1

ROUTING OF MULTIMODAL FREIGHT TRANSPORTATION USING A 2

CO-SIMULATION OPTIMIZATION APPROACH 3

4

5

6

7

Yanbo Zhao 8

Department of Electrical Engineering, University of Sothern California 9

3740 McClintock Ave. Los Angeles, CA 90089 10

Tel: 213-740-2244； Email: yanbozha@usc.edu 11

12

Petros Ioannou, Corresponding Author 13

Department of Electrical Engineering, University of Sothern California 14

3740 McClintock Ave. Los Angeles, CA 90089 15

Tel: 213-740-4452； Email: ioannou@usc.edu 16

17

Maged Dessouky 18

Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern 19

California, Los Angeles, CA 90089 20

Tel: 213-740-4891; Email: maged@usc.edu 21

22

23

Word count: 5962 words text + 6 tables/figures x 250 words (each) = 7462 words 24

25

26

27

28

29

30

Submission Date: November 15, 2016 31

Zhao, Ioannou, Dessouky 2

ABSTRACT 1

The complexity and dynamics of multimodal freight transportation with the unpredictability of the 2

effects of incidents, disruptions and demand changes make the optimum routing of freight traffic a 3

challenging task. Making routing decisions in a multimodal transportation environment to 4

minimize a certain objective relies on estimating the dynamical states of the multimodal traffic 5

network. The purpose of this paper is to formulate a multimodal freight routing problem and then 6

propose a COSMO (CO-SiMulation Optimization) approach that can lead to more efficient 7

decisions in freight routing by exploiting the availability of powerful computational software tools 8

in the state estimations of complex and dynamic multimodal transportation networks. In the 9

proposed approach, we develop a novel load balancing methodology to optimize routing decisions 10

from an overall system perspective. A simulation testbed consisting of a road traffic simulation 11

model and a rail simulation model for the Los Angeles/Long Beach Port area has been developed 12

and applied to demonstrate the efficiency of the proposed approach. 13

14

15

16

17

Keywords: Freight Routing, Multimodal, Simulation, Optimization 18

Zhao, Ioannou, Dessouky 3

1. INTRODUCTION 1

Efficient freight movement is an essential factor not only in urban transportation but also in social 2

and economic development as well as environmental considerations (1-3). The growth of 3

worldwide trade will significantly increase traffic congestion and air pollution due to existing 4

congestion in the urban transportation infrastructure especially in metropolitan areas with major 5

ports such as Los Angeles/Long Beach where there is a high concentration of both freight and 6

passenger traffic that share the same infrastructure. One of the biggest challenges for freight 7

transport efficiency in such a multimodal environment arises from the fact that the same rail and 8

road networks are used for moving people in addition to goods which leads to non-homogeneous 9

traffic. This non-homogeneity has a detrimental impact on the transportation system performance 10

because of the differences of vehicle sizes and dynamics between passenger and freight transport. 11

The freight vehicles such as freight trains and trucks take longer distances to stop and time to 12

accelerate from a stopping position, consume more fuel and generate more air pollution compared 13

to passenger vehicles. The situation becomes even worse during incidents and disruptions that lead 14

to network changes such as road or railway closures that require rapid response and distribution of 15

freight traffic across the multimodal network. Without efficient routing of the freight transport, the 16

transportation network will face severe capacity shortages, inefficiencies, and route load 17

imbalances across the network in space and time. Therefore a more efficient freight routing system 18

could save transport costs and contribute to sustainability and efficiency of the entire urban 19

transportation network. 20

Due to the important role of freight transportation, numerous researchers have addressed 21

the issue of multimodal freight transportation routing and scheduling (1-27). Jourquin and Beuthe 22

presented a multimodal freight model based on a digitized geographic network (1). Southworth 23

and Peterson developed a multi-layer intermodal shipment routing model in (2). The intermodal 24

freight transport between rail and road has been described in (4-6). As a fundamental issue for 25

optimum routing, the multicriteria shortest path problem in a multimodal network has been studied 26

by many researchers. Modesti and Sciomachen applied a link utility measure approach to solve the 27

multiobjective shortest path problem (7). Lozano and Storchi considered the impact of modal 28

transfer costs when finding the shortest multimodal path (8). Dynamic and stochastic routing for a 29

multimodal transportation environment was studied in (9) and (10). Speed-up techniques for the 30

shortest path algorithm have also been analyzed including Core-Based routing (11), label-setting 31

and label-correcting methods (12) and the improved label setting algorithms (13). Optimization 32

techniques have been commonly used to solve the multimodal transport routing and scheduling 33

problem such as in (14-18). Guelat and Florian proposed a linear approximation algorithm to solve 34

the multimodal multiproduct freight assignment problem (14). Castelli et al. used a 35

Lagrangian-based heuristic procedure to solve the signal line and general network scheduling 36

problem (15). Ham, Kim and Boyce showed the application of Wilson’s iterative balancing 37

method in interregional multimodal shipments (16). Zografos et al. developed a dynamic 38

programming based algorithm for multimodal time-scheduling with a shortest path algorithm (17). 39

Moccia et al. solved a multimodal routing problem with timetables and time windows by 40

integrating heuristics and a column generation algorithm (18). The main difficulty is that these 41

classical approaches of using mathematical models breaks down when faced with the control and 42

optimization of complex networks such as multimodal freight networks that exhibit nonlinear 43

travel time and cost functions which are difficult to mathematically represent with respect to the 44

routing decisions and to find a closed form solution. The availability of fast computers and 45

Zhao, Ioannou, Dessouky 4

software tools open the way for new approaches that go beyond the limitations of network 1

complexity. The traffic flows and states can be better predicted using simulation models that are 2

more complex and can capture phenomena that cannot be formulated with simple models (28). 3

Some researchers tried to solve the multimodal routing problem from the aspect of user 4

equilibrium in dynamic traffic assignment as in the unimodal road scheduling problem (19-25). 5

The common idea of user equilibrium based traffic assignment is to search the routing decision 6

such that the trip costs on the used routes for the same demand are equal and the costs of the used 7

routes are less than the costs of the other unselected routes. Peeta and Mahmassani formulated and 8

developed a simulation-based method for the dynamic traffic assignment problem in which the trip 9

costs for each origin-destination demand are in equilibrium (19). The proposed simulation model 10

based method was also applied in (20)-(22) to solve the intermodal routing problems. The 11

equilibrium model between supply and demand has been also studied in which all trips for all 12

suppliers share the same trip cost (23)-(24). A solution algorithm based on the cross entropy model 13

was proposed and compared to the moving successive average method that was widely used in the 14

user equilibrium problem (25). Afshin et al. developed a coordinated multimodal load balancing 15

system in which the suppliers share route information so that it could lead to better solutions for 16

the overall system (26). Moreover, Russ et al. and Yamada et al showed the applications of routing 17

and scheduling approach in multimodal freight network design (27)-(28). 18

The simulation models can also be integrated with control and optimization techniques to 19

provide better and more robust decisions (26). The purpose of this paper is to formulate and solve 20

the dynamical multimodal freight routing problem by exploiting the availability of powerful 21

computational software tools. We therefore propose a method we refer to as COSMO 22

(CO-SiMulation Optimization) as a potential innovation in dealing with multimodal transportation 23

routing that cannot be handled by the traditional way. We are proposing a novel load balancing 24

algorithm in the COSMO approach with estimates of the route states from outputs of the system 25

simulators based on work in (26). 26

The paper is organized as follows. Section 2 gives a formulation of the optimum routing 27

problem for multimodal freight transport. Section 3 proposes the COSMO approach and 28

demonstrates how to solve the formulated problem with the proposed approach. Section 4 shows 29

the experimental results of the proposed approach on portions of the transportation network in 30

Southern California. Finally the conclusions are discussed in Section 5. 31

2. PROBLEM FORMULATION 32

A multimodal freight transportation network G can be represented as a directed graph network 33

consisting of a set of nodes (N) with a set of directed links (L) connecting the nodes. A link in the 34

network could be one segment of a roadway or railway track while a node with zero length 35

connects multiple links. All passenger and freight traffic start and end at certain network nodes. 36

Let I and J be the sets of origin nodes and destination nodes respectively. Both I and J are a subset 37

of the node set N. In this paper, we are dealing with the routing of freight traffic flows that are 38

container flows between origin nodes and destination nodes. The analysis time horizon is 39

discretized into |K| small time intervals to formulate the problem. The notations that will be used 40

throughout the paper are defined as follows: 41

i The index of an origin node, iI; 42

j The index of a destination node, jJ; 43

k The index of time, kK where K = {0, 1, …, |K|}; 44

Zhao, Ioannou, Dessouky 5

l The index of a link in the network G, lL; 1

Pi, j The set of all paths from an origin i to a destination j; 2

p The index of a path from an origin i to a destination j, p Pi, j; 3

t The length of a time interval (unit: hour); 4

,i jd The total demand in number of containers departing from origin node i to 5

destination node j; 6

.

p

i jx k The number of containers departing from origin node i to destination node j 7

using path p with a departure time of k; 8

ly k The traffic volume of link l at time k; 9

lw k The travel time of link l at time k; 10

We next describe the constraints for our multimodal freight transport routing problem. 11

The first constraint ensures that the total amount of shipped containers throughout the analysis 12

time horizon for each origin/destination pair equals the required demand amount. 13

,

, , , for ,i j

p

i j i j

k p P

x k d i I j J

(1) 14

Let 1 2 | |, ,..., LY k y k y k y k

be the vector of traffic volumes (unit: 15

vehicles/hour) on links 1 to |L| at time k. Then the relationship of the traffic volume on a link l with 16

the departure container traffic and other parameters in the network can be expressed as a nonlinear 17

dynamical equation: 18

1 , , , , for ,l l l ly k f y k a k X k k l L k K (2)

19

20

where

21

, ,: , ,p

i j i jX k x k i I j J p P

(3)

22

In (2), fl is a nonlinear and time-dependent function of the traffic volume of a link lL. 23

The impact of the traffic volumes from adjacent links at time k is denoted by al(k) and X(k) is the 24

vector of departure freight traffic volumes from all origin nodes at time k as in (3). Since yl(k) and 25

al(k) contain the impact of the previous departure container traffic before time k (i.e., X(r) for r < 26

k) so only X(k) is included in equation (2). The link volumes in the transportation network are 27

time-dependent due to various factors such as time-dependent passenger traffic, network changes, 28

accidents and incidents. 29

Let 1 2 | |, ,..., LW k w k w k w k

be the vector of travel time (unit: t) of links 1 to 30

|L| at time k. lw k specifies the time length that a container takes to travel on link l if it enters link 31

l at time k. The link travel time is a function of the link volume at time k which is time-dependent 32

because of the impact of the time-dependent passenger traffic, network incidents and railway 33

dispatching decisions. The travel time of a link is dependent on not only the link flow on itself but 34

also on the flows of the other links, therefore, 35

, , for W k g Y k k k K

(4) 36

Let ,

p

i jT k be the travel time (unit: t) of a path p from an origin node i to destination 37

node j if a container departs from origin i at time k. Assume a path p contains links 38

Zhao, Ioannou, Dessouky 6

,1 ,...pp p Nl l where Np is the number of links on this path p. Define

,p nple k as the entering 1

time at link , pp nl for a container on path p with a departure time of k at the origin. Then the path 2

travel time can be computed as follows: 3

, ,,

1

p

p n p np p

p

N

p

i j l l

n

T k w e k

(5) 4

where 5

,1 , +1 , , ,, and , for 1,..., 1

p p n p n p n p np p p pl l l l l p pe k k e k e k w e k n N

(6) 6

7

Let ,

p

i jS k be the average cost for shipping one container from origin i to destination j 8

with a departure time of k, then the objective function can be expressed as 9

.

, ,

i j

p p

i j i j

k K i i j J p P

S k x k

(7) 10

where 11

, , , ,+ , for , , ,p p p

i j i j p i j i jS k C k T k i I j J p P k K (8) 12

In (8), ,

p

i jC k is the non-time independent cost (unit: dollar) generated by vehicle setup, 13

distance cost, etc. per container from origin i to destination j at departure time k with path p. This 14

cost could be obtained directly based on the information such as path distance and used vehicle 15

type. Kp is the weight value of the travel time on path p. 16

In summary, the multimodal routing problem can be expressed as follows. 17

,

, ,mini j

p p

i j i j

k K i i j J p P

S k x k

(9) 18

subject to constraints (1) – (6) and 19

, ,0 for , , ,p

i j i jx k i I j J p P k K (10) 20

0 ( ) for ,l ly k u k l L k K (11) 21

given ,i jd , ( )lu k for , , ,i I j J l L k K (12) 22

where ( )lu k is the capacity of link l at time k, i.e. the maximum amount of vehicles that 23

could be on link l at time k. 24

The explicit forms of the dynamical functions in (2) and (4) are difficult to 25

mathematically express directly due to the nonlinearities and complex variable interactions. 26

Therefore, we propose a COSMO approach in which the traffic network simulation models are 27

used to replace the mathematical functions of (2) and (4) to generate more accurate link volumes 28

and link/path travel times to solve the above optimization problem iteratively. 29

3. SOLUTION TECHNIQUE 30

3.1 The COSMO Approach Introduction 31

Zhao, Ioannou, Dessouky 7

1 2

FIGURE 1 COSMO approach for optimum multimodal routing 3

4

The proposed COSMO approach as shown in Figure 1 works as follows: The freight network 5

represents the physical network used for multimodal freight transport, i.e. the transportation 6

network including road network and rail network as well as their interactions. Traffic data 7

including link states, network incident and disruption information, and expected passenger traffic 8

demands are fed into the simulation models that are developed to describe the characteristics of the 9

actual freight transportation network. The predicted future freight network states from the running 10

of the simulation models with the traffic data inputs and the candidate routing decision are used to 11

update the states of the links and network. The link states include traffic volumes, travel time, 12

congestion status, fuel consumption, pollution emission etc. Then the optimization block finds the 13

new candidate routes that can reduce the most total cost. If one of the stopping criteria is not 14

satisfied, the load balancing block redistributes the freight flows from the current used routes to the 15

new found routes to reduce the total cost. This load balancing operation leads to new freight 16

network states, updated link states and possible new minimum cost routes. Therefore this iterative 17

procedure continues until one of following stopping criteria is satisfied: 1) The maximum number 18

of iterations is reached; 2) The change of the total cost is less than a predefined value between two 19

consecutive iterations. Once the stopping criterion is satisfied the final solution (i.e. the routing 20

results) are given to the actual freight network to implement. We next discuss the approach of 21

Figure 1 in detail. 22

Let , ,

p

i j k X be the predicted average cost per container of ,

p

i jx k in a routing decision 23

X from the updated link states based on the simulation outputs where X is a routing decision from 24

all the origins to the destinations for the entire analysis horizon, i.e. 25

, ,, , , ,p

i j i jX x k i I j J p P k K

(13) 26

Since the network simulation models can predict time-dependent link volumes and travel 27

times under the constraints of the traffic flow dynamics (2) - (6) and link capacities (11), the 28

original optimization problem can be rewritten as: 29

,

, , ,min i j

p p

i j k i j

k K i I j J p P

TC X X x k

(14) 30

subject to 31

,

, , , for ,i j

p

i j i j

k p P

x k d i I j J

(15) 32

, ,0, for , , ,p

i j i jx k i I j J p P k K (16) 33

given ,i jd for ,i I j J (17) 34

Assume that the cost function TC is differentiable with respect to the routing decision, the 35

first-order necessary conditions for an optimal solution *X of the above problem are: 36

Zhao, Ioannou, Dessouky 8

*', '

*'

', ' ,'' ' ' ' ', '

'

', '

' 0, '

for '

i j

p p

i j i jpk K i I j J p P i j X X

p

i j

TC Xx k x k

x k

x k

(18) 1

where is the feasible solution set given by constraints (15)-(17). The first order derivative in (18) 2

gives the change in the total cost by adding one more unit of containers on path p’ with a departure 3

time of k’ from origin i’ to destination j’, i.e. the marginal cost of a route. The conditions (18) mean 4

that the total cost cannot be reduced further by changing the optimal solution *X to another 5

solution locally. 6

The conditions (18) also state that at the optimal solution, the marginal route costs of any 7

used paths connecting the same OD pair are equal or less than the marginal cost of any other 8

unused paths connecting this OD pair. Otherwise, there exists another solution such that the total 9

cost can be reduced further. In other words, by redistributing some containers from the routes 10

having greater marginal route costs to other routes having smaller marginal path costs, the total 11

cost may be reduced. The idea of the proposed COSMO approach is to iteratively search the routes 12

with most reduced cost then conduct the load balancing from the current used routes to the new 13

found routes that could reduce the most total cost. The overall steps of the COSMO approach can 14

be described as follows: 15

Step 1: Obtain an initial solution 16

Set the iteration counter m = 0. Assign the given freight demands to a subset of predefined 17

routes in the transportation network, and obtain an initial routing decision (0)X . As an example, the 18

predefined route for each OD pair can be the route with minimum shipping cost without adding the 19

freight demands. (0)

,i jP k is the set of the used routes in the initial decision at time k. 20

Step 2: Update the link states 21

Set the current routing decision ( )mX into the freight network simulation models and run 22

the simulation models to obtain updated link volumes, travel times as well as other link states. 23

Step 3: Search for new minimum route 24

With the updated link states, find new time-dependent routes with the minimum marginal 25

cost for the OD pairs. Then determine the candidate route that can reduce the most total cost by 26

evaluating the minimum cost routes of all OD pairs. Let rp be a new found route with the most 27

reduced cost for an OD pair (i, j) and its departure time be rk . 28

( 1) ( )

, , if m m

i j i j r rP k P k p k k (19) 29

30

Step 4: Check for Convergence 31

Check whether the convergence criteria is satisfied, stop the algorithm if the cost 32

reduction between two consecutive iterations is less than a predefined threshold value or the 33

maximum number of iterations has been achieved. Otherwise, go to the load balancing step 5. 34

Step 5: Perform load balancing 35

For the OD pair with the new minimum route, conduct load balancing by redistributing 36

the freight loads from the current used routes connecting this OD pair to the new route with the 37

minimum marginal cost. 38

Considering the fact that it is difficult to find the explicit functional form of TC(X), the 39

marginal costs of the different routes cannot be computed directly. A possible way to find the route 40

marginal cost is via running simulation models by adding one unit of container to the objective 41

route and checking the change in total cost. However this is impracticable for a large scale 42

Zhao, Ioannou, Dessouky 9

transportation network considering the fact the amount of possible routes grows exponentially 1

with respect to the network size. An alternative way is to estimate the route marginal costs with the 2

updated link states from the simulation models. We next describe the estimation model of the 3

marginal route cost from the simulation results and how to find the route with most reduced cost. 4

5

3.2 Finding the Minimum Route using Simulation Models 6

Assume we have a current routing decision X and its corresponding , ,

p

i j k X can be obtained from 7

running the simulation models. Then, the marginal costs of the different routes can be computed by 8

the following equation: 9

,

,

, , ,

, ,

,' '

', ' ', ' ,

,

+

i j

i j

p p

i j k i j

k K i I j J p P

p p

i j i j

p

i jp p p

i j p i j p i j pk K i I j J p P i j

X x kTC X

x k x k

T kC k T k x k

x k

(20) 10

Equation (20) shows the change of total cost if '

', ' 'p

i jx k is changed by one unit of 11

container. The first term in (20) is the non-travel time cost from the vehicle usage, distance cost, 12

etc. which is available directly for a given route without running the simulation models and the 13

second term is the cost of travel time of route p’ with a departure time of k’ that can be computed 14

using model (5) and (6) with the predicted link volumes and travel times that are the outputs of the 15

simulation models. Thus, the values of these two terms are obtained from the route information 16

and the outputs of the simulation models. The third term describes the change in the total travel 17

time cost when changing one unit of container on path p’ with a departure time of k’ from origin i’ 18

to j’, which is difficult to mathematically express directly since the travel time of a given route is a 19

complicated function of the link traffic volumes on this route and other network factors due to the 20

nonlinear dynamical characteristics and route interactions in the traffic network. 21

From the simulation outputs of the current routing decisions, we obtain the link states22

ly k , lw k , and path travel time ,

p

i jT k for

,, , ,i ji I j J p P k K . By the derivative 23

chain rule and equations (4)-(6), 24

, ,

, ,,

,

,

,

1, ,

1 ,

pp n p np p

p

pp n p np n p pp

p np

p p np

p Nl l

i j

p pni j i j

Nl ll

l pn l i j

w e kT k

x k x k

y e kwe k

y x k

(21) 25

,for , , ,i ji I j J p P k K 26

The term l

l

w

y

in (21) is the derivative of the link travel time with respect to the link 27

volume. It can be approximately determined using the simulated link traffic volumes ly k and the 28

link capacities lu k . The calculations of the derivative of the link travel time term for road links 29

and railway links are different due to their different characteristics. . 30

For the road links, the travel time derivative can be obtained using the fundamental 31

Zhao, Ioannou, Dessouky 10

diagram of traffic flow (29) with the observed link volume and average travel time. The travel time 1

derivative can also be determined using a road travel time model such as the Bureau of Public 2

Roads (BPR) function in (30) or other estimated functions in (31). Take the BPR function as an 3

example, 4

_ 1 ll l free

l

yw w

u

(22) 5

where lw is the link travel time, _l freew is the link free-flow travel time that is determined by the link 6

length and speed limit, ly is the link volume and lu is the link capacity. 0, 0 are model 7

parameters that can be estimated from historical data. With this function, the link travel time 8

derivative in (21) for a road link can be computed by the following equation, 9

1

_l free ll

l l l

w y kwk

y u k u k

for ,l L k K (23) 10

For the railway links, considering the impact of the passenger train schedule and the 11

freight dispatching decisions, the travel time of a link is not an explicit function of link volumes so 12

the corresponding travel time derivative cannot be estimated easily. Therefore the travel time 13

derivatives of the rail links are estimated from running the rail simulation models repeatedly or 14

using historical operational data. 15

The term

, ,

,

p n p np pl l

p

i j

y e k

x k

in (21) describes the derivative of the traffic volume of link 16

, pp nl at time ,p np

le k when ,

p

i jx k

changes by one unit of container. Ignoring the link 17

interactions, we can estimate this term using the following simple equation, 18

, ,, ,, ,

,

1, and

0, otherwise

p n p np pp p p n p np p

l lp n p n l l

p

i j

y e k l l e k e kt

x k

(24)

19

,for , , ,i ji I j J p P k K

20

21

Finally, the marginal costs of a route by (20)-(24) can be approximately computed by,

22

', ,' '

, , , , , , ,' ' ' '

' , '1,

+p

n p n p np p p

p n p n p n p n p n p n p np p p p p p p

p p np

Nl l l

l l l l l l lpni j l

z e k wTC Xc e k w e k e k

x k t y

(25) 23

24

,for , , ,i ji I j J p P k K 25

where , ,'p n p np pl lc e k

is the non-travel time cost (unit: dollar) of link '', pp nl at time

,p nple k

26

generated by vehicle usage, distance cost, etc. , 'p np

l is the value of travel time on link

'', pp nl .27

, ,'p n p np pl lz e k

is the total number of containers on link '', pp nl at time

,p nple k

. All the data 28

required in (25) can be obtained directly or computed approximately from the simulation model. 29

Then the marginal costs of a route is the sum of the time-dependent link costs if a link cost 30

Zhao, Ioannou, Dessouky 11

is set as, 1

+l l

l l l l

l

z k wc k w k k

t y

(26) 2

where lc k is the non-travel time cost (unit: dollar) of link l at time k’ generated by vehicle 3

usage, distance cost, etc. Therefore the problem of finding the routes with minimum marginal costs 4

can be converted into an elementary time-dependent shortest path problem where the link costs are 5

set as in (26). The time-dependent shortest path algorithms in references (8-11) can be applied to 6

find the shortest routes with minimum marginal cost for all OD pairs. Then, the rp in equation 7

(19) (i.e. the route with the most reduced) cost can be determined. 8

9

3.3 Load Balancing Algorithm 10

For an OD pair where a new route was found, the load balancing algorithm redistributes the freight 11

loads among ( 1)

,

m

i jP k

containing the current used routes and the new route based on the 12

marginal costs for the OD pairs with the new found routes. One possible way to do the loading 13

balancing is moving the freight loads between two routes iteratively until the marginal costs of all 14

the routes are equal as done in (28). However this load balancing algorithm faces a slow 15

convergence problem for large demand sizes or large network sizes. Here we propose a load 16

balancing algorithm with quicker convergence based on solving a linear programming problem 17

using an auxiliary routing solution m

AuxX , 18

min m

AuxTC X (27) 19

, , , , ,where for , , ,m mp p

i j k Aux i j k i jX X i I j J p P k K

(28) 20

subject to 21

1

,

, , , for ,m

i j

p

i j i j

k p P

x k d i I j I

(29) 22

1

, ,0, for , , ,mp

i j i jx k i I j J p P k K

(30) 23

given ,i jd for ,i I j J 24

25

Then the new routing decision 1mX

can be generated using a step size method [0,1]m , 26

1m m m m

m AuxX X X X (31) 27

The most widely applied method of step size selection is the Method of Successive 28

Averages (MSA) in which the step size is selected as 1/(m + 1) as in the user equilibrium 29

algorithms such as in (19,21). Although MSA works well for small scale networks, its convergence 30

becomes slow for large networks. In this paper, we decide the optimal step size by solving the 31

following optimization problem (32) in which the total cost of a new possible routing decision is 32

evaluated by running the simulation model. Due to the fact it is time consuming to evaluate all the 33

possible step sizes in the feasible set, we build a discrete set of candidate step sizes to be evaluated 34

in each iteration. 35

[0,1]

arg minm

m m m

m m AuxTC X X X

(32) 36

We next compare three different step size selection algorithms (i.e., enumeration as in 37

Abadi et al. (28), MSA, and the optimization approach using (32)) on a simple example. There are 38

Zhao, Ioannou, Dessouky 12

three possible routes connecting one OD pair whose characteristics and conditions during three 1

time intervals (one-hour each in length) are shown in Table 1. 2

3

TABLE 1 Route Characteristics and Traffic Conditions of Simple Example 4

Time

Interval Route

Length

(mile)

Capacity

(veh/hour)

Current

Demand

(veh/hour)

Travel Time

(min)

1

1 12 1100 1200 24

2 10 1000 1000 18

3 11 1050 1500 31

2

1 12 1100 1000 20

2 10 1000 950 17

3 11 1050 1100 21

3

1 12 1100 800 17

2 10 1000 900 17

3 11 1050 700 15

5

The number of containers between this OD pair is 1200 and the total cost is the sum of the 6

travel times in minutes of all the vehicles. Figure 2 shows the convergence of the three step size 7

algorithms. The x-axis is the iteration number and the y-axis is the total cost for that iteration. The 8

required numbers of iterations to stop for the three algorithms are about 750 for the enumeration 9

algorithm of Abadi et al., 20 for the MSA algorithm, and 6 for the optimal step size algorithm. 10

Therefore, the optimal step size method provides the best convergence speed although the three 11

algorithms find the nearly same reduced cost. 12

13

14 (a) (b) (c) 15

FIGURE 2 Performances of different load balancing algorithms 16

a) Enumeration Algorithm in Abadi et al. (25); b) MSA Algorithm; c) Optimal Step Size 17

Algorithm 18

19

0 200 400 600 800 1000 12002

2.5

3

3.5

4

4.5

5x 10

4

Iteration

Tota

l cost

5 10 15 20 25 30 35 40 45 502

2.5

3

3.5

4

4.5

5x 10

4

Iteration

Tota

l cost

1 2 3 4 5 6 72

2.5

3

3.5

4

4.5

5x 10

4

Iteration

Tota

l cost

Zhao, Ioannou, Dessouky 13

4. SIMULATION MODELS AND CASE STUDY 1

4.1 Simulation Models 2

The simulation models used in the COSMO approach of this paper consist of a macroscopic road 3

network model and a rail simulation model. In this paper we use the macroscopic traffic simulator 4

VISUM to develop the road network model to achieve fast network state predictions 5

computationally. The simulator parameters such as lane number, length and capacity etc. of the 6

links are configured based on a practical transportation network. The inputs including passenger 7

and freight traffic for the road network are expressed as number of trips between zones that are 8

origins and destinations within the road network. The passenger traffic data will be filled into the 9

network based on available historical data obtained from the Southern California Association of 10

Governments. 11

For the rail simulator, we use the railway simulation system of Lu et al. in (32) which was 12

developed based on the ARENA simulation software. The rail simulator is used to evaluate the 13

dynamical train movements for a complex rail network. The track network is divided into different 14

segments based on their speed limits, length, and locations. Then, an abstract track graph is 15

constructed with these segments. The inputs for the rail simulator are the passenger and freight 16

train schedules including their planned departure times, origin stations, and destinations. Then the 17

train movements in the track network are simulated to calculate the travel times and delays of all 18

involved trains. 19

The integration of the two models has been realized by sharing the OD demands and 20

simulation outputs. The road network simulator sends the freight traffic that will be delivered 21

through trains to the rail simulator. Then, the rail simulator creates the freight train schedule based 22

on the train capacity and simulates the train movements with the planned passenger trains together 23

to output the predicted train arrival times. After receiving the outputs of the rail simulator, the road 24

network simulator will generate necessary truck flows to dispatch containers from the rail stations 25

to their final destinations. 26

27

4.2 Case Study and Results 28

We evaluated the routing between six main destinations (D1 – D6) and three terminals (A, B, C) in 29

the region with different demands as shown in Figure 3. We assume that there are five trains with 30

homogenous capacities of 50 containers between the port terminals and two rail stations nearby the 31

destinations. The average weight of all the containers is assumed to be 25 tons and transportation 32

costs per unit (price/ton-mile) are assumed to be 8 cents for road and 3 cents for railway (34). The 33

time values of a road link and a rail link are set to be 40 dollars and 100 dollars per hour 34

respectively. The values of travel time are estimated based on truck/train usage costs, driver 35

salaries and possible loading/unloading costs. The demands of the six destinations are provided in 36

Table 2. Three shippers communicate their load demands to a coordinator who runs the COSMO 37

approach to generate routes for their demands by minimizing the overall cost. 38

Zhao, Ioannou, Dessouky 14

1

FIGURE 3 Example region of study 2

3

4

TABLE 2 Demands of Destinations 5

6

Destination 1 2 3 4 5 6

Supply

from A 0 60 400 0 0 560

Supply

from B 0 390 0 0 630 0

Supply

from C 350 0 0 600 70 0

Total

Demand 350 450 400 600 700 560

7

Three traffic conditions are evaluated including normal traffic, congested traffic and 8

traffic with road closure. In the normal traffic, the passenger traffic is set as the average daily 9

volumes obtained from the Southern California Association of Governments while in the 10

congested condition the passenger traffic is increased by 50% above the average daily volumes. In 11

the third case, the lane closures are introduced at two locations on the main freeways I-710 and 12

I-110 causing the capacities of the two freeway segments to reduce by a half. Figure 4 shows the 13

average cost of transferring all the containers from their origins to the assigned destinations via the 14

multimodal transportation network for the three traffic cases (normal traffic, congested traffic, and 15

congested traffic with road closure). As shown in the figure, the final solutions after using the 16

COSMO approach reduces the average costs compared to the initial solutions in which the 17

minimum cost route before adding the freight is selected as the route to transfer the containers for 18

each OD pair. The main reason for the cost improvements is the proposed approach reduces the 19

congestions of the main freeway bottlenecks by balancing the loads across the network. Moreover, 20

Coordinator

C

B A

SC SC

Train Station

Train Station

SC

D1 D2 D3

D4 D5 D6

Zhao, Ioannou, Dessouky 15

the COSMO approach provides a better optimization performance during congested traffic and 1

when a lane closure on the freeway occurs than normal traffic conditions. 2

3

4

5 FIGURE 4 Evaluation of the results of the three traffic conditions 6

7

8

5. CONCLUSION 9

In this paper, we formulated a multimodal freight routing problem and then developed a COSMO 10

approach that can lead to more efficient decisions in freight routing by using simulation models in 11

the state estimations of complex and dynamic transportation networks. A load balancing 12

methodology that optimizes routing decisions from an overall system perspective is proposed 13

based on estimations of route marginal costs from the simulation outputs. The performance of the 14

proposed approach is demonstrated with an example in the Los Angeles/Long Beach area. The 15

computational results showed that the effectiveness of the proposed approach in reducing the 16

average costs under different traffic conditions. The proposed methodology relies on the 17

estimation model of route marginal costs. As future work, the performance of the estimation model 18

under different traffic conditions will be evaluated and the scalability of the proposed approach 19

will also be analyzed. 20

6. ACKNOWLEDGEMENT 21

The research reported in this paper was partially supported by the National Science Foundation 22

under grant 1545130. 23

24

0

20

40

60

80

100

120

140

160

180

Normal

Traffic

Congested

Traffic

Lane

Closure

Average Cost of Initial

Solution

Average Cost of Final

Solution

Zhao, Ioannou, Dessouky 16

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