In this paper, a multi-periodic discrete interval scheduling problem is considered. This problem stems from an application to improve customer service in the branch offices of the Internal Revenue Service of Ecuador. A set of jobs must be scheduled non-preemptively on a set of parallel identical machines such that each job is processed in a time interval of a given length between its arrival time and its deadline. Moreover, the planning horizon is divided into several time periods and the state of a machine can either be active or inactive at each of these periods. Machines may perform jobs only while they are active, and each machine can process at most one job at the same time. The task consists of (i) selecting feasible processing intervals for the jobs, (ii) determining which machines are active at each period, and (iii) assigning selected job intervals to active machines. The goal is to minimize the total number of machine periods required to schedule all jobs. Two integer programming models are proposed for this problem, along with an asymptotically optimal algorithm for the special case in which all jobs have equal processing times. The algorithms are evaluated on both real-world and simulated instances.
This paper proposes new heuristics for the classic non-preemptive scheduling problem of assigning n jobs on m identical parallel machines, with the objective of minimizing the makespan. Starting from the list scheduling (LS) method, used for example in longest-processing-time-first (LPT) (Graham in SIAM J Appl Math 17(2):416–429, 1969) or SLACK (Della Croce and Scatamacchia in J Sched 23(2):163–176, 2020) heuristics, we derive a branching strategy that also considers assigning a job to the second-best machine, in addition to the first one, the only one usually dealt with in the literature. Throughout the exploration of solutions, we retain only the best solution. This strategy leads to two heuristics, designed to speed up the solution search. The first, called branch-and-parallelize LS (BPLS), parallelizes the two alternatives considered on each job. The second, branch-and-bound list scheduling (BBLS), applies a branch-and-bound method to widely prune the solution tree. As a trade-off between computation time and solution quality, these heuristics are parameterized in order to assign only a subset of jobs according to the branching strategy. Out of this subset, the assignment is made by the classic LS, that is, each time on the first available machine. We show from instances in the literature that our heuristics outperform many well-known algorithms on the vast majority of the cases considered. We also investigate the subspace of instances for which our heuristics are not able to beat all of these algorithms. Finally, we propose an ad hoc heuristic named MULTI-BBLS (MBBLS) which consists of multiple calls to BBLS that allow it to rank first even on this instance subspace.
Advanced cancer treatment requires the collaboration of highly specialized medical experts in a chain of information-intensive strategies; one way to alleviate this burden and better inform future trials is to build reliable models for drug administration. We design a dynamic model that incorporates both treatment-enhancing optimization and relaxation session scheduling based on model parameters to aid in drug delivery. A control strategy was devised in each treatment phase to optimize drug delivery, while metaheuristic algorithms were used to determine the length of each treatment and relaxation phase. Through several simulation experiments, we show that our framework’s treatment phases balance the risk of local tumor recurrence and chemotherapy-induced side effects better than current practices, shortening the treatment period and reducing side effects. Our study adds to the literature by improving healthcare delivery practices through the development of useful therapy schedules alternating between treatment and relaxation to enable cells to recover from the adverse effects of chemotherapy.
In this paper, we study the problem of minimizing the maximum late work with the popular option of job rejection on a single machine. We provide two fundamental properties regarding an optimal schedule, prove that the problem is NP-hard, and suggest pseudo-polynomial dynamic programming (DP), establishing that the problem is ordinary NP-hard. We also provide an extensive numerical study. Next, we leverage the provided DP to introduce an algorithm that maps the Pareto-optimal frontier.
Serverless computing has emerged as a dominant paradigm for deploying scalable, event-driven applications, offering abstraction from infrastructure management. However, despite their advantages, serverless platforms continue to face persistent challenges, including high cold start latency and inefficient resource utilization, particularly under bursty or unpredictable workloads. Traditional mitigation techniques, such as static pre-warming or function-specific prediction models, often lead to excessive resource consumption or suboptimal scheduling decisions. This paper proposes a novel tepid-aware scheduling framework that introduces an intermediate provisioning state—tepid containers—positioned between conventional cold and warm states. These containers are lightweight, runtime-ready environments that can be rapidly customized with dependencies based on predicted function invocation probabilities. We model the scheduling task as a bi-objective optimization problem, jointly minimizing response latency and power consumption under dynamic resource constraints. A heuristic algorithm is developed to assign each function request to one of the three provisioning modes—cold, tepid, or warm—based on system availability and prediction confidence. The proposed method is implemented on Fission.io, and its performance is evaluated across ten diverse scenarios, including synthetic workloads and real-world deployment cases. Experimental results show that tepid-aware scheduling reduces cold start latency by up to 57
This paper introduces a new scheduling model with a battery constraint to minimize the total rejection cost for non-just-in-time jobs, equivalently, to maximize the weighted number of just-in-time jobs. The battery constraint stipulates that a job can be processed on a machine only if the battery is sufficiently charged. We first analyze how the computational complexity varies with respect to key parameters, including job weights, processing times, the battery-charging function, and the number of machines. Subsequently, we propose a mixed-integer linear programming formulation of the problem and develop two heuristic algorithms. The performance of the proposed methods is validated through extensive numerical experiments.
We investigate a variant of the classic knapsack problem in the context of single-machine scheduling with controllable processing times under a makespan constraint. In this problem, the processing time of each job is a bounded linear decreasing function of the amount of resource allocated to its job processing operation. The task is to choose some jobs for processing with the goal to maximize the total profit while satisfying a given makespan constraint, where the total profit is the total revenue minus the total resource consumption cost. We show that the studied problem is NP-hard. By exploring the structural properties on the optimal solution, we propose a constant factor approximation algorithm with a performance ratio of 1/5 . Subsequently, we design a dynamic programming exact algorithm and a fully strongly polynomial time approximation scheme.
In a flow shop, jobs are serially processed on a set of machines and the machine order is the same for all the jobs. In a permutation flow shop, there is an additional assumption that the order in which jobs enter the machines is the same on each machine. While the meaning of "permutation" is clear for a flow shop, it is more ambiguous for a reentrant flow shop. In a reentrant flow shop, jobs are processed on some machines more than once, leading to several ways of understanding the meaning of permutation. We indicate that different researchers use the term permutation for different assumptions. We clear up this ambiguity by identifying four definitions of permutation scheduling in the context of reentrant permutation flow shops with the cyclic pattern. In addition, we analyze the impact of these definitions on optimization and on heuristic performance with respect to the makespan objective function. We first compare optimal solutions across different permutation types, identifying cases where the best solution follows a specific permutation structure. We then extend known permutation flow shop heuristics to all four permutation definitions. Through computational experiments, we evaluate the effectiveness of these heuristics.
Creating more robust rosters that offer medium-term planning security for employees is a desired goal in the public transportation sector. To tackle this problem, we introduce a new approach in the context of tram driver rostering called time frame rostering. In this approach, instead of directly assigning shifts to roster positions, time frames are first allocated to roster positions. These time frames are intervals wide enough to accommodate a variety of shifts. The shift assignment takes place only a few days before the actual workday. Thus, time frame rosters provide medium-term planning security, as tram drivers are only assigned shifts within their designated time frames. The goal of the time frame rostering problem is then to optimally assign time frames to a roster such that several constraints are met. In this paper, we formally define the time frame rostering problem and provide a solver-independent model of the problem. Furthermore, we compare two state-of-the-art solvers on real-world instances and demonstrate that optimal or almost optimal solutions can be found in a reasonable amount of time. Additionally, we verify these solutions by simulating absences and subsequent shift assignment.
The efficient use of energy can offer substantial cost savings to companies. Precise scheduling is an effective way to improve energy efficiency, since it typically requires little capital investment and has a short payback period. However, making optimal scheduling decisions can be complicated by frequent price changes in the marketplace. This creates a large-scale energy cost minimization problem, which can be difficult to solve. We study this problem for a manufacturing or computer resource with discretely variable speeds, where energy usage increases with speed. We provide a scalable optimal algorithm for minimizing energy cost with discrete speeds under frequently changing energy prices. The result is a more precise, faster and more scalable planning tool for scheduling to control energy costs than is currently available.
This paper considers the flexible job shop scheduling problem with nonlinear routes, a production environment with a wide range of relevant practical applications, especially in today’s on-demand printing industry. In order to approximate the problem of real-world applications, we consider the influence of a position-based learning effect on the processing time of the operations. The goal is to minimize makespan. In the present work, we are concerned with the development of effective and efficient methods for its solution. For this purpose, a local search method and four trajectory metaheuristics are considered. In the local search, we show that the classical strategy of reallocating only those operations that are part of the critical path can miss better-quality neighbors, as opposed to what happens in the case where there is no learning effect. Consequently, we introduce an alternative type of neighborhood reduction that eliminates only neighbors that are not better than the current solution. Additionally, we analyze the application of the classical strategy on top of the new reduction. Through experimentation, we verify that it significantly shrinks the size of the neighborhood, thereby increasing efficiency, with minimal loss of effectiveness. Extensive numerical experiments are performed. Statistical tests confirm that tabu search based on the reduced neighborhood, when applied to large-sized instances, outperforms the other three metaheuristics, namely iterated local search, greedy randomized adaptive search, and simulated annealing. Experiments on classical instances with linear routes only show that the introduced methods also stand out in relation to methods from the literature. All methods, instances, and solutions are freely available.
We consider a one-machine scheduling problem where the temperature of a job rises during processing and cools down when not being processed according to given linear heating and cooling rates. No job’s temperature is allowed to rise above a given threshold, and no job’s temperature can cool below 0. Another crucial property of our problem is that jobs can be preempted an arbitrary number of times, and even more, we allow that a job is processed for an infinitely small amount of time. We consider two objectives: minimize the makespan and minimize the sum of completion times. Our results are as follows. We show how to compactly represent a solution. Further, we prove that the problem of minimizing the sum of completion times can be solved in polynomial time by formulating it as a linear program and deriving a structural property. This result can be extended to hold for any number of machines. Further, we show that a minimum makespan can be found in O(n) time, even when heating and cooling rates are job-dependent.
The invigilator assignment problem provides an assignment of invigilators, also called proctors, to a predetermined exam timetable with given invigilator demand. We provide three problem variants with a focus on invigilator-timeslot preferences and socialization aspects. We prove that the variant with timeslot preferences is polynomially solvable, whereas incorporating socialization aspects makes the problem 𝒩𝒫 -hard. Hence, we also provide a heuristic approach and show its effectiveness based on real-life benchmark instances, arising from a case study conducted at Ghent University. Furthermore, we introduce and evaluate artificial instances based on well-known exam timetabling instances from the literature. Finally, we discuss the practical implications and key insights derived from the case study.
Mission planning algorithms typically solve multi-robot-multi-mission problems by converting them into mixed-integer programming. Then, the mission plan is made through an assignment algorithm based on the calculated cost, such as a sequential greedy algorithm. Generally, the cost used for assignment is set to the actual increasing cost when assigning one more mission from the current assignment state. To derive an exact solution, it is often necessary not to select the case with the lowest cost in the current assignment state. However, it cannot be judged whether the case with the lowest cost should be chosen just by the increasing cost when assigning one more mission from the current assignment state. In other words, it is difficult to minimize the time taken to complete all missions using existing algorithms. To solve this problem, a mission planning algorithm inspired by temporal difference-based reinforcement learning is proposed in this paper. The proposed algorithm consists of three parts: assignment policy decision, cost calculation based on the policy, and mission planning using the policy. The sequential greedy algorithm is selected as the policy, and the performance of the proposed algorithm is compared with other mission planning methods, including brute-force.
When independent jobs are to be scheduled onto identical machines, the typical goal is to minimize the makespan. A simple and efficient heuristic consists in scheduling first the job with the longest processing time (LPT heuristic), and to plan its execution as soon as possible. While the performance of LPT has already been largely studied, in particular its asymptotic performance, we revisit results and propose a novel analysis for the case of jobs generated through uniform integer compositions. Also, we perform extensive simulations to compare and empirically assess the asymptotic performance of five classical heuristics, including LPT. The results show that the absolute error rapidly tends to zero for several distributions of job processing times, including distributions studied by theoretical models, and realistic distributions coming from benchmarks.
The scheduling problems with job prohibitions are investigated. Here we are given a system of job subsets for positions of machines. For each position of a machine the subset indicates jobs, which can be performed in this position. The considered problems arise in the case of technological requisitions in production systems and multi-processor computer systems, where the order of job execution is influenced by fixed routes and structural constraints. At the same time, such problems play an important role in evolutionary algorithms, when the optimal recombination problem is solved in a crossover operator, and other iterative techniques based on solving a series of subproblems with job prohibitions. We analyze the computational complexity of the problem for various regular criteria in single-stage statements and multi-stage systems. We also discuss a general method to proving the NP-hardness of the specified class of scheduling problems using the polynomial reduction of the Ordered Partition Problem to the decision version of the problem, when an instance with the non-idle property is constructed. A solving approach based on the enumeration scheme and mixed integer linear programming models is proposed. This approach allows us to show that "almost all" instances are polynomially solvable, and it is important for evolutionary computation and other techniques based on enumerating partially mapped solutions.
This paper deals with a general open-shop scheduling problem where jobs have release and due dates, transferring between machines incurs a machine–pair-dependent delay, and any regular objective defined over job completion times is minimized. Such problems are widely encountered in practice, e.g., in industrial testing and maintenance processes and just-in-time logistics. Despite its relevance, exact solution methods from the literature have so far focused on the makespan objective or very specific special cases. We hence present the first exact method for the general open-shop scheduling problem with any regular minsum objective with respect to job completion times. Moreover, we demonstrate how branch-cut-and-price methods can be applied to the open shop. Finally, we also adapt the famous subset-row inequalities, originally proposed for the vehicle routing problem, to the open shop. Our computational study on both newly generated instances as well as those from the literature demonstrates good performance on a broad set of different objective functions, finding tight bounds even for the largest problems.
We consider scheduling problems of processing a given set of independent jobs on parallel p-batch machines subject to arbitrary processing set restrictions (PSR). Two types of p-batch machines are considered: equal-speed and uniform. For the corresponding P- and Q-problems, efficient (strongly polynomial) approximation algorithms with a priori and a posteriori absolute and relative accuracy bounds have been designed. The presented results are the first attempt of an approximate solution of a that general problem (with arbitrary PSR and individual batch capacities of machines) in a strongly polynomial time and with performance ratios strictly less than two. As an auxiliary result, we have developed a new (linear-time) rounding algorithm for a given fractional distribution of jobs by machines in a tree-like graph.
This paper examines an online integrated production and transportation scheduling problem in a make-to-order environment with time-dependent waiting and inventory holding costs. In this problem, the manufacturer has to produce the orders arriving over time and then deliver them to customers, resulting in production costs and transportation costs associated with the number of vehicles used. If orders are not immediately produced or transported upon arrival, waiting costs will be incurred during the interval between order receipt and transportation, along with inventory holding costs for storing products preceding their transportation will be incurred. The objective is to minimize the total cost of production, waiting, inventory holding, and transportation. We propose a primal–dual based heuristic algorithm with a competitive ratio of 3 and establish a lower bound of 2 on the competitiveness of any deterministic online algorithm for the problem. Furthermore, we investigate a general problem with considering both limited production capacity and fixed-plus-linear production cost structures.
