*3.1. Overview of Handan City*

The city of Handan is located in the southernmost part of Hebei Province, China, at the eastern foot of Taihang Mountain, bordering Shandong in the east, Henan in the south, Shanxi Province in the west and Xingtai City in the north. Its jurisdiction covers 6 districts, 1 county-level city and 11 counties. Its geographical location ranges 36◦040~37◦010 N and, 113◦280~115◦280 E with warm temperate semi-humid and semi-arid continental monsoon climate. The location of the area is shown in Figure 1.

**Figure 1.** Location of Handan city. **Figure 1.** Location of Handan city.

*3.2. Application of FIDP Model.*  In order to primely solve the problems mentioned above, FIDP is applied to optimize the allocation of water resources in Handan city. In detail, the established FIDP model would not only considers multiple objectives, such as the maximum economic benefit, the maximum overall satisfaction of water users, and the maximum environmental benefit, but also take the satisfaction of each water users into account. Meanwhile, the constraints would refer to the water supply capacity, the minimum guaranteed water demand, the ideal water demand, the water delivery capacity, and the COD emission limit. In addition, the uncertain factors involved in this model (e.g., water use benefit coefficient, ideal water demand, minimum guaranteed water demand, weight coefficient, COD discharge coefficient, maximum COD discharge, available water supply, water inflow at different stages, and water delivery capacity) can be expressed as interval parameters. Moreover, the dynamic factors in the process of water resources optimization, such as the water users' ideal water demand, guaranteed water demand, available water supply and water allocation changing with the stage, would be reflected by dynamic programming. The frame diagram of constructed FIDP model can be seen in Figure 2. In order to facilitate managers to make decisions, each stage is divided equally by the planning year, in which, January-March is the first stage, April-June is the second stage, the third stage is from July to Sep-At present, water resource managers in Handan are facing with many water resource problems, such as water resource shortage, uneven distribution of precipitation, and serious water pollution. For example, according to the Water Resources Bulletin [46], the per capita water consumption of the city in 2019 is 2.02 <sup>×</sup> <sup>10</sup><sup>2</sup> <sup>m</sup><sup>3</sup> with a population of 9.55 million. However, the water supply in 2019 is only 1927.84 <sup>×</sup> <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> , and the water shortage is 1.26 <sup>×</sup> <sup>10</sup><sup>6</sup> <sup>m</sup><sup>3</sup> . In addition, 61.30%~76.50% of the annual precipitation falls between June and September, which is extremely inconsistent with the needs from various water users. Actually, each user's water demand, especially the agricultural water demand, is different with the season changes. The growing period of crops in Handan mainly ranges from March to August, with the largest water demand occurring at the second stage which would account for about 50% of the annual water consumption. It is noted that the development of agricultural cultivation is paid the most attention in Handan City, and its water consumption accounts for about 55% of the total water consumption. So how to provide periodic water allocation for each user is a challenge for managers. Moreover, due to the uncertainties existing in water supply and the temporal variations of the planning horizon, the water resources system also has a number of uncertain factors, such as the water inflows at different stages, water efficiency, water demand, and pollutant discharge, which should be fully considered. Therefore, how to allocate water resources reasonably to ensure the sustainable development of this region is an urgent problem for

The goal of this model is to maximize its membership function:

*Max* <sup>±</sup> (5a)

Economic benefit constraint (maximize the economic benefit):

3.2.1. Objective Functions

λ

3.2.2. Constraints

managers to solve under condition of discordant water supply and demand, as well as various uncertain factors. where <sup>±</sup> *Cti* is the inflow of water source *i* in stage *t* (m3). Water transporting capacity constraint:

*SS C a x* (5g)

±± ± ± − −− = =

16 4 ( 1) ( 1) ( 1) 1 1 *ti t i ti t ijk t ijk j k*

= +−

#### *3.2. Application of FIDP Model* The total amount of water used in each region would be limited by the water trans-

In order to primely solve the problems mentioned above, FIDP is applied to optimize the allocation of water resources in Handan city. In detail, the established FIDP model would not only considers multiple objectives, such as the maximum economic benefit, the maximum overall satisfaction of water users, and the maximum environmental benefit, but also take the satisfaction of each water users into account. Meanwhile, the constraints would refer to the water supply capacity, the minimum guaranteed water demand, the ideal water demand, the water delivery capacity, and the COD emission limit. In addition, the uncertain factors involved in this model (e.g., water use benefit coefficient, ideal water demand, minimum guaranteed water demand, weight coefficient, COD discharge coefficient, maximum COD discharge, available water supply, water inflow at different stages, and water delivery capacity) can be expressed as interval parameters. Moreover, the dynamic factors in the process of water resources optimization, such as the water users' ideal water demand, guaranteed water demand, available water supply and water allocation changing with the stage, would be reflected by dynamic programming. The frame diagram of constructed FIDP model can be seen in Figure 2. In order to facilitate managers to make decisions, each stage is divided equally by the planning year, in which, January-March is the first stage, April-June is the second stage, the third stage is from July to September, and the fourth stage is from October to December. Its formulation would be expressed in the following form: porting capacity in the region. ± ± = <sup>≤</sup> 4 1 *tijk tij k x Q* (5h) where <sup>±</sup> *<sup>Q</sup> tij* denotes the maximum capacity of water source *i* transporting to the region *j* in stage *t* (m3). The COD emission constraint: Due to serious damages of human activities to the ecological environment in recent years, more and more managers begin to pay attention to the impact of ecological environment with the development of economy. Accordingly, each region has formulated the discharge capacity of pollutant COD to control environmental pollution. Therefore, the optimal allocation of water resources should meet this requirement. ±± ± == = <sup>≤</sup> 444 111 *tjk tijk j tik dx F* (5i) where <sup>±</sup> *<sup>j</sup> F* is the rated of COD emission in region *j* (kg). Nonnegative constrains: <sup>±</sup> ≥ 0 *tijk x* (5g) <sup>±</sup> = <sup>0</sup> 0 *<sup>i</sup> S* (5k) λ<sup>±</sup> 1 0 ≥ ≥ (5l)

**Figure 2.** Framework of the fuzzy-interval dynamic programming (FIDP) model.
