Showing papers in "Transportation Science in 2010"

Optimal Motorway Traffic Flow Control Involving Variable Speed Limits and Ramp Metering

Abstract: The impact of variable speed limits (VSL) on aggregate traffic flow behaviour on motorways is shown to bear similarities to the impact of ramp metering, in particular, when addressing potentially active bottlenecks. A quantitative model of the VSL impact is proposed that allows for VSL to be incorporated in a macroscopic second-order traffic flow model as an additional control component. The integrated motorway network traffic control problem involving ramp metering and VSL control measures is formulated as a constrained discrete-time optimal control problem and is solved efficiently even for large-scale networks by a suitable feasible direction algorithm. An illustrative example of a hypothetical motorway stretch is investigated under different control scenarios, and it is shown that traffic flow efficiency can be substantially improved when VSL control measures are used, particularly in integration with coordinated ramp metering.

The Effect of Supply Disruptions on Supply Chain Design Decisions

Abstract: We study an integrated supply chain design problem that determines the locations of retailers and the assignments of customers to retailers to minimize the expected costs of location, transportation, and inventory. The system is subject to random supply disruptions that may occur at either the supplier or the retailers. Analytical and numerical studies reveal the effects of these disruptions on retailer locations and customer allocations. In addition, we demonstrate numerically that the cost savings from considering supply disruptions at the supply chain design phase (rather than at the tactical or operational phase) are usually significant.

A Branch-and-Price Method for a Liquefied Natural Gas Inventory Routing Problem

Abstract: We consider a maritime inventory routing problem in the liquefied natural gas (LNG) business, called the LNG inventory routing problem (LNG-IRP). Here, an actor is responsible for the routing of the fleet of special purpose ships, and the inventories both at the liquefaction plants and the regasification terminals. Compared to many other maritime inventory routing problems, the LNG-IRP includes some complicating aspects such as (1) a constant rate of the cargo evaporates each day and is used as fuel during transportation; (2) variable production and consumption of LNG, and (3) a variable number of tanks unloaded at the regasification terminals. The problem is solved by a branch-and-price method. In the column generation approach, the master problem handles the inventory management and the port capacity constraints, while the subproblems generate the ship route columns. Different accelerating strategies are implemented. The proposed method is tested on instances inspired from real-world problems faced by a major energy company.

To Wait or Not to Wait---And Who Goes First? Delay Management with Priority Decisions

Abstract: Delay management is an important issue in the daily operations of any railway company. The task is to update the planned timetable to a disposition timetable in such a way that the inconvenience for the passengers is as small as possible. The two main decisions that have to be made in this respect are the wait-depart decisions, to decide which connections should be maintained in case of delays, and the priority decisions, which determine the order in which trains are allowed to pass a specific piece of track. The latter are necessary to take the limited capacity of the track system into account. While the wait-depart decisions have been intensively studied in the literature, the priority decisions in the capacitated case have been neglected so far in delay management optimization models. In the current paper, we add the priority decisions to the integer programming formulation of the delay management problem and are hence able to deal with the capacitated case. The corresponding constraints are disjunctive constraints leading to cycles in the resulting event-activity network. Nevertheless, we are able to derive reduction techniques for the network that enable us to extend the formulation of the never-meet property from the uncapacitated delay management problem to the capacitated case. We then use our results to derive exact and heuristic solution procedures for solving the delay management problem. The results of the algorithms are evaluated both from a theoretical and a numerical point of view. The latter has been done within a case study using the railway network in the region of Harz, Germany.

A Model and Algorithm for the Courier Delivery Problem with Uncertainty

Abstract: We consider the courier delivery problem (CDP), a variant of the vehicle routing problem with time windows (VRPTW) in which customers appear probabilistically and their service times are uncertain. We use scenario-based stochastic programming with recourse to model the uncertainty in customers and robust optimization for the uncertainty in service times. Our proposed model generates a master plan and daily schedules by maximizing the coverage of customers and the similarity of routes in each scenario, while minimizing the total time spent by the couriers and the total earliness and lateness penalty. To solve large-scale problem instances, we develop an insertion-based solution heuristic, called master and daily scheduler (MADS), and a tabu search improvement procedure. The computational results show that our heuristic improves the similarity of routes and the lateness penalty at the expense of increased total time spent when compared to a solution of independently scheduling each day. Our experimental results also show improvements over current industry practice on two real-world data sets.

A Dynamic Programming Heuristic for the Vehicle Routing Problem with Time Windows and European Community Social Legislation

Abstract: In practice, apart from the problem of vehicle routing, schedulers also face the problem of finding feasible driver schedules complying with complex restrictions on drivers' driving and working hours. To address this complex interdependent problem of vehicle routing and break scheduling, we propose a restricted dynamic programming heuristic for the vehicle routing problem with time windows and the full European social legislation on drivers' driving and working hours. The problem we consider includes all rules in this legislation, whereas in the literature only a basic set of rules has been addressed. In addition to this basic set of rules, the legislation contains a set of modifications that allow for more flexibility. To include the legislation in the restricted dynamic programming heuristic, we propose a break scheduling heuristic. Computational results show that our method finds solutions to benchmark instances---which only consider the basic set of rules---with 18% fewer vehicles and 5% less travel distance than state-of-the-art approaches. Moreover, our results are obtained with significantly less computational effort. Furthermore, the results show that including a set of rules on drivers' working hours---which has been generally ignored in the literature---has a significant impact on the resulting vehicle schedules: 3.9% more vehicle routes and 1.0% more travel distances are needed. Finally, using the modified rules of the legislation leads to an additional reduction of 4% in the number of vehicles and of 1.5% in travel distances. Therefore, the modified rules should be exploited in practice.

An Adaptive Large Neighbourhood Search Heuristic for the Capacitated Arc-Routing Problem with Stochastic Demands

Abstract: The capacitated arc-routing problem with stochastic demands (CARPSD) is an extension of the well-known capacitated arc-routing problem (CARP) in which demands are stochastic. This leads to the possibility of route failures whenever the realized demand exceeds the vehicle capacity. This paper presents the CARPSD in the context of garbage collection. It describes an adaptive large-scale neighbourhood search heuristic for the problem. Computational results show the superiority of this algorithm over an alternative solution approach.

A Time-Space Network Approach for the Integrated Vehicle-and Crew-Scheduling Problem with Multiple Depots

Abstract: This paper discusses the integrated vehicle-and crew-scheduling problem in public transit with multiple depots. It is well known that the integration of both planning steps discloses additional flexibility that can lead to gains in efficiency, compared to sequential planning. We present a new modeling approach that is based on a time-space network representation of the underlying vehicle-scheduling problem. The integrated problem is solved with column generation in combination with Lagrangian relaxation. The column generation subproblem is modeled as a resource-constrained shortest-path problem based on a novel time-space network formulation. Feasible solutions are generated by a heuristic branch-and-price method that involves fixing service trips to depots. Numerical results show that our approach outperforms other methods from the literature for well-known test problems.

Dynamic Revenue Management in Airline Alliances

Abstract: Major airlines are selling increasing numbers of interline itineraries in which flights operated by two or more airlines are combined and sold together. One reason for this increase is the rapid growth of airline alliances, which promote the purchase of interline itineraries and, therefore, virtually extend the reach of each alliance member's network. This practice, however, creates a difficult coordination problem: Each member of the alliance makes revenue management decisions to maximize its own revenue and the resulting behavior may produce suboptimal revenue for the alliance as a whole. Airline industry researchers and consultants have proposed a variety of static and dynamic mechanisms to control revenue management decisions across alliances (a dynamic mechanism adjusts its parameters as the number of available seats in the network changes and time passes). In this paper, we formulate a Markov game model of a two-partner alliance that can be used to analyze the effects of these mechanisms on each partner's behavior. We begin by showing that no Markovian transfer pricing mechanism can coordinate an arbitrary alliance. Next, we examine three dynamic schemes as well as three forms of the static scheme widely used in practice. We derive the equilibrium acceptance policies under each scheme and use analytical techniques as well as numerical analyses of sample alliances to generate fundamental insights about partner behavior under each scheme. The analysis and numerical examples also illustrate how certain transfer price schemes are likely to perform in networks with particular characteristics.

Column Generation with Dynamic Duty Selection for Railway Crew Rescheduling

Abstract: The Dutch railway network experiences about three large disruptions per day on average. In this paper, we present an algorithm to reschedule the crews when such a disruption occurs. The algorithm is based on column generation techniques combined with Lagrangian heuristics. Since the number of duties is very large in practical instances, we first define a core problem of tractable size. If some tasks remain uncovered in the solution of the core problem, we perform a neighborhood exploration to improve the solution. Computational experiments with real-life instances show that our method is capable of producing good solutions within a couple of minutes of computation time.

An Information-Theoretic Sensor Location Model for Traffic Origin-Destination Demand Estimation Applications

Abstract: To design a transportation sensor network, the decision maker needs to determine what sensor investments should be made, as well as when, how, where, and with what technologies. This paper focuses on locating a limited set of traffic counting stations and automatic vehicle identification (AVI) readers in a network, so as to maximize the expected information gain for the subsequent origin-destination (OD) demand estimation problem. The proposed sensor design model explicitly takes into account several important error sources in traffic OD demand estimation, such as the uncertainty in historical demand information, sensor measurement errors, as well as approximation errors associated with link proportions. Based on a mean square measure, this paper derives analytical formulations to describe estimation variance propagation for a set of linear measurement equations. A scenario-based (SB) stochastic optimization procedure and a beam search algorithm are developed to find suboptimal point and point-to-point sensor locations subject to budget constraints. This paper also provides a number of illustrative examples to demonstrate the effectiveness of the proposed methodology.

Sequencing Two Cooperating Automated Stacking Cranes in a Container Terminal

Abstract: The containerized trade market is growing rapidly with the uprising of the Far East. Container ports worldwide should be responsive by developing tools to handle these massive volumes of containers in order to retain their level of competitiveness. One of the areas in a container terminal that is highly affected by the increase in the demand of containers is the stack. The stack is used to temporarily store containers upon further transport to their destination. This study is concerned with scheduling sea and landside storages and retrievals in a stack with two cooperating automated stacking cranes working in a single block. We present a mathematical model to minimize the makespan for both cranes. Both an algorithm to derive a lower bound for the makespan and a simulated-annealing based heuristic are proposed to efficiently solve the problem. Numerical experiments show that the solutions of the heuristic method are within 2% of the lower bound for large instances.

The Vehicle Routing Problem with Stochastic Demand and Duration Constraints

Abstract: Time considerations have been largely ignored in the study of vehicle routing problems with stochastic demands, even though they are crucial in practice. We show that tour duration limits can effectively and efficiently be incorporated in solution approaches that build fixed, or a priori, tours for such problems. We do so by assuming that each tour must be duration feasible for all demand realizations, and determine the maximum duration of a given delivery tour by solving the optimization problem of an adversary. A computational study demonstrates the approach, and shows that enforcing tour duration limits impacts the structure of nearly-best solutions and may create the need for additional tours. However, for the instances considered, the price paid for robustness is small as the increase in total expected tour duration is modest.

Ground Delay Program Planning Under Uncertainty Based on the Ration-by-Distance Principle

Abstract: This paper presents ration-by-distance (RBD), a new allocation method to be used in planning ground delay programs (GDPs) for traffic flow management. It is shown that RBD minimizes total expected delay, under certain assumptions related to the manner in which GDPs are dynamically controlled. On the other hand, RBD taken to the extreme has poor characteristics with respect to the equity of the allocation it produces. To address this issue, we propose a constrained version of RBD as a practical alternative to allocation procedures used in operations today. It is shown that this algorithm has superior overall performance in terms of efficiency and equity relative to existing procedures.

Risk-Based Policies for Airport Security Checkpoint Screening

Abstract: Passenger screening is an important component of aviation security that incorporates real-time passenger screening strategies designed to maximize effectiveness in identifying potential terrorist attacks. This paper identifies a methodology that can be used to sequentially and optimally assign passengers to aviation security resources. An automated prescreening system determines passengers' perceived risk levels, which become known as passengers check in. The levels are available for determining security class assignments sequentially as passengers enter security screening. A passenger is then assigned to one of several available security classes, each of which corresponds to a particular set of screening devices. The objective is to use the passengers' perceived risk levels to determine the optimal policy for passenger screening assignments that maximize the expected total security, subject to capacity and assignment constraints. The sequential passenger assignment problem is formulated as a Markov decision process, and an optimal policy is found using dynamic programming. The general result from the sequential stochastic assignment problem is adapted to provide a heuristic for assigning passengers to security classes in real time. A condition is provided under which this heuristic yields the optimal policy. The model is illustrated with an example that incorporates data extracted from the Official Airline Guide (Official Airline Guide. 1998. OAG Business Travel Planner: North American Edition. Official Airline Guides, Bedfordshire, UK).

A Hybrid Monte Carlo Local Branching Algorithm for the Single Vehicle Routing Problem with Stochastic Demands

Abstract: We present a new algorithm that uses both local branching and Monte Carlo sampling in a multidescent search strategy for solving 0-1 integer stochastic programming problems. This procedure is applied to the single-vehicle routing problem with stochastic demands. Computational results show the effectiveness of this new approach to solving hard instances of the problem. This paper was accepted by former Editor-in-Chief Hani Mahmassani.

Truck Driver Scheduling in the European Union

Abstract: Since April 2007 working hours of truck drivers in the European Union are controlled by regulation (EC) No. 561/2006. According to the new regulation, road transport undertakings must organise the work of drivers in a way that drivers are able to comply with the regulations and can be made liable for infringements committed by the drivers. Although of particular importance in long-distance haulage, regulations on working hours of truck drivers have received very little attention in the scheduling literature. This paper presents a method for scheduling driving and working hours of truck drivers with respect to regulation (EC) No. 561/2006. Given a sequence of locations to be visited within specified time windows, the approach is guaranteed to find a schedule complying with the regulation if such a schedule exists.

European Driver Rules in Vehicle Routing with Time Windows

Abstract: As of April 2007, the European Union has new regulations concerning driver working hours. These rules force the placement of breaks and rests into vehicle routes when consecutive driving or working time exceeds certain limits. This paper proposes a large neighborhood search method for the vehicle routing problem with time windows and driver regulations. In this method, neighborhoods are explored using a column generation heuristic that relies on a tabu search algorithm for generating new columns (routes). Checking route feasibility after inserting a customer into a route in the tabu search algorithm is not an easy task. To do so, we model all feasibility rules as resource constraints, develop a label-setting algorithm to perform this check, and show how it can be used efficiently to validate multiple customer insertions into a given existing route. We test the overall solution method on modified Solomon instances and report computational results that clearly show the efficiency of our method compared to two other existing heuristics.

Wardrop Equilibria with Risk-Averse Users

Abstract: Network games can be used to model competitive situations in which agents select routes to minimize their cost. Common applications include traffic, telecommunication, and distribution networks. Although traditional network models have assumed that realized costs only depend on congestion, in most applications they also have an uncertain component. We extend the traffic assignment problem first proposed by Wardrop in 1952 by adding random deviations, which are independent of the flow, to the cost functions that model congestion in each arc. We map these uncertainties into a Wardrop equilibrium model with nonadditive path costs. The cost on a path is given by the sum of the congestion on its arcs plus a constant safety margin that represents the agents' risk aversion. First, we prove that an equilibrium for this game always exists and is essentially unique. Then, we introduce three specific equilibrium models that fall within this framework: the percentile equilibrium where agents select paths that minimize a specified percentile of the uncertain cost; the added-variability equilibrium where agents add a multiple of the variability of the cost of each arc to the expected cost; and the robust equilibrium where agents select paths by solving a robust optimization problem that imposes a limit on the number of arcs that can deviate from the mean. The percentile equilibrium is difficult to compute because minimizing a percentile among all paths is computationally hard. Instead, the added-variability and robust Wardrop equilibria can be computed efficiently in practice: The former reduces to a standard Wardrop equilibrium problem and the latter is found using a column generation approach that repeatedly solves robust shortest path problems, which are polynomially solvable. Through computational experiments of some random and some realistic instances, we explore the benefits and trade-offs of the proposed solution concepts. We show that when agents are risk averse, both the robust and added-variability equilibria better approximate percentile equilibria than the classic Wardrop equilibrium.

Calibrating Steady-State Traffic Stream and Car-Following Models Using Loop Detector Data

Abstract: The research reported in this paper develops a heuristic automated tool (SPD_CAL) for calibrating steady-state traffic stream and car-following models using loop detector data. The performance of the automated procedure is then compared to off-the-shelf optimization software parameter estimates, including the MINOS and Branch and Reduce Optimization Navigator (BARON) solvers. The model structure and optimization procedure is shown to fit data from different roadway types and traffic regimes (uncongested and congested conditions) with a high quality of fit (within 1% of the optimum objective function). Furthermore, the selected functional form is consistent with multiregime models, without the need to deal with the complexities associated with the selection of regime breakpoints. The heuristic SPD_CAL solver, which is available for free, is demonstrated to perform better than the MINOS and BARON solvers in terms of execution time (at least 10 times faster), computational efficiency (better match to field data), and algorithm robustness (always produces a valid and reasonable solution).

The Stochastic Multiperiod Location Transportation Problem

Abstract: This paper studies a stochastic multiperiod location-transportation problem (SMLTP) characterized by multiple transportation options, multiple demand periods, and a stochastic demand. We consider the determination of the number and location of the depots required to satisfy customer demand as well as the mission of these depots in terms of the subset of customers they must supply. The problem is formulated as a stochastic program with recourse, and a hierarchical heuristic solution approach is proposed. It incorporates a tabu search procedure, an approximate route length formula, and a modified procedure of Clarke and Wright (Clarke, G., J. W. Wright. 1964. Scheduling of vehicles from a central depot to a number of delivery points. Oper. Res.12 568--581). Three neighbourhood exploration strategies are proposed and compared with extensive experiments based on realistic problems.

A Real Options Approach for the Valuation of Highway Concessions

Abstract: The theory of real options offers an approach for the valuation of investments in real assets, based on the methodology developed for financial options. This approach is especially appropriate in the context of strategic decision making under conditions of uncertainty. This is the case for the valuation of highway concessions, where real options arise from certain clauses of the contracts, for example, a minimum traffic guarantee. The possible exercise of these kinds of rights means an added value for the project that cannot be easily captured using traditional procedures for investment valuation. In this paper, we develop a methodology to value these clauses of highway concessions and, for that purpose, we consider the traffic on the highway as the underlying asset in an option contract, taking into account that, when a nonfinancial asset is used, some adjustments have to be made to the options approach. This methodology is applied to the case of an already operating highway concession, using real data, and the authors obtain an estimate of the value of a minimum traffic guarantee, which depends on several parameters that are analyzed. The authors conclude that this methodology is an appropriate tool for the valuation of highway concessions that have operational flexibility.

An Efficient Greedy Heuristic for Warehouse-Retailer Network Design Optimization

Abstract: In this paper, we study a warehouse-retailer network design (WRND) model that simultaneously makes the location, distribution, and warehouse-retailer echelon inventory replenishment decisions. Although a column generation algorithm was proposed recently in the literature, it remains a challenge to solve large size instances of this NP-hard problem efficiently and effectively. The purpose of this paper is to present a greedy algorithm. Computational results demonstrate that our greedy algorithm can solve large-scale WRND problems efficiently with errors within 3%--4% on average.

Multiship Crane Sequencing with Yard Congestion Constraints

Abstract: Crane sequencing in container terminals determines the order of ship discharging and loading jobs that quay cranes (QCs) perform, so that the duration of a vessel's stay is minimized. The ship's load profile, berthing time, number of available bays, and QCs are considered. More important, clearance and yard congestion constraints need to be included, which, respectively, ensure that a minimum distance between adjacent QCs is observed and yard storage blocks are not overly accessed at any point in time. In sequencing for a single ship, a mixed-integer programming (MIP) model is proposed, and a heuristic approach based on the model is developed that produces good solutions. The model is then reformulated as a generalized set covering problem and solved exactly by branch and price (B&P). For multiship sequencing, the yard congestion constraints are relaxed in the spirit of Lagrangian relaxation, so that the problem decomposes by vessel into smaller subproblems solved by B&P. An efficient primal heuristic is also designed. Computational experiments reveal that large-scale problems can be solved in a reasonable computational time.

The Traveling Salesman Problem with Pickups, Deliveries, and Handling Costs

Abstract: This paper introduces a new variant of the one-to-many-to-one single vehicle pickup and delivery problems (SVPDP) that incorporates the handling cost incurred when rearranging the load at the customer locations. The simultaneous optimization of routing and handling costs is difficult, and the resulting loading patterns are hard to implement in practice. However, this option makes economical sense in contexts where the routing cost dominates the handling cost. We have proposed some simplified policies applicable to such contexts. The first is a two-phase heuristic in which the tour having minimum routing cost is initially determined by optimally solving an SVPDP, and the optimal handling policy is then determined for that tour. In addition, branch-and-cut algorithms based on integer linear programming formulations are proposed, in which routing and handling decisions are simultaneously optimized, but the handling decisions are restricted to three simplified policies. The formulations are strengthened by means of problem specific valid inequalities. The proposed methods have been extensively tested on instances involving up to 25 customers and hundreds of items. Our results show the impact of the handling aspect on the customer sequencing and indicate that the simplified handling policies favorably compare with the optimal one.

Estimating Nonparametric Random Utility Models with an Application to the Value of Time in Heterogeneous Populations

Abstract: The estimation of random parameters by means of mixed logit models is now current practice for the analysis of transportation behaviour. One of the most straightforward applications is the derivation of willingness-to-pay distribution over a heterogeneous population, an important element for dynamic tolling strategies on congested networks. In numerous practical cases, the underlying discrete choice models involve parametric distributions that are a priori specified and whose parameters are estimated. This approach can however lead to many problems for realistic interpretation, such as negative value of time, etc. In this paper, we propose to capture the randomness present in the model by using a new nonparametric estimation method, based on the approximation of inverse cumulative distribution functions. This technique is applied to simulated data, and the ability to recover both parametric and nonparametric random vectors is tested. The nonparametric mixed logit model is also used on real data derived from a stated preference survey conducted in the region of Brussels (Belgium). The model presents multiple choices and is estimated on repeated observations. The obtained results provide a more realistic interpretation of the observed behaviours.

Computing Time-Dependent Bid Prices in Network Revenue Management Problems

Abstract: We propose a new method to compute bid prices in network revenue management problems. The novel aspect of our method is that it naturally provides dynamic bid prices that depend on how much time is left until departure. We show that our method provides an upper bound on the optimal total expected revenue and that this upper bound is tighter than the one provided by the widely known deterministic linear programming approach. Furthermore, it is possible to use the bid prices computed by our method as a starting point in a dynamic programming decomposition-like idea to decompose the network revenue management problem by the flight legs and to obtain dynamic and capacity-dependent bid prices. Our computational experiments indicate that the proposed method improves on many standard benchmarks.

Split-Delivery Capacitated Arc-Routing Problem: Lower Bound and Metaheuristic

Abstract: This paper proposes lower and upper bounds for the split-delivery capacitated arc-routing problem (SDCARP), a variant of the capacitated arc-routing problem in which an edge can be serviced by several vehicles. Recent papers on related problems in node routing have shown that this policy can bring significant savings. It is also more realistic in applications such as urban refuse collection, where a vehicle can become full in the middle of a street segment. This work presents the first lower bound for the SDCARP, computed with a cutting plane algorithm and an evolutionary local search reinforced by a multistart procedure and a variable neighborhood descent. Tests on 126 instances show that the new metaheuristic outperforms on average a published memetic algorithm; achieves small deviations to the lower bound; and finds 44 optima, including 10 new ones.

Equilibrium Results for Dynamic Congestion Games

Abstract: Consider the following game. Given a network with a continuum of users at some origins, suppose users wish to reach specific destinations but they are not indifferent to the cost to reach them. They may have multiple possible routes but their choices modify the travel costs on the network. Hence, each user faces the following problem: Given a pattern of travel costs for the different possible routes that reach the destination, find a path of minimal cost. This kind of game belongs to the class of congestion games. In the traditional static approach, travel times are assumed constant during the period of the game. In this paper, we consider the so-called dynamic case where the time-varying nature of traffic conditions is explicitly taken into account. In transportation science, the question of whether there is an equilibrium and how to compute it for such a model is referred to as the dynamic user equilibrium problem. Until now, there was no general model for this problem. Our paper attempts to resolve this issue. We define a new class of games, dynamic congestion games, which capture this time-dependency aspect. Moreover, we prove that under some natural assumptions there is a Nash equilibrium. When we apply this result to the dynamic user equilibrium problem, we get most of the previous known results.

Cross-National Differences in Aviation Safety Records

Abstract: Data about the mortality risk of scheduled passenger air travel over 2000--2007 around the world is examined in this paper. Worldwide, the average passenger death risk per scheduled flight over 2000--2007 was about one in 3.0 million. However, much as the center of mass of a doughnut is the center of the hole---where there is no mass---the worldwide average represents the actual risk level in few if any countries. The data support a three-population risk model across nations, in which the differences in death risk are not statistically significant within groups but are highly significant across groups. The safest nations are the traditional first-world countries (e.g., Canada, Japan), with a death risk per flight of about 1 in 14 million. Next safest are those developing-world nations that have either have recently attained first-world status (e.g., Singapore, South Korea) or are classified by experts as newly industrialized (e.g., Brazil, China) Their aggregrate death risk per flight was about 1 in 2 million. The least safe nations statistically are remaining developing-world countries, with a death risk per flight of about 1 in 800,000. In terms of relative risk, divergences within the developing world are modest compared to the overall difference between the first and developing worlds. The observed risk pattern might reflect a confluence of economic and cultural factors.