Best Possible Approximation Algorithms for Single Machine Scheduling with Increasing Linear Maintenance Durations

An improved on-line algorithm for single parallel-batch machine scheduling with delivery times

The research on the theory and method of single machine scheduling is a difficult subject, but it is very important for the companies to improve the production efficiency and effectiveness. The study on single machine scheduling problem has the history of 50 years, but there is still a gap between the classical scheduling theories and practical scheduling problems. According to this characteristic, the problems in practical scheduling area and the various factors need to be concerned are mentioned, and the main methods to solving the single machine scheduling problem and their applications are presented in details. Finally, the directions and suggestions of future work in single machine scheduling problem are summarized.

This paper considers the single machine scheduling problem with cost constraints based on customer driven using noncooperative game theory. The single machine scheduling problem with cost constraints based on customer driven is introduced, and customers bidding strategy is analyzed using noncooperative game method. Then noncooperative game model of single machine scheduling with cost constraints is defined using noncooperative game theory, in which the customer is mapped to the player, customer quotation corresponds to player strategy, and a payoff function of players is defined on condition of two main customer objectives. The satisfactory result of scheduling problem can be achieved by seeking Nash equilibrium solution, and the steps of seeking Nash equilibrium solution for the proposed scheduling game model are given. A single scheduling case is analyzed and solved, and the result verifies the validity of the proposed game model and the solving method.

Approximability of the Ground State Problem for Certain Ising Spin Glasses

In this chapter, we consider some practical scheduling problems under unavailability constraints (breakdown periods, maintenance durations and/or setup times). Such problems can be met in different industrial environments and be associated to numerous real-life applications. This explains why many researchers have become interested in this subject. We aim to present the recent approaches proposed to solve these problems and to discuss their performances. This family of scheduling problems, addressed in this chapter, has been intensively studied (Kacem [8], Lee [17], Schmidt [24]). The studied criteria in this chapter are related to the flowtime minimization (the weighted and unweighted cases). The chapter is organized in two main parts. The first part focuses on the single machine scheduling problem (see Section 2). The second part is devoted to the parallel machine scheduling problem (see Section 3). In each part, we present the main contributions and explain their principles (complexity results, heuristic algorithms and their worstcase performance, existing approximation schemes, exact methods, branch-and-bound algorithms, dynamic programming, integer linear models, lower bounds. . .). Finally, Section 4 concludes the paper.

This paper considers the single batch-processing machine scheduling problem with aging effect and multi-maintenance activity concept. Real processing time of jobs increases due to machine’s aging effect. Maintenance time is constant and the machine returns to its initial condition after each maintenance activity and the aging effect starts anew. Our scheduling problem is under assumptions: non identical job sizes and dynamic job arrivals. We have to determine jointly (1) the maintenance frequencies, (2) the position of the maintenance activities on the machine, (3) the grouping of the jobs into the batches, and (4) the sequence of the batches on the machine, to minimize makespan. At first, we formulate the problem as a mathematical model. Since this problem has been proved to be NP-hard; therefore, a heuristic and two meta-heuristic algorithms are offered for the real-sized problems. The performance of the proposed algorithms is analyzed on the various test problems based on the considered measures.

We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith’s rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k ( k − 1)/( k + 1) , denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time.

An improved approximation algorithm for single machine scheduling with job delivery

Single machine scheduling (SMS) problems seem to have received substantial attention because of several reasons. These type of problems are important both because of their own intrinsic value, as well as their role as building blocks for more generalized and complex problems. In a multi-processor environment single processor schedules may be used in bottlenecks, or to organize task assignment to an expensive processor; sometimes an entire production line may be treated as a single processor for scheduling purposes. Also, compared to multiple processor scheduling, SMS problems are mathematically more tractable. Hence more problem classes can be solved in polynomial time, and a larger variety of model parameters, such as various types of cost functions, or an introduction of change-over cost, can be analyzed. Single processor problems are thus of rather fundamental character and allow for some insight and development of ideas when treating more general scheduling problems.

Scheduling jobs under decreasing linear deterioration

We discuss the problem of sequencing precedence-constrained jobs on a single machine to minimize the average weighted completion time. This problem has attracted much attention in the mathematical programming community since Sidney’s pioneering work in 1975. We look at the problem from a polyhedral perspective and uncover a relation between Sidney’s decomposition theorem and different linear programming relaxations. More specifically, we present a generalization of Sidney’s result, which particularly allows us to reason that virtually all known 2-approximation algorithms comply with his decomposition. Moreover, we establish a connection between the single-machine scheduling problem and the vertex cover problem. Indeed, in the special case of series-parallel precedence constraints, we prove that the sequencing problem can be seen as a special case of vertex cover. We also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem. Finally, we provide a characterization of the active inequalities of a linear programming relaxation in completion time variables.KeywordsSchedule ProblemCompletion TimeSingle MachinePrecedence ConstraintVertex CoverThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A grasp for single machine scheduling with sequence dependent setup costs and linear delay penalties

This study considers a single machine scheduling problem with the objective of minimizing the total weighted tardiness of the jobs. This problem is one of the most famous problems in single machine scheduling theory and it is NP-hard. In this paper, we propose a hybrid heuristic which combines GRASP with Path Relinking to find good quality solutions for the considered problem. The performance of the hybrid GRASP heuristic is tested over multiple benchmark problems from the OR-Library with up to 100 jobs and experimental results show that it is very competitive with the existing good performing algorithms.

In this paper we consider the single machine scheduling problems with sum-of-logarithm-processing-times based and position based learning effects, i.e., the actual job processing time of a job is a function of the sum of the logarithms of the processing times of the jobs already processed and its position in a sequence. The logarithm function is used to model the phenomenon that learning as a human activity is subject to the law of diminishing return. We show that even with the introduction of the proposed model to job processing times, several single machine problems remain polynomially solvable.

This paper is concerned with single machine scheduling problem with fuzzy due dates where satisfaction based benefit are relevant. It gives an analysis and proof to the character of optimal solution for the scheduling problem. A property is proposed based on the analysis and it is also extended to some special cases.