A Fuzzy Multi Objective Approach to Waste Management Department of Applied Mathematics and Computational Sc. Shri G.S.Institute of Technology & Science,Indore,Madhya Pradesh,India Abstract: Nowadays, waste management is a problem of major relevance for all societies. Finding acceptable strategies to cope with such a problem is becoming a quite hard task, owing to the increasing awareness of environmental issues by population and authorities. In general, this awareness has led to the development of enhanced pollution, control technologies and to a more rigorous legislation on waste handling and disposal, to minimize the related environmental impact. Waste management is a problem that is even more felt at the municipal level, where decision makers should plan an effective strategy, taking simultaneously into account conflicting objectives. In addition, the problem is characterized by an intrinsic uncertainty of the estimates of costs and environmental impacts. These reasons have led several authors to propose multi-criteria decision approaches. In this paper, Waste management system, including one treatment plant of organic material and other maintenance & running cost, has been considered. A multi objective approach to support decision makers in the planning of their waste management system is described. Key words: Fuzzy set theory, optimization problems, waste treatment plant, maintenance & running cost, multi objective decision model, decision making. INTRODUCTION Classical decision making generally deals with a set of alternatives comprising the decision space, a set of states of nature comprising the state space, a relation indicating the state or outcome to be expected from each alternative action and finally a utility or objective function, which orders these outcomes according to their desirability. A decision is said to be made conditions of certainty when the outcome for each action can be determine and ordered precisely. In this case, the alternatives that leads to the outcome yielding the highest utility is chosen. A decision is made under conditions of risk, on the other hand, when the only available knowledge concerning the outcome states is their probability distributions. Again this information can be used to optimize the utility function. When knowledge of the probabilities of the outcome states is unknown, decisions must be made under conditions of uncertainty. In this case, Fuzzy decision theories may be used to accommodate this vagueness. Applications of fuzzy sets within the field of decision making have, for the most part, consisted of extensions or fuzzifications of the classical theories of decision making. While decision making under conditions of risk and uncertainty have been modeled by probabilistic decision theories and by game theories, fuzzy decision theories attempt to deal with the vagueness of fuzziness inherent in subjective are imprecise determinations of preferences, constraints, and goals. In this paper, we briefly introduce some of the simple applications of fuzzy sets in the area of selecting a waste treatment plant having the maximum reliability, minimum cost and minimum maintenance and operating cost. FUZZY SETS AND PROBABILITY The characteristic function of a crisp set assigns a value of either or 0 to each individual in the universal set thereby discriminating between members and non-members of the crisp set under consideration. However, the function can be generalized such that the value assigned to the elements of the universal set fall within a specified range and indicate the membership grade of these elements in the set in question. Larger values denote higher degrees of set membership. Such a function is called a membership function and the set defined by it a fuzzy set. 40
Let X denote a universal set. Then, the membership function µ A by which a fuzzy set A is usually defined has the form µ A : X 0,, Where[0,] denotes the interval of real numbers from 0 to, inclusive. For example, we can define a possible membership function for the fuzzy set of real numbers close to 0 as follows µ A x = + 20x 2 The graph of this function is shown in Figure. µ A x 2 2 x Figure : A possible membership function of the fuzzy set of real numbers. Using this function, we can determine the membership grade of each real number in this fuzzy sets, which signifies the degree to which that number is close to 0. A fuzzy set approach, pioneered by [7], is useful for uncertainty analysis where a probabilistic database is not available and/or when (interval) values of input variables are uncertain. The fuzzy sets and fuzzy logic theory can also take into account the linguistic terms generally used to describe uncertainty in waste management. There have been limited attempts to exploit fuzzy logic within the risk management domain. Fuzzy set theory was developed exactly based on the premise that the key elements in human thinking are not numbers, but linguistic terms or labels of fuzzy sets. Decision-making [2][5] methods using fuzzy sets theory have gradually gained acceptance because of their capabilities in handling the impreciseness that is common in system specifications, states as alternatives ratings. Historically, the mathematical theory of probability has been the most widely used uncertainty-reasoning tool. In addition to probability theory, these are other numerical calculi for the explicit representation of uncertainty or techniques to manage uncertainty using logic or other symbolic formalism [6]. Most of these uncertainty reasoning techniques are data intensive. They required significant data collection. Formulation of mathematical representation assessment of conditional probability, or definition of probability density functions. However, uncertainties involved in waste management projects are difficult to assess using 4
traditional tools such as probability theory. Probability theory cannot deal with important aspect of uncertainty and cannot explain some important aspect of observed waste management practice. Fuzzy sets are often incorrectly assume to indicate some form of probability despite the fact that they can take on similar value, it is important to realize that membership grades are not probabilities. One immediately apparent difference is that the summation of probabilities on a finite universal set must equal, while there is no such requirement for membership grades. SIMPLE FUZZY MODEL OF DECISIONS A fuzzy model of decisions is presented that accommodates certain constraints C and goals G. Here both constraints and goals are treated as fuzzy sets characterized by membership functions [4][3]. µ C : X [0,] and µ G : X [0,], where X is the universal set of alternative actions. The expected outcomes that result from these actions in many cases can be assumed to remain deterministic or probabilistic, thus restricting the introduction of fuzziness only to the goals and constraints themselves. The fuzziness which may more accurately reflect the actual state of knowledge or preference are normally vague and are represented in linguistic terms for which the decision maker should frame the goals and constraints accordingly. The membership function of the fuzzy goal in this case serves much the same purpose as a utility or objective function that orders the outcomes according to preferability. Unlike the classical theory of decision making under constraints, however, in which constraints are defined on the set X of alternatives and goals are modeled as performance functions from X to another space, the symmetry between the goals and constraints under this fuzzy model allows them to be treated in exactly the same manner. This model can be extended to allow goals and constraints to be defined on different universal sets, for instance, the set X of possible actions and the set Y of possible effects or outcomes. In this case, the fuzzy constraints may be defined on the set X and the fuzzy goals on the set Y such that µ C : X [0,] and µ G : Y [0,]. A function f can then be defined as a mapping from the set of actions X to the set of outcomes Y, f : X Y, such that a fuzzy goal G defined on the set Y induces a corresponding fuzzy goal G on set X. Thus, µ G (x): µ G (f(x)). A fuzzy decision D may then be defined as the choice that satisfies both the goals G and the constraints C. If we interpret this as a logical and, we can model it with the intersection of the fuzzy sets G and C, D = G C, Which can easily be extended for any number of goals and constraints. If the classical fuzzy set intersection is used, the fuzzy decision D is then specified by the membership function where x ε X. µ D (x) = min[µ G (x), µ C (x)]. 42
Membership Value Membership Value International Journal of Engineering Technology, Management and Applied Sciences APPLICATION The fuzzy model of decision making proposed by [] is illustrated by a simple example. Suppose there are many alternatives and the problem is to choose the best alternatives. For example there are four different possible waste treatment plants a, b, c and d, the initial cost of which are given by the function f such that f(a) = 6,00,000 f(b) = 4,50,000 f(c) = 3,00,000 f(d) = 2,00,000 Our goal is to choose the treatment plant that will give us a least initial cost given the constraints that the plant is reliable and having reasonable maintenance and running cost. This first constraint of reliability can be represented by the fuzzy set C defined on our universal set of alternative treatment plants as follows: C =.8/a +.6/b + /c +.4/d. The membership function for the constraint C is shown in Figure 2..8.6.4 a b c d Treatment Plant Figure 2: A membership function for the constraint C Our second constraint concerning the maintenance and running cost to each treatment plant is defined by the fuzzy set C 2 such that C 2 =.9/a +.6/b + /c +/d. The membership function for the constraint C 2 is shown in Figure 3..9.6 a b c d Maintenance and running cost Figure 3: A membership function for the constraint C 2 43
Membership Value International Journal of Engineering Technology, Management and Applied Sciences The fuzzy goal G o initial cost is defined on the universal set X of initial costs by the membership function 0 for x > 7,80,000 µ G (x) = -.00025 ( x /000-78 ) 2 + for,80,000 x 7,80,000 for x <,80,000, where x ε X. The membership function of the fuzzy set G is shown in Figure 4. -.00025(x /000-78 ) 2 +.999.964.87 59 80000 200000 300000 450000 600000 780000 Membership function for initial cost Figure 4: A membership function of the fuzzy set G The corresponding goal G induced on the set of alternative treatment plants by the function f is given by G = 59/a +.87/b +.964/c +.999/d Taking the standard fuzzy set intersection of these three fuzzy sets, we obtain the fuzzy decision D, where D = G C C 2 = 59/a +.6/b + /c +.4/d. From the above we take the minimum of above set and alternative d as the choice that satisfy our goal and constraints. CONCLUSION The field of management, including the areas of decision making, operations research and risk analysis has been an active area for application of fuzzy set theory and the theory of evidence. Within the field of management, application of fuzzy set theory have been made for inventory scheduling, personal management, new product development, investment, public policy, planning, management decision support expert systems and system management. The proposed comprehensive DSS model allows municipal decision makers to plan the treatment plants that must be used in management system which optimise the cost. A management system is formalized in a constrained non-linear optimization problem, where decision variables are both integer and continuous. As in many environmental related problems, the decision problem is multi objective. Then, a suitable technique can be applied interactively with the decision maker to obtain a solution which represents a compromise acceptable to decision maker. 44
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