Evgeny R. Gafarov

Single Machine Scheduling Problems with Financial Resource Constraints: Some Complexity Results and Properties

We consider single machine scheduling problems with a non-renewable resource. These types of problems have not been intensively investigated in the literature so far. For several problems of these types with standard objective functions (namely the minimization of makespan, total tardiness, number of tardy jobs, total completion time and maximum lateness), we present some complexity results. Particular attention is given to the problem of minimizing total tardiness. In addition, for the so-called budget scheduling problem with minimizing the makespan, we present some properties of feasible schedules.

A special case of the single-machine total tardiness problem is NP-hard

In this paper, it is shown that the special case B-1 of the single-machine total tardiness problem 1 ∥ ΣTj is NP-hard in the ordinary sense. For this case, there exists a pseudo-polynomial algorithm with run time O(n σpj).

A hybrid algorithm for the single-machine total tardiness problem

We propose a hybrid algorithm based on the ant colony optimization (ACO) meta-heuristic, in conjunction with four well-known elimination rules, to tackle the NP-hard single-machine scheduling problem to minimize the total job tardiness. The hybrid algorithm has the same running time as that of ACO. We conducted extensive computational experiments to test the performance of the hybrid algorithm and ACO. The computational results show that the hybrid algorithm can produce optimal or near-optimal solutions quickly, and its performance compares favourably with that of ACO for handling standard instances of the problem.

Two-station single-track railway scheduling problem with trains of equal speed

A single track railway scheduling for 2 stations and several segments is considered.Often this situation concerns the case of maintenance of one track of a double track line.A reduction to the single machine scheduling problems with setup-times is presented.Polynomial time solution algorithms were developed.Can serve as a basis to develop efficient algorithms for decision support systems. In this paper, the single-track railway scheduling problem with two stations and several segments of the track is considered. Two subsets of trains are given, where trains from the first subset go from the first station to the second station, and trains from the second subset go in the opposite direction. The speed of trains over each segment is the same. A polynomial time reduction from the problem under consideration to a special case of the single-machine equal-processing-time scheduling problem with setup times is presented. Different polynomial time algorithms are developed for special cases with divers objective functions under various constraints. Moreover, several theoretical results which can be ranked in a series of similar investigations of NP-hardness of equal-processing-time single-machine scheduling problems without precedence relations are obtained.

A new graphical approach for solving single-machine scheduling problems approximately

Often the problem of determining an optimal or approximate production schedule in a company can be reduced to the problem of solving a scheduling problem on a bottleneck machine. However, even the majority of the resulting single-machine scheduling problems are NP-hard from a computational point of view and, therefore, it is difficult to solve large instances of such problems exactly. In this paper, we consider five such single-machine problems, where a dynamic programming algorithm of pseudo-polynomial complexity exists. The running time of such an algorithm can often be improved by so-called graphical algorithms which do not need to consider all states in a dynamic programming algorithm separately. Based on such graphical algorithms, we present fully polynomial-time approximation schemes (FPTASes) for five single-machine total tardiness problems. The new FPTASes have the best running time among the known approximation schemes for these problems. The presented approach is rather general and can be applied to many other scheduling and combinatorial optimisation problems as well.

A note on a single machine scheduling problem with generalized total tardiness objective function

In this note, we consider a single machine scheduling problem with generalized total tardiness objective function. A pseudo-polynomial time solution algorithm is proposed for a special case of this problem. Moreover, we present a new graphical algorithm for another special case, which corresponds to the classical problem of minimizing the weighted number of tardy jobs on a single machine. The latter algorithm improves the complexity of an existing pseudo-polynomial algorithm by Lawler. Computational results are presented for both special cases considered.

On project scheduling problem

Consideration was given to the resource-constrained project scheduling problem and its special cases. The existing lower estimates of the objective function—minimization of the project time—were compared. It was hypothesized that the optimal value of the objective function of the nonpreemptive resource-constrained project scheduling problem is at most twice as great as that of the objective function with preemption. The hypothesis was proved for the cases of parallel machines and no precedence relation.

Algorithms for some maximization scheduling problems on a single machine

In this paper, we consider two scheduling problems on a single machine, where a specific objective function has to be maximized in contrast to usual minimization problems. We propose exact algorithms for the single machine problem of maximizing total tardiness 1‖max-ΣTj and for the problem of maximizing the number of tardy jobs 1‖maxΣUj. In both cases, it is assumed that the processing of the first job starts at time zero and there is no idle time between the jobs. We show that problem 1‖max-ΣTj is polynomially solvable. For several special cases of problem 1‖maxΣTj, we present exact polynomial algorithms. Moreover, we give an exact pseudo-polynomial algorithm for the general case of the latter problem and an alternative exact algorithm.

A new effective dynamic program for an investment optimization problem

After a series of publications of T.E. O’Neil et al. (e.g. in 2010), dynamic programming seems to be the most promising way to solve knapsack problems. Some techniques are known to make dynamic programming algorithms (DPA) faster. One of them is the graphical method that deals with piecewise linear Bellman functions. For some problems, it was previously shown that the graphical algorithm has a smaller running time in comparison with the classical DPA and also some other advantages. In this paper, an exact graphical algorithm (GrA) and a fully polynomial-time approximation scheme based on it are presented for an investment optimization problem having the best known running time. The algorithms are based on new Bellman functional equations and a new way of implementing the GrA.

Two-dedicated-machine scheduling problem with precedence relations to minimize makespan

Two-dedicated-parallel-machine scheduling problem with precedence constraints to minimize makespan is considered. This problem originally appeared as a sub-problem in assembly line balancing but it has also its own applications. Complexity and approximation results for this scheduling problem and its special cases with chains of jobs or equal-processing-times are presented.