Optimization formulation objective function

This chapter includes a discussion of how to formulate objective functions involved in economic analysis, an explanation of the important concept of the time value of money, and an examination of the various ways of carrying out a profitability analysis. In Appendix B we cover, in more detail, ways of estimating the capital and operating costs in the process industries, components that are included in the objective function. For examples of objective functions other than economic ones, refer to the applications of optimization in Chapters 11 to 16. 

In the optimization, the objective function is to maximize the product generated versus process time. The desired product was defined by endpoint equalities and inequalities, such as amount of unreacted components. In addition, the safety conditions required certain path-constraints for the state variables such as temperature. Unfortunately, we experienced optimization problems with the above formulation. The problems stem from getting stuck in infeasible regions due to complexity of the process and the nonlinearity of the objective function. At the moment, we are working to overcome these problems so that we can test the runaway behavior and cooler limitations with respect to optimization. 

Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifac t performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this sec tion. 

Formulation of the Objective Function The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. You must be able to translate the desired objective into mathematical terms. In the chemical process industries, the obective function often is expressed in units of currency (e.g., U.S. dollars) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. 

Various criteria were proposed for the optimal selection of the equipment configuration and the number and sizes of units. In grass-root design, the capital cost of equipment is mostly used as the optimization criterion. In retrofit design a more appropriate objective function seems to be the net profit, which has to be maximized. Papageorgaki and Reklaitis (1993) formulated the criterion as follows ... 

The method above does not account for differences in the profitability of various products. Reinhardt and Rippin formulated an objective function that takes these into account. They proposed to use the Here and Now design including that objective function. The Here and Now design comprise three steps (1) optimization of the design and operating variables for the worst possible realization of the uncertain parameters ... 

Publications on optimal design of tree networks are further divided into single-branch trees or pipelines (C6, F4, L3, L6, S8) and many-branch trees (B7, C7, F4, Kl, K2, M3, M9, Nl, R5, W10, Y1, Zl). For our purposes, since the pipeline problems can always be solved using the optimization methods developed for the many-branch tree networks, we need to dwell no further on this special case. On the other hand, it is important to note that the form of the objective function could influence the applicability of a given optimization method. For the sake of concreteness, problem formulations and optimization techniques will be discussed in the context of applications. 

The solution was found in a total time of 522.07 CPU seconds and the resulting formulation had 162 binary variables. The objective function had an optimal value of 1.869x 106c.u. with 9 time points. The total effluent generated was 504.63 t. As with the previous solution, had recycle/reuse not been considered the total effluent would have been 562 t of water for the same production. This relates to a 10.2% decrease in the amount of effluent generated by recycling/reusing wastewater. 

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continu-ous optimization problems can be written as the following mixed integer optimization problem ... 

Process synthesis is a task of formulating the process configuration for a purpose by defining which operations or equipment are used and how they are connected together. There are two basic approaches for process synthesis 1) classical process synthesis, analysis and evaluation, and 2) optimization of process structure by using a suitable objective function. 

Steps 1, 2, and 3 deal with the mathematical definition of the problem, that is, identification of variables, specification of the objective function, and statement of the constraints. We devote considerable attention to problem formulation in the remainder of this chapter, as well as in Chapters 2 and 3. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. 

If the objective function and constraints in an optimization problem are nicely behaved, optimization presents no great difficulty. In particular, if the objective function and constraints are all linear, a powerful method known as linear programming can be used to solve the optimization problem (refer to Chapter 7). For this specific type of problem it is known that a unique solution exists if any solution exists. However, most optimization problems in their natural formulation are not linear. 

For each of the following six problems, formulate the objective function, the equality constraints (if any), and the inequality constraints (if any). Specify and list the independent variables, the number of degrees of freedom, and the coefficients in the optimization problem. 

The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. As discussed in Chapter 1, you must be able to translate a verbal statement or concept of the desired objective into mathematical terms. In the chemical industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the goal of the enterprise is to minimize costs or maximize profits subject to a variety of constraints. In other cases the problem to be solved is the maximization of the yield of a component in a reactor, or minimization of the use of utilities in a heat exchanger network, or minimization of the volume of a packed column, or minimizing the differences between a model and some data, and so on. Keep in mind that when formulating the mathematical statement of the objective, functions that are more complex or more nonlinear are more difficult to solve in optimization. Fortunately, modem optimization software has improved to the point that problems involving many highly nonlinear functions can be solved. 

Suppose you wanted to find the configuration that minimizes the capital costs of a cylindrical pressure vessel. To select the best dimensions (length L and diameter D) of the vessel, formulate a suitable objective function for the capital costs and find the optimal (LID) that minimizes the cost function. Let the tank volume be V, which is fixed. Compare your result with the design rule-of-thumb used in practice, (L/D)opt = 3.0. 

To understand the strategy of optimization procedures, certain basic concepts must be described. In this chapter we examine the properties of objective functions and constraints to establish a basis for analyzing optimization problems. We identify those features that are desirable (and also undesirable) in the formulation of an optimization problem. Both qualitative and quantitative characteristics of functions are described. In addition, we present the necessary and sufficient conditions to guarantee that a supposed extremum is indeed a minimum or a maximum. 

One method of handling just one or two linear or nonlinear equality constraints is to solve explicitly for one variable and eliminate that variable from the problem formulation. This is done by direct substitution in the objective function and constraint equations in the problem. In many problems elimination of a single equality constraint is often superior to an approach in which the constraint is retained and some constrained optimization procedure is executed. For example, suppose you want to minimize the following objective function that is subject to a single equality constraint... 

This example focuses on the design and optimization of a steady-state staged column. Figure El 2.1 shows a typical column and some of the notation we will use, and Table El2.1 A lists the other variables and parameters. Feed is denoted by superscript F. Withdrawals take the subscripts of the withdrawal stage. Superscripts V for vapor and L for liquid are used as needed to distinguish between phases. If we number the stages from tihe bottom of the column (the reboiler) upward with k= 1, then V0 = L1 = 0, and at the top of the column, or the condenser, Vn = Ln+l = 0. We first formulate the equality constraints, then the inequality constraints, and lastly the objective function. 

Optimization in the design and operation of a reactor focuses on formulating a suitable objective function plus a mathematical description of the reactor the latter forms a set of constraints. Reactors in chemical engineering are usually, but not always, represented by one or a combination of... 

The problem is to allocate optimally the crudes between the two processes, subject to the supply and demand constraints, so that profits per week are maximized. The objective function and all constraints are linear, yielding a linear programming problem (LP). To set up the LP you must (1) formulate the objective function and (2) formulate the constraints for the refinery operation. You can see from Figure El6.1 that nine variables are involved, namely, the flow rates of each of the crude oils and the four products. 

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 Chapter 3 we discussed the formulation of objective functions without going into much detail about how the terms in an objective function are obtained in practice. The purpose of this appendix is to provide some brief information that can be used to obtain the coefficients in objective functions in economic optimization problems. Various methods and sources of information are outlined that help establish values for the revenues and costs involved in practical problems in design and operations. After we describe ways of estimating capital costs, operating costs, and revenues, we look at the matter of project evaluation and discuss the many contributions that make up the net income from a project, including interest, depreciation, and taxes. Cash flow is distinguished from income. Finally, some examples illustrate the application of the basic principles. 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

To apply the procedure, the nonlinear constraints Taylor series expansion and an optimization problem is resolved to find the solution, d, that minimizes a quadratic objective function subject to linear constraints. The QP subproblem is formulated as follows ... 

Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation, using nonlinear programming techniques, is discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities and thus requires efficient and novel solution techniques. 

An optimum seeking method is a systematic way of manipulating a set of variables to find the values of the variables to maximize or minimize some criterial. Their most popular uses have been economic ones such as profitability or costs or technical criterial such as conversion of raw materials or product recovery. For the applications described in this paper, the optimization criterial are the minimization of the squared deviations from zero of the equations chosen to constitute the objective function. The equations chosen to formulate the objective function can be written as... 

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