The linear programming model of water resource utilizationin the irrigated area of Huaibei
Zhenlong Wang
(Institute of Water Resource of Huai River Committee and Anhui Province, Bengbu)
ABSTRACT: This paper applies the theory of system engineering to Weidian irrigated area in Huaibei
in order to get the maximum net profit. Restricted by the available water both of surface and ofgroundwater, the water quantity which is needed by crops, the ability that well yields, the pumping
ability of pump, and the requirement of waterlogging prevention, the better planting pattern which isbased on annual water supply and distribution of surface water and ground water are put forward, andthd optimal from of irrigation is provided as well. All these make scientific plan and management of
the irrigated area possible.
1 INTRODUCTION
The total infield of Huaibei area in Anhui province is 2.067* 106hm2, and the cultivatable area, which is
the important base of grain and cotton, is 50.6% of the infield of Anhui province. In recent years, the increasing
population, the increasing industrial and agricultural water demand, and the bad-distributed water resource leadto the conflict between the agricultural supply and demand. Especially in dry year, the groundwater level
decreases, and the cost of irrigation increases. Severely, the pumps are left unused, and the benefit of irrigating
engineer cannot be achieved. In order to get the maximum profit of irrigation, the problems of Huaibei Area thatneed to be solved exigently are how to distribute the resource of surface water and ground water, how to arrange
the layout of crops and how to choose the best system of water-saving irrigating engineer.

2 THE LINEAR PRIGRAMMING MODEL
2.1 The general situation of drainage and irrigation in irrigated area
The irrigated area of Weidian is 55km2,equal to 4064.5hm2, and the main crops are winter wheat, cole,summer corn, bean, cotton, peanut, poi, and a little bit of paddy. The re-growing index is 1.85. In order to make afull use of the irrigation resource, integrated drainage systems are built in irrigated areas according to thetechniques of water-logging prevention and soil prohibition. The water-transporting system ( including thetrenches and PVC pipelines), pumping-well (including the shallow well, small well, small available well),reserve clough and other water resource reserving projects are also built, In recent years, as the mobile spray andsemi-fixed spray are developed, the integrated system of draining, irrigating, managing and using the irrigation
resource comes into being. The shallow groundwater and channel-regulating water are used as the main
resources for irrigation.

2.2 The comprehensive consideration of building mathematical model(1) The tow water sources i.e. the ground water and channel-regulating water, can work at the same time.
In this paper, four kinds of typical years (75%,80%,90%,95%) are adopted. The regulating cycle is year and thecalculating period is month.
(2) The change of the water in the aquifer equals to the gaining minus the losing. It is prescribed that thelosing amount should be less than the exploiting should be less than the max permitted depth and meanwhilemore than the depth of water-logging prevention and soil prohibition.
(3) The increment of the production due to the irrigation is calculated by the model of Jensen Multiplying.
(4) Many factors, which are dynamic and nonlinear, such as the hydrology, hydrogenlogy, agriculture,economy and produce function, are involved in this model. Fox example, the moisture produce function, therunning cost function, the function of consuming energy by pumping, the coefficient of precipitation recharge
and so on. The author linearizes the model and achieves the result by the way of linear programming.

2.3 The mathematical model
2.3.1 The object function(1) The annual gross profit of irrigation B’yields:
JB’=∑ εr (g − g ) AJ =1

jj

oj

(1)

oj

Where ε is apportionment of irrigation profit, whose value is between 0.25～0.45 according to the localsituation; Aoj is the irrigated area of crop j; rj is the price of crop j; gi is yield of j after irrigating; goj is yield of jwithout irrigating.
(2) The annual cost of irrigation includes the management cost (Z1) and cost of pumping energy (Z2). Z1
consists of the depreciated cost of infield engineer, pumping-well engineer and electro mechanic pump, therepair fee and the salary of manipulators. In this article the combination of different irrigation engineer systems
are taken into account. The cost of pumping energy (Z2) increases due to the increase of pumping and thedecrease of ground-water level, so the author’s opinion is that Z2 is the function of pumping quantity andpumping intake level, i.e. Z2= f(m,H). As for different types of pump ( centrifugal pump and pump of unconfinedgroundwater), the unit energy of consuming is different. The net profit of irrigation is:
JmaxB=∑ εA r (g − g ) − Z − f(m, H)
oj jj

oj

(2)

1

j=1

2.3.2 The limitary conditions(1)The ratio of a certain kind crop to other kinds in the irrigated area should be less than the corresponding
max ratio. It is:(3)

Xj≤Xmj(2) The water that the crops need in the infield at every period comes from pumped groundwater, theregulating water in the trench, the recharge of groundwater and the effective precipitation. The formula is:I

η1iwi+η2iwi+∑I

Gy +i =1∑

IA oj α si Pij ≥i =1

∑A E
ij

(4)

ij

i =1

where η1 is coefficient of utilized water in the trench; w is max of pumped water of the trench; η2 is thecoefficient of utilized well water; m is extraction of groundwater; α s is the coefficient of effective precipitation;
Pij is the precipitation during the period of the crops growing; Eij is demand of water of the crop j; G is recharge
to the groundwater; i is period i.
(3) The annual extraction of groundwater should be no more than allowed:
I∑
i =1

Imi +∑
i =1

IA0Ei −∑i =1I

α g Pi A 0 −

∑
i =1I

β1m i −

∑ β w − ∆u ≤02i(5)i =1Where E is evaporation of unconfined groundwater; α g is the coefficient of precipitation recharge; β1,β2
are tropical coefficients of groundwater and trench water respectively.
(4) The regulating water (W) should be no more than the max of possible supply (wm). It is:wi≤wmi(6)(5) The pumped water (m) should be less than the yield of the well (Mm). It is:mi≤Mmi(7)(6) The restriction of the change of the groundwater level
Hi=Ki-α g Piµ+miEβ mβ w+ i − 1 i − 2 1µA 0µµA 0µA 0(8)

Where Hi and Hi-1 are the depths of groundwater at the beginning and the end of period respetively; μis
specific yield of the aquifer. Others are meanings just like what have been mentioned.
(7) The restriction of groundwater level for waterlogging prevention and soil prohibition.
(9)

Hi≥Hmin

Where Hmin is the minimum depth of groundwater table to prevent undesired water-logging and soilprohibition. It equals to 0.5m.
(8) The depth of groundwater should be less than the max permitted decline. It is:(10)

Hi≤Hmax

Where Hmax is maximum permitted groundwater table decline. For centrifugal pump, Hmax equals to 7～8m.
(9) The positive restrict:Xj≥0(j=1, 2 …J) mi, wi≥0

(i=1, 2 … I)

(11)

3 RESULTAND ANALYSIS
3.1 Result
The author input the basic data, 75%,80%,90%,95%, of the typical years, including the data of agriculture,
economy, hydrology, hydrogeology, the river flow and water consuming, then which were calculated by themethod of Excel

linear programming. The progress is just to fill the blank cell with the corresponding

coefficient of the decisive variant in the restriction equation, and the optimized results can be achieved easily.
The optimized results of growing ratio are shown in the Table 1, and the results of max net profit of the engineer
system and the optimized water-supplies in every period are computed also. The restriction equations of 75%
year are shown in formula (12).
56.9X1+54.9X2+45.7X5-1.0m10-1.0w10≤44.6
50.3X1+38.1X2-1.0m11-1.0w11≤32.2
38.1X2+28.2X2-1.0m12-1.0w12≤23.1
33.2X1+27.4X2-1.0m1-1.0w1≤9.9
46.2X1+89.2X2-1.0m2-1.0w2≤6.3
131.5X1+113.0X2-1.0m3-1.0w3≤71.0
190.5X1+153.5X2+47.2X8-1.0m4-1.0w4≤143.8
143.6X1+73.2X2+88.7X8-1.0m5-1.0w5≤91.8
123.3X3+79.7X4+128.2X5+83.8X6+213.7X7+132.9X8+91.4X9-1.0m6-1.0w6≤13.7
239.4X3+209.7X4+170.1X5+150.1X6+222.4X7+152.7X8+137.2X9-1.0m7-1.0w7≤193.8
263.9X3+259.4X4+259.1X5+204.5X6+271.1X7+183.0X8+152.2X9-1.0m8-1.0w8≤166.0
69.5X3+104.4X4+183.8X5+152.4X6+229.7X7+33.5X8+76.2X9-1.0m9-1.0w9≤92.0(12)

Table 1. The result of optimizing the ratio of different crops in different typical years
TylicalWinterYear(%)

Item

75
809095

Summerwheat

Coleratio(%)

80.0

5.0

2area(hm )3217.5

ratio(%)
2

Beancotton

Poi

Paddy

PeanutOtherTotal

50.0

20.0

5.0

5.0

5.0

10.0

5.0

185

202.0

2030.8813.4

200.0202.0

202.0

406.7

202.0

7476.4

71.7

5.0

50.0

20.0

5.0

3.0

5.0

10.0

5.0

175

area(hm )2913.5202.0

2030.8

813.4

202.0

122.7

202.0

406.7

202.0

7095.1

ratio(%)

79.6

5.0

50.0

0.0

5.0

0.0

5.0

10.0

5.0

160

2

area(hm )3236.2202.0

2030.8

0.0

202.0

0.0

202.0

406.7

202.0

6481.7

ratio(%)

80.0

5.0

49.7

0.0

5.0

0.0

0.0

10.0

5.0

155

2

3250.8202.0

2021.4

0.0

202.0

0.0

0.0

406.7

202.0

6284.9area(hm )cornSummer

3.2 Analysis of the result
We can know from the Table 1, the optimized index of re-grow in 75% year, 80% year, 90% year, 95%year are correspondingly 1.85, 1.75, 1.60, 1.55, i.e. the max ratio of growing in 75% year. In 80～90% years, thegrowing ratios of bean, poi and paddy are limited to some degree. For example, the ratio is zero in 95% year. So
we can infer that local area is good for the winter wheat, cole, summer corn, summer cotton and peanut’sgrowing, but the growing of poi, bean and paddy is limited.
We can know from Table 2, the system with the max net profit is the system of small shallow
well-mechanical pump－semi-fixed sprayed irrigation, and the second is small shallow well － mechanicalpump-PVC pipe. Under the precondition of assurance of water source, the system of small original well －electric pump-original trench can be used as supplement. So, what we should develop is the system of small
shallow well － mechanical pump-semi-fixed sprayed irrigation or PVC pipe, not the system of shallow well－ electric pump-original trench.
We can know from the Table 3, the peak of water-supply is between Apr. when the spring crops need water
most and the summer-autumn crops also need water to grow. And this is consistent with the practical situation.
According to the result of optimization, in 75% typical year, the supply of groundwater is enough. But in the80～95% dry year, the supply of water is not sufficient, so the crops only depend on the groundwater, then thesurface water should be reserved as supplement and used with groundwater together to achieve the mosteconomic profit.

4 CONCLUSIONS
The agricultural produce and engineer systems are involved in the model. The author put forward the best
distribution of irrigation water and the best layout of agriculture in irrigating areas. It is significant to regulate thestructure of growing reasonably, to improve the manage level, to achieve the economic profit of water resource
and to accelerate the develop-ment of water-saving agriculture. The results show that the groundwater cannot
fully satisfy the demand of irrigation, and the surface water should be reserved for supplement; i.e. under theprecondition that the drainage of the infield not be affected, the measures, such as building the dam and gate in
the big trench and channel, can be taken. By controlling the groundwater, increasing the reserved of surface
water and the recharge to the groundwater, we can make the irrigating water cycle well.