In this section, we present a bi-objective nonlinear programming model (P) for managing rail-truck intermodal transportation of hazardous materials.
Sets:
Set of shippers, indexed by i
Set of origin terminals, indexed by j Set of destination terminals, indexed by k Set of receivers, indexed by l
Set of traffic-classes, indexed by z. The elements of this set are derived from
pairing every shipper i∊ I = {1, 2, . . . ,a} with the receiver l∊ L = {1, 2, . . . , f} Figure 3-2: A view of queuing at origin terminal j and destination terminal k with
respectively three and two pieces of equipment Origin terminal j 𝜆̅ 𝑗,𝜆 𝑗 𝜆̅ 𝑗,𝜆 𝑗 𝜆̅ 𝑗,𝜆 𝑗 𝜆̅ 𝑘,𝜆 𝑘 𝜆̅ 𝑘,𝜆 𝑘 Destination terminal k
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Set of inbound drayage between each shipper i∊ I = {1, 2, . . . ,a} and each
origin terminal j∊ J = {1, 2, . . . ,b}, indexed by p.
Set of outbound drayage between each destination terminal k∊ K = {1, 2, . . . ,e}
and each receiver l∊ L= {1, 2, . . . , f}, indexed by q.
Set of intermodal train services between each terminal pair j-k, where j∊ J = {1,
2, . . . ,b}and k∊ K = {1, 2, . . . ,e}, indexed by v.
Set of train service legs for intermodal train service type v operating between
terminals j∊ J= {1, 2, . . . ,b} and k∊ K = {1, 2, . . . ,e}, indexed by s Set of equipment under consideration at origin terminal j, indexed by m
Set of equipment under consideration at destination terminal k, indexed by m′ Input parameters:
A large number
Cost of moving one hazmat container on path p for inbound drayage ̅ Cost of moving one regular container on path p for inbound drayage
Cost of moving one hazmat container using intermodal train service of type v ̅ Cost of moving one regular container using intermodal train service of type v
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̅ Cost of moving one regular container on path q for outbound drayage
Purchase cost of an equipment at origin terminal j Purchase cost of an equipment at origin terminal k
Fixed cost of operating intermodal train service of type v
Population exposure due to moving one hazmat container on path p for inbound drayage.
Population exposure due to moving one hazmat container on intermodal train service of type v.
Population exposure due to moving one hazmat container on path q for outbound drayage.
The exposure caused by a unit of hazmat container in the queue of an equipment in origin terminal j
The exposure caused by a unit of hazmat request in the queue of an equipment in destination terminal k
Inbound drayage time using path p
Travel time of intermodal train service of type v Outbound drayage time using path q
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Maximum number of containers that can be loaded on intermodal train service of
type v
Service rate at each equipment at origin terminal j Service rate at each equipment at origin terminal k
Expected demand for hazmat containers in traffic-class z ̅ Expected demand for regular containers in traffic-class z Decision variables:
Expected hazmat containers of traffic-class z using path p for inbound drayage ̅ Expected regular containers of traffic-class z using path p for inbound drayage
Expected hazmat containers of traffic-class z on train service of type v. ̅ Expected regular containers of traffic-class z on train service of type v.
Expected hazmat containers of traffic-class z using path q for outbound drayage ̅ Expected regular containers of traffic-class z using path q for outbound drayage
1 if ; 0 otherwise 1 if ; 0 otherwise 1 if ; 0 otherwise
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̅ 1 if ̅ ; 0 otherwise
̅ 1 if ̅ ; 0 otherwise
̅ 1 if ̅ ; 0 otherwise
Number of intermodal train service of type v
1 if new equipment m in origin terminal j is acquired; 0 otherwise ́ 1 if new equipment m′ in destination terminal k is acquired; 0 otherwise Arrival rate of hazmat containers to equipment m in origin terminal j
̅ Arrival rate of regular containers to equipment m in origin terminal j
Arrival rate of hazmat containers to equipment ́ in destination terminal k
̅ ́ Arrival rate of regular containers to equipment ́ in destination terminal k
∑ ∑ [ ̅ ̅ ] ∑ ∑ [ ̅ ̅ ] ∑ ∑ ̅ ̅ ∑ ∑ ∑ ∑ ∑ ́ ́ (3-1) ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ̅ ∑ ∑ ́ ́ ̅ ́ ́ ́ (3-2) s.t.
69 ∑ ∑ (a) (3-3) ∑ ̅ ∑ ̅ (b) ∑ ∑ (c) ∑ ̅ ∑ ̅ (d) ∑ (a) (3-4) ∑ ̅ ̅ (b) ∑ ̅ (3-5) (a) (3-6) (b) (c) ̅ ̅ (d) ̅ ̅ (e) ̅ ̅ (f) ∑ ∑ ∑ (a) (3-7) ∑ ̅ ∑ ∑ ̅ (b) ∑ ́ ́ ∑ ∑ (c) ∑ ̅ ́ ́ ∑ ∑ ̅ (d) ̅ ( ) ́ ̅ ́ ́ (3-8) ́ (a)
70 ̅ ̅ ̅ ̅ ( ̅ )( ) ́ ̅ ́ ( ́ ̅ ́ ) ́ ́ (b) ̅ (a) (3-9) ́ ̅ ́ ́ (b) ̅ (a) (3-10) ́ ̅ ́ ́ (b) ̅ ̅ ̅ ̅ ́ ̅ ́ (3-11) ̅ ̅ ̅ ́ { }
This bi-objective model aims to minimize the total cost and the total risk. The cost objective (3-1) contains inbound and outbound drayage costs, rail-haul cost, the fixed cost to operate different types of train services, and the equipment acquisition cost at multimodal terminals. The risk objective (3-2) represents the population exposure due to inbound and outbound drayage, intermodal trains in the network, and congestion of hazmat containers at intermodal terminals. To evaluate the congestion risk, we have used average number of hazmat containers waiting to be served, which is equal to hazmat arrival rate multiplied by average hazmat waiting time. Constraint (3-3) represents the transshipment function being performed by different terminals, while accounting for different types of intermodal train service in the network. Constraint (3-4) guarantees that each receiver’s demands are satisfied. Constraint (3-5) evaluates the number of each train service needed. UvNv represents the capacity of a service type v, which is equal to the maximum number of containers hauled over each of its legs. For example, if a service has
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one intermediate stop, and therefore, is composed of two legs, each carrying 50 and 100 containers respectively, then UvNv = max (50, 100) = 100. Assuming the maximum length for each train (Uv) is 20 containers, five trains for that service are required to carry the containers. In other words, the number of trains for a particular service is determined by the service leg on which maximum number of railcars would have to be moved. Constraint (3-6) sets the indicator variables associated with different links, and this information is used in (3-8) to evaluate the feasibility of including that link in forming an intermodal chain. Constraint (3-7) ensures that the sum of the rates of hazmat and regular containers served on all equipment at each terminal are equal to related arrival rate. The nonlinear constraint (3-8) ensures that all shipments arrive at the customer location by the specified delivery-times. The travel time for a shipment is composed of inbound and outbound drayage time, travel time of intermodal train, average waiting and service time in terminals. Constraint (3-9) ensures that a request (hazmat or regular) can be allocated to a piece of equipment only if that equipment is purchased, while constraint (3-10) enforces queue steady-state conditions. The feasible domains of the decision variables are defined in (3-11).