Uncertain Processing Times Research Articles

Optimality Region for Job Permutation in Single-Machine Scheduling with Uncertain Processing Times

The problem of scheduling optimally a given set of jobs on a single machine is studied. The lower and upper bounds on the admissible duration of each job are known. The optimality criterion of the schedule is the minimum total completion time of a given set of jobs. Some properties of the optimality region for a job permutation are investigated. Polynomial algorithms for constructing the optimality region for a job permutation and also for calculating the volume of this region are developed. The existence conditions of an empty optimality region for a job permutation are determined. A criterion for the existence of a job permutation with the maximum possible volume of the optimality region is established.

Minimizing maximum cost for a single machine under uncertainty of processing times

One of the most known results in the machine scheduling is Lawler’s algorithm to minimize the maximum cost of jobs processed by a single machine subject to precedence constraints. We consider an uncertain version of the same min-max cost scheduling problem. The cost function of each job depends on the job completion time and on an additional generalized numerical parameter, which may be a tuple of parameters. For each job, both, its processing time and the additional parameter are uncertain, only intervals of possible values of these parameters are known.We analyse certain classes of cost functions and develop polynomial algorithms which construct min-max regret solutions. The considered problems cover the most general range of studied cases of interval uncertainty.In the only two papers that present algorithms for minimizing the maximum regret for the problem with uncertain job processing times, the algorithms are based on extremal scenarios, where some uncertain parameters take their maximum values, while all others take their minimum possible values.We show that it is impossible to always limit the search to extremal scenarios. Our approach is based on new ideas different from those underlying previous work.Finally, we show that our approach outperforms all known results for constructing min-max regret solutions for the min-max cost scheduling problem under uncertainty of job processing times.

Algorithms for four-machine flowshop scheduling problem with uncertain processing times to minimize makespan

We consider the four-machine flowshop scheduling problem to minimize makespan where processing times are uncertain. The processing times are within some intervals, where the only available information is the lower and upper bounds of job processing times. Some dominance relations are developed, and twelve algorithms are proposed. The proposed algorithms first convert the four-machine problem into two stages, then, use the well-known Johnson’s algorithm, known to yield the optimal solution for the two-stage problem. The algorithms also use the developed dominance relations. The proposed algorithms are extensively evaluated through randomly generated data for different numbers of jobs and different gaps between the lower and upper bounds of processing times. Computational experiments indicate that the proposed algorithms perform well. Moreover, the computational experiments reveal that one of the proposed algorithms, Algorithm A7, performs significantly better than the other eleven algorithms for all possible combinations of the number of jobs and the gaps between the lower and upper bounds. More specifically, error percentages of the other eleven algorithms range from 2.3 to 27.7 times that of Algorithm A7. The results have been confirmed by constructing 99% confidence intervals and tests of hypotheses using a significance level of 0.01.

A New Approach Generating Robust and Stable Schedules in m-Machine Flow Shop Scheduling Problems: A Case Study

This paper considers a scheduling problem with uncertain processing times and machine breakdowns in industriall/office workplaces and solves it via a novel robust optimization method. In the traditional robust optimization, the solution robustness is maintained only for a specific set of scenarios, which may worsen the situation for new scenarios. Thus, a two-stage predictive algorithm is proposed to efficiently handle the uncertainties and find robust and stable solutions. The first stage creates robust solutions and ensures their stability in the new scenarios. The second stage proposes a novel stability measure to proactively offset the effects of the machine breakdowns of the former stage. Moreover, a tri-component measure based on efficiency, robustness, and stability is proposed which aims to create a realistic schedule to satisfy the customers, manufacturers, and the staff. To meet the customer’s requirements, the robustness measure is defined based on the tardiness and the delivery dates of jobs. Finally, the proposed algorithm is applied to a case study, and the findings are compared with the empirical data. The results emphasize the superiority of the proposed technique in satisfying the customers, staff, and increasing the profitability and accountability of the company.

Profit-oriented distributionally robust chance constrained flowshop scheduling considering credit risk

Customer credit risk or payment probability, influenced by factors such as financial conditions and bank policies, has hindered fast Asia-Pacific economic growth. Besides, the working time is usually limited due to regulations and limited resources. Driven by profit, some jobs may be rejected on tactical level and the accepted jobs are scheduled on operational level, respecting the allowed working time. This paper studies a stochastic flowshop scheduling problem, assuming that only the mean and covariance matrix of uncertain payment probabilities and processing times are known. The objective is to maximise the profit level, i.e. the probability of the profit no less than the planned one, while controlling the risk of surpassing the limited working time. A new distributionally robust chance constrained model is proposed. The sample average approximation (SAA) method, the robust SAA method and a hierarchical approach, based on an approximated mixed integer second-order conic program, are developed. Numerical experiments show that the hierarchical approach is more efficient. Moreover, some managerial insights are drawn.

A Neutrosophic Number-Based Memetic Algorithm for the Integrated Process Planning and Scheduling Problem With Uncertain Processing Times

Process planning and scheduling are two crucial components in a flexible manufacturing system. Lots of novel meta-heuristics have been applied to the integrated process planning and scheduling (IPPS) problem for an efficient utilization of manufacturing resources; nevertheless, the tricky part in real life stems from the uncertainty in processing times. Existing publications regarding IPPS problems mainly focus on cases with nominal or fixed processing times; nevertheless, processing time fluctuations will certainly result in an intolerable deviation between the actual makespan and the nominal one. This research focuses on the IPPS problem with uncertain processing times to hedge against the uncertainty in makespan. The novel neutrosophic numbers are first introduced to model the uncertain processing times. A neutrosophic number based mixed integer linear programming (MILP) model is established; due to the non-deterministic polynomial (NP)-hardness and the complexity in solving the model, a variable neighborhood search (VNS) incorporated memetic algorithm (MA) is then developed to facilitate more robust solutions. In the proposed algorithm, the nominal makespan criterion and the deviation (robustness) criterion have been considered in a weighted sum manner. The well-known Kim's benchmark is adopted to test the performance of the proposed algorithm and different degrees of fluctuations are also defined in experiments. Computational results reveal that the VNS based local search method is powerful in capturing promising solutions; competitive solutions with superior nominal makespan and robustness have been obtained. This research presents a novel perspective or methodology to seek more robust solutions for the uncertain IPPS problem.

A Single Machine Scheduling with Periodic Maintenance and Uncertain Processing Time

A single machine scheduling problem with periodic maintenance is studied in this paper. Due to many uncertainties in reality, the processing time is recognized as an uncertain variable. The aim is to minimize the makespan at a confidence level. An uncertain chance-constrained programming model is developed to delve into the impact of uncertainties on decision variables, and an algorithm for calculating the objective function is proposed. According to the theoretical analysis, a novel method named longest shortest processing time (LSPT) rule is proposed. Subsequently, an improved genetic algorithm (GA-M) combined with LSPT rule is proposed. Numerical experiments are used to verify the feasibility of this model and algorithm.

Fuzzy project scheduling with critical path including risk and resource constraints using linear programming

Project scheduling is one of the important issues of project management which has raised the interest of the researches, and several methods have been developed for solving this problem. While certain models are used in most studies, uncertainty is one of the intrinsic properties of most projects in real world which consist some activities with uncertain processing times and resource usages. In this paper, a fuzzy linear programming model is proposed for project scheduling considering risk and resources constraints under uncertain environment in which activity duration and the amount of resources used by each activity is defined as a fuzzy membership function. The proposed model is simulated in MATLAB R2009a software and four test cases adopted from the literature are implemented. The computational results show that the proposed model decreases the critical path length about 4% in competition with similar methods.

Fuzzy project scheduling with critical path including risk and resource constraints using linear programming

Project scheduling is one of the important issues of project management which has raised the interest of the researches, and several methods have been developed for solving this problem. While certain models are used in most studies, uncertainty is one of the intrinsic properties of most projects in real world which consist some activities with uncertain processing times and resource usages. In this paper, a fuzzy linear programming model is proposed for project scheduling considering risk and resources constraints under uncertain environment in which activity duration and the amount of resources used by each activity is defined as a fuzzy membership function. The proposed model is simulated in MATLAB R2009a software and four test cases adopted from the literature are implemented. The computational results show that the proposed model decreases the critical path length about 4% in competition with similar methods.

Synthesis and implementation of supervisory control for manufacturing systems under processing uncertainties and time constraints

Abstract In this paper, we exploit the supervisory control theory to synthesize a responsive control logic that assures non-blocking, safety, and time constraints in a manufacturing system subject to uncertain processing times arising from direct human intervention and flexible operations. A modular approach is used to compute a set of timed supervisors and a reduced coordinator that forces machine operations or inhibits human interventions in a minimally restrictive way. A new architecture for structured implementation of timed supervisors in programmable logic controllers is presented and applied to a real-world test-bench.

Exploiting partial correlations in distributionally robust optimization

In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set $$\{(\mathbf x , \mathbf x {} \mathbf x '): \mathbf x \in \mathcal {X}\}$$ where $$\mathcal {X}$$ is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in the newsvendor problem, project evaluation and review technique networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled. We also discuss extensions where the random coefficients are assumed to be non-negative and additional overlapping correlation information is available.

Proactive and reactive scheduling of the steelmaking and continuous casting process through adaptive robust optimization

The schedule of a steelmaking and continuous casting (SCC) process defines the unit assignments of casts and ladles, and the processing sequence and timing in each unit. These decisions have to meet strict production requirements such as limited storage duration and continuous operation under various disruptions. Uncertain processing time is a common factor that may invalidate the initial schedule. Traditional deterministic reactive scheduling implements corrective measures for every disruption event by either repairing the deterministic schedule or rescheduling from scratch. In this work, we present a novel adaptive robust optimization method for SCC scheduling under uncertainty. Compared to the deterministic scheduling method, adaptive robust proactive scheduling generates an initial schedule immune to most major disruptions eliminating the need for most changes. Adaptive robust hybrid scheduling presents an attractive trade-off between solution conservatism and excessive scheduling modifications.

Energy-aware fuzzy job-shop scheduling for engine remanufacturing at the multi-machine level

The rise of the engine remanufacturing industry has resulted in increased possibilities of energy conservation during the remanufacturing process, and scheduling could exert significant effects on the energy performance of manufacturing systems. However, only a few studies have specifically addressed energy-efficient scheduling for remanufacturing. Considering the uncertain processing time and routes and the operation characteristics of remanufacturing, we used the crankshaft as an illustrative case and built a fuzzy job-shop scheduling model to minimize the energy consumption during remanufacturing. An improved adaptive genetic algorithm was developed by using the hormone modulation mechanism to deal with the scheduling problem that simultaneously involves parallel machines, batch machines, and uncertain processing routes and time. The algorithm demonstrated superior performance in terms of optimal value, run time, and convergent generation in comparison with other algorithms. Computational results indicated that the optimal scheduling scheme is expected to generate 1.7 kW · h of energy saving for the investigated problem size. In addition, the scheme could improve the energy efficiency of the crankshaft remanufacturing process by approximately 5%. This study provides a basis for production managers to improve the sustainability of remanufacturing through energy-aware scheduling.

Dynamic Schedule Method Based on Rolling Horizon Procedure for a Discrete Manufacturing Shop with Uncertain Processing Time

To solve the rescheduling problem of discrete manufacturing shop with uncertain processing time, an improved ant algorithm based on the rolling horizon procedure is proposed. The dynamic scheduling strategy is based on a hybrid scrolling mechanism driven by events and cycles. Analysing disturbances in the process, we divide them into explicit and implicit disturbances. The length deviation tolerance (LDT) of processing time is designed and proposed to filter out redundant rescheduling. The dynamic scheduling algorithm, based on improved ant colony algorithms, is considered with the resource constraint of the shop, and with the use of a customized state transition rule, it helps to overcome the drawbacks of long ant path searching prone to stagnation. Using simulation, the performance of the dynamic scheduling strategy and scheduling algorithm are analysed and verified, and better rolling scheduling policy parameters are obtained.

Fuzzysequencing problemwith uncertain processing time

In this paper, we suggested a new approach to obtained an optimal sequence “for fuzzy sequencing problem with trapezoidal fuzzy number.” Using new fuzzy arithmetic operation and ranking method, an improved algorithm is proposed to obtain the fuzzy total elapsed time for processing all the jobs without converting problem into classical form. The approach is efficiently illustrated with a numerical example.

Robust disassembly line balancing with ambiguous task processing times

Disassembly line balancing problem (DLBP), which is to select disassembly process, open workstations and assign selected tasks to opened workstations, plays an important role in the recycling of End Of Life products. In real-world disassembly operations, task processing times are usually stochastic due to various factors. Most related works address the uncertain processing times by assuming that the probability distribution is known and the task processing times are independent of each other. In practice, however, it is difficult to get the complete distributional information and there is always underlying correlation between the uncertain processing times. This paper investigates the DLBP with partial uncertain knowledge, i.e. the mean and covariance matrix of task processing times. A new distributionally robust formulation with a joint chance constraint is proposed. To solve the problem, an approximated mixed integer second-order cone programming (MI-SOCP) model is proposed, and a two-stage parameter-adjusting heuristic is further developed. Numerical experiments are conducted, to evaluate the performance of the proposed method. We also draw some managerial insights and consider an extension problem.

Scenario-based multi-objective robust scheduling for a semiconductor production line

Scheduling problems of semiconductor manufacturing systems (SMS) with the goal of optimising some classical performance indices (NP-hard), tend to be increasingly complicated due to stochastic uncertainties. This paper targets the robust scheduling problem of an SMS with uncertain processing times. A three-stage multi-objective robust optimisation (MORO) approach is proposed, that can collaboratively optimise the performance indices and their robustness measures. In the first stage, this paper studies the scheduling problem in the deterministic environment and obtains feasible scheduling strategies that perform well in four performance indices (the average cycle time (CT), the on-time delivery rate (ODR), the throughput (TP), and the total movement amount of wafers (MOV)). Then, in the second stage, the uncertainties are introduced into the production system. In the third stage, this paper proposes a hybrid method consisting of scenario planning, discrete simulation, and multi-objective optimisation to obtain an approximately and more robust optimal solution from the feasible scheduling strategy set. The proposed MORO approach is tested in a semiconductor experiment production line and makes a full analysis to illustrate the effectiveness of our method. The results show that our MORO is superior concerning the total robustness with multi-objective.

Robotic-cell scheduling with pick-up constraints and uncertain processing times

Technological developments have propelled the deployment of robots in many applications, which has led to the trend to integrate an increasing number of uncertain processes into robotic and automated equipment. We contribute to this domain by considering the scheduling of a dual-gripper robotic cell. For systems with one potential bottleneck, we determine conditions under which the widely used swap sequence does not guarantee optimality or even feasibility and prove that optimal schedules can be derived under certain conditions when building on two types of slack we introduce. With the addition of a third type of slack and the concept of fixed partial schedules, we develop an offline-online scheduling approach that, in contrast with previous work, is able to deal with uncertainty in all process steps and robot handling tasks, even under pick-up constraints. The approach can deal with single- or multiple-bottleneck systems, and is the first approach that is not restricted to a single predefined sequence such as the swap sequence. Our approach is well suited for real-world applications, since it generates cyclic schedules and allows integration into commonly-used frameworks for robotic-cell scheduling and control.We demonstrate the applicability of our approach to cluster tools in semiconductor manufacturing, showing that our approach generates feasible results for all tested levels of uncertainty and optimal or near-optimal results for low levels of uncertainty. With additional symmetry-breaking constraints, the model can be efficiently applied to industrial-scale test instances. We show that reducing uncertainty to below 10% of the processing time would yield significantly improved cycle lengths and throughput. We also demonstrate that the widely used swap sequence only finds solutions for less than 1% of the instances when strict pick-up constraints are enforced and processing times are heterogeneous. As our approach finds feasible solutions to all of these instances, it enables the application of robotic cells to a significantly broader application environment.

Scheduling with uncertain processing times in mixed-criticality systems

Many scheduling problems that can be identified inside safety-critical applications, such as in autonomous cars, tend to be mixed-critical. Such scheduling problems consider tasks to have different criticalities depending on the safety levels (activation of brakes vs. activation of air-conditioning). The biggest challenge in those scheduling problems arises from the uncertainty of processing times as it disturbs the predictability of the system and thus makes the certification of the system difficult. To overcome this uncertainty, we model the tasks to have multiple processing times concerning their criticality. This approach converts these scheduling problems into a deterministic scheduling with alternative processing times.Here, we study an NP-hard single machine scheduling problem with makespan minimization, where the non-preemptive tasks can have multiple processing times. To solve the problem, we propose an approximation algorithm, a novel mixed-integer linear programming block formulation, and an efficient exact branch-and-price decomposition for two criticality levels. Furthermore, we demonstrate that the optimal schedules are represented as trees, which enables to formulate an exact algorithm for the problem with three criticality levels. The efficiency of the proposed method is demonstrated for difficult problem instances with up to 1000 tasks. The experimental evaluation demonstrates that our algorithms have improved the results of the best-known method by nearly two orders of magnitude.

Robust min\u2013max regret scheduling to minimize the weighted number of late jobs with interval processing times

We consider the robust version of single machine scheduling problem with the objective to minimize the weighted number of jobs completed after their due-dates. The jobs have uncertain processing times represented by intervals, and decision-maker must determine their execution sequence that minimizes the maximum regret. We develop an exact solution algorithm based on a specialized branch and bound method, using mixed-integer linear programming formulations for a common due-date and for job-dependent due-dates. Finally, we examine the solution algorithm in a series of computational experiments.