5. Fuzzy Optimization

Transcription

1 5. Fuzzy Optimization 1. Fuzzine: An Introduction Fuzzy Memberhip Function Memberhip Function Operation Optimization in Fuzzy Environment Fuzzy Set for Water Allocation Fuzzy Set for Reervoir Storage and Releae Target Fuzzy Set for Water Quality Management Summary Additional Reference (Further Reading) 144

2 135 5 Fuzzy Optimization The precie quantification of many ytem performance criteria and parameter and deciion variable i not alway poible, nor i it alway neceary. When the value of variable cannot be preciely pecified, they are aid to be uncertain or fuzzy. If the value are uncertain, probability ditribution may be ued to quantify them. Alternatively, if they are bet decribed by qualitative adjective, uch a dry or wet, hot or cold, clean or dirty, and high or low, fuzzy memberhip function can be ued to quantify them. Both probability ditribution and fuzzy memberhip function of thee uncertain or qualitative variable can be included in quantitative optimization model. Thi chapter introduce fuzzy optimization modelling, again for the preliminary creening of alternative water reource plan and management policie. 1. Fuzzine: An Introduction Large, mall, pure, polluted, atifactory, unatifactory, ufficient, inufficient, excellent, good, fair, poor and o on are word often ued to decribe variou attribute or performance meaure of water reource ytem. Thee decriptor do not have crip, well-defined boundarie that eparate them from other. A particular mix of economic and environmental impact may be more acceptable to ome and le acceptable to other. Plan A i better than Plan B. The water quality and temperature i good for wimming. Thee qualitative, or fuzzy, tatement convey information depite the impreciion of the italicized adjective. Thi chapter illutrate how fuzzy decriptor can be incorporated into optimization model of water reource ytem. Before thi can be done ome definition are needed Fuzzy Memberhip Function Conider a et A of real or integer number ranging from ay 18 to 25. Thu A [18, 25]. In claical (crip) et theory, any number x i either in or not in the et A. The tatement x belong to A i either true or fale depending on the value of x. The et A i referred to a a crip et.if one i not able to ay for certain whether or not any number x i in the et, then the et A could be referred to a fuzzy. The degree of truth attached to that tatement i defined by a memberhip function. Thi function range from 0 (completely fale) to 1 (completely true). Conider the tatement, The water temperature hould be uitable for wimming. Jut what temperature are uitable will depend on the peron aked. It would be difficult for anyone to define preciely thoe temperature that are uitable if it i undertood that temperature outide that range are abolutely not uitable. A memberhip function defining the interval or range of water temperature uitable for wimming i hown in Figure 5.1. Such function may be defined on the bai of the repone of many potential wimmer. There i a zone of impreciion or diagreement at both end of the range. The form or hape of a memberhip function depend on the individual ubjective feeling of the member or individual who are aked their opinion. To define thi particular memberhip function, each individual i could be aked to define hi or her comfortable water temperature interval (T 1i, T 2i ). The memberhip value aociated with any temperature value T equal the number of

3 136 Water Reource Sytem Planning and Management 1 f 1 m A B (x) E020108d C o E C o Figure 5.1. A fuzzy memberhip function for uitability of water temperature for wimming. Figure 5.3. Memberhip function for water temperature that are conidered too cold or too hot. E020108e 1 et A too cold et B too hot C o Figure 5.2. Two memberhip function relating to wimming water temperature. Set A i the et defining the fraction of all individual who think the water temperature i too cold, and Set B define the fraction of all individual who think the water temperature i too hot. individual who place that T within their range (T 1i, T 2i ), divided by the number of individual opinion obtained. The aignment of memberhip value i baed on ubjective judgement, but uch judgement eem to be ufficient for much of human communication Memberhip Function Operation Denote the memberhip function aociated with a fuzzy et A a m A (x). It define the degree or extent to which any value of x belong to the et A. Now conider two fuzzy et, A and B. Set A could be the range of temperature that are conidered too cold, and et B could be the range of temperature that are conidered too hot. Aume thee two et are a hown in Figure 5.2. The degree or extent that a value of x belong to either of two et A or B i the maximum of the two individual memberhip function value. Thi union memberhip function i defined a: m A B (x) maximum(m A (x), m B (x)) (5.1) Thi union et would repreent the range of temperature that are either too cold or too hot, a illutrated in Figure 5.3. The degree or extent that a value of a variable x i imultaneouly in both et A and B i the minimum of the two individual memberhip function value. Thi interection memberhip function i defined a: m A B (x) minimum (m A (x), m B (x)) (5.2) Thi interection et would define the range of temperature that are conidered both too cold and too hot. Of coure it could be an empty et, a indeed it i in thi cae, baed on the two memberhip function hown in Figure 5.2. The minimum of either function for any value of x i 0. The complement of the memberhip function for fuzzy et A i the memberhip function, m A c (x), of A c. m A c (x) 1 m A (x) (5.3) The complement of et A (defined in Figure 5.2) would repreent the range of temperature conidered not too cold for wimming. The complement of et B (alo defined in Figure 5.2) would repreent the range of temperature conidered not too hot for wimming. The complement of the union et a hown in Figure 5.3 would be the range of temperature conidered jut right. Thi complement et i the ame a hown in Figure Optimization in Fuzzy Environment Conider the problem of finding the maximum value of x given that x cannot exceed 11. Thi i written a: Maximize U x (5.4)

4 Fuzzy Optimization 137 U 20 m C 15 x = < 11 U = x 10 E020108j x E020108g maximum feaible value X of U, and x Figure 5.6. Memberhip function repreenting the vicinity of 11. m G (x) 1/{1 [1/(x 10) 2 ]} if x 10 m G (x) 0 otherwie (5.6) Figure 5.4. A plot of the crip optimization problem defined by Equation 5.4 and 5.5. Thi function i hown in Figure 5.5. The contraint on x i that it hould be in the vicinity of 11. Suppoe the reult of a poll aking individual to tate what they conider to be in the vicinity of 11 reult in the following contraint memberhip function, m C (x): m G m C (x) 1/[1 (x 11) 4 ] (5.7) E020108h x Figure 5.5. Memberhip function defining the fraction of individual who think a particular value of x i ubtantially greater than 10. ubject to: x 11 (5.5) The obviou optimal olution, x 11, i hown in Figure 5.4. Now uppoe the objective i to obtain a value of x ubtantially larger than 10 while making ure that the maximum value of x hould be in the vicinity of 11. Thi i no longer a crip optimization problem; rather, it i a fuzzy one. What i perceived to be ubtantially larger than 10 could be defined by a memberhip function, again repreenting the reult of an opinion poll of what individual think i ubtantially larger than 10. Suppoe the memberhip function for thi goal, m G (x), reflecting the reult of uch a poll, can be defined a: Thi memberhip function i hown in Figure 5.6. Recall the objective i to obtain a value of x ubtantially larger than 10 while making ure that the maximum value of x hould be in the vicinity of 11. In thi fuzzy environment the objective i to maximize the extent to which x exceed 10 while keeping x in the vicinity of 11. The olution can be viewed a finding the value of x that maximize the minimum value of both memberhip function. Thu, we can define the interection of both memberhip function and find the value of x that maximize that interection memberhip function. The interection memberhip function i: m D (x) minimum{m G (x), m C (x)} minimum{1/(1 [1/(x 10) 2 ]), 1/(1 (x 11) 4 )} if x 10 0 otherwie (5.8) Thi interection et, and the value of x that maximize it value, i hown in Figure 5.7. Thi fuzzy deciion i the value of x that maximize the interection memberhip function m D (x), or equivalently: Maximize m D (x) max min{m G (x), m C (x)} (5.9) Uing LINGO, the optimal olution i x and m D (x)

5 138 Water Reource Sytem Planning and Management E020108k m D contraint x 3. Fuzzy Set for Water Allocation Next conider the application of fuzzy modelling to the water allocation problem illutrated in Figure 5.8. Aume, a in the previou ue of thi example problem, the problem i to find the allocation of water to each firm that maximize the total benefit TB(X): Maximize TB(X) 6x 1 x 2 1 7x 2 1.5x 2 2 8x 3 x 2 3 (5.10) Thee allocation cannot exceed the amount of water available, Q, le any that mut remain in the river, R. Auming the available flow for allocation, Q R, i 6, the crip optimization problem i to maximize Equation 5.10 ubject to the reource contraint: x 1 x 2 x 3 6 (5.11) The optimal olution i x 1 1, x 2 1, and x 3 4 a previouly obtained in Chapter 4 uing everal different objective Figure 5.7. The interection memberhip function and the value of x that repreent a fuzzy optimal deciion optimization method. The maximum total benefit, TB(X), from Equation 5.10, equal To create a fuzzy equivalent of thi crip model, the objective can be expreed a a memberhip function of the et of all poible objective value. The higher the objective value the greater the memberhip function value. Since memberhip function range from 0 to 1, the objective need to be caled o that it alo range from 0 to 1. The highet value of the objective occur when there i ufficient water to maximize each firm benefit. Thi uncontrained olution would reult in a total benefit of and thi happen when x 1 3, x , and x 3 8. Thu, the objective memberhip function can be expreed by: m(x) 6x 1 x 2 1 7x 2 1.5x 2 2 8x 3 x (5.12) It i obviou that the two function (Equation 5.10 and 5.12) are equivalent. However, the goal of maximizing objective function 5.10 i changed to that of maximizing the degree of reaching the objective target. The optimization problem become: maximize m(x) 6x 1 x 2 1 7x 2 1.5x 2 2 8x 3 x ubject to: x 1 x 2 x 3 6 (5.13) The optimal olution of (5.13) i the ame a (5.10 and 5.11). The optimal degree of atifaction i m(x) Figure 5.8. Three waterconuming firm i obtain benefit B i from their allocation x i of water from a river whoe flow i Q. Q firm 1 B 1 = 6x x x1 x2 firm 3 B 3 = 8x 3 x3 2 x3 firm 2 B 2 = 7x 2 1.5x 2 2 E020108m

6 Fuzzy Optimization 139 m C variable value T 15.6 remark target torage for each period E020108n Q- R Figure 5.9. Memberhip function for about 6 unit more or le. E020827q S 1 S 2 S 3 R1 R2 R reervoir torage volume at beginning of period 1 reervoir torage volume at beginning of period 2 reervoir torage volume at beginning of period 3 reervoir releae during period 1 reervoir releae during period 2 reervoir releae during period 3 Next, aume the amount of reource available to be allocated i limited to about 6 unit more or le, which i a fuzzy contraint. Aume the memberhip function decribing thi contraint i defined by Equation 5.14 and i hown in Figure 5.9. m c (X) 1 if x 1 x 2 x 3 5 m c (X) [7 (x 1 x 2 x 3 )]/2 if 5 x 1 x 2 x 3 7 m c (X) 0 if x 1 x 2 x 3 7 (5.14) The fuzzy optimization problem become: Maximize minimum (m G (X), m C (X)) ubject to: m G (X) 6x 1 x 2 1 7x 2 1.5x 2 2 8x 3 x m C (X) [7 (x 1 x 2 x 3 )]/2 (5.15) Solving (5.15) uing LINGO to find the maximum of a lower bound on each of the two objective, the optimal fuzzy deciion are x , x , x , m(x) 0.67, and the total net benefit, Equation 5.10, i TB(X) Compare thi with the crip olution of x 1 1, x 2 1, x 3 4, and the total net benefit of Fuzzy Set for Reervoir Storage and Releae Target Conider the problem of trying to identify a reervoir torage volume target, T S, for the planning of recreation facilitie given a known minimum releae target, T R, and reervoir capacity K. Aume, in thi imple example, thee known releae and unknown torage target mut apply in each of the three eaon in a year. The objective will be to find the highet value of the torage target, T S, Table 5.1. The LINGO olution to the reervoir optimization problem. that minimize the um of quared deviation from actual torage volume and releae le than the minimum releae target. Given a equence of inflow, Q t, the optimization model i: 3 Minimize D (T S S t ) 2 DR 2 t 01T S (5.16) t ubject to: S t Q t R t S t 1 t 1, 2, 3; if t 3, t 1 1 (5.17) S t K t 1, 2, 3 (5.18) R t T R DR t t 1, 2, 3 (5.19) Aume K 20, T R 25 and the inflow Q t are 5, 50 and 20 for period t 1, 2 and 3. The optimal olution, yielding an objective value of 184.4, obtained by LINGO i lited in Table 5.1. Now conider changing the objective function into maximizing the weighted degree of atifying the reervoir torage volume and releae target. Maximize (w S m St w R m Rt ) (5.20) t where w S and w R are weight indicating the relative importance of torage volume target and releae target repectively. The variable m St are the degree of atifying torage volume target in the three period t, expreed by Equation The variable m Rt are the degree of atifying releae target in period t, expreed by Equation m S = { S t /T S for S t T S (5.21) (K S t )/(K T S ) for T S S t m R = R t /T R for R t T R (5.22) { 1 for Rt T R

7 140 Water Reource Sytem Planning and Management m S variabl e value degree 2.48 remark total weighted um memberhip function value E T 20 target torage volume E020108o 0 T K torage St S 1 S 2 S torage volume at beginning of period 1 torage volume at beginning of period 2 torage volume at beginning of period 3 Figure Memberhip function for torage volume. R1 R2 R reervoir releae in period 1 reervoir releae in period 2 reervoir releae in period 3 E020108p m R T R Figure Memberhip function for releae releae Rt M1 M2 M3 M1 M 2 M3 R M1 R M 2 R M um weighted memberhip value period 1 um weighted memberhip value period 2 um weighted memberhip value period 3 torage volume memberhip value period 1 torage volume memberhip value period 2 torage volume memberhip value period 3 reervoir releae memberhip value period 1 reervoir releae memberhip value period 2 reervoir releae memberhip value period 3 Box 5.1. Reervoir model written for olution by LINGO SETS: PERIODS /1..3/: I, R, m, m, mr, 1, 2, m1, m2; NUMBERS /1..4/: S; ENDSETS!*** OBJECTIVE ***; max = degree + 01*TS;!Initial condition; (1) = (TN + 1);!Total degree of atifaction; degree m(t));!weighted degree in period (PERIODS(t): m(t) = w*m(t) + wr*mr(t); S(t) = 1(t) + 2(t); 1(t) < TS ; 2(t) < K TS ;!m(t) = (1(t)/TS) (2(t)/(K TS)) = rewritten in cae dividing by 0; m1(t)*ts = 1(t); m2(t)*(k TS) = 2(t); m(t) = m1(t) m2(t); mr(t) < R(t)/TR ; mr(t) < 1; S(t+1) = S(t) + I(t) R(t);); DATA: TN = 3; K = 20; w =?; wr =?; I = 5, 50, 20; TR = 25; ENDDATA Equation 5.21 and 5.22 are hown in Figure 5.10 and 5.11, repectively. Thi optimization problem written for olution uing LINGO i a hown in Box 5.1. Given weight w S 0.4 and w R 0.6, the optimal olution obtained from olving the model hown in Box 5.1 uing LINGO i lited in Table 5.2. E020903d Table 5.2. Solution of fuzzy model for reervoir torage volume and releae baed on objective If the objective Equation 5.20 i changed to one of maximizing the minimum memberhip function value, the objective become: Maximize m min maximize minimum {m St, m Rt } (5.23) A common lower bound i et on each memberhip function, m St and m Rt, and thi variable i maximized. The optimal olution change omewhat and i a hown in Table 5.3. Thi olution differ from that hown in Table 5.2 primarily in the value of the memberhip function. The target torage volume operating variable value, T S, tay the ame in thi example. 5. Fuzzy Set for Water Quality Management Conider the tream pollution problem illutrated in Figure The tream receive wate from ource

8 Fuzzy Optimization 141 located at Site 1 and 2. Without ome wate treatment at thee ite, the pollutant concentration at Site 2 and 3 will exceed the maximum deired concentration. The problem i to find the level, x i, of watewater treatment (fraction of wate removed) at Site i 1 and 2 required to meet the quality tandard at Site 2 and 3 at a variable value remark MMF T S 1 S 2 S 3 R1 R2 R3 M1 M2 M3 R M1 R M2 R M minimum memberhip function value target torage volume torage volume at beginning of period 1 torage volume at beginning of period 2 torage volume at beginning of period 3 reervoir releae in period 1 reervoir releae in period 2 reervoir releae in period 3 torage volume memberhip value period 1 torage volume memberhip value period 2 torage volume memberhip value period 3 reervoir releae memberhip value period 1 reervoir releae memberhip value period 2 reervoir releae memberhip value period 3 Table 5.3. Optimal olution of reervoir operation model baed on objective E020827t minimum total cot. The data ued for the problem hown in Figure 5.12 are lited in Table 5.4. The crip model for thi problem, a dicued in the previou chapter, i: Minimize C 1 (x 1 ) C 2 (x 2 ) (5.24) ubject to: Water quality contraint at ite 2: max [P 1 Q 1 W 1 (1 x 1 )]a 12 /Q 2 P 2 [(32)(10) (1 x 1 )/86.4] 0.25/12 20 which, when implified, i: x Water quality contraint at ite 3: {[P 1 Q 1 W 1 (1 x 1 )]a 13 max [W 2 (1 x 2 )]a 23 }/Q 3 P 3 {[(32)(10) (1 x 1 )/86.4] 0.15 [80000(1 x 2 )/86.4] 0.60}/13 20 which, when implified, i: x x (5.25) (5.26) Retriction on fraction of wate removal: 0 x i for ite i 1 and 2 (5.27) For a wide range of reaonable cot, the optimal olution found uing linear programming wa 0.78 and 0.79, or eentially 80% removal efficiencie at Site 1 and 2. Compare thi olution with that of the following fuzzy model. To develop a fuzzy verion of thi problem, uppoe the maximum allowable pollutant concentration in the tream at Site 2 and 3 were expreed a about 20 mg/l or le. Obtaining opinion of individual of what firm 1 recreation producing W1 park treamflow Q ite 1 W1 ( 1 x1) ite 2 W2 ( 1 x2) ite 3 Figure A tream pollution problem of finding the wate removal efficiencie (x 1, x 2 ) that meet the tream quality tandard at leat cot. firm 2 producing W 2 E020108q

9 142 Water Reource Sytem Planning and Management Table 5.4. Parameter value elected for the water quality management problem illutrated in Figure parameter flow wate Q1 m 3 / 10 flow jut uptream of ite 1 Q 2 Q 3 W1 kg/day 250,000 pollutant ma produced at ite 1 W 2 unit m 3 / m 3 / kg/day value remark 12 flow jut uptream of ite 2 13 flow at park 80,000 pollutant ma produced at ite 2 E020827u pollutant conc. P1 mg/l 32 concentration jut uptream of ite 1 P 2 P 3 mg/l mg/l 20 maximum allowable concentration uptream of 2 20 maximum allowable concentration at ite 3 decay fraction a 12 a 13 a fraction of ite 1 pollutant ma at ite fraction of ite 1 pollutant ma at ite fraction of ite 2 pollutant ma at ite 2 m m τ E020108r concentration E % treatment efficiency x i Figure Memberhip function for about 20 mg/l or le. Figure Memberhip function for bet available treatment technology. they conider to be 20 mg/l or le, a memberhip function can be defined. Aume it i a hown in Figure Next, aume that the government environmental agency expect each polluter to intall bet available technology (BAT) or to carry out bet management practice (BMP) regardle of whether or not thi i required to meet tream-quality tandard. Aking expert jut what BAT or BMP mean with repect to treatment efficiencie could reult in a variety of anwer. Thee repone can be ued to define memberhip function for each of the two firm in thi example. Aume thee memberhip function for both firm are a hown in Figure Finally, aume there i a third concern that ha to do with equity. It i expected that no polluter hould be required to treat at a much higher efficiency than any other polluter. A memberhip function defining jut what difference are acceptable or equitable could quantify thi concern. Aume uch a memberhip function i a hown in Figure Conidering each of thee memberhip function a objective, a number of fuzzy optimization model can be defined. One i to find the treatment efficiencie that maximize the minimum value of each of thee memberhip function. Maximize m max min{m P, m T, m E } (5.28)

10 Fuzzy Optimization 143 m E variable M value 0.93 remark minimum memberhip value E020827v E020108t If we aume that the pollutant concentration at ite j 2 and 3 will not exceed 23 mg/l, the pollutant concentration memberhip function m Pj are: m Pj 1 p 2j /5 (5.29) The pollutant concentration at each ite j i the um of two component: p j p 1j p 2j (5.30) where p 1j 18 (5.31) p 2j (23 18) (5.32) If we aume the treatment plant efficiencie will be between 70 and 90% at both Site i 1 and 2, the treatment technology memberhip function m Ti are: m Ti (x 2i /5) (x 4i /0.10) (5.33) and the treatment efficiencie are: x i 0.70 x 2i x 3i x 4i (5.34) where x 2i 5 (5.35) x 3i 5 (5.36) x 4i 0.10 (5.37) Finally, auming the difference between treatment efficiencie will be no greater than 14, the equity memberhip function, m E, i: m E Z (/5) D 1 (1 Z) (/(0.14 5)) D 2 (5.38) where D 1 5Z (5.39) x 1 - x 2 Figure Equity memberhip function in term of the abolute difference between the two treatment efficiencie. 14 X1 X2 P2 P3 p M 2 p M treatment efficiency at ite 1 treatment efficiency at ite 2 pollutant concentration jut uptream of ite 2 pollutant concentration jut uptream of ite 3 memberhip value for pollutant concentration ite 2 memberhip value for pollutant concentration ite 3 MT memberhip value for treatment level ite 1 MT memberhip value for treatment level ite 2 ME 0 memberhip value for difference in treatment Table 5.5. Solution to fuzzy water quality management model Equation 5.28 to D 2 (0.14 5) (1 Z) (5.40) x 1 x 2 DP DM (5.41) DP DM D 1 5(1 Z) D 2 (5.42) Z i a binary 0, 1 variable. (5.43) The remainder of the water quality model remain the ame: Water quality contraint at ite 2: [P 1 Q 1 W 1 (1 x 1 )] a 12 /Q 2 P 2 (5.44) [(32)(10) (1 x 1 )/86.4] 0.25/12 P 2 Water quality contraint at ite 3: {[P 1 Q 1 W 1 (1 x 1 )] a 13 [W 2 (1 x 2 )] a 23 }/Q 3 P 3 (5.45) {[(32)(10) (1 x 1 )/86.4] 0.15 [80000(1 x 2 )/86.4] 0.60}/13 P 3 Retriction on fraction of wate removal: 0 x i for ite i 1 and 2. (5.46) Solving thi fuzzy model uing LINGO yield the reult hown in Table 5.5. Thi olution confirm the aumption made when contructing the repreentation of the memberhip function in the model. It i alo very imilar to the leat-cot olution found from olving the crip LP model.

11 144 Water Reource Sytem Planning and Management 6. Summary Optimization model incorporating fuzzy memberhip function are ometime appropriate when only qualitative tatement are made when tating objective and/or contraint of a particular water management problem or iue. Thi chapter ha hown how fuzzy optimization can be applied to ome imple example problem aociated with water allocation, reervoir operation, and pollution control. Thi ha been only an introduction. Thoe intereted in more detailed explanation and application may refer to any of the additional reference lited in the next ection. 7. Additional Reference (Further Reading) BARDOSSY, A. and DUCKSTEIN, L Fuzzy rulebaed modeling with application to geophyical, biological, and engineering ytem. Boca Raton, Fla., CRC Pre. CHEN, S.Y Theory and application of fuzzy ytem deciion-making. Dalian, China, Dalian Univerity of Technology Pre. (In Chinee.). KINDLER, J Rationalizing water requirement with aid of fuzzy allocation model. Journal of Water Reource Planning and Management, ASCE, Vol. 118, No. 3, pp KUNDZEWICZ, W. (ed.) New uncertainty concept in hydrology and water reource. Cambridge, UK, Cambridge Univerity Pre. LOOTSMA, F.A Fuzzy logic for planning and deciion-making. Boton, Ma., Kluwer Academic. TERANO, T.; ASAI, K. and SUGENO, M Fuzzy ytem theory and it application. San Diego, Calif., Academic Pre. TILMANT, A.; VANCLOSSTER, M.; DUCKSTEIN, L. and PERSOONNS, E Comparion of fuzzy and nonfuzzy optimal reervoir operating policie, Journal of Water Reource Planning and Management, ASCE, Vol. 128, No. 6, pp ZHOU, H.-C Note on fuzzy optimization and application. Dalian, China, Dalian Univerity of Technology Pre. ZIMMERMANN, H.-J Fuzzy et, deciion-making, and expert ytem. Boton, Ma., Kluwer Academic.

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