Processing Times Of Operations Research Articles

Comparing deterministic, robust and online scheduling using entropy

We conjecture that when the uncertainty of scheduling information increases, one should change from deterministic, through robust to, finally, online scheduling techniques. Previously, extensive mathematical investigations have been carried out on the stability of a deterministic schedule for uncertain operation processing times. In this paper, we will use an empirical approach and an entropy measure to justify the transition between deterministic, robust and online scheduling. The use of an entropy measure in our context can be perceived, in a broader sense, as a pro-active approach to deal with changes in the level of information uncertainty and relative importance of each term in the total schedule execution cost. The level of information uncertainty may change due to the performance deterioration of processors (machines or human) and the replacement of old machines with new ones; and the changes in relative importance of cost elements may be due to changes in shop floor priorities and pressure from partners in the supply chain network. One can decide upon the scheduling strategies to be employed based on the latest entropy value of the information considered and the relative importance of each cost term.

A Lagrangian Relaxation Based Approach to Schedule Asset Overhaul and Repair Services

Overhaul and repair services are important segments of the remanufacturing industry, and are characterized by complicated disassembly, repair and assembly process plans, stochastic operations, and the usage of rotable inventory. In view of today's time-based competition, effectively scheduling such services and managing rotable inventory and uncertainties are becoming imperative to achieve on-time deliveries and low overall costs. In this paper, a novel formulation for overhaul and repair services is presented where key characteristics, such as uncertain asset arrivals and operation processing times, and rotable parts are abstracted to model an overhaul center and multiple repair shops in a distributed framework to reflect organizational structures. Interactions between the overhaul center and repair shops are described by sets of coupling constraints across the organizations. Rotable inventory dynamics is formulated in terms of repair operation completion times and asset assembly beginning times to facilitate minimization of inventory holding costs through scheduling. A solution methodology combining Lagrangian relaxation, stochastic dynamic programming, and heuristics is developed to schedule operations in a coordinated manner to minimize total tardiness, earliness, and inventory holding costs. Additionally, penalty terms associated with coupling constraint violations are introduced to the objective function to improve algorithm convergence and schedule quality, and a surrogate optimization framework is used to overcome the inseparability difficulty caused by the penalty terms. Numerical testing results show that the new approach is computationally effective to handle rotable inventory and uncertainties, and provides high quality schedules with low overall costs for stochastic remanufacturing systems. Note to Practitioners-Overhaul and repair services for jet engines, helicopters, airplanes, are important segments of the remanufacturing industry, and are characterized by complicated disassembly, repair and assembly process plans, stochastic operations, and the usage of rotable inventory. In view of today's highly competitive business climate, effectively scheduling such services and managing rotable inventory and uncertainties are becoming critical to achieve on-time deliveries and low overall costs. In this paper, a novel formulation for overhaul and repair services is presented where key characteristics, such as uncertain asset arrivals and operation processing times, and rotable parts are abstracted to model an overhaul center and multiple repair shops in a distributed framework to reflect organizational structures. A solution methodology based on decomposition and coordination is developed to schedule operations to minimize total tardiness, earliness, and inventory holding costs. Numerical testing results show that the method is computationally efficient for managing rotable inventory and uncertainties, and generates high quality schedules with low overall costs. The value of rotable inventory to reduce tardiness costs and buffer uncertainties is demonstrated, and the robustness of the new method is evaluated by cases with different settings of machine utilization levels and uncertainty levels. The scalability of the method to solve large problems with hundreds of assets is also demonstrated.

Scheduling jobs under decreasing linear deterioration

This paper considers the scheduling problems under decreasing linear deterioration. Deterioration of a job means that its processing time is a function of its execution start time. Optimal algorithms are presented respectively for single machine scheduling of minimizing the makespan, maximum lateness, maximum cost and number of late jobs. For two-machine flow shop scheduling problem to minimize the makespan, it is proved that the optimal schedule can be obtained by Johnson's rule. If the processing times of operations are equal for each job, flow shop scheduling problems can be transformed into single machine scheduling problems.

Concurrent open shop scheduling to minimize the weighted number of tardy jobs

We consider a relaxed version of the open shop scheduling problem--the "concurrent open shop" scheduling problem, in which any two operations of the same job on distinct machines are allowed to be processed concurrently. The completion time of a job is the maximum completion time of its operations. The objective is to schedule the jobs so as to minimize the weighted number of tardy jobs, with 0-1 operation processing times and a common due date d. We show that, even when the weights are identical, the problem has no (1-e)ln m-approximation algorithm for any e > 0 if NP is not a subset of DTIME(nlog log n), and has no c ċ ln m-approximation algorithm for some constant c > 0 if P ≠ NP, where m is the number of machines. This also implies that the problem is strongly NP-hard. We also give a (1+d)- approximation algorithm for the problem.

System for Upper Airway Segmentation and Measurement with MR Imaging and Fuzzy Connectedness

Rationale and Objectives The purpose of this study was to evaluate whether a computerized system developed to help delineate the upper airway and surrounding structures with magnetic resonance (MR) imaging was effective for aiding in the diagnosis of upper airway disorders in children. Materials and Methods The authors performed axial T2-weighted MR imaging to gather information about different aspects of the airway and its surrounding soft-tissue structures, including the adenoid and palatine tonsils, tongue, and soft palate. Images were processed and segmented to compute the architectural parameters of the airway (eg, surface description, volume, central [medial] line, and cross-sectional areas at planes perpendicular to the central line). The authors built a software package for the visualization, segmentation, registration, prefiltering, interpolation, standardization, and quantitative analysis of the airway and tonsils. Results The system was tested with 40 patient studies. For every study, the system segmented and displayed a smooth three-dimensional rendition of the airway and its central line and a plot of the cross-sectional area of the airway orthogonal to the central line as a function of the distance from one end of the central line. The precision and accuracy for segmentation was 97%. The mean time taken per study was about 4 minutes and included the operator interaction time and processing time. Conclusion This method provides a robust and fast means of assessing the airway size, shape, and level of restriction, as well as a structural data set suitable for use in modeling studies of airflow and mechanics.

Computer aided optimization of scheduling and machining parameters in job shop manufacturing systems

A nonlinear mathematical model for joint determination of optimal job shop schedule and process parameters selection is presented. The proposed model extends the production schedule (n parts and m machines) to accommodate the process parameters since both the process planning and the production schedule are highly related. The model developed relaxes the assumption that processing times are deterministic, since the processing times of operations depend on the sequence of the operations. The present work shows that the process parameters selection has a significant influence on the optimal schedule. The model developed has been tested and validated using numerical examples.

Job Shop Scheduling With Two Jobs And Nonregular Objective Functions

We consider the job shop scheduling problem with two jobs. We consider a broad class of non-regular, quasi-convex functions of the completion time of the two jobs. We show that the optimal solution, for this class of objective functions, can be computed in O(r log r + log H) time, where r is the number of operation pairs using the same machine, and H is the maximum operation processing time.

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

This paper is concerned with reliable multistation series queueing networks. Items arrive at the first station according to a Poisson distribution and an operation is performed on each item by a server at each station. Every station is allowed to have more than one server with the same characteristics. The processing times at each station are exponentially distributed. Buffers of nonidentical finite capacities are allowed between successive stations. The structure of the transition matrices of these specific type of queueing networks is examined and a recursive algorithm is developed for generating them. The transition matrices are block-structured and very sparse. By applying the proposed algorithm the transition matrix of a K-station network can be created for any K. This process allows one to obtain the exact solution of the large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures can be calculated. Scope and purpose The exact analysis of queueing networks with multiple servers at each workstation and finite capacities of the intermediate queues is extremely difficult as for even the case of exponential operation (service or processing) times the Markovian chain that models the system consists of a huge number of states which grows exponentially with the number of stations, the number of servers at each station and the queue capacity of each intermediate queue of the resulting system. The scope and purpose of the present paper is to analyze and provide a recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks. By applying the proposed algorithm one may create the transition matrix of a K-station queueing network for any K. This process allows one to obtain the exact solution of the resulting large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures of the system can be obtained.

Machine loading problem of FMS: A fuzzy-based heuristic approach

A fuzzy-based solution methodology has been formulated to address the machine-loading problem in a flexible manufacturing system. The objectives considered are minimization of system unbalance and maximization of throughput, whereas the systems technological constraints are posed by availability of machining time and tool slots. The job ordering/job sequence determination before loading is carried out by evaluating the membership contribution of each job to its characteristics such as batch size, essential operation processing time and optional operation processing time. The operation?machine allocation decisions are made based on the evaluation of membership contribution of operation machine allocation vector. The formulation of membership function is based on logical derivations and enjoys reasonable analytical support. The proposed heuristic is tested on 10 problems adopted from literatures and the results reveal substantial improvement in solution quality over, some of existing heuristic-based approaches.

Scheduling batches with simultaneous job processing for two-machine shop problems

We consider the problem of scheduling independent jobs on two machines in an open shop, a job shop and a flow shop environment. Both machines are batching machines, which means that several operations can be combined into a batch and processed simultaneously on a machine. The batch processing time is the maximum processing time of operations in the batch, and all operations in a batch complete at the same time. Such a situation may occur, for instance, during the final testing stage of circuit board manufacturing, where burn-in operations are performed in ovens. We consider cases in which there is no restriction on the size of a batch on a machine, and in which a machine can process only a bounded number of operations in one batch. For most of the possible combinations of restrictions, we establish the complexity status of the problem.

Evaluating the impact of process changes on cluster tool performance

Cluster tools are highly integrated machines that can perform a sequence of semiconductor manufacturing processes. Their integrated nature can complicate analysis when evaluating how process changes affect the overall tool performance. This paper presents two integrated models for understanding the behavior of a simple, single loadlock cluster tool. The first model is a network model that evaluates the total lot processing time for a given sequence of activities. By including a manufacturing process model (in the form of a response surface model, or RSM), the model calculates the lot makespan, the total time to process a lot of wafers, as a function of the process parameter values and other operation times. This model allows us to quantify the sensitivity of total lot processing time with respect to process parameters and times. In addition, we present an integrated simulation model that includes a process model. For a given scheduling rule that the cluster tool uses to sequence wafer movements, we can use the simulation to evaluate the impact of process changes, including changes to product characteristics and changes to process parameter values. In addition, we can construct an integrated network model to quantify the sensitivity of total lot processing time with respect to process times and process parameters in a specific scenario. We also present an evaluation of the effectiveness of two different scheduling rules, push and pull. The examples presented here illustrate the types of insights that we can gain from using such methods. Namely, the lot makespan is a function not simply of each operation's process time, but specifically of the chosen process parameter values. Modifying the process parameter values may also have significant impacts on the manufacturing system performance, a consequence of importance that is not readily obvious to a process engineer when tuning a process. This result can be seen either with the decrease of raw process time causing little change to the makespan, or the extreme example in which this could cause an increase in makespan because of an inefficient scheduling rule. Additionally, because the cluster tool's maximum throughput, which is the inverse of the lot makespan, depends on the process parameters, the tradeoffs between process performance and throughput should be considered when evaluating potential process changes and their manufacturing impact.

A permutation flow-shop scheduling problem with convex models of operation processing times

The paper is an extension of the classical permutation flow-shop scheduling problem to the case where some of the job operation processing times are convex decreasing functions of the amounts of resources (e.g., financial outlay, energy, raw material) allocated to the operations (or machines on which they are performed). Some precedence constraints among the jobs are given. For this extended permutation flow-shop problem, the objective is to find a processing order of the jobs (which will be the same on each machine) and an allocation of a constrained resource so as to minimize the duration required to complete all jobs (i.e., the makespan). A computational complexity analysis of the problem shows that the problem is NP-hard. An analysis of the structure of the optimal solutions provides some elimination properties, which are exploited in a branch-and-bound solution scheme. Three approximate algorithms, together with the results of some computational experiments conducted to test the effectiveness of the algorithms, are also presented.

Scheduling multi‐operation jobs on a single machine

We consider the problem of scheduling n multi‐operationjobs on a single machine. Eachjob comprises up to F operations that belong to different families.Changeovers of production from one family to another have associated set‐up times.A job completes when all of its operations have been processed. This type of problem arisesin the manufacture of food products. It combines the batching aspect of the well‐knownfamily scheduling models with an assembly element (where the job's operationsare assembled to make the final product).Our analysis covers three classic optimality criteria: the maximum lateness, theweighted number of late jobs, and the sum of job completion times. We show that the problemof minimizing the maximum lateness is equivalent to its counterpart without assembly.This enables us to derive extensions of known complexity results and to indicate appropriatealgorithms. The number of late jobs problem is shown to be binary NP‐hard when thereare two families, and unary NP‐hard when there are an arbitrary number of families,even when all set‐up times are identical. For a fixed number of families, we give adynamic programming algorithm to minimize the weighted number of late jobs, which requirespseudo‐polynomial running time. A similar algorithm solves the sum of completion timesproblem in polynomial time, under the additional assumption that the processing times ofoperations between families are agreeable.

Ultrasonic flow polishing

Dies and moulds used in the fabrication of metal, glass and plastic products require a high quality surface finish in terms of both appearance and integrity. Current manual polishing methods of surface finishing are time consuming and expensive. Consequently, a reduction in operator input and processing time has the potential to significantly reduce the cost of the finished mould. An ultrasonically energised polishing process, suitable for the polishing of complex dies and moulds has been successfully developed and patented. The process is based upon the combination of the two ‘non-traditional’ techniques of abrasive flow machining and ultrasonic machining. It has proved to be capable of improving surface finishes by factors of up to 10:1. This paper describes the development of the process and reports results achieved.

Nonstrict vector summationin multi-operation scheduling

We consider several multi-operation scheduling problems with m machines and n jobs, including flow shop, open shop, assembly line, and a few special cases of job shop with the minimum makespan criterion. It is demonstrated that the problems in question can be efficiently solved by approximation algorithms with fairly good performance guarantee in the worst case. The algorithms constructed are based on a geometric technique called “nonstrict vector summation“. It consists of assigning an (m−1)-dimensional vector to each job and then finding an order in which these vectors should be summed so that all partial sums would lie within a given family of half-spaces (specified for a given scheduling problem).The partial sums are allowed sometimes to go out of this or that half-space of the family, which explains the term “nonstrict” in the title of the paper. For the open shop problem, this technique guarantees its polynomial-time solution, provided that the maximum machine load l max (i.e., the maximum over all machines of the total processing time of operations of a machine) is large enough. In the case of three machines and l max as large as at least 7 times the maximum processing time of an operation, we can find the optimal schedule in O(nlog n) time.

Scheduling of flexible manufacturing systems with the consideration of tool set-up times

This paper presents a scheduling method that uses Petri net modeling and heuristic search to handle the tool setup. In manufacturing systems, a tool is attached to a particular machine to process a particular operation. The activity to attach a tool to a particular machine and detach the tool from the machine requires time. The processing time of operations varies according to the attached tool and the machine used. The method proposed in this paper uses Petri net to model these characteristics and applies a search algorithm to the reachability graph of the Petri net model to generate an optimal or near-optimal schedule. New heuristic functions are developed for efficient search. The experimental results that show the effectiveness of the proposed method are presented.

Stability of an optimal schedule in a job shop

This paper is devoted to the calculation of the stability radius of an optimal schedule for a job shop problem, when the objective is to minimize mean or maximum flow times. The approach used may be regarded as an a posteriori analysis, in which an optimal schedule has already been constructed and the question is to determine such changes in the processing times of operations as do not destroy the optimality of the schedule at hand. More precisely, the stability radius denotes the largest quantity of independent variations of the processing times of the operations such that an optimal schedule of the problem remains optimal. Although in scheduling theory mainly deterministic problems are considered and the processing times are supposed to be given in advance, such scheduling problems do not often arise in practice. Even if the processing times are know before applying a scheduling procedure, OR workers are forced to take into account possible changes and errors within the practical realization of a schedule, e.g. additionally arriving jobs, machine breakdowns, the precision of equipment which is used to calculate the processing times, and so on. In other words, usually in practice a schedule has to be realized under conditions of uncertainty. This paper investigates the influence of errors and possible changes of the processing times on the optimality of a schedule. To this end, extensive numerical experiments with randomly generated job shop scheduling problems are performed and discussed. The developed software provides the possibility of comparing the values of the stability radii, the numbers of optimal schedules and some other “numbers” for two often used criteria. The main question we try to answer is how large the stability radius is, on average, for randomly generated job shop problems.

Impact of Routing Flexibility on the Performance of an FMS—A Simulation Study

The evolving manufacturing environment is characterized by a drive toward increasing flexibility. One possible manifestation of flexibility within an FMS is in the form of routing flexibility. Providing this typically is an expensive proposition, and system designers therefore aim to provide only the required levels commensurate with a given set of operating conditions. This paper presents a framework based on a Taguchi experimental design for studying the nature of the impact of varying levels of routing flexibility on the performance of an FMS. Simulation results indicate that increases in routing flexibility, when made available at the cost of an associated penalty on operation processing time, is not always beneficial. There is an optimal flexibility level, beyond which system performance deteriorates, as judged by the makespan measure of performance. It is suggested that the proposed methodology can be used in practice for not only setting priorities on specific design and control factors but also for highlighting likely factor level combinations that could yield near-optimal shop performance.

Influence of conditions of manufacture and storage time on the textural characteristics of Xixona turrón

Texture is one of the main quality characteristics of Xixona turrón (an almond and sugar based confectionery product). During its production process a soft and complex W/O emulsion becomes a ‘semihard’ O/W system. Because of the homogenization difficulties with the highly viscous mass during the inversion process step, a fraction of non-inverted sample remains trapped in the final product, thereby affecting its overall texture. In this paper, hardness and other textural properties of turrón, in relation to the extent of inversion in the mass, were characterized. Texture changes during the product's storage were also analysed and correlated with the operation variables of the process and storage time. Texture of the turrón is mostly affected by the degree of inversion of the product which is, at the same time, a function of the operation variables (temperature (T), stirring speed (ω) and process time: inversion time + homogenization time ( t h)). A great advance of the inversion implies an increase in the product's hardness with loss of elasticity and cohesiveness. Nevertheless, in these conditions the structure becomes more stable during the storage period. From the established model in this paper, it is possible to select the process conditions to obtain the desired texture for the product on the basis of consumer preferences and/or its structural stability.

PRODUCTION PLANNING DECISIONS IN FLEXIBLE MANUFACTURING SYSTEMS WITH RANDOM MATERIAL FLOWS

Abstract In this paper, we consider the FMS planning problem of determining optimal machine workload assignments in order to rninimize mean part flow time. We decompose this problem into the subproblems of first forming machine groups and next assigning operations to these groups. Three types of grouping configurations—no grouping, partial grouping and total grouping—are considered. In both no grouping and partial grouping, each machine is tooled differently. While each operation is assigned to only one machine in no grouping, partial grouping permits multiple operation assignments. On the other hand, total grouping partitions the machines into groups of identically-tooled machines; each machine within a group is capable of performing the same set of operations. Within this grouping framework, we consider three machine loading objectives—minimizing the total deviation from the optimal group utilization levels, minimizing part travel and maximizing routing flexibility, for generating a variety of system configurations. A queueing network model of an FMS is used to determine the optimal configurations and machine workload assignments for the no grouping and total grouping cases. It is shown that under total grouping, the configuration of M machines into G groups that minimizes flow time is one in which the sizes of the machine groups are maximally unbalanced and the workload per machine in the larger groups is higher. This extends previous results on the optimality of unbalancing both machine group sizes and machine workload to the mean flow time criterion. A simulation experiment is next conducted to evaluate the alternative machine configurations to understand how their relative performance depends upon the underlying system characteristics, such as system utilization level and variation among operation processing times. We also investigate the robustness of these configurations against disruptions, such as machine unreliability and variation in processing batch sizes. While different configurations minimize mean flow time under different parameter values, partial grouping with state-dependent part routing performs well across a wide range of these values. Experimental results also show that the impact of disruptions can be reduced by several means, such as aggregating operations of a part to be performed at the same machine, in addition to providing routing flexibility.