Processing Time Distributions Research Articles

Distributionally robust single machine scheduling with release and due dates over Wasserstein balls

Single machine scheduling aims at determining the job sequence with the best desired performance, and provides the basic building block for more advanced scheduling problems. In the present study, a single machine scheduling model with uncertain processing time is considered by incorporating the job release time and due date. The job processing time follows unknown probability distribution, and can be estimated via the historical data. To model the uncertainty, the processing time distribution is defined over a Wasserstein ball ambiguity set, which covers all feasible probability distributions within the confidence radius of the empirical distribution, known as the center of the ball. Then a data-driven distributionally robust scheduling model is constructed with individual chance constraints. In particular, two equivalent reformulations are derived with respect to the ℓ1-norm and ℓ2-norm metrics of the Wasserstein ball, namely, a mixed-integer linear programming and a mixed-integer second order cone programming model, respectively. To accelerate the solving of large-scale instances, a tailored constraint generation algorithm is introduced. In the numerical analysis, the proposed distributionally robust scheduling approach is compared with the state-of-the-art methods in terms of out-of-sample performance.

Process time distribution simulation in robotic assembly line balancing

Feedback control systems utilised in car body construction cause process time variance when compensating for external disturbances. By considering these in robotic assembly line balancing, the risk of cycle time violations can be controlled. This requires knowledge of the underlying process time distributions, which are not known in advance. Therefore, a simulation method is proposed to assess the impact of varying process time distributions on the balancing of robotic assembly lines. The initial step involves acquiring the process times of existing production processes. In the subsequent simulation, these are randomly and repeatedly selected as substitutes for the process times in the balancing of a new robotic assembly line. The impact of process time distribution variations on the result is investigated, and a single solution can be selected. The proposed method is evaluated based on the balancing of a robotic assembly line for a body-in-white rear compartment. Results are compared to normally distributed process times, which is a common assumption for modelling uncertain process times. Both approaches are evaluated utilising actual process time distributions. It is demonstrated that the proposed method yields fewer and less severe underestimations of cycle times, thereby reducing the number of uncontrolled violations of cycle times.

Co-Evolutionary Algorithm for Two-Stage Hybrid Flow Shop Scheduling Problem with Suspension Shifts

Demand fluctuates in actual production. When manufacturers face demand under their maximum capacity, suspension shifts are crucial for cost reduction and on-time delivery. In this case, suspension shifts are needed to minimize idle time and prevent inventory buildup. Thus, it is essential to integrate suspension shifts with scheduling under an uncertain production environment. This paper addresses the two-stage hybrid flow shop scheduling problem (THFSP) with suspension shifts under uncertain processing times, aiming to minimize the weighted sum of earliness and tardiness. We develop a stochastic integer programming model and validate it using the Gurobi solver. Additionally, we propose a dual-space co-evolutionary biased random key genetic algorithm (DCE-BRKGA) with parallel evolution of solutions and scenarios. Considering decision-makers’ risk preferences, we use both average and pessimistic criteria for fitness evaluation, generating two types of solutions and scenario populations. Testing with 28 datasets, we use the value of the stochastic solution (VSS) and the expected value of perfect information (EVPI) to quantify benefits. Compared to the average scenario, the VSS shows that the proposed algorithm achieves additional value gains of 0.9% to 69.9%. Furthermore, the EVPI indicates that after eliminating uncertainty, the algorithm yields potential improvements of 2.4% to 20.3%. These findings indicate that DCE-BRKGA effectively supports varying decision-making risk preferences, providing robust solutions even without known processing time distributions.

Variability propagation in manufacturing systems: the impact of the processing time distribution

ABSTRACT In manufacturing and service systems, the negative effects of variability propagation are usually addressed through approximation models and other tools, such as simulation and optimization algorithms. This paper investigates the conditions in which these approaches are ineffective, and considering only the mean and variance of the processing time distribution is misleading. A scenario analysis deepens the impacts of considering the entire processing time distribution (beyond its mean and variance) on the inter-departure times in balanced and unpaced lines modeled through Discrete Event Simulation. The results correlate the impacts on variability propagation of approximating the processing time distribution with system characteristics such as utilization levels, line sizes, and inter-arrival and processing time variability. Production planning approaches can benefit from these results to reduce variability propagation, particularly in flexible and reconfigurable manufacturing systems, largely adopted in Industry 4.0, that can highly influence processing time distributions by varying product mix and line configuration.

Optimal Rate-Matrix Pruning For Heterogeneous Systems

We consider large-scale load balancing systems where processing time distribution of tasks depend on both task and server types. We analyze the system in the asymptotic regime where the number of task and server types tend to infinity proportionally to each other. In such heterogeneous setting, popular policies like Join Fastest Idle Queue (JFIQ), Join Fastest Shortest Queue (JFSQ) are known to perform poorly and they even shrink the stability region. Moreover, to the best of our knowledge, in this setup, finding a scalable policy with provable performance guarantee has been an open question prior to this work. In this paper, we propose and analyze two asymptotically delay-optimal dynamic load balancing approaches: (a) one that efficiently reserves the processing capacity of each server for "good" tasks and route tasks under the Join Idle Queue policy; and (b) a speed-priority policy that increases the probability of servers processing tasks at a high speed. Introducing a novel analytical framework and using the mean-field method and stochastic coupling arguments, we prove that both policies above achieve asymptotic zero queueing, whereby the probability that a typical task is assigned to an idle server tends to 1 as the system scales.

An improved deep Q-learning algorithm for a trade-off between energy consumption and productivity in batch scheduling

The single-batch machine, commonly found in industrial manufacturing, can concurrently process a group of jobs in variable-speed batches, leading to fluctuating levels of both energy consumption and processing time. Identifying an optimal balance between total energy consumption and makespan presents a challenge due to the intricate nature of their relationship in real-world scenarios. This study introduces a mixed-integer programming model designed to determine optimal machine operating states for diverse workloads, meeting customer requirements while concurrently achieving energy savings for sustainability. An energy-aware batch scheduling deep Q-learning network (EBSDQN) framework has been created, encompassing sequencing policies, batching rules, and speed adjustment policies. This framework is explicitly crafted to tackle the NP-hard problem, a challenge that commonly perplexes commercial solvers like Gurobi when seeking optimal solutions within a 360-second time frame for small-scale instances. The EBSDQN design is fortified with efficient action decoding policies, refined reward assessments, exploitative neighborhood rules, and a streamlined buffer training process. This approach not only conserves training time but also adapts dynamically in accordance with the principles of the Markov Decision Process (MDP) and deep neural network. The developed algorithm exhibits impressive robustness, reaching convergence after 600 episodes of training. In separate comparisons with a commercial solver, two single-objective algorithms, and two multi-objective algorithms, our algorithm consistently demonstrates superior overall performance across the same instances. In the worst-case analysis, further examination delves into the influences of job features. In summary, this study underscores the high sensitivity of both TEC and Cmax to job processing time distribution, suggesting the prioritization of TEC over Cmax in optimization for sustainable planning and operations. Furthermore, this research has the potential to accelerate the integration of artificial intelligence into the manufacturing sector.

LRMP: Layer Replication with Mixed Precision for spatial in-memory DNN accelerators.

In-memory computing (IMC) with non-volatile memories (NVMs) has emerged as a promising approach to address the rapidly growing computational demands of Deep Neural Networks (DNNs). Mapping DNN layers spatially onto NVM-based IMC accelerators achieves high degrees of parallelism. However, two challenges that arise in this approach are the highly non-uniform distribution of layer processing times and high area requirements. We propose LRMP, a method to jointly apply layer replication and mixed precision quantization to improve the performance of DNNs when mapped to area-constrained IMC accelerators. LRMP uses a combination of reinforcement learning and mixed integer linear programming to search the replication-quantization design space using a model that is closely informed by the target hardware architecture. Across five DNN benchmarks, LRMP achieves 2.6-9.3× latency and 8-18× throughput improvement at minimal (<1%) degradation in accuracy.

Scheduling with Testing of Heterogeneous Jobs

This paper studies a canonical general scheduling model that captures the fundamental trade-off between processing jobs and performing diagnostics (testing). In particular, testing reveals the required processing time and urgency of need-to-schedule jobs to inform future scheduling decisions. The model captures a range of important applications. Prior work focused on special cases (e.g., jobs with independent and identically distributed processing time) to devise optimal policies. In contrast, the current paper studies the most general form of the model and describes two simple heuristics to solve it; adaptive weighted shortest processing time is an adaptive generalization of Smith’s rule that optimally solves several important extensions of previously studied models, whereas index policy optimally solves a closely related stochastic optimization bandit problem. The latter achieves an approximation guarantee that quickly approaches a constant factor that is bounded by two as the number of jobs grows and approaches optimally when the testing time decreases. Extensive numerical experiments suggest that our policies effectively solve the general setting (under 0.1% from optimal on average and under 10% from optimal in rare, worst-case instances). This paper was accepted by Jeannette Song, operations management. Supplemental Material: The data files and online appendix are available at https://doi.org/10.1287/mnsc.2023.4833 .

Wasserstein distance‐based distributionally robust parallel‐machine scheduling

Recent research on distributionally robust (DR) machine scheduling has used a variety of approaches to describe the region of ambiguity of uncertain processing times by imposing constraints on the moments of the probability distributions. One approach that has been employed outside machine scheduling research is the use of statistical metrics to define a distance function between two probability distributions. Adopting such an approach, we study Wasserstein distance-based DR parallel-machine scheduling, where the ambiguity set is defined as a Wasserstein ball around an empirical distribution of uncertain processing times corresponding to finitely many samples. The objective is to minimize a DR objective that concerns the worst-case expected total completion time-related cost over all the distributions arising from the Wasserstein ambiguity set, subject to DR chance constraints on the machine service capacity. We show that the problem can be equivalently re-formulated as a mixed-integer linear program (MILP), which has a more simplified formulation when the bounded support set reduces to a left bounded one. To solve the resulting model, we develop a tailored branch-and-Benders-cut algorithm incorporating some enhancement strategies, including in-out Benders cut generation, aggregated sample group cut generation, and two-stage Benders cut generation, which significantly outperforms the CPLEX solver. Experiment results on comparing our model with the deterministic and stochastic counterparts and the model with first-order moment ambiguity set illustrate the benefits of considering distributional ambiguity and Wasserstein ambiguity set.

Process Modeling a Radiation Oncology Clinic Workflow From Therapeutic Simulation to Treatment: Identifying Impending Strain and Possible Treatment Delays

The administration of safe, high-quality radiation therapy requires the systematic completion of a series of steps from computed tomography simulation, physician contouring, dosimetric treatment planning, pretreatment quality assurance, plan verification, and, ultimately, treatment delivery. Nevertheless, due consideration to the cumulative time required to complete each of these steps is often not given sufficient attention when determining patient start date. We set out to understand the systemic dynamics as to how varying patient arrival rate can affect treatment turnaround times using Monte Carlo simulations. We developed a process model workflow for a single physician, single linear accelerator clinic that simulated arrival rates and processing times for patients undergoing radiation treatment using the AnyLogic Simulation Modeling software (AnyLogic 8 University edition, v8.7.9). We varied the new patient arrival rate from 1 to 10 patients per week to understand the effect of treatment turnaround times from simulation to treatment. We used processing-time estimates determined in prior focus studies for each of the required steps. Altering the number of patients simulated from 1 per week to 10 per week resulted in a corresponding increase in average processing time from simulation to treatment from 4 to 7 days. The maximum processing time for patients from simulation to treatment ranged from 6 to 12 days. To compare individual distributions, we used the Kolmogorov-Smirnov statistical test.We found that altering the arrival rate from 4 patients per week to 5 patients per week resulted in a statistically significant change in the distributions of processing times (P=.03). The results of this simulation-based modeling study confirm the appropriateness of current staffing levels to ensure timely patient delivery while minimizing staff burnout. Simulation modeling can help guide staffing and workflow models to ensure timely treatment delivery while ensuring quality and safety.

Morphological characteristics and atomic evolution behavior of nanojoints in Ag nanowire interconnect network

Ag nanowires (AgNWs) have shown great application value in the field of flexible electronics due to their excellent optical and electrical properties, and the quality of its joints of AgNWs in the thin film network directly plays a key role in its performance. In order to further improve the joint quality of AgNWs under thermal excitation, the thermal welding process and atomic evolution behavior of AgNWs were investigated through a combination of in situ experimental and molecular dynamics simulations. The influence of processing time, temperature, and stress distribution due to spatial arrangement on nanojoints was systematically explored. What is more, the failure mechanisms and their atomic interface behavior of the nanojoints were also investigated.

Transient Behavior of a Queueing Model with Hyper-Exponentially Distributed Processing Times and Finite Buffer Capacity.

In the paper, a finite-capacity queueing model is considered in which jobs arrive according to a Poisson process and are being served according to hyper-exponential service times. A system of equations for the time-sensitive queue-size distribution is established by applying the paradigm of embedded Markov chain and total probability law. The solution of the corresponding system written for Laplace transforms is obtained via an algebraic approach in a compact form. Numerical illustration results are attached as well.

Integrated optimisation of pricing, manufacturing, and procurement decisions of a make-to-stock system operating in a fluctuating environment

ABSTRACT Manufacturers experience random environmental fluctuations that influence their supply and demand processes directly. To cope with these environmental fluctuations, they typically utilise operational hedging strategies in terms of pricing, manufacturing, and procurement decisions. We focus on this challenging problem by proposing an analytical model. Specifically, we study an integrated problem of procurement, manufacturing, and pricing strategies for a continuous-review make-to-stock system operating in a randomly fluctuating environment with exponentially distributed processing times. The environmental changes are driven by a continuous-time discrete state-space Markov chain, and they directly affect the system's procurement price, raw material flow rate, and price-sensitive demand rate. We formulate the system as an infinite-horizon Markov decision process with a long-run average profit criterion and show that the optimal procurement and manufacturing strategies are of state-dependent threshold policies. Besides that, we provide several analytical results on the optimal pricing strategies. We introduce a linear programming formulation to numerically obtain the system's optimal decisions. We, particularly, investigate how production rate, holding cost, procurement price and demand variabilities, customers' price sensitivity, and interaction between supply and demand processes affect the system's performance measures through an extensive numerical study. Furthermore, our numerical results demonstrate the potential benefits of using dynamic pricing compared to that of static pricing. In particular, the profit enhancement being achieved with dynamic pricing can reach up to 15%, depending on the problem parameters.

Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism

In the paper a finite-capacity discrete-time queueing system with geometric interarrival times and generally distributed processing times is studied. Every time when the service station becomes idle it goes for a vacation of random duration that can be treated as a power-saving mechanism. Application of a single vacation policy is one way for the system to achieve symmetry in terms of system operating costs. A system of differential equations for the transient conditional queue-size distribution is established. The solution of the corresponding system written for double probability generating functions is found using the analytical method based on a linear algebraic approach. Moreover, the representation for the probability-generating function of the stationary queue-size distribution is obtained. Numerical study illustrating theoretical results is attached as well.

Process modeling radiation oncology clinic workflow from therapeutic simulation to treatment: Identifying impending strain and possible treatment delays.

39 Background: The administration of safe, high-quality radiotherapy requires the systematic completion of a series of steps from CT simulation, physician contouring, dosimetric treatment planning, pre-treatment quality assurance, plan verification, and ultimately treatment delivery. Nevertheless, due consideration to the cumulative time required to complete each of these steps is often not given sufficient attention when determining patient start date. On one hand, the nature of cancer therapy relies on timely treatment delivery. On the other hand, overly ambitious treatment turnaround times can lead to staff burnout, and result in medical errors. We sought to better understand how changes in patient volume could impact turnaround time through a simulation-based study. Methods: We developed a process model workflow for a single physician, single linear-accelerator clinic that simulated arrival rates and processing times for patients undergoing radiation treatment using AnyLogic Simulation Modeling software (AnyLogic 8 University edition, v8.7.9). We varied the new patient arrival rate from 1 to 10 patients per week to understand the impact treatment turnaround times from simulation to treatment. We utilized processing time estimates for each of the required steps as was determined in a departmental focus group study. We assumed an eight hour workday, and that contouring, treatment planning, verification, and treatment delivery followed the following distributions, respectively (in hours): triangular (2, 4, 16); triangular (8, 16, 60); triangular (0.25, 0.5, 1). Results: Altering the number of patients simulated per week from 1-10 patients resulted in a corresponding increase average processing time from simulation to treatment from 4 to 7 days. The maximum processing time for patient from simulation to treatment ranged from 6-14 days. To compare individual distributions, we utilized the Kolmogorov-Smirnov (K-S) statistical test. We found that altering the arrival rate from 4 patients per week to 5 patients per week resulted in a statistically significant change in the distributions of processing times (p = 0.03) and resulted in a corresponding increase in the maximum patient processing time from approximately 6 days to 12 days, or from approximately 1 week to 2 weeks. Conclusions: The results of this simulation-based modeling study confirm the appropriateness of current staffing levels to ensure timely patient delivery while minimizing staff burnout. However, sustained increases in volume could require increased staffing to ensure timely treatment delivery while minimizing staff burnout. Simulation modeling can help guide staffing and workflow models to ensure timely treatment delivery.

Transverse liquid composite moulding processes for advanced composites material manufacturing

ABSTRACT The hydrodynamic compaction behaviour in transverse (through-thickness) impregnation variants of Liquid Composite Moulding (LCM) results in a non-homogeneous fibre volume fraction distribution. A fully coupled simulation model of the entire family of transverse resin infusion/compression processes is presented, which simulates the hydrodynamic compaction behaviour and predicts the process time, fluid pressure, effective stress, and fibre volume fraction distribution. The model is applied to the compression of a saturated preform stack, resin injection into a dry preform, and Compression Resin Transfer Moulding. The simulations are verified using analytic solutions and validated against experimental results from the literature. Special forms of the constitutive equations are constructed to lead to a constant diffusion coefficient, and the resulting model is solved semi-analytically. The results are used to verify the full non-linear simulation model. Additionally, it is shown that assumptions of quasi-steady flow and uniform deformation introduce significant errors in processes involving direct piston compaction of the fibre preform.

Heavy traffic scaling limits for shortest remaining processing time queues with heavy tailed processing time distributions

We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, the server preemptively serves the job with the shortest remaining processing time first. Since one needs to keep track of the remaining processing times of all jobs in the system in order to describe the evolution, a natural state descriptor for an SRPT queue is a measure valued process in which the state of the system at a given time is the finite nonnegative Borel measure on the nonnegative real line that puts a unit atom at the remaining processing time of each job in system. In this work we are interested in studying the asymptotic behavior of the suitably scaled measure valued state descriptors for a sequence of SRPT queuing systems. Gromoll, Kruk and Puha (Stoch. Syst. 1 (2011) 1–16) have studied this problem under diffusive scaling (time is scaled by r2 and the mass of the measure normalized by r, where r is a scaling parameter approaching infinity). In the setting where the processing time distributions have bounded support, under suitable conditions, they show that the measure valued state descriptors converge in distribution to the process that at any given time is a single atom located at the right edge of the support of the processing time distribution with the size of the atom fluctuating randomly in time. In the setting where the processing time distributions have unbounded support, under suitable conditions, they show that the diffusion scaled measure valued state descriptors converge in distribution to the process that is identically zero. In Puha (Ann. Appl. Probab. 25 (2015) 3381–3404) for the setting where the processing time distributions have unbounded support and light tails, a nonstandard scaling of the queue length process is shown to give rise to a form of state space collapse that results in a nonzero limit. In the current work we consider the case where processing time distributions have finite second moments and regularly varying tails. Results of Puha (Ann. Appl. Probab. 25 (2015) 3381–3404) suggest that the right scaling for the measure valued process is governed by a parameter cr that is given as a certain inverse function related to the tails of the first moment of the processing time distribution. Using this parameter we consider a novel scaling for the measure valued process in which the time is scaled by a factor of r2, the mass is scaled by the factor cr/r and the space (representing the remaining processing times) is scaled by the factor 1/cr. We show that the scaled measure valued process converges in distribution (in the space of paths of measures). In a sharp contrast to results for bounded support and light tailed service time distributions, this time there is no state space collapse and the limiting measures are not concentrated on a single atom. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain R+ valued random field which is determined from a single Brownian motion. Along the way we establish convergence of suitably scaled workload and queue length processes. We also show that as the tail of the distribution of job processing times becomes lighter in an appropriate fashion, the difference between the limiting queue length process and the limiting workload process converges to zero, thereby approaching the behavior of state space collapse.

Distributionally robust scheduling algorithms for total flow time minimization on parallel machines using norm regularizations

In this paper, we study a distributionally robust parallel machines scheduling problem, minimizing the total flow time criterion. The distribution of uncertain processing times is subject to ambiguity belonging to a set of distributions with constrained mean and covariance. We show that the problem can be cast as a deterministic optimization problem, with the objective function composed of an expectation and a regularization term given as an ℓp norm. The main question we ask and answer is whether the particular choice of the used ℓp norm affects the computational complexity of the problem and the robustness of its solution. We prove that if durations of the jobs are independent, the solution in terms of any ℓp norm can be solved in a pseudopolynomial time, by the reduction to a non-linear bipartite matching problem. We also show an efficient, polynomial-time algorithm for ℓ1 case. Furthermore, for instances with dependent durations of the jobs, we propose computationally efficient formulation and an algorithm that uses ℓ1 norm. Moreover, we identify a class of covariance matrices admitting a faster, polynomial-time algorithm. The computational experiments show that the proposed algorithms provide solutions with a similar quality to the existing algorithms while having significantly better computational complexities.

Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

Scheduling jobs with normally distributed processing times on parallel machines

We consider a stochastic parallel machine scheduling problem, where the jobs have uncertain processing time described by a normal probability distribution. The objective is to maximize the probability that all the jobs are completed before a common due date. The considered problem has many practical applications, but it is notoriously known to be difficult as it involves several non-linearities which complicates its analysis and solution.In this work, we have developed novel lower and upper bounds on the objective function. The upper bound is computed via a solution to a problem where a subset of machines is represented as a single machine having a modified due date. Furthermore, we study lower and upper bounds on the number of jobs that must be scheduled on a machine in an optimal schedule. Subsequently, we use the bounds to construct an efficient branch-and-price algorithm where the pricing problem is found to be related to an inflatable stochastic Knapsack problem. An advantage of the branch-and-price algorithm is a constraint branching mechanism that mitigates symmetries in the solution space. The performance evaluation of the proposed algorithm shows that our algorithm outperforms the state-of-the-art method.In this paper, we also study a special case of the problem assuming two machines. We developed a scalable method whose efficiency arises from the concavity of the relaxed objective function and a fast procedure to recover optimal integer solution from it. These improvements allowed us to solve instances with 500 jobs within a few seconds.