Solving Interval Investment Problem in Vague Environment Using Dynamic Programming Approach

- PAVAN KUMAR, PhD,
- Hamiden Abd,
- El-Wahed Khalifa,
- W A Afifi,
- Pavan Kumar

El-Wahed Khalifa

Department of Mathematics, College of Science and Arts, Qassim University, Department of Operations Research, Faculty of Graduate Studies for Statistical Research, Cairo University

W A Afifi

Mathematics and Statistics Department, College of Science, Taibah University

Pavan Kumar

Division of Mathematics, School of Advanced Science and Languages, VIT Bhopal University

## Abstract

In financial planning problems, the determination of the best investment is one of the interesting optimization models. In the proposed work, an investment problem (IP) is introduced in vague environment. The vagueness in return parameter is characterized by normalized heptagonal fuzzy number (HFN). One of the suitable interval approximations, namely, an inexact rough interval of a normalized HFN is utilized. Afterward, the inexact rough interval investment problem is considered. A dynamic programming (DP) approach is developed, which is applied for optimizing the fuzzy investment problem. The ideology of ''rough interval number'' is suggested in the mathematical modeling framework of the proposed problem to show the rough data as an inexact rough interval of piecewise quadratic fuzzy numbers. Afterward, the DP approach is applied to solve and compute a rough interval solution. Finally, a numerical example is yielded for the utility of the approach to apply on real-world problem for the decision-maker. The obtained results consist of the total optimal return with inexact rough intervals on a $ 10 million investments is as follows: $ [[1.69, 2.08]: [1.75, 1.91]] millions.