Soft Time Windows Research Articles

A Graph-Refinement Algorithm to Minimize Squared Delivery Delays Using Parcel Robots

In recent years, parcel volumes have reached record highs, prompting the logistics industry to explore innovative solutions to meet growing demand. In densely populated areas, delivery robots offer a promising alternative to traditional truck-based delivery systems. These autonomous electric robots operate on sidewalks and deliver time-sensitive goods, such as express parcels, medicine and meals. However, their limited cargo capacity and battery life require a return to a depot after each delivery. This challenge can be modeled as an electric vehicle-routing problem with soft time windows and single-unit capacity constraints. The objective is to serve all customers while minimizing the quadratic sum of delivery delays and ensuring each vehicle operates within its battery limitations. To address this problem, we propose a mixed-integer quadratic programming model and introduce an enhanced formulation using a layered graph structure. For this layered graph, we present two solution approaches based on relaxations that reduce the number of nodes and arcs compared to the expanded formulation. The first approach, Iterative Refinement, solves the current relaxation to optimality and refines the graph when the solution is infeasible for the expanded formulation. This process continues until a proven optimal solution is obtained. The second approach, Branch and Refine, integrates graph refinement into a branch-and-bound framework, eliminating the need for restarts. Computational experiments on modified Solomon instances demonstrate the effectiveness of our solution approaches, with Branch and Refine consistently outperforming Iterative Refinement across all tested parameter configurations.

Vehicle routing problem for omnichannel retailing including multiple types of time windows and products

This paper addresses the capacitated vehicle routing problem for omnichannel retailing with multiple types of time windows (hard time windows and soft time windows) and multiple types of products simultaneously. The problem aims to transfer multiple products from a central warehouse to stores and satellites, where split delivery is allowed since the demands of stores or satellites are large and the different product types have different volumes. Based on different characteristics of purchasing channels, the time window of a store is a hard constraint while the time window of a satellite is a soft one. This type of vehicle routing problem exists extensively in omnichannel retail distribution systems in the logistics industry of China, which considers multiple purchasing channels for customers. Since this type of vehicle routing problem is an NP-hard problem and more complicated than the conventional vehicle routing problem with time windows, we design an adaptive large neighborhood search (ALNS) method to solve it. Finally, numerical experiments are conducted to evaluate the effectiveness of the proposed algorithm and reveal some managerial insights. The numerical experiments show that the proposed algorithm can obtain a high-quality solution under a shorter computation time compared with the state-of-the-art MIP solver (Gurobi).

Just-in-time scheduling of unrelated parallel machines with family setups and soft time window constraints

ABSTRACT This study addresses an unrelated parallel machine scheduling problem with family setups and soft time windows, which includes considerations for machine eligibility. The objective is to minimize the total weighted tardiness and earliness, and to maximize the number of Just-in-Time jobs. Four different metaheuristic algorithms are compared to determine which type – single-point (simulated annealing), population-based evolutionary (artificial immune system, genetic algorithm), or swarm intelligence algorithms (ant colony optimization) – is more effective in solving this problem. Two strategies, namely penalty and repair, are proposed to handle precedence constraints. Experimental results demonstrate that ant colony optimization with local search which employs the repair strategy, as a swarm intelligence-based algorithm outperforms its rivals in terms of objective function values and robustness. The performance of this algorithm has shown an improvement in objective function values ranging from 2.12% to 8.52% on average, with a relative percentage deviation of 10.35.

Modeling a Multimodal Routing Problem with Flexible Time Window in a Multi-Uncertainty Environment

In this study, we extend the research on the multimodal routing problem by considering flexible time window and multi-uncertainty environment. A multi-uncertainty environment includes uncertainty regarding the demand for goods, the travel speed of the transportation mode, and the transfer time between different transportation modes. This environment further results in uncertainty regarding the delivery time of goods at their destination and the earliness and lateness caused by time window violations. This study adopts triangular fuzzy numbers to model the uncertain parameters and the resulting uncertain variables. Then, a fuzzy mixed integer nonlinear programming model is established to formulate the specific problem, including both fuzzy parameters and fuzzy variables. To make the problem easily solvable, this study employs chance-constrained programming and linearization to process the proposed model to obtain an equivalent credibilistic chance-constrained linear programming reformulation with an attainable global optimum solution. A numerical case study based on a commonly used multimodal network structure is presented to demonstrate the feasibility of the proposed method. Compared to hard and soft time windows, the numerical case analysis reveals the advantages of the flexible time window in reducing the total costs, avoiding low reliability regarding timeliness, and providing confidence level-sensitive route schemes to achieve flexible routing decision-making under uncertainty. Furthermore, the numerical case analysis verifies that it is necessary to model the multi-uncertainty environment to satisfy the improved customer requirements for timeliness and enhance the flexibility of the routing, and multimodal transportation is better than unimodal transportation when routing goods in an uncertain environment. The sensitivity analysis in the numerical case study shows the conflicting relationship between the economic objective and the reliability regarding the timeliness of the routing, and the result provides a reference for the customer to find a balance between them.

Two-Phase Fuzzy Real-Time Approach for Fuzzy Demand Electric Vehicle Routing Problem with Soft Time Windows

Environmental concerns have called for several measures to be taken within the logistics and transportation fields. Among these measures is the adoption of electric vehicles instead of diesel-operated vehicles for personal and commercial-delivery use. The optimized routing of electric vehicles for the commercial delivery of products is the focus of this paper. We study the effect of several practical challenges that are faced when routing electric vehicles. Electric vehicle routing faces the additional challenge of the potential need for recharging while en route, leading to more travel time, and hence cost. Therefore, in this work, we address the issue of electric vehicle routing problem, allowing for partial recharging while en route. In addition, the practical mandate of the time windows set by customers is also considered, where electric vehicle routing problems with soft time windows are studied. Real-life experience shows that the delivery of customers’ demands might be uncertain. In addition, real-time traffic conditions are usually uncertain due to congestion. Therefore, in this work, uncertainties in customers’ demands and traffic conditions are modeled and solved using fuzzy methods. The problems of fuzzy real-time, fuzzy demand, and electric vehicle routing problems with soft time windows are addressed. A mixed-integer programming mathematical model to represent the problem is developed. A novel two-phase solution approach is proposed to solve the problem. In phase I, the classical genetic algorithm (GA) is utilized to obtain an optimum/near-optimum solution for the fuzzy demand electric vehicle routing problem with soft time windows (FD-EVRPSTW). In phase II, a novel fuzzy real-time-adaptive optimizer (FRTAO) is developed to overcome the challenges of recharging and real-time traffic conditions facing FD-EVRPSTW. The proposed solution approach is tested on several modified benchmark instances, and the results show the significance of recharging and congestion challenges for routing costs. In addition, the results show the efficiency of the proposed two-phase approach in overcoming the challenges and reducing the total costs.

Path Optimisation of Electric Vehicles for Collection and Delivery based on Intelligent Waste Bins

This paper proposes a dynamic electric vehicle path optimization method for intelligent garbage collection and transportation systems to address the increasing uncertainty of urban household waste and the carbon emissions caused by vehicles in traditional garbage collection and transportation processes. With the goal of minimizing total cost and maximizing garbage collection and transportation, adjust the soft time window in a timely manner based on whether the amount of garbage generated by the intelligent garbage bin reaches the collection and transportation threshold. Taking into account the position of the garbage bin to be cleared, the position of the garbage collection and transportation vehicle, the amount of electricity, and the loading capacity, dynamically adjust the vehicle path. Due to the fact that genetic algorithms require a long time to converge to an acceptable solution, this paper uses greedy genetic algorithms to comprehensively utilize their advantages in local search and global search capabilities to achieve faster convergence solutions. The research results indicate that compared with traditional static garbage collection and transportation modes, dynamic garbage collection and transportation schemes can reduce total costs and increase garbage collection and transportation volume, and reasonable planning of the path of electric collection and transportation vehicles can effectively reduce carbon emissions. Finally, the effectiveness and efficiency of the algorithm were verified through comparative analysis of numerical examples.

Path Optimization of Container Multimodal Transportation Considering Differences in Cargo Time Sensitivity

To optimize the container multimodal transportation path selection problem with differences in cargo time sensitivity, this paper introduces the concept of mixed time windows and establishes an integer programming model from the perspective of carriers. The objective is to minimize the total transportation cost while meeting the customer’s on-time delivery requirements. The model imposes hard time window constraints on cargoes with on-time delivery requirements and soft time window constraints on general cargoes. A bi-level genetic algorithm is employed to solve the model. A case study is conducted to demonstrate the effectiveness of the proposed approach. Experimental results show that the bi-level genetic algorithm’s convergence ability, efficiency, stability, and global search capabilities are far superior to those of traditional single-level genetic algorithms. This indicates that the use of multi-stage solution approaches can mitigate the shortcomings of genetic algorithms in dealing with high-dimensional problems. In addition, when analyzing the effects of time penalty intensity and time window on decision-making results, it is found that the more time-sensitive cargoes are more likely to be affected by the time window setting, and will tend to sacrifice costs to ensure that the time demand is met. At the same time an increase in demand for time-sensitive services is accompanied by an increase in operating costs. Accurate assessment and classification of time sensitivity of cargoes can effectively prevent wasting resources in unnecessary areas, and ultimately maximize decision outcomes and ensure the most efficient use of resources.

A Sustainable Multi-Objective Model for Capacitated-Electric-Vehicle-Routing-Problem Considering Hard and Soft Time Windows as Well as Partial Recharging.

Due to the high pollution of the transportation sector, nowadays the role of electric vehicles has been noticed more and more by governments, organizations, and environmentally friendly people. On the other hand, the problem of electric vehicle routing (EVRP) has been widely studied in recent years. This paper deals with an extended version of EVRP, in which electric vehicles (EVs) deliver goods to customers. The limited battery capacity of EVs causes their operational domains to be less than those of gasoline vehicles. For this purpose, several charging stations are considered in this study for EVs. In addition, depending on the operational domain, a full charge may not be needed, which reduces the operation time. Therefore, partial recharging is also taken into account in the present research. This problem is formulated as a multi-objective integer linear programming model, whose objective functions include economic, environmental, and social aspects. Then, the preemptive fuzzy goal programming method (PFGP) is exploited as an exact method to solve small-sized problems. Also, two hybrid meta-heuristic algorithms inspired by nature, including MOSA, MOGWO, MOPSO, and NSGAII_TLBO, are utilized to solve large-sized problems. The results obtained from solving the numerous test problems demonstrate that the hybrid meta-heuristic algorithm can provide efficient solutions in terms of quality and non-dominated solutions in all test problems. In addition, the performance of the algorithms was compared in terms of four indexes: time, MID, MOCV, and HV. Moreover, statistical analysis is performed to investigate whether there is a significant difference between the performance of the algorithms. The results indicate that the MOSA algorithm performs better in terms of the time index. On the other hand, the NSGA-II-TLBO algorithm outperforms in terms of the MID, MOCV, and HV indexes.

生鲜农产品冷链集货车辆路径与货物配载联合优化研究

As a bridge connecting agricultural production and consumption, the circulation of agricultural products has the function of connecting supply and demand, guiding production and promoting consumption. However, the development of rural logistics in China is slow, and most logistics centers still rely on experience to plan the pick-up vehicle routings, resulting in long transport time and high cost. In order to improve the efficiency of pick-up and reduce transportation costs, a joint optimization model of cold-chain pick-up vehicle routing and cargo allocation for fresh agricultural products was proposed in this study. Soft time window constraint and three-dimensional loading constraints were considered, and the lowest pick-up cost was used as optimization goals in this model. In addition, adaptive large neighborhood search algorithm (ALNS) and heuristic depth-first search algorithm (HDFS) were combined to solve the model. A case study of Kunming International Flower Auction Center was conducted to compare the schemes of pick-up vehicle routing before and after optimization. Results demonstrate that the pick-up cost after optimization decreases by 9.6 %, the number of vehicles decreases by one, the total volume utilization rate of vehicles increases by 23 %, and the total load utilization rate of vehicles increases by 15 %. This study provides a model reference and solution method for enterprise operators to formulate schemes of pick-up vehicle routing quickly and reasonably.

Demand-Responsive Transit Service With Soft Time Windows Considering Real-Time Disruptions Based on Bounded Rationality

Demand-responsive transit (DRT) with smartphone-based applications is emerging as a flexible and sustainable mobility service, transforming urban transportation. Nevertheless, to satisfy the real-time and inconsistent demand, it is becoming increasingly important to capture the decision-making psychology of order cancellations. In this study, a two-phase optimization framework is presented in response to real-time disruptions, including order cancellations and the insertion of new real-time passengers. In contrast to random real-time demand, this paper is more concerned about the impacts of the feedback information on order cancellations. Bounded rationality is incorporated into the model to discuss the decision-making process of cancellation behaviors. With regard to the soft window, a compensation strategy is proposed to promote the profit while encouraging passengers for a long-term use. Additionally, solution algorithm based on variable neighborhood search (VNS) and rolling horizon is constructed to approach the Pareto solutions set. To testify the validity of the proposed algorithm, small-scale experiments in simplified Sioux Falls network are investigated for multiple runs. Meanwhile, a real-world case study in Beijing is explored to evaluate the system performance considering real-time disruptions. The results indicate that the dynamic DRT service can substantially improve the system profit but increase the penalty cost. The profit presents a significant improvement to 940 (renminbi) RMB as a result of the insert of real-time passengers. This study, therefore, not only provides a deeper insight into the analysis of passenger cancellation behavior but also contributes to construct a more flexible DRT service.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

Core-based cost allocation for collaborative multi-stop truckload shipping problem

With the recently emerged digital platforms in logistics, shippers can easily collaborate by bundling their heavy less-than-truckload orders via multi-stop truckload shipping to reduce transportation cost. The platform has responsibility for planning the shipping routes, for bundling the orders and fairly allocating the cost to shippers. To address this challenging problem in practice, we propose a new cooperative game based on a variant of a pickup and delivery model with soft time windows for shipper collaboration. The centralized optimization model is NP-hard and the core of the game may be empty. We adopt the least-core concept and simplify the core stability constraints as route-wise conditions. Based on theoretical results, we propose an innovative route-generation joint searching algorithm that iteratively solves the centralized optimization and least-core allocation problems at the same time, where the route-generation subproblem is solved by a customized multi-start local search subroutine. Extensive computational experiments on a real-world case demonstrate that the proposed algorithm can quickly generate a near-optimal solution with minor optimality gap and a least-core allocation with small stability deviation. With our algorithm, the shippers also receive substantial cost savings from collaboration.

Optimization of Electric Vehicle Routes Considering Multi-Temperature Co-Distribution in Cold Chain Logistics with Soft Time Windows

Inspired by the practice of urban distribution of fresh products, we introduce a new electric vehicle routing problem with soft time windows. In this problem, goods with different temperature layers can be distributed in ordinary electric vehicles simultaneously based on the cold storage insulation box. The primary objective is to devise optimized distribution routes for logistics companies to minimize distribution costs, including transportation, refrigeration, and charging costs. To address this, we present a mathematical model for the problem and propose an improved ant colony optimization algorithm combined with a 2-opt algorithm. Based on Solomon dataset, we conduct numerical experiments to verify the effectiveness of the proposed model and algorithm. The numerical results demonstrate that multi-temperature co-distribution can lead to a reduction in distribution cost and an improvement in distribution efficiency.

Research on Path Optimization of Vehicle-Drone Joint Distribution considering Customer Priority

To meet the personalized distribution needs of customers, comprehensively consider customer value, the urgency of customer needs, and the impact of priority distribution to the customer on the enterprise, and based on regional restrictions, put forward vehicle-drone joint distribution path optimization problem considering customer priority. First, the goal is to minimize the sum of total distribution cost and customer priority cost integrating soft time windows and constructing a path optimization model of the vehicle-drone joint distribution. Second, a two-stage hybrid algorithm is proposed for the problem model. In the first stage, the deep neural network and the grid search improved support vector machine algorithm (DNN-GSM-SVM) are used to screen and classify customer priority features, and in the second stage, adaptive large-scale neighborhood search improved genetic algorithms (ALNS-GA) are used to solve the problem of vehicle-drone joint distribution path planning problem. Finally, combined with the numerical example, the optimization scheme of the vehicle-drone joint distribution path considering priority is analyzed. Compared with the three algorithms and error analysis, the effectiveness of the model and the two-stage algorithm was verified. Compared with the results of the scheme that does not consider priority, the results show that priority can significantly improve customer satisfaction. The efficiency of the vehicle-drone joint distribution was verified by comparing the three scenarios.

Vehicle routing in the case of uncertain customer demands and soft time windows: A neuro-fuzzy logic approach

Vehicle routing, with its many variants, is one of the most important and frequently solved problems in transportation engineering. The aim of this paper is to develop a decision-making support tool for addressing the issue of dispatching vehicles in scenarios characterized by uncertain demands within soft time windows. In real-world scenarios, it is not uncommon for customer demands to exhibit flexibility, where certain early arrivals or delays may be deemed acceptable. Therefore, this paper introduces vehicle routing in more realistic contexts, offering potential practical implementations. The methodology for solving the problem is based on a fuzzy logic system whose membership functions are additionally adjusted using a neural network. Such a tool, neuro-fuzzy logic, is suitable for solving a defined routing problem since it can consider all the mentioned uncertainties in the distribution systems. Each user is assigned a performance index that considers travel time, demand, and delivery time windows. Then, the performance index is used as input data in the proposed vehicle routing tool based on the Clarke-Wright algorithm. The described approach has been tested on a concrete example, mimicking a distribution network resembling real world conditions, incorporating estimated travel times between customers. The results demonstrate that the proposed approach can effectively handle customer demands, with an average delay of 5.05 minutes during the 80-minute distribution. In future research, some environmental factors could be included in the proposed model. In addition, one of the directions of future research could be vehicle re-routing using the ideas from this paper.

Fuzzy Superposition Operation and Knowledge-driven Co-evolutionary Algorithm for Integrated Production Scheduling and Vehicle Routing Problem with Soft Time Windows and Fuzzy Travel Times

Fuzzy Superposition Operation and Knowledge-driven Co-evolutionary Algorithm for Integrated Production Scheduling and Vehicle Routing Problem with Soft Time Windows and Fuzzy Travel Times

Optimization algorithm for the time-dependent vehicle routing problem with softtime windows: A research based on mixed integer model and mixed tabu search

Optimization algorithm for the time-dependent vehicle routing problem with softtime windows: A research based on mixed integer model and mixed tabu search

考虑碳排放成本的多车型农资配送路径优化研究

Agriculture is the foundation of the national economy, and agricultural materials are the basis of agricultural development. As the three rural issues (agriculture, countryside, and farmers) become increasingly important, the distribution of agricultural materials attracts extensive attention. Given the slow development of rural logistics, the traditional agricultural material distribution process encounters many problems, such as cumbersome distribution links, high distribution costs, and low profit for enterprises, which in turn cause high production costs and low income for farmers. In consideration of battery energy consumption and soft time window constraints, this study adopted the agricultural material distribution route as the study object and established an optimization model of the agricultural material distribution route with fixed , transportation, energy consumption, time window penalty, and carbon emission costs as the objective functions. With regard to the algorithm, the operation of differential update and chaotic disturbance was innovatively enhanced and applied to the improved ant colony algorithm to simulate the model and obtain the optimal distribution route optimization model. Results show that the traditional ant colony algorithm improved by differential updating and chaotic disturbance has the advantages of low distribution cost, reasonable route, small number of activated vehicles, and short convergence time. Compared with the traditional ant colony algorithm, the improved ant colony algorithm can converge to the global optimum faster. This study provides guidance and suggestions on route selection and vehicle configuration to reduce costs and increase efficiency and offers certain theoretical support to alleviate urban traffic pollution and implement carbon trading policies in the future.

Formulation and heuristic method for urban cold-chain logistics systems with path flexibility – The case of China

The focus of this paper is on achieving a win-win situation regarding the economic, environmental, and social impacts of the cold chain logistics terminal distribution system. This paper proposes three multi-objective models to investigate the above effects by incorporating soft time windows, heterogeneous fleets, and path flexibility, with defining the objectives of examining logistics costs, fuel consumption, carbon emissions, quality damage to perishable commodities, and customer satisfaction using six evaluation functions. To solve the proposed models, an efficient optimization framework is developed by combining domain operators with versatile multi-objective evolutionary algorithms (MOEA) to obtain Pareto solutions. Extensive experiments are conducted to test the validity of the concerned model and algorithms. The results demonstrate that: (1) the proposed algorithm is effective in solving the proposed model; (2) the proposed multi-path strategy can effectively improve the performance of cold-chain logistics systems compared to single-path strategies; (3) evaluation functions that assess customer satisfaction greatly affect the performance of cold-chain logistics systems; and (4) the trade-off relationship between the objectives should be investigated to define the model. The paper also provides valuable managerial insights for improving the efficiency and sustainability of cold-chain logistics operations.

Optimizing Electric Truck Routing and Charging with Soft Time Windows Using Vehicle-to-Grid Technology

Due to the rapid increase in the use of electric vehicles and instability in energy supply, the application of vehicle-to-grid (V2G) technology has gained attention in the freight transportation sector. V2G has the potential to increase the efficiency of power grid and make additional profits by utilizing surplus power from electric vehicle batteries. This paper proposes an optimization model for electric trucks (ETs) to provide operational decision-making support for the freight transportation sector. The objective of the model is to minimize the total net cost, which includes charging cost, discharging reward, and time penalties, while considering changes in ET charging cost and the system marginal price. Furthermore, we conduct sensitivity analysis in the vehicle routing problem with soft time windows using ETs in the V2G system.