Identical Machines Research Articles

Harmony search algorithm to enhance approach to permutation flow shop scheduling problem

The permutation flow-shop scheduling problem (PFSP) is one of the widely analysed and investigated problems within the general field of scheduling, as its major focus is on the allocation and sequencing of a set of jobs across a set of machines in order to minimize the makespan or satisfy other criteria. This paper further generalizes the problem to the distributed permutation flow-shop scheduling problem (DPFSP), namely in a manufacturing system with multiple factories, where every factory employs identical machines; with the objective of achieving the least makespan at the worst factory in terms of processing time. We resolve the issue through a Harmony Search Algorithm (HSA), a metaheuristic model developed from the musical harmonisation concept, together with constraint programming techniques, including interval variables and non-adjacent constraints. The problemsolving algorithm incorporates a lower bound estimation heuristic for efficiently directing the search in the right possible solution zone. Performance evaluation on both the small benchmark and the large benchmark data set showed the ability of the HSA to perform the different assignment in various problem scenarios that include the different number of jobs, machines and factories to support them. Comparisons with the best-known values established the performance of the HSA algorithm in identifying high quality solutions for the PFSP in less computation time.

Scheduling Splitable Jobs on Identical Parallel Machines to Minimize Makespan using Mixed Integer Linear

The scheduling of parallel machines with and without a job-splitting property, deterministic demand,and sequence-independent setup time with the goal of minimizing makespan is examined in this work.For simultaneous processing by multiple machines, single-stage splitable jobs are broken into random(job) sections. When a job starts to be processed on a machine, an operator has to setup the machinefor an hour. By creating two Mixed Integer Linear Programming models, this work proposes amathematical programming strategy (MILP). A MILP model takes the job-splitting property intoaccount. Another model, however, does not include the job-splitting property. This study investigatesthe performance of the proposed models using Gurobi solver. These programs' numerical calculationsare based on actual problems in the Indonesian city of Bandung's plastics industry. On four identicalparallel injection molding machines, 318 jobs must be finished in 22 periods. The real schedulingmethod is contrasted with these two MILP models. The maximum workload imbalance, the maximumrelative percentage of imbalance, and the makespan of these three scheduling systems are used toevaluate their effectiveness. Without the job-splitting property, MILP can handle the real issue ofscheduling identical parallel machines on injection molding machines to reduce makespan, resultingin a 36% average decrease. The MILP model's job-splitting property can reduce makespan by anadditional 2.40%. The order of relative ranking is MILP with job-splitting property, MILP withoutjob-splitting property, and actual scheduling based on the makespan minimization, workloadimbalance, and relative percentage of imbalance.

Minimizing the maximum lateness for scheduling with release times and job rejection

We study scheduling problems with release times and rejection costs with the objective function of minimizing the maximum lateness. Our main result is a PTAS for the single machine problem with an upper bound on the rejection costs. This result is extended to parallel, identical machines. The corresponding problem of minimizing the rejection costs with an upper bound on the lateness is also examined. We show how to compute a PTAS for determining an approximation of the Pareto frontier on both objective functions on parallel, identical machines. Moreover, we present an FPTAS with strongly polynomial time for the maximum lateness problem without release times on identical machines when the number of machines is constant. Finally, we extend this FPTAS to the case of unrelated machines.

Minimizing total completion time and makespan for a multi-scenario bi-criteria parallel machine scheduling problem

Multi-criteria scheduling problems under uncertainty remain a relatively unexplored topic in theoretical computer science despite substantial practical interests. This work studies a bi-objective identical parallel machine scheduling problem under uncertainty, in which the first objective is to minimize the total completion time, and the second is to minimize the makespan. Especially a job’s processing time is assumed to be represented by a polynomial function with respect to scenario u∈U, where U⊂R+ is an interval containing an infinite number of scenarios.In this work, we are looking for a compact and complete description of the set of possibly optimal solutions, along with their objective function values, over the set of scenarios or an approximation with a performance guarantee. First, to better understand the characteristics of the studied problem, we consider two single-objective problems: parallel machine scheduling problem under uncertainty with the total completion time and makespan criterion, respectively. For the problem with the total completion time criterion, we demonstrate that the set of possibly optimal schedules can be found in polynomial time. In contrast, for the problem with the makespan criterion, we provide a (1+ϵ)-approximation algorithm. For the bi-objective problem, we provide a 2-approximation algorithm for any number of parallel machines and a (1+ϵ)-approximation algorithm where a fixed number of parallel machines is considered.

The dual-resource-constrained re-entrant flexible flow shop a constraint programming approach and a hybrid genetic algorithm

ABSTRACT The dual-resource-constrained re-entrant flexible flow shop scheduling problem represents a specialised variant of the flow shop scheduling problem, inspired by real-world scenarios in screen printing industries. Besides the well-known flow shop structure, stages consist of identical parallel machines and operations may re-enter the same stage multiple times before completion. Moreover, each machine must be operated by a skilled worker, making it a dual-resource-constrained problem according to the existing literature. The objective is to minimise the total length of the production schedule. To address this problem, our study employs two methods: a constraint programming model and a hybrid genetic algorithm with a single-level solution representation and an efficient decoding heuristic. To evaluate the performance of our methods, we conducted a computational study using different problem instances. Our findings demonstrate that the proposed hybrid genetic algorithm consistently delivers high-quality solutions, particularly for large instances, while also maintaining a short computational time. Additionally, our methods improve existing benchmark results for instances from the literature for a subclass of the problem. Furthermore, we provide managerial insights into how dual-resource constraints affect the solution quality and the efficiency associated with different workforce configurations in the described production setting.

Energy-aware production lot-sizing and parallel machine scheduling with the product-specific machining tools and power requirements

This study addresses a multi-product lot-sizing and scheduling problem with sequence-dependent setup times, considering that the machining operations cause energy consumption. The production facility comprises identical parallel machines under which the production of each product requires a certain set of tools. The energy requirement of production depends on the product-specific machining tools. The problem deals with determining the minimum cost lot-sizing and scheduling plan considering the energy capacity of the production facility. We formulate the problem as a mixed integer linear programming model by introducing energy consumption-related costs and constraints. We perform a case study on CNC milling and turning workshops. Further, we propose an heuristic approach combining a decomposition-based Simulated Annealing heuristic and Fix&Optimise algorithms to handle larger-sized problem instances. The computational performance of the proposed heuristic approach is evaluated against the proposed mixed integer linear programming model on a numerical study. Our numerical experiments reveal that the proposed heuristic approach is capable of providing cost-efficient solutions without compromising time efficiency.

Bleeding us dry: The financial impact of full blood examinations in the immediate postoperative period.

Full blood examinations, often referred to as FBE, are commonly ordered postoperatively, despite limited utility in many of its markers in the acute phase. It is estimated that in the 2022-2023 financial year, the Australian healthcare system billed over $13 million for full blood examinations (FBEs) to Medicare. This study aims to assess the cost of using FBE following surgery. We explore potential cost savings by using a haemoglobin examination (HE) in replace of FBE, with both tests run on identical machines, producing the same result, but at a fraction of the cost. A retrospective analysis was conducted at a single institution, including all patients who underwent minimally invasive laparoscopic pelvic surgeries between 1/7/2017 and 30/6/2019. Patient records were examined to identify patient demographics, postoperative pathology tests and interventions. Patients who received FBE in the first 24 h following surgery were identified and included in the study. Using national surgery and admission statistics, a potential cost-saving analysis will be performed. Among 519 men who underwent robotic-assisted pelvic surgery, 325 patients had routine postoperative investigations, with 323 receiving FBE and 2 receiving HE. Abnormal results were found in the majority of patients that underwent FBE. Eight patients received packed red blood cell transfusion, none of these meeting the hospital-specific criteria for transfusion protocol. Twelve patients received antibiotics, none were in response to abnormal FBE, with all patients experiencing a fever, given prophylactically or in the days following the surgery. FBE and HE are both listed on the Medicare Benefits Scheme at $16.95 and $7.85, respectively, the difference being $9.10. Extrapolating the existing data, within the first 24 h following surgery, the estimated savings were $8818, with savings increasing accumulatively with longer observation intervals following surgery. When similar savings are applied to national figures, the potential savings to the Australian Public Healthcare system likely exceeds millions. Our study revealed that over half of the patients who underwent a RARP received FBE within the first 24 h postoperatively, the vast majority of which exhibited abnormal results that were not acted upon. These findings substantiate the limited utility of FBE in the postoperative setting. Cell markers observed in FBE are predominantly subjective, but consensus exists regarding the importance of evaluating haemoglobin levels postoperatively. Considering that one in four hospital admissions involves surgical procedures, and a $9.10 price differential between FBE and HE, the potential annual economic impact of utilising routine FBEs for assessing haemoglobin levels immediately after surgery is likely to reach millions. Although having obvious flaws, these results underscore the potential accumulative cost arising from everyday clinical judgement and the importance of thoughtful consideration when ordering pathology. The routine ordering of FBE postoperatively, without properly considering its indication, incurs significant costs. This study highlights the potential cost savings by HE instead, emphasising the need for revaluation and appropriate utilisation of pathology tests in the immediate postoperative period given the physiological acute phase response expected postoperatively.

MODEL PENJADWALAN BACKWARD HYBRID FLOW SHOP 2 STAGE UNTUK MEMINIMASI MEAN FLOWTIME

The scheduling problem is a common challenge when it comes to executing tasks, making the resolution of this issue crucial for decision-making in production activities. The problem addressed in this research is the company's desire to conduct production without waiting times between stages to maintain the quality of pharmaceutical raw materials. Additionally, the production results must meet predefined due dates. The production system formed is a Hybrid Flow Shop system, which is an extension of the Flow Shop concept where one or all stages have identical parallel machines. The objective of this study is to minimize the time between job start and job completion, defined in scheduling problems as the minimization of mean Flow time. The research collected data from 5 products in the company to model according to the research objectives. Process times for each stage and due dates for each product have been predetermined. After modeling and processing using the LINGO application, the optimal result obtained is backward scheduling with an average flow time of 12.6. This demonstrates that the model in this research effectively addresses the Hybrid Flow Shop problem using backward scheduling to minimize mean flow time

Parallel machine scheduling with a total energy consumption limitation for minimizing total completion time

This article investigates an energy-efficient, identical parallel machine scheduling problem. The objective is to minimize the total completion time while total energy costs do not exceed a given limitation. First, a reasonable range of total energy cost limits and the NP-hardness of this problem are analysed. Then, a mixed-integer programming model is presented. Afterwards, an improved simulated annealing (ISA) algorithm is devised. The ISA algorithm incorporates four effective neighbourhood operators, a method for adaptive selection of operators, and an initial feasible solution generation method. Comparison with genetic algorithms (GAs) and simplified swarm optimization (SSO) algorithms verifies the effectiveness and efficiency of the ISA algorithm. Results demonstrate that the ISA algorithm can provide near-optimal or better solutions than CPLEX ® in small- and medium-scale instances. Experimental results on large-scale instances show that ISA significantly outperforms GAs and SSOs. ISA, in particular, can find the best values for all instances of 500 jobs except one.

Minimising the makespan on parallel identical machines with log-linear position-dependent processing times

This article studies the minimum makespan scheduling problem on parallel identical machines with the log-linear learning/ageing effect, also known as polynomial positional learning/ageing effect. To develop a Fully Polynomial Time Approximation Scheme to the problem, we start with an intermediate artificial variant that rounds the values to integers and restricts the solutions to instances sorted in the shortest/longest processing time order. To this end, we propose a dynamic programming algorithm and show that the difference between its returned value and the minimum makespan of the original problem is independent of the processing times. This then leads to an algorithm with provable guaranteed ϵ-additive approximation and pseudo-polynomial running time algorithm, resulting in the desired fully polynomial time approximation solution to the original, not restricted problem.

Scheduling two interfering job sets on identical parallel machines with makespan and total completion time minimization

Scheduling two interfering job sets on identical parallel machines with makespan and total completion time minimization

Competitive kill-and-restart and preemptive strategies for non-clairvoyant scheduling

AbstractWe study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of 3 for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any $$b > 1$$ b > 1 a tight analysis for the natural b-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of $$(1+3\sqrt{3})\approx 6.197$$ ( 1 + 3 3 ) ≈ 6.197 for the deterministic and of $$\approx 3.032$$ ≈ 3.032 for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is 2-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than 10 for adaptions of the b-scaling strategy to online release dates and unweighted jobs on identical parallel machines.

Advanced constraint programming formulations for additive manufacturing machine scheduling problems

In comparison with traditional subtractive manufacturing techniques, additive manufacturing (AM) enables fabricating complex parts through a layer-by-layer process. AM makes it possible to produce one-piece and lightweight functional products, which are traditionally made from several parts. This paper introduces constraint programming (CP) models to minimise makespan in single, parallel identical and parallel unrelated AM machine scheduling environments for selective laser melting. Alternative CP formulations are explored to improve efficiency. The proposed CP model significantly benefits from the introduction of interval variables to replace binary assignment variables, and pre-definitions to narrow the search space, resulting in increased search performance. A computational study has been conducted to compare the performance of our proposed CP model with both a mixed-integer programming and a genetic algorithm from existing literature, evaluating improvements made to its search capability. Computational results indicate that the proposed CP model can obtain high-quality solutions in a timely manner even for several large-size instances.

Scheduling jobs with shared additional operations on parallel identical machines

The paper is concerned with assigning jobs, each with associated additional operations, to identical machines. A machine, allocated to a job, must also process all the additional operations associated with this job. An additional operation that is associated with several jobs assigned to the same machine needs to be processed by this machine only once. The goal is to minimise the time needed for the completion of all jobs and their additional operations. It is shown that even very restricted particular cases of the considered problem remain NP-hard in the strong sense. For the general case, the paper introduces two mixed integer linear programs as well as a broad class of approximation algorithms and a performance guarantee that is valid for any algorithm in this class. It is shown that, for one of the above-mentioned NP-hard particular cases, the considered class contains the best possible approximation algorithm. The performance of the mixed integer linear programs and several approximation algorithms is compared by means of computational experiments.

Predictive control of a series-connected two five-phase synchronous magnet permanent machines

The work presented in this paper focuses on studying the concept of two five-phase permanent magnet synchronous machines connected in series, powered by a single five-arm converter, in order to considerably minimize circulating currents. Firstly, we modeled a system composed of two identical five-phase synchronous machines, whose stator windings are connected in series. This modeling aims to demonstrate that the control of each machine in the group can be independent of the other, even though both machines are powered by a single five-phase converter. Then, we carried out the predictive control of the two five-phase machines connected in series. This control is based on the prediction of future switching states, which are managed via the discrete model of the system. The optimal switching state is obtained according to a predetermined cost function. To demonstrate the features of the presented control algorithm, the results of the simulation conducted using MATLAB/Simulink are presented and discussed to evaluate the performance and efficiency of the independent control of the two five-phase synchronous machines connected in series.

Energy-efficient scheduling in an identical parallel machine environment with peak power consumption and deadline constraints

Energy-efficient scheduling is an essential means of achieving sustainability in manufacturing systems. This paper defines and addresses the problem of scheduling n jobs on m identical parallel machines in which peak power consumption and deadline constraints exist simultaneously. The objective is to maximize the total value of the selected jobs. We show that this problem is equivalent to a special case of the rectangular knapsack problem, based on which four properties are observed. To solve the problem, an effective mixed integer linear programming model is proposed based on the properties, and it is much more efficient compared to the performance of modeling methods inspired by other works. Furthermore, three effective decoding methods are proposed and embedded into genetic algorithms (GAs). Comparisons with the exact algorithm (i.e., the proposed model) show that our GAs can lead to good-quality solutions within one second for small instances. Meanwhile, experimental results for large instances indicate that the proposed GAs can obtain near-optimal or satisfactory solutions. Finally, results also show that the proposed GA-DD significantly outperforms the existing matheuristic algorithm.

An adapted constraint‐programming formulation of the resource‐constrained project scheduling problem applied to the identical parallel machines group shop and mixed shop scheduling problems

AbstractIn this paper, we study the group shop and the mixed shop scheduling problems with single and identical parallel machines at each workstation with the makespan criterion. We adapted a constraint programming formulation previously presented for the classical resource‐constrained project scheduling problem. The effectiveness of our approach is evident in the fact that it achieved optimality in 107 out of 130 classical group shop scheduling problem instances and in 320 classical mixed shop scheduling problem instances. In the last set, we obtained 13 new optimal solutions.

A note on scheduling identical parallel machines with preemptions and setup times

We correct, extend and improve recent algorithmic and computational complexity results for the identical parallel machine scheduling problem with preemptions and setup times preceding uninterrupted job parts. The objective is to minimise the makespan. Our results include a proof of strong NP-hardness for an arbitrary number of machines, and a pseudo-polynomial algorithm and an FPTAS for the two machines case.

A mathematical model and solution algorithms for optimising system-level configurations of reconfigurable single part flow lines

This study addresses a system-level configuration selection problem for reconfigurable single part flow lines (R-SPFLs) with parallel identical flexible machines at each stage. The problem is to determine the number of stages, the number of machines of a certain type at each stage and the assignment of operations to each stage in order to satisfy the demand of a part type. Precedence relations among the operations and space limitations are also considered. For the objective of minimising the sum of machine purchase and operation processing costs, a nonlinear integer programming model is developed that gives the optimal solutions. Then, due to the problem's complexity, a basic variable neighbourhood search algorithm is proposed that constructs an initial solution using a greedy-type heuristic and improves it using a shaking and a local search improvement methods. Moreover, the basic algorithm is extended to a general one that uses a variable neighbourhood descent method. Computational results show that the two algorithms outperform the previous one significantly in both solution quality and computation times, and the general algorithm improves the basic one significantly. Finally, the applicability of the variable neighbourhood search algorithms is shown by reporting a brief case study.

Heuristic algorithm for lot sizing and scheduling on identical parallel machines

In this paper a newly developed heuristic algorithm for lotsizing and scheduling on identical parallel machines is presented. A basic concept and an aim of lotsizing and scheduling are described and clarified. Furthermore, models for considered problems for fixed and variable demand as well as for one and many identical parallel machines are presented. A concept and a formal description of newly developed heuristic algorithm is described. The paper includes in details the obtained results and solutions of conducted experiments and the description of used data sets.