Stochastic Single Machine Scheduling Research Articles

Stochastic single-machine scheduling with workload-dependent maintenance activities

Stochastic single-machine scheduling with workload-dependent maintenance activities

Stochastic single machine scheduling with time-dependent deterioration or position-dependent learning effect

Stochastic single machine scheduling with time-dependent deterioration or position-dependent learning effect

Stochastic single machine scheduling problem as a multi-stage dynamic random decision process

In this work, we study a stochastic single machine scheduling problem in which the features of learning effect on processing times, sequence-dependent setup times, and machine configuration selection are considered simultaneously. More precisely, the machine works under a set of configurations and requires stochastic sequence-dependent setup times to switch from one configuration to another. Also, the stochastic processing time of a job is a function of its position and the machine configuration. The objective is to find the sequence of jobs and choose a configuration to process each job to minimize the makespan. We first show that the proposed problem can be formulated through two-stage and multi-stage Stochastic Programming models, which are challenging from the computational point of view. Then, by looking at the problem as a multi-stage dynamic random decision process, a new deterministic approximation-based formulation is developed. The method first derives a mixed-integer non-linear model based on the concept of accessibility to all possible and available alternatives at each stage of the decision-making process. Then, to efficiently solve the problem, a new accessibility measure is defined to convert the model into the search of a shortest path throughout the stages. Extensive computational experiments are carried out on various sets of instances. We discuss and compare the results found by the resolution of plain stochastic models with those obtained by the deterministic approximation approach. Our approximation shows excellent performances both in terms of solution accuracy and computational time.

Single machine scheduling problem with stochastic sequence-dependent setup times

In this study, we consider stochastic single machine scheduling problem. We assume that setup times are both sequence dependent and uncertain while processing times and due dates are deterministic. In the literature, most of the studies consider the uncertainty on processing times or due dates. However, in the real-world applications (i.e. plastic moulding industry, appliance assembly, etc.), it is common to see varying setup times due to labour or setup tools availability. In order to cover this fact in machine scheduling, we set our objective as to minimise the total expected tardiness under uncertain sequence-dependent setup times. For the solution of this NP-hard problem, several heuristics and some dynamic programming algorithms have been developed. However, none of these approaches provide an exact solution for the problem. In this study, a two-stage stochastic-programming method is utilised for the optimal solution of the problem. In addition, a Genetic Algorithm approach is proposed to solve the large-size problems approximately. Finally, the results of the stochastic approach are compared with the deterministic one to demonstrate the value of the stochastic solution.

A stochastic scheduling approach to minimise the number of risky jobs and total probability of tardiness

In this paper, single machine stochastic scheduling problem is addressed where jobs have probabilistic processing times and deterministic due dates. In classical scheduling literature, a job is called 'tardy' if it is completed after its due date; otherwise it's called 'early'. However, in probabilistic concept, a job can have a non-zero probability of tardiness. To capture this situation, a stochastic nonlinear mathematical model is developed. The objective is to minimise the expected number of tardy jobs and the total probability of tardiness. Experimentation is performed with datasets having varying number of jobs from 10 to 100. Results are compared with deterministic model. The proposed approach resulted in a significant decrease in the number of risky jobs and the total probability of tardiness. In addition, as the variance of processing times increased, the proposed stochastic approach provided safer schedules in terms of the total probability of tardiness.

Stochastic single-machine scheduling with random resource arrival times

Scheduling in an uncertain environment remains a meaningful yet challenging direction of research. In this paper, we consider a new scheduling setting from practical complex business applications, where resources (e.g., raw materials) used for processing jobs arrive randomly, due to reasons such as unstable transportation service caused by extreme weather conditions, unreliable suppliers, unpredictable industrial actions, etc. Further, jobs must be processed one by one and preemption is not allowed. The processing times of jobs are not known but their distribution. We incorporate these factors into a stochastic single-machine scheduling model and examine two different common types of objectives: minimizing total expected weighted completion time and minimizing total expected weighted squared completion time. We derive and prove a natural and intuitive optimal policy for the model with the first objective. Besides, we find that, under some mild conditions, the well-known policy in stochastic scheduling, WSEPT (weighted shortest expected processing time), still holds optimal for achieving either of objectives. The numerical example further supports and illustrates our results, which provide decision-makers insights into tricky uncertain scheduling problems.

A Dynamic Programming Approach for Emergency Vehicle Dispatching Problems

In this research, emergency vehicle dispatching problems faced with in the wake of massive natural disasters are considered. Here, the emergency vehicle dispatching problems can be regarded as a single machine stochastic scheduling problems, where the processing times are independently and identically distributed random variables, are considered. The objective of minimizing the expected number of tardy jobs, with distinct job due dates that are independently and arbitrarily distributed random variables, is dealt with. For these problems, optimal static-list policies can be found by solving corresponding assignment problems. However, for the special cases where due dates are exponentially distributed random variables, using a proposed dynamic programming approach is found to be relatively faster than solving the corresponding assignment problems. This so-called Pivot Dynamic Programming approach exploits necessary optimality conditions derived for ordering the jobs partially.

Minimizing earliness and tardiness costs in stochastic scheduling

We address the single-machine stochastic scheduling problem with an objective of minimizing total expected earliness and tardiness costs, assuming that processing times follow normal distributions and due dates are decisions. We develop a branch and bound algorithm to find optimal solutions to this problem and report the results of computational experiments. We also test some heuristic procedures and find that surprisingly good performance can be achieved by a list schedule followed by an adjacent pairwise interchange procedure.

Stochastic scheduling problems with general position-based learning effects and stochastic breakdowns

The focus of this study is to analyze position-based learning effects in single-machine stochastic scheduling problems. The optimal permutation policies for the stochastic scheduling problems with and without machine breakdowns are examined, where the performance measures are the expectation and variance of the makespan, the expected total completion time, the expected total weighted completion time, the expected weighted sum of the discounted completion times, the maximum lateness and the maximum tardiness.

Stochastic single machine scheduling with random common due date

This paper studies the single machine scheduling problem for minimising the expected total weighted deviations of completion times from random common due date. Jobs have exponentially distributed processing times and the common due date is a generalised Erlang distribution. The optimal schedules are shown to be ∧-shaped. Moreover, we give the optimal schedules when the machine is subject to stochastic breakdowns with independent and exponentially distributed uptimes and downtimes.

A branch and bound for single machine stochastic scheduling to minimize the maximum lateness

Article history: Received 20 September 2011 Received in revised form November, 2, 2011 Accepted November, 3 2011 Available online 11 November 2011 This paper studies the problem of single machine stochastic scheduling with random processing times, deterministic due dates and an independent setup time. The jobs are also deteriorated based on the position, which their processes are done. The objective function is to find a schedule of jobs, which minimizes the expected value of maximum lateness. A branch and bound scheme is presented to solve the problem analytically and a simulated annealing metaheuristic (SA) is also provided for solving the problem in larger scales. Computational experiments demonstrate that the proposed SA is capable of finding near optimal solutions with very low gap. © 2012 Growing Science Ltd. All rights reserved

Solving a single machine stochastic scheduling problem using a branch and bound algorithm and simulated annealing

This paper tackles a stochastic version of single machine scheduling with random processing times, deterministic due dates, independent setup times and precedence constraints. The jobs are also deteriorated based on the position at which their processes are done. The objective function is to find a schedule of jobs which minimizes the expected value of total weighted tardiness. A branch and bound scheme is presented to solve the problem analytically. Since the problem is NP-hard, it is not possible to find global optimum values for larger scales of the problem in a reasonable run time, therefore, a simulated annealing meta-heuristic (SA) is also provided to solve the larger scale problems. Computational experiments demonstrate that the proposed SA is strongly capable of finding near optimal solutions with a very low gap.

Minimizing total tardiness in a stochastic single machine scheduling problem using approximate dynamic programming

This paper addresses the non-preemptive single machine scheduling problem to minimize total tardiness. We are interested in the online version of this problem, where orders arrive at the system at random times. Jobs have to be scheduled without knowledge of what jobs will come afterwards. The processing times and the due dates become known when the order is placed. The order release date occurs only at the beginning of periodic intervals. A customized approximate dynamic programming method is introduced for this problem. The authors also present numerical experiments that assess the reliability of the new approach and show that it performs better than a myopic policy.

Safe scheduling: Setting due dates in single-machine problems

We consider single-machine stochastic scheduling models with due dates as decisions. In addition to showing how to satisfy given service-level requirements, we examine variations of a model in which the tightness of due-dates conflicts with the desire to minimize tardiness. We show that a general form of the trade-off includes the stochastic E/T model and gives rise to a challenging scheduling problem. We present heuristic solution methods based on static and dynamic sorting procedures. Our computational evidence identifies a static heuristic that routinely produces good solutions and a dynamic rule that is nearly always optimal. The dynamic sorting procedure is also asymptotically optimal, meaning that it can be recommended for problems of any size.

Minimising a weighted quadratic function of job lateness in the stochastic single machine scheduling problem

We study a static single machine scheduling problem in which processing times, due-dates, and penalties for not completing jobs on time are distinct arbitrary random variables. The objective is to identify an optimal sequence, which minimises the expected weighted sum of a quadratic function of job lateness. The problem is NP-hard to solve; however, based on a precedence relation structure among adjacent jobs, we develop an exact algorithm. Our computational results demonstrate that the algorithm solves large problem instances quickly. Furthermore, the proposed problem is general in the sense that its special cases reduce to some classical deterministic or new stochastic single machine models.

Stochastic single machine scheduling subject to machines breakdowns with quadratic early-tardy penalties for the preemptive-repeat model

In this paper we research the problem in which the objective is to minimize the sum of squared deviations of job expected completion times from the due date, and the job processing times are stochastic. In the problem the machine is subject to stochastic breakdowns and all jobs are preempt-repeat. In order to show that the replacing ESSD by SSDE is reasonable, we discuss difference between ESSD function and SSDE function. We first give an express of the expected completion times for both cases without resampling and with resampling. Then we show that the optimal sequence of the problem V-shaped with respect to expected occupying time. A dynamic programming algorithm based on the V-shape property of the optimal sequence is suggested. The time complexity of the algorithm is pseudopolynomial.

Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem

We study a static stochastic single machine scheduling problem in which jobs have random processing times with arbitrary distributions, due dates are known with certainty, and fixed individual penalties (or weights) are imposed on both early and tardy jobs. The objective is to find an optimal sequence that minimizes the expected total weighted number of early and tardy jobs. The general problem is NP-hard to solve; however, in this paper, we develop certain conditions under which the problem is solvable exactly. An efficient heuristic is also introduced to find a candidate for the optimal sequence of the general problem. Our illustrative examples and computational results demonstrate that the heuristic performs well in identifying either optimal sequences or good candidates with low errors. Furthermore, we show that special cases of the problem studied here reduce to some classical stochastic single machine scheduling problems including the problem of minimizing the expected weighted number of early jobs and the problem of minimizing the expected weighted number of tardy jobs which are both solvable by the proposed exact or heuristic methods.

Single machine stochastic scheduling to minimize the expected number of tardy jobs using mathematical programming models

This paper studies the single machine scheduling problem for the objective of minimizing the expected number of tardy jobs. Jobs have normally distributed processing times and a common deterministic due date. We develop new approaches for this problem that generate near optimal solutions. The original stochastic problem is transformed into a non-linear integer programming model and its relaxations. Computational study validates their effectiveness by comparison with optimal solutions.

Single-Machine Scheduling with Exponential Processing Times and General Stochastic Cost Functions

We study a single-machine stochastic scheduling problem with n jobs, in which each job has a random processing time and a general stochastic cost function which may include a random due date and weight. The processing times are exponentially distributed, whereas the stochastic cost functions and the due dates may follow any distributions. The objective is to minimize the expected sum of the cost functions. We prove that a sequence in an order based on the product of the rate of processing time with the expected cost function is optimal, and under certain conditions, a sequence with the weighted shortest expected processing time first (WSEPT) structure is optimal. We show that this generalizes previous known results to more general situations. Examples of applications to practical problems are also discussed.

The single machine stochastic scheduling with the weighted job tardiness minimization

This paper considers scheduling n jobs on a single machine where the job processing times and due dates are independent random variables with arbitrary distribution functions. We consider the case that the weighted job tardiness in expectation is minimized. It is assumed that job’s due dates are compatible with processing times and weights. We show that the jobs should be sequenced in decreasing stochastic order of their due dates.