Single-Machine Scheduling Problems with the General Sum-of-Processing-Time and Position-Dependent Effect Function

The paper deals with some single-machine scheduling problems with setup time considerations where the processing time of a job is given as a function of its starting times and position in a 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 (\( \delta \geqslant 0 \)) power of job completion times, the total weighted completion time, the maximum lateness and the number of tardy jobs. We show that the makespan minimization problem, the total completion time minimization problem, and the sum of the δth power of job completion times minimization problem can be solved in polynomial time, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.

Single-machine scheduling problems with deteriorating jobs and learning effects

We consider the problem of rescheduling a facility modeled as a single machine in the face of newly arrived jobs with part-type dependent setup times. The facility contains a number of jobs that have been assigned due dates and scheduled so as to meet them. We wish to insert the new jobs into the existing schedule in a manner that will minimize the disruption of the jobs in the system and minimize the total weighted completion time or the maximum completion time of the new jobs. We provide a polynomial-time algorithm for the maximum completion time problem, prove that the total weighted completion time problem is NP-hard in the strong sense and study several of its special cases. In particular, we show that the case with reverse-agreeable weights (of which the unit weight problem is a special case) can be solved in polynomial time when the number of part types is fixed. We also present two heuristics for the problem with arbitrary weights and develop data-dependent worst-case error bounds. Extensive computational experiments show that the heuristics consistently obtain near-optimal solutions in very reasonable CPU times.

We study a problem of scheduling customized jobs on a single-machine. Each job requires two operations: one standard and one specific. Standard operations are processed in batches under item availability, and each batch requires a set-up time. Based on structural properties of the optimal solution, we introduce a generic dynamic programming scheme that builds an optimal schedule by alternately inserting blocks of operations of two distinct types. Our approach yields efficient algorithms for the sum of completion times problem with agreeable processing times and the maximum lateness problem. The number of late jobs problem is shown to be NP-hard in the ordinary sense, but is pseudo-polynomially solvable. A polynomial algorithm is also given for a special case of this problem. Our results indicate the differences between this problem and its counterpart under batch availability.

Scheduling with learning effects has received growing attention nowadays. A well-known learning model is called ‘position-based learning’ in which the actual processing time of a job is a non-increasing function of its position to be processed. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases. Motivated by this observation, we propose two truncated learning models in single-machine scheduling problems and two-machine flowshop scheduling problems with ordered job processing times, respectively, where the actual processing time of a job is a function of its position and a control parameter. Under the proposed learning models, we show that some scheduling problems can be solved in polynomial time. In addition, we further analyse the worst-case error bounds for the problems to minimize the total weighted completion time, discounted total weighted completion time and maximum lateness.

Worst-case and numerical analysis of heuristic algorithms for flowshop scheduling problems with a time-dependent learning effect

Recently, learning scheduling problems have received increasing attention. However, the majority of the research assume that the actual job processing time is a function of its position. This paper deals with the single-machine scheduling problem with a sum-of-processing-time-based learning effect. By the effect of sum-of-processing-time-based learning, we mean that the processing time of a job is defined by total normal processing time of jobs in front of it in the sequence. We show that the single-machine makespan problem remains polynomially solvable under the proposed model. We show that the total completion time minimization problem for a≥1 remains polynomially solvable under the proposed model. For the case of 0<a<1, we show that an optimal schedule of the total completion time minimization problem is V-shaped with respect to normal job processing times.

In this paper, we develop branch-and-bound algorithms for several hard, two-agent scheduling problems, i.e., problems in which each agent has an objective function which depends on the completion times of its jobs only. Our bounding approach is based on the fact that, for all problems considered, the Lagrangian dual gives a good bound and can be solved exactly in strongly polynomial time. The problems addressed here consist in minimizing the total weighted completion time of the jobs of agent A, subject to a bound on the cost function of agent B, which may be: (i) total weighted completion time, (ii) maximum lateness, (iii) maximum completion time. An extensive computational experience shows the effectiveness of the approach.

Single-machine scheduling problems with both deteriorating jobs and learning effects

This paper deals with a single machine scheduling problem with general past-sequence-dependent (psd) setup time and log-linear learning in which the setup times and learning effects are job-dependent. The setup times are unique functions of the length of already processed jobs, and the learning effects show that processing times are unique decreasing functions of job positions. The goal is to find the optimal sequences that minimise objective functions such as the makespan, total completion time, total lateness, total waiting cost, total waiting time, total absolute differences in completion times, and the sum of earliness, tardiness and common due date penalty. Special cases of the resulting problems are solvable in polynomial time; however, the general problems are difficult to solve. We propose branch-and-bound (B&B) methods to derive the optimal sequences for such problems. Computational results show that the proposed methods can solve relatively large problem instances in reasonable amounts of time.

In this paper we deal with the single machine scheduling problem with one non-availability interval to minimize the maximum lateness where jobs have positive tails. Two cases are considered. In the first one, the non-availability interval is due to the machine maintenance. In the second case, the non-availability interval is related to the operator who is organizing the execution of jobs on the machine. The contribution of this paper consists in an improved fully polynomial time approximation scheme (FPTAS) for the maintenance non-availability interval case and the elaboration of the first FPTAS for the operator non-availability interval case. The two FPTASs are strongly polynomial.

Pareto optimization for the two-agent scheduling problems with linear non-increasing deterioration based on Internet of Things

Single-machine scheduling under the job rejection constraint

We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith’s rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k ( k − 1)/( k + 1) , denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time.

Single-machine scheduling problems with past-sequence-dependent delivery times