Optimization continuous variables

All of these variables must be varied in order to minimize the total cost or maximize the economic potential (see Chapter 2). This is a complex optimization problem involving both continuous variables (e.g. batch size) and integer variables (e.g. number of units in parallel) and is outside the scope of the present text9. 

The results for this scenario were obtained using GAMS 2.5/CPLEX. The overall mathematical formulation entails 385 constraints, 175 continuous variables and 36 binary/discrete variables. Only 4 nodes were explored in the branch and bound algorithm leading to an optimal value of 215 t (fresh- and waste-water) in 0.17 CPU seconds. Figure 4.5 shows the water reuse/recycle network corresponding to fixed outlet concentration and variable water quantity for the literature example. It is worth noting that the quantity of water to processes 1 and 3 has been reduced by 5 and 12.5 t, respectively, from the specified quantity in order to maintain the outlet concentration at the maximum level. The overall water requirement has been reduced by almost 35% from the initial amount of 165 t. 

The formulation for this scenario entails 1411 constraints, 511 continuous and 120 binary variables. The reduction in continuous variables compared to scenario 1 is due to the absence of linearization variables, since no attempt was made to linearize the scenario 2 model as explained in Section 4.3. An average of 1100 nodes were explored in the branch and bound search tree during the three major iterations between the MILP master problem and the NLP subproblem. The problem was solved in 6.54 CPU seconds resulting in an optimal objective of 2052.31 kg, which corresponds to 13% reduction in freshwater requirement. The corresponding water recycle/reuse network is shown in Fig. 4.10. 

We start with continuous variable optimization and consider in the next section the solution of NLP problems with differentiable objective and constraint functions. If only local solutions are required for the NLP problem, then very efficient large-scale methods can be considered. This is followed by methods that are not based on local optimality criteria we consider direct search optimization methods that do not require derivatives as well as deterministic global optimization methods. Following this, we consider the solution of mixed integer problems and outline the main characteristics of algorithms for their solution. Finally, we conclude with a discussion of optimization modeling software and its implementation on engineering models. 

For continuous variable optimization we consider (3-84) without discrete variable y. The general NLP problem (3-85) is presented here ... 

In order to overcome the previous limitations and generate data-independent models, a wide variety of optimization approaches employ a continuous-time representation. In these formulations, timing decisions are explicitly represented as a set of continuous variables defining the exact times at which the events take place. 

In our case study, the problem instances for SNP optimization result from master data of approximately 10000 location-products, 10000 recipes, 400 production resources and 10 resources relevant for campaign production with 3-10 different products per campaign resource. This translates into a MIP model with about 700 000 continuous variables, 1000 binary variables and 300 000 linear constraints. 

Continuous variables can assume any value within an interval discrete variables can take only distinct values. An example of a discrete variable is one that assumes integer values only. Often in chemical engineering discrete variables and continuous variables occur simultaneously in a problem. If you wish to optimize a compressor system, for example, you must select the number of compressor stages (an integer) in addition to the suction and production pressure of each stage (positive continuous variables). Optimization problems without discrete variables are far easier to solve than those with even one discrete variable. Refer to Chapter 9 for more information about the effect of discrete variables in optimization. 

An engineer typically strives to treat discrete variables as continuous even at the cost of achieving a suboptimal solution when the continuous variable is rounded off. Consider the variation of the cost of insulation of various thickness as shown in Figure El.l. Although insulation is only available in 0.5-in. increments, continuous approximation for the thickness can be used to facilitate the solution to this optimization-problem. 

In addition to the Premium Excel Solver and Optquest, there are many other software systems for constrained global optimization see Pinter (1996b), Horst and Pardalos (1995), and Pinter (1999) for further information. Perhaps the most widely used of these is LGO (Pinter, 1999), (Pinter, 2000), which is intended for smooth problems with continuous variables. It is available as an interactive development environment with a graphical user interface under Microsoft Windows, or as a callable library, which can be invoked from an application written by the user in... 

Continuous- Variable Optimal Design Standard integer sizes ... 

The problem involves nine optimization variables (jcc, c — 1 to 5 Qp, p = 1 to 4) in the preceding formulation. All are continuous variables. The objective function is a linear function of these variables, and so are Equations (a) and (b), hence the problem is a linear programming problem and has a globally optimal solution. 

In sub-problem 4 the process model constraints (function of both integer and continuous variables) are considered along with the objective function. The optimal solvent is identified by either solving a smaller MINLP problem (if the number of feasible solutions is large) or a set of NLP problems (if the number of feasible solutions is small) by fixing the values of integer variables. 

As reported [see problem 7, Chem, Eng. ScL, 45, 595-614 (1990)], the attack on this problem used 2077 continuous variables, 204 integer variables, 2108 constraints, and gave as an optimal solution the design shown in Fig. PlO.ll. 

Continuous improvement PAT framework provides many options and opportunities including research data collection in production Manufacturing Science WG creating regulatory flexibility through ICH Q8 CGMP initiative objective Product is in specification Acceptance criteria—variable/continuous data Evolutionary, incremental process optimization, continuous, daily activity Carried out by plant and quality staff... 

A wide range of nonlinear optimization problems involve integer or discrete variables in addition to the continuous variables. These classes of optimization problems arise from a variety of applications and are denoted as Mixed-Integer Nonlinear Programming MINLP problems. 

The general mathematical model of the superstructure presented in step 2 of the outline, and indicated as (7.1), has a mixed set of 0 - 1 and continuous variables and as a result is a mixed-integer optimization model. If any of the objective function and constraints is nonlinear, then (7.1) is classified as mixed- integer nonlinear programming MINLP problem. 

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 ... 

Remark 4 The presented optimization model is an MINLP problem. The binary variables select the process stream matches, while the continuous variables represent the utility loads, the heat loads of the heat exchangers, the heat residuals, the flow rates and temperatures of the interconnecting streams in the hyperstructure, and the area of each exchanger. Note that by substituting the areas from the constraints (B) into the objective function we eliminate them from the variable set. The nonlinearities in the in the proposed model arise because of the objective function and the energy balances in the mixers and heat exchangers. As a result we have nonconvexities present in both the objective function and constraints. The solution of the MINLP model will provide simultaneously the... 

Remark 1 The resulting optimization model is an MINLP problem. The objective function is linear for this illustrative example (note that it can be nonlinear in the general case) and does not involve any binary variables. Constraints (i), (v), and (vi) are linear in the continuous variables and the binary variables participate separably and linearly in (vi). Constraints (ii), (iii), and (iv) are nonlinear and take the form of bilinear equalities for (ii) and (iii), while (iv) can take any nonlinear form dictated by the reaction rates. If we have first-order reaction, then (iv) has bilinear terms. Trilinear terms will appear for second-order kinetics. Due to this type of nonlinear equality constraints, the feasible domain is nonconvex, and hence the solution of the above formulation will be regarded as a local optimum. 

The paper simplifies the analysis required for each configuration by eliminating all continuous variable optimization problems. (This sounds familiar, does it not ) Most notably they require all evaporator areas to be equal and add other specifications sufficient in number to absorb all the degrees of freedom. One might view some of these added specifications as "heuristics."... 

It is useful at this point to realize that with the composition of the stationary phase being a continuous variable and with retention and selectivity being strong functions of temperature, the optimum composition may also be expected to vary with temperature. Ideally therefore, temperature and stationary phase composition should be optimized simultaneously (see section 5.1.1). Moreover, once different lengths of columns with the individual stationary phases are applied instead of real mixtures, it is in theory feasible to optimize the temperature of each of the columns, as well as the ratio of column lengths simultaneously. 

These IF statements are really a form of discrete decision making embedded within the model. One possible approach to remove the difficulties it caused is to move the discrete decisions to the outside of the model and the continuous variable optimizer. For example, the friction factor equation can be selected to be the laminar one irrespective of the Reynolds number that is computed later. Constraints can be added to forbid movement outside the laminar region or to forbid movement too far outside the laminar region. If the solution to the well-behaved continuous variable optimization problem (it is solved with few iterations) is on such a constraint boundary, tests can be made to see if crossing the constraint boundary can improve the objective function. If so, the boundary is crossed—i.e., a new value is given to the discrete decision, etc. 

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