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Chapter 11

Optimal Cropping Pattern

Abstract

The cropping pattern plays an important role in planning an irrigation project as it decides the water requirement and, ultimately, the capacity of the reservoir and canal system. Selection of crop, based on water availability, climate, soil, affinity, market, income, etc., following the existing cropping pattern should be adequately considered during planning. In addition, selection of optimal crop is important to improve the utilization potential of the existing irrigation projects. Considering the importance of the subject, this chapter describes the procedure of selecting an optimal cropping pattern for the irrigation command using the linear programming technique. It also includes conjunctive-use planning in the command area.

Keywords

Conjunctive use; Cropping pattern; Irrigation potential; Linear programming; Optimal crop planning; Water productivity

Selection of cropping pattern plays an important role while planning an irrigation project, which decides the water requirement at the head and ultimately helps in the estimation of required capacity of the reservoir and the canal system. Selection of crop, based on water availability, climate, soil, affinity, market, income, etc., following the existing cropping pattern, should be adequately considered while planning. In addition, optimal cropping pattern is also important for increased demands of agricultural products, due to population growth, under limited water availability. It also helps in the effective utilization of water and land resources, and improves the water productivity and water-use efficiency of the system. In the agricultural sector, many crops cultivated in a particular region are not suitable for that region. For example, rice and sugarcane are being cultivated in semi-arid regions. In view of this, optimal cropping pattern for planning a sustainable irrigation project is highly required.

In addition, significant deviation in the cropping pattern also affects the existing irrigation projects resulting into reduced irrigation intensity and project duty. For example, the irrigation project in a semi-arid region designed for a cropping pattern comprising 35% wheat crop will in due course result in an uneven supply of water in the command area, if the wheat-cropped area is increased up to 50%–70%. Such deviations in the cropping pattern also reduce the utilization of actual irrigation potential of the designed system. Keeping these considerations in view, it is necessary to identify an optimal cropping pattern for a particular project under various irrigation water availability constraints (i.e., dependable water yield of the project).

An optimal cropping pattern is the allocation of the cropped area with different crops to the culturable command with maximum return for the available water stored in the reservoir. To achieve this, groups of farmers, Water Users Associations (WUAs) or project authority decide the optimum cropping pattern for the command area in the beginning of the season, depending upon the water availability in the reservoir, with the objective of meeting the maximum irrigation potential as well as the maximum economic return. To arrive at an efficient solution of these problems, mathematical models and irrigation management methodologies can be integrated for optimum utilization of resources. For this purpose, linear programming (LP) can be efficiently used. This approach is widely used in the previous studies for achieving different objectives (Loucks et al., 1981; Kumar and Pathak, 1989; Vedula and Mujumdar, 1992; Sethi et al., 2002; Raju and Kumar, 2004). This chapter presents an application of LP as a tool for optimum cropping pattern for the irrigation project under available live storage capacity of the irrigation reservoir.

11.1. Linear Programming

If the objective function and constraints of an optimization model are all linear, then the LP procedure can be applied for planning. In spite of its power and popularity, for most real-time water resources planning and management problems, LP is best viewed as a preliminary tool. Like other optimization methods, LP can provide initial design and operating policy information that simulation models require before they can simulate design and operating policies. The general structure of an LP model is given as follows (Eqs. 11.1–11.3):

$Max or Min z = \sum_{j = 1}^{n} c_{j} x_{j}$ $Max or Min z = \sum_{j = 1}^{n} c_{j} x_{j}$

(11.1)

subject to:

$\sum_{j = 1}^{n} a_{i j} x_{j} \leq b_{i}, for i = 1,2, \dots, m$ $\sum_{j = 1}^{n} a_{i j} x_{j} \leq b_{i}, for i = 1,2, \dots, m$

(11.2)

$x_{j} \geq 0, for all j = 1,2, \dots, n$ $x_{j} \geq 0, for all j = 1,2, \dots, n$

(11.3)

If any model fits this general form, where the constraints can be any combination of equalities (=) and inequalities (≥ or ≤), LP algorithm can be used to find the “optimal” values of unknown decision variables (i.e., x_j).

11.2. LP Formulation for Optimal Crop Planning

The development of an LP model to investigate the optimal cropping pattern is explained further using the following variables:

1. Culturable command area (CCA): A

2. Number of crops sown in the CCA for the crop calendar year, n: 4

3. Type of Rabi crops: wheat, mustard, green gram, and barley

4. Available water in the reservoir for irrigation supply: S

5. System efficiencies: η

6. Irrigation water requirement, average yield, and price of crops

In this formulation, it is assumed that the net input (expenses) is proportional to net return, and the minimum support price (MSP) for the crops is based on the average yield, average return, and average expenses incurred on the crop.

The LP problem for optimal cropping pattern can be formulated as follows (Eq. 11.4):

Objective function:

$Max z = α \times {\sum_{i = 1}^{n} Y_{i} {MSP}_{i} A_{i} |}_{Rabi} + β \times {\sum_{j = 1}^{m} Y_{j} {MSP}_{j} A_{j} |}_{Kharif}$ $Max z = α \times {\sum_{i = 1}^{n} Y_{i} {MSP}_{i} A_{i} |}_{Rabi} + β \times {\sum_{j = 1}^{m} Y_{j} {MSP}_{j} A_{j} |}_{Kharif}$

(11.4)

where i is the index for the number of Rabi crops; j is the index for the number of Kharif crops; Y is the average yield of the crop (kg/ha); MSP is the minimum support price of the crop (Rs/kg); A is the area under crop (ha); and α and β are the integer values for Rabi and Kharif seasons, respectively. For priority crops such as Kharif protection, value of β should be kept high enough as compared to the value of α.

The conveyance and field efficiency of the system can be considered appropriately based on the condition of conveyance system and irrigation method, respectively.

Subject to constraints:

1. Water availability constraint

$\sum_{i = 1}^{n} [10^{- 5} (A_{i} \times {CWR}_{i})] \leq η_{c} \cdot η_{f} \cdot S$ $\sum_{i = 1}^{n} [10^{- 5} (A_{i} \times {CWR}_{i})] \leq η_{c} \cdot η_{f} \cdot S$

(11.5)

where CWR is the crop water requirement during the growing period of crop (mm); η_c is the conveyance efficiency; η_f is the field application efficiency; and S is the water availability for irrigation supply (MCM).

The water available for irrigation supply is computed from the net catchment yield or reservoir yield, by deducting evaporation and seepage losses from the reservoir during the irrigation season. Since optimal cropping pattern is identified under various scenarios of water availability, the dependable reservoir yield term is used. The effective reservoir yield from the most dry to most wet year is considered; in terms of dependable reservoir yield, the 2%, 10%, 20%, 25%, 50%, 75%, 80%, and 90% dependable year yield is considered.

2. Crop area constraint

$\sum_{i = 1}^{n} A_{i} \leq CCA$ $\sum_{i = 1}^{n} A_{i} \leq CCA$

(11.6)

where CCA is in ha.

3. Nonnegative constraints

$A_{i} \geq 0; \forall i$ $A_{i} \geq 0; \forall i$

(11.7)

4. Crop diversity constraint

$A_{i} \geq (f_{i} / 100) \times CCA; \forall i$ $A_{i} \geq (f_{i} / 100) \times CCA; \forall i$

(11.8)

where f_i is the minimum percentage of the crop area required to maintain crop diversity.

The LP problem is solved using the simplex method, which yields the optimal area of crops to be cultivated under the available storage. In this formulation, the cost of production will not be considered to estimate the net return from the production. It will be based on the general assumption that gross return is relative to the cost of production, i.e., higher the cost of production, higher will be the gross income and vice versa. CWR can be replaced with the net irrigation requirement (IWR_net) after deducting effective rainfall (ER) from CWR.

To solve these problems easier, “Solver” of MS Excel can be used.

Example 11.1:

Determine the optimal cropping pattern for the irrigation project having CCA and ICA as 4395 and 3077 ha, respectively. The project is designed to irrigate Rabi crops as well as protect Kharif crop during its maturity. The crops supposed to be grown during Rabi are wheat, barley, gram, mustard, and Rabi fodder; whereas for Kharif protection only maize crop is taken. The CWR for these crops are given as follows:

Components	Crop
	Kharif	Rabi
	Maize^a	Wheat	Barley	Gram	Mustard	Fodder
CWR (mm)	133.99	409.17	429.06	350.91	319.71	535.61
GIR (mm)^b	225.19	687.68	721.11	589.76	537.32	900.18

The field application efficiency, η_f and conveyance efficiency η_c will be assumed as 75% and 85%, respectively. Other important inputs of the formulation are summarized in Table 11.1.

Water availability in the reservoir for various dependable years is given as follows:

Dependability (%)	Year	LC (MCM)	Irrigation Supply (MCM)
90	1998	0	0
80	2010	1.17	1.17
75	1997	4.02	3.64
50	2011	12.48	11.44
25	1994	16.79	15.47
20	1990	16.85	15.62
10	2013	21.52	19.89
2	2004	21.52	19.89

Table 11.1

Basic Input Required for Estimating Optimal Cropping Pattern

S. No.	Crops	CWR (mm)	Average Yield (kg/ha)	MSP (Rs/kg)
1	Maize	133.99	1386	1175
2	Wheat	409.17	2912	1350
3	Barley	429.06	2515	1100
4	Gram	350.91	955	3000
5	Mustard	319.71	1178	3000
6	Rabi fodder	535.61	750	500

Solution:

The LP formulation is developed on MS Excel to use the “Solver” program as follows. The basic input is prepared and summarized in Table 11.2.

CCA = 4395 ha; ICA = 3077 ha.

Case I: Optimal cropping pattern for 80% dependable year yield (S = 1.17 MCM):

$\begin{matrix} Max z = α \times (2912 \times 1350 \times A_{wheat} + 2515 \times 1100 \times A_{Barley} + 955 \times 3000 \times A_{Gram} + 1178 \times 3000 \times A_{Mustard} + 750 \times 500 \times A_{Fodder}) + β \times (1386 \times 1175 \times A_{Maize}) \end{matrix}$ $\begin{matrix} Max z = α \times (2912 \times 1350 \times A_{wheat} + 2515 \times 1100 \times A_{Barley} + 955 \times 3000 \times A_{Gram} + 1178 \times 3000 \times A_{Mustard} + 750 \times 500 \times A_{Fodder}) + β \times (1386 \times 1175 \times A_{Maize}) \end{matrix}$

subject to:

1. Water availability constraint

$\sum_{i = 1}^{n} [10^{- 5} (A_{i} \times {CWR}_{i})] \leq η_{c} \cdot η_{f} \cdot S$ $\sum_{i = 1}^{n} [10^{- 5} (A_{i} \times {CWR}_{i})] \leq η_{c} \cdot η_{f} \cdot S$

$\begin{matrix} [10^{- 5} (133.99 A_{Maize} + 409.17 A_{Wheat} + 429.06 A_{Barley} + 350.91 A_{Gram} + 319.71 A_{Mustard} \\ + 535.61 A_{Fodder})] \leq [0.75 \times 0.85 \times 1.17] \end{matrix}$ $\begin{matrix} [10^{- 5} (133.99 A_{Maize} + 409.17 A_{Wheat} + 429.06 A_{Barley} + 350.91 A_{Gram} + 319.71 A_{Mustard} \\ + 535.61 A_{Fodder})] \leq [0.75 \times 0.85 \times 1.17] \end{matrix}$

2. Crop area constraints

A_Maize ≤ 4395

A_Wheat ≤ 4395

A_Barley ≤ 4395

A_Gram ≤ 4395

A_Mustard ≤ 4395

A_Fodder ≤ 4395

Table 11.2

Basic Input Required for Estimating the Optimal Cropping Pattern

Components	Maize	Wheat	Barley	Gram	Mustard	Fodder	Total
CWR (mm)	133.99	409.17	429.06	350.91	319.71	535.61
GIR (mm)	225.19	687.68	721.11	589.76	537.32	900.18
Area under crop (ha)	?	?	?	?	?	?	?
Cropping pattern		?	?	?	?	?	?
Average yield (kg/ha)	1386	2912	2515	955	1178	750
MSP (Rs/qt)	1175	1350	1100	3000	3000	500
Gross return (lakh Rs)	?	?	?	?	?	?	Objective function, z
GIR (MCM)	?	?	?	?	?	?	?

3. Nonnegative constraints

A_Maize ≥ 0

A_Wheat ≥ 0

A_Barley ≥ 0

A_Gram ≥ 0

A_Mustard ≥ 0

A_Fodder ≥ 0

4. Crop diversity constraints

A_Wheat ≥ 30%

A_Barley ≥ 5%

A_Gram ≥ 5%

A_Mustard ≥ 5%

A_Fodder ≥ 5%

All these constraints are summarized as below:

Constraint-1: Area	(Area)	0.00	≤	4395
Constraint-2: Area	(Area)	3003.63	≤	4395
Constraint-3: LC	(LC)	19.89	≤	19.89
Nonnegative	A₁	0.00	>	0
	A₂	1042.0	>	0
	A₃	578.9	>	0
	A₄	578.9	>	0
	A₅	578.9	>	0
	A₆	225.1	>	0
Crop diversity	A₁	0.0	>	0
	A₂	1042.0	>	923.1
	A₃	578.9	>	153.85
	A₄	578.9	>	153.85
	A₅	578.9	>	153.85
	A₆	225.1	>	153.85

The value of α was considered lower as compared to β for higher value of dependability (i.e., 75%–90%). Solving the LP problem, the following cropping pattern is obtained for 80% dependable year:

$A_{Maize} = 519.6 ha; A_{Wheat} = 0; A_{Barley} = 0; A_{Gram} = 0; A_{Mustard} = 0; A_{Fodder} = 0$ $A_{Maize} = 519.6 ha; A_{Wheat} = 0; A_{Barley} = 0; A_{Gram} = 0; A_{Mustard} = 0; A_{Fodder} = 0$

For other dependable year yields the cropping patterns are:

Depend-ability (%)	Year	LC (MCM)	Irrigation Supply (MCM)	Maize (ha)	Wheat (ha)	Barley (ha)	Gram (ha)	Mustard (ha)	Fodder (ha)	Total Irrigation (ha)
90	1998	0	0							0.0
80	2010	1.17	1.17	519.6	0.0	0.0	0.0	0.0	0.0	519.6
75	1997	4.02	3.64	307.7	0.0	307.7	0.0	135.5	0.0	750.9
50	2011	12.48	11.44	0.0	252.2	0.0	615.4	615.4	307.7	1790.7
25	1994	16.79	15.47	0.0	615.4	212.5	615.4	615.4	307.7	2366.4
20	1990	16.85	15.62	0.0	636.4	454.6	454.6	454.6	315.7	2315.9
10	2013	21.52	19.89	0.0	810.4	578.9	578.9	578.9	402.0	2949.0
2	2004	21.52	19.89	0.0	1042.0	578.9	578.9	578.9	225.1	3003.6

The suggested cropping pattern for Rabi is:

Dependability (%)	LC (MCM)	Total Irrigated Area in Rabi (ha)	Economical and Optimal Cropping Pattern (%)
Dependability (%)	LC (MCM)	Total Irrigated Area in Rabi (ha)	Wheat	Barley	Gram	Mustard	Fodder
90	0	0.0	0.00	0.00	0.00	0.00	0.00
80	1.17	0.0	0.00	0.00	0.00	0.00	0.00
75	4.02	443.2	0.00	69.42	0.00	30.58	0.00
50	12.48	1790.7	14.08	0.00	34.37	34.37	17.18
25	16.79	2366.4	26.01	8.98	26.01	26.01	13.00
20	16.85	2315.9	27.48	19.63	19.63	19.63	13.63
10	21.52	2949.0	27.48	19.63	19.63	19.63	13.63
2	21.52	3003.6	34.69	19.27	19.27	19.27	7.49

Example 11.2:

Determine the optimal cropping pattern using the following data matrix:

Objective and Constraints	X₁	X₂	X₃	X₄	X₅	X₆	Condition
Objective and Constraints	Wheat	Barley	Paddy	Maize	Alfalfa	Onion	Condition
Net return (×10⁶ Rs)	8.74	7.01	18.98	29.96	8.77	19.11	Max
Land use (ha)	1	1	1	1	1	1	≤12,000
Seasonality (rotation)	+1	+1	−1	−1	−1	+1	≥0
Capital (×10⁶ Rs)	6.06	5.59	23.02	21.04	16.53	35.49	≤160,000

The monthly water requirement/constraints and water availability (×100 m³) are as follows:

Water Use (×10² m³)	Monthly Water Requirement (×100 m³)						Water Availability (×100 m³)
	X₁	X₂	X₃	X₄	X₅	X₆
	Wheat	Barley	Paddy	Maize	Alfalfa	Onion
Apr.	13.8	13.8	0	0	9.9	12.9	≤97,656.78
May.	15.6	11.3	0	0	3.4	17.1	≤119,834.4
Jun.	2.2	0	12.32	0	15.8	5.1	≤85,770.0
Jul.	0	0	32.28	10.62	16.5	0	≤90,216.3
Aug.	0	0	33.67	19.4	15.7	0	≤103,979.3
Sep.	0	0	35.38	20.01	12.6	0	≤96,897.0
Oct.	0	0	28.06	13.21	8.0	4.1	≤62,574.0
Nov.	0.9	0.4	4.22	0	4.5	4.0	≤50,488.6
Dec.	1.4	1.2	0	0	1.7	2.8	≤26,368.0
Mar.	8.1	8.1			6.1	7.4	≤58,863.2

Solution:

The LP formulation can be developed as follows.

The objective function will be defined as:

$Max [8.74 X_{1} + 7.01 X_{2} + 18.98 X_{3} + 29.96 X_{4} + 8.77 X_{5} + 19.11 X_{6}]$ $Max [8.74 X_{1} + 7.01 X_{2} + 18.98 X_{3} + 29.96 X_{4} + 8.77 X_{5} + 19.11 X_{6}]$

subject to:

1. Land availability constraints

$\begin{matrix} Winter : X_{1} + X_{2} + X_{6} \leq 12,000 (ha) \\ Spring : X_{3} + X_{4} + X_{5} \leq 12,000 (ha) \end{matrix}$ $\begin{matrix} Winter : X_{1} + X_{2} + X_{6} \leq 12,000 (ha) \\ Spring : X_{3} + X_{4} + X_{5} \leq 12,000 (ha) \end{matrix}$

2. Monthly water requirement

$Apr . : 13.8 X_{1} + 13.8 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 9.9 X_{5} + 12.9 X_{6} \leq 97,656.78$ $Apr . : 13.8 X_{1} + 13.8 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 9.9 X_{5} + 12.9 X_{6} \leq 97,656.78$

$May : 15.6 X_{1} + 11.3 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 3.4 X_{5} + 17.1 X_{6} \leq 119,834.4$ $May : 15.6 X_{1} + 11.3 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 3.4 X_{5} + 17.1 X_{6} \leq 119,834.4$

$Jun . : 2.2 X_{1} + 0.0 X_{2} + 12.32 X_{3} + 0.0 X_{4} + 15.8 X_{5} + 5.1 X_{6} \leq 85,770.0$ $Jun . : 2.2 X_{1} + 0.0 X_{2} + 12.32 X_{3} + 0.0 X_{4} + 15.8 X_{5} + 5.1 X_{6} \leq 85,770.0$

$Jul . : 0.0 X_{1} + 0.0 X_{2} + 32.28 X_{3} + 10.62 X_{4} + 16.5 X_{5} + 0.0 X_{6} \leq 90,216.3$ $Jul . : 0.0 X_{1} + 0.0 X_{2} + 32.28 X_{3} + 10.62 X_{4} + 16.5 X_{5} + 0.0 X_{6} \leq 90,216.3$

$Aug . : 0.0 X_{1} + 0.0 X_{2} + 33.67 X_{3} + 19.4 X_{4} + 15.7 X_{5} + 0.0 X_{6} \leq 103,979.3$ $Aug . : 0.0 X_{1} + 0.0 X_{2} + 33.67 X_{3} + 19.4 X_{4} + 15.7 X_{5} + 0.0 X_{6} \leq 103,979.3$

$Sep . : 0.0 X_{1} + 0.0 X_{2} + 35.38 X_{3} + 20.01 X_{4} + 12.6 X_{5} + 0.0 X_{6} \leq 96,897.0$ $Sep . : 0.0 X_{1} + 0.0 X_{2} + 35.38 X_{3} + 20.01 X_{4} + 12.6 X_{5} + 0.0 X_{6} \leq 96,897.0$

$Oct . : 0.0 X_{1} + 0.0 X_{2} + 28.06 X_{3} + 13.21 X_{4} + 8.0 X_{5} + 4.1 X_{6} \leq 62,574.0$ $Oct . : 0.0 X_{1} + 0.0 X_{2} + 28.06 X_{3} + 13.21 X_{4} + 8.0 X_{5} + 4.1 X_{6} \leq 62,574.0$

$Nov . : 0.9 X_{1} + 0.4 X_{2} + 4.22 X_{3} + 0.0 X_{4} + 4.5 X_{5} + 4.0 X_{6} \leq 50,488.6$ $Nov . : 0.9 X_{1} + 0.4 X_{2} + 4.22 X_{3} + 0.0 X_{4} + 4.5 X_{5} + 4.0 X_{6} \leq 50,488.6$

$Dec . : 1.4 X_{1} + 1.2 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 1.7 X_{5} + 2.8 X_{6} \leq 26,368.0$ $Dec . : 1.4 X_{1} + 1.2 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 1.7 X_{5} + 2.8 X_{6} \leq 26,368.0$

$Mar . : 8.1 X_{1} + 8.1 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 6.1 X_{5} + 7.4 X_{6} \leq 58,863.2$ $Mar . : 8.1 X_{1} + 8.1 X_{2} + 0.0 X_{3} + 0.0 X_{4} + 6.1 X_{5} + 7.4 X_{6} \leq 58,863.2$

3. Seasonality constraint

$X_{1} + X_{2} + X_{6} + X_{3} + X_{4} + X_{5} \leq 0$ $X_{1} + X_{2} + X_{6} + X_{3} + X_{4} + X_{5} \leq 0$

4. Capital constraints

$6.06 X_{1} + 5.59 X_{2} + 23.02 X_{3} + 21.04 X_{4} + 16.53 X_{5} + 35.49 X_{6} \leq 160,000$ $6.06 X_{1} + 5.59 X_{2} + 23.02 X_{3} + 21.04 X_{4} + 16.53 X_{5} + 35.49 X_{6} \leq 160,000$

5. Nonnegative constraints

$X_{1} \geq 0; X_{2} \geq 0; X_{3} \geq 0; X_{4} \geq 0; X_{5} \geq 0; X_{6} \geq 0$ $X_{1} \geq 0; X_{2} \geq 0; X_{3} \geq 0; X_{4} \geq 0; X_{5} \geq 0; X_{6} \geq 0$

$and 0 \leq X_{i} \leq 12,000, \forall i$ $and 0 \leq X_{i} \leq 12,000, \forall i$

Solving the above LP problem, the cropping pattern obtained is

Objective	X₁	X₂	X₃	X₄	X₅	X₆	Condition
Objective	Wheat	Barley	Paddy	Maize	Alfalfa	Onion	Condition
Net return (×10⁶ Rs)	8.74	7.01	18.98	29.96	8.77	19.11	20,499.5
Area under crop (ha)	6,376.25	0.0	0.0	4,504.34	0.0	749.18	≤12,000
Return per unit water use							0.346

Example 11.3:

Irrigation in the agricultural land is being met by run-of-river diversion project having mean monthly inflow as given below. The CCA of the area diversion scheme is 600 ha for which the cropping pattern need to be optimized between three crops, namely wheat, alfalfa, and cotton, having a net water demand as given below. Assume irrigation efficiency to be 65%. Also, determine the monthly diversion or water allocation.

Month	Monthly Inflow, I (m³/s)	Net Water Demand (mm)
Month	Monthly Inflow, I (m³/s)	Wheat	Alfalfa	Cotton
Sep.	0.20	0	0	114
Oct.	0.35	14.0	14.0	61
Nov.	1.00	5.0	5.0
Dec.	1.50	4.0	5.0
Jan.	1.70	2.0	1.0
Feb.	0.80	1.5	2.0
Mar.	0.75	33.0	33.0
Table Continued

Month	Monthly Inflow, I (m³/s)	Net Water Demand (mm)
Month	Monthly Inflow, I (m³/s)	Wheat	Alfalfa	Cotton
Apr.	0.60	142.0	141.0	24
May	0.36	138.0	97.0	82
Jun.	0.25	2.0	0	190
Jul.	0.20	0	0	225
Aug.	0.15	0	0	186
Total		341.5	298.0	882.0

Solution:

The LP formulation will be as follows.

Let A₁, A₂, and A₃ be the areas under wheat, alfalfa, and cotton crops. Before proceeding further into the LP formulation the following computations are to be made:

Crop	Net Water Demand (mm)			No. of Days	Monthly Inflow, I (m³/s)	Monthly Inflow, I (ha m)
Crop	Wheat	Alfalfa	Cotton
Area (ha)	A₁	A₂	A₃
Month
Sep.	0	0	114.0	30	0.20	51.84
Oct.	14.0	14.0	61.0	31	0.35	93.74
Nov.	5.0	5.0		30	1.00	259.20
Dec.	4.0	5.0		31	1.50	401.76
Jan.	2.0	1.0		31	1.70	455.33
Feb.	1.5	2.0		28	0.80	193.54
Mar.	33.0	33.0		31	0.75	200.88
Apr.	142.0	141.0	24.0	30	0.60	155.52
May	138.0	97.0	82.0	31	0.36	96.42
Jun.	2.0	0	190.0	30	0.25	64.80
Jul.	0	0	225.0	31	0.20	53.57
Aug.	0	0	186.0	31	0.15	40.18
Total	341.5	298.0	882.0			2066.77

Let Q_j be the monthly allocation of water to the agricultural land having a total CCA of 600 ha. The objective function will be defined as to satisfy the gross irrigation requirement on the field, considering the scheme irrigation efficiency of 65%.

$Min [\sum_{j = 1}^{m} η Q_{j} - (0.3414 A_{1} + 0.2980 A_{2} + 0.882 A_{3})]$ $Min [\sum_{j = 1}^{m} η Q_{j} - (0.3414 A_{1} + 0.2980 A_{2} + 0.882 A_{3})]$

where j is the index for the month and m is number of months during the irrigation season.

This is subject to following constraints:

1. Water availability constraints

$Sep . : (0.0 A_{1} + 0.0 A_{2} + 0.114 A_{3}) / 0.65 \leq 51.84 Oct . : (0.014 A_{1} + 0.014 A_{2} + 0.061 A_{3}) / 0.65 \leq 93.74 Nov . : (0.005 A_{1} + 0.005 A_{2} + 0.0 A_{3}) / 0.65 \leq 259.2 Dec . : (0.004 A_{1} + 0.005 A_{2} + 0.0 A_{3}) / 0.65 \leq 401.76 Jan . : (0.002 A_{1} + 0.001 A_{2} + 0.0 A_{3}) / 0.65 \leq 455.33 Feb . : (0.0015 A_{1} + 0.002 A_{2} + 0.0 A_{3}) / 0.65 \leq 193.54$ $Sep . : (0.0 A_{1} + 0.0 A_{2} + 0.114 A_{3}) / 0.65 \leq 51.84 Oct . : (0.014 A_{1} + 0.014 A_{2} + 0.061 A_{3}) / 0.65 \leq 93.74 Nov . : (0.005 A_{1} + 0.005 A_{2} + 0.0 A_{3}) / 0.65 \leq 259.2 Dec . : (0.004 A_{1} + 0.005 A_{2} + 0.0 A_{3}) / 0.65 \leq 401.76 Jan . : (0.002 A_{1} + 0.001 A_{2} + 0.0 A_{3}) / 0.65 \leq 455.33 Feb . : (0.0015 A_{1} + 0.002 A_{2} + 0.0 A_{3}) / 0.65 \leq 193.54$

$Mar . : (0.033 A_{1} + 0.033 A_{2} + 0.0 A_{3}) / 0.65 \leq 200.88 Apr . : (0.142 A_{1} + 0.141 A_{2} + 0.024 A_{3}) / 0.65 \leq 155.52 May : (0.138 A_{1} + 0.097 A_{2} + 0.082 A_{3}) / 0.65 \leq 96.42 Jun . : (0.002 A_{1} + 0.0 A_{2} + 0.190 A_{3}) / 0.65 \leq 64.8 Jul . : (0.0 A_{1} + 0.0 A_{2} + 0.225 A_{3}) / 0.65 \leq 53.57 Aug . : (0.0 A_{1} + 0.0 A_{2} + 0.186 A_{3}) / 0.65 \leq 40.18$ $Mar . : (0.033 A_{1} + 0.033 A_{2} + 0.0 A_{3}) / 0.65 \leq 200.88 Apr . : (0.142 A_{1} + 0.141 A_{2} + 0.024 A_{3}) / 0.65 \leq 155.52 May : (0.138 A_{1} + 0.097 A_{2} + 0.082 A_{3}) / 0.65 \leq 96.42 Jun . : (0.002 A_{1} + 0.0 A_{2} + 0.190 A_{3}) / 0.65 \leq 64.8 Jul . : (0.0 A_{1} + 0.0 A_{2} + 0.225 A_{3}) / 0.65 \leq 53.57 Aug . : (0.0 A_{1} + 0.0 A_{2} + 0.186 A_{3}) / 0.65 \leq 40.18$

2. Water allocation constraints

$Q_{j} \leq I_{j}; \forall j$ $Q_{j} \leq I_{j}; \forall j$

3. Land availability constraint

$A_{1} + A_{2} + A_{3} \leq 600 ha$ $A_{1} + A_{2} + A_{3} \leq 600 ha$

4. Nonnegative constraints

$Q_{j} \geq 0; \forall j$ $Q_{j} \geq 0; \forall j$

$A_{i} \geq 0; \forall i$ $A_{i} \geq 0; \forall i$

where i is the index for number of crops.

Figure 11.1 Monthly inflow and diversion from head work for irrigating crops.

Solving the above LP problem, the areas under wheat (A₁), alfalfa (A₂), and cotton (A₃) are determined as 160.5, 299.1, and 140.4 ha, respectively. The water to be diverted for irrigation in each month will be (Fig. 11.1):

Crop	Net Water Demand (mm)			No. of Days	Monthly Inflow, I (ha m)	Monthly Diversion, Q (ha m)
Crop	Wheat	Alfalfa	Cotton
Area (ha)	A₁ = 160.5 ha	A₂ = 299.1 ha	A₃ = 140.4 ha
Month
Sep.	0	0	114.0	30	51.84	24.62
Oct.	14.0	14.0	61.0	31	93.74	23.08
Nov.	5.0	5.0		30	259.20	3.54
Dec.	4.0	5.0		31	401.76	3.29
Jan.	2.0	1.0		31	455.33	0.95
Feb.	1.5	2.0		28	193.54	1.29
Mar.	33.0	33.0		31	200.88	23.33
Apr.	142.0	141.0	24.0	30	155.52	105.13
May	138.0	97.0	82.0	31	96.42	96.42
Jun.	2.0	0	190.0	30	64.80	41.53
Jul.	0	0	225.0	31	53.57	48.60
Aug.	0	0	186.0	31	40.18	40.18
Total	341.5	298.0	882.0		2066.77	411.96

11.3. LP-Based Conjunctive Use of Surface and Groundwater Resources

Conjunctive use implies a planned and coordinated management of surface and groundwater resources to maximize the efficient use of total water resources. Subsurface water can be used as a supplement to surface water during dry periods and peak demand periods as well as during periods of low rainfall. However, the conjunctive use of both surface water and subsurface water can avoid the overexploitation of any of these two resources. Optimization techniques could be very useful tool for conjunctive-use modeling. In this section, we will describe the conjunctive-use modeling using LP.

To start with, let us divide the total time duration (normally a water year) into discrete periods. The temporal resolution is problem specific and it could be in days, weeks, or months, or seasons like Rabi, Kharif, etc. Let us assume that we have M discrete periods. Also, assume that we have N feasible crops in the command area.

11.3.1. Decision Variables

For the conjunctive-use modeling the decision variables are:

1. Area of all feasible crops A_j; j = 1, 2, 3,…, N in ha.

2. Canal release QC_k; k = 1, 2, 3, …, M in ha m.

3. Groundwater withdrawal QG_k; k = 1, 2, 3, …, M in ha m.

11.3.2. Objective Function

In this case, we have taken the objective function as the maximization of net benefit from agricultural activity. However, the objective function depends upon the problem at hand. The net benefit would be the total market return minus the total cost of input. The cost of input here is divided into three sections: the cost of canal water, the cost of groundwater, and the cost other than that of water, which may include other expenditures, such as seeds, fertilizers, labor, transportation, etc. The objective function can be written as Eq. (11.9).

$Max z = \sum_{j = 1}^{N} (B_{j} A_{j} Y_{j} - C_{j} A_{j}) - C C \sum_{k = 1}^{M} Q C_{k} - C G \sum_{k = 1}^{M} Q G_{k}$ $Max z = \sum_{j = 1}^{N} (B_{j} A_{j} Y_{j} - C_{j} A_{j}) - C C \sum_{k = 1}^{M} Q C_{k} - C G \sum_{k = 1}^{M} Q G_{k}$

(11.9)

We can see from the objective function equation that it is a linear function of decision variables.

B_j is the market return for the j^th crop in Rs/ton; Y_j is the yield per hectare for j^th crop in ton/ha; C_j is the cost of input other than that of water in Rs/ha; CC is the cost of unit volume of canal water in Rs/ha m; and CG is the cost of unit volume of groundwater in Rs/ha m.

11.3.3. Constraints

The LP problem cannot be unconstrained; however, in this case, crop area, canal release, and groundwater withdrawals are the decision variables. Therefore the constraints are to be imposed to restrict excessive mining of these resources. The following constraints are used.

11.3.3.1. Water Availability Constraint

This constraint states that the total irrigation water requirement should be less than or equal to the available water. For the k^th period, where k = 1, 2, 3, …, M, the constraint can be written as Eq. (11.10).

$\sum_{j = 1}^{N} (A_{j} δ_{j k}) \leq η_{c} Q C_{k} + η_{g} Q G_{k}$ $\sum_{j = 1}^{N} (A_{j} δ_{j k}) \leq η_{c} Q C_{k} + η_{g} Q G_{k}$

(11.10)

where δ_jk is the irrigation water requirement for j^th in m for k^th period, η_c and η_g are the conveyance efficiency w.r.t. canal water and groundwater, respectively. We would have one constraint for each period; therefore we would have total M such constraints.

11.3.3.2. Land Availability Constraint

This constraint states that the total area occupied by crops at any time period should be less than the CCA or actual area available for agriculture during that period. For the k^th period where k = 1, 2, 3, …, M, the constraint can be written as Eq. (11.11)

$\sum_{j = 1}^{N} (A_{j} θ_{j k}) \leq CCA$ $\sum_{j = 1}^{N} (A_{j} θ_{j k}) \leq CCA$

(11.11)

where θ_jk is the area occupied by one unit area of the j^th in the k^th period. Its value varies from 0 to 1: 0 when the crop is absent and 1 when the crop is present.

11.3.3.3. Crop Area Limits

There could be a constraint on the maximum and minimum permissible areas under any crop. This constraint can be written as Eq. (11.12).

$A_{max j} \geq A_{j} \geq A_{min j}; j = 1,2,3, \dots, N$ $A_{max j} \geq A_{j} \geq A_{min j}; j = 1,2,3, \dots, N$

(11.12)

11.3.3.4. Surface Water–Resources Constraint

There could be constraints on the maximum water that can be released from the canal (Eqs. 11.13 and 11.14).

$Q C_{k} \leq Canal conveyance capacity$ $Q C_{k} \leq Canal conveyance capacity$

(11.13)

$\sum_{k = 1}^{m} Q C_{k} \leq Reservoir capacity allocated to irrigation$ $\sum_{k = 1}^{m} Q C_{k} \leq Reservoir capacity allocated to irrigation$

(11.14)

where k = 1, 2, 3, …, m; and m is number of season or period (in months).

11.3.3.5. Groundwater Resources Constraint

This constraint restricts the maximum and minimum amount of water that can be withdrawn from the groundwater source. A certain minimum quantity of water is to be withdrawn from the groundwater to avoid water logging in the area. On the contrary, the water withdrawn from the groundwater should not lead to excessive lowering of water table. This constraint is written as Eq. (11.15).

$η_{g} \sum_{k = 1}^{m} Q G_{k} \geq PGW$ $η_{g} \sum_{k = 1}^{m} Q G_{k} \geq PGW$

(11.15)

On the contrary, the water withdrawn from groundwater should not lead to excessive lowering of the water table. This constraint is written as Eq. (11.16).

${\hat{η}}_{g} \sum_{k = 1}^{m} Q G_{k} \leq PGW + M$ ${\hat{η}}_{g} \sum_{k = 1}^{m} Q G_{k} \leq PGW + M$

(11.16)

where PGW is post-project groundwater resource in ha m; M is the permissible mining in ha m; and

{\hat{η}}_{g}

${\hat{η}}_{g}$

is the overall efficiency in respect to groundwater.

Example 11.4:

In an area of 170 sq. km having CCA of 80%, three feasible crops, namely wheat, sugarcane, and paddy, are grown. The annual surface water available and annual groundwater recharges are 5000 and 2000 ha m, respectively. Recharge from the canal system is 20% of the annual canal water release. The net benefits (excluding cost of water) in Rs/ha for wheat, sugarcane, and paddy are 4000, 9500, and 2150 Rs/ha, respectively. The cost of groundwater and cost of canal water are Rs. 1000 and Rs. 650/ha m. The maximum permissible areas for wheat, sugarcane, and paddy are 5800, 7500, and 5200 ha, respectively.

The irrigation efficiency for groundwater and canal water are 0.8 and 0.6, respectively. The canal conveyance capacity is 750 ha m/month and the maximum permissible monthly pumpage is 700 ha m. The net irrigation requirement (mm) is

Crop	Jan.	Feb.	Mar.	Apr.	May	Jun.	Jul.	Aug.	Sep.	Oct.	Nov.	Dec.
Wheat	46.7	108	204	12.1	0	0	0	0	0	0	2.73	23.6
Sugarcane	0	76	138	218	306	216	0	27.6	49	130	103	47.4
Paddy	0	0	0	0	0	164	48.4	198	247	315	0	0

Compute the optimal conjunctive-use policy using LP, which maximizes the net benefit.

Solution:

Let A₁, A_2, and A₃ be the areas under wheat, sugarcane and paddy, respectively. The above LP problem can be written as follows.

The objective function can be written as

Maximize 4000A₁ + 9500A₂ + 2150A₃ − 1000QG₁ − 1000QG₂ − 1000QG₃ − 1000QG₄

−1000QG₅ − 1000QG₆ − 1000QG₇ − 1000QG₈ − 1000QG₉ − 1000QG₁₀ − 1000QG₁₁

−1000QG₁₂ − 650QC₁ − 650QC₂ − 650QC₃ − 650QC₄ − 650QC₅ − 650QC₆

−650QC₇ − 650QC₈ − 650QC₉ − 650QC₁₀ − 650QC₁₁ − 650QC₁₂

subject to the following constraints.

1. Water availability constraint

0.0467A₁ + 0A₂ + 0A₃ − 0.6QC₁ − 0.8QW₁ ≤ 0

0.1076A₁ + 0.076A₂ + 0A₃ − 0.6QC₂ − 0.8QG₂ ≤ 0

0.2036A₁ + 0.138A₂ + 0A₃ − 0.6QC₃ − 0.8QG₃ ≤ 0

0.01214A₁ + 0.218A₂ + 0A₃ − 0.6QC₄ − 0.8QG₄ ≤ 0

0.0A₁ + 0.306A₂ + 0A₃ − 0.6QC₅ − 0.8QG₅ ≤ 0

0.0A₁ + 0.216A₂ + 0.1642A₃ − 0.6QC₆ − 0.8QG₆ ≤ 0

0.0A₁ + 0.0A₂ + 0.0484A₃ − 0.6QC₇ − 0.8QG₇ ≤ 0

0.0A₁ + 0.0276A₂ + 0.198A₃ − 0.6QC₈ − 0.8QG₈ ≤ 0

0.0A₁ + 0.049A₂ + 0.247A₃ − 0.6QC₉ − 0.8QG₉ ≤ 0

0.0A₁ + 0.1302A₂ + 0.315A₃ − 0.6QC₁₀ − 0.8QG₁₀ ≤ 0

0.00273A₁ + 0.103A₂ + 0A₃ − 0.6QC₁₁ − 0.8QG₁₁ ≤ 0

0.0236A₁ + 0.0474A₂ + 0A₃ − 0.6QC₁₂ − 0.8QG₁₂ ≤ 0

2. Land availability constraints

A₁ + A₂ ≤ 13,600

A₂ + A₃ ≤ 13,600

3. Surface water–resources constraint

QC₁ + QC₂ + QC₃ + QC₄ + QC₅ + QC₆ + QC₇ + QC₈ + QC₉ + QC₁₀ + QC₁₁+ QC₁₂ ≤ 5000.

4. Conveyance capacity constraints

QC₁ ≤ 750

QC₂ ≤ 750

QC₃ ≤ 750

QC₄ ≤ 750

QC₅ ≤ 750

QC₆ ≤ 750

QC₇ ≤ 750

QC₈ ≤ 750

QC₉ ≤ 750

QC₁₀ ≤ 750

QC₁₁ ≤ 750

QC₁₂ ≤ 750

5. Groundwater constraints

QG₁ + QG₂ + QG₃ + QG₄ + QG₅ + QG₆ + QG₇ + QG₈ + QG₉ + QG₁₀+ QG₁₁ + QG₁₂ − 0.30QC₁ − 0.30QC₂ − 0.30QC₃ − 0.30QC₄− 0.30QC₅ − 0.30QC₆ − 0.30QC₇ − 0.30QC₈ − 0.30QC₉− 0.30QC₁₀ − 0.30QC₁₁ − 0.30QC₁₂ = 2000

6. Monthly pumpage constraints

QG₁ ≤ 700

QG₂ ≤ 700

QG₃ ≤ 700

QG₄ ≤ 700

QG₅ ≤ 700

QG₆ ≤ 700

QG₇ ≤ 700

QG₈ ≤ 700

QG₉ ≤ 700

QG₁₀ ≤ 700

QG₁₁ ≤ 700

QG₁₂ ≤ 700

7. Maximum permissible area constraints

A₁ ≤ 5800

A₂ ≤ 7500

A₃ ≤ 5200

The above problem contains 43 constraints and 28 decision variables. Therefore this problem is solved using Lindo, and the final solution obtained is provided in Table 11.3.

It can be seen from Figs. 11.2 and 11.3 that there would be more dependency on groundwater in the nonmonsoon periods, i.e., from October to June. From October to January more utilization of surface water resources takes place, whereas from March to June maximum utilization of both surface and groundwater resources takes place (Figs. 11.2 and 11.3).

Figure 11.2 Monthwise groundwater required.

Figure 11.3 Monthwise canal water required.

Table 11.3

Optimal Values of Decision Variables in Example 11.4

Variable	Value	Unit
A₁	2723.52	ha
A₂	3300.65	ha
A₃	403.72	ha
QG₁	0	ha m
QG₂	215.11	ha m
QG₃	700	ha m
QG₄	700	ha m
QG₅	700	ha m
QG₆	700	ha m
QG₇	24.42	ha m
QG₈	0	ha m
QG₉	326.81	ha m
QG₁₀	133.64	ha m
Table Continued

Variable	Value	Unit
QG₁₁	0	ha m
QG₁₂	0	ha m
QC₁	211.98	ha m
QC₂	619.68	ha m
QC₃	750	ha m
QC₄	321.01	ha m
QC₅	750	ha m
QC₆	365.38	ha m
QC₇	0	ha m
QC₈	285.05	ha m
QC₉	0	ha m
QC₁₀	750	ha m
QC₁₁	579.00	ha m
QC₁₂	367.87	ha m

11.4. Concluding Remarks

The chapter discusses optimal crop planning for irrigation projects using the LP technique, considering necessary constraints like water availability, crop types, crop affinity, financial cost and benefits, etc., with the ultimate objective of maximizing the net return, equitable distribution of water, and water-use efficiency. The chapter also includes the conjunctive-use planning in the command area.

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Table of Contents for Chapter 11. Optimal Cropping Pattern

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Abstract

Keywords

11.1. Linear Programming

11.2. LP Formulation for Optimal Crop Planning

11.3. LP-Based Conjunctive Use of Surface and Groundwater Resources

11.3.1. Decision Variables

11.3.2. Objective Function

11.3.3. Constraints

11.3.3.1. Water Availability Constraint

11.3.3.2. Land Availability Constraint

11.3.3.3. Crop Area Limits

11.3.3.4. Surface Water–Resources Constraint

11.3.3.5. Groundwater Resources Constraint

11.4. Concluding Remarks

Table of Contents for
Chapter 11. Optimal Cropping Pattern