Single machine total tardiness maximization problems: complexity and algorithms

This paper explores the characteristics of solutions when scheduling jobs in a shop with parallel machines. Three classical objective functions were considered: makespan, total completion time, and total tardiness. These three criteria were combined in pairs, resulting in three bi-objective formulations. These formulations were solved using the ε-constraint method to obtain a Pareto frontier for each pair. The objective of the research is to evaluate the Pareto set of efficient schedules to characterize the solution sets. The characterization of the solutions sets is based on two performance metrics: the span of the objective functions' values for the points in the frontier and their closeness to the ideal point. Results that consider four experimental factors indicate that when the makespan is one of the objective functions, the range of the processing times among jobs has a significant influence on the characteristics of the Pareto frontier. Simultaneously, the slack of due dates is the most relevant factor when total tardiness is considered.

In this paper, we study an integrated production and outbound distribution scheduling model with one manufacturer and one customer. The manufacturer has to process a set of jobs on a single machine and deliver them in batches to the customer. Each job has a release date and a delivery deadline. The objective of the problem is to issue a feasible integrated production and distribution schedule minimizing the transportation cost subject to the delivery deadline constraints. We consider three problems with different ways how a job can be produced and delivered: non-splittable production and delivery (NSP-NSD) problem, splittable production and non-splittable delivery (SP-NSD) problem and splittable production and delivery (SP-SD) problem. We provide a polynomial-time algorithm that solves two special cases of SP-NSD and SP-SD problems. Solving these problems allows us to compute a lower bound for the NP-hard problem NSP-NSD, which we use in a branch and bound (B&B) algorithm to solve problem NSP-NSD. The computational results show that the B&B algorithm outperforms a MILP formulation of the problem implemented on a commercial solver. keywords: single machine scheduling production and delivery release dates deadlines transportation costs branch and bound.

Scheduling with controllable release dates and processing times: Total completion time minimization

Just-in-time JIT (Just In Time) is an important procedure in scheduling problems which aims at minimizing minimising both earliness and tardiness at the same time. In this case, the problem is a multi-objective scheduling problem. A candidate solution method for this type of problem is finding the Pareto-optima. The present research investigates the single single machine scheduling problem in which the three objects: number of tardy job, maximum earliness, and maximum tardiness must be minimized minimised

Bi-criteria scheduling with machine assignment costs

Order acceptance and scheduling (OAS) problems are realistic for enterprises. They have to select the appropriate orders according to their capacity limitations and profit consideration, and then complete these orders by their due dates or no later than their deadlines. OAS problems have attracted significant attention in supply chain management. However, there is an issue that has not been studied well. To our best knowledge, no prior research examines the carbon emission cost and the time-of-use electricity cost in the OAS problems. The carbon emission during the on-peak hours is lower than the one in mid-peak and off-peak hours. However, the electricity cost during the on-peak hours is higher than the one during mid-peak and off-peak hours when time-of-use electricity (TOU) tariff is used. There is a trade-off between sustainable scheduling and the electricity cost. To calculate the objective value, a carbon tax and carbon dioxide emission factor are included when we evaluate the carbon emission cost. The objective function is to maximize the total revenue of the accepted orders and then subtract the carbon emission cost and the electricity cost under different time intervals on a single machine with sequence-dependent setup times and release date. This research proposes a mixed-integer linear programming model (MILP) and a relaxation method of MILP model to solve this problem. It is of importance because the OAS problems are practical in industry. This paper could attract the attention of academic researchers as well as the practitioners.

This paper considers a classic model of weighted single machine scheduling problem and aims to improve it to a real-world application through fuzzy set theory. For this purpose, processing times and due dates of jobs are defined as fuzzy numbers. In the proposed model, two objectives are considered to be minimized: average tardiness and number of tardy jobs. The objectives are converted into fuzzy statement through fuzzy arithmetic. Because the problem is NP-hard, it is proposed to solve the model through three well-known meta-heuristic algorithms as Simulated Annealing, Tabu Search, and Genetic Algorithm albeit with some modifications. Comparative analysis of algorithms is attainable through different experimented results on generated benchmark problems with different sizes and difficulties. Efficiency of the developed algorithms is analyzed on the different versions of the model which come from the assumption of different parameters and/or components. Algorithms' behaviors to managerial insights are also considered in some sensitivity analysis experiments. The obtained results show the applicability of the proposed model in real-world scheduling problems.

In many industries including metallurgy, petrochemicals, paper, aerospace, pharmaceutical, electronics, ceramic and automotive manufacturing, the production system can contain a machine bottleneck which processes, in some cases, all the jobs. The management of this machine is crucial for the enterprise because it is responsible for delays in customers' orders delivery. The jobs scheduling on this machine is a way of approaching the problem and treats the whole production system. The single machine model provides a basis for more complicated environment such those seen in many real cases. Also, several studies have shown that several jobs, if not all, have time-dependent setup times. So, decision makers must therefore organize the jobs scheduling on this single machine by trying to minimize downtime while respecting the different deadlines. On the other hand, several resolution methods have been developed in literature to solve this kind of problem using approaching methods like metaheuristics or hybrid methods integrating exact methods. This thesis is in this direction. In this research, we propose several effective approaches to solve the single machine scheduling problem with sequence-dependent setup times with the objective to minimize total tardiness of the jobs. First, a new crossover operator in a genetic algorithm is proposed. This algorithm shows its ability to be competitive with the best approaches found in the literature on benchmark problems. We thus highlight the importance of incorporating specific problem knowledge into genetic operators, even if classical genetic operators could be used. However, the results obtained show that the proposed genetic algorithm is less efficient in comparison to the best resolution method found in the literature. Indeed, we noticed that the proposed algorithm lacks an intensification process for enhancing its. To overcome this shortcoming, we explore resolution class methods that showed promising prospects in the last decade. Even, the hybrid algorithms have achieved very good results in a wide variety of problems. In recent years, several researchers have introduced the hybridization of metaheuristics with exact methods. In fact, this kind of hybridization can be an attractive alternative because both methods have very different characteristics that can be combined to produce better results. Thus we explore the design of hybrid genetic algorithms improving the intensification process of the proposed genetic algorithm by integrating an exact method and mechanisms from other resolution methods to improve the performance. However, we find few approaches hybridizing metaheuristics and constraint programming in the literature to solve scheduling problems. In a second step, we propose a modeling for the total tardiness single machine with setup-dependent times scheduling problem using the constraint-based scheduling and the ILOG C++ API. The related experiments show that algorithm efficiency improves with specific tree search strategies and heuristic scheduling variables. Under prohibitive time computing, the proposed approach results are generally far from the best known solutions. However, we got interesting performance on small instances to consider allowing its hybridization with other resolution methods. In a third step, we introduce a hybrid approach based on the integration between genetic algorithm and constraint based scheduling approaches. This latter approach is integrated in the reproduction and the intensification processes of a genetic algorithm autonomously. Finally, we propose a hybrid algorithm which integrates several concepts from constraint programming, multi-objective optimization and ant colonies optimization in a crossover operator into a genetic algorithm. The proposed hybrid algorithms show excellent performance on benchmark problems from the literature by improving many of the best known solutions for some of them.

The paper presents a bicriterion approach to solve the single-machine scheduling problem in which the job release dates can be compressed while incurring additional costs. The two criteria are the makespan and the compression cost. For the case of equal job processing times, an O(n4) algorithm is developed to construct integer Pareto optimal points. We discuss how the algorithm developed can be modified to construct an e-approximation of noninteger Pareto optimal points. The complexity status of the problem with total weighted completion time criterion is also established.

The Oven Scheduling Problem (OSP) is a new parallel batch scheduling problem that arises in the area of electronic component manufacturing. Jobs need to be scheduled to one of several ovens and may be processed simultaneously in one batch if they have compatible requirements. The scheduling of jobs must respect several constraints concerning eligibility and availability of ovens, release dates of jobs, setup times between batches as well as oven capacities. Running the ovens is highly energy-intensive and thus the main objective, besides finishing jobs on time, is to minimize the cumulative batch processing time across all ovens. This objective distinguishes the OSP from other batch processing problems which typically minimize objectives related to makespan, tardiness or lateness. We propose to solve this NP-hard scheduling problem using exact techniques and present two different modelling approaches, one based on batch positions and another on representative jobs for batches. These models are formulated as constraint programming (CP) and integer linear programming (ILP) models and implemented both in the solver-independent modeling language MiniZinc and using interval variables in CP Optimizer. An extensive experimental evaluation of our solution methods is performed on a diverse set of problem instances. We evaluate the performance of several state-of-the-art solvers on the different models and on three variants of the objective function that reflect different real-life scenarios. We show that our models can find feasible solutions for instances of realistic size, many of those being provably optimal or nearly optimal solutions.

We introduce the Oven Scheduling Problem (OSP), a new parallel batch scheduling problem that arises in the area of electronic component manufacturing. Jobs need to be scheduled to one of several ovens and may be processed simultaneously in one batch if they have compatible requirements. The scheduling of jobs must respect several constraints concerning eligibility and availability of ovens, release dates of jobs, setup times between batches as well as oven capacities. Running the ovens is highly energy-intensive and thus the main objective, besides finishing jobs on time, is to minimize the cumulative batch processing time across all ovens. This objective distinguishes the OSP from other batch processing problems which typically minimize objectives related to makespan, tardiness or lateness. We propose to solve this NP-hard scheduling problem via constraint programming (CP) and integer linear programming (ILP) and present corresponding CP- and ILP-models. For an experimental evaluation, we introduce a multi-parameter random instance generator to provide a diverse set of problem instances. Using state-of-the-art solvers, we evaluate the quality and compare the performance of our CP- and ILP-models, which could find optimal solutions for many instances. Furthermore, using our models we are able to provide upper bounds for the whole benchmark set including large-scale instances.

In this paper we study a parallel machine scheduling model with different job release dates, where each job is either accepted and processed by a machine at or after its release date, or it is rejected and a certain penalty cost is paid. The objective is to minimize the makespan of the accepted job plus the total penalty of all rejected jobs. The scheduling problem is NP-hard in the strong sense. Zhang and Lu (4OR A Q J Oper Res 14:165–172, 2016) have proposed a 2-approximation for the problem, and a fully polynomial time approximation scheme (FPTAS) for the special case when the number of machines m is fixed. In this paper we present an improved 2-approximation and a polynomial time approximation scheme for the problem. We also propose an improved FPTAS for the case when m is fixed.

The problem of minimizing the weighted average time to complete jobs on a single machine is considered. The release dates of jobs are fixed. The preemptions are possible. The processing times of all jobs are equal. The computational complexity this problem is unknown. An approach to the analysis of the structure of the optimal solutions is proposed. Polynomial algorithms are described for some important classes.

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that nl jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m2Πl=1mnl2) dynamic programming algorithm coupled with numerical experiments is presented.

In this paper we consider a single machine preemptive scheduling problem to minimize the weighted number of late jobs. There are given n jobs and for each job we have a release date, a processing time and a due date. It is assumed that certain specified jobs have to be completed on time. The due dates for these jobs are considered as deadlines. while for the other jobs due dates may be violated. For the case of similarly ordered release and due dates (when there is no advantage to preemption) as well as for the case of embedded release and due date intervals and preemption allowed, we transform the initial problem into a reduced problem, where all jobs with deadlines are eliminated. It gives an opportunity to propose O(n log n) algorithms for some special cases of the considered problems.