Single Machine Scheduling and Selection to Minimize Total Flow Time with Minimum Number Tardy

Emmons considered the problem of sequencing N jobs on a single machine to minimize total flow time with the minimum number of tardy jobs. He proposed an effective branch-and-bound algorithm for this problem. In this paper, we show that Emmons' algorithm can be extended to a more difficult scheduling problem which includes an optimal selection of jobs as well.     

Single-machine scheduling of proportionally deteriorating jobs by two agents

We consider a problem of scheduling a set of independent jobs by two agents on a single machine. Every agent has its own subset of jobs to be scheduled and uses its own optimality criterion. The processing time of each job proportionally deteriorates with respect to the starting time of the job. The problem is to find a schedule that minimizes the total tardiness of the first agent, provided that no tardy job is allowed for the second agent. We prove basic properties of the problem and give a lower bound on the optimal value of the total tardiness criterion. On the basis of these results, we propose a branch-and-bound algorithm and an evolutionary algorithm for the problem. Computational experiments show that the exact algorithm solves instances up to 50 jobs in a reasonably short time and that solutions obtained by the metaheuristic are close to optimal ones.     

A Branch-and-Bound Approach for a Two-machine Flowshop Scheduling Problem

In this paper, we present a branch-and-bound approach for solving a two-machine flow shop scheduling problem, in which the objective is to minimize a weighted combination of job flowtime and schedule makespan. Experimental results show that the algorithm works very well for certain special cases and moderately well for others. In fact, it is able to produce optimal schedules for 500-job problems in which the second machine dominates the first machine. It is also shown that the algorithm developed to provide an upper bound for the branch-and-bound is optimal when processing times for jobs are the same on both machines. The primary reason for developing the branch-and-bound approach is that its results can be used to guide other heuristic techniques, such as simulated annealing, tabu search and genetic algorithms, in their search for optimal solutions for larger problems.     

Single machine earliness and tardiness scheduling

We examine the problem of scheduling a given set of jobs on a single machine to minimize total early and tardy costs without considering machine idle time. We decompose the problem into two subproblems with a simpler structure. Then the lower bound of the problem is the sum of the lower bounds of two subproblems. A lower bound of each subproblem is obtained by Lagrangian relaxation. Rather than using the well-known subgradient optimization approach, we develop two efficient multiplier adjustment procedures with complexity O(nlog n) to solve two Lagrangian dual subproblems. A branch-and-bound algorithm based on the two efficient procedures is presented, and is used to solve problems with up to 50 jobs, hence doubling the size of problems that can be solved by existing branch-and-bound algorithms. We also propose a heuristic procedure based on the neighborhood search approach. The computational results for problems with up to 3 000 jobs show that the heuristic procedure performs much better than known heuristics for this problem in terms of both solution efficiency and quality. In addition, the results establish the effectiveness of the heuristic procedure in solving realistic problems to optimality or near optimality.     

An improved branch and bound algorithm for single machine scheduling with deadlines to minimize total weighted completion time

We consider a scheduling problem in which n jobs with distinct deadlines are to be scheduled on a single machine. The objective is to find a feasible job sequence that minimizes the total weighted completion time. We present an efficient branch-and-bound algorithm that fully exploits the principle of optimality. Favorable numerical results are also reported on an extensive set of problem instances of 20-120 jobs.     

Optimal Scheduling of a Two-stage Hybrid Flow Shop

We present an exact branch-and-bound algorithm for the two-stage hybrid flow shop problem with multiple identical machines in each stage. The objective is to schedule a set of jobs so as to minimize the makespan. This is the first exact procedure which has been specifically designed for this strongly -hard problem. Among other features, our algorithm is based on the exact solution of identical parallel machine scheduling problems with heads and tails. We report the results of extensive computational experiments on instances which show that the proposed algorithm solves large-scale instances in moderate CPU time.     

Crane scheduling with non-crossing constraint

In this paper, we examine crane scheduling for ports. This important component of port operations management is studied when the non-crossing spatial constraint, which is common to crane operations, is considered. We assume that ships can be divided into holds and that cranes can move from hold to hold but jobs are not pre-emptive, so that only one crane can work on one hold or job to complete it. Our objective is to minimize the latest completion time for all jobs. We formulate this problem as an integer programming problem. We provide the proof that this problem is NP-complete and design a branch-and-bound algorithm to obtain optimal solutions. A simulated annealing meta-heuristic with effective neighbourhood search is designed to find good solutions in larger size instances. The elaborate experimental results show that the branch-and-bound algorithm runs much faster than CPLEX and the simulated annealing approach can obtain near optimal solutions for instances of various sizes.     

Application of an optimization problem in Max-Plus algebra to scheduling problems

The problem tackled in this paper deals with products of a finite number of triangular matrices in Max-Plus algebra, and more precisely with an optimization problem related to the product order. We propose a polynomial time optimization algorithm for 2×2 matrices products. We show that the problem under consideration generalizes numerous scheduling problems, like single machine problems or two-machine flow shop problems. Then, we show that for 3×3 matrices, the problem is NP-hard and we propose a branch-and-bound algorithm, lower bounds and upper bounds to solve it. We show that an important number of results in the literature can be obtained by solving the presented problem, which is a generalization of single machine problems, two- and three-machine flow shop scheduling problems. The branch-and-bound algorithm is tested in the general case and for a particular case and some computational experiments are presented and discussed.     

Minimizing total flow time in the single-machine scheduling problem with periodic maintenance

This paper deals with a single-machine scheduling problem with limited machine availability. The limited availability of machine results from periodic maintenance activities. In our research, a periodic maintenance schedule consists of several maintenance periods. Each maintenance period is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the total flow time subject to periodic maintenance and nonresumable jobs. Some important theorems are proved for the problem. A branch-and-bound algorithm that utilizes several theorems is proposed to find the optimal schedule. We also develop a heuristic to solve large sized problems. In this paper, computational results show that the proposed heuristic is highly accurate and efficient.     

A hybrid genetic algorithm for an identical parallel-machine problem with maintenance activity

The scheduling of maintenance activities has been extensively studied, with most studies focusing on single-machine problems. In real-world applications, however, multiple machines or assembly lines process numerous jobs simultaneously. In this paper, we study a parallel-machine scheduling problem in which the objective is to minimize the total tardiness given that there is a maintenance activity on each machine. We develop a branch-and-bound algorithm to solve the problem with a small problem size. In addition, we propose a hybrid genetic algorithm to obtain the approximate solutions when the number of jobs is large. The performance of the proposed algorithms is evaluated based mainly on computational results.     

Lagrangian domain reductions for the single machine earliness–tardiness problem with release dates

This paper presents new elimination rules for the single machine problem with general earliness and tardiness penalties subject to release dates. These rules, based on a Lagrangian decomposition, allow to drastically reduce the execution windows of the jobs. We measure the efficiency of these properties by integrating them in a branch-and-bound. Tests show that instances with up to 70 jobs without release dates, and up to 40 jobs with release dates, can be optimally solved within 1000 seconds.     

A branch-and-bound algorithm with fuzzy inference for a permutation flowshop scheduling problem

This paper considers an m-machine permutation flowshop scheduling problem of minimizing the makespan. This classical scheduling problem is still important in modern manufacturing systems, and is well known to be intractable (i.e., NP-hard). In fact branch-and-bound algorithms developed so far for this problem have not come to solve large scale problem instances with over a hundred jobs. In order to improve the performance of branch-and-bound algorithms this paper proposes a new dominance relation by which the search load could be reduced, and notices that it is based on a sufficient precondition. This suggests that the dominance relation holds with high possibility even if the precondition approximately holds, thus being more realistic. The branch-and-bound algorithm proposed here takes advantage of this possibility for obtaining an optimal solution as early as possible in the branch-and-bound search. For this purpose this paper utilizes membership functions in the context of the fuzzy inference. Extensive numerical experiments that were executed through Monte Carlo simulations and benchmark tests show that the developed branch-and-bound algorithm can solve 3-machine problem instances with up to 1000 jobs with probability of over 99%, and 4-machine ones with up to 900 jobs with over 97%.     

A branch-and-bound algorithm for single-machine scheduling with batch delivery minimizing flow times and delivery costs

This paper addresses scheduling a set of jobs on a single machine for delivery in batches to customers or to other machines for further processing. The problem is a natural extension of minimizing the sum of flow times 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 improvements over an existing dynamic programming algorithm.     

Single-machine batch scheduling minimizing weighted flow times and delivery costs

This paper addresses scheduling a set of jobs on a single machine for delivery in batches to one customer or to another machine for further processing. The problem is a natural extension of that of minimising the sum of weighted flow times, considering the possibility of delivering jobs in batches and introducing batch delivery costs. The scheduling objective adopted is that of minimising the sum of weighted 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 method. For the special case, when the maximum number of batches is fixed, the branch-and-bound scheme provided shows significant improvements over an existing dynamic-programming algorithm.     

带准备时间的单机指数时间学习效应排序问题

研究带有准备时间的单机学习效应模型,其中工件加工时间具有指数时间学习效应,即工件的实际加工时间是已经排好的工件加工时间的指数函数。学习效应模型考虑工件的实际加工时间同时依赖于工件本身的加工时间和已加工工件的累计加工时间,目标函数为最小化总完工时间。这个问题是NP-难的,提出了一个数学规划模型来求解该问题的最优解。通过分析几个优势性质和下界,提出分支定界算法来求解此问题,并设计启发式算法改进分支定界算法的上界值。通过仿真实验验证了分支定界算法在求解质量和时间方面的有效性。     

A class of on-line scheduling algorithms to minimize total completion time

We consider the problem of scheduling jobs on-line on a single machine and on identical machines with the objective to minimize total completion time. We assume that the jobs arrive over time. We give a general 2-competitive algorithm for the single machine problem. The algorithm is based on delaying the release time of the jobs, i.e., making the jobs artificially later available to the on-line scheduler than the actual release times. Our algorithm includes two known algorithms for this problem that apply delay of release times. The proposed algorithm is interesting since it gives the on-line scheduler a whole range of choices for the delays, each of which leading to 2-competitiveness.We also show that the algorithm is 2α competitive for the problem on identical machines where α is the performance ratio of the Shortest Remaining Processing Time first rule for the preemptive relaxation of the problem.     

Minimizing makespan on a single burn-in oven in semiconductor manufacturing

This paper considers a scheduling problem for a single burn-in oven in semiconductor manufacturing industry where the oven is a batch processing machine and each batch processing time is represented by the largest processing time among those of all the jobs contained in the batch. The objective measure of the problem is the maximum completion time (makespan) of all jobs. This paper investigates a static case in which all jobs are available to process at time zero, and also analyzes a dynamic case with different job-release times, for which a branch-and-bound algorithm and several heuristics are exploited. The worst case error performance ratios of the heuristics are also derived.     

Scheduling with unexpected machine breakdowns

We investigate an online version of a basic scheduling problem where a set of jobs has to be scheduled on a number of identical machines so as to minimize the makespan. The job processing times are known in advance and preemption of jobs is allowed. Machines are non-continuously available, i.e., they can break down and recover at arbitrary time instances not known in advance. New machines may be added as well. Thus machine availabilities change online. We first show that no online algorithm can construct optimal schedules. We also show that no online algorithm can achieve a bounded competitive ratio if there may be time intervals where no machine is available. Then we present an online algorithm that constructs schedules with an optimal makespan of C_max^OPT if a lookahead of one is given, i.e., the algorithm always knows the next point in time when the set of available machines changes. Finally, we give an online algorithm without lookahead that constructs schedules with a nearly optimal makespan of C_max^OPT+, for any >0, if at any time at least one machine is available. Our results demonstrate that not knowing machine availabilities in advance is of little harm.     

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.     

Branch-and-bound and weight-combination search algorithms for the total completion time problem with step-deteriorating jobs

In many real situations, it is found that if certain maintenance procedures fail to be completed prior to a pre-specified deterioration date, then the jobs will require extra time for successful completion. In this paper, a single-machine total completion time problem with step-deteriorating jobs is considered. A branch-and-bound method incorporated with several dominance properties and a lower bound is developed to derive the optimal solution for this problem. In addition, a weight-combination search algorithm is proposed to search for a near-optimal solution. Computational results indicate that the branch-and-bound algorithm can solve most of the problems with up to 24 jobs in a reasonable amount of time. Moreover, the proposed heuristic algorithm is accurate with mean deviations from the optimum value of less than 0.3%.