if the action considers only a change in the system's operation, p n (t n ) = 0 . Note that q n (t) should be carefully tuned according to the nature of the nth action.
Under the assumption that a new action is implemented only if the capacity of the previous action has reached its maximum limit, the total capacity at time t > t n can be written as: where n * = N(t) and q n * (t x ) ≤ p n * − p n * , . Flexibility divides the system's ability to respond to an event into two groups. The first includes all actions that do not require significant use of resources either because the system is equipped with the required capabilities to absorb changes in demand, or because changes can be implemented with a relatively small effort. The second group consists of all those actions that imply large modifications of the system. Investments (cost of changes) in the first group are smaller than in the second. Junca and Sánchez-Silva [38] state that the important question behind flexibility is not what is the possibility to change, since any system can change, but whether it is worth the cost of change.

Cost-benefit evaluation of action paths
The action's space ( A ) may have two forms: It is either a continuous space or consists of a set of well-defined possible interventions. The execution of every action has a cost associated. Then, for a given time mission, it is possible to compute the discounted cost of all investments incurred by the owner of the project; this is, where (t n ) is the discount function with the discount rate. The common form of the discount rate is (t) = e − ⋅t for small values of . In addition, C[⋅] is the cost associated with each intervention. For new projects, the initial investment C 0 is the cost of building the system capabilities at time t = 0 (initial construction). The times and sizes of interventions, or the selection of an action from the set of feasible options, depend on the evolution of the system, the demand, and the stakeholder's decisions.
The total benefit derived from the existence and operation of the system in the timeframe [0, t m ] could be computed as: where z(t) are the annual gains received by stakeholders. The gains are frequently expressed in terms of an annual rate r b , in this case, B = (r b ∕ )e − t m . Then, the objective function for the life cycle cost analysis becomes, with g n (t) the density of the time to the nth intervention; in this case it is assumed that the annual benefits are constant. Note that the financial analysis requires computing the expected value of both the cost of investments and the benefits, which can be obtained by simulation. To do this, the system is subjected to a large set of demand realizations, and the discounted cost-benefit relationship (i.e., Z) of every response history is recorded. This information can be used to make statistical inferences (e.g., [Z] , and 2 Z ). Similar objective functions have been proposed by other authors [22,33,40].

Systems subjected to random failures
Infrastructure failures may result from the relationship between capacity demand and the occurrence of extreme events, some of which have grown in frequency and intensity due to climate change. The failure probability could be written as: where M is a mechanical failure (function of capacity and demand) and E is the failure caused by an extreme event. The probability of a mechanical failure depends on changes in capacity and demand and are computed as P(M) = P(V(t) − D(t) < 0) ; in this case, V(t) and D(t) represent the system capacity and demand at a given time t. If the two failure modes are independent, P(M, E) = P(M)P(E) with P(E) the annual failure probability caused by an extreme event (e.g., flooding, earthquakes). In the model, only extreme events that may cause failure are considered. In the case of extreme events, P(E) will decrease as additional capabilities are added to the system. In the case of floods or wildfires, P(E) is a combination of the action taken by increasing system capabilities and the increase of the event frequency and intensity. Failure caused by extreme events can be modeled in terms of the occurrence rate, (t) , which is constant for events such as earthquakes, or an increasing function for events highly influenced by climate change, such as hurricanes or floods.
Then, cost function can be written as:

(8) P(F) = P(F|M, E)P(M, E) + P(F|M,Ē)P(M,Ē)
where C[F j (t)] is the cost of the jth failure, t j the time at which it occurs, and Q(t) the number of failures in the time mission t m , which is random. Note that in order to evaluate the objective function Z, it is necessary to evaluate the total expected discounted costs, then, with g n (t) and f j (t) the densities of the nth intervention and the jth failure.

Resilience within the context of evolving systems
Resiliency is frequently described as the system's capacity to recover from or bounce back from some undesirable state to reach a new state that could be its original (or improved) condition, e.g., see [1,19]. The term resilience is used extensively in many contexts beyond physical infrastructure, such as biology and ecosystems, organizational dynamics, and individual psychology. The value of a quick recovery after a damaging event is undoubtedly a desired property of any system. For example, in the case of events such as earthquakes or hurricanes that cause extensive damage to largely populated areas, prompt and efficient response and recovery are paramount. Depending upon the system, recovery capacity combines the system properties and strategic decisions, along with the uniqueness of every damaging scenario, which makes it challenging to define an "optimal" recovery strategy.
Several aspects are essential when considering the resilience of infrastructure systems [48]. First, a damaging event and the system recovery occur usually within very short periods compared with its time mission. Secondly, in practice, the response of a system frequently leads to a new system state with new system properties that change previous design and management assumptions. Finally, from the perspective of stakeholders, the value of resilience should be looked at not only regarding the current event but within the context of all future events that might occur throughout its lifetime. In summary, resilience is a system property whose value should be considered within the context of current and future decisions; i.e., the cost and value of making systems resilient depend on assessing all events that might occur during the project's lifetime. Focusing on the immediate response only goes against the fundamentals of (10) decision-making, which suggest that decisions should be based on long-term expected values [29]. Bankes [1] argues that "one of the great challenges before our technological culture is creating systems and institutions that are highly resilient in the face of complexity and deep uncertainty. To play its role, it is necessary to adapt to changes both frequent and rare and be able to contend with future situations that cannot now be anticipated." It can be noticed that failures depend on the processes in the upper levels of a hierarchical structure since they are responsible for guaranteeing that the system fulfills its purpose. Once there is a failure, processes in the lower parts of the hierarchy become relevant, i.e., immediate response and planning, recovery execution, and final adjustment (Fig. 5). In real problems, a strategy to manage this process should be part of the entire system management policy.

Resilience model
System resilience analyses are usually directed toward identifying the recovery function of a system that has been damaged. The recovery function depends on the nature and characteristics of the system, the level of damage, and how the recovery process is managed. Overall, it is reasonable to assume that the shape of the recovery function can be modeled as a scaled sigmoid function: where the parameter b controls the extent of the recovery (in capability units) and depends on the level of damage. The value of a is the velocity of this process. This formulation differentiates three processes (Fig. 5). The initial stage describes the immediate response, recovery preparation, and (11) planning; the intermediate groups all main recovery tasks; and the last part is the time it takes for the system to make the final adjustments. For resilient systems, the parameter a is large, while smaller values of a define poor recoveries.
Including resilience in the model that describes the evolution of the system throughout its lifetime (see Sect. 6) implies that recovery after failure is not instantaneous. This penalizes the cost-benefit relationship (objective function Z) by reducing the gains B and incurring some additional costs. If the system functionality required for receiving some income occurs only when the system capability reaches a threshold V r , it is reasonable to assume that the benefit function during the recovery process is: with r b the benefit rate, for example, US$ per time unit. Therefore, the total benefits derived from all recoveries after some disruptions are computed as follows: with t r w the recovery time of the wth disruption. The factor K b is used to relate the level of recovery with the income. Note that when incorporated in the NPV evaluation, the expected discounted value should be evaluated. This approach can be used to simulate systems with various resilience levels and compare the impact of various strategies. Furthermore, this resilience model can be incorporated into the objective function Z by modifying the benefits.

Basic assumptions
Consider an infrastructure (e.g., highway, levee) subjected to a demand described by a stochastic process {D t } t≥0 (e.g., traffic, sea surge). In this example, three possible ways to approximate the design and management of the system are considered. First, there is the "standard" case, where the system is designed to withstand the maximum expected demand within its time mission. Then, the entire required capacity is built at time t = 0.
The second, called "fully flexible," is a system designed for an initial capacity V(t 0 ) based on an estimated demand at time t 1 ≤ t m (e.g., t 1 ≈ 0.2t m ). In addition, the system is equipped with some specific features (embedded flexibility) that allow making changes in capacity at a cost C f as long as V(t) ≤ ⋅ V(t 0 ) ; changes that take capacity beyond (i.e., V(t) > ⋅ V(t 0 ) ) may occur but a unit cost C e = ⋅ C f with > 1 . In practice, the selection of depends on the problem at hand and is usually referred to as available flexible range [38]. In this example, and for illustrative purposes, = 3 , = 4 and C f = 0.2 per unit of additional capacity. For instance, these values can be observed in the context of modular construction, which facilitates capacity expansion by adding new components. In these systems, a value of = 3 , i.e., a flexibility capacity range up to three times its original value, is reasonable. A typical infrastructure where these values are reported in the design and expansion of airports, e.g., see [42]. The value of was also taken from the context of modular design. For example, Sharma et al. [44] found that integrating modular construction and Building Information Modeling (BIM) could lead to time savings of approximately 74% compared to conventional construction techniques. Similarly, Lawson et al. [25] reported that adopting modular construction methods can substantially decrease construction-related accidents by 80%.
While in the first two cases, the capacity space is continuous, i.e., the system may increase its capacity when required to the value needed, in the third case, changes in capacity can only take values in a set of fixed and welldesigned actions (pathways). All other system properties are kept the same. These pathways have specific starting times and capacities, which cannot be modified. The total system capacity increases by adding the capacity provided by the contribution of the paths taken. Selecting a specific path depends on the system's management policy. Figure 6 presents the flow diagram of the model used to describe fully flexible systems. The flow diagram for the adaptation pathways and the standard cases are shown in Appendix, i.e., see Figs. 13 and 14.

Evaluation and selection of the best management policies
As mentioned in Sect. 4.3, a policy is a strategy to decide when to make an intervention and the size of that intervention. In this example, the criteria to decide whether or not to intervene the system is safety, which is evaluated in terms of some failure probability, e.g., P(V(t) − D(t) ≤ 0) or using any other system condition h(V(t), D(t)) = w.
In the case of the "standard" design, capacity is determined by evaluating the expected demand at the time mission, D e (t m ) (e.g., using some predictive model), and computing the required capacity to withstand that demand. The demand function D e (t) is a predictive model expressed usually as: D e (t) = (t)t , where (t) is the annual growth rate that may depend of time or not. The investment to build that capacity at time t = 0 corresponds to the initial cost C s (t = 0) . The initial capacity is maintained throughout the project's life unless a safety requirement is reached, case in which an investment is made to upgrade the system.
For the fully flexible system, the initial capacity V(0) is computed based on the expected demand at a time t 1 ≤ t m , i.e., D e (t 1 ) . The extent of interventions also depends on predictions of the demand for time windows of length t p ≤ t m . There is no rule to choose t 1 and t p , but it is reasonable to assume that t 1 ≈ t p . In the proposed model, demand predictions to implement system adaptations are defined by a regression D r (t) , which is computed using actual demand observations in the period [t q , t] with for example t q = t − t p , i.e., observations within the last five to ten-time intervals (e.g., years). Frequent interventions may affect the operation and the users' perception of the system; therefore, it is common to restrict the minimum time between interventions to a value u. The initial investment (cost) in this case is C 0,f = (1 + ) ⋅ C 0 with > 0 . The parameter corresponds to the cost of adding flexibility to the system at the outset, e.g., building a stronger foundation that allows further structure expansion. In this example, = 0.2 . Lawson et al., [25] argue that the fixed costs associated with establishing and operating a manufacturing facility that produces modular components can constitute up to 20% of the total construction cost.
In the case of pathways, jumps can only take the system to the capacity defined by one of the options (paths) available. These actions can be fully characterized by physical and operational properties obtained from technical studies. In the case of adaptation pathways, the system does not have any embedded flexibility; therefore, the unit cost of any intervention is the same everywhere in the action's space. The model assumes that a specific path is selected only if, within a reasonable time interval, there is not an alternative path that requires a smaller investment.
The objective is to find the policy that maximizes the system's long-term value. Then, several policies could be constructed by defining the values of (see Sect. 9.1), t p , P f (or w), and u. In the case of pathways, the policy may also include the paths available and the times at which they could be implemented. Note that policies do not define a specific plan of action but a guide on how to manage the system to maximize its performance within t m . Simulations were used to find the best policy. The simulation requires first the definition of a large set of possible demand paths. Afterward, each policy is implemented, and the system's response to every demand in terms of the net present value is recorded. In this example, every policy is tested for N = 2000 sample paths of the demand. In the end, it is possible to obtain the mean and standard deviation of the NPV for every policy. Then, policies can be compared in a space defined by the standard deviation and the mean of the NPV, which is a Page 13 of 20 144 Fig. 6 Flow diagram to evaluate a sample path of an infrastructure system whose performance is defined by a fully flexible strategy widely used technique, for example, in finance, to compare portfolios. In this space, it is possible to identify the policies that create an efficient frontier and select from them the one that best describes the stakeholders' interests.

Results of simulation and discussion
For the standard design case, the initial capacity was evaluated at the beginning of every simulation by making an initial prediction of the demand at time t m (see Sect. 9.2). Initial design capacities are within a range between 170 and 220 (in appropriate units) approximately. For the other two designs, the initial capacity depends on a prediction of the demand at time t 1 = 10 y leading to values between 60 and 80.
In Fig. 7, a sample path of the response of the three systems to a sample path of the demand is presented. Table 2 summarizes the policy that defines every case. It is interesting to note that the safety criteria (failure probability) that define whether an intervention is required is not always the same. It depends on other restrictions, such as the time of the previous intervention or the availability of operational resources (budget).
After subjecting the policies defined by the parameters in Table 2 to N = 2000 random demand scenarios, it is possible to compute the probability density of the net present value as shown in Fig. 8. It can be observed that the fully flexible design leads to a better cost-benefit relationship on average. Besides, the fully flexible and adaptation pathways models are far better than the standard design. This is explained by the fact that flexibility allows delaying investments until they are necessary, thus reducing the impact on the NPV. The difference between the fully flexible and the adaptation pathways is that the former makes better use of the available flexibility range. In contrast, the adaptation pathways strategy is restricted to a fixed set of alternatives (set of actions A = {a 1 , a 2 , a 3 , a 4 } ). It can also be noticed that the variance of the NPV is larger for the standard design and smaller for the fully flexible system. Regarding the parameters that   (Table 2), it is interesting to notice that in both flexible cases, as the safety criteria become more strict (i.e., larger values of w, of smaller failure probabilities), the NPV decreases. This is explained by the fact that safer systems require keeping higher capacities. Similar results were observed for all policies studied. Finally, notice that for a given policy, the fully flexible case is an upper limit to pathways. We explored 100 possible policies, making variations of the parameters shown in Table 2. As mentioned before, each policy was subjected to N = 2000 demand scenarios, and the statistics of the NPV were computed. In Fig. 9 every point is the result of a policy in the space [Z] -Z . The results show that regardless of the policy, fully flexible systems provide the best solutions; this is larger NPV with lower standard deviations. The results for adaptation pathways vary widely, mostly because pathway scenarios were selected randomly; in all cases, only four possible pathways were selected, and the starting times and contributions to capacity were randomly selected. This is justified by the fact that pathways cannot be fully characterized at the outset. They may surge as options as better knowledge of the system and new technology becomes available. In Fig. 9, an efficient frontier is drawn for the fully flexible case from which an acceptable operational policy could be chosen. This efficient frontier is not computed for pathways since their selection depends upon many external factors. Finally, note the inefficiency of the standard design, i.e., low expected NPV and high variability.

Example 2: Flexible resilient systems
Consider the same systems evaluated in example 1 (Sect. 9). In this case, they are subjected to extreme events that might cause random failures. Every time a failure occurs, a recovery process occurs according to the model presented in Eq. 11. Figure 10 shows a realization of the evolution with time of the three systems studied when the possibility of failures is included. The failure probability was computed using Eq. 8 assuming the demand and capacity have c.o.v D = 25% and c.o.v V = 12.5% respectively. Besides, the system is exposed to an extreme event with an annual occurrence rate of 1∕50 = 0.02.
For the particular case of the model studied, the parameter that sums up the recovery process is a in equation 11. It is beyond the scope of this paper to characterize all aspects that define a; then, it will be assumed that a is directly related to the level of damage. This means that the larger the damage, the longer the recovery times. Figure 11 describes in detail how the recovery process is modeled in the case shown in Fig. 10a, the three main recovery processes are indicated. The system is subjected to N = 2000 random demand paths. Figure 12a presents the histogram of the NPV for the three models considered. As in the case without failures, the fully flexible system is the best option, and both fully flexible and adaptation pathways are much better than the standard design. Note also that the fully flexible system has the smaller dispersion, while the standard case has the largest. As expected, compared to the case in which failures are not allowed, the expected NPV and its dispersion are smaller.
Although the number of failures during the lifetime (i.e., 60 years) is, on average, larger for flexible systems (i.e., 2.18), these systems have the smallest recovery times (left axis in Fig. 12b). In addition, the expected NPV (right axis in Fig. 12b) is significantly higher when flexibility is incorporated. These results show that flexible systems are more resilient in the long term since they have shorter recovery times and larger expected NPV than standard designs.

Conclusions
The paper explores alternative ways to examine the infrastructure management problem. It presents a conceptual framework, which materializes on a mathematical model that enhances traditional life cycle cost evaluations. The conceptual framework uses systems thinking and flexibility as the two main pillars for understanding how systems change over time. The life cycle analysis is used to find the best management policy based on a cost-benefit analysis of multiple policy options. To illustrate the proposed approach, three infrastructure design and management strategies are compared. These are a traditional approach in which the system is designed and built to withstand an expected demand at time t m , and two strategies with flexibility as the key component. These two strategies are called fully flexible and adaptation pathways. The benefits and limitations of   these three cases are compared in two illustrative examples that consider the system's performance with and without failures due to extreme events.
Infrastructure management, including resilience, should be addressed by recognizing the dynamic nature of systems and modeling their evolution over time. This means that understanding infrastructure requires thinking in terms of processes, which in turn means approaching the problem from a systems thinking perspective. Systems thinking is a framework that contributes to looking at the problem as interacting processes organized hierarchically. Flexibility facilitates changes in the route of a process or its final objective. Flexibility is key to managing uncertainty, facilitates the unavoidable need to deal with unexpected circumstances, and favors financial responsibility and sustainability.
The results of the model have shown that incorporating flexibility in the management of systems has a significant impact on long-term performance. As more flexibility is added to the system, there is a higher expected net present value with less variability. The differences in the expected NPV may be in the order of three or four times in comparison with the standard design. In addition, as systems become more flexible, the scatter of the NPV decreases because the possibility to change enhances the capacity to manage uncertainty, thus increasing the expected rewards (NPV). Incorporating flexibility into systems that might fail has a similar effect but have a smaller expected NPV and variability. Although the number of failures increases as the system is more flexible, the average recovery times are smaller. This is caused mainly because extremely robust systems lead to more expensive failures and significant recovery times. Adaptation pathways are an important strategy because they focus on feasible, relevant solutions. However, they do not recognize the importance of making minor adjustments to the system, the possibility that the selected pathways change over time, or the likelihood of new alternatives in the future. The results show that being fully flexible is an upper limit condition for other flexible strategies, particularly for adaptation pathways. Finally, it is important to stress that incorporating flexibility means defining a new system management strategy. Then, it requires a commitment on behalf of stakeholders and those responsible for the operation. Flexibility cannot be added to all systems in the same way, nor are all options always available.
The conceptual aspects of this paper and the results show that the proposed strategy is a step forward in the direction of creating a new understanding of infrastructure design and management. It facilitates to move toward a more efficient, resilient, and sustainable built environment. The assumptions presented in this paper should be tailored for each infrastructure type, but the concepts and the primary strategy are well grounded conceptually. Future work will move toward implementing this strategy in practical applications.