Completion Time Variance Research Articles

Time-flexible min completion time variance in a single machine by quadratic programming

In the context of job scheduling, the time required to complete a task is related not only to the intrinsic difficulty of the task, but also to the operator’s willingness to speed up (or slow down) its execution. In fact, service operators are sometimes authorized to flexibly calibrate job processing times when this is beneficial for the efficient design of services and production plans. In this paper, we show that some forms of time flexibility have major consequences on the operator’s ability to efficiently solve the problem of scheduling non-preemptive jobs to minimize the variance of their completion times. In fact, although this remains a challenging combinatorial problem, authorizing forms of processing time flexibility allows for solving it up to optimality by convex quadratic programming approaches, with a view to tackling large-scale instances, where no exact algorithm can be applied. Our contribution allows establishing a form of least flexibilization of job processing times while guaranteeing the solvability of the resulting problem by convex quadratic programming approaches. To this end, we provide novel bounding conditions for the characterization of an optimal sequence that strengthen and integrate state-of-the-art dominance properties. Our numerical tests indicate that this new methodology is capable of approaching the solution of the original min completion time variance problem with a max relative difference of about 0.05% (on average), with respect to the time-flexible solution.

A multi-machine scheduling solution for homogeneous processing: Asymptotic approximation and applications

In the context of job scheduling in parallel machines, we present a class of binary programs for the minimization of the τ-norm of completion time variances (a flexible measure of homogeneity in multi-machine job processing). Building on overlooked properties of the min completion time variance in a single machine and on an equivalent bilevel formulation, our approach provides asymptotic approximations (with quadratic convergence) in the form of lower and upper bounds. While resulting in almost identical solutions, these bounds can be computed more efficiently than the original τ-norm minimization problem. Further, dominance properties are enforced as linear constraints to improve the characterization of the exact solutions by these asymptotic approximations. On the empirical side, the proposed methodology is applied to three real contexts of processing services: nursing home management, IT troubleshooting service, and 311 call centers. Our numerical studies reveal that the proposed lower bound problem can be solved in less than one third of computation time, in comparison to the exact problem, while resulting in almost identical jobs sequences. Additionally, considering large-scale simulated instances with up to 450 jobs, our upper bound problem outperforms the state-of-the-art heuristic method in terms of solution quality in 100% of the analyzed instances.

Minimization of completion time variance in flowshops using genetic algorithms

The majority of the flowshop scheduling literature focuses on regular performance measures like makespan, flowtime etc. In this paper a flowshop scheduling problem is addressed where the objective is to minimize completion time variance (CTV). CTV is a non-regular performance measure that is closely related to just-in-time philosophy. A Microsoft Excel spreadsheet-based genetic algorithm (GA) is proposed to solve the problem. The proposed GA methodology is domain-independent and general purpose. The flowshop model is developed in the spreadsheet environment using the built-in formulae and function. Addition of jobs and machines can be catered for without the change in the basic GA routine and minimal change to the spreadsheet model. The proposed methodology offers an easy-to-handle framework whereby the practitioners can implement a heuristic-based optimization tool with the need for advanced programming tools. The performance of the proposed methodology is compared to previous studies for benchmark problems taken from the literature. Simulation experiments demonstrate that the proposed methodology solves the benchmark problems efficiently and effectively with a reasonable accuracy. The solutions are comparable to previous studies both in terms of computational time and solution quality.

Completion Time Variance and the Product Rate Variation Problem

Completion Time Variance and the Product Rate Variation Problem

Completion time variance and the product rate variation problem

Just-in-time manufacturing necessitates finding short repeated sequences of products. The literature addressed this mixed model scheduling objective using the product rate variation (PRV), which maximises the deviation between the actual proportions and ideal proportions of jobs in the sequence. Researchers have reported application of PRV minimisation in manufacturing as well as in service industry. In this paper, we address this problem considering the completion times of the jobs in the sequence instead of the PRV, and show instances where maximising the sum of completion time variances (SCTVs) is an effective alternative to PRV for the mixed model sequencing problem. We propose algorithms using tabu search and genetic algorithms to solve the SCTV problem and show computational results.

Time-Flexible Min Completion Time Variance in a Single Machine by Quadratic Programming

Time-Flexible Min Completion Time Variance in a Single Machine by Quadratic Programming

The Variance Learning Curve

The expansive learning curve literature in operations management has established how various facets of prior experience improve average performance. In this paper, we explore how increased cumulative experience affects performance variability or consistency. We use a two-stage estimation method of a heteroskedastic learning curve model to examine the relationship between experience and performance variability among paramedics at the London Ambulance Service. We find that, for paramedics with lower experience, an increase in experience of 500 jobs reduces the variance of task completion time by 8.7%, in addition to improving average completion times by 2.7%. Similar to prior results on the average learning curve, we find a diminishing impact of additional experience on the variance learning curve. We provide an evidence base for how to model the learning benefits of cumulative experience on performance in service systems. Our findings imply that the benefits of learning are substantially underestimated if the consistency effect is ignored. Specifically, our estimates indicate that queue lengths (or wait times) might be overestimated by as much as 4% by ignoring the impact of the variance learning curve in service systems. Furthermore, our results suggest that previously established drivers of productivity should be revisited to examine how they affect consistency, in addition to average performance. This paper was accepted by Charles Corbett, operations management.

An almost exact solution to the min completion time variance in a single machine

We consider a single machine scheduling problem to minimize the completion time variance of n jobs. This problem is known to be NP-hard and our contribution is to establish a novel bounding condition for a characterization of an optimal sequence. Specifically, we prove a necessary and sufficient condition (which can be verified in O(nlogn)) for the characterization of five scheduling positions in the optimal sequence. Applying this characterization, we propose a new approach to derive the highest lower bound for the minimal completion time variance, outperforming the existing bounds for this problem. The numerical tests indicate that the novel lower bound is, on average, less than 0.01% far away from the optimal solution, outperforming all the existing lower bounds on the minimum completion time variance.

Balancing trade-offs in one-stage production with processing time uncertainty

Production scheduling faces three challenges, two of which are trade-offs and the third is processing time uncertainty. The two sources of trade-offs are between inconsistent key performance indicators (KPIs), and between the expected return and the risk of KPI portfolios. Given the KPIs of total completion time (TCT) and variance of completion times (VCT) are inconsistent for one-stage production, we propose our trade-off balancing (ToB) heuristics. Based on comprehensive case studies, we show that our ToB heuristics efficiently and effectively balance the trade-offs from these two sources. Daniels and Kouvelis (DK) proposed a scheduling scheme to optimize the worst-case scenarios against processing time uncertainty, and they designed the endpoint product (EP) and endpoint sum (ES) heuristics for robust scheduling accordingly. Using 5 levels of coefficients of variation (CVs) to represent processing time uncertainty, we show that our ToB heuristics are robust as well, and even better than the EP and ES heuristics at high levels of processing time uncertainty. In addition, our ToB heuristics generate undominated solution spaces of KPIs, which provides a solid base in deciding control and specification limits for stochastic process control (SPC). Moreover, based on the normalized deviations from optima, our trade-off balancing scheme can be generalized to balance any inconsistent KPIs.

The Variance Learning Curve

The expansive learning curve literature in Operations Management has established how various facets of prior experience improve average performance. In this paper, we explore how increased cumulative experience affects performance variability, or consistency. We use a two-stage estimation method of a heteroskedastic learning curve model to examine the relationship between experience and performance variability among paramedics at the London Ambulance Service. We find that for paramedics with lower experience, an increase in experience of 500 jobs reduces the variance of task completion time by 8.7%, in addition to improving average completion times by 2.7%. Similar to prior results on the average learning curve, we find a diminishing impact of additional experience on the variance learning curve. We provide an evidence base for how to model the learning benefits of cumulative experience on performance in service systems. Our findings imply that the benefits of learning are substantially underestimated if the consistency effect is ignored. Specifically, our estimates indicate that queue lengths (or wait times) might be overestimated by as much as 4% by ignoring the impact of the variance learning curve in service systems. Furthermore, our results suggest that previously established drivers of productivity should be revisited to examine how they affect consistency, in addition to average performance.

Proxy Experience Replay: Federated Distillation for Distributed Reinforcement Learning

Traditional distributed deep reinforcement learning (RL) commonly relies on exchanging the experience replay memory (RM) of each agent. Since the RM contains all state observations and action policy history, it may incur huge communication overhead while violating the privacy of each agent. Alternatively, this article presents a communication-efficient and privacy-preserving distributed RL framework, coined federated reinforcement distillation (FRD). In FRD, each agent exchanges its proxy experience replay memory (ProxRM), in which policies are locally averaged with respect to proxy states clustering actual states. To provide FRD design insights, we present ablation studies on the impact of ProxRM structures, neural network architectures, and communication intervals. Furthermore, we propose an improved version of FRD, coined mixup augmented FRD (MixFRD), in which ProxRM is interpolated using the mixup data augmentation algorithm. Simulations in a Cartpole environment validate the effectiveness of MixFRD in reducing the variance of mission completion time and communication cost, compared to the benchmark schemes, vanilla FRD, federated reinforcement learning (FRL), and policy distillation (PD).

A two-phase simulated annealing algorithm to minimise the completion time variance of jobs in flowshops

In this paper, simulated annealing (SA) algorithm is employed to solve the flowshop scheduling problem with the objective of minimising the completion time variance (CTV) of jobs. Four variants of the two-phase SA algorithm (SA-I to SA-IV) are proposed to solve flowshop scheduling problem with the objective of minimising the CTV of jobs without considering the right shifting of completion time of jobs on the last machine. The proposed SA algorithms have been tested on 90 benchmark flowshop scheduling problems. The solutions yielded by the proposed SA variants are compared with the best CTV of jobs reported in the literature, and the proposed SA-III is found to perform well in minimising the chosen performance measure (CTV) particularly in medium and large size problems.

A two-phase simulated annealing algorithm to minimise the completion time variance of jobs in flowshops

A two-phase simulated annealing algorithm to minimise the completion time variance of jobs in flowshops

A branch and price algorithm for single-machine completion time variance

This paper studies a single machine scheduling problem to minimize the completion time variance (CTV) of all jobs. A new time-indexed quadratic programming formulation is proposed, in which the quadratic terms are further linearized into a mixed integer programming (MIP) with non-negative continuous variables. To solve the problem to optimality, a branch and price (BP) algorithm is designed, which combines the Lagrangian relaxation with clusters. Extensive computational experiments on three sets of benchmark problem instances and randomly generated ones are conducted and the results show that the proposed BP algorithm outperforms state-of-the-art approaches.

Simulated annealing algorithms to minimise the completion time variance of jobs in permutation flowshops

In this paper, the permutation flowshop scheduling problem with the objective of minimising the completion time variance (CTV) of jobs is considered, and a simulated annealing algorithm (SA) and two modified SAs (MSA1 and MSA2) are proposed. In the first phase, the proposed algorithms are used to minimise CTV of jobs without any right shifting of completion times of jobs on the last machine (RSCT). As followed in the literature some times, the RSCT (except that of the last job) is attempted in the second phase, and in the third phase, we convert sequences so as to follow the V-shaped property with respect to processing time of jobs on the last machine, followed by RSCT except that of the last job, so that the makespan and the machine utilisation in the shop floor remain the same. We present an extensive computational evaluation of the existing and the proposed heuristic algorithms.

Sustainable production through balancing trade-offs among three metrics in flow shop scheduling

In sustainable manufacturing, inconsistencies exist among objectives defined in triple-bottom-lines (TBL) of economy, society, and environment. Analogously, inconsistencies exist in flow shop scheduling among three objectives of minimizing total completion time (TCT), maximum completion time (MCT), and completion time variance (CTV), respectively. For continuous functions, the probability is zero to achieve the objectives at their optimal values, so is it at their worst values. Therefore, with inconsistencies among individual objectives of discrete functions, it is more meaningful and feasible to seek a solution with high probabilities that system performance varies within the control limits. We propose a trade-off balancing scheme for sustainable production in flow shop scheduling as the guidance of decision making. We model trade-offs (TO) as a function of TCT, MCT, and CTV, based on which we achieve stable performance on min(TO). Minimizing trade-offs provides a meaningful compromise among inconsistent objectives, by driving the system performance towards a point with minimum deviations from the ideal but infeasible optima. Statistical process control (SPC) analyses show that trade-off balancing provides a better control over individual objectives in terms of average, standard deviation, Cp and Cpk compared to those of single objective optimizations. Moreover, results of case studies show that trade-off balancing not only provides a stable control over individual objectives, but also leads to the highest probability for outputs within the specification limits. We also propose a flow shop scheduling sustainability index (F S S I). The results show that trade-off balancing provides the most sustainable solutions compared to those of the single objective optimizations.

Simulated annealing algorithms to minimise the completion time variance of jobs in permutation flowshops

In this paper, the permutation flowshop scheduling problem with the objective of minimising the completion time variance (CTV) of jobs is considered, and a simulated annealing algorithm (SA) and two modified SAs (MSA1 and MSA2) are proposed. In the first phase, the proposed algorithms are used to minimise CTV of jobs without any right shifting of completion times of jobs on the last machine (RSCT). As followed in the literature some times, the RSCT (except that of the last job) is attempted in the second phase, and in the third phase, we convert sequences so as to follow the V-shaped property with respect to processing time of jobs on the last machine, followed by RSCT except that of the last job, so that the makespan and the machine utilisation in the shop floor remain the same. We present an extensive computational evaluation of the existing and the proposed heuristic algorithms.

A dominance-based heuristic to minimise completion time variance in a single machine system

This paper addresses the problem of minimising the variance of job completion times in a deterministic single-machine scheduling system. Minimising completion time variance is an appropriate objective in scheduling environments where service uniformity is essential. Due to the NP-hard nature of the problem, various heuristics have been presented to obtain a near-optimal solution. In this paper, we introduce a new heuristic based on some powerful precedence (dominance) relation structures, including the concepts of permanent and temporary precedence (dominance) relationships, to determine the positions of adjacent and non-adjacent jobs in a sequence. Our computational experiments demonstrate that the proposed heuristic significantly outperforms the existing ones in deriving the optimal or near-optimal solutions for three well-known sets of benchmark problems and some large randomly generated instances. [Received: 5 August 2018; Revised: 24 December 2018; Accepted: 23 January 2019]

A dominance-based heuristic to minimise completion time variance in a single machine system

A dominance-based heuristic to minimise completion time variance in a single machine system

Combining PREM compilation and static scheduling for high-performance and predictable MPSoC execution

Many applications require both high performance and predictable timing. High-performance can be provided by COTS Multi-Core System on Chips (MPSoC), however, as cores in these systems share main memory, they are susceptible to interference from each other, which is a problem for timing predictability. We achieve predictability on multi-cores by employing the predictable execution model (PREM), which splits execution into a sequence of memory and compute phases, and schedules these such that only a single core is executing a memory phase at a time.We present a toolchain consisting of a compiler and a scheduling tool. Our compiler uses region and loop based analysis and performs tiling to transform application code into PREM-compliant binaries. In addition to enabling predictable execution, the compiler transformation optimizes accesses to the shared main memory. The scheduling tool uses a state-of-the-art heuristic algorithm and is able to schedule industrial-size instances. For smaller instances, we compare the results of the algorithm with optimal solutions found by solving an integer linear programming model. Furthermore, we solve the problem of scheduling execution on multiple cores while preventing interference of memory phases.We evaluate our toolchain on Advanced Driver Assistance System (ADAS) application workloads running on an NVIDIA Tegra X1 embedded system-on-chip (SoC). The results show that our approach maintains similar average performance to the original (unmodified) program code and execution, while reducing variance of completion times by a factor of 9 with the identified optimal solutions and by a factor of 5 with schedules generated by our heuristic scheduler.