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Dynamic and Uncertain Optimization Problems in Intelligent Systems

This page describes the project "Dynamic and Uncertain Optimization Problems in Intelligent Systems: A study of solutions and applications." (P07-TIC02970) from the Regional Government of Andalusia.

Introduction

The times are gone where solving problems with computational methods could relay only on a simple static model of the problem with a single objective and where all data was perfectly known. The kind of problems that appear in the current socio-technological context associated to intelligent systems presents two relevant characteristics: dynamism and uncertainty.

This new reality leads to a strong need for new methods and models that can cope with the dynamism and the uncertainty in all or some of the parts of the problems at hand. In this project we pretend to conduct our research on how the tools that Soft Computing provides can be the appropriate ones to obtain such methods and models. On one hand, the tools provided by fuzzy logic are specially fitted to deal with uncertainty. On the other hand, many concepts originated on the field of metaheuristics are perfect to deal with problems where time takes an important part.

The project is structured in relation with the potential scenarios that can appear when we combine the different components of the problems (objective function and restrictions) and the presence or absence of the dynamism and uncertainty characteristics in any of these components or both. Then, three main axis are defined to cover the spectrum of different research options: the models, the methods and the applications that emerge on each scenario.

We pretend, in first term, to obtain an updated view of the scenarios in every proposed axis to end up with a map of the available models, methods and applications. On second term, we want to develop cooperative computational strategies to solve the optimization problems in as many scenarios as possible. Finally, our ultimate goal is to try to produce advances in every possible map and scenario.

Background

Here we will describe the more important contributions in the areas of dynamism and uncertainty to give a good picture of the backgrounds that supports this project, its opportunity and feasibility. We will start with a review of the current state of the art on both subjects and then we will show our contributions to each of them.

The state of the art

Fuzzy Optimization

The concept of fuzzy set was introduced by Zadeh in 1965 to allow elements to belong to a set in a gradual rather than an abrupt way (i.e. permitting memberships valued in the interval [0,1] instead of in the set {0,1}). Ever since then, applications and developments based on this simple concept have evolved to such an extent that it is practically impossible nowadays to encounter any area or problem where applications, developments, products, etc. are not based on fuzzy sets. Today they constitute one of the fundamental parts of the new area called Soft Computing, that has became the basis of the modern Intelligent Systems.

In the Intelligent Systems context, an important type of problem in particular are optimization problems, which optimize the maximum or minimum value that a function may reach on a previously specified set, and these and everything relating to them are covered by the area known as Mathematical Programming, which includes an enormous variety of situations including the linear and non-linear cases, randomness, one or several decisors and so on.

The best studied model among all the ones included in Mathematical Programming and the one with more important practical repercussions is the single objective linear model and this is the field tackled by Linear Programming. The methods and models of Linear Programming have relevant applications on the different areas of Engineering, Business, Math, Intelligent Systems and other areas related in one or other way with optimization that form a theoretical framework very well suited to solve quite complex situations in an efficient and elegant way.

Despite that, as we have just said, the models and methods of Linear Programming are the most and better studied ones, or even because of that together with their known accuracy and efficiency, they can be easily adapted to new technological contexts. This also influences that they are taking a key role in the most recent scientific developments as is the case of their incorporation and implementation inside the generators of Decision Support Systems (DSSs).

In this manner these models are at the heart of one of the most promising research lines in the scope of Intelligent Systems. Consequently, despite its more than fifty years of life, they are still part of the state of the art on the field.

Usually, the description of a real problem is done in terms that, while being perfectly understandable, are difficult to be represented in an effective way: "the transport costs will be around 7€", "the profit is more or less a 30%",... When we have to deal with data of this nature, that obviously do not need to be probabilistic, the usual approach is to take the values that we think that are more representative of the true ones, 7 and 30% for instance. This leads to distorted problems where we can end up having solutions that despite being optimal for the problem used can be quite far of the true solution that the original problem could have if we had used the real appropriate values that could have been 7.15 and 29.8%.

Because of this problem, the selection of an adecuate representation of the information is a key task, that can serve to obtain a better guarantee of the correctness of the solutions being searched. It is also important because, depending on the imprecission modelling we do, we could have different concepts of optimum and, therefore, different concepts of optimization.

From this point on, we will be referring with imprecission to what is usually known as fuzziness, that is, a kind of linguistic uncertainty that has a perfect sense for the human beings despite the lack of exact information it provides ("i don't know how old is he, but he is young") and that can be modelled with the fuzzy set theory.

Generally, an optimization problem is formulated as $ Max \{z=f(x)/ g_i(x)) \leq 0, i\in M, x\geq 0\} $ where $ f $ y $ g_i $, $ i\in M $ are real functions that can have different properties. Depending on these properties a large spectrum of different problems can be obtained.

As it is clear, this formulation assumes that the decision maker has an exact information about the elements that take part in the problem. But even if this were true, it is usual that the decision makers feel more comfortable expressing their knowledge of the problem in linguistic terms, that is, using conventional linguistic labels rather than precise numerical data.

Therefore, it makes perfect sense to talk about optimization problems that are formulated using this kind of fuzzy statements. This fuzziness is produced by the way that the knowledge of the decision maker is expressed and not as a cause of a random nature that will not be considered in any manner. In a word, we are going to assume that the imprecision of the data that defines the problem is of fuzzy nature.

The first optimization problem with a fuzzy formulation that exists on the literature comes from almost four decades ago [1]. In this paper, the key concepts of fuzzy restrictions, goals and optimal decisions that now are already classic. Despite there exists a number of methods to the resolution of this problems [2][3][4][5][6], today three are considered to be the most important types, depending if the uncertainty is placed in the restrictions, in the coefficients of the functions or in both. Moreover, and independently of the practical precise applications, the case where $ f $ and $ g_i $, $ i\in M $ are linear functions stands over [4][7][8][9].

More precisely, when we consider a model with fuzzy restrictions, that is, when the decision makers assume that there can be some tolerance in the satisfaction of the restrictions defined as a margin of violation that they establish, the associated problem can be expressed as:

x \geq 0 \end{array} \] (1)

Each restriction $ g_i(x)\leq _f b_i,~i\in M=\{ 1,2, \ldots , m\} $ can be modelled by means of the membership function

\Re \rightarrow [0,1]~~ / ~~ \mu _i(g_i(x))= \left\{ \begin{array}{cl} 1 & \hbox{if } g_i(x) \leq b_i \\ h_i(g_i(x)) & \hbox{if } b_i \leq g_i(x) \leq b_i + t_i \\ 0 & \hbox{if } g_i(x)\geq b_i + t_i \\ \end{array} \right. \] (2)

These functions express that the decision makers tolerate violations on each restriction up to a $ b_i+t_i $, $ i\in M $ value. On the other hand, the $ h_i $ functions are assumed to be continuous and non-decreasing for these restrictions. The $ \mu_i $ function is defined for every $ x \in X $ and it gives the degree of satisfaction of the ith restriction for $ x \in X $.

Despite the origin of (1) can be found on [1], the problem was tackled on [10] and [11] where additional hipotheses on the nature of the objective function were considered that are not relevant here. To solve (1) three different approximations can be considered [10], [11] and [12]. Concretely, making use of the fuzzy set representation theorem, a demonstration on how to obtain a fuzzy solution for (1) was presented on [12]. It was also shown that the solutions proposed on [10] and [11] are particular cases of the proposed fuzzy solution. Later on, it was demonstrated [13] that the solution method presented on [12] was indepent of the problem being linear or not.

If we instead consider the coefficients of the objetive function (the costs) to be fuzzy, what we are assuming is that the parameters can be considered as defined by fuzzy number, that is, they are characterized by membership functions of the form

\Re\rightarrow [0,1],~ j\in N ~~ / ~~ \mu_j(v)=\left\{\begin{array}{ll}0 & \hbox{if }v\leq r_j \hbox{ or; }v\geq R_j \\ p_j(v) & \hbox{if }r_j\leq v\leq \underline{c}_j\\ q_j(v) & \hbox{if }\overline{c}_j\leq v\leq R_j\\ 1 & \hbox{if }\underline{c}_j\leq v\leq \overline{c}_j \end{array} \right \] (3)

where $ p_j(\cdot ) $ and $ q_j(\cdot ) $ are strictly increasing and decreasing continuous functions respectively, such that $ p_j(\underline{c}_j)=q_j(\overline{c}_j)=1 $, $ \forall j\in N $.

Although there exists a range of $ p_j $ and $ q_j $ functions (linear, exponential, logarithmic, convex and concave parabolic, ...), fuzzy numbers are usually considered as flat fuzzy numbers with linear $ p_j(\cdot ) $ and $ q_j(\cdot ) $ functions.

To solve these models there are different approximations [3][14][15]. Particularly, on [7] it is demonstrated that the proposed method of [3] gives a formal context to find a problem for them that includes the solutions of the methods proposed on [14][15][5].

On other hand, the case where the fuzziness is present at the coefficients of the functions that define the restrictions has only been solved when the problem is considered to be of a totally linear nature, having as formulation

\\ & a_{i}^fx\leq _f b_i^f,~i\in M \\ & x\geq 0\end{array} \] (4)

where for each $ i\in M $, $ a^f_i=(a^f_{i1},\ldots , a^f_{in}) $, $ a^f_{ij}\in F(\Re ),~j\in N $, $ b^f_i\in F(\Re ) $, and $ c\in\Re ^n $.

To solve (4) with the goal of obtaining a fuzzy solution and not a precise one, we can assume some violations are accepted on their restrictions up to a maximum wide of value $ t_i^f $, $ i\in M $ [4]. Notice that, on the contrary to the 1 case, $ t_i^f $ has to be a fuzzy number due to the nature of the coefficients that thake part on each restriction. From this poing of view, on [4], an operative method of resolution to the general model is proposed, and on [2], a solution to the originally formulate problem is obtained depending on the concept of comparison of fuzzy numbers used.

The third important group of problems appears when we consider that the uncertainty is present on both the objective and the set of restrictions. This interesting model have only been solved on the linear case, that is, when the formulation is

& \displaystyle \sum ^n_{j=1}a_{ij}^fx_j\leq _f b_i^f \\ & x_j\geq 0,~i\in M,~j\in N\end{array} \] (5)

where all the fuzzy elements that take part have a clear meaning. In [16] a method to solve this general model is presented.

Despite that the theoretical basis of the resolution methods for fuzzy optimization problems was developed between 1975 and 1995, the interest on the subject has returned on the last years with the development of a wide variety of applications that are aimed at solving real world problems and that can be included on the context that Soft Computing provides.

To serve as an example, at the end of this section there is a subsection that includes references to relevant applications and new theoretical developments on the last five years.

Dynamic Optimization

On the previous subsection we have done a summary of how it is possible to properly model some kind of uncertainty and which are the strategies to solve these models.

In this subsection we will focus on the so called Dynamic Optimization Problems (DOPs). Under this name we include the problems where the objective function, the variables, the environmental conditions and/or the structure of the problem can change while the problem is being solved. The basic formulation of a problem of this kind is:

\[ DOP = \left\{ \begin{array}{l} Optimize \hspace{3 mm} f(x,t) \\ s.t. \hspace{3 mm} x \in F(t) \subseteq S, \hspace{1 mm} t \in T \\\end{array}\right \} \]

where

  • $ S \in \Re^n $, $ S $ is the search space.
  • t is the time.
  • S \times T \to R $ is the objective function that assigns a numeric value for each possible solution ($ x \in S $) at the time t.
  • $ F(t) $ is the set of feasible solutions $ x \in F(t) \subseteq S $ at the time instant $ t $.

Therefore, the goal (in contrast with the static problems situation) is not finding the optimum for every different $ t $ value but to be able to "follow" this optimum as close as possible across the search space as time passes. And, summarizing, it is a scenario where one of the Soft Computing maxims perfectly applies: it is better to satisfy than to optimize.

The easiest natural way for solving a dynamic problem is assuming that each change gives birth to a new optimization problem that can be solved "from zero". This approach is only feasible if the time between two consecutive changes is large enough to allow for a full reoptimization.

A more intelligent alternative consists on reusing the knowledge obtained in the previous search to solve the new problem more quickly. This approach has sense if we assume that the changes are gradual and not abrupt. If changes were abrupt, then to use the previous information can be more perjuditial than benefitial because it will lead the search to non-promising regions of the search space.

Fortunately, the great majority of real problems vary on a gradual manner, and the correct use of the generated knowledge to solve the problem can lead to high quality solutions on less time than the time we would need without that additional information.

Together with assuming gradual changes and the usefulness of collecting data from the search and solutions previously obtained, the strategies applied to these kind of problems have also to include enough flexibility to react to changes in a proper way.

The interest on solving dynamic problems with the aforementioned characteristics is relatively new, and there are practically no references prior to 1990. This field of research has not been really established until the last decade when we can find solid literature on this subject and specially in the scope of evolutionary algorithms.

The level of activity can be verified with publications such as [17] and [18]. There have also been special numbers of relevant journal like [19] and specific workshops on the subject are done on conferences like GECCO, EVO-*, CEC, etc.

If we associate flexibility with adaption capability we can easily reach the following conclusion: the natural evolution is a mechanism that has served for the adaptation to an environment in constant change; therefore, a computational model that mimics natural evolution has to be appropriate for dynamic problems.

On the other side, it is reasonable to assume that it is easier to react to a change if we use a set of solutions instead of just one. These are the reasons why evolutionary algorithms have been (and are) the most used tools to the resolution of dynamic problems.

The main problem in the application of evolutionary or (more generally) population-based algorithms is the convergence, because once the population has lost diversity the adaption capability almost dissapears. It is because of this that there is a need to establish additional mechanisms to preserve or generate diversity. These mechanism can be grouped in 4 categories:

  1. Generate diversity after a change: when a change is detected the solutions on the population are mutated to ease the transition to a new optimum. As it is normal, there exists a balance problem with the quantity of mutation to apply. If the mutation is too much, the scheme obtained is very similar to a complete restart while if the mutation is too low, the convergence problem will not be solved.
  2. Maintain the diverity: The idea is to avoid complete convergence at every moment. It is possible to use mechanisms like adding random solutions [20], "sharing" [21], or other more complex ones like the mechanism proposed on [22] where the notion of "free energy" is used to measure the diversity.
  3. Memory-based schemes: the information that is generated during the execution is stored in a memory that can be explicit or implicit. In the first case we define strategies to store and retrieve information from the memory [23][22]. For instance, we could store solutions, cost history, etc. When we use implicit memory, we add additional information to the representation, and this redundance is also subject to the selection processes [24]. It is important to emphasize that there exists a close relation between memory and diversity and therefore, if memory is used it is also necesary to have mechanisms to maintain diversity.
  4. Multiple populations: several populations are used simultaneously, that explore different regions of the search space, potentially enabling for a better tracking of the optima of the problem to be solved. There also exists some interpretations that consider this approach as a distributed memory mechanism. Some works on this subject are [25][26][27].

On the last five years we can find other metaheuristics applied to this kind of problems such as ant colony optimization, particle swarm optimization with one or multiple swarms or even methods based on the inmune system. The review on [28] includes information about the specific works. A great number of the cited works make use of a set of test problems that is being assumed as "standard". Between them , the most important ones are Moving Peaks Benchmark [23], dynamic knapsack [22], dynamic bit-matching [29] and, more recently, there have been some relations between multi-objective and dynamic problems [30].

Sadly, it is still very difficult to replicate the published results, moreover when the metrics used to measure the performance of an algorithm are still in full development.

Previous works and results of our research team

Our research team has broad experience both in dealing with uncertainty and on the kind of metaheuristics that can be applied to solve dynamic problems. Moreover, we have started a new research line that is devoted to the combination of uncertainty + dynamism + heuristics that is starting to have fruits in terms of publications.

Now we will describe our own projects and publications that are strictly related with this project and that are serving as a good grounding for the successful accomplishment of our current research. More details of the research done by our group can be found at the the scientific history of the Models of Decision and Optimization (MODO) Research Group site.

Projects

  • Heurísticas Fuzzy: Aplicación de Metodologías Fuzzy en el Desarrollo de Sistemas de Ayuda a la Decisión basados en Metaheurísticas (TIC2002-04242-C03).
  • Nature Inspired Smart Information Systems and Task Force on Nature Inspired Cooperative Strategies for Optimization (FP6-2002-IST-1/13569).
  • Desarrollo de Heurísticas Cooperativas basadas en Soft Computing: Aplicaciones en Bioinformática (TIN2005-08404-C04-01).
  • RESCATE: Coordinación y Articulación de I+D+I Basada en Soft Computing (TIN2005-24790-E).
  • Modelos Inspirados en la Naturaleza: Aplicaciones en Soft Computing (TIC-00129-PE).
  • Tratamiento de los Términos Lingüísticos en Escalamiento y Soft Computing (TIN2005-02418).

Recent publications

  • J. Cadenas, D. Pelta, H. Pelta, J.L. Verdegay (2004). Application of Fuzzy Optimization to Diet Problems in Argentinean Farms, European Journal of Operational Research, Vol 158 (1), 218-228.
  • D. Pelta, N. Krasnogor, C. Bousono-Calzon, J. L. Verdegay, J. Hirst, E. Burke (2005). A fuzzy sets based generalization of contact maps for the overlap of protein structures, Fuzzy Sets and Systems, Vol 152, Nro 1, 103-123.
  • D. Pelta, A. Sancho-Royo, C. Cruz, J.L. Verdegay (2006). Using memory and fuzzy rules in a cooperative multi-thread strategy for optimization},  Information Sciences, Volume 176, Issue 13, 1849-1868.
  • F. Jiménez, J.M. Cadenas, G. Sánchez, A.F. Gómez-Skarmeta, J.L. Verdegay (2006). Multi-objective evolutionary
    computation and fuzzy optimization    . International Journal of Approximate Reasoning 43, 1, pp 59-75.
  • C. Cruz, D. Pelta, J.L. Verdegay (2007). Una Estrategia Cooperativa para la Resolución de Problemas Dinámicos de Optimización, En Actas del V Congreso sobre Metaheurísticas, Algoritmos Evolutivos, y Bioinspirados, 239 - 245.
  • C. Cruz, D. Pelta, F. Bonachela, A. Sancho Royo (2007). Analysis and Comparison of Two Nature-Inspired Cooperative Strategies for Dynamic Optimization Problems,  International Journal of Intelligent Systems (accepted).
  • B. Melián, J. Verdegay (2007), Fuzzy Optimization Models in the Design of WDM Networks. IEEE Transactions on Fuzzy Systems (accepted).
  • F. Hernandes, M.T. Lamata, J.L. Verdegay, A.Yamakami. The shortest path problem on networks with fuzzy parameters. Fuzzy Sets and Systems (in press).
  • D. Pelta, C. Cruz, A. Sancho-Royo (2007). Probabilistic vs. Fuzzy Rules in a Cooperative Strategy for
    Dynamic Optimization Problems    . Applied Soft Computing (submitted).

Goals

A great variety of optimization problems, with a high dimensionality, a single objective and where all the elements (variables, objective function, etc.) are well defined, can be solved reasonably well and with an acceptable computational effort, by using metaheuristics or problem-specific techniques for any of them. As an example we can cite the Travelling Salesman Problem that is at the core of many problems such as route design, sequencing, ... where the advances on the computational and algorithmic techniques allows to solve practically any problem instance.

But, on the contrary, it is assumed that the big challenge that we will face on the scope of intelligent systems in the next years will be associated to the set of problems that have several objectives that also can evolve through time and where there can exist non probabilistic uncertainty on the values that the variables or coefficients take or even where the objective function is not perfectly known and may need to be determined from a simulation or from subjective evaluations coming from experts. It is clear that static or inflexible resolution systems, that are not able to adapt or detect the changes, are not appropriate to solve this kind of problems. Summarizing, we can state that the kind of problems that appear in the current socio-technological context presents two relevant characteristics: dynamism and uncertainty.

Therefore, it is essential to develop methods and models that can face the dynamism and uncertainty that can appear in all or some of the parts of the problem at hand. On this project, the initial hypothese that we anticipate is that the tools that Soft Computing provides are the appropriate ones to obtain such methods and models. The reasons are quite clear: on one hand, the tools provided by fuzzy logic are specially fitten to deal with uncertainty, for example, by giving birth to models that are better fitted to the problem being solved. On the other hand, many concepts originated on the field of metaheuristics are perfect to deal with problems where time takes an important part. As an example we can mention the cooperative stratategies for optimization, where a set of solutions can co-evolve to obtain solutions to the problems that change with time.

The core of this project is structured around the potential scenarios that can appear when combining the components of the problems of interest, where we basically refer to the objetive function and the restrictions, and the characteristics of dynamism and uncertainty. That's to say, we will start with a matrix as the one that follows:

Project scenariosProject scenarios

where each cell can be interpreted as an scenario associated to these kinds of problems, and where the complexity (displayed as a gray scale) increases from top to bottom and from left to right. In this way, the most complex scenario is given by problems where both the objective function and the restrictions have some kind of uncertainty and they also vary with time. It must be notice that this last scenario can be subdivided on three parts, because it would be possible to find problems with dynamic cost function and uncertain restrictions or the opposite option as well as having both the cost function and the restrictions uncertain and dynamic. Having said this, we will refer to the diagram of the figure for the sake of simplicity.

Then, for each possible scenario, we define a map with three research axis that are designed to cover the full spectrum of developments: the models, the methods and the applications that there appear. Graphically, the map will be the one given on the next figure:

MapsMaps

On this context, the global goal of this project will be the study of dynamic and uncertain optimization models on intelligent systems for the resolution of problems in areas such as Logistics (cut, location, ...), Economics (portfolio selection, ...) etc.

More precisely, the specific goals that we want to achieve are:

  • To gather together and organize all the available information and the types of uncertainty and dynamism that can appear.
  • To develop automatic problem generators that allow to control the level of uncertainty and dynamism on both the objective and any possible restrictions.
  • To propose a comparison methodology for algorithms in problems with dynamism and uncertainty, that includes the design of experiments and performance metrics.
  • Design, implement and evaluate strategies for solving problems with uncertainty and dynamism.
  • Apply the proposed strategies to solve instances of real problems from the intelligent systems area.

An additional goal is the preparation of PhD thesis, the participation on the best conferences and the publication of results in important journals.

References

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  2. [Campos1989] Linear programming problems and ranking of fuzzy numbers, Campos, L., and Verdegay J. L. , Fuzzy Sets Syst., Volume 32, Number 1, Amsterdam, The Netherlands, The Netherlands, p.1-11, (1989)
  3. [Delgado1987] Imprecise costs in mathematical programming problems, Delgado, M., Verdegay J. L., and Vila MA , Control and Cybernetics, Volume 16, Number 2, p.113-121, (1987)
  4. [Delgado1989] A general model for fuzzy linear programming, Delgado, M., Verdegay J. L., and Vila MA , Fuzzy Sets and Systems, Volume 29, Number 1, p.21-29, (1989)
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  9. [Zimmermann1987] Fuzzy sets, decision making, and expert systems, Zimmermann, H. J. , (1987)
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