[insert Figure 3 here]
The computational difficulty of the MINLP problem depends on the number
of ingredient candidates and the complexity of the adopted models. A
large number of ingredient candidates often create a combinational
problem. Many rigorous models (e.g., thermodynamic and transport
phenomena models) involve nonlinear and nonconvex equations. Along with
complex surrogate models (e.g., neural network), the optimization
problem is prone to convergence failure if the problem is directly
solved using standard MINLP solvers. Some of these problems can be
handled with advanced algorithms. For instance, Schweidtmann and
Mitsos42,43 recently developed and applied an
efficient global solver for ANN embedded MINLP problems. Alternatively,
the problem can be resolved by reformulating the optimization
problem.44 Two techniques are proposed for enhancing
optimization convergence and finding better solutions (Figure 3). If the
problem can be directly solved, the solution is sent for experimental
validation. Otherwise, generalized disjunctive programming (GDP) can be
used because of the need to calculate the multiple intermediate
variables in the mechanistic models. If the GDP problem still cannot be
solved or better solutions are needed, the model(s) in use are replaced
with alternative model(s) and repeat the calculations.
GDP reformulation
As can be seen in Eq. 11, even if i -th ingredient candidate is
not selected, its intermediate variables (\(IM_{i}^{m}\)) must be
calculated. Also, forcing \(V_{i}\) to 0 may lead to singularity at\(V_{i}=0\) for some models (e.g., logarithmic function). Thus, when
mechanistic models are employed and multiple intermediate variables are
calculated through complex equations, the redundant constraints and
singularities can lead to convergence failure. Similar to the tray
selection problem in distillation column design,44cosmetic formulation problem can be formulated using GDP. As an
alternative way to program discrete-continuous problem, GDP is a
logic-based method containing Boolean and continuous variables.
Constraints are expressed as disjunctions, algebraic equations, and
logic propositions.44 The following disjunction can be
used to express the bounds and intermediate variables for Eq. 11.
\(\par
\begin{bmatrix}Y_{i}\\
VL_{i}\leq V_{i}\leq VU_{i}\\
\par
\begin{matrix}IM_{i}^{m}=IMG^{m}(V_{i},ms)\\
\end{matrix}\\
\end{bmatrix}\bigvee\par
\begin{bmatrix}\neg Y_{i}\\
V_{i}=0\\
\par
\begin{matrix}IM_{i}^{m}=0\\
\end{matrix}\\
\end{bmatrix}\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\) (12)
where \(Y_{i}\) is the Boolean variable for ingredient selection. If thei -th ingredient is selected, the bounds on \(V_{i}\) are
fulfilled and its intermediate variables \(IM_{i}^{m}\) are calculated.
Otherwise, they are not calculated and simply set as 0.
To solve a GDP problem, it is often transformed back into MINLP using
big-M or convex-hull relaxation to take advantage of standard MINLP
solvers. It is found that the big-M method is more appropriate in
solving mixture design problem since singularity issue can still occur
in the convex-hull relaxation.12 After transforming
the above disjunction using big-M approach, the cosmetic formulation
problem is reformulated below. \(S_{i}\) has a one-to-one correspondence
with \(Y_{i}\). bm is a sufficiently large parameter.
\(\operatorname{}{q=f(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms)}\)(13)
s.t. \(PL^{k}\leq P^{k}\leq PU^{k}\), \(k\in K=MM\cup SM\)
\(\left\{\par
\begin{matrix}VL_{i}-bm\bullet(1-S_{i})\leq V_{i}\leq VU_{i}+bm\bullet(1-S_{i})\\
-bm\bullet S_{i}\leq V_{i}\leq bm\bullet S_{i}\\
\par
\begin{matrix}\text{IM}G^{m}\left(V_{i},ms\right)-bm\left(1-S_{i}\right)\leq IM_{i}^{m}\leq IMG^{m}\left(V_{i},ms\right)+bm(1-S_{i})\\
-bm\bullet S_{i}\leq IM_{i}^{m}\leq bm\bullet S_{i}\\
\end{matrix}\\
\end{matrix}\right.\ \) Big-M constraint
\(P^{m}=G^{m}(IM_{I_{A,1}}^{m},IM_{I_{A,2}}^{m},\ldots,IM_{I_{Z,z}}^{m})\),\(m\in MM\)
\(P^{s}=g^{s}(V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}},ms)\),\(s\in SM\)
\(H\left(S_{I_{A,1}},S_{I_{A,2}},\ldots,S_{I_{Z,z}},V_{I_{A,1}},V_{I_{A,2}},\ldots,V_{I_{Z,z}}\right)\leq 0\)
\(msL\leq ms\leq msU\), \(S_{i}\in\left\{0,1\right\}^{i}\),\(\sum_{i}{V_{i}=1}\),\(i\in\left\{I_{A,1},I_{A,2},\ldots,I_{Z,z}\right\}\)
Model substitution
Some rigorous mechanistic models are too complicated to be directly used
for optimization even if they are programmed using GDP. There is always
a trade-off between model accuracy and traceability. In this case, the
complicated but accurate rigorous models can be replaced by simple
short-cut model or surrogate model to reduce the computational effort
and to seek out even better solutions. A surrogate model is a good
choice when it is relatively easy to generate simulation data as
training data from the rigorous model. Although the model accuracy is
reduced, it is easier to solve and to obtain the global
solution.42,45–47 After model substitution, the newly
generated optimization problem should be solved and the optimal solution
obtained can be denoted as \(V_{i}^{*}\). This solution must be
validated using the original rigorous mechanistic models. If the
validation fails, the newly generated optimization problem should be
re-solved by adding the equation below to remove this solution
(\(V_{i}^{*}\)) that fails validation. Otherwise, the solution can be
sent for experimental verification.
\(\sum_{i}\left(V_{i}^{**}-V_{i}^{*}\right)^{2}\geq tol\) (14)
Here, \(V_{i}^{**}\) is the solution of a new round optimization. The
parameter tol is a small tolerance.