Ferruh Erdogdu
CONTENTS
5.1 Introduction ... 111 References ... 1135.1 INTRODUCTION
Optimization is the choice of a best alternative from a specified set of alternatives. As reported by Evans (1982), Aris (1964) defines the optimization as ‘‘getting the best you can out of a given situation.’’ Optimization opens up the possibility of achieving the best case among the possible alternatives (Evans, 1982). Achieving optimization, therefore, requires some way of describing the potential alternatives and deciding to choose the best alternative (Norback, 1980; Evans, 1982).
In design, construction and maintenance of any engineering system, different technological and managerial decisions are required to be given at different stages of the process to either minimize the effort required or to maximize the benefit desired (Rao, 1996). The formal description of any optimization problem has three parts:
. Set of variables that the optimization method can control and use to specify the alternatives (e.g., applying different process temperature profiles during thermal processing to achieve a better processing for a given objective function)
. Set of requirements (e.g., the differential equations, boundary conditions, and integral equations specifying the constraints that the system and the variables are subjected to) that the optimization method must achieve or satisfy
. Measure of performance to compare one alternative to another (the objective function) (Norback, 1980)
The objective function, which may be continuous or in some cases discrete, is the function to be optimized (maximized or minimized). This can be accomplished by using either a mathematical model or byfitting an equation through the experimental data (Saguy, 1983). The optimization problems can be divided into continuous and discrete types depending on the nature of the objective function. Discrete problems
usually have afinite number of variables, each of which assumes exactly one value at an optimal solution. In continuous problems, the optimal values are functions of some parameter, and a solution to the problem requires the specification of this function over a set of parameters. Generally the continuous optimization problems require mathematical approaches for a solution (Norback, 1980; Saguy, 1983).
Thermal processing in food processing operations can be given as an example for a continuous problem. This results in a continuous function indicating the optimal path as the solution to a continuous time dynamic problem (Kamien and Schwartz, 1991). For this case, variation in process temperature can become one control variable, and maximization of a nutrient, for example, in overall volume or at the surface can be applied as an objective function. The constraints can be given as the lower and higher limits of the process temperature profile as well as the coldest point lethality or temperature obtained at the end of the process (Norback, 1980) where they might include differential equations and boundary conditions used in the solution of time–temperature profiles to be described.
The continuous optimization problems are reported to be difficult to solve due to the possible nonlinear and distributed nature of the system dynamics and presence of explicit and implicit constraints on both control variable and objective function (Banga et al., 2001). There have been numerous methodologies recommended for this problem in the literature (Erdogdu and Balaban, 2003).
Choosing the objective function, control variables, and constraints are significant steps in the application of any method that can be applied for the given situation since the global optimum point might not be achieved due to the insensitivities of the objective functions with respect to the control variable (Erdogdu and Balaban, 2003). Convergence difficulties might also happen because of possible high linearities and discontinuities of the interested systems. In addition, the economical impact or lack of it might also limit the applicability of an optimization methodology (Saguy, 1982). By introduction of economical-related constraints or objectives, a more useful set of operating conditions can be obtained (Rogers, 1985).
On the basis of the number of objective functions for the performance criteria describing the performance of the system, the objective tofind the optimal solution is called single-objective optimization when an optimization problem describing a physical system involves only one objective function (Deb, 2002). When the optimization problem involves more than one objective function, it is known as multiobjective optimization. Such problems are also known as multiple criterion decision-making problems (Deb, 2002).
Finding an optimal solution for a food processing problem by selecting the best condition from a large set of alternatives can be a difficult process. With the increase in the availability of higher computing abilities, there have been significant develop- ments in the optimization methodologies and different statistical and mathematical techniques proposed for this objective.
Therefore, in Part II, different optimization methodologies are discussed starting with the statistical optimization techniques. Pontryagin’s method, with its highly advanced mathematical nature is presented, dynamic optimization and mixed integer linear and nonlinear programming methodologies are discussed, and potential uses and limitations of artificial intelligence-genetic algorithms, dynamic optimization
and multiobjective optimization procedures as well as possible and expected appli- cations of computational fluid dynamics methodologies are covered. In addition, tabu search procedure and eigenvalue optimization techniques for nonlinear dynamic analysis and design are also presented.
REFERENCES
Aris, R., Discrete Dynamic Programming, Blaiesdell Publishing Co., New York, 1964. Banga, J.R., Pan, Z., and Singh, R.P., On the optimal control of contact cooking processes,
Trans. Ind. Chem. Eng., 79, 145, 2001.
Deb, K., Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, West Sussex, England, 2002.
Erdogdu, F. and Balaban, M.O., Complex method for nonlinear constrained multi-criteria (multi-objective function) optimization of thermal processing, J. Food Process Eng., 26, 357, 2003.
Evans, L.B., Optimization theory and its application in food processing, Food Tech., 36, 88, 1982.
Kamien, M.I. and Schwartz, N.L., Dynamic Optimization—The Calculus of Variations and Optimal Control in Economics and Management, Elsevier Science Publishing Co. Inc., New York, 1991.
Norback, J.B., Techniques for optimization of food processes, Food Tech., 34, 86, 1980. Rao, S.S., Engineering Optimization—Theory and Practice, John Wiley & Sons, Inc.,
New York, 1996.
Rogers, J.A., Optimizing process and economic aims, Chem. Eng., 9, 23, 1985.
Saguy, I., Optimization theory, techniques, and their implementation in the food industry: Introduction, Food Tech., 36, 87, 1982.
Saguy, I., Optimization methods and applications, in Saguy, I. (Ed.), Computer Aided Techniques in Food Technology, Marcel Dekker, New York, 1983, p. 268.