The two-stage assembly scheduling problem to minimize total completion time with setup times

We address the two-stage assembly scheduling problem where there are m machines at the first stage and an assembly machine at the second stage. The objective is to schedule the available n jobs so that total completion time of all n jobs is minimized. Setup times are treated as separate from processing times. This problem is NP-hard, and therefore we present a dominance relation and propose three heuristics. The heuristics are evaluated based on randomly generated data. One of the proposed heuristics is known to be the best heuristic for the case of zero setup times while another heuristic is known to perform well for such problems. A new version of the latter heuristic, which utilizes the dominance relation, is proposed and shown to perform much better than the other two heuristics.     

The 2-stage assembly flowshop scheduling problem with total completion time: Efficient constructive heuristic and metaheuristic

In this paper, we address the 2-stage assembly scheduling problem where there are m machines in the first stage to manufacture the components of a product and one assembly station (machine) in the second stage. The objective considered is the minimisation of the total completion time. Since the NP-hard nature of this problem is well-established, most previous research has focused on finding approximate solutions in reasonable computation time. In our paper, we first review and derive a number of problem properties and, based on these ideas, we develop a constructive heuristic that outperforms the existing constructive heuristics for the problem, providing solutions almost in real-time. Finally, for the cases where extremely high-quality solutions are required, a variable local search algorithm is proposed. The computational experience carried out shows that the algorithm outperforms the best existing metaheuristic for the problem. As a summary, the heuristics presented in the paper substantially modify the state-of-the-art of the approximate methods for the 2-stage assembly scheduling problem with total completion time objective.     

Lot streaming in a two-stage assembly system

In this paper, we address a scheduling problem belonging to a two-stage assembly system that can also be viewed as a mass customization system. The first stage of this system consists of a set of subassembly machines, each of which produces a component type. These components are then transferred in sublots to Stage 2, where they are assembled into finished products. Stage 2 consists of multiple parallel machines. Each of these machines represents an assembly facility devoted to a product type. We represent this configuration as a m+n system, where there are m parallel machines at Stage 1 and n parallel machines at Stage 2. Lot-attached and sublot-attached setups, and transfer times/costs are encountered on the machines at both the stages. We assume that the subassemblies are transferred in equal-sized sublots to Stage 2. Given a number of lots (jobs), the problem is to determine the number of sublots to use for each lot and the sequence in which to process the lots. We consider two different objective functions, namely, minimize makespan, and minimize the total cost incurred. In view of the fact that both of these problems are NP-hard, we develop two column generation-based heuristic methods and show their efficacy over direct solution of the mixed integer programming formulations of the underlying problems. In fact, the results of our computational investigation on the use of these methods on large-sized problems reveal attainment of almost optimal solutions within a few seconds of CPU time. We also present some managerial insights.     

Setting a common due date in a constrained flowshop: A variable neighbourhood search approach

In this paper we study a due date setting problem in a flowshop layout. The problem consists of scheduling a set of jobs arriving to the system together with jobs already present (denoted as old jobs), in order to set a common due date for the new jobs. Since the old jobs have a common due date that must not be violated, our problem is a rescheduling problem with the objective of minimising the makespan of the new jobs (thus obtaining the tightest possible due date for the new jobs) and a constraint since the maximum tardiness of the old jobs must be equal to zero. This approach leads to an interesting scheduling problem in which two different objectives are considered, each one for a subset of the jobs that must be scheduled. To the best of our knowledge, this type of problems have been scarcely considered in the literature, and only for very specific purposes. Since our problem is clearly NP-hard, a new heuristic based on variable neighbourhood search (VNS) has been designed. The computational results show that our proposed heuristic outperforms two existing heuristic methods for similar problems in the literature.     

A two-stage hybrid flowshop scheduling problem in machine breakdown condition

This paper considers a two-stage hybrid flowshop scheduling problem in machine breakdown condition. By machine breakdown condition we mean that the machine may not always be available during the scheduling period. Machine failure may occur with a known probability after completing a job. Probability of machine failure depends on the previous processed job. The problem to be studied has one machine at the first stage and M parallel identical machines at the second stage. The objective is to find the optimal job combinations and the optimal job schedule such that the makespan is minimized. The proposed problem is compatible with a large scope of real world situations. To solve the problem, first, we introduce one optimal approach for job precedence when there is one machine in both stages and then provide a heuristic algorithm when there are M machines in stage two. To examine the performance of the heuristic, some experiments used are provided as well.     

Scheduling of multi-component products in a two-stage flexible flow shop

In this research, the problem of scheduling and sequencing of two-stage assembly-type flexible flow shop with dedicated assembly lines, which produce different products according to requested demand during the planning horizon with the aim of minimizing maximum completion time of products is investigated. The first stage consists of several parallel machines in site I with different speeds in processing components and one machine in site II, and the second stage consists of two dedicated assembly lines. Each product requires several kinds of components with different sizes. Each component has its own structure which leading to difference processing times to assemble. Products composed of only single-process components are assigned to the first assembly line and products composed of at least a two-process component are assigned to the second assembly line. Components are placed on the related dedicated assembly line in the second phase after being completed on the assigned machines in the first phase and final products will be produced by assembling the components. The main contribution of our work is development of a new mathematical model in flexible flow shop scheduling problem and presentation of a new methodology for solving the proposed model. Flexible flow shop problems being an NP-hard problem, therefore we proposed a hybrid meta-heuristic method as a combination of simulated annealing (SA) and imperialist competitive algorithms (ICA). We implement our obtained algorithm and the ones obtained by the LINGO9 software package. Various parameters and operators of the proposed Meta-heuristic algorithm are discussed and calibrated by means of Taguchi statistical technique.     

Minimizing makespan for flow shop scheduling problem with intermediate buffers by using hybrid approach of artificial immune system

In this study, three new meta-heuristic algorithms artificial immune system (AIS), iterated greedy algorithm (IG) and a hybrid approach of artificial immune system (AIS-IG) are proposed to minimize maximum completion time (makespan) for the permutation flow shop scheduling problem with the limited buffers between consecutive machines. As known, this category of scheduling problem has wide application in the manufacturing and has attracted much attention in academic fields. Different from basic artificial immune systems, the proposed AIS-IG algorithm is combined with destruction and construction phases of iterated greedy algorithm to improve the local search ability. The performances of these three approaches were evaluated over Taillard, Carlier and Reeves benchmark problems. It is shown that the AIS-IG and AIS algorithms not only generate better solutions than all of the well-known meta heuristic approaches but also can maintain their quality for large scale problems.     

Single machine scheduling problem with interval processing times to minimize mean weighted completion time

The single resource scheduling problem is commonly applicable in practice not only when there is a single resource but also in some multiple-resource production systems where only one of the resources is bottle neck. Thus, the single resource (machine) scheduling problem has been widely addressed in the scheduling literature. In this paper, the single machine scheduling problem with uncertain and interval processing times is addressed. The objective is to minimize mean weighted completion time. The problem has been addressed in the literature and efficient heuristics have been presented. In this paper, some new polynomial time heuristics, utilizing the bounds of processing times, are proposed. The proposed and existing heuristics are compared by extensive computational experiments. The conducted experiments include a generalized simulation environment and several additional representative distributions in addition to the restricted experiments used in the literature. The results indicate that the proposed heuristics perform significantly better than the existing heuristics. Specifically, the best performing proposed heuristic reduces the error of the best existing heuristic in the literature by more than 75% while the computational time of the best performing proposed heuristic is less than that of the best existing heuristic. Moreover, the absolute error of the best performing heuristic is only about 1% of the optimal solution. Having a very small absolute error along with a negligible computational time indicates the superiority of the proposed heuristics.     

Heuristics for hybrid flow shops with controllable processing times and assignable due dates

This paper considers a generalization of the permutation flow shop problem that combines the scheduling function with the planning stage. In this problem, each work center consists of parallel identical machines. Each job has a different release date and consists of ordered operations that have to be processed on machines from different machine centers in the same order. In addition, the processing times of the operations on some machines may vary between a minimum and a maximum value depending on the use of a continuously divisible resource. We consider a nonregular optimization criterion based on due dates which are not a priori given but can be fixed by a decision-maker. A due date assignment cost is included into the objective function. For this type of problems, we generalize well-known approaches for the heuristic solution of classical problems and propose constructive algorithms based on job insertion techniques and iterative algorithms based on local search. For the latter, we deal with the design of appropriate neighborhoods to find better quality solution. Computational results for problems with up to 20 jobs and 10 machine centers are given.Scope and purposeTraditional research to solve multi-stage scheduling problems has focused on regular measures of performance based on a single criterion and assumes that several decisions related to due dates and processing times have already been made. However, in many industrial scheduling practices, managers develop schedules based on multicriteria and have to decide the due dates and processing times as part of the scheduling activities. Further, in practical scheduling situations, there are multiple machines at each stage and the objective function often reflects the total cost of processing, earliness and tardiness. Such scheduling problems require significantly more effort in finding acceptable solutions and hence have not received much attention in the literature. For this reason, this paper considers one such hybrid flow shop scheduling problem involving nonregular measures of performance, controllable processing times, and assignable due dates. We combine and generalize the existing approaches for the classical flow shop problem to the problem under consideration. Computational experiments are used to evaluate the utility of the proposed algorithms for the generalized scheduling problems. Brah and Hunsucker (European Journal of Operational Research 1991;51:88–99) and Nowicki and Smutnicki (European Journal of Operational Research 1998;106:226–253) describe branch and bound and tabu search algorithms for the approach used in the development of heuristic algorithms can also be adapted to several other complex practical scheduling problems.     

Two-stage hybrid flowshop scheduling with simultaneous processing machines

Simultaneous processing machines, common in processing industries such as steel and food production, can process several jobs simultaneously in the first-in, first-out manner. However, they are often highly energy-consuming. In this paper, we study a new two-stage hybrid flowshop scheduling problem, with simultaneous processing machines at the first stage and a single no-idle machine with predetermined job sequence at the second stage. A mixed integer programming model is proposed with the objective of minimizing the total processing time to reduce energy consumption and improve production efficiency. We give a sufficient and necessary condition to construct feasible sequencing solutions and present an effective approach to calculate the time variables for a feasible sequencing solution. Based on these results, we design a list scheduling heuristic algorithm and its improvement. Both heuristics can find an optimal solution under certain conditions with complexity O(nlogn), where n is the number of jobs. Our experiments verify the efficiency of these heuristics compared with classical heuristics in the literature and investigate the impacts of problem size and processing times.     

The production scheduling problem in a multi-page invoice printing system

This paper addresses the production scheduling problem in a multi-page invoice printing system. The system comprises three stages: the stencil preparation stage, the page printing stage and the invoice assembly stage. Since each page can be considered as a component and the invoice as the finished product, the production system for multi-page invoices can be treated as an assembly-type flowshop with parallel machines at the last two stages. Moreover, two types of sequence-dependent setup operations are considered at the second stage. The objective is to minimize the makespan for all the invoice orders. We first formulate this problem into a mixed-integer linear programming (MILP). Then a hybrid genetic algorithm (HGA) is proposed for solving it due to its NP-hardness. To evaluate the performance of the HGA heuristic, a lower bound for the makespan is developed. Numerical experiment indicates that our algorithm can solve the problem efficiently and effectively.     

An efficient heuristic for a two-stage assembly scheduling problem with batch setup times to minimize makespan

This paper considers a two-stage assembly scheduling problem of N products with setup times to minimize the makespan. In this problem, there is a machining machine which produces components in the first stage. When the required components are available, a single assembly machine can assemble these components into products in the second stage. A setup time is needed whenever the machining machine starts processing components, or the item of component is switched on the machine. The problem is formulated as a mixed integer programming model, and several properties for finding optimal solutions are developed. Moreover, an efficient heuristic based on these optimal properties is proposed. A lower bound is derived to evaluate the performance of the proposed heuristic. Computational results show that the proposed heuristic can obtain a near optimal solution in almost zero time and the average percentage deviation is only 0.478.     

A two-stage hybrid flowshop scheduling problem with a function constraint and unrelated alternative machines

This article addresses a two-stage hybrid flowshop scheduling problem with unrelated alternative machines. The problem to be studied has m unrelated alternative machines at the first machine center followed by a second machine center with a common processing machine in the system. The objective is to minimize the makespan of the system. For the processing of any job, it is assumed that the operation can be partially substituted by other machines in the first center, depending on its machining constraints. Such scheduling problems occur in certain practical applications such as semiconductors, electronics manufacturing, airplane engine production, and petrochemical production. We demonstrate that the addressed problem is NP-hard and then provide some heuristic algorithms to solve the problem efficiently. The experimental results show that the combination of the modified Johnson's rule and the First-Fit rule provides the best solutions within all proposed heuristics.Scope and purpose     

Multiobjective pseudo‐variable neighborhood descent for a bicriteria parallel machine scheduling problem with setup time

A multiobjective variable neighborhood descent (VND) based heuristic is developed to solve a bicriteria parallel machine scheduling problem. The problem considers two objectives, one related to the makespan and the other to the flow time, where the setup time depends on the sequence, and the machines are identical. The heuristic has a set of neighborhood structures based on swap, remove, and insertion moves. We propose changing the local search inside the VND to a sequential search through the neighborhoods to obtain nondominated points for the Pareto‐front quickly. In the numerical tests, we consider a single‐objective version of the heuristic, comparing the results on 510 benchmark instances to show that it is quite effective. Moreover, new instances are generated in accordance with the literature for the bicriteria problem, showing the ability of the proposed heuristic to return an efficient set of nondominate solutions compared with the well‐known nondominated sorting genetic algorithm II.     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

Glass container production scheduling through hybrid multi-population based evolutionary algorithm

Driven by a real-world application in the capital-intensive glass container industry, this paper provides the design of a new hybrid evolutionary algorithm to tackle the short-term production planning and scheduling problem. The challenge consists of sizing and scheduling the lots in the most cost-effective manner on a set of parallel molding machines that are fed by a furnace that melts the glass. The solution procedure combines a multi-population hierarchically structured genetic algorithm (GA) with a simulated annealing (SA), and a tailor-made heuristic named cavity heuristic (CH). The SA is applied to intensify the search for solutions in the neighborhood of the best individuals found by the GA, while the CH determines quickly values for a relevant decision variable of the problem: the processing speed of each machine. The results indicate the superior performance of the proposed approach against a state-of-the-art commercial solver, and compared to a non-hybridized multi-population GA.     

A note: Simple heuristics for scheduling a maintenance activity on unrelated machines

Following several recent papers discussing various problems of scheduling a maintenance activity, we focus here on scheduling a maintenance activity on unrelated parallel machines. The objective is to minimize flow-time. In the basic setting, we assume that all the machines must be maintained simultaneously. The problem is known to be NP-hard, and we introduce and test numerically an efficient heuristic and a lower bound, both based on a solution of a matching problem. We also study the relaxed version, where the machines are not restricted to be maintained at the same time. Similar heuristic and lower bound are proposed and tested.     

Nonpreemptive Coordination Mechanisms for Identical Machines

We focus on the problem of scheduling n weighted selfish tasks on m identical parallel machines and we study the performance of nonpreemptive coordination mechanisms. A nonpreemptive coordination mechanism consists of m local scheduling policies that decide the processing order of the tasks on each machine without delays or interruptions. We discuss the existence of Nash equilibria for this setting and show that it is not a guaranteed property of all nonpreemptive coordination mechanisms. Next, we focus on the wider class of randomized Nash equilibria and prove lower bounds on the price of anarchy. Our lower bounds are presented in comparison to the currently best known coordination mechanism (which uses delays) and lead to the conclusion that preemption or delays are required in order to improve on the currently best known solution.     

Hierarchical minimization of completion time variance and makespan in jobshops

This paper addresses the problem of minimizing makespan for a given set of n jobs to be processed on each of m machines in a static jobshop, subject to the minimum completion time variance (CTV). A lower bound on CTV is developed for the static jobshop problem. A backward scheduling approach is proposed using the observations on the development of lower bound for hierarchical minimization of CTV and makespan. A lower bound on makespan subject to minimum CTV is also presented for this problem. Finally, we present two simulated annealing heuristic approaches using the concepts of forward and backward scheduling. Their performances are compared against each other through the use of the lower bounds established in this work. The simulated annealing heuristic based on backward scheduling is shown to perform well by evaluating the developed heuristics on 82 jobshop problems taken from literature.