Defining the Objective Function

Define the objective function, the independent and the dependent variables, and the constraints first. Then set this problem up, and list all of the steps to solve it. You... 

OBJ defines the objective function, MBALSPL1,MBALSPL2,MBALSPL3, MBALMIXl, MBALMIX2,... 

First, we need to define the objective function, which is the fimction that we will attempt to minimize or maximize. In this example, we want to maximize storage capacity. We can represent this requirement mathematically as... 

Overlay objective function. Define the objective function and overlay this onto the AR to see where it intersects the AR boundary (easy) Once the AR has been determined, the limits of achievability by the system—for the kinetics and the feed point—are known. This information may be used to answer one or more design or optimization questions related to the system. Sections of the objective function that intersect the AR are optimal points, relative to the objective function specified. This task is often straightforward to carry out in practice. 

Defining the object function in terms of the rotated coordinate system, for the projection along lines of constant r, we have... 

The dimensions of the columns/shear walls are denoted as dimx and dimy corresponding to X and y axis, respectively. The formulation of the minimum initial construction cost leads to designs with eccentricities (ecM-CR, ecM-cv) larger than one meter. The two formulations implemented for minimizing the torsional response improve only the associated eccentricity defines the objective function of the problem When the ecM-CR is minimized, ecM-cv is increased, and vice versa. In Table 8 the cross-sections of the columns/shear walls for all optimum solutions are given. It is clear that the sizes of the cross-sections for those formulations that C/n is the dominant criterion are smaller the smallest ones. 

First, we define the objective function (see Figure 8.22b) such as energy consumption, the number of trays of a column, and the conversion of a reactor. We have up to three optimization algorithms for this case SQR, general reduced gradient, and simultaneous modular SQP. The particularities of the methods can be found elsewhere [44,45]. We define the independent variables and the constraints over variables from the streams or the equipment just by selecting the unit or the stream and the variable and the range of values of operation and an initial value. 

Finally, we define the objective function given by a simplified production cost involving the product obtained minus the cost of operation of the separation stages and the reactors. 

Let us define the objective function the inequality condition and the bounds for bothD andL. Figure 10.45 shows the MATLAB definition for the objective function given by Eq. tl0.96L... 

The next step in constructing an aggregate planning model is to define the objective function. 

In Step 3 of the calibration, we will use Aspen HYSYS to vary several activity factors in order to minimize the objective function. We define the objective function as the weighting sum of the absolute deviations from the model prediction and measure data. We can select terms in the objective function by going to the Objective section of the Calibration Control Tab. We show this interface in Figure 5.84. 

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function... 

Finding the best solution when a large number of variables are involved is a fundamental engineering activity. The optimal solution is with respect to some critical resource, most often the cost (or profit) measured in doUars. For some problems, the optimum may be defined as, eg, minimum solvent recovery. The calculated variable that is maximized or minimized is called the objective or the objective function. 

Let w, be the weight we wish to assign to point x, y,. The objective function is now defined... 

Linear Programming.28—A linear programming problem as defined in matrix notation requires that a vector x 0 (non-negativity constraints) be found that satisfies the constraints Ax <, b, and maximizes the linear function cx. Here x = (xx, , xn), A = [aiy] (i = 1,- -,m j = 1,- , ), b - (61 - -,bm), and c = (cu- -,c ) is the cost vector. With the original (the primal) problem is associated the dual problem yA > c, y > 0, bij = minimum, where y yx,- , ym)-A duality theorem 29 asserts that if either the primal or the dual has a solution then the values of the objective functions of both problems at the optimum are the same. It is a relatively easy matter to obtain the solution vector of one problem from that of the other. 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

Step 3. Learning Step Solve the objective function minimization problem with respect to the weights c (and the coefficients, w, if not defined in the previous step). 

Note that the set X% is defined such that it contains all solutions that have not been dominated by any other solution. Property 4 guarantees that we will not miss an element of the optimal set, since the objective function value of X is lower than that of y. 

With the objective functional, J, being defined as in Eq. (1) and using the penalty term/[e(f)] = the gradient of the objective functional with respect to... 

The objective functional for the optimal control of photodissociation may be defined as ... 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

Optimal for single-hatch operation. For the sake of simplicity suppose that (1) the performance of an equipment unit is the fraction of the feed material converted to the material that is suitable for the next stage (e.g. the yield of the desired intermediate or final product), and (2) that the objective function is the amount of suitable material produced per unit time. Let us consider the situation shown in Fig. 7.4-5. On completion of cleaning at time ta processing of a batch begins. This processing is characterized by the performance curve/(r), e.g. the yield or conversion versus time relationship. The objective function F is defined as ... 

It can be shown that this can be generalized to the case of more than two variables. The standard solution of a linear programming problem is then to define the comer points of the convex set and to select the one that yields the best value for the objective function. This is called the Simplex method. 

In ESL, the initial conditions for concentrations A, B, C, and D are followed by the table of measured data, time (min.) versus titrated volume (mL). In the DYNAMIC section, the program statements are identical to those of the model equations. Here, a function generator, FGENl, is used to describe the tabular data. The objective function to be minimised is defined as follows... 

In the parameter estimation, the product components P and MB were included. The objective function was defined as... 

Froa the above discussion it should be obvious that the selection of an appropriate objective function is a difficult task. The choice of the objective function is a critical factor in automated aethods development, since it is used to define the response surface. It is highly likely that different objective functions will result in the production of different response surfaces and the location of different optimum ejqaerimental conditions for the separation. Yet, it is not possible to set hard guidelines for the selection of the objective function, which must be chosen by practical experience keeping the objectives of the separation in mind. 

Usually the space over which the objective function is minimized is not defined as the p-dimensional space of p continuously variable parameters. Instead it is a discrete configuration space of very high dimensionality. In general the number of elements in the configuration space is exceptionally large so that they cannot be fully explored with a reasonable computation time. 

Copp and Everet (1953) have presented 33 experimental VLE data points at three temperatures. The diethylamine-water system demonstrates the problem that may arise when using the simplified constrained least squares estimation due to inadequate number of data. In such case there is a need to interpolate the data points and to perform the minimization subject to constraint of Equation 14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First, unconstrained LS estimation was performed by using the objective function defined by Equation 14.23. The parameter values together with their standard deviations that were obtained are shown in Table 14.5. The covariances are also given in the table. The other parameter values are zero. 

---