Mathematical Model

1. Model Assumptions

The basic modeling assumptions are as follows:

(1) Given all attributes of links within the studied network including cost, time, and capacities.

(2) We assume that the information of available routes and terminals is prior known in the pre-disruption phase

(3) The freights of each shipment or order are detachable. The model can transport the partial freights of each order, but it can’t transport by different paths.

(4) The freights with identical shipment only can be delivered by one path. The identical shipment owns the same origin and destination.

(4) The renting time contain the dispatching time from the carriers.

(5) The schedules of different modes can link up smoothly. We don’t consider the connecting or waiting time when the schedules are not fit.

2. Network Presentation

The network which we consider is presented as an intermodal logistic network with multihub, multimode,multicarrier and multicommodity. It is composed of service centers, airport and seaport terminals and links connecting these terminals.

The network is given by G=(N,L), where N={1,…,n} is set of nodes and L={(i.j)| i,j

∈ N } is the set of links. Except for the international express network, the other networks are considered in the recovery option.

The freight demand to be satisfied is given by a set of orders b ϵ B. For each order, b, the freights have specific origin, destination and cargo time value function depending on the freight characteristic. The freights form different orders may have the same origin and destination (O-D pair) or same cargo time value function. A path is defined as an

3. Failure type

We divide the failure types into three categories which are link failure, mode failure , and node failure. The corresponding capacity reduced factor are 𝛼_𝑖𝑗^𝑚 , 𝛽_𝑖 , 𝑎𝑛𝑑 𝛾_𝑖.

We consider the situation of aircrafts and trucks damaged to the mode failure.

Although it may belong to a slight disruption that is not the primary focus in this study, but we also can add it to the other failure type to discuss them together.

4. Symbol explanation

Notations employed in the problem formulation are synopsized as follows:

notation meaning

G (N, L)

N is the set of nodes in the process network. i,j ∈ N L is the set of arcs in the process network. (i,j) ∈ L

B is the set of shipment numbers. It can inform that freights are from which origin to which destination. b ∈ B

M is the set of modes. m ∈ M Tr is the set of trucks. Tr ∈ M Ai is the set of aircrafts. Ai ∈ M Sh is the set of ships. Sh ∈ M

Ra is the set of high speed trains. Ra ∈ M

E is the set of multi-carrier recovery activities that include renting

partner’s mode capacity and renting non-partner’s mode capacity. e ∈ E E={ using partner’s modes, using non-partner’s modes}

o is the set of origin. o ∈ OS

d is the set of destination. d ∈ DS

OS is the set of service center in the origin country. OS∈N OP is the set of seaport or airport in the origin country. OP∈ N DS is the set of service center in the destination country. DS∈ N

Decision Variables are defined below:

𝑓_𝑖𝑗^𝑏𝑚 the amount of freights with shipment number b shipped on arc(i,j) by mode m during post-disaster

𝑊𝐿_𝑖𝑗 the capacities of private trucks which are re-allocated to arc(i,j).

𝐾𝑅_𝑖𝑗^𝑚𝑒 rented capacity of mode m on arc (i,j) from partner e

𝐴^𝑏 the total shipping time from the origin to the destination for the cargos with shipment number b

The given parameters are defined as below:

1. capacity

𝐾_𝑖𝑗^𝑚 = maximum allowable capacities of arc (i,j) for mode m 𝐾𝐿_𝑖 = the available capacities of own trucks at node i

𝑊_𝑖𝑗 = the capacities of original trucks allocation on arc (i,j) before the disruption occurs.

𝑊𝐴_𝑖𝑗 = available flight capacities on arc (i,j).

𝐴𝐾_𝑖𝑗^𝑚𝑒 = available renting capacities on arc (i,j) of mode m by carrier e (partner)

2. capacity reduced factor

𝛼_𝑖𝑗^𝑚 = capacity reduction factor on arc(i,j) of mode m due to the damage of links

𝛽_𝑖 = capacity reduction factor at node i due to the damage of modes 𝛾_𝑖 = capacity reduction factor at node i due to the damage of nodes

3. cost

𝑅𝐶^𝑚𝑒 = cost of renting capacity from carrier e for mode m on arc (i, DP is the seaport or airport in the destination country DS is the service

center in the destination country. DP∈ N

TP is the seaport or airport in the transshipment country, including the hub airport. TP∈ N

j) ,including transportation cost and dispatching cost

𝑉𝐶 = the cost of changing the allocation of own trucks

4. time

𝑆_𝑖 = service time and sorting time at node i.

𝑇_𝑖𝑗^𝑚 = travel time from node i to node j by mode m

𝑅𝑇_𝑖^𝑚𝑒 = contacting, dispatching and preparing time for mode m from the carrier e at node i if company rents partner’s or non-partners’ mode capacities on arc (i,j).

𝐸𝑇 = additional time spending on changing the allocation of trucks

5. others

𝐷𝑀_𝑜𝑑^𝑏 = total demands of cargos with shipment number b that can be satisfied from origin o to destination d during pre-disaster

𝜇(𝑡)^𝑏= cargo time value function of commodities with shipment number b (cargo time value per unit freight)

𝛿_𝑖𝑗^𝑏𝑚= indicator variable；𝛿_𝑖𝑗^𝑏𝑚 = 1 if cargo flow with shipment number b passes through arc (i,j) using mode m and 𝛿_𝑖𝑗^𝑏𝑚 = 0 otherwise.

𝑦_𝑖𝑗^𝑚𝑒 = indicator variable；𝑦_𝑖𝑗^𝑚𝑒 = 1 if recovery capacity (𝐾𝑅_𝑖𝑗^𝑚𝑒) is not zero and 𝑦_𝑖𝑗^𝑚𝑒 = 0 otherwise.

5. Mathematical model

Based on the notation and variables, the resilient model is presented as following.

Max ∑ [𝜇^𝑏(𝐴^𝑏) (∑ 𝑓_𝑖𝑗^𝑏𝑚

∑ 𝑊𝐿_𝑖𝑗

To reach the maximized customers’ satisfaction, we want the cargos be delivered as more as possible even during the disruption, but the cargos may not be delivered totally. Thus, we hope the express company to transport the cargos with higher value first and the cargo value depends on the time which can reflect the characteristic of the express industry. The first objective function (1) is the maximization of the product of total cargo time value and the corresponding throughput. On the other hand, when we want to deliver the cargos during the disruption, the cost of recovery must increases. This is not the expectation of the express company. It wants to lower cost. Thus, the second objective function (2) is minimization of the total incremental resilient costs including the renting cost and cost of changing truck allocation. The express company needs to balance the customers’ satisfaction and the recovery cost at the same time.

It is worth noting that when the value of second objective function is equal to zero, the system may still be able to deliver cargos. Because we are not going to minimize the total transportation cost. We only minimize the additional cost caused by the recovery activities so even if we don’t adopt any recovery strategies, the model can transport the cargos.

The constraints are explained as below. Constraint (3), (4) and (5) are the link capacity constraints. If link fails, constraint (3) will be affected. If node fails, constraint (4) and (5) will be affected. Constraint (6) and (7) are the mode capacity constraints for ground transportation.

The freight flow doesn’t exceed the own trucks capacities and renting capacities. If the freights are transported by train, the flows are less than or equal to the rent capacities (constraint (7)). Constraint (8) and (9) are the same with (6) and (7), but they are the mode capacity constraints for the transnational transportation. 𝑊𝐴_𝑖𝑗^𝑚𝑒 is the own flight capacities.

Constraint (10) and (11) require that the truck capacities and rent capacities don’t exceed the available capacities.

Constraint (12) and (13) are the flow conservation constraints. Demand constraints (14) and (15) guarantee that the total number of shipments flowing out the origin and in the destination don’t exceed the demand for the O-D pair. Constraint (17) specifies that at most one path can be chosen by the identical shipment and (18) and (19) define the binary variables of 𝛿_𝑖𝑗^𝑏𝑚 and 𝑦_𝑖𝑗^𝑚𝑒.

The total shipping time of each shipment represents the cargo value. Any adopted recovery activities will increase the recovery time and reflect on the shipping time. Thus, the constraint (20) calculates the total shipping time, including handling time at nodes ,

transportation time , renting time, and re-allocating time. The renting time includes the transshipping time.