The significance of Operations Research (OR) in current human life in a business context cannot be overstated. OR consultants or analysts undertake detailed assessments of company issues, processes, and vulnerabilities, leveraging advanced pattern and trend analysis to construct models. According to Churchman, C. W [1], "OR has been described as a method, an approach, a set of techniques, a team activity, a combination of many disciplines, an extension of particular disciplines (mathematics, engineering, economics), a new discipline, a vocation, even a religion. It is perhaps some of all these things." The field of OR involves using diverse techniques and algorithms to solve particular mathematical models. These models demand finding the values of decision variables that achieve the best possible outcome for the objective function while adhering to a system of inequality and equality constraints. These models yield critical insights and strategic interventions to resolve organizational problems efficiently.
OR plays a pivotal role across various domains. It has various real-life applications, such as the Traveling Salesman problem [2], inventory [3], traffic management [4], network flow optimization [5], Linear Programming (LP) [6], and so on. In the realm of RO, decision-making and problem-solving play crucial roles across diverse industries. Among the topics within RO, Linear Programming Problems (LPPs) emerge as a widely discussed subject, which will be explored in greater detail in the following paragraph.
LP has diverse applications, including different areas but not limited to addressing transportation challenges, optimizing vehicle routes, public transportation schedules, and airline operations to enhance efficiency and reduce costs. Researchers have explored various aspects of LP, such as Tripathi and Kumar [7], which discussed the LPP under uncertainty environments. Later, Tripathi and Kumar [8] and Tripathi et al. [9] conducted a short literature review on the LPP to explore Neutrosophic LP within advanced extended fuzzy contexts.
Additionally, OR contributes to communication network design, improving data transmission efficiency and call center management. Meanwhile, network analysis, which integrates the Critical Path Method (CPM) [10] and Program Evaluation and Review Technique (PERT) [11], offers valuable insights for project management, considering time constraints, resource allocation, and risk analysis. Pratyusha et al. [12] reviewed the extended uncertainty principle for Neutrosophic critical path problems. This first time, Miriyala and Kumar [13] used Python programming in Neutrosophic environments to solve critical path problems in extended uncertain environments. Additionally, Miriyala and Kumar [14] discussed another approach to solving critical path problems in a Neutrosophic environment using Python.
Furthermore, OR models optimize inventory levels, pricing strategies, and the entire supply chain, balancing costs, profits, and service levels. In inventory management, Dubey and Kumar [15] and Dubey et al. [16] reviewed the recent trends and advancements of the inventory model under the extended uncertainty principle. Moreover, with sensitivity analysis, Dubey and Kumar [17] investigated Neutrosophic inventory models in advanced fuzzy contexts.
This initiative centers around achieving specific objectives: enhancing efficiency, reducing costs, and improving overall effectiveness. Noteworthy contributions to the field of Operational Research are summarized in
Operational research impact on real-life applications: insights from researchers Authors Areas Applications Significance
This table is shown in the PDF version of the article.
Lapidoth investigated by the author, with the exploration of capacity Duality theory Multiple-Antenna and Moser bounds through duality and consideration of practical Systems [18] applications for multiple-antenna systems on flat-fading channels.
The author proposed a novel operator selection framework Scheduling Grid scheduling Fu et al. [19] using Maximum Spread (MS) and Generational Distance problem problems (GD) to enhance diversity while maintaining convergence. The objective of this paper is to expand upon a multi-Multi-objective Hossein et objective programming methodology by introducing an LPP optimization al. [20] approach for solving LPP involving grey data and problems variables.
Many to Many In this paper, an enhanced approach to address the Zhu et al.
Assignment assignments assignment problem is presented by incorporating [21] problem backtracking into the Kuhn–Munkres algorithm.
Simaiakis A novel analytical queuing model for airport departure Airport Departure and processes is introduced and validated in this paper. The Queuing theory Process Balakrishnan model incorporates stochastic and dynamic features [22] through transient analysis of the D/E/1 queuing system.
The figure labeled as
Pictorial representation of Table 1.
LP is an important optimization method. Soviet mathematician Leonid Kantorovich introduced the concept of LP in the late 1930s, followed independently by American mathematician George Dantzig [23]. Its main goal is to find the best possible value for decision variables while meeting all the constraints. These constraints can be things like resource limitations or budget restrictions. Additionally, LP helps us maximize or minimize an objective function representing our ultimate goal.
It has many applications in various fields, including engineering, efficient home energy management [24], finance, transportation, and so on. Furthermore, we have explored significant contributions to the LPP in
Researchers in various fields have explored the impact of LPPs on real-life applications.
This table is shown in the PDF version of the article.
simplex programming solving quadratic programming problems by reducing method the symmetric basis matrix, akin to the revised simplex method in LP objective functions.
Abaffy and 2013 ABS Mixed integer The author demonstrates the integration of Gomory's Fodor [26] Algorithm LP original cutting-plane approach into the ABS class of algorithms.
Chopra et al. [27] 2017 Hungarian Multirobot The author suggests a distributed adaptation of the Method assignment Hungarian method to efficiently tackle the well-problem. known multi-robot assignment.
Darvishi and 2020 Dual Grey LPPs To bridge grey system theory with optimization, the Nasseri [28] simplex author introduces a dual LPP, thereby extending its method foundational definitions and concepts.
Singaravelan et 2021 Two-Phase Efficient The author presents an innovative Two-Phase al. [24] Method- Energy Simplex Method-based demand response program based Management customized for residential customers, aiming to reduce peak demand, optimize electricity consumption costs, enhance consumer comfort, and improve computational efficiency.
In the survey, we examined the contributions of different authors to LPPs. This underscores the crucial role that LPPs play in the field of OR. Researchers and authors delve into strategies for maximizing profits while minimizing costs and losses within these LPPs. The decision variables in this model represent the quantities to be determined.
However, significant gaps exist in the current understanding of LPP, particularly regarding its application in real-world scenarios. The main objective of this paper is to delve into the contemporary trends and applications of LPPs. Therefore, we aim to bridge these gaps by examining network flow problems. Network problems primarily involve three distinct types of network problems: the shortest path, maximal cost flow, and Minimal Cost Flow (MCF) problems. This review article extensively examines crucial elements concerning network flow problems.