Total Weighted Completion Time Minimization Research Articles

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.

Research on Delivery Times Scheduling with Sum of Logarithm Processing Times-Based Learning Effect

In this paper, we address single-machine scheduling with sum of logarithm processing times-based learning effect. Under exponential past-sequence-dependent delivery times, we proved that the makespan and total completion time minimizations can be solved by the smallest processing time (SPT) first rule. Under the agreeable weight condition and agreeable due date condition, we show that the total weighted completion time minimization and maximum tardiness minimization can be solved in polynomial time, respectively.

Minimizing Total Weighted Completion Times for Semi-online Single Machine Scheduling

This paper addresses the semi-online scheduling problem for total weighted completion time minimization 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, called MCSWPT (Modified Critical Shortest Weighted Processing Time), that makes use of the available pieces of information in order to produce better schedules compared to its online peers is presented. A numerical analysis is presented, showing the impact of having the considered partial information.

Ant colony optimization algorithm for total weighted completion time minimization on non-identical batch machines

This paper aims at the problem of scheduling a set of jobs with arbitrary sizes on parallel batch processing machines with arbitrary capacities. The optimization objective is to minimize the total weighted completion time of jobs. After describing the studied problem, we analyze its complexity and present a lower bound of the problem. A heuristic is provided to solve the problem firstly. Then, with the proposed first job selection strategy based on the weights of jobs, two algorithms based on the ant system and the max-min ant system, respectively, are designed to address the problem. Through extensive experiments, the performance of the proposed algorithms is compared with several state-of-the-art algorithms. The comparative results verify effectiveness and efficiency of the proposed algorithms.

A Heuristic’s Job Order Gain in Pyramidal Preemptive Job Scheduling Problems for Total Weighted Completion Time Minimization

A possibility of speeding up the job scheduling by a heuristic based on the shortest processing period approach is studied in the paper. The scheduling problem is such that the job volume and job priority weight are increasing as the job release date increases. Job preemptions are allowed. Within this model, the input for the heuristic is formed by either ascending or descending job order. Therefore, an estimator of relative difference in duration of finding an approximate schedule by these job orders is designed. It is ascertained that the job order results in different time of computations when scheduling at least a few hundred jobs. The ascending-order solving becomes on average by 1 % to 2.5 % faster when job volumes increase steeply. As the steepness of job volumes decreases, this gain vanishes and, eventually, the descending-order solving becomes on average faster by up to 4 %. The gain trends of both job orders slowly increase as the number of jobs increases.

Mathematical formulations for scheduling jobs on identical parallel machines with family setup times and total weighted completion time minimization

This paper addresses the parallel machine scheduling problem with family dependent setup times and total weighted completion time minimization. In this problem, when two jobs j and k are scheduled consecutively on the same machine, a setup time is performed between the finishing time of j and the starting time of k if and only if j and k belong to different families. The problem is strongly NP-hard and is commonly addressed in the literature by heuristic approaches and by branch-and-bound algorithms. Achieving proven optimal solution is a challenging task even for small size instances. Our contribution is to introduce five novel mixed integer linear programs based on concepts derived from one-commodity, arc-flow and set covering formulations. Numerical experiments on more than 13000 benchmark instances show that one of the arc-flow models and the set covering model are quite efficient, as they provide on average better solutions than state-of-the-art approaches, with shorter computation times, and solve to proven optimality a large number of open instances from the literature.

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The scheduling theory studying, in particular, minimization of the total weighted completion time, which refers to planning, organizing, and executing complex or multistep processes of assembling, manufacturing, building, dispatching, computing, etc., possesses both exact and heuristic approaches to the schedule computation. The computation time of the exact schedule approach grows immensely when the number of jobs is increased off just a few jobs (roughly, off 6 to 9, depending also on how jobs are divided into job parts). Therefore, a lot of heuristics are used to find the approximate schedule but to obtain it much faster. The heuristics’ approximate schedule is not always executed in the exactly minimal total weighted completion time, but the loss is commonly not so great. Moreover, when the number of jobs is of order of hundreds, the scheduling problems become intractable by any exact schedule approaches, and so the heuristics remain the single way to find a schedule. Considering the preemptive scheduling problem by subsequent length-equal job importance growth, there are two ways to input the job release dates and the respective priority weights. On the one hand, the release dates can be given in ascending order; then the respective priority weights will be a set of, generally speaking, non-decreasing values. On the other hand, the release dates can be given in descending order; and then the respective priority weights will be a set of, generally speaking, non-increasing values. Having estimated the averaged time of obtaining the approximate schedule by both ascending and descending orders of inputting the job release dates, the heuristic’s job order is revealed to be very significant. Its significance grows as the number of jobs increases. The influence of the heuristic’s job order also grows as the number of job parts increases. The descending job order has the growing advantage for scheduling about 300 jobs and more. In particular, the descending job order’s advantage in scheduling 100000 jobs divided in two parts each is almost 42 %. So, for total weighted completion time minimization in the preemptive scheduling problem by subsequent length-equal job importance growth, the job release dates are to be input in the descending order. However, the heuristic’s job order gain in scheduling a lesser number of jobs (a few tens and up to 100, 200, 300) remains uncertain due to considerable fluctuations of the much shorter computation time.

Complexity of Scheduling with Proportional Deterioration and Release Dates

In this paper, we analyze the computational complexity of single-machine scheduling problems with deteriorating jobs having non-zero release dates. The processing time of each job is a proportional increasing function of its starting time. The objectives are to minimize the total weighted completion time, the number of tardy jobs and the total completion time, respectively. We first show that the total weighted completion time minimization problem and the number of tardy jobs minimization problem are binary NP-hard. Then, we present some optimal properties for the two problems under the restriction that the jobs have identical deteriorating rates. Finally, we show that the total completion time minimization problem with deadline is binary NP-hard, and in the case when the jobs have identical deadlines, agreeable release dates and deterioration rates, it can be solved in \(O(n\log n)\) time.

Strongly Fully Polynomial Time Approximation Scheme for the weighted completion time minimization problem on two-parallel capacitated machines

We consider the total weighted completion time minimization for the two-parallel capacitated machines scheduling problem. In this problem, one of the machines can process jobs until a certain time T1 after which it is no longer available. The other machine is continuously available for performing jobs at any time. We prove the existence of a strongly Fully Polynomial Time Approximation Scheme (FPTAS) for the studied problem, which extends the results for the unweighted version (see [I. Kacem, Y. Lanuel and M. Sahnoune, Strongly Fully Polynomial Time Approximation Scheme for the two-parallel capacitated machines scheduling problem, Int. J. Plann. Sched. 1 (2011) 32–41]). Our FPTAS is based on the simplification of a dynamic programming algorithm. Moreover, we present a set of numerical experiments and we discuss the results.

Strongly Fully Polynomial Time Approximation Scheme for the Weighted Completion Time Minimization Problem on Two-Parallel Capacitated Machines

We consider the total weighted completion time minimization for the two-parallel capacitated machines scheduling problem. In this problem, one of the machines can process jobs until a certain time T1 after which it is no longer available. The other machine is continuously available for performing jobs at any time. We prove the existence of a strongly Fully Polynomial Time Approximation Scheme (FPTAS) for the studied problem, which extends the results for the unweighted version (see Kacem, Lanuel and Sahnoune (2011)). Our FPTAS is based on the simplification of a dynamic programming algorithm.

Scheduling Jobs with Truncated Exponential Sum-of-Logarithm-Processing-Times Based and Position-based Learning Effects

In this study, we consider a scheduling problem with truncated exponential sum-of-logarithm-processing-times based and position-based learning effects on a single machine. We prove that the shortest processing time (SPT) rule is optimal for the makespan minimization problem, the sum of the θth power of job completion times minimization problem, and the total lateness minimization problem, respectively. For the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, we present heuristic algorithms (the worst-case bound of these heuristic algorithms are also given) according to the corresponding single machine scheduling problems without learning considerations. It also shows that the problems of minimizing the total tardiness, the total weighted completion time and the discounted total weighted completion time are polynomially solvable under some agreeable conditions on the problem parameters.

Parallel-Machine Scheduling with Time-Dependent and Machine Availability Constraints

We consider the parallel-machine scheduling problem in which the machines have availability constraints and the processing time of each job is simple linear increasing function of its starting times. For the makespan minimization problem, which is NP-hard in the strong sense, we discuss the Longest Deteriorating Rate algorithm and List Scheduling algorithm; we also provide a lower bound of any optimal schedule. For the total completion time minimization problem, we analyze the strong NP-hardness, and we present a dynamic programming algorithm and a fully polynomial time approximation scheme for the two-machine problem. Furthermore, we extended the dynamic programming algorithm to the total weighted completion time minimization problem.

Minimizing the total weighted completion time of fully parallel jobs with integer parallel units

We consider the total weighted completion time minimization in the following scheduling problem. There are m identical resources available at each time unit, and n jobs. Each job requires a number si of resources and one resource can only be assigned to one job at each time unit. Each job is also called fully parallel such that the job is satisfied once it receives enough resources no matter how the resources distribute. The objective is to find a schedule that minimizes ∑wiCi, where wi is the weight of job Ji and Ci is the time when job Ji receives si resources. We show that the total weighted completion time minimization is NP-hard when m is an input of the problem. We then give a simple greedy algorithm with an approximation ratio 2. Finally, we present a polynomial time algorithm with complexity O(nd+1) to solve this problem when the number of different resource requirements that are not multiples of m is at most d.

Solution algorithms for the total weighted completion time minimization flow shop scheduling with decreasing linear deterioration

In this paper, we consider a two-machine flow shop scheduling problem with decreasing linear deterioration. By decreasing linear deterioration, we mean that the processing time is a decreasing function of its execution start time. The objective is to find a sequence that minimizes the total weighted completion time. Several dominance properties and some lower bounds are derived to speed up the elimination process of a branch-and-bound algorithm. Two heuristic algorithms are also proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational results for randomly generated problem instances are presented.

Single machine past-sequence-dependent delivery times scheduling with general position-dependent and time-dependent learning effects

This paper studies the single machine past-sequence-dependent (p-s-d) delivery times scheduling with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ⩾0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of minimization of the makespan, the total completion time, the sum of the θth (θ⩾0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, the maximum tardiness minimization problem and the total tardiness minimization problem can be solved in polynomial time under certain conditions.

Scheduling problems with past-sequence-dependent setup times and general effects of deterioration and learning

In this paper we consider the single-machine setup times scheduling with general effects of deterioration and learning. By the general effects of deterioration and learning, we mean that the actual job processing time is a general function of the processing times of the jobs already processed and its scheduled position. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We show that the problems to minimize the makespan, the sum of the δth (δ>0) power of job completion times, the total lateness are polynomially solvable. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness (tardiness) minimization problem, the total tardiness minimization problem can be solved in polynomial time under certain conditions.

Single-machine scheduling problems with an actual time-dependent deterioration

In this paper, we consider the single machine scheduling problems with an actual time-dependent deterioration effect. By the actual time-dependent deterioration effect, we mean that the processing time of a job is defined by increasing function of total actual processing time of jobs in front of it in the sequence. We show that even with the introduction of an actual time-dependent deterioration to job processing times, makespan minimization problem, total completion time minimization problem, the total lateness, and the sum of the quadratic job completion times minimization problem remain polynomially solvable, respectively. We also show that the total weighted completion time minimization problem, the discounted total weighted completion time minimization problem, the maximum lateness minimization problem, and the total tardiness minimization problem can be solved in polynomial time under certain conditions.

Single machine scheduling with general time-dependent deterioration, position-dependent learning and past-sequence-dependent setup times

The paper deals with single machine scheduling problems with setup time considerations where the actual processing time of a job is not only a non-decreasing function of the total normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We consider the following objective functions: the makespan, the total completion time, the sum of the δth (δ ≥ 0) power of job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the δ th (δ ≥ 0) power of job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.

Several single-machine scheduling problems with general learning effects

In this paper we consider several single-machine scheduling problems with general learning effects. By general learning effects, we mean that the processing time of a job depends not only on its scheduled position, but also on the total normal processing time of the jobs already processed. We show that the scheduling problems of minimization of the makespan, the total completion time and the sum of the θth (θ⩾0) power of job completion times can be solved in polynomial time under the proposed models. We also prove that some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time.

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ⩾ 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.