Two-machine Case Research Articles

Flow-shop scheduling with exact delays to minimize makespan

The flow-shop scheduling problem with exact delays is generalization of no-wait flow-shop scheduling in which an exact delay exists between the consecutive tasks of each job. The problem with distinct delays to minimize the makespan is strongly NP-hard even for the two-machine case with unit execution time tasks. Providing polynomial-time solutions for special cases of the problem, we show that the two-machine permutation flow-shop case is solvable in O(nlogn) time, while the case with more than two machines is strongly NP-hard. We also show that the multi-machine case with delays following the ordered structure possesses the pyramidal-shaped property and propose an O(n2)-time dynamic program to solve it. We further improve the time complexity of the solution algorithm to O(nlogn) under certain conditions.

Parameter Identification for Bernoulli Serial Production Line Model

Model-based analysis of production systems is one of the main areas in manufacturing research. The foundation of the successful application of these theoretical studies is the availability of valid and high-fidelity mathematical models that are capable of capturing the behavior of job flow in production systems. The modeling process of a production system, however, may require a significant amount of nonstandardized work that can only be done properly by someone with solid training in the area and extensive experience through real case studies. This poses a critical challenge in the effective implementation of these valuable theoretical results in the Industry 4.0 era. To overcome this, we propose a new production systems modeling paradigm inspired by system identification: calculate production system model parameters that best match the standard system performance metrics measured on the factory floor. Specifically, in this article, we consider production lines characterized by the Bernoulli serial line model and develop algorithms that identify model parameters to fit the system throughput and work-in-process. Analytical algorithms are derived to solve this problem in a two-machine line case and then extended to multi-machine lines. The accuracy and computational efficiency of the algorithms are demonstrated through extensive numerical experiments. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —A high-fidelity mathematical model is of critical importance to the implementation of any model-based production system analysis method. Currently, the construction of such models is carried out in an ad hoc manner. The quality of the resulting models may heavily depend on the training, experience, intuition, and personal preference of the modeler. The proposed model parameter identification method focuses on standard key performance indices commonly measured on the factory floor. The advantage is twofold. First, these standard performance metrics are consistently defined regardless of industry, thus avoiding any data-ambiguity issue that may occur when using complex machine/equipment status data. Second, measuring these performance metrics in real time is typically convenient and cost effective, even for manufacturing plants without high-end IT infrastructure, thus making the technology accessible to not only large but also small- and mid-sized manufacturers. Using the algorithms developed in this article, a practitioner can quickly construct a serial production line model and then utilize it to access the rich library of production analysis, design, and control methods available in the literature.

Minimizing maximum delivery completion time for order scheduling with rejection

We study an order scheduling problem with rejection, in which each order consists of multiple product types and each product type should be manufactured on a dedicated machine. The aim is to find a solution to minimize a linear sum of the maximum delivery completion time of the accepted orders and the total penalty of the rejected orders. Even if the delivery times of all orders are zero, the problem is shown to be binary $$\mathcal{NP}$$ -hard in the two-machine case and it is shown to be unary $$\mathcal{NP}$$ -hard when the number of machines is arbitrary. Three approximation algorithms are proposed and their worst-case performance ratios are analyzed. For the scenario where the number of machines is fixed, a pseudo-polynomial dynamic programming algorithm and a fully polynomial time approximation scheme are devised for it.

Approximation algorithms for makespan minimization on identical parallel machines under resource constraints

The problem of minimizing the makespan on parallel identical machines is considered in the presence of additional resources, provided that some jobs at any time of their processing require one unit of a particular resource. We establish a lower bound on the worst-case performance of any group technology algorithm, which schedules the composite jobs formed of the original jobs that require the same resource. A simple group technology algorithm is given such that in the worst case no group technology algorithm performs better. An algorithm for the two-machine case is presented which guarantees a tight worst-case performance ratio of 6/5.

Buffer allocation in asynchronous serial production systems with Bernoulli machines during transients

Some asynchronous production systems can never be truly balanced. This phenomenon demands the incorporation of asynchronism and unreliability in buffer allocation problems (BAPs). This paper reports a BAP in an asynchronous serial production system with Bernoulli machines. A profit-based integer programming model is formulated to maximise the throughput and minimise the buffer capacity simultaneously. Because the production run is finite in the current mass customisation environment, a transient analysis is used to estimate the system performance. Piecewise closed-form expressions are derived for a two-machine case based on Markovian structures. For M > 2 machine lines, an efficient recursive algorithm based on aggregation is developed. Considering the nonlinear characteristic of this BAP, a dynamic-objective particle swarm optimisation with neighbourhood searching is developed. This approach is verified by a case from an assembly plant. The results demonstrate that the programming method can increase the system profit by 1.85%.

Fully polynomial time approximation scheme to maximize early work on parallel machines with common due date

We study the scheduling problem on parallel identical machines in order to maximize the total early work, i.e. the parts of non-preemptive jobs executed before a common due date, and investigate mainly the model with a fixed number of machines, for which a dynamic programming approach and a fully polynomial time approximation scheme (FPTAS) are proposed. The proposal of these methods allowed us to establish the complexity and approximability status of this problem more exactly. Moreover, since our FPTAS can be also applied for the two-machine case, we improve considerably the result known in the literature for this model, in which a polynomial time approximation scheme (PTAS) was given. The new FPTAS has not only the best computational complexity, but also the much better approximation ratio than the PTAS. Finally, the theoretical studies are completed with computational experiments, performed for dynamic programming, PTAS and FPTAS, showing the high efficiencies of FPTAS both in terms of time consumption and solution quality.

On the parameterized tractability of the just-in-time flow-shop scheduling problem

We study the parameterized complexity of a set of flow-shop scheduling problems in which the objective is to maximize the weighted number of just-in-time jobs. Our analysis focuses on the case where the number of due dates is considerably smaller than the number of jobs and thus can be considered as a parameter. We prove that the two-machine problem is W[1]-hard with respect to this parameter, even if all processing times on the second machine are of unit length, while the problem is in XP when parameterized by the number of machines combined with the number of different due dates. We then move on to study the tractability of the problem when combining the number of different due dates with either the number of different weights or the number of different processing times on the first machine. We prove that in both cases the problem is fixed-parameter tractable for the two-machine case and is W[1]-hard for three or more machines.

Makespan optimization in a no-wait flowline manufacturing cell with sequence-dependent family setup times

The no-wait flowline manufacturing cell scheduling problem with sequence-dependent family setup times and the objective of minimizing makespan is NP-hard in the strong sense even in the two-machine case. This work proposes an efficient matheuristic for solving this problem to optimality in a reasonable computational time. To demonstrate the effectiveness and efficiency of the proposed matheuristic, computational experiments are performed using a well-known benchmark problem set. The computational results reveal that all test instances of the problem are solved to optimality using the proposed matheuristic, and that the computational times of the matheuristic are comparable with those of state-of-the-art meta-heuristics for finding approximate solutions. This work contributes significantly to the research into algorithms for solving this computationally intractable problem.

Minimizing Total Idle Energy Consumption in the Permutation Flow Shop Scheduling Problem

In this paper, we investigate the well-known permutation flow shop (PFS) scheduling problem with a particular objective, the minimization of total idle energy consumption of the machines. The problem considers the energy waste induced by the machine idling, in which the idle energy consumption is evaluated by the multiplication of the idle time and power level of each machine. Since the problem considered is NP-hard, theoretical results are given for several basic cases. For the two-machine case, we prove that the optimal schedule can be found by employing a relaxed Johnson’s algorithm within O([Formula: see text]) time complexity. For the cases with multiple machines (not less than 3), we propose a novel NEH heuristic algorithm to obtain an approximate energy-saving schedule. The heuristic algorithms are validated by comparison with NEH on a typical PFS problem and a case study for tire manufacturing shows an energy consumption reduction of approximately [Formula: see text] by applying the energy-saving scheduling and the proposed algorithms.

Non-permutation flowshop scheduling problem with minimal and maximal time lags: theoretical study and heuristic

In this paper, we address the non-permutation flowshop scheduling problem with minimal and maximal time lags between successive operations of each job. For this problem, the set of permutation schedules is not a dominant set but not all non-permutation schedules are feasible because they are not always able to satisfy all time lag constraints. We present a theoretical study, limited to the two-machine case, and related to the change on one machine of the order of two successive jobs of a permutation schedule. This study gives first the necessary conditions to make such move with regard to the feasibility of the schedule; secondly the necessary conditions to make such move interesting with regard to either the makespan or the number of tardy jobs. Through this analysis, we obtain new properties of dominance of permutation schedules. The results of the study are incorporated into a heuristic algorithm which starts the search with optimal permutation schedules and tries to improve them so as to obtain better non-permutation schedules. We also propose a mixed integer linear programming model. The objective function is to minimize lexicographically the number of tardy jobs as primary criterion and the makespan as secondary one. Computational experiments are performed to compare permutation with non-permutation schedules.

Identical parallel machine scheduling with time-dependent processing times

This paper introduces an exact algorithm which minimizes the total completion time for a two parallel machine problem with time dependent deterioration effect (2|rj=t0,pj=bj×tj|Cmax). By deterioration effect, we mean that the processing time of a job is defined by an increasing function of its starting time. The proposed algorithm combines a lexicographic search and a list scheduling rule based on a largest deteriorating rate first strategy to provide the optimal solution.This algorithm is extended to solve the general version of this problem (P|rj=t0,pj=bj×tj|Cmax). This extension is obtained by transforming the original m-machine problem into a series of two-machine sub-problems, and solving them repeatedly using the optimal algorithm cited above.A numerical experiment study is performed to evaluate the effectiveness of the proposed algorithm by comparison with three heuristic methods inspired by the literature. The obtained results show that for the two-machine case, the proposed optimal algorithm performs very well even for large instances. For the general case with m>2, the obtained results show that the extended version of the algorithm outperforms significantly the heuristic methods.

Notes on a hierarchical scheduling problem on identical machines

For the hierarchical scheduling problem on identical machines to minimize the maximum T-time of all machines under the condition that the total completion time of all jobs is minimum, where the T-time of a machine is defined as the total completion time of jobs scheduled on the machine, it is NP-hard if the number of the machines is fixed, and strongly NP-hard otherwise. When the number of the machines is fixed, a forward dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS) have been presented in a literature. In the literature, it is showed that the worst-case ratio of the classical algorithm SPT is at most 116 and at least 53. In this paper, we give an improved worst-case ratio, which is at most 95 and at least 74, of the algorithm. Another algorithm, whose worst-case ratio is at most 76 and at least 3533, is provided for the two-machine case. On the other hand, we present a backward dynamic programming algorithm and an FPTAS with the better time complexities.

Copula-based Randomized Mechanisms for Truthful Scheduling on Two Unrelated Machines

We design a Copula-based generic randomized truthful mechanism for scheduling on two unrelated machines with approximation ratio within [1.5852,1.58606], offering an improved upper bound for the two-machine case. Moreover, we provide an upper bound 1.5067711 for the two-machine two-task case, which is almost tight in view of the known lower bound of 1.506 for the scale-free truthful mechanisms. Of independent interest is the explicit incorporation of the concept of Copula in the design and analysis of the proposed approximation algorithm. We hope that techniques like this one will also prove useful in solving other related problems in the future.

순서 독립적인 셋업타임을 가진 동일작업의 병렬기계 배치스케줄링

We consider the problem of scheduling identical jobs with sequence-independent setup times on parallel machines. The objective is to minimize total completion times. We present the pseudopolynomial-time algorithm for the case with a fixed number of machines and an efficient approximation algorithm for our problem with identical setup times, which is known to be NP-hard even for the two-machine case.

Split scheduling with uniform setup times

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup, a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine- and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue, why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with the objective to minimise total weighted completion time, we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine- and sequence-independent setup times.

Bernoulli serial lines with deteriorating product quality: performance evaluation and system-theoretic properties

Manufacturing systems with perishable products are widely seen in practice (e.g. food, metal processing, etc.). In such systems, the quality of a part is highly dependent on its residence time within the system. However, the behaviour and properties of these systems have not been studied systematically and, therefore, it is carried out in this paper. Specifically, we assume that the probability that each unfinished part is of good quality is a decreasing function of its residence time in the preceding buffer. Then, in the framework of serial production lines with machines having Bernoulli reliability model, we derive closed-form formulas for performance evaluation in the two-machine line case, and develop an aggregation-based procedure to approximate the performance measures in -machine lines. In addition, we study monotonicity properties of these production lines using numerical experiments. A case study in an automotive stamping plant is described to illustrate the theoretical results obtained.

Heuristics for parallel machine scheduling with batch delivery consideration

We consider the parallel machine scheduling problem in which thefinished jobs need to be delivered to a customer in batches by asingle vehicle. The goal is to minimize the makespan, i.e., thetime by which the vehicle has delivered all jobs and returned toits initial location. We distinguish two types of batchingstrategies. The strategy of Type 1 permits the jobs processed ondifferent machines to compose a delivery batch, and the strategyof Type 2 assumes that only the jobs processed on the same machinecan compose a batch. For both types of the $m$-machine case, wepropose $(2-\frac{1}{m})$-approximation algorithms respectively.For both types of the two-machine case, we obtain two improved$\frac{4}{3}$-approximation algorithms.

Performance Analysis of Serial Production Lines with Deteriorating Product Quality

In manufacturing systems with perishable products (e.g., food, metal processing, etc.), the quality of a part is highly dependent on its residence time within the system. However, behavior and properties of these systems have not been studied systematically, and, therefore, is carried out in this paper. Specifically, we assume that the probability that each unfinished part is of good quality is a decreasing function of its residence time in the preceding buffer. Then, in the framework of serial production lines with machines having Bernoulli reliability model, we derive closed-form formulas for performance evaluation in the two-machine line case, and develop an aggregation-based procedure to approximate the performance measures in M > 2-machine lines. In addition, we study monotonicity properties of these production lines using numerical experiments. Finally, a case study in an automotive stamping plant is discussed.

Comparing the minimum completion times of two longest-first scheduling-heuristics

For the basic problem of non-preemptively scheduling n independent jobs on m identical parallel machines so that the minimum (or earliest) machine completion time is maximized, we compare the performance relationship between two well-known longest-first heuristics—the LPT- (longest processing time) and the RLPT-heuristic (restricted LPT). We provide insights into the solution structure of these two sequencing heuristics and prove that the minimum completion time of the LPT-schedule is at least as long as the minimum completion time of the RLPT-schedule. Furthermore, we show that the minimum completion time of the RLPT-heuristic always remains within a factor of 1/m of the optimal minimum completion time. The paper finishes with a comprehensive experimental study of the probabilistic behavior of RLPT-schedules compared to LPT-schedules in the two-machine case.

A closed-form solution for the optimal release times for the [formula omitted] deteriorating jobs [formula omitted] problem

We consider job scheduling on a flow-line production system, which covers a wide range of real-world manufacturing situations from plastic molding, steel milling to machine maintenance, and the service industry, where the duration of a task performed on a job is an arbitrary monotone non-decreasing function of the time the job has spent in the system. Our model is set in a deterministic environment with the initial conditions (i.e., job release times r j ) as decision variables (determined by the parameters γ 1 , γ 2 , … , γ n , which control the time elapsed since the first machine becomes available). The main feature of the problem to minimize the sum of weighted completion times–as compared to, say, the problem to minimize the makespan considered earlier (Wagneur and Sriskandarajah (1993) [23])–is that its solution depends on the rate of growth of the processing time functions. We confine our study to the two-machine case for the sake of simplicity. We derive a closed-form formula for the optimal job release times for a finite set of jobs. This result also applies to the problem to minimize the flow time as a special case.