Framework for Water and Food Security Assessment and Management in Arid Regions

A Framework for Water and Food Security Assessment and Management in Arid Regions
Supplemental file

Contents:

1. Section S1

S1.1 Development level and aridity by region

S1.2 National water resources development scenarios in Egypt

S1.3 Optimization method parameterization

2. Section S2

S2.1 Egypt’s historical cropping pattern

S2.2 Average global price scenarios for selected crops

S2.3 Regression models linking global and local prices
S2.4 NWFT model variables and constraints

3. Section S3

S3.1 Objective functions and filtering criteria for selected cropping patterns under future scenarios

S3.2 Uncertainty analysis for filtered alternative cropping patterns

Section S1

S1.1 Development level and aridity by region

Figure S1.1: Climatic areas in developed and developing countries; developed countries include countries in transition. Data for aridity are from FAO [2009], and the country development classification is from the UN [2017].

S1.2 National water resources development scenarios in Egypt

Table S1.1 Egypt’s water resources development scenarios until 2050 [Abdelkader et al., 2018].

S1.3 Optimization method parameterization

In this study, the ACPAR framework solves the national cropping pattern planning as a multi-objective optimization problem, where four objective functions were determined to reflect the conflicting interests of decision makers. In such problems, the optimization process does not result in a single “optimum” or “best” solution as the case of single-objective optimization, but a set of solutions known as Pareto optimal solutions (PF, Pareto optimal frontier). This set represents the tradeoffs that exist among the conflicted objectives, as it is not possible to improve one objective without degrading one or more other objectives. Determining this PF is the target of a multi-objective optimization process, which can be achieved by finding the group of solutions that have minimum (or maximum) values in all objectives compared to any other possible solution, assuming minimization (or maximization) problem. Although this might seem easy for trivial problems, it is not the case for complex and real-world problems, as it would be computationally intensive and inefficient to find all possible solutions until the true (global) PF is found. Rather, the multi-objective optimization methods seek high quality approximation to the PF that can be found with the least possible computations. In this regard, multi-objective evolutionary optimization algorithms (MOEAs) are found to be of great value and advantage [Goldberg, 1989]. Nonetheless, MOEAs require rigorous parameterization procedures to guarantee finding high quality approximation of the PF.  In this section, a brief generic description of the optimization method used in ACPAR framework is introduced, followed by an explanation of the parametrization exercise performed to insure its efficiency in finding high quality approximation to the PF.

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ACPAR framework incorporates an optimization method named Uniform Spacing Multi-objective Differential Evolution [USMDE; Chichakly and Eppstein, 2013]. The method simply depends on sampling and search techniques to explore the space of all possible solutions until it finds the PF. In doing so, the method uses a population of solutions that are simultaneously evaluated at each single run of the search algorithm [Storn & Price, 1997]. USMDE is similar to other MOEAs methods, in which they are all inspired by biological evolution processes like reproduction, mutation, recombination, and selection. Those evolution-like processes are performed in USMDE based on four general conceptual steps: (i) Initialization:  using Latin hypercube sampling (LHS) technique to assign initial solutions to the members of the population of size N; (ii) Mutation: expanding the search space by creating random changes in one or more members of the current population, yielding new solutions that might be better or worse than existing population members; (iii) Crossover or Recombination: inspired by the crossover of DNA strands that occurs in reproduction, the algorithm attempts to find better solutions by combining some of the current solutions; and (iv) Selection:inspired by natural evolution,the algorithm performs a selection process in which the ‘fit’ members of the population survive and contribute to the PF, and the ‘least fit’ members are partially eliminated, except few  that are kept in the PF as they might lead to better solutions in following steps. This selection process is performed such that it keeps the size of the PF less than or equal to the population size N throughout the search procedures. To reach a final PF, the three steps of Mutation, Recombination, and selection are repeated iteratively to improve the PF quality, until a stopping criterion is met, which is usually maximum number of iterations (i.e. generations).

The USMDE method has five major parameters that govern these four conceptual steps, namely: the seed number (S), which controls the initial values of the population in the initialization step; Population size (N), which determines the size of the PF and the count of search zones to be explored throughout the four steps; Scaling factor (F), which controls how far the search would expand and cover new zones in the mutation process; Recombination Probability (Cr), which controls the diversity of solutions selected for recombination process; and Maximum number of iterations (Gmax), which causes the algorithm to stop. The values of those parameters should be selected carefully, as they influence the search capabilities and deepness of exploration of USMDE of the possible solutions, and thus, eventually affect the overall quality of the reached PF.

In this study, to ensure that USMDE was used efficiently in the ACPAR framework, a diagnostic assessment framework, which is originally used to evaluate and compare different MOEAs, was adopted to find the best parameters set [Reed et al., 2013]. In this parameterization exercise, there are six steps that were performed: (a) Latin hypercube sampling: to create 1,000 parameters value combinations for the USMDE’s parameters of (N, F, Cr, and Gmax) considering their full range indicated in Table S1.2; (b) Running USMDE for the case study of Egypt under each of those 1000 generated samples and finding the PF for each of them; (c) Considering the influence of the randomness of initial population on the produced PFs by generating random 50 seed numbers and rerunning each of the 1000 samples under those 50 seed values to produce their PFs (i.e. 50,000 PFs in total); (d) Evaluating each of the produced 50,000 PFs by comparing them with a reference PF (RPF) using three evaluation metrics (explained later); (e) Summarizing the evaluation results for the 50,000 PFs and building expressive figures named control maps (explained later); and finally (f) Selecting the best parameters set guided by those control maps.

In the parameterization of MOEAs, the identification of a reference PF (RPF) is an essential step, as this RPF acts as a benchmark for the highest possible quality a PF can achieve. Nonetheless, identifying this RPF is a real challenge for a complex problem like the one using the ACPAR framework, as the true (global) PF that acts as a RPF is unknown. Alternatively, the best known PF is usually considered as the RPF [Reed et al., 2013], which is followed in this study, as we considered the best PF of the generated 50,000 PFs to be the RPF(see Figure S1.2). This best PF happened to be the one with the highest computational demand. In parameterization exercise, our goal is to select the parameters set that yields a PF of a quality that approaches the RPF’s quality with the least computational demand possible. Although many metrics exist to evaluate the PF’s quality, they eventually measure one or more of three main characteristics, namely: (a) convergence, which refers to the proximity of the solutions of the PF to those of the RPF; (b) consistency, which refers to the degree of coverage of  the PF to all the zones existing in the RPF (i.e. express the existence of  any gaps in the PF); and (c) diversity, which refers to the degree of extent of the PF to represent the full range of tradeoffs as represented by the RPF. In this study, we used three different evaluation metrics that are extensively used and recommended in the literature. The generational distance (IG) mainly measures convergence (the lower, the better); it is estimated by averaging the Euclidian distance between each PF solution and its nearest neighbor RPF solution over all the solutions of a PF [Van Veldhuizen and Lamont, 1998]. The additive ɛ-indicator (Iɛ) is a good measure for the consistency (the lower, the better), it expresses the existence of gaps by estimating the largest distance required by any PF solution to dominate (i.e. be better than) its nearest neighbor in the RPF solution [Zitzler et al., 2002]. The last metric is the hypervolume (Hv), which is an overall measure of convergence and diversity but less sensitive to consistency changes (the higher, the better). Hv is estimated as the volume of the objective space dominated by a PF relative to that dominated by RPF; the more dominance reflects more convergence and diversity [Zitzler et al., 2002]. The Hv is the most challenging metric to estimate, especially for multi-objective problems, as in our case study.  However, there are some methods that approximate the Hv calculations and evaluate it with good accuracy for such high dimensional cases; the HypE is one of those methods that was used in this exercise for this purpose [Bader and Zitzler, 2011].

In the parametrization exercise performed in this study, each of the 1000 parameter set samples was evaluated using the three evaluation metrics mentioned above, and repeated for 50 random seeding for each sample. To summarize the huge results of this evaluation, there are figures known as control maps that are typically used [see Figure S1.3; Reed et al., 2013]. A control map provides information about the 1000 parameter sample values, their evaluation metric values averaged over the 50 random seeds. It also indicates the number of function evaluations (NFE), which reflects the computational demand corresponding to each parameter set, and estimated by the multiplication of the population size (N) by the maximum number of iterations (Gmax), it acts as an effectiveness measure because it implicitly reflects the time required to perform the optimization procedure. In those maps, the population size (N) is used as a proxy for the parameter set samples, as this is more convenient and consistent with the literature [Salazar et al., 2016], where generally, the quality of a PF is mainly driven by the population size (N). The purpose of a control map is to illustrate the “sweet spots” in the parameter space that yield PF of high quality [Goldberg, 2002].

Figure S1.3 shows that this “sweet spot” for the three evaluation metrics exists for a wide range of parameters sets. Figure S1.3a reflects the ability of a PF to converge, and generally, it can be noticed that the majority of parameter sets of N > 100 tend to converge with the RPF.  In contrast, the consistency in USMDE application for ACPAR seems to be more governed by NFE than N, as indicated in figure S1.3b. Generally, any NFE less than 40,000 would yield consistent PF compared to RPF. The hypervolume is the most expressive evaluation metric and most commonly used for PF evaluation, thus, it would be decisive in selecting parameter sets as figure S1.3c shows the majority of parameter sets of N > 200 tend to have Hv value > 0.8, which reflects a high degree of convergence and diversity compared to the RPF. In conclusion, the best parameter set that fulfills the three evaluation metrics and produces high quality PF should has N > 200 and NFE < 40000, which was followed in this study. The N was taken as 200 and the Gmax was taken as 150 (i.e. NFE = 200*150 = 30000). For the Cr and F parameters, it was found that their sweet spot range is [0.05- 0.2] and [0.3- 0.65], respectively. Thus, Cr was assigned a value of 0.2 and F a value of 0.6.

Table S1.2.  USMDE parameters’ range used in the Latin Hypercube sampling.

* Seed number (S) changed 50 times then combined with each of the 1000 samples generated for (N, F, Cr, and Gmax).

Figure S1.2: 3D plot for the best known approximate Pareto optimal frontier produced by ACPAR, which is considered as the reference Pareto optimal frontier. Each of the plotted points represents a normalized Pareto optimal solution that is described by four values ranges between 0 and 1. The four objective functions are Agriculture gross margin (GM), Virtual water imports (VWI), Water demand for agriculture (WDA), and Economic costs of import (ECI), respectively.

(a)   Generational Distance (IG)

(b)    Additive ɛ-indicator (Iɛ)

(c) Hypervolume (Hv)
 

Figure S1.3: Control maps showing the average evaluation metrics of: (a) Generational distance, (b) Additive ɛ-indicator, and (c) Hypervolume, averaged over 50 seeding for 1000 parameter samples. The horizontal axis represents the population size (N) as an indicator of the parameter sets and the vertical axis represents the number of function evaluations (NFE) to reflect the computational demand, and the colors reflect the corresponding averaged evaluation metric value. The hypervolume is calculated relative to the maximum value that belongs to the reference Pareto frontier (RPF) (i.e. hypervolume of RPF = 1).

Section S2

S2.1 Egypt’s Historical Cropping Pattern

Table S2.1: Major crops cultivated in Egypt and the decadal average cropping pattern during the period between 1986 and 2013.

* Total agricultural area is larger than the physical land area, as the same land is cultivated more than one time each year.

S2.2 Average global price scenarios for selected crops:

(a) Cereals

(b) Oil crops

(c) Roots

(d) Pulses

(e) Fruits and Vegetables

Figure S2.1: The average global price scenarios of selected crops. NoCC means no climate change is considered in this scenario. GFDL, HGEM, IPSL, MIROC are all climate change scenarios. See Robinson et al. [2015] for more information.

S2.3 Regression models linking global and local prices for selected crops

Figure S2.2: Regression models linking global and local crop prices for selected crops, during the baseline period (1986-2013).

S2.4 NWFT model variables and constrains

Table S2.2: Key variables in NWFT model and their sources [Abdelkader et al., 2018].

Land, water, trade, and cropping pattern are variables that are constrained in the NWFT model. For every year of simulation, the total area of the cropping pattern is not allowed to exceed the land available for agriculture. The water allocated for all uses in a given year (t) (i.e. industrial, municipal, and agriculture) should not exceed the national water resources available from all sources in the same year. For municipal and industrial sectors, water supply should equal exactly the demand. Agriculture water supply cannot exceed the agriculture water demand but can be less. The trade constraints guarantee that the import and export of food are according to food shortages and surpluses, so national food balance remains valid for every year for each crop. The cropping pattern non-negativity constraint assures that none of the decision variables has negative value. The equations of these constraints are stated below:

Land: $\sum _{i=1}^n\mathrm{xi}*\mathrm{A}\left(\mathrm{t}\right)$

= $\mathrm{A}\left(\mathrm{t}\right)$

                                                                         ∀ t                      (1)

Water: WSA(t)+WSM(t)+WSI(t) = WDA(t)+WDM(t)+WDI(t)                  ∀ t      (2a)

            WSA(t) ≤ WDA(t), WSM(t) = WDM(t), WSI(t) =WDI(t)               ∀ t                 ( 2b,c,d)      

Trade: PROD i (t) + IMP i(t) = EXP i(t)+ CONS i(t)+ LOSS i(t)+ ∆S i(t)   ∀ t, i                   (3)

Non-Negativity:   xi ≥ 0                                       ∀ i                      (4)

Where,

A(t): the national land available for agriculture in a given year t

WSA(t), WDA(t): national water supply, and demand for agriculture in a given year t

WSM(t), WDM (t): national water supply, and demand form municipal in a given year t

WSI(t), WDI (t): national water supply, and demand for industry in a given year t

PROD i (t): national food production for crop i in a given year t

CONS i (t): national food consumption for crop i in a given year t

LOSS i (t): national food losses for crop i in a given year t

IMP_i(t): is the national imported quantity of crop i in year t (tonnes)
EXP_i(t) is the national exported quantity of crop i in year t (tonnes)
∆S i (t): national food stock variations for crop i in a given year t
xi: Decision variables (i.e. each crop ratio of the national land available for agriculture)

Section S3

S3.1 Objective functions and filtering criteria for selected cropping patterns under future scenarios

Table S3.1: Projected values of objective functions and filtering criteria for selected solutions under future scenarios.

* Values are averaged over the four price scenarios.

S3.2 Uncertainty analysis for filtered alternative cropping patterns

In the ACPAR framework there are some important variables like crop yield and crop losses that were not considered to change under future scenarios, which might impact the conclusions made for future planning as presented in section 4.2.2 in the paper. Hence, here we introduced uncertainty analysis to address this issue. In this analysis, the 19 filtered ACPs were reevaluated along with the historical cropping pattern (HCP), in which the uncertainty in their objective functions and filtering criteria were estimated based on the uncertainty in ACPAR variables. We considered the range of variables stated in the national scenarios (Table S1.1) to be wide enough to address the uncertainty of future national development states (except in neglecting crop yields and crop losses changes), likewise the range of global price scenarios (i.e. SSP1, SSP2, SSP3, and SSP2-HGEM). Hence, those ranges are sampled using Latin hypercube sampling, along with the ranges of crop yields and crop losses indicated in Table S3.2. A total of 1200 samples are generated that were then used to produce the uncertainty bounds represented in Figure S3.

Table S3.2: Uncertainty ranges for possible future changes of crop yield and crop losses.

* 2050 multiplication factor is a factor used to generate the crop yield and crop losses values of year 2050 by multiplying it with the value of 2013, then linearly interpolate the values in the period in between.

** The minimum and maximum possible future crop yield multiplication factors for different crops were compiled from [Jaggard et al., 2010].

Figure S3: Uncertainty bounds for the 19 filtered alternative cropping patterns (ACPs) and the historical cropping pattern (HCP), for the four objective functions of (a) Agriculture gross margin (GM), (b) Virtual water import, (c) Water demand for agriculture (WDA), (d) Economic costs of import (ECI), and the filtering criteria of (e) National food self-sufficiency (NFSS).

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