A STEP Method based multiple objective methodology for irrigation water management to model preferences and tradeoffs

Transcription

1 A STEP Method baed multiple obective methodology or irrigation water management to model preerence and tradeo Xevi, E.,3 and S. Khan,2,3 CSIRO Land and Water, Griith, NSW 2 Charle Sturt niverity, Wagga Wagga, NSW 3 Cooperative Reearch Centre or Irrigation Future emmanuel.xevi@ciro.au Keyword: Spatial ield, imilarity, error, tolerance, comparion EXTENDED ABSTRACT Management deciion relating to river bain are diicult due to the complex interaction o biophyical, economic and ocial variable. Moreover, conlicting obective o dierent taeholder have to be conidered. Lac o adequate inormation to determine water allocation, environmental low requirement and the increaing intenity o cropping ytem require a better eaonal ditribution o water to atiy conumptive and in-tream environmental demand. Water demand management that conider ytem contraint on water conveyance and loe in addition to environment requirement will reult in optimum productivity o irrigation area and better management o river low. Many deciion upport ytem in agricultural enterprie ue conventional linear programming approach to optimize a ingle obective unction uch a total gro margin. However, a agricultural ytem become more complex, multiple obective that are in conlict with each other need to be addreed. Mathematical programming technique are required to ormulate the problem and ind a compromie olution. A methodology baed on the technique o STEM (Step Method) i provided that allow or the progreive articulation o preerence. Thi i an iterative procedure that narrow the region on the eiciency rontier in which the inal compromie olution i ound. The procedure involve iteration where preerence are baed on the obective pace o previou iteration. The deciion maing proce i entirely in the obective pace and reult are preented in the orm o graph and table o obective value and utility value. A utility unction i ued to elect the bet obective unction at each iteration and can tae variou orm (linear, convex or concave curve). The choice o the utility unction i ubective and there i cope or invetigating variou utility unction. We have adopted a linear utility unction in thi tudy. The technique i applied to a hypothetical nodal networ hown in Figure. For the purpoe o thi paper, the irrigated area i divided into eight region with a total irrigable land o 2,808 ha and a potential or growing ourteen crop (rice, wheat, oat, barley, maize, canola, oybean, winter pature, ummer pature, lucerne, vine, ummer vegetable, winter vegetable, citru and tone ruit). In the current analyi groundwater pumping under the irrigable area i permitted to atiy crop water demand i urace water upplie are not uicient. Thereore the model ha the potential to develop conunctive water management option or achieving a better demand pattern rom the urace water. The deciion maer participate ully in the proce by tipulating her preerence and accepting a compromie olution. Berembed Weir Murrumbidgee River D D2 D3 D4 D8 D7 D6 Figure : Conceptual nodal networ repreentation D5 49

2 . INTRODCTION A the world natural reource dwindle, competition or them increae at a rapid pace. In the cae or water reource, competition tem rom arm level to regional and even national level oten with conlicting goal and obective. Sutainable irrigation management remain a daunting ta (Khan et al. 2006). Irrigation induced waterlogging and alinity reduce agricultural production and impoe other economi ocial, and the environmental cot (ee Hou et al. 2005; Wicheln 2002; Ragan et al. 2000; Wicheln 999; Huain et al. 2004). The cot are oten not conined to a paddoc or private property but pread over a wider cale. There can be a range o o-arm impact including damage to road and inratructure, lo o aethetic value and reduced biodiverity etc (ee Characli et al. 2005). Other impact include diplacement o labour, out migration, income diparity, and overall reduction in ood output. Thee o-ite cot or externalitie have important policy implication regarding land and water reource management and how tax dollar are deployed to mitigate alinity and create ocial beneit or all. Many deciion upport ytem in agricultural enterprie ue conventional linear programming approach to optimize a ingle obective unction uch a total gro margin. However, a agricultural ytem become more complex, multiple obective that are in conlict with each other need to be addreed. Irrigation demand management that conider urace and ground water contraint on water conveyance and loe in addition to environment requirement will reult in optimum productivity o irrigation area and better management o river low. Conequently multi-criteria deciion maing technique (MCDM) are neceary to adequately addre thee complexitie. A multi-criteria approach ha been ued extenively to olve divere deciion problem including ri aement in agricultural ytem (Berbel, 993). Furthermore, Tecle, (998) ued Compromie Programming (CP) to develop a multi-obective deciion upport ytem or analying multi-reource oret management problem. A method nown a the STEM (Step Method) decribed in Alan and Shamir (980) and due to Benayoun et al. (97) are nown a the generating method where ingle obective optimiation are contructed and point are generated on the non-inerior et and one o thee point i elected a a compromie olution. ing thi method the concept o optimal olution in ingle obective problem that i generally unique i replaced by the concept o nondominated olution (i.e. eaible olution or which no improvement in any obective unction i poible without acriicing at leat one o the other obective unction). The multi-obective problem that thi model addree comprie three obective unction: maximizing net return (NR), minimizing variable cot (VC) and minimizing total upplementary groundwater pumping requirement to meet crop demand rom the irrigated area. The management option to achieve the above obective conit o election o an appropriate mix o crop, optimum level o groundwater pumping and appropriate allocation o water or irrigation and environment. Contraint impoed on the ytem are: continuity total arm area monthly water allocation monthly environmental low requirement monthly groundwater pumping In addition, water allocation rule and pumping target or each month are contraint impoed on the ytem. Input variable required conit o: monthly rainall monthly crop water requirement crop growth duration crop actor yield, price and variable cot o crop In previou wor (Xevi and Khan 2003, Xevi and Khan (2005)), we attempted to ind olution to the multiple obective problem uing goal programming with weight attached to the obective unction. In thi paper we decribe a method (STEM) that ee a compromie olution to multiple obective problem that include bioeconomic obective with the optimum ue o water reource under conlicting demand. nlie other method ued to olve multi-obective problem, STEM i interactive and involve the deciion maer cloely. While other method tend to ind olution at the extreme point o the eaible region, STEM ind non-extreme point eicient olution by exploring the entire eaible region. A compromie olution acceptable to the deciion maer can be ound in thi interactive 50

3 method. The method wa applied to a conceptual nodal networ repreentation o part o the Murrumbidgee irrigation area in Autralia. Further detail about the data ued in the model can be ound in Xevi and Khan (2003), Xevi and Khan (2005). 2. METHOD AND THEORY In general, multiple obective problem can be ormulated with the ollowing equation (ee Alan and Shamir, 980) Max ( x) = [ ( x) = c x;...; ubect to Ax b and x 0 p ( x) = c () where x i an n-dimenional vector o deciion variable, A i matrix o technology variable, b i a vector o orcing variable and i a vector o p-dimenional obective unction and and c i a vector o coeicient. Taing x a the eaible region in the deciion pace or equation and M a the maximum value attainable by olving only obective ubect to the contraint, then M { ( x) } = Max (2) x X The optimal deciion variable at thi point i given by { x ( x M } x = ) = (3) The other obective unction will tae on value given by: = x ) (4) ( The method o the STEM procedure involve the ollowing tep. Firt, p ingle obective unction are olved ubect the contraint. Each olution produce an optimal deciion point x and an obective value ( x ). Second, deine the minimum value m that i attainable by obective a: p x] m = Min ( ) (5) [m, M ] give the range o value in which the inal value o obective i expected to lie ubect to the contraint and the eect o all the other competing obective. Third, deine a utility unction,, over the range [m, M ] or each obective. Thee utility unction can tae dierent hape (ee Figure 2) and i largely dependent on the preerence o the deciion maer. They relect the weight aigned by the deciion maer to achieve dierent proportion o the range [m, M ]. The range o i taen to be [0,] and i deined a: (m ) = 0 (6) (M ) =. (7) m b tility, ( ) a c M Figure 2: Shape o the utility unction,. Adopting the linear utility unction, the utility can be deined a: m ( = M m (8) = ) We aume that the above orm o the utility unction i a true repreentation o the utility and urther aume that they are additive. The bet olution among the p ingle obective olution i elected uing the ollowing relationhip: Max ( T ( )) = Max( [ ( x )]) (9) p = I we denote the obective unction that reult rom equation 9 by then T = T ( ) (0) 5

4 T range rom 0 to p and repreent the maximiation o the obective unction that produce the maximum utility according to equation 8. The obective unction reache it maximum poible value M and the other obective have value given by. We now begin an iterative procedure where p ingle obective olution are ought with additional contraint. Except or the obective, the ollowing contraint are added to equation. [ () ( x )] n + [ And or the obective : ( x [ ( x)] + x ] (2) )] n [ ( )] n [ Δ where n i the iteration number and Δ i an appropriate adutment to the obective value to allow room or the other obective to improve. Δ i elected and revied at each iteration to atiy eaibility o the optimiation procedure a well a conidering the preerence o the deciion maer. n n 3. OBJECTIVE FNCTIONS AND CONSTRAINTS The multi-obective problem decribed in thi paper conit o three obective unction: maximizing net return (NR), minimizing variable cot (VC) and minimizing total upplementary groundwater pumping requirement (TP) to meet crop demand rom the irrigated area. Conceptually, NR and VC may repreent the view o agriculturalit while minimizing total pumping may be the deired goal to avoid groundwater mining and pollution o aquier in ituation where vertical egregation o aquier alinity occur. The management option to achieve the above obective conit o election o an appropriate mix o crop, optimum level o groundwater pumping and appropriate allocation o water or irrigation and environment. Contraint impoed on the ytem include eaonal environmental low target. In addition, water allocation rule and pumping target or each month are contraint impoed on the ytem. The three obective unction were ormulated a ollow: Max NR = CGM ( c) X ( c) { WREQ( c c m X ( c) C } C P( (3) w p c m Min VC = ( X ( c) WREQ( C ) cm w + X ( c) Vcot( c) (4) c Min TP = P( cm (5) where X(c) = area o crop c (Ha), CGM(c) = gro margin or crop c ($), WREQ( = water requirement or crop c in month m (ML), C w = total cot o water per unit volume ($/ML), C p = cot o groundwater pumping and delivery ($/ML), Vcot = variable cot (uch a ertilizer and peticide application) per hectare other than water cot or crop c and P( = volume o ground water pumped rom irrigation area or crop c in month m (ML). Detailed decription o contraint can be ound in Xevi and Khan (2005). 3.. THE HYPOTHETICAL IRRIGATION AREA For the purpoe o thi paper, the irrigated area i divided into eight region (Figure 3) with a total irrigable land o 2,808 ha and a potential or growing ourteen crop (rice, wheat, oat, barley, maize, canola, oybean, winter pature, ummer pature, lucerne, vine, ummer vegetable, winter vegetable, citru and tone ruit). In the current analyi groundwater pumping rom the irrigable area i permitted to atiy crop water demand i urace water upplie are not uicient. Licened bore are located within the ub-catchment that ue water or to dometic ue and irrigation Detailed decription o inlow, environmental requirement, crop and arm area can be ound in (Xevi and Khan, 2003; Xevi and Khan, 2005) 4. RESLTS Three linear programming olution were obtained at each iteration o the STEM proce uing GAMS otware (Brooe et al., 998), which correpond to the three obective unction: maximizing net return (NR), minimizing variable cot (VC) and minimizing total upplementary groundwater pumping requirement (TP). 52

5 Table. Initial pay-o matrix o ingle optimiation o obective. Obective ymbol 2 3 NR(million$) VC(million$) TP (ML) M m D Table how the reult o the initial iteration (n=0) o ingle obective olution and their correponding. Alo, hown in the table i the maximum M, minimum m and the range o obective pace D = M m. Table 2 how the value o the utility unction given in equation 8 and the ummation a i given in equation 9. From table 2, obective 3 (TP) ha the highet ummation value o.9 and i thereore the bet obective choen or the next iteration. Table 2: tility at initial tage o iteration (n=0). Table 3 how the ingle obective olution with contraint and 2 impoed. The adutment impoed on obective 3 (TP) i 4% o it range, D = i.e. Δ = 4.3. Table 4 how the correponding value o the utility unction. At thi tage obective (NR) i the bet obective according to the criterion peciied in equation 9 and 0. Thi proce wa continued to the third iteration and the progre i hown in Figure 4, 5 and 6 a the iteration marche toward a compromie olution. At iteration 3 (not hown) the inal compromie olution wa NR = $5.95 million, TC = $66.52 million and TP = ML and reult rom minimiation o total pumping (TP). At thi point the net revenue (NR) only reached 7.5% o it poible range; TC reached 82% o it poible range. Table 3. Pay-o matrix at econd iteration o ingle optimiation o obective. Value o obective and 2 are in million while obective 3 i in thouand o dollar Obective ymbol 2 3 NR(million$) Obective / = ( m ) D 2 VC(million$) TP(ML) ymbol 2 3 M NR 0 0 m D VC TP

6 Table 4. tility at econd iteration (n=) Obective / = ( m ) D ymbol 2 3 NR Total Pumping (ML) Thouand (TP) Iteration # 0 2 VC TP Figure 6. Iteration progre or TP (dotted line repreent upper limit or obective unction value M and the olid line repreent lower limit o obective unction value, 5. CONCLSION Total Net Proit Million 98 (NR) Iteration # Figure 4. Iteration progre or NR (dotted line repreent upper limit or obective unction value M and the olid line repreent lower limit o obective unction value, (m ) Total Cot Million (TC) Iteration # Figure 5. Iteration progre or TC (dotted line repreent upper limit or obective unction value M and the olid line repreent lower limit o obective unction value, m ) A ramewor i preented rom which multiple deciion that are in conlict can be made with tradeo. The tradeo are entirely under the control o the deciion maer and i able to direct preerence or one or the other obective. Point on the non-inerior et are generated through a et o ingle obective unction and contraint and one o thee point i elected a the compromie olution. The procedure involve iteration where preerence are baed on the obective pace o previou iteration. The deciion maing proce i entirely in the obective pace and reult are preented in the orm graph and table o obective value and utility value. The choice o the value or aduting the obective value i completely ubective and need taeholder input which aect the inal compromied olution. A linear utility unction wa arbitrarily adopted in thi tudy but there i cope or invetigating and comparing the reult rom other utility unction. There i an extenive deciion pace that i not included in thi paper. The deciion pace may be ued by deciion maer to aect their articulation o preerence but by ar the obective pace predominate in arriving at thee preerence partly becaue their dimenion are coniderably lower and can be analyed more eaily. 6. ACKNOWLEDGEMENTS The author wih to acnowledge unding upport rom the CRC or Irrigation Future and the Water or a Healthy Country Flaghip. 54

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