Alexander A. Lazarev

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).

SPECIAL CASE OF THE SINGLE MACHINE TOTAL TARDINESS PROBLEM IS NP-HARD

Abstract In this paper we show that the special case B-1 (Lazarev et al., 2004) of the single machine total tardiness problem 1||Σ T j is NP-hard in the ordinary sense. For the case we have constructed pseudo-polynomial algorithm O ( n Σ p j ) time.

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.

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.

Solution of the NP-hard total tardiness minimization problem in scheduling theory

The classical NP-hard (in the ordinary sense) problem of scheduling jobs in order to minimize the total tardiness for a single machine 1‖ΣTj is considered. An NP-hard instance of the problem is completely analyzed. A procedure for partitioning the initial set of jobs into subsets is proposed. Algorithms are constructed for finding an optimal schedule depending on the number of subsets. The complexity of the algorithms is O(n2Σpj), where n is the number of jobs and pj is the processing time of the jth job (j = 1, 2, …, n).

The pareto-optimal set of the NP-hard problem of minimization of the maximum lateness for a single machine

The classical problem of scheduling theory that is NP-hard in the strong sense 1|rj|Lmax is considered. New properties of optimal schedules are found. A polynomially resolved case of the problem is selected, when the release times (rj), the processing time (pj), and due dates of completion of processing (dj) of jobs satisfy the constraints d1 ≤ ... ≤ dn and d1 − r1 − p1 ≥ ... ≥ dn − rn − pn. An algorithm of run time O(n3logn) finds Pareto-optimal sets of schedules according to the criteria Lmax and Cmax that contains no more than n variants.

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.

Estimates of the Absolute Error and a Scheme for an Approximate Solution to Scheduling Problems

An approach is proposed for estimating absolute errors and finding approximate solutions to classical NP-hard scheduling problems of minimizing the maximum lateness for one or many machines and makespan is minimized. The concept of a metric (distance) between instances of the problem is introduced. The idea behind the approach is, given the problem instance, to construct another instance for which an optimal or approximate solution can be found at the minimum distance from the initial instance in the metric introduced. Instead of solving the original problem (instance), a set of approximating polynomially/pseudopolynomially solvable problems (instances) are considered, an instance at the minimum distance from the given one is chosen, and the resulting schedule is then applied to the original instance.

Graphic approach to combinatorial optimization

Consideration was given to a graphic realization of the method of dynamic programming. Its concept was demonstrated by the examples of the partition and knapsack problems. The proposed method was compared with the existing algorithms to solve these problems.