(733g) LP Reformulation to Approximate Non-Convex Blending in MILP Scheduling Problems Using Factors

Blend scheduling operations in chemical process industries, as crude-oil refineries and metal processing facilities, involves logistics and quality problems in large scale non-convex models. However, these problems are prohibitively expensive to be solved in full space Mixed-Integer Non-Linear Programming (MINLP) approaches since the initial steps of the binary variables relaxation in a Non-Linear Problem (NLP). To overcome this, the main strategy found in the literature is to decompose the quantity-logic-quality phenomena into a two-stage solution [1,2,3]. First, scheduling assignments in quantity-logic relationships are solved in a Mixed-Integer Linear Problem (MILP) by neglecting the non-convex blending constraints also referred to as pooling. Then, a quantity-quality NLP problem is solved with fixed discrete decisions. The major drawback from this partition in MILP and NLP models is the possible distance or gap between their objective functions. In addition, infeasibilities in the NLP solution may arise by setting up undesired discrete decisions in the MILP.

We develop a Linear Programing (LP) reformulation to proxy or approximate NLP quality constraints (from the blending of streams) in the MILP stage using parameters or coefficients of qualities, what we call factors. In this approach, factors are augmented equality balance constraints for each property or intensive variable (factor = f_i) between raw material sources (x_i) and product sinks, using slacks (for <= constraints) or surpluses (for >= constraints) to guarantee the balance and replacing the blended material by the desired specification (f_s) in the formula: sum(i,f_i*x_i) = f_s*sum(i,x_i) + (f_slackor f_surplus). This potentially reduces the MILP-NLP gap as more quality information is programmatically transferred to the logistics optimization. If the quality information is neglected in the MILP problem, several degenerated binary solutions will be found, all with the same MILP objective for different sets of sources to sinks assignment, which unfortunately may substantially increase the time to reduce the MILP-NLP gap.

A similar proposition for calculating overall contaminant flow balances is applied in water network systems to significantly improve the strength of the lower bound in global optimum models [4,5] though this is considered as an NLP by the bilinear terms of blending of water stream flows with different contaminant concentrations. In our proposition, the concentrations are parameters instead of variables, therefore this approach is valid only for the mixing point between raw materials and products whereby properties or other intensive variables cannot propagate throughout the network. Although analogous reformulation is commonplace in non-linear planning problems by simplifying or aggregating the real topology of process networks, it can be included in scheduling problems under certain circumstances as in product blend-shops with pseudo-constant concentration of the raw materials and also in the preparation of feed processing of crude-oils to oil-refineries.

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[2] Z. Jia, M. Ierapetritou, J.D. Kelly, 2003, Refinery Short-Term Scheduling Using Continuous Time Formulation: Crude-Oil Operations. Industrial & Engineering Chemistry Research, 42, 3085-3097.

[3] J.D. Kelly, B.C. Menezes, F. Engineer, I.E. Grossmann, 2017, Crude-oil Blend Scheduling Optimization of an Industrial-Sized Refinery: a Discrete-Time Benchmark. In FOCAPO, Tucson, AR.

[4] R. Karuppiah, I.E. Grossmann, 2006, Global Optimization for the Synthesis of Integrated Water Systems in Chemical Processes. Computers Aided Chemical Engineering, 30, 650-673.

[5] E. Ahmetovic, I.E. Grossmann, 2011, Global Optimization for the Synthesis of Integrated Water Systems in Chemical Processes. AICHE Journal, 57(2), 434-457.