Yuri N. Sotskov

Some concepts of stability analysis in combinatorial optimization

Abstract This paper surveys the recent results in stability analysis for discrete optimization problems, such as a traveling salesman problem, an assignment problem, a shortest path problem, a Steiner problem, a scheduling problem and so on. The terms “stability”, “sensitivity” or “postoptimal analysis” are generally used for the phase of an algorithm at which a solution (or solutions) of the problem has been already found, and additional calculations are also performed in order to investigate how this solution depends on changes in the problem data. In this paper, the main attention is paid to the stability region and to the stability ball of optimal or approximate solutions. A short sketch of some other close results has been added to emphasize the differences in approach surveyed.

Heuristics for hybrid flow shops with controllable processing times and assignable due dates

This paper considers a generalization of the permutation flow shop problem that combines the scheduling function with the planning stage. In this problem, each work center consists of parallel identical machines. Each job has a different release date and consists of ordered operations that have to be processed on machines from different machine centers in the same order. In addition, the processing times of the operations on some machines may vary between a minimum and a maximum value depending on the use of a continuously divisible resource. We consider a nonregular optimization criterion based on due dates which are not a priori given but can be fixed by a decision-maker. A due date assignment cost is included into the objective function. For this type of problems, we generalize well-known approaches for the heuristic solution of classical problems and propose constructive algorithms based on job insertion techniques and iterative algorithms based on local search. For the latter, we deal with the design of appropriate neighborhoods to find better quality solution. Computational results for problems with up to 20 jobs and 10 machine centers are given.

NP-hardness of shop-scheduling problems with three jobs

This paper deals with the problem of scheduling n jobs on m machines in order to minimize the maximum completion time or mean flow time of jobs. We extend the results obtained in Sotskov (1989, 1990, 1991) on the complexity of shop-scheduling problems with n = 3. The main result of this paper is an NP-hardness proof for scheduling 3 jobs on 3 machines, whether preemptions of operations are allowed or forbidden.

Stability Radius of an Optimal Schedule: A Survey and Recent Developments

The usual assumption that the processing times of the operations are known in advance is the strictest one in deterministic scheduling theory and it essentially restricts its practical aspects. Indeed, this assumption is not valid for the most real-world processes. This survey is devoted to a stability analysis of an optimal schedule which may help to extend the significance of scheduling theory for some production scheduling problems. The terms ‘stability’, ‘sensitivity’ or ‘postoptimal analysis’ are generally used for the phase of an algorithm at which a solution (or solutions) of an optimization problem has already been found, and additional calculations are performed in order to investigate how this solution depends on the problem data. We survey some recent results in the calculation of the stability radius of an optimal schedule for a general shop scheduling problem which denotes the largest quantity of independent variations of the processing times of the operations such that this schedule remains optimal. We present formulas for the calculation of the stability radius, when the objective is to minimize mean or maximum flow time. The extreme values of the stability radius are of particular importance, and these cases are considered more in detail. Moreover, computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are briefly discussed. We also show that the well-known test problem with 6 jobs and 6 machines has both stable and unstable optimal makespan schedules.

Complexity of mixed shop scheduling problems: A survey

We survey recent results on the computational complexity of mixed shop scheduling problems. In a mixed shop, some jobs have fixed machine orders (as in the job shop), while the operations of the other jobs may be processed in arbitrary order (as in the open shop). The main attention is devoted to establishing the boundary between polynomially solvable and NP-hard problems. When the number of operations per job is unlimited, we focus on problems with a fixed number of jobs.

On the calculation of the stability radiusof an optimal or an approximate schedule

The main objective of this paper is to stimulate interest in stability analysis for scheduling problems. In spite of impressive theoretical results in sequencing and scheduling, up to now the implementation of scheduling algorithms with a rather deep mathematical back-ground in production planning, scheduling and control, and in other real-life problems with sequencing aspects is limited. In classical scheduling theory, mainly deterministic systems are considered and the processing times of all operations are supposed to be given in advance. Such problems do not often arise in practice: Even if the processing times are known before applying a scheduling procedure, OR workers are forced to take into account the precision of equipment, which is used to calculate the processing times, round-off errors in the calculation of a schedule, errors within the practical realization of a schedule, machine breakdowns, additional jobs, and so on. This paper is devoted to the calculation of the stability radius of an optimal or an approximate schedule. We survey some recent results in this field and derive new results in order to make this approach more suitable for practical use. Computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are presented. The extreme values of the stability radius are considered in more detail. The new results are amply illustrated with examples.

Schedule execution for two-machine flow-shop with interval processing times

This paper addresses the issue of how to best execute the schedule in a two-phase scheduling decision framework by considering a two-machine flow-shop scheduling problem in which each uncertain processing time of a job on a machine may take any value between a lower and upper bound. The scheduling objective is to minimize the makespan. There are two phases in the scheduling process: the off-line phase (the schedule planning phase) and the on-line phase (the schedule execution phase). The information of the lower and upper bound for each uncertain processing time is available at the beginning of the off-line phase while the local information on the realization (the actual value) of each uncertain processing time is available once the corresponding operation (of a job on a machine) is completed. In the off-line phase, a scheduler prepares a minimal set of dominant schedules, which is derived based on a set of sufficient conditions for schedule domination that we develop in this paper. This set of dominant schedules enables a scheduler to quickly make an on-line scheduling decision whenever additional local information on realization of an uncertain processing time is available. This set of dominant schedules can also optimally cover all feasible realizations of the uncertain processing times in the sense that for any feasible realizations of the uncertain processing times there exists at least one schedule in this dominant set which is optimal. Our approach enables a scheduler to best execute a schedule and may end up with executing the schedule optimally in many instances according to our extensive computational experiments which are based on randomly generated data up to 1000 jobs. The algorithm for testing the set of sufficient conditions of schedule domination is not only theoretically appealing (i.e., polynomial in the number of jobs) but also empirically fast, as our extensive computational experiments indicate.

Complexity of shop-scheduling problems with fixed number of jobs: a survey

The paper surveys the complexity results for job shop, flow shop, open shop and mixed shop scheduling problems when the number n of jobs is fixed while the number r of operations per job is not restricted. In such cases, the asymptotical complexity of scheduling algorithms depends on the number m of machines for a flow shop and an open shop problem, and on the numbers m and r for a job shop problem. It is shown that almost all shop-scheduling problems with two jobs can be solved in polynomial time for any regular criterion, while those with three jobs are NP-hard. The only exceptions are the two-job, m-machine mixed shop problem without operation preemptions (which is NP-hard for any non-trivial regular criterion) and the n-job, m-machine open shop problem with allowed operation preemptions (which is polynomially solvable for minimizing makespan).

Stability of Johnson's schedule with respect to limited machine availability

The paper deals with the scheduling problem of minimizing the makespan in the two -machine n -job flow-shop with w non-availability intervals on each of the two machines. This problem is binary NP-hard even if there is only one non-availability interval ( w = 1) either on the first machine or on the second machine. If there are no non-availability intervals on any machine ( w = 0), the two-machine flow-shop problem may be easily solved using Johnsons permutation of n jobs. We derived sufficient conditions for optimality of Johnsons permutation in the case of the given w S 1 non-availability intervals. The instrument we use is stability analysis, which answers the question of how stable an optimal schedule is if there are independent changes in the processing times of the jobs. The stability analysis is demonstrated on a huge number of randomly generated two -machine flow-shop problems with 5 h n h 10 000 and 1 h w h 1000.

Mean flow time minimization with given bounds of processing times

Abstract We consider a job shop scheduling problem under uncertain processing times and fixed precedence and capacity constraints. Each of the random processing times can take any real value between given lower and upper bounds. The goal is to find a set of schedules which contains at least one optimal schedule (with mean flow time criterion) for any admissible realization of the random processing times. In order to compute such a set of schedules efficiently and keep it as small as possible, we develop several exact and heuristic algorithms and report computational experience based on randomly generated instances.