Blend-scheduling optimisation for continuous and batch mixtures: modelling and solving algorithms

Simultaneous blending and scheduling optimisation represents a mixed-integer nonlinear programming (MINLP) problem, whereby the binary variable relaxation that forms a nonlinear programming (NLP) in the first stage of a full-space algorithm (without dropping any decision variable) may lead to convergence issues. Furthermore, there is no guarantee to reach a global optimal solution since it solves an NLP problem to then address the mixed-integer linear programming (MILP) problem in outer-approximation algorithms. On the other hand, by neglecting quality variables and constraint of the blends or mixtures in the MILP problem, in a decomposed or non-full space problem (when dropping nonlinear constraints), infeasibilities or local optimal solutions may be found in the second stage NLP programs. In this case, there will be a change that from the MILP resulted assignments of the components to be blended, quality to specify the final blended material in the NLP problem may not be suffice. Or even if so, local optimal solution may occur. To skip these issues, an MILP-NLP decomposition is tailored to solve, in the first stage, the MILP logistics problem of the blend-scheduling optimisation whereby the blending relationship for each property quality is approached as amounts of quality balances. In such constraints, the blended quality variable is replaced by the bounds of the specifications plus the slack variable or less the surplus variable to close the linear quality balance. However, this MILP-NLP decomposition with a blending approximation in the MILP still cannot avoid convergence issues or local optimal inherent to the NLP stage and is dependent on the volume- or mass-based material flow and blended property governing rules. Then, we apply an optimisation-simulation algorithm to converge the MILP solution to a global optimal by considering a substitution of the current blending error successively in the next MILP solution. Nevertheless, this algorithm can be applied only in continuous mixtures, but we introduce the novel strategy to solve batch mixtures by using component quantities found in a continuous mixture topology as inputs.