An algorithm for single machine sequencing with release dates to minimize total weighted completion time

Each of n jobs is to be processed without interruption on a single machine which can handle only one job at a time. Each job becomes available for processing at its release date, requires a processing time and has a positive weight. Given a processing order of the jobs, the earliest completion time for each job can be computed. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times. In this paper a branch and bound algorithm for the problem is derived. Firstly a heuristic is presented which is used in calculating the lower bound. Then the lower bound is obtained by performing a Lagrangean relaxation of the release date constraints; the Lagrange multipliers are chosen so that the sequence generated by the heuristic is an optimum solution of the relaxed problem thus yielding a lower bound. A method to increase the lower bound by deriving improved constraints to replace the original release date constraints is given. The algorithm, which includes several dominance rules, is tested on problems with up to fifty jobs. The computational results indicate that the version of the lower bound using improved constraints is superior to the original version.     

Algorithm for minimizing weighted earliness penalty in single-machine problem

In this paper, the problem of minimizing the weighted earliness penalty in a single-machine scheduling problem is addressed. For this problem, every job is assumed to be available at time zero and must be completed before or on its deadline. No tardy job is allowed. Each job has its own earliness penalty and deadline. The paper identifies several local optimality conditions for sequencing of adjacent jobs. A heuristic algorithm is developed based on these local optimality conditions. Sample problems are solved and the solutions obtained from the heuristic are compared to solutions obtained from the heuristics developed by Chand and Schneeberger. Also, comparisons are performed between the solutions obtained from the heuristic and the optimal solutions obtained from a mathematical modeling approach for problems involving 10 and 15 jobs. The results show that the developed heuristic produces solutions close to optimal in small size problems, and it also outperforms the Chand and Schneeberger's method.     

A hybrid two-stage transportation and batch scheduling problem

We study the coordinated scheduling problem of hybrid batch production on a single batching machine and two-stage transportation connecting the production, where there is a crane available in the first-stage transportation that transports jobs from the warehouse to the machine and there is a vehicle available in the second-stage transportation to deliver jobs from the machine to the customer. As the job to be carried out is big and heavy in the steel industry, it is reasonable assumed that both the crane and the vehicle have unit capacity. The batching machine processes a batch of jobs simultaneously. Each batch occur a setup cost. The objective is to minimize the sum of the makespan and the total setup cost. We prove that this problem is strongly NP-hard. A polynomial time algorithm is proposed for a case where the job transportation times are identical on the crane or the vehicle. An efficient heuristic algorithm for the general problem is constructed and its tight worst-case bound is analyzed. In order to further verify the performance of the proposed heuristics, we develop a lower bound on the optimal objective function. Computational experiments show that the heuristic algorithm performs well on randomly generated problem instances.     

A branch and bound algorithm to minimize total tardiness of jobs in a two identical-parallel-machine scheduling problem with a machine availability constraint

This research focuses on the problem of scheduling jobs on two identical parallel machines that are not continuously available with the objective of minimizing total tardiness. After processing a given number of jobs, each machine requires a preventive maintenance task, during which the machine cannot process jobs. We present dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as an upper bound obtained from a heuristic algorithm. Performance of the algorithm is evaluated through a series of computational experiments on randomly generated instances and results are reported.     

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.     

A heuristic for scheduling in a two-stage hybrid flowshop with renewable resources shared among the stages

In this paper we propose a heuristic for solving the problem of resource constrained preemptive scheduling in the two-stage flowshop with one machine at the first stage and parallel unrelated machines at the second stage, where renewable resources are shared among the stages, so some quantities of the same resource can be used at different stages at the same time. Availability of every resource at any moment is limited and resource requirements of jobs are arbitrary. The objective is minimization of makespan. The problem is NP-hard. The heuristic first sequences jobs on the machine at stage 1 and then solves the preemptive scheduling problem at stage 2. Priority rules which depend on processing times and resource requirements of jobs are proposed for sequencing jobs at stage 1. A column generation algorithm which involves linear programming, a tabu search algorithm and a greedy procedure is proposed to minimize the makespan at stage 2. A lower bound on the optimal makespan in which sharing of the resources between the stages is taken into account is also derived. The performance of the heuristic evaluated experimentally by comparing the solutions to the lower bound is satisfactory.     

Scheduling on parallel identical machines to minimize total tardiness

This paper focuses on the problem of scheduling n independent jobs on m identical parallel machines for the objective of minimizing total tardiness of the jobs. We develop dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as upper bounds obtained from a heuristic algorithm. Computational experiments are performed on randomly generated test problems and results show that the algorithm solves problems with moderate sizes in a reasonable amount of computation time.     

需要安装时间的平行多功能机排序问题的启发式算法

考虑需要安装时间的平行多功能机排序问题。在该模型中,每个工件对应机器集合的一个子集,其只能在这个子集中的任一台机器上加工,称这个子集为该工件的加工集合;工件分组,同组工件具有相同的加工时间和加工集合,不同组中的工件在同一台机器上连续加工需要安装时间,目标函数为极小化最大完工时间。对该问题NP-难的一般情况设计启发式算法:首先按照特定的规则将所有工件组都整组地安排到各台机器上,然后通过在各机器间转移工件不断改进当前最大完工时间。通过与下界的比较检验算法的性能,大量的计算实验表明,算法是实用而有效的。     

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.     

Scheduling preemptive open shops to minimize total tardiness

This article considers the problem of scheduling preemptive open shops to minimize total tardiness. The problem is known to be NP-hard. An efficient constructive heuristic is developed for solving large-sized problems. A branch-and-bound algorithm that incorporates a lower bound scheme based on the solution of an assignment problem as well as various dominance rules are presented for solving medium-sized problems. Computational results for the 2-machine case are reported. The branch-and-bound algorithm can handle problems of up to 30 jobs in size within a reasonable amount of time. The solution obtained by the heuristic has an average deviation of less than 2% from the optimal value, while the initial lower bound has an average deviation of less than 11% from the optimal value. Moreover, the heuristic finds approved optimal solutions for over 65% of the problems actually solved.     

A dual algorithm for the one-machine scheduling problem

A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn ² and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.     

Backtracking and its Amoebic Properties in One-dimensional Machine Location Problems

This paper studies the assignment of M unique machines to M equally spaced locations along a linear material handling track with the objective of minimizing the cost of (jobs) backtracking (i.e. moving upstream). Because of the arrangement of machines and restrictions imposed by the sequence of operations for each job, some jobs may have to backtrack to complete required processing on different machines. This problem is formulated as a quadratic assignment problem. An optimal solution to a problem with large M is computationally intractable. The backtracking distance matrix in problems involving equally-spaced machine locations in one dimension is seen to possess some unique characteristics called amoebic properties. Ten amoebic properties have been identified and exploited to devise a heuristic and a lower bound on the optimal solution. Results which describe the performance of the heuristic and the lower bound are presented.     

Two-machine flow shop total tardiness scheduling problem with deteriorating jobs

This paper studies a two-machine scheduling problem with deteriorating jobs which their processing times depend on their waiting time. We develop a branch and bound algorithm to minimize the total tardiness criteria. A lower bound, several dominance properties and an initial upper bound derived from a heuristic algorithm are used to increase the speed of branch and bound algorithm and decrease its required memory space. Computational results are presented to evaluate effectiveness and efficiency of the algorithms.     

Single machine scheduling subject to deadlines and resource dependent processing times

The problem of scheduling n jobs on a single machine is studied. Each job has a deadline and a processing time which is a linear decreasing function of the amount of a common resource allocated to the job. The objective is to find simultaneously a sequence of the jobs and a resource allocation so as the deadlines are satisfied and the total weighted resource consumption is minimized. The problem is shown to be solvable in O(n log n) time if the resource is continuously divisible. If the resource is discrete, then the problem is proved to be binary NP-hard. Some special cases are solvable in O(n log n) time. A fully polynomial approximation scheme is presented for the general problem with discrete resource.     

Minimizing Completion Time Variance with Compressible Processing Times

We introduce a new formulation of the standard completion time variance (CTV) problem with n jobs and one machine, in which the job sequence and the processing times of the jobs are all decision variables. The processing time of job i (i=1, ,n) can be compressed by an amount within [l_i, u_i], which however will incur a compression cost. The compression cost is a general convex non-decreasing function of the amount of the job processing time compressed. The objective is to minimize a weighted combination of the completion time variance and the total compression cost. We show that, under an agreeable condition on the bounds of the processing time compressions, a pseudo-polynomial algorithm can be derived to find an optimal solution for the problem. Based on the pseudo-polynomial time algorithm, two heuristic algorithms H1 and H2 are proposed for the general problem. The relative errors of both heuristic algorithms are guaranteed to be no more than , where is a measure of deviation from the agreeable condition. While H1 can find an optimal solution for the agreeable problem, H2 is dominant for solving the general problem. We also derive a tight lower bound for the optimal solution of the general problem. The performance of H2 is evaluated by complete enumeration for small n, and by comparison with this tight lower bound for large n. Computational results (with n up to 80) are reported, which show that the heuristic algorithm H2 in general can efficiently yield near optimal solutions, when n is large.     

加工时间和尺寸成正比单机排序和工件运输的最小化最大完工时间问题

在单机排序和工件运输的最小化最大完工时间问题中,工件首先在一台机器上加工,然后被一辆有容量限制的汽车运送到一个顾客．当工件的加工时间和尺寸无关时, Chang和Lee已经证明该问题是强NP困难的．他们也给出了一个启发式算法,它的最差执行比为5／3,并且这个界是紧的．本文考虑工件的加工时间和尺寸成正比的情形,证明了Chang和Lee的算法有更好的最差执行比53／35,并提供了一个新的启发式算法,它的最差执行比是3／2,并且这个界是最好的．     

Resource constraint scheduling with a fractional shared resource

We consider a scheduling problem motivated by mining in remote off-grid areas. In this model, mines have pre-assigned mineral processing jobs and their own machine for executing these jobs. Each job also needs a certain amount of electricity in order to get completed. The electricity, on the other hand, is of limited supply and must be shared between the mines. We present a mathematical formulation of the problem and a Lagrangian relaxation based heuristic. Computational results which compares our heuristic with genetic algorithm and simulated annealing are also presented.     

The coordination of transportation and batching scheduling

We study a coordinated scheduling problem of production and transportation in which each job is transported to a single batching machine for further processing. There are m vehicles that transport jobs from the holding area to the batching machine. Each vehicle can transport only one job at a time. The batching machine can process a batch of jobs simultaneously where there is an upper limit on the batch size. Each batch to be processed occurs a processing cost. The problem is to find a joint schedule of production and transportation such that the sum of the total completion time and the total processing cost is optimized. For a special case of the problem where the job assignment to the vehicles is predetermined, we provide a polynomial time algorithm. For the general problem, we prove that it is NP-hard (in the ordinary sense) and present a pseudo-polynomial time algorithm. A fully polynomial time approximation scheme for the general problem is obtained by converting an especially designed pseudo-polynomial dynamic programming algorithm.     

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

In this paper, we consider the problem of scheduling n jobs on m machines in an open shop environment so that the sum of completion times or mean flow time becomes minimal. For this strongly NP-hard problem, we develop and discuss different constructive heuristic algorithms. Extensive computational results are presented for problems with up to 50 jobs and 50 machines, respectively. The quality of the solutions is evaluated by a lower bound for the corresponding preemptive open shop problem and by an alternative estimate of mean flow time. We observe that the recommendation of an appropriate constructive algorithm strongly depends on the ratio n/m.     

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.