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Fuzzy optimal search methods
Fuzzy Sets and Systems 46 (1992) 331-337 North-Holland
331
Fuzzy optimal search methods C a n I~ik Electrical and Computer Engineering Department, Syracuse University, Syracuse, NY 13244-1240, USA
Salwa A m m a r Business Administration Department, LeMoyne College, Syracuse, NY 13205, USA Received January 1990 Revised September 1990
Abstract: This paper gives analytical approaches and algorithms for decision tree search techniques. The decision trees are assumed to have a fixed number of stages and predefined possible states at every stage. The costs of traversing the tree are characterized as approximate and defined as fuzzy numbers. Two search methods, each drawing from an existing non-fuzzy search algorithm, are described. The first method is a dynamic programming search, where the principle of optimality with fuzzy costs is addressed. The second method is an A* search for which the notion of a lower bound estimate of costs is utilized to increase the efficiency of the search. Algorithms for each method are included. Theorems and proofs which complete the analytical development of these techniques are also included. Finally, a numerical example illustrating the procedure is given.
Keywords: Search methods; operations research; fuzzy dynamic programming; fuzzy A* search; fuzzy cost.
1. Introduction Search methods have been extensively studied to solve multi-stage (combinatorial) decision problems. Dynamic programming and the A* algorithm are two closely related search methods that are efficient and yield the optimum (minimum cost) solution if one exists. These two and other graph search methods are extensively discussed by Nilsson [1] in a non-fuzzy setting. The efficiency aspect of dynamic programming and the A* algorithm has attracted the attention of researchers working on multi-stage decision problems with fuzzy aspects. In one of the earliest contributions to the problem, Bellman
and Zadeh [2] dealt with multistage decision making in a fuzzy environment. Kacprzyk [3] also applied dynamic programming and branch and bound algorithms to systems in a fuzzy environment and extended the results for fuzzy termination time. Baldwin and Pilsworth [4] solved a multistage decision problem in a fuzzy environment with fuzzy system dynamics and a cost function that evaluates fuzzy states, control and goal constraints into a membership grade. The outcome is hence non-fuzzy. This approach is computationally complicated and is not well suited for practical applications. Esogbue et al. [5] investigated the multi-stage control of a stochastic system in a fuzzy environment using two different formulations when the system under control had a stochastic model. A major portion of a book by Kacprzyk is devoted to tree search algorithms where the system and/or the environment are fuzzy [6]. What is common in the above work is that although certain components of the problem are modeled using fuzzy set theory, the costs or the outcomes of the decisions are non-fuzzy. Yager [7] proposed the use of search trees in assessing the possibility of reaching the goal state in a production system where each rule is assigned a weight, and he later used an A* algorithm to find the path of least resistance in a similar production system [8]. The costs were composed using a min operation and a more general T-norm operation, and hence the approach differs greatly from what is given in the rest of this paper. Dubois et al. [9] concentrated on problems where costs are modeled as fuzzy intervals and utilized possibility theory in ranking alternatives. Optimality of the ranking was implied when the intervals did not overlap. An excellent treatment of fuzzy programming as well as an extensive survey of heuristic fuzzy tree search methods are given by Dubois and Prade [10]. In this paper we will deal with multi-stage decision problems where the cost associated with each decision or decision sequence is known only approximately and can be modeled as a fuzzy
0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved
332
C. lfik, S. Ammar / Fuzzy optimal search methods
number. In order to concentrate our efforts on this new aspect of fuzzy multi-stage decision making problem, we will assume that the cost of a decision is somehow determined and will not elaborate on that process. We will further assume that the future costs or outcomes are not discounted. A practical example of the problem described by these assumptions is mobile robot path planning. Usually, a path from an initial position to a goal position is planned in an abstacle-strewn environment, using data from maps and sensory feedback, which are of different scale, accuracy and reliability [11]. We first discuss some issues involved in extending the principle of optimality to problems with fuzzy costs. We also show that, when there is an admissible underestimate of the remaining cost, optimal A* search can be extended to problems with fuzzy costs. The ranking of alternatives with fuzzy costs is performed using the approach discussed in [12], which is based on the extended min operation. There are two basic reasons for this choice. First, the solution found in this manner includes an optimal solution in the crisp-set sense, if there is one. Since a major concern of the paper is to generalize the two aforementioned search methods by simply replacing the deterministic costs with fuzzy ones, it is important to have this property. Second, in solving problems such as robot path planning we associate inexactness with risk, and treat all possible alternatives with membership grades larger than a certain minimum level as of acceptable risk. In this regard, having as the 'best' decision a fuzzy set of decisions is quite convenient. Following this approach, all unexplored nodes which have a cost that overlaps with the fuzzy set 'minimum of costs at all nodes' are simultaneously expanded. Therefore the approach may not look very efficient. However, it lends itself to conveniently proving the theorems on the optimality of search methods with fuzzy costs. Nevertheless, the efficiency of the algorithms can be improved by heuristically pruning the search tree, at the expense of analytically proven optimality. If the computational complexity of search is a major concern, ranking with possibility theory [9, 10] may be used, since based on 4-step criteria only one node of the graph is expanded at a time.
2. Dynamic decision trees The aim of the search method described in this section is to determine the 'best' decision sequence for a multistage decision problem. At every stage the outcome of a given decision is described as a fuzzy number representing the approximate cost associated with that decision. The objective is to minimize the overall cost of all stages. The composition of the fuzzy costs over the successive stages is performed using the extended sum [13], due to the nature of intended applications. For example, distance, fuel, travel time are all additive costs. The extended operations of fuzzy arithmetic, in general, increase fuzziness. That is, their results are fuzzier than their arguments. This can be observed in Figures 2-5, which have sets with exaggerated fuzziness for the purpose of demonstration. In most problems, however, the 'magnitude' of costs increases along with their fuzziness; hence their relative uncertainty does not change much. Similarly, the sum of two random numbers is a random number whose probability density function is the convolution of the first two, with a variance larger than those of the original random numbers. Nevertheless, the 'increasing fuzziness' of the extended sum can be of concern to the reader, especially if costs are linguistic variables that assume values from finite universes. In such a case the fuzziness can be kept at a minimum if the result of a fuzzy arithmetic operation is mapped to the closest linguistic value. Similar techniques to reduce fuzziness are routinely used in pattern matching with a good degree of success (e.g., see Nafarieh and Keller [14]).
2.1. Definitions and notation Consider an m-stage decision tree. Let Di be the set of possible decisions at stage i = 1. . . . , m. Also let fl(d) be the fuzzy number denoting the cost of decision d e Di. Define di:J
=
(d i, d i+l . . . . .
d j)
to be the sequence of decisions from stages i to j and fl(d iq) to be the composite cost of stages i to j. Using iterative backward composition, fl(d i :m) m fl(d) ~ fl(d i+l :m)
333
C. l#ik, S. Ammar / Fuzzy optimal search methods
and
d i : m = ( d , d i÷l:m) f o r i = m t o l , where ~ denotes the sum of two fuzzy sets using the extension principle. Initialization is given by
fl(d,,,+l:m)
=
{1/0}
and
d m+~:m = 0,
introduced to allow for the definition of iterative algorithms.
2. 2. Determining the 'best' decision sequence The best decision sequence can be determined exhaustively by first evaluating fl(d ~:m) for every d I :m • Dx X/92 X • • • X D,, and then by comparing the fuzzy numbers with the minimization objective in mind. H e r e we consider comparison procedures described in [12]. Let the minimum cost be defined as
The above condition for the existence of a crisp single sequence of decisions is shown in [12], named 'strong match'.
2. 3. Search method Thus far we have described an exhaustive search which composes fuzzy costs using the extended sum and compares them using the extended minimum. To justify an efficient search technique parallel to dynamic programming, we need to show that the principle of optimality holds over the operations of extended sum and extended minimum.
Theorem
1. Let X, Y, Z be convex fuzzy sets with upper semicontinuous membership functions defined on the same ordered universe U. (Note that fuzzy numbers are a special case and obey the same property.) Then X ~) m~n(Y, Z) -- m~n((X ~ Y), (X ~ Z)).
fl(d,:,~), =
mln ~.
fl(d' :m)
dl:m~Dix...XDm
where ml~'n is obtained using the extension principle [13]. In general fl(dl:m) * can be a different fuzzy set from all f l ( d l : m ) . Then, the minimum cost may not crisply and uniquely correspond to a particular sequence of decisions. The best sequence 8~:m is hence defined as a fuzzy set on D~ x . . . x Dm where the membership level of a given sequence of decisions is the level at which the corresponding composite cost fl(d ~:'~) is said to match the minimum cost fl(dl:m) *. The concept of match is defined in [12]. In a special case, the minimum cost fl(dl:m) * may in fact correspond to one of the fuzzy numbers fl(d ~:m). Then, the best decision sequence is defined as a crisp single sequence of decisions. This special case occurs in the presence of dominance defined as follows. Let fl(d) and fl(d') be fuzzy sets representing costs of decisions d and d' respectively. Also let the comparison be based on a minimization objective. The decision d is said to dominate d' or equivalently fl(d) <~fl(d') if for all o: • (0, 1], inf[fl(d)]~ ~< inf[fl(d')]~ and
This theorem is in fact an application of a fundamental property of fuzzy interval calculus to extended sum and extended min. A specific form of that property given by Dubois and Prade [13] is below: Let f ( x ) and g'(x', x") be continuous mappings from ~ to ~ and from ~2 to ~ , respectively, and monotone increasing in each of their arguments such that f ( x ) = g(x, x) for all x. Then f ( X ) = g(X, X ) where X is a fuzzy set as defined in T h e o r e m 1. The above property can be used to prove Theorem 1 by simply making the following substitution:
f ( x ) = x + rain(y, z), g(x', x") = min(x'y, x " + z); since f ( x ) = g(x, x), the result follows. The above theorem will allow us to construct dynamic programming search where at every stage i = m - 1, m - 2 . . . . . 1, d i:m : (d, d i+l:m)
for d • Di,
fl(d':m)*= m~nfl(d':m) deDi
sup[fl(d)]o~ ~< sup[fl(d')]~
and
where [-],~ is the o~-cut of a fuzzy set.
fl(d' :m) = Jr(d) ~ fl(d i+' :m),].
C. l#ik, S. Ammar / Fuzzy optimal search methods
334
In the general case the best decision sequence 6 i:m is defined by its membership function, which can be constructed iteratively as follows: ~(d ~:m) = m i n { a ¢ / , ( d ~÷1:m)}, where tr is the level at which fl(d i:m) is said to match fl(dl:m) *. In the presence of dominance tr is set equal to 1 for the dominating sequence and 0 for all other sequences.
2. 4. Algorithm At stage m Initialize dm+l :m = O,
I~(d m+l:m)
fll(d2:2) * = m~n[fl(c), fl(e)]. From Figure 2, we obtain a~c = 0.8 and c~, = 1, and hence/~(c) = 0.8 and/*(e) = 1. Then 65:2= {0.8/c, l/e}. Note that the subscripts denote the state. Moving to the second state in stage 2, assume that the cost of decisions f and g are such that f dominates g (see Figure 3). Then at state 2, f12(d2:2) * =
=
{/*(d/:') ] d i:" • Di x Di+ 1 x . . .
m~n[fl(f), fl(g)]
and # ( f ) = 1 and #(g) -- 0. Then, 622:2= { l / f } . At stage 1, the cost of decisions a and b are given in Figure 4. The composite costs are shown in Figure 5, which shows the dominance of branch a over branch b.
fl ( d' :m) * r nd,o, fl ( d ' :m) Find c~d, the level at which fl(d":") matches f l ( d i : ' ) * /~(d ':m) = min{ o~a, , ( d '+1 :m)} The best decision sequence is =
Consider a two-stage decision tree, shown in Figure 1. The first stage has two decisions a and b, the second stage has two states each with two decision alternatives, c and e, and f and g, respectively. Assume that the cost of each decision at every stage and state are fuzzy numbers and the objective is to minimize the composite cost of two successive decisions. Using a backward search we start at state 1 in stage 2. Assume that the costs of decisions c and e are such that dominance does not exist (see Figure 2). Then, at state 1,
= 1;
f l ( d ' + ' : " ) * = {1/0} For every stage i = m to 1 For every state I s in stage i For every decision d • De Construct d i:m = {d, d i ÷1:"} Determine fl(d 1: ' ) = fl(d) ~ fl(d i÷1:m), Check for dominance 1. If dominance exists (i.e., there exists d' • Di such that fl(d', d ~+':") ~< fl(d, d i+l:m) V d • D,) then d' dominates all other decisions at stage i Set fl(d~:") * = fl(d') ~ fl(d i+' :m), Set/~(d', d ~+l:m) = i~(di+, :m) and/z(d, d i+1 :m) = 0 Vd • D~, d ¢: di 2. If dominance does not exist
6 i:m
2.5. Numerical example
fl3(d' :2), = fl(a) ~ ill(d2:2) *, /u(a, c) =/u(c) = 0.8
x Din}
The solution of the algorithm is given by the fuzzy set of decision sequences, & : ' .
and
/u(a, e) = #(e) = 1.
Finally, 6~ :2= {0.8/(a, c), 1/(a, e)}.
a ///£~teb3
stage1 {
~tate 1 ~state 2 stage2 {
7\ Fig. 1. A two-stage decision tree.
1 The subscript for state is omitted to simplify the notation.
C. l~ik, S. Ammar / Fuzzy optimal search methods
3. A* search The A* search is performed on a decision tree as described above with the introduction of the concept of lower bound of costs [1]. The premise is that, if the cost of a path is less than the lower bound estimates of the costs of all other paths, then that path can be declared to be an optimal solution without fully expanding the graph and
335
computing the actual costs. In the fuzzy set theoretic context, we define a lower bound as follows: Definition. Let .~" and X be two fuzzy numbers. is a lower bound of X (denoted as .~ ~
~t13)
1.0 0.8
R Fig. 2. Costs at state 1 at stage 2.
1.0
R Fig. 3. Costs at state 2 at stage 2.
1.0
~/~ (b) R Fig. 4. Costs at state 3 at stage 1.
~(a)(9 ~I (d2:2)*
~ (b)(9 ~i (d2:2)*
1.0
R Fig. 5. Composite costs at stage 1.
C. lfik, S. Ammar / Fuzzy optimal search methods
336
Note that the above strict definition for the lower bound is chosen so that m~n(X, X ) = X. That is, the underestimate is smaller than the set itself according to most fuzzy set ranking schemes, including that used in this paper. Using the backward composition of costs in an m-stage tree, we need to evaluate at every stage i the lower bound estimate of the overall cost of stages 1 through i - 1. It is by comparing the underestimates of the overall costs that the A* search determines the node(s) that needs to be further explored and the decision sequences to be expanded.
Proof. If Z ~<2( (9 Y then
3.1. Optimality of A* search with fuzzy costs
3.2. Algorithm
To use the A* method with fuzzy costs, we need to show that (1) the sum of the minimum cost of a partial decision sequence and a lower bound cost estimate of the remaining sequence is a lower bound estimate of the overall cost, and (2) if an actual cost of a complete decision sequence is less than or equal to the lower bound cost of a second decision sequence, it is also less than or equal to the actual cost of the second decision sequence. The following theorem and corollary prove these properties for the above given definition of a lower bound. A generic notation with no subscripts or superscripts is used for the sake of clarity. Theorem 2. Let )(, X and Y be fuzzy numbers.
If f( <~X, then f( (g Y <~X (g Y. Proof. By definition of the lower bound, A" ~
Using Theorem 1, we
min(X (9 Y, X (9 Y) = min(X, X) (9 Y = .~" (9 Y. Hence,
inf[Z]~ ~
[]
Based on the above theorem and corollary, the A* search procedure for a backward composition case can be given as follows: 1. List-open = all decision sequences at stage m
(din ~Om). 2. Determine underestimate of total cost for new di:m~ List-open as follows: Obtain fl(di), and evaluate fl(d i:m) = [fl(di) (9/~(di+l'm)] where d i:m= [di, (d i+l:m)*]. Obtain the underestimate/~(d 1:i-1) and evaluate ~ ( a 1 :m) = [ ~ ( d ' :i--1) (9 fl(d~:,,,)]. 3. Find/~(d' :m), ~__. m~n, : m ~ L i s t - o p e n ~(dl :m). 4. Find the fuzzy set of best decision sequence b* as in dynamic programming. 5. If i = 1 for all sequences with ~ . ( d i : ' ) > 0 Then b* is the result. Else Expand in List-open all sequences with/,a.(d i : ' ) > 0 as follows: d i - l : m .= [di_l, (di:m)*]. Goto Step 2. Since for a two-stage problem it is difficult to demonstrate the advantages of an A* algorithm over dynamic programming, a numeric example is not given in this section. Most operations, however, are similar to those shown in Section 2.
inf[~ (9 Y]~ ~
[]
Corollary. Let X, f(, Y, and Z be fuzzy numbers and f( <~X. l f Z <~f( (g Y then Z <<-X(g Y.
4. Conclusion
A major contribution of this paper is the simple generalization of search problems to a case where costs are modeled as convex fuzzy sets over an ordered universe (or more
C. l~ik, S. Ammar / Fuzzy optimal search methods
specifically, as fuzzy numbers). In order to keep the non-fuzzy search problem as a special case of the generalized one, the overall cost of a sequence of decisions is found using the extended sum, and the smallest cost is found based on the extended min operation, at the expense of computational complexity. It is, however, straightforward to simplify the algorithms if strict correspondence to the non-fuzzy counterparts is not necessary. Fuzzy dynamic programming and A* algorithms that are based on the above premise are admissible, that is, they find the optimal solution if one exists. It is trivial to show that the fuzzy optimal solution includes the optimal solution of all crisp search problems obtained by selecting one element of the 1-cut (i.e., o:-cut with tr = 1) of each fuzzy cost to be the crisp cost of the same decision. References [1] N. Nilsson, Principles of Artificial Intelligence (Tioga, Palo Alto, CA, 1980). [2] R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (1970) 141164. [3] J. Kacprzyk, Decision making in a fuzzy environment with fuzzy termination time, Fuzzy Sets and Systems 1 (1978) 169-179.
337
[4] J.F. Baldwin and B.W. Pilsworth, Dynamic programming for fuzzy systems with fuzzy environment, J. Math. Anal. Appl. 85 (1982) 1-23. [5] A.O. Esogbue, M. Fedrizzi, and J. Kacprzyk, Fuzzy dynamic programming with stochastic system, in: J. Kacprzyk and M. Fedrizzi, Eds., Combining Fuzzy Imprecision with Probabilistic Uncertainty In Decision Making (Springer-Verlag, New York, 1988) 266-285. [6] J. Kacprzyk, Multistage Decision Making Under Fuzziness (Verlag T1]V Rheinland, Cologne, 1983). [7] R.R. Yager, Robot planning with fuzzy sets, Robotica 1 (1983) 41-50. [8] R.R. Yager, Paths of least resistance in possibilistic production systems, Fuzzy Sets and Systems 19 (1986) 121-132. [9] D. Dubois, H. Farreny and H. Prade, Combinatorial search with fuzzy estimates, in: J. Kacprzyk and S. Orlowski, Eds., Optimization Models Using Fuzzy Sets and Possibility Theory (Reidel, Dordrecht, 1987) 171-185. [10] D. Dubois and H. Prade, Possibility Theory (Plenum Press, New York, 1988). [11] C. I§ik and A. Meystel, Pilot level of a hierarchical controller for an unmanned robot, IEEE Trans. Robotics and Automation 4 (1988) 241-255. [12] S. Ammar, Determining the 'best' decision in the presence of imprecise information, Fuzzy Sets and Systems 29 (1989) 293-302. [13] D. Dubois and H. Prade, Fuzzy Sets and Systems-Theory and Applications (Academic Press, New York, 1980). [14] A. Nafarieh and J.M. Keller, A fuzzy logic rule-based automatic targer recognizer, Internat. J. Intelligent Systems 6 (1991) 295-312.
331
Fuzzy optimal search methods C a n I~ik Electrical and Computer Engineering Department, Syracuse University, Syracuse, NY 13244-1240, USA
Salwa A m m a r Business Administration Department, LeMoyne College, Syracuse, NY 13205, USA Received January 1990 Revised September 1990
Abstract: This paper gives analytical approaches and algorithms for decision tree search techniques. The decision trees are assumed to have a fixed number of stages and predefined possible states at every stage. The costs of traversing the tree are characterized as approximate and defined as fuzzy numbers. Two search methods, each drawing from an existing non-fuzzy search algorithm, are described. The first method is a dynamic programming search, where the principle of optimality with fuzzy costs is addressed. The second method is an A* search for which the notion of a lower bound estimate of costs is utilized to increase the efficiency of the search. Algorithms for each method are included. Theorems and proofs which complete the analytical development of these techniques are also included. Finally, a numerical example illustrating the procedure is given.
Keywords: Search methods; operations research; fuzzy dynamic programming; fuzzy A* search; fuzzy cost.
1. Introduction Search methods have been extensively studied to solve multi-stage (combinatorial) decision problems. Dynamic programming and the A* algorithm are two closely related search methods that are efficient and yield the optimum (minimum cost) solution if one exists. These two and other graph search methods are extensively discussed by Nilsson [1] in a non-fuzzy setting. The efficiency aspect of dynamic programming and the A* algorithm has attracted the attention of researchers working on multi-stage decision problems with fuzzy aspects. In one of the earliest contributions to the problem, Bellman
and Zadeh [2] dealt with multistage decision making in a fuzzy environment. Kacprzyk [3] also applied dynamic programming and branch and bound algorithms to systems in a fuzzy environment and extended the results for fuzzy termination time. Baldwin and Pilsworth [4] solved a multistage decision problem in a fuzzy environment with fuzzy system dynamics and a cost function that evaluates fuzzy states, control and goal constraints into a membership grade. The outcome is hence non-fuzzy. This approach is computationally complicated and is not well suited for practical applications. Esogbue et al. [5] investigated the multi-stage control of a stochastic system in a fuzzy environment using two different formulations when the system under control had a stochastic model. A major portion of a book by Kacprzyk is devoted to tree search algorithms where the system and/or the environment are fuzzy [6]. What is common in the above work is that although certain components of the problem are modeled using fuzzy set theory, the costs or the outcomes of the decisions are non-fuzzy. Yager [7] proposed the use of search trees in assessing the possibility of reaching the goal state in a production system where each rule is assigned a weight, and he later used an A* algorithm to find the path of least resistance in a similar production system [8]. The costs were composed using a min operation and a more general T-norm operation, and hence the approach differs greatly from what is given in the rest of this paper. Dubois et al. [9] concentrated on problems where costs are modeled as fuzzy intervals and utilized possibility theory in ranking alternatives. Optimality of the ranking was implied when the intervals did not overlap. An excellent treatment of fuzzy programming as well as an extensive survey of heuristic fuzzy tree search methods are given by Dubois and Prade [10]. In this paper we will deal with multi-stage decision problems where the cost associated with each decision or decision sequence is known only approximately and can be modeled as a fuzzy
0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved
332
C. lfik, S. Ammar / Fuzzy optimal search methods
number. In order to concentrate our efforts on this new aspect of fuzzy multi-stage decision making problem, we will assume that the cost of a decision is somehow determined and will not elaborate on that process. We will further assume that the future costs or outcomes are not discounted. A practical example of the problem described by these assumptions is mobile robot path planning. Usually, a path from an initial position to a goal position is planned in an abstacle-strewn environment, using data from maps and sensory feedback, which are of different scale, accuracy and reliability [11]. We first discuss some issues involved in extending the principle of optimality to problems with fuzzy costs. We also show that, when there is an admissible underestimate of the remaining cost, optimal A* search can be extended to problems with fuzzy costs. The ranking of alternatives with fuzzy costs is performed using the approach discussed in [12], which is based on the extended min operation. There are two basic reasons for this choice. First, the solution found in this manner includes an optimal solution in the crisp-set sense, if there is one. Since a major concern of the paper is to generalize the two aforementioned search methods by simply replacing the deterministic costs with fuzzy ones, it is important to have this property. Second, in solving problems such as robot path planning we associate inexactness with risk, and treat all possible alternatives with membership grades larger than a certain minimum level as of acceptable risk. In this regard, having as the 'best' decision a fuzzy set of decisions is quite convenient. Following this approach, all unexplored nodes which have a cost that overlaps with the fuzzy set 'minimum of costs at all nodes' are simultaneously expanded. Therefore the approach may not look very efficient. However, it lends itself to conveniently proving the theorems on the optimality of search methods with fuzzy costs. Nevertheless, the efficiency of the algorithms can be improved by heuristically pruning the search tree, at the expense of analytically proven optimality. If the computational complexity of search is a major concern, ranking with possibility theory [9, 10] may be used, since based on 4-step criteria only one node of the graph is expanded at a time.
2. Dynamic decision trees The aim of the search method described in this section is to determine the 'best' decision sequence for a multistage decision problem. At every stage the outcome of a given decision is described as a fuzzy number representing the approximate cost associated with that decision. The objective is to minimize the overall cost of all stages. The composition of the fuzzy costs over the successive stages is performed using the extended sum [13], due to the nature of intended applications. For example, distance, fuel, travel time are all additive costs. The extended operations of fuzzy arithmetic, in general, increase fuzziness. That is, their results are fuzzier than their arguments. This can be observed in Figures 2-5, which have sets with exaggerated fuzziness for the purpose of demonstration. In most problems, however, the 'magnitude' of costs increases along with their fuzziness; hence their relative uncertainty does not change much. Similarly, the sum of two random numbers is a random number whose probability density function is the convolution of the first two, with a variance larger than those of the original random numbers. Nevertheless, the 'increasing fuzziness' of the extended sum can be of concern to the reader, especially if costs are linguistic variables that assume values from finite universes. In such a case the fuzziness can be kept at a minimum if the result of a fuzzy arithmetic operation is mapped to the closest linguistic value. Similar techniques to reduce fuzziness are routinely used in pattern matching with a good degree of success (e.g., see Nafarieh and Keller [14]).
2.1. Definitions and notation Consider an m-stage decision tree. Let Di be the set of possible decisions at stage i = 1. . . . , m. Also let fl(d) be the fuzzy number denoting the cost of decision d e Di. Define di:J
=
(d i, d i+l . . . . .
d j)
to be the sequence of decisions from stages i to j and fl(d iq) to be the composite cost of stages i to j. Using iterative backward composition, fl(d i :m) m fl(d) ~ fl(d i+l :m)
333
C. l#ik, S. Ammar / Fuzzy optimal search methods
and
d i : m = ( d , d i÷l:m) f o r i = m t o l , where ~ denotes the sum of two fuzzy sets using the extension principle. Initialization is given by
fl(d,,,+l:m)
=
{1/0}
and
d m+~:m = 0,
introduced to allow for the definition of iterative algorithms.
2. 2. Determining the 'best' decision sequence The best decision sequence can be determined exhaustively by first evaluating fl(d ~:m) for every d I :m • Dx X/92 X • • • X D,, and then by comparing the fuzzy numbers with the minimization objective in mind. H e r e we consider comparison procedures described in [12]. Let the minimum cost be defined as
The above condition for the existence of a crisp single sequence of decisions is shown in [12], named 'strong match'.
2. 3. Search method Thus far we have described an exhaustive search which composes fuzzy costs using the extended sum and compares them using the extended minimum. To justify an efficient search technique parallel to dynamic programming, we need to show that the principle of optimality holds over the operations of extended sum and extended minimum.
Theorem
1. Let X, Y, Z be convex fuzzy sets with upper semicontinuous membership functions defined on the same ordered universe U. (Note that fuzzy numbers are a special case and obey the same property.) Then X ~) m~n(Y, Z) -- m~n((X ~ Y), (X ~ Z)).
fl(d,:,~), =
mln ~.
fl(d' :m)
dl:m~Dix...XDm
where ml~'n is obtained using the extension principle [13]. In general fl(dl:m) * can be a different fuzzy set from all f l ( d l : m ) . Then, the minimum cost may not crisply and uniquely correspond to a particular sequence of decisions. The best sequence 8~:m is hence defined as a fuzzy set on D~ x . . . x Dm where the membership level of a given sequence of decisions is the level at which the corresponding composite cost fl(d ~:'~) is said to match the minimum cost fl(dl:m) *. The concept of match is defined in [12]. In a special case, the minimum cost fl(dl:m) * may in fact correspond to one of the fuzzy numbers fl(d ~:m). Then, the best decision sequence is defined as a crisp single sequence of decisions. This special case occurs in the presence of dominance defined as follows. Let fl(d) and fl(d') be fuzzy sets representing costs of decisions d and d' respectively. Also let the comparison be based on a minimization objective. The decision d is said to dominate d' or equivalently fl(d) <~fl(d') if for all o: • (0, 1], inf[fl(d)]~ ~< inf[fl(d')]~ and
This theorem is in fact an application of a fundamental property of fuzzy interval calculus to extended sum and extended min. A specific form of that property given by Dubois and Prade [13] is below: Let f ( x ) and g'(x', x") be continuous mappings from ~ to ~ and from ~2 to ~ , respectively, and monotone increasing in each of their arguments such that f ( x ) = g(x, x) for all x. Then f ( X ) = g(X, X ) where X is a fuzzy set as defined in T h e o r e m 1. The above property can be used to prove Theorem 1 by simply making the following substitution:
f ( x ) = x + rain(y, z), g(x', x") = min(x'y, x " + z); since f ( x ) = g(x, x), the result follows. The above theorem will allow us to construct dynamic programming search where at every stage i = m - 1, m - 2 . . . . . 1, d i:m : (d, d i+l:m)
for d • Di,
fl(d':m)*= m~nfl(d':m) deDi
sup[fl(d)]o~ ~< sup[fl(d')]~
and
where [-],~ is the o~-cut of a fuzzy set.
fl(d' :m) = Jr(d) ~ fl(d i+' :m),].
C. l#ik, S. Ammar / Fuzzy optimal search methods
334
In the general case the best decision sequence 6 i:m is defined by its membership function, which can be constructed iteratively as follows: ~(d ~:m) = m i n { a ¢ / , ( d ~÷1:m)}, where tr is the level at which fl(d i:m) is said to match fl(dl:m) *. In the presence of dominance tr is set equal to 1 for the dominating sequence and 0 for all other sequences.
2. 4. Algorithm At stage m Initialize dm+l :m = O,
I~(d m+l:m)
fll(d2:2) * = m~n[fl(c), fl(e)]. From Figure 2, we obtain a~c = 0.8 and c~, = 1, and hence/~(c) = 0.8 and/*(e) = 1. Then 65:2= {0.8/c, l/e}. Note that the subscripts denote the state. Moving to the second state in stage 2, assume that the cost of decisions f and g are such that f dominates g (see Figure 3). Then at state 2, f12(d2:2) * =
=
{/*(d/:') ] d i:" • Di x Di+ 1 x . . .
m~n[fl(f), fl(g)]
and # ( f ) = 1 and #(g) -- 0. Then, 622:2= { l / f } . At stage 1, the cost of decisions a and b are given in Figure 4. The composite costs are shown in Figure 5, which shows the dominance of branch a over branch b.
fl ( d' :m) * r nd,o, fl ( d ' :m) Find c~d, the level at which fl(d":") matches f l ( d i : ' ) * /~(d ':m) = min{ o~a, , ( d '+1 :m)} The best decision sequence is =
Consider a two-stage decision tree, shown in Figure 1. The first stage has two decisions a and b, the second stage has two states each with two decision alternatives, c and e, and f and g, respectively. Assume that the cost of each decision at every stage and state are fuzzy numbers and the objective is to minimize the composite cost of two successive decisions. Using a backward search we start at state 1 in stage 2. Assume that the costs of decisions c and e are such that dominance does not exist (see Figure 2). Then, at state 1,
= 1;
f l ( d ' + ' : " ) * = {1/0} For every stage i = m to 1 For every state I s in stage i For every decision d • De Construct d i:m = {d, d i ÷1:"} Determine fl(d 1: ' ) = fl(d) ~ fl(d i÷1:m), Check for dominance 1. If dominance exists (i.e., there exists d' • Di such that fl(d', d ~+':") ~< fl(d, d i+l:m) V d • D,) then d' dominates all other decisions at stage i Set fl(d~:") * = fl(d') ~ fl(d i+' :m), Set/~(d', d ~+l:m) = i~(di+, :m) and/z(d, d i+1 :m) = 0 Vd • D~, d ¢: di 2. If dominance does not exist
6 i:m
2.5. Numerical example
fl3(d' :2), = fl(a) ~ ill(d2:2) *, /u(a, c) =/u(c) = 0.8
x Din}
The solution of the algorithm is given by the fuzzy set of decision sequences, & : ' .
and
/u(a, e) = #(e) = 1.
Finally, 6~ :2= {0.8/(a, c), 1/(a, e)}.
a ///£~teb3
stage1 {
~tate 1 ~state 2 stage2 {
7\ Fig. 1. A two-stage decision tree.
1 The subscript for state is omitted to simplify the notation.
C. l~ik, S. Ammar / Fuzzy optimal search methods
3. A* search The A* search is performed on a decision tree as described above with the introduction of the concept of lower bound of costs [1]. The premise is that, if the cost of a path is less than the lower bound estimates of the costs of all other paths, then that path can be declared to be an optimal solution without fully expanding the graph and
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computing the actual costs. In the fuzzy set theoretic context, we define a lower bound as follows: Definition. Let .~" and X be two fuzzy numbers. is a lower bound of X (denoted as .~ ~
~t13)
1.0 0.8
R Fig. 2. Costs at state 1 at stage 2.
1.0
R Fig. 3. Costs at state 2 at stage 2.
1.0
~/~ (b) R Fig. 4. Costs at state 3 at stage 1.
~(a)(9 ~I (d2:2)*
~ (b)(9 ~i (d2:2)*
1.0
R Fig. 5. Composite costs at stage 1.
C. lfik, S. Ammar / Fuzzy optimal search methods
336
Note that the above strict definition for the lower bound is chosen so that m~n(X, X ) = X. That is, the underestimate is smaller than the set itself according to most fuzzy set ranking schemes, including that used in this paper. Using the backward composition of costs in an m-stage tree, we need to evaluate at every stage i the lower bound estimate of the overall cost of stages 1 through i - 1. It is by comparing the underestimates of the overall costs that the A* search determines the node(s) that needs to be further explored and the decision sequences to be expanded.
Proof. If Z ~<2( (9 Y then
3.1. Optimality of A* search with fuzzy costs
3.2. Algorithm
To use the A* method with fuzzy costs, we need to show that (1) the sum of the minimum cost of a partial decision sequence and a lower bound cost estimate of the remaining sequence is a lower bound estimate of the overall cost, and (2) if an actual cost of a complete decision sequence is less than or equal to the lower bound cost of a second decision sequence, it is also less than or equal to the actual cost of the second decision sequence. The following theorem and corollary prove these properties for the above given definition of a lower bound. A generic notation with no subscripts or superscripts is used for the sake of clarity. Theorem 2. Let )(, X and Y be fuzzy numbers.
If f( <~X, then f( (g Y <~X (g Y. Proof. By definition of the lower bound, A" ~
Using Theorem 1, we
min(X (9 Y, X (9 Y) = min(X, X) (9 Y = .~" (9 Y. Hence,
inf[Z]~ ~
[]
Based on the above theorem and corollary, the A* search procedure for a backward composition case can be given as follows: 1. List-open = all decision sequences at stage m
(din ~Om). 2. Determine underestimate of total cost for new di:m~ List-open as follows: Obtain fl(di), and evaluate fl(d i:m) = [fl(di) (9/~(di+l'm)] where d i:m= [di, (d i+l:m)*]. Obtain the underestimate/~(d 1:i-1) and evaluate ~ ( a 1 :m) = [ ~ ( d ' :i--1) (9 fl(d~:,,,)]. 3. Find/~(d' :m), ~__. m~n, : m ~ L i s t - o p e n ~(dl :m). 4. Find the fuzzy set of best decision sequence b* as in dynamic programming. 5. If i = 1 for all sequences with ~ . ( d i : ' ) > 0 Then b* is the result. Else Expand in List-open all sequences with/,a.(d i : ' ) > 0 as follows: d i - l : m .= [di_l, (di:m)*]. Goto Step 2. Since for a two-stage problem it is difficult to demonstrate the advantages of an A* algorithm over dynamic programming, a numeric example is not given in this section. Most operations, however, are similar to those shown in Section 2.
inf[~ (9 Y]~ ~
[]
Corollary. Let X, f(, Y, and Z be fuzzy numbers and f( <~X. l f Z <~f( (g Y then Z <<-X(g Y.
4. Conclusion
A major contribution of this paper is the simple generalization of search problems to a case where costs are modeled as convex fuzzy sets over an ordered universe (or more
C. l~ik, S. Ammar / Fuzzy optimal search methods
specifically, as fuzzy numbers). In order to keep the non-fuzzy search problem as a special case of the generalized one, the overall cost of a sequence of decisions is found using the extended sum, and the smallest cost is found based on the extended min operation, at the expense of computational complexity. It is, however, straightforward to simplify the algorithms if strict correspondence to the non-fuzzy counterparts is not necessary. Fuzzy dynamic programming and A* algorithms that are based on the above premise are admissible, that is, they find the optimal solution if one exists. It is trivial to show that the fuzzy optimal solution includes the optimal solution of all crisp search problems obtained by selecting one element of the 1-cut (i.e., o:-cut with tr = 1) of each fuzzy cost to be the crisp cost of the same decision. References [1] N. Nilsson, Principles of Artificial Intelligence (Tioga, Palo Alto, CA, 1980). [2] R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (1970) 141164. [3] J. Kacprzyk, Decision making in a fuzzy environment with fuzzy termination time, Fuzzy Sets and Systems 1 (1978) 169-179.
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[4] J.F. Baldwin and B.W. Pilsworth, Dynamic programming for fuzzy systems with fuzzy environment, J. Math. Anal. Appl. 85 (1982) 1-23. [5] A.O. Esogbue, M. Fedrizzi, and J. Kacprzyk, Fuzzy dynamic programming with stochastic system, in: J. Kacprzyk and M. Fedrizzi, Eds., Combining Fuzzy Imprecision with Probabilistic Uncertainty In Decision Making (Springer-Verlag, New York, 1988) 266-285. [6] J. Kacprzyk, Multistage Decision Making Under Fuzziness (Verlag T1]V Rheinland, Cologne, 1983). [7] R.R. Yager, Robot planning with fuzzy sets, Robotica 1 (1983) 41-50. [8] R.R. Yager, Paths of least resistance in possibilistic production systems, Fuzzy Sets and Systems 19 (1986) 121-132. [9] D. Dubois, H. Farreny and H. Prade, Combinatorial search with fuzzy estimates, in: J. Kacprzyk and S. Orlowski, Eds., Optimization Models Using Fuzzy Sets and Possibility Theory (Reidel, Dordrecht, 1987) 171-185. [10] D. Dubois and H. Prade, Possibility Theory (Plenum Press, New York, 1988). [11] C. I§ik and A. Meystel, Pilot level of a hierarchical controller for an unmanned robot, IEEE Trans. Robotics and Automation 4 (1988) 241-255. [12] S. Ammar, Determining the 'best' decision in the presence of imprecise information, Fuzzy Sets and Systems 29 (1989) 293-302. [13] D. Dubois and H. Prade, Fuzzy Sets and Systems-Theory and Applications (Academic Press, New York, 1980). [14] A. Nafarieh and J.M. Keller, A fuzzy logic rule-based automatic targer recognizer, Internat. J. Intelligent Systems 6 (1991) 295-312.