Prioritizing cropping alternatives based on attribute ...

Journal of the Saudi Society of Agricultural Sciences xxx (2017) xxx–xxx

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Prioritizing cropping alternatives based on attribute specification and comparison using MADM models C.P. Devatha, Arun Kumar Thalla ⇑ Department of Civil Engineering, National Institute of Technology-Karnataka, Surathkal, Mangalore 575025, Karnataka, India

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 July 2017 Revised 19 September 2017 Accepted 27 September 2017 Available online xxxx

This paper presents a logical procedure and its effectiveness to handle set of alternatives for attaining rabi cropping pattern by multiple attribute decision making (MADM) approach which includes methods viz. simple additive weighting (SAW), weighted product method (WPM), technique for order preference by similarity to ideal solution (TOPSIS) and preference ranking organization method for enrichment evaluations (PROMEHTEE). The study area is concerned with banahil distributary of akaltara branch canal of Hasdeo Bango irrigation command, Janjgir-Champa district, Chattisgarh, India. It covers 8 villages of Akaltara Block and 14 villages of Pamgarh Block in Janjgir-Champa District. Information on various attributes/criteria like type of crops (wheat, mustard, gram, safflower, sunflower), type of soil (clay, clay loam, sandy loam) cropped area, water usage, cost of production, cost of cultivation (including irrigation cost) and sale price of crops had been collected from various government departments (Agriculture and Water Resources) etc. and group of farmers from the local command area. Results obtained with MADM approach is compared with the non-linear optimization model (NLP) developed using LINGO standard optimization package. Based on the above decision making method and LINGO model results, wheat is found to be most profitable crop followed by sunflower. Performance of MADM is found to be satisfactory and ranking had been obtained for the crops considered in the study. Ó 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Cropping pattern Decision making models Water management Ranking

1. Introduction The use and management of available water resources are a critical concern in the developing scarce scenario due to rising demand in agriculture, domestic and industrial needs. These demands still increase under the challenging environment viz. climate change, urbanization, greenhouse effect, etc. The scarcity of water in many parts of the world emphasizes on the need to maximize the benefits from the irrigation system by adopting effective & efficient water management systems/policies. Irrigation management is essential to meet the growing demands by appropriate allocation of water for irrigation projects, optimal cropping pattern subjected to various constraints like type of soil, type of crop, water ⇑ Corresponding author. E-mail addresses: (A.K. Thalla).

[email protected]

(C.P.

Devatha),

[email protected]

Peer review under responsibility of King Saud University.

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use, crop area etc., and also to obtain maximum benefits. Optimization models are most widely accepted in the field of irrigation system planning and management to achieve the said goals (Mosleh et al., 2017). Optimization models using Linear Programming (LP) technique had been extensively used in the field of water resources like ground water management, sea water intrusion, irrigation management, reservoir system operation management and hydro power management (Singh 2012). But to incorporate simultaneously varying constraints involved in the problems related to irrigation, Dynamic programming (DP) (Dai and Li 2013; Tran et al. 2011; Alaya et al., 2003; Vedula and Mujumdar 1992) is one of the well known techniques used in the field of irrigation water management (Singh 2012). DP involves non-linear variables in the model. However, non-linear programming (NLP) technique generally involves complex mathematical equations to be solved hence its use is limited. Some of the research work carried out using NLP model are to address deficit irrigation to maximize net returns with the available water resources, integrated soil-water balance to determine the optimal reservoir release policies, farm optimization with adequate and limited water supplies, achieving optimal cropping pattern (Garg and Dadhich, 2014;

https://doi.org/10.1016/j.jssas.2017.09.007 1658-077X/Ó 2017 The Authors. Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Devatha, C.P., Thalla, A.K. Prioritizing cropping alternatives based on attribute specification and comparison using MADM models. Journal of the Saudi Society of Agricultural Sciences (2017), https://doi.org/10.1016/j.jssas.2017.09.007

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C.P. Devatha, A.K. Thalla / Journal of the Saudi Society of Agricultural Sciences xxx (2017) xxx–xxx

Georgiou and papamichail 2008; Alaya et al. 2003; Ghahraman and Sepaskhah 2002). It is seen from the literature that almost all the optimization models were developed for specific conditions involving specific constraints like economical benefits, crop production, water requirments etc., though with well-known overall objective being the maximization of benefits. However, to achieve the goal i.e. maximization of benefits, proper planning and decision making process plays a key role. Hence there is a need to develop a mathematical tool that is simple, logical, so as to guide the decision makers in selecting a suitable alternative among various possibilities. The multiple attribute decision making (MADM) tool can handle concurrently the various criteria which may be dimensional or non-dimensional in nature. It is an excellent decision making tool for evaluating and ranking/prioritizing the alternatives even when the criteria involved is complex(Geng and Wardlaw, 2013; Venkata Rao, 2008; Rao and Patel, 2010). Recently MCDM is applied for assessing risk involved in transport of trace metals in river Ganges by Srinivas et al. (2017); ground water vulnerability assessment for Unnao and Rae Bareli district, Uttar Pradesh, India by combining Geographical Information Systems (GIS) & MCDM Approach (Agarwal and Garg, 2016) and spatial flood vulnerability mapping in view of climate change effect by MCDM approach (Song and Chung, 2016). MADM methods find application in various fields of water resources management like flood management, irrigation water pricing, integrated water resources management, irrigation system evaluation, assessing flood risk, operational management of water supply system, sustainable water resource planning, water loss reduction in supply lines, importance of land use and seasonal factors on ground water pollution by various researchers (Chitsaz and Banihabib 2015; Azarnivand et al., 2015; Geng and Wardlaw, 2013; Carroll et al., 2013; Roozbahani et al., 2012; Mutikanga et al., 2011; Behzadian et al., 2010; Latinopoulos, 2008; Raju and Vasan, 2007; Srdjevic et al., 2004; Raju et al., 2000). MADM is also referred as discrete multiple criteria decision making MCDM (Korhonen et al., 1992). Commonly used MCDM approaches include Compromise Programming (CP) (Zeleny, 1973), Goal Programming (GP) (Charnes and Cooper, 1977), Simple Additive method (SAW), Weighted Product Method (WPM), Analytical Hierarchy Process (AHP) (Saaty, 1980), technique for order preference by similarity to ideal solution (TOPSIS), Multiple attribute utility theory (MAUT), elimination and choice translating reality (ELECTRE) and Preference Ranking Organisation Method for Enrichment Evaluations (PROMTETHEE). These methods not only finds application in water resources but also in various fields like bio-degradable waste management options (Karagiannidis and Perkoulidis, 2009), selection of feed stock for anaerobic digestion (Rao and Baral, 2011), manufacturing environment (Rao and Patel, 2010), thermal power plants (Garg et al., 2007), electroplating systems (Kumar and Agrawal, 2009) and selection of robots (Bhangale et al., 2004). Hence, in this study an attempt is made to implement some of the MADM methods to obtain rabi cropping pattern. SAW, WPM, TOPSIS and PROMETHEE methods were used in the prioritization and ranking of different alternatives considered. The result thus obtained in MADM approach is compared with the well accepted standard optimization package using LINGO (LINGO, 2006). 2. Materials and methods 2.1. Study area Minimata (Hasdeo) Bango multipurpose project (Irrigation & Hydel Project) is situated near the village Bango on Hasdeo river,

the largest tributary of Mahanadi river in Korba district of Chhattisgarh, India. The command area lies between 21° 510 to 21° 590 North latitude and 82° 170 to 82° 260 East longitude. The length of Banahil distributary considered in the present study is 12.78 km. The banahil distributary started in 1991 to irrigate 11,106.43 hectare of kharif paddy crop that covers 8 villages of akaltara block and 14 villages of pamgarh block in janjgirchampa district with a design discharge of 10.43 m3/sec. The map of banahil distributary with benefited village boundary and is shown in Fig. 1(a and b). The climate of the study area is hot and semi-humid. Normally, the temperature is maximum in the month of May and minimum in the month of January. The monthly average maximum and minimum temperature varies from 27.3 °C to 41.7 °C and the relative humidity of the area varies from 25.3 to 95.1% and its average value is always greater than 38%. The study area receives rainfall mainly from the south west monsoon of about 72% from midJune to third week of September. The average annual rainfall of the area is 1350 mm. The data on crops, weather, soils, canal and cost of cultivation pertaining to the study area is collected from various state government departments/agencies like water resources, rural development (Panchayat), agriculture and agricultural research station and also from farmers of the command area. Data on cropping pattern and agronomic management for rabi-summer seasons for years (1995–2011) is obtained from the Agricultural department of Janjgir-Champa district, Chhattisgarh. The soil of the command area is classified as, sandy clay loam, clay loam and clay. The cost of cultivation of different crops grown in different soils of the study area is estimated in consultation with agricultural department and agricultural water management scientists and group of farmers, working in the same command area. Cost of cultivation includes the following operations (i.e. plant production and plant protection). The experimental data, ‘‘water applied vs. yield” of different crops in different soils, were collected from the annual progress reports, Indian Council of Agricultural Research (ICAR), Bilaspur, chhattisgarh, India. 2.2. Model formulation In the subsequent section the formulation of decision matrix is discussed to choose the best alternative for achieving maximum returns with five most commonly grown rabi crops (mustard, sunflower, wheat, gram and safflower) subjected to the various dimensional and non-dimensional attribute/criteria (crop area, water usage, sale price of crop, cost of cultivation, crop production) based on the significance of the objective considered. The minimum available canal water during the study period has been considered as 35.34 Mm3. The land area constraints for certain crops are fixed as minimum so that the most profitable crops should not be dominant over the entire command area, which will also fulfill the basic food requirement of local people. The limitation of minimum area for each crop has been fixed as per the present cropping pattern (wheat 30%, sunflower 4%, mustard 16%, gram 8%, and safflower 4%) of cultivated area in rabi season. The entire command area comprises of three types of soil namely clay (539.0 Ha), clay loam (1795.7 Ha) and sandy clay loam (4017.0 Ha). In each soil type, above mentioned crops and cropping percentage were considered to grow. Total crop area is calculated under mustard, sunflower, wheat, gram and safflower according to the mentioned percentage classification. The influencing factor of the model is water production function (relationship between crop yield and water application is termed as WPF). The WPF of mustard, sunflower, wheat, gram, safflower


3


Fig. 1. (a) Banahil distributary (b) Village boundary. Source: ICAR water management project IGAU Bilaspur (C.G.)

Table 1 Formulation of model attributes. Crops used (Alternatives)

Crop Area (CA) (Ha)

Crop production (CP) (kg/Ha)

Cost of Cultivation (CC) INR Per Ha

Sale Price (SP) INR per kg

Water usage (WU)

Mustard (A1) Sunflower (A2) Wheat (A3) Safflower (A4) Gram (A5)

1777.03 1795.71 6200.92 444.26 888.52

1500 1500 3200 1200 1500

13420 13470 18920 10269 13941

25 28 12.85 25 28

Below average Average Above Average Very low Low

and summer rice were developed (Hexem and Heady 1978). Based on the above information, water usage by crop for better yield is determined within lower and upper limits and it results into quantitative attributes. The sale price of crops is considered as per the guidelines of state agricultural department. The constraints like crop area, crop production, sale price and cost of cultivation were qualitative attributes (Value based) and water usage (WU) is quantitative attribute. Based on expert judgement from the command area, inter relationships between the attributes have been established. Crop area and crop production are beneficial attributes whereas cost of cultivation and sale price are non-beneficial attributes. In this case, qualitative attributes were subjected to ranked value judgment on a fuzzy conversion scale. By using fuzzy set theory, the scale of the attributes were converted to corresponding fuzzy numbers and then converted to the crisp scores as proposed by Chen and Hwang (1992). Based on the data collection and information from the various departments/agencies and expert judgements, data obtained is developed in the form of decision matrix presented in Table 1.

3. MADM methods 3.1. SAW method This is simple method and it is recommended by Edwards et al. (1982). The equation is of the form.

Pi ¼

M X W j ðmij Þnormal

ð1Þ

j¼1

Where mij is the measure of performance of jth decision criterion, (mij)normal represents the normalized value of mij, Wj is the weight of importance of jth criterion and Pi is the overall or composite score of the crop alternative (Ai). The alternative with the highest value of Pi is considered as the best alternative. Previously this method is suitable only for identical units of measures and it can be extended to non identical units of measures once elements in the decision matrix is normalized. 3.2. WPM method Each alternative is compared with others by multiplying a number of ratios, one for each criterion. Each ratio is raised to the power of the relative weight of the corresponding criterion dimensionless (Chen and Hwang, 1992). Since it is raised to powers, all the attributes becomes generally, in order to compare the two alternatives

Pi ¼

M Y ½ðmij Þnormal wj

ð2Þ

j¼1

Each normalized value of an alternative with respect to an attribute, i.e. (mij) normal, is raised to the power of the relative weight of the corresponding attribute. The alternative with the highest Pi value is considered as the best alternative.


4


ratio, CR should be less than 0.1). Relative importance matrix developed for the present study is shown in Eq. (5)

3.3. TOPSIS method This technique is developed by Hwang and Yoon (1981) based on the concept that the best alternative will be chosen by the shortest distance from the ideal solution. Steps involved in TOPSIS method is explained in the following section. 1. Development of decision matrix: Data collected for the various attributes/criteria are put together and it is represented in the form of matrix which is known as decision matrix. 2. Development of relative importance matrix: This matrix is developed based on the judgments obtained corresponding to relative importance of one attribute to the other attribute considered in the study from the field experts. 3. Normalised decision matrix: It is obtained by normalizing the decision matrix. This step is to take care of all dimensional and non dimensional units. 4. Weighted normalized matrix: Information from weight matrix is incorporated into normalized matrix to obtain weighted normalized matrix. 5. Determination of ideal (best) I+i and negative ideal (worst) I-i solutions: It is associated with beneficial and non-beneficial attributes. After determining the weighted normalised matrix, ideal best I+i and negative ideal worst I i solutions were estimated. For I+i , in case of beneficial attribute, I+i indicates higher value of the attribute else if it is non-beneficial attribute, I+i indicates lower value of the attribute. For I i , in case of beneficial attribute, Ii indicates lower value of the attribute else if it is non-beneficial attribute, I+i indicates higher value of the attribute. 6. Calculation of separation of measures: Separation measure for positive ideal and negative ideal solutions is obtained based on the Euclidean distance between any alternative and ideal solution. 7. Determination of suitability index: Suitability index is calculated to find the relative closeness to the ideal solution. Based on this index values, ranking has been performed. Decision matrix developed in Section 2.2 (Table 1) is used for the formulation of normalized matrix as shown in Eq. (3) which is carried out to make the elements of decision matrix to vary from 0 to 1 so that it will eliminate the problem of dimensionality of the attributes.

CA CP 2 A1 0:287 0:469 6 A2 6 0:290 0:469 6 A3 6 6 1:000 1:000 6 A4 4 0:072 0:375 A5 0:143 0:469

CC SP WU 3 0:765 0:514 0:817 7 0:762 0:459 0:670 7 7 0:543 1:000 0:568 7 7 7 1:000 0:514 0:761 5 0:737 0:459 1:000

CA CP CC SP WU

2

CA CP 1 4

6 6 1=4 6 6 1=3 6 6 4 1=2 1=5

SP WU 3 2 5 7 1 1=3 1=2 2 7 7 3 1 4 1=3 7 7 7 2 1=4 1 1=2 5 1=2 1=3 2 1

0:263 0:349 0:420 0:458 0:422

Wj ¼ ð0:4724 0:1103 0:1921 0:1196 0:1055Þ

ð6Þ

For which CR value is 0.09 which is less than 0.1. Hence it shows good consistency with the judgements. Next step is to obtain weighted normalized matrix and it is calculated by incorporating weights into the normalized matrix and is shown in Eq. (7).

2

0:1240 0:0385 0:0807

6 6 0:1253 6 6 0:4328 6 6 4 0:0620 0:0310

0:0548 0:0445

3

7 0:0613 0:0543 7 7 0:0822 0:1138 0:0281 0:0641 7 7 7 0:0385 0:0839 0:0613 0:0277 5 0:0308 0:0618 0:0548 0:0364

0:0385 0:0810

ð7Þ

The next step is to obtain the ideal (best) and negative-ideal (worst) solutions and is shown in Table 2.The next step is to obtain the separation measures and these are shown in Table 3.The relative closeness of particular alternative to the ideal solution is calculated and given below:

CMustard ¼ 0:2501 CWheat ¼ 0:8497 CSunflower ¼ 0:2545 CGram ¼ 0:1501 CSafflower ¼ 0:1379 Example: To calculate the separation measures (positive and negative) shown in Table 3, a sample calculation is illustrated below. Normalized weight matrix as shown in Eq. (4) is of the form

2

a11

a12

a13

a14

a15

e11

e12

e13

e14

e15

3

7 6 6 b11 b12 b13 b14 b15 7 7 6 6 c11 c12 c13 c14 c15 7 7 6 7 6 4 d11 d12 d13 d14 d15 5

ð3Þ Table 2 Best and Worst Solutions for the attributes.

3

7 6 6 0:265 0:349 0:422 0:513 0:515 7 7 6 6 0:916 0:745 0:593 0:235 0:607 7 7 6 7 6 4 0:131 0:349 0:437 0:513 0:263 5 0:066 0:280 0:322 0:458 0:345

ð5Þ

The weights of each criteria are calculated as proposed by Saaty (1980) and shown in Eq. (6)

After developing the decision matrix, normalized matrix is obtained as shown in Eq. (4)

2

CC 3

Ideal Best

Ideal Worst

I+CA = 0.4328 I+CP = 0.0822 I+CC = 0.0618 I+SP = 0.0613 I+WU = 0.0277

I CA = 0.0310 I CP = 0.0308 I CC = 0.1138 I SP = 0.0281 I WU = 0.0641

ð4Þ

Importance of different attributes with respect to the problem statement, pair wise comparison matrix with a scale of relative importance is developed. Judgement scale of representation based on analytic hierarchy process (AHP) proposed by Saaty (1980) is adopted in the present study. The developed matrix is verified with the consistency indices and it is satisfying the limits (Consistency

Table 3 Separation measure for positive ideal and negative ideal solutions. Positive Separation Measure

Negative Separation Measure

P+Mustard = 0.3129 P+Sunflower = 0.3123 P+Wheat = 0.0717 P+Gram = 0.3740 P+Safflower = 0.4045

P Mustard = 0.1044 P Sunflower = 0.1060 P Wheat = 0.4051 P Gram = 0.0659 P Safflower = 0.0647



For Ii, in case of beneficial attribute, I+i indicates higher value of the attribute else if it is non-beneficial attribute, I+i indicates lower value of the attribute. Now ideal best is higher value of first attribute (I+CA) and ideal worst (I CA) is lower value of first attribute in the mentioned matrix. Similarly ideal best and ideal worst for all other considered attributes are identified. Accordingly Table 2 is obtained.Then, Positive Separation measure is calculated as per the expression mentioned below.

PþMustard ¼ PþMustard

5

where G(i) is a non-decreasing function of the observed deviation (d) between two alternatives a1 and a2 over the criterion c(i). 3. Development of preference index: It determines the outranking relation on the set of actions. After specifying a preference function Pi and weight wi for each criterion ci(i = 1, 2,. . ., M) of the Q problem, the multiple criteria preference index a1a2 is then defined as the weighted average of the preference functions Pi and is expressed in Eq. (9)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 ða11 IþCA Þ þ ða12 IþCP Þ þ ða13 IþCC Þ þ ða14 IþSP Þ þ ða15 IþWU Þ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ð0:124 0:4328Þ2 þ ð0:0385 0:0822Þ2 þ ð0:0807 0:0618Þ2 ¼ 0:3129 ¼t þð0:0548 0:0613Þ2 þ ð0:0445 0:0277Þ2

In the same lines, Negative Separation measure was calculated by the expression

PMustard ¼

PMustard

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða11 ICA Þ2 þ ða12 ICP Þ2 þ ða13 ICC Þ2 þ ða14 ISP Þ2 þ ða15 IWU Þ2

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u ð0:124 0:0310Þ2 þ ð0:0385 0:0308Þ2 þ ð0:0807 0:1138Þ2 ¼ 0:1044 ¼t þð0:0548 0:0281Þ2 þ ð0:0445 0:0641Þ2

The relative closeness of particular alternative for example Mustard with its ideal solution is calculated as

CMustard ¼

PMustard þ ðP Mustard þ PMustard Þ

Y

a1a2 ¼

M X

ð9Þ

WiPi;a1a2

1

¼

0:1044 ¼ 0:2501 ð0:3129 þ 0:1044Þ

Similarly others viz. CWheat, CSunflower, CGram and CSafflower are computed. 3.4. PROMETHEE method This method is chosen for selection criteria due to its wide application in almost all fields of science and technology (Roozbahani et al., 2012; Venkata Rao and Patel, 2010). It provides better idea of choosing the alternatives from best to worst possible cases. Steps involved in this method is explained in the following section. 1. First two steps (i.e. Formulation of decision matrix and relative importance matrix) is same as dealt in Sections 2.2 and 3.3 in other methods. 2. Development of preference function (Pi): This function is used by the decision maker for comparing the alternatives in terms of each separate criterion. Six types of preference functions (Brans and Vincke, 1985) can be used. It translates the difference between the evaluations obtained by two alternatives (a1 and a2) in terms of a particular criterion, into a preference degree ranging from 0 to 1. In this study, preference ‘‘usual function” is considered for all the crireia. It is difference between the values of criteria for alternatives a1 and a2. If Pi, a1a2 be the preference function associated to the criterion ci. Then it is shown in Eq. (8)

Pi;a1a2 ¼ GðiÞ½cðiÞa1 cðiÞa20 6 Pi;a1a2 6 1

ð8Þ

Q a1a2 represents the intensity of preference of the decision maker of alternative a1 over alternative a2, when considering simultaneously all the criteria. Its value ranges from 0 to 1. 4. Development of outranking relations: It is determined in terms of leaving flow, entering flow and the net flow for an alternative a belonging to a set of alternatives A are defined by the following Eq. (10):

uþ ðaÞ ¼ u ðaÞ ¼

XQ x eA X Q x eA

xa ax

uðaÞ ¼ uþ ðaÞ u ðaÞ

9 > > > > = > > > > ;

ð10Þ

Where uþ ðaÞ is the leaving flow, u ðaÞ is the entering flow and uðaÞ is the net flow. uþ ðaÞis the measure of the outranking character of a (i.e. dominance of alternative a over all other alternatives) and u ðaÞ gives the outranked character of a (i.e. degree to which alternative a is dominated by all other alternatives). 5. Based on the netflows, prioritization of alternatives has been carried out. After obtaining weights explained in Section 3.3, next step is to have information on decision maker preference function. In the present study, preference usual function is used for all the criteria. If two alternatives have a difference d – 0 in criterion ci, then a preference value ranging between 0 and 1 is assigned to the ‘better’ alternative crop whereas the ‘worse’ alternative crop receives a value 0. If d = 0, then they are indifferent which results in an assignment of 0 to both alternatives and it is explained in step 2. The pairwise comparison of criterion (CC) matrix is developed. As


6


Table 4 Preference values P resulting from the pairwise comparisons of the five alternative crops with respect to criterion cost of cultivation.

Wheat Sunflower Mustard Gram Safflower

Wheat

Sunflower

Mustard

Gram

Safflower

_ 0 0 0 0

1 _ 0 1 0

1 1 _ 1 0

1 0 0 _ 0

1 1 1 1 _

an example, Table 4 shows the preference matrix with respect to cost of cultivation. CC is a non-beneficial criterion and hence lower values are desired. The crop having a comparatively low value of CC is said to be ‘better’ than the other. Similarly preference matrix has been developed with respect to every criterion considered (with respect to crop area, sale price, crop production and water usage) in the present study. Preference indices, Leaving flow, entering flow and the net flow values for different alternatives are calculated as explained in step 3 & 4 and is shown in Table 5. 3.5. NLP model using LINGO A NLP model is formulated to maximize the profit subjected to the restriction of water availability and is solved using Lingo (Shrivastava et al., 2012). The objective function is expressed as shown in Eq. (11)

MaxZ ¼ ðPY CÞA

ð11Þ

Where, Z is the net return, Rs. (Indian currency); P is the sale price of crop, Rs/kg; Y is the yield of the crop, kg/ha; C is the cost of cultivation, Rs/ha; and A is the cultivated area, ha. The model considered the WPF, sale price of crops, crop area, land availability, cost of cultivation including irrigation cost, water usage as model constraints. The total area is sub divided into a number of sub areas on the basis of soils and land availability constraints. The minimum available canal water during the study period is taken as the water availability constraints. The land area constraint for certain crops is fixed as minimum so that the most profitable crop should not dominant over the entire command area. The decision variables such as water depth and cropped area considered as positive. 4. Results and discussion MADM approach is used to obtain the ranking of the alternatives for the crops considered with the different criteria. In order to anlayse the alternatives, four method (SAW, WPM, TOPSIS and PROMETHEE) of MADM approach is used in the present study. Based on the selection index values, ranking/prioritization is carried out for obtaining optimal cropping pattern. Index values obtained for all the methods were shown in Table 6. Ranking is carried out based on the highest index value corresponds to rank 1 and lowest index value corresponds to rank 5. It is seen from the results, small variation may occur when we apply different methods due to its mathematical formulations for

Table 6 Index Values obtained by SAW, WPM, TOPSIS and PROMEHTEE method.

PMustard PSunflower PWheat PGram PSafflower

SAW

WPM

TOPSIS

PROMETHEE

0.4818 0.4606 0.8666 0.4213 0.4091

0.4376 0.4246 0.8377 0.3156 0.2318

0.2501 0.2545 0.8497 0.1501 0.1379

0.2693 0.6915 3.7890 1.5983 2.6128

the same set of parameters. SAW and WPM methods shows wheat is followed by mustard. It may be attributed to the linear and exponential combination of weights for SAW and WPM respectively. In case of TOPSIS and PROMETHEE, wheat is followed by sunflower. The reason may be due to incorporation of logical interrelationship between the criterias considered by calculating the separation measures to the ideal solution and by pairwise comparisons. The selection index values obtained using MADM approach is shown in Table 6. TOPSIS and PROMETHEE had performed satisfactorily in comparison with NLP model results. In NLP model, wheat is the most profitable crop followed by sunflower based on the results obtained uisng LINGO model. The net benefit obtained for wheat is more compared to other crops within the considered constraints. The prioritization/ranking of crops considered is shown for the methods used in the present study. But in deciding optimal cropping pattern, social, land availability (in this study it is considered as qualitative attribute), environmental factors may also be considered to arrive at suitable alternatives. 4.1. RANKING SAW Wheat-Mustard-Sunflower-Gram-Safflower WPM Wheat-Mustard-Sunflower-Gram-Safflower TOPSIS Wheat-Sunflower-Mustard-Gram-Safflower PROMETHE Wheat-Sunflower-Mustard-Gram-Safflower 5. Conclusion This paper attempted to achieve rabi cropping pattern using MADM approach. It deals with types of attributes which are quite conflicting in nature. The results obtained through MADM approach is evaluated and compared with LINGO model results. Summarization of present research conclusions includes: 1. Various types of attributes viz. crop area, water usage, sale price of crop, cost of cultivation, crop production were decided for each crop related to the significance of the optimal cropping pattern from the collected information. 2. Four methodologies namely SAW, WPM, TOPSIS and PROMETHEE of MADM approach were employed for achieving cropping pattern. 3. The results obtained for SAW and WPM were slightly differed may be due to the weightages been assigned to each attribute but not in comparison of attributes.

Table 5 Resulting preference indices as well as leaving, entering and net flow values.

Wheat Sunflower Mustard Gram Safflower

Wheat

Sunflower

Mustard

Gram

Safflower

u+(a)

u(a)

u(a)

0.0000 0.1055 0.0000 0.0000 0.0000

0.8945 0.0000 0.1196 0.1921 0.1196

1.0000 0.6645 0.0000 0.3024 0.0000

1.0000 0.4724 0.6976 0.0000 0.2252

1.0000 0.7748 0.8804 0.3024 0.0000

3.8945 2.0173 1.6976 0.7969 0.3448

0.1055 1.3259 1.9669 2.3952 2.9576

3.7890 0.6915 0.2693 1.5983 2.6128



4. Results obtained from TOPSIS and PROMETHEE showed good judgment in ranking the alternatives. These methods found to be effective in acquiring various attribute information into account for comparison, ranking and selection of optimum cropping pattern. This method allows adding or neglecting any attribute based on the significance of the criteria. It allows the decision maker to change or select the importance of the criteria with respect to the objective of the study. 5. In TOPSIS method, generation of separation measures for obtaining ideal worst and ideal best alternatives which helped to obtain nearest hypothetical best solution. 6. IN PROMETHEE method, preference function is developed and outranking relationships have been established between the alternatives through which ranking of alternatives is carried out from best to worst ones using net flow values. 7. MADM results has been evaluated in comparison with LINGO model results and it is satisfactory. Quick decisions can be taken based on the MADM approach because computation time required is less compared to NLP.

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