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The Variable Neighborhood Search Heuristic for the Containers Drayage Problem with Time Windows D. Popović, M. Vidović, M. Nikolić DEPARTMENT OF LOGISTICS DEPARTMENT OF OPERATIONS RESEARCH IN TRAFFIC

The Variable Neighborhood Search Heuristic for the Containers Drayage Problem with Time Windows D. Popović, M. Vidović, M. Nikolić DEPARTMENT OF LOGISTICS

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The Variable Neighborhood Search Heuristic for the

Containers Drayage Problem with Time Windows

D. Popović, M. Vidović, M. Nikolić

DEPARTMENT OF LOGISTICSDEPARTMENT OF OPERATIONS RESEARCH IN TRAFFIC

Containers Drayage Problem with Time Windows (CDPTW)

The problem is typical for the intermodal transportation systems where containers are distributed by trucks to customers located in the area oriented to a container sea port or an inland container terminal.

Drayage includes regional movements of loaded and empty equipment (trailers and containers) by tractors between terminals, shippers, consignees, and equipment yards. Here we have limited research scope only to empty and loaded containers.

Drayage operations and especially container truck transportation incurs significant transportation cost. Therefore it is very important to improve the efficiency of container transportation through solving the truck scheduling problem in container drayage operation (Zhang et al. 2010).

slide: 2 / 13

On the routing problem in the CDPTW

Containers transportation problem belong to a pickup and delivery problems, and because of the nature of the problem, drayage operations also corresponds to multi-stop Vehicle Routing Problems with Backhauls.

slide: 3 / 13

In the case when combined chassis for transporting two 20ft containers or one 40ft container is used the objective of the routing problem is optimal matching of pickup and delivery nodes (up to four nodes per route).

On the routing problem in the CDPTW

Empty and loaded containers from terminal should be distributed to costumers, and empty and loaded containers should be picked up at customers’ sites and hauled back to the terminal. Unlimited fleet of 2 twenty-foot equivalent units vehicle capacity. Time windows.Split deliveries.

slide: 4 / 13

There are 15 possible matchings of task nodes into merged routes, and 4 direct pickup or delivery routes.

Mathematical formulation of the CDPTW

Vidović et al. (2012) presented two mathematical formulations for the CDPTW: multiple assignment formulation, and general mixed integer programming (MIP) formulation.

Large scale problem instances (100 pickup/delivery locations) could not be optimally solved in reasonable CPU time. The CPU time needed to obtain the solution for the CDPTW is very important for real life applications. The reason lies in the fact that a routing plan for all pickup/delivery tasks needs to be obtained on day to day basis. Therefore, we developed the Variable Neighborhood Search (VNS) heuristic for solving large scale real-life problem instances.

The results from two above mentioned optimal mathematical models are used as benchmarks for the evaluation of the VNS heuristic.

slide: 5 / 13

The VNS heuristic

Mladenovic and Hansen (1997) developed the VNS algorithm as a new metaheuristic concept with the basic idea of a systematic change of a neighborhood within a local search algorithm. After the initial solution construction, the VNS heuristic uses local search and shaking procedure to iteratively improve the current best solution until a stopping criterion is met.

slide: 6 / 13

The initial solution in proposed VNS heuristic is obtained by the sweep method (biggest “polar gap”).

All solutions must be feasible regarding time windows and possible matchings of task nodes into routes.

max angle

starting node

The VNS heuristic – local searchslide: 7 / 13

Task reallocating from one route to another:

Tasks interchange between two routes:

reallocate to PURPLE ROUTE

reallocate to RED ROUTE

interchange one task from GRAY ROUTE with one task from GREEN ROUTE

interchange one task from BLUE ROUTE with one task from RED ROUTE

The VNS heuristic – shaking

For the shaking procedure, we use a method of the “destruction” of randomly chosen route/routes with multiple stops and reconstruction of single stop (direct) routes to each of those stops.

The number of routes to be “destructed” is denoted as Sh_size, and the number of repeated passes for given Sh_size is denoted by Sh_pass. These values define the total number of shaking neighborhoods.

slide: 8 / 13

Sh_size = 2“Destruction” of

PURPLE and RED routes

The VNS heuristic - algorithmslide: 9 / 13

NO

Construction of the initial solution

Init.: Sh_size ← 0; Sh_pass ← 0;

The local search procedure

Temp_obj_val < Current_obj_val

FINAL SOLUTION = Current_solYES

The shaking procedure

The local search procedure

Calc.: Current_obj_val; Current_sol

Calc.: Temp_obj_val; Temp_sol

YES Current_obj_val ← Temp_obj_valCurrent_sol ← Temp_sol

NO

NOYES

Sh_pass == Sh_pass_max

Sh_pass ← Sh_pass + 1

Sh_size == Sh_size_max

Sh_size ← Sh_size + 1

Test instances - Solomon VRPTW benchmark (http://web.cba.neu.edu/~msolomon/problems.htm) slide: 10 / 13

We tested the VNS heuristic on three group of instances presented in paper by Vidović et al. (2012) for which optimal solutions exists. Additionally, we generated fourth group of large scale problem instances that cannot be solved optimally in reasonable CPU time and that are solved only by VNS heuristics.

There are six different sets of problem instances. Task nodes are randomly generated in problem sets R1 and R2, clustered in problem sets C1 and C2, and both, randomized and clustered in problem sets RC1 and RC2. Also, problem sets R1, C1 and RC1 have a short scheduling horizon (few possible costumers per route) while problem sets R2, C2 and RC2 have a long scheduling horizon.

Computational Results Intel i3 CPU M380 2.53 GHz, 6 GB RAM CPLEX 12.2. , C++ Visual Studio 2010 (64-bit). slide: 11 / 13

N’

Multiple assignment formulation

General MIPformulation VNS heuristic

Solution CPU time [sec] Solution CPU time

[sec]Average solution

StDev [%]

Average Error [%]

Avg. CPU time [sec]

10 34833.5 0.024 34833.5 519.578 34981.5 0.32 0.30 0.013

15 56089.4 0.071 *56308.9 899.258 56332.0 0.49 0.45 0.032

50 198225.8 30.908 - - 200217.5 0.52 0.97 0.566

N' Instance N Avg. solution

StDev [%]

Avg. CPU time [sec]

100

R1 01 123 371052 0.55 2.901R1 02 123 338771 0.61 2.465R2 01 123 337175 0.56 2.869R2 02 123 297029 0.61 3.882C1 01 124 379768 0.88 3.364C1 02 124 348699 0.55 2.291C2 01 124 393768 0.47 2.103C2 02 124 360699 0.42 3.544RC1 01 125 441104 0.54 2.363RC1 02 125 410643 0.56 2.555RC2 01 125 416730 0.68 2.868RC2 02 125 416326 0.23 2.351

Average 375980.4 0.55 2.796

Total number of pickup/delivery locations (nodes) is denoted with N', and total number of containers (tasks) is denoted with N. For all four cases, each instance is solved in 50 iterations by the VNS heuristic with the following shaking parameters: Sh_size_max=5 ; Sh_pass_max=10 .

Conclusionsslide: 12 / 13

Results from small and medium scale problems shows that the VNS heuristic has relatively small average error compared to optimal solutions (0.30%, 0.45%, and 0.97% for cases with 10, 15, and 50 nodes respectively), with a considerably lower CPU time.

Wang and Regan (2002) stated that the typical containers drayage problem handles at most 75 containers a day. From this statement and the obtained results we can say that the VNS heuristic is capable of finding solutions for real life scale problems in a few seconds.

Although we observe 20ft and 40ft containers in our model (because they are the most frequently used ones), the proposed VNS heuristic can be applied to the CDPTW with other container dimensions where a modular concept vehicle can carry up to two containers of any dimension.

Further research should focus on heterogeneous vehicles, multiple use of vehicles, additional improvement of the VNS heuristic.

T H A N KYO U !