Consider an agricultural land-water resource allocation problem, in which yields are spatial dependent and stochastically correlated. To achieve sustainability, we formulate a multi objective optimization problem, in which the decision maker determines the cultivation areas and the supplemental irrigation water levels at different locations, with social, economic and environmental goals in mind. For the social goal, we minimize the root mean squared difference of incomes among locations. For the economic goal, we minimize the production risk. We show that minimizing production risk is equivalent to maximizing the service level, when demand is normally distributed. For the environmental goal, we minimize the resource utilization. Assume that the yield vector at different locations follows a multivariate normal distribution. We formulate the multi-objective optimization problem using a weight global criterion method, and we provide a sufficient condition for convex quadratic programming. We demonstrate the applicability of our proposed framework in the case study of sugarcane production in Thailand. To capture yield response to water, we propose several models including linear and nonlinear regressions, and we obtain the closed-form expression for the linear and probit yield response models. The numerical experiment reveals that our solution significantly improves the social and economic goals, compared to the current policy. Finally, we illustrate how to apply our model to quantify the monetary value from reducing yield variability, which could be resulted from smart irrigation or precision agriculture.

1. Multi-objective land-water allocation model
for sustainable agriculture with
predictive stochastic yield response
Kannapha Amaruchkul
Graduate School of Applied Statistics,
National Institute of Development Administration (NIDA), Bangkok, Thailand
The 47th International Congress on Science, Technology and Technology-based Innovation (STT47)
Session B: Math Stat ComSc for Sustainable Development Goals (SDGs)
K. Amaruchkul (2021). Multiobjective land–water allocation model for sustainable agriculture with predictive stochastic yield response.
International Transactions in Operations Research. https://doi.org/10.1111/itor.13015
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2. Introduction
• Agricultural production requirements are growing, as the world’s
population continues to increase (United Nations, 2019)
• Since agriculture is the biggest water user globally and a primary
source of land use, improving agricultural productivity needs to be
coupled with conserving natural resources (Food and Agriculture
Organization, 2017).
• Being the largest component of the GDP of many developing nations,
agriculture has to generate enough jobs and incomes to achieve rural
economic growth and poverty eradication.
• Poor land-water resource allocation in agriculture could lead to
significant social, economic and environmental costs.
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3. Objective
To propose a multi-objective optimization model for land-water
allocations with
• social
• economic
• environment goals
when there are uncertainties in
• crop demand
• yields
• precipitations at different locations
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4. 4
Introduction Allocation model Yield response model Case study
Literature review of multi-objective optimization models under uncertainties for agricultural resource planning.
(1=“Yes” and 0=“No”)

5. 5
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1. Min production risk
2. Min income inequality
3. Min env. impact
Literature review of multi-objective optimization models under uncertainties for agricultural resource planning.
(1=“Yes” and 0=“No”)

6. 6
Introduction Allocation model Yield response model Case study
Literature review of yield response models. (1=“Yes” and 0=“No”)

7. Sustainable agriculture model formulation
Decision variables
• Consider a decision maker (DM), endowed with 𝑚 cultivation areas,
uses the total production from 𝑚 locations to satisfy a random crop
demand 𝐷𝑇.
• At the beginning of a season, the DM needs to decide
• 𝑥 = 𝑥1, … , 𝑥𝑚 for cultivation areas
• 𝑤 = (𝑤1, … , 𝑤𝑚) for irrigation water levels where 𝑤𝑖 = (𝑤𝑖,1, … , 𝑤𝑖,𝐿𝑖
)
where 𝐿𝑖 is the number of growing months during the entire season.
• Cultivation area may include both DM’s own and leased lands.
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8. Sustainable agriculture model formulation
Conjuctive use of water
Let 𝑅𝑖 = (𝑅𝑖,1, … , 𝑅𝑖,𝐿𝑖
)
be the random rainfall.
Assume rainfed 𝑅𝑖ℓ and
irrigation water 𝑤𝑖ℓ
substitute.
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https://groundwaterexchange.org/conjunctive-use/
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9. Sustainable agriculture model formulation
Reference crop evapotranspiration ETo mm/day
Penman-Monteith equation
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Tmean = mean daily temperature (°C)
p = mean daily % of annual daytime hrs
http://www.fao.org/3/s2022e/s2022e07.htm#3.1.3%20blaney%20criddle%20method
Blaney-Criddle method
Radiation term + Aerodynamic term
Introduction Allocation model Yield response model Case study

10. Sustainable agriculture model formulation
Crop water need ETx mm/day
• Crop water need (ETcrop mm/day) is defined as the depth (or amount) of water needed to meet
the water loss through evapotranspiration
• Crop water need depends on a) climate; b) crop type; c) growth stage
Chaibandit, K., Konyai, S., Slack, D., 2017. Evaluation of the water footprint in sugarcane in eastern Thailand. Engineering Journal 21, 5.
https://www.researchgate.net/publication/233726388_Trees_and_water_smallh
older_agroforestry_on_irrigated_lands_in_Northern_India/figures?lo=1
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11. Sustainable agriculture model formulation
Crop water requirement (CWR)
The crop water requirement at location 𝑖
𝐸𝑇𝑥,𝑖ℓ the maximum daily evapotranspiration in month ℓ and 𝑛𝑖ℓ
𝑔
the number of growing days in month ℓ
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12. Sustainable agriculture model formulation
Crop yields
• Crop yields are stochastic and spatial dependence.
• Let 𝑈 = (𝑈1, … , 𝑈𝑚) denote a random vector of crop yields, where 𝑈𝑖
is a crop yield at location 𝑖
• Assume 𝑈~ multivariate normal distribution with
mean vector (𝜇1 𝑤1 , … , 𝜇𝑚 𝑤𝑚 ) and
covariance matrix Σ = [𝛾𝑖𝑗] where 𝛾𝑖𝑗 = 𝑐𝑜𝑣 𝑈𝑖, 𝑈𝑗
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Other approaches to model dependency include copulas (Nelsen, 1999) and scenario representation in stochastic programming
(Birge and Louveaux, 1997). Copula approach allows greater flexibility, but we need to estimate the marginal densities and the
copula. The correlation approach involves only the first two moments, easily estimated from historical data.
Introduction Allocation model Yield response model Case study

13. Sustainable agriculture model formulation
Crop yields
• Crop yields are stochastic and spatial dependence.
• Let 𝑈 = (𝑈1, … , 𝑈𝑚) denote a random vector of crop yields, where 𝑈𝑖
is a crop yield at location 𝑖
• Assume 𝑈~ multivariate normal distribution with
mean vector (𝜇1 𝑤1 , … , 𝜇𝑚 𝑤𝑚 ) and
covariance matrix Σ = [𝛾𝑖𝑗] where 𝛾𝑖𝑗 = 𝑐𝑜𝑣 𝑈𝑖, 𝑈𝑗
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14. The biomass production at location 𝑖
𝑄𝑖 = 𝑈𝑖𝑥𝑖
Total production
Mean
Variance
where 𝑣𝑎𝑟 𝑈𝑖 = 𝜎𝑖
2
and 𝑐𝑜𝑟 𝑈𝑖, 𝑈𝑗 = 𝜌𝑖𝑗
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Sustainable agriculture model formulation
Biomass production
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15. Sustainable agriculture model formulation
Average household income
• Let 𝑝𝑖 be the per-unit selling price
• Let 𝑛𝑖 be the number of farmer households in area 𝑖
• For shorthand, let 𝑡𝑖 = 𝑝𝑖/𝑛𝑖
• The household income from the agricultural produce is
• The average household income across 𝑚 locations is
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16. Sustainable agriculture model formulation
Feasible region
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17. Sustainable agriculture model formulation
Feasible region
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Let 𝑟𝑝 be the required mean production

18. Sustainable agriculture model formulation
Feasible region
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Let 𝑟𝑐 be the required average household income

19. Sustainable agriculture model formulation
Feasible region
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bounds on cultivation area

20. Sustainable agriculture model formulation
Feasible region
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bounds on water level

21. Sustainable agriculture model formulation
Feasible region
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bounds on water volume
Let 𝜏 be a unit conversion.
If the unit of the water volume to be
𝑚3/day, conversion factor is 𝜏 =
0.001 when the irrigation water level
(depth) 𝑤𝑖ℓ is given in mm/day and
the area 𝑥𝑖 is given in 𝑚2
.

22. Sustainable agriculture model formulation
The “three pillars of sustainability”
• Social goal
Minimizing income inequality
• Economic goal
Minimizing production risk
• Environmental goal
Minimizing resource usage
22
https://twitter.com/carbonbit_/status/1321771385253933056
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23. Sustainable agriculture model formulation
The “three pillars of sustainability”
• Social goal: Minimizing income inequality
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Gini = A/(A+B)
Expected sum of squared differences (SSD)
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24. Sustainable agriculture model formulation
The “three pillars of sustainability”
Economic goal: Minimizing production risk
Note that if the production target is achieved, i.e., constraint (3) is binding, then minimizing (10) is also equivalent
to minimizing Value at Risk (VaR) and Tail-Value-at-Risk (TVaR).
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https://link.springer.com/chapter/10.1007/978-0-387-77117-5_10
Although increasing the cultivation area always increases the mean of the total
production, it can sometimes increase the production risk.
The production risk also increases with the correlation of yields.
Markowitz’ mean-variance portfolio optimization
Introduction Allocation model Yield response model Case study

25. Sustainable agriculture model formulation
The “three pillars of sustainability”
• Environmental goal: Minimizing resource usage
• For the land resource,
• For the water resource,
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www.resources.org/archives/federal-climate-policy-toolkit-land-use-forestry-and-agriculture/
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26. • Social goal (𝑖 = 1) income inequality: ℎ1 𝑥, 𝑤 =
𝜈
2𝑚
(THB)
• Economic goal (𝑖 = 2) stdev of production: ℎ2 𝑥, 𝑤 = 𝑣𝑎𝑟 𝑄𝑇 (ton)
Normalization: For each goal 𝑖 = 1,2, let
• Environmental goal (𝑖 = 3)
• Land utilization
• Water utilization
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Sustainable agriculture model formulation
Multi-objective for sustainable resource allocation
Introduction Allocation model Yield response model Case study

27. • To formulate a multi-objective optimization, we use a weighted global criterion
method, one of the most common general scalarization methods, and it is
sufficient for Pareto optimality (Marler and Arora (2004) and Bokrantz and
Fredriksson (2017)).
• Let 𝜔𝑖 ∈ [0,1] be the weight of goal 𝑖 where 𝑖=1
4
𝜔𝑖 = 1
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Sustainable agriculture model formulation
Multi-objective for sustainable resource allocation
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28. Sustainable agriculture model analysis
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29. Sustainable agriculture model analysis
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30. Yield response model
FAO66’s Aquacrop model (Food and Agriculture
Organization, 2012)
where 𝑌𝑥 and 𝑌𝑎 are the maximum and actual yield,
ETx and ETa are the maximum and actual
evapotranspiration, and 𝐾𝑦 is crop coefficient.
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Introduction Allocation model Yield response model Case study
Note that 𝐾𝑦 > 1 implies that crop is very sensitive to water deficit;
1% in water deficit leads to more than 1% yield reduction.
Yield loss = 𝐾𝑦 × (Water deficit)

31. Yield response model
• Unconstrained yield loss
• Constrained yield loss
• The actual yield is
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32. Yield response model
• Dependent variable
yield loss 𝑔𝑖
• Independent variable
For Blaney-Criddle method, temperature is used for calculation of 𝐸𝑇𝑥,𝑖ℓ
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33. Yield response model
1. FAO model
2. Restricted linear model
3. Linear model
4. Probit model
5. Nonlinear probit
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34. Yield response model analysis
Theorem 5.
For a linear model, the yield response to water is
For a probit mode, the yield response to water is
where and
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35. Case study: Sugarcane production in Thailand
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36. Case study (con.)
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Average MAEs and RMSEs for the FAO and our proposed yield response models, across all 𝑚 = 14 locations
Slope and intercept parameters for yield response models

37. 37
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Spider chart of social, economic and environmental scores under different policies
𝜔1, 𝜔2, 𝜔3, 𝜔4 =
(0.3, 0.5, 0.1, 0.1)

38. 38
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39. Conclusions
• Summary
• Formulate the land-water resource allocation with social, economic and
environmental goals
• Yield ~ MVN
• Mean yield depends on irrigation water and rainfall in each month
• Extend FAO66’Aquacrop model to several yield loss models
• Extensions
• Site-specific management such as allocating fertilizers and resources for pest control
among all farms
• Multiple decision periods. In some settings such as contract farming, the field and
crop conditions (e.g., typical agronomic data and remote-sensing data) can be
monitored periodically
• For a strategic decision, we can employ a mechanism design model to improve the
efficiency of the entire supply chain by constructing an optimal contract
• Allow the agricultural prices and the outputs to be correlated
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