Table IV depicts the EC values for the 36 problem instances (i.e., six on each transportation corridor), when rail was the mode of transportation. As indicated earlier, three different types of train make-up options have been considered. For example, the first option entailed using 10 trains each of which carried 10 hazmat railcars to meet the demand, whereas the third scenario implied using a hazmat-unittrain. We next provide the details associated with determining the EC on the first track segment in *Corridor A* (i.e., italicized), which runs from Montreal to Albany, and note that other risk numbers could be determined similarly.

###### 4.2.1.1. Derailment probability for position i

Probability of train derailment, , on the first track segment in *Corridor A* (Table II) was determined using the technique outlined in Section 'Train Derailment' On the basis of the existing literature,[29] we assumed that approximately 25% of all derailments can be classified as train-mile caused, and the remaining as car-mile caused. We also know from Table II that the length of this segment is 240 miles, and that it belongs to track class 4. The latter lets us estimate *RC* and *RT* from Table I. Finally, we assume a 100-railcar train, and replace the various terms in Equation (5) with the respective parameters to get 0.000172 as the probability of train derailment on this segment.

Probability of point of derailment starting at position *i*, , is estimated using the best-fit distributions from Bagheri *et al*.,[17] and, as outlined in Section 'Point of Derailment', Fig. 4 depicts the resulting point of derailment probabilities for each of the 100 railcar positions for the track section between Montreal and Albany, and their exact values are indicated in Table AI. For example, the probability that point of derailment starts at the very first position is 0.0432, which together with the train derailment probability is fed into Equation (4) to determine the probability of derailment of the first slot in the train, that is, .

It is clear from Fig. 4 (and Table AI) that the third quarter of the train has the lowest derailment probabilities, and hence may be the safest position to place the hazmat railcars. Note that this observation is consistent with another work in the area,[12] and hence we choose this train-quarter to place the hazmat railcars under the first and the second make-up plans. More specifically, for a 100-railcar train, hazmat railcars would be placed starting at the 51st slot and until the 75th under the second plan, and until the 60th under the first. This implies that for the problem instance being analyzed, all the 25 hazmat railcars are placed in the third quarter of the train, while the other slots contain regular freight.

###### 4.2.1.2. Consequence for position i

Recall that consequence associated with any position in a train would require information on conditional probability of derailment of hazmat railcars, conditional probability of derailed hazmat railcars releasing, and the population exposure associated with the number of railcars releasing. We next outline the computation of each term for the problem instance being analyzed. For expositional reasons, we focus on the computation for the *first* slot in the train, and note that values for other slots can be determined similarly.

Probability that a given number of railcars will derail given that the derailment started at the first position (i.e., ) was estimated using the technique proposed in Saccomanno and El-Hage,[30] and outlined in Section 'Number of Railcars Derailed' Fig. 5 depicts the respective probabilities for a given number of railcars derailing when derailment starts at the first position of the train, and the values range from a high of 0.0292 to a low of 0.0017 (Table AII). Note that there is no hazmat risk if less than 51 railcars derail—since hazmat railcars are in the 51st through the 75th slots. On the other hand, hazmat risk will accrue if the number of railcars derailed exceeds 50, and that all the hazmat railcars will pose a risk if . Furthermore, and as highlighted in Table AII, the probability that the first hazmat railcar will derail is 0.0071—which is the probability that *m = 51*—whereas the probability of all hazmat railcars derailing is 0.0036 or less.

As indicated in Section 'Number of Derailed Railcars Releasing', we intend to take advantage of the FRA database containing the rail accident records from 1997 to 2006. Hence, we assume a conditional release probability of 0.0903 given that a hazmat railcar has derailed. Furthermore, given that the conditional release probability from each hazmat railcar is independent of each other, the probability that *j = 2* is 0.008649 (i.e., 0.0903 × 0.0903), and so on for other values of *j*.

Population exposure was estimated using the technique outlined in Section 'Population Exposure' To that end, two pieces of information were required, that is, the number of hazmat railcars releasing, and the quantity of hazmat released from each railcar. The first issue was addressed by building scenarios where the number of railcars releasing ranged from the minimum to the maximum possible value for the given setting. For example, for the problem instance being analyzed, *j* will range from 1 to 25—since each derailed hazmat may or may not release. The second element was estimated based on the existing literature that specified loss of entire lading within 10 minutes, which was less than the anticipated response time for emergency service providers.[12] Given the response time, this assumption is reasonable since it is not possible to contain (or stop) hazmat release from ruptured containers in less than 10 minutes. For each value *j* can assume, the atmospheric parameters—likely to result in the worst case scenario—were fed into ALOHA to generate the release rate from the ruptured hazmat containers, followed by the substitution of resulting values into Equation (8) for aggregate concentration level estimation. The IDLH level for chlorine (i.e., 10 ppm), as specified by the Centers for Disease Control and Prevention,[32] was used to estimate the threat zone (or evacuation distance) using Equation (9)—which in turn was used in ArcView GIS to generate exposure bands around the transport segment. Finally, avenue programming in ArcView GIS was used to estimate the number of people inside the band, which is the value of population exposure for the given number of hazmat railcars releasing. For example, 50,801 individuals were exposed along the track segment from Montreal to Albany if one railcar containing chlorine released, while 84,278 if two railcars released. Table AIII reports the conditional release probabilities and population exposure numbers for all possible values *j* can assume on this track segment.

As indicated earlier, and given that the derailment starts at the first train slot, at least 51 railcars have to derail to pose any hazmat risk. For this setting, Table AIII lists the conditional probabilities of , conditional probabilities that , population exposure, and the consequence associated with each *j*. For instance, the conditional probability that 51 railcars would derail is 0.0072; the conditional probability of release from the only derailed hazmat railcar is 0.0903, which in turn will expose 50, 801 individuals along this segment. Hence, = , and associated consequence is On the other hand, if 52 railcars derail then both hazmat railcars pose a risk—and hence one needs to take this additional situation into consideration. To that end, the conditional probability of release (i.e., 0.008649) and the associated population exposure from two hazmat railcars (84, 278) would also be used, which will yield a consequence value of Similarly, we can compute the consequence associated with each position in the train given that derailment started at the first train slot, and the resulting value is (Table AIII). As evident in Table AIII, the total consequence was determined by summing over possible values of *j*. Note that although the fourth train-quarter does not contain any hazmat railcars, transport risk would still accrue if the number of railcars derailed is more than 75—because such a derailment would involve all the hazmat railcars placed in the third train-quarter.

For the track segment between Montreal and Albany in *Corridor A*, the EC risk associated with the first train slot is . Similarly, EC risk for all other slots can be computed, and they are listed in Table AIV. For the second train make-up plan, the EC risk from a single train is 0.3716. Since four trains of this type are needed to meet the demand, the EC risk from the second train make-up plan is (i.e., boldfaced in Table IV). The proposed assessment approach was used similarly to estimate the equivalent for the second segment of *Corridor A* (i.e., from Albany to New York), and also the other values in Table IV.

###### 4.2.1.3. Other risk measures

In the interest of providing general conclusions, we also report results using two other most popular measures of hazmat transport risk. Table V depicts the solutions for the 36 problem instances when *population exposure* was the measure of risk, wherein the assessment methodology as proposed in Verma and Verter[18] was followed. For instance, around 34.8 million people would be exposed in *Corridor A*, if the second train make-up plan would be used to meet the demand for chlorine. It is important to note that, for any given corridor, population exposure risk is the highest under the first train make-up plan and the lowest under the third. This is resulting from the nonlinearity of the aggregate concentration curve (i.e., Equations (8) and (9)), and favors sending fewer but larger over more frequent but smaller hazmat loads. On the other hand, results using the *incident probability* approach as proposed in Bagheri *et al*.[17] are reported in Table VI, which for the corridor in question was 1.15 × 10^{−4}. Since consequence is being captured using a simple binary variable, there is a single value for each train make-up plan for both chlorine and ammonia. Furthermore, the incident probability risk for a single train is the lowest for the first train make-up plan, since fewer number of hazmat railcars in a train imply fewer sources of risk—and there are no nonlinear curves to be exploited.

Table V. Population Exposure for Rail | Train Type |
---|

| Chlorine | Ammonia |
---|

Corridors | 1st | 2nd | 3rd | 1st | 2nd | 3rd |
---|

*A* | 45,701,830 | 34,838,452 | 22,713,949 | 9,270,148 | 7,066,623 | 4,607,292 |

*B* | 8,781,361 | 6,694,021 | 4,364,363 | 1,781,209 | 1,357,814 | 885,266 |

*C* | 11,687,808 | 8,909,602 | 5,808,876 | 2,370,752 | 1,807,222 | 1,178,271 |

*D* | 8,291,260 | 6,320,418 | 4,120,782 | 1,681,797 | 1,282,032 | 835,858 |

*E* | 12,790,033 | 9,749,828 | 6,356,686 | 2,594,327 | 1,977,653 | 1,289,389 |

*F* | 3,260,237 | 2,485,275 | 1,620,348 | 661,306 | 504,113 | 328,671 |

Table VI. Incident Probability for Rail | Train Type |
---|

Corridors | 1st | 2nd | 3rd |
---|

*A* | 0.46 × 10^{−4} | 1.15 × 10^{−4} | 3.95 × 10^{−4} |

*B* | 0.53 × 10^{−4} | 1.31 × 10^{−4} | 4.44 × 10^{−4} |

*C* | 0.61 × 10^{−4} | 1.51 × 10^{−4} | 5.15 × 10^{−4} |

*D* | 0.85 × 10^{−4} | 2.11 × 10^{−4} | 7.40 × 10^{−4} |

*E* | 1.13 × 10^{−4} | 2.80 × 10^{−4} | 9.49 × 10^{−4} |

*F* | 0.33 × 10^{−4} | 0.81 × 10^{−4} | 2.87 × 10^{−4} |