Lot-sizing and scheduling problem with earliness tardiness and setup penalties

We consider the problem of determining the production lot sizes and their schedules so that the sum of setup cost, holding cost, and tardy cost is minimized. There are n orders waiting to be processed. Each order has its own due date, earliness penalty and tardiness penalty. The production is done in lots, which is the manufacturing of the same product continuously. No early delivery is allowed. Each order is delivered once, on the due date or immediately after the production of the order is completed. We present an algorithm, based on the use of the assignment problem, to optimally solve the problem with zero setup cost. For the problem with setup cost, the Backward Search algorithm is proposed to solve the small size problem. To deal with the large size problem, two heuristics which are Sequencing with Optimal Timing and Genetic Algorithm are presented.     

Solving production scheduling with earliness/tardiness penalties by constraint programming

This paper deals with an application of constraint programming in production scheduling with earliness and tardiness penalties that reflects the scheduling part of the Just-In-Time inventory strategy. Two scheduling problems are studied, an industrial case study problem of lacquer production scheduling, and also the job-shop scheduling problem with earliness/tardiness costs. The paper presents two algorithms that help the constraint programming solver to find solutions of these complex problems. The first algorithm, called the cost directed initialization, performs a greedy initialization of the search tree. The second one, called the time reversing transformation and designed for lacquer production scheduling, reformulates the problem to be more easily searchable when the default search or the cost directed initialization is used. The conducted experiments, using case study instances and randomly generated problem instances, show that our algorithms outperform generic approaches, and on average give better results than other nontrivial algorithms.     

Heuristics for the single machine scheduling problem with quadratic earliness and tardiness penalties

In this paper, we consider the single machine scheduling problem with quadratic earliness and tardiness costs, and no machine idle time. We propose several dispatching heuristics, and analyse their performance on a wide range of instances. The heuristics include simple and widely used scheduling rules, as well as adaptations of those rules to a quadratic objective function. We also propose heuristic procedures that specifically address both the earliness and the tardiness penalties, as well as the quadratic cost function. Several improvement procedures were also analysed. These procedures are applied as an improvement step, once the heuristics have generated a schedule.     

A branch-and-bound algorithm for single machine scheduling with quadratic earliness and tardiness penalties

This paper considers the problem of scheduling a single machine, in which the objective function is to minimize the weighted quadratic earliness and tardiness penalties and no machine idle time is allowed. We develop a branch and bound algorithm involving the implementation of lower and upper bounding procedures as well as some dominance rules. The lower bound is designed based on a lagrangian relaxation method and the upper bound includes two phases, one for constructing initial schedules and the other for improving them. Computational experiments on a set of randomly generated instances show that one of the proposed heuristics, used as an upper bound, has an average gap less than 1.3% for instances optimally solved. The results indicate that both the lower and upper bounds are very tight and the branch-and-bound algorithm is the first algorithm that is able to optimally solve problems with up to 30 jobs in a reasonable amount of time.     

A branch and bound approach for single machine scheduling with earliness and tardiness penalties

An n job, single machine scheduling problem in which each job has a distinct due date, d_i, is studied in this paper. The objective is to determine an optimal schedule π⁰_s for a set of jobs, S, such that the total absolute deviation of the schedule is minimized. This objective function is based on the due date value and on the earliness or tardiness of each job in the selected sequence. This paper presents a bounding scheme for the calculation of different lower bounds based on the overlap elimination procedure on a Just-In-Time schedule. Properties and theorems of the overlap elimination procedure are also provided. Finally, a numerical example is illustrated and some extensions of the approach are also discussed.     

Exact and heuristic procedures for single machine scheduling with quadratic earliness and tardiness penalties

The present paper studies the single machine, no-idle-time scheduling problem with a weighted quadratic earliness and tardiness objective. We investigate the relationship between this problem and the assignment problem, and we derive two lower bounds and several heuristic procedures based on this relationship. Furthermore, the applicability of the time-indexed integer programming formulation is investigated. The results of a computational experiment on a set of randomly generated instances show (1) the high-quality results of the proposed heuristics, (2) the low optimality gap of one of the proposed lower bounds and (3) the applicability of the integer programming formulation to small and medium size cases of the problem.     

Algorithms for job scheduling problems with distinct time windows and general earliness/tardiness penalties

This paper addresses the single machine scheduling problem with distinct time windows and sequence-dependent setup times. The objective is to minimize the total weighted earliness and tardiness. The problem involves determining the job execution sequence and the starting time for each job in the sequence. An implicit enumeration algorithm denoted IE and a general variable neighborhood search algorithm denoted GVNS are proposed to determine the job scheduling. IE is an exact algorithm, whereas GVNS is a heuristic algorithm. In order to define the starting times, an O(n²) idle time insertion algorithm (ITIA) is proposed. IE and GVNS use the ITIA algorithm to determine the starting time for each job. However, the IE algorithm is only valid for instances with sequence-independent setup times, and takes advantage of theoretical results generated for this problem. Computational experiments show that the ITIA algorithm is more efficient than the only other equivalent algorithm found in the literature. The IE algorithm allows the optimal solutions of all instances with up to 15 jobs to be determined within a feasible computational time. For larger instances, GVNS produces better-quality solutions requiring less computational time compared with the other algorithm from the literature.     

A genetic algorithm approach for the single machine scheduling problem with linear earliness and quadratic tardiness penalties

In this paper, we consider the single machine scheduling problem with linear earliness and quadratic tardiness costs, and no machine idle time. We propose a genetic approach based on a random key alphabet. Several genetic algorithms based on this approach are presented. These versions differ on the generation of the initial population, as well as on the use of local search. The proposed procedures are compared with existing heuristics, as well as with optimal solutions for the smaller instance sizes.     

A discrete colonial competitive algorithm for hybrid flowshop scheduling to minimize earliness and quadratic tardiness penalties

Recently introduced colonial competitive algorithm (CCA) has shown its excellent capability on different optimization problems. The aim of this paper is to propose a discrete version of this method to determine a schedule that minimizes sum of the linear earliness and quadratic tardiness in the hybrid flowshops scheduling problem with simultaneously considering effects of sequence-dependent setup times and limited waiting time. In other word we assume that the waiting time for each job between two consecutive stages cannot be greater than a given upper bound. Also for this problem, a mixed integer program is formulated. Computational results are presented to evaluate the performance of the proposed algorithms for problems with different structures.     

Parallel machine earliness and tardiness scheduling with proportional weights

In this paper we study the problem of scheduling n jobs with a common due date and proportional early and tardy penalties on m identical parallel machines. We show that the problem is NP-hard and propose a dynamic programming algorithm to solve it. We also propose two heuristics to tackle the problem and analyze their worst-case error bounds.Scope and purposeScheduling problems to minimize the total weighted earliness and tardiness (WET) arise in Just-in-time manufacturing systems, where one of the objectives is to complete each job as close to its due date as possible. The earliness and tardiness weights of a job in WET tend to increase with the value of the job. Because processing time is often a good surrogate for the value of a job, it is reasonable to consider weights that are proportional to job processing times. In this paper we study the parallel identical machine WET problem with proportional weights. We propose both exact and approximation algorithms to tackle the problem.     

Preemption in single machine earliness/tardiness scheduling

We consider a single machine earliness/tardiness scheduling problem with general weights, ready times and due dates. Our solution approach is based on a time-indexed preemptive relaxation of the problem. For the objective function of this relaxation, we characterize cost coefficients that are the best among those with a piecewise linear structure with two segments. From the solution to the relaxation with these best objective function coefficients, we generate feasible solutions for the original non-preemptive problem. We report extensive computational results demonstrating the speed and effectiveness of this approach.     

Crossdock truck scheduling with time windows: earliness,tardiness and storage policies

This article proposes to simultaneously plan inbound and outbound truck arrivals and departures in a cross-docking platform, as well as the internal pallet handling. The objective is to minimize both the total number of pallets put in storage and the dissatisfaction of the transportation providers, by creating a truck schedule as close as possible to the wished schedule they communicate in advance. The problem is modeled with an integer program tested on generated instances to assess its performance, especially regarding the computation time. The problem is proven to be np-hard in the strong sense. Since the execution takes too long to be used on a daily basis by platform managers, three heuristics are also proposed and tested. Two are based on integer programs solved sequentially, the third one is a tabu search in which the storage part of the objective function is evaluated by a maximum flow model in a graph. Numerical experiments show in which conditions each heuristic performs best, which can help choosing a solution method when confronted to a real-life problem.     

Minimization of earliness, tardiness and due date penalties on uniform parallel machines with identical jobs

We consider a problem of scheduling n identical nonpreemptive jobs with a common due date on m uniform parallel machines. The objective is to determine an optimal value of the due date and an optimal allocation of jobs onto machines so as to minimize a total cost function, which is the function of earliness, tardiness and due date values. For the problem under study, we establish a set of properties of an optimal solution and suggest a two-phase algorithm to tackle the problem. First, we limit the number of due dates one needs to consider in pursuit of optimality. Next, we provide a polynomial-time algorithm to build an optimal schedule for a fixed due date. The key result is an O(m² log m) algorithm that solves the main problem to optimality.Scope and purpose: To extend the existing research on cost minimization with earliness, tardiness and due date penalties to the case of uniform parallel machines.     

Single machine scheduling with controllable processing times to minimize total tardiness and earliness

In most deterministic scheduling problems job processing times are considered as invariable and known in advance. Single machine scheduling problem with controllable processing times with no inserted idle time is presented in this study. Job processing times are controllable to some extent that they can be reduced or increased, up to a certain limit, at a cost proportional to the reduction or increase. In this study, our objective is determining a set of compression/expansion of processing times in addition to a sequence of jobs simultaneously, so that total tardiness and earliness are minimized. A mathematical model is proposed firstly and afterward a net benefit compression–net benefit expansion (NBC–NBE) heuristic is presented so as to acquire a set of amounts of compression and expansion of jobs processing times in a given sequence. Three heuristic techniques in small problems and in medium-to-large instances two meta-heuristic approaches, as effective local search methods, as well as these heuristics are employed to solve test examples. The single machine total tardiness problem (SMTTP) is already NP-hard, so the considered problem is NP-hard obviously. The computational experiments demonstrate that our proposed heuristic is efficient approach for such just-in-time (JIT) problem, especially equipped with competent heuristics.     

Multiagent scheduling method with earliness and tardiness objectives in flexible job shops.

Flexible job-shop scheduling problems are an important extension of the classical job-shop scheduling problems and present additional complexity. Such problems are mainly due to the existence of a considerable amount of overlapping capacities with modern machines. Classical scheduling methods are generally incapable of addressing such capacity overlapping. We propose a multiagent scheduling method with job earliness and tardiness objectives in a flexible job-shop environment. The earliness and tardiness objectives are consistent with the just-in-time production philosophy which has attracted significant attention in both industry and academic community. A new job-routing and sequencing mechanism is proposed. In this mechanism, two kinds of jobs are defined to distinguish jobs with one operation left from jobs with more than one operation left. Different criteria are proposed to route these two kinds of jobs. Job sequencing enables to hold a job that may be completed too early. Two heuristic algorithms for job sequencing are developed to deal with these two kinds of jobs. The computational experiments show that the proposed multiagent scheduling method significantly outperforms the existing scheduling methods in the literature. In addition, the proposed method is quite fast. In fact, the simulation time to find a complete schedule with over 2000 jobs on ten machines is less than 1.5 min.     

飞机地面作业提前/拖期调度研究

对具有不同时间窗的提前/拖期调度方法应用于飞机地面作业调度进行了研究,用模糊规划建立数学模型,用三角模糊数表示了地面作业中时间的不确定性,应用粒子群优化算法对模型进行了仿真。实验结果表明,在不同情况下该算法都能取得较为满意的结果。     

Single-machine batch delivery scheduling with job release dates,due windows and earliness,tardiness, holding and delivery costs

This paper deals with a single-machine scheduling problem in which jobs are released in different points in time but delivered to customers in batches. A due window is associated with each job. The objective is to schedule the jobs, to form them into batches and to decide the delivery date of each batch so as to minimize the sum of earliness, tardiness, holding, and delivery costs. A mathematical model of the problem is presented, and a set of dominance properties is established. To solve this NP-hard problem efficiently, a solution method is then proposed by incorporating the dominance properties with an imperialist competitive algorithm. Unforced idleness and forming discontinuous batches are allowed in the proposed algorithm. Moreover, the delivery date of a batch may be decided to be later than the completion time of the last job in the batch. Finally, computational experiments are conducted to evaluate the proposed model and solution procedure, and results are discussed.     

Ant colony systems for the single-machine total weighted earliness tardiness scheduling problem

Single-machine weighted earliness tardiness scheduling is a prevalent problem in just-in-time production environments. Yet, the case with distinct due dates is strongly NP-hard. Herein, it is approximately solved using ASV, an ant colony-based system with a reduced number of ants and of colonies and with daemon actions that explore the search space around the ants using a variable neighborhood search (VNS). The numerical investigation provides computational proof of the utility of the daemon actions. In addition, it infers that these latter can be applied either to the initial or to subsequent colonies. Furthermore, it highlights the importance of using ant colony optimization as the multiple restart engine of VNS. Finally, it shows that ASV obtains the optimum for most small-sized instances. It has a 0.2 % average deviation from the optimum over all benchmark instances.     

Due date assignments and scheduling a single machine with a general earliness/tardiness cost function

We study three different due date assignment problems in scheduling a single machine which differ from each other based upon the objective function and due date assignment method being used. Two different objective functions are considered. The first is a cost function that includes earliness, tardiness and due date assignment penalties and the second is a function that includes penalties due to the number of tardy jobs and due date assignments. We assume that the earliness, tardiness and due date assignment penalties are continuous and non-decreasing functions of the corresponding duration. The goal is to minimize each objective function for two different due date assignment methods. The first is a method in which the assigned due dates are restricted to be equal while the second is a method that allows us to assign different due dates to different jobs.