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.     

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.     

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.     

Dispatching heuristics for the single machine weighted quadratic tardiness scheduling problem

In this paper, we consider the single machine scheduling problem with weighted quadratic tardiness costs. Several efficient dispatching rules are proposed. These include existing heuristics for the linear problem, as well as procedures suitably adapted to the quadratic objective function. Also, both forward and backward scheduling procedures are considered.The computational results show that the heuristics that specifically take into account the quadratic objective significantly outperform their linear counterparts. Also, the backward scheduling approach proves to be superior, and the difference in performance is even more noticeable for the harder instances.The best of the backward scheduling heuristics is both quite efficient and effective. Indeed, this procedure can quickly generate a schedule even for large instances. Also, its relative deviation from the optimum is usually rather low, and it performs adequately even for the more difficult instances.     

Metaheuristics for the single machine weighted quadratic tardiness scheduling problem

This paper considers the single machine scheduling problem with weighted quadratic tardiness costs. Three metaheuristics are presented, namely iterated local search, variable greedy and steady-state genetic algorithm procedures. These address a gap in the existing literature, which includes branch-and-bound algorithms (which can provide optimal solutions for small problems only) and dispatching rules (which are efficient and capable of providing adequate solutions for even quite large instances). A simple local search procedure which incorporates problem specific information is also proposed.The computational results show that the proposed metaheuristics clearly outperform the best of the existing procedures. Also, they provide an optimal solution for all (or nearly all, in the case of the variable greedy heuristic) the smaller size problems. The metaheuristics are quite close in what regards solution quality. Nevertheless, the iterated local search method provides the best solution, though at the expense of additional computational time. The exact opposite is true for the variable greedy procedure, while the genetic algorithm is a good all-around performer.     

具有提前ö拖期惩罚的热轧钢管批调度问题研究

建立了具有提前／拖期惩罚的热轧钢管批调度问题的混合整数非线性规划模型，提出并证明了给定合同排序下的最优组批方式，从而将原问题转化为易求解的合同排序问题．同时，建立了转化问题的数学模型并设计了遗传算法．仿真实验验证了模型和算法的有效性．     

A variable neighborhood search for minimizing single machine weighted earliness and tardiness with common due date

Although the concept of just-in-time (JIT) production systems has been proposed for over two decades, it is still important in real-world production systems. In this paper, we consider minimizing the total weighted earliness and tardiness with a restrictive common due date in a single machine environment, which has been proved as an NP-hard problem. Due to the complexity of the problem, metaheuristics, including simulated annealing, genetic algorithm, tabu search, among others, have been proposed for searching good solutions in reasonable computation times. In this paper, we propose a hybrid metaheuristic that uses tabu search within variable neighborhood search (VNS/TS). There are several distinctive features in the VNS/TS algorithm, including different ratio of the two neighborhoods, generating five points simultaneously in a neighborhood, implementation of the B/F local search, and combination of TS with VNS. By examining the 280 benchmark problem instances, the algorithm shows an excellent performance in not only the solution quality but also the computation time. The results obtained are better than those reported previously in the literature.     

An efficient memetic algorithm for total weighted tardiness minimization in a single machine with setups

The single machine scheduling problem with sequence-dependent setup times with the objective of minimizing the total weighted tardiness is a challenging problem due to its complexity, and has a huge number of applications in real production environments. In this paper, we propose a memetic algorithm that combines and extends several ideas from the literature, including a crossover operator that respects both the absolute and relative position of the tasks, a replacement strategy that improves the diversity of the population, and an effective but computationally expensive neighborhood structure. We propose a new decomposition of this neighborhood that can be used by a variable neighborhood descent framework, and also some speed-up methods for evaluating the neighbors. In this way we can obtain competitive running times. We conduct an experimental study to analyze the proposed algorithm and prove that it is significantly better than the state-of-the-art in standard benchmarks.     

Minimizing the weighted sum of squared tardiness on a single machine

This paper considers a problem in which there is a set of jobs to be sequenced on a single machine. Each job has a weight and the objective is to sequence the jobs to minimize total weighted squared tardiness. A branch-and-bound algorithm is developed for optimally solving the problem. Several dominance conditions are presented for possible inclusion in the branch-and-bound algorithm. The dominance conditions are included in the branch-and-bound algorithm, which is tested on randomly generated problems of various numbers of jobs, due date tightness and due date ranges. The results show that the dominance conditions dramatically improve the efficiency of the branch-and-bound algorithm.     

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.     

Minimizing earliness and tardiness subject to total completion time in an identical parallel machine system

This study addresses the identical parallel machine scheduling problem in which the total earliness and tardiness about a common due date are minimized subject to minimum total flowtime, P∥(E+T)/∑_Ci$P\parallel (E+T)/\sum C_i$. The problem is demonstrated to be transformed into a simplified version of the parallel machine problem with the objective of minimizing makespan subject to minimum total flowtime, P∥_Cmax/∑_Ci$P\parallel C_{\max }/\sum C_i$. Several properties are considered to solve optimally the restricted version of the problem. A streamlined binary integer programming model is developed to solve the P∥_Cmax/∑_Ci$P\parallel C_{\max }/\sum C_i$ problem and the P∥(E+T)/∑_Ci$P\parallel (E+T)/\sum C_i$ problem. Computational experiments indicate that the binary integer programming model is superior to the existing optimization algorithm for the P∥_Cmax/∑_Ci$P\parallel C_{\max }/\sum C_i$ problem in the literature.     

约束满足混合算法求解提前/拖期Job Shop调度问题

针对提前/拖期Job Shop调度问题,建立其约束满足优化问题模型,提出了一种约束满足与禁忌搜索结合的混合算法。该算法基于约束满足思想,通过约束传播技术和启发式修复算法,得到可行调度作为禁忌搜索算法的初始解;再进行关键路径上的邻域变换,优化当前解;并采用一种全局邻域交换策略,扩大搜索空间,改善优化结果。数据实验表明了该混合算法的可行性和有效性。     

A branch-and-bound algorithm for single-machine scheduling with batch delivery and job release times

This paper addresses scheduling a set of jobs with specified release times on a single machine for delivery in batches to customers or to other machines for further processing. This problem is a natural extension of minimizing the sum of flow times in the presence of release time by considering the possibility of delivering jobs in batches and introducing batch delivery costs. The scheduling objective adopted is that of minimizing the sum of flow times and delivery costs. The extended problem arises in the context of coordination between machine scheduling and a distribution system in a supply chain network. Structural properties of the problem are investigated and used to devise a branch-and-bound solution scheme. Computational experiments show significant improvement over an existing dynamic programming algorithm.     

A genetic algorithm for minimizing total tardiness/earliness of weighted jobs in a batched delivery system

This paper endeavors to solve a novel complex single-machine scheduling problem using two different approaches. One approach exploits mathematical modeling, and the other is based upon genetic algorithms. The problem involves earliness, tardiness, and inventory costs and considers a batched delivery system. The same conditions might apply to some real supply chains, in which delivery of products is conducted in a batched form and with some costs. In such delivery systems, the act of buffering the products can have both positive effects (i.e., decreasing the delivery costs and early jobs) and negative ones (i.e., increasing the number of tardy and holding costs). Accordingly, the proposed solution takes into account both effects and tries to find a trade-off between them to hold the total costs low. The suggestions are compared to existing solutions for older non-batched systems and have illustrated outperformance.     

Weighted tardiness for the single machine scheduling problem:An examination of precedence theorem productivity

Earlier research by Kanet [11] has provided a number of new theorems for deciding precedence between pairs of jobs for 1∣∣Σw_jT_j. The theorems supplant those of Rinnooy Kan, Lageweg, and Lenstra [16]. Presented here are the results of an analysis of the marginal benefit these new theorems provide over the earlier versions of Rinnooy Kan et al. Results show that the new theorems can provide noteworthy improvements in the ability to discover precedence relations between job pairs. For a large set of problem instances the new theorems uncovered up to 8% more precedence relations than the original theorems of Rinnooy Kan et al. The improvement in the productivity in discovering precedence relations shows to be dependent on the coefficient of variation of the distribution of job weights. Logical application of the theorems is to include them in search procedures and/or heuristic approaches to 1||Σw_jT_j. One such heuristic based on the theorems is provided here in which the solutions to a large set of sample problems are within 8–12% of the optimum.     

A heuristic algorithm to minimize total weighted tardiness on a single machine with release times

This paper attempts to solve a single machine‐scheduling problem, in which the objective function is to minimize the total weighted tardiness with different release dates of jobs. To address this scheduling problem, a heuristic scheduling algorithm is presented. A mathematical programming formulation is also formulated to validate the performance of the heuristic scheduling algorithm proposed herein. Experimental results show that the proposed heuristic algorithm can solve this problem rapidly and accurately. Overall, this algorithm can find the optimal solutions for 2200 out of 2400 randomly generated problems (91.67%). For the most complicated 20 job cases, it requires less than 0.0016 s to obtain an ultimate or even optimal solution. This heuristic scheduling algorithm can therefore efficiently solve this kind of problem.     

Minimizing total earliness and tardiness on a single machine using a hybrid heuristic

This paper focuses on scheduling jobs with different processing times and distinct due dates on a single machine with no inserted idle time as to minimize the sum of total earliness and tardiness. This scheduling problem is a very important and frequent industrial problem that is common to most just-in-time production environments. This NP hard scheduling problem is herein solved using a hybrid heuristic which combines local search heuristics (dispatching rules, hill climbing and simulated annealing) and an evolutionary algorithm based on genetic algorithms. The heuristic involves low and high, relay and teamwork hybridization. Computational results reflect the sizeable solution quality improvement induced by hybridization, and assess the impact of each type of hybridization on the efficiency of the hybrid heuristic.     

Minimizing total earliness and tardiness through unrelated parallel machine scheduling using distributed release time control

Using unrelated parallel machine scheduling to minimize the total earliness and tardiness of jobs with distinct due dates is a nondeterministic polynomial-hard problem. Delayed customer orders may result in penalties and reduce customer satisfaction. On the other hand, early completion creates inventory storage costs, which increase the total cost. Although parallel machines can increase productivity, machine assignments also increase the complexity of production. Therefore, the challenge in parallel machine scheduling is to dynamically adjust the machine assignment to complete the job within the shortest possible time. In this paper, we address an unrelated parallel machine scheduling problem for jobs with distinct due dates and dedicated machines. The objective is to dynamically allocate jobs to unrelated parallel machines in order to minimize the total earliness and tardiness time. We formulate the problem as a mixed integer linear programming (MILP) model and develop a modified genetic algorithm (GA) with a distributed release time control (GARTC) mechanism to obtain the near-optimal solution. A preliminary computational study indicates that the developed GARTC not only provides good quality solutions within a reasonable amount of time, but also outperforms the MILP model, a classic GA and heuristic approaches described in the literature.     

An improved earliness–tardiness timing algorithm

Earliness–tardiness criteria with distinct due dates usually induce NP-complete problems. Researchers have focused on particular cases like the timing problem, which is to look for the optimal schedule when the jobs sequence is already known. These timing algorithms are very useful since they can be used in more complex procedures. In the first part of this paper we provide the most efficient and fairly general algorithm to solve the one-machine timing problem. It is then adapted to a permutation flow shop problem.     

最小化提前和延误惩罚的批处理问题

考虑机器容量有限的同时加工排序问题, 为享有公共交货期窗口[e; d]的n个工件分批并排序以最小化总的赋权提前/延误惩罚. 本文把窗时排序与同时加工排序结合起来研究, 假设每个批的容量是b(< n, 其中n为工件的个数), 而且最早交货期e 和最晚交货期d已知. 但该问题是NP–完备的, 首先给出最优排序的几条性质, 进而解决了两类特殊情况.