Recent Advances in Mathematical Programming Techniques for the Optimization of Process Systems under Uncertainty

Abstract

Optimization under uncertainty has been an active area of research for many years. However, its application in Process Synthesis has faced a number of important barriers that have prevented its effective application. Barriers include availability of information on the uncertainty of the data (ad-hoc or historical), determination of the nature of the uncertainties (exogenous vs. endogenous), selection of an appropriate strategy for hedging against uncertainty (robust optimization vs. stochastic programming), large computational expense (often orders of magnitude larger than deterministic models), and difficulty in the interpretation of the results by non-expert users. In this paper, we describe recent advances that have addressed some of these barriers.

Data-driven stochastic robust optimization: General computational framework and algorithm leveraging machine learning for optimization under uncertainty in the big data era

“…The objective is to maximize the NPV, which is given in (17). The constraints include capacity expansion constraints (18)- (19), budget constraints (20) The objective function is defined by (17…”

“…Optimization under uncertainty has attracted tremendous attention from both academia and industry [16][17][18][19][20][21]. Uncertain parameters, if not being accounted for, could render the solution of an optimization problem suboptimal or even infeasible [22].…”

Joint capacity planning and distribution network optimization of coal supply chains under uncertainty

“…Meanwhile, many general and tailored algorithms have been developed in the past two decades for solving two-stage stochastic optimization problems. For several excellent reviews on those models and algorithms, the reader is referred to the excellent works by Verderame, 18 Grossmann, 19 and Grossmann et al 20 It should be noted that most of the available models and algorithms are developed based on two-stage stochastic programming with convex recourse, which can be efficiently solved using duality-based decomposition methods. Li et al, [21][22][23][24] Sundaramoorthy et al, 25 and Chen et al 26 developed several NGBDs to solve TSSIP models with nonconvex function in the objective function and constraints.…”

“…As the demand in each zone follows a normal distribution, the total demand of all zones assigned to DF j during the lead time in planning period t for scenario s also has a normal distribution with mean l t j P k2K l k;t;s Z j;k and standard deviation ffiffiffi l t j q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P k2K Z j;k r 2 k;t;s q . Similar to constraints (8) and (9), a penalty term with prohibiting unit cost is added to the right-hand side of constraints (20) and (21) so that these two constraints are guaranteed to be feasible. Constraints (22) and (23) state that if DF j is selected (i.e., Y j 51), the capacity for that facility is nonzero and must be smaller than the upper bound of the selected region, and if facility j is not selected (i.e., Y j 50), U 1 j;r1 50 for all r1 2 R1 and the capacity of facility j is set to 0.…”

Multistage adaptive optimization using hybrid scenario and decision rule formulation

“…In this method, a hierarchy of increasing adaptability bridges the gap between static robust formulations and the fully adaptable formulation. For a review on recent advances in optimization under uncertainty, the reader can refer to these references …”