A Hybrid Multiobjective Evolutionary Algorithm for Solving Vehicle Routing Problem with Time Windows

Transcription

1 A Hybrid Multiobjective Evolutionary Algorithm for Solving Vehicle Routing Problem with Time Windows Abstract. Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of customers with known demands and predefined time windows. Routes for the vehicles must meet all given constraints as well as minimize the objectives of travel distance and vehicles numbers. We proposed hybrid multiobjective evolutionary algorithm (HMOEA) incorporates simple heuristics for local exploitation and the concept of Pareto's optimality for solving multiobjective optimization in VRPTW. It is featured with specialized evolutionary operators and variable-length chromosome representation to accommodate the sequence oriented optimization in VRPTW. The HMOEA improves the routing solutions in many aspects, such as lower routing cost, better population distribution, wider scattering area and good convergence trace. 1 Introduction Vehicle routing problems with time window (VRPTW) is an example of the popular extension from VRP. In VRPTW, a set of vehicles with limited capacity is to be routed from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. A vehicle arriving earlier than the earliest service time of a node will incur waiting time. This penalizes the transport management either in the direct waiting cost or the increased number of vehicle due to the fact that a vehicle will service fewer nodes when waiting time is longer. Due to its inherent complexities and usefulness in real life, the VRPTW continues to draw attentions from researchers and has become a well-known problem in network optimization. Surveys about VRPTW can be found in [1][2][3], etc. Many heuristic approaches and exact methods have been applied to the problem of VRPTW [1][4][5][6][7]. Using exact methods, the computation time required to obtain such solutions is prohibitive if the problem size grows large. Some heuristic approaches are lacking of robustness or sensitive to datasets given. Evolutionary algorithm (EA) that emulates the Darwinian-Wallace principle of survival-of-the-fittest in natural selection and genetics have been applied to produce near-optimal solutions for VRPTW [8][9][10]. Prinetto et al.[11] proposed a hybrid genetic algorithm for traveling salesman problem, in which 2-opt and Or-opt were incorporated with the genetic algorithm. Blanton and Wainwright[12] presented two new crossover operators, Merge Cross#1 and Merge Cross#2, and showed that the new operators are superior to traditional crossover operators. Homberger and Gehring [9] proposed the approach of sub-dividing the optimization problem into phases based on the optimization objectives in VRPTW. In their approach, the optimization was performed in two independent evolution phases for different objectives. VRPTW involves the routing optimization for multiple vehicles so as to meet all given constraints. It minimizes multiple objectives such as the travel distance and number of vehicles concurrently, which has made it best solved by multiobjective optimization approaches. Most of the existing routing techniques, however, are single objective optimization based approaches which apply the method of penalty function or combine the different criteria via a weighting function. Such methods often require a set of precise settings of weights, which are usually not manageable nor understood. This paper thus proposes a hybrid multiobjective evolutionary algorithm (HMOEA) that incorporates simple heuristics for local exploitation and the concept of Pareto's optimality for solving multiobjective optimization in VRPTW. It is featured with modified evolutionary operators and variable length chromosome representation. The design of the algorithm has its focus on the need of VRPTW that integrates vehicle routing sequence with the considerations of timings, costs, and vehicle numbers. Without aggregating multiple criteria into a compromise function, the HMOEA optimizes all routing constraints and objectives concurrently, which greatly improves the routing solutions in many aspects, such as lower routing cost, better population distribution, wider scattering area, and good convergence trace. This paper is organized as follows: Section 2 gives the brief problem model and the test problems for VRPTW. The program flowchart of HMOEA is described in detailed in Section 3. Section 4 presents the extensive simulations and comparison results of HMOEA based upon the Solomon s 56 data sets. Conclusions are drawn in Section 5. 2 Problem Formulation Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. A route for vehicle is described by the sequence of customers that the vehicle are going to visit. Time window (the earliest service time and latest service time) are given for every customer site. Vehicle must arrive before or within the time window of each customer. Vehicle that arrives earlier must wait until earliest service time starts. Fig. 1 shows a graphical model of VRPTW and its solution. This simple example has two routes, R1 and R2, and every customer is given a number as its identity. The arrows

2 connecting the customers show the sequences of vehicle visit, where every route must start and end at the depot. Travel cost between customers i and j is denoted by c ij. The cost is calculated with the following equation: c ij = ( i j ) + ( i j ) 2 x x 2 y y where i x is the coordinate x for customer i; and i y is the coordinate y for customer i.. (1) Depot R1 R2 Fig. 1. Example solution for routing plan The mathematical model for VRPTW can be found in Solomon[3] where six benchmark problems for the vehicle routing problem with time window constraints (VRPTW) have been suggested. The Solomon s problems consist of 56 data sets. Each problem comes with specify number of customers, maximum number of vehicles, maximum capacity of a vehicle and details of every customer. Customers details are given in sequence of customer index, location in x coordinate and y coordinate, demand for load, ready time, due date and service time required. All the 56 problems are divided into 6 categories based on the pattern of customers locations and time windows. Accordingly, these 6 categories are named C1, C2, R1, R2, RC1 and RC2. The different test cases validates the performance of proposed HMOEA. 3 Multiobjective Evolutionary Algorithm 3.1 Program Flowchart of HMOEA and representation To provide an overall view of the proposed algorithm, the program flowchart of HMOEA is shown in Fig. 2. Each of the features such as chromosome representation, evolutionary operators, and local exploitation algorithms will be described in detailed in the following sections. The initial population is built with the requirement that each individual must at least be a feasible candidate solution, i.e., every individual and every route must be feasible. Infeasible route has to be discarded. The initialization is a random procedure; it starts by inserting customers to an empty route in random order. Any customer that voids any constraints is deleted and a new route is added to serve the remaining customers. This process continues until all customers are routed and a feasible initial population is built as depicted in Fig. 3.

3 Start Pre-processing Build initial population Cost function evaluations Stopping criteria met? End Selection Route-exchange Crossover Multimode mutation Local search exploitation Elitism Fig. 2. The program flowchart of proposed HMOEA Initialize Population Shuffle all Customers Initiate new route Delete customer End Insert a customer to route. Feasible route? NO YES YES All customers routed? NO Fig. 3. Steps of building the initial population As soon as all individuals are created, they will be evaluated according to the fitness functions and subsequently be ranked based upon Pareto ranking scheme. The evolution continues with tournament selection process where good individuals will be retained for reproduction, as well as the elitism process for which a number of good individuals will be kept for next generation without going through the genetic operations. Evolutionary operators consist of route-exchange crossover and multi-mode mutation will then be applied to modify the selected individuals to form a new feasible generation. Worse individuals in the new generation will be replaced by the elite individuals reserved from the elitism process. To further improve the internal routing of these new individuals, local exploitation algorithms with different heuristics are applied at the end of every 10 generations (the period of 10 generations are taken after considering the trade-off between optimization performance and the extra simulation time needed for local exploitations). All individuals in the new generation will be evaluated based on the fitness functions and the evolution process repeats until the stopping criterion is met. The stopping criterion in HMOEA depends on performance improvements over the generations, i.e., the simulation stops when no significant improvement is recorded over the last 10 generations.

4 In HMOEA, a variable length chromosome representation is applied where each chromosome encodes a complete solution including the number of routes/vehicles and the customers served by these vehicles. Every chromosome can have different numbers of routes for the same data set. As shown in Fig. 4, a chromosome may consist of several routes, and each route may consist of a sequence of customers to be served. Unlike conventional coding approach, a route or a gene in the chromosome here is not a constant but consists of the sequence of customers to be served. Such representation allows the number of vehicles to be manipulated directly or minimized. Therefore, it is suitable for multiobjective optimization in VRPTW. Route-exchange crossover, multi-mode mutation, and Pareto ranking scheme-designed for multiobjective optimization in VRPTW, are described in the following sub-sections. Chromosome A chromosome encodes a complete routing solution A route (vehicle) contains sequence of customers Fig. 4. Data structure of chromosome representation in HMOEA 3.2 Route-exchange Crossover The classical one-point crossover produced infeasible sequence because of duplication and omission of vertices. In this literature, we propose a unique but simple crossover operation that allows the good sequence of routes or genes in a chromosome to be shared with other individuals in the population. Good routes in VRPTW would be those with customers/nodes arranged in sequence where cost of routing (distance) is small and the time window is perfectly fit one after another. Once a pair of chromosomes is selected for crossover, the best route in each of the two chromosomes will be exchanged. The best route may be chosen based on several criteria, e.g., low average cost over nodes, which could be computed easily from the variable length chromosome representation in HMOEA. To ensure that all chromosomes are feasible routing solutions after crossover, any duplication will be deleted. 3.3 Multi-mode Mutation The mutation for our HMOEA is designed to perform multiple modes of operation. It complements the crossover operation, which does not provide good facility to optimize local route information. The mutation rate is considerably small because its operation could be destructive and the chromosome structure and information of routes might be destructed. In order to trigger more possible moves with better routing solutions, a few operations including Partial_Swap, Split_Route and Merge_Routes are implemented and governed by three parameters the mutation rate (PM), elastic rate (PE), and squeeze rate (PS). The elastic rate determines the operation of Partial_Swap, which picks two routes in a chromosome and swaps the two routes at a random point. The squeeze rate determines the operation of splitting or merging a route. Many constraints to be considered when the Merge_Routes is performed, eg., the merging operation should avoid violation against all hard constraints, such as time windows and vehicle capacity. Detail of operation of the three mutation modes is given in Table 1. Table 1. The multi mode mutation incorporated in HMOEA Mutation mode Merge_Routes Split_Route Partial_Swap Description Two routes in a chromosome are selected and named route A and route B. The algorithm tries to move all nodes in route B to route A. The shifting is carried out a node at a time. When all the nodes in route B have been completely shifted to route A, route B will be deleted. This algorithm splits a route into two new routes. The splitting is performed based on a random number as the cut point. This algorithm swaps 2 routes at a random point. The swapping must be feasible or else the original routes will be restored. The cut point must be a number smaller or equal to the shortest size of the two chosen routes.

5 3.4 Pareto Ranking The VRPTW is solved as a multiobjective optimization problem where two objectives, that is the number of vehicles (NV) and cost of routing (CR), need to be optimized concurrently subject to a number of constraints like time window and vehicle capacity. The Pareto ranking scheme is adopted here to assign the relative fitness of each individual in a population. The ranking approach assigns the same smallest rank (highest fitness) for all non-dominated individuals, while the dominated individuals are inversely ranked according to how many individuals in the population dominating them based on the criteria below: Smaller number of vehicles but equal cost of routing Smaller routing cost but equal of number of vehicles Smaller routing cost and smaller number of vehicles After the ranking process, tournament selection with a tournament size of 2 is performed where all individuals in the population are randomly grouped into pairs, and the individual with a lower rank (higher fitness) will be selected for reproduction. A simple elitism mechanism is employed in HMOEA for faster convergence and better routing solutions, where a small number of good individuals (0.5% of the population size) are kept and replace the worst individuals in next generation, without going through the evolutionary operations. 3.5 Local Exploitation in HMOEA The role of local search is vital in multiobjective evolutionary optimization to provide better convergence as well as to discover any gaps or missing trade-off regions along an evolution. To save expensive computation, HMOEA incorporates 3 simple local heuristics to search for better routing solutions in VRPTW, which includes Intra_Route, Lambda_Interchange [13], and Shortest_pf. Descriptions of these heuristics are given in Table 2. No preference is made among the local heuristics, and one of them will be randomly executed at the end of generation regularly to all individuals. Table 2. The various local heuristics incorporated in HMOEA LOCAL HEURISTICS Intra_Route Lambda_Interchange Shortest_pf DESCRIPTION Two routes are picked. This algorithm generates two different numbers based on number of nodes of both routes. It then swaps two nodes from each route. Feasibility is validated on both new generated routes. If both routes are acceptable, they will be updated as part of the solution, otherwise original routes will be recovered. A number of nodes will be moved from a route to another route. Assume two routes of B and A are chosen, the algorithm starts by scanning through nodes in route A, and moves feasible node into route B. The procedure repeats until a pre-defined number of nodes are achieved or until the scanning ends at the last node in route A. The algorithm tries to rearrange the order of nodes in one route. Given a route that involves several customers, the first node is chosen based on its distance from the depot; the second node is again chosen based on its distance from the first customer node. The process repeats until all nodes in the original routes are re-routed. Route will be restored if the new route is infeasible. 4 Simulation Results and Comparisons In all simulations, the following parameter settings have been chosen after some preliminary testing and base on historical experience: Crossover rate = 0.7 Mutation rate = 0.1 to 0.3 Elastic rate = 0.5 Squeeze rate = 0.7 to 0.8 Population size = Multiobjective Optimization Results for VRPTW This section presents the routing performances of HMOEA, particularly on its multiobjective optimization effort that offers the some advantages over the single objective based vehicle routing methods. To investigate the difference, three types of simulations with similar settings but different of optimization criteria have been performed. The first type of simulations only concern the optimization criterion of cost of routing (CR); the second type of simulations are performed based upon the criterion

6 of number of vehicles (NV); and the third type of simulations considers multiple objectives (MO), i.e., both criteria of CR and NV are considered concurrently Better Population Distribution and Wider Scattering Area Scattering area is measured by dividing the entire interested region of the objective domain into grid. If any individual exists in a grid, one scattering point is counted regardless of the number of individuals in the particular grid. A large scattering area is generally preferred in MO optimization since it allows a broader coverage of solutions along the trade-off surface such that decision maker could choose one particular solution depending on his/her preference. As shown in Table 3, MO significantly outperforms CR and NV by giving the largest scattering areas for all 5 categories of Solomon s data sets. For example, in category RC1-07, MO scored 60 scattering points while CR and NV only scored 36 and 34 points, respectively Good Convergence Trace Fig. 5 shows the convergence trace of the best routing solution at each generation along the evolution for the test data of R1. Since the figure plots the cost of routing against the generation number, the performance of NV is poor for all the 6 categories, as expected. Meanwhile, the convergence of CR appears to be comparable to MO as shown in the graphs. One observation from the simulations is that CR often reduces the cost of routing rapidly which hinders its ability to minimize the number of vehicles. Table 3. Comparison of scattering points for CR, NV and MO Category Total Grids CR NV MO C R R RC RC Fig. 5. Convergence trace for CR, NV and MO 4.2 Performance Comparisons In this section, the results obtained from HMOEA are compared to the best-known routing solutions compiled from different heuristics published in literature according to our best knowledge. The proposed HMOEA has produced excellent routing results, where 22 data sets (out of 56) achieve a lower routing cost as compared to the best-known solutions. Besides, the HMOEA also gives competitive routing solutions for 20 instances with similar or smaller number of vehicles and slightly higher routing cost (1%-2% in average) as compared to the best-known results.

7 Table 4. Comparison between different heuristics and HMOEA Problem Class Rochat and Taillard, [14] C C R R RC RC All Potvin and Bengio, [15] Taillard et al., [16] Chiang and Russell, [17] Homberger and Gehring, [9] Tan et al., [18] HMOEA Table 4 compares the average number of vehicles and average cost of routing obtained from each category for six popular heuristics and HMOEA. In each grid, there are two numbers representing the average vehicle numbers (upper) and average cost of routing (lower), respectively. For example, in category C1, the number pair (10.00, ) means that over the 9 instances in C1, the average vehicle numbers deployed is 10 and the average travel distance is The last row is the total accumulated sum indicating the total number of vehicles and total travel distance for all 56 instances. As can be seen, HMOEA leads to new best average results with the smallest CR and NV for category C1. Besides, it also produces the smallest average routing cost for the categories of R1, RC1 and RC2, with minor increase in the average number of vehicles, e.g., 2.7%-2.9% higher as compared to heuristics giving the second best average routing cost. As can be seen from the last row in Table 5, HMOEA also produces the best-cumulated total routing cost for the 6 categories of data sets. To study the consistency and reliability of the results obtained by HMOEA, 15 repeated simulations with randomly generated initial population have been performed for data instances of C1-09, C2-01, R1-07, R2-07, RC1-07 and RC2-07. The simulation results are represented in box plot format to visualize the distribution of simulation data efficiently. Each box plot represents the distribution of a sample population where a thick horizontal line within the box encodes the median, while the upper and lower ends of the box are the upper and lower quartiles. Dashed appendages illustrate the spread and shape of distribution, and dots represent the outside values. As shown in Fig. 6, the results obtained by HMOEA for the 15 simulation runs are rather consistent and all variance is found to be within 5%-15% from the mean value. It is observed that the category of type 1 (C1, R1, RC1) gives a much smaller variance as compared to the category of type 2 (C2, R2, RC2). The reason is that the number of customers per route (length of route) is shorter for the category of type 1 Fig. 6. Variance in box plot for one data instance in each category 5 Conclusion Vehicle routing problem with time windows involves the optimization of routes for multiple vehicles so as to meet all constraints and to minimize the objectives of travel distance and number of vehicles, which is inherently a multiobjective routing problem. Unlike existing routing heuristics, this paper is among the first to incorporate multiobjective optimization paradigm in solving VRPTW. A multiobjective evolutionary algorithm (HMOEA) has been proposed, which incorporates

8 variable length chromosome representation and modified evolutionary operators to accommodate the sequence-oriented optimization in VRPTW. Without the need of aggregating multiple criteria in VRPTW into a compromise function, the HMOEA optimizes all routing constraints and objectives concurrently, which greatly improves the routing solutions in many aspects, such as lower routing cost, better population distribution, wider scattering area and good convergence trace. References 1. Desrosier, J., Dumas, Y., Solomon, M., and Soumis, F., Time constraint routing and scheduling, In Handbooks in Operations Research and Management Science 8: Network Routing, M. Ball (eds.). Elsevier Science Publishers, Amsterdam, pp , Golden, B. L. and Assad, A. A., Vehicle Routing: Methods and Studies, Amsterdam; New York : North-Holland, Solomon, M. M., Algorithms for vehicle routing and scheduling problem with time window constraints, Operations Research, vol.35, no.2, pp , Breedam, A. V., Comparing descent heuristic and metaheuristic for the vehicle routing problem, Computer & Operations Research, vol. 28, no. 4, pp , Laporte, G., The vehicle routing problem: an overview of exact and approximate algorithms, Europe Journal of Operational Research, vol. 59, no.3, pp , Potvin, J. Y., Kervahut, T, Garcia and B, Rousseau, J. M., The vehicle routing problem with time windows - part I: Tabu search, INFORMS Journal on Computing, vol. 8, no. 2, pp , Tan, K. C., Lee, L. H., Zhu, Q. L. and Ou, K., Heuristic methods for vehicle routing problem with time windows, Artificial Intelligence in Engineering, vol. 15, pp , 2001a. 8. Grefenstette, J. J., Gopal, R., Rosmaita, B. and Van Gucht, D., Genetic algorithms for the traveling salesman problem, Proc. First Intl. Conf. Genetic Algorithms and Their Applications, pp , Homberger, J. and Gehring, H., Two evolutionary metaheuristic for the vehicle routing problem with time windows, INFOR, vol. 37, no. 1, pp , Thangiah, S. R., Osman, I. H., and Sun, T., Hybrid genetic algorithm, simulated annealing and Tabu search methods for vehicle routing problems with time windows, Technical Report SRU CpSc-TR-94-27, Computer Science Department, Slippery Rock University, Prinetoo, P. M. R. and Reorda, M. S., Hybrid genetic algorithms for the traveling salesman problem, Proceedings of the Fifth International Conference on Genetic Algorithms, pp , Blanton, J. L. Jr. and Wainwright, R. L., Multiple vehicles routing with time and capacity constraints using genetic algorithms, Proceedings of the Fifth International Conference on Genetic Algorithms, pp , Osman IH, Christofides N., Simulated annealing and descent algorithms for capacitated clustering problem, Research Report, Imperial College, University of London, Rochat, Y. and Tailard, E. D., Probabilistic diversification and intensification in local search for vehicle routing problem, Journal of Heuristic, vol. 1, no. 1, pp , Potvin, J. Y. and Bengio, S., The vehicle routing problem with time windows - part II: genetic search., INFORMS Journal on Computing, vol. 8, no. 2, pp , Taillard, E., Badeau, P., Gendreau, M., Guertin, F. and Potvin, J. Y., A Tabu search heuristic for the vehicle routing problem with soft time windows, Transportation Science, vol. 31, no.2, pp , Chiang, W. C. and Ruseel, R. A., A reactive tabu search metaheuristics for the vehicle routing problem with time windows, INFORMS Journal on Computing, vol. 9, pp , Tan, K. C., Lee, T. H. and Ou, K. and Lee, L. H., A messy genetic algorithm for the vehicle routing problem with time window constraints, IEEE Congress on Evolutionary Computation, pp , 2001b.

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10 Truck and trailer vehicle routing problem (TTVRP) TTVRP is a transportation problem for moving empty or laden container for a logistic company in a city with local port. Owing to limited resource of its own fleet of vehicles (trucks and trailers), the company might have to subcontract certain job orders. The factors that have been into account include the number of trucks, number of trailers, trailer exchange points, time window constraints and routing cost. We introduce a model for this routing problem i.e. vehicle routing with time window and trailer constraints. An enhanced multiobjective evolutionary algorithm is proposed to optimize the solution of this problem, and further analysis has enabled us to derive decision-making information from the results. 1 BACKGROUND Singapore ranks among the top international maritime centers of the world. Singapore's sheltered and deep-water harbor lies strategically at the crossroads of major sea routes in South-east Asia. It is the focal point for some 400 shipping lines with links to more than 740 ports worldwide. The Republic's standing as an international maritime centre rests on its port, which is the world's busiest, in terms of shipping tonnage. In 2002, a total of 142,745 vessels with a shipping tonnage of million gross tons called at the port, which handled million tones of cargo. The port is also one of the busiest in the world in terms of container throughput. In 2002, the port handled a total of million twenty-foot equivalent units (TEUs), (Maritime, 2002). From the facts, an efficient transportation solution is essential to support the port activities in lieu with the extremely high throughput at the port. Trucks and Trailers Vehicle Routing Problem (TTVRP) includes the time windows constraint and trailers availability. The idea of solving this problem instigated from the observation on prosperous logistic industries in Singapore. We present a model of a logistic company that provides transportation service for container movements within the country. Ultimately, the results from optimization are utilized as input to planning such as number of trucks required for long-term operation. Conventional routing optimization strives to obtain a single best solution that solves the problem with minimum routing cost or minimum number of trucks. In this problem model, the solution of optimization is a set of multiobjective solution that can be useful for planning purpose. Besides that, jobs can be considered for outsourced in certain cases to achieve the desired optimized routing. The actual routing for the containers involves many intricate factors. Number of trucks in a fleet regulates maximum number of jobs that can be handled internally within certain period of time. All jobs must be serviced within a given time window. However, jobs could be considered for outsourcing if necessary, which the jobs are not handled by internal fleet of trucks. The routing plan needs to cater different type of trailers that are available at every trailer exchange points (TEP). Number of trailer exchange points, distributed in the region is another issue to consider in solving this real life transportation problem. Having said that, there are a wide variation of job orders that may have diverse requirements towards the trailers type, time window constraints and moreover locations of source-destination. The purpose of the model is to optimize routing cost and number of trucks, both the objectives concurrently. The information from optimized routing result is useful to management decision especially for their planning purpose. In order to model the essential features in the real problem, multiple constraints and objectives are being incorporated into the model. Using exact algorithm to solve TTVRP would be hard and require intensive time due to the problem is NP-hard. Hence, a better option would be solving the TTVRP with metaheuristics algorithm. In addition to that, the proposed TTVRP model is a multiobjective problem. It implies that the algorithm performed must be able to handle multiobjective optimization efficiently. Most of those conventional single objective optimization algorithms might not be able to serve the purpose well. Consequently, we applied the hybrid multiobjective evolutionary algorithm (HMOEA) to optimize this transportation model. This evolutionary algorithm uses a unique representation to encode the solution of problem. Consequently, its evolution needs special evolutionary operators to take place. The population improves over generations until it finally yields a set of solutions at Pareto frontier. It is then desired to extract many useful information from near optimum solutions, such as the best fleet size to accommodate a variety of jobs orders, the effect of increasing number of trailer exchange points, the trend for different number of trailers available at TEPs and so on. All the information is crucial to a decision maker to make brilliant judgment or to avoid making incorrect conclusion. The management team should find the information greatly helpful especially when they try to get the picture of those complex correlations among existing parameters in the routing problem. The test cases are generated with some random variables that simulate long-term operation of business activities. The management can thus formulate the planning on certain parameters such as number of trucks (long term capital cost) so that day-to-day operation cost can be held at minimum.

11 1.1 PROBLEM SCENARIO In standard vehicle routing problem, we have to find out the best-optimized route assignment and scheduling for vehicle to visit all the customers. In many instances the VRP modeling may not reflect the scenario in real application. The model that we propose is derived from a real world application, which consists of a number of job orders to be served by trucks and trailers daily. This model extends from previous VCPS system (Lee et al., accepted) with detail maneuver of the trailers in a routing plan. Trailers are resources with certain limitation as what happen in real world and the allocation of trailers in different locations affects the routing as well. The movement of containers among customers, depots and the port are major transportation job orders considered. A container load is handled just like a normal truckload. The only difference is that these loads use containers (with a possible chassis) instead of trailers only. On the equipment assignment point of view, correct trailer type is an essential. Container loads can be inbound which refers to a loaded container is to be taken from a vessel to a customer and returned empty to depot. It can also be an outbound job, which an empty container is to be picked up from the depot, taken to the customer, and returned loaded to the vessel. Every job order contains information about source of location and destination of location and other customers information. Besides, load requirement and time windows are specified as hard-constraints implied. There are altogether 6 types of job orders, vary from each other in these factors: source and destination (port, warehouse, depot or trailer exchange), time windows (tight or loose), loaded trip (or empty) and type of trailers (20 or 40). To be specific, possible type of jobs are listed as below: 1. Import with trailer type Import with trailer type Export with trailer type Export with trailer type Empty container movement with trailer type Empty container movement with trailer type 40 The company owns a number of trucks and a number of trailers. The maximum number of truck is 40 but it is not rather a fixed parameter. Instead the number of trucks is one of the objectives of optimization. In addition to that, number of trailers is larger than number of trucks. A truck must be accompanied with a trailer when servicing a customer. In another words, the routing involves not only truck but trailer location is an equivalent important issue here. An export job order works this way: from its initial location, a truck has to pick up a correct trailer at trailer exchange point, followed by a container at the depot. Then, it proceeds to warehouse and it will leave the trailer and container there for about 2 days where the container is filled. After that, a truck (may not be the same truck as earlier) will be assigned to move the loaded container using that trailer to port. The truck finally has to leave the container at the port and departs together with its trailer. An import job order works the other way. From its initial location, a truck has to pick up a correct trailer at TEP before it proceeds to port. The trailer is used to carry loaded container at the port. Then, the truck moves the container to warehouse and leave it there for 2 days approximately. Finally a truck (may not be the same truck) will move this empty container from warehouse to depot (using a trailer). Again the truck finally has to leave the depot with its trailer unloaded. In fact there are times when a truck has had a correct trailer type with it, that it can serve a job without going to trailer exchange point. Otherwise, a truck need to pick a trailer when it has mismatch trailer type or it does not carry a trailer with it at that moment. Of course, it will choose the nearest TEP where trailer is available to exchange and to pickup. Number of trailers available at an exchange point depends on how many trailers were picked up and returned to it. Constraint that imposed in this model is time window at source and destination of job orders. Basically assumption is made that all trailer exchange points have operating hours same as truck drivers working hours (8:00 am to 8:pm). 1.2 MODEL OF SCENARIO Base on the scenario above, we make some refinement on the model. The problem is modeled on a daily basis, which is the planning horizon only span one day. Note that, all import and export jobs consist of two sub-trips and a two-days interval at customer warehouses. As a result, the 2 days interval at customer warehouse divides a job into two separate planning horizons nicely (one day each). Import jobs and export jobs can now be split into two independent tasks which each of them fall into different planning horizon periods. By doing this, we break up job orders into sub-job type precisely: (Hereinafter the this is referred as sub-job or a task). In other words, a job orders can possibly divided into 2 tasks. Generally a task involves travel from a point (source) to another point (destination) as in table 1.

12 Table 5 Tasks type and description Task type Task description Source Destination Trailer type 1 Sub-trip of import job Port Warehouse 20 2 Sub-trip of import job Port Warehouse 40 3 Sub-trip of import job Warehouse Depot 20 4 Sub-trip of import job Warehouse Depot 40 5 Sub-trip of export job Depot Warehouse 20 6 Sub-trip of export job Depot Warehouse 40 7 Sub-trip of export job Warehouse Port 20 8 Sub-trip of export job Warehouse Port 40 9 Empty container movement Port Depot Empty container movement Depot Port / Depot Empty container movement Port Depot Empty container movement Depot Port/ Depot 40 Everyday, the number of trailers at every trailer exchange point is randomly generated. Number of trailers at the TEPs can be due to trailers that are left over from the previous planning horizon. All the pickup, return and exchange activities can result change to number of trailers available. Besides, there is possible a number of trailers that park at the customer warehouses too. All these undetermined factors suggest that the trailer resource available at every TEPs in the initial of planning horizon is random. Trailer number generally refers to all trailers park at trailer exchange points that also implies number of trailers available for pickup to use. A truck has to pick up a correct trailer from nearest TEP if it serves task type 1,2,5,6, 9,10,11,12 and it does not have a trailer with it OR the trailer it has is not the correct type. For task 3/4 and task 7/8, trailer need not visit a TEP before servicing the task because the correct trailer has been brought to the place beforehand (two days before). Contrastingly, the truck that serves subjob 3,4,7 and 8 must not have any trailer with it. If a trailer is attached to the truck, it must be returned to trailer exchange point beforehand. For an example, a truck that serves sub-job 7 leaves the destination (port) of previous task together with a trailer. If the same truck is to serve another task type 3,4, 7 or 8, it must travel to a TEP to drop the trailer (that it has from previous task). In short, a truck needs to visit a trailer exchange point for these conditions: It needs a trailer for task 1,2,5,6,9,10,11,12 and it does not have one now. It needs a trailer for task 1,2,5,6,9,10,11,12 and it has wrong trailer type with it. It has a trailer with it but it has to service sub-job 3,4,7,8 (The truck hence needs to travel to TEP and drop the current trailer.) Trailer availability at TEP needs frequent update. Other than initial number of trailer at the beginning of the day, trailers number will change due to pick-up and return activities. A trailer that is returned earlier in a day would be available for pick-up later on the same day. In order to do so, we perform the following approach: Time segmentation for trailer resource: Working hours per day: 12 hours X 60 minutes = 720 minutes Time per segment: 10 minutes Number of time slots available: 720 minutes / 10 minutes / slot = 72 slots Number of trailer available for pick-up in a particular time slot is equal to number of trailer of previous time slot, added with trailers returned in previous time slot and minus with trailers picked-up in previous time slot. Note that, different trailer types are managed in separate list. Let s say TEP have 3 trailers (type20) now and in current time slot the following events have happened: 1 trailer (type 20) is returned followed by 2 trailers (type 20) is picked up. In the next time slot, the trailer exchange point should have 2 trailers (type 20) available for pick up. (Note: = 2 )

13 2 COMPUTATIONAL RESULTS AND DISCUSSION ON EVOLUTIONARY ALGORITHM FOR TTVRP In order to verify the algorithm and model presented, 29 test cases have been generated. The problem instances are optimized using HMOEA. The results are presented in the following sections. 2.1 GENERAL TRUCK VS COST OVER GENERATION The convergence trend of any optimization algorithm is a useful indication to validate the functioning of optimization process. In TTVRP, there are two objectives, the number of trucks and routing cost in general. Both objectives were chosen as measure to show the convergence trend of HMOEA. As evolutionary algorithms are generation based, hence we have chose to show how minimization of both objectives occurs throughout the generations. Figure 1 shows the changes of routing cost for one of the test cases. Every individual in a population has a routing cost that is computed based on its planning. The average routing cost refers to total routing cost of all individuals in a population divided by number of individuals in the population. Figure 1 shows that its average routing cost reduces throughout every generation. The decline is faster at the earlier generations as compared to later. Although it is difficult to prove that optimal solution has been found, we can strongly believe that the algorithm has optimized the routing cost effectively. The reduction trend over generation shows HMOEA able to minimize routing cost-one of the major objectives in TTVRP problem. As the second objective in TTVRP model, number of trucks is another important target that must be monitored. For this purpose, similar analysis has been done on the optimization of number of trucks required in routing plan. The number of trucks required in one individual is evaluated from chromosome. In this case, the average truck number refers to total truck number of all individuals in a population divided by the population size. Similar to routing cost progress analysis, the average value of truck number for all the individuals in population is computed for statistical analysis. Figures 2 shows the minimization of total number of trucks in average from initial generation and so on. The rapid reduction for number of truck is understandable as an initial population is generated randomly and it leaves much space to be improved. In fact, from the analysis, we are convinced that both objectives have been minimized concurrently. Detail about the interdependent analysis to number of trucks as well as routing costs would be brought up in following sections. It is here to show that the HMOEA algorithm can optimize the TTVRP optimization problem Cost improvement for test case 100_3_4 Average routing cost 50 Average truck number for 100_3_4 Average truck number Average routing cost (a) Truck number Iteration Generation Figure 1 and 2 Graph for routing cost and truck number over 50 generations

14 2.2 MOVEMENT OF ELITIST POINTS Truck vs. Cost : Pareto First Int 1 Int 2 Final Cost of routing Number of truck Figure 2 Plot of elitist points HMOEA accomplishes optimization through evolution over generation. In TTVRP, the two main objectives are to be minimized concurrently. Figure 3 shows the elitist plot on objectives domain. The elite solutions are solution points that give the lowest routing cost for each truck number. It is to stress that, a non-dominated solution must be an elitist point but a elitist may not be non-dominated solution. In other words, non-dominated solution is the subset of elitist solution. The searching algorithm is struggling in between two distinct goals: (i) To discover solutions that fall on Pareto-optimal the closer the better, and (ii) To find solutions as scattered as possible in the obtained non-dominated front. The chance to find the best solution is higher when solutions in a population distributed well in search space. In some sense, the two goals are orthogonal to each other. The first goal requires a deep search towards the Pareto-optimal region, while the second goal requires a broad search along the Paretooptimal front. All elitist solutions are plotted on graph using their truck numbers and routing costs as coordinate respectively. To ease comparison, we consider the vehicle number ranges from 0 to 40 only. Moreover, to show the distinction for performance only initial generation, final generation and two other intermediate generations are plotted with different markers. A line is drawn connecting all solution points of last generation to emphasize the pseudo Pareto curve clearly. (A Pareto front comprises of non-dominated solutions only) The figure shows that the algorithm had performed optimization on to TTVRP problem and able to find solutions near to Pareto front as well. Besides, observation tells that, the initial solution points obtained from HMOEA seems congested to partial of solution space. The length of the curve is shorter in initial generation as number of solutions is fewer too. However, as evolution continues, improvement of performance is observed. The diversity of solution points increases compare to earlier generation. This is clearly shown in figure 4, as the elitist curves for different generations are moving towards better position. The arrow points to the direction of better elitist curve. Subsequently, the final elitist curve consists of the most largest number of solution points that are non-dominated by any other existing solutions in population. These solutions are known as near-optimal points, which qualify themselves the candidates for best routing plan. Although some solutions are apparently not practical to be adopted in real industry, for example with zero truck number, the routing plan subcontracts all tasks to external company. This policy is strongly against the will of logistic management. It is important to remember that, optimization algorithm applied to multiobjective problem can never substitute human s expertise, but it assists we to make correct decision.

15 Truck vs. Cost : Pareto First Int 1 Int 2 Final Cost of routing Number of truck Figure 3 Movement of best solutions for several different generations

16 2.3 HOW TO USE TRUCK VS COST INFORMATION The optimization of TTVRP returns a collection of solutions for routing plan. The solutions that are non-dominated by other points in solution space can be obtained from a collection of elitists, which each of them yields different routing cost and number of trucks. These data could be very useful especially during the planning stage of business. During the experiment, the value of routing cost and truck number for all non-dominated solution is archived. These values are then plotted on Pareto graph as shown in figure 5. The results of Pareto front from different test cases have been plotted for analysis. The TTVRP problem is a multiobjective problem. It is not desired to determine which the most optimized solution is. In many practical situations, the decision must be made on the trade off between the two objectives, as the management may have other factors or constraints too. The use of multiobjective algorithm, return a set of non-dominated solutions without repetition optimization process for different scenario. Consider the following example, where the management knows quantity of tasks they have to service, and rough information about number of trailers available at each trailer exchange points. The manager would like to find out suitable parameter values so that the day-to-day routing performance can be optimized. In the first situation, the manager can utilize the Pareto curve from result to find out estimated total cost incurred in operation. Assuming the number of trucks available in company is fixed, it is possible to estimate routing cost from the Pareto curve simply by reading the matching cost value from y-axis. Contrastingly, if the manager is given a budget to cover all the routing cost in the project. He has to determine minimum number of internal trucks to be allocated for a project so that the spending can be kept below the budget allowed. The Pareto curve can provide required information by estimating the truck number value from x-axis in graph Routing cost vs. truck number routing cost truck number Figure 5 Routing cost vs. truck number for different type of test cases. 3 CONCLUSION Efficient routing of transportation can reduce business cost and thus increases profitability. However, when size of routing problem is getting larger, computation time required to solve this problem using exact methods increase exponentially. In order to optimize the proposed routing mode, a hybrid multiobjective evolutionary algorithm has been devised to solve the TTVRP. Some unique features such as representation of solution, evolutionary operators and other the local search heuristics have been designed specifically to handle the TTVRP problem. After simulation, analysis and statistics figures are presented to show the results of optimization. Upon all, some important information from routing solution has been derived from the analysis. Generally, the simulation has proven its ability to optimize routing solution efficiently under all the constraints applied. The improvement of both objective functions throughout the optimization process has been very promising. From the results, set of non-dominated solutions can be obtained.