Total Weighted Completion Time Research Articles

Scheduling with a weight-modifying activity to minimize the total weighted completion time

This paper considers a single-machine scheduling problem to minimize the total weighted completion time with a weight modifying activity, after which the job weights are discounted by a given factor. The problem is known to be ordinary NP-hard. We propose two mixed integer linear programs (MILPs) and a dynamic programming algorithm to optimally solve the problem. Optimality properties are established and then formulated as pruning constraints to improve the problem-solving efficiency of the MILPs. Special cases are discussed and shown to be solvable by polynomial time algorithms. Complexity status of the studied problem with several instance characteristics is shown. Computational experiments indicate that the optimality properties can reduce the computing efforts and that one of the proposed MILPs can solve instances of 200 jobs in a few seconds.

Scheduling on parallel dedicated machines with job rejection

We study scheduling problems on parallel dedicated machines. Thus, each job can be processed on one specific machine only. The option of job-rejection is considered, and the total permitted rejection cost of all the jobs is bounded. Six scheduling problems are solved: ( i ) minimising makespan, ( ii ) minimising makespan with release-dates, ( iii ) minimising total completion time, ( iv ) minimising total weighted completion time, ( v ) minimising total load, and ( vi ) minimising maximum tardiness. Pseudo-polynomial dynamic programming algorithms are introduced for all these NP-hard problems.

Rescheduling with New Orders Under Bounded Disruption

Rescheduling problems arise when unpredicted events occur, such as the arrival of new orders. These new jobs should be integrated in a proper way in the existing schedule of the so-called old jobs, with the aim of minimizing an objective function for the joint set of jobs. To avoid a major disruption of the original schedule, each old job is not allowed to deviate from its original completion time by more than a certain threshold. Filling a gap in the existing literature, we consider the minimization of the total weighted completion time. The resulting rescheduling problem is shown to be weakly NP-hard and several properties of the structure of an optimal schedule are derived. These can be used for the construction of an exact dynamic programming algorithm with pseudo-polynomial running time. A fully polynomial time approximation scheme is obtained from the dynamic program by three different scaling and reduction steps. Finally, for the minimization of the number of late jobs a strong NP-hardness result is derived. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was partially supported by the Ministero dell’Istruzione, dell’Università e della Ricerca [Award TESUN-83486178370409 finanziamento dipartimenti di eccellenza CAP. 1694 TIT. 232 ART. 6]. U. Pferschy acknowledges support by the Field of Excellence COLIBRI at the University of Graz.

A branch-and-price algorithm for unrelated parallel machine scheduling with machine usage costs

This paper considers unrelated parallel machine scheduling involving machine usage costs, in addition to classic job completion time-related costs. The usage cost of each machine is made up of a fixed usage cost and a variable usage cost proportional to the total processing time of the jobs assigned to it. These features model many practical situations where machine usage costs include, for example, rental fees when the machines are not owned but rented. To tackle this problem, four mathematical models based on the Shortest Weighted Processing Time (SWPT) rule are introduced. Additionally, the problem is formulated into a set-partitioning model, for which a branch-and-price algorithm is proposed with an appropriate branching strategy. This facilitates the development of an efficient pseudo-polynomial dynamic programming algorithm and a polynomial-time heuristic to solve the pricing problem. Extensive numerical experiments demonstrate the superior performance of the proposed branch-and-price algorithm over the four SWPT-based mathematical formulations and an existing branch-and-price algorithm designed for a special case. Notably, it can optimally solve instances involving up to 225 jobs and 15 machines within one hour. Moreover, statistical analyses reveal that the proposed polynomial-time heuristic significantly reduces the computation time, and the mathematical model based on the contribution of every job to the total weighted completion time exhibits the best overall performance.

Price of Anarchy in uniform parallel machines scheduling game with weighted completion time as social goal

This article investigates the efficiency of Nash equilibria in a utilitarian scheduling game, where each job, acting as an agent, strategically selects a machine for its processing. The study focuses on a uniform parallel machine environment, employing the Weighted Shortest Processing Time rule as the local policy and the total weighted completion time as the social goal. We establish upper and lower bounds for the Price of Anarchy, offering valuable insights for this scheduling game.

Approximation of the Objective Function of Single-Machine Scheduling Problem

The problem of the approximation of the coefficients of the objective function of a scheduling problem for a single machine is considered. It is necessary to minimize the total weighted completion times of jobs with unknown weight coefficients when a set of problem instances with known optimal schedules is given. It is shown that the approximation problem can be reduced to finding a solution to a system of linear inequalities for weight coefficients. For the case of simultaneous job release times, a method for solving the corresponding system of inequalities has been developed. Based on it, a polynomial algorithm for finding values of weight coefficients that satisfy the given optimal schedules was constructed. The complexity of the algorithm is O(n2(N+n)) operations, where n is the number of jobs and N is the number of given instances with known optimal schedules. The accuracy of the algorithm is estimated by experimentally measuring the function ε(N,n)=1n∑j=1n∣wj−wj0∣wj0, which is an indicator of the average modulus of the relative deviation of the found values wj from the true values wj0. An analysis of the results shows a high correlation between the dependence ε(N,n) and a function of the form α(n)/N, where α(n) is a decreasing function of n.

Single Machine Scheduling Proportionally Deteriorating Jobs with Ready Times Subject to the Total Weighted Completion Time Minimization

In this paper, we investigate a single machine scheduling problem with a proportional job deterioration. Under release times (dates) of jobs, the objective is to minimize the total weighted completion time. For the general condition, some dominance properties, a lower bound and an upper bound are given, then a branch-and-bound algorithm is proposed. In addition, some meta-heuristic algorithms (including the tabu search (TS), simulated annealing (SA) and heuristic (NEH) algorithms) are proposed. Finally, experimental results are provided to compare the branch-and-bound algorithm and another three algorithms, which indicate that the branch-and-bound algorithm can solve instances of 40 jobs within a reasonable time and that the NEH and SA are more accurate than the TS.

New Results on Single-Machine Scheduling with Rejection to Minimize the Total Weighted Completion Time

Abstract In this paper, we study eight single-machine scheduling problems with rejection and position-dependent parameters. We consider two position-dependent parameters as follows: (1) position-dependent weights and (2) position-dependent processing times. In addition, we also introduce a weight-modifying activity or a rate-modifying activity into our problems. In the first six problems, the task is to minimize the sum of the total weighted completion time of accepted jobs and the total rejection cost of rejected jobs. We show that all six problems can be solved in polynomial time. In the last two problems, the task is to minimize the total weighted completion time of accepted jobs under the constraint that the total rejection cost of rejected jobs can not exceed a given upper bound. We show that these two problems are binary NP-hard and each problem admits an FPTAS.

Efficient Approximation Algorithms for Scheduling Coflows With Total Weighted Completion Time in Identical Parallel Networks

Efficient Approximation Algorithms for Scheduling Coflows With Total Weighted Completion Time in Identical Parallel Networks

Solving Preemptive Single Machine Scheduling with Lovebird Algorithm and Constructive Heuristics for Equal-Length Jobs

Scheduling is essential in manufacturing to maintain machinery and increase revenue. Machines that are routinely maintained will boost production, whereas scheduling based on total weighted completion time (TWCT) can improve the performance of production equipment. The Lovebird Algorithm (LBA), Constructive Heuristics (CH), and combination of LBA-CH are used in this study to schedule a single machine with equal processing times. LBA, a population-based algorithm, represents job parts as birds for jobs that can be preempted. The experiments, which used ten jobs divided into dozens of job parts, show that LBA-CH can effectively reduce TWCT compared to CH by 18.16%.

Developing New Bounds for the Performance Guarantee of the Jump Neighborhood for Scheduling Jobs on Uniformly Related Machines

This study investigates the worst-case performance guarantee of locally optimal solutions to minimize the total weighted completion time on uniformly related parallel machines. The investigated neighborhood structure is Jump, also called insertion or move. This research focused on establishing the local optimality condition expressed as an inequality and mapping that maps a schedule into an inner product space so that the norm of the mapping is closely related to the total weighted completion time of the schedule. We determine two new upper bounds for the performance guarantee, which take the form of an expression based on parameters that describe the family of instances: the speed of the fastest machine, the speed of the slowest machine, and the number of machines. These new bounds outperform the parametric upper bound previously established in the existing literature and enable a better understanding of the performance of the solutions obtained for the Jump neighborhood in this scheduling problem, according to parameters that describe the family of instances.

MATHEMATICAL MODELS AND METHODS OF COORDINATED PLANNING

Modern processes of globalization and economic competition require a significant increase in the requirements for the professional degree of top-level managers who manage the activities of international corporations, regional economies, branch ministries, etc. Their efficient operation is impossible without the use of basic scientific developments and appropriate software which implement the main qualitative law of complex organizational and production systems management: the law of coordinated management (planning), when management decisions at the top level take into account interests that may not coincide, or even be antagonistic in organizational and production subsystems connected by a certain structure of mutual relations within a single organizational and production complex system. In this work, we consider a two-level organizational and production system, which in terms of the generally known theory of active systems is defined as “decision-making center → elements (of an organizational and production subsystem)”. We consider formal models of elements of two classes, linear continuous and discrete, aggregated production models which belong to the same class of NP-hard single-stage scheduling problems. For both types of element models, we give compromise criteria and corresponding methods of constructing compromise solutions based on the results of Prof. A. A. Pavlov for multi-objective linear programming, as a result of his theoretical research for discrete optimization problems under uncertainty, and the theory of PSC-algorithms created by him and his students, that is, algorithms containing polynomial complexity subalgorithms for constructing feasible solutions that satisfy theoretically substantiated sufficient signs of optimality. In this work, we use the PSC-algorithm for the NP-hard scheduling problem “Minimization of the total weighted completion time of jobs on a single machine with precedence relations given by a directed acyclic graph”.

Single machine adversarial bilevel scheduling problems

We consider single machine scheduling problems in the context of adversarial bilevel optimization where two agents, the leader and the follower, take decisions on the same jobset and the leader acts first with the aim of inducing the worst possible solution for the follower. Thus, the follower schedules the jobs in order to optimize a given criterion. The considered criteria are the total completion time, the total weighted completion time, the maximum lateness and the number of tardy jobs. We focus on adversarial bilevel scheduling with job selection and data modification. In the case with job selection, the leader selects a fixed cardinality subset of the jobs that the follower schedules next. In the case with data modification, the leader can modify some of the data (processing times, due dates, weights), given a limited budget Q. Thus, the follower schedules the set of jobs with modified data. For all the considered criteria either we provide polynomial-time algorithms or show that they can be solved in the worst-case in pseudo-polynomial time.

Communication-aware scheduling of precedence-constrained tasks on related machines

Scheduling precedence-constrained tasks is a classical problem that has been studied for more than fifty years. However, little progress has been made in the setting where there are non-uniform communication delays between tasks. In this work, we propose a new scheduler, Generalized Earliest Time First (GETF), and provide the first provable, worst-case approximation guarantees for the goals of minimizing both the makespan and total weighted completion time of tasks with precedence constraints on related machines with machine-dependent communication speeds.

Particle swarm optimization for integrated scheduling problem with batch additive manufacturing and batch direct-shipping delivery

This study addresses a novel proRblem for the integrated scheduling of additive manufacturing and delivery processes with the objective of minimizing the total weighted completion time of the ordered parts. In the manufacturing process, batch production on non-identical parallel additive manufacturing machines is considered. A single batch of production, which is referred to as a build, consists of parts requiring the same material. Sequence-dependent setup times occur between two adjacent builds. Once the production run is completed, the parts in the builds are regrouped into delivery batches within the same customer. In the delivery process, parts are delivered by homogeneous vehicles to the corresponding customers through direct-shipping method. To optimally solve the small problem, a mixed-integer linear programming model is formulated. For solving large-scale problems, three rule-based heuristics that are embedded in particle swarm optimization (PSO) are proposed. A tight lower bound is developed to evaluate the performance of the proposed algorithms. The computational experiment shows that the proposed PSO with delivery waiting time minimization rule is effective to solve small-scale problems, while in large-scale problems, capacity usage maximization rule performs the best and both outperform another existing metaheuristic. Several sensitivity analyses are also performed to provide additional managerial insights.

Controllable Processing Time Scheduling with Total Weighted Completion Time Objective and Deteriorating Jobs

In this paper, we consider two single-machine scheduling problems with deteriorating jobs and controllable processing time. The job processing time is a function of its starting time and resource allocation. For the convex resource function, a bicriteria analysis is provided, where the first (respectively, second) criteria is to minimize total weighted completion time (respectively, total resource consumption cost). The first problem is to minimize the weighted sum of the total weighted completion time and the total resource consumption cost. The second problem is to minimize the total weighted completion time subject to the total resource consumption cost is bound. These both problems are NP-hard, we propose some heuristic algorithms and a branch-and-bound algorithm to solve the problems.

Scheduling coflows for minimizing the total weighted completion time in heterogeneous parallel networks

Coflow is a network abstraction used to represent communication patterns in data centers. The coflow scheduling problem encountered in large data centers is a challenging NP-hard problem. Many previous studies on coflow scheduling mainly focus on the single-core model. However, with the growth of data centers, this single-core model is no longer sufficient. This paper addresses the coflow scheduling problem within heterogeneous parallel networks, which feature an architecture consisting of multiple network cores running in parallel. In this paper, two polynomial-time approximation algorithms are developed for the flow-level scheduling problem and the coflow-level scheduling problem in heterogeneous parallel networks, respectively. For the flow-level scheduling problem, the proposed algorithm achieves an approximation ratio of O(log⁡m/log⁡log⁡m) when all coflows are released at arbitrary times, where m represents the number of network cores. On the other hand, in the coflow-level scheduling problem, the proposed algorithm achieves an approximation ratio of O(m(log⁡m/log⁡log⁡m)2) when all coflows are released at arbitrary times. Moreover, we propose a heuristic algorithm for the flow-level scheduling problem. Simulation results using synthetic traffic traces validate the performance of our algorithms and show improvements over the previous algorithm.

A cooperative adaptive genetic algorithm for reentrant hybrid flow shop scheduling with sequence-dependent setup time and limited buffers

This paper deals with a reentrant hybrid flow shop problem with sequence-dependent setup time and limited buffers where there are multiple unrelated parallel machines at each stage. A mathematical model with the minimization of total weighted completion time is constructed for this problem. Considering the complexity of the problem at hand, an effective cooperative adaptive genetic algorithm (CAGA) is proposed. In the algorithm, a dual chain coding scheme and a staged-hierarchical decoding approach are, respectively, designed to encode and decode each solution. Six dispatch heuristics and a dynamic adjustment method are introduced to define initial population. To balance the exploration and exploitation abilities, three effective operations are implemented: (1) two new crossover and mutation operators with collaborative mechanism are imposed on genetic algorithm; (2) an adaptive adjustment strategy is introduced to re-optimize better solutions after mutation operations, where ant colony search algorithm and modified greedy heuristic are intelligently switched; (3) a reset strategy with dynamic variable step strategy is embedded to re-generate some non-improved solutions. A Taguchi method of design of experiment is adopted to calibrate the parameter values in the presented algorithm. Comparison experiments are executed on test instances with different scale problems. Computational results show that the proposed CAGA is more effective and efficient than several well-known algorithms in solving the studied problem.

Privacy Attacks on Schedule-Driven Data

Schedules define how resources process jobs in diverse domains, reaching from healthcare to transportation, and, therefore, denote a valuable starting point for analysis of the underlying system. However, publishing a schedule may disclose private information on the considered jobs. In this paper, we provide a first threat model for published schedules, thereby defining a completely new class of data privacy problems. We then propose distance-based measures to assess the privacy loss incurred by a published schedule, and show their theoretical properties for an uninformed adversary, which can be used as a benchmark for informed attacks. We show how an informed attack on a published schedule can be phrased as an inverse scheduling problem. We instantiate this idea by formulating the inverse of a well-studied single-machine scheduling problem, namely minimizing the total weighted completion times. An empirical evaluation for synthetic scheduling problems shows the effectiveness of informed privacy attacks and compares the results to theoretical bounds on uninformed attacks.

Minimising total weighted completion time for semi-online single machine scheduling with known arrivals and bounded processing times

This paper addresses the semi-online scheduling problem of minimising the total weighted completion time on a single machine, where a combination of information on jobs release dates and processing times is considered. In this study, jobs can only arrive at known future times and a lower bound on jobs processing times is known in advance. A new semi-online algorithm is presented and is shown to be the best possible for the considered problem. In order to make this statement, a new lower bound on the competitive ratio of any semi-online algorithm for the problem is developed and, using competitive analysis, the proposed semi-online algorithm is shown to have a competitive ratio that matches the lower bound.