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Integrated fleet mix and routing decision for hazmat transportation: A developing country perspective
Accepted Manuscript
Integrated Fleet Mix and Routing Decision for Hazmat Transportation: A Developing Country Perspective Anand Kumar, Debjit Roy, Vedat Verter, Dheeraj Sharma PII: DOI: Reference:
S0377-2217(17)30530-1 10.1016/j.ejor.2017.06.012 EOR 14494
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
8 June 2016 5 June 2017 6 June 2017
Please cite this article as: Anand Kumar, Debjit Roy, Vedat Verter, Dheeraj Sharma, Integrated Fleet Mix and Routing Decision for Hazmat Transportation: A Developing Country Perspective, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.012
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Integrated Fleet Mix and Routing Decision for Hazmat Transportation: A Developing Country Perspective Anand Kumar Indian Institute of Technology Kharagpur, Debjit Roy
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West Bengal, India Indian Institute of Management Ahmedabad, Gujarat 380015, India, and
Rotterdam School of Management, Erasmus University, 3062PA Rotterdam, The Netherlands
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Vedat Verter
Desautels Faculty of Management-McGill University, Montreal QC, Canada Dheeraj Sharma
Indian Institute of Management Ahmedabad, Gujarat 380015, India and
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Indian Institute of Management Rohtak,
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Abstract
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Haryana 124001, India
In developing countries, truck purchase cost is the dominant criteria for fleet acquisition-
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related decisions. However, we contend that other cost factors such as loss due to the number of en route truck stoppages based on a truck type and recovery cost associated with a route choice decision, should also be considered for deciding the fleet mix and minimizing
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the overall costs for long-haul shipments. The resulting non-linear model, with integer variables for the number and type of trucks, and the route choices, is solved via genetic algorithm. Using real data from a bulk liquid hazmat transporter, the trade-offs between 1
Corresponding Author, Email: [email protected], Phone: +91-7966324823, Fax: +91-7966326896
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the cost of travel, loss due to number of truck stoppages, and the long-term recovery cost associated with the risk of exposure due to a hazmat carrier accident are discussed. The numerical experiments show that when factors related to public safety and truck stoppages are taken into account for transportation, the lowest total cost and optimal route choice do not align with the cheapest truck type option; rather, the optimal solution corresponds to
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another truck type and route with total costs significantly less than the total costs associated with the cheapest truck type. Our model challenges the current truck purchasing strategy adopted in developing countries using the cheapest truck criteria.
Keywords
Logistics, OR in Societal Problem Analysis, Routing, Risk Analysis
Introduction
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”Over 3 billion tons of hazardous materials (HAZMAT) are transported by commercial carriers in the United States each year. Such hazmat shipments can be highly risky and highly sensitive and if improperly handled, labeled, or packaged could result in the loss of life, property damage, and harm to national security interests,” United States Government
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Accountability Office (2014). The most common hazardous materials (hazmats) for transportation include cooking gas, fuel oil, and chemicals such as ethyl alcohol, and gasoline
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(Kara and Verter (2004)). Due to flammable, corrosive, explosive, poisonous or radioactive nature of dangerous goods, the carriers of hazmats, when involved in a road accident, may lead to disastrous consequences such as fire, explosion, spillage, and leakage, resulting in
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a large number of fatalities and injuries besides property loss and environmental pollution (Yang et al. (2010)).
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Road network, the most used transportation network in developed countries, is highly regulated in terms of hazmat movement. The regulatory body controls the maximum amount of long-haul drive time (without stoppages), limit choice of routes, and capacity
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of truck for hazmat movement (Toumazis and Kwon (2013)). Such regulations include US Department of Transportation, HOS (Hours of Service) Regulations for long-haul and regional operations of transporters. It mandates a driver carrying any shipment to drive not more than 11 hours after 10 consecutive hours of off-duty. Further, drivers must take a mandatory 30 minutes break by their eighth hour of coming on duty. The 14-hour duty 2
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period may not be extended with off-duty time for breaks, meal and fuel stops. Limitation on the choice of routes also exists in the real world scenario. For example, Canadian city of Brandon in Manitoba province can serve as an illustration where a website gives a map of the city which shows the roads that can be used by the trucks carrying dangerous goods (www.brandon.ca). However, developing countries such as India, are yet to develop regulation or lax implementation of existing regulations.
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a holistic regulatory regime of hazmat transportation while facing issues of either lack of Hence, there is a potential threat to public safety in developing countries where lax regulations on movement of hazardous goods may allow transport companies to choose minimal distance routes, while completely ignoring vulnerable zones of high density areas leading to high transport risk. Transport risk, which is defined as a measure of expected number of
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people, who would suffer the consequence of a possible hazmat accident, receives minimal consideration during route choice decisions. Chakrabarti and Parikh (2013b) confirms that about 90% of hazmat incidents in India occur on highways and majority of them occur in close vicinity to rural areas. Factoring in the potential threats posed by transportation of hazards, the paper attempts to bring about an optimal balance between minimizing cost of transport and cost of recovery (remedies) associated with hazmat transportation.
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Among the more recent transporter challenges, driver-related issues have gained little attention (Zhou and Chen (2015)). Major challenges faced by the drivers (especially in developing countries) are: poor vehicle operating conditions, nonstandard working hours,
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no standard resting places between origin and destination points, and delays due to policy and regulatory needs (border/custom delays). To improve the vehicle operating condition, commercial vehicle manufacturers are designing trucks with ergonomic chairs and seats
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that have been found to increase productivity and comfort for the drivers (Rosecrance et al. (1993); De Looze et al. (1993); Hedge and Sakr (2005); Eikhout et al. (2005); Vink
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(2005)). In concurrence with research on ergonomic design, we contend that the choice of truck (brand) has an effect on the comfort of the driver’s seat, resting seat, and the number of truck stoppages per kilometer of travel (beyond the necessary stoppages such as
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mandatory rest-breaks). In long-haul transport, there could be potential cost advantages with trucks that include an ergonomic seat with an air-conditioned driver cabin against a cheaper truck with a non-ergonomic seat, and without an air-conditioned truck cabin. In such a scenario, the combined effect of cost advantages with less number of stops per kilometer (km) may out-weigh the difference in acquisition cost between the cheaper and 3
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the expensive truck types for long distance shipments. Hence, the decision with respect to the fleet mix (number and type of trucks) is affected by the relative position of the origin and destination nodes in the network. Several studies in literature determine the optimal fleet mix by integrating it with the vehicle routing problem (Golden et al. (1984); Salhi and Rand (1993); Desrochers and Ver-
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hoo (1991)). However, existing models that are mostly developed from a developed country perspective do not capture the nuances with respect to long-haul transport in developing countries. The differences are:
1. Large variety of the truck types (make and capacity): As discussed earlier, the transporters can select from a variety of truck brands (with different make, driver-cabin comfort,
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and capacity). All truck types in developed countries have a good basic operating condition for the drivers (such as air blower or air conditioner in the cabin). From a developing country’s perspective, this criteria becomes special because the basic amenities may be absent in low cost truck types. Hence, different amount of driver discomfort may cause the number of truck stoppages between two places to vary from one truck to another. 2. Population-based risk : In developing countries, the road network may have limited route
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choices for hazmat transport and the transporter is often compelled to use a road that passes through a dense neighborhood, which carries a high population risk. Further, the restriction over transportation of hazardous materials over highly populated road network
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is minimal or absent. Hence, population-based risk should be assessed for fleet mix and the route choice decision (Chakrabarti and Parikh (2013a), Chakrabarti and Parikh (2013b)). 3. Criteria for fleet acquisition: Truck purchase cost is the dominant factor considered
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during fleet acquisition. The authors also surveyed 50 fleet owners in India and asked them to identify the most important factor for a new truck purchase. The factors included
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after-sales service, fuel efficiency, total cost of ownership, spares availability, brand image, ongoing relation with brand, modern and new brand, durability and reliability, strong dealer support, purchase price, finance arrangement by brand, relation with truck supplier, driver
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recommendation and mechanic recommendation. Out of 50 fleet owners, 24 responded that purchase price is the most important factor for truck purchase decision. These issues have been studied separately in the prevailing literature. One of the pri-
mary contributions of this paper is to incorporate all three nuances in an integrated model. 4
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In addition, our approach in capturing the truck type-dependent stoppages and representing the population risk faced by the people residing along the highway segments as well as the population centers, is quite novel. Hence, the model built in this paper is more relevant and compelling from a developing country perspective.
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Trucking companies expect their hired drivers to ensure a minimum daily travel distance irrespective of the truck make. Since a poor truck make may lead to excess number of truck stoppages, the driver may lag behind his schedule. To make up the lost time, the driver takes shorter routes associated with higher risks (in a unrestricted road network). Hence, the three factors: truck type, truck’s load capacity and number of trucks used in a shipment, together referred to as ‘fleet mix’ decides the optimal route choice. We argue that studies
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that have ignored the losses suffered due to truck stoppage in deciding the ‘fleet mix’ may not be optimal and accounting for such losses presents a much more realistic scenario compared to the existing studies. In this research, we study the trade-offs between the truck purchase finance loan costs (trucks type in the shipment) and the costs associated with the routing decisions (i.e., the costs of transportation, loss due to truck stoppages, and costs associated with accident recovery). We also develop a population-based measure of risk to quantify recovery cost in the event of an accident. The hazmat transportation risk is incorporated in
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the objective function of the model as the expected accident recovery costs. The proposed risk measure not only captures the population at risk at the source and destination nodes,
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but also includes the ones along the travel route. Using this definition of risk, we attempt to answer the following research questions: a) If the transport carrier intends to purchase a fleet of trucks with different make (truck
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type), capacities, and load type, what should be the fleet size and type for every shipment? b) For a particular shipment, how should the transporter select the optimal route among a set of available routes between origin and destination nodes?
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The decision variables are the appropriate fleet of vehicles serving a particular shipment and the route index. The objective of this problem is to minimize the sum of travel cost of shipment over the network, the cost of transport risk, and the loss due to truck stoppages.
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We specifically examine the problem that involves the transportation of multiple types of dangerous goods (chemical goods) to be shipped over multiple origin-destination pairs. The resulting mathematical formulation is a non-linear optimization model with integer decision variables.
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The remainder of the paper is organized as follows. Section 2 reviews literature on modeling hazmat transportation problems. Section 3 describes model specification with discussion on the transportation network design problem and the model formulation to determine the optimal fleet mix. Section 4 discusses the solution methodology adopted to solve the non-linear optimization formulation. Section 5 first presents an illustrative
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example to demonstrate the main trade offs in the problem, and a real life case to highlight the complexities and the effectiveness of the proposed approach to tackle such problem instances. Section 6 compares the integrated solution approach with a lowest truck purchase cost and two-stage sequential solution approach. Section 7 presents our research conclusions and the scope for future work.
Literature Review
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Looking at the importance and seriousness of the issue, transportation of hazardous materials has been studied extensively in operation research in last twenty years. Transportation of hazardous materials has been studied in context of modeling for road network (Bianco et al. (2009); Erkut and Alp (2007); Erkut and Gzara (2008); Kara and Verter (2004);
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Zhang et al. (2000), emergency response network models (Berman et al. (2007)), multilocation routing methodology (Xie et al. (2012); Leonelli et al. (2000); Verter and Erkut (1997)), decision modeling for risk assessment (Clarke and Sacre (2009), Sun et al. (2016)),
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routing of hazmat trucks (Bell (2006); Akgün et al. (2007); Bell (2007)), factor affecting on accidents in hazmat transportation (Zhau et al. (2012)), cost related models (Glickman (2006)).
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et al. (2007); Revelle et al. (1991)) and in context of fuzzy graph modeling (Boulmakoul Previous studies in the field of hazmat risk concentrated upon the two specific problems
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namely assessment of transport risk for hazmat shipments (Erkut and Ingolfsson (2005); Erkut and Verter (1998)) and identification of route with minimal transport risk (Zografos and Androutsopoulos (2004), Reniers et al. (2010), Bronfman et al. (2015), Szeto et al.
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(2017)). Possible solution to reduce shipment risk can be the appropriate methodology in identification of route by keeping various costs into consideration (Leonelli et al. (2000); Erkut and Verter, 1997). Erkut and Ingolfsson (2000) defines risk in the hazardous materials transportation as “the most-widely used definition of literature is the expected consequence 6
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of an incident (accident resulting in a release), which, for each edge of the network, is equal to the product of the incident probability and a quantifiable consequence (such as number of people evacuated).” Verter and Erkut (1997) evaluated routing policies for a manageable number of alternate routes and concluded that the route selected for delivery of hazmats should be determined
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on the basis of combined increase in cost of transportation and increase in insurance costs due to a possible accident. Indirectly, the use of geographic information system (GIS) can also help to deal with risk related to hazmat transportation and also in network designing (Bianco et al. (2009); Zhang et al. (2000)). On the contrary, literature on hazmat suggest that the number and location of containers with hazardous cargo varied considerably among trains and consequently it is important to define a boundary of exposure zone as a function
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of the volume of hazmat being shipped (Verma and Verter (2007); Kara and Verter (2004)). In quantification of risk in hazmat transportation, Kara and Verter (2004) quantified risk as the total number of people exposed within a threshold distance from a given road link whereas Verma and Verter (2007) quantified risk as the population exposed from multiple release sources of hazmat (attached as railcar compartments) as a function of rupture diameter, volume, and nature of hazmat carried. The most appropriate measure of risk used in the literature is a measure of the population exposed to the route used for the
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shipment and also it has been considered as a better measure of transport risk assessment than occurrence of an incident (Batta and Chiu (1988)).
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Several studies considered the likelihood of encountering a hazmat accident during transportation as their risk measure (Abkowitz et al. (1992); Saccomanno and Chan (1985)). Erkut and Verter (1995) as well as Alp (1995) observed that the transport risk is a measure
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of the expected number of people who would suffer the consequence of a possible hazmat accident. Erkut and Alp (2007) proposed an integrated routing and scheduling model for
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hazmat transportation that minimizes the en route risk (accident probability multiplied by exposure) subject to a constraint on the total duration of the trip. Instead of rerouting hazmat movement to minimize transport risk, Kawprasert and Barkan (2008) presented an
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optimization model for rail route rationalization. Garrido (2008) developed a road-pricing scheme for safe transportation or disposal of hazmat. List and Mirchandani (1991) developed a detailed model for routing hazardous materials
and locating hazardous waste treatment facilities based on minimal total cost, total societal 7
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risk, and the maximum risk exposed to an individual. Kara and Verter (2004) presented a bilevel programming formulation that identified the minimum risk design for the road network and demonstrated that there is a significant chance of risk reduction in transportation of hazmats if the government intervenes in the decision of making road links for transportation of hazmats Among a number of possible routes between destinations it is possible that
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routes with shorter length are exposed to more transport risk (i.e., prone to accidents, heavily populated, high traffic density etc.) while other longer routes may possess larger cost along with an acceptable risk of transport (Marcotte et al. (2009); Akgün et al. (2007); Bell (2006)). Kang et al. (2014) developed a value-at-risk model to generate route choices for a hazmat shipment based on specific risk confidence level. Various route selections were made depending on the choice of confidence level. Iakovou et al. (2015) provided a multi-commodity network flow model for the problem of routing hazardous vessels. Instead
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of the traditional shortest path formulation model, they employed network flow model to capture shipments of multiple commodities over multiple origin-destination pairs. Fan et al. (2015) proposed a bi-objective programming model and a heuristic algorithm to optimize the routing of hazardous materials transportation with road closure considerations using a case study from China. Van Raemdonck et al. (2013) presented a comprehensive survey on risk analysis systems for the transport of hazardous materials and developed a methodology
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for assessing the local accident risk.
As discussed in this section, several attempts have been made towards study of optimal
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fleet and route mix of hazardous materials shipment over complex networks. The approaches of the authors vary in their consideration of design parameters such as hazmat variability, route preferences and availability, truck type variability (in terms of capacity, compatibility
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with hazmat shipment and cost of travel per kilometer). Table 1 presents a selection of representative papers that analyzes the fleet mix problem for hazmat transportation.
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We contribute to the literature in fleet mix decision making for hazmat transportation by presenting an integrated framework that considers the truck type, population-based risk,
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and truck stoppages. We describe the model in the next section.
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Table 1: Representative studies for fleet mix decision in hazmat transportation Travel Distance Minimization
Kara and Verter (2004) Verter and Kara (2008) Pooley (1993) Golden et al. (1984) Chakrabarti and Parikh (2013a);Chakrabarti and Parikh (2013b) This Paper
X X X
X X X
X
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Populationbased Risk X X
Truck Stoppages and Corresponding Fleet Acquisition
X
X X
X
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Large Variety of Truck Types (Make and Capacity)
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Citation
Model Description
The transporter needs to design the flow of shipments in the network in such a manner that not only travel cost is reduced but also the risk of transportation is minimized. In practice,
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the route corresponding to the minimum travel cost may not always be the route associated with the minimum risk. Hence, a model is formulated to select the shipment route as well as the fleet mix with minimum risk associated with transporting a shipment. The key features
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of our model lie in considering the effect of truck types on the monetary loss due to truck stoppages and the effect of route selection on the transport risk. We describe the modeling
Problem Background and Model Assumptions
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3.1
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premise, the model assumptions, and the cost parameters in this section.
The model prepared in this study deals with a single season demand of hazmats and it is assumed that the model is valid for a new carrier agency. It is also assumed that the
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agency does not own any trucks at the beginning of the planning period. In practice, the transport company either buys a truck or rents them from the market. The rental period is assumed to be a single planning period. Let us denote the existing road network topology using G = (N, A) where N is the set of nodes (vertices) and A is the set of links among the 9
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nodes (arcs). The following notation is used for the development of the model. I : Set of Nodes, indexed by i M : Set of hazmat types, indexed by m S : Set of shipments in order summary, indexed by s K : Set of truck types, indexed by k m(s)
to a shipment s with hazmat m is denoted by ρi
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The set of road links are denoted by (i, j). The number of people affected at node i due . Meanwhile, the number of people
affected in link (i, j) due to transport of shipment s carrying hazmat m is represented by m(s)
ρij
. Let us define the distance between node i and node j as lij , while the algebraic sum
of the lengths of individual links forms the total length of the route. The other modeling assumptions are listed below:
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• The model is developed for a single period demand.
• The edges of the network are assumed to be bi-directional. (i.e., lij = lji ) • All routes on the network are available for the shipment of all types of hazmats. • There is a single period truck procurement budget.
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• Each type of truck is designated to carry specific types of hazmats only. • There is no backlog of orders. All orders scheduled for a planning period will be
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shipped in the same period only.
• Trucks with same capacity are assumed to have same number of tires and associated
Modeling Monetary Loss due to Truck Stoppages
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3.2
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tire maintenance charges.
Truck drivers typically halt for taking rest after driving a stretch of a few hundred kilometers. We use the following approach to estimate the loss due to truck stoppages during
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transport. For every stop due to fatigue of a truck driver, we consider a fixed amount of essential expense incurred by the transporter. In this paper, we are particularly concerned with the time and monetary loss due to additional (non-essential) truck stoppages. Non-essential truck stoppages occur due to driver fatigue and cause monetary loss to the 10
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transporter. Based on interactions with truck drivers, we found that almost all drivers experience fatigue and stop more than once during their journey. The number of stoppages is affected by the type of truck used for transport. Different types of truck have different number of stoppages depending upon health issues of drivers such as fatigue. Therefore, we use the term called ‘loss due to truck stoppages’, which can be measured by the number of
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stoppages made per 1000 kilometers depending on a truck type. It is observed that trucks with fewer number of truck stoppages have better ergonomic seat design and can run at longer continuous stretches. The associated cost with every essential stop is fixed with respect to every truck type. The product of the number of stoppages and the associated cost provides the cost due to loss of truck stoppages along a route. It is important to note that the monetary loss due to truck stoppages is independent of the type of hazmat delivered
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by the truck and depends only on the type of truck.
It was also observed that trucks with driver-friendly design features (such as luxury driver seat with a larger cushion size, higher tolerance to mechanical vibrations during driving, and larger backrest adjustment angle) had higher acquisition cost but could run larger lengths without stopping. On the other hand, trucks with lower acquisition cost may have fewer driver-friendly design features (such as multi-purpose driver seat and lower
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tolerance to mechanical vibrations during driving) and may need the driver to make several
Modeling Population at Risk
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3.3
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essential stops before reaching the destination.
Since the routing decision and choice of fleet jointly affects the amount of risk posed by a particular shipment, our first goal is to assess the risk of accidents pertaining to all the
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routes available in the network between any connected pair of nodes. We assume that every route is susceptible to an accident; however, the level of severity would be dependent on the population density and hazmat carried along the route. Since a route on the network
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is composed of several interlinks, the total risk associated with a route will be dependent on the risk associated with the individual interlinks (Abkowitz et al. (2001); Chakrabarti and Parikh (2011)) For any origin destination pair (i, j), the estimate of the total population exposed to 11
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Figure 1: Illustration of Population-based risk measure
hazmat between the nodes i and j is defined using Equation 1 (illustrated in Figure 1). hρ + ρ i i j + ρij lij
(1)
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ρtij =
where ρtij is an estimate of the total population between nodes i and j. ρi and ρj represent the actual population of nodes i and j. ρij represents the estimate of population lying on the connecting link between nodes i and j.
Note that the hazmat accident can occur either at the nodes or in between the links connecting the nodes. Hence, in an event of an accident of a vehicle carrying a hazmat, the population residing around the place of accident will be at risk. Consider hazmat m is
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carried along a link consisting of two nodes i and j for a shipment s. In case an accident occurs in the link (i, j) during transportation where the nodes i and j are located in close
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proximity (i.e., shorter links), the population at both nodes would bear a larger risk of exposure. On the other hand, if the nodes are separated by a large distance, the risk of exposure to the population at the nodes is minimal. In case of a large interlink distance; m(s)
ρij
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a significant portion of the population at risk is the one residing along the link. To find for a link carrying shipment s , we multiply ρtij with a parameter severity coefficient
(γm(s) ) of the hazmat m, which results to the amount of people affected on link (i, j) due
m(s)
ρij
= ρtij γm(s)
(2)
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to any adverse incident during transportation of carried hazmat m.
We introduce a term, βij , which represents the probability of an accident in the link (i, j).
The severity parameter, γm(s) which denotes the fraction of the total exposed population actually affected by the hazmat accident, is less than or equal to 1. We account for the population risk from the transport firm’s perspective by including the cost of recovery 12
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component in the objective function.
3.4
Problem Formulation
We define a binary decision variable Xijs that attains a value of 1 if shipment s passes
Xijs =
(
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through the link (i, j). In short,
1 if a shipment s passes through link (i, j) 0 otherwise
(3)
Xijs is a decision variable for the model, which represents the decision about the selection of a particular link (i, j) for shipment s. Since throughout the network there are several
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possible routes available between the origin and destination, the choice of route not only affects the total travel cost but also the recovery cost. Moreover, the number of trucks (nsk ) of each truck type k chosen for the shipment s is also a decision variable, which takes only integer non-negative values. There exists a choice for the transporters to either carry multiple small trucks with same capacity or a single large truck with higher capacity. This choice is reflected by the trade-off between the purchase finance loan cost and the travel
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cost of the trucks.
Let Yijs be another parameter that defines the hazmat network over G = (N, A) and shows
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if the link (i, j) is feasible for transportation of shipment s (roads could be inaccessible due to vehicle type restrictions or topological reasons). By definition, =
(
1 if link (i,j) is accessible to shipment s 0 otherwise
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Yijs
)
(4)
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Let the total number of trucks used for the shipment s is N s where, Ns =
X
nsk
(5)
k∈K
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Here, nsk represents the number of trucks of type k selected for delivering shipment s. Let ds be the demand in metric tons of shipment s. Let em k be a binary parameter, which is
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defined as follows: em k =
(
1 if truck type k can carry hazmat m 0 otherwise
)
(6)
The term qk denotes the truck load capacity of truck type k. We denote the cost per
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kilometer (km) in the link for a truck of type k on link (i, j) as ckij . This cost ckij (unit variable cost of travel of truck type k on link (i, j)) includes fuel cost per kilometer for a truck type, standby cost of vehicle per kilometer, maintenance cost per kilometer, and employee wages. Apart from the travel cost per unit kilometer and the truck purchase finance loan (or lease) cost per time period, there are two additional costs associated with the total cost of shipment: 1) the monetary loss due to truck stoppages, which is the total amount
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incurred on the number of essential stops during the travel period, and 2) the recovery cost per person, which is the amount spent on the total recovery of a victim exposed to hazmat. The cost components are discussed below.
Cost of Travel
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3.4.1
The average cost of travel per kilometer for truck type k on link (i, j) is denoted by ckij . The travel distance (considering both onward and return journey) for delivering a shipment, is
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twice the length of route per trip. Since the number of trucks of type allotted to a shipment s is nsk , the cost of transportation is 2ckij lij nsk . This cost is incurred only over the link (i, j) selected for the transport of shipment s. Such trucks return empty to the origin since the
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trucks are specialized to carry hazmats only and in a realistic scenario, hazmat shipments flow uni-directionally from the refineries to the interior locations of the continent. The
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transporter specifically does not enjoy any advantage of using the truck during return journey for other purposes/shipments mainly due to the specialized nature of trucks as well as unidirectional flow of hazmat shipments. The tankers are built using Petroleum
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& Explosives Safety Organization (PESO) guidelines and are not allowed to carry other commodities. Hence the total cost of travel of shipment s during the planning period is P P expressed by k∈K (i,j)∈A 2ckij lij nsk Xijs .
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3.4.2
Purchase Finance Loan (Lease) Cost of Trucks
Let fk be the purchase finance loan (lease) cost for truck type k for the planning period.
3.4.3
Monetary Loss due to Truck Stoppages
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Since there are nsk trucks of type k in shipment s, the total fixed cost of all trucks of P shipment s in the planning period is given by k∈K fk nsk .
Monetary loss due to truck stoppages is incurred as a result of the total number of essential stops made during the onward and the return journey. Let us denote the number of stops per 1000 kilometers for truck type k as uk and the cost incurred due to one such stop for a truck type k is Bk . Since the actual distance travelled during the shipment delivery between
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every link (i, j) is 2lij and the number of trucks of type k used for shipment s is nsk , the total number of stops on a link (i, j) is 2uk lij Xijs nsk . We multiplied this cost with a binary decision variable Xijs to ensure that the cost is incurred only on the selected shipment route.
3.4.4
Cost of Recovery
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Hence, the expression for loss due to truck stoppages of shipment s in the planning period P P is given by k∈K (i,j)∈A 2uk lij Xijs nsk Bk .
m(s)
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When a vehicle is delivering shipment s along a link (i, j), the population denoted by ρij
carries a risk of hazmat exposure in the event of an accident. Very dangerous type of hazmat commodities such as chlorine pose a substantial transport risk and can expose a
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large number of people in the event of an accident. Such chemicals have a higher severity γm coefficient. We assume that any hazmat m transported by a shipment s poses an identical risk to the population irrespective of the quantity of hazmat it carries (also
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see Verma and Verter (2007) where they assume economies of risk associated with hazmat transportation in unit trains). For example, the evacuation distance upon a hazmat incident is defined in terms of hazmat type and not truck capacity (see, CANUTEC,
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https://www.tc.gc.ca/eng/canutec). Therefore, when multiple vehicles are used for delivering a single shipment, the risk introduced by individual vehicles is added to obtain the total transport risk. If we denote the number of trucks of type k used for shipment s by nsk ,
the cost of long term recovery per person exposed to hazmat m by ϕm(s) , and the probability 15
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of an accident on link (i, j) by βij , we can represent the total expected recovery cost of shipP P m(s) ment s in the planning period using the expression k∈K (i,j)∈A ϕm(s) ρij nsk Xijs βij γm(s) . The objective function is now expressed as a minimization of the sum of the four cost terms. Let the total cost of shipment s be defined by 2ckij lij nsk Xijs
+
k∈K (i,j)∈A
X
k∈K
fk nsk
+
!
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Z(s) = min
" X X
X X
2uk lij Xijs nsk Bk
k∈K (i,j)∈A
!# X X m(s) m(s) s s + ϕ ρij nk Xij βij γm(s) s.t.
X
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k∈K (i,j)∈A
m(s)
nsk ek
k∈K
qk ≥ ds
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Xijs ≤ Yijs ∀ s ∈ S, (i, j) ∈ A +1 if i = O(s) X X s s Xij − Xji = −1 if i = D(s) ∀i ∈ I , s ∈ S (i,j)∈A (j,i)∈A 0 otherwise Xijs ∈ 0, 1 ∀ s ∈ S, (i, j) ∈ A
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(7)
∀s∈S
(8)
(9) (10)
Equation (7) guarantees that there are enough trucks of type k with capacity qk allotted
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to fulfil the demand for a particular hazmat. Equation (8) ensures that non-permissible links are not chosen, while Equation (9) ensures the flow constraint of the route chosen for a particular shipment s. Moreover, the decision to choose a particular route for delivering
Solution Methodology
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4
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a shipment is modeled using a binary decision variable, refer Equation (10).
Amaldi et al. (2011) point out that any version of the hazmat transport network design problem where a subset of arcs are forbidden is NP-hard even when a single commodity 16
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has to be shipped. Palmer and Kershenbaum (1995) and Li and Bouchebaba (2000) have used genetic algorithm for the Optimal Communication Spanning Tree problem, a variant of hazmat network design problem. We resort to the Genetic Algorithm also because the proposed model is a nonlinear mixed integer program. The objective function involves multiplication of decision variables nsk and Xijs in three of the four cost terms. A neighbor-
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hood search meta-heuristic would require the computation of the value of such a complex objective function a large number of times. Thus, we avoided using neighborhood search algorithms in tackling our problem. Since there exist a small number of constraints, however, the primary handicap of genetic algorithms concerning the need to eventually repair solutions to ensure feasibility is much less of an issue here. Therefore, we opted for using the genetic algorithm as our meta-heuristics domain. Since our aim in this paper is to demonstrate the problem dynamics and study the main trade-offs, possible improvements the scope of this paper.
4.1
Chromosome Formation
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to the solution approach we outline below by extending it to a memetic approach is out of
The chromosome here has been defined as the string of decision variables corresponding to
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one particular shipment. The decision variables for truck type selection (ns1 , ns2 , ns3 , ..., nsk ), s , X s , ..., X s , X s , X s , ..., X s , ..., X s , X s , . . . , X s ) and the decision variables for route selection (X11 12 21 22 1I 2I I1 I2 II
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have been conjoined to form one string of length K + I 2 for a shipment. The length of the chromosome increases with second order exponential along with the computation time and memory requirement as the number of nodes increase. Hence one chromosome for
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shipment s can be expressed with length of chromosome as K + I 2 and g(s) is defined the
4.2
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cost associated with one solution.
Population Size
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The initial population size was taken for 200 chromosomes following the trend of setting the minimum population size to be greater than the number of decision variables. The initial population was chosen from a set of feasible values. A range of population sizes were tested for the algorithm and a population size of 500 provided to optimal results, both in 17
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the sample as well as the real example (see Sections 5.1 and 5.2).
4.3
Fitness Function
shall be a transformed one as per the following rule: f (s) =
(
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Since the objective function is a cost minimization function the fitness function chosen here
Cmax − g(s) if g(s) < Cmax 0
otherwise
)
(13)
where g(s) is the cost function as shown in Section 3.4 as an aggregation of four other cost
4.4
Crossover and Mutation
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functions. Cmax is chosen as the maximum value of g(s) observed in the current population.
Crossover convergence operation intends to pull the population towards a local minimum/maximum, while mutation is a variance operation. Mutation is intended to occasionally pull out one or more chromosomes of a population from a local minimum/maximum
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space and potentially discover a better minimum/maximum space.
Since the end goal is to bring the population to convergence, selection/crossover happens more frequently (typically every generation). Mutation, being a variance operation, should
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happen less frequently, and typically only affects a few chromosomes of a population (if any) in any given generation. A range of mutation and crossover probabilities were determined after several empirical runs while observing the fitness function. For the purpose of
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this study, the crossover and mutation probabilities are fixed at 0.1 and 0.01 respectively. We observe that lowering the values of mutation probability does not change the optimal
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solution. However, a higher mutation probability shifts the solution to a sub-optimal point, at which total incurred cost is relatively higher than the optimal cost (see the real case in Section 5.2). Mostly, higher mutation probability affected the choice of truck types in case
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of shipments carrying hazmat 1 and hazmat 4, while the solution remained unchanged for all shipments carrying hazmat 2 and hazmat 3.
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4.5
Termination Criteria
The termination criteria used in the current evaluation is limited by the number of iterations or, the change in fitness value less than 10−6 for 20 successive iterations, whichever is encountered earlier. In the numerical examples, the GA terminates after reaching a solution
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when there is no appreciable change (10-5) in the objective function value over 5 successive generations. The time taken for running of the solution with the design parameters with 16 nodes was 315 seconds. Longer runs of the algorithm do not lead to a better solution. However, the design parameters have been empirically determined to allow a range of mutation and crossover probabilities so that global minimum can be obtained. The pseudocode for
5
Numerical Experiments
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the GA is included in Appendix A.
To illustrate the application of the proposed model, we use two sets of data. The first data set corresponds to a small scale example and is used for the simplicity of exposition. We later illustrate the model using actual data from a bulk hazmat transporter in western
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India. We consider all cost values in Indian Rupees where 1 Indian Rupee = 0.015 US
5.1
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Dollar.
Small Scale Example
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Consider the transportation network shown in Figure 2, which includes five cities (A, B, C, D, and E) and six road links (AB, AC, AD, BE, CE, and DE ). The length of a road link is indicated along the link. First, all possible routes are identified between an origin
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destination pair. Note that this is an illustrative example and therefore, we enumerate the routes.
We consider the total population at risk along the available routes in the network. Let
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this matrix be represented by P . The population figures in Table 2 are expressed as lakh units (×105 ). We consider one shipment to be completed in the planning period (50 T of
Hazmat 1, Acetone, is sent between between source node A and destination node E ). For the purpose of discussion, we shall solve the optimization problem by considering only one 19
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Figure 2: Network nodes and links along with the length of links (in kms) the network (×105 ) people E M 9 8 9 10
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Table 2: Population of Nodes and interlinks of ρij A B C D A 5 4 12 8 B 4 3 M M C 12 M 10 M D 8 M M 6 E M 9 8 9
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shipment route and fleet. A similar procedure can be followed for the remaining shipments that belong to the same planning period. Table 3 gives a description of truck types and
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their capacities that are available for procurement. As discussed earlier, the variable cost of transportation includes fuel consumption cost, administrative expenses per km (wages), and tire cost per km (Table 3). Each truck type
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may have a different cost coefficient associated with the loss due to truck stoppage per stop during the travel operation. In our example, we assume that brand 1 trucks require one stop after running 400 kilometers and brand 2 trucks require one essential stop after
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running 600 kilometers.
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We assume identical administrative cost (driver wages and supervision cost) across all truck types. Trucks with same carrying capacity have same number of tires. Since the cost of changing tires depends on the number of tires in the carrier truck, the cost of changing tires for trucks with identical capacities are the same. The amount of fuel consumed over a distance by different truck types may be different. For instance, it is observed that truck 20
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types 1 and 3 have identical number of tires and equal cost of changing tires, but a different fuel cost per kilometer. Likewise, truck types 2 and 4 incur equal expenditure on changing tires due to a similar reason (Table 3). Table 3: Variable cost components per kilometer for all links in the network Fuel cost/km
Administrative Cost/km
Cost of changing tires/km
Total Variable Cost/km
Capacity (in metric tons) (qk )
Type Type Type Type
10 20 9 18
1.5 1.5 1.5 1.5
0.5 1.5 0.5 1.5
12 23 11 21
10 20 10 20
T T T T
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1 2 3 4
Cost of Purchase Finance Loan (Lease) 10,000 18,000 12,000 19,000
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Truck Type
We also assume that the variable cost of travel per kilometer (km) for a truck type across different links is the same. Other additional travel costs such as toll fees are not considered in this example. It is assumed that for every stop during the travel in the route, a standby cost is incurred (Table 4).
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Table 3 includes the travel cost (variable cost per km.) incurred for the four truck types Table 4: Standby cost/stop and number of stops made for every truck type on different routes
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1 2 3 4
200 250 200 250
No. of stops per 1000 km. 2.50 2.50 1.67 1.67
Route-1
Route-2
Route-3
650.00 812.50 325.00 406.25
650.00 812.50 325.00 406.25
800.00 1000.00 400.00 500.00
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Type Type Type Type
Standby Cost
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Routes/Truck Type
during the delivery of the shipment. It is also assumed that every essential stop is of equal length of time and corresponds to the standby cost structure provided in the Table 4. We
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now determine the total population on the routes (ρtij ) for all links (i, j) using the formula
given in Equation (1) for the network shown in Figure 2. The parameters included in Table 2 are used to estimate the population at risk during delivery of shipments. Here P (i) ∀i ∈ {A, B, C, D, E} is analogous to ρi , P (i − j) ∀ i 6= j is analogous to ρij as
21
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explained in Section 3.3. The unit of these figures is the same as mentioned in Table 2. ρtAB
"
# P (A) + P (B) 5+3 = + P (A − B) = 4 + = 4.32 AB 25
Following the same procedure for other links, we obtain, −
4.32 12.43 8.27
− 4.32 t ρij = 12.43 − − 8.27 − 9.32
−
−
−
8.67
−
−
−
9.4
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−
9.32 8.67 9.4 −
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We assume that out of the four types of trucks, truck types 1 and 2 incur more losses due to frequent stops in comparison to truck types 3 and 4. It is also assumed that the hazmat (acetone) has a severity, (γm ), value of 0.2. It implies that out of a total of 100 people exposed to the hazmat, the effect of hazmat can be seen actually on only 20 of them. This assumption is made to justify the low severity values of industrial chemicals that are transferred to different cities by hazmat carriers. Many industrial chemicals naturally exist
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as liquids while a few exist as gases. Toxic gases (such as chlorine) are assumed to have very high severity values whereas acids and ketones have low severity values. Therefore,
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radioactive chemicals or toxic gases could have a high long term recovery cost. Let the solution for the shipment be represented by an array where { Xij1 , (n1 , n2 , n3 , n4 } represents number of trucks of type i used in the shipment and Xij1 represents the optimal links. The
results show that the best route is route 2 (AC-CE). The number of truck used per truck
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type in the optimal solution comes out to be {1, 2, 0, 0}. The value of objective function at
this solution is Rs. 58090.
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The solutions to the sample problem described in Figure 2 are investigated to study the trade-offs among the various components of the total cost expression. We calculated the individual components of the total cost expression (i.e., the loss due to truck stoppages, cost
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of recovery, cost of purchase finance loan of trucks, and the variable cost of transportation) for all solutions and examined the trade-offs among them. Particularly, we examined the trade-off between:
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1. loss due to truck stoppages and cost of purchase finance loan (lease) of trucks, and 2. total variable cost and recovery cost. It is important to observe that the acquisition costs of trucks belonging to brands 3 and 4 are higher than that of brands 1 and 2 in order to visualize the trade-off between loss
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due to truck stoppages and the procurement cost of the trucks. For a shipment, if the transporter procures a truck belonging to either brand 3 or brand 4, the losses associated with truck stoppages will be less (owning to fewer stops during the journey). On the other hand, if the transporter includes a brand 1 or brand 2 (lower acquisition cost) in the fleet, the inclusion of such trucks would increase the loss due to truck stoppages (due to frequent stops). If the company is delivering shipments over very long distances, the loss
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due to truck stoppages significantly increases when the shipment is sent through brand 1 trucks (owning to a large number of stops). Hence, for shipment over longer distances, we advise the transporters to use trucks which incur low monetary loss due to stoppages (Xie et al. (2012)). This trade-off can be observed in the Figure 3. The solutions compared are {Xij1 , (1, 2, 0, 0)}, {Xij1 , (0, 2, 1, 0)}, {Xij1 , (1, 1, 0, 1)}, {Xij1 , (0, 1, 1, 1)}, {Xij1 , (1, 0, 0, 2)},
and {Xij1 , (0, 0, 1, 2)}. This set of solutions is chosen to explain the trade-offs at a constant
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ED
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risk level.
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Figure 3: Trade-off between purchase finance loan (lease cost) of the trucks and monetary loss due to truck stoppages for six solutions Table 5 corresponds to the good quality solutions of the small scale example mentioned
in Section 5.1 of the paper. The solutions mentioned along with in table 5 and table 6 show 23
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non-dominated solutions with corresponding total cost and population at risk. The trade-off between the variable cost of travel and the total population at risk involves the choice of routes and the total population affected with each route selection. If a shipment is delivered using a large number of low capacity trucks, it incurs high variable cost and high cost of recovery. On the other hand, if the same quantity is transported by a few larger
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capacity trucks, it incurs relatively low variable cost of travel and low cost of recovery. Note that the cost of recovery depends on the number of trucks and is not affected by the capacity of the trucks. In Table 6, we see that all solutions with respect to a population-based risk value, contain the same number of vehicles in their fleet along with the same route choice. The remaining solutions either contain a large number of trucks in their fleet or contain a different route choice. Table 5 and Table 6 show the total cost with and without considering
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the losses due to truck stoppages respectively.
The results show that for the same level of exposure to the population, a number of combinations of fleet qualify; however, the total costs of shipments are unequal. Currently, {Xij1 , (1, 2, 0, 0)} is the best solution; however, if the distances between the nodes are large,
the benefit (savings) of including driver-productive trucks may dominate the additional
expenses of the cost of acquisition of driver-productive trucks. In those cases, the optimal
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solution will shift towards the other end (i.e., {Xij1 , (0, 0, 1, 2)}).
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CE
PT
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Table 5: Total cost vs. population at risk for good quality solutions to the sample problem when losses due to frequent truck stoppages are included; the cells denote the cost figures corresponding to a solution and a risk level Population at Risk Solution Level 1: 13.64(×105 ) Level 2: 21.09(×105 ) {1,(1,2,0,0)} 58,090 {1,(0,0,1,2)} 59,923 {1,(1,0,0,2)} 58,486 {1,(0,2,1,0)} 59,526 {1,(1,1,0,1)} 58,288 {1,(0,1,1,1)} 59,725 {2,(1,1,0,1)} 58,288 {2,(1,0,0,2)} 58,486 To estimate the total recovery cost, we consider the population at risk in the origin and
destination nodes as well as the population that lies between the nodes. Non-uniformity of 24
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population distribution in the network creates the uncertainty that shorter routes may/may not be economically feasible for transportation. Longer routes may incur high transportation costs and low recovery costs (if routes traverse through a sparsely populated region) while shorter routes may incur low transportation costs and high recovery costs (if routes traverse through a densely populated zone). Hence a trade-off is observed between the
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total variable cost of travel (depending on the length of path taken) and the recovery cost (depending on the total population exposed to the hazmat). The variable cost of shipment also increase with the increase in length of the route.
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Table 6: Total cost (excluding truck stoppages) vs. population at risk for good quality solutions to the sample problem; the cells denote the cost figures corresponding to a solution and a risk level Population at Risk Levels Solutions Level 1: 13.64×105 Level 2: 17.67 ×105 Level 3: 21.09 ×105 {1,(1,2,0,0)} 53,540 {1,(0,0,1,2)} 56,890 {1,(1,0,0,2)} 55,020 {1,(0,2,1,0)} 55,410 {1,(1,1,0,1)} 54,280 {1,(0,1,1,1)} 56,150 {1,(1,2,0,0)} 53,540 {1,(1,0,0,2)} 55,020 {1,(1,1,0,1)} 54,280 {3,(1,2,0,0)} 55,280 {3,(0,0,1,2)} 58,480 {3,(1,0,0,2)} 56,640 {3,(0,2,1,0)} 57,120 {3,(1,1,0,1)} 55,960 {3,(0,1,1,1)} 57,800 {2,(1,2,0,0)} 53,540 {2,(1,0,0,2)} 55,020 {2,(1,1,0,1)} 54,280
The total population at risk is only affected by the length of the route and the popu-
lation density along the route. Table 5 denotes good quality solutions as per the network mentioned in Figure 2. The solution taken into consideration ensures that the total amount transported is not less than the total available demand at different nodes. 25
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5.2
Realistic Case from a Bulk Hazmat Transporter
We now illustrate the model using the operational data from a mid-size bulk transporter in Ahmedabad, Gujarat, India. The company owns a total of 30 trucks of different capacities ranging from 10 to 20 metric tons. The trucks operate between 16 nodes. Moreover, in
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periods of excess demand, it procures trucks on lease from third parties to provide services. During the discussions, several interesting facts were obtained. The decision for the choice of the route was done by the transport company itself. However, they also allowed some flexibility to the driver to choose an alternate route. In general, the routes choice included state/national highways, service lanes and excluded routes that are in proximity to the schools, hospitals, and other densely populated areas. In case of an accident, the driver is well trained to limit the hazmat exposure with safety and caution. The order summary
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of the data is shown in Table 7. Table 8 shows a list of available truck types that can be procured for transportation. The value of the parameter em k is shown in Table 9. Table 10 highlights the severity and recovery cost of the four hazmats in the real world transportation problem.
The estimation of the parameters used for the study (such as variable cost of travel, Standby Cost and Number of essential stops per 1000 kilometer), provided by the transport
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company, considers several factors including number of tires, trailer capacity load, truck capacity, make of truck and frequency of replacement of tires. The set of parameters related
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to truck type in the study is based on averages for every truck type irrespective of the age of the truck. The parameters related to the hazmat are empirically determined to cover a range of severity and cost of mitigation values. Census of India, 2011 remains the source of
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the population data of the nodes used in the study. Probability of road accident in a link, which has been estimated as 35 per 10000 vehicles (see https://data.gov.in/keywords/
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transport-research-wing). In all the cases, the GA terminates after reaching a solution when there is no appreciable change (10−5 ) in the objective function value over 5 successive generations. The time taken
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for running of the solution with the design parameters with 16 nodes is 315 seconds.
26
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hazmat transporter Origin Destination Node Node O(s) D(s) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1)
Tarapur(13) Tarapur(13) Tarapur(13) Tarapur(13) Dabasa(14) Dabasa(14) Ankleshwar(15) Ankleshwar(15) Ankleshwar(15) Ankleshwar(15) Ankleshwar(15) Mandideep(16) Mandideep(16)
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1 2 3 4 5 6 7 8 9 10 11 12 13
Table 7: Order summary of the bulk Order Quan- Hazmat Carried tity (in tonnes) (m(s)) (ds ) 20 1 16 2 10 2 16 4 20 1 10 2 20 1 20 2 16 2 26 2 26 1 20 1 16 3
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Shipment number s
Four distinct origin destination pairs are identified in the order summary of the data.
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These pairs are Kandla-Ankleshwar, Kandla-Mandideep, Kandla-Dabasa, Kandla-Tarapur (see Figure 4a for maps of the Gujarat state and possible network of nodes between Kandla and Ankleshwar). The origin corresponds to the Port of Kandla. The length of the links
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for the network corresponding to each origin destination pair and the population of nodes and links are shown in Table 11 and 12 respectively. These parameter values are obtained by interpreting the Google maps and data from the Census of India (2011) (Figure 4b).
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For the purpose of estimating population along the road links, it is assumed that the population density in a district is uniformly distributed. A one kilometer rectangular region is considered for calculating the total population along the link on both sides of the road.
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For route links lying between two different districts, it is assumed that the population figure in the first half of the link belongs to one district and the remaining half belong to the other district. The result for this problem is shown in the Table 13. Figure 4c shows the feasible
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links in the network of 16 nodes considered in the study. The actual problem was modeled in a similar fashion where Table 13 describes the total
number of orders that were received for a particular period. For all the orders shown in
27
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the Table 13, the origin node of the shipment is Kandla (denoted as Node 1), while the destination of the shipment could be any of the following: Tarapur (denoted by Node 13), Dabasa (denoted by Node 14), Ankleshwar (denoted by Node 15), and Mandideep (denoted by Node 16). Hazmats 1 to 4 are MDC, Acetic Acid, MEK, and Acetone, respectively. The total cost corresponds to the sum of individual cost components as mentioned in section the shipment must pass in order to incur the least cost.
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3.4. The route shown in Table 13 is the optimal route mentioning the nodes through which It can be observed that the same truck type is not always the optimal choice for carrying shipments with the same hazmat over the same route. For instance, truck type 4 is chosen for transporting shipment with hazmat 1 and order quantities less than 20 tonnes (shipment numbers 1, 5, 7, and 12). However, the truck types 2 and 3 are chosen to transport the
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same hazmat 1 with an order quantity of 26 tonne (see Shipment 11). This choice is clearly due to variation in different cost components influencing the choice of trucks even between the same origin and destination pair. Further, consider the shipments 10 and 11. These two shipments correspond to different hazmat types but have same order quantities of 26 tonnes. Though the optimal choice of route and the number of trucks for carrying both shipments are identical, the truck types differ (3 and 9 for shipment 10 vs. 2 and 3 for shipment 11). Since hazmat types 1 and 2 differ in the recovery cost parameters, different
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choice of trucks minimize the other truck-related costs over the same route. Also note that the solutions in Table 13 correspond to truck types 3, 4, 8, and 9, which does not include
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the cheapest truck type option 2. Truck type 8 is 85% more expensive than Truck type 2; however, the additional purchase cost offsets the reduction in other cost components.
Comparative Study of Different Solution Approaches
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6
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In this section, we compare three different solution approaches to fleet mix bulk hazmat transportation problem and compare their solution quality in terms of cost. The three
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different solution approaches include: 1. Lowest truck purchase cost approach 2. Two-stage sequential solution approach 3. Integrated solution approach 28
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Lowest truck purchase cost approach is most commonly used to decide fleet mix among hazmat transporters in India. In this approach, the lowest cost truck is selected to fulfill the demand and the shortest route is selected between the source and destination pair corresponding to a shipment. The transporters typically choose the shortest route to minimize cost. The two-stage sequential solution approach is performed to see if the integrated prob-
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lem can be decomposed into two simpler subproblems and solved to optimality. In the first stage, we chose to first select the route corresponding to minimum cost of recovery. Note that we did not consider the travel cost criterion to select the route because the shorter distance routes often correspond to high density population centers. In the second stage, we determine the optimal fleet mix corresponding to the chosen route in the first stage. In the second stage, we minimize the travel cost, purchase cost, and loss due to truck stoppages. tify the optimal routes and fleet mix.
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The integrated solution approach presented in this paper attempts to simultaneously iden-
Table 14 compares the cost components corresponding to the optimal solution obtained for the bulk hazmat transporter using the three stated approaches. In approach 1 (lowest truck purchase cost with the shortest route), the travel cost is quite low but the cost of recovery is high due to high population risk. In this case, the cost of recovery exceeds the
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cost due to truck stoppages. In the two-stage sequential approach, we observe that the fleet mix choices for all the 13 solutions are same as the optimal fleet mix obtained by
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the integrated solution (approach 3). However, the route choices obtained in the two-stage approach differed from the optimal choice obtained from the integrated solution approach.
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The difference in optimal costs obtained from approaches 1 and 2 in comparison to integrated approach suggests that the integrated solution approach yields better quality
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solutions. Based on the parameters considered in the study, average cost savings of approach 3 over approach 1 is 14.04% (average cost difference of Rs. 3,90,823) and average cost saving of approach 1 over approach 2 is 0.43% (with an average cost difference of Rs.
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10,790). Table 14 also highlights that though some solutions obtained from integrated solution approach are common with solutions obtained from approaches 1 and 2, there is no such instance where integrated solution approach performs worse than solutions obtained from either of the approaches 1 and 2. The solutions obtained from the integrated solution approach have a few common instances with the lowest truck cost approach (highlighted 29
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in bold) and with the two-stage sequential solution approach (highlighted in italics), and an instance common to all the three approaches (highlighted using an underline). We infer that integrated solution approaches provides best solutions irrespective of the nature of the shipments which include distance between source and destination nodes for a shipment and
7
Conclusion and Future Work
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quantity and severity of the hazmat carried in the shipment.
Transporters are currently seeking alternate methods to improve their loss due to truck stoppages, minimize transportation risk, and deliver shipments with lower travel costs. While the fleet mix affects the number of truck stoppages and the travel costs, the choice of
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the routes affects the transportation risk. Through a real case from a mid-size transporter, we show that the choice of the fleet can affect the selected route for shipment delivery. In particular, our results show that the truck type with the lowest purchase cost is not the optimal choice. We believe that this paper is the first attempt to capture the effect of truck type on loss due to truck stoppages and fleet mix decision. Using a non-linear optimization minimum cost solutions.
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based integer programming model, we capture the three-way cost interaction and obtain
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In our case study, we assume that the fixed cost of acquisition corresponds to the cost of hiring a truck for a single shipment delivery. It is likely that the distribution of shipments over different planning periods might be irregular. It rests on the management of the
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carrier company to ensure that they do bring sufficient demand receipts over the network so that none of their trucks remain idle. This research could be extended by including time-based stoppages and hour-of-service regulations in truck movements. It means that
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truck drivers may commute longer distances during the night time than the day time due to reduced traffic and less congestion and some necessary stoppages may be necessary for driver safety. Hence, the departure of truck may be scheduled in such a manner that it
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avoids the peak hours of traffic during shipment delivery. The model may also be adjusted to consider time-based variations in truck-speed parameters and its effect on travel cost and route selection. Acknowledgements: We thank the editor and the reviewers for their detailed comments, 30
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which have led to improved exposition of this paper.
References Abkowitz, M., DeLorenzo, J., Duych, R., Greenberg, A., McSweeney, T., 2001. Comparative Record, TRB 2001 Annual Meeting, 1–27.
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519.
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Input: Instance Π, Rate c of Elitism, Rate m of Mutation,
Output: Solution X
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// Initialization
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Number iter of iterations
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Size P of Population,
Generate P feasible solutions randomly; Save them in the population P op ;
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// Loop until the terminal condition for i = 1 to iter do
// Elitism based selection
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Number of elitism ne = P.c;
Select the best ne solutions in P op and save them in P op1 ; // Crossover Number of crossover nc = (P − ne )/2; 36
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for j = 1 to nc do Randomly select two solutions XA and XB from P op; Generate XC and XD by one-point crossover to XA and XB ; Floor or Ceil to nearest integer and check bounds Save XC and XD to P op2 ; // Mutation for j = 1 to nc do Select a solution Xj from P op2;
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endfor
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Mutate each bit of Xj under the rate m and generate a new solution Xj ; 0
if Xj is infeasible 0
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endif 0
update Xj with Xj in P op2; endfor // Updating update P = P op1 + P op2 ; endfor
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// Returning the best solution
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Update Xj with a feasible solution by repairing Xj ;
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Return the best solution X in P op;
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Table 8: Various truck types available (and the associated cost components) to the transporter for lease Capacity (in MT) qk
Standby Cost
Cost of Acquisition (in Rs.)
1 2 3 4 5 6 7 8 9
Brand Brand Brand Brand Brand Brand Brand Brand Brand
10 10 16 20 20 20 20 20 10
300 200 300 500 500 300 400 500 300
1,350,000 1,300,000 1,800,000 2,200,000 2,250,000 2,400,000 2,500,000 2,400,000 1,350,000
1 2 2 2 4 3 1 3 1
No. Stops 1000 uk 7.5 10.0 10.0 10.0 8.0 8.0 7.5 8.0 7.5
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Table 9: Parameter values of em k for all possible values Hazmat (m)/ Methylene Acetic Methyl Truck Type (k) Dichloride Acid Ethyl (MDC) Ketone (MEK) Type 1 1 0 1 Type 2 1 0 1 Type 3 1 1 0 Type 4 1 0 1 Type 5 1 0 1 Type 6 1 0 1 Type 7 1 0 1 Type 8 1 1 1 Type 9 1 1 0
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of per km,
Fuel Cost per km
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Brand Manufacturer
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Truck Type k
13.2 12.5 17.8 22.0 24.0 24.5 25.0 28.0 13.2
of k and m Acetone
1 1 1 1 1 1 1 1 0
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Table 10: Severity and recovery cost parameter values associated with hazmat m Hazmat (m) Methylene Acetic Acid Methyl Ethyl Acetone Dichloride Ketone (MEK) (MDC) Severity 0.05 0.3 0.36 0.75 Recovery Cost (in Rs.) 2000 106 500 240
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39 Figure 4: a) Average population density of all districts of Gujarat (Census of India 2011) b) Network from Kandla to Ankleshwar along with the nodes falling on the routes (Google Maps), and c) Available links within the network
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Table 11: Length of links within the network as described in Figure 4(b) Kandla 0 62.5 -
Samkhiyali 62.5 0 43 165 -
Maliya 43 0 37.8 41.1 -
Morbi 37.8 0 46.2 105 -
Halvad 41.1 46.2 0 73.2 97 157 206 -
Jalalabad 165 0 149 578 -
Sayla 105 73.2 0 35.2 -
Limbdi 97 35.2 0 101 -
Vatva 157 149 0 52.4 -
Nadiad 206 52.4 0 391 62.7 -
Dewas 578 391 0 175
Vadodra 62.7 0 20 90.8 -
Tarapur 101 0 58.3 -
Dabasa 20 58.3 0 -
Ankleshwar 90.8 0 -
Mandideep 175 0
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Distance Kandla(1) Samkhiyali(2) Maliya(3) Morbi(4) Halvad(5) Jalalabad(6) Sayla(7) Limbdi(8) Vatva(9) Nadiad(10) Dewas(11) Vadodra(12) Tarapur(13) Dabasa(14) Ankleshwar(15) Mandideep(16)
Maliya 8299 14022 12852 10418.85 -
Morbi 12852 178055 11711.7 26617.5 -
Halvad 10418.85 11711.7 24325 12224.4 16199 82974.5 72924 -
Jalalabad 23100 606 83738 132073 -
Sayla 26617.5 12224.4 13185 5878.4 -
Limbdi 16199 5878.4 40071 44389.5 -
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Samkhiyali 2875 5897 8299 23100 -
Vatva 82974.5 83738 4525013 37492.2 -
Nadiad 72924 37492.2 196793 149362 34265.55 -
Dewas 132073 149362 231672 33250
Vadodra 34265.55 1491045 11040 35866 -
Tarapur 44389.5 14934 36845.6 -
Dabasa 11040 36845.6 7240 -
Ankleshwar 35866 112643 -
Mandideep 33250 39859
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Kandla 14695 2875 -
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Table 12: Population of links within the network as described in Figure 4(b) Distance Kandla(1) Samkhiyali(2) Maliya(3) Morbi(4) Halvad(5) Jalalabad(6) Sayla(7) Limbdi(8) Vatva(9) Nadiad(10) Dewas(11) Vadodra(12) Tarapur(13) Dabasa(14) Ankleshwar(15) Mandideep(16)
Order Quantity (in tonnes) 20 16 10 16 20 10 20 20 16 26 26 20 16
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Shipment No. 1 2 3 4 5 6 7 8 9 10 11 12 13
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Table 13: Route and truck choices made for delivering the shipments of bulk hazmat transporter Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Origin 1 1 1 1 1 1 1 1 1 1 1 1 1
Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
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Total Cost 2266645.51 1849183.85 1390844.45 1867415.60 2287151.50 1404423.16 2363293.71 2554627.05 1920711.25 3371106.59 3319635.12 2432094.28 2432800.72
Optimal Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-10-12-14 1-2-3-5-10-12-14 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-11-16 1-2-6-9-10-11-16
Optimal Trucks Number (Type) 1(4) 1(3) 1(9) 1(3) 1(4) 1(9) 1(4) 1(8) 1(3) 1(3),1(9) 1(2),1(3) 1(4) 1(4)
AC Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Quantity 20 16 10 16 20 10 20 20 16 26 26 20 16
Quantity 20 16 10 16 20 10 20 20 16 26 26 20 16
41 Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
No. of Truck(Type) 1(4) 1(3) 1(9) 1(3) 1(4) 1(9) 1(4) 1(8) 1(3) 1(3),1(9) 1(2),1(3) 1(4) 1(4)
Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-10-12-14 1-2-3-5-10-12-14 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-11-16 1-2-6-9-10-11-16
Cost of Travel 15162.40 12267.80 9097.40 12267.80 19153.20 11491.90 36938.00 47012.00 29886.20 52049.00 50873.70 53649.20 53649.20
Integrated Problem Solution Purchase Finance Loan Cost 2200000 1800000 1350000 1800000 2200000 1350000 2200000 2400000 1800000 3150000 3100000 2200000 2200000
Cost of Recovery 17023.11 16240.05 16240.05 34471.80 24468.30 23342.76 42405.71 40455.05 40455.05 80910.09 84811.42 56515.08 57221.52
Total Cost 2266645.51 1849183.85 1390844.45 1867415.60 2287151.50 1404423.16 2363293.71 2554627.05 1920711.25 3371106.59 3319635.12 2432094.28 2432800.72
Total Cost 2266645.51 1849183.81 1390844.49 1867415.56 2306908.61 1425409.08 2363524.74 2555306.20 1934859.76 3407471.91 3362572.33 2434395.50 2435671.96
Total Cost 2678844.22 2781688.99 1390844.45 2713741.60 2705525.60 1404423.16 2760523.42 2864076.80 2864076.80 3371106.59 3319635.12 2842367.16 2843780.03
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Loss due to Stoppages 34460.00 20676.00 15507.00 20676.00 43530.00 19588.50 83950.00 67160.00 50370.00 88147.50 83950.00 121930.00 121930.00
Cost of Recovery 17023.11 16240.05 16240.05 34471.80 48891.01 46642.02 90646.34 86476.60 86476.60 172953.21 181292.67 102117.10 103393.56
Two-Stage Sequential Problem Solution Cost of Travel Purchase Finance Loan Cost Loss due to Stoppages 15162.40 2200000 34460.00 12267.76 1800000 20676.00 9097.44 1350000 15507.00 12267.76 1800000 20676.00 17727.60 2200000 40290.00 10636.56 1350000 18130.50 22268.40 2200000 50610.00 28341.60 2400000 40488.00 18017.16 1800000 30366.00 31378.20 3150000 53140.50 30669.66 3100000 50610.00 40418.40 2200000 91860.00 40418.40 2200000 91860.00
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Cost of Recovery 34046.22 32480.09 16240.05 68943.60 48936.60 23342.76 84811.42 80910.10 80910.10 80910.09 84811.42 113030.16 114443.03
Lowest Truck Cost Solution Purchase Finance Loan Cost 2600000 2700000 1350000 2600000 2600000 1350000 2600000 2700000 2700000 3150000 3100000 2600000 2600000 Loss due to Stoppages 27568.00 31014.00 15507.00 27568.00 34824.00 19588.50 46592.00 52416.00 52416.00 88147.50 83950.00 79592.00 79592.00
Cost of Travel 17230.00 18194.90 9097.40 17230.00 21765.00 11491.90 29120.00 30750.70 30750.70 52049.00 50873.70 49745.00 49745.00
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Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13-14 1-2-3-5-8-13-14 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-11-16 1-2-3-5-10-11-16
Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-10-12-14 1-2-3-5-10-12-14 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-11-16 1-2-6-9-10-11-16
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No. of Truck(Type) 1(4) 1(3) 1(9) 1(3) 1(4) 1(9) 1(4) 1(8) 1(3) 1(3),1(9) 1(2),1(3) 1(4) 1(4)
No. of Truck(Type) 2(2) 2(9) 1(9) 2(2) 2(2) 1(9) 2(2) 2(9) 2(9) 1(3),1(9) 1(2),1(3) 2(2) 2(2)
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CE Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Quantity 20 16 10 16 20 10 20 20 16 26 26 20 16
Table 14: Appendix A.2.: Comparison of solution methods
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Integrated Fleet Mix and Routing Decision for Hazmat Transportation: A Developing Country Perspective Anand Kumar, Debjit Roy, Vedat Verter, Dheeraj Sharma PII: DOI: Reference:
S0377-2217(17)30530-1 10.1016/j.ejor.2017.06.012 EOR 14494
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
8 June 2016 5 June 2017 6 June 2017
Please cite this article as: Anand Kumar, Debjit Roy, Vedat Verter, Dheeraj Sharma, Integrated Fleet Mix and Routing Decision for Hazmat Transportation: A Developing Country Perspective, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.06.012
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Integrated Fleet Mix and Routing Decision for Hazmat Transportation: A Developing Country Perspective Anand Kumar Indian Institute of Technology Kharagpur, Debjit Roy
1
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West Bengal, India Indian Institute of Management Ahmedabad, Gujarat 380015, India, and
Rotterdam School of Management, Erasmus University, 3062PA Rotterdam, The Netherlands
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Vedat Verter
Desautels Faculty of Management-McGill University, Montreal QC, Canada Dheeraj Sharma
Indian Institute of Management Ahmedabad, Gujarat 380015, India and
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Indian Institute of Management Rohtak,
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Abstract
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Haryana 124001, India
In developing countries, truck purchase cost is the dominant criteria for fleet acquisition-
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related decisions. However, we contend that other cost factors such as loss due to the number of en route truck stoppages based on a truck type and recovery cost associated with a route choice decision, should also be considered for deciding the fleet mix and minimizing
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the overall costs for long-haul shipments. The resulting non-linear model, with integer variables for the number and type of trucks, and the route choices, is solved via genetic algorithm. Using real data from a bulk liquid hazmat transporter, the trade-offs between 1
Corresponding Author, Email: [email protected], Phone: +91-7966324823, Fax: +91-7966326896
1
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the cost of travel, loss due to number of truck stoppages, and the long-term recovery cost associated with the risk of exposure due to a hazmat carrier accident are discussed. The numerical experiments show that when factors related to public safety and truck stoppages are taken into account for transportation, the lowest total cost and optimal route choice do not align with the cheapest truck type option; rather, the optimal solution corresponds to
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another truck type and route with total costs significantly less than the total costs associated with the cheapest truck type. Our model challenges the current truck purchasing strategy adopted in developing countries using the cheapest truck criteria.
Keywords
Logistics, OR in Societal Problem Analysis, Routing, Risk Analysis
Introduction
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”Over 3 billion tons of hazardous materials (HAZMAT) are transported by commercial carriers in the United States each year. Such hazmat shipments can be highly risky and highly sensitive and if improperly handled, labeled, or packaged could result in the loss of life, property damage, and harm to national security interests,” United States Government
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Accountability Office (2014). The most common hazardous materials (hazmats) for transportation include cooking gas, fuel oil, and chemicals such as ethyl alcohol, and gasoline
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(Kara and Verter (2004)). Due to flammable, corrosive, explosive, poisonous or radioactive nature of dangerous goods, the carriers of hazmats, when involved in a road accident, may lead to disastrous consequences such as fire, explosion, spillage, and leakage, resulting in
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a large number of fatalities and injuries besides property loss and environmental pollution (Yang et al. (2010)).
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Road network, the most used transportation network in developed countries, is highly regulated in terms of hazmat movement. The regulatory body controls the maximum amount of long-haul drive time (without stoppages), limit choice of routes, and capacity
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of truck for hazmat movement (Toumazis and Kwon (2013)). Such regulations include US Department of Transportation, HOS (Hours of Service) Regulations for long-haul and regional operations of transporters. It mandates a driver carrying any shipment to drive not more than 11 hours after 10 consecutive hours of off-duty. Further, drivers must take a mandatory 30 minutes break by their eighth hour of coming on duty. The 14-hour duty 2
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period may not be extended with off-duty time for breaks, meal and fuel stops. Limitation on the choice of routes also exists in the real world scenario. For example, Canadian city of Brandon in Manitoba province can serve as an illustration where a website gives a map of the city which shows the roads that can be used by the trucks carrying dangerous goods (www.brandon.ca). However, developing countries such as India, are yet to develop regulation or lax implementation of existing regulations.
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a holistic regulatory regime of hazmat transportation while facing issues of either lack of Hence, there is a potential threat to public safety in developing countries where lax regulations on movement of hazardous goods may allow transport companies to choose minimal distance routes, while completely ignoring vulnerable zones of high density areas leading to high transport risk. Transport risk, which is defined as a measure of expected number of
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people, who would suffer the consequence of a possible hazmat accident, receives minimal consideration during route choice decisions. Chakrabarti and Parikh (2013b) confirms that about 90% of hazmat incidents in India occur on highways and majority of them occur in close vicinity to rural areas. Factoring in the potential threats posed by transportation of hazards, the paper attempts to bring about an optimal balance between minimizing cost of transport and cost of recovery (remedies) associated with hazmat transportation.
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Among the more recent transporter challenges, driver-related issues have gained little attention (Zhou and Chen (2015)). Major challenges faced by the drivers (especially in developing countries) are: poor vehicle operating conditions, nonstandard working hours,
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no standard resting places between origin and destination points, and delays due to policy and regulatory needs (border/custom delays). To improve the vehicle operating condition, commercial vehicle manufacturers are designing trucks with ergonomic chairs and seats
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that have been found to increase productivity and comfort for the drivers (Rosecrance et al. (1993); De Looze et al. (1993); Hedge and Sakr (2005); Eikhout et al. (2005); Vink
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(2005)). In concurrence with research on ergonomic design, we contend that the choice of truck (brand) has an effect on the comfort of the driver’s seat, resting seat, and the number of truck stoppages per kilometer of travel (beyond the necessary stoppages such as
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mandatory rest-breaks). In long-haul transport, there could be potential cost advantages with trucks that include an ergonomic seat with an air-conditioned driver cabin against a cheaper truck with a non-ergonomic seat, and without an air-conditioned truck cabin. In such a scenario, the combined effect of cost advantages with less number of stops per kilometer (km) may out-weigh the difference in acquisition cost between the cheaper and 3
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the expensive truck types for long distance shipments. Hence, the decision with respect to the fleet mix (number and type of trucks) is affected by the relative position of the origin and destination nodes in the network. Several studies in literature determine the optimal fleet mix by integrating it with the vehicle routing problem (Golden et al. (1984); Salhi and Rand (1993); Desrochers and Ver-
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hoo (1991)). However, existing models that are mostly developed from a developed country perspective do not capture the nuances with respect to long-haul transport in developing countries. The differences are:
1. Large variety of the truck types (make and capacity): As discussed earlier, the transporters can select from a variety of truck brands (with different make, driver-cabin comfort,
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and capacity). All truck types in developed countries have a good basic operating condition for the drivers (such as air blower or air conditioner in the cabin). From a developing country’s perspective, this criteria becomes special because the basic amenities may be absent in low cost truck types. Hence, different amount of driver discomfort may cause the number of truck stoppages between two places to vary from one truck to another. 2. Population-based risk : In developing countries, the road network may have limited route
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choices for hazmat transport and the transporter is often compelled to use a road that passes through a dense neighborhood, which carries a high population risk. Further, the restriction over transportation of hazardous materials over highly populated road network
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is minimal or absent. Hence, population-based risk should be assessed for fleet mix and the route choice decision (Chakrabarti and Parikh (2013a), Chakrabarti and Parikh (2013b)). 3. Criteria for fleet acquisition: Truck purchase cost is the dominant factor considered
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during fleet acquisition. The authors also surveyed 50 fleet owners in India and asked them to identify the most important factor for a new truck purchase. The factors included
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after-sales service, fuel efficiency, total cost of ownership, spares availability, brand image, ongoing relation with brand, modern and new brand, durability and reliability, strong dealer support, purchase price, finance arrangement by brand, relation with truck supplier, driver
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recommendation and mechanic recommendation. Out of 50 fleet owners, 24 responded that purchase price is the most important factor for truck purchase decision. These issues have been studied separately in the prevailing literature. One of the pri-
mary contributions of this paper is to incorporate all three nuances in an integrated model. 4
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In addition, our approach in capturing the truck type-dependent stoppages and representing the population risk faced by the people residing along the highway segments as well as the population centers, is quite novel. Hence, the model built in this paper is more relevant and compelling from a developing country perspective.
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Trucking companies expect their hired drivers to ensure a minimum daily travel distance irrespective of the truck make. Since a poor truck make may lead to excess number of truck stoppages, the driver may lag behind his schedule. To make up the lost time, the driver takes shorter routes associated with higher risks (in a unrestricted road network). Hence, the three factors: truck type, truck’s load capacity and number of trucks used in a shipment, together referred to as ‘fleet mix’ decides the optimal route choice. We argue that studies
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that have ignored the losses suffered due to truck stoppage in deciding the ‘fleet mix’ may not be optimal and accounting for such losses presents a much more realistic scenario compared to the existing studies. In this research, we study the trade-offs between the truck purchase finance loan costs (trucks type in the shipment) and the costs associated with the routing decisions (i.e., the costs of transportation, loss due to truck stoppages, and costs associated with accident recovery). We also develop a population-based measure of risk to quantify recovery cost in the event of an accident. The hazmat transportation risk is incorporated in
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the objective function of the model as the expected accident recovery costs. The proposed risk measure not only captures the population at risk at the source and destination nodes,
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but also includes the ones along the travel route. Using this definition of risk, we attempt to answer the following research questions: a) If the transport carrier intends to purchase a fleet of trucks with different make (truck
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type), capacities, and load type, what should be the fleet size and type for every shipment? b) For a particular shipment, how should the transporter select the optimal route among a set of available routes between origin and destination nodes?
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The decision variables are the appropriate fleet of vehicles serving a particular shipment and the route index. The objective of this problem is to minimize the sum of travel cost of shipment over the network, the cost of transport risk, and the loss due to truck stoppages.
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We specifically examine the problem that involves the transportation of multiple types of dangerous goods (chemical goods) to be shipped over multiple origin-destination pairs. The resulting mathematical formulation is a non-linear optimization model with integer decision variables.
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The remainder of the paper is organized as follows. Section 2 reviews literature on modeling hazmat transportation problems. Section 3 describes model specification with discussion on the transportation network design problem and the model formulation to determine the optimal fleet mix. Section 4 discusses the solution methodology adopted to solve the non-linear optimization formulation. Section 5 first presents an illustrative
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example to demonstrate the main trade offs in the problem, and a real life case to highlight the complexities and the effectiveness of the proposed approach to tackle such problem instances. Section 6 compares the integrated solution approach with a lowest truck purchase cost and two-stage sequential solution approach. Section 7 presents our research conclusions and the scope for future work.
Literature Review
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2
Looking at the importance and seriousness of the issue, transportation of hazardous materials has been studied extensively in operation research in last twenty years. Transportation of hazardous materials has been studied in context of modeling for road network (Bianco et al. (2009); Erkut and Alp (2007); Erkut and Gzara (2008); Kara and Verter (2004);
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Zhang et al. (2000), emergency response network models (Berman et al. (2007)), multilocation routing methodology (Xie et al. (2012); Leonelli et al. (2000); Verter and Erkut (1997)), decision modeling for risk assessment (Clarke and Sacre (2009), Sun et al. (2016)),
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routing of hazmat trucks (Bell (2006); Akgün et al. (2007); Bell (2007)), factor affecting on accidents in hazmat transportation (Zhau et al. (2012)), cost related models (Glickman (2006)).
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et al. (2007); Revelle et al. (1991)) and in context of fuzzy graph modeling (Boulmakoul Previous studies in the field of hazmat risk concentrated upon the two specific problems
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namely assessment of transport risk for hazmat shipments (Erkut and Ingolfsson (2005); Erkut and Verter (1998)) and identification of route with minimal transport risk (Zografos and Androutsopoulos (2004), Reniers et al. (2010), Bronfman et al. (2015), Szeto et al.
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(2017)). Possible solution to reduce shipment risk can be the appropriate methodology in identification of route by keeping various costs into consideration (Leonelli et al. (2000); Erkut and Verter, 1997). Erkut and Ingolfsson (2000) defines risk in the hazardous materials transportation as “the most-widely used definition of literature is the expected consequence 6
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of an incident (accident resulting in a release), which, for each edge of the network, is equal to the product of the incident probability and a quantifiable consequence (such as number of people evacuated).” Verter and Erkut (1997) evaluated routing policies for a manageable number of alternate routes and concluded that the route selected for delivery of hazmats should be determined
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on the basis of combined increase in cost of transportation and increase in insurance costs due to a possible accident. Indirectly, the use of geographic information system (GIS) can also help to deal with risk related to hazmat transportation and also in network designing (Bianco et al. (2009); Zhang et al. (2000)). On the contrary, literature on hazmat suggest that the number and location of containers with hazardous cargo varied considerably among trains and consequently it is important to define a boundary of exposure zone as a function
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of the volume of hazmat being shipped (Verma and Verter (2007); Kara and Verter (2004)). In quantification of risk in hazmat transportation, Kara and Verter (2004) quantified risk as the total number of people exposed within a threshold distance from a given road link whereas Verma and Verter (2007) quantified risk as the population exposed from multiple release sources of hazmat (attached as railcar compartments) as a function of rupture diameter, volume, and nature of hazmat carried. The most appropriate measure of risk used in the literature is a measure of the population exposed to the route used for the
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shipment and also it has been considered as a better measure of transport risk assessment than occurrence of an incident (Batta and Chiu (1988)).
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Several studies considered the likelihood of encountering a hazmat accident during transportation as their risk measure (Abkowitz et al. (1992); Saccomanno and Chan (1985)). Erkut and Verter (1995) as well as Alp (1995) observed that the transport risk is a measure
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of the expected number of people who would suffer the consequence of a possible hazmat accident. Erkut and Alp (2007) proposed an integrated routing and scheduling model for
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hazmat transportation that minimizes the en route risk (accident probability multiplied by exposure) subject to a constraint on the total duration of the trip. Instead of rerouting hazmat movement to minimize transport risk, Kawprasert and Barkan (2008) presented an
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optimization model for rail route rationalization. Garrido (2008) developed a road-pricing scheme for safe transportation or disposal of hazmat. List and Mirchandani (1991) developed a detailed model for routing hazardous materials
and locating hazardous waste treatment facilities based on minimal total cost, total societal 7
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risk, and the maximum risk exposed to an individual. Kara and Verter (2004) presented a bilevel programming formulation that identified the minimum risk design for the road network and demonstrated that there is a significant chance of risk reduction in transportation of hazmats if the government intervenes in the decision of making road links for transportation of hazmats Among a number of possible routes between destinations it is possible that
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routes with shorter length are exposed to more transport risk (i.e., prone to accidents, heavily populated, high traffic density etc.) while other longer routes may possess larger cost along with an acceptable risk of transport (Marcotte et al. (2009); Akgün et al. (2007); Bell (2006)). Kang et al. (2014) developed a value-at-risk model to generate route choices for a hazmat shipment based on specific risk confidence level. Various route selections were made depending on the choice of confidence level. Iakovou et al. (2015) provided a multi-commodity network flow model for the problem of routing hazardous vessels. Instead
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of the traditional shortest path formulation model, they employed network flow model to capture shipments of multiple commodities over multiple origin-destination pairs. Fan et al. (2015) proposed a bi-objective programming model and a heuristic algorithm to optimize the routing of hazardous materials transportation with road closure considerations using a case study from China. Van Raemdonck et al. (2013) presented a comprehensive survey on risk analysis systems for the transport of hazardous materials and developed a methodology
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for assessing the local accident risk.
As discussed in this section, several attempts have been made towards study of optimal
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fleet and route mix of hazardous materials shipment over complex networks. The approaches of the authors vary in their consideration of design parameters such as hazmat variability, route preferences and availability, truck type variability (in terms of capacity, compatibility
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with hazmat shipment and cost of travel per kilometer). Table 1 presents a selection of representative papers that analyzes the fleet mix problem for hazmat transportation.
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We contribute to the literature in fleet mix decision making for hazmat transportation by presenting an integrated framework that considers the truck type, population-based risk,
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and truck stoppages. We describe the model in the next section.
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Table 1: Representative studies for fleet mix decision in hazmat transportation Travel Distance Minimization
Kara and Verter (2004) Verter and Kara (2008) Pooley (1993) Golden et al. (1984) Chakrabarti and Parikh (2013a);Chakrabarti and Parikh (2013b) This Paper
X X X
X X X
X
X
Populationbased Risk X X
Truck Stoppages and Corresponding Fleet Acquisition
X
X X
X
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3
Large Variety of Truck Types (Make and Capacity)
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Citation
Model Description
The transporter needs to design the flow of shipments in the network in such a manner that not only travel cost is reduced but also the risk of transportation is minimized. In practice,
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the route corresponding to the minimum travel cost may not always be the route associated with the minimum risk. Hence, a model is formulated to select the shipment route as well as the fleet mix with minimum risk associated with transporting a shipment. The key features
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of our model lie in considering the effect of truck types on the monetary loss due to truck stoppages and the effect of route selection on the transport risk. We describe the modeling
Problem Background and Model Assumptions
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3.1
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premise, the model assumptions, and the cost parameters in this section.
The model prepared in this study deals with a single season demand of hazmats and it is assumed that the model is valid for a new carrier agency. It is also assumed that the
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agency does not own any trucks at the beginning of the planning period. In practice, the transport company either buys a truck or rents them from the market. The rental period is assumed to be a single planning period. Let us denote the existing road network topology using G = (N, A) where N is the set of nodes (vertices) and A is the set of links among the 9
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nodes (arcs). The following notation is used for the development of the model. I : Set of Nodes, indexed by i M : Set of hazmat types, indexed by m S : Set of shipments in order summary, indexed by s K : Set of truck types, indexed by k m(s)
to a shipment s with hazmat m is denoted by ρi
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The set of road links are denoted by (i, j). The number of people affected at node i due . Meanwhile, the number of people
affected in link (i, j) due to transport of shipment s carrying hazmat m is represented by m(s)
ρij
. Let us define the distance between node i and node j as lij , while the algebraic sum
of the lengths of individual links forms the total length of the route. The other modeling assumptions are listed below:
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• The model is developed for a single period demand.
• The edges of the network are assumed to be bi-directional. (i.e., lij = lji ) • All routes on the network are available for the shipment of all types of hazmats. • There is a single period truck procurement budget.
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• Each type of truck is designated to carry specific types of hazmats only. • There is no backlog of orders. All orders scheduled for a planning period will be
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shipped in the same period only.
• Trucks with same capacity are assumed to have same number of tires and associated
Modeling Monetary Loss due to Truck Stoppages
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3.2
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tire maintenance charges.
Truck drivers typically halt for taking rest after driving a stretch of a few hundred kilometers. We use the following approach to estimate the loss due to truck stoppages during
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transport. For every stop due to fatigue of a truck driver, we consider a fixed amount of essential expense incurred by the transporter. In this paper, we are particularly concerned with the time and monetary loss due to additional (non-essential) truck stoppages. Non-essential truck stoppages occur due to driver fatigue and cause monetary loss to the 10
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transporter. Based on interactions with truck drivers, we found that almost all drivers experience fatigue and stop more than once during their journey. The number of stoppages is affected by the type of truck used for transport. Different types of truck have different number of stoppages depending upon health issues of drivers such as fatigue. Therefore, we use the term called ‘loss due to truck stoppages’, which can be measured by the number of
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stoppages made per 1000 kilometers depending on a truck type. It is observed that trucks with fewer number of truck stoppages have better ergonomic seat design and can run at longer continuous stretches. The associated cost with every essential stop is fixed with respect to every truck type. The product of the number of stoppages and the associated cost provides the cost due to loss of truck stoppages along a route. It is important to note that the monetary loss due to truck stoppages is independent of the type of hazmat delivered
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by the truck and depends only on the type of truck.
It was also observed that trucks with driver-friendly design features (such as luxury driver seat with a larger cushion size, higher tolerance to mechanical vibrations during driving, and larger backrest adjustment angle) had higher acquisition cost but could run larger lengths without stopping. On the other hand, trucks with lower acquisition cost may have fewer driver-friendly design features (such as multi-purpose driver seat and lower
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tolerance to mechanical vibrations during driving) and may need the driver to make several
Modeling Population at Risk
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3.3
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essential stops before reaching the destination.
Since the routing decision and choice of fleet jointly affects the amount of risk posed by a particular shipment, our first goal is to assess the risk of accidents pertaining to all the
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routes available in the network between any connected pair of nodes. We assume that every route is susceptible to an accident; however, the level of severity would be dependent on the population density and hazmat carried along the route. Since a route on the network
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is composed of several interlinks, the total risk associated with a route will be dependent on the risk associated with the individual interlinks (Abkowitz et al. (2001); Chakrabarti and Parikh (2011)) For any origin destination pair (i, j), the estimate of the total population exposed to 11
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Figure 1: Illustration of Population-based risk measure
hazmat between the nodes i and j is defined using Equation 1 (illustrated in Figure 1). hρ + ρ i i j + ρij lij
(1)
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ρtij =
where ρtij is an estimate of the total population between nodes i and j. ρi and ρj represent the actual population of nodes i and j. ρij represents the estimate of population lying on the connecting link between nodes i and j.
Note that the hazmat accident can occur either at the nodes or in between the links connecting the nodes. Hence, in an event of an accident of a vehicle carrying a hazmat, the population residing around the place of accident will be at risk. Consider hazmat m is
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carried along a link consisting of two nodes i and j for a shipment s. In case an accident occurs in the link (i, j) during transportation where the nodes i and j are located in close
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proximity (i.e., shorter links), the population at both nodes would bear a larger risk of exposure. On the other hand, if the nodes are separated by a large distance, the risk of exposure to the population at the nodes is minimal. In case of a large interlink distance; m(s)
ρij
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a significant portion of the population at risk is the one residing along the link. To find for a link carrying shipment s , we multiply ρtij with a parameter severity coefficient
(γm(s) ) of the hazmat m, which results to the amount of people affected on link (i, j) due
m(s)
ρij
= ρtij γm(s)
(2)
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to any adverse incident during transportation of carried hazmat m.
We introduce a term, βij , which represents the probability of an accident in the link (i, j).
The severity parameter, γm(s) which denotes the fraction of the total exposed population actually affected by the hazmat accident, is less than or equal to 1. We account for the population risk from the transport firm’s perspective by including the cost of recovery 12
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component in the objective function.
3.4
Problem Formulation
We define a binary decision variable Xijs that attains a value of 1 if shipment s passes
Xijs =
(
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through the link (i, j). In short,
1 if a shipment s passes through link (i, j) 0 otherwise
(3)
Xijs is a decision variable for the model, which represents the decision about the selection of a particular link (i, j) for shipment s. Since throughout the network there are several
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possible routes available between the origin and destination, the choice of route not only affects the total travel cost but also the recovery cost. Moreover, the number of trucks (nsk ) of each truck type k chosen for the shipment s is also a decision variable, which takes only integer non-negative values. There exists a choice for the transporters to either carry multiple small trucks with same capacity or a single large truck with higher capacity. This choice is reflected by the trade-off between the purchase finance loan cost and the travel
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cost of the trucks.
Let Yijs be another parameter that defines the hazmat network over G = (N, A) and shows
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if the link (i, j) is feasible for transportation of shipment s (roads could be inaccessible due to vehicle type restrictions or topological reasons). By definition, =
(
1 if link (i,j) is accessible to shipment s 0 otherwise
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Yijs
)
(4)
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Let the total number of trucks used for the shipment s is N s where, Ns =
X
nsk
(5)
k∈K
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Here, nsk represents the number of trucks of type k selected for delivering shipment s. Let ds be the demand in metric tons of shipment s. Let em k be a binary parameter, which is
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defined as follows: em k =
(
1 if truck type k can carry hazmat m 0 otherwise
)
(6)
The term qk denotes the truck load capacity of truck type k. We denote the cost per
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kilometer (km) in the link for a truck of type k on link (i, j) as ckij . This cost ckij (unit variable cost of travel of truck type k on link (i, j)) includes fuel cost per kilometer for a truck type, standby cost of vehicle per kilometer, maintenance cost per kilometer, and employee wages. Apart from the travel cost per unit kilometer and the truck purchase finance loan (or lease) cost per time period, there are two additional costs associated with the total cost of shipment: 1) the monetary loss due to truck stoppages, which is the total amount
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incurred on the number of essential stops during the travel period, and 2) the recovery cost per person, which is the amount spent on the total recovery of a victim exposed to hazmat. The cost components are discussed below.
Cost of Travel
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3.4.1
The average cost of travel per kilometer for truck type k on link (i, j) is denoted by ckij . The travel distance (considering both onward and return journey) for delivering a shipment, is
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twice the length of route per trip. Since the number of trucks of type allotted to a shipment s is nsk , the cost of transportation is 2ckij lij nsk . This cost is incurred only over the link (i, j) selected for the transport of shipment s. Such trucks return empty to the origin since the
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trucks are specialized to carry hazmats only and in a realistic scenario, hazmat shipments flow uni-directionally from the refineries to the interior locations of the continent. The
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transporter specifically does not enjoy any advantage of using the truck during return journey for other purposes/shipments mainly due to the specialized nature of trucks as well as unidirectional flow of hazmat shipments. The tankers are built using Petroleum
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& Explosives Safety Organization (PESO) guidelines and are not allowed to carry other commodities. Hence the total cost of travel of shipment s during the planning period is P P expressed by k∈K (i,j)∈A 2ckij lij nsk Xijs .
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3.4.2
Purchase Finance Loan (Lease) Cost of Trucks
Let fk be the purchase finance loan (lease) cost for truck type k for the planning period.
3.4.3
Monetary Loss due to Truck Stoppages
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Since there are nsk trucks of type k in shipment s, the total fixed cost of all trucks of P shipment s in the planning period is given by k∈K fk nsk .
Monetary loss due to truck stoppages is incurred as a result of the total number of essential stops made during the onward and the return journey. Let us denote the number of stops per 1000 kilometers for truck type k as uk and the cost incurred due to one such stop for a truck type k is Bk . Since the actual distance travelled during the shipment delivery between
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every link (i, j) is 2lij and the number of trucks of type k used for shipment s is nsk , the total number of stops on a link (i, j) is 2uk lij Xijs nsk . We multiplied this cost with a binary decision variable Xijs to ensure that the cost is incurred only on the selected shipment route.
3.4.4
Cost of Recovery
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Hence, the expression for loss due to truck stoppages of shipment s in the planning period P P is given by k∈K (i,j)∈A 2uk lij Xijs nsk Bk .
m(s)
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When a vehicle is delivering shipment s along a link (i, j), the population denoted by ρij
carries a risk of hazmat exposure in the event of an accident. Very dangerous type of hazmat commodities such as chlorine pose a substantial transport risk and can expose a
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large number of people in the event of an accident. Such chemicals have a higher severity γm coefficient. We assume that any hazmat m transported by a shipment s poses an identical risk to the population irrespective of the quantity of hazmat it carries (also
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see Verma and Verter (2007) where they assume economies of risk associated with hazmat transportation in unit trains). For example, the evacuation distance upon a hazmat incident is defined in terms of hazmat type and not truck capacity (see, CANUTEC,
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https://www.tc.gc.ca/eng/canutec). Therefore, when multiple vehicles are used for delivering a single shipment, the risk introduced by individual vehicles is added to obtain the total transport risk. If we denote the number of trucks of type k used for shipment s by nsk ,
the cost of long term recovery per person exposed to hazmat m by ϕm(s) , and the probability 15
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of an accident on link (i, j) by βij , we can represent the total expected recovery cost of shipP P m(s) ment s in the planning period using the expression k∈K (i,j)∈A ϕm(s) ρij nsk Xijs βij γm(s) . The objective function is now expressed as a minimization of the sum of the four cost terms. Let the total cost of shipment s be defined by 2ckij lij nsk Xijs
+
k∈K (i,j)∈A
X
k∈K
fk nsk
+
!
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Z(s) = min
" X X
X X
2uk lij Xijs nsk Bk
k∈K (i,j)∈A
!# X X m(s) m(s) s s + ϕ ρij nk Xij βij γm(s) s.t.
X
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k∈K (i,j)∈A
m(s)
nsk ek
k∈K
qk ≥ ds
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Xijs ≤ Yijs ∀ s ∈ S, (i, j) ∈ A +1 if i = O(s) X X s s Xij − Xji = −1 if i = D(s) ∀i ∈ I , s ∈ S (i,j)∈A (j,i)∈A 0 otherwise Xijs ∈ 0, 1 ∀ s ∈ S, (i, j) ∈ A
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(7)
∀s∈S
(8)
(9) (10)
Equation (7) guarantees that there are enough trucks of type k with capacity qk allotted
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to fulfil the demand for a particular hazmat. Equation (8) ensures that non-permissible links are not chosen, while Equation (9) ensures the flow constraint of the route chosen for a particular shipment s. Moreover, the decision to choose a particular route for delivering
Solution Methodology
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4
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a shipment is modeled using a binary decision variable, refer Equation (10).
Amaldi et al. (2011) point out that any version of the hazmat transport network design problem where a subset of arcs are forbidden is NP-hard even when a single commodity 16
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has to be shipped. Palmer and Kershenbaum (1995) and Li and Bouchebaba (2000) have used genetic algorithm for the Optimal Communication Spanning Tree problem, a variant of hazmat network design problem. We resort to the Genetic Algorithm also because the proposed model is a nonlinear mixed integer program. The objective function involves multiplication of decision variables nsk and Xijs in three of the four cost terms. A neighbor-
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hood search meta-heuristic would require the computation of the value of such a complex objective function a large number of times. Thus, we avoided using neighborhood search algorithms in tackling our problem. Since there exist a small number of constraints, however, the primary handicap of genetic algorithms concerning the need to eventually repair solutions to ensure feasibility is much less of an issue here. Therefore, we opted for using the genetic algorithm as our meta-heuristics domain. Since our aim in this paper is to demonstrate the problem dynamics and study the main trade-offs, possible improvements the scope of this paper.
4.1
Chromosome Formation
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to the solution approach we outline below by extending it to a memetic approach is out of
The chromosome here has been defined as the string of decision variables corresponding to
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one particular shipment. The decision variables for truck type selection (ns1 , ns2 , ns3 , ..., nsk ), s , X s , ..., X s , X s , X s , ..., X s , ..., X s , X s , . . . , X s ) and the decision variables for route selection (X11 12 21 22 1I 2I I1 I2 II
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have been conjoined to form one string of length K + I 2 for a shipment. The length of the chromosome increases with second order exponential along with the computation time and memory requirement as the number of nodes increase. Hence one chromosome for
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shipment s can be expressed with length of chromosome as K + I 2 and g(s) is defined the
4.2
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cost associated with one solution.
Population Size
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The initial population size was taken for 200 chromosomes following the trend of setting the minimum population size to be greater than the number of decision variables. The initial population was chosen from a set of feasible values. A range of population sizes were tested for the algorithm and a population size of 500 provided to optimal results, both in 17
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the sample as well as the real example (see Sections 5.1 and 5.2).
4.3
Fitness Function
shall be a transformed one as per the following rule: f (s) =
(
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Since the objective function is a cost minimization function the fitness function chosen here
Cmax − g(s) if g(s) < Cmax 0
otherwise
)
(13)
where g(s) is the cost function as shown in Section 3.4 as an aggregation of four other cost
4.4
Crossover and Mutation
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functions. Cmax is chosen as the maximum value of g(s) observed in the current population.
Crossover convergence operation intends to pull the population towards a local minimum/maximum, while mutation is a variance operation. Mutation is intended to occasionally pull out one or more chromosomes of a population from a local minimum/maximum
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space and potentially discover a better minimum/maximum space.
Since the end goal is to bring the population to convergence, selection/crossover happens more frequently (typically every generation). Mutation, being a variance operation, should
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happen less frequently, and typically only affects a few chromosomes of a population (if any) in any given generation. A range of mutation and crossover probabilities were determined after several empirical runs while observing the fitness function. For the purpose of
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this study, the crossover and mutation probabilities are fixed at 0.1 and 0.01 respectively. We observe that lowering the values of mutation probability does not change the optimal
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solution. However, a higher mutation probability shifts the solution to a sub-optimal point, at which total incurred cost is relatively higher than the optimal cost (see the real case in Section 5.2). Mostly, higher mutation probability affected the choice of truck types in case
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of shipments carrying hazmat 1 and hazmat 4, while the solution remained unchanged for all shipments carrying hazmat 2 and hazmat 3.
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4.5
Termination Criteria
The termination criteria used in the current evaluation is limited by the number of iterations or, the change in fitness value less than 10−6 for 20 successive iterations, whichever is encountered earlier. In the numerical examples, the GA terminates after reaching a solution
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when there is no appreciable change (10-5) in the objective function value over 5 successive generations. The time taken for running of the solution with the design parameters with 16 nodes was 315 seconds. Longer runs of the algorithm do not lead to a better solution. However, the design parameters have been empirically determined to allow a range of mutation and crossover probabilities so that global minimum can be obtained. The pseudocode for
5
Numerical Experiments
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the GA is included in Appendix A.
To illustrate the application of the proposed model, we use two sets of data. The first data set corresponds to a small scale example and is used for the simplicity of exposition. We later illustrate the model using actual data from a bulk hazmat transporter in western
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India. We consider all cost values in Indian Rupees where 1 Indian Rupee = 0.015 US
5.1
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Dollar.
Small Scale Example
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Consider the transportation network shown in Figure 2, which includes five cities (A, B, C, D, and E) and six road links (AB, AC, AD, BE, CE, and DE ). The length of a road link is indicated along the link. First, all possible routes are identified between an origin
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destination pair. Note that this is an illustrative example and therefore, we enumerate the routes.
We consider the total population at risk along the available routes in the network. Let
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this matrix be represented by P . The population figures in Table 2 are expressed as lakh units (×105 ). We consider one shipment to be completed in the planning period (50 T of
Hazmat 1, Acetone, is sent between between source node A and destination node E ). For the purpose of discussion, we shall solve the optimization problem by considering only one 19
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Figure 2: Network nodes and links along with the length of links (in kms) the network (×105 ) people E M 9 8 9 10
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Table 2: Population of Nodes and interlinks of ρij A B C D A 5 4 12 8 B 4 3 M M C 12 M 10 M D 8 M M 6 E M 9 8 9
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shipment route and fleet. A similar procedure can be followed for the remaining shipments that belong to the same planning period. Table 3 gives a description of truck types and
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their capacities that are available for procurement. As discussed earlier, the variable cost of transportation includes fuel consumption cost, administrative expenses per km (wages), and tire cost per km (Table 3). Each truck type
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may have a different cost coefficient associated with the loss due to truck stoppage per stop during the travel operation. In our example, we assume that brand 1 trucks require one stop after running 400 kilometers and brand 2 trucks require one essential stop after
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running 600 kilometers.
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We assume identical administrative cost (driver wages and supervision cost) across all truck types. Trucks with same carrying capacity have same number of tires. Since the cost of changing tires depends on the number of tires in the carrier truck, the cost of changing tires for trucks with identical capacities are the same. The amount of fuel consumed over a distance by different truck types may be different. For instance, it is observed that truck 20
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types 1 and 3 have identical number of tires and equal cost of changing tires, but a different fuel cost per kilometer. Likewise, truck types 2 and 4 incur equal expenditure on changing tires due to a similar reason (Table 3). Table 3: Variable cost components per kilometer for all links in the network Fuel cost/km
Administrative Cost/km
Cost of changing tires/km
Total Variable Cost/km
Capacity (in metric tons) (qk )
Type Type Type Type
10 20 9 18
1.5 1.5 1.5 1.5
0.5 1.5 0.5 1.5
12 23 11 21
10 20 10 20
T T T T
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1 2 3 4
Cost of Purchase Finance Loan (Lease) 10,000 18,000 12,000 19,000
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Truck Type
We also assume that the variable cost of travel per kilometer (km) for a truck type across different links is the same. Other additional travel costs such as toll fees are not considered in this example. It is assumed that for every stop during the travel in the route, a standby cost is incurred (Table 4).
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Table 3 includes the travel cost (variable cost per km.) incurred for the four truck types Table 4: Standby cost/stop and number of stops made for every truck type on different routes
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1 2 3 4
200 250 200 250
No. of stops per 1000 km. 2.50 2.50 1.67 1.67
Route-1
Route-2
Route-3
650.00 812.50 325.00 406.25
650.00 812.50 325.00 406.25
800.00 1000.00 400.00 500.00
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Type Type Type Type
Standby Cost
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Routes/Truck Type
during the delivery of the shipment. It is also assumed that every essential stop is of equal length of time and corresponds to the standby cost structure provided in the Table 4. We
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now determine the total population on the routes (ρtij ) for all links (i, j) using the formula
given in Equation (1) for the network shown in Figure 2. The parameters included in Table 2 are used to estimate the population at risk during delivery of shipments. Here P (i) ∀i ∈ {A, B, C, D, E} is analogous to ρi , P (i − j) ∀ i 6= j is analogous to ρij as
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explained in Section 3.3. The unit of these figures is the same as mentioned in Table 2. ρtAB
"
# P (A) + P (B) 5+3 = + P (A − B) = 4 + = 4.32 AB 25
Following the same procedure for other links, we obtain, −
4.32 12.43 8.27
− 4.32 t ρij = 12.43 − − 8.27 − 9.32
−
−
−
8.67
−
−
−
9.4
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−
9.32 8.67 9.4 −
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We assume that out of the four types of trucks, truck types 1 and 2 incur more losses due to frequent stops in comparison to truck types 3 and 4. It is also assumed that the hazmat (acetone) has a severity, (γm ), value of 0.2. It implies that out of a total of 100 people exposed to the hazmat, the effect of hazmat can be seen actually on only 20 of them. This assumption is made to justify the low severity values of industrial chemicals that are transferred to different cities by hazmat carriers. Many industrial chemicals naturally exist
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as liquids while a few exist as gases. Toxic gases (such as chlorine) are assumed to have very high severity values whereas acids and ketones have low severity values. Therefore,
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radioactive chemicals or toxic gases could have a high long term recovery cost. Let the solution for the shipment be represented by an array where { Xij1 , (n1 , n2 , n3 , n4 } represents number of trucks of type i used in the shipment and Xij1 represents the optimal links. The
results show that the best route is route 2 (AC-CE). The number of truck used per truck
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type in the optimal solution comes out to be {1, 2, 0, 0}. The value of objective function at
this solution is Rs. 58090.
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The solutions to the sample problem described in Figure 2 are investigated to study the trade-offs among the various components of the total cost expression. We calculated the individual components of the total cost expression (i.e., the loss due to truck stoppages, cost
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of recovery, cost of purchase finance loan of trucks, and the variable cost of transportation) for all solutions and examined the trade-offs among them. Particularly, we examined the trade-off between:
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1. loss due to truck stoppages and cost of purchase finance loan (lease) of trucks, and 2. total variable cost and recovery cost. It is important to observe that the acquisition costs of trucks belonging to brands 3 and 4 are higher than that of brands 1 and 2 in order to visualize the trade-off between loss
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due to truck stoppages and the procurement cost of the trucks. For a shipment, if the transporter procures a truck belonging to either brand 3 or brand 4, the losses associated with truck stoppages will be less (owning to fewer stops during the journey). On the other hand, if the transporter includes a brand 1 or brand 2 (lower acquisition cost) in the fleet, the inclusion of such trucks would increase the loss due to truck stoppages (due to frequent stops). If the company is delivering shipments over very long distances, the loss
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due to truck stoppages significantly increases when the shipment is sent through brand 1 trucks (owning to a large number of stops). Hence, for shipment over longer distances, we advise the transporters to use trucks which incur low monetary loss due to stoppages (Xie et al. (2012)). This trade-off can be observed in the Figure 3. The solutions compared are {Xij1 , (1, 2, 0, 0)}, {Xij1 , (0, 2, 1, 0)}, {Xij1 , (1, 1, 0, 1)}, {Xij1 , (0, 1, 1, 1)}, {Xij1 , (1, 0, 0, 2)},
and {Xij1 , (0, 0, 1, 2)}. This set of solutions is chosen to explain the trade-offs at a constant
CE
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risk level.
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Figure 3: Trade-off between purchase finance loan (lease cost) of the trucks and monetary loss due to truck stoppages for six solutions Table 5 corresponds to the good quality solutions of the small scale example mentioned
in Section 5.1 of the paper. The solutions mentioned along with in table 5 and table 6 show 23
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non-dominated solutions with corresponding total cost and population at risk. The trade-off between the variable cost of travel and the total population at risk involves the choice of routes and the total population affected with each route selection. If a shipment is delivered using a large number of low capacity trucks, it incurs high variable cost and high cost of recovery. On the other hand, if the same quantity is transported by a few larger
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capacity trucks, it incurs relatively low variable cost of travel and low cost of recovery. Note that the cost of recovery depends on the number of trucks and is not affected by the capacity of the trucks. In Table 6, we see that all solutions with respect to a population-based risk value, contain the same number of vehicles in their fleet along with the same route choice. The remaining solutions either contain a large number of trucks in their fleet or contain a different route choice. Table 5 and Table 6 show the total cost with and without considering
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the losses due to truck stoppages respectively.
The results show that for the same level of exposure to the population, a number of combinations of fleet qualify; however, the total costs of shipments are unequal. Currently, {Xij1 , (1, 2, 0, 0)} is the best solution; however, if the distances between the nodes are large,
the benefit (savings) of including driver-productive trucks may dominate the additional
expenses of the cost of acquisition of driver-productive trucks. In those cases, the optimal
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solution will shift towards the other end (i.e., {Xij1 , (0, 0, 1, 2)}).
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CE
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Table 5: Total cost vs. population at risk for good quality solutions to the sample problem when losses due to frequent truck stoppages are included; the cells denote the cost figures corresponding to a solution and a risk level Population at Risk Solution Level 1: 13.64(×105 ) Level 2: 21.09(×105 ) {1,(1,2,0,0)} 58,090 {1,(0,0,1,2)} 59,923 {1,(1,0,0,2)} 58,486 {1,(0,2,1,0)} 59,526 {1,(1,1,0,1)} 58,288 {1,(0,1,1,1)} 59,725 {2,(1,1,0,1)} 58,288 {2,(1,0,0,2)} 58,486 To estimate the total recovery cost, we consider the population at risk in the origin and
destination nodes as well as the population that lies between the nodes. Non-uniformity of 24
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population distribution in the network creates the uncertainty that shorter routes may/may not be economically feasible for transportation. Longer routes may incur high transportation costs and low recovery costs (if routes traverse through a sparsely populated region) while shorter routes may incur low transportation costs and high recovery costs (if routes traverse through a densely populated zone). Hence a trade-off is observed between the
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total variable cost of travel (depending on the length of path taken) and the recovery cost (depending on the total population exposed to the hazmat). The variable cost of shipment also increase with the increase in length of the route.
AC
CE
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ED
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Table 6: Total cost (excluding truck stoppages) vs. population at risk for good quality solutions to the sample problem; the cells denote the cost figures corresponding to a solution and a risk level Population at Risk Levels Solutions Level 1: 13.64×105 Level 2: 17.67 ×105 Level 3: 21.09 ×105 {1,(1,2,0,0)} 53,540 {1,(0,0,1,2)} 56,890 {1,(1,0,0,2)} 55,020 {1,(0,2,1,0)} 55,410 {1,(1,1,0,1)} 54,280 {1,(0,1,1,1)} 56,150 {1,(1,2,0,0)} 53,540 {1,(1,0,0,2)} 55,020 {1,(1,1,0,1)} 54,280 {3,(1,2,0,0)} 55,280 {3,(0,0,1,2)} 58,480 {3,(1,0,0,2)} 56,640 {3,(0,2,1,0)} 57,120 {3,(1,1,0,1)} 55,960 {3,(0,1,1,1)} 57,800 {2,(1,2,0,0)} 53,540 {2,(1,0,0,2)} 55,020 {2,(1,1,0,1)} 54,280
The total population at risk is only affected by the length of the route and the popu-
lation density along the route. Table 5 denotes good quality solutions as per the network mentioned in Figure 2. The solution taken into consideration ensures that the total amount transported is not less than the total available demand at different nodes. 25
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5.2
Realistic Case from a Bulk Hazmat Transporter
We now illustrate the model using the operational data from a mid-size bulk transporter in Ahmedabad, Gujarat, India. The company owns a total of 30 trucks of different capacities ranging from 10 to 20 metric tons. The trucks operate between 16 nodes. Moreover, in
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periods of excess demand, it procures trucks on lease from third parties to provide services. During the discussions, several interesting facts were obtained. The decision for the choice of the route was done by the transport company itself. However, they also allowed some flexibility to the driver to choose an alternate route. In general, the routes choice included state/national highways, service lanes and excluded routes that are in proximity to the schools, hospitals, and other densely populated areas. In case of an accident, the driver is well trained to limit the hazmat exposure with safety and caution. The order summary
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of the data is shown in Table 7. Table 8 shows a list of available truck types that can be procured for transportation. The value of the parameter em k is shown in Table 9. Table 10 highlights the severity and recovery cost of the four hazmats in the real world transportation problem.
The estimation of the parameters used for the study (such as variable cost of travel, Standby Cost and Number of essential stops per 1000 kilometer), provided by the transport
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company, considers several factors including number of tires, trailer capacity load, truck capacity, make of truck and frequency of replacement of tires. The set of parameters related
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to truck type in the study is based on averages for every truck type irrespective of the age of the truck. The parameters related to the hazmat are empirically determined to cover a range of severity and cost of mitigation values. Census of India, 2011 remains the source of
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the population data of the nodes used in the study. Probability of road accident in a link, which has been estimated as 35 per 10000 vehicles (see https://data.gov.in/keywords/
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transport-research-wing). In all the cases, the GA terminates after reaching a solution when there is no appreciable change (10−5 ) in the objective function value over 5 successive generations. The time taken
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for running of the solution with the design parameters with 16 nodes is 315 seconds.
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hazmat transporter Origin Destination Node Node O(s) D(s) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1) Kandla(1)
Tarapur(13) Tarapur(13) Tarapur(13) Tarapur(13) Dabasa(14) Dabasa(14) Ankleshwar(15) Ankleshwar(15) Ankleshwar(15) Ankleshwar(15) Ankleshwar(15) Mandideep(16) Mandideep(16)
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1 2 3 4 5 6 7 8 9 10 11 12 13
Table 7: Order summary of the bulk Order Quan- Hazmat Carried tity (in tonnes) (m(s)) (ds ) 20 1 16 2 10 2 16 4 20 1 10 2 20 1 20 2 16 2 26 2 26 1 20 1 16 3
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Shipment number s
Four distinct origin destination pairs are identified in the order summary of the data.
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These pairs are Kandla-Ankleshwar, Kandla-Mandideep, Kandla-Dabasa, Kandla-Tarapur (see Figure 4a for maps of the Gujarat state and possible network of nodes between Kandla and Ankleshwar). The origin corresponds to the Port of Kandla. The length of the links
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for the network corresponding to each origin destination pair and the population of nodes and links are shown in Table 11 and 12 respectively. These parameter values are obtained by interpreting the Google maps and data from the Census of India (2011) (Figure 4b).
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For the purpose of estimating population along the road links, it is assumed that the population density in a district is uniformly distributed. A one kilometer rectangular region is considered for calculating the total population along the link on both sides of the road.
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For route links lying between two different districts, it is assumed that the population figure in the first half of the link belongs to one district and the remaining half belong to the other district. The result for this problem is shown in the Table 13. Figure 4c shows the feasible
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links in the network of 16 nodes considered in the study. The actual problem was modeled in a similar fashion where Table 13 describes the total
number of orders that were received for a particular period. For all the orders shown in
27
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the Table 13, the origin node of the shipment is Kandla (denoted as Node 1), while the destination of the shipment could be any of the following: Tarapur (denoted by Node 13), Dabasa (denoted by Node 14), Ankleshwar (denoted by Node 15), and Mandideep (denoted by Node 16). Hazmats 1 to 4 are MDC, Acetic Acid, MEK, and Acetone, respectively. The total cost corresponds to the sum of individual cost components as mentioned in section the shipment must pass in order to incur the least cost.
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3.4. The route shown in Table 13 is the optimal route mentioning the nodes through which It can be observed that the same truck type is not always the optimal choice for carrying shipments with the same hazmat over the same route. For instance, truck type 4 is chosen for transporting shipment with hazmat 1 and order quantities less than 20 tonnes (shipment numbers 1, 5, 7, and 12). However, the truck types 2 and 3 are chosen to transport the
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same hazmat 1 with an order quantity of 26 tonne (see Shipment 11). This choice is clearly due to variation in different cost components influencing the choice of trucks even between the same origin and destination pair. Further, consider the shipments 10 and 11. These two shipments correspond to different hazmat types but have same order quantities of 26 tonnes. Though the optimal choice of route and the number of trucks for carrying both shipments are identical, the truck types differ (3 and 9 for shipment 10 vs. 2 and 3 for shipment 11). Since hazmat types 1 and 2 differ in the recovery cost parameters, different
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choice of trucks minimize the other truck-related costs over the same route. Also note that the solutions in Table 13 correspond to truck types 3, 4, 8, and 9, which does not include
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the cheapest truck type option 2. Truck type 8 is 85% more expensive than Truck type 2; however, the additional purchase cost offsets the reduction in other cost components.
Comparative Study of Different Solution Approaches
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6
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In this section, we compare three different solution approaches to fleet mix bulk hazmat transportation problem and compare their solution quality in terms of cost. The three
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different solution approaches include: 1. Lowest truck purchase cost approach 2. Two-stage sequential solution approach 3. Integrated solution approach 28
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Lowest truck purchase cost approach is most commonly used to decide fleet mix among hazmat transporters in India. In this approach, the lowest cost truck is selected to fulfill the demand and the shortest route is selected between the source and destination pair corresponding to a shipment. The transporters typically choose the shortest route to minimize cost. The two-stage sequential solution approach is performed to see if the integrated prob-
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lem can be decomposed into two simpler subproblems and solved to optimality. In the first stage, we chose to first select the route corresponding to minimum cost of recovery. Note that we did not consider the travel cost criterion to select the route because the shorter distance routes often correspond to high density population centers. In the second stage, we determine the optimal fleet mix corresponding to the chosen route in the first stage. In the second stage, we minimize the travel cost, purchase cost, and loss due to truck stoppages. tify the optimal routes and fleet mix.
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The integrated solution approach presented in this paper attempts to simultaneously iden-
Table 14 compares the cost components corresponding to the optimal solution obtained for the bulk hazmat transporter using the three stated approaches. In approach 1 (lowest truck purchase cost with the shortest route), the travel cost is quite low but the cost of recovery is high due to high population risk. In this case, the cost of recovery exceeds the
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cost due to truck stoppages. In the two-stage sequential approach, we observe that the fleet mix choices for all the 13 solutions are same as the optimal fleet mix obtained by
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the integrated solution (approach 3). However, the route choices obtained in the two-stage approach differed from the optimal choice obtained from the integrated solution approach.
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The difference in optimal costs obtained from approaches 1 and 2 in comparison to integrated approach suggests that the integrated solution approach yields better quality
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solutions. Based on the parameters considered in the study, average cost savings of approach 3 over approach 1 is 14.04% (average cost difference of Rs. 3,90,823) and average cost saving of approach 1 over approach 2 is 0.43% (with an average cost difference of Rs.
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10,790). Table 14 also highlights that though some solutions obtained from integrated solution approach are common with solutions obtained from approaches 1 and 2, there is no such instance where integrated solution approach performs worse than solutions obtained from either of the approaches 1 and 2. The solutions obtained from the integrated solution approach have a few common instances with the lowest truck cost approach (highlighted 29
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in bold) and with the two-stage sequential solution approach (highlighted in italics), and an instance common to all the three approaches (highlighted using an underline). We infer that integrated solution approaches provides best solutions irrespective of the nature of the shipments which include distance between source and destination nodes for a shipment and
7
Conclusion and Future Work
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quantity and severity of the hazmat carried in the shipment.
Transporters are currently seeking alternate methods to improve their loss due to truck stoppages, minimize transportation risk, and deliver shipments with lower travel costs. While the fleet mix affects the number of truck stoppages and the travel costs, the choice of
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the routes affects the transportation risk. Through a real case from a mid-size transporter, we show that the choice of the fleet can affect the selected route for shipment delivery. In particular, our results show that the truck type with the lowest purchase cost is not the optimal choice. We believe that this paper is the first attempt to capture the effect of truck type on loss due to truck stoppages and fleet mix decision. Using a non-linear optimization minimum cost solutions.
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based integer programming model, we capture the three-way cost interaction and obtain
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In our case study, we assume that the fixed cost of acquisition corresponds to the cost of hiring a truck for a single shipment delivery. It is likely that the distribution of shipments over different planning periods might be irregular. It rests on the management of the
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carrier company to ensure that they do bring sufficient demand receipts over the network so that none of their trucks remain idle. This research could be extended by including time-based stoppages and hour-of-service regulations in truck movements. It means that
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truck drivers may commute longer distances during the night time than the day time due to reduced traffic and less congestion and some necessary stoppages may be necessary for driver safety. Hence, the departure of truck may be scheduled in such a manner that it
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avoids the peak hours of traffic during shipment delivery. The model may also be adjusted to consider time-based variations in truck-speed parameters and its effect on travel cost and route selection. Acknowledgements: We thank the editor and the reviewers for their detailed comments, 30
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which have led to improved exposition of this paper.
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519.
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Input: Instance Π, Rate c of Elitism, Rate m of Mutation,
Output: Solution X
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// Initialization
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Number iter of iterations
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Size P of Population,
Generate P feasible solutions randomly; Save them in the population P op ;
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// Loop until the terminal condition for i = 1 to iter do
// Elitism based selection
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Number of elitism ne = P.c;
Select the best ne solutions in P op and save them in P op1 ; // Crossover Number of crossover nc = (P − ne )/2; 36
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for j = 1 to nc do Randomly select two solutions XA and XB from P op; Generate XC and XD by one-point crossover to XA and XB ; Floor or Ceil to nearest integer and check bounds Save XC and XD to P op2 ; // Mutation for j = 1 to nc do Select a solution Xj from P op2;
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endfor
0
Mutate each bit of Xj under the rate m and generate a new solution Xj ; 0
if Xj is infeasible 0
0
endif 0
update Xj with Xj in P op2; endfor // Updating update P = P op1 + P op2 ; endfor
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// Returning the best solution
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Update Xj with a feasible solution by repairing Xj ;
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Return the best solution X in P op;
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Table 8: Various truck types available (and the associated cost components) to the transporter for lease Capacity (in MT) qk
Standby Cost
Cost of Acquisition (in Rs.)
1 2 3 4 5 6 7 8 9
Brand Brand Brand Brand Brand Brand Brand Brand Brand
10 10 16 20 20 20 20 20 10
300 200 300 500 500 300 400 500 300
1,350,000 1,300,000 1,800,000 2,200,000 2,250,000 2,400,000 2,500,000 2,400,000 1,350,000
1 2 2 2 4 3 1 3 1
No. Stops 1000 uk 7.5 10.0 10.0 10.0 8.0 8.0 7.5 8.0 7.5
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Table 9: Parameter values of em k for all possible values Hazmat (m)/ Methylene Acetic Methyl Truck Type (k) Dichloride Acid Ethyl (MDC) Ketone (MEK) Type 1 1 0 1 Type 2 1 0 1 Type 3 1 1 0 Type 4 1 0 1 Type 5 1 0 1 Type 6 1 0 1 Type 7 1 0 1 Type 8 1 1 1 Type 9 1 1 0
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of per km,
Fuel Cost per km
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Brand Manufacturer
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Truck Type k
13.2 12.5 17.8 22.0 24.0 24.5 25.0 28.0 13.2
of k and m Acetone
1 1 1 1 1 1 1 1 0
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Table 10: Severity and recovery cost parameter values associated with hazmat m Hazmat (m) Methylene Acetic Acid Methyl Ethyl Acetone Dichloride Ketone (MEK) (MDC) Severity 0.05 0.3 0.36 0.75 Recovery Cost (in Rs.) 2000 106 500 240
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39 Figure 4: a) Average population density of all districts of Gujarat (Census of India 2011) b) Network from Kandla to Ankleshwar along with the nodes falling on the routes (Google Maps), and c) Available links within the network
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Table 11: Length of links within the network as described in Figure 4(b) Kandla 0 62.5 -
Samkhiyali 62.5 0 43 165 -
Maliya 43 0 37.8 41.1 -
Morbi 37.8 0 46.2 105 -
Halvad 41.1 46.2 0 73.2 97 157 206 -
Jalalabad 165 0 149 578 -
Sayla 105 73.2 0 35.2 -
Limbdi 97 35.2 0 101 -
Vatva 157 149 0 52.4 -
Nadiad 206 52.4 0 391 62.7 -
Dewas 578 391 0 175
Vadodra 62.7 0 20 90.8 -
Tarapur 101 0 58.3 -
Dabasa 20 58.3 0 -
Ankleshwar 90.8 0 -
Mandideep 175 0
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Distance Kandla(1) Samkhiyali(2) Maliya(3) Morbi(4) Halvad(5) Jalalabad(6) Sayla(7) Limbdi(8) Vatva(9) Nadiad(10) Dewas(11) Vadodra(12) Tarapur(13) Dabasa(14) Ankleshwar(15) Mandideep(16)
Maliya 8299 14022 12852 10418.85 -
Morbi 12852 178055 11711.7 26617.5 -
Halvad 10418.85 11711.7 24325 12224.4 16199 82974.5 72924 -
Jalalabad 23100 606 83738 132073 -
Sayla 26617.5 12224.4 13185 5878.4 -
Limbdi 16199 5878.4 40071 44389.5 -
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Samkhiyali 2875 5897 8299 23100 -
Vatva 82974.5 83738 4525013 37492.2 -
Nadiad 72924 37492.2 196793 149362 34265.55 -
Dewas 132073 149362 231672 33250
Vadodra 34265.55 1491045 11040 35866 -
Tarapur 44389.5 14934 36845.6 -
Dabasa 11040 36845.6 7240 -
Ankleshwar 35866 112643 -
Mandideep 33250 39859
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Kandla 14695 2875 -
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Table 12: Population of links within the network as described in Figure 4(b) Distance Kandla(1) Samkhiyali(2) Maliya(3) Morbi(4) Halvad(5) Jalalabad(6) Sayla(7) Limbdi(8) Vatva(9) Nadiad(10) Dewas(11) Vadodra(12) Tarapur(13) Dabasa(14) Ankleshwar(15) Mandideep(16)
Order Quantity (in tonnes) 20 16 10 16 20 10 20 20 16 26 26 20 16
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Shipment No. 1 2 3 4 5 6 7 8 9 10 11 12 13
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Table 13: Route and truck choices made for delivering the shipments of bulk hazmat transporter Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Origin 1 1 1 1 1 1 1 1 1 1 1 1 1
Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
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Total Cost 2266645.51 1849183.85 1390844.45 1867415.60 2287151.50 1404423.16 2363293.71 2554627.05 1920711.25 3371106.59 3319635.12 2432094.28 2432800.72
Optimal Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-10-12-14 1-2-3-5-10-12-14 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-11-16 1-2-6-9-10-11-16
Optimal Trucks Number (Type) 1(4) 1(3) 1(9) 1(3) 1(4) 1(9) 1(4) 1(8) 1(3) 1(3),1(9) 1(2),1(3) 1(4) 1(4)
AC Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Quantity 20 16 10 16 20 10 20 20 16 26 26 20 16
Quantity 20 16 10 16 20 10 20 20 16 26 26 20 16
41 Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
Destination 13 13 13 13 14 14 15 15 15 15 15 16 16
No. of Truck(Type) 1(4) 1(3) 1(9) 1(3) 1(4) 1(9) 1(4) 1(8) 1(3) 1(3),1(9) 1(2),1(3) 1(4) 1(4)
Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-10-12-14 1-2-3-5-10-12-14 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-11-16 1-2-6-9-10-11-16
Cost of Travel 15162.40 12267.80 9097.40 12267.80 19153.20 11491.90 36938.00 47012.00 29886.20 52049.00 50873.70 53649.20 53649.20
Integrated Problem Solution Purchase Finance Loan Cost 2200000 1800000 1350000 1800000 2200000 1350000 2200000 2400000 1800000 3150000 3100000 2200000 2200000
Cost of Recovery 17023.11 16240.05 16240.05 34471.80 24468.30 23342.76 42405.71 40455.05 40455.05 80910.09 84811.42 56515.08 57221.52
Total Cost 2266645.51 1849183.85 1390844.45 1867415.60 2287151.50 1404423.16 2363293.71 2554627.05 1920711.25 3371106.59 3319635.12 2432094.28 2432800.72
Total Cost 2266645.51 1849183.81 1390844.49 1867415.56 2306908.61 1425409.08 2363524.74 2555306.20 1934859.76 3407471.91 3362572.33 2434395.50 2435671.96
Total Cost 2678844.22 2781688.99 1390844.45 2713741.60 2705525.60 1404423.16 2760523.42 2864076.80 2864076.80 3371106.59 3319635.12 2842367.16 2843780.03
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Loss due to Stoppages 34460.00 20676.00 15507.00 20676.00 43530.00 19588.50 83950.00 67160.00 50370.00 88147.50 83950.00 121930.00 121930.00
Cost of Recovery 17023.11 16240.05 16240.05 34471.80 48891.01 46642.02 90646.34 86476.60 86476.60 172953.21 181292.67 102117.10 103393.56
Two-Stage Sequential Problem Solution Cost of Travel Purchase Finance Loan Cost Loss due to Stoppages 15162.40 2200000 34460.00 12267.76 1800000 20676.00 9097.44 1350000 15507.00 12267.76 1800000 20676.00 17727.60 2200000 40290.00 10636.56 1350000 18130.50 22268.40 2200000 50610.00 28341.60 2400000 40488.00 18017.16 1800000 30366.00 31378.20 3150000 53140.50 30669.66 3100000 50610.00 40418.40 2200000 91860.00 40418.40 2200000 91860.00
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Cost of Recovery 34046.22 32480.09 16240.05 68943.60 48936.60 23342.76 84811.42 80910.10 80910.10 80910.09 84811.42 113030.16 114443.03
Lowest Truck Cost Solution Purchase Finance Loan Cost 2600000 2700000 1350000 2600000 2600000 1350000 2600000 2700000 2700000 3150000 3100000 2600000 2600000 Loss due to Stoppages 27568.00 31014.00 15507.00 27568.00 34824.00 19588.50 46592.00 52416.00 52416.00 88147.50 83950.00 79592.00 79592.00
Cost of Travel 17230.00 18194.90 9097.40 17230.00 21765.00 11491.90 29120.00 30750.70 30750.70 52049.00 50873.70 49745.00 49745.00
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Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13-14 1-2-3-5-8-13-14 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-12-15 1-2-3-5-10-11-16 1-2-3-5-10-11-16
Route 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-8-13 1-2-3-5-10-12-14 1-2-3-5-10-12-14 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-12-15 1-2-6-9-10-11-16 1-2-6-9-10-11-16
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No. of Truck(Type) 1(4) 1(3) 1(9) 1(3) 1(4) 1(9) 1(4) 1(8) 1(3) 1(3),1(9) 1(2),1(3) 1(4) 1(4)
No. of Truck(Type) 2(2) 2(9) 1(9) 2(2) 2(2) 1(9) 2(2) 2(9) 2(9) 1(3),1(9) 1(2),1(3) 2(2) 2(2)
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CE Hazmat 1 2 2 4 1 2 1 2 2 2 1 1 3
Quantity 20 16 10 16 20 10 20 20 16 26 26 20 16
Table 14: Appendix A.2.: Comparison of solution methods
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