Decision models for optimal cropping patterns in irrigations based on crop water production functions

Agricultural Water Management, 3 (1980) 65--76

65

Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

DECISION MODELS FOR OPTIMAL CROPPING PATTERNS IN IRRIGATIONS BASED ON CROP WATER PRODUCTION FUNCTIONS

R. KUMAR* and S.D. KHEPAR**

*Department of Agricultural Engineering, Haryana Agricultural University, Hissar (India) **Department of Soil and Water Engineering, Punjab Agricultural University, Ludhiana (India) (Accepted 13 December 1979)

ABSTRACT Kumar, R. and Khepar, S.D., 1980. Decision models for optimal cropping patterns in irrigations based on crop water production functions. Agric. Water Manage., 3: 65--76. A study was conducted to demonstrate the usefulness of alternative levels of water use over the fixed yield approach when there is a constraint on water. In the multi-crop farm models used, a water production function for each crop could be included so that one has the choice of selecting alternative levels of water use depending upon water availability. Water production functions (square root and quadratic type) for seven crops, viz. wheat, gram, mustard, berseem, sugarcane, paddy and cotton, based on experimental data from irrigated crops were used. The fixed yield model was modified incorporating the stepwise water production functions using a separable programming technique. The models were applied on a selected canal co m m an d area and optimal cropping patterns determined. Sensitivity analysis for land and water resources was also conducted. The water production function approach gives better possibilities of deciding upon land and water resources.

INTRODUCTION

In modern agriculture, water plays a vital role. It is, therefore, essential to have the o p t i m u m use of land and water resources. In irrigated agriculture, where various crops are competing for a limited quantity of water, linear programming is one of the best tools for the optimal allocation of water resources (Anonymous, 1969~ Maji and Sarkar, 1976; Singh and Sirohi, 1976; Sirohi et al., 1976; Gulati, 1977). With these linear programming models, it is possible to choose among crops, b u t alternative levels of water use for each crop could n o t be found. When water becomes scarcer or the cost of irrigation water increases, the farmer would like to adjust the cropping pattern b y decreasing the area under crops that demand more water or by applying less water to the crops. Pomareda (1977) carried o u t a theoretical analysis of water production functions using linear programming. 0 3 7 8 - - 3 7 7 4 / 8 0 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 2 5 © 1980 Elsevier Scientific Publishing Company

66

The objective of this paper is to demonstrate the usefulness of alternative levels of water use over the fixed yield approach when there is a constraint on water. The approach used is an extension of the theoretical analysis by Pomareda (1977) to a canal c o m m a n d area. The models used for the study are described below. DESCRIPTION OF MODELS

The fixed yield model The farm objective function can be specified as the maximization of the annual net returns, Z, subject to constraints on the availability of water and other inputs. The objective function then becomes Max Z = Z P i X i Y i -

ZAijXiCij

(1)

where i = an integer pertaining to crop (1,2 ... n);] = an integer pertaining to input (1,2 ... r); P = unit price of crop; X = area under crop; Y = crop yield/ unit area; A = a m o u n t of input; C = cost of input. The function is subject to the water availability constraint (2)

WiXi <~ w

where W = a m o u n t of water/unit area to produce a crop yield Y; w = total a m o u n t of available water. Constraints on other inputs is

ZAijXi <~Sj

(3)

where S = m a x i m u m availability of input. Further, the constraint must be positive here

Xi ~ 0

(4)

This model is a linear programming formulation, except the objective function which can be made linear by approximating the production functions by linear segments. The problem can be formulated as a modified separable programming model. The objective function is separable into a sum of convex functions of individual variables (Hillier and Lieberman, 1967).

Model with alternative levels o f water use A generalized crop water stepwise production is shown duction function has been divided into k linear segments, (1,2 ... s). A decrease in the a m o u n t of water from W0 to implies a decrease in yield from Y0 to Yk, i.e. A Yik. Two were considered in the production function. (i) (W0, Y0) is the point which identifies the m a x i m u m the corresponding water use.

in Fig. 1. The prowhere k is an integer Wk, i.e. A Wik, reference points yield attainable and

67

i Y1 Y2

Y =F(W)

o

/

J

~r

-v s C3

7 WS --WATER

W2 W1 W0 APPLIED ( m m } ~

Fig. 1. Generalized stepwise water production function.

(ii) (Ws, Ys) is the point which identifies the lowest crop yield and corresponding water use. The crop which does n o t at least give a yield equal to 7"8 should n o t be considered. The zone between Ys and Yo is the Rational Zone for resource allocation. It starts from a point where the average product/unit of resource is maximum and ends at the point where the maximum: p r o d u c t is attained. The level of water use m a y be chosen anywhere between s and o, where the marginal value of the p r o d u c t equals the price of water. The model given by eqns. 1--4 is modified n o w by taking into account stepwise production functions for the crops under consideration. The objective function then becomes M a x Z = [ ~2Xio Yio - ~ ~-,XikA Yik ]Pi - ~ A i y ( X i o - X i k ) Cij

(5)

subject to following constraints (a) Water availability constraint Y'XioWio - Y-,Y-,XikAWik < w

(6)

(b) Constraint for other inputs

ZY, Aiy(Xio - Xik ) < S1

(7)

(c) Requirement that total area equals total planted area X i k - Xio < 0

(8)

(d) Non-negativity constraint

Xio >~ 0

(9)

Xik >~ O The model is basically broken down into four blocks of activities: producing activities, production functions, p r o d u c t selling activities, and input purchasing activities (Table I).

Irrigation Constraints

Objective function Yield balances Input Requirements Water balance

Rows

-1

1--1

-AW~

-~W~k 1-

1

-AWn~

-1

W~o--Wno

AY~k A Yn,

Z~Yll

Water production functions X~ 1- - X l k Xn 1

-Yno AI 1 An i Aii--Anl

-Yl0

Cropping activities Xio - - X n o

Columns

Linear programming table to incorporate production function

TABLE 1

-AWnk

A Ynl~

Xn k

PI 1

1

Pn

-1

-Ci ~

Product sales Input S~ Sn purchase

-1

< 0

< W

< Sj

-Cij Max < 0

RHS

00

69

The producing activities are d e n o t e d by Xi (i = 1 ... n). The yields and level of water use assigned to these activities are the m a x i m u m attainable values (W0, Y0 ), as shown in Fig. 1. Production functions are included in the model using the second group of activities. Each production function has s steps. The A Wik for the i-th crop and the k-th step indicates the a m o u n t of water saved when the yield decreases by the a m o u n t A Yik. All the water savings are accounted for in the water balance row and yield decreases in block diagonal form in the yield balance row. The model also includes the requirement t h a t the total area from which water is saved, i.e. total area planted, equals total area irrigated. Input purchasing and product selling activities are assumed to take place at a fixed price and supply of water is assumed fixed. To safeguard the interest of the farmers as well as of the country, the following additional constraints were considered in the model: (a) the minimum area under berseem should be equal to 10% of the total in the winter season; (b) the m a x i m u m area under mustard should not exceed 15% of the total area in the winter season; (c) m a x i m u m area under sugarcane should not exceed 10% of the total area in the winter season. RESULTS AND DISCUSSION

The models considering fixed yields and alternative levels of water use were applied on a selected canal c o m m a n d area (Kotkapura distributory, Ferozepur district, Punjab) having a discharge of 45 1/s to determine the optimal cropping patterns. The different crops considered were wheat, gram, mustard, berseem, sugarcane, c o t t o n and paddy. Wheat, gram, mustard and berseem are winter season crops, and c o t t o n and paddy are monsoon season crops. Sugarcane, being an annual crop, is considered both in the winter and m o n s o o n seasons. Maximum culturable c o m m a n d areas in winter and monsoon seasons were 173 and 139 ha, respectively. Maximum water availabilities were considered under three sets of conditions: (a) water losses in main water courses and field channels are 25%; (b) water losses are reduced to 10% by lining main water courses; (c) water losses are negligible by lining all channels.

Optimal cropping patterns (using fixed yields) Market price of crops, gross return/unit area corresponding to m a x i m u m yield for each crop, variable costs/unit area (except the cost of water) for each crop, and the a m o u n t o f water to be applied to each crop are given in Annex 1 The prevailing price of water was Rs. 0.423]ha-mm. Net returns per unit area corresponding to m a x i m u m yield for each crop are given in Annex 1. The model was run for three sets of water availability conditions, and area constraints and optimal cropping patterns given in Table II determined. The re-

70 TABLE II Optimal areas under different crops for different sets of conditions of water availability for fixed yields (in ha) a Sample Crop No.

1 2 3 4 5 6 7

Wheat Gram Mustard Berseem Sugarcane Cotton Paddy

Maximum availability of water (ha-mm) 84 457 (water loss is 25%)

100 178 (water loss is 10%)

11 275 (no loss)

0 113.0 26.00 17.00 17.00 76.40 0

0 113.00 26.00 17.00 17.00 108.19 0

20.452 92.547 26.00 17.00 17.00 122.00 0

Total area under crops for different seasons (ha) (i) Winter 173.00 173.00 season

(ii) Monsoon 93.40 season Total net returns (Rs.) 647 627.16

173.00

125.19

139.00

741 157.33

785 061.11

a Water use and yields correspond to maximum limit.

sults show that during the winter season, the total land area will be occupied for all three water availabilities. In the monsoon season, a certain area is left unused at lower availabilities of water, i.e. 94.9 and 125.2 ha are occupied against 139 ha at water availability of 83 457 and 100 178 ha-mm. Wheat has been fully replaced by gram except in the last case. This is probably due to the higher water requirement of wheat. The total net returns increase from Rs 647 627.16 to Rs. 785 061.11 with the increase in the a m o u n t of water from 83 457 to 111 275 ha-mm. Optimal cropping patterns (using alternative levels of water use)

Quadratic and square root type production functions for wheat, gram, mustard, berseem, sugarcane, cotton and paddy, derived from experimental data on irrigation of crops (Dastare et al., 1970; Anonymous, 1972--1977), were used. The empirical relationships for the production functions are expressed as Square r o o t function Y = a - b w + c w °'s

(10)

where, Y = crop yield (qt/ha); w = a m o u n t of water/unit area to produce yield Y (qt/ha); a, b, c, = empirical coefficients.

71 Quadratic function Y= a + bw-

cw 2

(11)

The estimated values of the coefficients are given in Table III. The functions were linearized as described above. Rational zones of resource allocation were defined for each function. Using these functions, the a m o u n t of water for minimum and m a x i m u m yields, and m a x i m u m yield for each crop given in Table III were estimated. The production functions were divided into linear segments using the technique given in Fig. 1. Water and yield b~l~nces for each crop were determined from stepwise production functions. The estimated values of water and yield balances for different segments are given in Table IV. Market price of crops, price of water, variable costs/unit area (except the cost of water), amounts of water available under three sets of conditions and area constraints were introduced. The data were fed into a computer and optimal cropping given in Table V determined. It can be observed from Table V that the area under different crops for different water availabilities is the same. Gram and p a d d y crops do n o t figure in all the three cases. During the winter and monsoon seasons, total area is occupied. The total net returns increased from Rs. 792 611.22 to Rs. 800 652.57 with the increase in water availability. Table VI shows the water use and yield obtained/unit area for different crops for three sets of water availability conditions. Water use and yields for wheat, mustard and sugarcane are the same for all three sets of conditions. The water use for wheat and mustard are 200 and 190 m m / h a against the m a x i m u m values of 307.20 and 320.71 mm/ha, respectively, and the corresponding yields are 36.28 and 18.30 qt/ha against maximum yields of 36.58 and 18.44 qt/ha. In the case of sugarcane, water use and yield are at m a x i m u m limit. The water use for berseem increases from 550 to 608.15 m m / h a against 716.05 ram/ha, while the water use for c o t t o n increases from 306.10 to 526.05 m m / h a against 526.05 m m / h a with the increase in a m o u n t of water. Corresponding to the increase in water use, the yields for berseem and cotton increase from 780.00 to 783.90 qt/ha and from 13.36 to 13.76 qt/ha, respectively. The model adjusts the water use when the availability of water changes. Although the optimal areas under different crops is the same for the given amounts of water available, the total net returns increase with the increase in water. Comparison of the results in Tables II and V reveals t h a t the area under mustard, berseem, sugarcane and p a d d y is the same in both models. The area under c o t t o n is more at lower water availabilities in the case of alternative levels of water use as compared to fixed yields, whereas, in the last case, the area is equal in both models. Alternative levels of water use allows higher net returns. The difference in the net returns for both models decreases with increase in the a m o u n t of water.

Wheat Mustard Bemeem Cotton Paddy

26.5235 14.7430 25.5379 6.6033 5.9384

Gram 15.4759 Sugarcane - 1 1 . 5 4 4 1

0.04561 2.92837

0.03274 0.011537 1.0692 0.013607 0.035206 0.00019 0.0027

1.14767 0.41322 57.2238 0.62418 2.412043

c

0.8611 0.999

0.645 0.9006 0.979 0.953 0.296

R2

0 65.388

0 0 0 0 0

Value of W at which Y is m i n i m u m (ram/ha)

119.76 542.30

307.20 "320.71 716.16 526.05 1173.48

Value of W at which Y is m a x i m u m (mm/ha)

Wheat

Gram

Mustard

Berseem

Sugarcane

Cotton

Paddy

1

2

3

4

5

6

7

307.20 -36.58

119.76 -18.20

320.71 -18.44

716.05 -791.20

542.30 -782.50

526.05 -13.76

1173.48 -47.25

W.B. Y.B.

W.B. Y.B.

W.B. Y.B.

W.B. Y.B.

W.B. Y.B.

W.B. Y.B.

W.B. Y.B.

0

-173.48 0.25

-126.05 0.16

-62.30 12.50

-166.05 11.20

-130.71 0.14

-39.76 0.20

-107.20 0.30

1

N u m b e r of segments

-473.48 2.05

-246.05 0.46

-142.30 52.50

-266.05 36.20

-190.71 0.44

--69.76 0.90

-182.20 1.50

2

W.B. = W a t e r b a l a n c e ( r a m / h a ) ; Y.B. = y i e l d b a l a n c e ( q t / b a ) .

Crop

Sample No.

-673.48 5.05

-346.05 1.20

-242.30 157.50

-366.05 71.20

-245.71 0.94

-94.76 1.70

-207.20 2.00

3

-773.48 7.25

-406.05 1.96

-342.30 317.50

-466.05 126.20

-295.71 1.89

-119.76 2.72

-232.20 2.50

4

-873.48 10.25

-446.05 2.66

-392.30 420.00

-566.05 226.20

-308.71 2.44

-257.20 3.58

5

W a t e r a n d yield b a l a n c e s f o r d i f f e r e n t c r o p s ( f r o m s t e p w i s e w a t e r p r o d u c t i o n f u n c t i o n s )

TABLE IV

6 7

(ii) Quadratic function

1 2 3 4 5

b

Coefficients of the functions

a

(i) Square root function

Sample Crop No.

Production functions for different crops used in this study

TABLE III

-522.05 5.96

-666.05 416.20

-297.20 6.78

7

-526.05 7.16

-696.05 531.20

-307.20 10.06

8

-716.05 765.20

9

-973.48 -1073.48 -1123.48 -1173.48 14.25 20.75 26.05 41.31

-486.05 3.66

-477.30 615.00

-616.05 301.20

-320.71 3.69

-282.20 5.18

6

18.2072 782.50

36.58 18.44 791.20 13.76 47.25

Maximum value of Y (qt/ha)

73 TABLE V Optimal areas under different crops for different sets of conditions of water availability for alternative levels of water use Sample Crop No.

Maximum availability of water (ha-mm) 83 457

1 2 3 4 5 6 7

Wheat 113.00 Gram 0 Mustard 26.00 Berseem 17.00 Sugarcane 17.00 Cotton 122.00 Paddy 0

Total area under crops for different (i) Winter 173.00 season (ii) Monsoon 139.00 season Total net return (Rs) 792 611.22

100 178

111 275

113.00 0 26.00 17.00 17.00 122.00 0

113.00 0 26.00 17.00 17.00 122.00 0

seasons (ha) 173.00

173.00

139.00

139.00

799 725.59

800 652.57

TABLE VI Water use and yield for alternative levels of water use Sample Crop No.

Maximum availability of water (ha-mm) 83 457

100 178

Water use (ram/ha)

Yield (qt/ha) 36.28 (36.58)

Water use (ram/ha)

1

Wheat

200.00 (307.20)

2

Gram

.

3

Mustard

190.00 (320.71)

18.30 (18.44)

190.00 (320.71)

4

Berseem

550.00 (716.05)

780.00 (791.20)

5

542.30 Sugarcane (542.30)

6

Cotton

306.10 (526.05)

7

Paddy

.

.

.

111 275 Yield (qt/ha)

36.28 (36.58)

18.30 (18.44)

190.00 (320.71)

18.30 (18.44)

550.00 (716.05)

780.00 (791.20)

608.15 (716.05)

783.90 (791.20)

782.50 (782.50)

542.30 (542.30)

782.50 (782.50)

542.30 (542.30)

782.50 (782.50)

13.36 (13.76)

443.27 (526.05)

13.65 (13.76)

526.05 (526.05)

13.76 (13.76)

.

Values in parentheses pertain to fixed yields.

.

.

36.28 (36.58)

Yield (qt/ha)

200.00 (307.20)

.

200.00 (307.20)

Water use (ram/ha)

.

.

.

.

74

Sensitivity analysis for water and land resources The sensitivity analysis was carried out to s t u d y the effect of changes in water and land resources on the total net returns. The results for fixed yields and alternative levels of water use given in Table VII show that in the case of fixed yields, at different levels o f available water each unit of reduction in water from 83 457 to 43 263; 100 178 to 43 263; and 111 275 to 107 441 ha-mm will reduce the net returns by Rs. 5.59, 5.59 and 0.85/ha-mm, respectively. Each unit of water added beyond 83 457 to 107 441; 100 178 to 107 441 and 111 275 to 128 622 ha-mm will add the same amounts to the value of n e t returns. In the case of alternative levels of water use, each unit of reduction in water from 83 457 to 80 269; 100 178 to 9 4 9 0 9 and 111 275 to 110 287 ha-ram decreases the net returns by Rs. 0.58, 0.087 and 0.05/ha-mm, respectively. Similarly, each unit of water added beyond 83 457 to 94 909; 100 178 to 110 287 and 111 275 to 113 110 ha-mm adds the same amounts to the net returns. Beyond these limits, the decrease or increase in net returns may be more. As the water is a scarce resource, the addition is difficult. The reduction in net returns is much less in the case of alternative levels of water use as compared to fixed yields. In the case of fixed yields, each unit of reduction in land in the winter season from 173.00 to 60.00, 112.35 and 116.53 decreases the net returns by Rs. 853.96, 853.96 and 1421.52/ha, respectively, for the three sets of water availabilities, and addition of each unit in land in the winter season b e y o n d 173.00 to 508.61, 648.23 and 205.01 ha increases the net returns by the same amounts. Each unit of reduction in land in the monsoon season from 93.40 to 53.14; 125.19 to 84.92 and 139.00 to 106.02 decreases the net returns by Rs. 2493.03/ha in all three cases, and addition of each unit in land in the monsoon season b e y o n d 93.40 to 119.13; 125.19 to 150.91 and 139.00 to 146.28 ha will still decrease the net returns, except in the last case when the net returns increase by Rs. 2493.03/ha. As in the first two cases, a certain area in the monsoon season is left unused; the addition of land, instead of increasing, decreases the net returns. In alternative levels of water use, each unit of reduction in land in the winter season from 173.00 to 115.73, 122.45 and 163.84 reduces the net returns by Rs. 1 5 7 3 . 6 5 , 1672.52 and 1680.07/ha, respectively, and addition of each unit of land increases the net returns by the same amount. In the monsoon season, the reduction of each unit of land from 139.00 to 110.36, 119.78 and 135.51 ha reduces the net returns by 2698.92, 2896.66 and 2916.57/ha, respectively, and addition of each unit of land b e y o n d 139.00 to 150.38, 152.17 and 140.87 adds the same amounts to the net returns. It is always desirable to bring more area under cultivation. The alternative levels of water use allow much higher returns by addition of land as compared to fixed yields. From these conclusions, it is obvious that alternative levels of water use is superior to fixed yields for optimal utilization o f land and water resources.

173.00 a (173.00)

83457 (83457)

Land (monsoon 93.40 a season) (139.00)

Land (winter season)

Water

-2493.03 (-2698.92) -3751.06 (+2698.92)

--853.96 (-1573.65) +853.96 (+1573.65)

60.00 (115.74) 508.61 (188.94)

53.14 (110.36) 119.13 (150.38)

-5.59 (-0.581) +5.59 (+0.58)

43263 (80269) 107441 (94909)

125.20 a (139.00)

173.00a (173.00)

100178 (100178}

Resource used

Shadow price

Resource used

Range

100 178

83 457

Maximum availability of water (ha-mm)

a Figures are in ha. Figures in parentheses pertain to alternative levels of water use.

1

Sample Resource No.

84.92 (119.78) 150.91 (152.17)

112.35 (122.46) 648.23 (199.34)

43263 (94909) 107441 (110287)

Range

-2493.03 (-2896.66) -3751.06 (+2896.66)

--853.96 (-1672.52) +853.96 (+1672.52)

-5.59 (-0.087) +5.59 (+0.087)

Shadow price

Sensitivity analysis for resources (using fixed yields and alternative levels of water use)

TABLE VII

139.00 a (139.00)

173.00 a (173.00)

111275 (111275)

Resource used

111 275

146.02 (140.87)

(135.51)

106.02

116.53 (163.82) 205.01 (177.94)

107441 (110287) 128622 (113110)

Range

-2493.07 (-2916.57) +2493.03 (+2916.57)

--1421.52 (-1680.07) +1421.52 (+1680.07)

-0.85 (-0.05) +0.85 (+0.05)

Shadow price

c~

76 REFERENCES Anonymous, 1969. Efficient water use and farm management study. Joint Indo-American Report, The Ralph M. Parsons Company, Los Angeles, CA, New York, NY. Gulati, H.S., 1977. A production function based model for canal water optimization. Thesis for the degree of M. Tech. (Agric. Eng. ), Punjab Agricultural University, Ludhiana, India, pp. 1--100. HiUier, F.S. and Lieberman, G.J., 1967. Introduction to Operations Research, Holden-Day Inc., San Francisco, CA. Maji, C.C. and Sarkar, T.K., 1976. Optimal cropping system and irrigation efficiency in existing canal-command area. All India Symposium on Agricultural Systems Theory and Applications, Punjab Agricultural University, Ludhiana, pp. 56--61. Pomareda, C., 1977. Economic analysis of irrigation production functions - - a n application of linear-programming. Unpublished paper, World Bank, Washington, DC, pp. 1--11. Singh, L and Sirohi, A.S., 1976. An application of linear programming for optimal allocation of canal and tubewell water among crop areas. All India Symposium on Agri cultural Systems Theory and Applications, Punjab Agricultural University, Ludhiana,

pp. 111--116. Sirohi, A.S., Hiremath, K.C. and Sharma, V.K., 1976. Use of linear programming and simulation for temporal and spatial allocation of irrigation water of reservoir projects. All India Symposium on Agricultural Systems Theory and Applications, Punjab Agricultural University, Ludhiana, pp. 34--40.

ANNEX I Cost'.' and returns (Rs/ha) in the case of fixed yield approach Sample No.

Crop

Unit price of crop (Rs/qt)

Gross return (Rs/ha)

Costs except the cost of water (Rs/ha)

M a x i m u m Cost of water water applied (Rs/ha) (ram/ha)

Total cost (Rs/ha)

Net return (Rs/ha)

1 2 3 4 5 6 7

Wheat Gram Mustard Berseem Sugarcane Cotton Paddy

122.50 147.80 341.40 7.00 13.50 401.70 89.00

4483.80 2691.50 6295.28 5538.40 10563.75 5527.60 4205.25

2669.80 1117.00 1699.55 2558.60 5090.48 2362.55 2439.68

307.20 119.76 320.71 716.05 542.30 526.05 1173.48

2799.80 1167.65 1835.15 2861.60 5320.48 2585.15 2935.68

1684.00 1523.85 4460.13 2676.80 5243.27 2942.55 1269.57

a a a

a a

a Price considering income from straw of the crops.

130.00 50.65 135.60 303.00 230.00 222.50 496.00