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THE EFFICACY OF TRAPEZOIDAL FUZZY NUMBERS
AND ITS APPLICATION
Ajjaz Maqbool Dar1, Aafaq A. Rather2, *, Rushika Kinjawadekar3, Abhay Deshpande4, Maryam Mohiuddin5, Rashid A. Ganaie6, Khursheed Ahmad7
department of Mathematics, Govt College for Women's Nawakadal Srinagar j&k-190002, India 2, "Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune-411004, India 3Department of Mathematics & Statistics, Faculty of Science & Technology, Vishwakarma University,
Pune, India
5,6Department of Statistics, Annamalai University, Annamalai nagar, Tamil Nadu, India 7Department of Public Administration, Government Degree College Tangmarg, Kashmir, India 1ajjazmaqbool013@gmail.com, 2, *aafaq7741@gmail.com, 3rushika.kinjawadekar@vupune.ac.in, 4abhaypdeshpande18@gmail.com, 5masmariam7@gmail.com, 6rashidau7745@gmail.com,
7dkbutt99@gmail.com
Abstract
Numerous fields, including engineering, agriculture, and management sciences, have been using trapezoidal fuzzy numbers. In this study, we first develop Trapezoidal Fuzzy Number (TFN) and then attempt to formulate a model to handle element uncertainty in order to solve a linear programming problem. Making good decisions will only require this type of approximation.
Keywords: Trapezoidal Fuzzy Number, linear programming problem, initial problem and membership function.
1. Introduction
Optimization problem is a one of the most important operation research techniques, and it is used in many areas in agriculture planning, science, Technology and engineering which may be important in both economic and social point of view. We formulate the problem mathematically which may arise in our daily life our aim is to minimize cost and maximize the profit, with certain constrains or restrictions are to be considered. In order to get the best possible result of those problems faced by Agriculturalists to allocate the optimum number of vegetation in their farmhouses. To increase the area under cultivation there are numerous ways to achieving high productivity. If we utilize the resources in a proper way which may be helpful to increase the crop production. Many operation research techniques have been used in planning agriculture activity one of the technique is linear programming. In 1947 George Danzig was given the concept of Linear Programming problem. If we have a limited number of resources, we use linear programming method to optimize the problem. The Zimmermann [1] presented the concept of formulation of fuzzy linear programming problem. Orlovsky [2] made several attempts to investigate the potential of fuzzy set theory as a valuable tool for comprehensive mathematical analysis of practical problems. To address the many kinds of FLP problems, numerous authors utilize various techniques. In almost all areas of decision-making problems, fuzzy approaches
have been developed. Particularly Tamiz [3], and Ross [4]. Delgado and Verdegay [5] construct a broad model of fuzzy linear programming within the fuzzy and fuzzy right side of technical coefficients and also demonstrates that it is possible to solve the dual problem using the same programme. Fung and Hu [6] introduced the fuzzy number-based approach coefficients. Kumar and Rajendra [7] solved a fuzzy linear programming problem with fuzzy variables in parametric form. By utilizing a ranking function and defining a crisp model, Verdegray [8] and Maleki [9] ranking function can be identified in comparisons between fuzzy numbers. In order to determine a workable and ideal solution, we study the linear programming issue in this essay in its conventional form. To solve the linear fuzzy linear programming problem, we utilize the algorithm by trapezoidal fuzzy number is considered.
2. Model Formulation
Maxmize Z = CY
AY < B (1)
Y > 0
Where C is vector component, A is coefficient Matrix, B is crisp parameters and Y is decision variable. 2.1 Generalized Trapezoidal Fuzzy Number (TFN)
The generalized Fuzzy Trapezoidal number T = , t2, t3, t4, w) is a fuzzy subset of real line R, whose membership function satisfies the following postulates:
t1 < x < t2, is a continuous mapping from R to the closed interval [0, 1] ju~ (x) = 0, -x>< x < tj
jU~ (x) is strictly increasing with constant rate on ^ < x < t2
(x) is strictly decreasing with constant rate on t3 < x < t4 j~(x) = 0, t4 < x < <»
Membership function is given by
(y)=
u -1
w(-
,t 4 - Y .
t A — to
t1 < Y < 12
(2)
elesewhere
12 < Y < to
w
13 < Y < 14
0
Where, tj < t2 < t3 < t4 andw e (0,1]
If w=1, the generalized TFN can be written as
T = (tj, t2, t3, t4 ) and the membership function is given by
(y)=
w(-
y - h ^ 2 - tl
w(
t4 - Y
' 4_N
tA — t-i
ti < Y < 12
12 < Y < t3
13 < Y < 14
elesewhere
(3)
1
0
Now we can take ordered pair of parametric of fuzzy numbers with left-hand alpha- cut and right-hand alpha-cut, which are bounded left non-decreasing and bounded right non-increasing functions over [0,1],
i.e. T = {(t1 +a(t2 -11), t4 +a(t4 -13)}
The above mathematical model can be formulated as given below
Maximize Z = C1 (yLi, yRi) + C2 (yL2 , yR2 ).........+ Cn (yLn , yRn )
Subject to
ah1(yL1> yR1) + ah2(yL2> yR2) +............+ ahn (yLn = yRn ) < (bLn = bRn ) (4)
yLj,yRj — 0, forallh = 1,2,3,......m and j = 1,2,3,....n.
3. Applications
A 20 hectares of land is under cultivation of three different crops such as wheat, corn and pulses with certain requirement for capital (in euros) and working hours as shown below:
_Table 1: 20 hectares of land under cultivation_
Crops per acre_Capital (€)_Workers (hours)
Wheat 50 10
Corn 33 8
Pulses 27 4
In this problem the profit of the above three different crops are wheat €38/ acre, Corn €32/ acre and pulses €28/ acre acres of land. The amount and working hours are respectively €460 and around 52 hours. Now, we decide how many hectares of land are required for each crop in order to maximize the profit. Let y1 be the cultivated area with wheat, y2 be the cultivated area with corn and y3 be the cultivated area with pulses. We can have characterized the rough data by a trapezoidal fuzzy number as: 22 hectares = (18, 22, 24, 25) about €460 = (380, 410, 440, 450); around 55 hours = (46, 48, 50, 55). The problem can be written as:
Maximize Z = 38y1 + 32y2 + 28y3 Subject to
50y1 + 33y2 + 27y3 < (380,410,440,450) (5)
10y1 + 8y2 + 4y3 < (46,48,50,55) y1 + y2 + y3 < (18,22,24,25)
The crisp model of the above problem
Maximize Z = 38y£l + 32yL2 + 28yL3 and Z 2 = 38 yR1 + 32 yR2 + 28 yR3
Subject to
50+ 33yL2 + 27yL3 < 380 + 30« (6)
50y^i + 33yR2 + 27yR3 < 450 + 10« 10y£i + 8yl2 + 4yL3 < 46 + 2« 10yRi + 8yR2 + 4yR3 < 55 + 5«
yz.i + yL2 + yL3 < 18 + 4« yR1 + yR2 + yR3 < 25 + «
LINGO 12.0 [10] can be used to acquire the results for the various values of presented in table (2) below. The ideal response to the initial problem is y£1 = (0, 0,0,0), y£2 = (0, 0, 0, 0), and yi3 = (11.62, 11.75,
11.87, 12.0) and is the ideal value = (325.50,329,323.50,336).
Table 2: Cropping combination provides best overall solution
a 0.25 0.50 0.75 1.0
yn 0 0 0 0
yL 2 0 0 0 0
y L3 11.62 11.75 11.87 12
yRl 0 0 0 0
yR 2 0 0 0 0
yR 3 14.06 14.37 14.68 15
z, 325.50 329 323.50 336
Z 2 393.75 402.50 411.25 420
4. Conclusion
The application of fuzzy linear programming to resolve a production planning problem in agriculture has been covered in this study. The paper finishes by explaining how FLPP is transformed into clear multi-objective linear programming problems and how the farmer achieves the best possible outcomes while using constrained resources. Only the trapezoidal membership function is taken into account in this paper.
References
[1] Zimmermann, H.J., (1991). Fuzzy Set Theory and Its Applications, (2nd rev. ed). Boston:
Kulwer.
[2] Orlovsky, S. A. (1980). Fuzzy Sets and Systems, 3, 311-321.
[3] Tamiz, M., (1996). Multi-objective programming and goal programming theories and Applications, Germany: Springer-Verlag.
[4] Ross, T.J., (1995). Fuzzy logic with engineering Applications, New York: McGraw-Hill.
[5] Deldago, M., Verdegay, J.L., Vila, M.A. (1989). A General Model for Fuzzy Linear Programming, Fuzzy Set and System, 29: 21-29.
[6] Fang, S.C., Hu, C.F., Wu, S.Y. & Wang, H.F. (1999). Linear Programming with Fuzzy Coefficients in Constraint, Computers and Mathematics with Applications, 37: 63-76.
[7] P. Senthilkumar and G. Rajendran, (2010). On the solution of Fuzzy linear programming Problem, International journal of computational Cognition, 8(3) 45-47.
[8] Campos, L. and Verdegay, J. L., (1989). Fuzzy Sets and Systems, 32, 1-11.
[9] Maleki, H.R., Tata, M., Mashinchi, M., (2000). Fuzzy Sets and Systems, 9, 21-33.
[10] Lingo 12.0, LINDO Inc. Ltd.
