Multistage Stochastic Programming to Support Water Allocation Decision-Making Process in Agriculture: A Literature Review

: Water represents a crucial resource to support agricultural production and the world’s rising food needs. However, the intervention of various factors intricates the proper water allocation, adding uncertainty and increasing risk in the decision-making process. Multistage stochastic programming (MSP) is a widely used programming technique for supporting water allocation problems governed by uncertainty. Therefore, this study performs a literature review on agricultural water allocation based on MSP, identifying crop yield as the principal farmers’ beneﬁts of proper water allocation, four main water allocation problem concerns, and four different uncertain sources. In addition, the study exhibits the advantages of multistage stochastic, interval, and fuzzy programming mixtures to provide better water allocation schemes.


Introduction
There is significant pressure for proper irrigation water management planning, since irrigated agriculture is currently the primary user of freshwater worldwide [1,2], and water represents a central input for crop production and agriculture development [3]. However, due to the increase in human activities and user demands, the availability of water resources quality and quantity has decreased [4,5], which causes conflicts between users in various locations worldwide [6][7][8]. Such situations induce multiple uncertainties that interact and lead to a complex water allocation and scheduling decision-making process. Inexact optimization techniques under uncertainty involve a set of strategies that allow one to face these problems on agricultural water allocation [9]. Multistage programming is a highly used technique that provides stage-structured decision-making schemes for supporting water decision-making based on scenario analysis, modelling uncertain parameters as random variables [10]. MSP establishes an optimization procedure comprising two or more stages. The first stage corresponds to crucial decisions at the beginning of the planning horizon. Other stages incorporate scenario-dependent decisions that let planning corrections reduce the system's total cost [11], allowing proper allocation schemes. Therefore, this work performs a literature review that discloses the primary considerations in water allocation in agriculture and supports a description of agricultural water allocation addressed through MSP, answering the following guiding questions:

1.
What are the implications of proper water resources allocation in improving farmers' benefits?
1. What are the implications of proper water resources allocation in improving farmers' benefits? 2. What are the main challenges faced in the water allocation decision-making process? 3. What are the main uncertain modelling strategies related to MSP?

Material and Methods
This study uses the Scopus and Web of Science databases, since they support exploring and selecting high-impact and peer-reviewed papers with extended coverage [12,13]. The search equation includes three layers. The first and second layers contain stochastic modelling and stage stochastic programming schemes. The third layer includes the study object. The equation avoids the agriculture term, due to its effect of about 78% reduction in the documents obtained. We used a hybrid methodology between Preferred Reporting Items for Systematic Reviews and Meta-Analyzes (PRISMA) and snowball sampling methodology for the final retrieved papers (Figure 1). On 8 October 2021 , the search equation retrieved 379 documents, identifying 62 article-type documents in the 2000-2021 timeline through the inclusion-exclusion criteria. The selected articles section follows a concept-matrix review [14], using the three guiding questions' answers.

Results
A broad description shows China as the country with the most agricultural water allocation research, probably due to the resource scarcity and the high demand from users in the region [15]. Environmental sciences, engineering, and agricultural sciences comprise the main subareas, grouping 66% of the works. The primary authors related to the problem of water allocation using the MSP technique are Huang G. H, Li Y. P, and Luocks D. P. In addition, Zhang et al. state a categorization for water decision-makers, considering their role in the allocation and their impact on the entire supply chain [16]. 1. The water managers decide the allocation between the primary water users in the region, i.e., industry, municipality, agriculture, and ecological users. 2. The reservoir managers must determine the proper water allocation regarding different zones, farms, districts. 3. The farmers must decide the water distribution strategy among different crops. If the farmers grow

Results
A broad description shows China as the country with the most agricultural water allocation research, probably due to the resource scarcity and the high demand from users in the region [15]. Environmental sciences, engineering, and agricultural sciences comprise the main subareas, grouping 66% of the works. The primary authors related to the problem of water allocation using the MSP technique are Huang G. H, Li Y. P, and Luocks D. P. In addition, Zhang et al. state a categorization for water decision-makers, considering their role in the allocation and their impact on the entire supply chain [16]. 1. The water managers decide the allocation between the primary water users in the region, i.e., industry, municipality, agriculture, and ecological users. 2. The reservoir managers must determine the proper water allocation regarding different zones, farms, districts. 3. The farmers must decide the water distribution strategy among different crops. If the farmers grow more than one crop, they might face a water allocation and scheduling problem, but they might face a water scheduling problem if they grow a single crop [17,18]. For simplicity, in Appendix A, Table A1 contains a summary of the retrieved works.

First Guiding-Question Answer
The decision-makers' dependency relationship for water allocation establishes a scheme where lower decision-makers' demands ascend to the first decision-maker, which must decide the water quota to fulfil all the requirements. However, each decision-maker seeks different objectives. The first decision-maker pursues the highest system maximum benefit and environmental sustainability. The second decision-maker protects all lowerlevel users' rights and supports distribution efficiency. Finally, the final decision-makers (farmers) focus on obtaining the most significant benefit possible. Therefore, farmers have the highest risk levels in the system [16]. There are few but significant effects of a proper water allocation plan on the farmer's benefits [9,[19][20][21], regarding that most of the works address the first and second-level decision-makers. Then, an appropriate water allocation of upper levels allows the farmer to achieve the following: 1. To satisfy his water demands primarily and crop yield goals. 2. To prioritize the most flexible crops with maximum net benefits through less water consumption. 3. To plan future production and address proper crop-pattern schemes. 4. To avoid excessive farmers' investments in irrigation and production systems, incurring high costs since a lower water release occur. 5. To promote water-saving and reuse processed water sources. Such implications ensure the agricultural region's development, support the farmer's benefits, maximize the system benefit, assure food, and conserve the natural resources available in the area.

Second Guiding-Question Answer
Multiple researchers agree that the main problems faced by decision-makers regarding water allocation vary and are inherent to the region where the problem occurs. However, the different case studies highlight the following four main problems: 1. the multiple users' water demands. 2. The available water resources scarcity. 3. The climate change effects. 4. The detriment of the quality of water sources to fulfil water users' needs [20,[22][23][24]. At the same time, these problems are associated to factors governed by uncertainty that are classified into four main classes. The hydrological factors relate to the water cycle and the availability of water resources. Climatic factors are associated with elements that characterize climatic weather. Socio-economic factors link the behaviour of prices and the social environment of the region. The productive factors are related to the productive capacities, production schemes, and decision-makers infrastructure. Hydrological and climatic factors represent the primary uncertain sources in the reviewed works ( Figure 2). Water flow levels from available sources are the main uncertain parameter. Such levels are strongly associated with climatic conditions [25,26], implying a critical importance of applying techniques for climatic conditions modelling, which then allows the decisionmaker to deduce the availability of future resources. Nevertheless, parameter modelling also lies in the volume of available data, the quality and reliability, and the vagueness and ambiguity [27,28]. Although in different magnitudes, all decision-makers must face these situations to generate proper water allocation plans.

Third Guiding-Question Answer
Proper allocation of water resources at the agricultural level presents complexness that requires careful treatment of the case studies' situations. The problem definition al-

Third Guiding-Question Answer
Proper allocation of water resources at the agricultural level presents complexness that requires careful treatment of the case studies' situations. The problem definition allows for the specifying of aspects of the modelling process as required data, the available strategies for parameters modelling, and the suitable types of mathematical programming for every case study. According to the uncertain parameter modelling strategies used, two programming strategies under uncertainty are linked to MSP, interval parameter programming (IPP) and fuzzy programming (FP). The IPP allows water resource allocation considering intervals to express inherent uncertainty, while FP uses the fuzzy set theory. Each optimization strategy relates to different application situations according to the most suitable method to tackle uncertain parameters. However, due to the complexity of water allocation systems, these techniques have been integrated, exploiting their benefits in reflecting the complexities and multiple uncertainties in the model, allowing for higher and more efficient water allocation schemes [29,30]. Additionally, there are difficulties, such as non-linearity behavior in the model [31] and the number of objectives to fulfil [32], which provide a more reasonably realistic model. Figure 3 summarizes all the optimization techniques used in the studies.

Third Guiding-Question Answer
Proper allocation of water resources at the agricultural level presents complexness that requires careful treatment of the case studies' situations. The problem definition allows for the specifying of aspects of the modelling process as required data, the available strategies for parameters modelling, and the suitable types of mathematical programming for every case study. According to the uncertain parameter modelling strategies used, two programming strategies under uncertainty are linked to MSP, interval parameter programming (IPP) and fuzzy programming (FP). The IPP allows water resource allocation considering intervals to express inherent uncertainty, while FP uses the fuzzy set theory. Each optimization strategy relates to different application situations according to the most suitable method to tackle uncertain parameters. However, due to the complexity of water allocation systems, these techniques have been integrated, exploiting their benefits in reflecting the complexities and multiple uncertainties in the model, allowing for higher and more efficient water allocation schemes [29,30]. Additionally, there are difficulties, such as non-linearity behavior in the model [31] and the number of objectives to fulfil [32], which provide a more reasonably realistic model. Figure 3 summarizes all the optimization techniques used in the studies.

Conclusions
This study addresses a literature review to identify the works that use MSP techniques for proper water allocation through an agricultural emphasis. The general findings disclose the complexity of water allocation processes in agriculture, the significant effects of adequate water allocation systems on farmers' benefits, and the matter of implementing

Conclusions
This study addresses a literature review to identify the works that use MSP techniques for proper water allocation through an agricultural emphasis. The general findings disclose the complexity of water allocation processes in agriculture, the significant effects of adequate water allocation systems on farmers' benefits, and the matter of implementing advanced modelling techniques that provide suitable water planning schemes. At the same time, the study allows the identification of a less frequent use of MSP techniques aimed at the final decision-maker, without considering the significance of supporting a proper allocation at the farm scale. Even if proper allocation represents reducing water needs at the upper levels, the decrease in errors on setting water requirements, and systems' penalty reductions, more studies should be carried out because this is not a topic under intense research. Therefore, future studies should evaluate the interactions in the crop production processes to define the water requirements, allowing one to scale reliable information to higher levels.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The Table A1 shows the summary of the selected works according to the decisionmaker (i.e., first, second, and third decision-maker), the type of study (Applied Case Studies-ACS and Hypothetical Case Studies-HCS), and the optimization strategy used. If the optimization strategy is mixed (mixed-1: researchers used MSP merged with interval or fuzzy techniques, mixed-2: researchers used MSP merged with interval and fuzzy techniques) or straightforward (i.e., only use MSP techniques).