Successive LP Approximation for Nonconvex Blending in MILP Scheduling Optimization Using Factors for Qualities in the Process Industry

Abstract

We develop a linear programming (LP) approach for nonlinear (NLP) blending of streams to approximate nonconvex quality constraints by considering property variables as constants, parameters, or coefficients of qualities that we call factors. In a blend shop, these intensive properties of streams can be extended by multiplying the material flow carrying out these amounts of qualities. Our proposition augments equality balance constraints as essentially cuts of quality material flow for each property specification in a mixing point between feed sources and product sinks. In the LP factor formulation, the product blend quality is replaced by its property specification and variables of slacks and/or surpluses are included to close the balance; these are called factor f lows and are well known in industry as product giveaways. Examples highlight the usefulness of factors in successive substitution by correcting nonlinear blending deltas in mixed-integer linear models (MILP) and to control product quality giveaways or premium specifications in blend shops.

Blend Scheduling Solutions in Petroleum Refineries towards Automated Decision-Making in Industrial-like Blend-Shops

“…NLP constraint relaxations or approximation as LP formulas in MILPs as in Lee et al [13]; Castro and Grossmann [14]; Kelly et al [15]; 3.…”

“…The authors present industrial-size examples of scheduling optimization in a petroleum refinery in Asia. It includes the production of lubes and asphalts with sequence-dependent switchovers between modes of operation, as well as gasoline blend scheduling operations using decomposition strategies and factor-flow cuts based on nominal amounts of qualities presented in Kelly et al [15]. The paper discusses the benefits of optimization-based decisions using mixed-integer linear programming (MILP), which include optimized schedules, improved coordination with sales and marketing, enhanced stewardship in feedstock selection and operations planning, therefore, capturing spot market opportunities, and aligning blend schedules with product quality specifications.…”

“…To overcome the burden of exploding sets of solutions in the MILP-NLP-IVV combinations (MILP-NLP pairs multiplied by the number of IVV datasets), with or without the support of the solution pool settings found in the CPLEX, GUROBI and XPRESS solvers, Kelly et al [15] propose that instead of totally neglecting quality information in the MILP-NLP hierarchical decomposition, it can be approximated or considered the following fundamental blending of streams-for-stocks (SFS) in blend-shops to be made. For the parameters or fixed/given qualities of the component-quality inputs and specifications (component yields, blended properties, etc.)…”

Scheduling and Feed Quality Optimization of Concentrate Raw Materials in the Copper Refining Industry

“…However, in the literature, the main strategy of blending and scheduling optimization for considering the several stages of storage, mixing, and movement of raw materials involved in the logistics and feed quality operations is to decompose the MINLP model into the solutions of MILP and NLP programs. − In such an MILP–NLP decomposition approach, the quality information from the mixing of streams is neglected in the MILP step, which might produce infeasibilities in the NLP subproblems if the discrete assignments found in the MILP do not allow the matching of the quality constraints of the NLP. Alternatives in the literature for approximating the nonlinearities of blending in the MILP before solving the NLPs make use of (a) piecewise McCormick envelopes to linearly under- and overestimate the bilinear terms, − (b) multiparametric disaggregation, and (c) augmented equality balances as cuts of quality material flows . Successive substitution to iteratively correct the MILP predictions , have also been used to start the NLPs.…”

“…Alternatives in the literature for approximating the nonlinearities of blending in the MILP before solving the NLPs make use of (a) piecewise McCormick envelopes to linearly under- and overestimate the bilinear terms, − (b) multiparametric disaggregation, and (c) augmented equality balances as cuts of quality material flows . Successive substitution to iteratively correct the MILP predictions , have also been used to start the NLPs.…”