Optimal crop allocation including market trends and water availability

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Optimal crop allocation including market trends and water availability Maritza E. Cervantes-Gaxiola , Erik F. Sosa-Niebla , ´ ´ , Jose´ M. Ponce-Ortega , Oscar M. Hernandez-Calder on Jesus ´ Raul ´ Ortiz-del-Castillo , Eusiel Rubio-Castro PII: DOI: Reference:

S0377-2217(20)30130-2 https://doi.org/10.1016/j.ejor.2020.02.012 EOR 16332

To appear in:

European Journal of Operational Research

Received date: Accepted date:

26 June 2019 7 February 2020

Please cite this article as: Maritza E. Cervantes-Gaxiola , Erik F. Sosa-Niebla , ´ ´ , Oscar M. Hernandez-Calder on Jose´ M. Ponce-Ortega , Jesus ´ Raul ´ Ortiz-del-Castillo , Eusiel Rubio-Castro , Optimal crop allocation including market trends and water availability, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.02.012

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HIGHLIGHTS Optimal crop allocation including market trends and water availability 

An optimization approach for the optimal crop allocation is proposed.



The model involves future prices and water availability as well as use, reuse and recycling of water.



The Autoregressive Moving Average model is implemented to predict prices.



The process retrofit between cycles is considered.

Optimal crop allocation including market trends and water availability Maritza E. Cervantes-Gaxiolaa, Erik F. Sosa-Nieblaa, Oscar M. Hernández-Calderóna, José M. Ponce-Ortegab, Jesús Raúl Ortiz-del-Castilloa, Eusiel Rubio-Castroa,* Full name and email address - Maritza E. Cervantes-Gaxiola - [email protected] - Erik F. Sosa-Niebla - [email protected] - Oscar M. Hernández-Calderón - [email protected] - José M. Ponce-Ortega - [email protected] - Jesús Raúl Ortiz-del-Castillo - [email protected] - Eusiel Rubio-Castro - [email protected]

* Corresponding author. Email: [email protected] Tel. +52 667 7137860 ext. 177.

Affiliation a

Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa 80000, México a Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa 80000, México a Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa 80000, México b Chemical Engineering Department, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58060, México a Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa 80000, México a Chemical and Biological Sciences Department, Universidad Autónoma de Sinaloa, Culiacán, Sinaloa 80000, México

ABSTRACT This paper describes a multi-period mixed integer nonlinear programming model for determining the optimal crop allocation for several planting cycles based on future crop prices and fresh water availability. An autoregressive moving average model is implemented to predict prices, and a new superstructure that includes all configurations of interest for the reuse, recycling, storage, and regeneration of water is employed. Two characteristic examples considering maize, wheat, alfalfa, and beans and their optimal scheduling for the years 2020, 2021, and 2022 are solved with the objective of maximizing the total annual profit. In determining the profit, the operating costs include fresh water, fresh fertilizer, and pumping and the capital costs include storage tanks, treatment units, pipelines, and pumps.

Keywords: Combinatorial optimization, assignment, water integration, price prediction, mixedinteger nonlinear programming.

1. INTRODUCTION Over the past two decades, population growth and quality of life demands have led to increased requirements for water. In agriculture, 70% of freshwater withdrawals from surface and groundwater are used for crop production (Molden, 2007). As climate change continues to affect the global temperature, precipitation, and water cycles (Allen and Ingram, 2002; Lenderink and VanMeijgaard, 2010; Reder et al., 2016), its greatest impact will be on agriculture, especially in semi-arid and arid areas with already limited natural water resources (Fischer et al., 2005; Calzadilla et al., 2010). In addition, the world population is projected to reach 9.5 billion by the year 2050, and this will have a significant effect on food security (Singh, 2014). To feed the growing population, the global irrigated area should increase by 30 million hectares, entailing an increase of 40% in demand for water and energy over the next 20 years (FAO et al., 2015). This emphasizes the need for the optimal allocation of resources to balance socioeconomic development and water conservation (Li et al., 2016). To address the abovementioned problems, several strategies have been proposed, such as increasing the efficiency of water use (Gohari et al., 2017), desalination (Erfani et al., 2014), water transactions (Erfani et al., 2014), infrastructure for water storage (Iglesias et al., 2017), irrigation deficit levels (Garg and Dadhich, 2014), regeneration and reuse of certain fertilizers (Woltersdorf et al., 2018), effect of climatic change over the usage of groundwater irrigation (Robert et al., 2018), environmental crop impacts (Montemayor et al., 2019), uncertainties with respect to supply and demand of crops (Golmohammadi and Hassini, 2019), and resource optimization (Maneta et al., 2009; Adeyemo and Otieno, 2010; Rubio-Castro et al., 2016; Khare et al., 2019). The problem of annual crop planning was recently considered by Adewumi and Chetty (2017). In most modeling approaches, the crop plan decision is represented as the best combination of land and crop that allows the optimal use of farm resources (Dogliotti et al., 2003; Haneveld and Stegeman, 2005; Bachinger and Zander, 2007). The objective of achieving an adequate crop plan is often based on a paradigm of complete rationality using a single monetary optimization criterion and, in some cases, the optimization of multiple attributes (Annetts and Audsley, 2002). In this regard, the optimal allocation of land and water resources has been addressed (Dai and Li, 2013). Adekanmbi and Olugbara (2015) proposed a multiobjective optimization method (profit, crop production, and planting area) to address the crop planning problem, and López-Mata et al. (2016) developed a direct solution algorithm capable of determining the crop plan that maximizes the profitability of an irrigation farm. Ridier et al. (2016) addressed the relationship between crop rotation and risks associated to prices and production. Hao et al. (2017) presented an integrated model of the agricultural crop pattern considering the uncertainty of water availability and the

potential for water saving in the future, with the objective of maximizing the net agricultural benefit per unit of irrigation water. Adewumi et al. (2017) proposed a mathematical formulation for the annual allocation of crops based on economic factors, and Aljanabi et al. (2018) developed a model for optimizing the allocation of recovered wastewater to ensure the optimal allocation of crops and reclaimed wastewater on cultivated agricultural land with the aim of maximizing the net benefit. Ikudayisi et al. (2018) applied an evolutionary algorithm to optimize the allocation of irrigation water and the distribution of crops under the constraint of limited water availability for specific crops. Mellaku et al. (2018) proposed a linear programming model that estimates the area of land that should be allocated for certain crops in a specific growing season so as to maximize the benefit subject to a given set of ecological, financial, and food crop production self-sufficiency constraints. Fereidoon and Koch (2018) developed an optimization tool for determining the optimum crop pattern that maximizes the total annual benefits in five agricultural plains, while Mohammadian and Heydari (2019) established a model that determines the optimal crop production for maximizing efficiency and employment and minimizing costs and fertilizer use. Regis Mauri (2019) developed a solution approach to improve a mathematical model previously reported for the crop rotation scheduling problem. Several authors have argued that crop allocation must consider dynamic changes in aspects like water availability, prices, and climatic conditions (Aubry et al., 1998; Nuthall, 2010), issues that have not been addressed in previous optimization approaches. Therefore, this study develops a model for determining the optimal allocation of a certain number of crops into a certain number of parcels. The future prices and availability of water are considered, and the objective function aims to maximize the total annual profit. 2. MODEL FORMULATION The model proposed in this paper is based on two superstructures, which are shown in Figs. 1 and 2. These superstructures represent the assignation of crops into several parcels and the mass water integration, respectively. The description and mathematical representation of the superstructures are as follows. 2.1 Crop assignation to parcels The assignation of crops to each parcel of land is illustrated in Fig. 1, which shows a general representation of four crops and four parcels. Here, all possible assignations are considered because each crop can be cultivated on each parcel. The following disjunction is used to model this situation:

  Z c1, p   Acc , p  A p     anual,min anual anual,max p c,p  Pc , p  pc,p     w f     Zc,p,t Zl,c,p,t     max min in max  li min  Li   li  fc  Ffc   fc    c , p ,t c,p,t  l,c,p,t l , c , p ,t l,c,p,t    c,p,t 

(1)

  ¬Z1c,p   Acc , p  0     anual Pc , p  0     f w   ¬Zc,p,t    ¬Zl,c,p,t        Li  0   Ffclin,c , p ,t  0    c , p , t   

Fig. 1. Proposed superstructure for allocating crops W L In the above expression, Z c1, p , Zc,p,t , and Zl,c,p,t are Boolean variables used to determine the

assignation of crop c to parcel p, the depth of water needed to irrigate crop c in parcel p during time period t ( Lic, p,t ), and the fertilizer flowrate l for crop c in parcel p ( Ffclin,c , p ,t ), respectively. Acc, p is the crop assigned to parcel p, Ap is the area of parcel p, Pcanual is the production of crop c in parcel ,p max min p, lic,p,t and lic,p,t are the upper and lower limits of water depth used for crop c in parcel p max min during time period t, and fcl,c,p,t and fcl,c,p,t are the upper and lower limits of fertilizer used for

crop c in parcel p during time period t.

Note that when the Boolean variable Z c1, p is true, parcel p is assigned to cultivate crop c to obtain W L the corresponding production; therefore, Boolean variables Zc,p,t and Zl,c,p,t are also true. This

means that both water and fertilizer are required for crop c. In the opposite case, if Z c1, p is false, then parcel p is not assigned to crop c and the crop production is zero; thus, the other Boolean variables are also false. Finally, to transform this disjunction into algebraic equations, the convex hull reformulation (Vecchietti et al., 2003) is used to obtain the following: Acc, p  zc1, p Ap ,

c  C; p  P

(2)

anual,min anual,max z1c, p pc,p  Pcanual  z1c, p pc,p , ,p

z cC

1 c, p

 1,

c  C; p  P

pP

(3) (4)

z1c, p  zcw, p,t ,

c  C; p  P, t T

(5)

z1c, p  zlf,c, p,t ,

l  L; c  C; p  P, t T

(6)

min max zcw, p,t lic,t  Lic, p,t  zcw, p ,t lic,t , min max zlf,c, p,t fcl,c,t  Ffclin,c, p,t  zlf,c , p,t fcl,c,t ,

c  C; p  P, t T l  L; c  C; p  P, t T

(7) (8)

min max min max where Ap , panual,min , panual,max , lic,t , lic,t , fcl,c,t , and fcl,c,t are known quantities c,p c,p

provided by the producers. The limits for the water depth are estimated from the following relationships: min lic,p,t  Ccs p Drc,t , max lic,p,t  Cs p Drc,t ,

p  P; c  C; t T p  P; c  C; t T

(9) (10)

where Ccs p is the critical capacity of the soil and Cs p is the soil capacity, which is the root depth in each period of irrigation. In particular, Ccs p is related to the permanent wilting point (  pwp ) and irrigation criterion ( Crp ),which depend on the soil type (see Table S1 of the Supplementary material). Ccs p  Cs p  Crp  Cs p   pwp   ,

pP

(11)

After the water depth for each period has been calculated and with a given value for the irrigation in efficiency ( EFcirrig ,t ), the inlet water to parcel p during each period ( Fwcc, p ,t ) can be estimated as

follows:

 1 Fwcc,inp ,t    EF irrig  c ,t

  Lic , p ,t Acc , p , 

c  C; p  P; t  T

(12)

In this work, the irrigation efficiency is assumed to be 0.70 (Fundación PRODUCE Sinaloa A.C. et al., 2011). 2.2 Mass water integration The mass water integration is based on the superstructure illustrated in Fig. 2, where all configurations of interest are considered. In other words, reuse, recycling, regeneration, and storage are options for the distribution of water. The modeling of these options is described in the following subsections.

Fig. 2. Proposed superstructure for water integration in parcels 2.2.1 Mass balance in the inlet parcels The inlet water ( Fwp inp,t ) and fertilizer ( Ffplin, p ,t ) for parcel p are the summations of the sources assigned for each crop: Fwpinp ,t   Fwccin, p ,t ,

p  P; t  T

(13)

Ffplin, p ,t   Ffclin,c, p ,t ,

l  L; p  P; t  T

(14)

cC

cC

Note that only one crop can be allocated to each parcel, and the inlet water and fertilizer flowrates are defined by the needs of each crop. In this way, the total inlet flowrate on each parcel ( Fptot inp,t ) is equal to the inlet water ( Fwp inp,t ) and fertilizer ( Ffplin, p ,t ) flowrates, i.e.,

Fptot inp ,t   H 2O Fwpinp ,t   Ffplin, p ,t ,

p  P; t  T

(15)

lL

where  H 2O is the water density and the other terms are calculated as follows: Fwpinp,t 

 Fwfs

wW

Ffplin, p ,t  Fff l , p ,t 

w, p ,t



 Fwpp

p1P

 Ffpp

p1P

l , p1 , p , t

p1, p ,t

 sS

Fwstps ,t , p

H O

Fwprp ,t ,

p  P; t  T

(16)

2

  Ffstpl , s ,t , p ,

l  L; p  P ; t  T

(17)

sS

In the above equations, Fwfsw, p,t is the fresh water flowrate, Fwpp p, p,t is the water recirculated between parcels, Fwstps,t , p is the water flowrate from storage tanks to parcels, Fwprp,t is the precipitated water, Fffl , p,t is the fresh fertilizer flowrate, Ffppl , p1 , p,t is the fertilizer recirculated between parcels, and Ffstpl ,s,t , p is the fertilizer flowrate from storage tanks to parcels. 2.2.2 Mass balance in the outlet of parcels out The outlet flowrate for each parcel ( Fptot out p ,t ) is composed of water ( Fwp p ,t ) and fertilizer

( Ffplout , p ,t ). These terms are determined in the same way as for the inlet flowrates: out out Fptot out p ,t  Fwp p ,t  H 2O   Ffpl , p ,t ,

p  P; t  T

(18)

lL

out Fwp out p ,t   Fwcc , p ,t ,

p  P; t  T

(19)

out Ffplout , p ,t   Ffcl ,c , p ,t ,

l  L; p  P; t  T

(20)

cC

cC

out In the above equations, Fwccout , p ,t is the outlet water flowrate in parcel p growing crop c, and Ffcl , c , p ,t

is the outlet fertilizer flowrate for crop c in parcel p. These are calculated as follows: in ev Fwccout , p ,t  Fwcc, p ,t  Fwcc , p ,t  Lic , p ,t Acc , in ab Ffclout ,c , p ,t  Ffcl ,c , p ,t  Ffcl , c , p ,t , in Ffclab,c, p,t  EFl ,abs c Ffcl ,c , p ,t ,

c  C; p  P; t T

l  L; c  C; p  P; t T

l  L; c  C; p  P; t T

(21) (22) (23)

where Fwccev, p ,t is the water lost by evapotranspiration, Ffclab,c , p ,t is the fertilizer absorbed by crops, and EFl ,abs c is the fertilizer absorption efficiency, taken to be 0.65 in this study (Salvagiotti et al., 2000). In Fig. 2, it is assumed that the outlet parcel flowrate can be directed to other parcels ( Fpp p, p1 ,t ), to the storage tanks ( Fpst p , s ,t ), to the treatment units ( Fptu p,u ,t ), and/or to environmental discharge ( Fpep,t ):

Fptot out p ,t 

 Fpp

p1P

p , p1 ,t

  Fpst p , s ,t   Fptu p ,u ,t  Fpe p ,t , sS

p  P; t  T

(24)

uU

Note that, to determine the quantities of water and fertilizer in the flowrates included in equation (24), it is necessary to stablish the following component balances: Fpp p , p1 ,t   H 2O Fwpp p , p1 ,t   Ffppl , p , p1 ,t ,

p  P; p1  P; t  T

lL

Cfplout , p ,t Fpp p , p1 ,t  Ffppl , p , p1 ,t ,

(25)

l  L; p  P; p1  P; t T

Fpe p ,t   H 2O Fwpe p ,t   Ffpel , p ,t ,

(26)

p  P; t  T

(27)

lL

Cfplout , p ,t Fpe p ,t  Ffpel , p ,t ,

l  L; p  P; t  T

Fptu p ,u ,t   H 2O Fwpu p ,u ,t   Ffpul , p ,u ,t ,

(28) p  P; u U ; t  T

(29)

lL

Cfplout , p ,t Fptu p ,u ,t  Ffpul , p ,u ,t ,

l  L; p  P; u U ; t T

Fpst p , s ,t   H 2O Fwps p , s ,t   Ffpsl , p , s ,t ,

(30)

p  P; s  S ; t  T

(32)

lL

Cfplout , p ,t Fpst p , s ,t  Ffpsl , p , s ,t ,

l  L; p  P; s  S ; t T

(33)

where Fwpe p ,t is the water flowrate from parcels to environmental discharge, Ffpel , p ,t is the fertilizer flowrate from parcels to environmental discharge, Fwpu p,u ,t is the water flowrate from parcels to treatment units, Ffpul , p,u ,t is the fertilizer flowrate from parcels to treatment units,

Fwps p , s ,t is the water flowrate from parcels to storage tanks, Ffpsl , p,s,t is the fertilizer flowrate from parcels to storage tanks, and Cfplout , p ,t is the concentration of fertilizer in the outlet parcel flowrate, determined as: out out Cfplout , p ,t Fptot p ,t  Ffpl , p ,t ,

l  L; p  P; t T

(34)

2.2.3 Mass balance in the storage tanks The total flowrate in the storage tanks ( Fstot s ,t ) is equal to the total flowrate in the storage tanks in time period t-1 ( Fstot s ,t 1 ) plus the flowrates sent from the parcels to the storage tanks (  Fpst p , s ,t ) pP

and the fresh water flowrate (  Fwfstw, s ,t ), minus the flowrates directed from the storage tanks to wW

parcels (  Fstps , p ,t ): pP

Fstots ,t  Fstots ,t 1   Fpst p , s ,t  pP

 Fwfst

wW

w , s ,t

  Fstps , p ,t , pP

s  S;t T

(35)

Therefore, the fertilizer concentration in the storage tanks is determined by the following component mass balance:

Cfstl , s ,t Fstots ,t  Cfstl , s ,t 1 Fstots ,t 1   Cfplout , p ,t Fpst p , s ,t    Cfwl , w Fwfst w, s , t  pP

 Cfst

l , s ,t

pP

wW

l  L; s  S ; t  T

Fstps , p ,t ,

(36)

Combining equations (35) and (36), it is possible to determine the quantity of water for each period in the storage tanks: Fstots ,t  Cfstl , s ,t Fstots ,t   Fstots ,t 1  Cfstl , s ,t 1 Fstots ,t 1     out   Fpst p , s ,t   Cfpl , p ,t Fpst p , s ,t   pP  pP      Fstps , p ,t   Cfstl , s ,t Fstps , p ,t  , pP  pP 

(37) l  L; s  S ; t  T

Finally, the water and fertilizer sent from the storage tanks to the parcels can be calculated as:

Ffstpl ,s, p,t  Cfstl ,s,t Fstps, p,t ,

l  L; s  S; p  P; t T

Fwstps , p ,t  Fstps , p ,t   Ffstpl , s , p ,t ,

s  S ; p  P; t  T

(38) (39)

lL

2.2.4 Mass balance in the treatment units The inlet flowrate in the treatment unit ( Ftutotuin,t ) is determined by the flowrates from the parcels (  Fptu p ,u ,t ): pP

Ftutotuin,t   Fptu p ,u ,t ,

u U ; t  T

(40)

pP

This flowrate is composed of water ( Fwtuuin,t ) and fertilizer ( Fftulin,u ,t ) components: Ftutotuin,t  Fwtuuin,t   Fftulin,u ,t ,

u U ; t  T

(41)

pP

Fwtuuin,t   Fwpu p ,u ,t ,

u U ; t  T

(42)

pP

Fftulin,u ,t   Ffpul , p ,u ,t ,

u U ; t  T

(43)

pP

To determine the inlet fertilizer concentration in the treatment units ( Cftulin,u ,t ), the following mass component balance is used: Cftulin,u ,t Ftutotuin,t   Cfplout , p ,t Fptu p ,u ,t , pP

l  L; u U ; t  T

(44)

In addition, from Fig. 2, once the flowrate in the treatment units meets the environmental constraints, the following flowrate is discarded as environmental discharge ( Ftueu ,t ):

Ftutotuout,t  Ftueu ,t ,

u U ; t T

(45)

That segregated flowrate is composed of water ( Fwuuout,t ) and fertilizer ( Fftulout ,u ,t ) components: out Ftutotuout,t   Fftulout ,u ,t  Fwuu ,t ,

u U ; t  T

(46)

lL

where the outlet fertilizer flowrate is related to the removal capacity EFl ,rem of the treatment units u (in this study, EFl ,rem u  0.95 following Rubio-Castro et al., 2010): rem in Fftulout ,u ,t  EFl ,u Fftul ,u ,t ,

l  L; u U ; t T

(47)

To determine the outlet fertilizer concentrations in the treatment units, the following component balance is needed: out out Cftulout ,u ,t Ftutotu ,t  Fftul ,u ,t

(48)

2.2.5 Mass balance in the environmental discharge The flowrate in the environmental discharge ( Fetott ) is determined by the flowrates coming from the parcels and/or from the treatment units: Fetott   Fpe p ,t   Ftueu ,t , pP

t T

(49)

uU

The fertilizer concentrations in the environmental discharge ( Cfel ,t ) are calculated using the following component balance: out Cfel ,t Fetott   Cfplout , p ,t Fpe p ,t   Cftul ,u ,t Ftueu ,t , pP

l  L; t  T

(50)

uU

To satisfy the environmental constraints, it is necessary to satisfy the following constraint with respect to the maximum permissible concentration Cfelmax (in this study, Ωcfel,tmax  50 ppm ,t according to NOM-001-SEMARNAT-1996):

Cfel ,t  Ωcfel,tmax ,

l  L; t T

(51)

2.3 Feasible constraints and crop production In agricultural activities, there are constraints related to the lower and upper limits of crop production and on water use. Hence, the next logical constraints are needed: available Fwfreshw,t  Afww,t ,

w W ; t T

Fwfreshw,t   Fwfsw, p ,t   Fwfstw, s ,t , pP

(52) w W ; t  T

(53)

sS

panual,min  Pccanual  pccanual,max c,p

(54)

pccper,min  Pccper  pccper,max

(55)

available where Afww,t is the available fresh water in time period t, pc anual,min is the lower limit for the c

annual production of crop c, pc anual,max is the upper limit for the annual production of crop c, c

Pccanual is the annual production of crop c, pc cper,min is the lower limit for the per-year production of perennial crop c, pc cper,max is the upper limit for the per-year production of perennial crop c, and Pccper is the annual production of perennial crop c. With respect to annual and perennial production,

the following expressions allow these values to be determined: Pccanual   Ac , p Pccanual ,p ,

c C

(56)

pP

Pccper   Ac , p Pccper , p ,t ,

c C

(57)

pP tT

anual,set Pccanual  zc1, p Ωpcc,p , ,p 1 per,set Pccper , p ,t  zc , p Ωpcc,p,t ,

c  C; p  P

(58)

c  C; p  P; t T

(59)

where Pcc, p is the production per hectare of annual crop c in parcel p, Pccper , p ,t is the production per hectare of perennial crop c in parcel p in period t, Ωpcanual,set is a known parameter used to define c,p per,set the performance per hectare of annual crop c in parcel p, and Ωpcc,p,t is a known parameter used to

define the performance per hectare of perennial crop c in parcel p during time period t. 2.4 Annual operating time The annual operating time ( HY ) is the quotient of the time required to irrigate crops during each period ( Timei preq,t ) and the number of parcels ( Nup ):

HY   Timei preq,t

(60)

Nup

pP tT

The required time period is determined by the irrigation time in periods where the crops require water ( Ωtimei t ). This is because not all crops require water in each time period. In particular, the irrigation time is calculated as follows: Fwhd p,t  Fwpinp,t Ωtimei t ,

p  P; t T

(61)

where Fwhd p ,t is the water flowrate. The next logical expression is used to activate or deactivate the required time period:



Timei preq,t  Ωtimei t 1  e

 1 Fwpin p ,t



(62)

In equation (62), 1 is a known parameter with a small value (for example, 1×10-3). Therefore, the following options are possible for the value of e

 1 Fwpin p ,t

:

   Fwp inp ,t  0  Fwp inp ,t  0      Fwhd    Fwhd  e 1 p ,t  0   e 1 p ,t  1      1 Fwpin p ,t   Fwpin p ,t  1  1  e 1  0  1 e     req req Timei p ,t  Ωtimei t   Timei p ,t  0 

(63)

2.5 Objective function The objective function aims to maximize the profit from crop sales ( Pft ). This is calculated as the income from the sale of crops ( Inc ) minus the outgoings related to the operating and capital costs ( Opc , Cac ). Pft  Inc  Opc  Cac

(64)

where: ,per Inc   Punitcsell ,anual Pccanual   Punitcsell Ac , p Pccper ,t , p ,t

(65)

Opc  Cfreshw  Cfreshf  Pmp op  Piping op  Stwop  Tru op

(66)

Cac  Pmp cap  Piping cap  Stwcap  Tru cap

(67)

cC

cC pP tT

,per In the above equations, Punitcsell , anual is the unitary sale price of annual crops, Punitcsell is the ,t

unitary sale price of perennial crops, Cfreshw is the fresh water cost, Cfreshf is the fresh fertilizer cost, Pmp op is the operating cost of pumping, Piping op is the operating cost of piping, Stwop is the operating cost of the storage tanks, Tru op is the operating cost of the treatment units, Pmp cap is the capital cost of the pumps, Piping cap is the capital cost of the pipes, Stwcap is the capital cost of the storage tanks, and Tru cap is the capital cost of the treatment units. The components for the operating costs are described as follows: Cfreshw 

  Cunit

wW pP tT

op w ,t

Fwfsw, p ,t

Cfreshf   Cunitlop,t ( Fffl , p ,t / 0.46) lL pP tT

(68) (69)

op kw Pmp op  H Y  Cuniteop,t Pwkw p , s ,t  H Y  Cunite ,t Pws , p ,t  pP sS tT

HY

sS pP tT

 Cunit

pP p1P tT

op e ,t

op kw Pwkw p , p1 ,t  H Y  Cunite ,t Pw p ,u ,t

(70)

pP uU tT

op Piping op  H Y  Cunit op pi ,t Fpst p , s ,t  H Y  Cunit pi , t Fstps , p , t  pP sS tT

HY

 Cunit

sS pP tT

op pi ,t

Fpp p , p1 ,t  H Y  Cunit op pi ,t Fptu p ,u ,t

p,p1ÎP tT

(71)

pP uU tT

Stwop  HY  Cunitsop,t Fstots ,t

(72)

Tru op  HY  Cunituop,t Ftutotuin,t

(73)

sS tT

uU tT

where Cunit is the unitary cost of the different raw materials (fresh water, fertilizers, and electricity) and maintenance (pipelines, storage tanks, and treatment units), Pw is the power demand of the pipelines used to connect the components of the superstructure (parcels, storage tanks, and treatment units). Pwckw, s ,t , Pwskw,c,t , Pwckw,c1 ,t , and Pwckw,u ,t are determined as follows:

Pwkw p , s ,t

  Fpst p , s ,t      He p , s  3.048   g   3600Ωtimei t    1x103  gc  EFppump ,s    

(74)

Pwskw, p ,t

  Fstps , p ,t      Hes , p  3.048   g   3600Ωtimei t    1x103  gc  EFs ,pump p    

(75)

Pwkw p , p1 ,t

  Fpp p , p1 ,t    He p , p1  3.048 3600Ωtimei t  3 g    1x10 gc  EFppump , p1  

Pwkw p ,u ,t

  Fptu p ,u ,t      He p ,u  3.048   g   3600Ωtimei t    1x103  gc  EFppump ,u    





    

(76)

(77)

where EF pump is the efficiency of the pumps used to send water between processing units (in this study, EF pump  0.6 following Leeper, 1981), He is the equivalent height between the process units, g is acceleration due to gravity, and gc is a conversion factor for acceleration due to gravity. It is also necessary to calculate the components of the capital cost ( Pmp cap , Piping cap , Stwcap , and

Tru cap ). These are given by the following expressions:

Pmp cap

 Cfix pump p,s  pP  sS  pump   Cfix s,p s  S  pP  KF   Cfix pump p,p1   pp1PP    Cfix pump p,u  pP  uU

Piping cap

   Cva

1  e

 2 Fpst capc p ,s

1  e

 2 Fstpscapc ,p



 2 Fpp capc p,p



 2 Fptuccapc ,u





1 e

1  e

1









  3 Fca capc pip p ,s   Cfix p,s 1  e p  P  sS    3 Fcascapc pip ,p   Cfix s,p 1  e  psSP  KF    3 Fca capc pip p , p1   Cfix p,p1 1  e  ppPP 1   3 Fca capc p ,u   Cfix pip p,u 1  e  pP  uU







 Fpst 

     pump   Cva s,p  Fstpscapc ,p    pP  sS   pump capc   Cva p,p1 Fpp p , p1   pP  p1P   pump    Cva p,u  Fptuccapc  ,u  pP uU  pP sS

pump p,s

capc  p,s



  3600  v p , s pP  sS  capc  Fca D s, p s, p pip   Cva s,p   3600  vs , p pP  sS   Fca capc p , p1 D p , p1 pip   Cva p,p1  3600  v p , p1  pP p1P   Fca capc p ,u D p ,u pip   Cva p,u  3600  v p ,u pP  uU  pip   Cva p,s

Fca capc p,s Dp ,s



1  m 

n



(79)



(80)





(81)

 capc  Tru cap  K F   Cfix utu 1  e 5 Fcau   Cva utu  Fcaucapc   uU uU 

KF 

(78)



 capc  Stwcap  K F  Cfix sw 1  e 4 Fcas   Cva ssw  Fcascapc   s sS  sS 

m 1  m 



n

1

(82)

In the above equations, Cfix is the fixed cost of the process components, Cva is the unit variable cost of the process components, K F is a factor used to annualize the investment,  is the flowrate density (in this study,   100 kg m 3 ), v is the velocity (in this study, v  1 m s ), D is the distance between process components, m is the annual interest (in this study, m  5% ), n is the

number of years required to recuperate the investment (in this study, n  5 ), Fca capc is the capacity of the pipeline segments for the mass integration, storage tanks, and treatment units,  is a parameter reflecting the economies of scale ( 0    1 ; in this study,   0.6 ), and  is a parameter with a small value (for example, 1×10-3). In addition, to activate or deactivate the fixed capital cost in equations (78)–(82), the following logical expression is applied; this is analogous to equation (63):

 Fcaxcapc  0   Fcaxcapc  0      capc   Fca capc  e x x  0   e  x Fcax  0       x Fcaxcapc   Fca capc 1  e  1   1  e x x  1 Cfix  Cfix know   Cfix  0  x  x x   

(83)

Note that, in equation (83), Cfix know is a known parameter, and the subscript x represents the x process components (p, s, and u) and their possible combinations. In other words, in the computational code, equation (83) must be repeated as many times as required to activate or deactivate the fixed capital costs. Finally, the capacities are determined as follows:

Fststot,t  Fcascapc , Fcascapc   max Fst ,

Ftuutot,t  Fcaucapc , Fcaucapc   max Ftu ,

Fpp p, p1 ,t  Fcacapc p , p1 , max Fcacapc p , p1  Fpp ,

Fstps , p,t  Fcascapc ,p , max Fcascapc , p  Fstp ,

Fpst p, s ,t  Fcacapc p,s , max Fcacapc p , s  Fpst ,

Fptu p,u ,t  Fcacapc p ,u , max Fcacapc p ,u  Fptu ,

s  S ; t T

(84)

sS

(85)

u U ; t T u U

(86) (87)

p, p1  P; t T

p, p1  P s  S ; p  P; t T s  S; p  P p  P; s  S ; t T p  P; s  S p  P; u U ; t T p  P; u U

(88) (89) (90) (91) (92) (93) (94) (95)

where max is an upper limit for the capacities of the process components (pipelines, storage tanks, and treatment units; a value of 22,000 m3 is used). Note that equations (63) and (83) do not use

binary variables. For the allocation problem described in this paper, this is an advantage in terms of finding good, feasible solutions. 2.6 Retrofit of pipes, pumps, storage tanks, and treatment units When the crop allocation is made for more than one cycle, the process equipment (pipes, pumps, storage tanks, and treatment units) acquired in the first cycle can be used as existing equipment in the second cycle. The equipment selected in the second cycle may need to be retrofitted in the third cycle, and so on. The following disjunction is employed to model this situation:   Z yRET    Z yRET  RET ,1 RET ,2      Zy Zy    min    Fcap y  0   exist max exist   y  Fcap y  fcap y       fcap y  Fcap y   y  RET     Cfix y  0   RET RET Cfix y  0 Cfix y  0       RET      Fcap y  0   RET exist RET Fcap y  0     Fcap y  Fcap y  fcap y  

(96)

In this expression, the superscript RET denotes retrofitting, y represents the process components (storage tank, treatment unit, pipes, and pumps) in the proposed superstructure, and the superscript exist represents an existing process component. Z yRET , Z yRET ,1 , and Z yRET ,2 are Boolean variables for determining the fixed and variable costs for pipes, pumps, storage tanks, and treatment units after the first cycle;  min is a lower limit for the capacity of the process components, max is an upper y y limit for the capacity of the process components, Fcapy is the retrofitted capacity of the process components, CfixyRET is the retrofitted fixed cost, FcapyRET is the retrofitted capacity to calculate the variable cost, and fcapexist is the capacity of existing process components. To model the above y disjunction, the convex hull reformulation (Vecchietti et al., 2003) is used; and it is given in the section III of the Supplementary material. 2.7 Price prediction method The sale prices of agricultural products were statistically predicted using the Autoregressive Moving Average (ARMA) model, also called the Box–Jenkins model, applied to time series data. The ARMA model is a tool for understanding and even predicting future values of the time series. The model consists of two parts: an autoregressive (AR) part and a moving (MA) part. The model is denoted as ARMA(p, q), where p is the order of the AR part and q is the order of the MA part. The AR model of order p, AR(p), estimates time series data xt through a model of the form: p

xt  k xt k   t k 1

(97)

where k 1  k  p  corresponds to the parameters of the model and  t is the error associated with the estimation of xt . It is necessary to impose certain restrictions on the parameter values to ensure that the model works properly (i.e., we have a stationary process). The MA model of order q, MA(q), represents time series data through the model: q

xt   t  k  t k

(98)

i 1

where k 1  k  q  corresponds to the parameters of the model and  t k  0  k  q  is the error term. Thus, ARMA(p, q) refers to a model with p AR terms and q MA terms. This model combines the AR and MA models as follows: p

q

k 1

k 1

xt  k xt k  k  t k   t

(99)

It is generally assumed that the error terms (  t ) are independent and identically distributed random variables, taken from a sample with a zero-mean normal distribution, i.e.,  t

N  0,  2  , where  2

is the variance. These assumptions may be weak and, if they are not satisfied, the properties of the model may change. In fact, a change in the assumption of independence and identical distribution could result in considerable differences. In general, a temporary data series Fk  0  k  m can be described by two components, the general trend, represented by a continuous function

f k  f  tk   0  k  m , and the fluctuations

xk  0  k  m . The function can be obtained by applying some nonlinear least-squares method to

the discrete data Fk (see Fig. S1 in the Supplementary material), and then the fluctuations can be determined as xk  Fk  f k (see Fig. S2 in the Supplementary material). After selecting an adequate function f k  f  tk  (i.e., one that captures the general trend of the time series Fk ), the ARMA model is applied to the fluctuations xk instead of the time series Fk , and the model adjusts according to the following relationship: p

q

i 1

i 1

Fk  f (tk )  i xk i  i k i   k

(100)

where f (tk ) is fully defined and the coefficients to be adjusted correspond to the sets i 1  i  p  and i 1  i  q  . Note that the coefficients  t k  0  k  q  represent the error of the adjustment between the time series

Ft and its estimated value:

p

q

i 1

i 1

f (tk )  i xk i  i k i

(101)

To apply this methodology, we consider the monthly price of maize over the period August 1, 1985–August 1, 2015, as shown in Fig. S3 in the Supplementary material (Farmdoc, 2015). Thus, it is possible to establish the general holding through the exponential function: f (tk )  80.26exp  8.463  105 tk 

(102)

where t k is given in days. This function establishes the temporary price fluctuations xk (see Fig. S4 in the Supplementary material). Considering that the market price of maize is influenced by the production cycle, it was established that the price fluctuations were affected by historical information from the 11 previous price fluctuations. Thus, maize prices were predicted using ARMA(p = 11, q = 11). Model predictions after the period of reported data (i.e., after August 1, 2015) were made by simulating the error values (  k ) through a normal random (distribution) function using a zero mean (  = 0) and a variance (  2 )obtained from the historical errors between the real data and the predicted data (  k in the period August 1, 1985–August 1, 2015). Parameter adjustments were made by applying the linear least-squares method to the historical prices. Fig. S5 of the Supplementary material shows maize price forecasts for the period September 1, 2015–January 1, 2040. Table S2 in the Supplementary material reports the parameters i 1  i  p  and i 1  i  q  obtained from the historical prices, which were then used to make the price predictions. The same strategy was applied to the historical prices of wheat, beans, and alfalfa (Farmdoc, 2015); the results are presented in Figs. S6–S8 of the Supplementary material, and the adjusted parameters of the model in each case are listed in Table S3. Finally, the nomenclature of the proposed model is presented in the Supplementary material. 3. RESULTS AND DISCUSSION This paper describes the use of the proposed model to solve two examples from Mexico. The scenarios are illustrated in Fig. S9 of the Supplementary material. Example 1 considers two crops (maize and wheat) that must be allocated to four parcels measuring five hectares each; the distribution is shown in Fig. S10 of the Supplementary material. This allocation is realized under two scenarios (unlimited fresh water and limited water), each considering free production and the minimal demands of crops. Example 2 considers the allocation of four crops (maize, wheat, alfalfa, and beans) to 12 parcels. In Example 2, only the limited water and minimum/maximum production conditions are considered. The distribution of parcels is presented in Fig. S10 of the Supplementary

material. Moreover, the crop prices for the years 2020, 2021, and 2022 were considered (see Table S4 in the Supplementary material) in determining the optimal allocation. These crop prices were obtained from the prediction method proposed in this paper. The proposed model was coded using the GAMS software (Brooke et al., 2014) and solved using the CONOPT, CPLEX, and DICOPT solvers. Example 1 represents a problem with 4,658 equations, 3,895 continuous variables, and 200 binary variables; Example 2 has 4,320 equations, 34,820 continuous variables, and 2,916 binary variables. Finally, the CPU time for Example 1 was 47 s and that for Example 2 was 315 s. Here, the proposed model is a highly non linear and

non convex problem that involves several bilinear and exponential terms. Therefore, a solution approach was employed in order to ensure near global optimal solutions. This strategy is presented in the section V of the supplementary material. 3.1 Analysis of results for Example 1 Tables S5–S8 in the Supplementary material present the parameters for Example 1, where the operational and economic data are representative of Sinaloa State in México. Table S9 in the Supplementary material shows the irrigation scheduling for this example, which was suggested by the local producers. Considering this, Fig. 3 and Figs. S11–S12 in the Supplementary material show the optimal configurations for Scenario 1 (unlimited fresh water) and free production for three cycles. In particular, for the first cycle, maize was assigned in all parcels and the water and fertilizer are reused in the irrigation periods (see Fig. 3). This reuse allows 60,599 m3 of fresh water and 4,043 kg of fresh fertilizer to be saved. For the second and third cycles, wheat is assigned in all parcels, making it possible to save 41,753 m3 and 2,312 kg of water and fertilizer, respectively. As the reuse from parcel 1 to 3 and from parcel 2 to 4 relies on gravity, neither pipes nor pumps are required; for the reuse from parcel 3 to 1 and from parcel 4 to 1, pipes and pumps are selected (see Fig. 3). From Figs. S11–S12 in the Supplementary material (optimal configurations for the second and third cycles), the reuse from parcel 1 to 3, from parcel 2 to 4, from parcel 3 to 4, and from parcel 4 to 3 saves 83,506 m3 of fresh water and 4,625 kg of fresh fertilizer. Here, the selected pipes and pumps in the first cycle represent options for retrofitting in the second cycle, and the same happens from the second to the third cycle. This retrofitting means that, in the third cycle, the piping and pumping capital costs are equal to zero. This is important because the assignation of crops is conducted with respect to the optimal relationship between crop prices and the production and demands of fresh sources. In addition, Table S10 of the Supplementary material lists the economic aspects related to the configurations of Fig. 3 and Figs. S11–S12. It can be seen that the total profit of the scheduling

under the criterion of Scenario 1 and unlimited fresh water is equal to 152,907.00 US$, which is incorporates profits from the first, second, and third cycles of 47,112 US$, 58,122 US$, and 47,673 US$, respectively.

Fig. 3. Optimal configuration for Example 1 of the first cycle (Scenario 1: free production)

Figure S13 in the Supplementary material shows the optimal configuration for example 1 for the minimum and maximum production of maize and wheat. Specifically, the lower and upper limits of maize production are 100 tons and 200 tons and those for wheat are 70 tons and 140 tons. This means that all crops must be assigned to at least one parcel. In Fig. S13, maize is assigned to parcels 1 and 3 and wheat is assigned to parcels 2 and 4. This assignation allows an annual production of 100 tons of maize and 70 tons of wheat; the total scheduling profit is 141,470 US$ (42,616 US$ in the first cycle, 51,762 US$ in the second cycle, and 47,092 US$ in the third cycle). Note that the above crop assignation allows the direct reuse of water and fertilizer between parcels 1 and 3 and between parcels 2 and 4; therefore, annual savings of 47,319 m3 of water and 2,800 kg of fertilizer are achieved. This, in turn, demands the pipes and pumps to handle the flowrate from parcels 3 and 4 to parcels 1 and 2; therefore, capital and operating costs of 1,276 US$ are required in the first

cycle. In the second and third cycles, these pipes and pumps operate with the same capacity as in the first cycle, and hence their fixed and capital costs are equal to zero. Regarding Scenario 2 (limited fresh water) and both free and restricted production, the optimal configuration for the three cycles is shown in Fig. S14 of the Supplementary material. For this scenario, maize is assigned to parcels 1 and 3, while wheat is assigned to parcels 2 and 4. This assignation is the same for all cycles and for both free production and restricted production, because this is the only configuration that is possible with the fresh water available. In other words, it is not possible to cultivate just one crop, because there is not enough water to irrigate four parcels in the same period. In this regard, the mass exchange between parcels (1 to 3, 2 to 4, 3 to 1, and 4 to 2) results in savings of 47,319 m3 of water and 2,800 kg of fertilizer. In addition, a tank is required to store 18,402 m3 of fresh water in period 1 for use in period 4. This storage tank has a cost of 5,436 US$ and the recirculation of water between parcels and from the storage tank to parcels has a capital piping and pumping cost of 1,805 US$; the pump operating cost is equal to 700 US$. These values are given in Table S10 of the Supplementary materials, and it can be observed that the costs for storage tanks, pumps, and pipelines are only considered in the first cycle. This is because this equipment and accessories are bought and installed in the first cycle of the period of the scheduling, and are used in the remaining cycles with the same capacities. The total scheduling profit for the configuration shown in Fig. S14 is equal to 134,328 US$. Finally, note that the best scenario for crop scheduling is that with unlimited fresh water and free production. However, in most agricultural areas, water is limited, and the production is based on the minimum and maximum demands of the market in order to satisfy consumers and regulate crop prices. Hence, Example 2 is solved under the conditions of Scenario 2 and limited fresh water. 3.2 Analysis of results for Example 2 For Example 2, four crops (maize, wheat, alfalfa, and bean), five storage tanks, and twelve parcels are considered (see Fig. S10 in the Supplementary material). Tables S11–S13 in the Supplementary material present the irrigation scheduling for cycle one (October 2019–September 2020), cycle two (October 2020–September 2021), and cycle three (October 2021–September 2022). These irrigation periods were provided by producers, and it is important to note that maize and bean seeds are sown in October, whereas wheat and alfalfa seeds are sown in November and December, respectively. As alfalfa is a perennial crop, the seed sowing is made in the first cycle only, and the plants can be harvested each month for the next three years. This characteristic of alfalfa means there are 15 periods in the first cycle and 17 in the remaining cycles. Tables S14–S15 in the Supplementary material present the lower and upper limits of water depth and fertilizer for Example 2. The values

pip pip pip pip of Cunitsop,t , Cuniteop,t , Cunitlop,t , Cunitsop,t , Cunituop,t , Cunit op pi ,t , Cfix p,s , Cfix s,p , Cva p,s , Cva s,p ,

Cfix pip p,u ,

Cfixssw , Cvassw , Cfixutu , Cvautu , Cfixpump Cva pip p,u , p,s ,

pump , Cfixpump Cfixs,p p,u ,

Cva pump p,p1 ,

pump pump pump ev , Cva pump Cva s,p p,s , Cva p,s , Cva s,p , Hes , p , He p , s , Hep ,u , and Fwcc , p ,t are as defined for Example

pump pip 1. The values for the remaining parameters ( Cfix pump p,p1 , Cfix p,p1 , Cva p,s , He p , p1 , Ds , p , Dp , s , Dp , p1 ,

available , set , Pcanual,min , Pc anual,max , Fwp p ,t , and Afww,t ) are listed in Tables S16–S19 of the Dp,u , Pccanual c c ,p

Supplementary material. The optimal crop allocation and its mass water integration are presented in Fig. 4 and Figs. S15– S16 of the Supplementary material. Alfalfa was assigned to parcels 1 and 2; maize to parcels 3, 4, 5, and 12; wheat to parcels 6 and 7; and beans to parcels 9, 10, and 11. In the first cycle, any crops could have been assigned to any parcels, but for the second and third cycles, the allocation of alfalfa is fixed to parcels 1 and 2; the remaining crops can be assigned to any of the other parcels (3–12). However, the optimal allocations are the same for all cycles, although there are some differences in the connections between crops. For example, in the first cycle, storage tank 1 is required and receives 6,740 m3 of fresh water in period 9; in the remaining cycles, the fresh water fed into this storage tank totals 17,872 m3. Hence, the storage capacity is equal to the second value, whereas the storage cost in the first cycle is determined by the fixed and variable cost of storing 6,740 m3. However, in the second cycle, the storage cost is determined only by the variable cost in terms of the difference between 17,872 m3 and 6,740 m3. The stored water is used to satisfy the water demand in parcels 1 and 2, where alfalfa has been allocated. Table 1 presents the results for Example 2 related to Fig. 4 and Figs. S15–S16. The total scheduling profit is equal to 1,134,365 US$, with profits of 261,708 US$, 429,098 US$, and 443,558 US$ in the first, second, and third cycles, respectively. Note that the difference between the annual profits is due to the incomes from crop sales, as the alfalfa is harvested and sold over twelve months and achieves the best production during the second and third cycles. In addition, the equipment and accessories purchased in the first cycle (pumps and pipelines used to send water and fertilizer between parcels 1 and 1, 2 and 1, 4 and 2, 7 and 6, and 12 and 5) are considered for retrofitting in the second and third cycles. This means that the piping, pumping, and storage capital costs in the third cycle are equal to zero. Finally, the mass water integration saves 601,083 m3 of water and 27,613 kg of fertilizer; the flowrates in Fig. 4 and Figs. S15–S16 are given in Tables S20–S28 of the Supplementary material. These savings not only represent a better economic scenario, but also make it possible to sow the

crops and meet the demands of food when there are not enough fresh resources (water and fertilizer). Furthermore, the environmental impact is reduced.

Table 1. Results for Example 2 Limited fresh water Concept Profit (US$ year -1) Incomes by crop sales (US$ year -1) Fresh water cost (US$ year -1) Fertilizer cost (US$ year -1) Storage cost (US$ year -1) Treatment cost (US$ year -1) Capital cost of pipelines (US$ year -1) Capital cost of pumps (US$ year -1) Cost of pipelines from parcel to parcel (US$ year -1) Cost of pipelines from parcels to storage tanks (US$ year -1) Cost of pipelines from storage tanks to parcels (US$ year -1) Cost of pumps from parcel to parcel (US$ year -1) Cost of pumps from parcels to storage tanks (US$ year -1) Cost of pumps from storage tanks to parcels ((US$ year -1) Operating cost of pumps (US$ year -1) Operating cost of pipelines (US$ year -1)

Maximum and minimum production (cycles) First Second Third 261,708 429,098 443,558 285,250 455,085 465,215 4,201 5,957 5,959 12,413 12,949 12,946 3,040 3,916 0 0 0 0 262 18 0 1,977 395 0 210 11 0 0 0 0 52 7 0 1,582 395 0 0 0 0 395 0 0 1,649 2,752 2,752 0 0 0

Total fresh water (m3 year-1)

280,077

397,109

397,261

Total fresh fertilizer (kg year -1)

18,342

19,134

19,130

46,462

61,991

61,991

177,253

211,886

211,944

7,658 200 70 960 40

9,866 200 70 1740 40

9,870 200 70 1740 40

3

-1

Total environmental discharge (m year ) 3

-1

Fresh water saving (m year ) -1

Fresh fertilizer saving (kg year ) Annual maize production (ton year -1) Annual wheat production (ton year -1) Annual alfalfa production (ton year -1) Annual bean production (ton year -1) Total scheduling profit (US$)

1,134,365

Fig. 4. Optimal configuration for Example 2 in the first cycle 4. CONCLUSIONS A multi-period mixed integer nonlinear programming model has been proposed for determining the optimal allocation of crops into a number of parcels. The objective function aimed to maximize the total annual profit, constituting the income from selling crops minus the operating and capital costs

required to reuse, regenerate, and store water. Specifically, crop allocation was set to be a function of available fresh water, future crop prices, and consumer demands. The correlations used to predict future crop prices were obtained from the application of an ARMA model to historical price data for maize, wheat, beans, and alfalfa; however, this method could be applied to any crops. The results demonstrate that the optimal allocation can be determined from the optimal relationship between the crop prices, water availability, and costs by employing mass water integration (purchase and operation of pipes, storage tanks, treatment units, and pumps). Additionally, the proposed model can be adapted to different cases in terms of costs, distances between crops, number and type of crops, number of parcels, and consumer demand. The simultaneous optimization of crop allocation and water integration enables reductions in water and fertilizer use of around 40–60%; this reduces both costs and environmental discharges. In addition, the scenario of free production and unlimited water represents the best economic scenario, with a scheduling profit some 8–13% greater than that in scenarios with restricted production. The greatest annual cost and the lowest annual profit are associated with the first cycle. This is because this cycle requires the greatest investments, although these investments result in savings of around 15–55% in example 1 and 6–10% in example 2. Finally, the total cost is made up of approximately 20% for fresh water, 60% for fertilizer, and 20% for the pipelines and pumps.

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