1. 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

2. Containers Drayage Problem with Time Windows ( 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

3. 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).

4. 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.

5. 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

6. 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

7. The VNS heuristic – local search slide: 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

8. 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

9. The VNS heuristic - algorithm slide: 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_sol YES 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_val Current_sol ← Temp_sol NO YES Sh_pass == Sh_pass_max Sh_pass ← Sh_pass + 1 Sh_size == Sh_size_max Sh_size ← Sh_size + 1

10. 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.

11. 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’N’ Multiple assignment formulation General MIP formulation VNS heuristic Solution CPU time [sec] Solution CPU time [sec] Average solution StDev [%] Average Error [%] Avg. CPU time [sec] 1034833.50.02434833.5519.57834981.50.320.300.013 1556089.40.071*56308.9899.25856332.00.490.450.032 50198225.830.908--200217.50.520.970.566 N'InstanceN Avg. solution StDev [%] Avg. CPU time [sec] 100 R1 011233710520.552.901 R1 021233387710.612.465 R2 011233371750.562.869 R2 021232970290.613.882 C1 011243797680.883.364 C1 021243486990.552.291 C2 011243937680.472.103 C2 021243606990.423.544 RC1 011254411040.542.363 RC1 021254106430.562.555 RC2 011254167300.682.868 RC2 021254163260.232.351 Average375980.40.552.796 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.

12. Conclusions slide: 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.

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