Vehicle Routing with Driver Learning for Real World CEP Problems

Transcription

1 th Hawaii International Conference on System Sciences Vehicle Routing with Driver Learning for Real World CEP Problems Marcel Kunkel PickPoint AG Ludwig-Eckes-Allee Nieder-Olm, Germany mk@pickpoint.de Michael Schwind Chair of IT-based Logistics Grüneburgplatz Frankfurt Main, Germany schwind@wiwi.uni-frankfurt.de Abstract Despite the fact that the vehicle routing problem (VRP) with its variants has been widely explored in operations research, there is very little published research on the VRP concerning real world constraint combinations and large problem sizes. In this work a heuristic solution approach for the VRP with real world constraints is presented driven by the requirements defined by clients in the courier, express and parcel (CEP) delivery industry in order to support their routing plan decisions and driver assignments. The solution algorithm used combines several local-search-based heuristics with constructive elements to solve the VRP with driver learning (VRPDL). As conceptual proof large instances for the capacitated VRP (CVRP) including 560 to 1200 customers are tested and compared to known benchmark results. From those instances new sub-instances are created and sequentially tested adding the driver learning constraint. Finally, the solver is applied to real world CEP instances with driver learning. I. INTRODUCTION Since its introduction by Dantzig and Ramser [7] the vehicle routing problem (VRP) has developed to one of the most studied problems in operations research. The basic VRP tries to find the shortest route for servicing a certain number of customers while using as less as possible vehicles in the service fleet. VRP problems are combinatorial optimization problems which are NP-hard [20]. Adding further constraints or increasing the problem size of such makes them even more complex and time consuming to solve and require heuristics find a satisfying solution in a reasonable time span. Solution approaches can be grouped into population-based heuristics such as genetic algorithms (GA) [2] or ant colony optimizations (ANT) [8] where what is learned from one or several initial route plans is used to create new route plans and into local-search-based approaches like TABU search [16] or variable neighborhood search (VNS) [19] where a new solution is always generated out of the neighborhood of the current solution. Local search heuristics use several moves techniques in order to explore the neighborhood of a given solution changing one or a few edges or switching the position of a node within or between routes. All local-search-based approaches need intensification and diversification elements in order to deeply explore and diversify from a given solution to find near optimal solutions after a relevant number of iterations. The local search algorithm proposed here combines exchange and 2-opt moves with new targeted move patterns where edge structures are chosen to be changed in order to fulfill the optimization goals. It also uses nearest neighbor and expulsion procedures for speed improvement as well as intensification purposes. This work is influenced by real world problems coming from the courier, express and parcel (CEP) delivery industry, where several thousand customers have to be visited from each depot during one workday. The customer demands (quantity requirement) have to be fulfilled without exceeding the truckload of the vehicle (capacity constraint). Drivers develop a certain routine driving in similar areas and visiting the same customers on their routes regularly. This fact will be integrated as customer-based driver learning depending on the regularity with which a certain driver delivers to that customer or the surrounding area. This promises significant savings of time due to better orientation in delivery process, etc. Furthermore, a driver can only start his route in the morning if the sorting of the parcels is done and the vehicle is loaded, which is interpreted as a maximum route length constraint. Solving real world problems with good results means to assign knowledgeable drivers on improved route plans and therefore helping the CEP organization to increase their efficiency in their operations. The remainder of the article is structured as follows. After a short characterization of the VRP with driver learning [23] in the parcel delivery domain a short introduction to recent research into relevant VRP literature and a mathematical formulation is given for the VRPDL. The second chapter also gives insight into the local-search-based solution algorithm by presenting the corresponding pseudo-code together with the underlying ideas. In the third chapter comparisons with standard VRP instances introduced by Li, Golden and Wasil [18] are drawn. From those instances new sub-instances are randomly generated and sequentially tested using the our heuristics together with driver learning. Ultimately the setup is used for the time sequential optimization of real world CEP instances with driver learning. II. VEHICLE ROUTING PROBLEM WITH DRIVER LEARNING In this chapter some related literature concerning large and rich VRP jointly with driver learning is discussed. Later the /12 $ IEEE DOI /HICSS

2 mathematical formulation of the VRPDL together with its local-search-based solution approach is introduced. A. Related Literature The vehicle routing problem with its variants consisting of combinations of constraints has been broadly researched over the last twenty years. A good overview is given in the book on VRPs edited by Toth and Vigo [24] and in the work of Laporte [17]. For the topic of VRPs including time windows (VRPTWs) a good survey article is provided by Bräysy and Gendreau [2], [1] and recently from Gendreau and Tarantilis [9]. The relaxation of the VRPTW to the VRP with 1-sided deadlines (VRPD) is studied by Campbell, Park and Karlaftis [5], [6], [21], [13]. For the real world CEP application domain, approaches that are able to solve large problem instances are of interest. Golden [10] has developed such instances including 200 to 483 customers. Instances and solution approaches with 560 to 1200 customers have been proposed by Li, Golden and Wasil [18]. The authors also built a solver based on a modified record-to-record (RTR) travel approach and used the given test instances for benchmarking. Those benchmark instances will be used as a base for the analysis that will be presented in chapter III. Research on large scale VRPTW was done by Kytöjoki et al. [15] and Gehring and Homberger [11], both do not consider driver learning effects. Sometimes real world problems in the VRPTW domain are called rich vehicle routing problems (RVRP). These RVRPs are extensions of VRP incorporating important issues arising in real world applications. A good survey summarizing several variants of the RVRP is given by Bräysy et al. ([3], [4]). A typical representative of work on this problem type is presented by Kok et al. [14]. They regard the scheduling of driver recreation brakes according to European legislation in junction with an underlying VRPTW. The resulting problem is called VRPTW with driver rules (VRPTWDR). There are two recent approaches which try to solve the VRPTWDR. The Kok et al. approach employs a combination of a heuristic and a constrained dynamic programming approach to find solutions for a modified Solomon benchmark. The second approach presented by Prescott-Gagnon et al. [22] uses a neighborhood search heuristic that relies on a TABU search algorithm in order to generate new routes. Driver learning is implicitly addressed in the work of Wong [25] where from the customer perspective a driver to customer relationship is postulated, which was realized through the constraint, that the package is delivered to the customer by the same driver every time he gets a parcel. The research of Zhong et al. [26] is also driven by UPS and its delivery structures. He separates in his paper core areas where the same driver delivers the customer every day and flex zones around the depots where the driver with least cost (drive time + stop duration) on that specific day is used to deliver to that customer. min (t ij + τ jk + p j ) x ijk (1) ij N k K d j x ijk Q k k K (2) i,j N (t ij + τ jk ) x ijk T max k K (3) i,j N w j = (t ij + τ jk ) y ijk (4) ij N,k K { w j T j ifw j >T j p j = (5) 0 otherwise x ijk =1 j N \{N 0 0} (6) i N\{N j } k K x ilk x ljk =0 l N \{N 0 0} (7) i,j N\{N 0 } k K x 0jk + x i0k =2 k K (8) i,j N TABLE I MATHEMATICAL FORMULATION OF THE VRPDL B. Mathematical Formulation The mathematical formulation of the VRPDL is given in Tab. I. Given a complete directed graph G =(N, E) where N = {0,.., n} is a set of nodes and E = {(i, j) i, j N(i j)} are the edges. Node 0 is the depot and must be the start and end point of each route. The travel time t ij represents the time taken to travel from node i to node j. It is further assumed that there is a set of equal vehicles, driven by a set of drivers K = {1,.., k}. τ jk represents the driver individual stop time of node j. Under the hypothesis that in the real world express business the sum of demands of all customers is lower than the vehicle capacity for all tours which fulfill the drivers working time limit, capacity constraints are left out of the equation for all routes where total time does not exceed the maximum allowed drive time T max (3). This is the case for the real world instances. As the artificial test instances include capacity constraints, they will be retained in further analyses. The times τ jk are based on local experiences and follow basically the learning and forgetting curve model (LFCM) by Jaber and Bonney [12]. The LFCM varies the forgetting intercept and the forgetting slope, such that the forgetting slope is assumed to be dependent on the learning slope, the amount of equivalent units accumulated by the point of break, and the maximum break length at which total forgetting occurs. Although the LFCM is relevant for updating the individual drivers learning rates for every customer on every single day, the day to day optimization uses only the single value τ jk per customer and driver valid for that day which is multiplied with the initial stop duration R 0. The result is the individual stop duration of that customer valid for a specific driver stop 1316

3 combination. The matrix containing the learning rates τ jk is updated every day following the LFCM model. Note that for simplification reasons we assume that the the time gap between the visits to one single customer is not long enough that forgetting occurs, so the LFCM model will be reduced to τ jk = R 0 r f. f describes the number of visits of the individual customer and r describes the learning factor between two delivery attempts without forgetting. CEP experts 3 stated that the time reduction for the delivery stop after three successful deliveries is about 30% and that an initial stop duration of R 0 =5minutes is realistic. Applying these numbers makes a r value of 0.9 realistically achieving r f =0.72 after three attempts. This means that this delivery would consume 3.6 minutes. Introducing the binary x ijk the objective function (1) minimizes the total service time. The total service time is composed out of the drive time t ij from node i to j plus the driver dependent service time τ jk plus an additional penalty p j if a deadline T j is violated. As seen in (1) and (4) the formulation treats penalty violation equally to overall route length violation. It might be necessary to introduce a further factor to give the penalties more weight and ensure that deadlines are kept. (4) describes the arrival time w j on node j.x ijk is a binary indicating whether customers on route k are visited before j. p j only applies, if the delivery to customer j takes place after T j (5). (6) requires that customer j is only visited from one vehicle k. (7) ensures that one arc goes to the customer and one arc leaves the customer. (8) eliminates sub routes in combination with (7) as in every tour k the depot is visited exactly twice. C. Heuristic Solution Approach In order to create an initial solution and modified Solomon I2 2 is produced. Unlike to the classic Solomon approach only V Sol % of the maximum route length is assigned before switching to the next furthest customers creating a new route. The residual customers are then inserted at least cost using all routes until routes are closed when L max 1.3 of maximum route length L max is reached. Note that leaving that factor close to L max could create unassigned customers at the end of the process if a small number of vehicles are used. The number of vehicles are also a result of that initial solution and can be improved by eliminating vehicles during the solution process. Note that in this implementation all relevant information are stored within SQL-database tables. All ordering and lookups are also performed within SQL-statements. For further improvement during the route construction phase, the construction is intermediated by local search exchange and 2- opt moves, which are performed as intra route moves during the initial phase and as inter route moves during the end phase. For further improvement this work applies a modified TABU search similar to the granular TABU search presented by Toth and Vigo [24]. In prior analyses this approach delivered good 3 Interview with Marc Wenger, former CEO trans-o-flex, 2/12/ Farthest from depot first and then assigning least cost inserts until route length is reached, Solomon initialization with deadlines first to be done TABLE II VRPTW SOLVER #DEFINITIONS: M = {number of moves} F 1 =15; F 2 =30{number of nearest edges for long and short moves} N =10{length of tabu list M tabu } TP 1 =50;TP 2 = 200 {TP 1 node not touched, leave out for TP 2 moves} FARTHEST = 200 {number of moves till farthest move pattern is used.} NO IMP = 500 {accepted number of moves without improvement} #INITIALIZATIONS: 1: Create distance matrix D with (i, j,distance, time, F 1 binary, F 2 binary) and add indices i, j and i, j 2: Create driver learning matrix K with k, i, learning value 3: Run identification of F 1 and F 2 and tag in D 4: Run Solomon initialization and create Solution S 1 5: Create distance matrix D F 1 (and also D F 2 ) as: select from D as D 1, DasD 2 where D 1.j=D 2.i and D 1.i!= D 2.i and D 1.j!= D 2.j and D 1.tag=true and D 2.tag=true {create combined distance matrix} 6: Add index D 1.i, D 1.j,D 2.i, D 1 id 2 i {using this index reduces search to logarithmic effort} #LOCAL OPTIMIZATION: for a =1to M do 7: Evaluate decision variable exchange or 2-opt as COND 1 8: Evaluate decision variable longshort as COND 2 9: Evaluate decision variable farthest move as COND 3 10: Evaluate move matrix M result using COND 1, COND 2 and COND 3 {SQL Statement order by saving ascending} while b = true do Compare M result with M tabu, if valid move found b = false end while 11: Perform move 12: Evaluate route length violation as COND 4 if COND 4 then 13: Drop shortest route 14: Assign residual customers to remove routes 15: Split longest route end if 16: Eval. new route plan length L new if L new >L best then if MOV E curr MOV E Lupdate >NO IMP then 17: Set current route plan = best route plan else 18: Set best route plan = current route plan end if end if 19: Update M tabu end for results for larger problem sizes and interacts perfectly with the implementation of the routes and moves within SQL-database tables. After initialization the solution algorithm starts with exchange moves until optimum is reached. The solver oscillates between exchange and 2-opt moves. 2-opt moves are implemented as inter and intra route moves. Every move will be entered in a tabu table hindering the solution algorithm from recombining the edges used to a combination which has already been explored. As ultimately those local optima might be necessary for further exploration at a later stage, the number of the current move is also entered in the tabu table which allows to unfreeze those moves at a later stage. As exchange moves would need too many non-improving moves and big tabu tables the 2-opt move is used for solving crossed loops on the same routes with only one move. Note that the 2-opt move needs more work on route reconstruction than the exchange move and changes the direction of the route within 1317

4 the loop. This move is therefore cost intensive and creates further distortion of rich instances. Currently exchange and 2-opt moves perform their moves until they reach a local optimum. If a move creates a solution which violates route length or capacity constraints, the route with the smallest route length is dropped, their stops are inserted on lowest cost and the violating route is split into two routes. This is done by dividing the route into equal three parts according to the number of nodes. Within the second part of the route the split node is determined such that the distance to the depot is the smallest. In order to avoid route dropping and reorganizing too often, thresholds for this decision are set to the factor 1.7 at the beginning of the solution process, decreasing linearly to 1.2 at the end of the solution process. Accordingly lower bounds for route reorganization are set to 0.5 of allowed route length increasing to 0.7 during the solution process. The pseudo-code of our solution algorithm is given in Tab. II. At every move the solver creates an ordered table of all exchange and 2-opt move results. From that table the first best moves which are not tabu are randomly selected and performed. Real world instances have revealed that in areas where long (high cost, high mileage) edges are used further improvement potential can be realized. Therefore, the algorithm introduces the farthest move pattern which forces the solver to take the combination of the longest two edges which follow each other after a defined amount of moves and forces it to unbind them with the 1 out of 10 best inserts. As the tabu list prevents falling back to the former solution, this methods creates major distortion in a well defined area and has proven to further improve the overall solution. In order to speed up the solution process only the F 1,2 nearest edges of every node are considered. The solution algorithm works with two values of F 1 =15and F 2 =30. Note that this also helps to reduce the growth of the evaluation matrix when growing the problem sizes from exponential to linear growth. This advantage has to be carefully used as it has been shown that in clustered problems it could create problems in finding binding edges among the clusters. Therefore, edges including the depot and edges leading to and from a customer with an associated deadline are excluded from this granularity rule. A further reduction of the evaluation matrix is achieved by introducing a move age factor which is a product of the tabu list evaluation. If a node has not been chosen in the exchange move for a certain number of moves TP 1 then it would be left out of the evaluation matrix for the next TP 2 moves. Finally, if for a certain number of neighborhood moves no improvement to the best solution is found, the solver is started again with the best solution so far. Randomization and the tabu table lead the algorithm to other regions of the solution space. III. COMPUTATIONAL RESULTS The Solomon instances are the most common artificially produced benchmark instances for the CVRP and the VRPTW. Nearly every solution approach in this area has been tested using them. However, CEP problems with the characteristics More than 500 customers per depot Capacity constraints are dominated by tour length restrict. Time windows can be relaxed to one-sided time windows called deadlines Driver learning as a new constraint are not covered by the Solomon benchmark instances., this work focuses on testing against large VRP benchmark instances proposed by Li et al. [18]. Furthermore, those instances are adapted by adding driver learning on a level that is realistic for the typical CEP company. A. Golden Benchmark Instances The second set of Li et al. instances reach from 560 to 1200 customers with capacity and route length constraints. Note that those tests have been run to ensure that the algorithm is capable of solving instances without driver learning similar to existing benchmarks. On average the solution algorithm achieved an improvement of 0.4 % compared to the RTR presented by Li et al. [18]. Compared with the best known solutions the solver is 0.6 % above these graphically evaluated solutions. All results are listed in Tab. III. Fig. 1. Golden 1200 customer, solution value: Fig. 1 shows the optimized route plan of the 1200 customer instance presented by Li et al. with a result of 37,053 in route length compared to 37,410 by the RTR approach of Li et al. B. Driver Learning Benchmark Instances The VRP problem generator introduced in Li et al. [18] generates one unique instance for every number of customers entered, which made it impossible to use for generating different similar instances of the same problem size. Therefore, out of the 560 and 1200 customer instances subsets of 250 customers were created, by picking 250 customers in both cases randomly several times. In the case of the 560 (1200) customer 1318

5 Instance Best Known LI Golden TABU HYBRID Diff best known Diff. Li et al ,3% -1,2% ,2% 0,1% ,1% -0,1% ,3% -0,8% ,1% -0,1% ,3% 2,4% ,4% 0,2% % -0,1% ,1% -0,6% % -3% ,4% -1% Avg. 0,6% -0,4% TABLE III COMPARISON TABU HYBRID WITH LI GOLDEN SHOW ON AVERAGE AN IMPROVEMENT OF 0.4% degree of repetition were increased to 20 to ensure enough cases of recurrence. It can be seen that the degree of attraction increases within the sequential optimization of the instances. The resulting route plans of 250 out of 560 (1200) customers are presented as examples in Fig. 2 (Fig. 3). The corresponding route length of all 10 (20) sub instances are shown in Tab. VIII. Comparing Fig. 1 with Fig. 3 the route similarity is obvious. Fig. 2. Subset of 250 customers out of 560 customers (Golden instance) instance, 10 (20) 250 customer instances were created. Those instances naturally have many overlapping customers, which represents reality in the CEP industry. With this setup it will be shown that the solver uses the driver learning constraint and its implementation to assign recurring customers to the same driver more often than to other drivers. With the learning rate r =0.9 presented earlier, the 250 customers in 10 cycles had the distribution in Tab. IV, which can be read as follows: A recurrence of 4 in line 4 means that this group of customers has been part of the 250 customer instances 4 times out of 10. Within those 4 times a minimum learning rate of =0.65 were achieved. In this case this group realized an average learning rate of r = 0.76 which corresponds with a average recurrence of the same driver at this customer of 2.7. The number of customers, in this case 113, indicates how many customers have been chosen 4 times. Similar results were achieved for the 1200 customer instance with 20 sequences of 250 customers, which is shown in Tab. V. Note that the Fig. 3. Subset of 250 customers out of 1200 customers (Golden instance) C. Real World Benchmark Instances with Driver Learning As an representative application for the VRPDL solution algorithm a special CEP sub-domain, the In-night delivery, together with anonymized real world data from PickPoint AG will be used. The specific delivery requirement is that all parcels must be delivered by 7:00 AM in the morning and all deliveries be made using key solutions into the trunks of cars or lockers. The end customers are therefore not private individuals, but rather service engineers. The data set of the base instance contains German delivery data for approximately 2000 customer locations, delivered from 1319

6 Recurrence Min. Learning rate Avg. Learning rate Number of customers Avg. driver recurrence TABLE IV RECURRENCE OF CUSTOMERS AND REALIZED LEARNING RATES BASED ON THE 560 CUSTOMERS LI GOLDEN INSTANCE, THE MORE OFTEN A CUSTOMER WAS PART OF THE SUBSET, THE LOWER THE LEARNING RATE BECAME Recurrence Min. Learning rate Avg. Learning rate Number of customers Avg. driver recurrence TABLE V RECURRENCE OF CUSTOMERS AND REALIZED LEARNING RATES BASED ON THE 1200 CUSTOMERS LI GOLDEN INSTANCE, THE MORE OFTEN A CUSTOMER WAS PART OF THE SUBSET, THE LOWER THE LEARNING RATE BECAME Recurrence Min. Learning rate Avg. Learning rate Number of customers Avg. driver recurrence TABLE VI RECURRENCE OF CUSTOMERS AND REALIZED LEARNING RATES BASED ON THE 467 CUSTOMERS CEP INSTANCE IN FRANKFURT REGION, PROVING THE RECURRENCE OF CUSTOMER DRIVER COMBINATIONS Fig Real World CEP customers have been randomly selected and optimized in Frankfurt region about 20 regional depots (e.g. Frankfurt, Munich, Cologne, Rostock). From those depots and their assigned customers Frankfurt (467 customers) and Cologne (340 customers) have been chosen to run 10 optimization sequences of 250 customers in the case of Frankfurt and 160 customers (proportional!) in the case of Cologne. The results of one sample instance from each region are graphically presented in Fig. 4 and Fig. 5. The route lengths of all sequential instances in both regions are listed in Tab. VIII. The random selection of real world customers following geographical routing situations leads to bigger differences in route length than on the artificial sub-instances created from the Li et al. instances. Nevertheless the solution algorithm achieves a good level of driver recurrence similar to the recurrence in the artificial instances. For benchmarking the average driver recurrence of Tab. VI and Tab. VII can be compared with the average driver recurrence of Tab. IV and Tab. V. Eg. after 7 runs (line 7 in all tables) the artificial instances achieve an average driver recurrence of 3.7 and 3.5 and in the real world instances after 7 runs recurrence rates of 3.3 and 4.5 are 1320

7 Recurrence Min. Learning rate Avg. Learning rate Number of customers Avg. driver recurrence TABLE VII RECURRENCE OF CUSTOMERS AND REALIZED LEARNING RATES BASED ON THE 340 CUSTOMERS CEP INSTANCE IN COLOGNE REGION, PROVING THE RECURRENCE OF CUSTOMER DRIVER COMBINATIONS Run Golden 250/560 Golden 250/1200 Frankfurt Cologne TABLE VIII OVERVIEW OF ROUTE LENGTH ON ALL INSTANCES achieved. Note that this fact together with the high variance of the route lengths of the Frankfurt and Cologne instances (see Tab. VIII) show that, even if the route plans vary a lot among the sub-instances, the driver learning effect is retained well together with good results. IV. CONCLUSION &OUTLOOK This paper introduces a modified TABU search algorithm for the solution of a VRP problem that is combined with the new real world constraint driver learning. In order to test our algorithm we used the benchmark instances introduced by Li et al. [18]. The solution algorithm was successfully tested and achieved an average improvement of 0.4%. In order to evalute the driver learning effect following the LFCM model we generated sequential sub-instances derived from the Li et al. original instances as well as real world instances from delivery data collected by the PickPoint AG. For all sequential instances good values of so called driver recurrence (e.g 4.5 times the same driver has been chosen on average to deliver a customer which occurred 7 times in the Cologne instance) were achieved together with good results for overall route length. Integrating driver learning leads to a significant reduction of stop durations compared to a modeling approach without driver learning where learning only happens coincidentally. Work in progress will focus on the integration of the current solver in real world CEP operations and their daily route plan creation and on a further increase of the instance size for CEP instances from the parcel delivery domain. It will also be necessary to integrate the capability of respecting different delivery deadlines with our solution algorithm instead of reflecting a single 7:00 delivery constraint to make our approach applicable for the express parcel delivery domain. Parallel work will focus on the aspect of finding good estimates on the number of routes or vehicles needed for a given instance which is especially difficult with rich vehicle routing problems derived from real world instances. REFERENCES [1] O. Bräysy and M. Gendreau. Vehicle routing problem with time windows, Part I: Route construction and local search algorithms. Transportation Science, 39(1): , [2] O. Bräysy and M. Gendreau. Vehicle routing problem with time windows, Part II: Metaheuristics. Transportation Science, 39(1): , [3] O. Bräysy, M. Gendreau, G. Hasle, and A. Løkketangen. A survey of heuristics for the vehicle routing problem, Part I: Basic problems and supply side extensions. Technical report, SINTEF, Oslo, Norway, [4] O. Bräysy, M. Gendreau, G. Hasle, and A. Løkketangen. A survey of heuristics for the vehicle routing problem, part II: Demand side extensions. Technical report, SINTEF, Oslo, Norway, [5] A. M. Campbell and B. W. Thomas. Probabilistic traveling salesman problem with deadlines. Transportation Science, 42(1):1 21,

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