Fund Allocation for Civil Infrastructure Security Upgrading
Nikos D. Lagaros, Ph.D. Lecturer, School of Civil Engineering, Institute of Structural Analysis & Seismic Research, National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece, e-mail:
[email protected]
Konstantinos Kepaptsoglou, Ph.D. Post-doctoral Researcher, School of Civil Engineering, Department of Transportation Planning and Engineering, National Technical University of Athens, 5, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece, e-mail:
[email protected]
Matthew G. Karlaftis, Ph.D. Associate Professor, School of Civil Engineering, Department of Transportation Planning and Engineering, National Technical University of Athens, 5, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece, e-mail:
[email protected]
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Fund Allocation for Civil Infrastructure Security Upgrade
Abstract: Security of transportation infrastructure is of primary importance in recent years as a result of various breaches worldwide. Increased security concerns leads transportation authorities to improve, upgrade and enhance surveillance, prevention and response equipment in facilities, frequently under tight budget and operational constraints. In this context, we propose a generic selection and resource allocation (S&RA) model for security upgrade of civil infrastructure, along with a novel technique for efficiently solving the model for real-world instances. An application of the model is offered for the case of the Athens, Greece, metro system and results are discussed.
Keywords: Transportation security; selection and resource allocation; budget; particle swarm optimization
1.
Introduction
Project Selection and Resource Allocation (S&RA) is at the core of management tasks and activities encountered by organizations and authorities that deal with transportation networks and infrastructure. Indeed, organizations regularly face the need to allocate their limited funds to different programs/projects, often satisfying conflicting objectives and interests (Zanakis et al., 1995). Examples in the transportation sector include optimal programming of maintenance, repair and rehabilitation (MR&R) activities in pavements and bridges and the replacement of transit fleets. The corresponding selection and resource allocation (S&RA) problem can be formally described as follows (Zanakis et al., 1995): “Consider a set of alternative programs/projects, with a quantified benefit assigned to each program/project. The S&RA problem focuses on the decision on which project to fund and in what amount. This decision is subject to constraints and dictated by - often conflicting - objectives”. Security in transportation has become a topic of primary importance in recent years primarily due to attacks against transportation systems such as the cases of the Tokyo, London and Madrid metro systems, in Moscow’s Domodedovo airport and so on. Indeed, concentration of passengers in confined transportation facilities along with the significant role of transportation to modern societies and the economy, makes transportation systems attractive as potential targets for terrorist activities (Meyer, 2010; Cox et al., 2011). Worldwide, transportation authorities are increasingly concerned with security of their facilities and keep upgrading and expanding relevant surveillance, prevention and response equipment and tactics, often under operational difficulties and budget limitations. In this context, we focus on S&RA for the upgrade and expansion of security equipment
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in transportation infrastructures and propose a non-linear programming model and a metaheuristics algorithm for solving it. We also offer a comprehensive application of the model for the Athens, Greece, metro system. The remainder of the chapter is structured as follows: the next section offers a comprehensive overview of S&RA methods and techniques and their applications in different areas of transportation. Then, the S&RA model for the problem at hand along with the proposed solution method is presented. An application of the model for the case of the Athens metro system is provided and results are discussed. The chapter concludes with model application insights and proposals for future research in the area.
2.
Background
S&RA is the final step of the so-called project evaluation process; previous steps include identification of candidate projects, determination of evaluation criteria and estimation of benefits per alternative project. As a problem, S&RA has been extensively investigated in the literature with excellent surveys offered by Zanakis et al. (1995), Heidenberger and Stummer (1999) and a reference textbook by Bower (1986). In this section, we will first provide an overview of the methods and techniques used for S&RA of projects and then we review relevant work in the field of civil infrastructure and transportation.
2.1. S&RA Methods and Techniques Typically, S&RA consists of two tasks: i. measurement of benefits and, ii. optimal selection/allocation of resources. The first task pertains to determining the benefits associated with each alternative project; these benefits, in turn, refer to establishing and quantifying
various decision/evaluation criteria (Zanakis et al., 1995); the second task refers to optimally selecting projects and allocating funds.
2.1.1. Benefit Measurement Various approaches have been proposed in the literature for measuring project benefits; these may be categorized into four groups (Heidenberger and Stummer, 1999): comparative approaches, scoring approaches, traditional economic models, and group decision techniques. Comparative approaches refer to pair wise comparison of projects using either an additive or a ratio scale. While these methods often require a large number of ‘project-toproject’ comparisons and the need to repeat any measurement when a project is added, their advantage is the minimum impact of errors because of the large number of comparisons involved (Locket and Startford, 1987). The most popular among comparative methods is the analytical hierarchy process (AHP), introduced by Saaty (1980). Scoring methods assign an overall benefit measure to each project. Scores of alternative projects with respect to each decision criterion are determined and then combined to yield the overall project score. The simplest among scoring methods is the checklist technique, in which fulfillment of requirements is examined along with the level of conformation to that requirement. Traditional scoring models, on the other hand, use multiple value assessment scores and weights for each decision criterion. These models are widely applied for measuring project benefits since - in most cases - they succeed in offering a consistent ranking of projects (Krawiec, 1984). Utility analysis (often referred to as multiattribute utility analysis) is an advanced, axiomatically founded, scoring method (Heidenberger and Stummer, 1999). The method is based on aggregating evaluation criteria in a single
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utility function, which expresses a decision maker’s preference towards a project. Utility functions are less preferred by practitioners due to the complexity involved in deriving them, as well as because of their inferior performance (Schoemaker and Waid, 1982). Economic methods are based on the monetary perspective of projects and exploit capital budgeting techniques for estimating the economic performance of alternative projects. Relevant methods include the estimation of economic indices, discounted cash flow methods and “options approach” (Heidenberger and Stummer, 1999). As for group decision techniques, they combine knowledge and experience collected by experts to rate and rank projects. The Delphi method is probably the most popular group decision technique; it relies on a consensus by a panel of experts who offer their judgment for alternative projects based on structured questionnaires (Zanakis et al., 1995).
2.1.2. Optimal Selection of Projects and Allocation of Resources Once project benefits are measured, their ranking becomes readily available. However, the question regarding how to select projects under budgetary, time and other constraints still remains largely unaddressed. Further, the amount of funds to be allocated to each project often becomes an additional item to be derived. In this context, the prevailing approaches for optimal project selection (particularly in the area of transportation), and fund allocation, are based on optimization models while other methods involve decision and game theory, statistical models, and simulation (Heidenberger and Stummer, 1999). Traditional optimization-oriented approaches for S&RA are based on mathematical programming models. These models seek to maximize benefits from selected projects under various budgetary and other constraints. Depending upon the nature of the S&RA, a variety of linear, integer and non-linear specifications may be considered. For instance,
linear models assume both linear consumption of resources and benefit contribution of projects as well as independence of projects (Winston 2003). On the other hand, nonlinear models are used in cases of complex decision problems, implying that benefits and constraints are represented by non-linear objective functions. As for integer programming models, these are introduced in cases of explicit selection of projects because of their ability to represent “yes/no” and discrete decisions (Winston, 2003). While optimization models may offer a straightforward representation of S&RA problems, they often require increased computational power for obtaining optimal solutions, particularly when problem sizes increase. This is particularly true for discrete and integer programming models which are usually combinatorial in nature and thus difficult to solve (Winston, 2003). For this reason, a number of heuristic and metaheuristic techniques have been proposed by the literature to solve such problems (with lagrangean relaxation and genetic algorithms as frequently used examples). Ibaraki and Katoh (1988) present a detailed discussion on complexity and on methods for solving relevant resource allocation problems. Apart from straightforward mathematical programming models, techniques such as goal, dynamic and stochastic programming have been considered for S&RA. In goal programming, projects are selected so that a set of goals (in terms of project benefits) are achieved, while dynamic programming models focus on resource allocation problems where programming of projects in a subsequent time period is affected by S&RA decisions made in a previous period. Finally, stochastic models are used in cases where some inputs are uncertain and subject to variation while fuzzy programming in exploited in cases of broad (fuzzy) ranges of inputs and similar expectations in the S&RA results.
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As an alternative to mathematical programming models, the literature has proposed decision and game theory methods, simulation and cognitive emulation approaches (statistical models, expert systems, and so on) for S&RA (interested readers can refer to Heidenberger and Stummer (1999) for more details and references on these methods and their application in S&RA).
Table 1 summarizes tasks and techniques exploited for S&RA of projects [Insert Table 1 about here]
2.2. S&RA applications in transportation Transportation systems are vital lifelines in constant need for maintenance, improvement, upgrade and expansion. Given economic, social, organizational and other constraints, along with the size, cost and impact of transportation projects, optimal S&RA in transportation systems has attracted considerable interest in both academia and practice. Indeed, the literature includes a vast number of S&RA cases and applications in the transportation sector. Here we provide different S&RA areas of application in transportation and indicative references.
2.2.1. Transportation Fund Allocation at the Network Level Fund distribution in the network level is a common problem for governments and upper level decision makers; the scope is to allocate funds so that network performance is maintained to desirable levels. As such, Haghani and Wei (1993) proposed a binary non-linear programming model for capacity expansion of a network and used a linear transformation to solve it. Another study by Tao and Schonfeld (2005) considered a non-linear model specification for fund allocation in a network, where network impacts were evaluated
with the use of a user equilibrium model. Gorloo et al. (2009) proposed an integer programming model for S&RA which maximized network reliability and Ferguson et al. (2010) developed a bi-level, non linear optimization model for determining capacity improvement projects in a transportation network. Their model considered environmental impacts (emissions) as benefits with the use of appropriate utility functions.
2.2.2. Optimal Programming of New Highway Project Construction Programming of highway projects is among those strategic transportation issues that directly fit the context of the S&RA problem. For example, Leu et al. (1999) used fuzzy optimization for scheduling highway project construction under resource constraints and uncertainty duration. Among recent studies, a multi-objective integer programming model was proposed by Iniestra and Gutierrez (2007) for evaluating interdependent problems and Teng et al. (2010) applied an extension of the analytical hierarchy process technique (fuzzy AHP) for ranking and allocating funds for new highway projects in Taiwan. Following a different, decision theory oriented approach, Owens et al. (2011) used complexity maps and panels of experts for resource allocation among projects.
2.2.3. Infrastructure Management Infrastructure (asset) management in transportation refers to the integrated process for effectively operating, maintaining, upgrading and expanding transportation infrastructures (assets), such as pavements, bridges, rails and roadside elements (Sinha and Labi, 2007). Focus is given on improving the quality and performance of infrastructures and on minimizing their life-cycle costs. Project selection & resource allocation is obviously a
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critical task in infrastructure management since it sets optimal courses of action for maintaining and/or improving transportation infrastructures and determines required funding for that purpose. The literature exhibits a variety of models proposed and incorporated for S&RA actions in pavement and bridge management, while during the last decade the concept of highway asset management (including bridge, pavement and roadside elements) in S&RA has gained attention (Tiewater and Zimmerman, 2010). In the area of pavement management, S&RA examples include linear models (Hajek and Phang, 1988; Seyedshohadaie et al., 2010), dynamic programming (Feighan et al., 1988; Ouyang, 2007), goal programming (Sharaf and Youssef, 2001), multiobjective models (Chan et al., 2003), robust programming (Gao and Zhang, 2008), fuzzy programming (Mellano et al., 2009) and stochastic multi-objective models (Wu and Flintsch, 2009). Bridge management has exhibited a variety of S&RA optimization models, including linear specifications by Guigner and Madanat (2000), integer programming (AlSubhi et al., 1990), dynamic programming (Jiang and Sinha, 1989, Raqpazur et al. 1996), incremental benefit - cost approach (Farid et al., 1994a, 1994b; Robert et al., 2009) and multiobjective models (Hammad et al. 1996; Itoh et al. 1999, Liu and Frangopol, 2005a,b; Neves et al., 2006; Thompson et al, 2008). Asset management S&RA models in transportation have become popular in recent years; a notable earlier work was the integer programming model proposed by Ahmed (1983) for S&RA in highway maintenance management. Recently, Li and Murat (2006) and Li et al. (2010) proposed stochastic binary programming models for supporting highway asset investment decisions under budget uncertainty.
2.2.4. Transit Fleet Replacement
Frequently operating under strict budgetary constraints, decreased ridership and low fares, transit agencies constantly face the need for replacing their aging fleets; this comes as part of an effort for offering adequate level of service to passengers and maintaining market shares. In this context, optimal transit fleet replacement funding has been examined by some researchers: Kashnabis et al. (2002) developed a linear programming model for determining purchase and rebuilt of buses in a transit fleet. Mathew el al. (2010) and Mishra et al. (2010) introduced non-linear programming models for allocating funds to transit agencies based on fleet average remaining life and used both branch-and-bound and genetic algorithm techniques for solving the model.
2.2.5. Traffic Safety Improvements Safety improvements in highways involves the selection of appropriate countermeasures and allocation of resources in an effort to reduce traffic accident impacts. Brown et al. (1990) proposed integer and dynamic programming techniques for allocating safety highway grants and for achieving maximum benefits in terms of life and injury savings. Chowdhury and Garber (1996) and Chowdhury et al. (2000) used multiobjective analysis for optimally selecting safety countermeasures and Melachrinoudis and Kozanidis (2000) formulated a mixed integer model which considered both regular and ‘custom-made’ safety interventions in highways. In 2004, Kar and Datta formulated a linear model for allocating resources to safety projects for the State of Michigan based on a safety performance index.
2.2.6. Emergency Response and Security Following a disaster, recovery efforts are required to reduce impacts to local societies and to restore normal life activities. Transportation networks are vital lifelines and therefore 11
or primary importance in the overall recovery process and allocation of funds. In this context, Karlaftis et al. (2007) developed a three-stage integer programming model for allocating funds to transportation infrastructure recovery projects following a catastrophic event and solve it with the use of a genetic algorithm. Two papers by Orabi et al. (2009, 2010) focused on optimal resource allocation for reconstruction of infrastructures; for the purpose they proposed multi-objective resource allocation models for project programming and allocation of resources. A framework for minimizing the cost of hazard preparedness and appropriate allocation of resources was developed by El-Adaway and ElAnwar (2010). Finally, a recent report by Bier et al. (2009) derives analytical functions for allocating resources with the objective of transportation infrastructure protection.
Table 2 summarizes examples of S&RA applications in transportation [Insert Table 2 around here]
Overall, the literature on S&RA models and applications in transportation infrastructure expansion and management is rich. Most papers exploit mathematical programming models, while the stochastic nature of relevant problems is also considered in some cases. In this context, our objective is to contribute to existing knowledge by offering a generic model targeted towards transportation security upgrades.
3.
Problem Overview and Methodology
3.1. Problem Overview Consider a transportation facility i being at a level xi,initial with respect to the capacity and condition of its security infrastructure. This facility is assumed to have importance ratings with respect to its ridership (σtrans,i) and its location (σstrategic,i). The system/facility operator wishes to upgrade the security of the transportation facilities to the maximum possible level given existing equipment conditions, capacities and station importance, under the constraint of available funds; the main interest is to optimally allocate funds to stations for best overall security upgrades.
3.2. Methodology
3.2.1. Model Formulation Optimal fund allocation in this context is defined as a nonlinear programming problem formulated as follows: N
max trans ,i strategic,i ( xupgrade,i xinitial ,i )
(1)
i 1
s.t. N
K (Size , Age , x i 1
i
i
upgrade ,i
, xinitial ,i ) BTarget
xupgrade,i Z
(2) (3)
where I:
Set of N transportation facilities
i:
Transportation facility i I
σtrans,i:
Transportation importance of facility i 13
σstrategic,i:
Strategic importance of facility i
xupgrade,i:
Anticipated (post-upgrade) security level of facility i
xinitial,i:
Current security level of facility i
Sizei:
Size of facility i
Agei:
Age of of facility i
K:
Cost for upgrading facility i from level xinitial,i to level xupgrade,i
BTarget:
Available Budget
The objective function (Eq. 1) maximizes security upgrade levels for facilities; a premium is given to those facilities with increased importance ratings. The budget constraint in Eq. (2) implies that cost K per facility is a function of its characteristics (age, size) as well as its current and anticipated security level. It should be noted that the aforementioned model is a variant of the well-known knapsack problem (Winston, 2003) with non-linear constraints and a size of N integer decision variables.
3.2.2. Optimization Method In Particle Swarm Optimization (PSO), multiple candidate solutions coexist and collaborate simultaneously. Each solution is called “a particle” having a position and a velocity in the multidimensional design space while a population of particles is called a swarm. A particle “flies” in the problem search space looking for the optimal position. As “time” passes through its quest, a particle adjusts its velocity and position according to its own “experience” as well as the experience of other (neighbouring) particles. A particle's experience is built by tracking and memorizing the best position encountered. A PSO system combines local search (through self experience) with global search (through neighbouring experience), attempting to balance exploration and exploitation. Each particle maintains
its two basic characteristics, velocity and position, in the multi-dimensional search space that are updated as follows:
v j (t 1) wv j (t ) c1r1
s Pb,j s j (t ) c2r2 s Gb s j (t )
s j (t 1) s j (t ) v j (t 1)
(4) (5)
where vj(t) denotes the velocity vector of particle j at time t, sj(t) represents the position vector of particle j at time t, vector sPb,j is the personal best ever position of the jth particle, and vector sGb is the global best location found by the entire swarm. The acceleration coefficients c1 and c2 indicate the degree of confidence in the best solution found by the individual particle (c1 - cognitive parameter) and by the whole swarm (c2 - social parameter), respectively, while r1 and r2 are two random vectors uniformly distributed in the interval [0,1]. The symbol “ ” of Eq. (4) denotes the Hadamard product, i.e. the elementwise vector or matrix multiplication. Figure 1 shows the flowchart of the PSO algorithm, while Figure 2 depicts a particle’s movement in a two-dimensional design space. The particle’s current position sj(t) at time t is represented by the dotted circle at the lower left quadrant of the drawing, while the new position sj(t+1) at time t+1 is represented by the dotted bold circle at the upper right hand of the drawing. Figure 2 depicts how the particle’s movement is affected by: (i) it’s velocity vj(t); (ii) the personal best ever position of the particle, sPb,j, and (iii) the global best location found by the entire swarm, sGb.
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4.
Model Application
4.1. General Information We apply the model for the case of the Athens metro system (Figure 3) and particularly in its 52 metro stations, each with different size and age; five different test cases are examined with varying target budget (5.0, 10.0, 20.0, 40.0 and 80.0 million of monetary units (MU); we note that using the data in Tables 3 – 6, readers can replicated the results of this application, while MU are used instead of actual currency values for security purposes). The budget requirements for upgrading the ith metro station from its existing security level to different (higher) levels are given in Table 3. To consider the additional cost requirements imposed due to station age for the ith metro station, parameter σage is used (the parameter is the ratio of the budget for upgrading – to a predefined security level - a metro station of the same size and type to that of the same metro station constructed today; Table 4). Further, to consider the requirements for upgrading metro stations of different type-size parameter σsize is used (Table 5). The list of the 52 metro stations along with the age level, size-type, strategic importance factor (related to the location of the metro station), transportation importance (related to station demand) and the initial security level of the metro stations is given in Table 6.
4.2. PSO Parameters In order to examine the influence of the parameters of the PSO algorithm, a parametric study was performed to find the best parameters to be used for the PSO metaheuristic
algorithm. For this, a sample of 20 combinations of the following parameters was considered: NP (defined in the space [50,200]), inertia weight w (defined in the space [0.01,0.7]), cognitive parameter c1 (defined in the space [0.0,1.0]), and social parameter c2 (defined in the space [0.0,1.0]). Parameter combinations were generated by means of Latin hypercube sampling (see Figure 4). The resulting optimization runs for dealing with the fund allocation problem were equal to 20 parameter combinations × 100 random optimization runs = 2,000 optimization runs for BTarget = 5.0 millions MU. The mean and standard deviation values of the objective function value and the number of function evaluations are given in Tables 7 and 8 respectively. As can be seen in Table 7, the coefficient of variation of the mean values of the 20 parameter combinations is only 7.5%, while with reference to the function evaluations given in Table 8 the corresponding coefficient of variation is only 0.06%. From this we can conclude that the PSO algorithm is not sensitive to parameter values in the case of this fund allocation problem. Based on the maximum mean value of the objective function (Table 7), the 16th combination of parameters is used for the PSO metaheuristic algorithm. These values are as follows: number of particles NP = 190, inertia weight w = 0.0314, cognitive parameter c1 = 0.02, and social parameter c2 = 0.0151. For comparative purposes, the termination criterion is the same for all optimization runs; each procedure is terminated after 102 circles with no improvement.
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4.3. Results The resulting optimization runs for the fund allocation problem with five test cases, were equal to 5 test cases × 100 optimization runs = 500 optimization runs. Figures 5 to 9 depict the solutions obtained for the optimal fund allocation problem for the five target budgets used, when the PSO algorithm is implemented. As can be seen from these figures, for the cases of 5.0 to 40.0 million MU, all the available budget is used satisfying constraint (3), while on the other hand implementation of the algorithm for the case of BTarger = 80.0 million MU, results in 46.7 million MU used without achieving the upper security level for all metro stations (this is denoted as Step1 in Figure 9). For this reason, a second implementation of the algorithm was examined for the remaining 80.0 - 46.7 = 33.3 millions MU, considering that at metro stations security levels were below maximum (denoted as Step2 in Figure 9).
5.
Conclusions
This paper focuses on developing a generic fund allocation model and solution method for security upgrades of transportation infrastructures. The model is a variant of the knapsack problem with a non-linear budget constraint and it is solved with the use of a metaheuristics method (particle swarm optimization). The model was successfully applied to a real world case, the security upgrade of the 52 stations of the Athens metro system, and yielded feasible fund allocations for different budget scenarios. Our proposed approach can successfully lead to optimal fund allocation for any network of civil and transportation infrastructure.
6.
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Razaqpur, A.G., El-Halim A., Mohamed, H.A., 1996, Bridge management by dynamic programming and neural networks, Canadian Journal of Civil Engineering, 2(5), 1064-1069. Robert, W.E., Gurenich, D.I, Thompson, R.E. 2009. Multiperiod bridge investment optimization utilizing pontis results and budget constraints by work type. Proceedings of the 88th Transportation Research Board Annual Meeting, Washington DC, USA. Saaty, T.L. 1980. The analytical hierarchy process. McGraw-Hill, New York. Schoemaker, P.J., Waid, C.C. 1982. An experimental comparison of different approaches to determining weights in additive utility model. Management Science 28, 182-196. Seyedshohadaie, S.R., Damnjanovic, I., Butenko, S. 2010. Risk-based maintenance and rehabilitation decisions for transportation infrastructure networks. Transportation Research Part A 44(4), 236-248. Sharaf, E., Youssef, M.A. 2001. A two-fold optimization system for highway maintenance fund allocation. Proceedings of the Fifth International Conference on Managing Pavements, Seattle, Washington. Sinha, K., Labi, S. 2007. Transportation decision making: principles of project evaluation and programming. Wiley and Sons, Hoboken, NJ. Tao, X., Schonfeld, P. 2005. Lagrangian relaxation heuristic for selecting interdependent transportation projects under cost uncertainty Transportation Research Record: Journal of the Transportation Research Board 1931, 74-80.
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Teng, J-Y, Huang, W-C., Lin, M-C. (2010). Systematic budget allocation for transportation construction projects: A case in Taiwan. Transportation: Planning, Policy, Research, Practice, 37(2), 331-361. Thompson, P., Sinha, K.C., Labi, S., Patidar, V. 2008. Multiobjective optimization for bridge management systems. transportation research e-circular E-C128, Transportation Research Board, Washington DC. Tiewater, P., Zimmerman, K. 2010. Transportation asset management: definitions and principles. TR News 270, 7. Winston, W. 2003. Operations research: applications and algorithms. Duxbury Press, Pacific Grove, CA. Wu, Z., Flintsch, G.W. 2009. Pavement preservation optimization considering multiple objectives and budget variability. Journal of Transportation Engineering 135(5), 305-315. Zanakis, S.H., Mandakovic, T., Gupta, S.K., Sahay, S., Hong, S. 1995. A review of program evaluation and fund allocation methods within the service and government sectors. Socio-Economic Planning Sciences 29(1), 59-79.
Table 1. S&RA methods and techniques (based on Heidenberger and Stummer, 1999)
Comparative methods Analytical Hierarchy Process
Mathematical programming Linear models Non-linear models Integer models Goal programming Dynamic Programming Stochastic programming Fuzzy programming
Measurement of Benefits Scoring methods Economic methods Checklist Scoring Models Utility Analysis
Group Decision techniques Delphi
Economic Indices Discounted cash flow methods Options approach Optimal Project Selection and Resource Allocation Decision and Game theSimulation Congnitive emulation ory Decision trees Monte Carlo Simula- Statistical models tion Game Theory Expert Systems Heuristic Approaches
27
Table 2. S&RA example applications in transportation Area Transportation Fund Allocation in the Network Level
Optimal Programming of New Projects Pavement Management
Bridge Management
Asset Management
Transit Fleet Replacement Traffic Safety Improvement
Emergency Response and Security
Article Haghani and Wei (1993) Tao and Schonfeld (2005) Goorlo et al. (2009) Ferguson et al. (2010) Leu et al. (1999) Iniestra and Gutierrez (2007) Teng et al. (2010) Owens et al. (2011) Hajek and Phang (1988) Seyedshohadaie et al. (2010) Feighan et al. (1988) Ouyang (2007) Sharaf and Youssef (2011) Chan et al. (2003) Gao and Zhang (2008) Mellano et al. (2009) Wu and Flintsch (2009) Guigner and Madanat (2000) Al-Subhi et al. (1990) Jiang and Sinha (1989) Razaqpur et al (1996) Hammad et al. (1996) Itoh et al. (1997) Liu and Frangopol (2005a,b) Neves et al. (2006) Thompson et al (2008) Farid et al. (1994a,b) Robert et al. (2009) Ahmed (1983) Li and Murat (2006) Li et al.(2010) Khasnabis et al . (2002) Mathew et al. (2010) Mishra et al. (2010) Brown et al. (1990) Chowdhury and Garber (1996) Chowdhury et al. (2000) Melachrinoudis and Kozanidis (2000) Kar and Datta (2004) Karlaftis et al. (2007) Orabi et al. (2009) Orabi et al. (2010) Bier et al. (2009)
S&RA method Non linear model with linear transformation Non linear model combined with user equilibrium Integer model Bi-level non linear model Fuzzy programming Multi-objective integer model Fuzzy AHP Complexity maps – expert panel Linear model Linear model Dynamic programming Dynamic programming Goal programming Multiobjective model Robust programming Fuzzy programming Stochastic multiobjective model Linear model Integer model Dynamic programming Dynamic programming Multiobjective model Multiobjective model Multiobjective model Multiobjective model Multiobjective model Incremental Benefit- Cost Analysis Incremental Benefit- Cost Analysis Integer model Stochastic binary model Stochastic binary model Linear model Non-linear model Non-linear model Integer model and dynamic programming Multiobjective model Multiobjective model Mixed integer model Linear model Integer model Multiobjective model Multiobjective model Analytical models
Table 3. Budget upgrading in monetary units Security level 1 2 3 4 5
Budget requirement 1,000 5,000 10,000 100,000 500,000
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Table 4. Age levels of the metro stations Age period 2001-2011 1981-2000 1951-1980 1900-1950
σage 1.0 1.5 2.5 4.0
Table 5. Size of metro stations Age period Multilevel Large Medium Small
σsize 3.0 2.0 1.3 1.0
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Table 6. Station characteristics Station ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52
Station Name Kifisia KAT Marousi NeratziotissA Irini Irakleio Nea Ionia Pefkakia Perissos Ano Patisia Agios Eleftherios Patisia Agios Nikolao Attikin (old) Attiki (new) Viktoria Omonia Monastiraki (old) Monastiraki (new) Thisio Petralona Tavros Kallithea Moschato Faliro Piraeus Agios Antonios Sepolia Larisa Metaxourgio Panepistimio Syntagma Akropoli Fix Neos Kosmos Agios Ioannis Dafni Agios Dimitrios Aigaleo Elaionas Keramikos Evagelismos Megaro Mousikis Ampelokipi Panormou Katechaki Ethniki Amyna Holargos Nomismatokopeio Agia Paraskevi Halandri Doukissis Plakentias
Age 30-60 years 10-30 years 30-60 years Under 10 years 10-30 years 30-60 years 30-60 years 30-60 years 30-60 years 30-60 years 30-60 years 30-60 years 30-60 years 30-60 years Under 10 years 30-60 years Under 10 years Over 60 years Under 10 years Over 60 years Over 60 years 10-30 years 30-60 years 30-60 years 10-30 years Over 60 years Under 10 years 10-30 years 10-30 years 10-30 years 10-30 years 10-30 years 10-30 years 10-30 years 10-30 years 10-30 years 10-30 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years Under 10 years
Size Medium Small Small Small Medium Small Small Small Small Small Small Small Small Medium Large Small Multilevel Large Multilevel Small Small Medium Small Small Large Multilevel Large Medium Medium Large Large Multilevel Medium Medium Medium Medium Medium Large Large Medium Large Medium Medium Medium Large Medium Large Medium Medium Large Large Multilevel
σtrans 3 5 3 3 3 3 4 4 4 3 3 4 4 2 2 3 1 1 1 3 3 3 3 4 3 1 2 4 2 3 2 1 3 3 3 3 3 2 2 5 2 3 3 2 2 3 2 3 3 3 3 2
σstrategic 3 2 4 4 3 4 4 4 4 4 4 4 4 2 2 4 1 1 1 4 4 4 4 4 2 1 3 2 2 4 2 1 1 3 3 3 3 3 3 5 3 2 2 3 3 3 1 3 3 3 3 3
xinit 1 1 2 3 4 1 2 3 4 3 4 1 1 4 3 3 2 2 2 3 1 2 4 4 4 4 1 4 3 4 2 1 4 4 1 2 1 4 2 1 4 4 4 4 4 4 2 1 2 4 2 2
Table 7. Sensitivity analysis of PSO algorithm with reference to the objective function value Parameters Combination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Mean 3.89E+02 4.30E+02 4.12E+02 3.98E+02 3.89E+02 4.29E+02 4.26E+02 4.67E+02 3.89E+02 3.88E+02 4.52E+02 4.02E+02 4.43E+02 4.59E+02 4.53E+02 4.72E+02 4.70E+02 4.55E+02 3.98E+02 3.73E+02
Standard Deviation 1.49E+02 9.87E+01 1.03E+02 1.01E+02 1.11E+02 1.39E+02 1.07E+02 3.17E+01 8.10E+01 1.73E+02 9.17E+01 9.30E+01 1.06E+02 3.75E+01 1.21E+02 9.88E+01 3.08E+01 9.72E+01 1.35E+02 1.55E+02
Best
Worst
5.52E+02 5.52E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02 5.67E+02
1.78E+02 1.78E+02 1.78E+02 1.78E+02 1.78E+02 1.78E+02 1.78E+02 1.78E+02 1.78E+02 1.75E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02 1.60E+02
33
Table 8. Sensitivity analysis of PSO algorithm with reference to the number of function evaluations Parameters Combination 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Mean 1,000,224 1,002,144 1,001,572 1,001,280 1,000,188 1,000,440 1,000,080 1,000,110 1,001,000 1,000,640 1,000,512 1,000,890 1,000,552 1,000,350 1,000,960 1,000,350 1,001,300 1,001,880 1,000,286 1,000,400
Standard Deviation 9.91E-05 5.37E-05 6.06E-05 6.38E-05 7.38E-05 7.59E-05 5.93E-05 1.46E-05 5.37E-05 1.16E-04 4.52E-05 5.77E-05 5.42E-05 1.79E-05 5.96E-05 4.46E-05 1.40E-05 4.72E-05 8.58E-05 1.12E-04
Best
Worst
1,000,224 1,000,224 1,000,224 1,000,224 1,000,188 1,000,188 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080 1,000,080
1,000,224 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144 1,002,144
Step 1: Initialize Parameters F(s): objective function si: design variable n: number of design variables NP: number of particles w: inertia weight vmax: vector of maximum allowable velocity c1,c2: cognitive and social parameters TC: termination criterion
Step 2: Initialize Particles For i=1 to NP Random generation of the particle si(t=1) Calculate F(si(t=1))
Step 3: Generate t+1 swarm For i=1 to NP Generate velocity vector vi(t+1) according to Eq. (4) Generate position vector si(t+1) according to Eq. (5) Calculate F(si,t+1)
Step 4: Check of convergence Yes
Terminate Computations
Satisfied
No
Repeat Step 3
Figure 1. Flowchart of the particle swarm optimization algorithm
35
sGb sj(t+1)
v (t j
sGb,j
wv j (t)
c2 r
sj(t)
2 (s P b
-s j (
t))
v (j t)
) +1
j Pb,j -s (t))
c1r1(s
Figure 2. Visualization of the particle’s movement in a two-dimensional design space
Figure 3. The Athens metro system
37
r2 r2,up
r2,low r1,low
r1,up
r1
Figure 4. Latin hypercube sampling in the two-dimensional space
5,000 (103 MU) 6 Before After 5
Security Level
4
3
2
1
0
0
5
10
15
20
25
30
35
40
45
50
Metro Station
Figure 5. 5 million MU target budget
39
10,000 (103 MU) 6 Before After 5
Security Level
4
3
2
1
0
0
5
10
15
20
25
30
35
Metro Station
Figure 6. 10 million MU target budget
40
45
50
20,000 (103 MU) 6 Before After 5
Security Level
4
3
2
1
0
0
5
10
15
20
25
30
35
40
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50
Metro Station
Figure 7. 20 million MU target budget
41
40,000 (103 MU) 6 Before After 5
Security Level
4
3
2
1
0
0
5
10
15
20
25
30
35
Metro Station
Figure 8. 40 million MU target budget
40
45
50
80,000 (103 MU) 6 Before Afterstep 1 Afterstep 2
5
Security Level
4
3
2
1
0
0
5
10
15
20
25
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40
45
50
Metro Station
Figure 9. 80 million MU target budget
43
