Applying Genetic Algorithms to Solve the Fuzzy Optimal Profit Problem

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1 JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 8, () Applying Genetic Algoriths to Solve the Fuzzy Optial Profit Proble FENG-TSE LIN AND JING-SHING YAO Departent of Applied Matheatics Chinese Culture University Taipei, Taiwan E-ail: Departent of Matheatics National Taiwan University Taipei, 6 Taiwan This study investigated the application of genetic algoriths for solving a fuzzy optiization proble that arises in business and econoics. In this proble, a fuzzy price is deterined using a linear or a quadratic fuzzy deand function as well as a linear cost function. The obective is to find the optial fuzzy profit, which is derived fro the fuzzy price and the fuzzy cost. The traditional ethods for solving this proble are () using the etension principle, and () using the interval arithetic and α-cuts. However, we argue that the traditional ethods for solving this proble are too restrictive to produce an optial solution, and that an alternative approach is possibly needed. We use genetic algoriths to obtain an approiate solution for this fuzzy optial profit proble without using the ebership functions. We not only give epirical eaples to show the effectiveness of this approach, but also give theoretical proofs to validate the correctness of the algorith. We conclude that genetic algoriths can produce good approiate solutions when applied to solve fuzzy optiization probles. Keywords: genetic algoriths, fuzzy sets, fuzzy nubers, fuzzy optiization profit proble, fuzzy deand. INTRODUCTION In this study, we investigated the application of genetic algoriths to solve a fuzzy optiization proble that arises in business and econoics. This proble is often involved in the study of how changes in such variables as production or price will affect other variables such as revenue or profit [4,, ]. One proble instance is the fuzzy optial profit proble [], which is briefly depicted as follows. In a onopolist arket, producers can control arket prices and product quantities. The deand for a certain coodity is related to its price by a deand function P (. This eans that, as the price increases, deand usually falls, and that as the price falls, deand rises. Since revenue is equal to the price per unit ties the quantity sold, we can deterine the revenue received for selling units of the coodity as R ( P(. The cost of producing units of a certain coodity is given by a cost function C ( and the basic relation between profit, revenue and cost is forulated as N ( R( C(. The onopolist can thus easily obtain the aiu profit by doing soe siple calculations. Received August 9, ; revised October 5, ; accepted January 9,. Counicated by Chuen-Tsai Sun. 563

2 564 FENG-TSE LIN AND JING-SHING YAO On the other hand, in a perfect copetitive arket, the deand is no longer a fied value, even for the sae price function P (. The price, of course, will fluctuate at any tie in order to reflect various conditions in the arket. To deal with this kind of iprecise data, fuzzy sets provide a powerful tool for odeling and solving the optiization proble. Thus, this leads to the use of the fuzzy nuber to represent the fuzzy deand and the fuzzy profit. The optial price can then be obtained using the ebership functions and the centroid of the fuzzy profit. Nevertheless, the ain issue in this proble is as follows. Consider a quadratic algebraic epression a + b for a, b real paraeters and where is a real variable. Let y N(a, b, a + b. After substituting the triangular fuzzy nubers A, B, and into a + b for a, b, and, respectively, we obtain the fuzzy set A B +. The triangular fuzzy nubers are represented by A ( a, a, a + ), B ( b w, b, b + w ), and ( δ,, + δ ) together with their ebership functions µ(a A ), µ(b B ), and µ( ), respectively. Accordingly, there are two aor traditional ethods for evaluating the above fuzzy epression in the literature [, 3]. The first ethod for finding the value of A + B involves using the etension principle. Let Π(a, b, denote the iniu of µ(a A ), µ(b B ), and µ( ). Assue that we set Y equal to A + B. Then the ebership function for Y is defined by µ (y Y ) sup{π(a, b, N(a, b, y}. To find α-cuts of Y, let Φ(α) {N(a, b, a A ( α ), b B ( α ), (α)}, α. Then we have Φ(α) Y ( α ). Alternatively, the second ethod involves evaluating A + B using interval arithetic and α-cuts. For any triangular fuzzy nubers A and B, the following operations hold: () ( A B )(α) A ( α ) B ( α), and () ( A ± B )(α) A ( α ) ± B ( α ). A B and A ± B are coputed using the etension principle, and A ( α ) B ( α), A ( α ) ± B ( α ) are found using interval arithetic. Let Z ( α ) A ( α ) ( α) ( α) + B ( α ) ( α), for α. We can see that Y ( α ) is a subset of Z ( α ). Obviously, evaluating a fuzzy algebraic epression using interval arithetic and α-cuts can produce a larger fuzzy set than can evaluating using the etension principle. In addition, these traditional ethods for solving fuzzy equations are very often too restrictive to produce a solution [3]. The other disadvantages, as noted in [7], are that solving fuzzy equations is very coplicated due to the lack of inverse operators, and that the ultiple occurrence of paraeters in an epression results in increased iprecision. In this study, we applied a genetic algorith to obtain an approiate solution to the fuzzy optial profit optiization proble. When genetic algoriths are applied to solve this proble, the fuzzy equation coputation requires no etension principle or interval arithetic and α-cuts. The genetic algorith uses only the usual evolution. Assue that a is an arbitrary fuzzy set on the interval [, ], and that N ( ) is a b fuzzy profit. In the genetic algorith approach, we do not need to define the a ebership functions of N ( ). Instead, the interval [, ] is equally divided into b a partitions. Let,,,, be the partition points. Let ( ) [,] b µ,

3 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM 565,,,, be the ebership grade of in. Thus, we obtain a discrete fuzzy set ( µ, µ,..., µ ), where µ,,,,, is a rando nuber in [, ]. In other words, we wish to find a in [, ] by aiizing b N ( ) via the genetic algoriths. However, we cannot aiize N ( ) directly since it is a fuzzy set. Instead, we can copute the centroid of the fuzzy profit for the aiization proble. The centroid can be used as the fitness value for the evolution of the genetic algoriths. Therefore, the obective is to siply find a vector in [, ] + that aiizes the centroid. This paper is organized as follows. Section deals with the optial price for fuzzy profit based on the linear and quadratic deand functions. First, we discuss the traditional ethod for obtaining the optial solution using fuzzy optiization and then introduce proble discretization for the purpose of the genetic algorith application to obtain an approiate solution to this proble. In section 3, we design a genetic algorith for solving the fuzzy optial profit proble. Section 4 gives two illustrative eaples. Section 5 discusses the ain results of this work. Finally, we state our conclusions in section 6.. Linear Deand Function. OPTIMAL PRICE FOR FUZZY PROFIT First, a linear deand function is introduced. Assue that the linear function is P ( a b, a / b () and that the cost function is C ( e + g + k, () where a, b, e, g, k are known positive nubers, b ( a g) < a( b + k), and g < a. The profit function is defined by P( C( e + ( a g) ( b k), a / b N ( +. (3) In a onopolist arket, the profit function is surely (3). Since N ( a g ( b + k), a g we have ( ). According to the assuption that b ( a g) < a( b + k), we ( b + k) have a (, ]. Hence, the aiu profit is b N( a g ( a g) ) N e +. (4) ( b + k) 4( b + k) a P( However, in a perfect copetitive arket, the deand will vary, even for b the sae price P (. As a result, the deand can be represented by the fuzzy nuber.

4 566 FENG-TSE LIN AND JING-SHING YAO. Quadratic Deand Function Net, consider the quadratic deand function. Given the deand function a b + c,, (5) P( and the cost function C ( e + g + k,, (6) where all variables a, b, c, d, e, g, and k are known positive nubers, the relations of this quadratic function are b b 4ac () If b 4ac, then set. c b () If b 4ac <, then set c. Fro (5) and (6), we have the profit function as follows: 3 N ( e + ( a g) ( b + k) + c, Consider a onopolist arket.. (7) By differentiating (7), we obtain the following equation: N ( ( a g) ( b + k) + 3c,. The aiu is found to be 3c ( b + k) + ( a g). (8) The discriinant for (8) is D ( b + k) 3c( a ). g Case. If D >, then (8) has two roots, b + k D d and b + k + D d. 3c 3c We obtain N 3c( d )( ). Then, we have the following three possibilities: ( d () When < d < d <, a N( a( N( d), N( )). (9) () When < d < < d, a N( N( d). () (3) When < d < d, a N( N( ). () b + k Case. If D, then N ( 3c( ) > and a N( N( ) 3c. () Case 3. If D <, then N ( > < < and a N( N( ). (3) Assue that the optial deand is. Then the aiu profit is N ( ). Note that in a onopolist arket, the deand is uniquely deterined by the given price P(. However, in a perfect copetitive arket, the deand and the price do not necessary a P( depend on the deand function. Therefore, the deand b will vary, even for the sae price P(. This is a copletely different interpretation of the deand in a copetitive arket copared with that in a onopolist arket. As a result, the deand

5 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM 567 should be represented using linguistic variables, such as if the price is P(, then the deand is in the vicinity of, to obtain a reasonable description..3 Traditional Fuzzy Methods Obviously, the deand in a copetitive arket can be represented using the triangular fuzzy nuber. Fro (7), we obtain the fuzzy profit N ( ) e + (a g) (b + k). Let N ( y. Then (3) becoes (b + k) (a g) + e + y. (4) Fro (4), we can see that D ( y) ( a g) 4( b + k)( e + y), if y N( ). If D ( y) >, then (4) has two roots, which are quadratic forulas, shown as follows: ( a g D( y) r y), ( b + k) a g + D( y) r( y). (5) ( b + k) Otherwise, if D(y), (4) has a duplicate root, then a g r( y) r ( y). (6) ( b + k) After applying the etension principle ethod, we obtain the ebership function of the fuzzy profit N ( ) as N( ( y) y a[ ( r ( )), ( ( ))], ( sup ( ) y r y if y N N (, otherwise ), (7) where the ebership function of the triangular fuzzy nuber is ( ) and the ebership function of the fuzzy profit N ( ) is N ( )( y). Once we have N ( )( y), the centroid of N ( ), E(, can be obtained using the following equation: E( N ( ) N ( ) yn( ( y) dy. (8) N( ( y) dy Then the best solution of the fuzzy optial profit proble is obtained. Siilarly, consider the quadratic function. After we apply the etension principle, the ebership function of N ( ) is µ N ( ) ( y) sup µ ( if N - (y). Note N ( y) that N( y is a cubic equation. Clearly, producing N ( y ) for a cubic equation is not an easy task. (see section 5.3. for a detailed discussion)..4 Genetic Algoriths Approach The genetic algoriths approach is depicted as follows. Let the triangular fuzzy nuber (,, + ) be replaced by an arbitrary fuzzy set in an interval [, ], where is defined in (5) (see Fig. ).

6 568 FENG-TSE LIN AND JING-SHING YAO µ3 µ µ µ ( ) Fig.. Fuzzy set. The interval [, ] is divided into partitions,, where,,, are the partition points. Let the ebership grade of at be ( ) µ,,,, and µ [,]. Then, a discrete fuzzy set is obtained as follows: ( µ, µ,..., µ ) µ µ µ (9) Fro N ( y, if each N ( k ), k,,,, is different, then the fuzzy profit is defined by µ µ µ N ( ) N ( µ, µ,..., µ ) , () N ( ) N ( ) N ( ) and the centroid is defined by N N ( ) µ θ ( ( )). () µ Fro (7), let N ( y ; we obtain a cubic equation: 3 c ( b + k) + ( a g) ( e + y). Assue that there are three roots for this equation. Let N( ) N( ) N( ) y, 3 ( µ, µ,..., µ N )( y where 3. According to the etension principle, we obtain N ) sup ( a( ( ), ( ), ( )) a[,, ] y N ( µ 3 µ µ 3. Eq. () becoes a[ µ, µ ], µ. The centroid of ) Q 3 N ( is θ ( N ( )), () N( ) P where P µ + a( µ, µ, µ 3),, 3 and Q N( ) µ + N( )(a[ µ, µ, µ 3]).,, 3 Let the centroid of the fuzzy profit N ) N( µ, µ,..., µ ) be θ N ( )) centroidof N( µ, µ,..., µ ). ( (

7 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM 569 Finally, a vector is deterined in [, ] + (real nubers) in a population that aiizes )) by eans of genetic algoriths. 3. DESIGNING A GENETIC ALGORITHM FOR SOLVING THE FUZZY OPTIMAL PROFIT PROBLEM Genetic algoriths are stochastic search techniques based on the principles and echaniss of natural genetics and selection [8, 5]. Genetic algoriths grew out of Holland s [] study on adaptation in artificial and natural systes. The basic concept of genetic algoriths is that they start with a population of randoly generated candidates and evolve towards better solutions by applying genetic operators, such as crossover and utation, odeled on natural genetic inheritance and Darwinian survival-of-the-fittest [8, 5]. Therefore, genetic algoriths are theoretically and epirically have been proven to process robust search capabilities in cople spaces, thus offering a valid approach to probles requiring efficient and effective searching [6]. Genetic algoriths can be used to copute the ebership functions of fuzzy sets [, 3]. Given soe functional apping for a syste, soe ebership functions and their shapes are assued for the various fuzzy variables defined for a proble [6]. The ebership functions are coded as bit strings that are then concatenated. An evaluation function is used to evaluate the fitness of each set of ebership functions. There are two possible ways to integrate fuzzy logic and genetic algoriths [9]. One involves the application of genetic algoriths for solving optiization and search probles related to fuzzy systes [,, 7]. The other, is the use of fuzzy tools and fuzzy logic-based techniques for odeling different genetic algorith coponents and adapting genetic algorith control paraeters, with the goal of iproving perforance [9, 4, 8, 9]. Now, a genetic algorith for solving the fuzzy optial profit proble is given below. Step. Generate an initial population. An initial population of size n is randoly generated fro [, ] + according to the unifor distribution in the closed interval [,]. Let the population be ( µ, µ,..., ) µ µ µ µ , where,,, n and µ i is a real nuber in [, ], i,,,,. Each individual in a population is a chroosoe. Step. Calculate the fitness value for each chroosoe. For each chroosoe,,,, n, the centroid )) is calculated as the fitness value. The chroosoes in the population can be rated in ters of their fitness n values. Let the total fitness value of the population be T θ ( N ( )). The cuulative k fitness value (partial su) for each chroosoe, S θ ( N ( )), k,,, n, is k calculated. Net, intervals I [, S ], I [S -, S ],, 3,, n, and I n [S n-, S n ]

8 57 FENG-TSE LIN AND JING-SHING YAO are constructed for the purpose of selection. In our ipleentation, a roulette wheel approach [6, 8] is adopted as the selection echanis. Step 3. Selection and reproduction. Reproduction is a process in which each chroosoe is copied according to the selection process. The selection process begins with spinning of the roulette wheel n ties. Each tie, a single chroosoe is selected fro the current generation to create a new generation. The selection process is as follows. Each tie, a rando nuber r fro the range [, T] is generated. If r I, then chroosoe is selected; otherwise, the kth chroosoe k, k n, is selected if r I k. This selection process is continued until the new population has been created. Finally, the new population is renaed Y, Y, Y3, in the order they were picked. The probability of selection for each chroosoe )) such that it appears in the new population is, and its )) epected value is n T )) to go on into the net population. T. Thus, this procedure tends to choose ore with higher Step 4. Perfor crossover. Crossover is the key to genetic algoriths power. The purpose of crossover is to generate rearrangeents of coadapted groups of inforation fro high perforance structures [8]. The crossover ethod used here is the one-point ethod, which randoly selects one cut-point and echanges the right parts of two parents to generate offspring. Each pair ( Y, Y ), ( Y 3, Y 4 ),, ( Y n, Y n ), where n is even, produces two children via crossover. Let p be the probability of a crossover, p. Usually, p is between.6 and.9 [8], so we epect that, on average, 6% to 9% of the chroosoes will undergo crossover. Consider two chroosoes, Y and Y, for crossover. A rando nuber r is generated in the interval [, ]. If r p, then crossover is perfored on Y and Y. Otherwise, if r > p, the two children Y ' and Y ' are identical to their parents, Y and Y. Suppose that crossover needs to be perfored. Another rando integer nuber u is generated fro the range [, ]. Assue that u equals 3; the two chroosoes Y and Y are cut after the fourth position, and the ' offspring Y and Y ' are generated by echanging the right parts of each chroosoe. Step 5. Perfor utation. Mutation is a background operator that produces rando changes in various chroosoes [6]. Mutation resets one randoly selected position to a real nuber in [, ] or to zero with a probability equal to the utation rate (see Eaples in section 4). Let q be the probability of a utation, q. Usually q is a very sall value, around. to. [8], so we epect that, on average,.% to % of the total population will undergo utation. There are n positions in the whole population. We epect. n to. n utations per generation. For eaple, consider chroosoe ' Y. Let w i be a rando nuber in [, ], i. If w i < q, then the i+th position ' in Y is reset to a real rando nuber in [, ] or to zero. After the utation is copleted on the whole population, we let Y,,,, n. One iteration of the

9 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM 57 genetic algorith has been copleted. Step through step 5 is done K ties, where K is the aiu nuber of iterations. Finally, the algorith is terinated after K generations are produced. Let the last population be,,..., n. The centroid of each chroosoe, n, in the last population ust now be calculated. The aiu value of k )) is the best chroosoe in the population, i.e., a )) )) for soe k {, n,..., n}. The best chroosoe could be a sub-optial or the optial solution for the proble. Let the best chroosoe be (,,..., µ k k k k µ k µ k µ k ) µ µ If P ( i ) a bi + ci, for each i, where i,,,, are different values, then the fuzzy price function is P( k µ k k ) a b k + c k a b + c a b + c Therefore, the centroid of P ( ) is This is an estiate of the aial profit in the fuzzy sense. k k k µ µ +. (3) a b + c ( a b + c ) µ k E( P( k )). (4) µ k In particular, if P( ) P( ), i, then i,, µ k i µ k + of (3) will becoe P( ) P( ) i a( µ k, i, µ P( ) i k, ). In the following, two properties are given to show how good the final solution obtained by genetic algoriths is, and it is copared with the known crisp optial value. Property. The optial value k )) obtained using genetic algoriths is less than or equal to the crisp optial value N ( ). That is, )) N ( ). Proof. For any discrete fuzzy set, ( µ, µ,..., µ ) µ µ µ [, ], µ [,],,,, its fuzzy profit is N ( ) ) k , where N ( µ, µ,..., µ. When N ( k ), k,,,, are different values, the centroid is obtained by N N ( ) µ θ ( ( )). Note that N ( ) is the crisp optial solution (i.e. aiu µ profit). For each [, ],,,,, we have N ( ) N( ),.

10 57 FENG-TSE LIN AND JING-SHING YAO Since µ and, we obtain N( ) µ N( ) µ. Therefore, )) N( ). Furtherore, for each N ( ),,,,, if there eist three identical values, e.g. N ( ) N ( ) N ( 3), 3, then fro (), the centroid of Q N ( ) is θ ( N ( )), where P µ + a( µ, µ, µ 3) and P,, 3 Q N( ) µ + N( )(a[ µ, µ, µ 3]). We can see that )) N( ).,, 3 Therefore, we obtain k )) N( ). Q.E.D Property. The search space for genetic algoriths includes the crisp optial profit solution. However, the probability that the value of )) is equal to the crisp value N ( ) is sall. Proof. The crisp optial deand is in the interval [, ]. When the interval [, ] is divided into partitions,,,,,, we have the possibility that one of b the partition includes. Let i be, i {,,, }. Fro (9), we have µ µ µ , where µ is a rando nuber in [, ],,,,. We can see that there is a possibility of having µ for i and µ. Thus, we have µ i i i µ. Fro (), we obtain N,..., µ,,..., ) ( i i µ i N ) µ i. Then, the centroid θ ( N (,...,, µ,,..., ) i N ( i ) is equal to N ( ). N( ) Obviously, the genetic algorith search space theoretically includes the crisp optial profit solution. However, the probability that the value of )) equals the crisp value N ( ) is sall. Q.E.D. ( i 4. EMPIRICAL RESULTS Two epirical eaples are given below to illustrate the effectiveness of genetic algoriths when we applying the to solve the fuzzy optial profit proble. Eaple. The first instance is as follows: the price function, P(, 5; the cost function, C( + ; and the profit function, N( P( C( + 99, 5. Since N ( 99 4, we obtain Thus, the optial solutions for the crisp case are P(4.75) 5.5 and N(4.75) 5.5. Now, 5 consider that the deand is vague. Let. We have k k 5k, k,,,, which eans that ten partition points are given. Then, we obtain N( ) -, N( )

11 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM , N( ) 78, N( 3 ) 5, N( 4 ) 7, N( 5 ) 5, N( 6 ) 6, N( 7 ) 5, N( 8 ) 75, N( 9 ) 395, and N( ) 6. Fro () and (), we have N( ) µ N ( ) N( µ, µ,..., µ ) and the centroid )). The estiated price µ in the fuzzy sense is given by E( P( )) P( µ ) µ. The paraeters for genetic algoriths are () the probability of a crossover p.8 and () the probability of a utation q.3. An essential feature of genetic algoriths when they are used to solve the fuzzy optial profit proble is their effectiveness in solving the fuzzy proble. Two different approaches were taken in this study for the purpose of analyzing the effectiveness of the algorith. The first approach was based on using the sae nuber of generations but with different population sizes. The second approach was based on using the sae population size but with different nubers of generations. Each approach had five or si runs. In the first approach, the nuber of generations was, and the population size for each run was,, 3, 4, and 5, respectively. The best results for obtained in five runs are shown in Fig.. The partition points are 5,, 5,, 5,, etc. We can see that the genetic algorith was trying to approiate ( ). for, 5, and 3, and ( ). otherwise, where a bell-shaped discrete fuzzy set was obtained. In these runs, one of the best solutions obtained was (.4,.,.3,.,.9,.98,.87,.6,.4,.3,.) with the largest value being )) and the estiated price in the fuzzy sense being E ( P( )) A coparison of the above obtained result with that of the crisp case is % and % ebership partition points run run run 3 run 4 run 5 Fig.. A bell-shaped discrete fuzzy set obtained using the first approach. The nuber of generations is, and the population size in each run is,, 3, 4, and 5, respectively. As for the second approach, the population size was in all si runs. The nuber of generations in each run was,,,, 4,, 6,, 8,, and

12 574 FENG-TSE LIN AND JING-SHING YAO,, respectively. The best results for obtained in si runs are shown in Fig. 3. Once again, the genetic algorith was trying to approiate ( ). for, 5, and 3, and ( ). otherwise, in these runs. Note that the curve for run 6 shown in Fig. 3 has a bell-shaped discrete fuzzy set with ().86, (5).9, and (3).58. ebership partition points 5 run run run 3 run 4 run 5 run 6 Fig. 3. A bell-shaped discrete fuzzy set was obtained fro run 6, which had a population size of and, generations. If we let the utation reset one randoly selected position to zero (i.e. force the ebership grade to ), the best result obtained in each run becae (.,.,.,.,.,.8,.,,.). In this case, the aiu profit obtained by genetic algoriths was the discrete fuzzy set. 8 (where the ebership grade k 5 was.8 when was 5) with an optial value of 5 and an estiated price of 5. 5 Since the partition points k k 5k, k,,, did not include the optial deand 4.75 (the nearest point was 5 5.), the best profit obtained by the genetic algoriths was only 5 (the optial profit in the crisp case was 5.5). Finally, consider the situation when ore partition points are given. Let 5. The nuber of partition points was 5. Fig. 4 shows the ebership curves obtained fro si runs based on using the sae population size but different nubers of generations. The nuber of generations in each run was,,,, 4,, 6,, 8,, and,, respectively. All these curves are trying to approiate ( ). for 3.3 and 6.7. Note that the curve fro run 6 has a bell-shaped discrete fuzzy set with ().64, (3.33).86, (6.67).85, and (3).76. ebership partition points run run run 3 run 4 run 5 run 6 Fig. 4. The bell-shaped discrete fuzzy set obtained fro run 6 using the second approach with 5 partition points.

13 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM 575 Eaple. The proble instance is as follows: the price function is a quadratic, P( +, 5; the cost function is C( +, ; and the profit function is N( + + 3, 5. Since b 4ac 44 <, we have / 5 and N ( + ( 5) + 5 >. In the crisp case of the proble, when 5, the optial solutions are P(5) 76 and N(5) 365. Now, 5 consider that is vague. Let ; we have k k.5k, k,,,. We obtain N( ) -, N( ) 37.65, N( ) 8, N( 3 ).875, N( 4 ) 58, N( 5 ) 93.5, N( 6 ) 7, N( 7 ) 6.375, N( 8 ) 94, N( 9 ) 38.65, and N( ) 365. The centroid is E( P( )) P( µ )) N( µ ) µ, and the optial price in the fuzzy sense is ) µ. The paraeters and the two approaches used for genetic algoriths were all the sae as in Eaple. The best results for obtained in five runs using the first approach with the sae nuber of generations but with different population sizes are shown in Fig. 5. The partition points are.5,.,.5,.,.5,, etc. We can see that the genetic algorith was trying to approiate ( ). for 4.5, 5. and ( ). otherwise, where a half bell-shaped discrete fuzzy set was obtained. In these runs, one of the best solutions obtained was (.,.4,.7,.58,.38,.5,.6,.,.7,.849,.98) with the largest value being )) and the estiated price in the fuzzy sense being E ( P( )) A coparison of the results obtained with that of the crisp case is %.543%, and %.63% ebership partition points run run run 3 run 4 run 5 Fig. 5. A half bell-shaped discrete fuzzy set obtained using the first approach. For the second approach, using the sae population size but with different nubers of generations, the best results for obtained fro si runs are shown in Fig. 6.

14 576 FENG-TSE LIN AND JING-SHING YAO Fro Fig. 6, we can see that the genetic algorith was trying to approiate ( ). for 4.5, 5. and ( ). otherwise, where a half bell-shaped discrete fuzzy set was again obtained. Once again, if we let the utation reset one randoly selected position to zero, the best result obtained in each run becae (.,.,.,,.,.,.95). In this case, the aiu profit obtained by the.95 genetic algoriths was the discrete fuzzy set k 5. (the ebership grade was.95 when was 5.) with a value of ebership run run run 3 run 4 run 5 run 6 partition points Fig. 6. A half bell-shaped discrete fuzzy set obtained using the second approach. 5. DISCUSSION In this section, we will point out that our work has produced the following ain results based on the application of genetic algoriths to solve the fuzzy optial profit proble. 5. Genetic Algoriths Can Find the Optial Solution As eplained in Section, the crisp optial deand, was in (, ], and we can take as a suitable value to divide the interval (, ] into sall partitions, i.e.,,,,. According to Properties and, let i for soe i b {,,,, }. Fro (9), the rando nubers µ, µ,..., µ, generated fro the interval [, ], should have a possibility of µ, i and µ. Therefore, (9) becoes,...,, µ,,...,) ( i µ i µ i i. i i µ According to (), we have N (,...,, µ i,,..., ), where N ( ) is the crisp aiu profit. We conclude N( ) that the search space for genetic algoriths includes the crisp optial solution and crisp near-optial solutions.

15 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM Mutation Strategies In our ipleentation, two strategies are used to perfor utation in genetic algoriths. Norally, we use Strategy. Strategy resets one randoly selected position to a real nuber in [, ] (sets the ebership grade to the other value). On the other hand, Strategy resets one randoly selected position in a chroosoe to zero (forces the ebership grade to zero). Eaple in Section 4 shows that Strategy had in ore likely than Strategy to include the crisp optial solution in its search space. Epirical results show that the speed of convergence to the crisp optial solution using Strategy was uch faster than that achieved using Strategy. The underlying theore for Strategy is based on Property. 5.3 Coparison Between the Traditional Fuzzy Method and Genetic Algoriths The aor difference in obtaining the fuzzy optial profit between using the traditional fuzzy ethod [, ] and genetic algoriths is stated as follows Using the etension principle ethod As eplained in Section, when the deand is fuzzified into a triangular fuzzy nuber (,, + ), < <, >, obviously, the fuzzy profit is N ( ). By applying the etension principle ethod, we can obtain the ebership function of N ( ), µ N ( ) ( y) sup µ (. Since N( y is a cubic function, i.e., e + (a N ( y) g) (b + k) + c 3, it is difficult to obtain N ( y ). As a result, the ebership function µ N ( ) ( y) is difficult to obtain by applying the etension principle. Furtherore, if we let the three roots of N( y be t (y), t (y), and t 3 (y), then we can see that µ N ( ) ( y) sup in[ µ ( t( y )), µ ( t ( y)), µ ( t 3 ( y)) ], where t +, t + t t), t +, otherwise µ ( Obviously, it is also difficult to obtain µ N ( ) ( y) fro the above equation. In this case, the proposed genetic algorith approach can be considered as an alternative way to find a near optial solution Using genetic algoriths With the genetic algorith approach, however, no specific fuzzy set is needed (i.e. use of a triangular fuzzy nuber or other special fuzzy sets is not necessary). Initially, a

16 578 FENG-TSE LIN AND JING-SHING YAO population of discrete fuzzy sets, µ + µ + + µ,,,, n, is created based on the generation of rando nubers in [, ]. After the copletion of the evolution by genetic algoriths, the best discrete fuzzy set in the population is k µ + k µ + + µ k. The fuzzy profit is N ( ) k µ k + N( ) k µ k + N( ) + µ k. After the centroid ethod is used for defuzzification, the estiated profit in N( ) the fuzzy sense is obtained: )) k N( ) µ k µ k. Consider Eaple in section 4. The crisp optial solutions are P(5) 76 and N(5) 365. The best discrete fuzzy set obtained fro si runs is given in Fig. 6, represented by µ k. The fuzzy µ profit is N ( k ) N( N( µ ) µ ), N( ),,,, and the centroid is )) In addition, the best discrete fuzzy set in the population when using Strategy is and the centroid is )) k k.95. The fuzzy profit is N ( 5. ) k N(5.) k.95, CONCLUSIONS This study has investigated the genetic algorith approach to solving fuzzy optiization equations without using the ebership functions of fuzzy nubers. When genetic algoriths are applied to solve fuzzy equations, the coputation uses neither the etension principle nor the interval arithetic and α-cuts. The results of this study ay lead to the developent of effective genetic algoriths for solving general fuzzy optiization probles. In suary, the fuzzy concept of the proposed genetic algorith approach is different and gives alost the sae results as the traditional ethods. ACKNOWLEDGEMENT The authors are grateful to the anonyous referees, whose valuable coents helped to iprove the content of this paper.

17 APPLYING GENETIC ALGORITHMS TO SOLVE THE FUZZY OPTIMAL PROFIT PROBLEM 579 REFERENCES. J. J. Buckly and Y. Hayashi, Fuzzy genetic algorith and applications, Fuzzy Sets and Systes, Vol. 6, 994, pp J. J. Buckley and Y. Qu, On using α-cuts to evaluate fuzzy equations, Fuzzy Sets and Systes, Vol. 38, 99, pp J. J. Buckley and Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systes, Vol. 38, 99, pp J. J. Buckley, Solving fuzzy equations in econoics and finance, Fuzzy Sets and Systes, Vol. 48, 99, pp L. Davis, ed., Genetic Algoriths and Siulated Annealing, San Mateo, CA: Morgan Kaufann, M. Gen and R. Cheng, Genetic Algoriths & Engineering Design, John Wiley & Sons, Inc., New York, R. E. Giachetti, Evaluating engineering functions with iprecise quantities, 7 th International Fuzzy Systes Association Congress, Prague, Czech Republic, D. E. Goldberg, Genetic Algoriths in Search, Optiization, and Machine Learning, Addison-Wesley, Reading, MA, F. Herrera and M. Lozano, Fuzzy genetic algoriths: issues and odels, Dept. of Science and A. I., University of Granada, Technical Report No. 87, Granada, Spain, F. Herrera, M. Lozano, and J. L. Verdegay, Applying genetic algoriths in fuzzy optiization probles, Fuzzy Sets & Artificial Intelligence, Vol. 3, 994, pp J. Holland, Adaptation in Natural and Artificial Systes, University of Michigan Press, Ann Arbor, A. Hoaifar and E. McCorick, Siultaneous design of ebership functions and rule sets for fuzzy controllers using genetic algoriths, IEEE Transcations on Fuzzy Systes, Vol. 3, 995, pp C. L. Karr and E. J. Gentry, Fuzzy control of ph using genetic algoriths, IEEE Transactions of Fuzzy Systes, Vol., 993, pp M. A. Lee and H. Takagi, Dynaic control of genetic algoriths using fuzzy logic techniques, in Proceedings of Fifth International Conference on Genetic Algoriths (ICGA 93), 993, pp M. Mitchell, An Introduction to Genetic Algoriths, A Bradford Book, The MIT Press, Mass., T. J. Ross, Fuzzy Logic with Engineering Applications, McGraw-Hill Inc., New York, M. Sakawa, K. Kato, H. Sunada, and T. Shibano, Fuzzy prograing for ultiobective - prograing probles through revised genetic algoriths, European Journal of Operational Research, Vol. 97, 997, pp E. Sanchez, T. Shibata, and L. A. Zadeh, ed., Genetic Algoriths and Fuzzy Logic Systes, Soft Coputing Perspectives, World Scientific, C. H. Wang, T. P. Hong, and S. S. Tseng, Integrating fuzzy knowledge by genetic algoriths, IEEE Transactions on Evolutionary Coputation, Vol., 998, pp J. S. Yao and D. C. Lin, Optial fuzzy profit for price in fuzzy sense, Fuzzy Sets

18 58 FENG-TSE LIN AND JING-SHING YAO and Systes, Vol.,, pp J. S. Yao and S. C. Chang, Econoic principle of profit in fuzzy sense, Fuzzy Sets and Systes, Vol.,, pp Feng-Tse Lin ( 林豐澤 ) was born in Taipei, Taiwan. He received the M.S. degree in coputer engineering fro the National Chiao-Tung University, Hsinchu, Taiwan, in 984, and the Ph.D. degree in coputer science and inforation engineering fro the National Taiwan University, Taipei, Taiwan, in 99. Fro 984 to 987, he was a research assistant at Telecounication Laboratories, Chung-Li, Taiwan. Currently, he is an Associate Professor with the Departent of Applied Matheatics, Chinese Culture University, Yanginshan, Taipei, Taiwan. In 997, he was a Visiting Associate Professor, Division of Engineering and Applied Sciences, Harvard University, Cabridge, MA. His current research interests include fuzzy logic and its applications, cobinational optiization probles, genetic algoriths, learning classifier systes, and intelligent agents. Dr. Lin is a eber of Taiwanese Association for Artificial Intelligence (TAAI), Chinese Fuzzy Systes Association Taiwan (CFSAT), Coputer Society of the ROC, and IEEE Systes, Man, and Cybernetics Society. Jing-Shing Yao ( 姚景星 ) was born in Taipei, Taiwan. He received the Ph.D. degree in Matheatics fro National Kyushu University, Japan, in 973. He has been with the Departent of Matheatics, National Taiwan University, Taipei, Taiwan, since 965. He was an associate professor in and as a professor since 967. He headed the Departent of Matheatics, National Taiwan University and also served as Director of Matheatical center, National Science Council, R. O. C., fro 976 to 978. Fro 993 to 998, he was with Departent of Applied Matheatics, Chinese Culture University, Yanginshan, Taipei, Taiwan. He is currently a Professor Eeritus at National Taiwan University, Taipei, Taiwan, and also an Honorary Professor at Manageent College, Takang University, Tasui, Taiwan. His current research interests focus on operation research, fuzzy atheatics and statistics. Dr. Yao has published ore than 5 theoretical papers in various international ournals in the above areas.