3.2.6. Objective Function
(1) Social Benefit Target
Based on the principle of making water use a priority for domestic and ecological demand, the efficiency coefficient for domestic and ecological water use was assigned a large value. The industrial benefit coefficient was allocated using the gross industrial output value method [
34]. The other benefit coefficients relate to social, ecological, economic, and other factors, and were assigned a proportion of the industrial efficiency coefficient [
35]. The details of the coefficients are shown in
Table 4.
(2) Economic Benefit Target
In Tianjin, the price of domestic water is currently 4.9 yuan/m³, the price of industrial water is 7.85 yuan/m³, and the price of agricultural water is 0.21 yuan/m³. Assigning a price to the cost of ecological environment water is not straightforward. In most cases, the value of ecological water derives from public welfare as determined by government, rather than economic activity. Therefore, the direct economic benefits are small, and this study assumes a small value of 0.1 yuan/m³.
(3) Water Supply Cost Target
Local water resources are limited, and the cost of external water is high. Therefore, only the water supply cost of the external water is calculated. The price of raw Luanhe River water is currently 0.91 yuan/m³, and the price of raw River water is currently 2.16 yuan/m³.
3.2.8. Calculation Results and Analysis
(1) Analysis of Minimum Water Shortage
Figure 4 shows a graph of Tianjin’s minimum water shortage in 2020 for different water supply source schemes. It can be seen that, with the increase in the amount of Luanhe River water and River water supplied, the total water shortage is gradually reduced, and the minimum water shortage stabilizes at 2.82 × 10
8 m³. Based on a simple water supply and demand balance, Tianjin’s total water supply (35.98 × 10
8 m³) will have exceeded the predicted water demand (35.43 × 10
8 m³) when the cumulative amount of water from the Luanhe River water and River water exceeds 18 × 10
8 m³. However, the water demand has not been completely met, which shows that Tianjin’s water shortage is not a simple matter of water resource shortages.
A scheme was chosen with 7.5 × 10
8 m³ of Luanhe River water available at a 75% guarantee rate, and 12 × 10
8 m³ from River water available at an 80% guarantee rate. An analysis of the water distribution for water users in this scheme revealed that water shortages for agricultural users occurred at Baodi, Wuqing, Ninghe, and Beibu (
Table 5). An analysis of the capacity of the water pipeline in these four areas revealed that the maximum water transmission capacity has been reached. However, the two rules for external water supply allocation allow surplus water from River water to be available to meet all the water requirements of the users (
Table 6). Thus, the water shortages in Tianjin under the scheme chosen are the result of infrastructure limitations, rather than a lack of available water. In the four regions of Baodi, Wuqing, Ninghe, and Beibu, the water infrastructure connecting to the River water supplies is not adequate. At the same time, the pipelines’ capacity from the Luanhe River water to these four regions are limited, and cannot meet the requirements of all water users.
The water supply distribution and trends are shown in
Figure 5. Surface water, Luanhe River water, and River water make up the bulk of water supplies, with the largest amount of water being supplied by River water. Domestic water is mainly provided by Luanhe River water and River water. Industrial water supplies are more evenly spread among the different water sources, with River water providing the highest proportion. Agricultural water is mainly derived from local surface water, while ecological water is mainly provided by reclaimed water. The biggest users of external water are Zhongxinchengqu, followed by Binhaixinque. The consumption of external water in the suburban areas is very small, with only Jinghai suburban areas supplied by River water. The reason for this is that there is no water transmission infrastructure to distribute water from River water supply in areas other than Jinghai. This finding is consistent with the earlier findings regarding the effects of a lack of pipeline infrastructure in these areas. An analysis of the use of water sources in
Table 6 indicates that water shortages could be caused by both lack of water infrastructure capacity and lack of external water in Tianjin when there are insufficient supplies of Luanhe River water and River water. Therefore, it is necessary to either expand the maximum water transmission capacity of the pipeline from the Luanhe River to Tianjin, or add water transmission infrastructure in the water shortage areas so that more water from River water can be utilized. This will enable supplies to be augmented, while ensuring the stability of external water supply sources.
(2) Analysis of the Cost of Water Supply
All the schemes that meet the minimum water shortage of 2.82 × 10
8 m³ will have reached the minimum water requirement of the River water (greater than 4 × 10
8 m
3), irrespective of the two priority rules for external water. Therefore, the cost of water supply in Tianjin is not affected by the priority rules for external water. The cost of water supply for all the schemes (with water shortages of 2.82 × 10
8 m
3) was calculated by analyzing the relationship between the cost of water supply and two external water supplies. As can be seen from
Figure 6, the maximum cost of water supply is 29.53 × 10
8 yuan, with the amount of water supplied by the Luanhe River water and River water being 5.32 × 10
8 m³ and 11.43 × 10
8 m³, respectively. The minimum cost of water supply is 24.86 × 10
8 yuan, with the amount of water supplied by the Luanhe River and River water being 9.06 × 10
8 m³ and 7.69 × 10
8 m³, respectively. With the decrease in water from River water and the increase in water from the Luanhe River water, the cost of water gradually decreases. The reason for this is that the price of River water is higher than that of the Luanhe River water. When the amount of River water increases, the cost of water supply will inevitably increase. Therefore, the cost of water supply can be effectively controlled by setting a limit for River water.
(3) Competition between Social and Economic Benefit Targets
The main conflict between social benefit and economic benefit targets arises from the different water resource priorities of different water users. Domestic water demand is the highest priority of the two targets and, thus, water competition can be avoided. The remaining three water users have different priorities for the two targets. When the total available water cannot meet the total water demand, there will be strong competition for water. To more directly reflect the competition relationship between social and economic benefit targets, the schemes with 7.5 × 10
8 m³ of Luanhe River water at a 75% guarantee rate were selected for analysis. As can be seen from
Figure 7a, when the available amount of River water is low, and the social benefit target is optimal, the economic benefit is obviously lower than that when the economic benefit target is optimal. As the amount of River water increases, the gap between the two benefits gradually narrows and, at a minimum water shortage of 2.82 × 10
8 m
3, the benefits are almost same. The difference in values is caused by the priority of the water supply. The benefit coefficient of the two targets is different, resulting in different final values. Similarly, as shown in
Figure 7b, the gap between the social benefit when the economic benefit target is optimal and social benefit when the social benefit target is optimal gradually narrows as the amount of River water is increased, and the final trends are the same. In conclusion, when the total water supply to Tianjin cannot meet all the water requirements, the water supply priority of different water users will have a large impact on the social and economic benefit targets, especially when the water shortage is large. Therefore, reasonable setting of water supply priorities and water supply guarantee rates for water users can effectively maintain the balanced development of social and economic benefits.
(4) Trend Analysis of Objective Function and Water Shortage
The schemes with 7.5 × 10
8 m³ of diverted Luanhe River water, available at a 75% guarantee rate, were selected for analyzing the relationships between water shortage and social benefit, economic benefit, water supply cost, and comprehensive targets. As can be seen from
Figure 8, with a decrease in water shortage, all four target values show an increasing trend. When the water shortage is in the range of 2.82–4.57 × 10
8 m
3, the change trend for economic benefits and comprehensive targets is relatively smooth, and the change range for social benefits and water supply cost targets is large. The comprehensive target was obtained by weighted calculations after the standard deviation treatment of the four objective functions. The gentle trend within the 2.82–4.57 × 10
8 m³ water shortage range is mainly affected by the economic benefit target, and the overall trend increases as the water shortages decrease. It can be seen from
Figure 8d that there is an inflection point when the water shortage is 4.57 × 10
8 m
3. The water shortage on the left side of the inflection point is small, and the change trend is gentle, but the cost of water supply is high. The water shortage on the right side of the inflection point is large, and the change trend is obvious. Therefore, it is recommended that a water supply scheme on the left of the inflection point is chosen. If the cost of water supply is sufficient, it is recommended that a water supply scheme with a minimum water shortage of 2.82 × 10
8 m³ is chosen to meet the water demand of the region as far as possible. At this time, the diverted amounts of Luanhe River water and River water will be 7.5 × 10
8 m³ and 9.25 × 10
8 m³, respectively, and the cost of water supplied is 26.81 × 10
8 yuan. If the cost of water needs to be controlled, it is recommended that a water supply scheme in the middle of the 2.82–4.57 × 10
8 m³ water shortage range be chosen, such as the scheme with the water shortage at 3.32 × 10
8 m³. At this time, the diverted amounts of Luanhe River water and River water will be 7.5 × 10
8 m³ and 8.5 × 10
8 m³, respectively, and the cost of the water supplied will be 25.19 × 10
8 yuan.