An inexact optimization approach for river water-quality management

Transcription

1 Journa of Environmenta Management 81 (2006) An inexact optimization approach for river water-quaity management Subhankar Karmakar, P.P. Mujumdar Department of Civi Engineering, Indian Institute of Science, Bangaore , India Received 7 March 2005; received in revised form 19 Juy 2005; accepted 30 October 2005 Avaiabe onine 20 March 2006 Abstract A previousy deveoped fuzzy waste oad aocation mode (FWLAM) for a river system is extended to address uncertainty invoved in fixing the membership functions for the fuzzy goas of the poution contro agency (PCA) and the dischargers using the concept of grey systems. The mode provides fexibiity for the PCA and the dischargers to specify their goas independenty, as the parameters for membership functions are considered as interva grey numbers instead of deterministic rea numbers. An inexact or a grey fuzzy optimization mode is deveoped in a mutiobjective framework, to maximize the width of the interva vaued fractiona remova eves for providing atitude in decision-making and to minimize the width of the goa fufiment eve for reducing the system uncertainty. The concept of an acceptabiity index for order reation between two partiay or fuy overapping intervas is used to get a deterministic equivaent of the grey fuzzy optimization mode deveoped. The improvement of the optima soutions over a previousy deveoped grey fuzzy waste oad aocation mode (GFWLAM) is shown through an appication to a hypothetica river system. The fuzzy mutiobjective optimization and fuzzy goa programming techniques are used to sove the deterministic equivaent of the GFWLAM. r 2006 Esevier Ltd. A rights reserved. Keywords: Fuzzy mathematica programming; Grey systems theory; Uncertainty; Water-quaity management 1. Introduction Corresponding author. Te.: , ; fax: E-mai address: pradeep@civi.iisc.ernet.in (P.P. Mujumdar). URL: A waste oad aocation (WLA) mode for decisionmaking in water-quaity management of a river system, in genera, integrates a water-quaity simuation mode with an optimization mode to provide best compromise soutions acceptabe to both the poution contro agency (PCA) and dischargers. A number of WLA modes have been deveoped in the past for optima aocation of assimiative capacity of a river system (Loucks and Lynn, 1966; Loucks et a., 1981; Loucks, 1983; Fugiwara et a., 1988; Tung and Hathhorn, 1989). Uncertainty due to randomness has been addressed extensivey in the modes for water-quaity management of river systems, starting with the pioneering work of Loucks and Lynn (1966). Another type of uncertainty in water resources probems is the imprecision in management goas and mode parameters, which has been in genera addressed with fuzzy sets (Bogardi et a., 1983; Bardossy and Disse, 1993; Shreshta et a., 1996; Fontane et a., 1997; Teegavarapu and Simonovic, 1999). The concept of fuzzy decision (Beman and Zadeh, 1970) has been used in water-quaity management probems in recent work (e.g., Hathhorn and Tung, 1989; Chang et a., 1997; Sasikumar and Mujumdar, 1998, 2000; Cark, 2002; Mujumdar and Subbarao, 2004; Ning and Chang, 2004; Subbarao et a., 2004). Sasikumar and Mujumdar (1998) deveoped a fuzzy waste oad aocation mode (FWLAM) for water-quaity management of a river system, to incorporate the imprecisey defined conficting goas of PCA and dischargers in a fuzzy optimization framework. The soution provides a set of optima fractiona removas ( ^X) of the poutants and the maximum vaue of goa fufiment eve (^), which is a measure of the degree of fufiment of the fuzzy goas. A major imitation in the modes of Sasikumar and Mujumdar (1998, 2000), and Mujumdar and Sasikumar (2002) is that the membership parameters for management goas are assumed fixed and vaues are assigned based on some judgement (such as, for exampe, a ower bound of /$ - see front matter r 2006 Esevier Ltd. A rights reserved. doi: /j.jenvman

2 234 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) mg/l and an upper bound of 9 mg/l for DO concentration). Choice of appropriate vaues of membership parameters is an important issue in any fuzzy optimization mode, as these are highy subjective on the choice made by the decision-maker. In FWLAM, the parameters of the membership functions defining the goas of PCA depend on the desirabe and maximum permissibe eve of waterquaity. In practica situations different water-quaity standards for surface water are used for different uses for a water-quaity indicator. For exampe, standards for pubic water suppy, industria water suppy, agricutura water suppy, fish propagation and wid ife may a be different for the same water-quaity indicator, DO (Hammer, 1986). This resuts in an uncertainty in the membership parameters and eads to a second eve of fuzziness in the mode, with the membership functions themseves being imprecisey stated. Karmakar and Mujumdar (2004a, b) deveoped a grey fuzzy waste oad aocation mode (GFWLAM), which considers the uncertainty in the bounds and shape of the membership functions in fuzzy optimization modes for water-quaity management. The mode is aimed at reaxing the membership parameters by treating them as interva grey numbers. A terminoogy imprecise membership function is used to represent the membership functions with uncertain parameters. Consideration of imprecise membership functions in the fuzzy optimization mode imparts fexibiity in the soutions as the soutions are obtained as interva grey numbers. A set of interva optima fractiona remova eves ( ^X ) are obtained corresponding to a range of optima vaues of goa fufiment eve (^ ), whereas FWLAM gives ony a singe set of optima fractiona remova eves of poutants corresponding to the maximum goa fufiment eve. With the GFWLAM soutions, the fractiona remova eves can be adjusted within their optima grey intervas by the decision-maker in the fina decision scheme as required in a particuar situation. The width of the interva optima fractiona remova eve thus pays an important roe in the GFWLAM, as more width in the fractiona removas impies a wider choice to the decision-makers. In the present work a modified GFWLAM is deveoped to maximize the width of the interva vaued fractiona remova eves of poutants [i.e., (X þ X )] using an inexact or grey fuzzy optimization technique (Huang et a., 1993; Huang and Loucks, 2000) in a mutiobjective framework. This enhances the fexibiity and appicabiity in decision-making, as the decision-maker gets a wider range of optima fractiona removas than those of soutions obtained from GFWLAM (Karmakar and Mujumdar, 2004a, b). Simiar to GFWLAM, the upper and ower bounds of the goa fufiment eve (i.e., þ and ) are maximized, but in the modified GFWLAM the system uncertainty has aso been reduced by minimizing the width of the degree of goa fufiment eve [i.e., ( þ )]. In GFWLAM the order of consideration of the goas of PCA and the dischargers, aong with the seection of bounds of decision variabes create two different situations of mode formuation, which are termed as Cases 1 and 2. Each case is further subdivided into two submodes to obtain two extreme vaues of, which give the soutions for two extreme cases encompassing a intermediate possibiities. Therefore, in GFWLAM, four submodes are soved to obtain the range of optima fractiona remova eves. The present work deveops a mutiobjective optimization mode, which maximizes the width of the interva vaued fractiona remova eves and reduces the system uncertainty invoved in the optimization mode by minimizing the width of the interva vaued goa fufiment eve in a singe optimization mode, avoiding intermediate submodes. The next section presents a brief description of GFWLAM, deveoped in the earier work. 2. Grey fuzzy waste oad aocation mode (GFWLAM) A genera river system is considered for deveoping the GFWLAM. The system consists of a set of dischargers, who are aowed to reease poutants into the river after removing some fraction of the poutants. Fractiona remova is necessary to maintain acceptabe water-quaity conditions in the river as prescribed by the PCA. Acceptabe water-quaity conditions are ensured by checking the water-quaity in terms of water-quaity indicators (e.g., DO-deficit, hardness, nitrate nitrogen concentration) at a finite number of ocations, which are referred to as checkpoints. The goas of PCA are to ensure that poution is within an acceptabe imit by imposing water-quaity and effuent standards. On the other hand, the dischargers prefer to use the assimiative capacity of the river system to minimize the waste treatment cost, by assigning an aspiration eve (minimum desirabe treatment) and maximum fractiona remova eve for different poutants. These goas are imprecise and subjective, and are addressed in the mode through a fuzzy mathematica framework by assigning membership functions. The concentration of a water-quaity indicator is expressed as a function of the fractiona remova eves of the poutants using an appropriate poutant transport mode (such as, for exampe Streeter-Pheps mode, QUAL2E, WASP4, etc.). It is assumed, in the mode deveopment that: (1) steadystate fow conditions prevai; an instantaneous mixing of the poutants takes pace at the discharge point, (2) river characteristics are homogeneous and the river parameters for a river reach remain unchanged, (3) water-quaity indicators are such that the desirabe eve is ess than the permissibe eve (eg. DO-deficit, toxic poutant concentration, etc.), (4) a poutant affects one or more than one water-quaity indicator; but the poutants are chemicay non-reactive with each other, and (5) industria and municipa effuents are pretreated at the site prior to discharge into the river. Simiar to FWLAM, confict between the fuzzy goas of PCA and dischargers is modeed using the concept of fuzzy decision, but because of treating the membership parameters as interva grey numbers, the notion of fuzzy decision eads to the notion of grey

3 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) fuzzy decision. A terminoogy grey fuzzy decision is used to represent the fuzzy decision resuting from the imprecise membership functions. This terminoogy was used earier by Luo et a. (1999) to define a grey fuzzy motion decision combining grey prediction and fuzzy ogic contro theories. The notion of grey fuzzy decision presented in this paper is different from that used by Luo et a. (1999). An overview of the basic concepts of grey systems, fuzzy decision and grey fuzzy decision is given in the foowing subsections, as a prerequisite to the deveopment of GFWLAM Grey systems theory Grey systems theory was first proposed by Deng (1982). A grey system is a system other than white (system with competey known information) and back (system with competey unknown information) systems, and thus has partiay known and partiay unknown characteristics (Liu and Lin, 1998). In reaity, many processes of interest in environmenta management are in the grey stage due to inadequate and fuzzy information. A grey number is such a number whose exact vaue is unknown but a range within which the vaue ies is known (Liu and Lin, 1998; Yang and John, 2003). Let a denote a cosed and bounded set of rea numbers. A grey number (a 7 ) is defined as an interva with known ower (a ) and upper (a + ) bounds but unknown distribution information for a (Huang et a., 1995). a ¼ a ; a þ ¼ t 2 aja t a þ. (1) a 7 becomes a deterministic number or white number when a 7 ¼ a ¼ a +.Whena 7 ¼ [a, a + ] ¼ (-p, +p), a 7 is caed a back number. An interva number (Moore, 1979) or interva grey number (a 7 ¼ [a, a + ]) is one among severa casses of grey numbers (Liu and Lin, 1998). The grey degree is a measure, usefu for quantitativey evauating the quaity of uncertain input or output information for mathematica modes (Huang et a., 1995). The grey degree of an interva grey number is defined as its width [a o ¼ (a + a )] divided by its whitened mid vaue (WMV) [a m ¼ 1 2 (a +a + )] (Huang et a., 1995), and is expressed in percentage (%) as foows: Gdða Þ¼ða o =a m Þ100%, (2) where Gd(a 7 ) is the grey degree of a 7. Mode outputs with consideraby high grey degree have high width (a o ) of output variabes, which are considered as ess usefu and of poor quaity by the decision-makers. As the grey degree of the objective function of an optimization mode decreases, impying decreasing system uncertainties, the effectiveness of the grey mode increases. Therefore, a ower vaue of the grey degree of the optima objective function (e.g., goa fufiment eve,, of a fuzzy optimization mode) resuts in a more acceptabe grey soution. It shoud be noted for carification that the notion of grey systems modeing is different from the notion of sensitivity anaysis. The sensitivity anaysis is considered as a post-optimaity anaysis, where systematic variation of input parameters, considering variation of a singe parameter (i.e., univariate sensitivity anaysis) or a group of parameters (i.e., mutivariate sensitivity anaysis) at a time in a mode is performed to assess the effect of uncertainties or variation in these parameters on the mode output. But the grey systems modeing directy addresses the uncertainties of a uncertain mode parameters in a singe mathematica framework and gives the soutions as stabe intervas, which can be directy used for generating decision aternatives Fuzzy decision The fuzzy decision (Z) for a water-quaity management probem may be defined using the genera concept of fuzzy decision proposed by Beman and Zadeh (1970). They proposed, a broad definition of the fuzzy decision as a confuence of fuzzy goas and fuzzy constraints. The term confuence is context dependent and for a particuar situation a justifiabe aggregation shoud be performed to determine a meaningfu fuzzy decision. In the water-quaity management probem it is appropriate that the aggregating function shoud correspond to the aggregation operation ogica and, which corresponds to the set theoretic intersection. Noting that the decision space is defined by the intersection of different fuzzy goas, the fuzzy decision (Z) is written as foows: Z ¼ F 1 \ F 2, (3) where fuzzy sets F 1 and F 2 represent the two fuzzy goas. The membership function of the fuzzy decision (Z) is given by m Z ðxþ ¼ ¼ min½m F 1 ðxþ; m F 2 ðxþš, (4) where is the measuring variabe corresponding to the membership function of fuzzy decision (Z), which refects the degree of fufiment of the system goas. The term goa fufiment eve is used throughout the paper to represent the variabe. x is the argument of the fuzzy goas F 1 and F 2. The soution ^x corresponding to the maximum vaue of the membership function of the resuting Z is the optimum soution. That is m Z ð ^xþ ¼^ ¼ max ½m ZðxÞŠ, (5) x2z where ^ is the optima goa fufiment eve. Fig. 1 shows the concept of a fuzzy decision, where F 1 and F 2 are nonincreasing and non-decreasing membership functions, respectivey Grey fuzzy decision In the concept of a fuzzy decision, the arguments of F 1 and F 2 are deterministic rea numbers (x). When the goas F 1 and F 2 are imprecise fuzzy goas or grey fuzzy goas and the corresponding arguments are interva grey

4 236 numbers (x 7 ), the fuzzy decision eads to a grey fuzzy decision. Fig. 2 iustrates the concept of grey fuzzy decision considering the confuence of two imprecise membership functions for F 1 and F 2. Considering ogica and, corresponding to the set theoretic intersection as an aggregation operator, the grey fuzzy decision is determined. In Fig. 2 the decision Z 7 is not a fixed space (as shown in Fig. 1 for a fuzzy decision). It is a fexibe space, whose ower and upper boundaries are shown in Fig. 2 as A 00 FNGH 00 and A 00 ECMC 0 HH 00, respectivey. The soutions ^x corresponding to the maximum vaue of the membership function of the resuting grey fuzzy decision (Z 7 ) is an interva in the space CMC 0 N (Fig. 2). Mathematicay the grey fuzzy decision (Z 7 ) for F 1 and F 2 can be defined with the imprecise membership function: h n o m Z ðx Þ¼ ¼ min min m ðx Þ; m ðx Þ ; F 1 F 2 n oi min m ðx þ Þ; m ðx þ Þ, ð6þ F 1 F 2 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Fig. 1. Concept of fuzzy decision. h n o m þ ðx Þ¼ þ ¼ max min m þ ðx Þ; m þ ðx Þ ; Z F 1 F 2 n oi min m þ ðx þ Þ; m þ ðx þ Þ F 1 F 2 where m Z ðx Þ and m þ Z ðx Þ are ower and upper bounds of the imprecise membership functions for an interva [x, x + ], respectivey, encompassing a intermediate possibiities of membership vaues [i.e., m Z ðx Þ]. Eqs. (6) and (7) are vaid for a combinations of imprecise membership functions (i.e., non-increasing, non-decreasing, or a combination of the two). In GFWLAM two sets of imprecisey defined and conficting goas (i.e., goas of the PCA and dischargers) are addressed through an optimization mode by using the concept of grey fuzzy decision Goas of the PCA The PCA sets the desirabe concentration eve (c D ) and maximum permissibe concentration eve (c H ) of the water-quaity indicator j (e.g., DO-deficit, hardness, nitrate nitrogen concentration) at the water-quaity checkpoint (c D c H ). The goa E, of the PCA, is to make the concentration eve (c ) of water-quaity indicator j at the checkpoint as cose as possibe to the desirabe eve, c D, so that the water-quaity at the checkpoint is enhanced with respect to the water-quaity indicator j, for a j and. These goas are represented by a membership function. For exampe, if DO-deficit is the water-quaity indicator, a non-increasing membership function suitaby refects the goas of the PCA with respect to DO-deficit at a checkpoint. The uncertainty associated with membership parameters (c D and c H ) is addressed using interva grey numbers, and the membership parameters are expressed as c D and c H Using non-increasing imprecise membership functions, the grey fuzzy goas of PCA (i.e., E ) are ð7þ Fig. 2. Concept of grey fuzzy decision.

5 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) represented as 8 1 c þ ocd >< h i m ðc E Þ¼ ðc H c Þ=ðcH c D a Þ c D c c H >: 0 c 4cHþ (8) where c is the uncertain concentration eve of waterquaity indicator j at the checkpoint, represented as an interva grey number. The exponent a is a nonzero positive rea number. Assignment of a numerica vaue to this exponent is subject to the desired shape of the membership functions. A vaue of a ¼ 1 eads to a inear imprecise membership function, as shown in Fig Goas of the dischargers The goa of discharger m, F jmn, is to make the fractiona remova eve (x jmn ) as cose as possibe to the aspiration eve (x L mn ), to minimize the waste treatment cost of poutant n infuencing water-quaity indicator j. The grey fuzzy goas of the dischargers (i.e., F jmn ) are simiary represented as: m F jmmðx jmn Þ 8 1 x þ jmn oxl mn ; >< h i ¼ ðx M mn x jmn Þ=ðxM mn xl mn Þ bjmn x L mn x jmn xm >: 0 x jmn 4xMþ mn ; mn ; ð9þ where the aspiration eve and maximum acceptabe eve of the fractiona remova of the poutant n at discharger m are represented as x L mn and xm mn, respectivey (xl mn xm mn ). Simiar to the exponent a in Eq. (8), b jmn is a non zero positive rea number. A vaue of b jmn ¼ 1 eads to a inear imprecise membership function, as shown in Fig GFWLAM formuation In GFWLAM the grey fuzzy decision (Z 7 ) is expressed by using the concept of fuzzy decision [Eq. (3)]:! Z ¼ \ \!, (10) \ j; E F jmn j;m;n where E and F jmn represent the grey fuzzy goas of PCA and dischargers, respectivey (as shown in Figs. 3 and 4). The arguments of E and F jmn are c and x jmn, respectivey. The intersection of these non-increasing imprecise membership functions is shown in Fig. 5, where ABCDD 0 GH and AB 0 C 0 D 0 GH are ower and upper boundaries of the grey fuzzy decision, and the optima soution ^x is an interva in the space MC 0 NC. The concept of grey fuzzy decision is used to mode the conficting nature of the goas of PCA and dischargers in the optimization mode. The formuation of GFWLAM is written as Max (11) Subject to m F ðx jmn Þ¼ jmn c D m E h. i ðc Þ¼ ch c c H c D a 8j;, xm mn x jmn (12) h. i bjmn xl 8j; m; n, x M mn mn (13) c c H 8j;, (14) x L mn x jmn xm mn 8j; m; n, (15) Fig. 3. Linear imprecise membership function for the goas of PCA.

6 238 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Fig. 4. Linear imprecise membership function for the goas of dischargers. Fig. 5. Grey fuzzy decision resuting from confuence of the goas of PCA and dischargers (16) Constraints (12) and (13) are formuated from imprecise membership functions for the goas of PCA and dischargers, respectivey, and define the minimum goa fufiment eve ( 7 ). The crisp constraints (14) and (15) are based on the water-quaity requirements set by the PCA, and acceptabe fractiona remova eves by the dischargers, respectivey. Constraint (16) presents the bounds on the parameter 7. In the expression for the goas of PCA [constraint (12)], the concentration eve c, of waterquaity indicator j at checkpoint, may be expressed as c ¼ f ðx jmnþ, (17) where the transfer function (f) indicates the aggregate effect of a poutants and dischargers, ocated upstream of checkpoint on the water-quaity indicator j at that checkpoint. The transfer function can be evauated using appropriate mathematica modes that determine the spatia distribution of the water-quaity indicator due to poutant discharge into the river system from point sources (Fugiwara et a., 1987, 1988; Sasikumar and Mujumdar, 1998). For most water-quaity indicators, a high eve of fractiona remova of poutants (e.g., BOD oading, toxic poutant concentration, etc.) resuts in a ow eve of water-quaity indicator (e.g., DO-deficit, nitrate nitrogen concentration, etc.). The ower bound of waterquaity indicator (c ) is therefore expressed in terms of the upper bound of the fractiona remova eve (x þ jmn ) using Eq. (17). c ¼ f ðx þ jmnþ. (18) Simiary, c þ ¼ f ðx jmnþ. (19) Eqs. (18) and (19) are the deterministic equivaent of the Eq. (17), which contains interva grey numbers (i.e., c and x jmn ) in both the sides. The order of consideration of

7 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) constraints (12) and (13), aong with the seection of bounds of decision variabes (x jmn, xþ jmn ) creates two different cases of mode formuation, which are termed as Case 1 and Case 2, with each probem divided into two submodes. Submode 1 maximizes the upper bound, +, and Submode 2 maximizes the ower bound,. Lower and upper bounds of the decision (x jmn and xþ jmn ) are obtained from these two submodes. These four submodes (two each for Cases 1 and 2) are the deterministic equivaent of the grey fuzzy optimization mode given in (11) (16). The four submodes are soved to obtain the set of optima vaues of fractiona remova eves of the poutants (denoted by ^X ¼f^x jmng) as a set of fexibe poicies in the form of interva grey numbers. The present work is aso aimed at addressing the uncertainty in the assignment of membership functions for management goas of the PCA and dischargers as addressed in GFWLAM, but the mutiobjective approach adopted to obtain the deterministic equivaent of the grey fuzzy optimization mode is simper and the soutions are more usefu to the decision-makers as a wider choice in the interva vaued optima fractiona removas is obtained. The optima vaue of goa fufiment eve (^ ) may be viewed as a measure of compromise between the decisionmakers in the system (Kinder, 1992), and may aso be considered as a measure of confict existing in the system. A vaue of ^ ¼ 0 (i.e., ^ ¼ 0and^ þ ¼ 0) indicates a strong confict scenario, whereas a vaue of ^ ¼ 1 (i.e., ^ ¼ 1 and ^ þ ¼ 1) corresponds to a no-confict scenario. As is a measure of the degree of fufiment of the goas of PCA and dischargers, þ and are maximized to ensure the highest possibiity of fufiment of the management goas. Secondy, the width of the interva of fractiona remova eve [i.e., (x þ jmn x jmn )] is maximized, to ensure more fexibiity for the decision-makers in post-optimaity decision-making. Finay, the output uncertainty is reduced by minimizing the width of the degree of the goa fufiment eve [i.e., ( þ )]. A these objectives [viz., maximization of þ,,(x þ jmn x jmn ), and minimization of ( þ )] are addressed in a singe optimization mode in a mutiobjective framework aong with the uncertain information addressed through interva grey numbers. The next section gives a description of the modified GFWLAM. 3. Modified GFWLAM The grey fuzzy optimization mode given in (11) (16) forms the basis of the modified GFWLAM. The fuzzy inequaity constraints (12) and (13) address the goas of the PCA and dischargers in the optimization mode. These are the order reations (e.g., the reations greater than or equa to or ess than or equa to ) containing interva grey numbers on both the sides. Determination of meaningfu ranking between two partiay or fuy overapping intervas in the order reations is a potentia research area (e.g., Moore, 1979; Ishibuchi and Tanaka, 1990; Sengupta et a., 2001). Recenty, Sengupta et a. (2001) proposed a satisfactory deterministic equivaent form of inequaity constraints containing interva grey numbers by using the acceptabiity index (A). In the foowing subsection the formuation of the modified GFWLAM is presented using the concept of acceptabiity index for comparing the interva grey numbers Mode formuation Acceptabiity index (A) is defined as the grade of acceptabiity of the premise the first interva grey number (a 7 ) is inferior to the second (b 7 ), denoted as a 7 (o) b 7. Here, the term inferior to ( superior to ) is anaogous to ess than ( greater than ). The acceptabiity index (A) is expressed as (Sengupta et a., 2001) A a ðoþb ¼ mðb Þ mða Þ = wðb Þþwða Þ, (20) where [w(b 7 )+w(a 7 )]6¼0; w(a 7 ) is the haf-width of a 7 ¼ 1/2 (a + -a ); m(a 7 ) is the mean of a 7 ¼ 1/2 (a +a + ). Notations are simiary defined for the interva grey number b 7. The grade of acceptabiity of a 7 (o) b 7 may be cassified and interpreted further on the basis of the comparative position of the mean and haf-width of interva b 7 with respect to those of interva a 7. Let us consider an interva inequaity reation a x b, where x is a deterministic variabe. A satisfactory deterministic equivaent form of the interva inequaity reation a x b, is proposed as a x b ) a x b and A a xðoþb a 2 ½0; 1Š, (21) where a is interpreted as an optimistic threshod fixed by the decision-maker. A brief description of the concept of acceptabiity index is given in the appendix. The deterministic equivaent of the grey fuzzy optimization mode given in (11) (16) is formuated using the expression given in Eq. (21). By using the attributes mean, width and acceptabiity index of the interva grey numbers, the grey fuzzy optimization mode is reduced to a deterministic mutiobjective optimization mode, as foows Max þ, (22) Max, (23) Min þ þ þ (24) Subject to m E m ðx F jmn Þ¼ jmn A½ðc H h. ðc Þ¼ ch c þ h. xm mn c Þ=ðcH xþ jmn x Mþ mn c Hþ xl mn c D i 8j;, (25) i 8j; m; n, (26) c D ÞðoÞ a 1 2½0; 1Š 8j;, (27)

8 240 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) A½ðx M jmn x jmn Þ=ðxM jmn xl jmn ÞðoÞ a 2 2½0; 1Š 8j; m; n, (28) c D c þ c Hþ ; c D c c Hþ 8j;, (29) x L mn xþ jmn xmþ mn ; xl mn x jmn xmþ mn 8j; m; n, (30) c c þ 8j;, (31) x jmn xþ jmn 8j; m; n, (32) 0 þ 1; 0 1, (33) þ. (34) Constraints (25) (28) are the deterministic equivaent of constraints (12) (13). The acceptabiity index in constraints (27) and (28) compares the interva grey numbers in the inequaity constraints (12) and (13). In constraints (27) and (28), a 1 and a 2 are the optimistic threshods fixed by the decision-maker. The goas of the PCA and dischargers are represented by inear imprecise membership functions [i.e., substituting a and b jmn ¼ 1 in Eqs. (8) and (9), respectivey], as the Eq. (21) with acceptabiity index for ranking the interva grey numbers in the inequaity constraints is appicabe ony for inear programming probems (Sengupta et a., 2001). In water-quaity management probems, any continuous monotonicay non-increasing membership function wi address reasonaby we the imprecision in the management goas of the PCA and dischargers, but the pacement of membership parameters on the universe of discourse is important. The mathematica expression of acceptabiity index given in Eq. (20) is substituted in the constraints (27) and (28), and agebraic operations on the interva grey numbers (Moore, 1979; Liu and Lin, 1998) are performed to get a simpified form. The fina expressions of constraints (27) and (28) are given as ½fð þ þ Þ ðc H ðc Hþ þðc Hþ c Þ=ðcH c Þ=ðcH c þ Þ=ðcHþ c D Þ c Dþ Þg=fð þ Þ c Dþ Þ ðc H c þ Þ=ðcHþ c D ÞgŠ a 1 2½0; 1Š 8j; ð35þ ½fð þ þ Þ ðx M mn ðxmn Mþ þðxmn Mþ ðx M mn x jmn Þ=ðxM mn x jmn Þ=ðxM mn xþ jmn Þ=ðxMþ mn xþ jmn Þ=ðxMþ mn xl jmn Þ xlþ mn Þg=fðþ Þ xlþ mn Þ xl jmn ÞgŠ a 2 2½0; 1Š 8j; m; n. ð36þ Finay, (22) (34), repacing constraints (27) and (28) by (35) and (36), respectivey, represent the deterministic equivaent of the grey fuzzy optimization mode (11) (16) in a mutiobjective framework. A c and c þ terms in (22) (36) are expressed in terms of x þ jmn and x jmn, respectivey, using the expressions obtained from the water-quaity transport mode [simiar to Eq. (18) and (19)], which resuts in x þ jmn and x jmn as the ony decision variabes in the optimization mode. Objectives (22) and (23) maximize the upper and ower bound of the goa fufiment eve ( ), respectivey, which ensure the maximum possibiity of fufiment of the goas of PCA and dischargers. Objective (24) minimizes the width of the goa fufiment eve [ w ¼ð þ Þ]. This objective is incuded as the reduction of width of the goa fufiment eve impies reduction in system uncertainties and increase in effectiveness of the grey mode (Huang et a., 1995). Simiary, a higher fexibiity (i.e., higher width of the interva) of the decision variabes (x jmn ) is aways desirabe, as it aows a wider choice to the decision-makers. But objectives (22) (24) do not address the maximization of the width of decision variabes [i.e., (x þ jmn x jmn )]. The width of the decision variabes is maximized aong with the objectives (22) (24) whie soving the deterministic equivaent of the grey fuzzy optimization mode [i.e., (22) (34), repacing constraints (27) and (28) by (35) and (36)] using a fuzzy mutiobjective optimization technique (Sakawa et a., 1984). The procedure adopted is discussed in the section, Mode Appication. The probem is aso soved by using a fuzzy goa programming technique (Sakawa et a., 1987; Pa and Moitra, 2003) to examine the consistency of the soutions. A brief description of these optimization techniques is given in the appendix. 4. Mode appication The mutiobjective optimization mode [(22) (34)] for water-quaity management is appied to a hypothetica river system shown in Fig. 6, and soved by using fuzzy mutiobjective optimization and fuzzy goa programming techniques. The river network is discretized into four reaches. Four dischargers are considered as the point sources of poutants. The ony poutant considered in the system is BOD waste-oad due to point sources. The waterquaity indicator of interest is the DO-deficit at 18 checkpoints in the river system due to the point sources of BOD. The saturation DO concentration is taken as 10 mg/l for a the reaches. A deterministic streamfow of 7 Mcum/day is considered. Detais of the effuent fow and imprecise membership functions are given in Tabes 1 and 2, respectivey. A set of ower and upper bounds of membership parameters (c D ; c H ; x L m and xm m ) are fixed arbitrariy (Tabe 2) to obtain a set of optima soutions for demonstration. The notations of different variabes are simpified by retaining ony the suffixes (checkpoints) and m (dischargers). Optimization mode [(22) (34), with (35) and (36)] for water-quaity management of the hypothetica river system is now presented as foows: Max þ, (37) Max, (38)

9 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Fig. 6. Hypothetica river system. Tabe 1 Effuent fow data Discharger (1) Min½ð þ Þ=ð þ þ ÞŠ, (39) Subject to ðc Þ¼ c H c þ c Hþ c D 8, m E (40) m ðx F m Þ¼½ðxM m xþ m Þ=ðxMþ m xl m ÞŠ 8m, (41) m þ þ c H c þ c Hþ c D c Hþ c c H c Dþ þ þ c Hþ c c H c Dþ c H c þ c Hþ c D a 1 2½0; 1Š 8, ð42þ þ þ x M m xþ m x Mþ m xl m x Mþ m x m x M m xlþ m þ þ x Mþ m x m x M m xlþ m x M m xþ m x Mþ m xl m a 2 2½0; 1Š 8m, ð43þ c D c þ c Hþ x L m ; c D c c Hþ 8 (44) xþ m xmþ m ; xl m x m xmþ m 8m, (45) c c þ 8, (46) x m xþ m Effuent fow rate (10 4 m 3 /day) (2) BOD concentration (mg/l) (3) D D D D DO concentration (mg/l) (4) Source of data (Sasikumar, 1998; Mujumdar and Sasikumar, 2002). 8m, (47) 0 þ 1; 0 1, (48) þ, (49) where m ðc E Þ and m F mðx mþ are the imprecise membership functions, defining the grey fuzzy goas of PCA (E ) and dischargers (F m ). In constraints (42) and (43), a 1 and a 2 are optimistic threshods assumed equa to zero. The optimistic threshods a 1 and a 2 may be interpreted as the grades of acceptabiity of the premises m ðc E Þ is ess than and m F mðx m Þ is ess than, respectivey. Therefore, when a 1 ¼ a 2 ¼ 0, the mean vaues of both the intervas become equa to the mean vaue of, and thus a conservative optima soution or soution at no optimism eve for constraints (12) and (13) is obtained, whose deterministic equivaents are expressed through constraints (40) (43). Simiary, as the vaues of a 1 and a 2 increase towards unity a more optimistic optima soution is obtained. The decision-maker seects the vaues of a 1 and a 2 equa to zero when the water-quaity management issues in the river system are too critica and important, otherwise some optimistic strategy can be considered by choosing vaues of a 1 and a 2 cose to unity. A inear reationship, c ¼ f ðx mþ [anaogous to Eq. (17)], is deveoped, considering the method proposed by Fujiwara et a. (1987, 1988), and Sasikumar and Mujumdar (1998), based on the Streeter and Pheps equations (Streeter and Pheps, 1925), to express the DO deficit, c as a function of the eves of fractiona remova, x m. A c terms are expressed in terms of x m, which resuts in x m as the ony decision variabes in the optimization mode (37) (49). For further verification it shoud be noted that a detaied description of the transport mode used in the present hypothetica case study and the inear expressions [same as Eq. (17)] reating DO-deficit (c ) and BOD remova (x m ) are given in Sasikumar (1998) and Mujumdar and Sasikumar (2002). In the modified GFWLAM, objectives (37) and (38) are maximized, and objective (39) is minimized individuay in three separate subprobems aong with constraints (40) (49) to obtain the maximum and minimum possibe vaues of þ ; and ½ð þ Þ=ð þ þ ÞŠ (i. e., idea points Q 1, Q 2, Q 3 and worst possibe vaues q 1, q 2, q 3 of the fuzzy mutiobjective optimization technique as mentioned in the appendix), respectivey. As discussed earier, another objective of the river water-quaity management is to

10 242 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Tabe 2 Detais of imprecise membership functions River Reach (1) Check points (2) c D (mg/l) (3) c H (mg/l) (4) x L m (5) x M m (6) R (0.00) (3.00) (0.30) (0.85) R (0.10) (3.00) (0.30) (0.85) R (0.20) (3.50) (0.35) (0.85) R (0.20) (3.50) (0.35) (0.85) ( ) : deterministic case, : ower bound, + : upper bound, : source of data (Sasikumar, 1998; Mujumdar and Sasikumar, 2002). permit more fexibiity (i.e., more width of the interva) in the optima fractiona remova eve ( ^x m ). Thus maximization of the grey degree of x m [i.e., Gdðx m Þ¼ðxþ m x m Þ 0:5ðx þ m þ x mþš is considered as another objective aong with objectives (37) (39). The maximum and minimum vaues of the grey degree of x m are determined from the three subprobems, which are considered as the idea (i.e., Q 4m ) and worst (i.e., q 4m ) vaues, where subscript m denotes the discharger. To represent the objective of maximizing the vaue of Gd(x m ) a nondecreasing inear membership function, taking Q 4m and q 4m as the reference point and reservation eve, respectivey, is considered as m 4m ½Gdðx m 8 ÞŠ 1 Gdðx m Þ4Q 4m >< ¼ ½Gdðx m Þ q 4mŠ ½Q 4m q 4m Š q 4m Gdðx m ÞQ 4m 8m, >: 0 Gdðx m Þoq 4m ð50þ Simiary, maximization of objectives (37) and (38) is achieved by using non-decreasing inear membership functions. Objective (39) is a minimization type, and is addressed in the mode by the non-increasing inear membership function, given by where Q 3 and q 3 are the possibe individua maximum and minimum vaues, respectivey, of the third objective (39). The fuzzy decision concept with the minimum operator (i.e., ogica and ) is appied to aggregate the membership functions of objectives (37) (39) aong with four other objectives for maximizing Gd (x m ) for the four dischargers. The soution of the resuting probem gives a Paretooptima soution of the fuzzy mutiobjective optimization probem for fractiona remova eves of BOD (i.e., f ^X ¼fx m g). The water-quaity management probem with the same hypothetica river system is soved by using a fuzzy goa programming technique [given in the appendix], which confirms the consistency of soutions. The resuts obtained by appying the modified GFWLAM to the hypothetica river system are described in the foowing section. 5. Resuts and discussion The resuts obtained from the modified GFWLAM are summarized in Tabes 3 5, and faciitate a comparison between the deterministic case, where the membership parameters are deterministic numbers, and the grey uncertain case, where the membership parameters are uncertain. Tabe 3 shows the optima fractiona remova eves of the poutants by different dischargers (i.e., ^X ¼f^x mg) for the grey uncertain case (with average grey degree of 31.30%, as given in Tabe 2 for c D ; c H ; x L mn ; and xm mn ) obtained from the modified GFWLAM using the fuzzy mutiobjective optimization technique. In Tabe 3, coumns 1 3 show the resuts obtained from subprobems 1 3, i.e., maximization of +, maximization of - and minimization 8 >< 1 ½ð þ Þ=ð þ þ ÞŠoq 3 ; m 3 ½ð þ Þ=ð þ þ ÞŠ ¼ ½Q 3 fð þ Þ=ð þ þ ÞgŠ=½Q 3 q 3 Š; >: 0 q 3 ½ð þ Þ=ð þ þ ÞŠ Q 3 ; ½ð þ Þ=ð þ þ ÞŠ4Q 3 ; (51) of the width of (equivaent to minimizing ½ð þ Þ= ð þ þ ÞŠ), respectivey. The payoff matrix of the fuzzy mutiobjective optimization is formed, picking the minimum and maximum vaues of + [objective (37)], + [objective (38)], ð þ Þ=ð þ þ Þ [objective (39)] and Gdðx 1 Þ;...Gdðx 4 Þ from coumns 1 3; rows 6, 5, 8, and 9 12, respectivey, as the procedure mentioned in the

11 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Tabe 3 Detais of fractiona remova eves using fuzzy mutiobjective optimization technique (when a 1 and a 2 ¼ 0) Row No. Subprobem-1 (Max. þ ) (1) Subprobem-2 (Max. ) (2) Subprobem-3 [Min. ð þ Þ=ð þ þ Þ] (3) Modified GFWLAM (4) Deterministic mode (FWLAM) (5) GFWLAM (6) 1 x 1 [0.4845, ] [0.5955, ] [0.6060, ] [0.5198, ] [0.6150, ] [0.5970, ] 2 x 2 [0.4682, ] [0.5967, ] [0.4301, ] [0.5089, ] [0.6150, ] [0.5970, ] 3 x 3 [0.5190, ] [0.6124, ] [0.5517, ] [0.5446, ] [0.6360, ] [0.6120, ] 4 x 4 [0.5172, ] [0.6121, ] [0.6324, ] [0.5402, ] [0.6360, ] [0.6120, ] þ Gd( ) ð þ Þ= ð þ þ Þ Gd(x 1 ) Gd(x 2 ) Gd(x 3 ) Gd(x 4 ) Avg. Gd(X ) Tabe 4 Detais of DO-deficit at the checkpoints Checkpoints DO-deficit (mg/l) Gd(c 1 ) from modified GFWLAM Gd(c 1 ) from GFWLAM Lower bound (c 1 ) Upper bound (c 1 ) (1) (2) (3) (4) (5) appendix. For exampe, coumns 1 3, row 5 show the vaues of - obtained from sub probems 1 3, respectivey. The maximum vaue of - (i.e., Q 2 ¼ ) is obtained from sub probem 2 and the minimum vaue (i.e., q 2 ¼ ) is obtained from sub probem 1. The fuzzy set requirements of a the objectives are quantified by eiciting inear membership functions considering the eements in the payoff matrix as membership parameters as discussed in the appendix-fuzzy mutiobjective optimization. Coumn 4, rows 1 6 show the optima fractiona remova eves of the poutants by different dischargers (i.e., ^X ¼f^x m g) and corresponding ^ vaues. To evauate the quaity of input or output uncertain information, a measure of Grey degree is used [Eq. (2)]. The decrease in the grey degree of the optima vaue of the objective function indicates increasing effectiveness of the grey mode and decreasing system uncertainty. In coumn 4, row 7 the grey degree of (i.e., ) shows the quaity of the optima soutions (^ ¼ [0.1966, ]; where the WMV ( m ) ¼ , width ( w ) ¼ and grey degree of ^ ¼ (0.3562/0.3747) ¼ , i.e., 95.06%). In coumn 5 the deterministic case is presented, for which the average vaue of the grey degree of input parameters is zero. The resuting vaues of ^ and ^X in this case show the deterministic soutions, indicated by the zero vaues of grey degrees in rows 7 and The vaue of ^ in the deterministic case, [0.4277, ] ies in the cosed interva as obtained in the grey uncertain case, i.e., [0.1966, ], as the vaues of a membership parameters (i.e., c D ; c H ; x L mn ; and xm mn ) for the deterministic case ie within the ranges of intervas given in Tabe 2 for the grey uncertain case. Next in coumn 6, the grey uncertain case is presented as obtained from GFWLAM (Karmakar and Mujumdar, 2004a), for which the average vaue of the grey degree of input parameters (i.e., %) is the same as in the modified GFWLAM given in Tabe 2. Comparing the resuts shown in coumns 4 and 6, rows 1 4, it can be concuded that the widths of the optima fractiona remova eves of BOD ( ^X ) are more than those of GFWLAM, which aows a wider choice to the decisionmaker in decision-making. The same observation can be made more ceary by comparing the Gd( ^X ) vaues shown in rows 9 12 of coumns 4 and 6. The vaue of Gd(^ ) (given in row 7 of coumn 4) is, however, more than the vaue obtained from GFWLAM (row 7, coumn 6), which

12 244 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Tabe 5 Comparison of soutions obtained by appying fuzzy mutiobjective and fuzzy goa programming techniques Row no. Fuzzy mutiobjective optimization mode Fuzzy goa programming (minimax approach) a 1 and a 2 ¼ 0 (1) a 1 and a 2 ¼ 0.25 (2) a 1 and a 2 ¼ 0.5 (3) a 1 and a 2 ¼ 0 (4) a 1 and a 2 ¼ 0.25 (5) a 1 and a 2 ¼ 0.5 (6) 1 [0.1966, ] [0.2206, ] [0.2530, ] [0.2066, ] [0.2259, ] [0.2612, ] 2 Gd[ ] x 1 [0.5198, ] [0.5945, ] [0.3867, ] [0.5268, ] [0.5594, ] [0.6278, ] 4 x 2 [0.5089, ] [0.4889, ] [0.6030, ] [0.5302, ] [0.5383, ] [0.5753, ] 5 x 3 [0.5446, ] [0.5890, ] [0.5715, ] [0.5367, ] [0.5575, ] [0.5390, ] 6 x 4 [0.5402, ] [0.5441, ] [0.5973, ] [0.5357, ] [0.5291, ] [0.5662, ] 7 Avg. Gd[X ] Avg. Gd[C ] indicates more uncertainty in the system compared to that resuting from the GFWLAM soution. The resut obtained from the modified GFWLAM is more usefu to the decision-makers as it gives a wider range in the interva vaued optima fractiona remova eves of the poutants than GFWLAM, athough at the cost of increasing uncertainty. Tabe 4 gives the DO-deficit vaues for a 18 checkpoints resuting from the fractiona remova eves of BOD isted in Tabe 3 (coumn 4, rows 1 4). The reationships given in Eq. (18) and (19) are used to obtain the ower and upper bounds of the water-quaity indicators (c and c þ ) from the upper and ower bounds of the fractiona remova eve of poutant (x þ m and x m ), respectivey. Coumns 4 and 5 of the Tabe 4 compare the grey degrees of concentrations of DOdeficit obtained from the modified GFWLAM and GFWLAM. In Tabe 5, coumns 1 3 show the optima soutions obtained for a 1 ¼ 0, a 2 ¼ 0; a 1 ¼ 0.25, a 2 ¼ 0.25; and a 1 ¼ 0.50, a 2 ¼ 0.50; respectivey. As the vaues of a 1 and a 2 increase to unity, constraints (12) and (13), whose deterministic equivaents are expressed through constraints (40) (43), approach their imiting vaues and resut in more optimistic optima soutions. Comparing vaues of ^ in row 1 of coumns 1 3, it is seen that as the vaues of a 1 and a 2 increase, the vaues of both the bounds of increase, indicating an increased eve of optimistic optima soutions. The optima ranges of fractiona remova eves of BOD (i.e., ^X ) for a 1 ¼ 0, a 2 ¼ 0; a 1 ¼ 0.25, a 2 ¼ 0.25; and a 1 ¼ 0.50, a 2 ¼ 0.50 are given in rows 3 6 of coumns 1 3, respectivey. Coumns 4 6 show the soutions obtained from the fuzzy goa programming technique (minimax approach), which confirms the consistency of the resuts given in coumns 1 3. Simiary, comparing vaues of ^ in row 1 of coumns 4 6, it is seen that as the vaues of a 1 and a 2 increase the vaues of both the bounds of ^ increase, indicating the increased eve of optimistic optima soutions. The optima ranges of fractiona remova eves of BOD (i.e., ^X ) obtained by appying the fuzzy goa programming technique for a 1 ¼ 0, a 2 ¼ 0; a 1 ¼ 0.25, a 2 ¼ 0.25; and a 1 ¼ 0.50, a 2 ¼ 0.50 are given in rows 3 6 of coumn 4 6, respectivey. Rows 1 4 of coumn 5 in Tabe 3 present the optima fractiona remova eves of BOD for dischargers 1 4, i.e., x 1 E0.62, x 2 E0.62, x 3 E0.64, and x 4 E0.64, fixed by the decision-maker as obtained from the deterministic waterquaity management mode, FWLAM, deveoped earier by Sasikumar and Mujumdar (1998), which gives the optima goa fufiment eve as ¼ (rows 5 6 in coumn 5). As the vaues of fractiona remova eves are crisp or deterministic (i.e., white numbers), the decision-maker woud not get any fexibiity on adjusting these vaues in the fina decision scheme of poutant remova considering the technica and economic feasibity of these poutant treatment eves. In coumn 4, the optima vaues of the ower and upper bounds of (rows 5 6) correspond to different distributions of optima fractiona remova eves of BOD (rows 1 4). Therefore, the decision-maker gets a range of optima soutions in the fexibe decision space. For exampe, the interva vaued optima fractiona remova of BOD for discharger 1 ( ^x 1 ) is [0.5198, ]E[0.52, 0.67], which indicates that any vaue of BOD remova between 52% and 67% may be fixed for discharger 1, which ensures a -vaue within [0.1966, ]. Simiary, for other dischargers, i.e., ^x 2 E[0.51, 0.67]; ^x 3 E[0.54, 0.68]; and ^x 4 E[0.54, 0.68], wider ranges of optima fractiona remova eves are obtained than those given by the previousy deveoped GFWLAM, presented in rows 1-4, coumn 6 (i.e., ^x 1 E[0.60, 0.64]; ^x 2 E[0.60, 0.64]; ^x 3 E[0.61, 0.67]; and ^x 4 E[0.61, 0.67]). This indicates that the decision-maker gets a more fexibe optima soution space within which the fina decision for fractiona remova can be fixed. Seection of particuar vaues from the optima interva vaued fractiona remova eves of BOD ( ^x m ) for impementation in the fied ensures a -vaue within [0.1966, ]. From the mode formuation it can be concuded that panning for an upper bound of ^, i.e., ^þ represents an optimistic strategy ; whereas, panning for a ower bound of ^, i.e., ^ represents a conservative strategy.

13 S. Karmakar, P.P. Mujumdar / Journa of Environmenta Management 81 (2006) Concuding remarks This paper presents a modified grey fuzzy waste oad aocation mode (modified GFWLAM) for water-quaity management of a river system. A previousy deveoped fuzzy waste oad aocation mode (FWLAM) is extended to address uncertainty invoved in fixing the membership functions for conficting goas of the poution contro agency (PCA) and dischargers considering the membership parameters as interva grey numbers. The paper demonstrates modeing aspects of uncertain membership functions and shows the usefuness of soutions with a simpified hypothetica river system in which ony the BOD point sources are considered as poutants and the DO-deficit at a finite number of checkpoints is used as the water-quaity indicator. In a majority of waste oad aocation modes, the measure of performance is reated to the overa cost of poution contro, incuding the waste treatment cost. Incorporating the cost functions directy into the optimization modes poses difficuties due to uncertainty, ack of adequate data, and non-inearity associated with the cost functions. The FWLAM has the advantage that the cost functions are eiminated whie the fuzzy goas of the dischargers are incorporated through membership functions. The membership functions themseves are subjective statements of the perceptions of the decision-makers. The vaues of the parameters in the membership functions depend on the responses of a decision-maker to waterquaity and fractiona remova eves. In practica situations, different water-quaity standards for surface water intakes are used for different uses, which resuts in an uncertainty in the membership parameters for the goas of PCA. To address uncertainty in these bounds, the membership functions themseves are treated as fuzzy in the fuzzy optimization mode. Consideration of imprecise membership parameters in the fuzzy optimization mode imparts fexibiity in decision-making as the fractiona remova eves are obtained as interva grey numbers. The modified GFWLAM gives a new methodoogy to get a satisfactory deterministic equivaent of a grey fuzzy optimization probem, using the concept of acceptabiity index for a meaningfu ranking between two partiay or fuy overapping intervas. Athough the soutions obtained from the present mode provide more fexibiity than those given by the previousy deveoped GFWLAM (Karmakar and Mujumdar, 2004a), the appication of the modified GFWLAM is imited to grey fuzzy goas expressed by inear imprecise membership functions, whereas GFWLAM has the capabiity to sove the grey fuzzy optimization mode with monotonic non-inear imprecise membership functions with a and b jmn 6¼1, in Eqs. (8) and (9). The mode wi be more usefu with more reaistic soutions if it is appied to a critica case study where a dischargers pay a critica roe in infuencing the vaue of the mode performance measure,. A imitation of the mode presented in this paper, however, is that in situations where mutipe soutions exist, the optima vaues obtained from the deterministic mode may not ie within the optima interva vaues obtained from the modified GFWLAM, for some soutions. The present mode is not imited in appication to any particuar poutant or waterquaity indicator in the river system. Given appropriate transfer functions for spatia and tempora distribution of the poutants in the water body, it can be used for waterquaity management of any genera water system. The methodoogy described in the formuation of the modified GFWLAM to get a satisfactory deterministic equivaent of a grey fuzzy optimization mode is vaid ony for inear imprecise membership functions. Addressing the uncertainty in membership parameters with non-inear imprecise membership functions shoud provide a new direction for future research. A new grey fuzzy stochastic programming technique (Huang et a., 1995; Huang and Loucks, 2000) can be deveoped to get the optima vaues for fractiona remova in the form of interva grey numbers addressing aso the uncertainty due to randomness in variabes such as streamfow, effuent fow, reaeration coefficient, deoxygenation coefficient, and temperature. Appendix A.1. Acceptabiity index Sengupta et a. (2001) introduced an extended order reation between the intervas a ¼½a ; a þ Š and b ¼ ½b ; b þ Š on the rea ine R. A premise a ðoþb is formed to impy that a is inferior to b. Here, the term inferior to ( superior to ) is anaogous to ess than ( greater than ). Let I be the set of a cosed intervas on the rea ine R. The acceptabiity index (A) is defined as A :I I- [0, p) such that A a ðoþb ¼ mðb Þ mða Þ = wðb Þþwða Þ, (A.1) where ½wðb Þþwða ÞŠa0; wða Þ is the haf-width of a ¼ 1=2ða þ a Þ; mða Þ is the mean of a ¼ 1=2ða þ a þ Þ which is same as WMV in the grey systems (Huang et a., 1995). Notations are simiary defined for the interva grey number b. The grade of acceptabiity of a ðoþb may be cassified and interpreted further on the basis of the comparative position of the mean and haf-width of interva b with respect to those of interva a as foows: 8 >< ¼ 0 if mða Þ¼mðb Þ; A½a ðoþb Š¼ 40; o1 if mða Þomðb Þ and a þ 4b ; >: 1 if mða Þomðb Þ and a þ b : (A.2) If A ½a ðoþb Š¼0, then the premise a is inferior to b is not accepted. If 0oA ½a ðoþb Šo1, then the premise ½a ðoþb Š is accepted with different grades ranging from zero to one. If A ½a ðoþb Š1, the interpreter is absoutey satisfied with the premise ½a ðoþb Š.