Transactions on the Built Environment vol 33, 1998 WIT Press, ISSN

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1 Effects of designated time on pickup/delivery truck routing and scheduling E. Taniguchf, T. Yamada\ M. Tamaishi*, M. Noritake^ "Department of Civil Engineering, Kyoto University, Yoshidahonmachi, Sakyo-kyu, Kyoto, , Japan kuciv. kyoto-u. ac.jp *Department of Civil Engineering, Kansai University, Yamatecho, Suita, Osaka, , Japan Abstract This paper presents a mathematical model developed for investigating the effects of designated time windows on urban pickup/delivery truck routing and scheduling. Three heuristic techniques, Genetic Algorithms (GA), Simulated Annealing (SA) and Tabu Search (TS) were applied to obtain approximate optimal solutions for the urban pickup/delivery truck routing and scheduling problem with time windows. The performance of each of these techniques was compared using a small road network. Each of these techniques generally produced good solutions within a short computation time period. However, the TS technique consistently provided better solutions within shorter computation time periods, while the SA technique provided more stable solutions if longer computation times were allowed. The model was applied to investigate the effects of increasing the width of time windows. The model estimated that a considerable reduction in total delivery costs, travel and waiting time as well as the required number of trucks could be realised by relaxing time windows. The effects of relaxing time windows would not only provide benefits to shippers or freight carriers but also to society at large by alleviating traffic congestion and environmental problems in urban areas due to the reduction of truck traffic.

2 128 Urban Transport and the Environment for the 21st Century 1 Introduction The distribution of goods in urban areas is a major cause of road congestion and pollution. It also consumes a substantial amount of energy and human labour. Just-In-Time delivery systems tend to increase these problems, whereby small loads of goods are often carried by pickup/delivery trucks in urban areas to provide customers with a high level of service. Strict designated times for pickup and delivery are especially required to minimise inventory costs in modern supply chain management. A recent survey in Osaka and Kobe, Japan, estimated that around 50% of goods delivered and 45% of goods collected had either a predetermined designated time or time window. This has led to many shippers and freight carriers establishing sophisticated fleet management systems using advanced telematics to help rationalise their routing and scheduling of urban pickup/delivery trucks. This paper estimates the effects of advanced routing and scheduling systems. To investigate the impacts of such systems a mathematical model was developed to optimise routing and scheduling of pickup/delivery trucks. Several modern heuristic techniques, including genetic algorithms, simulated annealing and tabu search were evaluated. A model was then applied to investigate the effects of relaxing designated time windows. The vehicle routing problem with time windows (VRPTW) has been investigated by numerous researchers in the operations research field (e.g. Solomon'*). It is well known that the VRPTW is a complex and difficult NP-hard combinatorial optimisation problem. Some approaches have applied heuristic techniques such as genetic algorithms (e.g. Thangiah et al/), simulated annealing (e.g. Kokubugata et al.^) and tabu search (e.g. Potvin et al/) to obtain approximate solutions of the VRPTW. Gendreau et al.' reviewed the application of such modern heuristic techniques to the VRP and described the potential of such methods for tackling this type of complex problem. 2 Model 2.1 Formulation This section focuses on the vehicle routing problem with time windows. Here, a fleet of identical vehicles are required to collect goods from customers and deliver them to a depot. Each customer has a designated

3 Urban Transport and the Environment for the 21st Century 129 time window indicating the time period in which vehicles should arrive. Vehicles departing from the depot visit a subset of customers to pick up goods and return to the depot to unload them. Vehicles are allowed to make multiple return trips to the depot per day. Each customer must be assigned to exactly one route of a vehicle and all goods at each customer are loaded on the vehicle at one time. The total weight of all the goods for a trip must not exceed the capacity of vehicle. A mathematical model was formulated to represent the problem of determining the best solution for minimising the total cost of delivering the goods. The total cost is composed of three costs; (a) fixed cost for each vehicle used, (b) time cost for travelling and waiting, and (c) delay penalty for designated pickup/delivery time at customers. Let, m : number of vehicles n : number of customers C(X) : total cost (yen) X : representing the assignment and order of visiting customers for all vehicles A = Xj,X2,* ",X/,- **,X^ / x/: representing the assignment and order of visiting customers for vehicle / c,,: fixed cost of using vehicle / (yen /vehicle) S,(Xj): = 1; if vehicle / is used = 0; otherwise GI i: time cost of using vehicle / (yen /(min vehicle)) TI (x/): operating time for vehicle / (min) Cj, : delay penalty cost at customer / (yen/min) tg,(x); arrival time at customer / ^.: end of desired time for time window at customer i (see Figure 1) ff/(x,): load of vehicle / (kg) W<. i: capacity of vehicle / (kg) t,, : time limit of soft time window at customer / (see Figure 1).

4 130 Urban Transport and the Environment for the 21st Century The model can then be formulated as follows. mn in C(X) = X c,j 5, (x, ) + JT c,, 7) (x, ).max{<u,(x)-f.,} (1) subject to (2) (3) The problem specified by eqns (1) - (3) is to determine the optimal values of variables X, that represent the assignment of vehicles and their order in visiting customers. Figure 1 shows the penalty cost function for early and late vehicle arrivals. The time period (^. -/\,) defines the width of the soft time window. If a vehicle arrives at customer / earlier than time ^., it must wait until the designated time and a cost is incurred during waiting. If a vehicle is delayed, it must pay a penalty proportional to delay time. The delay cannot exceed t,, as shown in Figure 1. This type of penalty is typically used in Just-In-Time transportation systems. cost -* time Figure 1: Soft time window: costs for early and delayed vehicle arrivals

5 Urban Transport and the Environment for the 21st Century Heuristic techniques The performance of three heuristic techniques, genetic algorithms (GA), simulated annealing (SA) and tabu search (TS), for solving the problem specified by eqns (1) - (3) was investigated. The GA technique is based on generating a number of populations, where individuals within a population represent a solution to the problem. Subsequent generations are determined by procedures where parents are selected and new individuals (solutions) are produced based on processing characteristics of the parents. This involves multiplication, crossover and mutation. In this study 300 individuals were produced at random for the initial population and reproduced until the predetermined computation time was exhausted. Thirty elite individuals were preserved by evaluating their fitness, generally defined as the inverse of the value of objective function. The partially matched crossover was applied. There are three general methods for mutation; (1) Deletion and Insertion (DI), (2) Exchange (EX) and (3) Reversion (RE). Tests were used to choose best mutation methods as well as the mutation rate and crossover rate. SA is a neighbourhood search technique based on the analogy with the physical theory of how material cools within a heat bath. Here, a move corresponds to changing x, representing the assignment and the visiting order to customers of vehicle /. The same three methods (DI, EX, RE) of changing x/ were tested as described for the mutation methods in GA. The probability of replacing the current solution with neighbourhood solution is given by an exponential function, if the neighborfood solution has a larger objective function value than the current solution. A simple geometric cooling process was used for determining the control parameter in each iteration. TS is a neighbourhood search technique that makes systematic use of information on past variable exchanges to generate new solutions. The best solution in the neighborhood can be chosen unless it is in the tabu list. In this study three types of moves (DI, EX, RE) were tested while the tabu list keeps a record of the number of times that an exchange was selected to determine a new solution. Testing of both tabu tenure and aspiration criteria were undertaken.

6 132 Urban Transport and the Environment for the 21st Century 3 Application to a road network 3.1 Conditions for calculation Figure 2 shows a simple hypothetical road network used for comparing the performance of the heuristic techniques. Travel times between nodes (customers) for all vertical and horizontal links were set at 12 minutes and 18 minutes respectively. One depot is located in the centre of network and 10 customers were randomly selected from all other nodes in the network. The maximum number of trucks was limited to ten. The weight of goods to be picked up at each customer was randomly distributed between 250-2,000 kg. The width of time windows was 1 hour. The time limit of time window t, i was 1 hour after ^,. The delay penalty cost was set at 5 times than that of the waiting time cost of trucks, that is tana = Stan/? in Figure Comparison of heuristic algorithms Parameter estimates for the three heuristic techniques were determined using a benchmark problem whose exact optimal solution is known. The 18 minutes minutes I node depot Figure 2: Test road network

7 Urban Transport and the Environment for the 21st Century 133 following methods and parameters gave best solutions for the problem described above. (GA) Method of mutation: DI Crossover rate: 0.7 Mutation rate: 0.14 (SA) Methods of changing the array of number: EX Control parameter: (TS) Method of move: EX Tabu tenure: 20 These procedures and parameters were used to compare the three heuristic techniques. Figure 3 shows the average performance of the three techniques when applied to ten problems that were specified by randomly selecting ten customers (nodes) from the test network (Figure 2). The graph shows the average discrepancy for each technique versus computation time. The discrepancy is defined as the difference between the value of the objective function and the best known solution. A personal computer (MICRON MILLENNIA XKU, CPU Pentium II 300MHz, Memory 64 MB) was used to perform the calculations computation time (sec) Figure 3: Comparison of performance in three techniques

8 134 Urban Transport and the Environment for the 21st Century Good solutions were found by all three techniques. TS reached the best known solution with the shortest computation time. However, it is difficult with the TS technique to determine the appropriate tabu tenure that is suitable for a specific problem. Relatively small changes in the value of objective function were achieved for the GA. The solutions were largely effected by the selection of random numbers for generating the first population and the crossover and mutation procedures. SA gives more stable solutions with good accuracy, if the computation time is extended to 200 seconds. The selection of random numbers for probabilistic moves in the neighborhood search had almost no effect on the final solution. Therefore, while the accuracy of the TS technique was best for shorter computation time periods, SA is the best model when longer computation time were allowed. 3.3 Effects of relaxing time windows The effects of relaxing the time windows were also examined using the small network described in the previous section. Here, the width of the time window was set at 4 levels: 1,2,4 and 8 hours. For the first level, the starting time /\, was randomly selected from every hour between 9 a.m. to 4 p.m. For the second and subsequent levels, the time windows had either the same starting or finishing time as the first level but with extended width. Based on the comparison of the 3 heuristic techniques in the previous section, SA was used to investigate the effects of varying the width of time windows. Figure 4 shows the change in total cost, total travel time and the required number of trucks when the width of the time windows is increased. Each of these measures decreased as the width of time window increased. The width of time window at customers considerably affects the performance of urban pickup/delivery truck routing and scheduling. For example when comparing the costs where the width of time window is only 1 hour with that of 4 hours, the total operating costs and total travel time were reduced by 13 % and 9 % respectively. This would not only reduce the operating costs of shippers or freight carriers, but also provide many benefits for society at large due to the reduction in traffic congestion and environmental problem in urban areas. The model also estimated that the total waiting time at customers decreased from 29.4 to 0 minutes when the width of the time window was extended from 1 hour to 3 hours. This reduction in waiting time would also have a positive effect on traffic flow, since waiting trucks often impede traffic flow by occupying roadside space in the vicinity of customers.

9 Urban Transport and the Environment for the 21st Century 135 I I total cost BB total travel time -O- number of trucks width of time window (hours) Figure 4: Effects of increasing the width of time window In some cases in modern logistic systems, relaxation of the width of time windows is not allowed. This is common in the distribution of goods at large retailing store chains. However, it is considered important to quantify the effects of increasing the width of time windows on the total delivery costs and the number of trucks required and this information may provide shippers or freight carriers and customers with an incentive to relax the strict time windows in urban pickup/delivery trucks routing and scheduling. 4 Conclusions This paper presents a mathematical model developed to investigate the effects on varying time windows for urban pickup/delivery truck routing and scheduling. The performance of three heuristic techniques, GA, SA and TS, were evaluated. All of the techniques provided good solutions when tested on a small road network. TS performed the best for shorter computation time periods, while SA performed better if longer computation times were allowed (up to 200 sec.).

10 136 Urban Transport and the Environment for the 21st Century Considerable reductions in the total delivery cost, travel time, waiting time and the required number of trucks were estimated when the width of the time window was increased. As well as the reduced delivery costs for shippers or freight carriers, there would also be substantial benefits to society at large, due to the reduction of traffic congestion and environmental problems in urban areas if time windows were relaxed. Acknowledgments The authors would like to express their heartiest appreciation to Professor Y. lida, Kyoto University for his excellent advice to this study. References [1] Gendreau, ML, Laporte, G. & Potvin, J.-Y., Vehicle routing: modern heuristics, Chapter 9, Local search in combinatorial Optimization, eds. E. Aarts & J. K. Lenstra, John Wiley & Sons, pp , [2] Kokubugata, K, Itoyama, H. & Kawashima, H., Vehicle routing methods for city logistics operations, IFAC/IFIP/IFORS Symposium on Transportation Systems, Chania, Greece, eds. M. Papageorgiou & A. Pouliezos, pp , [3] Potvin, J.-Y., Kervahut, T, Garcia, B.-L. & Rousseau, J.-M, The vehicle routing problem with time windows; part I: tabu search, INFORMS Journal on Computing, 8, pp , [4] Solomon, M. M., Algorithms for the vehicle routing and scheduling problems with time window constraints, Operations Research, 35, pp , [5] Thangiah, S. R., Nygard, K. E. & Juell, P. L, GIDEON: a genetic algorithm system for vehicle routing with time windows, Seventh IEEE International Conference on Artificial Intelligence Applications, IEEE Computer Society Press, Los Alamitos, CA, pp , 1991.