An interactive inexact fuzzy bounded programming approach for agricultural water quality management

Agricultural Water Management 133 (2014) 104–111

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An interactive inexact fuzzy bounded programming approach for agricultural water quality management Y.M. Zhang a , H.W. Lu b , X.H. Nie c , L. He b,∗ , P. Du b a b c

Suzhou Research Academy, North China Electric Power University, Suzhou 215123, China College of Renewable Energy, North China Electric Power University, Beijing 102206, China Faculty of Engineering, University of Regina, Regina S4S 0A2, Canada

a r t i c l e

i n f o

Article history: Received 11 July 2013 Accepted 1 November 2013 Available online 24 December 2013 Keywords: Agricultural industry Fuzzy bounded interval Interactive Water quality Uncertainty

a b s t r a c t An interactive inexact fuzzy bounded programming (IFBP) approach is developed through introducing the concept of fuzzy bounded intervals into an interactive fuzzy compromise programming framework. It can provide decision support for decision makers with conflicting desires of greater objective value and higher safety levels of constraints. In this model, by determining a fuzzy goal associated with different feasibility degrees from a semantic correspondence, the degrees of satisfying each objective can be calculated. Decision makers can intervene in every step of the decision process through analyzing the degrees of approaching the aspiration levels and the risks of violating the constraints. The developed method is applied to an agricultural water quality management case for optimizing planting area, manure/fertilizer application amount, and livestock husbandry size. Results indicated that an increased feasibility degree would correspond to a reduced system benefit. Generally, by analyzing risks of violating the constraints in all solution processes, decision makers who have their own aspiration levels would be able to obtain a balanced solution considering the conflict between satisfying the aspiration levels and minimizing the violation risks. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Water is precious and very scarce in many countries due to an increase in industrial and agricultural demands (Qadir et al., 2006; Perez et al., 2007; Lu et al., 2011). The agricultural industry is a major water consumer in the world that contributes significantly to water pollution, soil erosion and ecological deterioration. Nonpoint source pollutions arising from the loss of unused nutrients can lead to high nitrogen and phosphorus concentrations in the water body, resulting in eutrophication (He et al., 2008a; Tan et al., 2011; Diaz et al., 2012). For decision makers (DM), it is challenging to maintain rapid economy development under depleting natural resources and degrading environmental conditions. Problems in water pollution, soil erosion and ecological deterioration could further hinder economic growth. Therefore, a sound system planning of agricultural industry, which can tackle the tradeoff between system benefits and environmental conditions, is desired. In water resources management, a variety of uncertainties exist from the random characteristics of natural events, the estimation errors in parameters, and the vagueness of planning objectives and constraints (Wu et al., 1997). In the past decades, a number of efforts were made by using mathematical programming techniques

∗ Corresponding author. Tel.: +86 10 61772939. E-mail address: [email protected] (L. He). 0378-3774/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.agwat.2013.11.003

to solve regional water quality problems under uncertainty (Ellis, 1987; Julien, 1994; Chaves et al., 2003; Ning and Chang, 2005; He et al., 2008b; Zhang et al., 2009; Lu et al., 2013). The majority of these methods were related to stochastic mathematical programming (SMP) (Shastri and Diwekar, 2006; Lu et al., 2009; He et al., 2010), fuzzy mathematical programming (FMP) (Chang et al., 2001; Ning and Chang, 2004; He et al., 2008; Zhang and Huang, 2010) and interval mathematical programming (IMP) (Liu et al., 2009; Zhang et al., 2009; Zhang and Huang, 2011; Lu et al., 2012). Actually, in practical problems, the quality of available information is often not satisfactory enough to be represented as deterministic numbers, probability distribution functions, or fuzzy membership functions. Instead, obtained data may often be represented as interval numbers which are intuitive. Moreover, sometimes, the lower and upper bounds of some interval parameters can rarely be acquired as deterministic values. In comparison, they may often be given as subjective information that can only be expressed as fuzzy sets, which derives the fuzzy bounded interval (FUBI) (Nie et al., 2007). In addition, decision makers often face conflicting desires of greater objective function value and higher safety level (in terms of constraint feasibility). In the sense recommended by Rommelfanger (1996), an interactive resolution for fuzzy linear programming was proposed by allowing the decision makers consider in an interactive way (Jimenez et al., 2007). It can solve the problems of how to (1) handle relationship between the fuzzy left- and right-hand sides of the constraints and (2) find the optimal value of the fuzzy

Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

105

objective function. To better account for both multi-uncertainty of data and interactions between objectives and constraints, one potential approach is to incorporate the concepts of fuzzy bounded interval into an inexact fuzzy programming framework. Therefore, the objective of this study is to develop such an interactive inexact fuzzy bounded programming (IFBP) approach and apply it to the planning for agricultural water quality management. Both deterministic and fuzzy-boundary-interval parameters will be used in the objective function and constraints to reflect various uncertainties. Interval solutions will be obtained with various degrees of satisfying the fuzzy goal and risks of violating constraints. An agricultural water quality model will be developed for optimizing planting area, manure/fertilizer application amount and livestock husbandry size, as well as maximizing total system benefit for local cropping and husbandry industry.

where fuzzy sets x− and x+ are the lower and upper bounds of x± , ˜ x− = ˜x+ , x± becomes a fuzzy set. ˜ respectively. When ˜ defined ˜ ˜ as follows: For x± , Sign (x± ) is ˜ ˜ Sign (x± ) = 1, if x± ≥0 (7) ˜ ˜

2. Interactive inexact fuzzy bounded programming

|x± |− = −x+ , ˜ ˜ and

2.1. Formulation The fuzzy sets theory was first proposed by Zadeh (1978) to model uncertainty in subjective information. It was developed for solving problems where descriptions of activities and observations were uncertain. The general notation of fuzzy sets can be presented as follows (Lai and Hwang, 1992): A(x) = {(x, A (x)), x ∈ X, A (x) ∈ [0, 1]},

(1)

where X = {x} is a universal set of elements, A(x) is a fuzzy set of X, and A (x) is the degree of membership for x in A. Usually A (x) is a number in the range of 0–1, where 0 represents the smallest possible grade and 1 is the largest one. The closer A (x) is to 1, the more likely element x belongs to A. Inversely, the closer A (x) is to 0, the less likely it is that x belongs to A. The membership function of any fuzzy set (A) may be conveniently expressed in a canonical form or all x ∈ X (Dubois and Prade, 1986):

A (x) =

⎧ fA (x) when x ∈ [a, b), ⎪ ⎪ ⎪ ⎨ when x ∈ [b, c),

1

gA (x) when x ∈ [c, d), ⎪ ⎪ ⎪ ⎩ 0

(2)

,

where a, b, c, d ∈ X and a ≤ b ≤ c ≤ d, fA is a real-valued function that is increasing and right-continuous, and gA is a real-valued function that is decreasing and left-continuous. The r-cut of fuzzy set a˜ is defined by a˜ r = {x ∈ ˝|a˜ (x)≥r}. The r-cuts are closed and boundary intervals and can be represented by a˜ r = [fa−1 (r), ga−1 (r)]. Heilpern (1992) defined the expected interval of fuzzy set a˜ , noted EI(˜a):





1

fa−1 (r) dr, 0



1

ga−1 (r) dr

(3)

0

The expected value of fuzzy set a˜ , noted EV (˜a), is defined as the half point of its expected interval: EV (˜a) =

E1a + E1a 2

.

(4)

For the trapezoidal fuzzy set, its expected interval and expected value are: EI(˜a) =

1 2

(a + b),



1 (c + d) ; 2

E v(˜a) =

1 (a + b + c + d). 4

(9)

|x± | = −x± , if x± ≤ 0 ˜ ˜ ˜ Thus, we have:

(10)

|x± |− = x− , ˜ ˜

(11)

|x± |+ = x+ , ˜ ˜

if x± ≥0, ˜ if x± ≤ 0; ˜

(12)

if x± ≥0, ˜

(13)

(14) |x± |+ = −x− , if x± ≤ 0. ˜ ˜ ˜ An interactive inexact fuzzy bounded programming (IFBP) model can thus be formulated as follows: ±

min f = C± X± ˜ ˜ subject to:

(15a)

A± X± ≤ B± ˜ ˜

(15b)

X± ≥0

(15c) ±

where A ∈ {} are matrices of FUBI with dimension m × n, a± is ˜ ˜ ij the element of matrix A± , C± ∈ {}n is the n-dimensional vector of FUBI, C± = (c ± , c± , . . ., ˜c ± ), ˜B± ∈ {}n is the m-dimensional vector of 1 2 ˜± ˜± ˜± ˜j ± ˜ ± FUBI, B = (b1 , b2 , . . ., bi ), X ∈  n is the n-dimensional vector of ˜ decision ˜ ˜ variables, ˜ and x± = (x± , x± , . . ., x± ), i = 1, 2, . . ., m, interval 1 2 j j = 1, 2, . . ., n. m×n

To solve the above IFBP problem where parameters are FUBI and variables are interval, it is first transformed into two sets of submodels, respectively corresponding to the lower and upper bounds of the formulated model. This transformation process is based on an interactive algorithm, which is different from normal interval analysis and best/worst case analysis. It was proposed and proven by Huang and Moore (1993) for solving linear programming problems involving interval parameters, and has been widely used by a large number of researcher tackling interval programming problems (Nie et al., 2007; Lu et al., 2009). According to Huang and Moore (1993), the submodel corresponding to the lower-bound objective-function value (i.e., f− ) is: −

For f : ˜



j=1 to k1

(6)

min



j=1 to k1

c− x− + ˜j j

subject to:

(5)

Based on the fuzzy sets theory, Nie et al. (2007) defined a fuzzy boundary interval (FUBI) x± where the lower and upper bounds of ˜ interval numbers are expressed by fuzzy sets: x± = [x− , x+ ] ˜ ˜ ˜

(8)

2.2. Solution method of IFBP

otherwise,

EI(˜a) = [E1a , E2a ] =

Sign (x± ) = −1, if x± ≤ 0 ˜ ˜ Its absolute value |x± | is: ˜ |x± | = x± , if x± ≥0 ˜ ˜ ˜

|a± |+ sign(a± )x− + ˜ ij ˜ ij j



j=k1 +1 to n

 j=k1 +1 to n

c− x+ ˜j j

(16a)

+

|a± |− sign(a± )x+ ≤ bi ˜ ij ˜ ij j ˜

(16b)

− + Solution: xj−S−opt for j = 1 to k1 , xj−S−opt for j = k1 + 1 to n. Based on this algorithm, the lower-bound submodel is formulated first. The optimal solutions of x corresponding to the lowest − objective function f are desired in˜ the lower bound submodel. ˜

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Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

In comparison, the submodel corresponding to the upper-bound objective-function value (i.e., f+ ) is: +

For f : ˜



min

j=1 to k1

c+ x+ + ˜j j

subject to:



j=1 to k1

|a± |− sign(a± )x+ + ˜ ij ˜ ij j



j=k1 +1 to n

 j=k1 +1 to n

c+ x− ˜j j

(17a)

−

|a± |+ sign(a± )x− ≤ bi ˜ ij ˜ ij j ˜

− − − − − xj+ ≥{xj−a−opt ||fj−a−opt = Min(fj−1−opt , fj−2−opt , . . ., fj−s−opt )}

(17b)

(17c)

+ − Solution: xj−S−opt for j = 1 to k1 , xj−S−opt for j = k1 + 1 to n. Each submodel is a fuzzy linear programming problem with fuzzy parameters and deterministic decision variables. Jimenez et al. (2007) proposed an interactive method to solve such problems. The main concern in this paper is the tradeoff between feasibility of decision vector x and optimality of objective function. Since both sides of constraints (16b) and (17b) are fuzzy sets, it is able to define the feasibility of the decision vectors in the constraints by fuzzy ranking methods. In this study, a fuzzy ranking method (Jimenez, 1996) was used to rank the fuzzy objective values and to deal with the inequality relation on constraints. For any ˜ the degree in which a˜ is bigger than b˜ is: pair of fuzzy sets a˜ and b,

˜ = M (˜a, b)

if E2a − E1b < 0, E2a − E1b

if 0 ∈ [E1a − E2b , E2a − E1b ] , (18) ⎪ E2a − E1b − (E1a − E2b ) ⎪ ⎪ ⎩ a b if E2 − E1 > 0,

1

˜ and are the expected intervals of a˜ and b. ˜ = 0.5, the fuzzy sets a˜ and b˜ are regarded as indifWhen M (˜a, b) ˜ it means a˜ is bigger than, or equal to, b˜ ferent. When M (˜a, b)≥˛, ˜ at least in a degree ˛ and can be represent by a˜ ≥˛ b. Therefore, the constraints with fuzzy parameters (˜ai x is bigger than, or equal to b˜ i at least in a degree ˛) can be defined by (Jimenez et al., 2007):

where

[E1a , E2a ]

a˜ i x≥˛ b˜ i ,

[E1b , E2b ]

i = 1, 2, . . ., m.

(19)

It can also be expressed as: ˜ = ˛, min {M (˜ai x, b)}

(20)

i=1,...,m

where ai is vector of aij , ai = (ai1 , ai1 , . . ., ain ).Recall Eq. (18), Eq. (19) is equivalent to: [(1 − ˛)E2ai

+ ˛E1ai ]x≥˛E2bi

+ (1 − ˛)E1bi

(21)

The set of decision vectors which are ˛-feasible are denoted by N(˛). Obviously, when ˛1 < ˛2 , then N(˛1 ) ⊃ N(˛2 ). This can be concluded that an increased feasibility ˛ corresponds to a decreased optimal objective function value. For the optimality of the objective function, this fuzzy ranking method can also be used to identify a decision vector x0 as an acceptable optimal solution: M (˜c  x, c˜  x0 )≥˛,

(22)

The vector x0 (˛) ∈ Rn is an ˛-acceptable solution if it is an optimal solution to the following model: min

n  j=1

EV (˜cj )xj

˛

Term

˛

Term

0 0.1 0.2 0.3 0.4 0.5

US Practically US Almost US Very US Quite US Neither As nor US

0.6 0.7 0.8 0.9 1

Quite AS Very AS Almost AS Practically AS Completely AS

US: unacceptable solution; AS: acceptable solution.

+ + + + + xj− ≤ {xj−a−opt ||fj−a−opt = Max(fj−1−opt , fj−2−opt , . . ., fj−s−opt )} (17d)

⎧ 0 ⎪ ⎪ ⎪ ⎨

Table 1 The 11 scales of linguistic terms.

(23a)

subject to: a˜ ij xj ≤ ˛ b˜ i

(23b)

xj ≥0,

(23c)

i = 1, 2, . . ., m, j = 1, 2, . . ., n Consider the trade-off between feasibility degree of a decision vector in satisfying the constraints and the objective. The bigger the feasibility degree, the worse the objective function value is. Therefore, decision makers would then face two conflicting objectives: to improve the objective function value or to improve the degree of satisfying the constraints. The membership function of a fuzzy set which was built in the decision space can represent the balance between feasibility degree of constraints and satisfaction degree of the goal. A sound solution is the one that has the biggest membership degree to this fuzzy set. In this study, this method was used to solve the upper- and lower-bound submodels separately. The best way to reflect DM preferences is to express them through linguistic term, establishing a semantic correspondence for different degrees of feasibility (Zadeh, 1978). Linguistic terms are encountered in the process of data acquisition since human subjective judgment is involved. Jimenez et al. (2007) established 11 scales for the semantic scale. The corresponding linguistic terms and preference degrees of DM to the objective values are showed in Table 1. Let ˛0 be the minimum constraint feasibility degree that DM is willing to accept. The acceptable discrete values of ˛ would be: M=



˛k = ˛0 + 0.1k|k = 0, 1, . . .,

1 − ˛0 0.1

⊂ [0, 1].

(24)

For each ˛k , the corresponding decision vectors x(˛k ) can be obtained by solving linear programming problem (23); objective values z˜ (˛k ) = c˜ x(˛k ) of the two submodels would also be calculated. Based on the information given by z˜ (˛k ), DM can determine ¯ its satisfaction tolerance G - and the tolerance threshold G by fuzzy ˜ is: ˜ The membership function of G set G. G˜ (z) =

⎧ ⎨1 ⎩

 ∈ [0, 1] 0

if z ≤ G ¯ decreasing on G - G

(25)

If DM wants to achieve a higher satisfaction degree to fuzzy goal (Max ), a lower level of fulfillment of constraints (˛-feasibility) would be derived. Therefore, DM can make a balance between the satisfaction degree to fuzzy goal and feasibility degree of con˜ can straints. The degree of satisfaction of each z˜ (˛k ) to fuzzy goal G be calculated through an index proposed by Yager (1979):

 +∞

KG˜ (˜z (˛k )) =

−∞

z˜ (˛k ) (z) · G˜ (z) dz

 +∞ −∞

(26)

z˜ (˛k ) (z) dz

where the denominator is the area under z˜ (˛k ) , and the numerator is the possibility of occurrence z˜ (˛k ) of each crisp value z weighted ˜ With both degrees of by its satisfaction degree  ˜ (z) of goal G. G

satisfaction (KG˜ ) and feasibility of constraints (˛), it is possible to

Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

Uncertain data

Dual uncertainties

Interval-parameter programming

Fuzzy boundary interval

IFP lower-bound submodel

IFP model

107

IFP upper-bound submodel

Fuzzy linear programming -feasible Solutions satisfaction degree Fuzzy decision from lower-bound IFP submodel

Solutions

Fuzzy decision from upper-bound IFP submodel Fig. 2. The study area.

Generation of decision alternatives

+

Fig. 1. Schematic of the IFBP methodology.

analyze the tradeoff between them and obtain a balanced solution. To make a decision between these two factors, F˜ is defined as a fuzzy set of feasibility of constraints with membership function ˛k = F˜ (x(˛k )), and S˜ is defined as a fuzzy set of degrees of satisfaction to goal with membership function KG˜ (˜z (˛k )) = S˜ (x(˛k )). Based on the definition (Bellman and Zadeh, 1970), fuzzy decision ˜ = F˜ ∗ S˜ is: D D˜ (x(˛k )) = ˛k ∗ KG˜ (˜z (˛k ))

(27)

where * represents a t-norm that can be minimum, the algebraic product, etc. The lower- and upper-bound submodels need to be resolved ˜− by this fuzzy interactive method (18)–(27), fuzzy decisions (D ˜ + ) of two submodels (16) and (17) would be obtained. The and D membership function (D˜ (x(˛k ))) built in the decision space can represent the balance between feasibility degree of constraints and satisfaction degree of the goal. Then the optimal solutions would correspond to the highest membership degrees according to fuzzy ˜ − and D ˜ + , respectively: decisions D D˜ (∗ x− ) = max {˛k ∗ [KG˜ − (˜z − (˛k ))]}, D˜ (∗ x+ ) ˛k ∈ M

= max {˛k ∗ [KG˜ + (˜z + (˛k ))]} ˛k ∈ M

(28)

Fig. 1 shows the scheme of the modeling methodology. The upper- and lower-bound submodels could be solved through this interactive approach. In general, the solution algorithm of the IFBP model is presented as follows: Step 1. Formulate IFBP model (15). − Step 2. Transform the IFBP model into two fuzzy submodels (f + ± ˜ and f ) where the lower bound of f is first desired since ± ˜ ˜ the objective is to minimize f . ˜ Step 3. Formulate lower-bound model (16) by Eq. (23) and obtain − a set of f (˛k ) by setting ˛0 based on Eq. (21). ˜ Step 4. Formulate upper-bound model (17) by Eq. (23) and obtain + a set of f (˛k ) in the same ˛0 . ˜ goal G ˜ − by f − (˛k ) and calculate its corresponding Step 5. Set a fuzzy satisfaction degree (K˜G˜ − ) by Eq. (26). ˜ + by f + (˛k ) and calculate its corresponding Step 6. Set a fuzzy goal G satisfaction degree (K˜G˜ + ) by Eq. (26).

˜ by Eq. (28) and obtain ∗ x± and ∗ f . Step 7. Set fuzzy decision D j ± − + ˜ Step 8. Solutions of the IFBP model are: ∗ f = [∗ f , ∗ f ], ∗ xj± = ˜ ˜ ˜ − + [∗ x , ∗ x ], ∀j. j

j

3. Application 3.1. Case study An agricultural water quality management problem is developed to illustrate applicability of the proposed IFBP model based on representative cost/benefit and technical data from the water management literature (Haith, 1982; Tisdale et al., 1993; Huang, 1996). In the study system, the main agricultural industry is crop farming (wheat, vegetables and potato), followed by animal husbandry (cattle, swine and poultry) (Fig. 2). There are three subareas with different manners of drawing water for irrigation and different crop distributions in the study area. The total area is 143.7 ha, with the area of tillable land being 124 ha. The three subareas contain tillable lands of 42, 43 and 39 ha, respectively. Irrigation water is drawn from a river passing the area through three canals. Agricultural production needs water for irrigation purpose, and generates nonpoint source pollution from manure/fertilizer applications (Haith, 1982). Due to spatial and temporal variations of different feeding sources, the river has a highly uncertain stream flow regime. The pattern of flow variability can be presented as fuzzy boundary intervals (FUBIs) using the form of mean, spread . In this system, three components, i.e. soil-crop, livestock, and human activities, are related to the water quality/quantity objectives, and water allocation for irrigation purpose is related to farming activities, channel flows, and economic returns (Huang, 1996). Moreover, nitrogen, in the form of nitrates such as NO3 -N, NO2 -N, and NH3 -N, can contaminate water and make it unsafe for drinking. The input parameters of pollutant loss and agricultural activities are listed in Tables 2 and 3. The main concern of decision makers is how to optimally allocate the water to both maximize the system benefit and to improve the degree of constraints satisfaction with limited water availability and increased water demand. Due to incompleteness and/or unavailability of required information, parameters in this system are expressed as interval numbers, as well as fuzzy boundary intervals when they are highly uncertain. Thus, an IFBP model will be formulated for the problem by optimizing planting area, manure/fertilizer application amount, and livestock husbandry size. The objective is to maximize total system benefit, with a number of resources and environmental constraints. It is assumed that the animals’ net energy and

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Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

m

crop i (t)); CFh± i=1 Hi± is the total cost of fertilizer application; CFh± is cost of fertilizer application ($/kg); Hi± is the amount of

Table 2 Input parameters of pollutant losses. Maximum allowance [38, 40] a± (kg/ha)

r m

Cost parameters: Gf± ($/t) [4, 5]

±

Gh± ($/kg)

b (kg/ha) [ 3750, 100 , 5500, 100 ] c˜1± (kg/ha) [ 9.5, 0.5 , 11, 0.4 ] c2± (kg/ha) [ 4.1, 0.20 , 5.0, 0.15 ] (kg/ha) [2.2, 2.3] u± 1 (kg/ha) [0.20, 0.22] u± 2 ± [10, 13] g (kg/t) Nitrogen content of manure g± (kg/t) [10, 13]

[0.7, 0.9]

Nutrient Content of soil h± (%) [0.0020, 0.0025] 1 h± (%) [0.0009, 0.0011] 2 Nitrogen volatilization rate ± p1 (%) [0.30, 0.35] p± (%) [0.10, 0.12] 2

fertilizer nitrogen applied to crop i (kg); CWit± W ± S± t=1 i=1 it it ± ˜ is the cost of water delivery; CWit is cost to deliver water to the area of crop i in subarea t ($/(m3 /s)). (2) Water quality constraints (a) Manure mass balance. The total generated manure by cattle, swine and poultry would be equal to the amount of manure applied to crops:

digestible protein requirements are principally supplied by onfarm crops such that no more cost would be paid for feeding animals. The detailed IFBP model is presented as follows: (1) Objective function Maximize f ˜

±

= total system benefit =

m r   t=1 i=1

PCi± Yi± Sit± +

r m

n  j=1

PLj± Tj± −

m r  

FCi± Sit± − CMf±

t=1 i=1

PL± T ± j=1 j j

is the total revenue from crops from livestock

(PLj±

is average return from livestock j ($/unit); Tj± is number of liver m stock j in the study area); FCi± Sit± is the total farming t=1 i=1 m cost of crops (FCi± is farming cost for crop i); CMf± i=1 Fi± is the total cost of manure collection/disposal (CMf± is cost of manure collection/disposal ($/t); Fi± is the amount of manure applied to







Fi± −

n 

i=1

Bj± Tj± = 0

(29a)

j=1

where Bj± is the amount of manures generated by livestock j that needs to be disposed (t/unit). (b) Crop nutrient balances. The total nitrogen demand of crops must be less than the available nitrogen from manure and fertilizer: m  i=1

In the objective functions, PCi± Yi± Sit± is the total revt=1 i=1 enue from crops (Yi± is Yield of crop i (kg/ha); Sit± is the area of crop i in subarea t (ha); PCi± is the price of crop i ($/kg).);

n

m 

Fi± − CFh±

m 

Hi± −

i=1

m r   t=1 i=1

CWit± W ± S± ˜ it it

(1 − p± )g ± Fi± + (1 − p± )Hi± ≥ 1 2

r 

NRi± Sit± , ∀i

(29)

(29b)

t=1

where g± is nitrogen content of manure (kg/t); p± is nitro1 is gen volatilization/denitrification rate of manure (%); p± 2 nitrogen volatilization/denitrification rate of nitrogen fertilizer (%); NRi± is nitrogen requirement of crop i (kg/ha).

Table 3 Input parameters of agricultural activities.

Yield of crop i (kg/ha) Farming cost for crop i ($/ha) Dissolved N in WS runoff (mg/L) Dissolved N in DS runoff (mg/L) Dissolved P in WS runoff (mg/L) Dissolved P in DS runoff (mg/L) Nitrogen requirement (kg/ha) WS runoff from crop land i (mm) DS runoff from crop land i (mm) Net energy content of crop i (Mcal/kg) Digestible protein content of crop i (%) Price of crop i ($/kg) Soil loss from crop land i (kg/ha)

Ci± Gi± ± N1i ± N2i ± P1i ± P2i q± i ± R1i ± R2i ˛± i ˇi± ı± i L± ˜i

Wheat

Vegetable

Potato

[5500, 6000] [90, 110] [1.5, 1.8] [0.8, 1.1] [0.15, 0.19] [0.08, 0.12] [90, 110] [65, 80] [45, 50] [3.3, 3.5] [0.108, 0.143] [0.11, 0.15] [ 2600, 50 , 2800, 50 ]

[20,000, 21,000] [5300, 5500] [2.7, 3.0] [1.7, 2.0] [0.26, 0.30] [0.22, 0.26] [140, 155] [82, 96] [51, 58] [0.22, 0.32] [0.01, 0.03] [0.44, 0.48] [ 9000, 50 , 9200, 50 ]

[14,500, 15,000] [1400, 1600] [2.2, 2.5] [1.4, 1.7] [0.23, 0.27] [0.21, 0.25] [80, 120] [86, 102] [55, 62] [0.8, 1.0] [0.018, 0.036] [0.18, 0.22] [ 3900, 50 , 4100, 50 ]

[ 0.051, 0.002 , 0.072, 0.001 ] [ 0.050, 0.002 , 0.071, 0.001 ] [ 0.052, 0.002 , 0.073, 0.001 ]

[ 0.043, 0.002 , 0.055, 0.001 ] [ 0.042, 0.002 , 0.054, 0.001 ] [ 0.044, 0.002 , 0.056, 0.001 ]

[1400, 1450] [2070, 2120] [2680, 2730]

[1375, 1425] [2050, 2100] [2650, 2700]

Quantity of irrigation water required (m3 /ha-s) W± [ 0.032, 0.002 , 0.041, 0.001 ] Subarea 1 (t = 1) i1 ˜± W [ 0.031, 0.002 , 0.040, 0.001 ] Subarea 2 (t = 2) i2 ± ˜ Subarea 3 (t = 3) W [ 0.033, 0.002 , 0.042, 0.001 ] i3 ˜ 3 Cost to deliver water to the farm of crop i in subarea t ($/(m /s)) v±i1 [1350, 1400] Subarea 1 (t = 1) Subarea 2 (t = 2) v±i2 [2020, 2070] Subarea 3 (t = 3) v±i3 [2630, 2680] Subarea 1

Subarea 2

Subarea 3

Maximum canal flow in subarea t (m3 /s)

Q± 1 ˜

[ 1.6, 0.09 , 1.8, 0.09 ] Cattle (j = 1)

[ 1.6, 0.09 , 1.8, 0.09 ] Swine (j = 2)

[ 0.9, 0.04 , 1.0, 0.04 ] Poultry (j = 3)

Manure generated (t/unit)

Bj±

[18, 19]

[1.8, 1.9]

[0.04, 0.06]

[330, 350] [4800, 5200] [480, 500]

[30, 35] [472, 515] [35, 40]

[1.6, 1.8] [175, 188] [2.0, 2.7]

Digestible protein requirement (kg/unit) Net energy requirement (Mcal/unit) Average return ($/unit) WS: wet season; DS: dry season.

ˇi± ˛± i j±

Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

(c) Energy and digestible protein requirements. The total energy and digestible protein supplies from the crops must be greater than the corresponding needs of livestock: m r  

Yi± NEi± Sit± ≥

t=1 i=1 m r  

n 

ERj± Tj±

(29c1)

j=1

Yi± NPi± Sit± −

t=1 i=1

n 

PRj± Tj± ≥0

(29c2)

j=1

where NEi± is net energy content of crop i (Mcal/kg); NPi± is digestible protein content of crop i (%); PRj± is digestible pro-

tein requirement of livestock j (kg/unit); ERj± is net energy requirement of livestock j (Mcal/unit). (d) Pollutant losses. (d1) Total nitrogen losses from agriculture must be less than the maximum allowance:

 m

 r

(g ± Fi± + Hi± − NRi±

Sit± ) ≤ MN ±

t=1

i=1

 r

Kt

(29d1)

t=1

where Kt is tillable area in subarea t (ha); MN± is the maximum allowable total nitrogen losses (kg/ha); (d2) Total soil losses from agriculture must be less than the maximum allowance: m r   t=1 i=1

S Li± Sit± ≤ M S ± ˜ ˜

r 

Kt

t=1 i=1

r 

Kt

(29d3)

t=1

where NC± is nitrogen content of soil (%); M N ± is maximum allowable solid-phase nitrogen loss˜ (kg/ha). (d4) Total solid-phase nitrogen loss from soil losses must be less than the maximum allowance: m r   t=1 i=1

NP ± S Li± Sit± ≤ M P ± ˜ ˜

r 

Kt

(29d4)

(RWi± NWi± + RDi± NDi± )Sit± ≤ MDN ±

t=1 i=1

r 

Kt

t=1

(29d5) NWi±

NDi±

where and are dissolved nitrogen concentration in wet- and dry-season runoff from land planted to crop i (mg/L); RWi± and RDi± are wet- and dry-season runoff from land planted to crop i (mm); MDN± is maximum allowable dissolved nitrogen loss by runoff (kg/ha). (d6) Total dissolved phosphorus loss from runoff must be less than the maximum allowance: m r   t=1 i=1

Objective value f (˛k ) ˜ + f (0.7) = (372,484, 372, 819, 373, 154) + f˜ (0.8) = (370,894, 371, 227, 371, 561) + f˜ (0.9) = (369,290, 369, 623, 369, 956) + f˜ (1) = (367,654, 367, 985, 368, 316) ˜

0.7 0.8 0.9 1

+

KG˜ + (f (˛k )) ˜ 0.59 0.54 0.48 0.43

D˜ + (x(˛k )) 0.413 0.430 0.435 0.429

where PWi± and PDi± are dissolved phosphorus concentration in wet- and dry-season runoff from land planted to crop i (mg/L); MDP± is maximum allowable dissolved phosphorus loss by runoff (kg/ha). (3) Water quantity constraints.The total water demand of each subarea must be less than the maximum water flow of cannels m 

W± S± ≤ Q ± , ∀t t ˜ it it ˜ i=1

(29e)

where W ± is quantity of irrigation water required by crop i in ˜ it subarea t (m3 /ha-s); Q ± is maximum canal flow within subarea t ˜ t (m3 /s). (4) Land constraints.The total area of crops in each subarea must be less than the maximum tillable land: m 

Sit± ≤ Kt ∀t

(29f)

(5) Technical constraints. Sit± , Fi± , Hi± , Tj± ≥0,

∀i, j, t

(29g)

The assumptions of the IFBP model includes: (1) the required animals’ net energy and digestible protein are principally supplied by on-farm crops such that no more cost would be paid for feeding animals; (2) the unit revenue and unit price of each crop do not vary spatially and temporally in developing water management decisions; (3) all of the generated manure by cattle, swine and poultry are applied to crops, with its loss capable of being ignored during the mass transfer process; (4) the protein requirement, soil loss, pollutant loss and water demand are linearly related to the area of each crop.

t=1

where NP± is phosphorus content of soil (%); M P ± is maximum allowable solid-phase phosphorus ˜loss (kg/ha). (d5) Total dissolved nitrogen loss from runoff must be less than the maximum allowance: m r  

+

˛k

i=1

t=1

NC ± S Li± Sit± ≤ M N ± ˜ ˜

Table 4 Optimal solutions of a set of ˛-acceptable upper-bound models.

(29d2)

where M S ± is maximum allowable soil loss (kg/ha); ˜ loss from land planted to crop i (kg/ha). S Li± is soil ˜ (d3) Total solid-phase nitrogen loss from soil losses must be less than the maximum allowance: m r  

109

(RWi± PWi± + RDi± PDi± )Sit± ≤ MDP ±

r 

3.2. Results analysis For decision makers, four semantic terms are chosen from Table 3 for analyzing feasibilities of the constraints (i.e. very, almost, practically and completely acceptable solutions, with feasibility degrees of 0.7, 0.8, 0.9 and 1, respectively). First, solutions of the upper-bound submodel under different feasibility degrees can be obtained (Table 4). The scheme for the upper-bound objectivefunction value represents an advantageous decision scheme. The objective values under ˛ = 0.7, 0.8, 0.9 and 1 would be $(372.5, 372.8, 373.2) × 103 , $(372.5, 372.8, 373.2) × 103 , $(369.3, 369.6, 370.0) × 103 and $(367.7, 368.0, 368.3) × 103 , respectively. Under advantageous conditions, the specified upper-bound fuzzy goal would be 385,000, with its tolerance level being 30,000. Therefore, ˜ + ) can be expressed by the following fuzzy subset: fuzzy (G

G˜ + (f ) = Kt

t=1

(29d6)

⎧ 0 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

if f ≤ 355, 000

z − 355,000 385, 000 − 355, 000

if 355, 000 < f ≤ 385, 000

1

if f > 385, 000 (30)

110

Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

Table 5 Optimal solutions of a set of ˛-acceptable lower-bound models. ˛k 0.7 0.8 0.9 1

−

Objective value f (˛k ) ˜ − f (0.7) = (178.0, 178.2, 178.4) − f˜ (0.8) = (177.0, 177.3, 177.5) − f˜ (0.9) = (176.2, 176.4, 176.6) − f˜ (1) = (175.3, 175.6, 175.8) ˜

−

KG˜ − (f (˛k )) ˜ 0.82 0.72 0.64 0.55

Table 6 Solutions of the IFBP model. D˜ − (x(˛k )) 0.571 0.578 0.574 0.552

Based on the upper-bound membership function of the fuzzy goal, the degrees of each solution satisfying this fuzzy goal are obtained. Obviously, an increase of ˛ corresponds to the decreases in net income and satisfaction degree to the fuzzy goal. The final decision thus inevitably involves a compromise between ˛ and KG˜ + , and consequently between higher objective function value and more acceptable solution. The final compromised decision would be the one that has the largest membership value of fuzzy decision considering both aspects. Therefore, the solution of the upper-bound submodel would be $(369.3, 369.6, 370.0) × 103 when ˛ = 0.9. In the lower-bound submodel, the scheme for the lower-bound objective function represents a demanding decision scheme. The solutions under different feasibility degrees are provided in Table 5. The objective values under ˛ = 0.7, 0.8, 0.9 and 1 would be $(178.0, 178.2, 178.4) × 103 , $(177.0, 177.3, 177.5) × 103 , $(176.2, 176.4, 176.6) × 103 , $(175.3, 175.6, 175.8) × 103 , respectively. The specified lower-bound fuzzy goal would be 180,000, with its tolerance level being 10,000. ˜ − ) can be expressed by the following Therefore, the fuzzy (G fuzzy subset:

G˜ − (f ) =

⎧ 0 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

if f ≤ 170, 000

z − 170,000 180, 000 − 170, 000

if 170, 000 < f ≤ 180, 000

1

if f > 180,000 (31)

Based on the lower-bound membership function of the fuzzy goal, the degrees of each solution satisfying this lower-bound fuzzy goal are obtained. Similarly, the higher ˛ (feasibility constraint), the lower the net income and satisfaction degree to the fuzzy goal. In the lower-bound submodel, the final compromising decision exists when ˛ = 0.9 and the total net income would be $(176.2, 176.4, 176.6) × 103 . Therefore, when the solution is practically acceptable (˛ = 0.9), this water quality model has its compromising solution (Table 6). Accordingly, the detailed solutions are listed in Table 6. In subarea 1, the area with planted vegetable would be [21.6, 34.7] ha. In subarea 2, wheat and vegetable would be planted by [10.3, 20.3] and [16.1, 22.7] ha. In subarea 3, vegetable would be planted ([8.3, 11.2] ha). The total cropping area would be [45.4, 71.7]% of the total tillable area. The percentage of cropping areas to the tillable land in the three subareas would be [51.4, 82.8], [61.3, 100] and [17.1, 22.8], respectively. The low percentage pertained to subarea 3 indicates that the majority of tillable area would be partly left idle. This is mainly because the strict irrigation water requirement and the high cost for delivering water. In all, most of the areas are planted with vegetables, accounting for [77.1, 87.1]% of the total cultivated area. This is because vegetables have larger yields and can be sold at higher prices. Since wheat is a feeding food for livestock that has high net energy and digestible protein contents, it would be planted by [10.3, 20.3] ha in subarea 2. It is noted that there is no need to plant potato, due to its high water requirements and pollutant losses. For livestock husbandry, the results demonstrate that sizes for cattle and swine should be limited at low levels, while poultry

Cropped Area

Crop

Solution

Subarea 1

Wheat Vegetable Potato

[0, 0] [21.6, 34.7] [0, 0]

Subarea 2

Wheat Vegetable Potato

[10.3, 20.3] [16.1, 22.7] [0, 0]

Subarea 3

Wheat Vegetable Potato

[0, 0] [8.3, 11.2] [0, 0]

Amount of manure application (t)

Wheat Vegetable Potato

[201.2, 201.2] [1059.9, 1093.4] [0, 0]

Amount of nitrogen fertiliser application (kg)

Wheat Vegetable Potato

0 [0, 27.1] 0

Size of livestock husbandry (unit)

Cattle Swine Poultry

[65, 66] 0 [994, 1829]

husbandry is allowed to be large. Due to the differences in energy and protein demands, market prices and manure-generation rates, the sizes for cattle, swine and poultry would respectively be [65, 66], 0 and [994, 1829]. The sizes of livestock husbandry are constrained by the allowance of manure/fertilizer to prevent agricultural nonpoint source pollution. The produced manure is used for cropping lands. All generated manure applied to the wheat, vegetable and potato farms would be 201.2, [1059.9, 1093.4], and 0 tonnes, respectively. Some expensive fertilizers would be purchased from external systems to meet the nutrient needs of the crops. The amount of purchased fertilizer for vegetable farms would be [0, 27.1] kg, which is very low compared to the generated manure. Generally, by analyzing risks of violating the constraints in all solution processes, decision makers who have their own aspiration levels would be able to obtain a balanced solution considering the conflict between satisfying the aspiration levels and minimizing the violation risks. The optimal solution of the objective function (i.e. f± = $[(176.2, 176.4, 176.6), (369.3, 369.6, 370.0)] × 103 ) represents a range for the net system benefit. The fuzzy boundary intervals are communicated into the optimization processes and resulting solutions. The obtained solutions can provide decision support under both demanding and advantageous system conditions. 4. Conclusions An interactive inexact fuzzy bounded programming approach was developed through introducing the concept of fuzzy bounded intervals into an interactive fuzzy compromise programming framework. Fuzzy bounded interval parameters were used to tackle independent uncertainties in the constraints’ left- and right-hand sides. In lower- and upper- bound submodels, the feasibility of a decision vector (with feasibility degree ˛) was defined by controlling the feasibility of the constraints. An agricultural water quality management case was studied for optimizing planting area, manure/fertilizer application amount, and livestock husbandry size. It was assumed that the animals’ net energy and digestible protein requirements were principally supplied by on-farm crops such that no more cost would be paid for feeding animals. Results of the case study indicated that useful solutions were generated. By determining a fuzzy goal associated with different feasibility degrees from a semantic correspondence, the degrees of each objective satisfying were calculated. In two submodels, flexibility in

Y.M. Zhang et al. / Agricultural Water Management 133 (2014) 104–111

reflecting potential system condition variations was caused by the existence of input uncertainties. Four semantic acceptable feasibilities of decision vectors were considered. An increased feasibility degree would correspond to a reduced system benefit. With the conflicting aspiration levels and violation risk, the final interval decisions were generated. It was indicated that, high system benefits would correspond to advantageous system conditions, while low system benefits would correspond to demanding system conditions. Generally, by analyzing risks of violating the constraints in all solution processes, the decision makers who have their own aspiration levels would be able to obtain a balanced solution considering the conflict between satisfying the aspiration levels and minimizing the violation risks. This study was an attempt to introduce the concept of fuzzy boundary intervals into an interactive fuzzy compromise programming framework. The results suggested that the proposed hybrid method was applicable to practical problems that were associated with highly complex and uncertain information. The IFBP can tackle interactive relationships not only between feasibility of decision vector and optimality of objective function, but also those between solutions of lower- and upper-bound objective function values. In future studies, it could be applied to other resources and environmental problems that were associated with interactive and conflicting complexities. Acknowledgements The authors are grateful to the editor and the anonymous reviewers for their insightful comments and suggestions. This research was supported by the China National Funds for Excellent Young Scientists (51222906), National Natural Science Foundation of China (41271540), Program for New Century Excellent Talents in University of China (NCET-13-0791), and Fundamental Research Funds for the Central Universities. References Bellman, R., Zadeh, L.A., 1970. Decision making in a fuzzy environment. Management Science 17, 141–164. Chang, N.B., Chen, H.W., Ning, S.K., 2001. Identification of river water quality using the fuzzy synthetic evaluation approach. Journal of Environmental Management 63 (3), 293–305. Chaves, P., Kojiri, T., Yamashiki, Y., 2003. Optimization of storage reservoir considering water quantity and quality. Hydrological Processes 17 (14), 2769–2793. Diaz, F.J., O’Geen, A.T., Dahlgren, R.A., 2012. Agricultural pollutant removal by constructed wetlands: implications for water management and design. Agricultural Water Management 104, 171–183. Dubois, D., Prade, H., 1986. Fuzzy sets and statistical data. European Journal of Operational Research 25, 345–356. Ellis, J.H., 1987. Stochastic water-quality optimization using imbedded chance constraints. Water Resources Research 23 (12), 2227–2238. Haith, D.A., 1982. Environmental Systems Optimization. Wiley, New York. He, L., Huang, G.H., Lu, H.W., Zeng, G.M., 2008a. Optimization of surfactant-enhanced aquifer remediation for a laboratory BTEX system under parameter uncertainty. Environmental Science and Technology 42 (6), 2009–2014. He, L., Huang, G.H., Lu, H.W., 2008b. A simulation-based fuzzy chance-constrained programming model for optimal groundwater remediation under uncertainty. Advances in Water Resources 31 (12), 1622–1635. He, L., Huang, G.H., Lu, H.W., 2010. A stochastic optimization model under modeling uncertainty and parameter certainty for groundwater remediation design—Part I. Model development. Journal of Hazardous Materials 176 (1), 521–526. Heilpern, S., 1992. The expected value of a fuzzy number. Fuzzy Sets and Systems 47 (1), 81–86.

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