MINLP modeling

Papalexandri, K. P., and Pistikopoulos, E. N. (1994). A multiperiod MINLP model for the synthesis of heat and mass exchange networks. Comput. Chem. Eng. 18(12), 1125-1139. 

The objective function, which in this case is the minimization of capital cost, is shown in constraint (3.51). Due to the non-linearity of this constraint the model becomes an MINLP model. 

MILP and MINLP models have been developed to take into account the PIS operational philosophy for assessing the efficacy of this philosophy and design, respectively. The MILP model is used to determine the effectiveness of the PIS operational philosophy by, firstly, solving the model with zero intermediate storage with and without the use of latent storage. In the illustrative example a 50% increase in the throughput was achieved. 

Secondly, the minimum amount of intermediate storage is determined with and without the PIS operational philosophy. In both cases the production goal was set to that which was achieved when the model was solved with infinite intermediate storage. In the illustrative example a 20% reduction in the amount of intermediate storage is achieved. The design model is an MINLP model due to the non-linear capital cost objective function. This model is applied to an illustrative problem and results in the flowsheet as well as determining the capacities of the required units. 

The second mathematical formulation presented, is a design model based on the PIS operational philosophy. This formulation is an MINLP model due to the capital cost objective function. The model is applied to a literature example and an improved design is achieved when compared to the flowsheet. The design model is then applied to an industrial case study from the phenols production facility to determine its effectiveness. The data for the case study are subject to a secrecy agreement and as such the names and details of the case study are altered. 

It must also be noted that the solution to the MILP with a zero relative gap was also infeasible, when substituted into the exact MINLP, and the value of the objective function was lower than the previous, it was 1.88 x 106c.u. Therefore, the usage of the higher relative gap solution is allowed. If the solution with a zero relative gap had been feasible, then one would have to use this as the starting point to the exact MINLP model. 

The specific problem characteristics are modeled most appropriately by a combination of concepts from various general modeling frameworks leading to a mixed-integer nonlinear programming (MINLP) model. 

The formulation of the engineered nonlinear short-term model presented is a variant of an MINLP model described in the dissertation by Schulz [5], In this subsection, all necessary indices, parameters and variables are introduced, and the constraints and the objective function are derived. In the following section, the nonlinear formulation is linearized yielding a MILP model. In order to keep track of the variables used in the MINLP and in the MILP formulation, they are displayed in Figure 7.3 along with some key parameters. 

The remaining variables and linear constraints from the MINLP-model are kept in the model. 

In this section, the numerical solutions of the MINLP-model and of the MILP-model as presented in Sections 7.4 and 7.5 are compared with respect to their solution quality (measured by the objective values) and the required solution effort (measured by the computing time). In order to compare the MILP-solution with the MINLP-solution, the optimized values for the start times of polymerizations tn, the recipe assignments W, and the total holdups Mnr are inserted into the MINLP-model and the objective is calculated. To guarantee comparability of the results, the models were stated with identical initial conditions, namely t° = 0, = 2 Vk, pf = 0 Vs, and ra = 0.4 Vs (i.e., the variables defined at the beginning of the corresponding time axes are fixed to the indicated values). For the algorithmic solution procedure, all variables were initialized by 1 (i.e., the search for optimal values starts at values of 1 ), and none of the solvers was specifically customized. 

The MINLP-model instances comprised 200 binary variables, 588 continuous variables and 1038 constraints. The linearization not only eliminates the nonlinearity but also leads to a reduced number of398 continuous variables and 830 constraints (the number of 200 binary variables is unchanged). The MINLP-problems were solved by the solver architecture DICOPT/CONOPT/CPLEX, and the MILP problems were solved by CP LEX, both on a Windows machine with an Intel Xeon 3 GHz CPU and 4 GB RAM. 

Viswanathan, J. and I. E. Grossmann. An Alternate MINLP Model for Finding the Number of Trays Required for a Specified Separation Objective. Comput Chem Eng 17 949-955 (1993). 

This CAMD single compound problem is formulated as an MINLP model as shown below. The performance objective function and the various property constraints in the model are discussed subsequently. 

The CAMD problem formulated as an MINLP model was solved using the decomposition approach and the results are presented below... 

In the above MINLP model, the potential recovery is the objective function (Eqn. 26) subject to other property and phase equilibrium constraints (Eqn s. 27-40). The decisions that need to be made are the identities of the solvent and anti-solvent (binary variables) and the composition of solvent and anti-solvent... 

The MINLP model for the retrofit design of a multiproduct batch plant takes the following form ... 

The above formulation is a mixed-integer nonlinear programming MINLP model and has the following characteristics. The binary variables appear linearly and separably from the continuous variables in both the objective and constraints, by defining a new set of variables Wj = tij/Tu and including the bilinear constraints WiTu = Uj- The continuous variables nitBi,Tu,Wi appear nonlinearly. In particular, we have bilinear terms of riiBi in the objective and constraints, bilinear terms of niT i and WiTu in the constraints. The rest of the continuous variables Vj, (V fc)ra, Vfjk appear linearly in the objective function and constraints. 

Another important applications of MINLP models have recently been reported for (i) the computer-aided molecular design aspects of selecting the best solvents (Odele and Macchietto, 1993) and (ii) the interaction of design and control of chemical processes (Luyben and Floudas (1994a), Luyben and Floudas (1994b)). 

A representative collection of algorithms developed for solving MINLP models of the form (6.2) or restricted classes of (6.2) includes, in chronological order of development, the following ... 

In the sequel, we concentrate on Y = 0, l due to our interest in MINLP models. Note also that the analysis includes the equality constraints h[x,y) = 0 which are not treated explicitly in Geoffrion (1972). 

The basic idea in Generalized Benders Decomposition GBD is the generation, at each iteration, of an upper bound and a lower bound on the sought solution of the MINLP model. The upper bound results from the primal problem, while the lower bound results from the master problem. The primal problem corresponds to problem (6.2) with fixed y-variables (i.e., it is in the jr-space only), and its solution provides information about the upper bound and the Lagrange... 

Remark 8 The master problem consists of valid linear supports, and hence relaxations of the nonlinear functions, for all points xk that result from fixing y = yk Y as stated by (6.20). As a result it represents a relaxation of the original MINLP model (6.13), and hence it is a lower bound on its solution, and it is identical to its solution if all supports are included. 

It should be noted, however, that we need to check whether Tkh(x) < 0 is quasi-convex at each iteration. If the quasi-convexity condition is not satisfied, then the obtained lower bound by OA/ER may not be a valid one that is, it may be above the global solution of the MINLP model. This may happen due to the potential invalid linearizations which may cut off part of the feasible region. 

This chapter introduces the fundamentals of mixed-integer nonlinear optimization. Section 6.1 presents the motivation and the application areas of MINLP models. Section 6.2 presents the mathematical description of MINLP problems, discusses the challenges and computational complexity of MINLP models, and provides an overview of the existing MINLP algorithms. 

The user specified subroutines allow for connections to various other programs such as process simulators and ordinary differential equation solvers. Currently, MINOPT is connected to the DASOLV (Jarvis and Pantelides, 1992) integrator, and can solve MINLP models with differential and algebraic constraints. 

Remark 4 The complete optimization model for the simultaneous matches-HEN problem consists of the objective function (note that Case I was used in Floudas and Ciric (1989) subject to the set of constraints presented in parts A,B and C). In this model we have binary variables yij denoting the potential existence of a match and continuous variables. As a result, the model is a MINLP problem. This MINLP model has a number of interesting features which are as follows ... 

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