IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

International Journal of Research in Information Technology (IJRIT) (IJRIT) www.ijrit.com

ISSN 2001-5569

Use of Zimmermann Multi Objective Fuzzy Linear Programming model to identify best areas of rice growing in the Dry and Intermediate zones in Sri Lanka S.M.N.S.K Seneviratne1 and W.B. Daundasekera2 Tutor, Department of Mathematics, University of Peradeniya, Kandy, Sri Lanka Email: [email protected] 2 Professor, Department of Mathematics, University of Peradeniya, Kandy, Sri Lanka Email: [email protected] 1

Abstract Use of Zimmermann Multi Objective Fuzzy Linear Programming (MOFLP) model to develop recommendations to increase rice production in Sri Lanka has been studied. Low country Dry zone and Intermediate zones of Sri Lanka are the most favorable Agro-ecological zones of Sri Lanka for rice production. Study locations were centered in these areas. Average yield, Gross income, Total cost, Average quantity of fertilizer, Land area cultivated by the farmer, Requirement of labour, Labour availability, Requirement of machinery and Machinery availability were used as the input data for the model. The model had identified that further increase in yield can be achieved with the use of optimum fertilizer, labour and machinery, in areas which already use high yielding varieties. The Zimmermann MOFLP model had been used in this study. The study had shown that this model can be successfully used in the management of Tropical Agricultural Systems, where vagueness is extremely high. Keywords- Zimmermann Multi Objective Fuzzy Linear Programming (MOFLP) model, Agricultural planning, Dry and Intermediate zones of Sri Lanka.

1. Introduction Proper agricultural management is an important necessity of the development of a country. Agricultural development can be enhanced by proper Agricultural planning. However planning of agriculture has become difficult due to the vagueness of the agricultural systems. Seneviratne and Daundesekara (2011) have shown that the vagueness of the agricultural systems can be properly managed by using Zimmermann Multi Objective Fuzzy Linear Programming (MOFLP). To use Multi Objective Fuzzy Linear Programming, a model has been developed by Zimmermann in 1985. This was done with the help of Fuzzy set theory put forward by Bellmann and Zadeh (1970). The work reported in this paper shows that the Zimmermann MOFLP model (1985) is extremely capable of handling agricultural management problems of developing countries. In these countries the vagueness is a severe barrier for proper agriculture development planning.

2. Methodology The study area is located in the low country Dry zone and Intermediate zone in Sri Lanka (Fig.1). The main soil type of the area is reddish brown earths and is the most widespread soil type of the country (Panabokke, 1996) .The purpose of the study is to identify best managed rice growing areas of the low

34 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

country Dry zone and Intermediate zone so that their management practices can be extrapolated to similar areas to boost rice crop production.

Fig. 1 The map showing the Dry and Intermediate zones in Sri Lanka (Panabokke and Kannangara, 1975)

35 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

Fig. 2 Map showing the distribution of the main soil type (Reddish Brown Earths) in the study area (Moormann and Panabokke, 1961)

Mathematical modeling for the three conflicting objectives is briefly explained below: Objective 1: Maximum Production (Z1) Production is maximized for meeting the demand. n

Maximize production =

∑ [ A ] ∗ [Y ] i =1

i

i

(1)

Ai = Area in the i th district (ha) Yi= Average yield in the i thdistrict (kg/ha)

36 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

Objective 2: Maximum Profit (Z2) The profit is obtained by subtracting total cost from the gross income.

Maximize profit =

 n   n   ∑ [ Ai ] * [GI i ]  −  ∑ [ Ai ] * [TCi ]   i =1   i =1 

(2)

GIi = Gross income in the i th district (Rs/ha) TCi= Total cost in the i th district (Rs/ha)

Objective 3: Optimum Fertilizer Utilization (Z3) In order to maintain the fertilizer of soil in proper manner one should concentrate on use of optimum fertilizer. n

Maximize Fertilizer utilization =

∑[ A ] ∗[F ] i =1

i

i

(3)

Fi = Average fertilizer utilization in the i th district (kg/ha) Constraints:

(i) Land n

∑ [ A ] ≤ CA i

i =1

(4)

i

CAi = Farmer’s cultivated area in the i th district (ha) (ii) Agricultural Labour n

∑ [ A ] ∗ [ L ] ≤ TL i

i =1

i

(5)

Li = Requirement of labour in the i th district (md/ha) TL = Labour availability in the i th district (md) (iii) Agricultural Machinery n

∑ [ A ] ∗ [M ] ≤ TM i =1

i

i

(6)

Mi = Requirement of machinery in the ith district (Rs/ha) TM = Machinery availability in the ith district (Rs) (iv) Non-negativity constraints Ai ≥ 0

for all i

(7)

37 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

Fuzzy linear programming algorithm (Zimmermann, 1985) is divided into six main steps: Step 1.

By taking only one objective at a time solve the problem as a Linear Programming problem.

Step 2. Determine the corresponding values of every objective at each solution derived, using the results of Step 1. Step 3. Calculate the maximum and minimum values for each objective function using data of Step 2. Step 4. Obtain the linear membership function. Step 5. For the Fuzzy Multi Objective problem obtain the equivalent Linear Programming model. Step 6. Obtain the compromise solution.

3. Results and Discussion

In this study, Zimmermann MOFLP model (1985) is used. The input data are given in the Table 1:

Table 1 Fertilizer, labour and machinery utilization by farmers for paddy cultivation

Study locations*

Average yield (Kg/ha)

Gross income (Rs/ha)

Total cost (Rs/ha)

Ampara Anuradhapura Kurunegala Mahaweli B Mahaweli C Mahaweli H Polonnaruwa

4,711.2 5,135.2 4,511.6 4,909.4 4,972.6 5,211.6 5,325.6

95,326 117,147 96,138 11,594 108,867 116,039 118,917

63,366 66,860 65,373 64,264 63,102 66,448 69,233

Average quantity of fertilizer (Kg/ha) 445.44 447.36 408.96 420.00 410.88 467.04 443.04

Area (ha)

Requirment of labour (md/ha)

Labour availability (md)

Requirment of machinery (Rs/ha)

Machinery availability (Rs)

1 1 1 1 1 1 1

56.6 80.2 83.5 74.4 78.2 81.6 67.2

62.3 88.2 91.9 81.8 86.0 89.8 73.9

22,726 16,586 16,949 17,400 16,293 15,888 19,894

24,998 18,245 18,644 19,140 17,922 17,476 21,884

* These locations are marked in the map (source - Cost of cultivation of Agricultural crops, Department of Agriculture (2005-2009) )

38 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

The following model is developed using the data given in the Table 1: Objective functions: Max Z1 =4,711.2 *A1 + 5,135.2*A2 + 4,511.6* A3+ 4,909.4* A4 + 4,972.6 *A5 + 5,211.6*A6 + 5,325.6*A7

(8)

Max Z2 =31,960.4 * A1+50,286.7* A2+30,765.3 * A3+47,329.5*A4 +45,765.1*A5 +49,591.3 *A6 + 49,684*A7

(9)

Max Z2 = 445.44*A1 + 447.36* A2 +408.96 * A3+420.00 * A4 +410.88* A5 +467.04 *A7 + 443.04 * A7

(10)

Constraints imposed on the objective functions: (1)Land in Yala season (ha)

A 1+ A 2+ A 3 + A 4 + A 5+ A 6+ A 7 ≤ 8

(11)

(2)Agricultural labour (man days)

56.6*A1 +80.2* A2 +83.5*A3+ 74.4* A4+78.2* A5+81.6*A6 +67.2*A7 ≤ 634.1

(12)

(3)Agricultural machinery (Rs)

22,725.6*A1+16,586.4 *A2+16,949.3*A3+ 17,399.5* A4 + 16,292.6 *A5 +15,887.5*A6+19,894.1 *A7 ≤ 167,217.6

(13)

(4)Non negativity constraints: A1 ,A2 ,A3 ,A4 ,A5 ,A6, A7 ≥ 0

(14)

39 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

The solution obtained by solving the above model is summarized in the Table 2: Table 2

Identification of best areas of rice growing from the Planning model and Multi Objective Fuzzy Linear Programming (MOFLP)

Area (ha) allocated by maximizing the objective function Districts and related parameters

Area (ha) allocated by MOFLP

Crop production (Z1)

Profit (Z2)

Fertilizer utilization (Z3)

Ampara

0

0

0.75

0

Anuradhapura

0

7.42

0

0

Kurunegala

0

0

0

0

Mahaweli B

0

0

0

0

Mahaweli C

0

0

0

0

Mahaweli H

0

0

7.25

4.60

Polonnaruwa

8

0.58

0

3.40

538

634

634

604

Machinery (Rs)

159,152.8

134,599.5

132,214.9

140,704.5

Crop production (kg)

42,604.80 *

41,192.03

41,317.5 **

42,079.89

397,472

401,945.89*

383,507.23**

397,045.2

3,544.32**

3,576.37

3,720.16 *

3,654.827

Labour (man days)

Profit (Rs) Fertilizer utilization(kg)

* maximum value ; **minimum value

40 S.M.N.S.K Seneviratne et al, IJRIT

IJRIT International Journal of Research in Information Technology, Volume 1, Issue 1, January 2013, Pg. 34-41

Table 3 Best areas of rice growing Land allocation

Hectares

Mahaweli H

4.60

Polonnaruwa

3.40

4. Conclusion The data in Table 2 and Table 3 show that the model has determined that Mahawali H and Polonnaruwa are the best areas of rice growing in the Dry and Intermediate zones in Sri Lanka. Thus, the solution of the model shows that these areas have utilized fertilizer at the optimum level which is according to the specifications recommended by the Department of Agriculture. It can be further observed that these two areas have also utilized labour and machinery most efficiently. The model has used these as the factors influencing income from rice cultivation. The data show that by using fertilizer, labour and machinery effectively in areas with similar soils, irrigation facilities, rainfall and temperature the income from rice grown in such areas could also be increased.

5. References [1] C.R. Panabokke and R. Kannangara, "The identification and demarcation of the Agro-ecological regions of Sri Lanka", Sri Lanka Association for the Advancement of Science, Vol. 31, No.3, 1975, pp.49. [2] C.R. Panabokke, Reddish Brown Earths In Soils and Agro-Ecological Environments of Sri Lanka, Colombo:NARESA publications,1996. [3] Department of Agriculture Cost of cultivation of Agricultural crops, Peradeniya: Agriculture press, 2005-2009. [4] F.R. Moormann and C.R. Panabokke , "Soils of Ceylon" ,Trop.Agric., Vol. 117,1961, pp1-69. [5] H.J. Zimmermann, "Application of fuzzy set theory to mathematical programming", Information Sciences Vol.36,1985, pp.25-58. [6] R.E. Bellman and L.A. Zadeh, "Decision-making in a fuzzy environment", Management Science, Vol.17, No.4,1970, pp.B141-B164. [7] S.M.N.S.K. Seneviratne and W.B. Daundasekera," Applicability of Zimmermann Multi objective Fuzzy Linear Programming Model Under Large Vagueness". Peradeniya University Research sessions (Purse), Vol.16, 2011, pp132.

41 S.M.N.S.K Seneviratne et al, IJRIT