Possibilistic Stochastic Water Management Model for Agricultural Nonpoint Source Pollution Xiaodong Zhang1; Guo H. Huang2; and Xianghui Nie3 Abstract: Agricultural activities are the main contributors of nonpoint source water pollution within agricultural systems. In this study, a possibilistic stochastic water management 共PSWM兲 model is developed and applied to a case study of water quality management within an agricultural system in China. This study is a first application of hybrid possibilistic chance-constrained programming approach to nonpoint source water quality management problems within an agricultural system. Hybrid uncertainties with the synergy of fuzzy and stochastic implications are effectively characterized by the PSWM model with the following advantages: 共1兲 it improves upon the existing possibilistic and chance-constrained programming methods through direct incorporation of fuzziness and randomness within a general optimization framework; 共2兲 it will not lead to more complicated intermediate models and thus have lower computational requirements; 共3兲 its solutions offer flexibility in interpreting the results and reflect the interactional effects of uncertain parameters on system conditions variations; and 共4兲 it can help examine the risk of violating system constraints and the associated consequences. The results of the case study show useful information for feasible decision schemes of agricultural activities, including the trade-offs between economic and environmental considerations. Moreover, a strong desire to acquire high agricultural income will run into the risk of potentially violating the related water quality standards, while willingness to accept low agricultural income will increase the risk of potential system failure 共violating system constraints兲. The results suggest that the developed approach is also applicable to many practical problems where hybrid uncertainties exist. DOI: 10.1061/共ASCE兲WR.1943-5452.0000096 CE Database subject headings: Water quality; Nonpoint pollution; Agriculture; Uncertainty principles; Decision making; Water management; China. Author keywords: Water quality; Nonpoint source pollution; Possibilistic programming; Chance-constrained programming; Agricultural system; Uncertainty; Decision making.
Introduction Nonpoint source 共NPS兲 water pollution resulted from agricultural activities is a major environmental concern in the world. The pollutants resulting from soil erosion and application of fertilizer and manure can cause serious deterioration of surface and groundwater quality through various routes 共U.S. EPA 1992; Charbonneau and Kondolf 1993; Malik et al. 1994; Huang 1996, 1998; Horan and Ribaudo 1999; Ning et al. 2006; Shrestha et al. 2006; Chen 2007; Chen et al. 2008; Nie et al. 2008兲. Due to its diffuse feature, NPS pollution is difficult to attribute to any single pollution event or source; it cannot be effectively mitigated or reduced relying on a single management practice. Sound management of agricultural activities for causing the NPS pollution provides an effective means of controlling the contamination 共Caruso 1 Faculty of Engineering and Applied Science, Univ. of Regina, Regina, Saskatchewan, Canada S4S 0A2. 2 Faculty of Engineering and Applied Science, Univ. of Regina, Regina, Saskatchewan, Canada S4S 0A2 共corresponding author兲. E-mail address:
[email protected] 3 Faculty of Engineering and Applied Science, Univ. of Regina, Regina, Saskatchewan, Canada S4S 0A2. Note. This manuscript was submitted on September 10, 2008; approved on April 3, 2010; published online on May 4, 2010. Discussion period open until June 1, 2011; separate discussions must be submitted for individual papers. This paper is part of the Journal of Water Resources Planning and Management, Vol. 137, No. 1, January 1, 2011. ©ASCE, ISSN 0733-9496/2011/1-101–112/$25.00.
2000; Berka et al. 2001; Huang and Xia 2001; Luo et al. 2005; Ripa et al. 2006; Dowd et al. 2008兲. However, there are trade-offs between economic and environmental considerations. It is thus desired that preferred decision schemes for managing agricultural activities within an agricultural system be addressed. To assist in solving the above problems, a mathematical model of the agricultural-environment system may be constructed. Mathematical models are useful tools for generating effective water quality management schemes since they can project consequences of alternative management, planning, or policy-level activities 共Haith 1982; Chang et al. 1995; Huang 1996, 1998; Ji and Chang 2005; Nie et al. 2008兲. Uncertainties inherently exist in related costs, impact factors and objectives, and effects on the system behavior. Previously, there were a substantial number of research efforts on water quality management under uncertainty using mathematical models. They were mainly related to stochastic linear programming 共SLP兲 derived from probability theory 共Lee and Kitanidis 1991; Cardwell and Ellis 1993; Huang 1998; Giri et al. 2001; Liu et al. 2008; Wang et al. 2009兲, and fuzzy linear programming 共FLP兲 based on fuzzy set theory 共Kindler 1992; Lee and Wen 1997; Chang et al. 1999; Sasikumar et al. 1999; Mpimpas et al. 2001; Saadatpour and Afshar 2007; Singh et al. 2007兲. A potential problem of SLP is that insufficient data may be available to obtain the probability density distributions 共PDFs兲 for some random parameters; even if these distributions are available, it may be hard to solve a large-scale stochastic water resources system planning problem with all uncertain data being expressed as PDFs 共Birge and Louveaux 1997; Luo et al. 2007兲.
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The concept of FLP was first developed by Bellman and Zadeh 共1970兲. Generally, FLP methods can be categorized into two major classes: flexible programming and possibilistic programming 共Inuiguchi and Ramik 2000; Inuiguchi et al. 2003; Torabi and Hassini 2008兲. In flexible programming, the flexibility in the given target values of the objective functions and the elasticity of the constraints are represented by fuzzy sets 共Chang et al. 1996; Mula et al. 2006兲. These membership functions of fuzzy objective and constraints are determined subjectively by the decision makers 共Lai and Hwang 1992; Chen and Chang 1998兲. Flexible programming methods do not greatly increase the complexity of the models. Nevertheless, in practical water quality management problems, many system parameters are imprecise, such as the unit costs/benefits of crops, crop yields, and the returns for livestock husbandry. Flexible programming cannot effectively incorporate imprecise coefficients of the objective function and constraints into the optimization management framework, thus leading to difficulties in generating sound decision schemes 共Huang et al. 1992, 1993兲. A possible approach for handling the imprecise coefficients of the objective function and constraints is to use possibilistic programming. Different from the probability theory, possibilistic programming focuses on the meaning of information collected rather than its measure 共Lai and Hwang 1992, 1993; Tanaka et al. 2000兲. Possibilistic programming may be a useful approach for including qualitative information in a decision model, but the information must still be quantified in numerical terms 共Dubois and Prade 1980; Buckley 1988; Inuiguchi and Ramik 2000兲. Since Zadeh 共1978兲 originally proposed possibility theory, a variety of research efforts have been made to further the development of possibilistic linear programming 共PLP兲 methods and solution algorithms. Lai and Hwang 共1992兲 proposed an auxiliary multiobjective linear programming 共MOLP兲 model to solve the PLP problem with imprecise coefficients of the objective and/or constraints. Hsu and Wang 共2001兲 employed Lai and Hwang’s approach in solving the proposed PLP model for managing the production planning problems. Wang and Liang 共2005兲 advanced an interactive PLP method to solve the multiproduct aggregate production planning 共APP兲 problem. Muela et al. 共2007兲 applied a fuzzy possibilistic model for the power generation planning. Torabi and Hassini 共2008兲 proposed a two-phase interactive fuzzy approach to solve a multiobjective PLP model for supply chain master planning problems. More studies can be found in Hsieh and Wu 共2000兲, Tang et al. 共2001兲, Liang 共2007兲, and Özgen et al. 共2008兲. However, few of the previous studies reported on development and application of PLP models for dealing with water quality management problems within the agricultural systems. Although PLP approaches can effectively handle the fuzzy coefficients of the objective and/or constraints restricted by possibility distributions, they may encounter difficulties under the conditions when uncertainties in right-hand side 共RHS兲 coefficients of the model’s constraints 共B兲 are represented as probability distributions. A multitude of stochastic optimization methods were proposed for treating the probability distribution information of B in water quality management problems 共Loucks et al. 1981; Maqsood et al. 2005; Luo et al. 2007; Tilmant and Kelman 2007兲, where chance-constrained programming 共CCP兲 is one of the major approaches 共Datta and Dhiman 1996; Jacobs et al. 1997; Huang 1998; Takyi and Lence 1999; Sethi et al. 2006兲. Using CCP, the original constraint of stochastic nature is converted into an equivalent linear constraint with the same size and structure as a deterministic version, especially when the left-hand side coefficients 共A兲 are deterministic and B are random. The only informa-
tion required about the uncertainty is the pi fractile for the unconditional distribution of B. The CCP is effective in reflecting probability distributions of B, but cannot address the fuzzy uncertainties of the coefficients of the objective and/or constraints. In comparison, PLP can deal with the imprecise coefficients, but has difficulties when the model’s RHS coefficients are only quantified as probability distributions. One potential approach for better accounting for uncertainties in the coefficients of the objective and constraints is to incorporate PLP within the CCP framework. Therefore, the objectives of this study are: 共1兲 development of a hybrid possibilistic CCP 共PCCP兲 approach for effectively handling uncertainties in the objective function’s coefficients 共C兲 presented as possibility distributions and those in the constraints’ RHS coefficients 共B兲 as probability distributions and 共2兲 provision of a case study related to water quality management within an agricultural system, where solutions for farming area, manure/ fertilizer application amount, and livestock husbandry size under different uncertainty levels will be obtained and interpreted. A possibilistic stochastic water management 共PSWM兲 model will be provided for reflecting the complex system features under hybrid uncertainty, where implications of water quality/quantity restrictions for achieving regional economic development objectives will be studied. The developed PSWM allows fuzzy information in the objective’s coefficients and probabilistic information in the constraints to be directly communicated into the optimization processes and resulting solutions, such that feasible decision schemes for different agricultural activities can be generated.
Model Development Mathematical Expression A general PLP problem with imprecise coefficients in the objective function can be formulated as follows: n
˜X = max Z = C
˜cixi 兺 i=1
共1a兲
Subject to: AX ⱕ B
共1b兲
Xⱖ0
共1c兲
˜ = vector of imprecise coefficients of the objective funcwhere C tion and has a possibility distribution. Although a variety of distributions exist, the triangular is the most common one for representing the imprecise nature of the ambiguous parameters due to its computational efficiency and simplicity in data acquisition 共Lai and Hwang 1992; Liang 2006; Özgen et al. 2008; Torabi and Hassini 2008兲. Without loss of generality, the triangular possibility distributions are adopted in this study as shown in Fig. 1. Cm is the most likely value with the highest possibility of 1 if normalized; C p 共the most pessimistic value兲 and Co 共the most optimistic value兲 are the least possible values. On the other hand, a general stochastic programming problem can be formulated as follows: max C共t兲X
共2a兲
A共t兲X ⱕ B共t兲
共2b兲
Subject to
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Ai共t兲X ⱕ B共t兲共P兲 ;
Ai共t兲 苸 A共t兲;
i = 1,2, . . . ,m
Xⱖ0 共P兲
˜ 关represented as 共C p, Fig. 1. Triangular possibility distribution of C m o C , C 兲兴
Xⱖ0
共2c兲
where A共t兲, B共t兲, and C共t兲 = sets with random elements defined on a probability space T; t 苸 T 共Charnes et al. 1972; Infanger and Morton 1996; Huang 1998兲; and X = vector of decision variables. Using the CCP approach, model 共2兲 can be converted into an “equivalent” deterministic version. First, a certain probability level pi 苸 关0 , 1兴 is fixed for each constraint i, and then the constraint should be satisfied with at least a probability of 1 − pi. As a result, the feasible solution set is restricted by the following constraints 共Charnes et al. 1972; Huang 1998兲: Pr关兵t兩Ai共t兲X ⱕ bi共t兲其兴 ⱖ 1 − pi,
Ai共t兲 苸 A共t兲,
bi共t兲 苸 B共t兲, 共3兲
i = 1,2, . . . ,m
The generated constraints are generally nonlinear, and the feasible constraints set is convex only for some particular distributions and certain levels of pi, such as the cases when: 共1兲 aij are deterministic and bi are random 共for all pi values兲; 共2兲 aij and bi are discrete random coefficients, with pi ⬎ maxr=1,2,. . .,R共1 − qr兲, where qr = probability associated with realization r; or 共3兲 aij and bi have Gaussian distributions with pi ⱖ 0.5 共Roubens and Teghem 1991; Huang 1998兲. When aij are deterministic and bi are random for model 共2兲, constraint 共3兲 can be specified as follows: Ai共t兲X ⱕ bi共t兲共pi兲,
∀i
共4兲
where bi共t兲共pi兲 = F−1 i 共pi兲, given the cumulative distribution function of bi 关i.e., Fi共bi兲兴 and the probability of violating constraints i共pi兲. However, the CCP method is unable to effectively handle the cases with imprecise coefficients of the objective functions and/or constraints. One potential approach for better accounting for uncertainties in the coefficients of the objective function is to integrate the possibilistic programming within the CCP framework, where possibility distributions are used for reflecting uncertainties in fuzzy coefficients of C and PDFs for B. This leads to a hybrid PCCP model as follows: ˜X max Z = C
共5a兲
Subject to Pr关兵t兩Ai共t兲X ⱕ bi共t兲其兴 ⱖ 1 − pi,
Ai共t兲 苸 A共t兲,
bi共t兲 苸 B共t兲, 共5b兲
i = 1,2, . . . ,m Xⱖ0
共5c兲
The constraints in model 共5兲 have been converted to an equivalent deterministic form, while the objective function coefficients are still considered imprecise 共Charnes et al. 1972; Huang 1998兲 ˜X max Z = C Subject to
共6a兲
共6b兲 共6c兲
共pi兲
where B共t兲 = 兵bi共t兲 兩 i = 1 , 2 , . . . , m其. The developed PCCP model is capable of dealing with various forms of parameters expressed as deterministic, possibilistic, and random variables. With the synergy of fuzzy and stochastic implications, it provides an effective tool to characterize the complex interrelations among the uncertain parameters. In the PCCP model, the great challenge is how to quantify the uncertainty in the decision making process. Fuzzy sets can be characterized based on semantic and cognitive vagueness in decision making 共Chen and Chang 2006兲. It may be difficult to identify fuzzy membership information for all system components in practical problems. Nested target tables can be established for eliciting membership functions of possibilistic variables 共Inuiguchi et al. 2000兲. In practices, a variety of linear and nonlinear functions can be suggested as membership functions based on the representative information in the literatures, reports, and other data sources. However, without losses of generality, linear triangular distributions are widely employed due to their simplicity and low computational requirements. Overview of the Study System The study area is located in the south of Jiangxi Province, China, and is close to the city of Longgang, China 共see Fig. 2兲. The Zhongzhou River, a tributary of the Ganjiang River, runs from southwest to northeast through the study area and finally enters into the Ganjiang River. The Zhongzhou River is the main water source for industrial and domestic uses in the city of Longgang, China and for agricultural development in the study area. Located in the upper reaches of the Zhongzhou River, the study area has a total area of 127.2 km2, with a population of approximately 24,000. It contains four subareas with different crop distributions and different manners of drawing water for irrigation. The area of the tillable land is 82.6 km2. Irrigation water for agricultural production is drawn from the Zhongzhou River through four canals. Crop farming is the main occupation of the area, besides livestock husbandry. The soils involve the dry and wet soils. Only certain soil/crop combinations are feasible in the study area. For example, the wet soils can only be planted with rice, while wheat, vegetables and sweet potato production may be allowed in the dry soils. The study area has a subtropical and monsoon climate, with high heat and rainfall in the summer season. The average daily temperature is about 18° C. The temperature difference over a year is approximately 23° C, and the number of frost-free days is 281. The climatic conditions provide advantages for planting rice, wheat, sweet potato, and other subtropical vegetables. Agricultural production needs water for irrigation, and generates NPS contaminants due to manure/fertilizer applications, leading to deterioration of water quality. The annual rainfall in the study area is 1,503 mm approximately. Most of precipitation occurs between April and July 共wet season兲. After the demands for industrial and domestic uses of the city of Longgang, China are satisfied, the remaining water 共3.2 m3 / s兲 is generally insufficient for agricultural production, especially in the dry season 共August to March兲 due to the spatiotemporal variations of precipitation. The conflict between limited water supply and increasing water demand for agricultural development is becoming a challenge to decision makers. In addition, agricultural activities significantly contribute
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Fig. 2. Study area
to NPS pollution in the Zhongzhou River. Thus, the problem under consideration is how to generate desired decision schemes for agricultural activities in order to maximize total benefit of the agricultural system under the given water quantity and quality restrictions. Data Analysis and Synthesis The model developed requires various forms of information from various data sources. The vagueness of management objectives resulting from the imprecision of the coefficients, and randomness of the natural hydrologic processes introduce uncertainty in water quality management systems 共Beck 1987兲. The fuzzy objective function involves four types of data consisting of crop yields, the economic values of crops and livestock, cost information, and related irrigation water quantity requirements. For modeling purposes, the decision makers focus more on variations of the agricultural incomes due to implementation of different schemes for agricultural activities. Fixed costs such as those for building, land, taxes, and so on are assumed to be unrelated to pollution control and are thereby not included in the model. Crop yields are assumed deterministic. However, unit costs and prices are treated as imprecise, bounded by their possibility distributions. These coefficients and related parameters are inherently uncertain over the planning horizon due to incompleteness and/or unavailability of required information. Assigning crisp values for such fuzzy parameters would lead to loss of useful information and may be inappropriate. Model parameters were determined based on representative data from field investigation and survey, consultation with local regulators and farmers, and related governmental reports 关Envi-
ronmental Informatics Laboratory 共EIL兲 2006; Jiangxi Water Resources Department 共JWRD兲 2007, 2008兴. The data required for the constraint set include water quality and quantity data. The maximum flows of four canals in the subareas are estimated as 0.8, 0.9, 0.8, and 0.7 m3 / s, respectively, due to the geographical characteristics of the targeted area 关Environmental Informatics Laboratory 共EIL兲 2006; Jiangxi Water Resources Department 共JWRD兲 2007, 2008兴. Hydrological and environmental information in the dry and wet seasons were identified. The pollutant loads can be restricted by maximum allowable amounts according to the guidelines in the study area. In this study, maximum allowable losses of total nitrogen, soil, solid-phase nitrogen and phosphorus, dissolved nitrogen and phosphorus by runoff are highly uncertain and contain a number of stochastic factors. These stochastic parameters can be determined through statistical analysis of information obtained from consultation with the decision makers, regulators, and experts in water quality management 关Environmental Informatics Laboratory 共EIL兲 2006; Jiangxi Water Resources Department 共JWRD兲 2007, 2008兴. Table 1 shows the fuzzy input parameters, and Table 2 shows the tillable area of dry and wet soils in the four subareas. Hybrid PSWM Model A hybrid PSWM model will be developed for the study system, which is mainly related to soil, crops, livestock, and human activities. The PSWM model is based on the following assumptions. 共1兲 The uncertain parameters expressed as possibilistic and probabilistic distributions are assumed to be independent of each another. 共2兲 In developing crop nutrient balance equation, only nitrogen is considered since potassium is not considered a poten-
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Table 1. Fuzzy Input Parameters Specified by Possibility Distributions Crop/soil distribution Rice 共i = 2 , j = 1兲
Wheat 共i = 1 , j = 2兲
Vegetable 共i = 1 , j = 3兲 Sweet potato 共i = 1 , j = 4兲
共0.18, 0.22, 0.24兲 共0.16, 0.18, 0.21兲 共0.35, 0.38, 0.40兲 共0.20, 0.24, 0.26兲 Price of crop j 共$/kg兲 共50, 52, 55兲 共60, 64, 66兲 共350, 390, 450兲 共150, 180, 200兲 Farming cost for soil i planted to crop j 共$1 , 000/ km2兲 Total cost to deliver water to the Sijt 关$ / 共m3 / s兲兴 Subarea 1 共960, 1,010, 1,060兲 共970, 1,020, 1,070兲 共980, 1,030, 1,080兲 共960, 1,010, 1,060兲 Subarea 2 共1,680, 1,730, 1,780兲 共1,740, 1,790, 1,840兲 共1,660, 1,710, 1,760兲 共1,660, 1,710, 1,760兲 Subarea 3 共1,770, 1,820, 1,870兲 共1,780, 1,830, 1,880兲 共1,760, 1,810, 1,860兲 共1,770, 1,820, 1,870兲 Subarea 4 共970, 1,020, 1,070兲 共950, 1,000, 1,050兲 共960, 1,010, 1,060兲 共950, 1,000, 1,050兲 Cost of manure collection and disposal 共$/t兲 共2.4, 3, 3.4兲 Cost of fertilizer application 共$/kg兲 共2.0, 2.4, 2.6兲 Livestock Ox 共k = 1兲 Swine 共k = 1兲 Poultry 共k = 1兲 共450, 500, 550兲 共50, 60, 65兲 共3.5, 4, 4.5兲 Average return from livestock k 共$/one兲 Note: Data are from Environmental Informatics Laboratory 共EIL兲 共2006兲 and Jiangxi Water Resources Department 共JWRD兲 共2007, 2008兲.
tial water pollutant and the total amount of phosphorus in the soil is generally not affected by fertilizer or manure application. The nitrogen requirement of different crops on different types of soils will vary with soil fertility 共Haith 1982; Huang 1996, 1998兲. 共3兲 The developed model is based on simplified relationships among the pollutant loads. A sophisticated water quality model is more effective for modeling the processes of contaminant transport and fate, i.e., taking into account the dilution capacity of the receiving water. However, its accuracy may be outweighed by the complexity created in the optimization procedure 共Chen and Chang 2006兲. 共4兲 When solving the auxiliary MOLP problem, only one intermediate variable is considered. This is based on an assumption that all the three auxiliary objectives have equal importance 共Lai and Hwang 1992, 1993兲. The objective of the PSWM model is to maximize total benefit of the agricultural system subject to a set of constraints for relationships between the decision variables and a number of water quantity and quality related restrictions. The decision variables represent farming area, manure/fertilizer application amount, and livestock husbandry size m
Maximize TSB = total system benefit = r
+
m
n
CP jCY ijSij 兺 兺 i=1 j=1
n
m
m
n
兺 兺 i=1 j=1
m
Hij −
n
p
W S 兺 兺 CW iju iju iju 兺 i=1 j=1 u=1
3
B kT k − B oT o = 0 兺 Fij − 兺 兺 i=1 j=1 k=1
共7b兲
where Bk = amount of manure generated by livestock k 共t/animal兲; Bo = amount of manure generated by humans 共t/capital兲; and To = population in the study area. The crop nutrient balance constraint is formulated as follows: ∀ i, j
where p1 = nitrogen volatilization/denitrification rate of manure 共%兲; p2 = nitrogen volatilization/denitrification rate of nitrogen fertilizer 共%兲; NM = nitrogen content of manure 共kg/t兲; and NRij = nitrogen requirement of crop j on soil i 共kg/ km2兲. Energy and digestible protein requirements are as follows:
Table 2. Tillable Soil Area in the Four Subareas 2
r
共7c兲
where TSB= total system benefit 共$兲; CP j = price of crop j 共$/kg兲; CY ij = yield of crop j on soil i 共kg/ km2兲; Sij = area of soil i planted RLk = average return from livestock k 共$/one兲; to crop j 共km2兲;
1
n
n
共7a兲
Subarea
m
共1 − p1兲 · NM · Fij + 共1 − p2兲 · Hij − NRij · Sij ⱖ 0
˜ S −G ˜ RLkTk − 兺 兺 G ij ij f 兺 兺 Fij 兺 i=1 j=1 k=1 i=1 j=1
˜ −G h
˜ = farming cost for soil i planted to Tk = number of livestock k; G ij ˜ = cost of manure collection/disposal 共$/t兲; G ˜ crop j 共$ / km2兲; G f h = cost of fertilizer application 共$/kg兲; Fij = amount of manure applied to soil i planted to crop j 共t兲; Hij = amount of fertilizer nitro = total cost to gen applied to soil i planted to crop j 共kg兲; CW iju deliver water to the Siju 关$ / 共m3 / s兲兴; Wiju = quantity of irrigation water required for soil i planted to crop j in subarea u 共m3 / km2 s兲; and Siju = area of soil i planted with crop j in subarea u 共km2兲. The water quality constraints include those for manure and crop nutrients, energy and digestible protein, and pollutant loads. The manure mass balance constraint is described as follows:
4
Wet soil 共km2兲 8.1 8.5 12.3 11.3 9.2 10.1 14.8 8.3 Dry soil 共km2兲 Note: Data are from Environmental Informatics Laboratory 共EIL兲 共2006兲 and Jiangxi Water Resources Department 共JWRD兲 共2007, 2008兲.
m
n
r
Ek · Tk − Eo · To ⱖ 0 兺 CY ij · ␣ j · Sij − 兺 兺 i=1 j=1 k=1 m
n
共7d兲
r
Dk · Tk − Do · To ⱖ 0 兺 CY ij ·  j · Sij − 兺 兺 i=1 j=1 k=1
共7e兲
where ␣ j = net energy content of crop j 共Mcal/kg兲;  j = digestible protein content of crop j 共%兲; Ek = net energy supplied to livestock k 共Mcal/animal兲; Eo = net energy supplied to humans 共Mcal/ capita兲; Dk = net digestible protein supplied to livestock k 共kg/ animal兲; and Do = net digestible protein supplied to humans 共kg/ capita兲.
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The pollutant loads constraints include those for soil loss, total nitrogen losses, solid-phase nitrogen/phosphorus losses, and dissolved nitrogen/phosphorus losses m
n
兺 兺 i=1 j=1
m
共NM · Fij + Hij − NRij · Sij兲 ⱕ a共t兲共pa兲 ·
n
m
兺 Lij · Sij ⱕ b共t兲 兺 i=1 j=1
Ki 兺 i=1
共7f兲
∀ i, j,u
共7o兲
where i represents the index of soil types; j represents the index of kinds of crops; k represents the index of species of livestock; u represents the index of subareas; m = number of soil types; n = number of kinds of crops; r = number of species of livestock; and p = number of subareas.
m
共pb兲
·
Ki 兺 i=1
共7g兲
Solution Method
共7h兲
Since a part of its coefficients have triangular possibility distributions, the objective function 共7a兲 is actually an imprecise one with a triangular distribution as well. It can be rewritten as follows:
n
Sij ⱕ Ki 兺 j=1
Fij, Hij, Siju, Tk ⱖ 0,
∀i
max共TSBm,TSBp,TSBo兲 = 关共Cm兲TX,共C p兲TX,共Co兲TX兴 m
n
兺 hi1 · Lij · Sij ⱕ c1共t兲 兺 i=1 j=1 m
Ki 兺 i=1
共7i兲
m
hi2 · Lij · Sij ⱕ c2共t兲共pc2兲 ·
兺 i=1
共7j兲
Ki
m
共R1ij · N1j + R2ij · N2j兲 · Sij ⱕ f 1共t兲共p f1兲 ·
n
兺 共R1ij · P1j + R2ij · P2j兲 · Sij ⱕ f 2共t兲 兺 i=1 j=1
Ki 兺 i=1
共7k兲
m
共p f2兲
·
Ki 兺 i=1
共7l兲
where a共t兲共pa兲 = maximum allowable total nitrogen losses corresponding to probability pa 共kg/ km2兲; b共t兲共pb兲 = maximum allowable soil losses corresponding to probability pb 共kg/ km2兲; Lij = soil loss from soil i planted to crop j 共kg/ km2兲; Ki = tillable area of soil i 共km2兲; c1共t兲共pc1兲 = maximum allowable solid-phase nitrogen loss corresponding to probability pc1 共kg/ km2兲; c2共t兲共pc2兲 = maximum allowable solid-phase phosphorus loss corresponding to probability pc2 共kg/ km2兲; f 1共t兲共p f1兲 = maximum allowable dissolved nitrogen loss by runoff corresponding to probability p f1 共kg/ km2兲; f 2共t兲共p f2兲 = maximum allowable dissolved phosphorus loss by runoff corresponding to probability p f2 共kg/ km2兲; hi1 = nitrogen content of soil i 共%兲; hi2 = phosphorus content of soil i 共%兲; R1ij = wet season runoff from soil i planted with crop j 共mm兲; R2ij = dry season runoff from soil i planted with crop j 共mm兲; N1j = dissolved nitrogen concentration in wet season runoff from land planted with crop j 共mg/L兲; N2j = dissolved nitrogen concentration in dry season runoff from land planted with crop j 共mg/L兲; P1j = dissolved phosphorus concentration in wet season runoff from land planted with crop j 共mg/L兲; P2j = dissolved phosphorus concentration in dry season runoff from land planted with crop j 共mg/L兲. The water quantity constraints are described as follows: m
n
or
n
兺 兺 i=1 j=1 m
·
n
兺 兺 i=1 j=1 m
m
共pc1兲
max
共cmi xi,cipxi,coi xi兲 兺 i=1
共8兲
where Cm = 共cm1 , cm2 , . . . , cmn 兲T, C p = 共c1p , c2p , . . . , cnp兲T, and Co = 共co1 , co2 , . . . , icon兲T. TSBm is the most likely value of the imprecise total system benefit, and TSBp 共the most pessimistic value兲 and TSBo 共the most optimistic value兲 are the least possible values. The imprecise objective function of the total system benefit can be converted into the following auxiliary three objective functions 共Lai and Hwang 1992, 1993; Özgen et al. 2008; Torabi and Hassini 2008兲: min TSB1 = 共TSBm − TSBp兲 = 共Cm − C p兲TX
共9a兲
max TSB2 = TSBm = CmTX
共9b兲
max TSB3 = 共TSBo − TSBm兲 = 共Co − Cm兲TX
共9c兲
In practical water quality management problem, Eqs. 共9a兲–共9c兲 are equivalent to simultaneously maximizing the most likely value of the total system benefit, maximizing the possibility of obtaining higher total system benefit, and minimizing the risk of obtaining lower total system benefit. A variety of MOLP methods can be used to solve the above problem 共Chang et al. 1995; Chen and Chang 1998, 2006; Li et al. 2006; Liang 2006, 2007; Zhang et al. 2010兲. In this study, Zimmermann’s fuzzy programming approach with normalization process is employed since it can explicitly measure the overall satisfaction degree of all objective functions 共Zimmermann 1978兲 max s.t.
ⱕ TSBi,
i = 1,2,3
x苸X
n
兺 Wiju · Siju ⱕ qu 兺 i=1 j=1
∀u
共7m兲
where qu = maximum canal flow within subarea u 共m3 / s兲. The technical constraints are as follows: p
Siju = Sij 兺 u=1
∀ i, j
共7n兲
0ⱕⱕ1
共10兲
where = overall satisfaction degree of the objective functions and TSBi = membership function for each objective function. In this study, linear membership functions are assumed because other nondecreasing or nonincreasing functions can be transferred into equivalent linear forms by variable transformations 共Lai and Hwang 1993兲. These are
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Table 3. Solutions of the PSWM Model Solution Symbol
Soil
Crop
Livestock
Cropping area 共km 兲 Wet Rice S211 Dry Wheat S121 Dry Vegetable S131 Dry Sweet potato S141 Wet Rice S212 Dry Wheat S122 Dry Vegetable S132 Dry Sweet potato S142 Wet Rice S213 Dry Wheat S123 Dry Vegetable S133 Dry Sweet potato S143 Wet Rice S214 Dry Wheat S124 Dry Vegetable S134 Dry Sweet potato S144 Amount of manure application 共103 t兲 Wet Rice F21 Dry Wheat F12 Dry Vegetable F13 Dry Sweet potato F14 Amount of nitrogen fertilizer application 共kg兲 Wet Rice H21 Dry Wheat H12 Dry Vegetable H13 Dry Sweet potato H14 Size of livestock husbandry 共103兲 Ox T1 Swine T2 Poultry T3 Net income 共$106兲
Subarea
P = 0.01
P = 0.05
P = 0.10
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4
8.1 6.9 2.3 0 0 10.1 0 0 0 8.7 0 0 6.7 0 8.3 0
8.1 0 9.2 0 0 10.1 0 0 0 14.8 0 0 5.6 4.7 3.6 0
8.1 0 9.2 0 0 10.1 0 0 0 14.8 0 0 8.0 4.0 4.3 0
25.8 70.6 27.4 0
23.9 81.4 33.9 0
28.1 79.4 36.3 0
0 640 0 0
0 0 0 0
0 0 0 0
2.6 0.8 143 共7.369, 8.125, 8.758兲 0.7215
3.5 0 150 共8.684, 9.499, 10.213兲 0.7250
3.7 0 166 共9.253, 10.131, 10.878兲 0.7096
2
TSB1 =
TSB2 =
冦 冦
1
if TSB1 ⬍ TSBPIS 1
TSBNIS 1 − TSB1 PIS TSBNIS 1 − TSB1
NIS if TSBPIS 1 ⱕ TSB1 ⱕ TSB1
0
if TSB1 ⬎ TSBNIS 1
冧
共11a兲 1
if TSB2 ⬎
TSB2 − TSBNIS 2 NIS TSBPIS 2 − TSB2
PIS if TSBNIS 2 ⱕ TSB2 ⱕ TSB2
0
if TSB2 ⬍ TSBNIS 2
TSBPIS 2
冧
m p TSBPIS 1 = min共TSB − TSB 兲;
m p TSBNIS 1 = max共TSB − TSB 兲
共12a兲 m TSBPIS 2 = max TSB ; o m TSBPIS 3 = max共TSB − TSB 兲;
m TSBNIS 2 = min TSB
共12b兲
o m TSBNIS 3 = min共TSB − TSB 兲
共12c兲
Result Analysis and Discussions
共11b兲
and TSBNIS represent the positive and negative where TSBPIS i i ideal solutions 共NIS兲, respectively. Similar to TSB2, TSB3 can be determined. The positive ideal solutions and NIS of the three objective functions can be determined by solving the corresponding six single-objective linear programming models as follows 共Hwang and Yoon 1981; Seo and Sakawa 1988; Lai and Hwang 1992; Wang and Liang 2005; Özgen et al. 2008; Torabi and Hassini 2008兲
Results Table 3 shows the solutions obtained from the PSWM model. The majority of dry soils in the study area should be cultivated. Most of the dry soils should be planted with wheat and vegetables due to their good market prices. The areas of wheat plantation for Subareas 1 to 4 should be 6.9, 10.1, 8.7, and 0 km2, respectively, when pi = 0.01 共or 0, 10.1, 14.8, and 4.7 km2, respectively, when pi = 0.05; or 0, 10.1, 14.8, and 4.0 km2, respectively, when pi
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= 0.10兲. The pi levels represent the probabilities that pollutant loads constraints would be violated. An increase of the pi level would lead to a decreased strictness of the constraints. For vegetable plantation, its area in Subarea 1 would be increased from 2.3 to 9.2 km2 with the increase of pi level from 0.01 to 0.05; in Subarea 4, its area would be decreased with the increase of pi level; while in Subareas 2 and 3, no soils should be planted with vegetables. Sweet potato production in the study area is totally limited due to its relatively high irrigation water demand and significant soil erosion and pollutant loss problems. In comparison, wet soils in Subareas 2, 3, and 4 should be totally or partly left idle 共see Table 2 for tillable areas in four subareas兲. Solutions for livestock husbandry indicate that sizes for ox, swine, and poultry should be 2.6⫻ 103, 0.8⫻ 103, and 143⫻ 103, respectively, when pi = 0.01 共3.5⫻ 103, 0, and 150⫻ 103, respectively, when pi = 0.05; 3.7⫻ 103, 0, and 166⫻ 103, respectively, when pi = 0.10兲. With the increase of pi level, the sizes for ox and poultry would be also increased, while the size for swine would be decreased to 0. Livestock husbandry is related to both direct income and manure supply. The results demonstrate that sizes for ox and swine should be limited to low levels, while that for poultry should be relatively large. This reflects the differences in their energy and protein demands, market prices and manure generation rates. The amounts of manure applied to rice, wheat and vegetable lands should be 25.8⫻ 103, 70.6⫻ 103, and 27.4⫻ 103 t, respectively, when pi = 0.01 共or 23.9⫻ 103, 81.4⫻ 103, and 33.9 ⫻ 103 t, respectively, when pi = 0.05; or 28.1⫻ 103, 79.4⫻ 103, and 36.3⫻ 103 t, respectively, when pi = 0.10兲. The pi level represents the probability that the constraint that pollutant loads including total nitrogen, soil, solid-phase nitrogen and phosphorus, dissolved nitrogen, and phosphorus cannot be greater than their maximum allowable amounts will be exceeded. In other words, it is not mandatory that these constraints should be always satisfied, but may be violated with certain confidence levels. A lower pi level means stricter environmental constraint. Fertilizer should only be applied to wheat land with 640 kg for pi = 0.01; no fertilizer applications are allowed for all of the agricultural lands when pi = 0.05 or 0.10. That is due to the higher nitrogen requirements of wheat and lower amount of manure application to wheat. A smaller fertilizer application amount is thus required for crop nutrient balance. Manure and fertilizer application can lead to significant pollutant losses from the agricultural lands. In order to satisfy water quality constraints for the river, only limited amount of manure and fertilizer can be used. Consequently, how to effectively allocate this limited “allowable amount” becomes an important issue. Manure is locally generated and is related to livestock husbandry which generates income when consuming this limited allowable amount. In comparison, fertilizer has to be purchased from external systems and consumes the limited allowable amount without any direct income. The above facts have been reflected in the PSWM solutions. Manure generated in the basin area should be applied to farmlands, while fertilizer application is limited to a low level. Generally, large manure application amount means a full use of wastes from livestock husbandry; and small fertilizer application amount will help to prevent NPS water pollution in the river. It is more environmentally and economically effective to have a relatively high livestock husbandry size 共and thus a large manure application amount兲 than to have a large fertilizer application amount directly. Table 3 also indicates that higher pi levels correspond to higher total system benefits. Since some of the coefficients of the objective are imprecise, the total system benefit is imprecise and
Fig. 3. Total system benefit given the different pi level
has a triangular possibility distribution. When pi = 0.01, the total system benefit= $共7.369, 8.125, 8.758兲 ⫻ 106. When pi is increased to 0.05 and 0.10, the corresponding total system benefits would be increased to $共8.684, 9.499, 10.213兲 ⫻ 106 and $共9.253, 10.131, 10.878兲 ⫻ 106, respectively. The overall satisfaction degree is 0.7215, 0.7250, and 0.7096, respectively, when pi = 0.01, 0.05, and 0.10. In total, the results from the PSWM model indicate that uncertain information caused by cognitive vagueness and randomness and their combinational effects on the system condition variations can be effectively communicated into the optimization management framework and resulting solutions. The solutions from PSWM offer flexibility in interpreting the results and generating decision schemes. Decision alternatives for farming area, manure/ fertilizer application amount, and livestock husbandry size under various uncertainty levels are obtained, reflecting complex system characteristics. The solution feature may be favored by decision makers due to increased applicability for identifying the final decision schemes under hybrid fuzzy and random uncertainties. Since the pi levels represent the probabilities of constraint violation, an increased pi level means a decreased strictness for the constraints and thus an expanded decision space, which may then result in an increased TSB value. Fig. 3 presents the relationship between the pi levels and total system benefits. Relation between total system benefits and pi levels demonstrates trade-offs between economic efficiency and system risk. Under advantageous conditions 共i.e., conditions with potentially higher allowable amount for pollutant emission兲, a higher total system benefit can be obtained, which corresponds to a higher pi level. That is, with an increased pi level, the amounts of maximum allowable NPS losses of soil, nitrogen, and phosphorus will be increased. A strong desire to acquire high agricultural income will run into the risk of potentially violating the related water quality standards. Willingness to accept low agricultural income will guarantee meeting the water quality standards; however, the risk of violating the constraints will increase due to lower allowable amount for pollutant emission. The research outputs were favored by the decision makers due to their flexibility and applicability for practical decision processes. The decision variables under different pi levels were useful for the decision makers to justify and/or adjust the decision schemes for the agricultural activities through incorporation of their implicit knowledge. Comparison between the PSWM and CCP Models The problem can also be solved through a CCP method by letting fuzzy coefficients in the PSWM model be equal to their most likely values 共values with the highest possibility degree兲. The CCP solutions are contained in Table 4. Solutions obtained from
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Table 4. Solutions Obtained from CCP Approach CCP solutions Symbol
Soil
Crop
Cropping area 共km 兲 Wet Rice S211 Dry Wheat S121 Dry Vegetable S131 Dry Sweet potato S141 Wet Rice S212 Dry Wheat S122 Dry Vegetable S132 Dry Sweet potato S142 Wet Rice S213 Dry Wheat S123 Dry Vegetable S133 Dry Sweet potato S143 Wet Rice S214 Dry Wheat S124 Dry Vegetable S134 Dry Sweet potato S144 Amount of manure application 共103 t兲 Wet Rice F21 Dry Wheat F12 Dry Vegetable F13 Dry Sweet potato F14 Amount of nitrogen fertilizer application 共kg兲 Wet Rice H21 Dry Wheat H12 Dry Vegetable H13 Dry Sweet potato H14 Size of livestock husbandry 共103兲 T1 T2 T3 Net income 共$106兲
Livestock
Subarea
P = 0.01
P = 0.05
P = 0.10
1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4
8.1 0 9.2 0 8.5 0 1.1 0 12.3 3.4 0 0 11.3 3.9 4.4 0
8.1 0 9.2 0 8.4 0 4.0 0 12.3 6.0 0 0 11.3 3.9 4.4 0
8.1 0 9.2 0 8.5 0 6.3 0 12.3 6.9 0 0 11.3 3.9 4.4 0
70.35 20.09 28.90 0.02
70.23 27.17 40.35 0
70.31 29.56 46.11 0
0 0 72.09 0
0 0 41.41 0
0 0 42.55 0
1.9 0.5 286 11.326
2.9 0.6 290 13.223
3.4 0 293 14.448
2
Ox Swine Poultry
the CCP method indicate that the cropping patterns are significantly different from those obtained from the PSWM model. Wet soils in the four subareas should be totally or almost totally planted with rice. Wheat would be planted to soils in Subareas 3 and 4, while vegetables would be planted in Subareas 1, 2, and 4. No sweet potato would be planted in the study area due to its relatively high irrigation water demand and significant soil erosion and pollutant loss problems. The total system benefit from the CCP solution is $11.326⫻ 106 when pi = 0.01 共or $13.223 ⫻ 106 and $14.448⫻ 106, respectively, when pi = 0.05 and 0.10兲. It is indicated that the agricultural income is greater than that obtained from the PSWM model under the same pi level. Using the CCP approach, only one set of deterministic solutions corresponding to each pi level is obtained, which represents a situation when all elements of C in model 共6兲 are equal to their most likely values. The obtained total system benefit is a deterministic value corresponding to only one of various interrelationships among the uncertain parameters. In addition, the CCP method overestimates the agricultural income compared with various scenarios from the PSWM model under same pi level. For example, when pi = 0.01, the agricultural income from the CCP solution is 39.40% higher than that under the most likely scenario, 53.70% than that under the most pessimistic scenario, and 29.32% than that under the
most optimistic scenario from the PSWM solution. When pi = 0.05 and 0.1, it reaches 39.20 and 42.61% higher than those from the most likely values of the PSWM solution, respectively. Thus, ignoring uncertainty in the objective function will significantly worsen the system performance and lead to unreal reflection of system conditions. The higher agricultural income from the CCP approach cannot be achieved in real-world cases under hybrid uncertainties of fuzziness and randomness due to their interacting impacts. Replacing fuzzy coefficients in the objective function with their most likely values will lead to loss of a number of significant information, and thus reduce the effectiveness of the obtained solution so that they cannot well reflect the complexity and uncertainty of the system. Although further sensitivity analyses can be undertaken, each of them can only represent a single response to one or several parameter variations.
Conclusions A PSWM model has been developed and applied to a real-world case study of agricultural NPS water quality management. The model is based on a hybrid PCCP approach. This study is a first
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application of PCCP approach to NPS water quality management problems within an agricultural system. Hybrid uncertainties with the synergy of fuzzy and stochastic implications can be effectively characterized. In practical water quality management, uncertainties caused by semantic and cognitive vagueness can be addressed through possibilistic distributions restricted by fuzzy sets while random uncertainties can be represented by probabilistic distributions; their combinational effects on the system condition variations can be effectively reflected through the developed PSWM model. It represents improvements of the existing PLP and CCP methods through effective incorporation of possibility distributions in C and probability distributions in B into a general optimization framework. The PSWM model has the following advantages: 共1兲 it will not lead to more complicated intermediate models and thus have lower computational requirements; 共2兲 its solutions offer flexibility in interpreting the results and reflect the interactional effects between fuzzy uncertainty from the objective and random uncertainty from the constraints; and 共3兲 it can help examine the risk of violating system constraints and the associated consequences. Different schemes for agricultural activities under different pi levels can be generated for reflecting trade-offs between economic and environmental considerations. A strong desire to acquire high agricultural income will run into the risk of potentially violating the related water quality standards, while willingness to accept low agricultural income will increase the risk of potential system failure 共violating system constraints兲. The decision variables under different pi levels are useful for the decision makers to justify and/or adjust the decision schemes for the agricultural activities through incorporation of their implicit knowledge. The developed PSWM model may be extended to more cases where the LHS and RHS coefficients in the constraints can be also expressed as fuzzy membership functions, although this study only handles a portion of fuzzy coefficients of the objective function. More fuzzy coefficients of A, B, C, and their combinations can be included and tackled in the optimization process. Another potential extension is to develop an interval possibilistic stochastic programming model for water quality management when parts of input parameters are expressed by interval bounds. In water quality management planning problems, defining lower and upper bounds on model parameters is more direct and may be more meaningful than specifying distributions. Thus, it is possible to develop such an interval PSWM model for water quality management planning under multiple uncertainties expressed by interval numbers, fuzzy membership distributions, and probabilistic distributions. While this study is a first attempt to solve water quality management issues within an agricultural system under hybrid fuzzy and random uncertainties, the method can be extended to other environmental management planning problems, such as waste management and air pollution control.
Acknowledgments This research was supported by the Major State Basic Research Development Program of MOST 共Grant No. 2006CB403307兲, and the Natural Science and Engineering Research Council of Canada. The writers would like to thank the editor, associate editor, and anonymous reviewers for their helpful comments and suggestions.
Notation The following symbols are used in this paper: a共t兲共pa兲 ⫽ maximum allowable total nitrogen losses corresponding to probability pa 共kg/ km2兲; Bk ⫽ amount of manure generated by livestock k 共t/animal兲; Bo ⫽ amount of manure generated by humans 共t/capital兲; b共t兲共pb兲 ⫽ maximum allowable soil losses corresponding to probability pb 共kg/ km2兲; CWiju ⫽ total cost to deliver water to the Siju 关$ / 共m3 / s兲兴; CY ij ⫽ yield of crop j on soil i 共kg/ km2兲; CP j ⫽ price of crop j 共$/kg兲; c1共t兲共pc1兲 ⫽ maximum allowable solid-phase nitrogen loss corresponding to probability pc1 共kg/ km2兲; c2共t兲共pc2兲 ⫽ maximum allowable solid-phase phosphorus loss corresponding to probability pc2 共kg/ km2兲; Dk ⫽ net digestible protein supplied to livestock k 共kg/animal兲; Do ⫽ net digestible protein supplied to humans 共kg/capita兲; Ek ⫽ net energy supplied to livestock k 共Mcal/animal兲; Eo ⫽ net energy supplied to humans 共Mcal/capita兲; Fij ⫽ amount of manure applied to soil i planted to crop j 共t兲; f 1共t兲共p f1兲 ⫽ maximum allowable dissolved nitrogen loss by runoff corresponding to probability p f1 共kg/ km2兲; f 2共t兲共p f2兲 ⫽ maximum allowable dissolved phosphorus loss by runoff corresponding to probability p f2 共kg/ km2兲; ˜ ⫽ cost of manure collection/disposal 共$/t兲; G f ˜ Gh ⫽ cost of fertilizer application 共$/kg兲; ˜ ⫽ farming cost for soil i planted to crop j G ij 共$ / km2兲; Hij ⫽ amount of fertilizer nitrogen applied to soil i planted to crop j 共kg兲; hi1 ⫽ nitrogen content of soil i 共%兲; hi2 ⫽ phosphorus content of soil i 共%兲; Ki ⫽ tillable area of soil i 共km2兲; Lij ⫽ soil loss from soil i planted to crop j 共kg/ km2兲; m ⫽ number of types of soils; N1j ⫽ dissolved nitrogen concentration in wet season runoff from land planted with crop j 共mg/L兲; N2j ⫽ dissolved nitrogen concentration in dry season runoff from land planted with crop j 共mg/L兲; NM ⫽ nitrogen content of manure 共kg/t兲; NRij ⫽ nitrogen requirement of crop j on soil i 共kg/ km2兲; n ⫽ number of kinds of crops; P1j ⫽ dissolved phosphorus concentration in wet season runoff from land planted with crop j 共mg/L兲; P2j ⫽ dissolved phosphorus concentration in dry season runoff from land planted with crop j 共mg/L兲;
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p1 ⫽ nitrogen volatilization/denitrification rate of manure 共%兲; p2 ⫽ nitrogen volatilization/denitrification rate of nitrogen fertilizer 共%兲; qu ⫽ maximum canal flow within subarea u 共m3 / s兲; R1ij ⫽ wet season runoff from soil i planted with crop j 共mm兲; R2ij ⫽ dry season runoff from soil i planted with crop j 共mm兲; RLk ⫽ average return from livestock k 共$/one兲; r ⫽ number of species of livestock; Sij ⫽ area of soil i planted to crop j 共km2兲; Siju ⫽ area of soil i planted with crop j in subarea u 共km2兲; Tk ⫽ number of livestock k; To ⫽ population in the study area; TSB ⫽ total system benefit 共$兲; Wiju ⫽ quantity of irrigation water required for soil i planted to crop j in subarea u 共m3 / km2 s兲; ␣ j ⫽ net energy content of crop j 共Mcal/kg兲; and  j ⫽ digestible protein content of crop j 共%兲.
References Beck, M. B. 共1987兲. “Water quality modeling: A review of the analysis of uncertainty.” Water Resour. Res., 23, 1393–1442. Bellman, R. E., and Zadeh, L. A. 共1970兲. “Decision making in a fuzzy environment.” Manage. Sci., 17, B-141–B-164. Berka, C., Schreier, H., and Hall, K. 共2001兲. “Linking water quality with agricultural intensification in a rural watershed.” Water Air Soil Pollut., 127, 389–401. Birge, J. R., and Louveaux, F. 共1997兲. Introduction to stochastic programming, Springer, New York. Buckley, J. J. 共1988兲. “Possibilistic linear programming with triangular fuzzy numbers.” Fuzzy Sets Syst., 26, 135–138. Cardwell, H., and Ellis, H. 共1993兲. “Stochastic dynamic-programming models for water-quality management.” Water Resour. Res., 29共4兲, 803–813. Caruso, B. S. 共2000兲. “Comparative analysis of New Zealand and US approaches for agricultural nonpoint source pollution management.” Environ. Manage. (N.Y.), 25共1兲, 9–22. Chang, N. B., Wen, C. G., Chen, Y. L., and Yong, Y. C. 共1996兲. “Optimal planning of the reservoir watershed by grey fuzzy multi-objective programming 共I兲: Theory.” Water Res., 30共10兲, 2329–2334. Chang, N. B., Wen, C. G., and Wu, S. L. 共1995兲. “Optimal management of environmental and land resources in a reservoir watershed by multi-objective programming.” J. Environ. Manage., 44共2兲, 144–161. Chang, N. B., Yeh, S. C., and Wu, G. C. 共1999兲. “Stability analysis of grey compromise programming and its applications.” Int. J. Syst. Sci., 30共6兲, 571–589. Charbonneau, R., and Kondolf, G. M. 共1993兲. “Land use change in California, USA: Nonpoint source water quality impacts.” Environ. Manage., 17共4兲, 453–460. Charnes, A., Cooper, W. W., and Kirby, P. 共1972兲. “Chance constrained programming: An extension of statistical method.” Optimizing methods in statistics, Academic, New York. Chen, B. 共2007兲. “Climate change and pesticide loss in watershed systems: A simulation modeling study.” J Environ. Inform., 10共2兲, 55– 67. Chen, H. W., and Chang, N. B. 共1998兲. “Water pollution control in the river basin by genetic algorithm-based fuzzy multi-objective programming modeling.” Water Sci. Technol., 37共8兲, 55–63. Chen, H. W., and Chang, N. B. 共2006兲. “Decision support for allocation of watershed pollution load using grey fuzzy multiobjective program-
ming.” J. Am. Water Resour. Assoc., 42共3兲, 725–745. Chen, M., Chen, J., and Sun, F. 共2008兲. “Agricultural phosphorus flow and its environmental impacts in China.” Sci. Total Environ., 405共1– 3兲, 140–152. Datta, B., and Dhiman, S. D. 共1996兲. “Chance-constrained optimal monitoring network design for pollutants in ground water.” J. Water Resour. Plann. Manage., 122共3兲, 180–188. Dowd, B. M., Press, D., and Los Huertos, M. 共2008兲. “Agricultural nonpoint source water pollution policy: The case of California’s Central Coast.” Agric., Ecosyst. Environ., 128共3兲, 151–161. Dubois, D., and Prade, H. 共1980兲. “Systems of linear fuzzy constraints.” Fuzzy Sets Syst., 3, 37–48. Environmental Informatics Laboratory 共EIL兲. 共2006兲. “An integrated decision support system for developing rural eco-environmental sustainability in the Mountain-River-Lake Region of Jiangxi Province, China.” Rep. Prepared for United Nations Development Program, Univ. of Regina, Regina, Sask., Canada. Giri, B. S., Karimi, I. A., and Ray, M. B. 共2001兲. “Modeling and Monte Carlo simulation of TCDD transport in a river.” Water Res., 35共5兲, 1263–1279. Haith, D. A. 共1982兲. Environmental systems optimization, Wiley, New York. Horan, R. D., and Ribaudo, M. O. 共1999兲. “Policy objectives and economic incentives for controlling agricultural sources of nonpoint pollution.” J. Am. Water Resour. Assoc., 35共5兲, 1023–1035. Hsieh, S. H., and Wu, M. S. 共2000兲. “Demand and cost forest error sensitivity analyses in aggregate production planning by possibilistic linear programming models.” J. Intell. Manuf., 11, 355–364. Hsu, H. M., and Wang, W. P. 共2001兲. “Possibilistic programming in production planning of assemble-to-order environments.” Fuzzy Sets Syst., 119, 59–70. Huang, G. H. 共1996兲. “IPWM: An interval parameter water quality management model.” Eng. Optimiz., 26, 79–103. Huang, G. H. 共1998兲. “A hybrid inexact-stochastic water management model.” Eur. J. Oper. Res., 107, 137–158. Huang, G. H., Baetz, B. W., and Patry, G. G. 共1992兲. “A grey linear programming approach for municipal solid waste management planning under uncertainty.” Civ. Eng. Syst., 9, 319–335. Huang, G. H., Baetz, B. W., and Patry, G. G. 共1993兲. “A grey fuzzy linear programming approach for waste management and planning under uncertainty.” Civ. Eng. Syst., 10, 123–146. Huang, G. H., and Xia, J. 共2001兲. “Barriers to sustainable water-quality management.” J. Environ. Manage., 61, 1–23. Hwang, C. L., and Yoon, K. 共1981兲. Multiple attribute decision making: Methods and applications, Springer, Berlin. Infanger, G., and Morton, D. P. 共1996兲. “Cut sharing for multistage stochastic linear programs with interstage dependency.” Math. Program., 75, 241–256. Inuiguchi, M., Greco, S., Slowinski, R., and Tanino, T. 共2003兲. “Possibility and necessity measure specification using modifiers for decision making under fuzziness.” Fuzzy Sets Syst., 137, 151–175. Inuiguchi, M., and Ramik, J. 共2000兲. “Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem.” Fuzzy Sets Syst., 111, 3–28. Inuiguchi, M., Tanino, T., and Sakawa, M. 共2000兲. “Membership function elicitation in possibilistic programming problems.” Fuzzy Sets Syst., 111, 29–45. Jacobs, T. L., Medina, M. A., and Ho, J. T. 共1997兲. “Chance constrained model for storm-water system design and rehabilitation.” J. Water Resour. Plann. Manage., 123共3兲, 163–168. Ji, J. H., and Chang, N. B. 共2005兲. “Risk assessment for optimal freshwater inflow in conjunction with sustainability indicators in a semiarid coastal bay.” Stochastic Environ. Res. Risk Assess., 19共2兲, 111– 124. Jiangxi Water Resources Department 共JWRD兲. 共2007兲. “Jiangxi tenth five-year planning of hydrological development.” Government Rep., Jiangxi, China 共in Chinese兲. Jiangxi Water Resources Department 共JWRD兲. 共2008兲. “Implementation
JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / JANUARY/FEBRUARY 2011 / 111
Downloaded 14 Jan 2011 to 142.3.100.23. Redistribution subject to ASCE license or copyright. Visit http://www.ascelibrary.org
report of water resources management of Ganjiang river basin.” Government Rep., Jiangxi, China 共in Chinese兲. Kindler, J. 共1992兲. “Rationalizing water requirements with aid of fuzzy allocation model.” J. Water Resour. Plann. Manage., 118, 308–318. Lai, Y. J., and Hwang, C. L. 共1992兲. “A new approach to some possibilistic linear programming problems.” Fuzzy Sets Syst., 49, 121–133. Lai, Y. J., and Hwang, C. L. 共1993兲. “Possibilistic linear programming for managing interest rate risk.” Fuzzy Sets Syst., 54, 135–146. Lee, C. S., and Wen, C. G. 共1997兲. “Fuzzy goal programming approach for water-quality management in a river basin.” Fuzzy Sets Syst., 89, 181–192. Lee, S. I., and Kitanidis, P. K. 共1991兲. “Optimal estimation and scheduling in aquifer remediation with incomplete information.” Water Resour. Res., 27共9兲, 2203–2217. Li, X. Q., Zhang, B., and Li, H. 共2006兲. “Computing efficient solutions to fuzzy multiple objective linear programming problems.” Fuzzy Sets Syst., 157, 1328–1332. Liang, T. F. 共2006兲. “Distribution planning decisions using interactive fuzzy multi-objective linear programming.” Fuzzy Sets Syst., 157, 1303–1316. Liang, T. F. 共2007兲. “Application of interactive possibilistic linear programming to aggregate production planning with multiple imprecise objectives.” Prod. Plann. Contr., 18共7兲, 548–560. Liu, Y., Yang, P. J., Hu, C., and Guo, H. C. 共2008兲. “Water quality modeling for load reduction under uncertainty: A Bayesian approach.” Water Res., 42共13兲, 3305–3314. Loucks, D. P., Stedinger, J. R., and Haith, D. A. 共1981兲. Water resource systems planning and analysis, Prentice-Hall, Englewood Cliffs, N.J. Luo, B., Maqsood, I., and Huang, G. H. 共2007兲. “Planning water resources systems with interval stochastic dynamic programming.” Water Resour. Manage., 21, 997–1014. Luo, B., Maqsood, I., Huang, G. H., Yin, Y. Y., and Han, D. J. 共2005兲. “An inexact fuzzy two-stage stochastic model for quantifying the efficiency of nonpoint source effluent trading under uncertainty.” Sci. Total Environ., 347, 21–34. Malik, A. S., Larson, B. A., and Ribaudo, M. 共1994兲. “Economic incentives for agricultural nonpoint-source pollution-control.” Water Resour. Bull., 30共3兲, 471–480. Maqsood, M., Huang, G. H., and Yeomans, J. S. 共2005兲. “An intervalparameter fuzzy two-stage stochastic program for water resources management under uncertainty.” Eur. J. Oper. Res., 167, 208–225. Mpimpas, H., Anagnostopoulos, P., and Ganoulis, J. 共2001兲. “Modelling of water pollution in the Thermaikos Gulf with fuzzy parameters.” Ecol. Modell., 142共1–2兲, 91–104. Muela, E., Schweickardt, G., and Garces, F. 共2007兲. “Fuzzy possibilistic model for medium-term power generation planning with environmental criteria.” Energy Policy, 35, 5643–5655. Mula, J., Poler, R., and Garcia, J. P. 共2006兲. “MRP with flexible constraints: A fuzzy mathematical programming approach.” Fuzzy Sets Syst., 157, 74–97. Nie, X. H., Huang, G. H., Wang, D., and Li, H. L. 共2008兲. “Robust optimisation for inexact water quality management under uncertainty.” Civ. Eng. Environ. Syst., 25共2兲, 167–184. Ning, S. K., Chang, N. B., Jeng, K. Y., and Tseng, Y. H. 共2006兲. “Soil erosion and non-point sources pollution impacts assessment with the aid of remote sensing.” J. Environ. Manage., 79共1兲, 88–101. Özgen, D., Önüt, S., Gülsün, B., Tuzkaya, U. R., and Tuzkaya, G. 共2008兲. “A two-phase possibilistic linear programming methodology for
multi-objective supplier evaluation and order allocation problems.” Inf. Sci. (N.Y.), 178, 485–500. Ripa, M. N., Leone, A., Garnier, M., and Lo Porto, A. 共2006兲. “Agricultural land use and best management practices to control nonpoint water pollution.” Environ. Manage., 38共2兲, 253–266. Roubens, M., and Teghem, J. 共1991兲. “Comparison of methodologies for fuzzy and stochastic multi-objective programming.” Fuzzy Sets Syst., 42, 119–132. Saadatpour, M., and Afshar, A. 共2007兲. “Waste load allocation modeling with fuzzy goals; simulation-optimization approach.” Water Resour. Manage., 21共7兲, 1207–1224. Sasikumar, K., Mujumdar, P. P., Kumar, A., and Minocha, V. K. 共1999兲. “Fuzzy optimization model for water-quality management of a river system.” J. Water Resour. Plann. Manage., 125, 180–187. Seo, F., and Sakawa, M. 共1988兲. Multiple criteria decision analysis in regional planning—Concepts, methods and applications, Reidel, Dordrecht, The Netherlands. Sethi, L. N., Panda, S. N., and Nayak, M. K. 共2006兲. “Optimal crop planning and water resources allocation in a coastal groundwater basin, Orissa, India.” Agric. Water Manage., 83, 209–220. Shrestha, S., Babel, M. S., Das Gupta, A., and Kazama, F. 共2006兲. “Evaluation of annualized agricultural nonpoint source model for a watershed in the Siwalik Hills of Nepal.” Environ. Modell. Software, 21共7兲, 961–975. Singh, A. P., Ghosh, S. K., and Sharma, P. 共2007兲. “Water quality management of a stretch of river Yamuna: An interactive fuzzy multiobjective approach.” Water Resour. Manage., 21共2兲, 515–532. Takyi, A. K., and Lence, B. J. 共1999兲. “Surface water quality management using a multiple-realization chance constraint method.” Water Resour. Res., 35共5兲, 1657–1670. Tanaka, H., Guo, P. J., and Zimmermann, H. J. 共2000兲. “Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems.” Fuzzy Sets Syst., 113, 323–332. Tang, J. F., Wang, D. W., and Fung, R. Y. K. 共2001兲. “Formulation of general possibilistic linear programming problems for complex industrial systems.” Fuzzy Sets Syst., 119, 41–48. Tilmant, A., and Kelman, R. 共2007兲. “A stochastic approach to analyze trade-offs and risks associated with large-scale water resources systems.” Water Resour. Res., 43共6兲, W06425. Torabi, S. A., and Hassini, E. 共2008兲. “An interactive possibilistic programming approach for multiple objective supply chain master planning.” Fuzzy Sets Syst., 159, 193–214. U.S. EPA. 共1992兲. “Managing nonpoint source pollution.” EPA-506/9-90, Office of Water, Washington, D.C. Wang, Q. J., Robertson, D. E., and Chiew, F. H. S. 共2009兲. “A Bayesian joint probability modeling approach for seasonal forecasting of streamflows at multiple sites.” Water Resour. Res., 45, W05407. Wang, R. C., and Liang, T. F. 共2005兲. “Applying possibilistic linear programming to aggregate production planning.” Int. J. Prod. Econ., 98, 328–341. Zadeh, L. A. 共1978兲. “Fuzzy sets as a basis for a theory of possibility.” Fuzzy Sets Syst., 1, 3–28. Zhang, X. D., Huang, G. H., Chan, C. W., Liu, Z. F., and Lin, Q. G. 共2010兲. “A fuzzy-robust stochastic multiobjective programming approach for petroleum waste management planning.” Appl. Math. Model., 34共10兲, 2778–2788. Zimmermann, H. J. 共1978兲. “Fuzzy programming and linear programming with several objective functions.” Fuzzy Sets Syst., 1, 45–55.
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