Journal of Optimization in Industrial Engineering Vol.14, Issue 1, Winter & Spring 2021, 121-131 DOI:10.22094/joie.2021.566423.1558 121 A Fractile Model for Stochastic Interval Linear Programming Problems S. Hadi Nasseri a, *, Salim Bavandi a a Department of Mathematics, University of Mazandaran, Babolsar, Iran Received 22 May 2018; Revised 09 January 2021; Accepted 10 January 2021 Abstract In this paper, we first introduce a new category of mathematical programming where the problem coefficients are interval random variables. These problems include two different kinds of ambiguity in the problem coefficients which are being interval and being random. We use Fractile method to solve these problems. In this method, using the existing method, we change the interval problem coefficients to the random mode and then we solve the random problem using Fractile method. Also, a numerical example is presented to show the effectiveness of this model. Finally, we emphasize that this approach can be useful for the model with multi-objective as a generalized model in the future study. Keywords: Random variable; Random interval variable; Random interval programming; Fractile model 1. Introduction The probability theory is one of the basic principles of modern mathematics which is related to other fields of mathematics such as algebra, topology, analysis, geometry, dynamical systems and it is one of the most important means to describe the complexity of uncertainty of the parameters. Also, its close relationships and commonalities with other fields of study such as computer sciences, ergodic theory, cryptography, game theory, analysis, differential equations, mathematics and physics, economics and statistical mechanics (Knill (2009)) has caused this theory to be practical in different fields including economics (Hildenbrand (1975)), random geometry (Matheron (1975)) or to be used in confronting with vague and imprecise information. In most decision makings, quantities used are not accurate data, but are dependent to the environmental conditions. Furthermore, collecting accurate information so that they are not dependent on the human diagnosis and judgment is very difficult or impossible in practice. This uncertainty could result from the incomplete, erroneous, missing or unknown data in different applications (Nasseri and Bavandi (2018)). This uncertainty sometimes could happen for a random variable which is called stochastic programming. Stochastic programming provides a framework for modeling the decision making problems which contain inaccurate data (S. H. Nasseri and S. Bavandi (2019), Bavandi, Nasseri and Triki (2020)). To formulate a stochastic programming problem, we should estimate a proper probability distribution which parameters obey. However, the estimation is not always a simple task because historical data of some parameters cannot be obtained easily especially when we face a new uncertain variable, and subjective probabilities cannot be specified easily when many parameters exist. Moreover, even if we succeeded to estimate the probability distribution from historical data, there is no guarantee that the current parameters obey the distribution actually. There are various approaches in the literature that can be used to solve the stochastic programming (Kall and Mayer (2004), Sakawa, Yano and Nishizaki (2013)). For decision problems under probabilistic uncertainty, from a different viewpoint, Charnes and Cooper (1959) proposed chance constrained programming which admits random data variations and permits constraint violations up to specified probability limits. Also Charnes and Cooper (1963) considered three types of decision rules, the minimum or maximum expected value model, the minimum variance model, and the maximum probability model for optimizing objective functions with random variables, which are referred to as the expectation model, the variance model, and the probability model, respectively. Moreover, Kataoka (1963) and Geoffrion (1967) individually proposed the fractile model. Sometimes, accurate measurement of the random data is impossible, therefore in these cases each random variable would be defined as an interval, so studying the Linear Programming models with Interval Coefficients (LPIC) (Suprajitno and Mohd (2008)) will be considered. Many researchers investigated interval linear programming problems on the basis of order relations between two intervals (Chanas and Kuchta (1996), Inuiguchi and Kume (1994), Jana and Panda (2014), Sengupta, Pal and Chakraborty (2001)). Interval linear programming problems have been studied by several authors, such as Bhurjee and Panda (2016), Ishibuchi and Tanaka (1989), Chanas and Kuchta (1996), Hladik (2015), Hladik (2014), Gen and Cheng (1997) and Wang *Corresponding author Email address: nhadi57@gmail.com