American Journal of Computational and Applied Mathematics 2013, 3(3): 186-194 DOI: 10.5923/j.ajcam.20130303.06 Second Degree Chance Constraints with Lognormal Random Variables – An Application to Fisher’s Discriminant Function for Separation of Populations Vas kar Sarkar 1,* , Kripasindhu Chaudhuri 2 , Rathindra Nath Mukherjee 3 1 Department of Mathematics, Aryabhatta Institute of Engineering and Management Durgapur, Panagarh, Dist.- Burdwan, Pin - 713148, West Bengal, India 2 Department of Mathematics, Jadavpur University, Jadavpur, Kolkata, 700032, West Bengal, India 3 Department of Mathematics, Burdwan University, Burdwan, West Bengal, India Abstract In this paper, we have discussed a transformation procedure of the second-degree chance constraints to the deterministic constraints for mathematical programming problems having general second degree chance constraints with lognormal random variables. We have used geometric inequality for this transformation. The transformed deterministic problem having non-linear constraints and linear or non-linear objective function can be solved using non-linear programming algorithm. Also we have applied this model to Fisher’s discriminant function for separation of populations. A numerical simulation have been considered along with a graphical representation of the reduced solution region. Keywords Stochastic Programming, Chance Constraint Programming, Lognormal Random Variable, Geometric Inequality, Deterministic Reduction, Non-linear Programming, Discriminant Function and Separation of Population 1. Introduction Stochastic or probabilistic programming[13] deals with the situations where some or all the parameters of the mathematical programming problem are described by stochastic or random variables rather than by deterministic ones. Several models have been presented in the field of stochastic programming[21]. Two major approaches to stochastic programming[13,14] are recognized as: 1. Chance constrained programming 2. Two-stage programming The chance constrained programming (CCP) can be used to solve problems involving chance constraints, i.e., constraints having finite probability of being violated. Its main feature is that the resulting decision ensures the probability of satisfying the constraints, i.e. the confidence level of being feasible. Thus, using CCP, the relationship between profitability and reliability can be quantified. The use of probabilistic constraints was initially introduced by Charnes et al.[6,7], while an early use of probabilistic programming in environmental economics is by Maler[17]. Since then, a number of environmental management case studies use probabilistic constraints[5,10,11,19,22,23]. Also the CCP, in recent years, has been generalized in several * Corresponding author: sarkar_vaskar@rediffmail.com (Vaskar Sarkar) Published online at http://journal.sapub.org/ajcam Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved directions and has various applications[21]. Many authors studied and developed this problem for the parameters having normal distribution, uniform distribution and exponential distribution, where the chance constraints are linear[3, 4, 21]. But lognormal distribution (with two parameters) has a significant role in human and ecological risk assessment for many reasons, of which the main three reasons are (i) many physical, chemical, biological, toxicological and statistical processes tend to create random variables that follow lognormal distributions[12], (ii) lognormal distributions are self-replicating under multiplication and division, i.e. products and quotients of lognormal random variables themselves follow lognormal distributions[1, 9] and (iii) when the conditions of the Central Limit Theorem hold[16], the mathematical process of multiplying a series of random variables will produce a new random variable (the product), which tends (in limit) to be lognormal in character, regardless of the distribution from which the input variables arise[2]. Also we may consider the fact that many environmental variables are non-negative means that they are generated by a skewed distribution. Several skewed probability models have been used to describe environmental data, including the Poisson, negative binomial, Weibull, gamma, exponential and the lognormal. Among these distributions, the lognormal has been the most widely applied[18]. In the year 1996, Cooper et.al.[8] noted the importance of using skewed distributions, to represent environmental variables within mathematical programming and lognormal distribution was