Thermal processing optimization through a modified adaptive random search

Abstract

This research suggests a modification of the adaptive random search method. The proposed modification is based on the utilization of the well-known logistic function or logistic curve in order to improve the random search adaptation characteristics. The algorithm tested results show the advantage of the random search modification over the previous random search organization, especially in the case of solving the multi-modal optimization problems. An interesting and important food industry optimization problem, such as thermal processing, was solved by the new organization of adaptive random search.

Introduction

A large number of real-life decision making problems arising in sciences, engineering and economics, can be formulated as a problem of optimization of various real-valued functions of real-valued parameters (objective function) with specific constraints, where the global optimum corresponds to the best solution to the initial problem (Horst and Pardalos, 1995). Such optimization problems may be nonlinear with many local solutions; moreover, a single computation of objective function with constraints, may take significant CPU-time. As an example, we can mention the dynamical optimization problem where single computations of optimization problems need to be integrated via a system of differential–algebraic equations over the whole time interval (Schlegel et al., 2005, Banga et al., 2005). Therefore, effective global optimization techniques with minimal optimization computations to reach global optimum location are needed.

Random search techniques are widely used to solve global optimization problems for various reasons. Firstly, random search techniques are easily used in practical implementations and they are independent of a priori information about the optimization problem to be solved. Secondly, the optimization problem could be considered as a “black-box” problem where no closed-form formulation and gradient information of the given problems are available.

Various classes of random search methods have been proposed over the past decades:

• Random Line Search (Rastrigin and Rubinstein, 1969, Zhensheng Yua and Dingguo Pu, 2008).

• Multiple random starts with local search utilization (multi-start methods) (Zhigljavsky, 1991, Zhigljavsky and Zilinskas, 2008).

• Controlled Random Search (Price, 1977, Banga and Casares, 1987, Tvrdík et al., 2007, Ali and Törn, 2004).

• Simulated annealing (Kirkpatrick et al., 1983).

• Tabu search (Glover, 1989).

• Ant colony optimizations (Dorigo, 1992, Dorigo and Gambardella, 1997, Dorigo and Stűtzle, 2004, Socha and Dorigo, 2008).

• Genetic algorithms (Rechenberg, 1973, Holland, 1975, Andersson, 2000, Neelesh et al., 2007).

Let the global optimization problem be formulated as follows, in which a continuous objective function of decision variables is to be minimized over a set of candidate decision variables:

$$\Phi (x)\to \underset{x\in X}{\min }\text{,}$$

where:

• $X=[0\text{,}1{]}^n\subset R^n$—is a non-empty set of feasible decisions (a proper subset of (Rⁿ),

• $x=〈x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n〉\in X$—is a real n—vector decision variable, and

• $\Phi :R^n\to R$—is a continuous objective function.

Supposing that all of the constraints are included in the objective function (1) by utilizing a penalty functions (Himmelblau, 1972).

The general algorithm scheme of random search (Zhigljavsky, 1991, Tikhomirov et al., 2007) could be presented as follows:

General algorithm scheme of random search:

• Set s = 1, x^s−1 = x^o, where x^o is the initial point of the searching process (x^o X) and s is the iteration number.

• Generate a new vector x′ $\in$ X from some probability distribution $P_s(x^{s-1})$.

• If $f(x^{\prime })<f(x^{s-1})$ then x^s = x′, else x^s = x^s−1.

• Set s = s + 1.

• If some stop condition is satisfied, then finish the search process, or else go to step 2.

Many modifications of random search can be constructed from the general algorithm scheme, mainly, by defining a concrete probability distribution P_s(x^s − 1,·) for each iteration number s. In the case where uniform distribution is chosen as the probability distribution P_s(x^s − 1,·) for all of the iteration numbers, the pure random search algorithm will be obtained (Zhigljavsky, 1991). It is well-known that the probability of finding a global solution of problem (1) in the case of pure random search utilization is equal to 1, but it takes too many computations of problem (1), therefore, a lot of modifications of the general algorithm have been proposed to significantly reduce the computations of problem (1).

The global stochastic optimization algorithm to be improved in this work belongs to several specific classes of random search algorithms (Zhigljavsky, 1991, Zhigljavsky and Zilinskas, 2008), namely adaptive random search algorithm (Sushkov, 1969). The term “adaptation” in this case consists of such modifications in the probability distribution P_s(x^s − 1,·), which throughout the whole search process, act as minimum computations of problem (1), locating global solutions. As an example of such algorithms we can mention here the Integrated Control Random Search algorithm (ICRS) (Banga and Casares, 1987). This method has been successfully utilized for a wide set of the practical optimization problems, especially for dynamical optimization of bioprocesses (Banga and Seider, 1996, Banga et al., 2005, García et al., 2005). The ICRS uses the Gaussian probability distribution P_s(x^s − 1,·) to generate new vectors $x^{\prime }\in X$, where adaptation consists of reducing and increasing the standard deviation of the Gaussian probability distribution around its mean.

The proposed modification here is based on logistic function implementation, in order to improve adaptive characteristics of random search. The logistic equation was first published by Verhulst (1845) as a model of population growth. The continuous version of a logistic model is the following differential equation:

$$\frac{\text{d}x}{\text{d}t}=\mu x(1-x)\text{,}$$

where μ is the Malthusian parameter (rate of maximum population growth). The solution of Eq. (2) could be written as:

$$x(t)=\frac{1}{1+\alpha e^{-\mu t}}\text{,}$$

where α and μ are the parameters of logistic curve. The function of x(t) is also called the sigmoid function (Krose, 1993).

Logistic functions or logistic curves find their application in a wide range of scientific fields. Some of the well-known applications are as follows:

• Artificial neural networks (ANN) (Krose, 1993), where sigmoid functions are often used for the ANN learning process.

• Statistics, where logistic functions are used as regression models, namely the logistic regression models (Hosmer and Lemeshow, 2000) and as a cumulative distribution function (Marin and Buckius, 1998).

• Sociology and Ecology: as models of population growth (Chapman, 1931; Mulligan and Gordon, 2006, Molles, 2004).

• Medicine: for simulation of tumor growths (Moate et al., 2004).

It should be noted that there is a class of global stochastic optimization methods – chaos optimization where logistic equation or, rather, logistic maps (May, 1976) were successfully applied (Yang et al., 2007, Li and Jiang, 1998). Logistic maps are modifications of the continuous logistic Eq. (2) and the well-known logistic map can be written as:

$$x^{n+1}=\mu x^n(1-x^n)\text{,}$$

where μ is the control of logistic map. The algorithms of chaos optimization use a logistic map as a pseudo-random vector x^j $\in$ Χ value generator. Optimization problem (1) needs to be calculated with respect to the generated vectors value and the best point (or points) should be chosen. The best point is assumed to be close to the global solution; therefore, the current best point is used as the center for the next decision’s vector generation. This process could be repeated until some stop condition of the search process is satisfied. It is known that for some control values μ, the values x^n + 1 behave chaotically (May, 1976, Devaney, 1992, Yang et al., 2007) and so this optimization technique was called “chaos optimization”.

In this work, the logistic curve is used differently. It reduces the same sub-domain I $\subseteq$ Χ, which is assumed to contain the global solution of problem (1), and iteration number of the random search algorithm or number of problem (1) computations. In other words, the logistic curve is used to reduce the deviation generated by the random search algorithm with values of x^j $\in$ Χ, j $\in$ 1:s, around the mean, and the mean is assumed to be the current best solution $x^l\in X\text{,}f(x^l)<f(x^j)\text{,}\forall j\in$ 1:s, of the search process.

Optimization of thermal processing of canned foods was solved by the proposed modification in order to show its advantage over the basic random search algorithm. Thermal processing is an important method of food preservation in the manufacture of shelf stable canned foods, and has been the cornerstone of the food processing industry for more than a century (Teixeira, 1992). The basic function of a thermal process is to inactivate bacterial spores of public health significance as well as food spoilage microorganisms in sealed containers of food, using heat treatments at temperatures well above the ambient boiling point of water in pressurized steam retorts (autoclaves). Excessive heat treatments should be avoided because they are detrimental to food quality, and underutilize plant capacity (Simpson et al., 2003).

Thermal process calculations, in which process times at specified retort temperatures are calculated in order to achieve safe levels of microbial inactivation (lethality), must be carried out carefully to assure public health safety and minimum probability of spoilage. Therefore, the accuracy of the methods used for this purpose is of importance to food science and engineering professionals working in this field (Holdsworth and Simpson, 2007). Considerable work has been reported in the literature, showing that variable retort temperatures (VRT) can be used to marginally improve the quality of canned food and significantly reduce processing times in comparison to traditional constant retort temperature (CRT) processing (Teixeria et al., 1975, Banga et al., 1991, Banga et al., 2003, Chen and Ramaswamy, 2002).

Optimization of thermal sterilization is an optimal control problem, where it is necessary to search for the best retort temperature as a function of process time. Banga et al. (1991) showed that this optimal control problem could be transformed into a nonlinear programming (NLP) problem, and in most cases the NLP problem became a multi-modal optimization problem with several types of constraints. These types of optimization problems make use of classical deterministic optimization methods within a local search domain, such as, Hooke-Jeeves, Nelder-Mead, and Quasi-Newton (Himmelblau, 1972); and are frequently limited in their effectiveness. In order to ensure a global solution of these problems, it would be more suitable to use global optimization methods.

This research suggests a modification of the adaptive random search method based on the utilization of the well-known logistic function or logistic curve in order to improve the random search adaptation characteristics. An interesting and important food industry optimization problem, such as thermal processing, was solved by the new organization of adaptive random search.