Scheduling within a limited budget is closely related to several much studied scheduling with compression and rescheduling problems, and is also a variant of the more recent model of scheduling with testing. In this problem, one seeks to minimize the makespan for a set of jobs that are to be processed on a number of parallel identical machines. Each job Jj is given an upper bound uj on its actual processing time pj, and a testing fee cj. In the offline case, the processing time pj is known to the scheduler; while in the oblivious case, pj is revealed to the scheduler only if the testing fee cj is paid. So the scheduler can choose to execute Jj on any one of the machines non-preemptively for uj time or pay for the testing and then execute the job for pj time. The scheduler is given a budget B to pay for testing and seeks to minimize the makespan within that budget. The offline problem is denoted as P∣uj, pj, cj, B∣Cmax, and the oblivious problem is denoted as P∣uj,−,cj,B∣Cmax, where P stands for multiple parallel identical machines with the number of machines being part of the input. We contribute a polynomial-time approximation scheme (PTAS) for the offline problem P∣uj, pj, cj, B∣Cmax, which leads to an almost tight (2+ϵ)-competitive algorithm for the oblivious problem P∣uj,−,cj,B∣Cmax.

The problem studied in this paper is to find a collection of vertex-disjoint paths in a given graph G = (V, E) such that each path has length at least k, called a long path, and the total number of edges on these paths is maximized. The problem is NP-hard for any fixed k or when k is part of the input, by a reduction from the Hamiltonian path problem. Berman and Karpinski presented a 7/6-approximation algorithm for k = 1, but for a general k >= 2, there is no approximation algorithm directly for the problem. We present the first local search (0.4394k + O (1))-approximation algorithm for any fixed k >= 1, and a 1.4254-approximation algorithm for k = 2 built on top of a maximum triangle-free path-cycle cover. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

The distribution of shared bikes is different from that of other goods. There are some demand stations which need a large number of shared bikes, such as bus stations, subway exits and business districts. The demand of these stations cannot be met in a single delivery, so the demand can be split into batches for distribution. Therefore, shared bikes need to be delivered from distribution centers to demand stations. However, these delivery vehicles generate carbon emissions during the process, which has an impact on environment. Thus, shared bikes distribution vehicle route selection with considering carbon emission under demand splitting is an important problem. The paper established a model for distribution vehicle route selection of shared bikes considering carbon emission which aims at minimization the sum of carbon emission cost and delivery cost, under demand splitting of the stations and delivery vehicles with load limit. Then an approximation algorithm GA is designed to solve it. The time complexity of GA was proved, and the upper and lower bounds of the approximate ratio of GA are discussed. Finally, an empirical example was facilitated by examining real shared bikes stations in the Yanta district of Xi'an, China, to verify the effectiveness of the model and algorithm. The approximation ratio of GA is 3.52 which shows that the approximate performance of the algorithm in the example is good. The results and conclusions yield a theoretical basis for decision-makers to optimize the delivery of shared bikes.

We investigate scheduling with testing in the multiprocessor environment. Scheduling with testing, disallowing job preemption, is a variant recently proposed by Dürr et al. and Albers et al. to model many real-life applications where the scheduler decides to either execute a job for a likely overestimated length of time, or test the job to obtain its exact processing time, followed by executing the job for the exact length of time on the same machine. We present an online algorithm with competitive ratio 2.8681, improving the previous best known guarantee of 2.9514; when every test operation takes one unit of time, we present an online algorithm with competitive ratio approaching 2.5276, improving the previous best known guarantee of 2.8081. Compared to the prior work, the novel design ideas in our algorithm for the general testing case are to set up multiple thresholds for job testing decision-making and to use multiple sorting/selection criteria to determine the job processing order. We also propose a way to construct a benchmark dataset from the one for the classic multiprocessor scheduling problem, and on this benchmark dataset we demonstrate the empirical performance of our algorithms against the prior algorithms.

Various new scheduling problems have been arising from practical production processes and spawning new research areas in the scheduling field. We study the parallel multi-stage open shops problem, which generalizes the classic open shop scheduling and parallel machine scheduling problems. Given m identical k-stage open shops and a set of n jobs, we aim to process all jobs on these open shops with the minimum makespan, i.e., the completion time of the last job, under the constraint that job preemption is not allowed. We present an efficient polynomial-time approximation scheme (EPTAS) for the case when both m and k are constant. The main idea for our EPTAS is the combination of several categorization, scaling, and linear programming rounding techniques. Jobs and/or operations are first scaled and then categorized carefully into multiple types so that different types of jobs and/or operations are scheduled appropriately without increasing the makespan too much.

Coflow scheduling is a challenging optimization problem that underlies many data transmission and parallel computing applications. In this paper, we study the indivisible coflow scheduling problem on parallel identical machines with the objective to minimize the makespan, i.e., the completion time of the last flow. In our problem setting, the number of the input/output ports in each machine is a fixed constant, each port has a unit capacity, and all the flows inside a coflow should be scheduled on the same machine. We present a (2 + ϵ ) -approximation algorithm for the problem, for any ϵ > 0 , in which the number of machines can be either a fixed constant or part of the input.

As the logistics industry modernizes, living standards improve, and consumption patterns shift, the demand for fresh food continues to grow, making cold chain logistics for perishable goods a critical component in ensuring food quality and safety. However, the presence of both soft and hard time windows among demand nodes can complicate the single-network distribution of perishable goods. In response to these challenges, this paper proposes an optimization model for multi-distribution center perishable goods delivery, considering both one-echelon and two-echelon network joint distributions. The model aims to minimize total costs, including transportation, fixed, refrigeration, goods damage, and penalty costs, while measuring customer satisfaction by the start time of service at each demand node. A two-stage heuristic algorithm is designed to solve the model. In the first stage, an initial solution is constructed using a greedy approach based on the principles of the k-medoids clustering algorithm, which considers both spatial and temporal distances. In the second stage, the initial routing solution is optimized using a linear programming approach from the Ortools solver combined with an Improved Adaptive Large Neighborhood Search (IALNS) algorithm. The effectiveness of the proposed model and algorithm is validated through a case study analysis. The results demonstrate that the initial solutions obtained through the k-medoids clustering algorithm based on spatio-temporal distance improved the overall cost optimization by 1.85% and 4.74% compared to the other two algorithms. Among the three two-stage heuristic algorithms, the Ortools-IALNS proposed here showed enhancements in the overall cost optimization over the IALNS, with improvements of 3.24%, 1.12%, and 0.41%, respectively. The two-stage heuristic algorithm designed in this study also converged faster than the other two heuristic algorithms, with overall optimization improvements of 1.55% and 1.28%, further validating the superior performance of the proposed heuristic algorithm.

The swift growth of e-commerce has intensified the pressure on the express delivery industry in terms of timely delivery. Therefore, it becomes imperative to explore the implementation of a joint distribution model that enables more efficient and sustainable delivery of express shipments. Considering factors such as the capacity of delivery stations and the number of transport vehicles, this study proposes a two-level delivery location selection model under the joint distribution mode, aiming to minimize the total cost and carbon emissions of the delivery station location selection. The proposed model is solved using the Improved Multi-objective Particle Swarm Optimization (IMOPSO) algorithm. According to the research results, by adopting the joint distribution model, the total cost can be significantly reduced to 63,867.05 yuan while also achieving a substantial reduction in carbon emissions to 997,945.81 kg. Furthermore, the facility utilization rate exceeds 90%. Notably, the IMOPSO algorithm exhibits notable advantages in terms of both accuracy and computational efficiency.

We investigate the seemingly untouched yet the most general parallel identical k -stage flow-shops scheduling, in which we are given an arbitrary number of indistinguishable k -stage flow-shops and a set of jobs each to be processed on one of the flow-shops, and the goal is to schedule these jobs to the flow-shops so as to minimize the makespan. Here the number k of stages is a fixed constant, but the number of flow-shops is part of the input. This scheduling problem is strongly NP-hard. To the best of our knowledge all previously presented approximation algorithms are for the special case where k = 2 , including a 17/6-approximation in 2017, a (2 + ϵ ) -approximation in 2018, and the most recent polynomial-time approximation scheme in 2020; they all take advantage of the number of stages being two. We deal with an arbitrary constant k ≥ 3 , where the k -stage flow-shop scheduling is already strongly NP-hard. To define a configuration that summarizes the key information about the job assignments in a feasible schedule, we present novel concepts of big job type, big job assignment pair type, and flow-shop type. We show that the total number of distinct configurations is a polynomial in the input number of flow-shops. We then present how to compute a schedule for each configuration that assigns all the big jobs, followed by how to allocate all the small jobs into the schedule at a cost only a fraction of the makespan. These together lead to a polynomial-time approximation scheme for the problem.

The rapid growth of e-commerce has heightened the importance for express delivery companies to ensure timely deliveries. Consequently, it is essential to explore ways to deliver more packages to customers while simultaneously reducing costs through the adoption of a joint distribution mode. This study presents a two-level delivery location selection model within the joint distribution mode, considering factors such as delivery station capacity and the number of transport vehicles, with the objective of minimizing the total cost associated with selecting delivery station locations. The proposed model is addressed using a combination of the k-means algorithm and the improved discrete firefly algorithm. In addition, to facilitate equitable cost allocation among enterprises, the Shapley value method is introduced in this study. A case study based on real data from an urban distribution network in the city of Hebei Province, China, is adopted to perform the experiments. The results of this study indicate that the improved algorithm not only improves solution accuracy but also reduces solution time when compared to both the particle swarm optimization and artificial bee colony methods. Furthermore, the application of the Shapley value method demonstrates the efficacy of a rational allocation of costs.

In order to recover the fault-sharing bicycle in the urban road network timely and effectively, considering the uncertain characteristics of the recovery demand on the single vehicle parking spot, a recovery periodic vehicle route selection model aiming at minimizing the total distance is established. The basis constrained robust optimization method is adopted, the uncertain recovery is described by bounded interval, and the disturbance coefficient and control coefficient are introduced to adjust the robustness and adaptability of the model. The approximation algorithm is designed to solve the model, and the upper and lower bounds of the approximation ratio of the approximation algorithm are analyzed. An example analysis is given to verify the effectiveness of the algorithm and the model.

We formulate the sharing bicycle relocating practice as a novel opti-mization problem,which can be regarded as a variant of the classic TSP problem while its objective function is no longer the length of the Hamiltonian tour but the carbon emission.A well-adopted carbon emission formula that is the product of the load of the vehicle and the travel distance is employed and we propose two heuristic algorithms Greedy and TSP-based,inside both of which we set the priority to reduce the load of the vehicle for minimizing carbon emission.The feasibility of both algorithms is proven and numerical experiments are conducted to validate their performance empirically.The promise of Greedy over TSP-based algorithm is shown to the sharing bicycle companies for their daily dispatching practice.

以分析西安市燃气型、电动型及甲醇型3种能源类型出租车在运营中的既有期望运营投入和非期望运营投入,既有期望产出和非期望产出为基础,本文构建基于DEA-Malmquist的考虑能源类型的出租车运营效率评价模型.从西安市出租车管理处提取西安市2020年9～12月期间14946辆出租车的GPS轨迹数据,并随机选取1000辆甲醇型出租车、1000辆电动型出租车及1000辆燃气型出租车作为样本进行实例分析.通过处理3000辆出租车GPS数据,得到乘客在工作日和双休日不同时段的出行需求和特征,分析3种不同能源类型出租车的运营效率差异.所得结论可为西安政府部门对不同能源类型出租车的管理运营提供决策支持.

针对应急救援物资紧缺难以满足所有需求的情形,以单个需求点最大缺货量最小、车辆运输费用最小为双目标,建立从配送中心到分发点再到需求点的两级配送路径选择模型,设计复杂性为O(n3)的近似算法GA进行求解,证明算法近似比的上下界并讨论影响因素,用数值验证算法GA的近似比接近于1,表明算法GA具有较好的性能.最后以雅安灾区配送实例验证模型和算法的有效性.

摘 要：出租汽车是城市交通系统中不可或缺的一部分，它能为出行者提供快捷、灵活的直达运输服务。

Abundant evidence has shown that the integration between urban rail transit system and urban land use planning is of key importance for cities' sustainable development. For planners and city managers, it is essential to scientifically understand the influencing mechanisms between the functional orientations as well as physical locations of the transit stations and the land use patterns alongside. We established a measurement system for the potential of land parcels. Further, we evaluated the land potential alongside a specific subway line, and conducted interaction analysis for between passenger volumes and station-surrounding land mixing degrees. In the end, we sited the subway stations according to the distance constrains and compared the planning outputs from our methods and the plans proposed by the government section. The result is they basically matched with each other, except for some minor differences that could be explained rationally, and the functional orientations of these stations are similar.

In the multiprocessor scheduling problem to minimize the total job completion time, an optimal schedule can be obtained by the shortest processing time rule and the completion time of each job in the schedule can be used as a guarantee for scheduling revenue. However, in practice, some jobs will not arrive at the beginning of the schedule but are delayed and their delayed arrival times are given to the decision-maker for possible rescheduling. The decision-maker can choose to reject some jobs in order to minimize the total operational cost that includes three cost components: the total rejection cost of the rejected jobs, the total completion time of the accepted jobs, and the penalty on the maximum tardiness for the accepted jobs, for which their completion times in the planned schedule are their virtual due dates. This novel rescheduling problem generalizes several classic NP-hard scheduling problems. We first design a pseudo-polynomial time dynamic programming exact algorithm and then, when the tardiness can be unbounded, we develop it into a fully polynomial time approximation scheme. The dynamic programming exact algorithm has a space complexity too high for truthful implementation; we propose an alternative to integrate the enumeration and the dynamic programming recurrences, followed by a depth-first-search walk in the reschedule space. We implemented the alternative exact algorithm in C and conducted numerical experiments to demonstrate its promising performance.

为了及时有效地回收城市道路网络中的故障共享单车,对分散于路网边上的故障单车进行聚类形成收集点,考虑聚类收集点上回收需求呈现的不确定性特征,建立以行驶总距离最小为目标的回收周期性车辆路径选择模型.采用基约束鲁棒优化方法,利用有界区间对不确定的回收量进行描述,引入扰动系数和控制系数调节模型的鲁棒性和适应性.针对模型设计近似算法求解,证明近似算法的时间复杂性,分析算法近似比的上下界,用实例验证算法的近似比,结果表明算法性能较好.最后,通过分析回收量发生波动时,即扰动系数和控制系数对目标函数和算法近似比的影响,进一步验证了算法和模型的有效性.

提出可卸货的移动在线背包问题,即一个装有货物的背包从起点出发对n个指定需求点提供服务,将所装货物在每个点按已知需求量卸下,并将该点数量无法预知的待取回货物装入背包带回起点,如何决策背包对需求点的服务次序及途经需求点是否取回货物,使得取回的货物数量尽可能的多.针对该问题,采用在线理论和方法,建立模型并设计在线算法F,分析需求点待取回的货物数量与背包将该需求点的货物卸下后剩余承载量的差的不同情形,证明F的竞争比并对竞争比的影响因素进行分析,结果表明载货下限越大、需求点个数越多、需求点待取回货物总数越多,算法F的执行效果越好.

Path cover is a well-known intractable problem that finds a minimum number of vertex disjoint paths in a given graph to cover all the vertices. We show that a variant, in which the objective is to minimize the number of length-0 paths, is polynomial-time solvable. We further show that another variant, to minimize the total number of length-0 and length-1 paths, is also polynomial-time solvable. Both variants find applications in approximating the two-machine flow-shop scheduling problem in which job processing has constraints that are formulated as a conflict graph. For the unit jobs, we present a 4/3-approximation for the scheduling problem with an arbitrary conflict graph, based on the exact algorithm for the above second variant of the path cover problem. For arbitrary jobs where the conflict graph is the union of two disjoint cliques, we present a simple 3/2-approximation algorithm.