An inexact multi-stage interval-parameter partial information programming model for water resources management under uncertainties Open Access

The world is increasingly considering water shortages. Because of urbanization, climate change, water pollution and so on, many countries are facing water shortage, which is a major challenge to economic and social progress (Yannopoulos et al. 2019; WWAP 2019). Cities are particularly vulnerable to the impact of water scarcity, which is directly affected by rapid urban growth and reduced freshwater supply (Ray & Shaw 2018). By 2050, 66% of the world's population will live in cities (54% in 2014), increasing the water scarcity crisis (Jensen & Wu 2018). Water problems are common throughout China. About two-thirds of Chinese cities face significant water shortages, and groundwater in some areas is over-utilized (Yang et al. 2019). The sustainability of water resources is a prerequisite for sustainable urban development (Wang et al. 2019). Driven by population growth, socio-economic development, and ever-changing consumption patterns, the world's water demand continues to increase (WWAP 2016). Several important aspects of water shortage need to be emphasized. Firstly, the available water varies greatly from season to season. Therefore, the amount of available water throughout the year is different from the amount that water users can rely on when there is only one season (Water Resources Group 2008). Furthermore, groundwater is being drained. The cultivating, drinking, industrial and environmental uses mean that make water resources are extricated almost every single day (Wada et al. 2010). Thirdly, most countries have enough water to meet the development needs of population growth and maintain rivers needed to protect natural environment. The problem is that we have done a terrible job in managing water resources (Water Resources Group 2008).

In urban water resources management, lots of framework parameters and their interrelationships may bring uncertainty and complexity. The complexity of various framework components and economic penalties for violation of guarantee objectives may complicate this uncertainty. In addition, due to the diversity of water supply and demand in time and space, the plan required for water distribution may undergo greater changes. Therefore, feasible optimization strategies must be developed to support urban water management (Li et al. 2008a, 2008b; Fan et al. 2011).

In the past few decades, many optimization techniques have been used to solve these water management problems, including fuzzy, stochastic, chance-constrained and interval programming methods (abbreviated as FMP, SMP, CCP and IMP) (Charnes et al. 1971; Huang 1996; Wang 1997; Jairaj & Vedula 2000; Ben & Masri 2005; Guo & Huang 2009; Li et al. 2010). Among them, many researchers have tried to solve these uncertainties through two-stage and multi-stage stochastic programming methods (TSP and MSP). TSP has powerful functions for problems that need to be analyzed for policy scenarios and whose coefficients on the right-side are random but have a known probability distribution (Li et al. 2006). In TSP, a decision is first made before the value of the random variable is known. Then the ‘penalty’ can be minimized from a second-stage decision after a random event occurs and its value is known. A penalty can be generated if promised water (decided in the first stage) is not reached (Huang & Loucks 2000). Extensive research has been conducted on the TSP method. For example, Schultz et al. (1996) studied two-stage stochastic integer programming. Barbarosoǧlu & Arda (2004) proposed a two-stage probabilistic stochastic programming model to coordinate the transportation of critical first-aid items during emergency response to disaster areas. Huang & Loucks (2000) proposed an inaccurate two-stage stochastic programming (ITSP) model. It overcomes uncertainties where parameter information cannot be expressed as a probability density function (PDF). Recently, Maqsood et al. (2005) developed the interval parameter fuzzy two-stage stochastic planning (IFTSP) method, which integrates the ITSP model and fuzzy programming technique for water resources management under uncertainties. In fact, many practical large-scale problems involve a series of decisions that interact with periodic results over time. The above TSP method cannot effectively represent multiple dynamic changes over time, especially when there is a sequential interaction.

As an extension of dynamic stochastic optimization techniques, various multi-stage stochastic programming methods have been developed to address dynamic features in water resources management. MSP extends TSP by allowing management measures to be taken based on updated information at each time-stage (Pereira & Pinto 1991). Recently, inexact multi-stage stochastic programming (IMSP) has been developed to manage water resources under uncertainty. Li & Huang (2007) developed inexact multistage stochastic quadratic programming (IMQP) to support water resources management in Heshui River Basin, China.

These above-mentioned methods still cannot resolve the problem of when information is only known partially. In previous inexact optimization programming techniques, probability distribution information was defined as certain values. However, for many practical problems, the probability distribution may not be clearly determined. This means that probability information can only be partially measured (Fan et al. 2011). To solve such problems, Ben & Masri (2005) used the α-cut technique to transform the problem into a stochastic program that contains linear partial information on probability distribution. Nevertheless, previous studies had not considered using the integrated method of incorporating interval programming, multi-stage stochastic programming, and linear partial information theory into one general optimization system to deal with complex water resource management problems.

Therefore, as an extension of previous studies, the purpose of this research is to establish an inexact multi-stage interval-parameter partial information programming model (IMIPM) and apply it to urban water resources management. This is the first attempt in which interval-parameter programming (IPP), multi-stage stochastic programming (MSP), and linear partial information (LPI) theory are integrated into one general framework system. It can support the process of water resources management and planning under uncertainties and randomness. To prove the applicability of the developed model, we take the actual water resources management in Harbin, China, as an example. The results of modeling will help water resource managers to choose the ideal water resource allocation plan under various system conditions.

It is assumed that the water resource manager is responsible for allocating water from different water resources to multiple water users during multiple time stages. The manager can describe the problem as expanding the normal income of the water allocation system in the city. On the one hand, if the guaranteed amount of water is conveyed, at that point it will produce benefits to the nearby economy; if this is not the case, the water will have to come from other more expensive sources or users will need to diminish advancement plans, which will bring about financial punishments on the local economy.

subject to:

Step 1: In model (4), water demand targets (⁠⁠) are first-stage decision variables, which need to be known before the water flow levels (random variables) are defined. In this study, the values of are identified by having be decision variables and this could relate to maximizing system benefit. Let ⁠, where and ⁠. Thus, when approaches upper bounds (⁠⁠), if the promised water is fulfilled, it will generate high system profits; however, if the promised water is not provided, a high penalty will need to be paid. Conversely, when reaches lower bounds (⁠⁠), the profit of the system may be lower, but the risk of violating the promised goal is also very low, so there will be lower penalties.

Step 3: The solution for model (5) provides the extreme upper bounds of system benefit. Let ⁠, and be solutions of submodel 1. Then the optimized water-allocation targets are and we should add the constraint to submodel 2.

Harbin is located between 125°42′—130°10′ east longitude and 44°04′—46°40′ north latitude. It is a transportation, political, and financial center in the northeast of China. Harbin is very short of water resources, with average annual rainfall of 538.80 mm, and annual per capita water resources share of only 1,350.00 m³, accounting for about 1/2 of the national per capita level. The urban per capita share for water is only 218.00 m³. It is one of the cities with severe water shortage in northern China. In this case, it is particularly important to optimize the allocation of limited water resources.

The developed method is applied to water resources management in Harbin, aiming at allocating water from multiple water sources to multiple water users over multiple time periods, and therefore achieving multiple planning goals. There are four users, these being municipal, industrial, agricultural and environmental users. Agriculture uses 70% of the world's available fresh water (Li & Qian 2018). The total area of Harbin is 53 × 10³ km², and the area of arable land is about 2.00 × 10⁶hm². Thus, the agricultural water consumption is huge in Harbin. The crops are ripe once a year, and in the summer half-year (May–October), warm-temperature crops are grown. In the winter half-year (November–April of the following year), because the temperature is below 0°C, no crops can be planted. Therefore, the planning time periods are divided into two parts, the first is the winter half-year and the second is the summer half-year. Figure 2 shows the general framework and application of the IMIPM method, which is based on IPP, MSP and LPI techniques. Table 1 is a summary of the related economic parameters in this water distribution system.

In order to solve the uncertainties in water flow level, this paper selects three time intervals, which are 1987–1996, 1997–2006, 2007–2016, and divides water inflow level into low, medium and high levels, which mean rainfall is 270–440 mm, 440–600 mm, and above 600 mm, respectively. The hydrological characteristic of Harbin is that the annual rainfall mainly occurs in the summer half-year, and the winter half-year has extremely low temperature and much less rainfall. The proportion of rainfall in the winter half-year to the annual rainfall is small, so for Harbin, it is feasible to take one year as a hydrological cycle. Therefore, for every ten years, the frequency of low, medium, and high water inflow level is estimated as the probability of water inflow level of Harbin. Figure 3 shows the water inflow level frequency of Harbin. Thus, in this study case, the general partial distribution constraints (3h) to (3j) can be expressed as follows: ⁠.

Table 2 presents the results of extreme points obtained by the above constraints. According to the probabilities of three water inflow levels fluctuating within three different intervals, only four combinations of these three probabilities will satisfy the constraints, which is represented by 3D graphics in Figure 4. Table 3 shows the total available water with different water inflow levels during two planning time periods.

Table 4 shows the water allocation targets from multi-water source to multi-users during multi-time periods. Table 5 presents the water conveyance capacity of the pipeline, the minimum water demand and the maximum water demand for users during each time-stage. To handle the dynamic uncertainties in this problem, a decision tree based on the two time-stages is generated (shown in Figure 5). Each node is branched with three nodes for the following planning period when the probability of the current water inflow (⁠⁠) is ⁠.

Tables 6–10 show the results generated through IMIPM. Most non-zero decision variables and solutions are interval numbers. In other words, relevant decisions must be sensitive to uncertain modeling inputs. For example, the interval solutions of reflect variations of system conditions caused by uncertain inputs of ⁠, ⁠, ⁠, ⁠, ⁠, and ⁠.

As shown in Table 6, for scenario A, when water inflow level is low (0.1) and reaches its upper bound in period 1, only the agricultural users have about [4.54,3.51] × 10⁸m³ of water shortage. When reaches its lower bound, every user has a certain degree of water shortage. When water inflow level is medium and high (0.6, 0.3, respectively) in period 1, water quantity is increasing. There is no water deficit for municipal and industrial users. Ecological users have a small amount of surface and ground water shortage, [0,0.11] × 10⁸m³ and [0,0.05] × 10⁸m³ respectively.

As shown in Table 7, for scenario A, under the worst situation (joint probability = 1%), when the water inflow levels are low during the whole planning horizon, the total allocated water for the four users would be ⁠, indicating a serious shortage. There is a big chance that water inflow levels are both mediums in two periods compared with other situations, with the probability being 36%. In this situation, the surface water shortage for agricultural and environmental users would be [3.31,4.85] × 10⁸m³, [0,0.30] × 10⁸m³ respectively. There is no other ground water shortage except [4.32,6.43] × 10⁸m³ for the agricultural user. The total allocated water would be ⁠, indicating a significant shortage. Under the best situation (joint probability is 9%), when the water inflow levels are high during the whole planning horizon, the total allocated water would be ⁠, but ⁠, indicating that the water demands of the four users would generally be satisfied. The objective function value under scenario A is ¥ ⁠. It suggests that the system benefit would alter correspondingly from to as decision variables vary within their lower and upper bounds.

For scenario B, the water allocation scheme is the same as Scenario A's in period 1 as shown in Table 6. This means that, during period 1, if the water inflow probability of low flow level is 0.10, the changes of medium and high water flow level probability have no effect on the water distribution scheme. When the water inflow level is M-M during two periods, the joint probability is 49% (Table 8), and at the same time, the optimized water allocation to the four users is reduced to compared with scenario A. The probability of M-M water inflow level in scenario B is higher than it is in scenario A, however, the probability of M-H water inflow level in scenario B is lower than it is in scenario A. This indicates that greater water quantity would be available in scenario A than in scenario B during period 2 when the water inflow level is medium in period 1. The objective function value under scenario B is ¥ ⁠.

For scenario C, values of decision variable are the same as those in scenario A, B and D during period 1 (Table 6). However, there are different water allocation schemes in each scenario since they have different water inflow probabilities of low, medium, and high levels. As shown in Table 9, when the water inflow level is high in period 1, the water shortage for the four users in scenario C (⁠⁠) is much higher than it is in scenario A and B (⁠⁠). That is because the water inflow probability of low level is high while the water inflow probability of medium and high levels are low compared with scenario A and B. Compared with scenario A and B, optimized water allocation from surface and ground water to agricultural users is smaller in scenario C during period 2. This means that the available water in scenario C is less than in scenario A and B. The objective function value under scenario C is ¥ ⁠.

The water distribution plans are similar under the four scenarios during period 1. The reason is that in period 1, the urban water demand and available water are both low, and especially the agricultural users' water demand is drastically reduced, so that in order to firstly meet the basic water demand of various users there is no big difference in water allocation scheme during period 1. For scenario D, when the water flow level is L-L and M-L (the joint probabilities are 4% and 10% respectively, shown in Table 10), that is, when the available water is low or moderate during period 1, as long as the available water is low during period 2, the optimal water allocation of surface water to industrial users is always ⁠. It can be seen that in scenario D, there is no significant increase in even though the probability of H-H (10%) increases. The reason may be that 10% is still a small probability event to happen. Therefore, the model chose a conservative water allocation plan. The objective function value under scenario D is ¥ ⁠.

According to the results, water allocations are sensitive to and ⁠. Municipal and industrial uses should be ensured because they provide the highest benefit (⁠⁠) when their water demand is met, whereas they are subject to the highest penalty (⁠⁠) if the promised water is not achieved. In comparison with that, agricultural and environmental uses correspond to lower benefits (⁠⁠) and penalties (⁠⁠). Consequently, the allocation water to agricultural users should be diminished firstly in the case of inadequate water supplies, followed by that to environment users when necessary. We can get the final objective function value of ¥ from Tables 6–10. The possible reasons that the range of varies greatly are as follows: (a) the water resources parameter interval values change too much in this area; (b) although the penalties of per unit water shortage for agricultural users are low, the large amount of water shortage leads to excessive penalties. In all of the four scenarios, there are always users having a certain degree of water shortage. It can be seen that the contradiction between supply and demand of water resources in Harbin is outstanding. It is of great significance to study how to improve the efficiency of water resources utilization (such as optimizing industrial structure).

In this study, an inexact multi-stage interval-parameter partial information programming model (IMIPM) has been proposed for urban water resources planning and management under various uncertainties. The model is a first attempt to combine optimization techniques of interval-parameter programming (IPP), multistage stochastic programming (MSP), linear partial information (LPI) theory, and two-stage stochastic programming into one framework. Compared with the traditional models, the uncertain parameters and their interactions in an urban water resources management system could be analyzed better. In addition, an interactive algorithm is introduced to deal with this model. Firstly, it brings out IMSP models according to every extreme point of the LPI; then each IMSP model is solved via two deterministic submodels (Huang & Loucks 2000). Finally, we could get the interval solutions for supporting managers in having multiple decision alternatives.

The developed model has been applied to a real-world case study to verify the rationality and practicality of this method. The system parameters are expressed as interval numbers. By tackling partial probability distributions through the LPI method, we have four flow level probability scenarios. The winter half-year and summer half-year can reflect time dynamics characters. Therefore, the final Harbin water resources water distribution system benefit obtained through the IMIPM model is ¥ ⁠. It suggests that the system benefit would alter correspondingly from to when the manager chooses different water allocation schemes.

Even though this model can consider multiple uncertainties in the water resources management system, there are still many problems that need to be overcome in future studies. For example, it is necessary to take into account the water resource losses generated in the process of water resources extraction, transmission and distribution to increase the data accuracy of the model parameters.

The authors would like to express sincere thanks to the Water Conservancy Research Office of Northeast Agricultural University.

All relevant data are included in the paper or its Supplementary Information.