A.V. Lotov, V.A. Bushenkov, and G.K. Kamenev

Feasible Goals Method.
Mathematical Foundations and Environmental Applications

The Edwin Mellen Press
Lewiston, NY USA, 1999
Synopsis
The book is devoted to the new graphic computer-based approach, the Feasible Goals Method (FGM), which helps to select preferred decisions from an infinite number of feasible decision alternatives. The main feature of the FGM is provided by the display of the variety of goals, which are feasible, i.e. such goals that may be a result of feasible decisions. The FGM develops the ideas of the goal method and of the multiple criteria decision making. 
Applying usual goal programming, one often meets a difficulty related to the fact that an identified goal (i.e. a combination of decision-related criterion values) is not feasible. Indeed, if the identified goal is distant from the variety of feasible goals, a feasible combination of criterion values computed for decision maker may disappoint him since he haven’t had any idea about how distant is the feasible goal from the identified goal. Moreover the feasible combination of criterion values provided to decision maker may depend more upon the technical parameters than on the identified goal and disregard preferences of decision maker. In contrast to the usual goal method, the variety of the feasible goals is displayed in the framework of the FGM. This feature solves the above problem. 
The FGM is based on the mathematical theory of approximation of multi-dimensional bodies by simple figures (polytopes or systems of balls). Exposition of feasible goals is based on a fast interactive or animated display of slices of the approximations in the form of decision maps, which are collections of efficiency frontiers among two criteria related to restrictions imposed on the values of other criteria. Since a decision map looks like the height contours of an usual topographical map, one can understand it fairly easily. By this, anyone may receive information about the potentialities of choice and tradeoffs among the criteria in a simple graphic form. After exploration of the variety of feasible goals, decision maker is can identify a preferred feasible goal by a simple click of a computer mouse. The strategy is computed automatically on the basis of the goal.
Applications of the FGM described in the book are mostly related to environmental decision problems. Reader can see how easily the FGM can be applied for search for smart strategies in environmental applications like the hypothetical problem of sea dumping of sluges in the Lower bight of the New York City, the problem of agricultural development of a small region in the Netherlands, in the decision and negotiation support system for developing of water quality plans in Russian rivers, in the negotiation support related to international strategies for atmosphere pollution abatement, in the demo version of computer support for developing smart response strategies to global climate change. Several economic applications are considered as well including development of strategies for long-time growth of a national economy taking environmental issues into account, development of feasible national economic-social goals, development of international trade strategies, development of economic reform strategies in Russia, etc. 
Current and future applications of the FGM via computer networks are considered. Since the FGM is implemented in the form of a colorful graphic computer software and can be easily assessed by any computer-literate person, it can be used as a basis for a new type of INTERNET resources, which provide objective information concerning feasible strategies that may solve important public problems such as environmental decision problems. In the framework of these resources, users can develop their own independent strategies. The FGM can be used in negotiation support as well as for group and collaborative decision making.
Mathematical foundations of the feasible goals method are related to numerical approximation of the set (variety) of the feasible goals. Three types of algorithms are considered in the book: 
•	algorithms based on the convolution of linear inequalities due to Fourier,
•	algorithms based on the approximation of convex bodies by polytopes,
•	algorithms based on the essential covering of non-convex sets by systems of simple figures. 
Moreover, algorithms for approximation of the Edgeworth-Pareto Hull of the set of the feasible goals are considered. Algorithms that display decision maps and algorithms that approximate reachable sets for dynamic systems are described, too. Two chapters are devoted to the theoretical aspects of the approximation of the sets of the feasible goals in abstract and functional spaces as well as to the disturbances of such sets. The last chapter of the book is devoted to the theory of convergence of methods for approximation of convex bodies with smooth and non-smooth frontiers.
A large part of the book (except three last chapters) is written in a simple form, and so most of the book can be assessed by any computer-literate person. The book can be used by graduate and postgraduate university students interested in the recent trends in the field of decision and negotiation support, in the application of computer networks and in modern decision methods in environmental issues. Mathematical part of the book may be of interest for specialists in applied mathematics. 
THE CONTENTS
INTRODUCTION. The role of the FGM technique in the framework of the search via computer networks for smart solutions of environmental problems is outlined.

CHAPTER 1. INTRODUCTION INTO THE FEASIBLE GOALS METHOD. The method is introduced on the basis of several simple models. Then, an example of the FGM-based search for a environmentally sound strategy of regional economic development is provided. The problem is related to non-source agricultural pollution.

CHAPTER 2. SEARCH FOR BALANCED ENVIRONMENTAL STRATEGIES. Examples of the application of the FGM technique in the framework of the search for smart environmental strategies are given. The following problems and software systems are discussed:
•	hypothetical problem of sea dumping of sluges in the Lower bight of the New York City,
•	problem of agricultural development of a small agricultural region in the Netherlands taking the water quality and the water quantity problems into account,
•	decision support system used by environmental engineers for the development of water quality plans for Russian rivers,
•	negotiation support system for the search for international strategies of atmosphere pollution abatement (exemplified by the case of Finland, Russia and Estonia),
•	demo version of computer support for the search for smart response strategies to global climate change (exemplified by the greenhouse effect).

CHAPTER 3. SEARCH FOR EFFICIENT STRATEGIES IN ECONOMIC TOPICS. Application of the FGM technique for exploration of following economic models is described in short: 
•	search for strategies for long-time growth of a national economy taking social and environmental issues into account (on the basis of an aggregated model of economic growth),
•	exploration of feasible national social-economic goals and a search for national economic strategies for medium time period (on the basis of a dynamic input-output model),
•	search for strategies of international trade (on the basis of a static input-output model of the world trade), 
•	search for strategies of short-time regulation of a market economy (on the basis of an econometric model of Finland),
•	search for smart strategies of economic reform in Russia (on the basis of a structured macro-economic model).

CHAPTER 4. METHODOLOGICAL PROBLEMS OF THE FGM. The chapter consists of eight sections, in which the following topics are discussed:
•	Interactive and Animated Decision Maps as well as their combination with the multimedia, virtual reality, geographical information systems and other new computer technologies,
•	psychology of decision making, mental models and their modification using the FGM technique,
•	application of the FGM technique in the framework of the search for concepts of technical systems: supersonic aircraft example,
•	FGM-based methodology for integrated assessment of environmental problems,
•	identification of mathematical models with the FGM technique,
•	application of the FGM technique in negotiation processes,
•	application of the FGM technique in computer networks,
•	negotiation support system used in the framework of the Russian federal program “Revival of the Volga River”.

CHAPTER 5. MATHEMATICS OF THE FGM TECHNIQUE. Mathematical issues related to the computer implementation of the FGM technique are discussed in short. This section provides software engineers with an understanding of the opportunities and limitations of the FGM technique. The FGM technique is based on algorithms for numerical approximation the set of feasible goals and on algorithms for displaying it in the form of interactive and animated decision maps. Algorithms are described in a simplified form. The chapter discusses the following topics:
•	algorithms based on the convolution of liner inequalities due to Fourier,
•	algorithms based on the approximating of convex bodies by polytopes,
•	algorithms based on the essential covering of non-convex sets by systems of simple figures,
•	algorithms for approximating the Edgeworth-Pareto Hulls,
•	algorithms for displaying the decision maps,
•	algorithms of the FGM technique in the case of dynamic systems,
•	algorithms of the FGM technique in the case of systems in partial derivatives.

CHAPTER 6. ON STABILITY AND APPROXIMATION OF THE SETS OF FEASIBLE GOALS IN CONVEX PROBLEMS. Theory of approximation of the sets of feasible goals in the problems with convex feasible sets and linear criteria is developed. The theory is based on the abstract theory of approximation of images of linear mappings in Banach spaces. The main results of the chapter include:
•	the theory, which relates stability and approximation;
•	the theory of stability and perturbations of linear systems in Banach spaces;
•	application of both theories to the differential inclusions.

CHAPTER 7. ON PERTURBATIONS OF THE SETS OF FEASIBLE GOALS IN LINEAR FINITE-DIMENSIONAL CASES. Theory of stability and perturbations of linear systems developed in chapter 6 is applied for the estimation of perturbations of the sets of feasible goals for linear finite-dimensional systems, including controlled dynamic multi-step systems.

CHAPTER 8. Efficient Algorithms FOR APPROXIMATION OF CONVEX bodies by polytopes. Theory of convergence of algorithms for approximation of convex sets of feasible goals is developed. It is shown that the methods developed by the authors have optimal speed of convergence for multi-dimensional bodies with both smooth and non-smooth frontiers.

CONCLUSION. THE FGM TECHNIQUE AT THE BEGINNING OF THE 21st CENTURY. The FGM-based active tools of future electronic mass media are outlined, which are aimed at the independent development of public strategies.