This work introduces a straight forward Python method for reducing the total time taken in a two-stage flow shop scheduling problem, incorporating job block criteria. Johnson’s Rule is applied to determine the optimal sequence of jobs across two machines. The paper’s goal is to create an efficient algorithm, developed using Python, aimed at minimizing the overall time machines are in use while taking into account the job block criteria for specific jobs. To confirm the effectiveness of this method, a numerical example is provided.

Sustainable manufacturing increasingly requires production schedules that balance environmental responsibility with delivery reliability. In flexible job shop environments, this challenge is especially difficult because machine assignment and sequencing decisions affect both the carbon footprint of production and the risk of missing job due dates. Motivated by this trade-off, this paper studies the Carbon-Aware Flexible Job Shop Scheduling Problem with Tardiness Penalty (CAFJSP-T), a flexible job shop formulation in which total carbon emissions and total tardiness penalty are treated as the two primary objectives, while energy consumption and makespan are retained as supporting performance indicators. To solve this problem, we propose a Policy-based Rough Optimization with Large Neighborhood Search (Pro-LNS) framework that combines Proximal Policy Optimization for fast, policy-guided construction of feasible schedules with an adaptive large neighborhood search procedure for targeted refinement. The two phases are aligned through a normalized scalarized objective that balances carbon emissions and tardiness penalty while preserving all precedence, eligibility, and machine-capacity constraints. Computational experiments on benchmark instances spanning small, medium, and large workcenter categories show that Pro-LNS produces high-quality schedules with strong due-date performance and controlled carbon emissions. Under equal objective weighting, the method achieves a median optimality gap of 6.12% relative to the exact formulation, with all reported instances remaining within 14%, while requiring only 4.08 seconds on average and at most 10.51 seconds. These results indicate that Pro-LNS is an effective and computationally practical approach for carbon-aware, tardiness-sensitive flexible job shop scheduling.

Scheduling problems represent a core challenge in the efficient management of industrial and service operations. Due to their structural complexity and significant practical relevance in both manufacturing and service sectors, Hybrid Flow Shop Scheduling Problems (HFSSPs) are widely recognized as NP-hard. Scheduling in contemporary manufacturing and production systems sometimes entails ambiguous and uncertain information, rendering classical deterministic methods less efficacious. This work presents a novel comparative analysis of the exact method Branch and Bound (BB) and heuristic algorithm Nawaz, Enscore, and Ham (NEH) for addressing the hybrid flow shop scheduling problem (HFSSP), where processing times are articulated via Interval-Valued Intuitionistic Fuzzy Sets (IVIFS). A ranking and scoring algorithm is utilised to convert IVIFS data into computationally manageable values, facilitating integration with BB and NEH methodologies. The results offer valuable insights for scheduling in uncertain and imprecise production environments, demonstrating how hybrid decision-making strategies that combine exact and heuristic methods can lead to more effective solutions.

Advancements in technology have broadened the possibility of solving flow-shop scheduling in multi-stations. Completing the waiting period of tasks in multi-stage with a task block is still a challenging job. Therefore, by considering two or more tasks together in one group, as it becomes significant when one task takes priority over another due to some scientific constraints or demand of the contract, it is a favourable domain in multistage flow shop scheduling for the modern era. This research work considers the multi-stage flow-shop scheduling problems with the task block. The objective of the proposed research is to minimize the complete awaiting duration of all the $n$ tasks on multi-stations, which also includes the equivalent task for a task block. The heuristic methodology has been applied to this study. The proposed algorithm is illustrated with numerical examples that have been implemented by MATLAB. The results demonstrate that the proposed scheme outperforms better, approximately 95 % and above, in comparison to the previous scheme. In addition, the proposed methodology supports industrial managers in better understanding the uncertainties of their selection or decision and provides more reliable and interpretable outcomes among multi-stage flow shop scheduling problems.

The paper addresses the classical two-stage FSSP with a single machine in the second stage and equipotential machines in the first. The uniqueness of this problem arises from the fact that the machine at the second stage is rented, with the objective being to minimize the rental cost. Efficient scheduling of jobs is critical in such environments to optimize resource usage and reduce operational costs. A distinguishing feature of this study is the representation of processing times on both stages using trapezoidal fuzzy numbers, which better capture uncertainty and variability in processing times compared to deterministic values. This fuzzy representation aligns well with real-world scenarios where exact processing times are often unavailable or subject to fluctuations. This paper's primary contribution is the creation of an optimization algorithm that uses the branch and bound (B&B) approach to tackle the issue. By breaking the problem space down into smaller subproblems and utilizing bounds to exclude less likely solutions, the B&B technique methodically explores the solution space. This method minimizes the expense of renting the secondstage machine while guaranteeing the identification of the ideal timetable. The fuzzy nature of the problem adds complexity to the scheduling task, as it requires handling the fuzziness in processing times while maintaining optimality. To ensure the robustness of the algorithm, it is implemented in MATLAB and tested against a variety of job sequences and machine configurations, along with the comparison of results with GA.

A flow shop is a workspace where machines, which could be humans or machines perform a range of tasks. It involves figuring out how to arrange several jobs in the most effective way possible. In the manufacturing sector, production scheduling is essential for several reasons, including lower product costs, higher productivity, customer happiness, and competitiveness. To adequately satisfy consumer needs and meet product demand, proper scheduling offers and promotes the proper usage of criteria such as available commodities, labour, and machines. This study illustrates the general algorithm and methodology comparison using fuzzy numbers, which is beneficial in figuring out the order of tasks. The aim is to provide the best way to minimize the makespan required to distribute shared resources over time to finish competing tasks. Furthermore, the machine processing times are not fixed; rather, they are interpreted as trapezoidal fuzzy numbers (TrFN). First, during the initial step, three parallel equipotential machines are taken and one machine is in the subsequent phase. Three parallel equipotential machines are taken initially and a single machine in the subsequent phase. Then a comparative study between branch and bound and heuristic methods like CDS (Campbell, Dudek, and Smith) and NEH (Nawaz, Enscore, and Ham) is done.

Scheduling, the method that deals with arranging the jobs on machines in industries or preparing a sequence of programs to be run on computers or in setting a time table and in many other such situations, is a powerful tool to handle these conditions easily. Usually, in industries for doing specific tasks, there are multiple machines instead of a single one. But all the machines are not having the same operational cost. So, jobs are assigned to these machines in a sequence so that the unit operational cost can be reduced to minimum along with reducing the total elapsed time. The present paper gives a brief description of a situation where the first stage of jobs is completed on equipotential machines and for the second stage the machine is taken on rent. So, the objective considered in this paper is to arrange the jobs in such a way that the rental cost can be minimized along with the total elapsed time.

The methodical process of organizing, managing, and maximizing work while ensuring the greatest possible use of both time and resources is known as scheduling. This paper indicates the desirable and necessary steps to discover the optimum solution for the three-phase flowshop scheduling problems. Here the multiple processors have been taken at the first phase and single processor at 2nd and 3rd stage along with transportation time between machines. Here the first methodologies B&B(Branch and Bound) is compared with the different heuristic methodologies like NEH(Nawaz Enscore Ham) and CDS(Campb ell Dudek Smith) to solve the mention problem. Comparative study considered to select the best methodology among the three with the help of numerical example.

The current paper investigates a two-stage flow shop scheduling model with no idle restriction, in which the time taken by machines to set-up is separately considered from the processing time. Owing to inherent usefulness as well as relevance in real-world situations, jobs' weight has additionally included. To eliminate machine idle time and cutting machine cost of rental, the reason for the conduct of the study is to provide a heuristic algorithm which, once put into practice, processes jobs in an optimal way, guarantees in smallest conceivable make span. Multiple computational examples generated in MATLAB 2019a serve as testament to the efficacy of the proposed strategy. The outcomes are contrasted with the current methods that Johnson, Palmer and NEH have demonstrated.

PurposeThis study aims to optimize blood donation drive efficiency by addressing operational bottlenecks and improving resource deployment, focusing on enhancing donor experience and reducing camp duration.Design/methodology/approachThe research uses a mixed-method approach, combining qualitative insights from blood banking officer interviews with quantitative data from 58 camp observations. “Simio” simulation software models various operational configurations, while mathematical techniques like queuing theory analyze key performance metrics. The process involves creating a baseline model, proposing optimizations and validating recommendations through real-world implementation.FindingsThe study achieved significant improvements: Reduced average donor time from 1.79 to 0.79 hours (56.4% improvement); Shortened camp operation time from 7.98 to 4.85 hours (39.2% improvement); Decreased waiting times at the medical check station from 45.23 to 0.60 minutes; Improved service rates across all stations, notably at registration (233.79% increase); and Streamlined processes through digitization and health check consolidation.Practical implicationsThe study provides actionable recommendations for blood bank managers, including digital pre-registration and optimized staff allocation, leading to substantial time and resource savings while enhancing donor experience.Social implicationsBy improving donation camp efficiency and experience, this research could increase donor retention rates and lead to a more stable blood supply, crucial for medical care.Originality/valueThis research presents a novel approach by combining discrete event simulation with real-world implementation and validation, offering an innovative solution to common bottlenecks in blood donation drives.

Scheduling is one of the many skills required for advancement in today’s modern industry. The flow-shop scheduling problem is a well-known combinatorial optimization challenge. Scheduling issues for flow shops are NP-hard and challenging. The present research investigates a two-stage flow shop scheduling problem with decoupled processing and setup times, where a correlation exists between probabilities, job processing times, and setup times. This study proposes a novel heuristic algorithm that optimally sequences jobs to minimize the makespan and eliminates machine idle time, thereby reducing machine rental costs. The proposed algorithm’s efficacy is demonstrated through several computational examples implemented in MATLAB 2021a. The results are compared with the existing approaches such as those by Johnson, Palmer, NEH, and Nailwal to highlight the proposed algorithm’s superior performance.

The paper demonstrates different sizes of wood logs are working on multistage flow shop scheduling problem in which stations are placed far away from each other, so transportation period is being associated between every two stations and due to a demand, the task block is used in the problem. The goal is to obtain the optimal sequence to minimize complete awaiting duration of pulping of paper on m stations. Thus, algorithm is supported with a numerical.

The paper presents the influence of the waiting time of jobs in a 2 machine k- job Flow Shop Scheduling (FSS) problem. The main intention of the study is to find a sequence of jobs that delivers the least sum of the time of waiting for jobs. A heuristic approach has been adopted to achieve the desired objective. The experiments are conducted for more than 2000 problems of various sizes for the problems with special structures and problems with random times of processing. The weighted mean absolute error (WMAE) for the average of the sum of the waiting times of jobs is computed for both kind of problems after comparing with the optimal solutions. WMAE has been obtained less than 0.0075 for problems with special structures and less than 0.087 for problems with random times of processing. The WMAE is also reducing significantly with the increase in job size. The results demonstrate that the presented step-by-step procedure of the heuristic delivers significantly close to optimal solutions.

Abstract Flow shop scheduling model (FSSM) is an important area of research in the field of Scheduling theory and it has many real applications in the industrial field. This study involves flow shop scheduling problem having machines on three levels in which utilization time of each task is considered as fuzzy triangular number. The concept of like parallel machines at each stage is also considered. Unit operational costs of jobs are also involved. The goal of this research is to recommend a heuristic approach inspired by Genetic algorithm (GA) which on implementation, provides an optimal or near-optimal schedule to reduce the make-span. Numerical example is also given to establish the usefulness of the proposed approach, and to approve the presentation, the results are compared with the existing methods like Branch and Bound (B & B).

This paper presents an effective fuzzy solution of complex priority bi-serial queue network.This study uses Zadeh's α -cut approach to transform fuzzy variables into crisp values in order to identify the membership solution of the system's performance measures.To establish the effectiveness of the proposed model Triangular Fuzzy numbers and their arithmetic operations are taken in consideration.The effectiveness of performance indicators for the purposed model are demonstrated numerically.