Process optimization discrete decision variables

In a retrofit batch design, we optimize the batch plant profitability defined as the total production value minus the cost of any new equipment. The objective is to obtain a modified batch plant structure, an operating strategy, the equipment sizes, and the batch processing parameters. Discrete decisions correspond to the selection of new units to add to each stage of the plant and their type of operation. Continuous decisions are represented by the volume of each new unit and the batch processing variables which are allowed to vary within certain bounds. 

In contrast to the sequential solution method, the simultaneous strategy solves the dynamic process model and the optimization problem at one step. This avoids solving the model equations at each iteration in the optimization algorithm as in the sequential approach. In this approach, the dynamic process model constraints in the optimal control problem are transformed to a set of algebraic equations which is treated as equality constraints in NLP problem [20], To apply the simultaneous strategy, both state and control variable profiles are discretized by approximating functions and treated as the decision variables in optimization algorithms. 

For the PDF model of the SMB process, full discretization was used, that is, both temporal and spatial variables were discretized leading to a huge system of algebraic equations. The standard SMB optimization problem has 33 997 decision variables and 33 992 equality constraints while the superstructure SMB optimization problem has 34 102 decision variables and 34 017 equality constraints. Note that there are many more degrees of freedom in the superstructure formulation (altogether 85) than in the standard SMB formulation (5 degrees of freedom). 

In Step 3, optimization of the structure and the parameters (Section 16.8.1), the two types of process optimizations are parameter and structural optimization. Parameter optimization is the process of determining the best value of a process unit parameter or stream quantity in terms of improving performance within a given set of constraints. Parameter optimization is usually a nonlinear continuous variable (over a range of variable values defined by upper and lower bounds). Structural optimization involves the determination of the best set of units and their interconnections such that the process configuration provides the best performance within a given set of constraints. Structural optimization requires discrete decisions. Pinch technology, described in Section 16.8.5, is a form of structural optimization. 

When there are only a few discrete-valued decision variables in a model, the most effective method of analysis is usually the most direct one total enumeration of all the possibilities. For example, a model with only eight 0-1 variables could be enumerated by trying all 2 = 256 combinations of values for the different variables. If the model is pure discrete, it is only necessary to check whether each possible assignment of values to discrete variables is feasible and to keep track of the feasible solution with best objective function value. For mixed models the process is more complicated because each choice of discrete values yields a residual optimization problem over the continuous variables. Each such continuous problem must be solved or shown infeasible to establish an optimtil solution for the full mixed problem. 

We consider a minimization rather than a maximization problem for the sake of notational convenience.) Here C R is a set of permissible values of the vector x of decision variables and is referred to as the feasible set of problem (11). Often x is defined by a (finite) number of smooth (or even linear) constraints. In some other situations the set x is finite. In that case problem (11) is called a discrete stochastic optimization problem (this should not be confused with the case of discrete probability distributions). Variable random vector, or in more involved cases as a random process. In the abstract fiamework we can view as an element of the probability space (fi, 5, P) with the known probability measure (distribution) P. 

With the feasible path approach the optimization algorithm automatically performs case studies by variing input data. There are several drawbacks the process equations (32c) have to be solved every time the performance index is evaluated. Efficient gradient-based optimization techniques can only be used with great difficulties because derivatives can only be evaluated by perturbing the entire flowsheet with respect to the decision variables. This is very time consuming. Second, process units are often described by discrete and discontinuous relations or by functions that may be nondifferentiable at certain points. To overcome these problems quadratic module models can be... 

The TMP design optimization is a MINLP (mixed-integer non-linear programming) problem since it has both a discrete, A Ref, and a continuous, Fu,nk, design parameter. The operational optimization subproblem has integer decision variables (number of active refiners in time) affecting the continuous state of intermediate tank volume through process dynamics. The tank volume is constrained to stay between a minimum and a maximum volume. 

For a typical flowsheet, such as the DME (dimethyl ether) PFD in Figure B.1.1 i Appendix B), there are many decision variables. The temperature and pressure of each unit can be varied. The size of each piece of equipment involves decision variables (usually several per unit). The reflux in tower T-201 and the purity of the distillate fromT-202 are decision variables. There are many more. Clearly, the simultaneous optimization of all of these decision variables is a difficult problem However, some subproblems are relatively easy. If Stream 4 (the exit from the methanol preheater) must be at 154°C, for example, the choice of which heat source to use (Ips, mps, or hps) is easy. There is only a sin e decision variable, there are only three discrete choices, and the choice has no direct impact on the rest of the process. The problem becomes more difficult if the temperature of Stream 4 is not constrained. 

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