Pablo Garcia-Herreros

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.

Evolution of concepts and models for quantifying resiliency and flexibility of chemical processes

Abstract This paper provides a historical perspective and an overview of the pioneering work that Manfred Morari developed in the area of resiliency for chemical processes. Motivated by unique counter-intuitive examples, we present a review of the early mathematical formulations and solution methods developed by Grossmann and co-workers for quantifying Static Resiliency (Flexibility). We also give a brief overview of some of the seminal ideas by Manfred Morari and co-workers in the area of Dynamic Resiliency. Finally, we provide a review of some of the recent developments that have taken place since that early work.

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 Systems Engineering 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/chance constrained optimization vs. stochastic programming), large computational expense (often orders of magnitude larger than deterministic models), and difficulty of interpretation of the results by non-expert users. In this paper, we describe recent advances that have addressed some of these barriers for mostly linear models.

Design of Supply Chains under the Risk of Facility Disruptions

Abstract The design of efficient supply chains is a major challenge for companies in the process industry. Supply chain performance is subject to different sources of uncertainty including reliability of the facilities. Facility disruptions are among the most critical events that supply chains can experience. In order to reduce the undesirable effects of disruptions, these events must be anticipated at the design phase of the supply chain. This work addresses the design of supply chains under the risk of facility disruptions by simultaneously considering decisions on the facility location and the inventory management. The proposed formulation is based on a two-stage stochastic programming framework where the scenarios are determined by the possible combinations of facility disruptions. The first stage decisions include the location of distribution centers and their storage capacity. The second stage decisions involve assigning customer demands to the distribution centers that are available in every scenario. The objective is to minimize the sum of investment cost and the expected cost of distribution during a finite time horizon. The formulation is implemented and compared with the optimal solution of the deterministic design problem though an illustrative example. The results show that the proposed formulation generates supply chain designs with the capability to adjust to adverse scenarios. This flexibility translates into significant savings when disruptions occur in the operation of supply chains.

Mixed-integer bilevel optimization for capacity planning with rational markets

Abstract We formulate the capacity expansion planning as a bilevel optimization to model the hierarchical decision structure involving industrial producers and consumers. The formulation is a mixed-integer bilevel linear program in which the upper level maximizes the profit of a producer and the lower level minimizes the cost paid by markets. The upper-level problem includes mixed-integer variables that establish the expansion plan; the lower level problem is an LP that decides demands assignments. We reformulate the bilevel optimization as a single-level problem using two different approaches: KKT reformulation and duality-based reformulation. We analyze the performance of these reformulations and compare their results with the expansion plans obtained from the traditional single-level formulation. For the solution of large-scale problems, we propose improvements on the duality-based reformulation that allows reducing the number of variables and constraints. The formulations and the solution methods are illustrated with examples from the air separation industry.

A cross-decomposition scheme with integrated primal---dual multi-cuts for two-stage stochastic programming investment planning problems

We describe a cross-decomposition algorithm that combines Benders and scenario-based Lagrangean decomposition for two-stage stochastic programming investment planning problems with complete recourse, where the first-stage variables are mixed-integer and the second-stage variables are continuous. The algorithm is a novel cross-decomposition scheme and fully integrates primal and dual information in terms of primal–dual multi-cuts added to the Benders and the Lagrangean master problems for each scenario. The potential benefits of the cross-decomposition scheme are demonstrated with numerical experiments on a number of instances of a facility location problem under disruptions. In the original formulation, where the underlying LP relaxation is weak, the cross-decomposition method outperforms multi-cut Benders decomposition. If the formulation is improved with the addition of tightening constraints, the performance of both decomposition methods improves but cross-decomposition clearly remains the best method for large-scale problems.

A Novel Cross-decomposition Multi-cut Scheme for Two-Stage Stochastic Programming

Abstract We describe a decomposition algorithm that combines Benders and scenario-based Lagrangean decomposition for two-stage stochastic programming investment planning problems with complete recourse. The first-stage variables are mixed-integer and the second-stage variables are continuous. The algorithm is based on the cross- decomposition scheme and fully integrates primal and dual information in terms of primal-dual multi-cuts added to the Benders and the Lagrangean master problems for each scenario. The benefits of the cross-decomposition scheme are demonstrated with an illustrative case study for a facility location problem with risk of disruptions.

Optimizing inventory policies in process networks under uncertainty

Abstract We address the inventory planning problem in process networks under uncertainty through stochastic programming models. Inventory planning requires the formulation of multiperiod models to represent the time-varying conditions of industrial process, but multistage stochastic programming formulations are often too large to solve. We propose a policy-based approximation of the multistage stochastic model that avoids anticipativity by enforcing the same decision rule for all scenarios. The proposed formulation includes the logic that models inventory policies, and it is used to find the parameters that offer the best expected performance. We propose policies for inventory planning in process networks with arrangements of inventories in parallel and in series. We compare the inventory planning strategies obtained from the policy-based formulation and the analogous two-stage approximation of the multistage stochastic program. Sequential implementation of the planning strategies in receding horizon simulations shows the advantages of the policy-based model, despite the increase in computational complexity.

Capacity planning with competitive decision-makers: Trilevel MILP formulation, degeneracy, and solution approaches

Capacity planning addresses the decision problem of an industrial producer investing on infrastructure to satisfy future demand with the highest profit. Traditional models neglect the rational behavior of some external decision-makers by assuming either static competition or captive markets. We propose a mathematical programing formulation with three levels of decision-makers to capture the dynamics of duopolistic markets. The trilevel model is transformed into a bilevel optimization problem with mixed-integer variables in both levels by replacing the third-level linear program with its optimality conditions. We introduce new definitions required for the analysis of degeneracy in multilevel models, and develop two novel algorithms to solve these challenging problems. Each algorithm is shown to converge to a different type of degenerate solution. The computational experiments for capacity expansion in industrial gas markets show that no algorithm is strictly superior in terms of performance.

Mathematical Programming Techniques for Optimization under Uncertainty and Their Application in Process Systems Engineering

In this paper we give an overview of some of the advances that have taken place to address challenges in the area of optimization under uncertainty. We first describe the incorporation of recourse in robust optimization to reduce the conservative results obtained with this approach, and illustrate it with interruptible load in demand side management. Second, we describe computational strategies for effectively solving two stage programming problems, which is illustrated with supply chains under the risk of disruption. Third, we consider the use of historical data in stochastic programming to generate the probabilities and outcomes, and illustrate it with an application to process networks. Finally, we briefly describe multistage stochastic programming with both exogenous and endogenous uncertainties, which is applied to the design of oilfield infrastructures.