Makespan Minimization Research Articles

Breaking symmetries to rescue sum of squares in the case of makespan scheduling

The Sum of Squares (SoS) hierarchy gives an automatized technique to create a family of increasingly tighter convex relaxations for binary programs. There are several problems for which a constant number of rounds give integrality gaps matching the best known approximation algorithm. In many other, however, ad-hoc techniques give significantly better approximation factors. The lower bounds instances, in many cases, are invariant under the action of a large permutation group. The main purpose of this paper is to study how the presence of symmetries on a formulation degrades the performance of the relaxation obtained by the SoS hierarchy. We do so for the special case of the minimum makespan problem on identical machines. Our first result is to show that a linear number of rounds of SoS applied over the configuration linear program yields an integrality gap of at least 1.0009. This improves on the recent work by Kurpisz et al. [30] that shows an analogous result for the weaker LS\(_+\) and SA hierarchies. Then, we consider the weaker assignment linear program and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying the SoS hierarchy for \(O_\varepsilon (1)\) rounds to this linear program reduces the integrality gap to \((1+\varepsilon )\). Our results suggest that the presence of symmetries were the main barrier preventing the SoS hierarchy to give tight relaxations.

Going to the core of hard resource-constrained project scheduling instances

The resource-constrained project scheduling problem (RCPSP) is one of the most studied problems in the project scheduling literature, and aims at constructing a project schedule with a minimum makespan that satisfies both the precedence relations of the network and the limited availability of the renewable resources. The problem has attracted attention due to its NP hardness status, and different algorithms have been proposed that solve a wide variety of RCPSP instances to optimality or near-optimality. In this paper, we analyse the hardness of this problem from an experimental point-of-view by testing different algorithms on a huge set of existing instances and detect which ones are difficult to solve. To that purpose, we propose a three-phased approach that makes use of five elementary blocks, well-performing algorithms and a huge amount of computational power to transform easy RCPSP instances into very hard ones. The purpose of this study is to create insight and understanding into what makes an RCPSP instance hard, and propose a new dataset that consists of a small set of instances that are impossible to solve with the algorithms currently existing in the literature. These instances should be as small as possible in terms of number of activities and resources, and should be as diverse as possible in terms of network structure and resource strictness. Such a dataset should enable researchers to focus their attention on the development of radically new algorithms to solve the RCPSP rather than gradually improving current algorithms that can solve the existing RCPSP instances only slightly better.

POP ≡ POCL, Right? Complexity Results for Partial Order (Causal Link) Makespan Minimization

We study PO and POCL plans with regard to their makespan – the execution time when allowing the parallel execution of causally independent actions. Partially ordered (PO) plans are often assumed to be equivalent to partial order causal link (POCL) plans, where the causal relationships between actions are explicitly represented via causal links. As a first contribution, we study the similarities and differences of PO and POCL plans, thereby clarifying a common misconception about their relationship: There are PO plans for which there does not exist a POCL plan with the same orderings. We prove that we can still always find a POCL plan with the same makespan in polynomial time. As another main result we prove that turning a PO or POCL plan into one with minimal makespan by only removing ordering constraints (called deordering) is NP-complete. We provide a series of further results on special cases and implications, such as reordering, where orderings can be changed arbitrarily.

Sustainable flexible flow shop scheduling optimization in flexible packaging industry using genetic algorithm

Flexible flow shop (FFS) scheduling optimization process has a high level of complexity and categorized as an NP-hard problem. X Co is a flexible packaging manufacturer that scheduled FFS manually with a low sustainability level. The objective of this research is to optimize FFS scheduling in X Co using Genetic Algorithm (GA) method to increase the sustainability level and measure its performance. The sustainability parameters contain three main aspects of sustainability. In this optimization process, makespan minimization used as the objective function, and elitism is applied in the GA method to make sure the makespan value is getting smaller in each generation. The result shows an increment of sustainability level in all parameters. Makespan value reduction by 135 hours, zero lateness, idle time reduction by 1755 (hour*machine), machine utility improvement by 5.58%, electricity consumption reduction by 23664 kWh which converted into a reduction of 16731 KgCO2e greenhouse gas, and also increase employee productivity as much as 0.066 jobs/shift. This research shows an optimal result with a higher level of sustainability in the implementation of optimization on FFS scheduling at X Co.

Bottleneck-Based Heuristic for Permutation Flowshop Scheduling

This paper addresses the permutation flowshop scheduling problem with objective of makespan minimization. It proposes a heuristic that apply bottleneck-based identification technique and Nawaz, Enscore and Ham (NEH) insertion technique. This experiment was carried out using measurement of four-machines and six-jobs simulated in Microsoft Excel Simple Programming. The computational result show that bottleneck-based (BNB) achieved 60 % and more of optimum solution and heuristic validation result prove that the proposed heuristic resulting to be much better than existing heuristic by achieving more best solution of 6 sets out of 10 sets replication.

Makespan minimization on unrelated parallel machines with a few bags

Let there be a set M of m parallel machines and a set J of n jobs, where each job j takes pi,j time units on machine Mi. In makespan minimization the goal is to schedule each job non-preemptively on a machine such that the length of the schedule, the makespan, is minimum. We investigate a generalization of makespan minimization on unrelated parallel machines (R||Cmax) where J is partitioned into b bags B=(B1,…,Bb), and no two jobs belonging to the same bag can be scheduled on the same machine. First we present a simple b-approximation algorithm for R||Cmax with bags (R|bag|Cmax). We also give a polynomial-time approximation scheme (PTAS) for R|bag|Cmax with machine types where both the number of machine types and bags are constant; two machines Mi and Mi′ have the same machine type if pi,j=pi′,j for all j∈J. This result infers the existence of a PTAS for uniform parallel machines (Q|bag|Cmax) when the number of machine speeds and number of bags are both constant. Then, we present a b/2-approximation algorithm for the graph balancing problem with b≥2 bags; the approximation ratio is tight for b=3, unless P=NP, and this algorithm solves the graph balancing problem with b=2 bags in polynomial time. In addition, we present a polynomial-time algorithm for the restricted assignment problem on uniform parallel machines with bags when all the jobs have unit length. To complement our algorithmic results, we show that when the jobs have lengths 1 or 2 it is NP-hard to approximate the optimum makespan within a factor smaller than 3/2 for both the restricted assignment and graph balancing problems with b=2 bags and b=3 bags, respectively. We also prove that Q|bag|Cmax with b=2 bags is strongly NP-hard.

A note: flowshop scheduling with linear deterioration and job-rejection

We study a scheduling problem on an $$m$$ -machine flowshop with linear deterioration of job processing times and job rejection. The objectives are minimum makespan and minimum total load, subject to an upper bound on the total permitted rejection cost. The problems are NP-hard (since the single machine makespan minimization version was shown to be hard), and we introduce pseudo-polynomial dynamic programming algorithms, thus proving that both problems are NP-hard in the ordinary sense.

Performance Analysis of Opposition Based Particle Swarm Optimization with Cauchy Distribution in Minimizing Makespan Time in Job Shop Scheduling

In the contemporary circumstances, manual solving of job shop scheduling problem (JSSP) is quite time consuming and inaccurate. The main intention of this paper is to analyze the performance of various optimization techniques in JSSP in order to minimize makespan time. This paper aims to analyze four optimization techniques viz, particle swarm optimization (PSO), genetic algorithm (GA), opposition based genetic algorithm(OGA) and opposition based particle swarm optimization with Cauchy distribution (OPSO CD) in addition to the existing optimization techniques applied in various research papers on combinatorial optimization problems viz., JSSP. A comparative study of these optimization techniques were conducted and analyzed to find out the most effective optimization technique on solving JSSP. Results show that OPSO CD is found to possess minimum makespan time in comparison with other algorithms for JSSP.

A Deterministic Flowshop Scheduling Problem to minimizing the Makespan using PA ACO

Scheduling problems are NP-hard in nature. Flowshop scheduling problems, are consist of sets of machines with number of resources. It matins the continuous flow of task with minimum time. There are various traditional algorithms to maintain the order of resources. Here, in this paper a new stochastic Ant Colony optimization technique based on Pareto optimal (PA-ACO) is implemented for solving the permutation flowshop scheduling problem (PFSP) sets. The proposed technique is employed with a novel local path search technique for initializing and pheromone trails. Pareto optimal mechanism is used to select the best optimal path solution form generated solution sets. A comparative study of the results obtained from simulations shows that the proposed PA-ACO provides minimum makespan and computational time for the Taillard dataset. This work will applied on large scale manufacturing production problem for efficient energy utilization.

Permutation flowshop scheduling with periodic maintenance and makespan objective

This paper addresses the permutation flowshop scheduling problem with availability constraints where the machines are not available cyclically and the starting and finishing times of the periodic unavailability intervals are known in advance. In our case, this constraint is due to the shift calendar in a company in the aerospace industry, where all machines must stop at the same time (shift changes, nights, weekends, etc.). In this company, shifts are formed by different teams of workers, and having one worker to complete the tasks of another worker is not desirable in terms of efficiency and quality of the operation. In the literature, this cyclical constraint is also reported due to preventive deterministic and fixed maintenance activities. For this reason, this constraint is known as periodic maintenance, and –to the best of our knowledge– it has been addressed for the single machine and parallel machines layouts, but not for the flowshop. For this layout, it can be observed that different scheduling problems may arise depending on the assumptions about the preemption of the operations. In our case, the preemption of operations is not allowed, and therefore, if an operation cannot be finished within the current availability period, then it should be scheduled in the next one. For this problem, the objective considered is the makespan minimisation. The structure and hardness of the problem depending on the size of the availability periods is studied using Mixed Integer Linear Programming and complete enumeration, in order to determine the range of values for the availability period that makes the problem under consideration to be substantially different than its classical (unconstrained) counterpart. For these cases, specific heuristics with different computational complexity are developed, and an extensive computational experience is carried out to establish the efficiency of the proposed heuristics. Additionally, an iterated greedy metaheuristic is employed, showing excellent results.

The no-wait flow shop with rejection

We consider the no-wait flow shop scheduling problem with rejection. We first show how to augment the travelling salesman problem formulation for the no-wait flow shop minimum makespan problem to incorporate the rejection option. We then focus on polynomially solvable cases by considering problems with ordered jobs. We present a third-order polynomial-time dynamic programming algorithm to minimise the sum of makespan and total rejection cost and faster quadratic algorithms for two special cases. We also present a third-order polynomial-time dynamic programming algorithm to minimise the sum of total completion time and total rejection cost with ordered jobs. We also exploit the duality between no-wait and no-idle flow shops to determine which operations to outsource in a no-idle flow shop with an outsourcing option.

Exact algorithms to minimize makespan on single and parallel batch processing machines

Batch-processing machine scheduling problem is one of the challenging problems in the machine scheduling literature where machines are capable of processing a batch of jobs simultaneously. In this paper, we tackle single and parallel batch processing machine scheduling problems with the objective of minimizing makespan. We propose a reformulation for parallel batch processing machine scheduling, which is based on decomposition in two levels, and an exact algorithm for its solution. To the best of our knowledge, there is no exact algorithm to solve this problem in the literature, except for its formulations solved by off-the-shelf solvers. In the first part of the proposed algorithm, we solve the single batch processing machine problem by a column-and-cut generation algorithm that provides a lower bound for the parallel machine problem. The second part of our proposed algorithm employs a search mechanism to find the minimum makespan for the parallel machine problem, which entails the solution of the reformulation of this problem by column generation at every iteration. The novel aspect of this column generation algorithm is the integration of batch generation and machine schedule generation in a single pricing subproblem. We test the performance of the proposed algorithms on randomly generated instances and show that, on average, they outperform the off-the-shelf solver. The major findings are that the single machine problem provides tight lower and upper bounds for the parallel machine problem, and the proposed algorithm for the parallel machine problem solves more instances to optimality than that for the single machine problem.

Iterative job splitting algorithms for parallel machine scheduling with job splitting and setup resource constraints

This paper examines a parallel machine scheduling problem with job splitting and setup resource constraints for makespan minimization. Jobs can be split into multiple sections, and such sections can be processed simultaneously on different machines. It is necessary to change setups between the processes of different jobs on a machine, and the number of setups that can be performed simultaneously is restricted due to limited setup operators. To solve this problem, we propose a mathematical programming model and develop iterative job splitting algorithms that improve a feasible initial solution step by step, taking into account job splitting, setup times, and setup resources. We derive a worst-case performance ratio of the algorithms and evaluate the performance of the proposed heuristics on a large number of randomly generated instances. We finally provide a case study of piston manufacturing in Korea.

A bio-inspired scheduling approach for machines and automated guided vehicles in flexible manufacturing system using hormone secretion principle

The product quality and production efficiency of a flexible manufacturing system have improved effectively by introducing the computer management and the material transportation system. The flexible manufacturing system performance greatly depends on the performance of the material transportation system. As a mobile robot controlled by a central controller, an automated guided vehicle has a strong ability for material transportation. This article studies a dynamic scheduling problem in a shop floor, where machines and automated guided vehicles run at a specified speed and specifies a mathematical model for the dynamic scheduling problem with the goal of makespan minimization. Meanwhile, inspired by the hormone secretion principle of the endocrine system, a bio-inspired scheduling optimization approach is developed to solve the dynamic scheduling problem in the flexible manufacturing system. To verify its practical application, the bio-inspired scheduling optimization approach and other scheduling approaches are tested, and the results illustrate that the bio-inspired scheduling optimization approach has better scheduling performance as well as optimizes the quality of integrated and real-time scheduling of machines and automated guided vehicles.

Examining additive manufacturing in supply chain context through an optimization model

This study explores the problem of integrated jobs and vehicles scheduling in a two-stage supply chain, where parts are processed on additive manufacturing (AM) machines and then distributed to customers. In this study, we develop an optimization model for the problem with the objective of makespan minimization. Besides, several structural properties and lower bound formulation are provided to explain the main characteristics of the problem. In this regard, this study also contributes to the existing academic literature by investigating the two-stage supply chain scheduling problem with the additive manufacturing technology. Because the addressed problem belongs to non-deterministic polynomial-time hardness (NP-hard) problem class, a best-fit heuristic-based approach and several selection rules are developed to solve the problem. Each selection rule is designed by considering a distinct structural property of the problem and, thus, each of which is considered to be a different algorithm. A comprehensive experimental study is conducted through random instances, which are generated for small- and large-sized problems by considering real-production data. According to the computational results, the best-fit capacity utilization based selection (BFCUBS) algorithm is superior to others and yields a substantial improvement in the makespan. The reason behind this fact is that the high utilization of capacity enables a large number of jobs to be completed in a short time. Besides, as the number of jobs decreases and the capacity utilization rates increases all algorithms provide better results.

Two Parallel Machines Scheduling with Two-Vehicle Job Delivery to Minimize Makespan

A problem of parallel machine scheduling with coordinated job deliveries is handled to minimize the makespan. Different jobs call for dissimilar sizes of storing space in the process of transportation. A range of jobs of one customer in the problem have priority to be processed on two identical parallel machines without preemption and then delivered to the customer by two vehicles in batches. For this NP-hard problem, we first prove that it is impossible to have a polynomial heuristic with a worst-case performance ratio bound less than 2 unless P = NP. Thereafter, we develop a polynomial heuristic for this problem, the worst-case ratio of which is bounded by 2.

A genetic algorithm with neighborhood search procedures for unrelated parallel machine scheduling problem with sequence-dependent setup times

PurposeThe purpose of this paper is to present a hybrid meta-heuristic based on genetic algorithms (GAs), simulated annealing, variable neighborhood descent and path relinking for solving the variant of the unrelated parallel machine scheduling problem considering sequence-dependent setup times.Design/methodology/approachThe authors carried out computational experiments on literature problem instances proposed by Vallada and Ruiz (2011) and Arnaout et al. (2010) to test the performance of the proposed meta-heuristic. The objective function adopted was makespan minimization, and the authors used relative deviation, average and population standard deviation as performance criteria.FindingsThe results indicate the competitivity of the proposed approach and its superiority in comparison with several other algorithms. In small instances proposed by Vallada and Ruiz (2011) and on small and large instances proposed by Arnaout et al. (2010), the proposed approach presented the best results in most tested problem instances.Practical implicationsIn small instances proposed by Vallada and Ruiz (2011) and on small and large instances proposed by Arnaout et al. (2010), the proposed approach presented the best results in most tested problem instances.Originality/valueThe proposed approach presented high-quality results, with an innovative hybridization of a GA and neighborhood search algorithms, tested in diverse instances of literature. Furthermore, the case study demonstrated that the proposed approach is recommended for solving real-world problems.

Mathematical Model and Evaluation Function for Conflict-Free Warranted Makespan Minimization of Mixed Blocking Constraint Job-Shop Problems

In this paper, we consider a job-shop scheduling problem with mixed blocking constraints. Contrary to most previous studies, where no blocking or only one type of blocking constraint was used among successive operations, we assume that, generally, we may address several different blocking constraints in the same scheduling problem depending on the intermediate storage among machines, the characteristics of the machines, the technical constraints, and even the jobs. Our objective was to schedule a set of jobs to minimize the makespan. Thus, we propose, for the first time, a mathematical model of the job-shop problem taking into account the general case of mixed blocking constraints, and the results were obtained using Mosel Xpress software. Then, after explaining why and how groups of jobs have to be processed, a blocking constraint conflict-free warranted evaluation function is proposed and tested with the particle swarm optimization and genetic algorithm methods. The results prove that we obtained a near-optimal solution to this problem in a very short time.

A Hybrid Estimation-of-Distribution Algorithm for Scheduling Flexible Job Shop With Limited Buffers Based on Petri Nets

This article focuses on the production scheduling problem in the flexible job shop (FJS) environment with limited buffers. Limited manufacturing resources and buffers may lead to blockage and deadlock phenomenon. In order to establish production scheduling with minimum makespan, the timed Petri net (PN) model of a production process is established. Based on this PN model, a novel Hybrid Estimation-of-Distribution Algorithm (HEDA) is proposed for solving the considered scheduling problem. A candidate solution for the problem is coded as an individual that consists of a route sequence for processing jobs and a permutation with repetition of jobs. A deadlock prevention policy is used to check the feasibility of individuals, such that it can be decoded into a feasible sequence of transitions, i.e., a feasible schedule. By using an effective voting procedure of elite individuals, two probability models in HEDA corresponding to different subsections of individuals are constructed. Based on the probability models, offspring individuals are then produced. As an improvement strategy, simulated-annealing-based local search is designed and incorporated into HEDA to enhance the entire algorithm’s search ability. The proposed hybrid HEDA is tested on FJS examples. The results show its feasibility and effectiveness.

A rule-based heuristic algorithm for on-line order batching and scheduling in an order picking warehouse with multiple pickers

In manual order picking systems, orders received from internal or external customers are collected by pickers that walk or ride through the warehouse. Generally, orders are grouped into several sub sets, i.e. batches to reduce picking time and cost. This paper considers the on-line order batching problem (OOBP) in an order picking warehouse with multiple pickers in which the maximum completion time of all batches (customer orders) has to be minimized. At first, a mathematical model is introduced for the off-line version of the problem. Then the on-line version of the problem is considered in which customer orders become available dynamically over time. Since the proposed model is NP-hard, a rule-based heuristic algorithm was proposed to solve the on-line problem. The main contributions of the present work is to propose a mathematical model for the off-line order batching problem with multiple pickers considering makespan minimization and to present a novel heuristic algorithm for solving the on-line version of the problem. To validate the proposed algorithm, it is proved that its competitive rate is equal to 2. Finally, the solution algorithm is evaluated through a series of experiments and the most appropriate routing, batching and selection policies are introduced.