**Preface to "Development and Optimization of Mathematical Models for Operations Research"**

## **1. Introduction**

The development of mathematical models and their optimization are fundamental for the effective resolution of many problems in operational research. In recent years, increased insights into real-world problems have led to the development of new mathematical models and new optimization algorithms, contributing to the development of a research area with increasing practical relevance. This Special Issue is dedicated to works at the interface of mathematical modeling, optimization, and operations research with a special focus on their real-world applications. The interest of the scientific community was significant, with submissions from authors from different countries from five continents, including Australia, China, Egypt, India, Israel, Portugal, Russia, Saudi Arabia, and the United States of America. Ten papers were accepted for publication after thorough peer-review by dedicated reviewers with expertise in the fields of the papers.

## **2. Description of Published Papers**

This section presents a brief overview of the published papers.

S. Singh et al. [1] proposed a multi-time generalized Nash equilibrium problem and proved its equivalence with a multi-time quasi-variational inequality problem. Moreover, the authors proved an existence theorem for equilibria. Next, they applied the considered model to a traffic network problem. Finally, they studied the multi-time generalized Nash equilibrium problem as a projected dynamical system and showed a numerical example with its approximated solution.

H. Zhou et al. [2] presented a genetic algorithm for optimizing the formation and schedule of heavy-haul trains, which is a special case of a more general Train Formation Problem. The authors established two integer programming models in stages involving the train service plan problem model and the train time-tabling problem model. They then implemented a number of experiments to illustrate the feasibility and effectiveness of the proposed approaches.

R. Mondal et al. [3] formulated an inventory model that combines two important components of inventory management: the demand of a product and the uncertainty of customers' behavior. They derived a mathematical formulation that maximized the profit of the inventory system. They considered three numerical examples to validate the model and solved them using different variants of quantum-behaved particle swarm optimization techniques in order to determine the duration of stock-in time and the preservation technology cost.

L. Alkhalifa and H. Mittelmann [4] introduced Piecewise Linear Approximation (PLA), one of most popular methods used to transform nonlinear problems into linear ones. Following a brief background and literature review, authors proposed two piecewise linear approximation helpers in mixed-integer nonlinear programming: one related to domain partitioning and another with a partial application of PLA to MINLP. They used quadratically constrained quadratic programming (QCQP) and MIQCQP to demonstrate that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

N. Krivulin et al. [5] presented an application of pairwise comparisons method on decision making: the determination of consumer preferences in hotel selection. They demonstrated several known methods on a sample of 202 university students, evaluating their preferred criteria when selecting a hotel for accommodation during a professional development program in a foreign country. The comparison of the solutions produced showed a high degree of similarity in results.

P. Ruan et al. [6] considered a two-echelon supply chain with an up-stream supplier and

down-stream retailer during the COVID-19 period. They constructed an inventory model considering the following four elements: ordering cost, holding cost, deterioration cost, and purchasing cost. A computer program provided a numerical solution indicating the minimum total cost per unit time.

R. Etgar and Y. Cohen [7] presented a novel approach to solve a current problem that R&D companies face: multi-annual project portfolio selection. The suggested approach was to expand and improve common meta-heuristic methods and, thus, solve this NP-hard problem of determining the roadmap of multi-annual portfolio planning. This study developed an efficient tool that can provide both practical and academic benefits.

A. Antunes et al. [8] focused on single-stage scheduling problems occurring in parallel machine environments. They applied a genetic algorithm (GA) to the scheduling problem of unrelated parallel machines in order to minimize the makespan of a set of tasks that was subject to varying setup times. A comparative statistical analysis of small and large instances of scheduling problems showed the advantage of the GA proposed.

K. Alnowibet et al. [9] presented modified algorithms based on conjugate gradient (CG) principles to solve local and global minimization problems. First, they improved the existing CG algorithm to enhance its global and local optimization capacities. Then, they obtained a hybrid stochastic conjugate gradient algorithm using the improved CG method. The performance profiles used to compare the proposed hybrid approach and four other hybrid stochastic conjugate gradient algorithms showed the competitiveness of the former. The authors tested both convex and non-convex problems.

B. Emambocus et al. [10] proposed an optimized discrete adapted Dragonfly Algorithm (DA) using the Steepest Ascent Hill-Climbing algorithm as a local search. They applied the proposed DA to a traveling salesman problem, modeling a package delivery system in Kuala Lumpur. The improved DA showed better performance than the discrete adapted DA and other studied swarm intelligence algorithms.

## **3. Conclusions**

As guest editors of the Special Issue 'Development and Optimization of Mathematical Models for Operations Research', we express our gratitude to all the authors who sent their articles for publication in this issue. We also cordially thank all anonymous referees and staff of MDPI for contributing to the creation of this Special Issue. Special thanks are due to the Managing Editor of the Special Issue, Ms. Linn Li, for her excellent collaboration and valuable assistance. We are confident that the papers selected for this Special Issue will attract a significant audience in the scientific community and further stimulate research involving the development of mathematical models and their optimization.

## **References**


**Humberto Rocha and Ana Maria A. C. Rocha** *Editors*
