International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Incorporating network considerations into pavement management systems Medury, Aditya; Madanat, Samer Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Incorporating Network Considerations into Pavement ManagementSystemsAditya MeduryPostdoctoral Scholar, Safe Transportation Research and Education Center, University ofCalifornia at Berkeley, USASamer MadanatProfessor, Department of Civil and Environmental Engineering, University of Californiaat Berkeley, USAABSTRACT: The objective of infrastructure management is to provide optimal maintenance, rehabilita-tion and replacement (MR&R) policies for a system of facilities over a planning horizon. While mostapproaches in the literature have studied the decision-making process as a finite resource allocation prob-lem, the impact of construction activities on the road network is often not accounted for. The state-of-the-art Markov decision process (MDP)-based optimization approaches in infrastructure management, whileoptimal for solving budget allocation problems, become internally inconsistent upon introducing networkconstraints. In comparison, approximate dynamic programming (ADP) enables solving complex problemformulations by using simulation techniques and lower dimension value function approximations. In thispaper, an ADP framework is proposed which provides better results than randomized policy frameworksin the presence of network constraints.1. INTRODUCTIONThe objective of transportation infrastructure man-agement is to provide optimal maintenance, reha-bilitation and replacement (MR&R) policies for asystem of facilities (roads, bridges, tunnels, etc.)over a planning horizon. While most approachesin the literature have studied it as a problem of opti-mal allocation of limited financial resources, the in-terdependence between facilities, as introduced bya unifying network configuration, is often not ac-counted for.The recognition of an over-arching network con-figuration introduces several challenges, as well asopportunities, for system-level MR&R decision-making. Existing system-level MR&R decision-making paradigms do not adequately account forthe impacts of construction activities on the roadtraffic. Consequently, the measures taken by agen-cies to mitigate the resulting travel time increasesoccur at the project-level, i.e., once the MR&R ac-tivities have already been determined. However,in order to effectively address these user concernsat the system-level, the network configuration canprovide insights into determining how each facilityaffects the system performance (such as the capac-ity or connectivity of the network). In this paper,network-induced inter-facility interactions are ex-amined in the context of a road network, whereinMR&R activities on individual road segments canlead to a cumulative effect on the capacity of thenetwork.2. LITERATURE REVIEWDiscrete-state, discrete-time Markov decision pro-cess (MDP)-based frameworks have been widelyused in infrastructure management, especially inthe context of uncertainty in the underlying facilityperformance models. One of the first instances ofusing MDP frameworks for infrastructure manage-ment was the development of the Arizona pavementmanagement system (Golabi et al., 1982). The LP-based approach utilized randomized policies to ef-fectively accommodate budget constraints within112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015the MR&R decision-making problem. Random-ized policies are probabilistic in nature, wherein theoptimal policy for a facility in a given conditionstate is defined as a probability distribution func-tion across two or more actions. The interpreta-tion of randomized policies in a network setting re-lies on the assumption that all the facilities in thesystem deteriorate homogeneously. Consequently,such an approach can also be referred to as a single-dimensional MDP problem.Randomized policies are shown to be optimalin a constrained MDP setting (Kallenberg, 1994).However, one of the limitations of the approach isthat due to the probabilistic nature of the solution,randomized policies do not directly translate intofacility-specific recommendations. In order to ad-dress this issue, Medury and Madanat (2014) ex-tended the LP-based approach to provide facility-specific policies, while retaining the optimality ofthe original problem formulation.Other MDP-based optimization frameworks havefocused on obtaining non-probabilistic, facility-specific policies. Such approaches rely on de-composing the system-level MDP problem intotwo-stage (facility-level and system-level) prob-lems (Yeo et al., 2013; Ohlmann and Bean, 2009).Since these techniques do not assume the facilitiesto be homogeneous, they can be classified as multi-dimensional MDP problems. However, since theseapproaches specifically cater to resource constraintsand employ heuristics to simplify the system-levelMDP problem, efficient solution procedures cannotalways be obtained for a general set of constraints.In recent times, reinforcement learn-ing/approximate dynamic programming (ADP)algorithms have also been applied to infrastructuremanagement problems. Durango-Cohen (2004)applied some learning techniques to facility-levelproblems under imperfect deterioration infor-mation scenarios. Gao and Zhang (2009) andKuhn (2010) utilized ADP frameworks to providesystem-level MR&R policies in the context ofbudget allocation problems. Through the use ofsimulation techniques and value function approx-imations, ADP seeks to solve multidimensionalMDP problems involving complex constraints.However, one of the limitations of the approachis that theoretical optimality guarantees cannotbe provided on the solutions thus obtained. Inaddition, while Gao and Zhang (2009) and Kuhn(2010) demonstrate the applicability of ADP-basedapproaches to infrastructure management, its per-formance vis-a-vis other MDP-based approacheswas not evaluated in these papers.Based on the overview of existing MDP-basedframeworks, it is observed that most system-levelMR&R decision-making methodologies considerthe resource allocation problem without recogniz-ing the presence of an underlying network config-uration. Dekker et al. (1997) suggests that interac-tions between individual components of an infras-tructure system can be classified into three differ-ent types: economic dependence (benefits/costs as-sociated with joint maintenance), structural depen-dence (set of facilities collectively determining sys-tem performance such as connectivity or capacity)and stochastic dependence (presence of correlateddeterioration factors like environment, loading).The work presented in this paper focuses on in-corporating the structural interdependence amongfacilities into the MR&R decision-making process.In particular, the use of ADP to model complexproblems in infrastructure management is moti-vated, since randomized policies, while optimal forbudget allocation problems, are restrictive in theirmodeling capabilities. In the following sections,the performances of ADP and a state-of-the-art ran-domized policy framework are presented with thehelp of a parametric study which utilizes a stylizednetwork to infer the impact of network-based con-straints on the MR&R decision-making.3. PARAMETRIC CASE STUDYIn this section, the network considerations are for-mally introduced into the MR&R decision-makingframework as mathematical constraints. Herein, thestructural interdependence between the individualfacilities is incorporated by imposing a network ca-pacity constraint on the MR&R activity selection.The use of capacity as a performance measure canbe interpreted as a supply-based criterion, whereinthe agency seeks to provide enough capacity duringMR&R activity implementation so that the associ-212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ated origin-destination demand can be met. Whileit is recognized that a capacity-based approach doesnot take into account the demand on individuallinks, it can be argued that the traffic can be re-routed to maximize capacity utilization, using ade-quate signage and real-time information dissemina-tion systems. The framework also allows for differ-ent construction work zone options to be consideredwhereby partial or complete closures can be imple-mented. This results in a trade-off between the lossin link capacity and the cost of an MR&R activity.3.1. Network Representation(a) network representation(b) individual link capacities; Cmax = 20 unitsFigure 1: Stylized 11-link pavement network for incor-porating structural interdependence (individual linkcapacities expressed in bracketsAs shown in figures 1(a) and 1(b), let us considera stylized network configuration consisting of 11road segments and 10 nodes connecting an originand a destination. Figure 1(b) represents the in-dividual link capacities, using which the resultingnetwork capacity, Cmax, can be obtained as 20 units.It is to be noted that Cmax represents the maximumcapacity available to the users in the absence ofMR&R activities.3.2. Approximate Dynamic ProgrammingApproximate dynamic programming is an MDP-based modeling framework which seeks to over-come the dimensionality issues associated with tra-ditional dynamic programming methods. It em-ploys an algorithmic strategy of stepping forwardthrough time, which obviates the need to loopthrough the entire state space in future time periods(Powell, 2007).In order to implement an ADP methodology, theforemost step is to model the value function ap-proximation for the network under investigation.Herein, the objective is to expoit the properties ofthe problem such that the learning process is re-duced to estimating a few key parameters of inter-est, which are also referred to as basis functions. Inthis paper, a set of linear, separable basis functionsare chosen, which can be summed up to provide anestimate of the future cost-to-go for a given state-action pair, (st ,at), in year t.In terms of notation, the pavement network is de-fined as a graph, G = (V,E), wherein the edges, E,represent the road segments, and the nodes, V , rep-resenting the points of intersection between any tworoad segments. An individual road segment is de-fined as a link, (q,r).For the network configuration of interest in Fig-ures 1(a) and 1(b), it is possible to identify groupsof road segments which respond similarly to thenetwork capacity constraints. For instance, it canbe argued that the individual road segments withinthe groups defined as, {1,2,3}, {4}, {5,6,7,8} and{9,10,11} have a near-identical policy response to-wards the network-based constraints. In the caseof links, 1-3 and 9-11, since all the road segmentsare in series, a MR&R activity/construction typechosen for any segment within the group leads tothe same loss in capacity of the network. For seg-ments 5-8, the symmetry in the network configura-tion is also taken into consideration while aggregat-ing the corresponding state-action space. The roadsegment, {4}, is considered as a singleton for thepurpose of approximating the value function.Finally, given the Markovian evolution of thesystem, it can also be assumed that for any givenyear, t, the optimal policy for a facility, (q,r) isonly a function of its current state, and the action-construction type pair under consideration. Conse-quently, the total number of basis functions chosenfor the value function approximation, |B|, can becalculated as follows:|B|= T |GN ||S||A||L|, (1)where,312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015T : length of the planning horizon,N: the total number of road segments,|GN |: number of groups having a homoge-neous response to the capacity con-straints,|S|: number of elements in the state space ofa single road segment,|A|: number of possible MR&R actionsavailable for a single road segment,|L|: number of construction options avail-able (partial and complete closures).The savings from using a value function approx-imation, in terms of memory requirement, can begauged from the fact that for a look-up table rep-resentation, the parameter space would comprise ofT |S|N |A|N |L|N combinations.With the help of these basis functions, a Q-function, which represents an estimate of the futurecost-to-go associated with implementing the action-construction type combinations, (at , lt), when thenetwork is in state, st , can be defined as follows:Q˜t(st ,at , lt) = ∑(q,r)∈Eθt,GN(q,r),st(q,r),at(q,r)lt(q,r),(2)where, GN(q,r) refers to the group number associ-ated with facilitiy, (q,r).It is important to acknowledge here that approx-imating the value function is not a precise science.For instance, a large state space adversely affectsthe convergence of ADP since it requires more pa-rameters to be learnt. In addition, there is alsoa greater need for exploration to adequately coverthe parameter space. Conversely, having too fewparameters may lead to a poor approximation ofthe future cost-to-go, leading to inefficient solution.The approach involves having some a-priori ex-pectations about the decision-making process, fol-lowed by modeling those beliefs using the approxi-mation function, and evaluating its performance us-ing simulations.Once the basis functions are defined, a standardADP approach can be broadly summarized as fol-lows:At first, the Q-functions are assigned an initialestimate of the future cost-to-go. On the basisof the current knowledge of the Q-functions, ac-tions which minimize the expected cost-to-go areselected. The optimal actions are then used to sim-ulate the future state of system, and the process isrepeated till the end of the planning horizon. Oncethe end of the planning horizon is reached, the cost-to-go predicted by the current estimate of the valuefunctions are compared with the actual costs in-curred through the simulation. Finally, the valuefunctions of the states visited in the simulation areupdated on the basis of the difference between therealized and predicted costs.Finally, in order to update the basis functions, atemporal difference (TD(λ )) learning algorithm isemployed. TD(λ ) learning is a two-stage procedurewhich relies on a forward pass to generate a samplepath, and a backward pass to update the parametersof the Q-function. For a detailed description aboutthe algorithm, please refer to Powell (2007).3.2.1. Problem FormulationThe objective of the MR&R optimization is to pro-vide optimal policies for each facility, (q,r), inthe network, while satisfying the budget restrictionsand the network capacity constraints:minx,x˜,z˜,y∑(q,r)∈E∑a∈A∑l∈L(c(st(q,r),a, l)+u(st(q,r))+α θt,GN(q,r),st(q,r),a,l)x(q,r),a,l,t , (3)subject to∑(q,r)∈E∑a∈A∑l∈Lc(st(q,r),a, l)x(q,r),a,l,t ≤ B (4)∑a∈A∑l∈Lx(q,r),a,l,t = 1∀(q,r) ∈ E, (5)52∑∆=1x˜(q,r),a,l,∆ ≥ x(q,r),a,l,t ,∀a ∈ A,∀l ∈ L,∀(q,r) ∈ E, (6)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015∑∆−da,l<∆′≤∆x˜(q,r),a,l,∆′ ≤ z˜(q,r),a,l,∆,∀a ∈ A,∀l ∈ L,∀∆ = 1, . . . ,52,∀(q,r) ∈ E, (7)y(D,O),∆ ≥ hCmax, ∀∆ = 1, . . . ,52, (8)C(q,r)[1−(∑a∈A∑l∈Lκ(q,r),a,l z˜(q,r),a,l,∆)]≥ y(q,r),∆∀(q,r) ∈ E,∀∆ = 1, . . . ,52, (9)∑r:(r,q)∈Ey(r,q),∆− ∑r:(q,r)∈Ey(q,r),∆ = 0∀(q,r) ∈ E,∀∆ = 1, . . . ,52, , (10)x(q,r),a,l,t , x˜(q,r),a,l,∆, z˜(q,r),a,l,∆ ∈ {0,1},y(q,r),∆ ∈ R+,∀a ∈ A,∀l ∈ L,∀∆ = 1, . . . ,52,∀(q,r) ∈ E,where,x(q,r),a,l,t : 1 if the MR&R action-construction type pair (a, l),is selected for facility (q,r) inyear t; 0 otherwise,x˜(q,r),a,l,∆: 1 if the action-construction typepair, (a, l), is started in week ∆for facility (q,r); 0 otherwise,z˜(q,r),a,l,∆: 1 if the action-construction typepair, (a, l), is under implementa-tion in week ∆ for facility(q,r); 0otherwise,y(D,O),∆: available network capacity in thepresence of MR&R activities inweek ∆, represented as a vir-tual edge connecting destinationD and origin O,y(q,r),∆: available link capacity for facility(q,r) in week ∆,u(st(q,r)): user costs linked with increasein vehicle wear-and-tear, fuel us-age, etc.,B: annual budget,d(q,r),a,l: duration of implementing theaction-construction type pair,(a, l),h: fraction of the maximum capac-ity of the network, Cmax, repre-senting a minimum network ca-pacity threshold,C(q,r): maximum link capacity associ-ated with facility (q,r), as definedwhen no MR&R activity is sched-uled,κ(q,r),a,l: loss in capacity associatedwith implementing the action-construction type pair, (a, l), forfacility (q,r),c(st(q,r),a, l): agency cost of undertaking theaction-construction type pair,(a, l).Herein, equation 3 represents the objective func-tion, defined as minimizing the expected system-level cost-to-go, as per the estimates of the Q-function; equation 4 indicates that the total amountspent on MR&R activities should be within the an-nual budget; equation 5 assigns exactly one action-construction type pair (including do-nothing) toeach facility. The scheduling constraints, as rep-resented by equations 6 - 7, are modeled as anon-preemptive scheduling problem (Sousa andWolsey, 1992), wherein the assigned activity iscompleted in one sequence; equation 6 ensuresthat only the chosen action-construction type pairis considered for assessing the feasibility of theMR&R scheduling, and equation 7 ensures that thechosen action-construction type pair undergoes acontinuous construction period of d(q,r),a,l weeks.Equation 8 guarantees that the network capacity inthe presence of MR&R activities does not violatethe minimum network capacity threshold; equation9 represents the loss in link capacity associated withimplementing an action-construction type pair onfacility(q,r); and equation 10 represents the flowconservation equation for every node in the net-work.It is important to note that the optimization rou-tine only imposes the constraints on the policies as-sociated with the current year of decision-making.However, since the sample path is generated bysolving the optimization problem in each time pe-riod, the underlying budget and network consider-ations are always satisfied along the sample path.Consequently, the resulting Q-function updates,which are also based on the sample path traversed,are consistent with the constraints of the problem.512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153.3. Problem Formulation: SNOMedury and Madanat (2014) proposed a si-multaneous network optimization (SNO) approach,which extends the linear programming-based ran-domized policy framework of Golabi et al. (1982)to allow for facility-specific policies for the firstyear of a multi-year MR&R decision-making prob-lem. The underlying problem formulation utilizesthe randomized policies for estimating the futurecosts, while associating facility-specific binary in-teger variables with each MR&R action for the cur-rent year of decision-making. The resulting op-timization formulation still remains optimal for abudget allocation problem. In addition, the pres-ence binary integer variables for the first year ofdecision-making allows the user to impose addi-tional constraints on the MR&R decisions, such asnetwork constraints.In order to add the network-based constraintsinto a problem formulation using SNO, the budgetallocation problem can be reformulated similar tothe ADP formulation using a facility-specific de-cision variable identical to the ADP formulation,x(q,r),a,l,t . Hence, the network constraints (equa-tions 5 - 10) can be modeled in an identical fashion.For more details on the SNO framework, please re-fer to Medury and Madanat (2014).The policies obtained using SNO also satisfythe network constraints for the current year of thedecision-making, t. However, the disadvantage ofSNO is that its future cost estimation relies onthe use of randomized policies which cannot ac-commodate the network constraints. Consequently,while SNO still provides feasible policies, the opti-mality of the solutions is no longer guaranteed.3.4. Scenario Generation: Supplementary Infor-mationThe condition state of the facilities is evaluated us-ing an eight point ordinal index, where 1 is the beststate and 8 is deemed to be an unacceptable stateby the agency. For the purpose of illustration, fourtypes of activities are considered: do-nothing, rou-tine maintenance, rehabilitation and reconstruction.It is assumed that maintenance activities can be im-plemented overnight and hence lead to no loss incapacity of a road segment. There exist two re-construction options: a partial road closure whichrequires 10 weeks of construction time and causes30% loss in capacity, while a complete road clo-sure can be completed in 2 weeks, but leads to a100% loss in capacity, i.e., the link is rendered in-accessible. These values are not based on empiricaldata, but are representative of the kind of trade-offswhich can be expected in real-life scenarios.The transition probability matrices, agency anduser cost structure is taken from Madanat (1993).Herein, maintenance and rehabilitation activitiesare prohibitively more expensive as the state wors-ens, whereas reconstruction incurs a fixed cost. Theuser cost also increases as the facility deteriorates,and a high penalty cost is imposed when the facil-ity is in the non-permissible condition state (s = 8).The planning horizon consists of 15 years and thediscount rate is chosen to be 5%. The salvage valueat the end of the planning horizon is set equal to theuser costs, wherein the user costs can be interpretedas a proxy for the quality of the terminating state ofthe facility.In order to compare ADP and SNO, three budgetlevels are considered, B = 50;100;150 units. Thenetwork capacity threshold, h, is chosen to be 0.75,such that the network constraints are active for thepurpose of the investigation. The initial conditionof the facilities is uniformly distributed between thestates, 1(good), 4(moderate), and 7(poor), withineach group, so as to capture a range of deteriora-tion levels.For each scenario, in order to learn the parame-ters of the Q-function, the ADP framework is firstimplemented for 1500 iterations. Thereafter, the Q-function is kept constant, and the performance ofthe two methodologies is compared on the basis ofsimulating the planning horizon 1000 times usingMonte Carlo simulations. The algorithms are pro-grammed in C++, and the optimization problemsare solved using CPLEX R© on a Windows-based OSwith a 3.10 GHz processor and 4GB RAM.3.5. ResultsFigure 2 represents the simulation-average of thesystem-level user-plus-agency costs incurred fromusing SNO and ADP. The results indicate that inthe presence of network constraints, ADP performs612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 2: Total costs comparison between SNO andADP for different budget levels (h=0.75)better than SNO (on an average) across all budgetlevels.Figure 3 shows a box plot representation that pro-vides information about the different quartiles ofthe realized cost distribution. The horizontal barsin each box plot represent the sample minimum,lower quartile, median, upper quartile and samplemaximum. In addition, a dot is marked on the plotto indicate the expected cost predicted by the op-timization approach at t=1. Using this representa-tion, figures 3(a) to 3(c) show that ADP’s predictedcosts align well with the median of the costs real-ized by the simulation. On the other hand, SNOprovides an inaccurate prediction of future costs asthe budget increases.The primary reason for the disparity in SNO’ssimulated and predicted costs is that the random-ized policies corresponding to the future years donot account for the network constraints. In the con-text of the parametric case study, reconstruction ac-tivities are required to be excluded for certain roadsegments due to capacity considerations. However,since the network constraints can only be imposedon the binary integer variables, SNO defers the ac-tivities to the future years, in the form of random-ized policies. As the sample path is simulated, thepolicies predicted for the future years are not re-alized, and the gap between the predicted and therealized costs widens. With an increase in the avail-able budget, a greater share of the randomized poli-cies are allocated for reconstruction activities, thus(a) B=50; h=0.75(b) B=100; h=0.75(c) B=150; h=0.75Figure 3: Box plots comparing ADP and SNO for vary-ing budget levels and h = 0.75leading to more inaccuracies in the estimation ofthe future costs.Figure 4 shows the convergence of the Q-function for B = 100 scenario. As can be observed,the realized and the predicted costs converge totheir final values after 500-600 iterations.4. CONCLUSIONS AND FUTURE WORKIn this paper, the state-of-the-art MDP-based ap-proaches in infrastructure management were ex-tended to incorporate network-based considerations712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 4: Convergence of the value function approxi-mation for B = 100 scenariointo MR&R decision-making. The structural inter-dependence problem was motivated to account forthe adverse impact of construction activities on theroad network. The results indicate that randomizedpolicies, while ideal for modeling budget allocationproblems, do not adequately capture these network-based constraints. In comparison, ADP is able toprovide better results by learning the value functionparameters using simulation techniques.In the context of infrastructure management, theuse of approximate dynamic programming is a re-cent phenomenon. While previous research estab-lished that MR&R decision-making problems canbe modeled using ADP for budget allocation prob-lems, it is shown here that ADP is beneficial formodeling problems with complex inter-facility dy-namics. The future work includes addressing someof the limitations associated with a capacity-basedframework, such as incorporating delay time met-rics, and scaling up the network size so as to betterassess the computational performance of ADP.5. REFERENCESDekker, R., Wildeman, R. E., and van derDuyn Schouten, F. A. (1997). “A review ofmulti-component maintenance models with eco-nomic dependence.” Mathematical Methods ofOperations Research, 45(3), 411–435.Durango-Cohen, P. L. (2004). “Maintenance and repairdecision making for infrastructure facilities withouta deterioration model.” Journal of Infrastructure Sys-tems, 10(1), 1–8.Gao, L. and Zhang, Z. (2009). “Approximate dy-namic programming approach to network-level bud-get planning and allocation for pavement infrastruc-ture.” Transportation Research Board 88th AnnualMeeting, Number 09-2344.Golabi, K., Kulkarni, R. B., and Way, G. B. (1982). “Astatewide pavement management system.” Interfaces,12(6), 5–21.Kallenberg, L. C. M. (1994). “Survey of linear program-ming for standard and nonstandard Markovian controlproblems. Part I: Theory.” Mathematical Methods ofOperations Research, 40(1), 1–42.Kuhn, K. D. (2010). “Network-level infrastructure man-agement using approximate dynamic programming.”Journal of Infrastructure Systems, 16(2), 103–111.Madanat, S. (1993). “Optimal infrastructure manage-ment decisions under uncertainty.” TransportationResearch Part C: Emerging Technologies, 1(1), 77–88.Medury, A. and Madanat, S. (2014). “A simultaneousnetwork optimization approach for pavement man-agement systems.” Journal of Infrastructure Systems,20(3) DOI:10.1061/(ASCE)IS.1943-555X.0000149.Ohlmann, J. W. and Bean, J. C. (2009). “Resource-constrained management of heterogeneous assetswith stochastic deterioration.” European Journal ofOperational Research, 199(1), 198–208.Powell, W. B. (2007). Approximate Dynamic Program-ming: Solving the curses of dimensionality. John Wi-ley & Sons, Inc.Sousa, J. and Wolsey, L. (1992). “A time indexed formu-lation of non-preemptive single machine schedulingproblems.” Mathematical Programming, 54(1), 353–367.Yeo, H., Yoon, Y., and Madanat, S. (2013). “Algorithmsfor bottom-up maintenance optimisation for heteroge-neous infrastructure systems.” Structure and Infras-tructure Engineering, 9(4), 317–328.8

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