4.1. Optimal Irrigation Water Allocation
The NITM model can effectively tackle uncertainties expressed as discrete intervals and probability distributions. Optimal solutions can reflect a tradeoff between economic benefits and related pre-regulated policy targets, and reflect the effects of marginal utility between unit benefit and irrigation water amount. When water availabilities do not satisfy irrigation targets, it will lead to water shortages and reduction of economic benefits (i.e., penalties due to infeasibilities) and further system-failure risks. The obtained solutions are expressed as deterministic or interval values. Generally, solutions expressed as intervals show that the decision variables are sensitive to uncertainties in the model [20
]. The interval solutions can provide many decision solutions for a fluctuating range and further investigate the relationship between economic benefits and violations of irrigation target risks. More practically, the greater the width of upper bound and lower bound, the greater degree of uncertainty. The smaller the width of upper bound and lower bound, the more reliable and accurate decision-making becomes.
a–d show optimal solutions for water allocation targets of the NITM model based on the improved plan (similar results can be obtained from the original plan). This result shows the improved water allocation to avoid unreasonable solutions from the original plan. The NITM model is capable of analyzing various policy scenarios for different irrigation targets. Water shortages occur when water availabilities do not satisfy irrigation targets. From scenarios 1 to 4, due to reductions of total water consumption, promised irrigation targets and optimal water allocation targets are diminishing accordingly, causing system losses and system-failure risk of water shortages to decline as well. This result also implies that different irrigation targets corresponds to different water shortages or surpluses, thereby affecting economic benefits and system-failure risks. Moreover, the optimal solutions for OC (i.e., oil-bearing crops) in the GZ would be [350.4, 380.5] × 104
, [357.8, 376.1] × 104
, [359.2, 381.5] × 104
, [376.4, 391.9] × 104
, and [402.4, 420.6] × 104
under inflow levels from h
= 1 to h
= 5, respectively. The optimized water allocation for EC (i.e., economic crops) in the GZ would be [54938.0, 55983.9] × 104
, [54791.4, 56179.7] × 104
, [54971.8, 56216.4] × 104
, [55321.4, 56669.2] × 104
, and [55671.0, 57835.7] × 104
under inflow levels from h
= 1 to h
= 5, respectively. Their absolute values of difference between upper bound and lower bound are 1045.9, 1388.3, 1244.6, 1347.8, 2164.7 × 104
, respectively. More importantly, the amount of these irrigation waters is also important for agricultural development and decision making in arid/semiarid regions. Because the results of water allocation are presented as gross irrigation water, thus, according to the model solution in the GZ, i.e., the amount of irrigation water loss depending on the water conveyance efficiency is [18391.4, 19852.4] × 104
, the total amount of water loss is [25747.9, 27793.3] × 104
, and the amount of water loss depending on the irrigation efficiency is [6356.5, 7940.9] × 104
. Similar results can be obtained in other subareas, according to the water conveyance efficiency and irrigation efficiency.
Similarly, optimal solutions for EC in the GT would be [9783.2, 15922.7] × 104 m3, [9578.7, 16070.5] × 104 m3, [9830.4, 16098.2] × 104 m3, [10318.0, 16439.8] × 104 m3, and [10805.6, 17319.9] × 104 m3 under the inflow levels from h = 1 to h = 5, respectively. According to , optimal targets would be identified by decision variables . For example, in the initial scenario (S1), , indicating that the optimized irrigation targets reach their upper bounds and no water shortage should occur. This result also shows that planners hold positive attitudes to water availabilities for the EC in the GZ and LZ. Other decision variables () approach their lower bounds and express conservative attitudes. Under such circumstances, water shortages are expressed as . As the decision variables are in the range 0–1, thus the satisfaction level of irrigation water requirements for different crops can be measured from . Therefore, resourse decisions can obtain improved overall solutions by realigning the first-stage decision with possible realizations of random uncertainty. On the one hand, if the promised water is not delivered, a second-stage decision can be made to adapt to irrigation target and thus minimize the system penalties because of infeasibility. On the other hand, if the runoff inflow is at high level, then, it may cause more waste of irrigation water. This is because a conservative irrigation target lead to irrigation water cannot be fully utilized. Therefore, the results reflect tradeoffs between irrigation targets and random runoff inflow.
The results show that scenarios 2, 3, and 4 present the same change trend as that of scenario 1 under different inflow levels associated with the total water availabilities. However, among the four scenarios, solutions present different patterns of optimal water allocation for subarea i to crop j under a certain level. For example, under h = 1 level, optimized water allocation for EC in the GZ would be [54938.0, 55983.9] × 104 m3, [49189.9, 50432.6] × 104 m3, [44790.0, 45783.7] × 104 m3, and [41246.3, 41924.1] × 104 m3 in scenarios 1–4, respectively. Therefore, among four scenarios, variation trends for both lower and upper bounds under a certain inflow level are the same (scenario 1 > scenario 2 > scenario 3 > scenario 4). Rather, both upper and lower bounds of optimized water allocation present significantly decreasing trends. However, there are also increasing in the lower bound values while decreasing in the upper bound values in the four scenarios (i.e., GT-GC, LZ-EC). This result indicates that interval variations and mid-values of scenario 1 are larger than others, verifying the effects of marginal utility of the objective function. Moreover, there are also increasing lower and upper bound interval values in four scenarios (i.e., LZ-GC, LZ-OC, and GT-EC). When lower bound values of optimal irrigation water allocations are equal to upper bound values, the system arrives at a minimum marginal utility and approaches maximum system benefits.
4.2. Economic Benefits Analysis
presents a system economic benefits comparison based on the original plan and improved plan for different scenarios. Different agricultural irrigation management policies associated with various irrigation water targets and water availabilities bring about different system economic benefits. In terms of the improved plan and original plan, system benefits of the improved plan are greater than those of the original plan for different scenarios. The higher level of water availability permitted for right-hand side constraints generates a loose constraint and thus expands the decision space. The expected system economic benefits are denoted as interval values
, indicating that benefits can fluctuate over a certain range, and thus decision alternatives can be generated. When objection function values approach their lower bounds, fewer system benefits can result while water shortages are lower corresponding to a lower risk of promised target violations. Otherwise, a higher system benefit can be obtained when water demands are satisfied, but water shortage levels may increase, and thus greater system failure risks can result. Therefore, the results can reflect interrelationships among promised irrigation targets, system economic benefits and risk levels.
Take the improved plan as an example to describe system economic benefits for four scenarios. The gross system benefits from irrigation water would be [2.36, 4.75] × 109 Yuan, [2.25, 4.21] × 109 Yuan, [2.14, 3.77] × 109 Yuan, and [1.99, 3.39] × 109 Yuan, respectively. Because the promised irrigation targets are reducing from scenarios 1 to 4, fewer system benefits would be achieved associated with a lower risk of violating water allocation constraints. The above analyses show that different policies on irrigation targets will lead to different system economic benefits and risk levels.
The main difference between the original plan and improved plan pertains to surface water availability under different inflow levels, which gives rise to different total water availabilities. Therefore, irrigation targets are progressively lower from scenarios 1 to 4. Only the lower bound of the total available water volume under h
= 2 based on the IP is lower than that of the original plan, but in other cases, both lower and upper bounds of the total available water based on the improved plan are larger than those of the original plan. Figure 6
a–c present the results for optimal water allocation under different scenarios based on the improved plan and original plan.
The results obtained from the improved plan are greater than or equal to those of the original plan. For example, for h = 1, the total optimized water allocation is [79918.4, 100681.6] × 104 m3 and [78069.7, 98730.3] × 104 m3 in scenario 1, [80468.9, 100131.1] × 104 m3 and [78388.4, 98411.6] × 104 m3 in scenario 2, [82851.7, 97748.3] × 104 m3 and [80742.0, 96058.0] × 104 m3 in scenario 3, [84000.0, 96600.0] × 104 m3 and [82050.3, 94749.7] × 104 m3 in scenario 4. With the total water consumption reducing, the lower bound of total optimal water allocation increases, while the upper bound values decrease. This result indicates that the interval range becomes much narrower, demonstrating the existence of marginal utility in the objective function. This result can also provide many decision alternatives for further analysis on the relationship between economic benefits and violations of irrigation target risks. Therefore, the improved plan corresponds to a lower water shortage level than the original plan under the same inflow level, and thus leads to less system-failure risk.
For a certain crop (taking EC in GZ as an example), optimal water allocation would be [54938.0, 55983.9] × 104 m3 and [54723.7, 55751.4] × 104 m3 in scenario 1; [49189.9, 50432.6] × 104 m3 and [48975.6, 50203.3] × 104 m3 in scenario 2; [44790.0, 45783.7] × 104 m3 and [44536.9, 45593.3] × 104 m3 in scenario 3; [41246.3, 41924.1] × 104 m3 and [41188.7, 41880.5] × 104 m3 in scenario 4. This result indicates that the former has more total available water, thus expanding decision-making space on the right-hand side constraints. For scenarios 1 to 4, optimal solutions decrease both lower and upper bounds values in the GZ-EC. However, in the LZ-GC, GT-GC and GT-EC, optimized solutions gradually increase both lower and upper bound values.
Generally, if more water is provided for agricultural irrigation, it will improve the crop yield and thus increase system benefits. More importantly, for different scenarios, the rate of economic growth based on the improved plan and original plan are not the same. For example, in terms of upper bound of economic benefits, the rate of growth based on the improved plan between scenario 1 and scenario 2, scenario 2 and scenario 3, scenario 3and scenario 4 are 12.7%, 11.9% and 11.1%, respectively, which is greater than 11.8%, 11.2% and 10.4% based on the original plan. Therefore, the NITM model can reflect how to achieve greater economic benefits in the case of considering the increase of water availabilities.