Vehicle Routing with Time Window for Regional Network Services ...

Vehicle Routing with Time Window for Regional Network Services - Practical Modelling Approach Iman Niroomand1, Amir H. Khataie2,Masoud Rahiminezhad Galankashi3 1

Department of Network Design, Canada Post, Ottawa, Ontario, Canada Department of Network Design, Canada Post, Ottawa, Ontario, Canada 3 Department of Material, Manufacturing and Industrial Engineering, UniversitiTeknologi Malaysia, Johor, Malaysia 2

Abstract- Vehicle Routing Problem with Time Window (VRPTW) has demonstrated an excessive application in postal service network design. In this paper we elaborate on how VRPTW modelling approach can benefit both firm and contractors. The business objective is to facilitate the process of retendering the contracts. Accordingly, the novel modeling approach has been developed in order to improve the practicality of the results. The Ant Colony Optimization (ACO) is used for solving the VRPTW for regional post office services. Keywords-Vehicle Routing Problem, Postal Service, Ant Colony Optimization I. INTRODUCTION Vehicle Routing Problem (VRP) is a combinatorial optimization problem that has been investigated for many years by scholars. VRP has extensive application in supporting decision making process in transportation, distribution, and logistics domains. The problemcan be defined based on different type and size of vehicles and it could vary from air to ground services. The basic VRP includes a distribution hub or center and a set of customers who needs to be visited. Vehicle starts the trip from the base (distribution center) to delivers committed itemsto each customer [1]. After the last visit, the vehicle must be repositioned at the base for the next day trip. Vehicle routing has several applications and can be customized for different type of transportation and logistics problems. Vehicle Routing with Time Windows (VRPTW) is auniversal problem in postal delivery service. In VRPTW, vehicle must visit each customer within the defined time window and must return to its based at a certain time [2].In order to solve VRPTW in real world not only the timing of visiting customer is important but also many other restrictions are imposed from both service provider and customers. These issues need to be fulfilled in order to enhance the practicality of the suggested solution [3]. In postal service distribution/collection network,each Postal Office (PO) might have different business physical and operational characteristics. For example, each PO has a different processing shifts, facility restrictions (e.g. high or low dock, parking limit),equipment limits (single stack vs. double stack possibility), and different road condition. Each of these restrictions can impact the truck type, truck size, and truck loading profile. Moreover, the suggested loop

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needs to respect certain legal and contractual obligations such as, the maximum routing length, minimum or maximum number of stops per route, and layover time between collection and delivery process. The VRP main algorithms could be divided to three categories: Exact, Heuristics and Meta Heuristics [5]. Exact algorithms mainly use branch and cut method. This type of algorithm requires asophisticated mathematical programming unless the problem is defined on small size (i.e. Hong Kong post office [6]) or the algorithm is used at strategic level.Such a problem can’t be solved easily at tactical or operational level. In addition to that and in a real worldmore assumptions are imposed by operation which makes the problem more complicatedto model or rarely ended in successful solving [7].Therefore, developing a comprehensive mathematical model which answers all the design aspects is almost impossible and such solution is neither cheap nor quick. Furthermore, the model needs to be modified time to time based on new rules and regulations.In reality and for successful implementation, many scenarios must be verified with operators and need to be approved by different shareholders either at operation level or at upper management level. In this matter, VRPTW model needs to be validated through different what-if scenario analysis and should also be solved during reasonable computational time. In this paper, the ACO algorithm is applied and modified for VRPTW problem. The criteria of developing such an algorithm are clearly discussed in order to have reliable operational network. In the end, the solution is compared with existing regional network to explain how the new route construction would benefit the postal corporation in retendering process.

II. PRACTICAL ROUTESUSING METAHEURISTIC ALGORITHM In general, there are two criteria to evaluate the superiority of a model;computational time vs. the solution accuracy. Computational time refers merely to methodology of solving the problem and quality of the solution refers to simplicity of implementation and flexibility of adapting new terms and constraints [4].

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Heuristic methods could be applied for larger size problems and through a constructive search approach from initial solution in its neighbourhood they precede till no further improvement is possible. Although these methods are finding solution faster but the quality of the solution might not be satisfactory since not the entire feasible area is investigated. Thus, algorithms are stuck in local optima at most of the time.In most cases, logistic firms prefer to employ experienced employee (local knowledge) to review transportation network and construct the routes manually through a trial and error process.

there is a specific route, there is a specific individual operational reason behind it. Therefore, the current network should not be totally way off from the optimal solution and constraints parameters are good starting point for restructuring the network. In this example one is able to extract the accurate constraints of each location.

In contrast to heuristic methods, Meta-heuristics methods are exploring the search space beyond the local optima and if they are used with improvement heuristics, they would find a promising solution. This method easily could be employed by logistic companies since the procedure of meta-heuristics are not problem specific and could be easily developed by any scripting language. Meta heuristic models in many cases could be a suitable substitution for mathematical model. For example, systematic randomized algorithms aregood alternative for mathematical model in solving combinatorial optimization problems [2] since they easily could be modified and utilized in practical problems. Among available randomized algorithms, Ant Colony Optimization (ACO) is more compatible with VRP. ACO was introduced in the early 1990’s [8]. Since then ACO has extensively used for modeling transportation problems such as Travel Sales Person (TSP) [9], Vehicle Routing Problem with Pickup and Delivery and Time Window [10], and Multiple Depot Vehicle Routing Problem [11]. ACO is a population search Meta-heuristic which is able to maintain a pool of solutions. By combining and replacing the solutions in the pool, ACO could cover different area of the search space (diversification). Moreover by applying the local search improvement techniques such as swap moves [12], and the 3-opt algorithm [13] through exchanging arcs of the routes, the optimal or near to optimal solution could be achieved. Therefore, the ACO could be a good candidate for applying the VRPTW to applicable transportation network. III. MODEL APPROACH For solving the VRPTW problem, first the existing network of current services (if there is any) to each location is mapped out. Mapping the existing network is an important step to visualize how the locations are scattered geographically (Figure 1) and how today’s routing has been structured (Figure 5). The existing network could be mapped through latitude and longitude of each location and the travel time between these locations could be extracted from the commercial map software such as PC-Miler or Google map. Mapping the current network helps the modeller to identify the existing constraints parameters. Usually the network in placehas been evolved over several years and if

Figure 1-locations distribution in the region Each location time window and facility constraint is pulled out as next step. In postal service industry each facility has certain delivery and collection time which all the mail volume should be delivered before the closed window time (hard constraint); however, in most cases there is no earliest required time to deliver. Basically, mail volume could be delivered as long as it meets the latest delivery time. Furthermore, PO is unique at each location and the postal service should employ different truck sizes for delivering the volume to each location. It is important to know that amount of volume could fluctuate based on the time of the week and month (Seasonality factor). The next step is calculating the appropriate cost to compare how new re-structuring will benefit the operational unit. In the literature cost parameters are mainly used for VRPTW optimizing are asset (truck) cost, total driven time, and total driven distant based on the kilometres (KM). Other hidden fees that need to be included in the route re-structuring such as driver layover time at the last stop before collection time, or route difficulty. In addition to these costs,we should also take into consideration the actual cost of restructuring (change). To find a practical solution we should always consider that operation unit is not in favour of a solution that makes the bidding process harder for contractors even though the solution reduces the internal operational cost. To be more specific, the contractors usually bid several routes at the same time and try to get the routes that are easy to manage daily. Accordingly, the solution similar to Figure 2 is not favourableto bid and normally contractor asks for much higher than anticipated cost for covering these routes. We should also consider the contractor business powerto find an optimal scenario.In regions that contractor has monopoly there is less room for negotiating.

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d route  G d Dagreed

Figure 2- Asymmetric route structure

G is considered as uncertainty parameter that makes sure the driven distant always follows the contract legal termswhen a disruption such as the road closures on some days, or weather conditions occurs. 3) The number of assigned location ( x ) per route should not be more than D locations and less than E locations. This constraint prevents defining asymmetric routes (either long or short). D dxdE

In Figure 3, although this re-structuring style is inevitable because of location timing or restriction but it should be minimized as much as possible because of zigzag patterns. This type of the restructuring might increase the service cost when the corporate sends the contracts for retendering. Taking into consideration all the facts mentioned above requires adding some new constraints to a typical VRPTW model to prevent the developing of poor route structure solution. Although by adding these new constraints, the retendering cost might be increased but the new solution is much more acceptable by both parties (corporation and contractors).

7) The total driven time per route should be less than the allowed driven time per day based on the contract minus J .

t route  G d Tagreed J is considered as uncertainty parameter that makes sure the driven time always follows the contract condition most of the time by considering the traffics, road condition, and etc.

Figure 3-Non-level route structure The total network cost is a function of total driving distance; the objective function is set to minimize the total driving distance while observing each location time window. After solving the algorithm all other cost parameters such as number of trucks is calculated automatically. The constraints are discussed as follow: 1) Each location should be visited at least once. 2) The delivery to each location should be earlier than location closing time window. 3) The total volume on each truck must be less than truck capacity. 4) A right truck size must be employed at each location. In addition to above constraints, other new set of constraints are added to VRPTW problem; 5) two consecutive nodes,excluding starting point (depot) and finish point (returning to depot), should not be more than ' KM far from each other. This constraint divides the search space to several small sub regions and emphasizes thatneighbour nodes being part of one route. 6) The total driven distant per route should be less than the allowed driven distant per day based on the contract minus G .

Figure 4 Network flowchart restructuring procedure

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The Figure 4 shows the flow chart of proposed route restructuring model. Three different phases are considered for developing new routing structure. First, the current network is mapped and all the traveling, locations, and timing constraints are extracted. Second, the optimization algorithm is run on these set of constraints and a new route structure is developed. In final phase, the best solution is mapped out (future map) and the cost of new network is compared with the current network cost. A new report is developed based on the new network schedule.

network is symmetric and all the routes more or less have equal number of the stops. Also, each route shares minimum common area with other routes although still there are a few numbers of routes that have to cross to the other routes territory due to timing or facility restriction issues.

There are an extensive researches on ACO model and the readers are referred to [8] for more detail of ACO algorithm. Local Search Improvement (LSI) is also an important part of any meta-heuristic algorithm. There are two LSI modules developed based on the node swap and route swap to improve the feasible constructed route as much as possible. These improvement modules are similar to [12] and [13]local improvement techniques. Figure 6-Future regional network structure IV. RE-STRUCTURING EXPERIMENT The developed model is employed for restructuring of a regional network consist of one distribution center and 99 locations to serve. The aim of re-structuring is assigning 15 new nodes (red color locations) to current network without having a major impact on each location delivery time. The main reason for minimum change is, each location has sensible operation schedule. Therefore, changing the visiting time of any location more than half an hour will create difficultyin current staff shift profile. The Figure 5 shows the existing network current route’s structure. By extracting the time window of each location and expanding each delivery time for half an hour we solved the model using ACO algorithm.

Comparing the cost of the future network with existing one shows that not only adding the new locations will not increase the total network cost but also by adjusting few locations working hours, there would be an opportunity to save $200K annually. V. DISCUSSION Practical VRPTW should take to account all the stakeholders concerns. Adding new rules and conditions to VRPTW model might increase the computational timing of the problem due to exhaustive search and covering many sub problems. But at the end it helps the new solution to be accepted easier by contractors and reduces the rework for the firm significantly. VI. CONCLUSION

Figure 5-Current regional network structure The condition for stopping the optimization is set to 500 non- improved solutions. Since ACO is using the probabilistic random search, it is better to run the algorithm for several times [1] in order to grasp optimal or near to optimal solution. For proposed network the optimization is run for 10 times. The minimum total driven distant came 2,043 KM while the worst total driven distant was 2,095 KM. As it shown in Figure 6 the structure of the new

In this paper the new conditions that help to develop a more practical VRPTW problem are discussed. This new rules are employed to make re-structuring routes more symmetric. An Ant Colony Algorithm (ACO) is developed to solve VRPTW problem considering all these new factors. The developed model is tested for a regional post offices network consist of 99 post offices. The goal of the restructuring was adding 15 new offices to current network. The solution reducessignificantly the total operational cost by small adjustment in certain offices working hour. VII. REFERENCES [1] M., Kheirkhahzadeh and A.A., Barforoush, “A hybrid algorithm for the vehicle routing problem”, IEEE Congress on Evolutionary Computation, PP.1791-1798, 2009

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[2] C. Qi and Y., Sun, “An Improved Ant Colony Algorithm for VRPTW”, International Conference on Computer Science and Software Engineering, PP.455-458, 2008 [3]T.G., Crainic, “Service network design in freight transportation”, European Journal of Operation Research, Vol.122, PP.272-288, 2000 [4]O., Braysy and M., Gendreau, “Vehicle Routing Problem with Time Windows, part I: Routing Construction and Local Search Algorithms”, Transportation Science, Vol.39, No.1, PP104-118, 2005 [5]G., Laporte,“What You Should Know about the Vehicle Routing Problem”, Naval Research Logistics, Vol.54, No.8, PP.811-819, 2007 [6]P., Ji and K., Chen, “The Vehicle Routing Problem: The Case of the Hong Kong Postal Service”, Transportation Planning and Technology, Vol.30, No.2-3, PP.167-182, 2007 [7]N.A., El-Sherbeny, “Vehicle routing with time windows: An overview of exact, heuristic and meta-heuristic methods”, Journal of King Saud University, Vol.22, No.3, PP.123-131, 2010 [8] C.,Blum, “Ant colony optimization: Introduction and recent trends”, Physics of Life Reviews, Vol.2,PP.353-373, 2005 [9]A.A.,Kazharov and V.M.,Kureichik, “Ant Colony Optimization Algorithm for Solving Transportation Problems”, Journal of Computer and Systems Sciences International, Vol.49, No.1, PP.30-43,2010 [10]E.G.,Carabetti, S.R., De Souza, M.C.P., Fraga, P.H.A., Gama, “An Application of the Ant Colony System Metaheuristic to the Vehicle Routing Problem with Pickup and Delivery and Time Windows”, Eleventh Brazilian Symposium on Neural Networks, PP.176-181, 2010 [11]J.,Ma and J.,Yuan, “Ant Colony Algorithm for Multiple-Depot Vehicle Routing Problem with Shortest Finish Time”, E-business Technology and Strategy Communication in Computer and Information Science, Vol.113,PP.114-123, 2010 [12]I.H., Osman, “Metastrategy simulated annealing and Tabu search algorithms for the vehicle routing problem”, Operations Research, Vol.41, PP.1165-1179, 1993 [13]B.Bullnheimer, R.F.,Hartl, C.Strauss, “An improved ant system algorithm for the vehicle routing problem”, Annals of Operations Research, Vol.89,PP.319-328, 1999