Mixed integer programs to improve solutions of vehicle routing problems with intra-route constraints

Recently, the quality and the diversity of transport services are more and more required. Moreover, in case of a great deal of services and selling goods, a significant part of price is transport cost. Thus, the design of models and applications which make possible efficient transport planning and scheduling becomes important. A great deal of real transport problems may be modelled by using Pickup and Delivery Problem with Time Windows (PDPTW) and capacity constraints, which is based on the realization of a set of transport requests by a fleet of vehicles with given capacities. Each request is described by pickup and delivery locations, time periods when pickup and delivery operations should be performed and needed load. Application of evolutionary approach has brought good results in case of another, simpler transport problem – the Vehicle Routing Problem with Time Windows (VRPTW). This paper is aimed at proposing a straightforward extension of VRPTW based heuristics for the PDPTW.KeywordsVehicle Rout ProblemTransport ServiceTotal Travel TimeDelivery ProblemVehicle Rout Problem With Time WindowThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Pickup and Delivery Problem with Time Windows (PDPTW) is a generalization of the well studied Vehicle Routing Problem with Time Windows (VRPTW). Since it models several typical planning situations in operational transportation logistics and public transit, the PDPTW has attracted growing interest in recent years. This paper proposes a Grouping Genetic Algorithm (GGA) for solving the PDPTW which features a group-oriented genetic encoding in which each gene represents a group of requests instead of a single request. The GGA is subject to a comparative test on the basis of two publicly available benchmark problem sets that comprise 9 and 56 PDPTW instances, respectively. The results show that the proposed GGA is competitive.

The Pickup and Delivery Problem with Time Windows (PDPTW) is a generalization of the well studied Vehicle Routing Problem with Time Windows (VRPTW). This paper studies a Grouping Genetic Algorithm for solving the PDPTW. The insertion-searching heuristics (in GGA) which can generate feasible solutions was improved, new data structures were built, and then three routing adjustment strategies were added to come up with the Multi-Strategy Grouping Genetic Algorithm (MSGGA). The PDPTW benchmark problems with 100 customers are calculated with MSGGA, and the comparison between the result and that of the reference shows that the new algorithm shortens the calculating time with its astringency, better solutions of four cases are obtained and stability is improved.

Urban development can compromise open space and many valued ecosystem functions such as biological diversity, forest carbon sequestration, and clean water. To maximize the protection of open space and ecosystems, community planners and conservation organizations can take preemptive action either by purchasing land before it is developed or by providing incentives to landowners not to sell their land for development. In either case, the parcels must be prioritized for retention. While a significant body of literature addresses this need for prioritization, optimal reserve selection models do not account for land-price feedback effects that arise in markets where open space conservation competes with development. In competitive land markets, conservation acquisitions can affect land prices either by increasing demand, and thereby shifting the competitive equilibrium, or by inducing amenity premiums. People are willing to pay more for lots that are adjacent to reserves. These price effects can lead to increased development outside of the reserves and unintended loss of open space and valued ecosystem functions. In “Dynamic Reserve Selection: Optimal Land Retention with Land-Price Feedbacks,” S. F. Tóth, R. G. Haight, and L. W. Rogers fill this gap by formulating a linear-integer programming model with adaptive cost coefficients that are endogenous to the parcel acquisition decisions. They show that it is not always optimal to buy as much land for conservation as possible early on in the land retention effort. Buying fewer, smaller but more expensive parcels that have high conservation value and higher risk of development appears to be optimal in many market scenarios. They also show that failure to account for these price effects can lead to significant losses in biological conservation. In the paper “Parameterized Supply Function Bidding: Equilibrium and Efficiency,” R. Johari and J. Tsitsiklis explore a fundamental trade-off in designing markets. The authors consider markets where firms compete to supply a good through a market mechanism. On one hand, sufficient flexibility must be granted to firms in declaring their supply functions to ensure that they can approximately declare their costs. On the other hand, as the strategic flexibility granted to firms increases, their temptation to misdeclare their cost increases as well. Indeed, while in principle arbitrary supply functions allow firms to declare all marginal cost information, in theory and practice we find that such strategic flexibility only encourages the exercise of market power. Indeed, such a phenomenon has been seen in various energy markets around the world. The authors shed light on this trade-off by studying a parameterized class of supply functions that allow firms enough flexibility to communicate information about their production cost, yet not enough flexibility to enable them to exercise market power and cripple the performance of the overall market. In other words, by partially restricting the range of possible supply functions firms can declare, the authors' mechanism design controls “gaming”; nevertheless, they demonstrate the resulting market-clearing mechanism is nearly efficient. Their analysis lends credence to the hypothesis that restricting the strategy space granted to firms can often improve allocative efficiency. A good partial flexible process structure has been proven to be an effective tool to match (random) demand and capacity in both manufacturing and service industries. While most existing literature focuses on discussing the average performance of flexible structures, other interesting questions are: How to design a good partial flexible process structure that performs close to full flexibility in all cases? Could a flexible structure designed by popular guidelines also perform well in the worst case? In “Process Flexibility Revisited: The Graph Expander and Its Applications,” M. C. Chou, G. A. Chua, C.-P. Teo, and H. Zheng study the worst-case performance of the flexible structure design problem under a more general setting, which considers a general class of objective functions. They propose a concept “Ψ-expander” (0 < Ψ ≤ 1), a variant from graph expander, to design partial flexible process structure, and they show that a 1-expander structure performs as well as full flexibility structure in all scenarios. A simple heuristic to design flexile structures in general nonsymmetric systems is developed based on this concept. Using numerical examples studied in previous literature, the structures designed by the heuristic performed favorably compared to other popular flexible structures. In customer service systems, such as call centers or hospital emergency departments, waiting customers are often unable to estimate their own delay. A long wait, coupled with feelings of uncertainty about the length of that wait, leads to poor service evaluation. For system managers, making delay announcements is a relatively inexpensive way of reducing customer uncertainty about delays, thereby improving customer satisfaction with the service provided. In “Wait-Time Predictors for Customer Service Systems with Time-Varying Demand and Capacity,” R. Ibrahim and W. Whitt investigate alternative ways to predict, in real time, the delay of an arriving customer in a service system with customer abandonment and time-varying demand and capacity. These delay predictions may be used to make delay announcements. The authors propose new, simple, and effective ways to generate better delay predictions in many-server queues with a time-varying arrival rate, a time-varying number of servers, and customer abandonment. Because a new project entails enormous uncertainty, it's advisable not to “bet the firm” on the new project. Instead, run a small-scale experiment before making an irreversible investment in project-specific assets. As the experiment proceeds, the likelihood of the project's success is slowly revealed; eventually the firm makes an expansion or an exit (abandonment) decision based on the profitability of the pilot project observed to date. According to real options theory, the decision criterion becomes more stringent (the decision rule requires stronger evidence before action) as uncertainty in the profitability increases, but it is not obvious if this means that it should take longer for the firm to make this decision when there is an increase in the uncertainty of the project's success. In “Acquisition of Project-Specific Assets with Bayesian Updating,” H. D. Kwon and S. A. Lippman examined this issue in a model with project-specific assets and a (pilot) project that is either highly profitable or unprofitable. While noise in the project's profit stream makes it impossible to determine the project's profitability, the firm takes a Bayesian approach to update its belief based on the observed profit stream. The authors show that the expected time it takes to make an expansion or an exit decision is not monotone in the uncertainty. Therefore, stringent expansion/exit criteria do not necessarily imply a long time to make a decision. The results have practical implications for firms faced with expansion via investment in project-specific assets or abandonment of the pilot project because the rules/policy for deciding when to expand and when to abandon are determined in advance, before psychological pressures mount. Moreover, in deciding whether or not to launch the pilot project, it is important to know how increases in uncertainty impact the time until the expansion/abandonment decision is made. Seasonal influenza (flu) is a major public health concern, and the first line of defense is the flu shot. Antigenic drifts and the high rate of influenza transmission require annual updates to the flu shot composition. The World Health Organization recommends which flu strains to include in the annual vaccine based on surveillance and epidemiological analysis. Thus far, the design and timing of the flu shot have been made in an ad hoc manner. In “Optimizing the Societal Benefits of the Annual Influenza Vaccine: A Stochastic Programming Approach,” O. Y. Özaltın, O. A. Prokopyev, A. J. Schaefer, and M. S. Roberts propose a multistage stochastic mixed-integer program to identify an optimal annual flu shot design. They calibrate their model with real-life data and incorporate risk sensitivity using mean-risk objective functions. The results provide valuable insights for pressing policy issues. Dynamic pricing, where price is adjusted over time to match supply with demand, has long been adopted in various industries. Though advanced information technologies further facilitate price changes, the costs of price adjustment prevail and sometimes could be quite significant. In “Integration of Inventory and Pricing Decisions with Costly Price Adjustments,” X. Chen, S. X. Zhou, and Y. Chen develop a stochastic, dynamic inventory system with price adjustment costs, which consist of both fixed and variable parts. To provide effective strategies for firms to manage inventory and set selling price of such systems, they characterize the optimal policies for two special scenarios: one with inventory carryover and no fixed price-change costs and the other with fixed price-change costs but no inventory carryover. For the general system, a heuristic is developed, and its effectiveness is demonstrated numerically. Deterministic fluid models can provide useful first-order approximations for the performance of stochastic queuing models of large service systems, because they des

The distribution of goods to a set of geographically dispersed customers is a common problem faced by carrier companies, well-known as the Vehicle Routing Problem (VRP). The VRP consists of finding an optimal set of routes that minimizes total travel times for a given number of vehicles with a fixed capacity. Given the demand of each customer and a depot, the optimal set of routes should adhere to the following conditions: ?? Each customer is visited exactly once by exactly one vehicle. ?? All vehicle routes start and end at the depot. ?? Every route has a total demand not exceeding the vehicle capacity. The travel times between any two potential locations are given as input to the problem. Consequently, the total travel is computed by summing up the travel time over the chosen routes. In reality, carrier companies are faced with a number of other issues not conveyed in the VRP. The research in this thesis introduces a number of realistic variants of the VRP. These variants consider the VRP as a core component and incorporate additional features. By definition the VRP is NP-hard. Throughout the years a vast amount of research was aimed at developing both exact and heuristic solution procedures. Building on this established literature, solution procedures are developed to fit the variants proposed in this thesis. The standard VRP considers that the travel time between any pair of locations is constant throughout the day. However, congestion is present in most road networks. Considering traffic congestion results in time-dependent travel times, where the travel time between two location depends not only on the distance between them but also on the time of day one chooses to traverse this distance. Time-dependent travel times are considered in Chapters 2 and 3 of this thesis. Thus, in these Chapters we incorporate the time dimension into the VRP. The standard VRP does not take into account any customer service aspect. The customers are presumed to be available to receive their goods upon arrival of the vehicles. However, a number of carrier companies quote their expected arrival time to their customers. We introduce the concept of self-imposed time windows (SITW). SITW reflect the fact that the carrier company decides on when to visit the customer and communicates this to the customer. Once a time window is quoted to a customer the carrier company strives to provide service within this time window. SITW differ from time windows in the widely studied VRP with time windows (VRPTW), as the latter are exogenous constraints. In Chapters 4 and 5 SITW are endogenous decisions in stochastic environments. Thus, in addition to the sequencings decisions required by the VRP further timing decisions are needed. This thesis extends the VRP in two major dimensions: time-dependent travel times and self-imposed time windows. In reality carrier companies are faced with various uncertainties. The presented models incorporated some of these uncertainties by addressing three stochastic aspects: (I) In Chapter 3 stochastic service times are considered. (II) In Chapter 4, stochasticity in travel time is modeled to describes variability caused by random events such as car accidents or vehicle break down. (III) Finally, in Chapter 5 the objective was to construct a long term plan for providing consistent service to reoccurring customers. Stochasticity in this thesis is treated in an a priori manner. The plan, consisting of routes and timing decisions where necessary, is determined beforehand and is not modified according to the realization of the random events. Chapter 2 addresses environmental concerns by studying CO2 emissions in a timedependent vehicle routing problem environment. In addition to the decisions required for the assignment and scheduling of customers to vehicles, the vehicle speed limit is considered. The emissions per kilometer as a function of speed, is a function with a unique minimum speed v*. However, we show that limiting vehicle speed to this v* might be sub-optimal, in terms of total emissions. We adapted a Tabu search procedure for the proposed model. Furthermore, upper and lower bounds on the total amount of emissions that may be saved are presented. Quantifying the tradeoff between minimizing travel time as opposed to CO2 emissions is an important contribution. Another important contribution lies in incorporating fuel costs in the optimization. As fuel costs are correlated with CO2 emissions, Chapter 2 shows that even in today’s cost structure limiting vehicle speeds is beneficial. Chapter 3 defines the perturbed time-dependent VRP (P-TDVRP) model which is designed to handle unexpected delays at the various customer locations. A solution method that combines disruptions in a Tabu Search procedure is proposed. In Chapter 3 we identify situations capable of absorbing delays. i.e. where inserting a delay will lead to an increase in travel time that is less than the delay length itself. Based on this, assumptions with respect to the solution structure of P-TDVRP are formulated and validated. Furthermore, most experiments showed that the additional travel time required by the P-TDVRP, when compared to the travel time required by the TDVRP, was justified. In Chapter 4 the notion of self imposed time windows is defined and embedded in the VRP-SITW model. The objective of this problem is to minimize delay costs (caused by late arrivals at customers) as well as traveling time. The problem is optimized under various disruptions in travel times. The basic mechanism of dealing with these disruptions is allocating time buffers throughout the routes. Thus, additional timing decisions are taken. The time buffers attempt to reduce potential damage of disruptions. The solution approach combines a linear programming model with a local search heuristic. In Chapter 4, two main types of experiments were conducted: one compares the VRP with VRP-SITW while the other compares VRPTW with VRPSITW. The first set of experiments assessed the increase in operational costs caused by incorporating SITW in the VRP. The second set of experiments enabled evaluating the savings in operational costs by using flexible time windows, when compared to the VRPTW. Chapter 5 extends the customer service dimension by considering the consistent vehicle routing problem. Consistency is defined by having the same driver visiting the same customers at roughly the same time. As such, two main dimensions of consistency are identified in the literature, driver- and temporal consistency. In Chapter 5, driver consistency is imposed by having the same driver visit the same customers. Furthermore, we impose temporal consistency by SITW. A stochastic programming formulation is presented for the consistent VRP with stochastic customers. An exact solution method is proposed by adapting the 0-1 integer L- shaped algorithm to the problem. The method was able to solve the majority of test instances to optimality.

‘Route balance’ is the difference between the longest and the shortest among all the route lengths. The route balance is frequently considered in a vehicle routing problem (VRP) for balancing the distance travelled among delivery vehicles. This is a common practice since in VRPs there are no time elements. However, in recent years, some research works have considered route balance in a vehicle routing problem with time windows (VRPTW). As there are more time elements in VRPTW, this paper refutes that route balance is sufficient for VRPTW by simultaneously optimising makespan and workload imbalance using ‘total time balance’ instead of route balance. On the other hand, ‘total time balance’ is the difference between the longest total time taken and the shortest total time taken among delivery vehicles. As such, makespan is the longest total time taken among vehicles. Total time taken for a vehicle is the sum of vehicle run time, waiting time and service time. In order to demonstrate the importance of using total time balance instead of route balance on VRPTW, three different multi-objective VRPTW models, namely, (1) only with general VRPTW objectives (i.e. ‘total distance travelled’ by all vehicles and ‘total number of vehicles’ used), (2) general VRPTW objectives with route balance and (3) general VRPTW objectives with total time balance are developed and solved by fitness aggregated genetic algorithm (FAGA) for 36 Solomon’s benchmark instances. By comparison of the makespan produced by the FAGA between the three cases, the importance of using total time balance instead of route balance on multi-objective VRPTW is demonstrated. Also, the makespan produced by the FAGA for the third case, i.e. general VRPTW objectives with total time balance is compared with the makespan produced by fitness aggregated differential evolution (FADE). By comparing makespan with statistical testing between FAGA and FADE, the outperformance of FAGA over FADE is verified. To check the practicality of the total time balance on multi-objective VRPTW, an instance with real time windows is also solved for the three cases by the FAGA and its makespan are compared and reported.

In this thesis we study three combinatorial optimization problems belonging to the classes of Network Design and Vehicle Routing problems that are strongly linked in the context of the design and management of transportation networks: the Non-Bifurcated Capacitated Network Design Problem (NBP), the Period Vehicle Routing Problem (PVRP) and the Pickup and Delivery Problem with Time Windows (PDPTW). These problems are NP-hard and contain as special cases some well known difficult problems such as the Traveling Salesman Problem and the Steiner Tree Problem. Moreover, they model the core structure of many practical problems arising in logistics and telecommunications. The NBP is the problem of designing the optimum network to satisfy a given set of traffic demands. Given a set of nodes, a set of potential links and a set of point-to-point demands called commodities, the objective is to select the links to install and dimension their capacities so that all the demands can be routed between their respective endpoints, and the sum of link fixed costs and commodity routing costs is minimized. The problem is called non- bifurcated because the solution network must allow each demand to follow a single path, i.e., the flow of each demand cannot be splitted. Although this is the case in many real applications, the NBP has received significantly less attention in the literature than other capacitated network design problems that allow bifurcation. We describe an exact algorithm for the NBP that is based on solving by an integer programming solver a formulation of the problem strengthened by simple valid inequalities and four new heuristic algorithms. One of these heuristics is an adaptive memory metaheuristic, based on partial enumeration, that could be applied to a wider class of structured combinatorial optimization problems. In the PVRP a fleet of vehicles of identical capacity must be used to service a set of customers over a planning period of several days. Each customer specifies a service frequency, a set of allowable day-combinations and a quantity of product that the customer must receive every time he is visited. For example, a customer may require to be visited twice during a 5-day period imposing that these visits take place on Monday-Thursday or Monday-Friday or Tuesday-Friday. The problem consists in simultaneously assigning a day- combination to each customer and in designing the vehicle routes for each day so that each customer is visited the required number of times, the number of routes on each day does not exceed the number of vehicles available, and the total cost of the routes over the period is minimized. We also consider a tactical variant of this problem, called Tactical Planning Vehicle Routing Problem, where customers require to be visited on a specific day of the period but a penalty cost, called service cost, can be paid to postpone the visit to a later day than that required. At our knowledge all the algorithms proposed in the literature for the PVRP are heuristics. In this thesis we present for the first time an exact algorithm for the PVRP that is based on different relaxations of a set partitioning-like formulation. The effectiveness of the proposed algorithm is tested on a set of instances from the literature and on a new set of instances. Finally, the PDPTW is to service a set of transportation requests using a fleet of identical vehicles of limited capacity located at a central depot. Each request specifies a pickup location and a delivery location and requires that a given quantity of load is transported from the pickup location to the delivery location. Moreover, each location can be visited only within an associated time window. Each vehicle can perform at most one route and the problem is to satisfy all the requests using the available vehicles so that each request is serviced by a single vehicle, the load on each vehicle does not exceed the capacity, and all locations are visited according to their time window. We formulate the PDPTW as a set partitioning-like problem with additional cuts and we propose an exact algorithm based on different relaxations of the mathematical formulation and a branch-and-cut-and-price algorithm. The new algorithm is tested on two classes of problems from the literature and compared with a recent branch-and-cut-and-price algorithm from the literature.

In this paper we present a parallel solver for the vehicle routing problem with time windows (VRPTW) and the pickup and delivery problem with time windows (PDPTW). The solver is based on parallel competition of particular solvers solving the given problem instance. The particular solvers are based on negotiation between a fleet of agents representing individual vehicles using a cost structure corresponding to the wellknown travel time savings insertion heuristic. The performance of the solver is assessed on the Homberger-Gehring and Li-Lim benchmark sets for the VRPTW and PDPTW case respectively. While both sets are widely used across the routing community, they were not addressed by previous agent-based studies. Thus the achieved average solution quality of 95% and 87% for the VRPTW and PDPTW cases represents a new best known result for these benchmark sets for agent-based approaches. An analysis of the solver's convergence, runtime and parameter sensitivity is provided within the experimental evaluation section, that has also not been provided by previous agent-based studies.

A general heuristic for vehicle routing problems

The classical vehicle routing problem (VRP) aims to find a set of routes at a minimal cost (finding the shortest path, minimizing the number of vehicles, etc) beginning and ending the route at the depot, so that the known demand of all nodes are fulfilled. Each node is visited only once, by only one vehicle, and each vehicle has a limited capacity. Some formulations also present constraints on the maximum traveling time. The VRPSD is a variation of the classical VRP, where each customer can be served by more than one vehicle. Thus, for the VRPSD, besides the delivery routes, the amount to be delivered to each customer in each vehicle must also be determined. The option of splitting a demand makes it possible to service a customer whose demand exceeds the vehicle capacity. Splitting may also allow decreasing costs. The vehicle routing problem with time windows and split deliveries (VRPTWSD) is an extension of the VRPSD, adding to it the time window restraints. Lenstra and Rinnooy Kan (1981) have analyzed the complexity of the vehicle routing problem and have concluded that practically all the vehicle routing problems are NP-hard (among them the classical vehicle routing problem), since they are not solved in polynomial time. According to Solomon and Desrosiers (1988), the vehicle routing problem with time windows (VRPTW) is also NP-hard because it is an extension of the VRP. Although the vehicle routing problem with split deliveries (VRPSD) is a relaxation of the VRP, it is still NP-hard (Dror and Trudeau, 1990, Archetti et al., 2005). Therefore, the VRPTWSD is NP-hard, since it is a combination of the vehicle routing problem with time windows (VRPTW) and the vehicle routing problem with split delivery (VRPSD), and that makes a strong point for applying heuristics and metaheuristic in order to solve the problem. This work develops a scatter search (SS) algorithm to solve a vehicle routing problem with time windows and split deliveries (VRPTWSD). To generate the initial solutions of SS we propose an adaptation of the sequential insertion heuristic of Solomon (1987). Ho and Haugland (2004) modified the customers’ demands of the Solomon’s test problems in order to perform split deliveries. Numerical results of SS are reported as well as comparisons with the Ho and Haugland algorithm. O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

We present an application of Ant Colony Optimization metaheuristic to the Pick-up and Delivery Problem with Time Windows (PDPTW), a variant of the Vehicle Routing Problem with Time Windows (VRPTW) with additional constraints on pairs of source-destination nodes.

In this paper, we introduce models for the optimization of the door-to-door freight transportation. The main thrust of these models stand in the allowance to forecast of freight amounts that will be transported daily considering capacity as well as time constraints (e.g. time availability of retailers and customers). We also have integrated real-world constraints to meet practical difficulties that may actually face transportation (e.g., fixed number of working hours of drivers, availability of vehicles). The transportation scheme is characterized by a consolidation center. We assumed, also, that freight transportation is mutualized. Thus, we have split the FDPTW (Pickup and Delivery Problem with Time Windows) into two problems: (i) is a Vehicle Routing Problem with Time Windows (VRPTW) with Pickup; (ii) is a VRPTW with Delivery. The proposed models are solved by LINGO. The output results are the optimal trucks in each model. Lastly, one of the elaborated models (VRPTW with delivery) is tested with Solomon's benchmark and for the other models we propose different numerical experiments to validate our contributions.

The Vehicle Routing Problem with Time windows (VRPTW) is an extension of the capacity constrained Vehicle Routing Problem (VRP). The VRPTW is NP-Complete and instances with 100 customers or more are very hard to solve optimally. We represent the VRPTW as a multi-objective problem and present a genetic algorithm solution using the Pareto ranking technique. We use a direct interpretation of the VRPTW as a multi-objective problem, in which the two objective dimensions are number of vehicles and total cost (distance). An advantage of this approach is that it is unnecessary to derive weights for a weighted sum scoring formula. This prevents the introduction of solution bias towards either of the problem dimensions. We argue that the VRPTW is most naturally viewed as a multi-objective problem, in which both vehicles and cost are of equal value, depending on the needs of the user. A result of our research is that the multi-objective optimization genetic algorithm returns a set of solutions that fairly consider both of these dimensions. Our approach is quite effective, as it provides solutions competitive with the best known in the literature, as well as new solutions that are not biased toward the number of vehicles. A set of well-known benchmark data are used to compare the effectiveness of the proposed method for solving the VRPTW.

One application of graph theory is to optimize the distribution problem. This problem can be solved using Vehicle Routing Problem with Time Window (VRPTW) model and its variants such as VRPTW, CVRPTW dan OVRPTW. This article comprehends the improvement of the solution with the local search method using perturbation on those variants. There are three parts in the method: generating an initial solution, improvement using local search, and perturbation. The initial solution was generated using the sequential insertion algorithm, the local search process used inter-route and intra-route operators, and the perturbation using ejection chain and double swap. Result of experiments showed that perturbation using double swap gave a better solution than ejection chain. This caused by two-times movement in the double swap that could examine all optimal solution possibilities. An example of implementation the VRPTW variant on distribution optimization is given in this article.

Vehicle Routing Problem with Time Windows (VRPTW) is a constrained NP-hard problem that designs the least cost routes from one depot to a set of geographically scattered points. Most of traditional techniques are difficult to solve this kind of problem. In this paper, we explored the validity of another swarm-intelligence-based model-Bacterial Foraging Optimization (BFO) for VRPTW solving. Original BFO, BFO with linear decreasing chemotaxis step (BFO-LDC) and BFO with non-linear decreasing chemotaxis step (BFO-NDC) are used to obtain the best solutions of a given VRPTW problem, respectively. The experimental results demonstrated that the proposed BFO algorithms have the potential to solve Vehicle Routing Problem with Time Windows (VRPTW) with rapid convergence rate and the good result accuracy.