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Fuzzy Goal Programming Model for the Unbalanced Transportation Problem with hyperbolic membership function
aBodkhe S. G, bM. H. Lohgaonkar
aDept. of Statistics A.S.C.College, Badnapur, Dist:- Jalna, Maharashtra, India.
bDept. of Statistics Shri Chhatrapati Shivaji Mahavidyalayae, Shrigonda, Maharashtra, India.
In this paper fuzzy goal programming modals can be formulated for unbalanced transportation problems by using the basic notions of fuzzy subsets and those problems can be solved by non-linear membership function. The max-min operator has been used to aggregate the fuzzy goals since this operator treats all the goals as equality important. The method is illustrated with a numerical example.
KEYWORDS- unbalanced transportation problem, fuzzy nonlinear membership function, multiple goal programming
1 Introduction
Goal programming (GP) is a suitable method to solve Multicriteria Decision Making (MCDM) problems with multiple conflicting objectives.
The detailed analysis of GP has been given by Lee [9], Ignizio [5,6,7], and Romero [12]. Goal Programming is being used in multicriteria decision making problems where the alternatives cannot be compared on the basis of a single performance criterion. In goal formulation, the aspiration level is fixed and the decision maker has no control over the deviation from the aspiration level. To overcome these difficulties, we have investigated Fuzzy Goal Programming (FGP) of an unbalanced transportation problem with multiple fuzzy goals where all the goals are equality important. To reflect the equal importance of the goals, equal weights are assigned to them. In most of the MCDM problems in the real world, the articulation of the goals and objectives of the decision maker are fuzzy in nature. Bellman and Zadeh [1] dealt with the problems involving decision making under fuzzy environment. The application of the fuzzy set theory to GP has been made by Narasimhan [11] and Hannan [3, 4].Tiwari et al. [14] have investigated how the preemptive priority structure can be used in FGP problems. Tiwari et al. [15] have developed an additive model for fuzzy goal programming problems.
In many instances, the decision maker finds multiple conflicting objectives in a transportation problem. In such cases, the usual transportation method cannot be used.
It is possible to apply the GP approach to such transportation problems. Lee and Moore [10] have shown the application of goal programming to multiobjective transportation problem. Kwak [8] considered a GP model for improved transportation problem. Weighted goal programming for unbalanced single objective transportation problem with budgetary constraint has been discussed by Singh and Kishore [13]. Bit et al [2] have presented a transportation problem model for the allocation of coal and its by products (soft coke, middling, and washed coal) from different sources and converting plants of coal mines to different consumption sites. The model has been formulated to meet the energy demand, and to minimize the transportation problem
Abstract
w w w . l i i r j . o r g I S S N 2 2 7 7 - 7 2 7 X Page 3 cost. A case study is presented and the problem is solved using goal-programming method in order to find a satisfactory solution.
In many decision-making situations, the decision maker faces an unbalanced transportation problem in which total supply is less than the total demand. Due to the scarcity of funds, the decision maker is not able to satisfy all the demand points fully with the existing availability. In such cases the solution is affected by the multiple objectives. This type of problem is faced by the government agencies like Food Corporation of India, (FCI) which supplies food grains from different warehouses (i.e., source points) to different distribution centers (i.e., demand points). Although the quantity of food grains supplied is not sufficient to meet the demand of each distribution center. It is not possible for the FCI to transport even the available capacity due to paucity of fund. It is very difficult to supply food grains even to all remote places where multiple objective values are very high due to lack of approachability to such places. But it is essential to supply a certain percentage of the demand to save the people from starvation.
2 Multi-objective Transportation problem (MOTP)
In decision making problems there often exist several non-commensurable criteria, which must be considered. This situation is formulated as a multicriterion optimization (also called multi-performance, Multiobjective, or vector optimization) problem in which the decision maker’s goal is to minimize and/or maximize not a single objective but several objectives simultaneously. λ is degree of membership function and its range is [0, 1]
3 Unbalanced Transportation Problem with Fuzzy Goals.
In a typical transportation problem, a homogeneous product is to be transported from each of m sources to n destinations. The sources are production facilities, warehouses, or supply point, characterized by available capacities ai ( i = 1,2,…, m ).
The destinations are consumption facilities, warehouses, or demand points, characterized by required levels of demand bj ( j = 1,2, …, n ), A penalty Cij is associated with transportation of a unit of the product from sources i to destination j corresponding to the p-th criterion. The penalty could represent transportation cost, delivery time, deterioration of total goods, quantity of goods delivered, under used capacity, etc. A variable Xij represents the unknown quantity to be transported from origin Oi to destination Dj. In the real would, however,
Multiobjective transportation problems are not balanced and objectives are not precisely specified (i.e., goals are fuzzy in nature). We have formulated the FGP model of an unbalanced multiobjective transportation problem in which the objectives and demands are specified imprecisely. The decision variables, supply constrains, fuzzy demand goals and fuzzy multiple goal are identified as follows:
Decision variable:
Decision variables for the model are defined as xij , i = 1,2,…,m; j = 1,2,…,n
where xij 0 for all i and j
System or supply constraints:
j 1n xij ai
, i = 1,2,…,m where ai > 0 for all i
Fuzzy demand goals:
m xij bj
i 1
, j = 1,2,…,n
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The symbol stands for approximately or fuzzily nearly greater than or equal to.
fuzzy budget goals:
Zp = n ij j
i 1 1
m p
c xi B,
j
Where B is the aspiration level of the objective, and the symbol stands for approximately or fuzzily nearly less than or equal to.
4 Mathematical model
Mathematical model of an unbalanced transportation problem with multiple fuzzy goals is stated as follows:
Find xij, i = 1,2,…, m; j = 1,2,…,n so as to satisfy the following constraints and fuzzy goals:
j 1n xij ai
, i = 1,2,..., m i 1m xij bj
, j = 1,2,…,n Zp = n ij j
i 1 1
m p
c xi B,
j
(1)
x 0 , a 0, b 0
ij i j for all i, and j
n m
bj ai
j 1 i 1
In an unbalanced problem, the total market requirements either exceed the total plant
capacity n j m i
j=1 i =1
i.e., b > a
( )or less than the total plant capacity n j m i
j=1 i =1
i.e., b < a
( ). In
our unbalanced transportation model, we have considered n j m i
j=1 i =1
i.e., b > a
( ).
5 Solution Procedure for the proposed model:
we convert the model to a linear programming model by using non-linear membership functions and max-min operator. The solution of linear programming model gives an efficient solution for the fuzzy goal-programming model.
Define a hyperbolic membership function of the fuzzy demand goals are as follows:
1
m b +b
1 j j 1
μAj(x) 2tanh i=1xij 2 αp 2
0
m
ifi=1xij bj m
if bj i=1xij bj m
ifi=1xij bj
,
(2) Where αp = 6/ (bj - bj*
),
bj* (j =1,2,…,n) is the lower tolerance limit of the j-th demand goal.
Hyperbolic membership functions of the fuzzy budget goals are defined as
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1,
1 B B m n p 1
μAn p(x) = 2tanh 2 -i 1 j 1c ij ijx αp + ,2
0,
m n p
ifi 1 j 1c ij ijx B
m n p
if B< c x B ij ij i 1 j 1
m n p
ifi 1 j 1c ij ijx B
(3) Where αp = 6/ (B* - B),
B*is the upper tolerance limit of the budget goal.
Use the max-min operator Problem (1) becomes.
Max λ
λ μ (x), j 1,2,...,n Aj
λ μ (x), p 1,2,...,p An p
Where
m b +bj j
1 1
λ Min 2tanh (i=1xij 2 ) α + ,p 2
1tanh B B -m n cpij ijx α +p 1
2 2 i 1j 1 2
(4)
An equivalent non linear programming model of the unbalanced transportation
problem with multiple fuzzy goals (1) can be stated as follow:
Max λ Subject to
m
xij a ,i i 1,2,...,m
j 1
λ μAj(x), j 1,2,...,n
λ μ (x), p 1,2,...,n An p
xij 0for all i, and j, λ 0
(5)
6. Numerical example:
There are four godowns (source points) from where the food grains are supplied to three different destination (demand points) cij are cost coefficients expressed in rupee per metric ton and ai and bj are expressed in lakhs of metric ton The transportation cost matrix is given as:
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Find x ,ij i =1,2,3; j=1,2,3,4
System or supply constraints:
3 3 3 3
x 10, x 4, x 4, x 12
1j 2j 3j 4j
j=1 j=1 j=1 j=1
Fuzzy demand goals:
i=14 xi112, i=14 xi214, i=14 xi314,
Fuzzy budget goals:
4 3 1
c x 110
ij ij i=1 j=1
x 0, for all i and j ij
(6)
z1 = 5x11 + 8x12 + 3x13 + 7x21 + 4x2 2+ 5x23 + 2x31 + 6x32 + 9x33 +4x41 + 6x42 + 6x43
The membership functions μ1(x), μ2(x), and μ3(x), of the demand goals, and the membership function μ4(x),of the budget goals, respectively, are defined as follows : If we use the hyperbolic membership function with
α =p (b -b )j6 j ,
6 6 6
α =1 (b - b )1 1 1, α =2 (b2 - b2 ) 7,
6 6 6
α =3 (b - b3 3 ) 7, α =4 (B4 - B4 ) 6
b + b1 1 = 9, b + b2 2 10.5,
2 2
b + b3 3 = 10.5, B + B4 4 = 100.5,
2 2
1,
1 1
μ (x)1 2tanh x11 x21 x31 9 + ,2
0,
if x11 x21 x31 12 if 6 x11 x21 x31 12 if x11 x21 x31 6
(7)
1,
1 6 1
μ (x)2 2tanh x12 x22 x32 10.5 7 +2 ,
0,
if x12+x22+x32 14 if 7 x12+x22+x32 14 if x12+x22+x32 7
D1 D2 D3 O1 5 8 3
10 O2 7 4 5 4 O3 2 6 9 4 O4 4 6 6
12 12 14
14
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,1,
1 6 1
μ (x)3 2tanh x13 x23 x33 10.5 7 +2
0,
if x13 x23 x33 14 if 7 x13 x23 x33 14 if x13 x23 x33 7
(9)
1, if z1 100
1 1
μ (x)4 2tanh 100.5-z1 6 + , if 100 z2 1 101
0, if z 101
1
(10)
An equivalent non-linear programming of the problem (6) is formulated as follows:
Max Xmn+1 Subject to
3 3 3 3
x1j 10, x2j 4, x3j 4, x4j 12 j=1 j=1 j=1 j=1
X11+X21+X31- Xmn+1 9
6X12+6X22+6X32-7 Xmn+1 63 (11)
6X13+6X23+6X33-7 Xmn+1 63 6Z1 + Xmn+1 603
ij mn
x 0, for all i j and X +1 0
Where,
Xmn+1= tanh-1(2λ-1)
Solution of the problem (11) is obtained by LINGO : λ =0 82.
X11 = 0.0, X21 = 0.0 X31 = 4, X41 = 2.49 X12 = 0.0, X22 = 4.0 X32 = 0.0,X42 = 3.58, X13 = 0.17X23 =0.0,X33 = 0.0 X43 = 5.93.
From the results, we observe the following points:
Amount of food grains supplied from source O1 = 0.17 metric tons.
Amount of food grains supplied from source O2 = 4.00 metric tons.
Amount of food grains supplied from source O3 = 4.00 metric tons Amount of food grains supplied from source O4 = 12 metric tons
Demand fulfilled at destination D1 = 6.49metric tons.
Demand fulfilled at destination D2 = 7.58metric tons.
Demand fulfilled at destination D3 = 6.1 metric tons.
w w w . l i i r j . o r g I S S N 2 2 7 7 - 7 2 7 X Page 8 Total demand fulfilled = 20.17 quintals.
Total supply = 20.17 quintals.
Transportation cost = 91.53 Rs.
Conclusion
In fuzzy goal programming formulation the aspiration levels of the fuzzy goals are fixed. The deviations from the goals in the fuzzy goal programming are under the control of the decision maker. In the numerical example, fuzzy formulations of the demand goals are within the reasonable limits. The transportation cost is Rs.91.53 Thus the present method is also suitable method for unbalanced transportation problem with fuzzy goals.
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