Total Weighted Completion Time Research Articles

Optimally rescheduling jobs with a Last-In-First-Out buffer

This paper considers single-machine scheduling problems in which a given solution, i.e., an ordered set of jobs, has to be improved as much as possible by re-sequencing the jobs. The need for rescheduling may arise in different contexts, e.g., due to changes in the job data or because of the local objective in a stage of a supply chain that is not aligned with the given sequence. A common production setting entails the movement of jobs (or parts) on a conveyor. This is reflected in our model by facilitating the re-sequencing of jobs via a buffer of limited capacity accessible by a LIFO policy. We consider the classical objective functions of total weighted completion time, maximum lateness and (weighted) number of late jobs and study their complexity. For three of these problems, we present strictly polynomial-time dynamic programming algorithms, while for the case of minimizing the weighted number of late jobs NP-hardness is proven and a pseudo-polynomial algorithm is given.

Multigraph Modeling for Urban Distribution of Emergency Commodities with Semisoft Time Windows under Uncertainty

In this paper, we studied a stochastic bi-objective mathematical model for effective and reliable rescue operations in multigraph network. The problem is addressed by a two-stage stochastic nonlinear mixed-integer program where the reliability of routes is explicitly traded-off with total weighted completion time. The underlying transportation network is able to keep a group of multiattribute parallel arcs between every pair of nodes. By this, the proposed model should consider the routing decision in logistic planning along with the path selection in an uncertain condition. The first stage of the model concerns with the vehicle routing decisions which is not involved with random parameters; besides, the second stage of the model involves with the departure time at each demand node and path finding decisions after observation of random vectors in the first stage considering a finite number of scenarios. To efficiently solve the presented model, an enhanced nondominated sorting genetic algorithm II (NSGA-II) is proposed. The effectiveness of the introduced method is then evaluated by conducting several numerical examples. The results implied the high performance of our method in comparison to the standard NSGA-II. In further analyses, we investigated the beneficiary of using multigraph setting and showed the applicability of the proposed model using a real transportation case.

Iterated greedy algorithms for a complex parallel machine scheduling problem

This paper addresses a complex parallel machine scheduling problem with jobs divided into operations and operations grouped in families. Non-anticipatory family setup times are held at the beginning of each batch, defined by the combination of one setup-time and a sequence of operations from a unique family. Other aspects are also considered in the problem, such as release dates for operations and machines, operation’s sizes, and machine’s eligibility and capacity. We consider item availability to define the completion times of the operations within the batches, to minimize the total weighted completion time of jobs. We developed Iterated Greedy (IG) algorithms combining destroy and repair operators with a Random Variable Neighborhood Descent (RVND) local search procedure, using four neighborhood structures to solve the problem. The best algorithm variant outperforms the current literature methods for the problem, in terms of average deviation for the best solutions and computational times, in a known benchmark set of 72 instances. New upper bounds are also provided for some instances within this set. Besides, computational experiments are conducted to evaluate the proposed methods’ performance in a more challenging set of instances introduced in this work. Two IG variants using a greedy repair operator showed superior performance with more than 70% of the best solutions found uniquely by these variants. Despite the simplicity, the method using the most common destruction and repair operators presented the best results in different evaluated criteria, highlighting its potential and applicability in solving a complex machine scheduling problem.

Minimizing the Total Weighted Waiting Times and Instability in a Rescheduling Problem with Dynamic Jobs Weight

Thanks to smart technological tools, customers can at any moment create or modify their commands. This reality forced many production firms to become sensitive in rescheduling processes. In the literature, most of rescheduling problems consider classical efficiency measures. However, some existing works also consider stability as a measure for limiting the deviation from initial schedule. In this work, we aim to bridge the gap in existing works on rescheduling by investigating a new approach to measure simultaneously efficiency by the total weighted waiting times and stability by the total weighted completion time deviation. This combination of criteria is very significant in industrial and hospital environments. In this paper, a single machine rescheduling problem with jobs arriving over time is considered. A mixed integer linear programming (MILP) model is designed for this problem and an iterative predictive-reactive strategy for dealing with the online part. Numerical results show that, at each time the jobs are rescheduled, the low weight ones move forward. Consequently, a new concept consisting in increasing the jobs weight as function of time is established. The effect of this new conception is evaluated by the variation of the average flowtime. Eventually, the computing time of the MILP resolution is studied to explore its limitations.

Total Completion Time Minimization for Scheduling with Incompatibility Cliques

This paper considers parallel machine scheduling with incompatibilities between jobs. The jobs form a graph equivalent to a collection of disjoint cliques. No two jobs in a clique are allowed to be assigned to the same machine. Scheduling with incompatibilities between jobs represents a well-established line of research in scheduling theory and the case of disjoint cliques has received increasing attention in recent years. While the research up to this point has been focused on the makespan objective, we broaden the scope and study the classical total completion time criterion. In the setting without incompatibilities, this objective is well-known to admit polynomial time algorithms even for unrelated machines via matching techniques. We show that the introduction of incompatibility cliques results in a richer, more interesting picture. We prove that scheduling on identical machines remains solvable in polynomial time, while scheduling on unrelated machines becomes APX-hard. Next, we study the problem under the paradigm of fixed-parameter tractable algorithms (FPT). In particular, we consider a problem variant with assignment restrictions for the cliques rather than the jobs. We prove that, despite still being APX-hard, it can be solved in FPT time with respect to the number of cliques. Moreover, we show that the problem on unrelated machines can be solved in FPT time for reasonable parameters, in particular, the parameter combination: maximum processing time, number of job kinds, and number of machines or maximum processing time, number of job kinds, and number of cliques. The latter results are extensions of known results for the case without incompatibilities, and can even be further extended to the case of total weighted completion time. All of the FPT results make use of n-fold Integer Programs that recently received great attention by proving their usefulness for scheduling problems.

A robust multi-response VNS-aiNet approach for solving scheduling problems under unrelated parallel machines environments

In the last few years, a lot of researchers addresses scheduling problems considering unrelated parallel machines to bridge theoretical problems and real-world manufacture environments. Most of these works consider objective function’s characteristics to assist the algorithms in exploiting the search space. This assistance usually leads to a better convergence of the algorithms, but it tends to leave these algorithms restricted only to the own optimization criterion studied. This feature hinders to extend these works to optimization criteria with less theoretical visibility. This work proposes an immuno-inspired algorithm for the resolution of scheduling problems under unrelated parallel environments, evaluating the hypothesis that it is possible to construct a robust approach that is extensible to different optimization criteria, achieving results similar to the state-of-art approaches in the literature for specific criteria. Four case studies are proposed, each one addressing an optimization criterion, which is a benchmark for sequencing problems. The criteria are Total Weighted Tardiness, Makespan, Total Tardiness, and Total Weighted Completion Time. The experiments indicate that the results achieved by the proposed approach are compatible and, in many cases, superior to the state-of-art approaches for the optimization criteria studied, using computational resources compatible with those presented by these approaches.

Predictive-reactive Strategy for Flowshop Rescheduling Problem: Minimizing the Total Weighted Waiting Times and Instability

Due to the fourth revolution experiencing, referred to as Industry 4.0, many production firms are devoted to integrating new technological tools to their manufacturing process. One of them, is rescheduling the tasks on the machines responding to disruptions. While, for static scheduling, the efficiency criteria measure the performance of scheduling systems, in dynamic environments, the stability criteria are also used to assess the impact of jobs deviation. In this paper, a new performance measure is investigated for a flowshop rescheduling problem. This one considers simultaneously the total weighted waiting time as the efficiency criterion, and the total weighted completion time deviation as the stability criterion. This fusion could be a very helpful and significant measure for real life industrial systems. Two disruption types are considered: jobs arrival and jobs cancellation. Thus, a Mixed Integer Linear Programming (MILP) model is developed, as well as an iterative predictive-reactive strategy for dealing with the online part. At last, two heuristic methods are proposed and discussed, in terms of solution quality and computing time.

Generating bicriteria schedules for correlated parallel machines involving tardy jobs and weighted completion time

This research focuses on studying the correlated parallel machine scheduling problem with release dates to minimize the number of tardy jobs and the total weighted completion time to find Pareto optima of all non-dominated solutions for both criteria. First, a mixed integer programming (MIP) model to find the entire efficient frontier for the studied problem has been proposed. Next, a bicriteria heuristic, named UTWC, and three artificial bee colony (ABC) variant algorithms have been proposed to tackle the studied problem. The three ABC variant algorithms are ABC, ABC with variable neighborhood search scheme (ABC_VNS), and ABC with simulated annealing scheme (ABC_SA). For small problem instances, we compare the proposed heuristic UTWC and three ABC variant algorithms with the efficient frontier generated by solving the MIP model. For large problem instances, we compare the UTWC and three ABC variant algorithms with an existing multi-objective tabu search algorithm (TSA). We create a reference set by combining the solutions from the three ABC variant algorithms and the TSA. The computational results indicate that the UTWC heuristic can find a set of non-dominated solutions that are uniformly distributed above the efficient frontier. The proposed ABC_VNS algorithm outperforms all the compared algorithms in both small and large problem instances.

Single-Machine Scheduling with Fixed Periodic Preventive Maintenance to Minimise the Total Weighted Completion Times

The single-machine scheduling problem with fixed periodic preventive maintenance, in which preventive maintenance is implemented periodically to maintain good machine operational status and decrease the cost caused by sudden machine failure, is studied in this paper. The adopted objective function is to minimise the total weighted completion time, which is representative of the minimisation of the global holding/inventory cost in the system. This problem is proven to be NP-hard; a position-based mixed integer programming model and an efficient heuristic algorithm with local improvement strategy are developed for the total weighted completion time problem. To evaluate the performances of the proposed heuristic algorithms, two new lower bounds are further developed. Computational experiments show that the proposed heuristic can rapidly achieve optimal results for small-sized problems and obtain near-optimal solutions with tight average relative percentage deviation for large-sized problems.

A tight approximation ratio of a list scheduling algorithm for a single-machine scheduling problem with a non-renewable resource

In this paper, we investigate an open problem by Gyorgyi and Kis for a single-machine scheduling problem with a non-renewable resource (NR-SSP) and total weighted completion time criterion. The problem is NP-hard, even if every job has the same processing time, and each weight is equal to its required amount of the resource. Gyorgyi and Kis prove that a simple list scheduling algorithm for this special case is a 3-approximation algorithm and conjecture that the algorithm for the case is a 2-approximation algorithm. We prove their conjecture. More strongly, we show that the approximation ratio is strictly less than 2 for any instance, and for any $$ \epsilon > 0 $$ , there exists an instance for which the approximation ratio is more than $$ 2-\epsilon $$ .

Scheduling of step-improving jobs with an identical improving rate

Job processing times change over time in real-life production and manufacturing systems due to various factors including machine or worker learning, machine deterioration, production system upgrades or technological shocks. For step-improving processing times, job processing times are reduced by a certain rate if they start to process at, or after, a common critical date, which has wide applicability in real-world settings, such as data gathering networks and production systems with part-time workers. This paper considers single machine scheduling of minimizing total weighted completion time with step-improving jobs. The problem is shown to be intractable. Both exact and heuristic algorithms are developed, and the approximability of the heuristic algorithm is shown for a special case of the problem. Finally, computational experiments show that the proposed algorithms provide very effective and efficient solutions.

New efficiency-stability criterion for flow shop rescheduling problem with mixed blocking constraints

Abstract New jobs arrival, jobs cancelation or machine breakdowns, are unexpected events that may occur during production run in real environments, thus disrupting the established schedule. Hence, rescheduling process is necessary for efficiently revising the initial schedule, preferably with little movements. In the literature, many scheduling problems consider unlimited buffer spaces between successive machines in flow shop environments. However, in real life systems, the capacity between machines may be limited or no-store existing. To fill the void, this paper considers a flow shop rescheduling problem with mixed blocking constraints, under disruptions caused by new jobs arrival. A new performance measure is investigated, considering simultaneously the schedule efficiency measured by the total weighted waiting time, and the schedule stability evaluated by the total weighted completion time deviation. This combination could be very helpful for real life industrial systems. A mixed integer linear programming model (MILP) is designed, as well as an iterative methodology based on the predictive reactive rescheduling strategy. At last, two heuristic methods are implemented and discussed, in terms of solution quality and computing time.

Energy-awareness scheduling of unrelated parallel machine problems with multiple resource constraints

The environmental and economic pressures caused by energy consumption arouse energy-saving consciousness of the manufacturing industry. To this end, the paper integrates energy-awareness into the research of the unrelated parallel machines scheduling problem with multiple auxiliary resources which is typical in the photolithography process of wafer fabrication. With a comprehensive consideration of jobs requiring different processing demands, setup times, different ready times, resource constraints and energy consumption, a scheduling model with a multi-objective function of minimising the total weighted completion time and total energy consumption of the system is developed. On the basis of the model, a modified multi-objective artificial immune algorithm integrated with non-domination sorting strategy is put forward to crack the problem. Furthermore, in order to improve the performance of the proposed algorithm, clone operators, neighbourhood search operators, elite-preservation operators are applied to the algorithm. Finally, the experimental results and analysis validate that the presented algorithm is efficient and effective.

Matheuristics for a parallel machine scheduling problem with non-anticipatory family setup times: Application in the offshore oil and gas industry

In this paper, we address a variant of a batch scheduling problem with identical parallel machines and non-anticipatory family setup times to minimize the total weighted completion time. We developed an ILS and a GRASP matheuristics to solve the problem using a constructive heuristic and two MIP-based neighborhood searches, considering two batch scheduling mathematical formulations. The problem derives from a ship scheduling problem related to offshore oil & gas logistics, the Pipe Laying Support Vessel Scheduling Problem (PLSVSP). The developed methods overcome the current solution approaches in the PLSVSP literature, according to experiments carried out on a benchmark of 72 instances, with different sizes and characteristics, in terms of computational time and solution quality. New best solutions are provided for all medium and large-sized instances, achieving a reduction of more than 10% in the objective function of the best case.

Two-Agent Pareto-Scheduling of Minimizing Total Weighted Completion Time and Total Weighted Late Work

We investigate the Pareto-scheduling problem with two competing agents on a single machine to minimize the total weighted completion time of agent A’s jobs and the total weighted late work of agent B’s jobs, the B-jobs having a common due date. Since this problem is known to be NP-hard, we present two pseudo-polynomial-time exact algorithms to generate the Pareto frontier and an approximation algorithm to generate a (1+ϵ)-approximate Pareto frontier. In addition, some numerical tests are undertaken to evaluate the effectiveness of our algorithms.

Pareto‐optimization of three‐agent scheduling to minimize the total weighted completion time, weighted number of tardy jobs, and total weighted late work

AbstractWe consider three‐agent scheduling on a single machine in which the criteria of the three agents are to minimize the total weighted completion time, the weighted number of tardy jobs, and the total weighted late work, respectively. The problem is to find the set of all the Pareto‐optimal points, that is, the Pareto frontier, and their corresponding Pareto‐optimal schedules. Since the above problem is unary NP‐hard, we study the problem under the restriction that the jobs of the first agent have inversely agreeable processing times and weights, that is, the smaller the processing time of a job is, the greater its weight is. For this restricted problem, which is NP‐hard, we present a pseudo‐polynomial‐time algorithm to find the Pareto frontier. We also show that, for various special versions, the time complexity of solving the problem can be further reduced.

주기적 납기일과 공통 납기일을 갖는 2-에이전트 스케줄링

We consider a single-machine scheduling problem with two agents. The objective is to minimize the objective function of agent 1 while that of agent 2 is below a given threshold. The objective function of agent 1 is the total weighted completion time, while that of agent 2 is the total weighted number of tardy jobs. Furthermore, the due date of agent 2 is assigned not to the specific job but to the position. Firstly, we show that the problem with the identical intervals between the consecutive due dates remains strongly NP-hard, even if the weights of jobs of agent 2 are identical. Finally, we proved the weak NP-hardness of the problem with a common due date.

Single machine lot scheduling with optional job-rejection

We consider single machine lot scheduling problems. A number of customer orders of different sizes may be processed in the same lot. Splitting orders between consecutive lots is allowed. Three scheduling measures are considered: makespan, total completion time and total weighted completion time. In all cases, the goal is minimizing the relevant scheduling measure, subject to an upper bound on the total permitted rejection cost. All three problems studied here are NP-hard. We introduce efficient pseudo-polynomial dynamic programming algorithms for all cases. Our numerical tests indicate that the proposed algorithms can solve efficiently large-size problems.

Single machine lot scheduling with non-uniform lot capacities and processing times

In this paper, we investigate the single machine lot scheduling problem in which each lot contains one or more jobs (or part of jobs). Jobs with different sizes are splittable and should be processed in consecutive lots. Any completed (part of a) job is delivered to the customer on its completion. We generalize the problem studied in Zhang et al. (Inf Process Lett 142:46–51, 2019) such that lots may have different capacities and processing times, and the sizes of jobs are unbounded. We show that for this generalized case, the WSPT (Weighted Shortest Processing Time first) rule is still optimal for both minimizing total weighted completion time and minimizing total weighted discounted completion time.

A robust bi-objective mathematical model for disaster rescue units allocation and scheduling with learning effect

This paper proposes a novel bi-objective mixed-integer linear programming (MILP) model for allocating and scheduling disaster rescue units considering the learning effect. When a natural phenomenon (e.g., earthquake or flood) occurs, the presented decision support model is expected to help decision-makers of emergency relief centers to provide efficient planning for rescue units to minimize the total weighted completion time of rescue operations, as well as the total delay in rescue operations. The problem has some features in common with unrelated parallel machine scheduling (UPMS) problem and traveling salesman problem (TSP). To deal with the inherent uncertainty and bi-objective nature of the problem, an uncertainty-set based robust optimization technique and multi-choice goal programming (MCGP) with utility functions are applied. To demonstrate the applicability of the proposed model, a real case study in Mazandaran province in Iran is presented. The computational results confirm the high complexity of the problem and the significant impacts of the uncertainty on the solution. Moreover, the analytical results provide useful insights to decision-makers for disastrous situations.