Computers and Chemical Engineering 28 (2004) 333–346 Sample average approximation methods for stochastic MINLPs Jing Wei, Matthew J. Realff ∗ School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Received 28 June 2002; received in revised form 20 March 2003; accepted 21 July 2003 Abstract One approach to process design with uncertain parameters is to formulate a stochastic MINLP. When there are many uncertain parameters, the number of samples becomes unmanageably large and computing the solution to the MINLP can be difficult and very time consuming. In this paper, two new algorithms (the optimality gap method (OGM) and the confidence level method (CLM)) are presented for solving convex stochastic MINLPs. At each iteration, the sample average approximation method is applied to the NLP sub-problem and MILP master problem. A smaller sample size problem is solved multiple times with different batches of i.i.d. samples to make decisions and a larger sample size problem (with continuous/discrete decision variables fixed) is solved to re-evaluate the objective values. In the first algorithm, the sample sizes are iteratively increased until the optimality gap intervals of the upper and lower bound are within a pre-specified tolerance. Instead of requiring a small optimality gap, the second algorithm uses tight bounds for comparing the objective values of NLP sub-problems and weak bounds for cutting off solutions in the MILP master problems, hence the confidence of finding the optimal discrete solution can be adjusted by the parameter used to tighten and weaken the bounds. The case studies show that the algorithms can significantly reduce the computational time required to find a solution with a given degree of confidence. © 2003 Elsevier Ltd. All rights reserved. Keywords: MINLP; Uncertainty; Process design; Stochastic programming; Sample average approximation 1. Introduction Process design under uncertainty can be formulated as multi-period or stochastic MINLPs (Halemane & Grossmann,1983; Paules & Floudas, 1992; Pistikopoulos, 1995). ν ∗ = min x,y,z i E θ [f(x,y,z i ,θ i )] s.t.g j (x,y,z i, θ i ) ≤ 0 ∀j ∈ J x ∈ X, z ∈ Z, y ∈{0, 1} m θ ∈ Θ (S-MINLP) where y is a vector of binary 0–1 variables denoting the choice of the units or the existence of the streams, x a vector of design variables such as unit sizes, z a vector of con- trol/state variables, which can vary over periods/scenarios, and θ a vector of uncertain parameters. The objective is of- ten to minimize the expected value of costs or maximize the expected value of profit. The constraint set J includes mass balances, unit design/operating models, design/operating ∗ Corresponding author. Tel.: +1-404-894-1834; fax: +1-404-894-2866. E-mail address: matthew.realff@chbe.gatech.edu (M.J. Realff). specifications, and some logical constraints. The exact eval- uation of the expected value is difficult or even impossible when the integral cannot be computed exactly or the objec- tive function f is not in a closed form. Then the expected value is often approximated through sample averaging. Un- der uncertain conditions, it is assumed that the design must remain feasible for every realization of the parameters con- sistent with their probability distributions. Therefore, the problem can become unmanageably large and its solution is time-consuming. Several papers in literature have addressed the various issues of solving such problems including: (i) integration methods, and (ii) sampling methods. For the integration methods, when the exact expected value cannot be computed, one usually uses a numerical integration technique or a sample average approximation method. Acevedo and Pistikopoulos (1996, 1998) compared the two approaches: Guassian quadrature formula for numer- ical integration (with both a full scan of uncertainty space and evaluation of feasible region) and Monte–Carlo (MC) sampling for sample average approximation. Novak and Kravanja (1999) suggested an approximation method using extreme points (vertices), in which the objective function is calculated by the weighted average over critical points and the feasibility of the design is ensured by the constraints at 0098-1354/$ – see front matter © 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00194-7