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

Resource allocation and uncertainty when modeling infrastructure networks as socio-technical systems Gomez, Camilo; Sánchez-Silva, Mauricio; Dueñas-Osorio, Leonardo Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Resource Allocation and Uncertainty when Modeling InfrastructureNetworks as Socio-Technical SystemsCamilo GomezPostdoctoral Researcher, Dept. of Civil & Environmental Engineering, StanfordUniversity, Stanford, USAMauricio Sánchez-SilvaAssociate Professor, Dept. of Civil & Environmental Engineering, Universidad de losAndes, Bogotá, ColombiaLeonardo Dueñas-OsorioAssociate Professor, Dept. of Civil & Environmental Engineering, Rice University,Houston, USAABSTRACT: This paper is in the context of risk-informed decision support and modeling infrastructurenetworks as socio-technical systems. The previously introduced Complex Distributed Agent NetworkFramework (CoDAN) framework is reviewed and several sources of uncertainty identified, along withpossible paths to overcome it in future research. One key problem is addressed, regarding the selectionof a decision horizon when engaging in resource allocation for disaster preparedness and its impact onthe ratio of installation costs and displacement costs. Three optimization formulations are presented andcompared for the resource allocation problem, considering several decision horizons. Results prove thecomputational efficiency of the proposed formulations and the effect of different decision horizons, whichlead to diverse allocation schemes.1. INTRODUCTIONInfrastructure networks are complex socio-technical systems, whose overall performance isthe result of the interaction of natural, physical,engineered, and social systems. Local actions,such as deterioration and maintenance, escalate toglobal behavior in non-intuitive ways due to thenonlinear relationships among the many compo-nents that comprise infrastructure networks. Such acomplex setting poses challenges for infrastructureengineers in the sense that system performanceresponds not only to physical processes (e.g.,deterioration) but also to organizational processes(e.g., human errors, risk criteria), along with theuncertainties associated to both.The authors have previously developed a de-cision support tool referred to as the ComplexDistributed Agent Network (CoDAN) framework(Gomez et al. (2014)), which enables the analysisof infrastructure networks across different levels,namely: the physical processes occurring at thecomponent level, the dynamics of connectivity andflow at the network level, the variability in deci-sions made by agents at the socio-economic level,and the interactions across these levels. CoDAN re-lies on network decomposition to identify relevantsub-systems which may be associated to decentral-ized decision problems and resources. For each ofthose sub-systems, performance assessment and de-cision supports techniques are applied to emulateinfrastructure operation.Several sources of uncertainty appear whenengaging in such wide-scope analysis and ac-counting for the inherent variability of socialprocesses (i.e., partial or asymmetric informa-tion, processing capability, errors), as well astraditional modeling of uncertainty in struc-tural engineering (Baker and Cornell (2008);112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Kiureghian and Ditlevsen (2009)). This paperhighlights major sources of uncertainty that chal-lenge the modeling of infrastructure networks associo-technical systems by reviewing key modulesof the CoDAN framework. Then, one specificproblem is addressed regarding resource allocationfor disaster preparedness and the effect of the ratioof the resource installation costs (i.e., the cost ofestablishing headquarters for infrastructure repaircrews) and displacement costs (i.e., the cost forcrews to actually move and repair distant nodesand/or links of an infrastructure network). Suchratio can be affected by the choice of a decisionhorizon in the sense that it could imply consideringmore displacements in the long run, or by asubjective valuation of the importance of effectivedisplacement (e.g., if quick displacements have animpact on prompt restoration and resilience).An overview of the CoDAN framework is pro-vided in Section 2, followed by a discussion on ma-jor sources of uncertainty in Section 3. Section 4describes the application of the CoDAN frameworkto the Chilean power network after the 2010 earth-quake and introduces three optimization programsfor the problem of resource allocation for disasterresponse. Section 5 provides an analysis of the im-pact of the choice of a decision horizon on the opti-mal scheme for resource allocation. Section 6 pro-vides conclusions and ideas for future work.2. THE CODAN FRAMEWORKThe CoDAN framework (Gomez et al. (2014)) is arisk-informed decision support tool for infrastruc-ture networks operation. CoDAN treats infrastruc-ture systems as the result of interacting physical,natural, and social systems, recognizing that:• decentralized actions and decisions (hence, theterm distributed)• are performed by humans in a social context(hence, the term agent)• to affect a set of interconnected components(hence, the term network)• whose behavior combines in a non-linear way(hence, the term complex).The latter implies that different decision prob-lems occur at different system description, or res-olution, levels, and that certain decentralizationschemes respond to each specific system and prob-lem. CoDAN is comprised of the three followingphases:• Phase I (“Knowing the system”) explores net-work properties (e.g., relative importance ofnetwork components) and embraces a human-machine interactive process to define relevantsub-systems in the context of a decision prob-lem, supported on supervised and unsuper-vised clustering algorithms;• Phase II (“Who’s in charge”) develops re-source allocation models in terms of sub-systems, signaling which of those are appro-priate for autonomous use of resources in thecontext of problems such as maintenance ordisaster relief throughout the network;• Phase III (“Behavior under control”) definesagents associated to key decision problems ofautonomous sub-systems, and provides themwith the capabilities to control variables of in-terest (e.g., connectivity, flow reliability) andpursue a target system state. Phase III isfurther divided into the modules of model-ing (graph theoretical models and deteriora-tion), assessment (performance and reliabilityevaluation), and intervention (decision supportfor maintenance actions) of infrastructure net-works.The integration of CoDAN phases enables agent-based modeling of infrastructure operation to sup-port decisions and policy-making, where agentsrepresent decision-makers who act in a decentral-ized yet coordinated way to keep up with aging,hazards, and increasing demand through local, cost-effective control processes. The overall behavior ofinfrastructure networks is, thus, the result of physi-cal processes, natural events, and interrelated deci-sions from multiple parties.A key underlying question in the CoDAN frame-work is how to define the sub-systems that are tobe modeled as agents. Consider the problem of al-locating resources for disaster preparedness (e.g.,repair crews to restore critical components in theaftermath of major disruptions). Repair crews areto be assigned in a way that minimizes installationcosts (i.e., physical resources and personnel) and212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015displacement costs (i.e., how long it takes for crewsto reach critical components to be restored).The relationship between the first and second setsof costs is not straightforward. Even though actualdisplacement costs may be small compared to in-stalling facilities, two considerations are notewor-thy: (a) rather than being a one-time cost, manydisplacements may occur for both preventive andcorrective maintenance throughout a system’s life-cycle, making displacement cost sensible to thechosen decision horizon; and (b) long displace-ments affect not only the actual transportation costbut also the time in which critical components canbe restored, thus, becoming an important factor forprompt recovery and resilience.3. UNCERTAINTY ACROSS CODANEach of the phases and modules in CoDAN are as-sociated to sources of uncertainty that pose chal-lenges for the practical use of CoDAN for realisticapplications. The major sources of uncertainty arelisted below.• Uncertainty on the network model itself, re-garding whether the “known model” matchesthe physical network and its attributes. Thiscan be addressed by taking advantage of avail-able GIS systems and information from en-gines such as Google Earth/Maps/Street View,as well as field work to enhance current mod-els.• Uncertainty on the network decompositionprocess, given the wide variety of availablealgorithms, the non-deterministic nature ofmost of them, and the selected parameters.The latter generates issues of reproducibil-ity and lack of confidence on the quality ofthe partitioning. Clustering validation metrics(Dunn (1973); Davies and Bouldin (1979)) areparamount to improve and standardize decom-position to some extent, as well as human-machine interaction to guarantee that detectedpatterns make sense in the decision-makingcontext.• Uncertainty on the attributes of nodes andlinks. Key features such as travel time, dis-tance, flow capacity, and reliability at the com-ponent level are valuable inputs for risk analy-sis, which are often only available through thesystem operator, and even then, are subjectedto variations due to deterioration, shocks,and/or congestion. Current implementationsof CoDAN rely on discrete event simulationand might benefit from a more comprehensiveprobabilistic risk assessment process involvingactual fragility curves and occurrence rates forspecific hazards.A second set of uncertainties also become rele-vant when embracing a holistic, socio-technical ap-proach appear at the agent level when accountingfor psychological and socio-economic aspects thataffect decision-makers, such as risk aversion, bi-ases, etc.:• Individual and organizational preferences mayinfluence decisions and are modeled in Co-DAN as weights for different variables withinthe optimization programs. Further improve-ments can be gained from multi-objective op-timization paradigms dealing explicitly withbalancing preferences.• Existing regulation and/or technical standardsdetermine target levels of safety and operationof specific physical sub-systems; such regula-tions may be affected by political postures andlobbying. These features are not straightfor-ward to incorporate into engineering models;within CoDAN, however, they could be intro-duced in the form of limiting constraints foragents’ optimization problems.• Demands, costs and capacities of infrastruc-ture related goods/services fluctuate accordingto socio-economic patterns, as well as the costsand actual effect of maintenance actions per-formed by operators. Stochastic optimizationSlyke and Wets (1969) is a natural response tosuch a challenge and is part of ongoing re-search regarding pre- and post-disaster deci-sion making.• The selected discount rate, valuation of promptrecovery and decision horizon may signifi-cantly affect the outcome of a decision anal-ysis, as will be discussed in this paper for thecase of installation and displacement costs forresource allocation in the context of disaster312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015response.Non-technical uncertainties are the most chal-lenging to quantify and incorporate into the modelsince they exceed the scope of hard sciences. How-ever, these uncertainties have significant effectson the decisions that drive physical infrastruc-ture performance, highlighting the need for cross-disciplinary research efforts.4. THE CHILEAN POWER NETWORK UN-DER THE 2010 EARTHQUAKEThe Chilean case allows to model the role of dis-tributed decision-making units observed in prac-tice from the perspective of the CoDAN frameworksince emergencies enforce decentralization due tolimitations in communication and coordination forprompt restoration of basic services, as highlightedin official reports (International-Energy-Agency(2012); of Energy (2012)). This section studiesthe Chilean electricity supply sub-network knownas Sistema Interconectado Central (SIC). The SICnetwork was chosen because it serves most of thepopulated areas and is located in the area that waslargely affected by the 2010 earthquake, includingthe region of Maule and important cities such asConcepcion, Valparaiso, and Santiago.4.1. Phase I: Knowing the systemA model of the SIC network was constructed basedon information available in government documents(of Energy (2012); International-Energy-Agency(2012)) and Google Maps. Then, topological in-dices are used to identify important elements thatmay provide insights about where clusters con-centrate. Topological importance is evaluated us-ing the degree, betweenness, and Page-Rank met-rics (Brandes and Erlebach (2005)), although moresophisticated metrics have recently become avail-able which may enhance the capabilities to detectrelevant nodes, such as the NWRank (Wang et al.(2014)) that ranks nodes and links simultaneously.Clustering algorithms and validation metrics ofclustering quality are then applied to perform thehierarchical network decomposition. The obtainedpotential centroids are fed into the supervised Ker-nel k-means algorithm (Dhillon et al. (2005)) usingthe distance matrix as the input kernel (i.e., equiv-Figure 1: Clustering of the Chilean network usingtopology-informed centroids based on betweenness(upper part) and Page Rank (lower part).alent to traditional k-means). The obtained clus-ters are shown in Figure 1 for the cases in whichk-means was informed by betweenness and PageRank.The Markov Clustering Algorithm (MCL)(van Dongen (2000)) was also applied to theChilean network varying input parameters throughtrial-and-error (human-machine interaction to ob-tain different numbers and sizes of clusters, lead-ing to only four distinguishable partitionings, twoof which are shown in Figure 1. These partition-ings along with those resulting from the topology-informed k-means are fed as initial solutions to aheuristic optimization algorithm (Particle SwarmClustering, or PSC) described in Gómez (2014),which seeks to maximize the Dunn index of cluster-ing quality (Dunn (1973)) based on an initial par-titioning as those provided by the topological in-dices.In this case, while the MCL-based initial so-lutions favor partitionings with big, well-definedclusters, the k-means-based initial solutions favorpartitionings with smaller clusters and more irreg-ular boundaries. Since the initializations obtainedduring the pre-processing stage do not show ev-idence of other relevant hierarchical levels (i.e.,there are no partitionings of higher or lower reso-lution than these), the hierarchical decompositioncan be concluded after running the PSC for the twomentioned levels. The hierarchical decompositionis, thus, comprised of four levels: at the higher412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015   100   25    39    36   5    10    8    7    16    8    8    14    14(a)(b) (c)Figure 2: Hierarchical decomposition of the ChileanSIC network (a); partitioning at hierarchical level 2,with 3 large clusters (b); partitioning at hierarchicallevel 3, with 10 small clusters (c).level, the whole 100-node network as a single clus-ter; next, a partitioning 3 large clusters; next, a par-titioning with 10 small clusters. Figure 2 showsthe hierarchical decomposition obtained throughthe combined action of topological indices, super-vised and unsupervised clustering, optimization ofthe clustering quality, and human-machine interac-tion, producing meaningful clusters (e.g., identify-ing the area of Concepcion) that will be useful forthe subsequent analysis.4.2. Phase II: Who’s in ChargePhase II deals with how resources are allocated andassociated to agents in charge of key sub-systemsand problems. Consider the case in which theobjective is to apply resource allocation through-out the clusters in the hierarchical decompositionin order to determine the location of repair crewsfor disaster response. Three Mixed Integer Pro-grams (MIPs) were devised to carry out the so-called Topology Informed Resource Allocation thatdefines an adequate decentralization scheme for re-pair crews for disaster response.The first formulation for the stated problemis through the archetype problem known asCapacitated Facility Location Problem (CFLP)(Levi and Shmoys (2004)); the second formulationis based on Gómez et al. (2011) and exploits the in-formation provided by the hierarchical decompo-sition from Phase I to obtain strategical and com-putational benefits without significantly sacrificingoptimality (moderate TI-CFLP, where TI stands forTopology-informed); and the third formulation isa more aggressive exploitation of the hierarchicalstructure (aggressive TI-CFLP), which may lead tosignificant error when the network does not respondto a community structure at all.4.2.1. Standard Resource AllocationThe standard resource allocation is performed byrunning a pure CFLP, which consists of minimizingthe costs of installation of facilities and displace-ment to satisfy demand at all nodes. The decisionvariables are: yih, which states whether a resourceof type h is to be placed at node i; and wi j, whichstates whether a user at node j is to be attended bya resource located at node i. Other parameters are:kh, the capacity of resource type h; di j, the distancefrom i to j; ρ j, the demand at node j; and ch, thecost of installing resource h. The objective functionis as follows:min∑i∈Vchyih+∑i, j∈Vdi jwi j (1)Subject to:∑i∈Vwi j = 1 (2)∑h∈Hyihkh ≥∑j∈Vwi jρ j (3)Constraints in Equations 2 and 3 are those properof the CFLP, namely: “every user must be served bya facility”, and “nodes must have sufficient installedcapacity to serve other nodes”.4.2.2. Moderate TI-CFLPA new set of decision variables (xqh) is introducedfor the moderate formulation, which states whethera resource of type h is to be placed at cluster q. Withsuch variable, the objective function is modified asfollows:min∑q∈Qchxqh+∑i, j∈Vdi jwi j (4)512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015A second modification is the inclusion of a hier-archy related constraint (on top of the two originalones) in order to capture the additional informationobtained through the clustering process.∑q∈QxqhRqi = yih (5)The additional constraint in Equation 5 uses thecentroid of the q-th cluster, Rqi, to map the decisionvariables for clusters (xqh) and the decision vari-ables for nodes (yih). This slight change has pos-itive impact on computational efficiency and strate-gic resource allocation as reported in subsequentsections.4.2.3. Aggressive TI-CFLPFollowing the same reasoning of exploiting the in-formation provided by the hierarchical decomposi-tion, a more aggressive simplification is proposed.The so-called aggressive Topology-informed Re-source Allocation assumes that the pursued re-source allocation must correspond vis-a-vis withthe found clusters in such a way that the optimizershould only find those clusters where resourcesneed to be deployed, and then all nodes in the clus-ter will be associated to such resource. Such strongassumption eliminates the need for the second termin the objective function:min∑q∈Qchxqh (6)The constraints in Equations 2 and 3 are re-statedin this formulation as shown in Equations 7 and 8,since this formulation does not contain the originaldecision variables yih and wi j, but only the one re-lated to clusters, xqh. To do so, it uses the parameterTqi, which states whether node i is in cluster q.∑h∈H∑q∈QTqixqh ≥ 1 (7)∑h∈H(kh−∑i∈VTqiρi)xqh ≤ 0 (8)4.3. Phase III: Behavior under controlIn Phase III, each repair crew is simulated as anagent running an optimization program in whicha multi-objective function is minimized, account-ing for key variables as connectivity and flow reli-ability, flow capacity, and cost of interventions tomaintain them within acceptable ranges. Details ofhow the repair crews are simulated can be foundin Gómez (2014); the focus of this paper, however,will be on sensitivity analysis for Phase II.5. SCENARIO ANALYSIS FOR RE-SOURCE ALLOCATIONOf the several identified uncertainties in socio-technical modeling of infrastructure networks, onethat affects the resource allocation process is howthe decision-maker weights displacement costs vs.installation costs. This relates to the high impactand wide span of tasks performed by maintenancecrews, not only under emergencies but during nor-mal operation. Alvear and Rodriguez (2006) sug-gest a cost of truck transportation by kilometer inChile of USD$2. However, the displacement costmay significantly grow either by considering a longdecision horizon (that translates into many trips inthe long term), or by a subjective valuation of shorttrips as a favorable condition for prompt recovery.The standard CFLP, as well as the moderate andaggressive TI-CFLPs will be evaluated for timehorizons of 1, 10, 20, and 35 years, thus, progres-sively increasing the impact of transportation on theobjective function. Potential facilities of two typesare considered, with capacities of 4000 and 2000(in order to make them feasible for the amounts ofnodes at the second and third level of the hierarchi-cal decomposition in Figure 2). Installation costsare set to USD$400000 and USD$200000, propor-tional to capacities and will remain fixed. Thecosts of transportation are based on theUSD$2/kmfrom Alvear and Rodriguez (2006), and the sce-nario analysis is such that the displacement costsare multiplied by the considered number of years,affecting the installation/displacement ratio by thesame factor.Table 1 shows comparisons of execution timeand objective function for the CFLP and TI-CFLPswith each of the stated time horizons. The values ofthe objective function decrease along with the timehorizon due to the smaller amount of trips consid-ered each time. The objective values for the mod-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015erate TI-CFLP are very close (with gaps of 0.49%or less) to those of the CFLP optima, while the ag-gressive TI-CFLP has larger errors (even reachingvalues near 30%); conversely, the aggressive TI-CFLP exhibits tremendous reductions in executiontime (using about 10−4 of the computation time toobtain a gap of 13%), while those of the moder-ate TI-CFLP are more modest but still significantlyfaster than the pure CFLP (using less than 4% of thecomputation time), with the advantage of remainingvery close to the optimum.Table 1: Performance comparison of the standard,moderate, and aggressive approaches considering trav-els over 35, 20, 10, and 1 years.Obj. ($) Error (%) Time (s)35 yr.S 2.77e+006 0 6.4294M 2.78e+006 0.23 0.2520A 3.26e+006 17.63 1×10−520 yr.S 2.08e+006 0 51.3689M 2.09e+006 0.49 0.3550A 2.58e+006 24 9.99×10−410 yr. .S 1.54e+006 0 68.1349M 1.54e+006 0.33 0.1460A 1.99e+006 29.17 9.99×10−41 yr.S 9.06e+005 0 580.8142M 9.06e+005 0 0.5700A 1.03e+006 13.07 9.99×10−4Figures 3 and 4 show the obtained solutions forthe three approaches and two of the four time hori-zons (1 year and 35 years): green diamonds de-note resources with capacity of 4000, blue squaresdenote resources with capacity of 2000, and redlines denote nodes assigned to a resource. The ob-tained allocation schemes show that small clustersare favored when the weight of transportation ishigh (i.e., when the time horizon is long) becausedistances within large clusters are less affordable.However, as the time horizon.is reduced more greendiamonds (hence, larger clusters) start to appear,mostly for the aggressive TI-CFLP.Designed networks (trips over 35 years)Figure 3: Resource allocation for the Chilean network:(left) CFLP; (center) Moderate TI-CFLP; (right) Ag-gressive TI-CFLPDesigned networks (trips over 1 year)Figure 4: Resource allocation for the Chilean network:(left) CFLP; (center) Moderate TI-CFLP; (right) Ag-gressive TI-CFLP712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20156. CONCLUSIONSThis paper highlighted major uncertainties thatemerge when modeling infrastructure networks associo-technical systems, following the previouslyintroduced CoDAN framework. While abundantresearch exists for uncertainties related to phys-ical processes, cross-disciplinary research effortsare paramount to deal with uncertainties on the sideof the decision-maker (discount rates, risk aver-sion). Specific approaches were recommended aspotential research lines to enhance CoDAN’s treat-ment of uncertainty, including multi-objective andstochastic programming. The impact of a choiceof decision horizon was studied for the problem ofallocating resources for disaster preparedness (re-sources being repair crews and/or facilities). Threeoptimization programs were devised for resourceallocation, including the Capacitated Facility Loca-tion Problem and two variations that take advantageof hierarchical network decomposition to improvecomputational efficiency (the moderate and aggres-sive Topology-informed CFLP). Different scenar-ios were analyzed for the decision horizon param-eter, which affects the ratio of installation costsand displacement costs in facility location prob-lems. Simulation results show that the aggressiveTI-CFLP reduces computation time dramaticallybut is not as accurate as the moderate TI-CFLP, sug-gesting promising results for their combined use inthe future. Finally, different ratios of installationcosts and displacement costs lead to different allo-cation schemes, but more importantly, such ratiosmay not only respond to different decision horizonbut to different valuation of short displacements bythe decision-maker if associated to prompt recov-ery.7. REFERENCESAlvear, S. and Rodriguez, P. (2006). “Estimacióndel Costo por Kilómetro de una Empresa de Trans-porte de C ayr gdae, lIonsd uMsátrrgiaenes Agrícola,Región del Maule, Chile.” Panor. Socioecon., 24(32),48–57.Baker, J. W. and Cornell, A. (2008). “Uncertainty prop-agation in probabilistic seismic loss estimation.pdf.”Struct. Saf., 30(3), 236–252.Brandes, U. and Erlebach, T. (2005). Network Analysis:Methodological Foundations (Lecture Notes in Com-puter Science / Theoretical Computer Science andGeneral Issues). (Paperback), Springer, Secaucus,NJ, USA, 1 edition (March).Davies, D. L. and Bouldin, D. W. (1979). “A Clus-ter Separation Measure.” Pattern Anal. Mach. Intell.IEEE Trans., (2), 224–227.Dhillon, I., Guan, Y., and Kulis, B. (2005). “A unifiedview of kernel k-means, spectral clustering and graphcuts.” Report No. 5.Dunn, J. C. (1973). “A Fuzzy Relative of the ISO-DATA Process and Its Use in Detecting CompactWell-Separated Clusters.” J. Cybern., 3(3), 32–57.Gómez, C. (2014). “Risk-informed decision support forinfrastructure network operation: the Complex Dis-tributed Agent Network (CoDAN) framework.” Ph.D.thesis, Universidad de los Andes, Universidad de losAndes.Gómez, C., Buriticá, J., Sánchez-Silva, M., and DueñasOsorio, L. (2011). “Optimisation-based decision-making for complex networks in disastrous events.”Int. J. Risk Assess. Manag., X(5/6), 1–20.Gomez, C., Sánchez-Silva, M., and Dueñas Osorio, L.(2014). “An applied complex systems frameworkfor risk-based decision-making in infrastructure en-gineering.” Struct. Saf., 50, 66–77.International-Energy-Agency (2012). “Oil and Gas Se-curity – emergency Response of IEA Countries.” Re-port no., International Energy Agency.Kiureghian, A. D. and Ditlevsen, O. (2009). “Aleatoryor epistemic ? Does it matter ?.” Struct. Saf., 31(2),105–112.Levi, R. and Shmoys, D. B. (2004). “LP-based ap-proximation algorithms for capacitated facility loca-tion.” Proc. 5th Annu. ACM-SIAM Symp. Discret. Al-gorithms (SODA, ACM Press, 206–218.of Energy, C.-M. (2012). “National Energy Strat-egy 2012–2030: Energy for the Future.” Report no.,Gobierno-de-Chile.Slyke, R. M. V. and Wets, R. (1969). “L-Shaped Lin-ear Programs with Applications to Optimal Controland Stochastic Programming.” SIAM J. Appl. Math.,17(4), pp. 638–663.van Dongen, S. M. v. (2000). “Graph Clustering by FlowSimulation.” Ph.D. thesis, University of Utrecht, Uni-versity of Utrecht (May).Wang, Z., Dueñas Osorio, L., and Padgett, J. E. (2014).“A new mutually reinforcing network node and linkranking algorithm.” Sci. Reports [In Rev.8

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