Mathematical modelling and optimisation of energy-conscious hybrid flow shop scheduling problem with unrelated parallel machines

This paper investigates an energy-conscious hybrid flow shop scheduling problem with unrelated parallel machines (HFSP-UPM) with the energy-saving strategy of turning off and on. We first analyse the energy consumption of HFSP-UPM and formulate five mixed integer linear programming (MILP) models based on two different modelling ideas namely idle time and idle energy. All the models are compared both in size and computational complexities. The results show that MILP models based on different modelling ideas vary dramatically in both size and computational complexities. HFSP-UPM is NP-Hard, thus, an improved genetic algorithm (IGA) is proposed. Specifically, a new energy-conscious decoding method is designed in IGA. To evaluate the proposed IGA, comparative experiments of different-sized instances are conducted. The results demonstrate that the IGA is more effective than the genetic algorithm (GA), simulating annealing algorithm (SA) and migrating birds optimisation algorithm (MBO). Compared with the best MILP model, the IGA can get the solution that is close to an optimal solution with the gap of no more than 2.17% for small-scale instances. For large-scale instances, the IGA can get a better solution than the best MILP model within no more than 10% of the running time of the best MILP model.     

Minimising makespan on parallel batch processing machines with non-identical ready time and arbitrary job sizes

This paper considers the parallel batch processing machine scheduling problem which involves the constraints of unequal ready times, non-identical job sizes, and batch dependent processing times in order to sequence batches on identical parallel batch processing machines with capacity restrictions. This scheduling problem is a practical generalisation of the classical parallel batch processing machine scheduling problem, which has many real-world applications, particularly, in the aging test operation of the module assembly stage in the manufacture of thin film transistor liquid crystal displays (TFT-LCD). The objective of this paper is to seek a schedule with a minimum total completion time for the parallel batch processing machine scheduling problem. A mixed integer linear programming (MILP) model is proposed to optimise the scheduling problem. In addition, to solve the MILP model more efficiently, an effective compound algorithm is proposed to determine the number of batches and to apply this number as one parameter in the MILP model in order to reduce the complexity of the problem. Finally, three efficient heuristic algorithms for solving the large-scale parallel batch processing machine scheduling problem are also provided.     

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.     

A tabu search algorithm for simultaneous machine/AGV scheduling problem

Machines and automated guided vehicles (AGVs) scheduling problems are two essential issues that need to be addressed for the efficiency of the overall production system. The purpose of this paper is to study the simultaneous scheduling problem of machines and AGVs in a flexible manufacturing system (FMS) since the global optimum cannot be reached by considering each of them individually. In this paper, a mixed integer linear programming (MILP) model is developed with the objective of makespan minimisation. The MILP model consists of the following two constraint sets: machines and AGVs scheduling sub-problems. As both sub-problems are known to be NP-hard, a heuristic algorithm based on tabu search (TS) is proposed to get optimal or near to optimal solution for large-size problems within reasonable computation time. The proposed algorithm includes a novel two-dimensional solution representation and the generation of two neighbour solutions, which are alternately and iteratively applied to improve solutions. Moreover, an improved lower bound calculation method is introduced for the large-size problems. Computational results show the superior performance of the TS algorithm for the simultaneous scheduling problem.     

Scheduling dynamically positioned tankers for offshore oil offloading

The operation of offshore oil fields entails transferring oil that accumulates in Floating Production Storage and Offloading Units to onshore terminals. To this end, a fleet of dynamically positioned tankers is deployed for transferring oil from large oil fields, which should be scheduled to meet operational constraints while minimising costs and economic losses. This work presents a mixed-integer linear programming (MILP) formulation for the problem of scheduling shuttle tankers that accounts for the essential constraints. Combined with an MILP solver, the model serves as a decision support tool to guide engineers in daily operations. A family of valid inequalities is proposed to strengthen the MILP formulation and reduce solving time with state-of-the-art solvers. Computational results are also reported on the application of the proposed model in tandem with a rolling horizon strategy.     

Two-machine flow shop problem with unit-time operations and conflict graph

This paper addresses the problem of scheduling, on a two-machine flow shop, a set of unit-time operations subject to the constraints that some conflicting jobs cannot be scheduled simultaneously on different machines. In the context of our study, these conflicts are modelled by general graphs. The problem of minimising the maximum completion time (makespan) is known to be NP-hard in the strong sense. We propose a mixed-integer linear programming (MILP) model. Then, we develop a branch and bound algorithm based on new lower and upper bound procedures. We further provide a computer simulation to measure the performance of the proposed approaches. The computational results show that the branch and bound algorithm outperforms the MILP model and can solve instances of size up to 20,000 jobs.     

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 novel branch-and-bound algorithm for the chance-constrained resource-constrained project scheduling problem

The resource-constrained project scheduling problem (RCPSP) has been widely studied during the last few decades. In real-world projects, however, not all information is known in advance and uncertainty is an inevitable part of these projects. The chance-constrained resource-constrained project scheduling problem (CC-RCPSP) has been recently introduced to deal with uncertainty in the RCPSP. In this paper, we propose a branch-and-bound (B&B) algorithm and a mixed integer linear programming (MILP) formulation that solve a sample average approximation of the CC-RCPSP. We introduce two different branching schemes and eight different priority rules for the proposed B&B algorithm. The computational results suggest that the proposed B&B procedure clearly outperforms both a proposed MILP formulation and a branch-and-cut algorithm from the literature.     

Using genetic algorithms (GA) and a coloured timed Petri net (CTPN) for modelling the optimization-based schedule generator of a generic production scheduling system

The semiconductor manufacturing industry is one of the most complicated manufacturing systems in the world. Considering its complex problem nature, such as the unrelated parallel machine environment, dynamic job arrival, non-pre-emption, inseparable sequence-dependent set-up time, multiple-resource requirements, general precedence constraint, and job recirculation, this study proposed the optimization-based schedule generator (OptSG) for solving the generalized scheduling problems arising from the semiconductor manufacturing environment. The separation of the problem structure and problem configuration in OptSG contributes to the structural independence, making OptSG robust and convenient in analysis and problem-solving in real settings with changing properties. Meanwhile, an MILP model was proposed as a benchmark to estimate the validity of OptSG. Inseparable sequence-dependent set-up time and multiple-resource requirements that have not been addressed simultaneously in the literature were considered in this model. By using different evaluation criteria, including makespan, total completion time and maximum tardiness, experiments were conducted to compare the solutions of the MILP model, OptSG and dispatching rule-based heuristics (DRBH). The results validated the solution quality of OptSG.     

A multi-objective optimisation model to integrating flexible process planning and scheduling based on hybrid multi-objective simulated annealing

Process planning and production scheduling play important roles in manufacturing systems. In this paper we present a mixed integer linear programming (MILP) scheduling model, that is to say a slot-based multi-objective multi-product, that readily accounts for sequence-dependent preparation times (transition and set up times or machine changeover time). The proposed scheduling model becomes computationally expensive to solve for long time horizons. The aim is to find a set of high-quality trade-off solutions. This is a combinatorial optimisation problem with substantially large solution space, suggesting that it is highly difficult to find the best solutions with the exact search method. To account for this, the hybrid multi-objective simulated annealing algorithm (MOHSA) is proposed by fully utilising the capability of the exploration search and fast convergence. Two numerical experiments have been performed to demonstrate the effectiveness and robustness of the proposed algorithm.     

More MILP models for hybrid flow shop scheduling problem and its extended problems

With the rapid development of computer technology and related softwares for mathematical models, mathematical modelling of scheduling problems is receiving growing attention from researchers. In this work, the hybrid flow shop scheduling problem with unrelated parallel machines (HFSP-UPM) with the objective aimed to minimise the makespan is studied. According to the characteristics of the HFSP-UPM, eight mixed integer linear programming (MILP) models are formulated in order to obtain optimal solutions based on different modelling ideas. Then, these models are extended to solve HFSP-UPM with sequence-dependent setup times (HFSP-UPM-SDST), no-wait HFSP-UPM (HFSP-UPM-NW) and HFSP-UPM with blocking (HFSP-UPM-B). All the proposed models and the existing model are detailedly compared and evaluated under three aspects namely modelling process, size complexity and computational complexity. Numerical experiments show that MILP models dependent on diverse modelling ideas perform very differently. The model developed based on stage precedence is the best one and should be given preference in future applications. In addition, the proposed models of HFSP-UPM-NW and HFSP-UPM-B improve several best known solutions for the test instances in the existing literature.     

Scheduling flexible manufacturing cells using Tabu Search

Scheduling is an important aspect in the overall control of a flexible manufacturing system. The research presented focuses on production scheduling of jobs within a flexible manufacturing cell (FMC)–one type of flexible manufacturing system. Due to the complexity of the FMC scheduling problem, a 0–1 mixed-integer linear programming (MILP) model is formulated for M machines and N jobs with alternative routings. Although small instances of the problem can be solved optimally with MILP models, a two-stage Tabu Search (TS² ) algorithm that minimises the manufacturing makespan (MS) is proposed to solve medium-to-large-scale problems more efficiently. During Stage I (construction phase), two heuristics are utilised to generate an initial feasible sequence and an initial MS solution. In Stage II (improvement phase), the acquired initial solutions from Stage I are combined with a Tabu Search meta-heuristic procedure that provides improved MS solutions. The TS² algorithm provides tremendous savings in computational time for medium/large-sized multi-machine FMC problems.     

The equipment maintenance scheduling problem in a coal production system

This paper addresses an equipment maintenance scheduling problem in a coal production system which includes three consecutive stages: the coal mining stage, the coal washing stage and the coal loading stage. Each stage is composed of different equipment that needs maintenance each day. There exists intermediate storage with finite capacities and the finished products are transported by train. Moreover, some equipment has a different preference for (aversion to) the start time of maintenance (STOM). The objective is to minimise the weighted sum of aversion about STOM, changeover times and train waiting time. We first formulate this problem into a mixed integer linear programming (MILP) model, then a hybrid genetic algorithm (HGA) is proposed to solve it. The proposed method has been tested on a practical coal enterprise in China and some randomly generated instances. Computational results indicate that our algorithm can produce near-optimal solutions efficiently.     

原油处理过程短期生产计划优化

在原油处理过程短期生产计划的递阶求解方法中,原油处理短期生产计划问题分为上下两层,上层根据市场需求产生一个目标炼油计划;在此基础上,下层得到一个详细生产计划以实现目标炼油计划。研究了在上层目标炼油计划已知的情况下,下层详细生产计划的求解问题。为该问题建立了基于离散时间表示的混合整数线性规划模型,分析了问题的特点并将其进行转化,给出了基于启发式的求解方法,在保证目标炼油计划实现的前提下,对原油转运过程中油品切换及不同油品的罐底混合进行了优化,取得了一定的成果。用一个工业实例验证了启发式规则的可行性和有效性。     

Scheduling in flowshops to minimize total tardiness of jobs

The problem of scheduling in static flowshops is considered with the objective of minimizing mean or total tardiness of jobs. A heuristic algorithm based on the simulated annealing (SA) technique is developed. The salient features of the proposed SA algorithm are the development of two new perturbation schemes for use in the proposed SA algorithm and a new improvement scheme to improve the quality of the solutions. The proposed algorithm is evaluated by using the benchmark problems available in the literature. The performance of the proposed SA algorithm is found to be very good, and the proposed heuristic performs better than the existing heuristics.     

An MILP Continuous-Time Framework for Short-Term Scheduling of Multipurpose Batch Processes Under Different Operation Strategies

This work introduces a novel MILP continuous-time framework to the optimal short-term scheduling of non-sequential multipurpose batch processes with intermediate storage vessels. It is based on a problem representation that describes the batch sequence at any processing/storage unit by providing the full set of predecessors for every batch. Different operation modes can be considered by making minor changes in the problem model. The proposed framework can also handle sequence-dependent changeovers as well as multiple storage tanks available to receive material from one or several processing units. Three example problems involving up to fifteen batches and six processing tasks were successfully solved. Compared with previous work, a drastic reduction in both problem size and CPU time has been achieved.     

Mathematical modelling and a meta-heuristic for flexible job shop scheduling

This study develops new solution methodologies for the flexible job shop scheduling problem (F-JSSP). As a first step towards dealing with this complex problem, mathematical modellings have been used; two novel effective position- and sequence-based mixed integer linear programming (MILP) models have been developed to fully characterise operations of the shop floor. The developed MILP models are capable of solving both partially and totally F-JSSPs. Size complexities, solution effectiveness and computational efficiencies of the developed MILPs are numerically explored and comprehensively compared vis-à-vis the makespan optimisation criterion. The acquired results demonstrate that the proposed MILPs, by virtue of its structural efficiencies, outperform the state-of-the-art MILPs in literature. The F-JSSP is strongly NP-hard; hence, it renders even the developed enhanced MILPs inefficient in generating schedules with the desired quality for industrial scale problems. Thus, a meta-heuristic that is a hybrid of Artificial Immune and Simulated Annealing (AISA) Algorithms has been proposed and developed for larger instances of the F-JSSP. Optimality gap is measured through comparison of AISA’s suboptimal solutions with its MILP exact optimal counterparts obtained for small- to medium-size benchmarks of F-JSSP. The AISA’s results were examined further by comparing them with seven of the best-performing meta-heuristics applied to the same benchmark. The performed comparative analysis demonstrated the superiority of the developed AISA algorithm. An industrial problem in a mould- and die-making shop was used for verification.     

Makespan minimisation on parallel batch processing machines with non-identical job sizes and release dates

The problem we study in this paper arises from the washing step of hospital sterilisation services. Washers in the washing step are capable of handling more than one medical device set as long as their capacity is not exceeded. The medical device set sizes and arrival times to the sterilisation service may be different, but they all have the same washing duration. Thus, we model the washing step as a batch scheduling problem where medical device sets are treated as jobs with non-identical sizes and release dates, but equal processing times. The main findings we present in this paper are the following. First, we study two special cases for which polynomial algorithms are presented. We then develop a 2-approximation algorithm for the general problem. Finally, we develop a MILP model and compare it with another MILP model from the literature. Computational results show that our MILP model outperforms the model from the literature.     

A constructive algorithm and a simulated annealing approach for solving flowshop problems with missing operations

In many practical cases of flowshop environments and especially in flowline manufacturing cells, some or all jobs may not require processing on all machines. Hence, this paper focuses on the flowshop scheduling problem with missing operations. A modification of the constructive NPS-set heuristic is proposed, which generates non-permutation schedules effectively. Furthermore, a two-phase simulated annealing (SA) method is presented that specifically considers missing operations in its procedure. The modified NPS-set heuristic and the two-phase SA are tested and statistically evaluated by an extensive computational study for total flow time criteria. The results show that the modified NPS-set heuristic as well as the specific consideration of missing operations can enhance the algorithms’ performance significantly.     

A two-stage three-machine assembly scheduling problem with deterioration effect

The two-stage assembly scheduling problem has received growing attention in the research community. Furthermore, in many two-stage assembly scheduling problems, the job processing times are commonly assumed as a constant over time. However, it is at odds with real production situations some times. In fact, the dynamic nature of processing time may occur when machines lose their performance during their execution times. In this case, the job that is processed later consumes more time than another one processed earlier. In view of these observations, we address the two-stage assembly linear deterioration scheduling problem in which there are two machines at the first stage and an assembly machine at the second stage. The objective is to complete all jobs as soon as possible (or to minimise the makespan, implies that the system can yield a better and efficient task planning to limited resources). Given the fact that this problem is NP-hard, we then derive some dominance relations and a lower bound used in the branch-and-bound method for finding the optimal solution. We also propose three metaheuristics, including dynamic differential evolution (DDE), simulated annealing (SA) algorithm, and cloud theory-based simulated annealing (CSA) algorithm for find near-optimal solutions. The performances of the proposed algorithms are reported as well.