Marc E. Posner

Earliness-Tardiness Scheduling Problems, I: Weighted Deviation of Completion Times About a Common Due Date

This paper and its companion Part II concern the scheduling of jobs with cost penalties for both early and late completion. In Part I, we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date, d. We assume that d is not early enough to constrain the scheduling decision. The weight of a job does not depend on whether the job is early or late, but weights may vary between jobs. We prove that the recognition version of this problem is NP-complete in the ordinary sense. We describe optimality conditions, and present a computationally efficient dynamic programming algorithm. When the weights are bounded by a polynomial function of the number of jobs, a fully polynomial approximation scheme is given. We also describe four special cases for which the problem is polynomially solvable. Part II provides similar results for the unweighted version of this problem, where d is arbitrary.

Generating Experimental Data for Computational Testing with Machine Scheduling Applications

The operations research literature provides little guidance about how data should be generated for the computational testing of algorithms or heuristic procedures. We discuss several widely used data generation schemes, and demonstrate that they may introduce biases into computational results. Moreover, such schemes are often not representative of the way data arises in practical situations. We address these deficiencies by describing several principles for data generation and several properties that are desirable in a generation scheme. This enables us to provide specific proposals for the generation of a variety of machine scheduling problems. We present a generation scheme for precedence constraints that achieves a target density which is uniform in the precedence constraint graph. We also present a generation scheme that explicitly considers the correlation of routings in a job shop. We identify several related issues that may influence the design of a data generation scheme. Finally, two case studies illustrate, for specific scheduling problems, how our proposals can be implemented to design a data generation scheme.

Scheduling with release dates on a single machine to minimize total weighted completion time

Abstract This paper considers the problem of scheduling jobs with release dates on a single machine to minimize the total weighted completion time. A branch and bound algorithm is proposed which incorporates three special features that contribute to its efficiency. Firstly, quickly computed lower bounds are obtained using a procedure which is based on job splitting. The job splitting methodology is shown to be applicable to a range of total weighted completion time scheduling problems. Secondly, the branching rule includes a release date adjustment mechanism which increases release dates at certain nodes of the tree with a view to tightening lower bounds. Thirdly, the branch and bound algorithm includes a new dominance rule for eliminating nodes of the search tree. Computational experience on problems with up to 50 jobs indicates that the proposed algorithm is superior to other known algorithms.

Maximizing the Lifetime of a Barrier of Wireless Sensors

To make a network last beyond the lifetime of an individual sensor node, redundant nodes must be deployed. What sleep-wake-up schedule can then be used for individual nodes so that the redundancy is appropriately exploited to maximize the network lifetime? We develop optimal solutions to both problems for the case when wireless sensor nodes are deployed to form an impenetrable barrier for detecting movements. In addition to being provably optimal, our algorithms work for nondisk sensing regions and heterogeneous sensing regions. Further, we provide an optimal solution for the more difficult case when the lifetimes of individual nodes are not equal. Developing optimal algorithms for both homogeneous and heterogeneous lifetimes allows us to obtain, by simulation, several interesting results. We show that even when an optimal number of sensor nodes has been deployed randomly, statistical redundancy can be exploited to extend the network lifetime by up to seven times. We also use simulation to show that the assumption of homogeneous lifetime can result in severe loss (two-thirds) of the network lifetime. Although these results are specifically for barrier coverage, they provide an indication of behavior for other coverage models.

Performance Measures and Schedules in Periodic Job Shops

This paper discusses the periodic job shop scheduling problem, a problem where an identical mixture of items, called a minimal part set MPS, is repetitively produced. The performance and behavior of schedules are discussed. Two basic performance measures, cycle time and makespan, are shown to be closely related. The minimum cycle time is identified as a circuit measure in a directed graph. We establish that there exists a class of schedules that minimizes cycle time and repeats an identical timing pattern every MPS. An algorithm is developed to construct such schedules. We show that minimizing the makespan as a secondary criterion, minimizes several other performance measures. For makespan minimization, we examine earliest starting schedules where each operation starts as soon as possible. We characterize the cases where after a finite number of MPSs, the earliest starting schedule repeats an identical timing pattern every fixed number of MPSs. We also develop a modification to an earliest starting schedule that repeats an identical timing pattern every MPS when the beginning operations on the machines are delayed.

Optimal sleep-wakeup algorithms for barriers of wireless sensors

The problem of sleep wakeup has been extensively studied for the full coverage model, where every point in the deployment region is covered by some sensor. Since the sleep-wakeup problem is NP-Hard for this model, several heuristics exist. For the model of barrier coverage, however, where sensors are deployed to form an impenetrable barrier for detecting moving objects (a flagship application of wireless sensor networks), design of an optimal sleep-wakeup algorithm is open. In this paper, we solve this open problem by proposing optimal algorithms not only for the often-used case of equal lifetime but also for the much harder case when sensor lifetimes are different. We prove the optimality of both algorithms. Our algorithms can be used to maintain not just barrier coverage but fault tolerant connectivity, as well, while maximizing the network lifetime. We use simulation to show that for random deployments, even when a minimal number of sensors have been deployed, our optimal algorithms can increase the network lifetime by 500% (from 10 weeks to more than a year). Finally, we show that using our optimal algorithms increases the network lifetime six times longer than that achievable using an existing sleep wake-up algorithm called Randomized Independent Sleeping (RIS).

Sensitivity Analysis for Scheduling Problems

This paper represents a first attempt at a systematic study of sensitivity analysis for scheduling problems. Because schedules contain both combinatorial and temporal structures, scheduling problems present unique issues for sensitivity analysis. Some of the issues that we discuss have not been considered before. Others, while studied before, have not been explored in the context of scheduling. The applicability of these issues is illustrated using well-known scheduling models. We provide fast methods to determine when a previously optimal schedule remains optimal. Other methods restore an optimal schedule after a parameter change. The value of studying the sensitivity of an optimal sequence instead of the sensitivity of an optimal schedule is demonstrated. We show that, for some problems, sensitivity analysis results depend on the positions of jobs with changed parameters. We identify scheduling problems where performing additional or different computations during optimization facilitates sensitivity analysis. To improve the robustness of an optimal schedule, selection among multiple optimal schedules is considered. We discuss which types of sensitivity analysis questions are intractable because the scheduling problem itself is intractable. We also study how heuristic error bounds vary when the data of a scheduling problem is continuously modified. Although we focus on scheduling problems, several of the issues we discuss and our classification scheme can be extended to other optimization problems.

Scheduling Parallel Machines for the Customer Order Problem

This paper considers scheduling problems where jobs are dispatched in batches. The objective is to minimize the sum of the completion times of the batches. While a machine can process only one job at a time, multiple machines can simultaneously process jobs in a batch. This simple environment has a variety of real world applications such as part kitting and customer order scheduling.A heuristic is presented for the parallel machine version of the problem. Also, a tight worst case bound on the relative error is found. For the case of two parallel machines, we examine two heuristics, which are based on simple scheduling rules. We find tight worst case bounds of 6/5 and 9/7 on the relative error and show that neither procedure is superior for all instances. Finally, we empirically evaluate these two heuristics. For large problems, the methods find solutions that are close to optimal.

Material Flow Optimization in By-Product Synergy Networks

By-product synergy (BPS) is an industrial ecology practice that involves utilization of industrial by-products as feedstocks for other industrial processes. A novel decision support tool is developed to analyze BPS networks that involve material processing and transport among regional clusters of companies. Mathematical programming techniques are used to determine the optimal network design and the material flows that minimize total cost or environmental impacts. This methodology is incorporated into a graphical software package called Eco-Flow. The tool has been applied to model and analyze available synergies in an existing BPS network centered in Kansas City, Missouri. A base case, which assumes no synergies, is compared with the optimal BPS solution found by Eco-Flow. The results for Kansas City suggest that when companies in the network cooperate to optimize the system profitability, up to

Parallel machine scheduling with high multiplicity

15 million per year of savings are possible. The findings also indicate that the BPS approach would result in 29% reduction in total cost, 25.8% reduction in average company cost, 30% reduction in carbon dioxide (CO) emissions, and 37% reduction in waste to landfill. The modeling approach is being extended to better represent the dynamics of industrial and ecological processes.