Minimizing weighted mean absolute deviation of job completion times from their weighted mean

We address a single-machine scheduling problem where the objective is to minimize the weighted mean absolute deviation of job completion times from their weighted mean. This problem and its precursors aim to achieve the maximum admissible level of service equity. It has been shown earlier that the unweighted version of this problem is NP-hard in the ordinary sense. For that version, a pseudo-polynomial time dynamic program and a 2-approximate algorithm are available. However, not much (except for an important solution property) exists for the weighted version. In this paper, we establish the relationship between the optimal solution to the weighted problem and a related one in which the deviations are measured from the weighted median (rather than the mean) of the job completion times; this generalizes the 2-approximation result mentioned above. We proceed to give a pseudo-polynomial time dynamic program, establishing the ordinary NP-hardness of the problem in general. We then present a fully-polynomial time approximation scheme as well. Finally, we report the findings from a limited computational study on the heuristic solution of the general problem. Our results specialize easily to the unweighted case; they also lead to an approximation of the set of schedules that are efficient with respect to both the weighted mean absolute deviation and the weighted mean completion time.     

An exact algorithm to minimize mean squared deviation of job completion times about a common due date

We consider a deterministic n-job, single machine scheduling problem with the objective of minimizing the Mean Squared Deviation (MSD) of job completion times about a common due date (d). The MSD measure is non-regular and its value can decrease when one or more completion times increases. MSD problem is connected with the Completion Time Variance (CTV) problem and has been proved to be NP-hard. This problem finds application in situations where uniformity of service is important. We present an exact algorithm of pseudo-polynomial complexity, using ideas from branch and bound and dynamic programming. We propose a dominance rule and also develop a lower bound on MSD. The dominance rule and lower bound are effectively combined and used in the development of the proposed algorithm. The search space is explored using the breadth first branching strategy. The asymptotic space complexity of the algorithm is O(nd). Irrespective of the version of the problem – tightly constrained, constrained or unconstrained – the proposed algorithm provides optimal solutions for problem instances up to 1000 jobs size under different due date settings.     

Minimizing total completion time and total deviation of job completion times from a restrictive due-date

We study a single-machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due-date. The due-date is assumed to be restrictive, i.e., it may be sufficiently small to have an impact on the optimal sequence. When more weight is allocated to total job completion times, the problem is shown to have a polynomial time solution. When more weight is allocated to total completion time deviations from the due-date, the problem is NP-hard in the ordinary sense. For the latter case, we introduce an efficient dynamic programming algorithm, which is shown numerically to perform well in all our tests.     

Minimizing total absolute deviation of job completion times: extensions to position-dependent processing times and parallel identical machines

As a measure of variation, total absolute deviation of job completion times (TADC) has received relatively little attention in scheduling literature. Minimizing TADC on a single machine was shown to have a polynomial time solution. In this note, we extend TADC in two directions: (i) we allow position-dependent processing times and (ii) we consider parallel identical machines. We show that each of these two more general problems, and TADC with both extensions remain polynomially solvable.     

Minimizing the average deviation of job completion times about a common due-date: An extension

This paper treats the same problem as considered by Kanet [Nav. Res. Logist. Q. 28, 643–651 (1981)] about sequencing n jobs on a single machine with penalties occuring when a job is completed early or late. The objective is to minimize the total penalty. The penalty function has the same form as assumed by Kanet, but the restrictive assumption on the due-dates is relaxed. The result is that the quoted common due-date is reduced, while the efficient algorithm proposed by Kanet can still be used to help determine the optimal job sequence.     

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.     

Minimizing the sum of maximum earliness and maximum tardiness in the single-machine scheduling problem with sequence-dependent setup time

This paper considers the problem of scheduling a given number of jobs on a single machine to minimize the sum of maximum earliness and maximum tardiness when sequence-dependent setup times exist (1∣ST _sd ∣ET_max). In this paper, an optimal branch-and-bound algorithm is developed that involves the implementation of lower and upper bounding procedures as well as three dominance rules. For solving problems containing large numbers of jobs, a polynomial time-bounded heuristic algorithm is also proposed. Computational experiments demonstrate the effectiveness of the bounding and dominance rules in achieving optimal solutions in more than 97% of the instances.     

Total completion time with makespan constraint in no-wait flowshops with setup times

The m-machine no-wait flowshop scheduling problem with the objective of minimizing total completion time subject to the constraint that the makespan value is not greater than a certain value is addressed in this paper. Setup times are considered non-zero values, and thus, setup times are treated as separate from processing times. Several recent algorithms, an insertion algorithm, two genetic algorithms, three simulated annealing algorithms, two cloud theory-based simulated annealing algorithms, and a differential evolution algorithm are adapted and proposed for the problem. An extensive computational analysis has been conducted for the evaluation of the proposed algorithms. The computational analysis indicates that one of the nine proposed algorithms, one of the simulated annealing algorithms (ISA-2), performs much better than the others under the same computational time. Moreover, the analysis indicates that the algorithm ISA-2 performs significantly better than the earlier existing best algorithm. Specifically, the best performing algorithm, ISA-2, proposed in this paper reduces the error of the existing best algorithm in the literature by at least 90% under the same computational time. All the results have been statistically tested.     

Minimizing the total completion time in a two-machine flowshop with sequence-independent setup times

We consider the problem of minimizing the sum of completion times in a two-machine permutation flowshop subject to setup times. We propose a new priority rule, several constructive heuristics, local search procedures, as well as an effective multiple crossover genetic algorithm. Computational experiments carried out on a large set of randomly generated instances provide evidence that a constructive heuristic based on newly derived priority rule dominates all the proposed constructive heuristics. More specifically, we show that one of our proposed constructive heuristics outperforms the best constructive heuristic in the literature in terms of both error and computational time. Furthermore, we show that one of our proposed local search-based heuristics outperforms the best local search heuristic in the literature in terms of again both error and computational time. We also show that, in terms of quality-to-CPU time ratio, the multiple crossover genetic algorithm performs consistently well.     

Minimizing the total completion time in a unit-time open shop with release times

We consider the problem of minimizing the total completion time in a unit-time open shop with release times where the number of machines is constant. Brucker and Krämer (1994) proved that this problem is solvable in polynomial time. However, the time complexity of the algorithm presented there is a polynom of a very high degree and, thus, the algorithm is not practicable even for a small number of machines. We give an O(n²) algorithm for the considered problem which is based on dynamic programming. The result is applied to solve a previously open problem with a special resource constraint raised by De Werra et al. (1991).     

Minimizing total completion time in a no-wait flowshop with sequence-dependent additive changeover times

This paper addresses the problem of minimizing total completion time in a two-machine no-wait flowshop where setup times of the jobs are sequence-dependent. Optimal solutions are obtained for two special flowshops and a dominance relation is developed for the general problem. Several heuristic algorithms with the computational complexity of O(n²) and O(n³) are constructed. The heuristics consist of two phases: in the first phase a starting list is developed and in the second a repeated insertion technique is applied. Computational experience demonstrates that the concept of repeated insertion application is quite useful for any starting list and that solutions for all starting lists converge to about the same value of less than 1% after a few iterations.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

Minimizing mean tardiness subject to unspecified minimum number tardy for a single machine

In this paper we propose a hybrid branch and bound algorithm for solving the problem of minimizing mean tardiness for a single machine problem subject to minimum number of tardy jobs. Although the minimum number of tardy jobs is known, the subset of tardy job is not known. The proposed algorithm uses traditional branch and bound scheme where lower bounds on mean tardiness are calculated coupled with using the information that the number of tardy jobs is known. It also uses an insertion algorithm which determines the optimal mean tardiness once the subset of tardy jobs is specified. An example is solved to illustrate the developed procedure.     

Minimizing total tardiness in parallel machine scheduling with setup times: An adaptive memory-based GRASP approach

This paper deals with the problem of scheduling jobs in uniform parallel machines with sequence-dependent setup times in order to minimize the total tardiness relative to job due dates. We propose GRASP versions that incorporate adaptive memory principles for solving this problem. Long-term memory is used in the construction of an initial solution and in a post-optimization procedure which connects high quality local optima by means of path relinking. Computational tests are carried out on a set of benchmark instances and the proposed GRASP versions are compared with heuristic methods from the literature.     

Minimizing the sum of weighted completion times in a concurrent open shop

We study minimizing the sum of weighted completion times in a concurrent open shop. We give a primal-dual 2-approximation algorithm for this problem. We also show that several natural linear programming relaxations for this problem have an integrality gap of 2. Finally, we show that this problem is inapproximable within a factor strictly less than 6/5 if P≠NP, or strictly less than 4/3 if the Unique Games Conjecture also holds.     

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.     

Minimizing maximum completion time in a proportionate flow shop with one machine of different speed

A flow shop with identical machines is called a proportionate flow shop. In this paper, we consider the variant of the n-job, m-machine proportionate flow shop scheduling problem in which only one machine is different and job processing times are inversely proportional to machine speeds. The objective is to minimize maximum completion time. We describe some optimality conditions and show that the problem is NP-complete. We provide two heuristic procedures whose worst-case performance ratio is less than two. Extensive experiments with various sizes are conducted to show the performance of the proposed heuristics.