Generating efficient schedules for identical parallel machines involving flow-time and tardy jobs

This paper considers the problem of generating a set of efficient (non-dominated) schedules on identical parallel machines involving total flow-time and total number of tardy jobs. In view of the NP-hard nature of this problem, heuristic algorithms are proposed for its solution. Two evaluation methods that provide a scheme by which multiple non-dominated sets can be compared are described and used to compare the performance of the proposed algorithms. Results of several experiments demonstrate the capability of the proposed algorithms and evaluation methodologies to address the problem at hand.     

Local search metaheuristics for discrete–continuous scheduling problems

Problems of scheduling nonpreemptable jobs which require simultaneously a machine from a set of parallel, identical machines and a continuous, renewable resource are considered. For each job there are known: its processing speed as a continuous, concave function of a continuous resource allotted at a time and its processing demand. The optimization criterion is the schedule length. The problem can be decomposed into two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to find an optimal (continuous) resource allocation among jobs already sequenced. Problem (ii) can be formulated as a convex programming problem with linear constraints and solved using proper solvers. Thus, the problem remains to generate a set of all feasible sequences of jobs on machines (this guarantees finding an optimal schedule in the general case). However, the cardinality of this set grows exponentially with the number of jobs. Thus, we propose to use heuristic search methods defined on the space of feasible sequences. Three metaheuristics: tabu search (TS), simulated annealing (SA) and genetic algorithm (GA) have been implemented and compared computationally with a random sampling technique. The computational experiment has been carried out on an SGI PowerChallenge XL computer with 12 RISC R8000 processors. Some directions for further research have been pointed out.     

A scheduling problem with job values given as a power function of their completion times

This paper deals with a problem of scheduling jobs on the identical parallel machines, where job values are given as a power function of the job completion times. Minimization of the total loss of job values is considered as a criterion. We establish the computational complexity of the problem – strong NP-hardness of its general version and NP-hardness of its single machine case. Moreover, we solve some special cases of the problem in polynomial time. Finally, we construct and experimentally test branch and bound algorithm (along with some elimination properties improving its efficiency) and several heuristic algorithms for the general case of the problem.     

Greedy scheduling with custom-made objectives

We present a methodology to automatically generate an online job scheduling method for a custom-made objective and real workloads. The scheduling problem comprises independent parallel jobs and parallel identical machines and occurs in Massively Parallel Processing systems and computational Grids. The system administrator defines the scheduling objective that may consider job properties and priorities of users or user groups. Our scheduling method combines a Greedy scheduling algorithm with the dynamic sorting of the waiting queue. This sorting algorithm uses a criterion that is modifiable by a set of parameters. Finding good parameter settings for the sorting criterion is viewed as a nonlinear optimization problem which is solved with the help of Evolution Strategies. We evaluate our scheduling method with real workload data and compare it to approximated optimal offline solutions and to the online results of the standard EASY backfill algorithm.     

Single Stage Minimum Absolute Lateness Problem with a Common Due Date on Non-Identical Machines

In this paper we consider scheduling n single operation jobs with a common due date on m non-identical machines (in parallel) so as to minimize the sum of the absolute lateness. We reduce the problem to a transportation problem that can be solved by a polynomial time algorithm. Furthermore, we consider the problem in the case of identical machines and we give a heuristic algorithm to minimize makespan among all schedules that minimize the absolute lateness problem.     

Fuzzy scheduling of job orders in a two-stage flowshop with batch-processing machines

In this paper, we present a mixed-integer fuzzy programming model and a genetic algorithm (GA) based solution approach to a scheduling problem of customer orders in a mass customizing furniture industry. Independent job orders are grouped into multiple classes based on similarity in style so that the required number of setups is minimized. The family of jobs can be partitioned into batches, where each batch consists of a set of consecutively processed jobs from the same class. If a batch is assigned to one of available parallel machines, a setup is required at the beginning of the first job in that batch. A schedule defines the way how the batches are created from the independent jobs and specifies the processing order of the batches and that of the jobs within the batches. A machine can only process one job at a time, and cannot perform any processing while undergoing a setup. The proposed formulation minimizes the total weighted flowtime while fulfilling due date requirements. The imprecision associated with estimation of setup and processing times are represented by fuzzy sets.     

Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs

In this paper, we present an algorithm for solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs. The path cost function considered is comprised of two attributes, travel time and toll, that are combined into a nonlinear generalized cost. Travel demand is determined endogenously according to a travel disutility function. Travelers choose routes with the minimum overall generalized costs. The algorithm involves two components: a bicriteria shortest path routine to implicitly generate the set of non-dominated paths and a projection and contraction method to solve the nonlinear complementarity problem (NCP) describing the traffic equilibrium problem. Numerical experiments are conducted to demonstrate the feasibility of the algorithm to this class of traffic equilibrium problems.     

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。     

New solution methods for single machine bicriteria scheduling problem: Minimization of average flowtime and number of tardy jobs

We consider the bicriteria scheduling problem of minimizing the number of tardy jobs and average flowtime on a single machine. This problem, which is known to be NP-hard, is important in practice, as the former criterion conveys the customer’s position, and the latter reflects the manufacturer’s perspective in the supply chain. We propose four new heuristics to solve this multiobjective scheduling problem. Two of these heuristics are constructive algorithms based on beam search methodology. The other two are metaheuristic approaches using a genetic algorithm and tabu-search. Our computational experiments indicate that the proposed beam search heuristics find efficient schedules optimally in most cases and perform better than the existing heuristics in the literature.     

A Heuristic for Common Due-date Assignment and Job Scheduling on Parallel Machines

In this paper we consider the problem of scheduling a set of simultaneously available jobs on several parallel and identical machines. The problem is to find the optimal due-date, assuming this to be the same for all jobs. We also seek to sequence the jobs such that some are early and some are late so as to minimize a penalty function. For the single-machine problem, we present a simple proof of the well-known optimality result that the optimal due-date coincides with one of the job completion times. We show that the optimal job sequence for the single-machine problem can be easily determined. We prove that the same optimal due-date result can be generalized to the parallel-machine problem. However, determination of the optimal job sequence for such a problem is much more complex, and we present a simple heuristic to find an approximate solution. On the basis of a limited experiment, we observe that the heuristic is very effective in obtaining near-optimal solutions.     

A Heuristic Solution Procedure to Minimize Makespan on a Single Machine with Non-linear Cost Functions

We consider the static single-facility scheduling problem where the processing times of jobs are a nondecreasing and differentiable function of their starting (waiting) times and the objective is to minimize the total elapsed time (makespan) in which all jobs complete their processing. We give a criterion for optimality of two jobs to be scheduled next to each other, and based on this criterion we propose a heuristic algorithm to solve the problem. The effectiveness of the algorithm is empirically evaluated for quadratic and exponential cost functions. In the quadratic case it is compared with the static heuristic algorithm proposed by Gupta and Gupta.     

A bicriteria m-machine flowshop scheduling with sequence-dependent setup times

In this study, a bicriteria m-machine flowshop scheduling with sequence-dependent setup times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and makespan. Only small size problems with up to 6 machines and 18 jobs can be solved by the proposed integer programming model. Also the model is tested on an example. We also proposed three heuristic approaches for solving large jobs problems. To solve the large sizes problems up to 100 jobs and 10 machines, special heuristics methods is used. Results of computational tests show that the proposed model is effective in solving problems.     

Improved MILP models for two-machine flowshop with batch processing machines

In this paper, we consider the problem of scheduling jobs in a flowshop with two batch processing machines such that the makespan is minimized. Batch processing machines are frequently encountered in many industrial environments such as heat treatment operations in a steel foundry and chemical processes performed in tanks or kilns. Improved Mixed Integer Linear Programming (MILP) models are presented for the flowshop problem with unlimited or zero intermediate storage. An MILP-based heuristic is also developed for the problem. Computational experiments show that the new MILP models can significantly improve the original ones. Also, the heuristic can obtain the optimal solutions for all the test problem instances.     

Resolution of a scheduling problem in a flowshop robotic cell

We develop in this paper a generic and precise identification of a scheduling problem in a flexible manufacturing system. We consider a flowshop robotic cell that processes several jobs. We assume that there is no intermediate buffer between machines. So, jobs may be blocked when downstream machines are busy. We present an integer programming model to determine the sequence of jobs that minimizes the makespan criterion. In order to solve large size problems, we propose a genetic algorithm (GA). Finally, computational experiments are proposed in order to compare the makespan returned by the GA to a lower bound.     

Integrated machine scheduling and vehicle routing with time windows

This paper integrates production and outbound distribution scheduling in order to minimize total tardiness. The overall problem consists of two subproblems. The first addresses scheduling a set of jobs on parallel machines with machine-dependent ready times. The second focusses on the delivery of completed jobs with a fleet of vehicles which may differ in their loading capacities and ready times. Job-dependent processing times, delivery time windows, service times, and destinations are taken into account. A genetic algorithm approach is introduced to solve the integrated problem as a whole. Two main questions are examined. Are the results of integrating machine scheduling and vehicle routing significantly better than those of classic decomposition approaches which break down the overall problem, solve the two subproblems successively, and merge the subsolutions to form a solution to the overall problem? And if so, is it possible to capitalize on these potentials despite the complexity of the integrated problem? Both questions are tackled by means of a numerical study. The genetic algorithm outperforms the classic decomposition approaches in case of small-size instances and is able to generate relatively good solutions for instances with up to 50 jobs, 5 machines, and 10 vehicles.     

Bicriteria train scheduling for high-speed passenger railroad planning applications

This paper is concerned with a double-track train scheduling problem for planning applications with multiple objectives. Focusing on a high-speed passenger rail line in an existing network, the problem is to minimize both (1) the expected waiting times for high-speed trains and (2) the total travel times of high-speed and medium-speed trains. By applying two practical priority rules, the problem with the second criterion is decomposed and formulated as a series of multi-mode resource constrained project scheduling problems in order to explicitly model acceleration and deceleration times. A branch-and-bound algorithm with effective dominance rules is developed to generate Pareto solutions for the bicriteria scheduling problem, and a beam search algorithm with utility evaluation rules is used to construct a representative set of non-dominated solutions. A case study based on Beijing–Shanghai high-speed railroad in China illustrates the methodology and compares the performance of the proposed algorithms.     

Scheduling parallel machines with inclusive processing set restrictions and job release times

We consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule. The machines have the same processing speed, but each job is compatible with only a subset of those machines. The machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process. We present an efficient algorithm for this problem with a worst-case performance ratio of 2. We also develop a polynomial time approximation scheme (PTAS) for the problem, as well as a fully polynomial time approximation scheme (FPTAS) for the case in which the number of machines is fixed.     

A bicriteria parallel machine scheduling with a learning effect of setup and removal times

In studies on scheduling problems, generally setup times and removal times of jobs have been neglected or by including those into processing times. However, in some production systems, setup times and removal times are very important such that they should be considered independent from processing times. Since, in general jobs are done according to automatic machine processes in production systems processing times do not differ according to process sequence. But, since human factor becomes influential when setup times and removal times are taken into consideration, setup times will be decreasing by repeating setup processes frequently. This fact is defined with learning effect in scheduling literature. In this study, a bicriteria m-identical parallel machines scheduling problem with a learning effect of setup times and removal times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and total tardiness. A mathematical programming model is developed for the problem which belongs to NP-hard class. Results of computational tests show that the proposed model is effective in solving problems with up to 15 jobs and five machines. We also proposed three heuristic approaches for solving large jobs problems. According to the best of our knowledge, no work exists on the minimization of the weighted sum of total completion time and total tardiness with a learning effect of setup times and removal times.     

The multicriteria p-facility median location problem on networks

In this paper we discuss the multicriteria p-facility median location problem on networks with positive and negative weights. We assume that the demand is located at the nodes and can be different for each criterion under consideration. The goal is to obtain the set of Pareto-optimal locations in the graph and the corresponding set of non-dominated objective values. To that end, we first characterize the linearity domains of the distance functions on the graph and compute the image of each linearity domain in the objective space. The lower envelope of a transformation of all these images then gives us the set of all non-dominated points in the objective space and its preimage corresponds to the set of all Pareto-optimal solutions on the graph. For the bicriteria 2-facility case we present a low order polynomial time algorithm. Also for the general case we propose an efficient algorithm, which is polynomial if the number of facilities and criteria is fixed.