Time-indexed Formulation Research Articles

Sparse dynamic discretization discovery via arc-dependent time discretizations

While many time-dependent network design problems can be formulated as time-indexed formulations with strong relaxations, the size of these formulations depends on the discretization of the time horizon and can become prohibitively large. The recently-developed dynamic discretization discovery (DDD) method allows many time-dependent problems to become more tractable by iteratively solving instances of the problem on smaller networks where each node has its own discrete set of departure times. However, in the current implementation of DDD, all arcs departing a common node share the same set of departure times. This causes DDD to be ineffective for solving problems where all near-optimal solutions require many distinct departure times at the majority of the high-degree nodes in the network. Region-based networks are one such structure that often leads to many high-degree nodes, and their increasing popularity underscores the importance of tailoring solution methods for these networks.To improve methods for solving problems that require many departure times at nodes, we develop a DDD framework where the set of departure times is determined on the arc level rather than the node level. We apply this arc-based DDD method to a temporal variant of the service network design problem (SND). We show that an arc-based approach is particularly advantageous when instances arise from region-based networks, and when candidate paths are fixed in the base graph for each commodity. Moreover, our algorithm builds upon the existing DDD framework and achieves these improvements with only benign modifications to the original implementation.

A MaxSAT approach for solving a new Dynamic Discretization Discovery model for train rescheduling problems

Train scheduling is a critical activity in rail traffic management, both off-line (timetabling) and on-line (dispatching). Time-Indexed formulations for scheduling problems are stronger than other classical formulations, like Big-M. Unfortunately, their size grows usually very large with the size of the scheduling instance, making even the linear relaxation hard to solve. Moreover, the approximation introduced by time discretization can lead to solutions which cannot be realized in practice. Dynamic Discretization Discovery (DDD), recently introduced by Boland et al. (2017) for the continuous-time service network design problem, is a technique to keep at bay the growth of Time-Indexed formulations and their response times and, at the same time, ensures the necessary modelling precision. By exploiting the DDD paradigm, we develop a novel approach to train dispatching and, more in general, to job-shop scheduling. The algorithm implemented represents the first application of a Maximum SATisfiability problem approach to the field. In our comparisons on real-life instances of train dispatching, our restricted Time-Indexed formulation solves faster on piece-wise constant objective functions, while the Big-M approach maintains the lead on linear continuous objectives.

A Flow-Based Formulation for Parallel Machine Scheduling Using Decision Diagrams

We present a new flow-based formulation for identical parallel machine scheduling with a regular objective function and without idle time. The formulation is constructed with the help of a decision diagram that represents all job sequences that respect specific ordering rules. These rules rely on a partition of the planning horizon into, generally nonuniform, periods and do not exclude all optimal solutions, but they constrain solutions to adhere to a canonical form. The new formulation has numerous variables and constraints, and hence we apply a Dantzig-Wolfe decomposition to compute the linear programming relaxation in reasonable time; the resulting lower bound is stronger than the bound from the classical time-indexed formulation. We develop a branch-and-price framework that solves three instances from the literature for the first time. We compare the new formulation with the time-indexed and arc time–indexed formulation by means of a series of computational experiments. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was partially funded by the European Union’s Horizon 2020 research and innovation program under [Marie Skłodowska-Curie Grant 754462]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0301 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0301 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

On time-indexed formulations for the parallel machine scheduling problem with a common server

This article studies the problem of scheduling independent jobs on parallel machines with a common server, the objective of which is to minimize the makespan. In this case, the common server is responsible for performing the setup operations and, therefore, there must be no conflicts while conducting them. Four alternative time-indexed formulations for the problem are considered and evaluated computationally. Moreover, two algorithms are presented that can significantly improve the performance of the best time-indexed formulation. The results obtained on two benchmark datasets involving up to 100 jobs suggest that the proposed improved algorithms are substantially better than existing approaches.

A branch-and-price-and-cut algorithm for operating room scheduling under human resource constraints

In this paper, we address an integrated operating room planning and scheduling problem that includes, with fine detail, constraints commonly encountered in practice (i.e., sequence, capacity and due date constraints) and for human resources other than surgeons, i.e., nurses. A new model of the sequence-dependent operating room cleaning times that arise because of surgeries with different infection levels is considered. To solve this difficult integrated planning and scheduling problem, we devise a branch-and-price-and-cut algorithm based on the time-indexed formulation of the problem. The basic column generation scheme relies on a label-correcting algorithm that we purposely developed for solving the pricing problems that are modeled as single operating room scheduling problems with time-dependent costs and sequence-dependent cleaning times. The pricing problems are strongly NP-Hard. The efficiency of the label-correcting algorithm is ensured by dominance rules among labels and by two algorithms for computing the upper and lower bound of labels. An effective cutting procedure, inspired by Benders’ decomposition and based on duality theory for linear programming, is developed for tightening the linear relaxation of the problem. With instances from the literature and that we generated, we conduct a numerical study to demonstrate the computational effectiveness of the solution method.

Branch-and-refine for solving time-expanded MILP formulations

One of the standard approaches for solving discrete optimization problems which include the aspect of time, such as the traveling salesman problem with time windows or the shortest path problem with time windows is to derive a so-called time-indexed formulation. If the problem has an underlying structure that can be described by a graph, the time-indexed formulation is usually based on a different, extended graph, commonly referred to as the time-expanded graph. The time-expanded graph can often be derived in such a way that all time constraints are incorporated in its topology, and therefore algorithms for the corresponding time-independent variant become applicable. The downside of this approach is, that the sets of vertices and arcs of the time-expanded graph are much larger than the ones of the original graph. In recent works, however, it has been shown that for many practical applications a partial graph expansion, that might contain time infeasible paths, often suffices to find a proven optimal solution. These approaches, instead, iteratively refine the original graph and solve a relaxation of the time-expanded formulation in each iteration. When the solution of the current relaxation is time feasible an optimal solution can be derived from it and the algorithm terminates. In this work we present new ideas, that allow for the propagation of information about the optimal solution of a coarser graph to a more refined graph and show how these can be used in algorithms, which are based on graph refinement. More precisely we present a new algorithm for solving Mixed Integer Linear Program (MILP) formulations that allows for the graph refinement to be carried out during the exploration of the branch-and-bound tree instead of restarting whenever the optimal solution was found to be infeasible. For demonstrating the practical relevance of this algorithm we present numerical results on its application to the shortest path problem with time windows and the traveling salesman problem with time windows.

An Efficient and Effective Approach for the Scheduling of Steelmaking-continuous Casting Process With Multi Different Refining Routes

With the rapid development of market-oriented demand of iron and steel enterprises towards multi variety, small batch and on-time delivery, the functional requirements of steelmaking-continuous casting (SCC) production process with multi different refining routes for the dynamic balance of logistics, resources and time increase sharply. An effective scheduling of SCC process is important significance for improving the efficiency of steelmaking production and reducing the material and energy consumption. This paper develops an improved Lagrangian relaxation (LR) approach to address this problem based on time-indexed formulation while considering SCC scheduling performance indexes: sojourn time and tardiness of each job in the last stage. A direction controllable target estimation level adjustment strategy with tighter lower bound is addressed to overcome the premature convergence and weaken the oscillation in LR iteration. Computational experiment is verified based on the actual steelmaking plant in China. The computational results show that the approach improves the schedule strategy for the actual SCC process, and promotes the development of the schedule theory with the separate structure characteristics.

Scheduling a single parallel-batching machine with non-identical job sizes and incompatible job families

We study the scheduling of jobs on a single parallel-batching machine with non-identical job sizes and incompatible job families. Jobs from the same family have the same processing time and can be loaded into a batch, as long as the batch size respects the machine capacity. The objective is to minimize the total weighted completion time; this common scheduling objective is equivalent with the minimization of in-process inventory when all jobs have the same release date. Our problem combines two classic combinatorial problems, namely bin packing and single machine scheduling. We develop three new mixed-integer linear-programming formulations, namely an assignment-based formulation, a time-indexed formulation (TIF), and a set-partitioning formulation (SPF). We also propose a column generation (CG) algorithm for the SPF, a branch-and-price (B&P) algorithm and a CG-based heuristic. A heuristic framework based on proximity search is also developed using the TIF. We examine how to reduce the size (number of variables) of the formulations in a preprocessing step. The SPF and B&P can solve instances with non-unit and unit job durations to optimality with up to 80 and 150 jobs within reasonable runtime limits, respectively. The proposed heuristics perform better than the methods available in the literature for the same problem.

Exact Approaches for Single Machine Total Weighted Tardiness Batch Scheduling

This paper addresses a single machine total weighted tardiness (TWT) batch-scheduling problem in which jobs have release dates, nonidentical sizes, and are compatible between each other. We propose two integer linear programming models: the first one is a time-indexed formulation (TIF), and the second is an innovative time-size-indexed formulation (TSIF). Although TIF clearly outperforms the existing formulation for the problem, TSIF is capable of producing much stronger bounds in practice. The latter also enables us to develop an efficient column-generation (CG) algorithm. The pricing subproblem corresponds to a resource-constrained shortest path problem that is solved using a bucket graph–based labeling algorithm. The solutions of such a subproblem may contain cycles (reprocessing of jobs), and thus, a memory mechanism called dynamic arc-based ng-sets is employed in the labeling with a view toward avoiding some of them. Moreover, we also implement a preprocessing scheme based on Lagrangian relaxation to perform variable fixing. Extensive computational experiments were carried out in 810 benchmark instances. The proposed CG algorithm is capable of solving instances with up to 100 jobs to optimality. In addition, we believe that this is the first exact approach for a TWT batch-scheduling variant capable of systematically solving instances with up to 50 jobs. High-quality results are also reported for three special cases of the problem—more precisely, when (i) the penalty weights are unitary, (ii) there are no release dates, and (iii) all due dates are set to zero and, hence, the objective becomes equivalent to minimizing the weighted completion time. Summary of Contribution: This paper provides the first exact algorithm for a standard variant of a batch-scheduling total weighted tardiness problem that can solve instances with up to 100 jobs to optimality, a considerable leap with respect to previous works. In particular, we propose a time-indexed formulation that has the advantage of being relatively simple to implement, and yet we show that it is not theoretically dominated by the other innovative formulation proposed in the paper referred to as the time-size-indexed formulation (TSIF). Moreover, we present a Lagrangian approach to quickly fix variables and an iterative column-generation (CG) procedure over a Dantzig–Wolfe decomposition of TSIF that combines an efficient pricing algorithm with a dynamic scheme to adjust the subproblem constraints. The proposed CG approach is capable of producing very strong bounds for the problem as well as for some of its special cases.

Just-in-time single-batch-processing machine scheduling

In this paper, we study a single-batch-processing machine (SBPM) scheduling problem by considering a just-in-time criterion. Our objective is to minimize the total weighted earliness and tardiness (WET) penalties of jobs, where the penalty rates of jobs are job-independent but the earliness and tardiness penalty rates are different, and different jobs may have different due dates. An SBPM can process several jobs simultaneously as a batch. We first model this scheduling problem using a mixed-integer linear model. We design two priority rules, i.e., earliest due date (EDD) and earliest starting time (EST), to sort jobs. Based on our analysis on the mathematical properties of this scheduling problem, we propose two heuristic algorithms AABF and ITSLS to construct and improve the near-optimal schedules. We also devise a lower bound method on a new approximate time-indexed formulation. Extensive numerical experiments demonstrate the effectiveness and efficiency of our algorithms.

Exact Methods and Heuristics for Order Acceptance Scheduling Problem under Time-of-Use Costs and Carbon Emissions

This research focuses on an Order Acceptance Scheduling (OAS) problem on a single machine under time-of-use (TOU) tariffs and taxed carbon emissions periods with the objective to maximize total profit minus tardiness penalties and environmental costs. Due to the NP-hardness of the considered problem especially in presence of sequence-dependent setup-times, two fix-and-relax (FR) heuristics based on different time-indexed (TI) formulations are proposed. A metaheuristic based on the Dynamic Island Model (DIM) framework is also employed to tackle this optimization problem. These approached methods show promising results both in terms of solution quality and solving time compared to state-of-the-art exact solving approaches.

Dealing with uncertain surgery times in operating room scheduling

The operating theater is one of the most expensive units in the hospital, representing up to 40% of the total expenses. Because of its importance, the operating room scheduling problem has been addressed from many different perspectives since the early 1960s. One of the main difficulties that has reduced the applicability of the current results is the high variability in surgery duration, making schedule recommendations hard to implement.In this work, we propose a time-indexed scheduling formulation to solve the operational problem. Our main contribution is that we propose the use of chance constraints related to the surgery duration’s probability distribution for each surgeon to improve the scheduling performance. We show how to implement these chance constraints as linear ones in our time-indexed formulation, enhancing the performance of the resulting schedules significantly.Through data analysis of real historical instances, we develop specific constraints that improve the schedule, reducing the need for overtime without affecting the utilization significantly. Furthermore, these constraints give the operating room manager the possibility of balancing overtime and utilization through a tuning parameter in our formulation. Finally, through simulations and the use of real instances, we report the performance for four different metrics, showing the importance of using historical data to get the right balance between the utilization and overtime.

Tactical runway scheduling for demand and delay management

PurposeAirports are limited in terms of capacity. Particularly, runways can only accommodate a certain number of movements (arrivals and departures) while ensuring safety and determined operational requirements. In such a constrained operating environment, any reduction in system capacity results in major delays with significant costs for airlines and passengers. Therefore, the efficient operation of airports is a critical cornerstone for demand and delay management of the whole air transportation system. Runway scheduling deals with the sequencing of arriving and departing aircraft at airports such that a predefined objective is optimized subject to several operational constraints, like the dependency of separation on the leading and trailing aircraft type or the runway occupancy time. This study aims to develop a model that acts as a tactical runway scheduling methodology for reducing delays while managing runway usage.Design/methodology/approachBy considering real airport performance data with scheduled and actual movements, as well as arrival/departure delays, this study presents a robust model together with an optimization algorithm, which incorporates the knowledge of uncertainty into the tactical operational step. The approach transforms the planning problem into an assignment problem with side constraints. The coupled landing/take-off problem is solved to optimality by exploiting a time-indexed (0, 1) formulation for the problem. The Binary Integer Linear Programming approach allows to include multi-criteria and multi-constraints levels and, even with some major simplifications, provides fewer sequence changes and target time updates, when compared to the usual approach in which the plan is simply updated in case of infeasibility. Thus, the use of robust optimization leads to a protection against tactical uncertainties, reduces delays and achieves more stable operations.FindingsThis model has been validated with real data from a large international European airport in different traffic scenarios. Results are compared to the actual sequencing of flights and show that the algorithm can significantly contribute to the reduction of delay, while adhering as much as possible to the operative procedures and constraints, and to the objectives of the airport stakeholders. Computational experiments performed on the case study illustrate the benefits of this arrival/departure integrated approach: the proposed algorithm significantly reduces weighted aircraft delay and computes efficient runway schedule solutions within a few seconds and with little computational effort. It can be adopted as a decision-making tool in the tactical stage. Furthermore, this study presents operational insights regarding demand and delay management based on the results of this work.Originality/valueScheduling arrivals and departures at runways is a complex problem that needs to address diverse and often competing considerations among involved flights. In the context of the Airport Collaborative Decision Making programme, airport operators and air navigation service providers require arrival and departure management tools that improve aircraft flows at airports. Airport runway optimization, as the main element that combines airside and groundside operations, is an ongoing challenge for air traffic management.

Solving a time-indexed formulation for an unrelated parallel machine scheduling problem by preprocessing and cutting planes

We consider a time-indexed formulation for the unrelated parallel machine scheduling problem. We show that all polyhedral knowledge known from the single machine problem (in particular, valid inequalities) is applicable to this formulation. We present new facet-inducing valid inequalities and a preprocessing technique involving fixing variables based on reduced costs. We combine both techniques in a basic cutting-plane algorithm and test the performance of the resulting algorithm by running it on randomly generated instances.

Air traffic flow management with layered workload constraints

Many regions of the world are currently struggling with congested airspace, and Europe is no exception. Motivated by our collaboration with relevant European authorities and companies in the Single European Sky ATM Research (SESAR) initiative, we investigate novel mathematical models and algorithms for supporting the Air Traffic Flow Management in Europe. In particular, we consider the problem of optimally choosing new (delayed) departure times for a set of scheduled flights to prevent en-route congestion and high workload for air traffic controllers while minimizing the total delay. This congestion is a function of the number of flights in a certain sector of the airspace, which in turn determines the workload of the air traffic controller(s) assigned to that sector. We present a MIP model that accurately captures the current definition of workload, and extend it to overcome some of the drawbacks of the current definition. The resulting scheduling problem makes use of a novel formulation, Path&Cycle, which is alternative to the classic big-M or time-indexed formulations. We describe a solution algorithm based on delayed variable and constraint generation to substantially speed up the computation. We conclude by showing the great potential of this approach on randomly generated, realistic instances.

Decision diagram-based integer programming for the paired job scheduling problem

The paired job scheduling problem seeks to schedule n jobs on a single machine, each job consisting of two tasks for which there is a mandatory minimum waiting time between the completion of the first task and the start of the second task. We provide complexity results for problems defined by three commonly used objective functions. We propose an integer programming formulation based on a decision diagram decomposition that models the objective function and some of the challenging constraints in the space of the flow variables stemming from the diagrams while enforcing the simpler constraints in the space of the original scheduling variables. We then show how to simplify our reformulation by projecting out a subset of the flow variables, resulting in a lifted reformulation for the problem that can be obtained without building the decision diagrams. Computational results show that our proposed model performs considerably better than a standard time-indexed formulation over a set of randomly generated instances.

A joint order acceptance and scheduling problem with earliness and tardiness penalties considering overtime

This paper considers a joint order acceptance and scheduling problem under a general scenario. A manufacturer receives multiple orders with a given revenue, processing time, release date, due date, deadline, and earliness and tardiness penalties. The manufacturer can be seen as a single-machine system. Due to limited capacity, the manufacturer cannot process every order and needs to determine the optimal set of accepted orders and corresponding production schedule such that the total profit is maximized. The manufacturer can extend its capacity with overtime by paying an additional cost. A time-indexed formulation is presented to model the problem. Two exact algorithms are proposed. The first algorithm, denoted by DPIA-GR, is a dynamic programming (DP)-based algorithm that starts by solving a relaxed version of the original model and successively recovers the relaxed constraint until an optimal solution to the original problem is achieved. The second algorithm, denoted by DPIA-LRGR, improves DPIA-GR by incorporating Lagrangian relaxation (LR). The subgradient method is employed to find the optimal Lagrangian multipliers. The relaxed model in DPIA-GR and the LR model in DPIA-LRGR can be represented using a weighted di-graph. Both algorithms are equivalent to finding the longest path in the graph and applying a graph reduction strategy to prevent unnecessary computational time and memory usage. A genetic algorithm (GA) is also proposed to solve large-scale versions of the problem. Numerical experiments show that both DPIA-GR and DPIA-LRGR solve the problem efficiently and outperform CPLEX and GA, but DPIA-LRGR offers better performance.

Mathematical Models of Energy-Conscious Bi-Objective Unrelated Parallel Machine Scheduling

The industrialization has led to the prosperity of human life. However, it causes the side effect that harms the environment. Moreover, the source of energy used to drive the industrialization comes from non-renewable resources that can be extinct. As the extensive energy user, the manufacturing sector can use energy efficiently by scheduling and planning. A scheduling system that incorporates environmental and the energy consumption is one of the initiatives to reduce energy consumption and reduce environmental effects. Therefore, this study addresses bi-objective unrelated parallel machine scheduling to minimize the total tardiness and energy consumption. The energy consumption follows the Time-Of-Use (TOU) tariffs price scheme. The problem is formulated as two mixed-integer programming (MIP) models, using the time-indexed and disjunctive formulation, and solved using the weighted sum method. We perform complexity and computational analysis to evaluate the performance of models. Numerical experiments show that the time-indexed formulation is more efficient than the disjunctive formulation. The results provide useful insights for decision-makers in the manufacturing sectors to be energy-conscious without neglecting the production efficiency.

Mathematical formulations for the parallel machine scheduling problem with a single server

This paper addresses the problem of scheduling independent jobs on identical parallel machines with a single server to minimise the makespan. We propose mixed integer programming (MIP) formulations to solve this problem. Each formulation reflects a specific concept on how the decision variables are defined. Moreover, we present inequalities that can be used to improve those formulations. A computational study is performed on benchmark instances from the literature to compare the proposed MIP formulations with other known formulations from the literature. It turns out that our proposed time-indexed variables formulation outperforms by far the other formulations. In addition, we propose a very efficient MIP formulation to solve a particular case of the problem with a regular job set. This formulation is able to solve all regular instances for the case of 500 jobs and 5 machines in less than 5.27 min, where all other formulations are not able to produce a feasible solution within 1 h.

Minimizing makespan on a single machine with release dates and inventory constraints

We consider a single-machine scheduling problem with release dates and inventory constraints. Each job has a deterministic processing time and has an impact (either positive or negative) on the central inventory level. We aim to find a sequence of jobs such that the makespan is minimized while all release dates and inventory constraints are met. We show that the problem is strongly NP-hard even when the capacity of the inventory is infinite. To solve the problem, we introduce a time-indexed formulation and a sequence-based formulation, a branch-and-bound algorithm, and a dynamic-programming-based guess-and-check (GC) algorithm. From extensive computational experiments, we find that the GC algorithm outperforms all other alternatives.