Kabir Rustogi

Simple matching vs linear assignment in scheduling models with positional effects: A critical review

This paper addresses scheduling models in which a contribution of an individual job to the objective function is represented by the product of its processing time and a certain positional weight. We review most of the known results in the area and demonstrate that a linear assignment algorithm as part of previously known solution procedures can be replaced by a faster matching algorithm that minimizes a linear form over permutations. Our approach reduces the running time of the resulting algorithms by up to two orders, and carries over to a wider range of models, with more general positional effects. Besides, the same approach works for the models with no prior history of study, e.g., parallel machine scheduling with deterioration and maintenance to minimize total flow time.

Approximation schemes for scheduling on a single machine subject to cumulative deterioration and maintenance

We consider a scheduling problem on a single machine to minimize the makespan. The processing conditions are subject to cumulative deterioration, but can be restored by a single maintenance. We link the problem to the Subset-sum problem (if the duration of maintenance is constant) and to the Half-Product Problem (if the duration of maintenance depends on its start time). For both versions of the problem, we adapt the existing fully polynomial-time approximation schemes to our problems by handling the additive constants.

Single Machine Scheduling with Time-Dependent Linear Deterioration and Rate-Modifying Maintenance

We study single machine scheduling problems with linear time-dependent deterioration effects and maintenance activities. Maintenance periods (MPs) are included into the schedule, so that the machine, that gets worse during the processing, can be restored to a better state. We deal with a job-independent version of the deterioration effects, that is, all jobs share a common deterioration rate. However, we introduce a novel extension to such models and allow the deterioration rates to change after every MP. We study several versions of this generalized problem and design a range of polynomial-time solution algorithms that enable the decision-maker to determine possible sequences of jobs and MPs in the schedule, so that the makespan objective can be minimized. We show that all problems reduce to a linear assignment problem with a product matrix and can be solved by methods very similar to those used for solving problems with positional effects.

Parallel Machine Scheduling: Impact of Adding Extra Machines

We consider the classical scheduling problems of processing jobs on identical parallel machines to minimize (i) the makespan (the maximum completion time) or (ii) the total flow time (the sum of the completion times). The focus of this study is on the impact that additional machines may have, if added to the system. We measure such a machine impact by the ratio of the value of the objective function computed with the original number of machines to the one computed with extra machines. We give tight bounds on the machine impact for the problem of minimizing the makespan, for both the preemptive and non-preemptive versions, as well as for the problem of minimizing the total flow time. We also present polynomial-time exact and approximation algorithms to make a cost-effective choice of the number of machines, provided that each machine incurs a cost and the objective function captures the trade-off between the cost of the used machines and a scheduling objective.

Scheduling with Time-Changing Effects and Rate-Modifying Activities

In scheduling theory, the models that have attracted considerable attention during the last two decades allow the processing times to be variable, i.e., to be subjected to various effects that make the actual processing time of a job dependent on its location in a schedule. The impact of these effects includes, but is not limited to, deterioration and learning. Under the first type of effect, the later a job is scheduled, the longer its actual processing time becomes. In the case of learning, delaying a job will result in shorter processing times. Scheduling with Times-Changing Effects and Rate-Modifying Activities covers and advances the state-of-the-art research in this area. The book focuses on single machine and parallel machine scheduling problems to minimize either the maximum completion time or the sum of completion times of all jobs, provided that the processing times are subject to various effects. Models that describe deterioration and learning effects to be considered include positional, start-time dependent, combined and cumulative, which cover most of the traditionally used models. The authors also consider more enhanced models in which the decision-maker may insert certain Rate-Modifying Activities (RMA) on processing machines, such as for example, maintenance or rest periods. In any case, the processing times of jobs are not only dependent on effects mentioned above but also on the place of a job in a schedule relative to an RMA. For most of the enhanced models described in the book, the polynomial time algorithms presented are based on similar algorithmic ideas such as reduction to linear assignment problems (in a full form or in a reduced form), discrete convexity, and controlled generation of options.

Single machine scheduling with a generalized job-dependent cumulative effect

We consider a single machine scheduling problem with changing processing times. The processing conditions are subject to a general cumulative effect, in which the processing time of a job depends on the sum of certain parameters associated with previously scheduled jobs. In previous papers, these parameters are assumed to be equal to the normal processing times of jobs, which seriously limits the practical application of this model. We further generalize this model by allowing every job to respond differently to these cumulative effects. For the introduced model, we solve the problem of minimizing the makespan, with and without precedence constraints. For the problem without precedence constraints, we also consider a situation in which a maintenance activity is included in the schedule, which can improve the processing conditions of the machine, not necessarily to its original state. The resulting problem is reformulated as a variant of a Boolean programming problem with a quadratic objective, known as a half-product, which allows us to develop a fully polynomial-time approximation scheme with the best possible running time.

Scheduling with Rate-Modifying Activities on Parallel Machines Under Various Effects

Unlike in the preceding chapters of this part, in this chapter we turn to the models on parallel machines. The jobs of a set \(N=\left\{ 1,2,\dots ,n\right\} \) have to be processed on m parallel machines \( M_{1},M_{2},\ldots ,M_{m}\), where \(m\le n\). Additionally, the decision-maker is presented with a list (RMP\(^{\left[ 1\right] }\), RMP\(^{ \left[ 2\right] },\ldots ,\) RMP\(^{\left[ K\right] }\)) of \(K\ge 1\) possible rate-modifying activities. The decision-maker may decide which of the listed RMPs to insert into a schedule, on which machine and in which order.

General Framework for Studying Models with Rate-Modifying Activities

The importance of the planning of machine maintenance for production enterprises and service organizations is widely recognized by both practitioners and management scientists.

Relevant Boolean Programming Problems

Quite often, an algorithm that finds either an exact or an approximate solution to a scheduling problem can be derived from a reformulation of the original problem in terms of another problem of combinatorial optimization, e.g., a Boolean programming problem.

Scheduling with Pure and Combined Additive Start-Time-Dependent Effects

In this chapter, we study the problems of minimizing the makespan and the total completion time on a single machine, provided that the actual processing times of the jobs are subject to a special form of a start-time-dependent effect. We also study effects in which such a start-time-dependent effect is combined with a positional effect.