Mathematical Models

Mathematical models have the potential to provide a cost-effective, objective, and flexible approach to assessing management decisions, particularly when these decisions are strategic alternatives. In some instances, mathematical model is the only means available for evaluating and testing alternatives. However, in order for this potential to be realized, models must be valid for the application and must provide results that are credible and reliable. The process of ensuring validity, credibility, and reliability typically consists of three elements: verification, validation, and calibration. Model verification, validation and calibration are essential tasks for the development of the models that can be used to make predictions with quantified confidence. Quantifying the confidence and predictive accuracy of model provides the decision-maker with the information necessary for making high-consequence decisions. There appears to be little uniformity in the definition of each of these three process elements. There also appears to be a lack of consensus among model developers and model users, regarding the actions required to carry out each process element and the division of responsibilities between the two groups. This paper attempts to provide mathematical model developers and users with a framework for verification, validation and calibration of these models. Furthermore, each process element is clearly defined as is the role of model developers and model users. In view of the increasingly important role that models play in the evaluation of alternatives, and in view of the significant levels of effort required to conduct these evaluations, it is important that a systematic procedure for the verification, validation and calibration of mathematical models be clearly defined and understood by both model developers and model users.

Finding the best way to optimize the portfolio after Markowitz's 1952 article has always been and will continue to be one of the concerns of activists in the investment management industry. Researchers have come up with different solutions to overcome this problem. The introduction of mathematical models and meta-heuristic models is one of the activities that has influenced portfolio optimization in recent decades. Along with the growing use of portfolios and despite its rich literature, there are still many unanswered issues and questions in this area. Also, Iranian capital markets, as emerging markets, require native research to answer these questions and issues. The purpose of this study is to provide a useful and effective tool to assist professionals and researchers in portfolio selection theory. This study, while comprehensively reviewing the literature on the subject and the developments and expansions made in the area of portfolio selection and optimization, reviews the types of problems and optimization methods.

Objective : Globally, road traffic accidents cause significant humanitarian, social, and economic costs, resulting in the need to have efficient and fast response mechanisms. Data-based tools can improve humanitarian aid's speed and equity using mathematical modeling, especially optimization, stochastic, fuzzy, and System Dynamics methods. This paper provides a systematic review of the role of these models in helping with post-accident humanitarian strategies and determining key factors that can affect the success of such models due to uncertainty. Methods : A systematic review was performed under PRISMA guidelines using the PICOS framework. Scopus and Web of Science literature were analyzed, focusing on peer-reviewed studies applying mathematical modeling to humanitarian response in road-accident contexts. Models were categorized by data type (stochastic, deterministic, fuzzy), method (exact vs. heuristic), and capability in managing uncertainty and feedback. Special attention was given to System Dynamics, which captures nonlinear feedback loops and time delays in prevention and response systems.  Results : Recent research highlights a shift toward predictive analytics, IoT, and machine learning to improve humanitarian logistics. Stochastic and fuzzy models effectively address real-world uncertainties, while dynamic and feedback-based models, particularly SD, outperform static ones by enhancing resource allocation, reducing response times, and strengthening decision-making. Conclusion : The mathematical modeling (in particular, with integration into the System Dynamics) demonstrates the possibility of humanitarian aid optimization in road accident handling. The paper highlights evidence-based, adaptive, and feedback-driven solutions through real-time information and uncertainty modeling to develop resilient, efficient, and scientific information-informed emergency response systems.