Sample Average Approximation SAA

It approximates the expectation of the stochastic formulation (usually called the true problem) and can be solved using deterministic algorithms. The SAA method was used among others by Shapiro and Homem-de-Mello (1998), Mark, Morton and 

Wood (1999), Linderoth, Shapiro and Wright (2002) for stochastic linear problems, Kleywegt, Shapiro and Homem-De-Mello (2001), Verweij et al. (2003) for stochastic integer problems, and Wei and Realff (2004), Goyal and Ierapetritou (2007) for MINLP problems. Problem (7.15) can be solved iteratively in order to provide statistical bounds on the optimality gap of the obj ective function value. For details and proofs see N orkin, Pflug and Ruszczysk (1998) and Mark, Morton and Wood (1999). The procedure consists of a number of steps as described in the following section. 

The objective values v, ...v of problem (7.16) and their corresponding solutions xh. Six are then obtained. 

According to Mark, Morton and Wood (1999) and Norkin, Pflug and Ruszczysk (1998), the value of vN in (7.17) is less than or equal to the true optimal value v obtained by solving the true problem, see Appendix C for proof. Therefore, 

Fix the solution value to the point obtained from the above minimization in (7.19) generate an independent sample N = ij1. and compute the value of the following objective function  

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In Chapter 3 of this book we discussed the problem of multisite refinery integration under deterministic conditions. In this chapter, we extend the analysis to account for different parameter uncertainty. Robustness is quantified based on both model robustness and solution robustness, where each measure is assigned a scaling factor to analyze the sensitivity of the refinery plan and integration network due to variations. We make use of the sample average approximation (SAA) method with statistical bounding techniques to generate different scenarios. 

Due to the complexity of numerical integration and the exponential increase in sample size with the increase of the random variables, we employ an approximation scheme know as the sample average approximation (SAA) method, also known as stochastic counterpart. The SAA problem can be written as ... 

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