Robust scheduling of a two-stage hybrid flow shop with uncertain interval processing times

This paper studies the makespan minimisation scheduling problem in a two-stage hybrid flow shop. The first stage has one machine and the second stage has m identical parallel machines. Neither the processing time nor probability distribution of the processing time of each job is uncertain. We propose a robust (min–max regret) scheduling model. To solve the robust scheduling problem, which is NP-hard, we first derive some properties of the worst-case scenario for a given schedule. We then propose both exact and heuristic algorithms to solve this problem. In addition, computational experiments are conducted to evaluate the performance of the proposed algorithms.     

Hedge against total flow time uncertainty of the uniform parallel machine scheduling problem with interval data

We consider a total flow time minimisation problem of uniform parallel machine scheduling when job processing times are only known to be bounded within certain given intervals. A minmax regret model is proposed to identify a robust schedule that minimises the maximum deviation from the optimal total flow time over all possible realisations of the job processing times. To solve this problem, we first prove that the maximal regret for any schedule can be obtained in polynomial time. Then, we derive a mixed-integer linear programming (MILP) formulation of our problem by exploiting its structural properties. To reduce the computational time, we further transform our problem into a robust single-machine scheduling problem and derive another MILP formulation with fewer variables and constraints. Moreover, we prove that the optimal schedule under the midpoint scenario is a 2-approximation for our minmax regret problem. Finally, computational experiments are conducted to evaluate the performance of the proposed methods.     

Bad-scenario-set robust scheduling for a job shop to hedge against processing time uncertainty

This paper proposed two robust scheduling formulations in real manufacturing systems based on the concept of bad scenario set to hedge against processing time uncertainty, which is described by discrete scenarios. Two proposed robust scheduling formulations are applied to an uncertain job-shop scheduling problem with the makespan as the performance criterion. The united-scenario neighbourhood (UN) structure is constructed based on bad scenario set for the scenario job-shop scheduling problem. A tabu search (TS) algorithm with the UN structure is developed to solve the proposed robust scheduling problem. An extensive experiment was conducted. The computational results show that the first robust scheduling formulation could be preferred to the second one for the discussed problem. It is also verified that the obtained robust solutions could hedge against the processing time uncertainty through decreasing the number of bad scenarios and the degree of performance degradation on bad scenarios. Moreover, the computational results demonstrate that the developed TS algorithm is competitive for the proposed robust scheduling formulations.     

Robust single machine scheduling with uncertain release times for minimising the maximum waiting time

We study a single machine scheduling problem (SMSP) with uncertain job release times (JRTs) under the maximum waiting time (MWT) criterion. To deal with the uncertainty, a robust model is established to find an optimal schedule, which minimises the worst-case MWT (W-MWT) when JRTs vary over given time intervals. Although infinite possible scenarios for JRTs exist, we show that only n scenarios are needed for calculating the W-MWT, where n is the number of jobs. Based on this property, the robust (SMSP) with uncertain JRTs to minimise the W-MWT is formulated as a mixed integer linear programming problem. To solve large-size problem instances, an efficient two-stage heuristic (TSH) is proposed. In the first stage, n near-optimal schedules are obtained by solving n deterministic scenario-based SMSPs, and their W-MWTs are evaluated. To speed up the solution and evaluation process, a modified Gusfield’s heuristic is proposed by exploiting the inner connections of these SMSPs. To further improve the schedule obtained in the first stage, the second stage consists of a variable neighbourhood search method by combining both swap neighbourhood search and insert neighbourhood search. We also develop a method to calculate the lower bound of the proposed model so that we can evaluate the performance of the solutions given by the TSH. Experimental results confirm the robustness of schedules produced and advantages of the proposed TSH over other algorithms in terms of solution quality and run time.     

Flexible job-shop scheduling problems with ‘AND’/‘OR’ precedence constraints

The purpose of this research is to solve flexible job-shop scheduling problems with ‘AND’/‘OR’ precedence constraints in the operations. We first formulate the problem as a Mixed-Integer Linear Program (MILP). The MILP can be used to compute optimal solutions for small-sized problems. We also developed a heuristic algorithm that can obtain a good solution for the problem regardless of its size. Moreover, we have developed a representation and schedule builder that always produces a legal and feasible solution for the problem, and developed genetic and tabu search algorithms based on the proposed schedule builder. The results of the computational experiments show that the developed meta-heuristics are very effective.     

A decomposition-based approach to flexible flow shop scheduling under machine breakdown

Manufacturing systems in real-world production are generally dynamic and often subject to a wide range of uncertainties. Recently, research on production scheduling under uncertainty has attracted substantial attention. Although some methods have been developed to address this problem, scheduling under uncertainty remains inherently difficult to solve by any single approach. This article considers makespan optimisation of a flexible flow shop (FFS) scheduling problem under machine breakdown. It proposes a novel decomposition-based approach to decompose an FFS scheduling problem into several cluster scheduling problems which can be solved more easily by different approaches. A neighbouring K-means clustering algorithm is developed to first group the machines of an FFS into an appropriate number of machine clusters, based on a proposed machine allocation algorithm and weighted cluster validity indices. Two optimal back propagation networks, corresponding to the scenarios of simultaneous and non-simultaneous job arrivals, are then selectively adopted to assign either the shortest processing time (SPT) or the genetic algorithm (GA) to each machine cluster to solve cluster scheduling problems. If two neighbouring machine clusters are allocated with the same approach, they are subsequently merged. After machine grouping and approach assignment, an overall schedule is generated by integrating the solutions to the sub-problems. Computation results reveal that the proposed approach is superior to SPT and GA alone for FFS scheduling under machine breakdown.     

Robust scheduling of a two-machine flow shop with uncertain processing times

This paper focuses on manufacturing environments where job processing times are uncertain. In these settings, scheduling decision makers are exposed to the risk that an optimal schedule with respect to a deterministic or stochastic model will perform poorly when evaluated relative to actual processing times. Since the quality of scheduling decisions is frequently judged as if processing times were known a priori, robust scheduling, i.e., determining a schedule whose performance (compared to the associated optimal schedule) is relatively insensitive to the potential realizations of job processing times, provides a reasonable mechanism for hedging against the prevailing processing time uncertainty. In this paper we focus on a two-machine flow shop environment in which the processing times of jobs are uncertain and the performance measure of interest is system makespan. We present a measure of schedule robustness that explicitly considers the risk of poor system performance over all potential realizations of job processing times. We discuss two alternative frameworks for structuring processing time uncertainty. For each case, we define the robust scheduling problem, establish problem complexity, discuss properties of robust schedules, and develop exact and heuristic solution approaches. Computational results indicate that robust schedules provide effective hedges against processing time uncertainty while maintaining excellent expected makespan performance     

Group production scheduling for minimum total tardiness Part(I)

This paper considers the problem of group scheduling on a single stage to minimize total tardiness. It is assumed that jobs are classified into several groups on the basis of group technology. Optimal decision as to scheduling sequences will be made as to product group and specific job. This paper proves basic theorems that establish the relative order in which pairs of groups are processed in an optimal schedule. In general, scheduling problems of moderate size may be at least partially ordered so that very few schedules remain to be searched. Two practical algorithms for determining the optimal group schedule and the near optimal group schedule are proposed. Numerical examples are presented in detail.     

Multi-rate power-controlled link scheduling for mesh broadband wireless access networks

The problem of multi-rate power-controlled collision-free scheduling in spatial time division multiple access (STDMA) wireless mesh networks is formulated as a mathematical program utilising cross layer information. As these mixed integer linear programs are intractable (NP-hard problems), optimal collision-free schedules can be found only for topologies consisting of a few nodes. To this end, approximation algorithms that are based on linear programming relaxation and randomised rounding are studied. The proposed framework, which aims to maximise the spatial timeslot reuse under predefined signal-to-interference noise ratio thresholds, is suitable for providing centralised scheduling in the mesh mode of the IEEE 802.16 standard. Performance aspects of the approximation algorithms under different scenarios are investigated.     

A comparison of stochastic programming and bi-objective optimisation approaches to robust airline crew scheduling

A prominent problem in airline crew scheduling is the pairings or Tour-of-Duty planning problem. The objective is to determine a set of pairings (or Tours-of-Duty) for a crew group to minimise the planned cost of operating a schedule of flights. However, due to unforeseen events the performance in operation can differ considerably from planning, sometimes causing significant additional recovery costs. In recent years there has been a growing interest in robust crew scheduling. Here, the aim is to find solutions that are “cheap” in terms of planned cost as well as being robust, meaning that they are less likely to be disrupted in case of delays. Taking the stochastic nature of delays into account, Yen and Birge (Transp Sci 40:3–14, 2006) formulate the problem as a two-stage stochastic integer programme and develop an algorithm to solve this problem. Based on the contradictory nature of the goals, Ehrgott and Ryan (J Multi-Criteria Decis Anal 11:139–150, 2002) formulate a bi-objective set partitioning model and employ elastic constraint scalarisation to enable the solution by set partitioning algorithms commercially used in crew scheduling software. In this study, we compare the two solution approaches. We improve the algorithm of Yen and Birge (Transp Sci 40:3–14, 2006) and implement both methods with a commercial crew scheduling software. The results of both methods are compared with respect to characteristics of robust solutions, such as the number of aircraft changes for crew. We also conduct experiments to simulate the performance of the obtained solutions. All experiments are performed using actual schedule data from Air New Zealand.     

Comparison of multi-objective evolutionary approaches for task scheduling in distributed computing systems

Parallel and distributed systems play an important part in the improvement of high performance computing. In these type of systems task scheduling is a key issue in achieving high performance of the system. In general, task scheduling problems have been shown to be NP-hard. As deterministic techniques consume much time in solving the problem, several heuristic methods are attempted in obtaining optimal solutions. This paper presents an application of Elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) and a Non-dominated Sorting Particle Swarm Optimization Algorithm (NSPSO) to schedule independent tasks in a distributed system comprising of heterogeneous processors. The problem is formulated as a multi-objective optimization problem, aiming to obtain schedules achieving minimum makespan and flowtime. The applied algorithms generate Pareto set of global optimal solutions for the considered multi-objective scheduling problem. The algorithms are validated against a set of benchmark instances and the performance of the algorithms evaluated using standard metrics. Experimental results and performance measures infer that NSGA-II produces quality schedules compared to NSPSO.     

Genetic algorithm for job-shop scheduling with machine unavailability and breakdowns

The job-shop scheduling problem (JSSP) is considered to be one of the most complex combinatorial optimisation problems. In our previous attempt, we hybridised a Genetic Algorithm (GA) with a local search technique to solve JSSPs. In this research, we propose an improved local search technique, Shifted Gap-Reduction (SGR), which improves the performance of GAs when solving relatively difficult test problems. We also modify the new algorithm for JSSPs with machine unavailability and breakdowns. We consider two scenarios of machine unavailability. First, where the unavailability information is available in advance (predictive) and, secondly, where the information is known after a real breakdown (reactive). We show that the revised schedule is mostly able to recover if the interruptions occur during the early stages of the schedules.     

Minimising total tardiness for the identical parallel machine scheduling problem with splitting jobs and sequence-dependent setup times

This paper focuses on an identical parallel machine scheduling problem with minimising total tardiness of jobs. There are two major issues involved in this scheduling problem; (1) jobs which can be split into multiple sub-jobs for being processed on parallel machines independently and (2) sequence-dependent setup times between the jobs with different part types. We present a novel mathematical model with meta-heuristic approaches to solve the problem. We propose two encoding schemes for meta-heuristic solutions and three decoding methods for obtaining a schedule from the meta-heuristic solutions. Six different simulated annealing algorithms and genetic algorithms, respectively, are developed with six combinations of two encoding schemes and three decoding methods. Computational experiments are performed to find the best combination from those encoding schemes and decoding methods. Our findings show that the suggested algorithm provides not only better solution quality, but also less computation time required than the commercial optimisation solvers.     

Bottleneck-based heuristics to minimize tardy jobs in a flexible flow line with unrelated parallel machines

This paper develops new bottleneck-based heuristics with machine selection rules to solve the flexible flow line problem with unrelated parallel machines in each stage and a bottleneck stage in the flow line. The objective is to minimize the number of tardy jobs in the problem. The heuristics consist of three steps: (1) identifying the bottleneck stage; (2) scheduling jobs at the bottleneck stage and the upstream stages ahead of the bottleneck stage; (3) using dispatching rules to schedule jobs at the downstream stages behind the bottleneck stage. A new approach is developed to find the arrival times of the jobs at the bottleneck stage, and two decision rules are developed to schedule the jobs on the bottleneck stage. This new approach neatly overcomes the difficulty of determining feasible arrival times of jobs at the bottleneck stage. In order to evaluate the performance of the proposed heuristics, six well-known dispatching rules are examined for comparison purposes. Six factors are used to design 729 production scenarios, and ten test problems are generated for each scenario. Computational results show that the proposed heuristics significantly outperform all the well-known dispatching rules. An analysis of the experimental factors is also performed and several interesting insights into the heuristics are discovered.     

Scheduling job shop associated with multiple routings with genetic and ant colony heuristics

The general job shop problem is one of the well known machine scheduling problems, in which the operation sequence of jobs are fixed that correspond to their optimal process plans and/or resource availability. Scheduling and sequencing problems, in general, are very difficult to solve to optimality and are well known as combinatorial optimisation problems. The existence of multiple job routings makes such problems more cumbersome and complicated. This paper addresses a job shop scheduling problem associated with multiple job routings, which belongs to the class of NP hard problems. To solve such NP-hard problems, metaheuristics have emerged as a promising alternative to the traditional mathematical approaches. Two metaheuristic approaches, a genetic algorithm and an ant colony algorithm are proposed for the optimal allocation of operations to the machines for minimum makespan time criterion. ILOG Solver, a scheduler package, is used to evaluate the performance of the proposed algorithms. The comparison reveals that both the algorithms are capable of providing solutions better than the solution obtained with ILOG Solver.     

Agent-based project scheduling

Agent technology offers a new way of thinking about many of the classic problems in operations research. Among these are problems such as project scheduling subject to resource constraints. In this paper, we develop and experimentally evaluate eight agent-based algorithms for solving the multimode, resource-constrained project scheduling problem. Our algorithms differ in the priority rules used to control agent access to resources. We apply our approach to a 51-activity project originally published by Maroto and Tormos [1]. We solve the problem using two types of agent-based systems: (i) a system of simple, reactive agents that we call basic agents; and (ii) a system of more complex, deliberative agents that we call enhanced agents. Of the eight priority rules tested, we find that priority based on shortest processing time performs best in terms of schedule quality when applied by basic agents while the priority based on earliest due date performs best when applied by enhanced agents. In comparing agents across priority rules, we find that enhanced agents generate much better schedules (with makespans up to 66% shorter in some cases) and require only slightly more computation time.     

Two FMS scheduling methods based on Petri nets: A global and a local approach

We are interested in the Flexible Manufacturing System (FMS) scheduling problem. Different methods have been explored to solve this problem and to master its combinatorial complexity, which is NP-hard in the general case. In this paper we will give two different scheduling methods based on Petri nets. The first one tends to solve the general scheduling problem (acyclic schedule) using the Constraint Programming method to avoid exhaustive search. The second method is a dedicated cyclic scheduling method. The aim is not to compare the methods' performances (computation time, results' quality) because they do not solve exactly the same problem, but to compare their application domains in terms of parts number.     

Multiple machine JIT scheduling: a tabu search approach

This paper presents an approach to solving the multiple machine, non-preemptive, earliness-tardiness scheduling problem with unequal due dates in a flow shop with machine tiers (FMT). In this variant of the flow shop problem, machines are arranged in tiers or groups, and the jobs must visit one machine in each tier. The processing times, machine assignments, and due dates are deterministic and known in advance. The objective is to find a permutation schedule that minimizes the total deviation of each job from its due date. A tabu search (TS) meta-heuristic combined with an LP evaluation function is applied to solve this problem and results are compared to optimal permutation solutions for small problems and the earliest due date schedule for large problems. Several neighborhood generation methods and two diversification strategies are examined to determine their effect on solution quality. Results show that the TS method works well for this problem. TS found the optimal solution in all but one of the small problem instances and improved the earliest due date solutions for larger instances where no optimal solutions could be found.     

Shiftwork scheduling algorithms with phase-delay feature

Shiftwork is common in manufacturing and service operations, but shiftwork has many negative effects on workers. Shiftwork researchers have suggested that shift changes should be in the clockwise (forward) direction. We present shiftwork scheduling algorithms that incorporate this phase-delay feature for two workweek scenarios, one under the 5-day workweek and the other under the 3- 4-day workweek. The proposed algorithms are designed to be implemented by hand and schedules generated will satisfy staffing and off-day requirements. Facility managers looking for better ways to schedule shiftwork may find the proposed scheduling algorithms useful.