Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries

We study a supply chain scheduling problem, where a common due date is assigned to all jobs and the number of jobs in delivery batches is constrained by the batch size. Our goal is to minimize the sum of the weighted number of tardy jobs, the due-date-assignment costs and the batch-delivery costs. We show that some well-known NPmathcal{NP}-hard problems reduce to our problem. Then we propose a pseudo-polynomial algorithm for the problem, establishing that it is NPmathcal{NP}-hard only in the ordinary sense. Finally, we convert the algorithm into an efficient fully polynomial time approximation scheme.     

Minimizing maximum tardiness and number of tardy jobs on parallel machines subject to minimum flow-time

This paper considers the problems of scheduling jobs on parallel identical machines where an optimal schedule is defined as one that gives the smallest maximum tardiness (or the minimum number of tardy jobs) among the set of schedules with optimal total flow-time (the sum of the completion times of all jobs). We show that these problems are unary NP-Hard, develop lower bounds for these two secondary criteria problems, and describe heuristic algorithms for their solution. Results of a computational study show that the proposed heuristic algorithms are quite effective and efficient in solving these hierarchical criteria scheduling problems.     

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries for multiple customers in supply chains

In this paper, an integrated due date assignment and production and batch delivery scheduling problem for make-to-order production system and multiple customers is addressed. Consider a supply chain scheduling problem in which n orders (jobs) have to be scheduled on a single machine and delivered to K customers or to other machines for further processing in batches. A common due date is assigned to all the jobs of each customer and the number of jobs in delivery batches is constrained by the batch size. The objective is to minimize the sum of the total weighted number of tardy jobs, the total due date assignment costs and the total batch delivery costs. The problem is NP-hard. We formulate the problem as an Integer Programming (IP) model. Also, in this paper, a Heuristic Algorithm (HA) and a Branch and Bound (B&B) method for solving this problem are presented. Computational tests are used to demonstrate the efficiency of the developed methods.     

Minimizing the weighted number of tardy jobs on parallel processors

We present a branch-and-bound algorithm to minimize the weighted number of tardy jobs on either identical or non-identical processors. Bounds come from a surrogate relaxation resulting in a multiple-choice knapsack. Extensive computational experiments indicate problems with 400 jobs and several machines can be solved quickly. The results also indicate what parameters affect solution difficulty for this algorithmic approach.     

Bi-criteria scheduling problems: Number of tardy jobs and maximum weighted tardiness

Consider a single machine and a set of n jobs that are available for processing at time 0. Job j has a processing time p_j, a due date d_j and a weight w_j. We consider bi-criteria scheduling problems involving the maximum weighted tardiness and the number of tardy jobs. We give NP-hardness proofs for the scheduling problems when either one of the two criteria is the primary criterion and the other one is the secondary criterion. These results answer two open questions posed by Lee and Vairaktarakis in 1993. We consider complexity relationships between the various problems, give polynomial-time algorithms for some special cases, and propose fast heuristics for the general case. The effectiveness of the heuristics is measured by empirical study. Our results show that one heuristic performs extremely well compared to optimal solutions.     

Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem

We study a static stochastic single machine scheduling problem in which jobs have random processing times with arbitrary distributions, due dates are known with certainty, and fixed individual penalties (or weights) are imposed on both early and tardy jobs. The objective is to find an optimal sequence that minimizes the expected total weighted number of early and tardy jobs. The general problem is NP-hard to solve; however, in this paper, we develop certain conditions under which the problem is solvable exactly. An efficient heuristic is also introduced to find a candidate for the optimal sequence of the general problem. Our illustrative examples and computational results demonstrate that the heuristic performs well in identifying either optimal sequences or good candidates with low errors. Furthermore, we show that special cases of the problem studied here reduce to some classical stochastic single machine scheduling problems including the problem of minimizing the expected weighted number of early jobs and the problem of minimizing the expected weighted number of tardy jobs which are both solvable by the proposed exact or heuristic methods.     

Minimizing the expected weighted number of tardy jobs in stochastic flow shops

This paper is devoted to two types of stochastic scheduling problems, one involving a single machine and the other involving a flow shop consisting of an arbitrary number of machines. In both problem types, all jobs to be processed have due dates, and the objective is to find a job sequence that minimizes the expected weighted number of tardy jobs. For the single-machine case, sufficient optimality conditions for job sequences are derived for various choices of due date and processing time distributions. For the case of a flow shop with an arbitrary number of machines and identically distributed due dates for all jobs, we prove the following intuitively appealing results: (i) when all jobs have the same processing time distributions, the expected weighted number of tardy jobs is minimized by sequencing the jobs in decreasing order of the weights, (ii) when all weights are equal, the jobs should be sequenced according to an increasing stochastic ordering of the processing time distributions.     

Minimizing the weighted number of tardy jobs on a single machine with release dates

In this paper, we describe an exact algorithm to minimize the weighted number of tardy jobs on a single machine with release dates. The algorithm uses branch-and-bound; a surrogate relaxation resulting in a multiple-choice knapsack provides the bounds. Extensive computational experiments indicate the proposed exact algorithm solves either weighted or unweighted problems. It solves the hardest problems to date. Indeed, it solves all previously unsolved instances. Its run time is the shortest to date.     

A faster algorithm for a due date assignment problem with tardy jobs

The single-machine due date assignment problem with the weighted number of tardy jobs objective, (the TWNTD problem), and its generalization with resource allocation decisions and controllable job processing times have been solved in O(n⁴) time by formulating and solving a series of assignment problems. In this note, a faster O(n²) dynamic programming algorithm is proposed for the TWNTD problem and for its controllable processing times generalization in the case of a convex resource consumption function.     

Minimizing earliness and tardiness penalties in a single-machine problem with a common due date

Scheduling problems involving both earliness and tardiness costs have received significant attention in recent years. This type of problem became important with the advent of the just-in-time (JIT) concept, where early or tardy deliveries are highly discouraged. In this paper we examine the single-machine scheduling problem with a common due date. Performance is measured by the minimization of the sum of earliness and tardiness penalties of the jobs. Since this problem is NP-hard, we propose a tabu search-based heuristic and a genetic algorithm which exploit specific properties of the optimal solution. Hybrid strategies are also analyzed to improve the performance of these methods. The proposed approaches are examined through a computational comparative study with 280 benchmark problems with up to 1000 jobs.     

Minimizing weighted earliness–tardiness and due-date cost with unit processing-time jobs

The classical single-machine scheduling and due-date assignment problem of Panwalker et al. [Panwalker, S.S., Smith, M.L., Seidmann, A., 1982. Common due date assignment to minimize total penalty for the one machine scheduling problem. Operations Research 30(2) (1982) 391–399] is the following: All n jobs share a common due-date, which is to be determined. Jobs completed prior to or after the due-date are penalized according to a cost function which is linear and job-independent. The objective is to minimize the total earliness–tardiness and due-date cost. We study a generalized version of this problem in which: (i) the earliness and tardiness costs are allowed to be job dependent and asymmetric and (ii) jobs are processed on parallel identical machines. We focus on the case of unit processing-time jobs. The problem is shown to be solved in polynomial (O(n⁴)) time. Then we study the special case with no due-date cost (a classical problem known in the literature as TWET). We introduce an O(n³) solution for this case. Finally, we study the minmax version of the problem, (i.e., the objective is to minimize the largest cost incurred by any of the jobs), which is shown to be solved in polynomial time as well.     

Minimizing number of tardy jobs on a batch processing machine with incompatible job families

In this paper we consider the problem of minimizing number of tardy jobs on a single batch processing machine. The batch processing machine is capable of processing up to B jobs simultaneously as a batch. We are given a set of n jobs which can be partitioned into m incompatible families such that the processing times of all jobs belonging to the same family are equal and jobs of different families cannot be processed together. We show that this problem is NP-hard and present a dynamic programming algorithm which has polynomial time complexity when the number of job families and the batch machine capacity are fixed. We also show that when the jobs of a family have a common due date the problem can be solved by a pseudo-polynomial time procedure.     

Minimizing the weighted number of late jobs in UET open shops

We consider a two-machine open shop problem where the jobs have release dates and due dates, and where all single operations have unit processing times. The goal is to minimize the weighted number of late jobs. We derive a polynomial time algorithm for this problem, thereby answering an open question posed in a recent paper by Brucker et al.This research was supported by the Christian Doppler Laboratorium für Diskrete Optimierung.     

A new branch and bound algorithm for minimizing the weighted number of tardy jobs

In this paper, the problem of sequencing jobs on a single machine to minimize the weighted number of tardy jobs is considered. Some new dominances between jobs are proposed and studied. A new branch and bound algorithm that can solve large problems, e.g. 85 jobs, is presented.     

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.     

带机器故障的两台机带权误工数排序问题

讨论机器带故障中断的两台平行机排序问题,工件加工时间均为单位时间,目标是极小化带权误工工件数.当转移时间t=0时给出了最优的算法.当t≠0时,给出了一个多项式时间的近似算法,并证明算法解与最优解至多相差一个带权误工数.     

Scheduling linear deteriorating jobs to minimize the number of tardy jobs

In this paper a problem of scheduling a single machine under linear deterioration which aims at minimizing the number of tardy jobs is considered. According to our assumption, processing time of each job is dependent on its starting time based on a linear function where all the jobs have the same deterioration rate. It is proved that the problem is NP-hard; hence a branch and bound procedure and a heuristic algorithm with O(n ²) is proposed where the heuristic one is utilized for obtaining the upper bound of the B&B procedure. Computational results for 1,800 sample problems demonstrate that the B&B method can solve problems with 28 jobs quickly and in some other groups larger problems are also solved. Generally, B&B method can optimally solve 85% of the samples which shows high performance of the proposed method. Also it is shown that the average value of the ratio of optimal solution to the heuristic algorithm result with the objective ??(1 ? Ui) is at most 1.11 which is more efficient in comparison to other proposed algorithms in related studies in the literature.     

Minimizing the number of machines with limited workload capacity for scheduling jobs with interval constraints

In this paper, we consider a parallel machine scheduling problem in which machines have a limited workload capacity and jobs have deadlines and release dates. The problem is motivated by the operation of energy storage management systems for microgrids under emergency conditions and generalizes some problems that have already been studied in the literature for their theoretical value. In this work, we propose heuristic and exact algorithms to solve the problem. The heuristics are adaptations of classical bin packing heuristics in which additional conditions on the feasibility of a solution are imposed, whereas the exact method is a branch-and-price approach. The results show that the branch-and-price approach is able to optimally solve random instances with up to 250 jobs within a time limit of one hour, while the heuristic procedures provide near optimal solution within reduced running times. Finally, we also provide additional complexity results for a special case of the problem.     

Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine

We study a flow-shop problem, where each of the jobs is limited to no more than two operations. One of these operations is common for all the jobs, and is performed on the same (”critical”) machine. Reflecting many applications, jobs are assumed to be processed in blocks on the critical machine. All the jobs share a common due-date, and the objective is minimum weighted number of tardy jobs. We prove that the problem is NP-hard. Then we formulate the problem as an integer program, and introduce a pseudo-polynomial dynamic programming algorithm, proving that the problem is NP-hard in the ordinary sense. We also propose an efficient heuristic, which is shown numerically to produce very close-to-optimal schedules. Finally, we show that the special case of identical weights is polynomially solvable.     

Single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs

Motivated by just-in-time manufacturing, we consider a single machine scheduling problem with dual criteria, i.e., the minimization of the total weighted earliness subject to minimum number of tardy jobs. We discuss several dominance properties of optimal solutions. We then develop a heuristic algorithm with time complexity O(n³) and a branch and bound algorithm to solve the problem. The computational experiments show that the heuristic algorithm is effective in terms of solution quality in many instances while the branch and bound algorithm is efficient for medium-size problems.