Vehicle Routing Problem With Soft Time Windows Research Articles

A unified Maximum Entropy Principle approach for a large class of routing problems

We present a novel Maximum Entropy Principle (MEP)-based modeling and algorithmic approach, for a large class of routing and scheduling problems including the Capacitated Vehicle Routing Problem (CVRP), the Vehicle Routing Problem with soft time-windows (VRPTW) and the Close-Enough Traveling Salesman Problem (CETSP). The MEP models routing and scheduling as ‘equivalent’ partitioning or clustering problems with side-constraints, and employs tools from statistical physics for assigning resources (routes/vehicles) to each node such that the resource allocation results in feasible, sub-optimal routes. The MEP can flexibly incorporate side-constraints related to minimum tour-lengths, capacities, schedules and reachability (like CETSP). Analytically, our model results in a second-order non-linear system of complex implicit equations. We show that an iterative approach effectively solves these equations, is equivalent to a gradient descent and converges to a local minimum. Despite the non-linear optimization model, the algorithm converges to an integer optimal solution. Computationally, we compare our approach to Simulated Annealing (SA), the CMT-14 benchmarks for VRP and benchmarks for CETSP. Our approach consistently outperforms SA for multiple variants of routing problems, specifically, the CVRP, VRPTW and CETSP. On the CMT-14 benchmark instances, our approach finds the optimal (when verifiable) number of vehicles, with a cumulative tour distance within 6.2% on average, and in comparable computation times of the best-known solutions (over all approaches for each instance). We also demonstrate the efficacy of our approach on benchmark instances of the CETSP and discuss our results. This demonstrates the potential of our MEP approach to be further embedded into hybridization heuristics for further improved results.

An Adaptive Variable Neighborhood Search Ant Colony Algorithm for Vehicle Routing Problem With Soft Time Windows

In this paper, an adaptive variable neighborhood search ant colony algorithm (AVNSACA) is proposed to solve the vehicle routing problem with soft time windows (VRPSTW). The ant colony algorithm’s pheromone update strategy is improved to make up for the lack of pheromone in the algorithm’s early stage. In order to avoid the algorithm falling into local optimum, two variable neighborhood search operators are designed, and the conditions for the algorithm to enter the variable neighborhood search are set. The effectiveness of AVNSACA in solving vehicle routing problem with soft time windows is verified by Solomon benchmark problem. Through the comparative analysis of the experimental results of the two algorithms, the advantages of the improved ant colony algorithm are illustrated. The experimental results show that the proposed algorithm can effectively obtain better solutions.

The opportunity cost of time window violations

This paper studies a variant of the vehicle routing problem with soft time windows (VRPSTW), inspired by real-world distribution problems. In applications, violations of the prescribed delivery time are commonly accepted. Customers’ inconvenience due to early or late arrival is typically modelled as a penalty cost included in the VRPSTW objective function, added to the routing costs. However, weighting routing costs against customer inconvenience is not straightforward for practitioners. In our problem definition, practitioners evaluate solutions by comparison with the hard time windows solution. The desired routing cost saving is set by the practitioners as a percentage of the nominal solution’s routing costs. The objective function minimizes the time window violations, or the customer inconvenience, with respect to the nominal solution. This allows practitioners to quantify the opportunity cost (i.e. the customer inconvenience), when a target routing cost saving is imposed. To solve the problem, we apply two exact algorithms: the first is based on a standard branch-and-cut-and-price (BCP), the second is a BCP nested in a bisection algorithm. Computational results demonstrate that the second algorithm outperforms the standard implementation. Solutions obtained with the opportunity cost interpretation of soft time windows are then compared with solutions obtained using both hard time windows and the standard interpretation of soft time windows.

Just-in-time delivery for green fleets: A feedback control approach

With increasing attention being paid to greenhouse gas (GHG) emissions, the transportation industry has become an important focus of approaches to reduce GHG emissions, especially carbon dioxide equivalent (CO2e) emissions. In this competitive industry, of course, any new emissions reduction technique must be economically attractive and contribute to good operational performance. In this paper, a continuous-variable feedback control algorithm called GEET (Greening via Energy and Emissions in Transportation) is developed; customer deliveries are assigned to a fleet of vehicles with the objective function of Just-in-Time (JIT) delivery and fuel performance metrics akin to the vehicle routing problem with soft time windows (VRPSTW). GEET simultaneously determines vehicle routing and sets cruising speeds that can be either fixed for the entire trip or varied dynamically based on anticipated performance. Dynamic models for controlling vehicle cruising speed and departure times are proposed, and the impact of cruising speed on JIT performance and fuel performance are evaluated. Allowing GEET to vary cruising speed is found to produce an average of 12.0–16.0% better performance in fuel cost, and −36.0% to +16.0% discrepancy in the overall transportation cost as compared to the Adaptive Large Neighborhood Search (ALNS) heuristic for a set of benchmark problems. GEET offers the advantage of extremely fast computational times, which is a substantial strength, especially in a dynamic transportation environment.

Solving the multi-objective Vehicle Routing Problem with Soft Time Windows with the help of bees

This paper presents a new model and solution for the multi-objective Vehicle Routing Problem with Soft Time Windows (VRPSTW) using a hybrid metaheuristic technique. The proposed methodology is developed on the basics of a new swarm based Artificial Bee Colony (ABC) algorithm combined with two-step constrained local search for neighborhood selection. VRPSTW involves computing the routes of a set of vehicles with fixed capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. Here, the time window constraints are relaxed into “soft”, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. The solution of routing problems with soft time windows has valuable practical applications. This paper uses a direct interpretation of the VRPSTW as a multi-objective optimization problem where the total traveling distance, number of window violations and number of required vehicles are minimized while capacity and time window constraints are met. Our work aims at using ABC inspired foraging behavior of honey bees which balances exploration and exploitation to avoid local optima and reach the global optima. The algorithm is applied to solve the well known benchmark Solomon׳s problem instances. Experimental results show that our suggested approach is quite effective, as it provides solutions that are competitive with the best known results in the literature. Finally, we present an analysis of our proposed algorithm in terms of computational time.

A Hybrid Algorithm for the Vehicle Routing Problem with Soft Time Windows and Hierarchical Objectives

This paper presents an algorithm for the vehicle routing problem with soft time windows (VRPSTW). It involves serving a set of customers, with earliest and latest time deadlines, which may be violated if a penalty is paid, and a constant service time at the customer site. Customer demands are served by capacitated vehicles. The purpose of this research is to develop a hybrid algorithm that includes an insertion heuristic, a local search algorithm and a meta-heuristic algorithm to solve VRPSTW problems with more than one objective. The first priority aims to find the minimum number of vehicles required and the second priority aims to search for the solution that minimizes the total travel time. Performance of the algorithmic approach is measured by two criteria: solution quality and run time. A set of well-known benchmark data and a genetic algorithm are used to compare the solution quality and running time of the algorithm. Results show a trade-off can be made between total cost and service when considering soft time windows. Running time results display that the hybrid algorithm has a higher performance than the genetic algorithm when the number of customers is less than 25.

Vehicle Routing Problem for Multiple Product Types, Compartments, and Trips with Soft Time Windows

This paper presents a mathematical model to solve the vehicle routing problem with soft time windows (VRPSTW) and distribution of products with multiple categories. In addition, we include multiple compartments and trips. Each compartment is dedicated to a single type of product. Each vehicle is allowed to have more than one trip, as long as it corresponds to the maximum distance allowed in a workday. Numerical results show the effectiveness of our model.

Yard crane scheduling in a container terminal for the trade-off between efficiency and energy consumption

Green transportation has recently been the focus of the transportation industry to sustain the development of global economy. Container terminals are key nodes in the global transportation network and energy-saving is a main goal for them. Yard crane (YC), as one type of handling equipment, plays an important role in the service efficiency and energy-saving of container terminals. However, traditional methods of YC scheduling solely aim to improve the efficiency of container terminals and do not refer to energy-saving. Therefore, it is imperative to seek an appropriate approach for YC scheduling that considers the trade-off between efficiency and energy consumption. In this paper, the YC scheduling problem is firstly converted into a vehicle routing problem with soft time windows (VRPSTW). This problem is formulated as a mixed integer programming (MIP) model, whose two objectives minimize the total completion delay of all task groups and the total energy consumption of all YCs. Subsequently, an integrated simulation optimization method is developed for solving the problem, where the simulation is designed for evaluating solutions and the optimization algorithm is designed for exploring the solution space. The optimization algorithm integrates the genetic algorithm (GA) and the particle swarm optimization (PSO) algorithm, where the GA is used for global search and the PSO is used for local search. Finally, computational experiments are conducted to validate the performance of the proposed method.

The trade-off between fixed vehicle costs and time-dependent arrival penalties in a routing problem

This paper introduces the vehicle routing problem with soft time windows (VRPSTW) in which problem definition differs from ones previously defined in literature. Branch-and-price approach is employed, resulting in a set partitioning master problem and its new subproblem. Novel techniques are consequently developed to solve this new subproblem. Experimental results report the comparisons of these solution techniques under the branch-and-price framework. The VRPSTW solutions have further been compared to the state-of-the-art literature, signifying the superiority of the VRPSTW on this issue.

A two-phase approach for jointly determining the lot size and delivery policy in a vendor-buyer integrated system with rework

This paper describes the development of an iterative heuristic algorithm utilizing graph coloring theory and cost minimization strategy to solve the vehicle routing problem with soft time windows (VRPSTWs) for a reverse logistic. The purpose of the proposed algorithm is to improve the computational eff iciency and fi nd the routing combinations graphically by applying the vertex coloring technique. Linkage among vertices was performed iteratively, taking the model's constraints into consideration. A Taiwanese waste collection network was utilized to demonstrate the implementation of the algorithm. Computational results revealed that the algorithm was eff icient and effective

Vehicle routing optimization with soft time windows in a fuzzy random environment

This paper is concerned with a vehicle routing problem with soft time windows (VRPSTW) in a fuzzy random environment. Two objectives are considered: (1) minimize the total travel cost and (2) maximize the average satisfaction level of all customers. After setting up the model for the VRPSTW in a fuzzy random environment, the fuzzy random expected value concept is used to deal with the constraints and its equivalent crisp model is derived. The global–local–neighbor particle swarm optimization with exchangeable particles (GLNPSO-ep) is employed to solve the equivalent crisp model. A case study is also presented to illustrate the effectiveness of the proposed approach.