Application of EDAS Method on Water Requirement in Agriculture

Application of EDAS Method on Water Requirement in Agriculture

Dr. A. Sahaya Sudha Assistant Professor, Department of Mathematics Nirmala College for Women, Coimbatore, India

Abstract Muticriteria Decision making is the latest developing mathematical technique which is being followed in management practices. Due to the complexity of a given situation the expected results are ambiguous and vague in the decision making process. This can be done by using the Mathematical analysis of a given situation by using one of the MCDM Methods called Evaluation based on Distance from Average solution (EDAS) method. In this paper an agricultural situation in a locality is considered where various parameters on water management in Agricultural crops are considered. So, to find a better alternative a study has been taken on different crops of different durations, water consumption, yield, market pricing and is mathematically evaluated by applying EDAS method for efficient water management practices for an agriculturist.

Keywords MCDM, water Management, EDAS, crop selection, irrigation

1. INTRODUCTION

   Multicriteria Decision making (MCDM) is a process of arriving at the best appropriate solution from a set of available alternatives in relation to a set of evaluation criteria. The MCDM methods have been discussed by various authors and has been published in a number of scientific and technical journals and proposed methods for solving various MCDM problems in areas of Economics[2,4], Management[8,9], production[10,11],sustainabledevelopment,[3,5], construction, [3,13,14] Logistics, [12]to name a few. The Evaluation Based on Distance from Average Solution (EDAS) is a new and efficient MCDM method. This was proposed and further extended EDAS method for multi

2. MATHEMATICAL ALGORITHM FOR THE PROPOSED METHOD

   The Mathematical procedure for the EDAS method for a decision making problem with n criteria and m alternatives can analyzed as per the following procedures:

   Step: 1 Select the available alternatives and the most important criteria which describes the alternatives to construct a Decision Making Matrix D as:

   C1 C2 C3 Cn

   A1 X11 X12 X13 X1n

   A2 x21 x22 x23 X2n

   D = A3 x31 x32 x33 X3n

   . . . . . .

   . . . . . .

   . . . . . .

   Am xm1 xm2 xm3 xmn

   Where w = [w1 , w2 , w3 ,…, wn] ———— (1)

   Let A1 , A2 , A3 ,…, Am are possible alternatives among which decision makers have to choose C1 , C2 , C3 ,…, Cn are criteria with which alternatives performance are measured, xij is the performance value of alternatives Ai with respect to criterion Cj , wj is the weight of criterion Cj and all xij are positive numbers

   Step 2: Determine the average solution (AVj) according to all the criteria as per the formula:

   n

   x

   i ij

   criteria Decision Making by Keshavarz Ghorabaee et.al., [1,6,7]. The desirability of alternatives in this method is

   AV j

   1 ——————- (2)

   n

   determined based on distances of them from an average solution (AV). Here we have two measures of dealing with the desirability of the alternatives. The first measure is the positive distance from average (PDA), and the second is the negative distance from average (NDA). These measures shows the difference between each solution (alternative) and the average solution. The evaluation of the alternatives is made according to higher values of PDA and lower values of NDA. Higher values of PDA and/or lower values of NDA represent that the solution (alternative) is better than average

   Step 3: The Positive Distance from average (PDA) is

   calculated according to the criteria which are beneficial or non

   beneficial:

   If jth criterion is beneficial PDA max(0,( X ij AVij )))

   j

   j

   ij AV

   If jth criterion is non beneficial

   solution.

   PDA max(0,( AVj X ij ))

   (3)

   j

   j

   ij AV

   Step 4: Calculate the Negative Distance from average (NDA) is calculated according to the criteria

   we have considered the main criteria as water requirement in this paper. The Criteria C1, C2 and C3 are Income, Water

   If jth criterion

   is beneficial

   Consumption and Water Cost .The above criteria is arrived

   max(0, AV X ))

   based on the availability and usage of water for a given crop

   NDAij

   j ij

   AVj

   and also the cost of water are worked out based on the yield and market price. It may also be noted that the other cost of

   If jth criterion is non beneficial

   inputs are not taken here since this is not significant in this

   NDAij

   max(0,( X ij AVij ))). (4)

   AVj

   study even though the other factors also influence the Economics of Farming activity.

   Step 5: The Weighted sum of PDA is obtained from the Average Matrix:

   m

   m

   SPi wj PDAi j

   j 1

   where wj denotes the weight of the criteria j

   Step 6: The Weighted sum of NDA is obtained from the Average Matrix:

   m

   TABLE I CRITERIA DESCRIPTION

   Criteria	Description
   Income	Income obtained from crop cultivation
   Water consumption	Quantity of Water required for the crop
   Water Cost	Cost involved in the water usage in crop cultivation

   Criteria	Description
   Income	Income obtained from crop cultivation
   Water consumption	Quantity of Water required for the crop
   Water Cost	Cost involved in the water usage in crop cultivation

   TABLE II

   Criteria
   Alternatives	Income(C1)	Water Consumption(C2)	Water Cost(C3)
   A1	188.57	13.71	0.07
   A2	4165.00	18.33	0.00
   A3	1894.23	23.08	0.01
   A4	489.09	15.15	0.03
   A5	188.57	11.43	0.06
   A6	1087.50	14.58	0.01
   A7	261.67	27.50	0.11
   A8	858.89	14.44	0.02
   A9	850.00	33.33	0.04

   Criteria
   Alternatives	Income(C1)	Water Consumption(C2)	Water Cost(C3)
   A1	188.57	13.71	0.07
   A2	4165.00	18.33	0.00
   A3	1894.23	23.08	0.01
   A4	489.0	15.15	0.03
   A5	188.57	11.43	0.06
   A6	1087.50	14.58	0.01
   A7	261.67	27.50	0.11
   A8	858.89	14.44	0.02
   A9	850.00	33.33	0.04

   DECISION MATRIX- DRIP IRRIGATION

   i

   i

   SNi

   j 1

   wj NDA j

   Step 7 : The Normalized values of SPi alternatives is calculated as follows:

   and SNi for all

   NSPi

   SPi

   max (SP )

   i i

   SN

   i

   i

   NSN i 1

   max i (SN i )

   Where NSPi and NSNi denote the normalized weighted sum of PDA and NDA respectively

   Step 8: The appraisal score ASi for all alternatives is obtained as:

   AS 1 (NSP NSN )

   Criteria
   Alternative	Income(C1)	Water Consumption(C2)	Water Cost(C3)
   A1	111.43	20.00	0.18
   A2	1726.67	26.67	0.02
   A3	1807.69	36.54	0.02
   A4	429.09	19.39	0.05
   A5	111.43	17.86	0.16
   A6	975.00	23.33	0.02
   A7	171.67	30.00	0.17
   A8	768.89	23.33	0.03
   A9	775.00	50.00	0.06

   Criteria
   Alternative	Income(C1)	Water Consumption(C2)	Water Cost(C3)
   A1	111.43	20.00	0.18
   A2	1726.67	26.67	0.02
   A3	1807.69	36.54	0.02
   A4	429.09	19.39	0.05
   A5	111.43	17.86	0.16
   A6	975.00	23.33	0.02
   A7	171.67	30.00	0.17
   A8	768.89	23.33	0.03
   A9	775.00	50.00	0.06

   TABLE III

   i 2 i

   i

   where 0 ASi 1

   DECISION MATRIX – FLOOD IRRIGATION

   Step 9: The alternatives are ranked according to the decreasing values of appraisal score (ASi). The alternative with the highest ASi is the best choice among the alternatives.

3. CASE ANALYSIS

   In this paper a decision making approach with an example of an agricultural activity is considered. An analysis of crops of different durations and different water requirement along with the production of economies is considered in a farming activity in a particular locality. A data pertaining to nine different types of crops based on their duration along with Income, Water Cost and the Quantity of water required using both Drip and Flood Irrigation methods is calculated as per the cropping pattern. According to the farmers requirement 9

   Step 1: The average solution for Drip Irrigation is calculated from the Table 2

   assigned alternatives A1, A2, A3, A4, A5, A6, A7, A8, A9 are

   ( AV

   9983.28

   )(C ) 1109.28 , (AV )(C )= 19.06,

   selected where the alternatives A1, A2 &, A3 are Long

   j 2

   j 1 9

   duration crops of 300-360 days , A4, A5,& A6 are medium duration crops of 120 -140 days and A7, A8,& A9 are short duration crops from 60-90 days. Here the economics of scale are worked out by giving the best option for a farmer to decide and cultivate a crop which is best suited along with the availability of resources where the main emphasis is on water conservation. Since water is the main source for agriculture

   ( AVj)(C3)= 0.038

   The average solution for Flood Irrigation is calculated from the Table III

   (AVj)(C1) =764.10 , (AVj) (C2) = 27.46, (AVj) (C3) = 0.08

   Step2: The Positive distance from average (PDA) for Drip Irrigation is

   TABLE V

   Alternatives	SPi	SNi	NSPi	NSNi	ASi	Ranking
   A1	0.0896	0.6983	0.1287	0.0000	0.0643	8
   A2	0.6910	0.0000	0.9919	1.0000	0.9959	1
   A3	0.6967	0.1091	1.0000	0.8437	0.9219	2
   A4	0.2389	0.1447	0.3430	0.7928	0.5679	5
   A5	0.1154	0.6183	0.1656	0.1145	0.1401	7
   A6	0.3711	0.0000	0.5327	1.0000	0.7664	3
   A7	0.0000	0.6832	0.0000	0.0217	0.0108	9
   A8	0.2554	0.0000	0.3667	1.0000	0.6833	4
   A9	0.0664	0.2709	0.0953	0.6120	0.3537	6

   Alternatives	SPi	SNi	NSPi	NSNi	ASi	Ranking
   A1	0.0896	0.6983	0.1287	0.0000	0.0643	8
   A2	0.6910	0.0000	0.9919	1.0000	0.9959	1
   A3	0.6967	0.1091	1.0000	0.8437	0.9219	2
   A4	0.2389	0.1447	0.3430	0.7928	0.5679	5
   A5	0.1154	0.6183	0.1656	0.1145	0.1401	7
   A6	0.3711	0.0000	0.5327	1.0000	0.7664	3
   A7	0.0000	0.6832	0.0000	0.0217	0.0108	9 /td>
   A8	0.2554	0.0000	0.3667	1.0000	0.6833	4
   A9	0.0664	0.2709	0.0953	0.6120	0.3537	6

   RANKING OF FLOOD IRRIGATION

   PDA (C

   max(0,(x11 AVj (c1))

   )

   max(0,0- 1109.28)

   0.0000

   11

   PDA (C

   11 Av j (C1)

   max(0,( AV j (c2 ) X12 ))

   )

   1109.28

   max(0,(19.06- 13.71)

   0.2806

   12 12

   Av j (C2 )

   19.06

   Similarly other values for Positive distance from average (PDA) for Drip Irrigation is calculated

   Step: 3 The Negative Distance from average (NDA) for Drip Irrigation is

   NDA (C )

   11 11

   max(0,( AVj (c1 ) X12 ))

   Av j (C2 )

   max(0,(1109.28- 188.57))

   1109.28

   0.8300

   NDA (C )

   12 12

   max(0,(x12 AVj (c1 ))

   Av j (C1 )

   max(0,(13.71- 19.06))

   0

   19.06

   Similarly the other values also calculated

   Step: 4 Weighted sum of PDA is calculated by multiplying the weights of each value in Step 2

   where w1=0.33, w2=0.33, w3=0.33

   Step: 5 Weighted sum of NDA (SNi)) is calculated by multiplying the weights of each value in Step 3 where w1=0.33, w2=0.33, w3=0.33

   Step: 6 The following table shows the Normalized values of SPi ,SNi and the Appraisal score (ASi) is obtained using the Formula given in the Step7and step8 of Algorithm

   TABLE IV

   NORMALIZED VALUES OF SPI & SNI DRIP IRRIGATION

   Alternatives	SPi	SNi	NSPi	NSNi	ASi	Ranking
   A1	0.0926	0.5516	0.0762	0.4171	0.2467	7
   A2	1.2149	0.0000	1.0000	1.0000	1.0000	8
   A3	0.4617	0.0695	0.3800	0.9266	0.6533	1
   A4	0.1389	0.1845	0.1143	0.8050	0.4597	4
   A5	0.1322	0.4503	0.1088	0.5242	0.3165	6
   A6	0.2955	0.0065	0.2432	0.9932	0.6182	2
   A7	0.0000	0.9464	0.0000	0.0000	0.0000	9
   A8	0.2694	0.0745	0.2218	0.9213	0.5715	3
   A9	0.0023	0.3242	0.0019	0.6575	0.3297	5

   Similarly we can calculate the values for Flood Irrigation also and the values are obtained in Table 5

   Fig 1: Pictorial Representation of Ranking Drip & Flood Irrigation

4. CONCLUSIONS

In this paper the EDAS method is applied to solve a multi criteria decision making problem of finding the best crop suited on the basis of minimum water requirement along with a higher income for a farmer. Here the best alternative of crop is suggested by considering lower cost and less usage of water with a higher income. The ranking of the alternatives for Drip Irrigation method in order are A3 > A6 > A8 > A4 > A9 > A5 > A1 > A2 > A7. Results indicate that A3 is the best alternative with ASi value of 0.6533.The ranking of the alternatives for Flood Irrigation in order are A2 > A3 > A6 > A8 > A4 > A9 > A5

> A1 > A7. Results indicate that A2 is the best alternative with ASi value of 0.9959 wherein A2 which is the best alternative in Flood Irrigation method. This method can be used to arrive at a decision for any number of crops also with number alternatives. Similarly this ranking also can be used for different durations based crops as per the above results, where Crop A3 is suited in long duration, A6 is suited in terms of medium duration and A8 is suited for short duration in the case of Drip Irrigation .In case of Flood Irrigation the best crop is A2, A6 and A8 for long duration, Medium duration and short duration crops respectively.

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