Optimization problem developed for different processes are very complicated. Mostly, the number of variables and constraints are very large. These complicated problems are difficult to solve by hand. Therefore, we take help of computer to solve those problems with very less effort that save our time and money. Computer programming solves these optimization problems using some numerical techniques. The algorithms are discussed in previous chapters. Some commercial software/software tools are available that can be used for optimization purpose. The most popular software/software tools are LINGO, MATLAB, MINITAB‚, GAMS etc. In this chapter, we will discuss these software/software tools with some examples.

LINGO

Introduction to LINGO

LINGO is modeling language and optimizer developed by LINDO Systems Inc. It is a simple tool for formulating large problems concisely, solve them, and analyze the solution using the power of linear and nonlinear optimization.

The outer window shown in Fig. 12.1, labeled Lingo 14.0, is the LINGO main frame window. All other sub-windows will be enclosed within this main frame window. The entire command menu and the command toolbar are placed at the top of the frame window. The status bar at the lower edge of the main frame window gives us various information related to the current state of LINGO. Both the status bar and the toolbar can be suppressed using the LINGO|Options command. The main window consists of a sub-window labeled Lingo Model-Lingo1 is used for Lingo Model. Here we will demonstrate how to enter and solve a small model in Windows platform. The text of the model's equations identical on all platforms, it is platform independent. Though, the procedure for entering a model is somewhat different on non-Windows platforms.

When we start LINGO in Windows platform, screen should look like the following:

Entering the Model

Example 12.1

We will now solve the problem 6.1 using LINGO.

The problem formulated in chapter 6 is given below:

subject to

After entering the above model within the Lingo Model-Lingo1 window, our model window should look like this:

Solving a LINGO Model

After entering the LINGO model into the LINGO Model window, the model can be solved by clicking the Solve button on the toolbar, by choosing LINGO|Solve from the menus, by using the keyboard shortcut “ctrl + U” or by pressing the button from toolbar menu as shown below […]

Optimization in the field of chemical engineering is required to utilize the resources in an efficient way as well as to reduce the environmental impact of a process. Application of optimization processes helps us achieve the most favorable operating conditions. Maximum profit is achievable if a process plant runs at optimum conditions. Knowledge of optimization theory as well its practical application is essential for all engineers.

The idea of this book came to my mind long back, perhaps six years ago. Then I started working on it; selecting topics to be included, collecting research papers, and preparing the manuscript. Many people helped me during this process, especially while collecting research articles from different sources: Sudip Banerjee, Arindam Chatterjee, D.K. Sandilya to name a few. I received very useful suggestions from reviewers of this manuscript.

This book contains detailed theory and applications of optimization in chemical engineering and related fields. Prerequisites for this book include some understanding of chemical engineering, biotechnology and mathematics. This book has been divided into twelve chapters. It contains various classical methods for optimization; it also introduces some of the recently developed topics in optimization. Examples from the field of chemical engineering and biochemical engineering are discussed throughout the book.

Chapter 1 discusses the classification and fundamentals of optimization methods. It also includes the salient features of optimization. This chapter also lists different types of objective functions and conditions for optimization. Chapter 2 gives emphasis to different chemical engineering processes and problem formulation procedures for optimization application. This chapter includes objective function formulation of fluid flow system, heat transfer equipments, mass transfer equipments, and reactors. One dimensional unconstrained problem formulation and optimization have been discussed in Chapter 3. This chapter includes different methods like Newton's method, Quasi-Newton method, Secant method etc. Chapter 4 discusses the Trust-Region methods for both constrained and unconstrained optimization problems. An overview of optimization of multivariable unconstrained functions is given in Chapter 5. This chapter comprises various search methods (i.e. random search, grid search), gradient method, Newton's method etc. Chapter 6 discusses the optimization methods for multivariable functions with constraints. This chapter contains both linear programming and non-linear programming. Optimization of staged and discrete processes has been discussed in Chapter 7. This includes dynamic programming, integer and mixed integer programming. Chapter 8 contains some advanced topics on optimization.

Large number of experimental run is required to develop the input-output relationship of any process. Utilization of statistical methods can reduce the number of the experimental run. This chapter elucidates the design of experiment for optimize use of experimental data. Response surface methodology is used to optimize the process output. A process plant can be optimized efficiently using these methods.

Design of Experiment

A huge number of experiments are required in research, development and optimization of any system. This research is carried out in labs, pilot plants or full-scale plants. An experiment may be a model based experiment or experiment with the physical system [Lazić, (2004)]. When the experimentation cost is very high, it is very difficult for the researcher to examine the numerous factors that have an effect on the processes using trial and error methods within a short time and limited resources. As an alternative, we need a technique that identifies some key factors in the most effective way. Then this leads the process to the best setting to satisfy the increasing demand for increased productivity with improved quality.

A one-factor-at-a-time experiment is used to study the effect of various factors during experimentation. This method involves changing a single factor at a time to analyze the impact of that factor on the product or process output. The main advantage of “one-factor-at-a-time” experiment is that we can perform the experiment easily. Despite this fact, these methods do not permit us to investigate of how a factor influences the process or a product in the presence of other factors. When the response of a process is changed due to the existence of one or more other factors, that relationship is called an interaction. Sometimes the effects of interaction terms are more significant compared to the individual effects. Because in the application environment of the process or product, most of the factors present together rather than the isolated incidents of single factors at different times. A chemical reacting system can be considered as an example of interaction between two factors. The reaction rate slightly increases when the temperature is increased and there is no effect of pressure on reaction rate. However, the reaction rate changes rapidly when both the temperature and pressure are changing simultaneously. In this system, an interaction does exist between the two factors that affect the chemical reaction.

Dynamic Programming

Multi-stage decision process arises during the study of different mathematical problems; the dynamic programming technique was created for taking care of these sorts of problems. Consider a physical system, at any time t, the state of this system is found out by a set of quantities that is called state variables or state parameters. At certain times, we are call upon to make a decision that will have an effect on the state of the system. This time may be prescribed in advanced or the process itself determines the time. The transformations of the state variables are equivalent to these decisions, the selection of a decision being identical with the choice of a transformation. The future state is guided by the outcome of the preceding decision, which maximizes some function of the parameters that describe the final state.

There are many practical problems where sequential decisions are made at various points in time, at various points in space, and at various levels, say, for a system, for a subsystem, or even for a component level. The problems for which the decisions are required to make sequentially are termed as sequential decision problems. They are also called multistage decision problems because for these problems decisions are to be made at a number of stages. Dynamic programming is a mathematical technique, which is most suitable for the optimization of multistage decision problems. Richard Bellman in the early 1950s [Bellman, (1953)] developed this technique. During application of dynamic programming method, a multi-dimensional decision problem was decomposed into a series of single stage decision problems. In this way, an N-variable problem can be expressed as a series of N single-variable problems. These N number of single-variable problems can be solved sequentially. Most of the time, these single-variable subproblems are quite easier to solve compared to the original problem. This decomposition to N single-variable subproblems is accomplished in such a way that the optimal solution of the original N-dimensional problem can be attained from the optimal solutions of the N one-dimensional problems. It is essential to make a note that the methods used to solve these one-dimensional stages do not affect the overall result. Any method can be used for this purpose; these methods may vary from an easy enumeration method to a relatively complicated technique like differential calculus or a nonlinear programming.

Introduction

Iterative methods for optimization are categorized into two classes. One class is called line search methods and the other class as trust region algorithms. Trust-Region methods are iterative method in which a model (mk) approximates the objective function (f) and this model is minimized in a neighborhood of the current iterate (the trust region). In case of a line-search method, the iterations are performed toward some particular directions; for example, the gradient directions are used to find the successive iterates in steepest descent [Liu and Chen, 2004]. However, in a Trust-Region algorithm, its iterates are derived by solving the corresponding optimization problem iteratively within an enclosed region. Therefore, we have more options to choose the iterates. Indeed, we can consider line-search methods as special cases of trust region methods [A. R. Conn et al., (2000)]. Trust-Region methods first introduced by M. J. D. Powel in 1970 [M. J. D. Powell, (1970)]. Powell [M. J. D. Powell, (1975)] also established the convergence result of unconstrained Trust-Region method optimization. Fletcher [R. Fletcher, (1972)] first recommended Trust-Region algorithms to solve linearly constrained optimization problems and non-smooth optimization problems [R. Fletcher, (1982)]. Trust-Region methods are very essential and effective methods in the area of nonlinear optimization. These methods are also useful for non-convex optimization problems and non-smooth optimization problems [Sun (2004)].

As most of the research works on trust region algorithms are mostly started in the 80s, trust region algorithms are less mature compare to line search algorithms, and the applications of trust region algorithms are limited as compared to line search algorithms. However, trust region methods have two major advantages. One is that they are reliable and robust; another is that they have very strong convergence properties. The key contents of any trust region algorithm are how to calculate the trust region trial step and how a decision can be made if a trial step should be accepted or not. An iteration of a trust region algorithm has the following form; a trust region is available at the beginning of the iteration. This is possible by considering an initial guess value X0 ∈ ℝn and trust region radius Δ0 > 0.

Heat Exchanger Network Optimization

An important task during the design of chemical processes is to changing the process streams from their available temperatures to the required temperatures without additional cost. Heat recovery is an important approach for reducing the cost by using the heat available from streams to be cooled to the streams to be heated. The distinguish between hot streams and cold streams are made based on whether the stream is cooled or heated not on the basis of temperature. Developing an efficient energy system is very essential in process industry to reduce waste heat available. This waste heat from one process can be recovered and reused for another process. Heat Exchanger Network (HEN) is used to optimize this heat recovery consequently the investment cost and energy consumption in the process plant. Synthesis of HEN is one of the most commonly discussed problems in a process industry. This method has importance during determination of energy expenditure for a process and improving the recovery of energy in industry. The first systematic method was introduced in 1970s to heat recovery with the concept of pinch analysis. Hohmann (1971), and Linnhoff and Flower (1978) introduced the pinch analysis for synthesis of HEN. A single task is decomposed into three different subtasks (i.e., targets) like minimum utility cost, minimum number of units, and minimum investment cost network configurations. The major benefit of decomposing this HEN synthesis problem is that handling of these sub-problems are much simpler than the original single-task problem. The decomposed sub-problems are given below:

1. Minimum utility cost This subtask or target is related to the amount of maximum energy recovery that can be attained in a feasible HEN with a constant heat recovery approach temperature (HRAT), that allows to eliminate various non-energy efficient HEN structures. Hohmann in 1971, first introduced the minimum utility cost and then Linnhoff and Flower [Linnhoff and Flower, (1978)] discussed this technique. Cerda et al. (1983) discussed this as an LP transportation model and is improved as LP transshipment model by Papoulias and Grossmann [Papoulias and Grossmann (1983)].

2. Minimum number of units For a fixed utility cost, this target finds the match combination with the minimum number of units and distribution of their load. The overall cost of the HEN depends on the number of units.

Stochastic Optimization

Global optimization algorithms are categorized into two major classes: Deterministic and stochastic (Dixon et al. 1975). In some chemical engineering optimization problems, the model cannot be entirely specified because, it depends on quantities that are random or probabilistic at the time of formulation. For instance, uncertainty governs the prices of fuels, the availability of electricity, and the demand for chemicals in the market [N. V. Sahinidis, (2004)]. These ever-changing market conditions require a high flexibility for chemical process industries under various product specifications and various feedstocks. Furthermore, the properties of processes change themselves during process operation, e.g., tray efficiencies and fouling of the equipment (heat exchanger, catalytic reactor), which leads to a decrease of product quality with the same operating conditions [Henrion et al. (2001)]. For deterministic optimization methods, the expected values of uncertain variables are generally used. In real life application, the values of uncertain variables deviate from their expected values. When we are introducing a stochastic element within deterministic algorithms, the deterministic assurance that the global optimum can be determined is relaxed into a confidence measure. Stochastic methods can be utilized to evaluate the probability of obtaining the global minimum. Stochastic ideas are generally used to development the stopping criteria, or for the approximation of the regions of attraction as utilized by some methods [Arora et al. 1995].

Uncertainties in process industries

Uncertainty is a part of real systems; in reality, every phenomenon has a certain degree of uncertainty. During their operation, chemical processes are frequently faced with uncertain conditions. From the perspective of chemical process operation, there exist two types of uncertainties. These uncertainties can occur due to variation either in external parameters, such as fluctuation of inlet flow, temperature, pressure, composition of the feedstock, supply of utilities and other environment parameters, or in internal process parameters for example heat transfer coefficients, tray efficiency and reaction rate constants, etc. The optimization problems become much more complex due to the existence of uncertain parameters and have considerable implications on process feasibility and other quality measures for instance controllability, safety, and environmental compliance [Hou et al. (2000)]. Uncertainty also arises in the planning problem. Uncertainty in planning problems can be categorized into two groups: endogenous uncertainty and exogenous uncertainty.