Flow shop scheduling algorithms for minimizing the completion time variance and the sum of squares of completion time deviations from a common due date

We consider two problems of m-machine flow shop scheduling in this paper: one, with the objective of minimizing the variance of completion times of jobs, and the other with the objective of minimizing the sum of squares of deviations of job completion times from a common due date. Lower bounds on the sum of squares of deviations of job completion times from the mean completion time of jobs for a given partial sequence are first presented. Using these lower bounds, a branch and bound algorithm based on breadth-first search procedure for scheduling n jobs on m-machines with the objective of minimizing completion time variance (CTV) is developed to obtain the best permutation sequence. We also present two lower bounds and thereafter, a branch and bound algorithm with the objective of minimizing the sum of squares of deviations of job completion times from a given common due date (called the MSD problem). The computational experience with the working of the two proposed branch and bound algorithms is also reported. Two heuristics, one for each of the two problems, are developed. The computational experience on the evaluation of the heuristics is discussed.     

A note on minimizing the sum of squares of machine completion times on two identical parallel machines

In this short note, we address the coherence between minimizing the sum of squares of machine completion times and minimizing makespan on two identical parallel machines. We show equivalence of the two objectives and identify interesting and useful relations which allow us to transfer worst-case ratios of approximation algorithms from one problem to the other.     

A memetic algorithm for minimizing the total weighted completion time on a single machine under linear deterioration

In this paper, we consider the problem of minimizing the total weighted completion time on a single machine. Jobs processing times are increasing linear function of start times. First, we present some new dominance properties for this NP-hard problem. And next, using these properties, we develop a memetic algorithm for the problem. The results of computational experiments show the good performance of the proposed algorithm.     

A sequential exchange approach for minimizing earliness–tardiness penalties of single-machine scheduling with a common due date

This study focuses on a class of single-machine scheduling problems with a common due date where the objective is to minimize the total earliness–tardiness penalty for the jobs. A sequential exchange approach utilizing a job exchange procedure and three previously established properties in common due date scheduling was developed and tested with a set of benchmark problems. The developed approach generates results better than not only those of the existing dedicated heuristics but also in many cases those of meta-heuristic approaches. And the developed approach performs consistently well in various job settings with respect to the number of jobs, processing time and earliness–tardiness penalties for the jobs.     

Meta-heuristic approaches for minimizing total earliness and tardiness penalties of single-machine scheduling with a common due date

This study addresses a class of single-machine scheduling problems involving a common due date where the objective is to minimize the total job earliness and tardiness penalties. A genetic algorithm (GA) approach and a simulated annealing (SA) approach utilizing a greedy local search and three well-known properties in the area of common due date scheduling are developed. The developed algorithms enable the starting time of the first job not at zero and were tested using a set of benchmark problems. From the viewpoints of solution quality and computational expenses, the proposed approaches are efficient and effective for problems involving different numbers of jobs, as well as different processing time, and earliness and tardiness penalties.     

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.     

Minimizing the weighted number of tardy jobs on a single machine with release dates

In this paper, we describe an exact algorithm to minimize the weighted number of tardy jobs on a single machine with release dates. The algorithm uses branch-and-bound; a surrogate relaxation resulting in a multiple-choice knapsack provides the bounds. Extensive computational experiments indicate the proposed exact algorithm solves either weighted or unweighted problems. It solves the hardest problems to date. Indeed, it solves all previously unsolved instances. Its run time is the shortest to date.     

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries

We study a supply chain scheduling problem, where a common due date is assigned to all jobs and the number of jobs in delivery batches is constrained by the batch size. Our goal is to minimize the sum of the weighted number of tardy jobs, the due-date-assignment costs and the batch-delivery costs. We show that some well-known NPmathcal{NP}-hard problems reduce to our problem. Then we propose a pseudo-polynomial algorithm for the problem, establishing that it is NPmathcal{NP}-hard only in the ordinary sense. Finally, we convert the algorithm into an efficient fully polynomial time approximation scheme.     

Minimizing earliness and tardiness penalties in a single-machine problem with a common due date

Scheduling problems involving both earliness and tardiness costs have received significant attention in recent years. This type of problem became important with the advent of the just-in-time (JIT) concept, where early or tardy deliveries are highly discouraged. In this paper we examine the single-machine scheduling problem with a common due date. Performance is measured by the minimization of the sum of earliness and tardiness penalties of the jobs. Since this problem is NP-hard, we propose a tabu search-based heuristic and a genetic algorithm which exploit specific properties of the optimal solution. Hybrid strategies are also analyzed to improve the performance of these methods. The proposed approaches are examined through a computational comparative study with 280 benchmark problems with up to 1000 jobs.     

A heuristic for single machine scheduling with early and tardy costs

The problem of scheduling jobs with distinct ready times and due dates in a single machine to minimise the total earliness and tardiness penalties is considered. A constructive heuristic, which determines the sequence of jobs and simultaneously inserts idle times, is proposed. Adjacent pairwise interchange is then applied to the schedule obtained. For problems involving at most 12 jobs the heuristic solutions are compared to optimal solutions. For larger problems with up to 80 jobs the heuristic is tested against a local search based on pairwise interchanges and four dispatching rules presented in the literature. In each case, idle times are optimally inserted.     

Scheduling about a common due date with kob-dependent asymmetric earliness and tardiness penalties

This paper is concerned with the problem of scheduling n jobs with a common due date on a single machine so as to minimize the total cost arising from earliness and tardiness. A general model is examined, in which earliness penalty and tardiness penalty are, respectively, arbitrary non-decreasing functions. Moreover, the model includes two important features that commonly appear in practical problems, namely, 1) earliness and tardiness are penalized with different weights which are job-dependent, and 2) the earliness (or tardiness) penalty consists of two parts, one is a variable cost dependent on the length of earliness (or tardiness), while the other is a fixed cost incurred when a job is early (or tardy). This model provides a general and flexible performance measure for earliness/tardiness scheduling, which has not been addressed before. We establish a number of results on the characterizations of optimal and sub-optimal solutions, and propose two algorithms based on these results. The first algorithm can find, under an agreeable weight condition, an optimum in time O(n² P_n), and the second algorithm can generate a sub-optimum in time O(nP_n), where P_n is the sum of the processing times. Further, we derive an upper bound on the relative error of the sub-optimal solution and show that, under certain conditions, the error tends to zero as n increases. Computational results are also reported to demonstrate the effectiveness of the algorithms proposed.     

Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε$2+\epsilon$.     

A note on the single machine scheduling to minimize the number of tardy jobs with deadlines

It is known that the single machine scheduling problem of minimizing the number of tardy jobs is polynomially solvable. However, it becomes NP-hard if each job has a deadline. Recently, Huo et al. solved some special cases by a backwards scheduling approach. In this note we present a dual approach—forwards greedy algorithms which may have better running time. For example, in the case that the due dates, deadlines, and processing times are agreeable, the running time of the backwards scheduling algorithm is O(n²)$O(n^2)$, while that of the forwards algorithm is O(nlogn)$O(n\log n)$.     

Fabrication scheduling on a single machine with due date constraints

There is a fabrication machine available for processing a set of jobs. Each job is associated with a due date and consists of two parts, one is common among all products and the other is unique to itself. The unique components are processed individually and the common parts are grouped into batches for processing. A constant setup time is incurred when each batch is formed. The completion time of a job is defined as the time when both of its unique and common components are completed. In this paper, we consider two different objectives. The first problem seeks to minimize the maximum tardiness, and the second problem is to minimize the number of tardy jobs. To minimize the maximum tardiness, we propose a dynamic programming algorithm that optimally solves the problem in polynomial time. Next, we show NP-hardness proof and design a pseudo-polynomial time dynamic programming algorithm for the problem of minimizing the number of tardy jobs.     

Two-machine flow shop scheduling with deteriorating jobs: minimizing the weighted sum of makespan and total completion time

This paper considers a two-machine flow shop scheduling problem with deteriorating jobs in which the processing times of jobs are dependent on their starting times in the sequence. The objective is to minimize the weighted sum of makespan and total completion time. To analyse the problem, we propose a mixed integer programming model, and discuss several polynomially solvable special cases. We also present a branch-and-bound algorithm with several dominance rules, an upper bound and a lower bound. Finally, we present results of computational experiments conducted to evaluate the performance of the proposed model and the exact algorithm.     

Minimizing the weighted number of tardy jobs and maximum tardiness in relocation problem with due date constraints

The relocation problem addressed in this paper is to determine a reconstruction sequence for a set of old buildings, under a limited budget, such that there is adequate temporary space to house the residents decanted during rehabilitation. It can be regarded as a resource-constrained scheduling problem where there is a set of jobs to be processed on a single machine. Each job demands a number of resources for processing and returns probably a different number of resources on its completion. Given a number of initial resources, the problem seeks to determine if there is a feasible sequence for the successful processing of all the jobs. Two generalizations of the relocation problem in the context of single machine scheduling with due date constraints are studied in this paper. The first problem is to minimize the weighted number of tardy jobs under a common due date. We show that it is NP-hard even when all the jobs have the same tardy weight and the same resource requirement. A dynamic programming algorithm with pseudo-polynomial computational time is proposed for the general case. In the second problem, the objective is to minimize the maximum tardiness when each job is associated with an individual due date. We prove that it is strongly NP-hard. We also propose a pseudo-polynomial time dynamic programming algorithm for the case where the number of possible due dates is predetermined.     

Two due date assignment problems in scheduling a single machine

We study two single-machine scheduling problems: minimizing the sum of weighted earliness, tardiness and due date assignment penalties and minimizing the weighted number of tardy jobs and due date assignment costs. We prove that both problems are strongly NP-hard and give polynomial solutions for some important special cases.     

A new branch and bound algorithm for minimizing the weighted number of tardy jobs

In this paper, the problem of sequencing jobs on a single machine to minimize the weighted number of tardy jobs is considered. Some new dominances between jobs are proposed and studied. A new branch and bound algorithm that can solve large problems, e.g. 85 jobs, is presented.     

Minimizing sum of completion times on a single machine with sequence-dependent family setup times

This paper presents a branch-and-bound (B&B) algorithm for minimizing the sum of completion times in a single-machine scheduling setting with sequence-dependent family setup times. The main feature of the B&B algorithm is a new lower bounding scheme that is based on a network formulation of the problem. With extensive computational tests, we demonstrate that the B&B algorithm can solve problems with up to 60 jobs and 12 families, where setup and processing times are uniformly distributed in various combinations of the [1,50] and [1,100] ranges.