A genetic algorithm for integrating lot-sizing and sequencing in scheduling a capacitated flow line

In this paper we present a genetic algorithm for solving an important but difficult scheduling problem: that of integrating the lot-sizing and sequencing decisions in scheduling a flow line involving sequence dependent setup times, capacity constraints, limited buffer capacity between machines, and due dates. The problem is based on a real world manufacturing facility that is also described. Novel crossover and mutation operators are presented for both the lot-sizing and sequencing parts of the scheduling problem and the performance of the genetic algorithm is compared to a heuristic approach of integration previously shown to have been effective.     

Capacitated lot sizing and sequence dependent setup scheduling: an iterative approach for integration

We consider here the lot sizing and scheduling problem in single-level manufacturing systems. The shop floor is composed of unrelated parallel machines with sequence dependent setup times. We propose an integer programming model embedding precise capacity information due to scheduling constraints in a classical lot-sizing model. We also propose an iterative approach to generate a production plan taking into account scheduling constraints due to changeover setup times. The procedure executes two decision modules per iteration: a lot-sizing module and a scheduling module. The capacitated lot-sizing problem is solved to optimality considering estimated data and aggregate information, and the scheduling problem is solved by a GRASP heuristic. In the proposed scheme the information flow connecting the two levels is managed in each iteration. We report a set of computational experiments and discuss future work.     

The multi-product,economic lot-sizing problem in flow shops: the powers-of-two heuristic

This paper presents a new and efficient heuristic to solve the multi-product, multi-stage, economic lot-sizing problem. The proposed heuristic, called the powers-of-two method, first determines sequencing decisions then lot sizing and scheduling decisions are determined. This method assumes that cycle times are integer multiples of a basic period and restricts these multiples to the powers of two. Once time multiples are chosen, we determine for each basic period of the global cycle the set of products to be produced and the production sequence to be used. Then a non-linear program is solved to simultaneously determine lot sizes and a feasible production schedule. To evaluate its performance, the powers-of-two method was compared to both the common cycle method and a reinforced version of the job-splitting heuristic. Numerical results show that the powers-of-two method outperforms both of these methods.Scope and purposeThe multi-product, multi-stage, economic lot-sizing problem studied in this paper is the problem of making sequencing, lot-sizing and scheduling decisions for several products manufactured through several stages in a flow shop environment so as to minimize the sum of setup and inventory holding costs while a given demand is fulfilled without backlogging. This problem and similar problems are met in many different industries like the food canning industry, the appliance assembly facilities or in beverage bottling companies. The most commonly used approach to deal with this problem is the common cycle approach where a lot of each product is produced each cycle. A few other approaches are also proposed. In this paper we propose a new and more efficient solution approach that assigns different cycle times to different products.     

Scheduling of parallel machines with sequence-dependent batches and product incompatibilities in an automotive glass facility

This application is motivated by a complex real-world scheduling problem found in the bottleneck workstation of the production line of an automotive safety glass manufacturing facility. The scheduling problem consists of scheduling jobs (glass parts) on a number of parallel batch processing machines (furnaces), assigning each job to a batch, and sequencing the batches on each machine. The two main objectives are to maximize the utilization of the parallel machines and to minimize the delay in the completion date of each job in relation to a required due date (specific for each job). Aside from the main objectives, the output batches should also produce a balanced workload on the parallel machines, balanced job due dates within each batch, and minimal capacity loss in the batches. The scheduling problem also considers a batch capacity constraint, sequence-dependent processing times, incompatible product families, additional resources, and machine capability. We propose a two-phase heuristic approach that combines exact methods with search heuristics. The first phase comprises a four-stage mixed-integer linear program for building the batches; the second phase is based on a Greedy Randomized Adaptive Search Procedure for sequencing the batches assigned to each machine. We conducted experiments on instances with up to 100 jobs built with real data from the manufacturing facility. The results are encouraging both in terms of computing time—5 min in average—and quality of the solutions—less than 10 % relative gap from the optimal solution in the first phase and less than 5 % in the second phase. Additional experiments were conducted on randomly generated instances of small, medium, and large size.     

A distributed shifting bottleneck heuristic for complex job shops

In this paper, we consider distributed versions of a modified shifting bottleneck heuristic for complex job shops. The considered job shop environment contains parallel batching machines, machines with sequence-dependent setup times and reentrant process flows. Semiconductor wafer fabrication facilities are typical examples for manufacturing systems with these characteristics. The used performance measure is total weighted tardiness (TWT). We suggest a two-layer hierarchical approach in order to decompose the overall scheduling problem. The upper (or top) layer works on an aggregated model. Based on appropriately aggregated routes it determines start dates and planned due dates for the jobs within each single work area, where a work area is defined as a set of parallel machine groups. The lower (or base) layer uses the start dates and planned due dates in order to apply shifting bottleneck heuristic type solution approaches for the jobs in each single work area. We conduct simulation experiments in a dynamic job shop environment in order to assess the performance of the heuristic. It turns out that the suggested approach outperforms a pure First In First Out (FIFO) dispatching scheme and provides a similar solution quality as the original modified shifting bottleneck heuristic.     

Minimization of squared deviation of completion times about a common due date

We discuss a non-preemptive single-machine job sequencing problem where the objective is to minimize the sum of squared deviation of completion times of jobs from a common due date. There are three versions of the problem—tightly restricted, restricted and unrestricted. Separate dynamic programming formulations have already been suggested for each of these versions, but no unified approach is available. We have proposed a pseudo-polynomial DP solution and a polynomial heuristic for general instance. Computational results show that tightly restricted instances of up to 600 jobs can be solved in less than 6 s. General instances of up to 80 jobs take less than 2 s.Statement of scope and purposeIn this paper, we have considered an NP-complete single-machine scheduling problem arising in JIT environment, a field of great importance in manufacturing industry. The objective of the problem is to schedule a set of given jobs to minimize the sum of squared deviation of their completion times from a common due date. This paper presents a number of precedence rules, a polynomial heuristic and more importantly a unified pseudo-polynomial dynamic programming formulation. Empirical results show that the dynamic programming formulation performs better than the existing approaches.     

Minimizing total tardiness and earliness on unrelated parallel machines with controllable processing times

Job scheduling has always been a challenging task in modern manufacturing and the most real life scheduling problems which involves multi-criteria and multi-machine environments. In this research our direction is largely motivated by the adoption of the Just-In-Time (JIT) philosophy in parallel machines system, where processing times of jobs are controllable. The goal of this paper is to minimize total weighted tardiness and earliness besides jobs compressing and expanding costs, depending on the amount of compression/expansion as well as maximum completion time called makespan simultaneously. Jobs due dates are distinct and no inserted idle time is allowed after starting machine processing. Also each machine is capable of processing only some predetermined jobs and operations with probably different speeds. A Mixed Integer Programming (MIP) model is proposed to formulate such a problem and is solved optimally in small size instances. A Parallel Net Benefit Compression-Net Benefit Expansion (PNBC–NBE) heuristic is then presented to acquire the optimal jobs set amount of compression and expansion processing times in a given sequence. To solve medium-to-large size cases, a proposed heuristic, two meta-heuristics and a hybrid technique are also employed. Experimental results demonstrate that our hybrid procedure is a proficient method and could efficiently solve such complicated problems.     

Scheduling Flexible Flow Shop in Labeling Companies to Minimize the Makespan

In the competitive global marketplace, production scheduling plays a vital role in planning in manufacturing. Scheduling deals directly with the time to output products quickly and with a low production cost. This research examines case study of a Radio-Frequency Identification labeling department at Avery Dennison. The main objective of the company is to have a method that allows for the sequencing and scheduling of a set of jobs so it can be completed on or before the customer’s due date to minimize the number of late orders. This study analyzes the flexible flow shop scheduling problem with a sequence dependent setup by modifying the processing time and setup time to minimize the makespan on multiple machines. Based on the defined mathematical model, this study includes an alternative approach and application of heuristic algorithm with the input being big data. Both optimization programs are used in this study and compared to determine which method can better solve the company’s problems. The proposed algorithm is able to improve machine utilization with large-scale problems.     

Reactive scheduling in a job shop where jobs arrive over time

The paper considers the dynamic job shop scheduling problem (DJSSP) with job release dates which arises widely in practical production systems. The principle characteristic of DJSSP considered in the paper is that the jobs arrive continuously in time and the attributes of the jobs, such as the release dates, routings and processing times are not known in advance, whereas in the classical job shop scheduling problem (CJSSP), it is assumed that all jobs to be processed are available at the beginning of the scheduling process. Reactive scheduling approach is one of the effective approaches for DJSSP. In the paper, a heuristic is proposed to implement the reactive scheduling of the jobs in the dynamic production environment. The proposed heuristic decomposes the original scheduling problem into a number of sub problems. Each sub problem, in fact, is a dynamic single machine scheduling problem with job release dates. The scheduling technique applied in theproposed heuristic is priority scheduling, which determines the next state of the system based on priority values of certain system elements. The system elements are prioritized with the help of scheduling rules (SRs). An approach based on gene expression programming (GEP) is also proposed in the paper to construct efficient SRs for DJSSP. The rules constructed by GEP are evaluated in the comparison of the rules constructed by GP and several prominent human made rules selected from literatures on extensive problem sets with respect to various measures of performance.     

Scheduling in a manufacturing shop with sequence-dependent setups

In this paper, the problem of scheduling multiple jobs in a flexible manufacturing cell with multiple machine stations is addressed. Due to the large capital investments that usually characterize flexible manufacturing systems (FMS), an area of control of great interest to system users is that of maximizing the system performance through the minimization of machine idle and setup times. The magnitude of total time spent on machine setups and idle times is influenced by the availability of jobs, job mix, similarities of jobs and job scheduling procedure used. Similar jobs on the same machine require less setup times. Similarly, the use of an adequate scheduling method also reduces total idle and setup times. Such reduction improves the flow times of jobs. In this paper, a heuristic algoritm for scheduling jobs with sequence dependent setup times in a FMS is presented. The measure of performance for evaluating schedule adequacy is the production makespan.     

Genetic algorithm for rotary machine scheduling with dependent processing times

This paper focuses on scheduling a rotary injection molding machine with dependent processing times. The injection machine has n pairs of positions to process n pairs of shoes. It is rotated after every cycle time. Cycle time is the maximum injection time of the jobs currently loaded in the machine. Thus, for all practical purposes, the processing time of a job depends on the combination of the jobs currently assigned to the machine. The uncertainty of processing time makes this problem more complicated than traditional parallel machine scheduling problems. Additionally, since switching jobs leads to mold changes, set-up time is also included in the analysis. We develop a Sequential Genetic Algorithm (SGA) to identify the best schedule with regard to makespan. In this approach, multiple GA evolvers are connected by using a feeding strategy, where each GA evolver identifies the best schedule with minimum makespan for the corresponding product family. A multi-segment (product lines) chromosome representation is applied to represent the product line sequence as well as the job sequence within a product family. Furthermore, an adaptive feeding strategy is also proposed to improve results and reduce computation times. Besides SGA, we also improve the performance of a traditional heuristic procedure by proposing a minimum ΔIT heuristic approach. The experimentation is performed by using four experimental data sets with different demand patterns and nine data sets from a shoe manufacturing plant. The results indicate that our SGA provides better schedule with respect to makespan value, while heuristic procedures take insignificant time to obtain results. Another observation is that adaptive feeding strategy helps to find good results in a shorter time.     

Selection of cutting conditions for power constrained parallel machine scheduling

Most previous studies on machining optimization focused on aspects related to machining efficiency and economics, without accounting for environmental considerations. Higher cutting speed is usually desirable to maximize machining productivity, but this requires a high power load peak. In Taiwan, electricity prices rise sharply if instantaneous power demand exceeds contract capacity. Many studies over the previous decades have examined production scheduling problems. However, most such studies focused on well-defined jobs with known processing times. In addition, traditional sequencing and scheduling models focus primarily on economic objectives and largely disregard environmental issues raised by production scheduling problems. This study investigates a parallel machine scheduling problem for a manufacturing system with a bounded power demand peak. The challenge is to simultaneously determine proper cutting conditions for various jobs and assign them to machines for processing under the condition that power consumption never exceed the electricity load limit. A two-stage heuristic approach is proposed to solve the parallel machine scheduling problem with the goal of minimizing makespan. The heuristic performance is tested by distributing 20 jobs over 3 machines with four possible cutting parameter settings.     

Deteriorating job scheduling to minimize the number of late jobs with setup times

Deteriorating jobs scheduling problems have been widely studied recently. However, research on scheduling problems with deteriorating jobs has rarely considered explicit setup times. With the current emphasis on customer service and meeting the promised delivery dates, we consider a single-machine scheduling problem to minimize the number of late jobs with deteriorating jobs and setup times in this paper. We derive some dominance properties, a lower bound, and an initial upper bound by using a heuristic algorithm to speed up the search process of the branch-and-bound algorithm. Computational experiments show that the algorithm can solve instances up to 1000 jobs in a reasonable amount of time.     

Decomposition heuristics for minimizing earliness–tardiness on parallel burn-in ovens with a common due date

This paper considers a scheduling problem for parallel burn-in ovens in the semiconductor manufacturing industry. An oven is a batch processing machine with restricted capacity. The batch processing time is set by the longest processing time among those of all the jobs contained in the batch. All jobs are assumed to have the same due date. The objective is to minimize the sum of the absolute deviations of completion times from the due date (earliness–tardiness) of all jobs. We suggest three decomposition heuristics. The first heuristic applies the exact algorithm due to Emmons and Hall (for the nonbatching problem) in order to assign the jobs to separate early and tardy job sets for each of the parallel burn-in ovens. Then, we use job sequencing rules and dynamic programming in order to form batches for the early and tardy job sets and sequence them optimally. The second proposed heuristic is based on genetic algorithms. We use a genetic algorithm in order to assign jobs to each single burn-in oven. Then, after forming early and tardy job sets for each oven we apply again sequencing rules and dynamic programming techniques to the early and tardy jobs sets on each single machine in order to form batches. The third heuristic assigns jobs to the m early job sets and m tardy jobs sets in case of m burn-in ovens in parallel via a genetic algorithm and applies again dynamic programming and sequencing rules. We report on computational experiments based on generated test data and compare the results of the heuristics with known exact solution for small size test instances obtained from a branch and bound scheme.     

A hybrid Lagrangian-simulated annealing-based heuristic for the parallel-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times

This paper examines the parallel-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times, time windows, machine eligibility and preference constraints. Such problems are quite common in the semiconductor manufacturing industry. In particular, this paper pays special attention to the chipset production in the semiconductor Assembly and Test Manufacturing (ATM) factory and constructs a Mixed Integer Programming (MIP) model for the problem. The primal problem is decomposed into a lot-sizing subproblem and a set of single-machine scheduling subproblems by Lagrangian decomposition. A Lagrangian-based heuristic algorithm, which incorporates the simulated annealing algorithm aimed at searching for a better solution during the feasibility construction stage, is proposed. Computational experiments show that the proposed hybrid algorithm outperforms other heuristic algorithms and meets the practical requirement for the tested ATM factory.     

Two-machine flowshop scheduling with bicriteria problem

This paper attempts to solve a two-machine flowshop bicriteria scheduling problem with release dates for the jobs, in which the objective function is to minimize a weighed sum of total flow time and makespan. To tackle this scheduling problem, an integer programming model with N²+3N variables and 5N constraints where N is the number of jobs, is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, a heuristic scheduling algorithm is presented. Experimental results show that the proposed heuristic algorithm can solve this problem rapidly and accurately. The average solution quality of the heuristic algorithm is above 99% and is much better than that of the SPT rule as a benchmark. A 15-job case requires only 0.018 s, on average, to obtain an ultimate or even optimal solution. The heuristic scheduling algorithm is a more practical approach to real world applications than the integer programming model.     

Dominance rules for single machine schedule with sequence dependent setup and due date

Some dominance rules are proposed for the problems of scheduling N jobs on a single machine with due dates, sequence dependent setup times and no preemption. Two algorithms based on Ragatz＇ s branch and bound scheme are developed including the dominance rules where the objective is to minimize the maximum tardiness or the total tardiness. Computational experiments demonstrate the effectiveness of the dominance rules.     

Dominance rules for single machine schedule with sequence dependent setup and due date

Some dominance rules are proposed for the problems of scheduling N jobs on a single machine with due dates, sequence dependent setup times and no preemption. Two algorithms based on Ragatz's branch and bound scheme are developed including the dominance rules where the objective is to minimize the maximum tardiness or the total tardiness. Computational experiments demonstrate the effectiveness of the dominance rules.     

A note on “the capacitated lot-sizing and scheduling problem with sequence-dependent setup costs and setup times

Gupta and Magnusson [The capacitated lot-sizing and scheduling problem with sequence-dependent setup costs and setup times. Computers and Operations Research 2005;32(4):727–47] develop a model for the single machine capacitated lot-sizing and scheduling problem (CLSP) with sequence dependent setup times and setup costs, incorporating all the usual features of setup carryovers. In this note we show that this model does not avoid disconnected subtours. A new set of constraints is added to the model to provide an exact formulation for this problem.     

Scheduling models for air traffic control in terminal areas

We propose a job-shop scheduling model with sequence dependent set-up times and release dates to coordinate both inbound and outbound traffic flows on all the prefixed routes of an airport terminal area and all aircraft operations at the runway complex. The proposed model is suitable for representing several operational constraints (e.g., longitudinal and diagonal separations in specific airspace regions), and different runway configurations (e.g., crossing, parallel, with or without dependent approaches) in a uniform framework. The complexity and the highly dynamic nature of the problem call for heuristic approaches. We propose a fast dynamic local search heuristic algorithm for the job-shop model suitable for considering one of the different performance criteria and embedding aircraft position shifting control technique to limit the controllers/pilots’ workload. Finally, we describe in detail the experimental analysis of the proposed model and algorithm applied to two real case studies of Milan-Malpensa and Rome-Fiumicino airport terminal areas. This work has been partially supported by grant CNR FPCCR021074 from the Italian National Research Council.