Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Volume 11, Number 3 (2016), pp. 305-316 Β© Research India Publications http://www.ripublication.com
Intuitionistic Fuzzy Two Stage Multiobjective Transportation Problems Rita Malhotra and S. K. Bharati1 Department of Mathematics, Kamala Nehru College University of Delhi, New Delhi-11049
Abstract Present paper proposes a new method to find the intuitionistic fuzzy optimal solution of two stage multiobjective transportation problems (MOTP). There are several methods are available in literature for the solution of such problem. But, there is no any methods are available for the solution of two stage MOTP with intuitionistic fuzzy (IF) parameters yet. Here, we have considered MOTP with triangular intuitionistic fuzzy numbers (TIFN) parameters. In this method problem is completed in two stages. Present method is very simple and easy to apply in real life transportation problem (TP). AMS Subject Classification: 90B06. Key Words: Intuitionistic Fuzzy Numbers, Triangular, Intuitionistic Fuzzy Numbers, Interval Numbers.
1. INTRODUCTION Transportation problem is one of the best optimization method applicable in various fields of human activity. TP deals with transportation of goods from a set of supply to a set of demand points so as minimize total transportation cost. Hitchcock [14] initiated and modelled basic transportation in form of standard linear programming problem. In beginning of decision making parameters of MOTP are assumed to be fixed in values. But due many uncertain situation like road conditions, traffic conditions, variation in diesel prices etc. and some other unpredicted factors like weather condition. Therefore 1
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due to such reason conventional models are not valid. Zadeh [30] introduced the concept of fuzzy sets for dealing uncertainty. Later Bellman and Zadeh [6] used it for decision making of real life problems. Verma et al. [25] applied fuzzy programming technique to solve MOTP via some non-linear membership functions. Das et al. [10] proposed solution methods for the solution of MOTP with interval cost, source and destinations parameters. Li and Lai [16] have presented a fuzzy compromise programming approach to multiobjective transportation problems. Wahed [27] studied the optimal compromise solution of MOTP and obtained solution is tested by using degree of closeness of the compromise solution to the ideal solution using family of distances. Further, Wahed and Lee [28] proposed interactive fuzzy goal programming approach for efficient solution of MOTP. Liu and Kao [18] have solved fuzzy transportation problems with fuzzy numbers cost, supply and demand based on extension principle. Further, Liu [19] considered the total cost bounds of the transportation problem with varying demand and supply. Ammar and Youness [1] investigated Ξ±-fuzzy efficient and efficient solutions of MOTP having fuzzy number parameters. Zangiabadi and Maleki [29] proposed fuzzy goal programming method for the solution of MOTP incorporating non-linear membership functions. Ritha and Vinotha [22] have solved fuzzy transportation problem in two stages. Peidro and Vasant [21] considered transportation planning with modified S- curve membership functions using interactive fuzzy multiobjective approach. Fuzzy set approach does not deal properly realistic MOTP due hesitation involved in information. Atanassov [2] generalized fuzzy sets by introducing the concept of intuitionistic fuzzy sets (IFS). It is a better tool for dealing problems in realistic way than fuzzy sets. IFS became very popular and applicable in management sciences, planning and various other fields. Angelov [4] investigated intuitionistic fuzzy optimization technique for the solution of TP. Later Bharati and Singh [7, 8, 9] used it for the solution of multiobjective linear programming problem and applied it in agricultural production planning. Jana and Roy [15] solved multiobjective intuitionistic fuzzy linear programming and applied it in transportation problem by using linear and non-linear membership functions. Recently, Antony et al [3] proposed method for the solution of transportation problem using triangular intuitionistic fuzzy numbers. Further, Singh and Yadav [24] investigated a new approach for the solution of intuitionistic fuzzy transportation problem of type-2. Kumar and Hussain [26] given method for the fully intuitionistic fuzzy real transportation problem based ranking method. Due to storage constraints destinations are unable to receive the quantity in excess of their minimum demand. Therefore we encounter a kind of problem. Sonia and Malhotra [23] resolved such problem by using polynomial approach and solved TP into two stages. Further, Gani and Rajak [11] studied two stage fuzzy transportation problem with triangular fuzzy numbers. Here, we have considered intuitionistic fuzzy solution of two stage multiobjective transportation problem (TSMOTP) in more realistic way.
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Definition 1. [2] Let π be universal sets. An intuitionistic fuzzy sets π΄Μ in π is a sets of the form π΄Μ = {(π₯, ππ΄Μ (π₯), ππ΄Μ (π₯)): π₯ β π}, where ππ΄Μ : π β [0, 1] and ππ΄Μ : π β [0, 1], define degrees of membership and non-membership of the element π₯ β π, respectively and for every π₯ β π, 0 β€ ππ΄Μ (π₯) + ππ΄Μ (π₯) β€ 1. The value of ππ΄Μ (π₯) = 1 β ππ΄Μ (π₯) β ππ΄Μ (π₯), is called the degree of non-determinacy (or uncertainty) of the element π₯ β π to the intuitionistic fuzzy set π΄Μ. Definition 2. An intuitionistic fuzzy set π΄Μ = {(π₯, ππ΄Μ (π₯), ππ΄Μ (π₯)): π₯ β β} is called intuitionistic fuzzy numbers if (i) There exists a real number π₯0 β β such that ππ΄Μ (π₯) = 1 and ππ΄Μ (π₯) = 0, (ii) Membership ππ΄Μ of π΄Μ is convex and non-membership ππ΄Μ is concave. (iii) ππ΄Μ is upper semi-continuous and ππ΄Μ is lower semi-continuous. {π₯ β β: ππ΄Μ (π₯) β€ 1}. (iv) Support(π΄Μ) = Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Μ
Definition 3. Let π and π be two universal sets, π: π β π be a function. Extension principle for intuitionistic fuzzy sets find membership and non-membership value π(π΄Μ) where π΄Μ is an intuitionistic fuzzy sets on π: ππ’π{ππ΄Μ (π₯), π₯ β π β1 {π¦}} ππ π¦ β π
ππππ(π) ππ(π΄Μ) (π¦) = { (1) 0, ππ π¦ β π
ππππ(π) πΌππ{ππ΄Μ (π₯), π₯ β π β1 {π¦}} ππ π¦ β π
ππππ(π) ππ(π΄Μ) (π¦) = { (2) 1, ππ π¦ β π
ππππ(π) Definition 4. A triangular intutionistic fuzzy number be given by π΄Μ = {(π1 , π2 , π3 ), (π1 , π2 , π3 )} where π1 , π2 , π3 , π1 , π2 , π3 β β such that π1 β€ π1 β€ π2 = π2 β€ π3 β€ π3 . Its membership and non-membership are given as: 0, π₯ = π2 1, π₯ = π2 0, π₯ β₯ π3 β§ π₯ β€ π1 1, π₯ β₯ π3 β§ π₯ β€ π1 ππ΄Μ (π₯) = { , ππ΄Μ (π₯) = { (3) π(π₯), π1 < π₯ < π2 π(π₯), π1 < π₯ < π2 π(π₯), π2 < π₯ < π3 π(π₯), π2 < π₯ < π3 where : (π1 , π2 ) β [0, 1], ΞΆ: (π2 , π3 ) β [0, 1], π: (π1 , π2 ) β [0, 1], π: (π2 , π3 ) β [0, 1]. Definition 5. Let triangular intuitionistic fuzzy number be given by π΄Μ = {(π1 , π2 , π3 ), (π1 , π2 , π3 )} where π1 β€ π1 β€ π2 = π2 β€ π3 β€ π3 . Then its parametric {[ π1 + πΌ( π2 β π1 ), π3 β πΌ( π3 β π1 )], [ π2 + (1 β πΌ)( π3 β form are π2 ), π2 β (1 β πΌ)( π2 β π1 )]}.
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Definition 6. [12, 13] Let πΜ = [ππΏ , ππ
] be an interval. The minimization problem with the interval-valued objective function is expressed as min {πΜ: πΜ β π}, where π is a set of constraints, which is equal to bi-objective mathematical programming Min {ππ
: π(πΜ) β π}.
2. THEORETICAL DEVELOPMENT Let minimum IF requirement of a homogeneous product at the destination π be πΜπ , IF availability be πΜπ of the same at source π and π π (π₯) is a vector of πΎ objectives. The TSMOTP with intuitionistic fuzzy time deals with supplying the destination their minimum πΌπΉ requirement is stage-I and πΌπΉquantity βπ πΜπ β πΜπ is supplied to the destination is stageII from the sources which have surplus IF quantity left after the completion of stage-I, Mathematically, stated the stage-I problem is: min
π π1π (π) = min [πππ₯πΌΓπ½ (π‘ππ (π₯ππ ))] , π = 1,2, β¦ , πΎ
(4)
Where βππ=1 π₯ππ β€ β
Μπ , π β πΌ, π = {βππ=1 π₯ππ β€ πΜπ , π β π½, π₯ππ β₯ 0, β(π, π) β πΌ Γ π½,
(5)
where β
Μπ , πΜπ , πππ πΌππ are TIFNs. Corresponding to a feasible solution πΜ = (π₯ππ ) of the stage-I problem, the set π(π) = π₯Μππ of the feasible solutions of stage-II problem is given by βππ=1 π₯Μππ β€ β
Μπ , π β πΌ, π = {βππ=1 π₯Μππ β€ πΜπ , π β π½, π₯Μππ β₯ 0, β(π, π) β πΌ Γ π½,
(6)
Μ β²π is the quantity available at the π π‘β source on the completion of the satge-I, where β
Μ β²π =β
π β βπ π₯ππ . that is β
Μ β²π =β
π - βπ ππ . Thus the TS-MOTP would be mathematically formulated as: Clearly βπΌ β
π ππππ₯βπ(π) πΜ2π (πΜ) = min πΜ β π(π)[max πΌ Γ π½ ( (π‘ππ (π₯ππ ))]
In this paper we have found feasible schedule π = (π₯ππ ) of the stage βI problem corresponding to which the optimal time for stage-II is such that the sum of the shipment times is the least. The TS-MOTP can, therefore, be stated as:
(7)
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ππππ₯βπ(π) πΜ1π (π) + ππππ₯βπ(π) πΜ2π (πΜ)
(8)
Problem (4) and (7) which are intuitionistic fuzzy transportation problem with TIFNs parameters becomes: ππ ππ ππ π ππ ππ ππ Minimize π§π = βπ π=1 βπ=1{(πππ1 , πππ2 , πππ3 ), (πππ1 , πππ2 , πππ3 )}π₯ππ
Such that π ππ
ππ
ππ
ππ
ππ
ππ
ππ ππ ππ β π₯ππ = {(πππ1 , πππ2 , πππ3 ), (πππ1 , πππ2 , πππ3 )}, π = 1, 2, β¦ , π, π=1 π ππ ππ ππ β π₯ππ = {(πππ1 , πππ2 , πππ3 ), (πππ1 , πππ2 , πππ3 )}, π = 1, 2, β¦ , π, π=1
π₯ππ β₯ 0, π = 1, 2, β¦ , π and π = 1, 2, β¦ , π.
1βπΌ
π 2πΌ βπ π=1 βπ=1 [
Minimize π§π =
ππ
ππ π (πππ1 +πππ3 )+πππ2
2
,
1βπΌ ππ ππ ππ π (πππ3 +πππ1 )+(πππ1 +πππ2 ) πΌ
2
] π₯ππ
Such that 1βπΌ
βππ=1 π₯ππ = [ 2πΌ
1βπΌ
2πΌ βπ π=1 π₯ππ = [
ππ
ππ π (πππ1 +πππ3 )+πππ2
2
,
ππ
ππ π (πππ1 +πππ3 )+πππ2
2
,
1βπΌ ππ ππ ππ π (πππ3 +πππ1 )+(πππ1 +πππ2 ) πΌ
2 1βπΌ ππ ππ ππ π (πππ3 +πππ1 )+(πππ1 +πππ2 ) πΌ
2
] , π = 1, 2, β¦ , π,
] , π = 1, 2, β¦ , π,
π₯ππ β₯ 0, 0 < πΌ < 1, π = 1, 2, β¦ , π and π = 1, 2, β¦ , π.
1βπΌ
π 2πΌ Minimize π§π = βπ π=1 βπ=1 [
ππ
ππ π (πππ1 +πππ3 )+πππ2
2
,
1+πΌ ππ 1βπΌ ππ ππ 1βπΌ ππ ππ π π + (π +π )+ (πππ3 +πππ1 )+2πππ2 2πΌ ππ1 2πΌ ππ3 ππ1 πΌ
4
] π₯ππ
Such that βππ=1 π₯ππ
1βπΌ ππ 1+πΌ ππ 1βπΌ ππ 1βπΌ ππ ππ π ππ ππ π (πππ1 +πππ3 )+πππ2 π + (πππ3 +πππ1 )+ (πππ3 +πππ1 )+2πππ2 2πΌ 2πΌ ππ1 2πΌ πΌ
=[
2
,
4
] , π = 1, 2, β¦ , π,
310
Rita Malhotra and S. K. Bharati 1βπΌ
2πΌ βπ π=1 π₯ππ = [
ππ
ππ π (πππ1 +πππ3 )+πππ2
2
,
1+πΌ ππ 1βπΌ ππ 1βπΌ ππ ππ ππ π π + (πππ3 +πππ1 )+ (πππ3 +πππ1 )+2πππ2 2πΌ ππ1 2πΌ πΌ
4
] , π = 1, 2, β¦ , π,
π₯ππ β₯ 0, 0 < πΌ < 1, π = 1, 2, β¦ , π and π = 1, 2, β¦ , π.
Minimize π§π =
3β3πΌ ππ πππ3
π πΌ βπ π=1 βπ=1 [
+
1+5πΌ ππ 1βπΌ ππ 3β3πΌ ππ π π +4πππ2 + π + πππ1 2πΌ ππ1 2πΌ ππ3 πΌ
8
] π₯ππ
Such that
βππ=1 π₯ππ βπ π=1 π₯ππ
= =
3β3πΌ ππ πππ3 πΌ
1+5πΌ ππ 1βπΌ ππ 3β3πΌ ππ π π +4πππ2 + π + πππ1 2πΌ ππ1 2πΌ ππ3 πΌ
+
8 3β3πΌ ππ πππ3 πΌ
+
1+5πΌ ππ 1βπΌ ππ 3β3πΌ ππ π π +4πππ2 + π + πππ1 2πΌ ππ1 2πΌ ππ3 πΌ
8
, π = 1, 2, β¦ , π,
, π = 1, 2, β¦ , π,
π₯ππ β₯ 0, 0 < πΌ < 1, π = 1, 2, β¦ , π and π = 1, 2, β¦ , π.
3. COMPUTATIONAL ALGORITHM A two stage multiobjective transportation problem can be solved in following manner: Step 1 Construct the multiobjective intuitionistic fuzzy transportation problems. Step 2 Convert all corresponding objectives and constraints into its crisp form. Step 3 Take one objective function out of given k objectives and solve it as a single objective subject to the given constraints. Form obtained solution vectors find the values of remaining (π β 1) objective functions. Step 4 Continue the step 3 for remaining (k β 1) objective functions. If all the solutions are same, then one of them is the optimal compromise solution. Step 5 Tabulate the solutions thus obtained in step 3 and step 4 to construct the Positive Ideal Solution (PIS) as given below.
Intuitionistic Fuzzy Two Stage Multiobjective Transportation Problems
Min π1 Min π2 Min π3 : :
π1 π2 β π1 π2 (π1 ) π1 (π2 ) π2β π1 (π3 ) π2 (π3 ) : :
Min ππ
π1 (ππ ) π2 (ππ ) π3 (ππ ) π1β²
π3 π3 (π1 ) π3 (π2 ) π3β
π2β²
311
... ... ... ..
ππ ππ (π1 ) ππ (π2 ) ππ (π3 )
...
ππβ ππβ²
π3β²
Max π β¦ Table 1.1. Positive Ideal Solution
X π1 π2 π3 : :
ππ
Step 6 From PIS, obtain the lower bounds and upper bounds for each objective functions, where ππβ and ππβ² are the maximum, minimum values respectively. Step 7 Set upper and lower bounds for each objective for degree of acceptance and degree of rejection corresponding to set of solutions obtained in step4. For membership functions: Upper and lower bound for membership functions π ππ = max(ππ (ππ )) π πΏπ = min(ππ (ππ )), 0β€πβ€πΎ For non -membership functions: π π π π πππ = ππ β π(ππ β πΏπ ), πππ = πΏπ , 0 < π < 1. Step 8 Consider the membership functions ππ (ππ (π₯)) and non-membership functions ππ (ππ (π₯)) As following linear functions: π 1, ππ (π₯) β€ πΏπ π
ππ βππ (π₯)
ππ (ππ (π₯)) =
π
π
ππ βπΏπ
π
,
π
πΏπ β€ ππ (π₯) β€ ππ
(9)
πΏππ β€ ππ (π₯) β€ πππ
(10)
π
{0, ππ (π₯) β₯ ππ 1, ππ (π₯) β₯ πππ ππ (ππ (π₯)) = {
ππ (π₯)βπΏππ πππ βπΏππ
,
0, ππ (π₯) β€ πΏππ
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Step 9 An intuitionistic fuzzy optimization technique for TS-MOLP problem with such membership and non-membership functions can be written as: Maximize πΌ β π½ Such that ππ (ππ (π₯)) β₯ πΌ, ππ (ππ (π₯)) β€ π½, πΌ + π½ β€ 1, πΌ β₯ π½, π½ β₯ 0, π
β π₯ππ = ππ , π = 1,2, β¦ , π, π=1 π
β π₯ππ = ππ , π = 1,2, β¦ , π, π=1
π₯ππ β₯ 0. Above linear programming problem can be solved by simplex method. Step 10 Repeat previous steps for stage-I and stage-II. Step 11 Calculate ππ‘ππ‘ππ = ππ π‘πππβπΌ + ππ π‘πππβπΌπΌ . Stage 12 Calculate acceptance and rejection degrees for ππ‘ππ‘ππ by using Zadeh extension principle. 4. ILLUSTRATION OF PROPOSED SOLUTION METHOD Consider the intuitionistic fuzzy transportation problem Min π§1 = 1Μπ₯11 + 2Μπ₯12 + 7Μπ₯13 + 7Μπ₯14 + 1Μπ₯21 + 9Μπ₯22 + 3Μπ₯23 + 4Μπ₯24 + 8Μπ₯31 + 9Μπ₯32 + 4Μπ₯33 + 6Μπ₯34 Μ π₯24 + 6Μπ₯31 + Min π§2 = 4Μπ₯11 + 4Μπ₯12 + 3Μπ₯13 + 4Μπ₯14 + 5Μπ₯21 + 8Μπ₯22 + 9Μπ₯23 + 10 2Μπ₯32 + 5Μπ₯33 + 1Μπ₯34 Such that π₯11 + π₯12 + π₯13 + π₯14 = 8 π₯21 + π₯22 + π₯23 + π₯24 = 19 π₯31 + π₯32 + π₯33 + π₯34 = 17 π₯11 + π₯21 + π₯31 = 11 π₯12 + π₯22 + π₯32 = 3 π₯13 + π₯23 + π₯33 = 14
Intuitionistic Fuzzy Two Stage Multiobjective Transportation Problems
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π₯14 + π₯24 + π₯34 = 16 π₯ππ β₯ 0 where, 1Μ = {(0.1,1,1.5), (0.5,1,1.2)}, 2Μ = {(1,2,2.5), (1.5,2,2)} 7Μ = {(3,7,9), (5,7,8)}, 7Μ = {(4,7,9), (6,7,8)} 1Μ = {(0.1,1,1.6), (0.6,1,1.5)}, 9Μ = {(7,9,11), (8,9,10)} 3Μ = {(1,3,5), (2,3,4)}, 4Μ = {(2,4,6), (3,4,5)} 8Μ = {(6,8,10), (7,8,9)}, 9Μ = {(7,9,12), (8,9,11)} 4Μ = {(2,4,7), (3,4,6)}, 6Μ = {(4,6,9), (5,6,8)} 4Μ = {(2,4,6), (3,4,5)}, 4Μ = {(3,4,7), (3,4,5)} 3Μ = {(1,3,5), (2,3,4)}, 4Μ = {(2,4,6), (3,4,5)} 4Μ = {(2,4,6), (3,4,5)}, 5Μ = {(3,5,7), (4,5,6)} 8Μ = {(5,8,10), (5.5,8,11)}, 9Μ = {(8,9,10), (8.5,9,9.5)} Μ = {(8,10,12), (9,10,10.5)}, 6Μ = {5,6,6.5), (5.5,6,6.1)} 10 2Μ = {(1,2,3), (1.2,2,2.8)}, 5Μ = {(4,5,6), (4.5,5,5.5)}. First stage optimal solutions are: π₯11 = 0.5, π₯12 = 1, π₯13 = 0, π₯14 = 0, π₯21 = 2, π₯22 = 0, π₯23 = 0.32, π₯24 = 0.17, π₯31 = 0, π₯32 = 0, π₯33 = 0.17, π₯34 = 1.82, Min π§1 = 28.64, Min π§2 = 34.43, πΌ = 0.78, π½ = 0. Second stage optimal solutions are: π₯11 = 4.5, π₯12 = 2, π₯13 = 0, π₯14 = 0, π₯21 = 4, π₯22 = 0, π₯23 = 12.06, π₯24 = 0.43, π₯31 = 0, π₯32 = 0, π₯33 = 1.43, π₯34 = 13.56, Min π§1 = 223.41, Min π§2 = 302.74, πΌ = 0.64, π½ = 0. Now using definition [3] we get total minimum values are given as: Min π1 = 252.05, Min π2 = 337.17 with degree of acceptance πΌ = 0.78 and degree of acceptance π½ = 0.
CONCLUSION In some situation due to limited capacity of storage, destinations are unable to receive the quantity in excess of their minimum demand. In this case, a single shipment is not possible. Therefore, items are shipped to destinations from the origins in two stages.
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Initially, the minimum demands of the destinations are shipped from origins to the destinations. After consuming part of whole of this initial shipment, they are prepared to receive the excess quantity in the second stage. The present method is based on intuitionistic fuzzy sets. Therefore, it will be perfect to handle real transportation problems.
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