Intuitionistic Fuzzy Two Stage Multiobjective Transportation Problems

Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Volume 11, Number 3 (2016), pp. 305-316 © 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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306

Rita Malhotra and S. K. Bharati

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 α-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.

Intuitionistic Fuzzy Two Stage Multiobjective Transportation Problems

307

Definition 1. [2] Let 𝑋 be universal sets. An intuitionistic fuzzy sets 𝐴̃ in 𝑋 is a sets of the form 𝐴̃ = {(𝑥, 𝜇𝐴̃ (𝑥), 𝜈𝐴̃ (𝑥)): 𝑥 ∈ 𝑋}, where 𝜇𝐴̃ : 𝑋 → [0, 1] and 𝜈𝐴̃ : 𝑋 → [0, 1], define degrees of membership and non-membership of the element 𝑥 ∈ 𝑋, respectively and for every 𝑥 ∈ 𝑋, 0 ≤ 𝜇𝐴̃ (𝑥) + 𝜈𝐴̃ (𝑥) ≤ 1. The value of 𝜋𝐴̃ (𝑥) = 1 − 𝜇𝐴̃ (𝑥) − 𝜈𝐴̃ (𝑥), is called the degree of non-determinacy (or uncertainty) of the element 𝑥 ∈ 𝑋 to the intuitionistic fuzzy set 𝐴̃. Definition 2. An intuitionistic fuzzy set 𝐴̃ = {(𝑥, 𝜇𝐴̃ (𝑥), 𝜈𝐴̃ (𝑥)): 𝑥 ∈ ℝ} is called intuitionistic fuzzy numbers if (i) There exists a real number 𝑥0 ∈ ℝ such that 𝜇𝐴̃ (𝑥) = 1 and 𝜈𝐴̃ (𝑥) = 0, (ii) Membership 𝜇𝐴̃ of 𝐴̃ is convex and non-membership 𝜈𝐴̃ is concave. (iii) 𝜇𝐴̃ is upper semi-continuous and 𝜈𝐴̃ is lower semi-continuous. {𝑥 ∈ ℝ: 𝜈𝐴̃ (𝑥) ≤ 1}. (iv) Support(𝐴̃) = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ Definition 3. Let 𝑋 and 𝑌 be two universal sets, 𝑓: 𝑋 → 𝑌 be a function. Extension principle for intuitionistic fuzzy sets find membership and non-membership value 𝑓(𝐴̃) where 𝐴̃ is an intuitionistic fuzzy sets on 𝑋: 𝑆𝑢𝑝{𝜇𝐴̃ (𝑥), 𝑥 ∈ 𝑓 −1 {𝑦}} 𝑖𝑓 𝑦 ∈ 𝑅𝑎𝑛𝑔𝑒(𝑓) 𝜇𝑓(𝐴̃) (𝑦) = { (1) 0, 𝑖𝑓 𝑦 ∉ 𝑅𝑎𝑛𝑔𝑒(𝑓) 𝐼𝑛𝑓{𝜈𝐴̃ (𝑥), 𝑥 ∈ 𝑓 −1 {𝑦}} 𝑖𝑓 𝑦 ∈ 𝑅𝑎𝑛𝑔𝑒(𝑓) 𝜈𝑓(𝐴̃) (𝑦) = { (2) 1, 𝑖𝑓 𝑦 ∉ 𝑅𝑎𝑛𝑔𝑒(𝑓) Definition 4. A triangular intutionistic fuzzy number be given by 𝐴̃ = {(𝑎1 , 𝑎2 , 𝑎3 ), (𝑏1 , 𝑏2 , 𝑏3 )} where 𝑎1 , 𝑎2 , 𝑎3 , 𝑏1 , 𝑏2 , 𝑏3 ∈ ℝ such that 𝑏1 ≤ 𝑎1 ≤ 𝑎2 = 𝑏2 ≤ 𝑎3 ≤ 𝑏3 . Its membership and non-membership are given as: 0, 𝑥 = 𝑎2 1, 𝑥 = 𝑎2 0, 𝑥 ≥ 𝑎3 ∧ 𝑥 ≤ 𝑎1 1, 𝑥 ≥ 𝑏3 ∧ 𝑥 ≤ 𝑏1 𝜇𝐴̃ (𝑥) = { , 𝜈𝐴̃ (𝑥) = { (3) 𝜙(𝑥), 𝑎1 < 𝑥 < 𝑎2 𝜂(𝑥), 𝑏1 < 𝑥 < 𝑎2 𝜁(𝑥), 𝑎2 < 𝑥 < 𝑎3 𝜉(𝑥), 𝑎2 < 𝑥 < 𝑏3 where : (𝑎1 , 𝑎2 ) → [0, 1], ζ: (𝑎2 , 𝑎3 ) → [0, 1], 𝜂: (𝑏1 , 𝑎2 ) → [0, 1], 𝜉: (𝑎2 , 𝑏3 ) → [0, 1]. Definition 5. Let triangular intuitionistic fuzzy number be given by 𝐴̃ = {(𝑎1 , 𝑎2 , 𝑎3 ), (𝑏1 , 𝑏2 , 𝑏3 )} where 𝑏1 ≤ 𝑎1 ≤ 𝑎2 = 𝑏2 ≤ 𝑎3 ≤ 𝑏3 . Then its parametric {[ 𝑎1 + 𝛼( 𝑎2 − 𝑎1 ), 𝑎3 − 𝛼( 𝑎3 − 𝑎1 )], [ 𝑎2 + (1 − 𝛼)( 𝑏3 − form are 𝑎2 ), 𝑎2 − (1 − 𝛼)( 𝑎2 − 𝑏1 )]}.

308


Definition 6. [12, 13] Let 𝑎̃ = [𝑎𝐿 , 𝑎𝑅 ] be an interval. The minimization problem with the interval-valued objective function is expressed as min {𝑎̃: 𝑎̃ ∈ 𝑆}, where 𝑆 is a set of constraints, which is equal to bi-objective mathematical programming Min {𝑎𝑅 : 𝑚(𝑎̃) ∈ 𝑆}.

2. THEORETICAL DEVELOPMENT Let minimum IF requirement of a homogeneous product at the destination 𝑗 be 𝑏̃𝑗 , IF availability be 𝑎̃𝑖 of the same at source 𝑖 and 𝑇 𝑘 (𝑥) is a vector of 𝐾 objectives. The TSMOTP with intuitionistic fuzzy time deals with supplying the destination their minimum 𝐼𝐹 requirement is stage-I and 𝐼𝐹quantity ∑𝑖 𝑎̃𝑖 − 𝑏̃𝑗 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

𝑘 𝑇1𝑘 (𝑋) = min [𝑚𝑎𝑥𝐼×𝐽 (𝑡𝑖𝑗 (𝑥𝑖𝑗 ))] , 𝑘 = 1,2, … , 𝐾

(4)

Where ∑𝑛𝑗=1 𝑥𝑖𝑗 ≤ ∅̃𝑖 , 𝑖 ∈ 𝐼, 𝑆 = {∑𝑛𝑗=1 𝑥𝑖𝑗 ≤ 𝜑̃𝑖 , 𝑗 ∈ 𝐽, 𝑥𝑖𝑗 ≥ 0, ∀(𝑖, 𝑗) ∈ 𝐼 × 𝐽,

(5)

where ∅̃𝑖 , 𝜑̃𝑖 , 𝑎𝑛𝑑 𝐼𝑖𝑗 are TIFNs. Corresponding to a feasible solution 𝑋̃ = (𝑥𝑖𝑗 ) of the stage-I problem, the set 𝑆(𝑋) = 𝑥̃𝑖𝑗 of the feasible solutions of stage-II problem is given by ∑𝑛𝑗=1 𝑥̃𝑖𝑗 ≤ ∅̃𝑖 , 𝑖 ∈ 𝐼, 𝑆 = {∑𝑛𝑗=1 𝑥̃𝑖𝑗 ≤ 𝜑̃𝑖 , 𝑗 ∈ 𝐽, 𝑥̃𝑖𝑗 ≥ 0, ∀(𝑖, 𝑗) ∈ 𝐼 × 𝐽,

(6)

̃ ′𝑖 is the quantity available at the 𝑖 𝑡ℎ source on the completion of the satge-I, where ∅ ̃ ′𝑖 =∅𝑖 − ∑𝑗 𝑥𝑖𝑗 . that is ∅ ̃ ′𝑖 =∅𝑖 - ∑𝑗 𝜑𝑗 . Thus the TS-MOTP would be mathematically formulated as: Clearly ∑𝐼 ∅ 𝑘 𝑀𝑖𝑛𝑥∈𝑆(𝑋) 𝑇̃2𝑘 (𝑋̃) = min 𝑋̃ ∈ 𝑆(𝑋)[max 𝐼 × 𝐽 ( (𝑡𝑖𝑗 (𝑥𝑖𝑗 ))]

In this paper we have found feasible schedule 𝑋 = (𝑥𝑖𝑗 ) of the stage –I 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)


309

𝑀𝑖𝑛𝑥∈𝑆(𝑋) 𝑇̃1𝑘 (𝑋) + 𝑀𝑖𝑛𝑥∈𝑆(𝑋) 𝑇̃2𝑘 (𝑋̃)

(8)

Problem (4) and (7) which are intuitionistic fuzzy transportation problem with TIFNs parameters becomes: 𝜇𝑘 𝜇𝑘 𝜇𝑘 𝑛 𝜈𝑘 𝜈𝑘 𝜈𝑘 Minimize 𝑧𝑘 = ∑𝑚 𝑖=1 ∑𝑗=1{(𝑐𝑖𝑗1 , 𝑐𝑖𝑗2 , 𝑐𝑖𝑗3 ), (𝑐𝑖𝑗1 , 𝑐𝑖𝑗2 , 𝑐𝑖𝑗3 )}𝑥𝑖𝑗

Such that 𝑛 𝜇𝑘

𝜇𝑘

𝜇𝑘

𝜇𝑘

𝜇𝑘

𝜇𝑘

𝜈𝑘 𝜈𝑘 𝜈𝑘 ∑ 𝑥𝑖𝑗 = {(𝑎𝑖𝑗1 , 𝑎𝑖𝑗2 , 𝑎𝑖𝑗3 ), (𝑎𝑖𝑗1 , 𝑎𝑖𝑗2 , 𝑎𝑖𝑗3 )}, 𝑖 = 1, 2, … , 𝑚, 𝑗=1 𝑚 𝜈𝑘 𝜈𝑘 𝜈𝑘 ∑ 𝑥𝑖𝑗 = {(𝑏𝑖𝑗1 , 𝑏𝑖𝑗2 , 𝑏𝑖𝑗3 ), (𝑏𝑖𝑗1 , 𝑏𝑖𝑗2 , 𝑏𝑖𝑗3 )}, 𝑗 = 1, 2, … , 𝑛, 𝑖=1

𝑥𝑖𝑗 ≥ 0, 𝑖 = 1, 2, … , 𝑚 and 𝑗 = 1, 2, … , 𝑛.

1−𝛼

𝑛 2𝛼 ∑𝑚 𝑖=1 ∑𝑗=1 [

Minimize 𝑧𝑘 =

𝜇𝑘

𝜈𝑘 𝑘 (𝑐𝑖𝑗1 +𝑐𝑖𝑗3 )+𝑐𝑖𝑗2

2

,

1−𝛼 𝜇𝑘 𝜇𝑘 𝜈𝑘 𝑘 (𝑐𝑖𝑗3 +𝑐𝑖𝑗1 )+(𝑐𝑖𝑗1 +𝑐𝑖𝑗2 ) 𝛼

2

] 𝑥𝑖𝑗

Such that 1−𝛼

∑𝑛𝑗=1 𝑥𝑖𝑗 = [ 2𝛼

1−𝛼

2𝛼 ∑𝑚 𝑖=1 𝑥𝑖𝑗 = [

𝜇𝑘

𝜈𝑘 𝑘 (𝑎𝑖𝑗1 +𝑐𝑖𝑗3 )+𝑎𝑖𝑗2

2

,

𝜇𝑘

𝜈𝑘 𝑘 (𝑏𝑖𝑗1 +𝑏𝑖𝑗3 )+𝑏𝑖𝑗2

2

,

1−𝛼 𝜇𝑘 𝜇𝑘 𝜈𝑘 𝑘 (𝑎𝑖𝑗3 +𝑎𝑖𝑗1 )+(𝑎𝑖𝑗1 +𝑎𝑖𝑗2 ) 𝛼

2 1−𝛼 𝜇𝑘 𝜇𝑘 𝜈𝑘 𝑘 (𝑏𝑖𝑗3 +𝑏𝑖𝑗1 )+(𝑏𝑖𝑗1 +𝑏𝑖𝑗2 ) 𝛼

2

] , 𝑖 = 1, 2, … , 𝑚,

] , 𝑗 = 1, 2, … , 𝑛,

𝑥𝑖𝑗 ≥ 0, 0 < 𝛼 < 1, 𝑖 = 1, 2, … , 𝑚 and 𝑗 = 1, 2, … , 𝑛.

1−𝛼

𝑛 2𝛼 Minimize 𝑧𝑘 = ∑𝑚 𝑖=1 ∑𝑗=1 [

𝜇𝑘

𝜈𝑘 𝑘 (𝑐𝑖𝑗1 +𝑐𝑖𝑗3 )+𝑐𝑖𝑗2

2

,

1+𝛼 𝜇𝑘 1−𝛼 𝜇𝑘 𝜈𝑘 1−𝛼 𝜇𝑘 𝜈𝑘 𝑘 𝑐 + (𝑐 +𝑐 )+ (𝑐𝑖𝑗3 +𝑐𝑖𝑗1 )+2𝑐𝑖𝑗2 2𝛼 𝑖𝑗1 2𝛼 𝑖𝑗3 𝑖𝑗1 𝛼

4

] 𝑥𝑖𝑗

Such that ∑𝑛𝑗=1 𝑥𝑖𝑗

1−𝛼 𝜇𝑘 1+𝛼 𝜇𝑘 1−𝛼 𝜇𝑘 1−𝛼 𝜇𝑘 𝜈𝑘 𝑘 𝜈𝑘 𝜈𝑘 𝑘 (𝑎𝑖𝑗1 +𝑎𝑖𝑗3 )+𝑎𝑖𝑗2 𝑎 + (𝑎𝑖𝑗3 +𝑎𝑖𝑗1 )+ (𝑎𝑖𝑗3 +𝑎𝑖𝑗1 )+2𝑎𝑖𝑗2 2𝛼 2𝛼 𝑖𝑗1 2𝛼 𝛼

=[

2

,

4

] , 𝑖 = 1, 2, … , 𝑚,

310

Rita Malhotra and S. K. Bharati 1−𝛼

2𝛼 ∑𝑚 𝑖=1 𝑥𝑖𝑗 = [

𝜇𝑘

𝜈𝑘 𝑘 (𝑏𝑖𝑗1 +𝑏𝑖𝑗3 )+𝑏𝑖𝑗2

2

,

1+𝛼 𝜇𝑘 1−𝛼 𝜇𝑘 1−𝛼 𝜇𝑘 𝜈𝑘 𝜈𝑘 𝑘 𝑏 + (𝑏𝑖𝑗3 +𝑏𝑖𝑗1 )+ (𝑏𝑖𝑗3 +𝑏𝑖𝑗1 )+2𝑏𝑖𝑗2 2𝛼 𝑖𝑗1 2𝛼 𝛼

4

] , 𝑗 = 1, 2, … , 𝑛,

𝑥𝑖𝑗 ≥ 0, 0 < 𝛼 < 1, 𝑖 = 1, 2, … , 𝑚 and 𝑗 = 1, 2, … , 𝑛.

Minimize 𝑧𝑘 =

3−3𝛼 𝜇𝑘 𝑐𝑖𝑗3

𝑛 𝛼 ∑𝑚 𝑖=1 ∑𝑗=1 [

+

1+5𝛼 𝜇𝑘 1−𝛼 𝜈𝑘 3−3𝛼 𝜈𝑘 𝑘 𝑐 +4𝑐𝑖𝑗2 + 𝑐 + 𝑐𝑖𝑗1 2𝛼 𝑖𝑗1 2𝛼 𝑖𝑗3 𝛼

8

] 𝑥𝑖𝑗

Such that

∑𝑛𝑗=1 𝑥𝑖𝑗 ∑𝑚 𝑖=1 𝑥𝑖𝑗

= =

3−3𝛼 𝜇𝑘 𝑎𝑖𝑗3 𝛼

1+5𝛼 𝜇𝑘 1−𝛼 𝜈𝑘 3−3𝛼 𝜈𝑘 𝑘 𝑎 +4𝑎𝑖𝑗2 + 𝑎 + 𝑎𝑖𝑗1 2𝛼 𝑖𝑗1 2𝛼 𝑖𝑗3 𝛼

+

8 3−3𝛼 𝜇𝑘 𝑏𝑖𝑗3 𝛼

+

1+5𝛼 𝜇𝑘 1−𝛼 𝜈𝑘 3−3𝛼 𝜈𝑘 𝑘 𝑏 +4𝑏𝑖𝑗2 + 𝑏 + 𝑏𝑖𝑗1 2𝛼 𝑖𝑗1 2𝛼 𝑖𝑗3 𝛼

8

, 𝑖 = 1, 2, … , 𝑚,

, 𝑗 = 1, 2, … , 𝑛,

𝑥𝑖𝑗 ≥ 0, 0 < 𝛼 < 1, 𝑖 = 1, 2, … , 𝑚 and 𝑗 = 1, 2, … , 𝑛.

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 (𝑘 − 1) objective functions. Step 4 Continue the step 3 for remaining (k − 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.


Min 𝑓1 Min 𝑓2 Min 𝑓3 : :

𝑓1 𝑓2 ∗ 𝑓1 𝑓2 (𝑋1 ) 𝑓1 (𝑋2 ) 𝑓2∗ 𝑓1 (𝑋3 ) 𝑓2 (𝑋3 ) : :

Min 𝑓𝑘

𝑓1 (𝑋𝑘 ) 𝑓2 (𝑋𝑘 ) 𝑓3 (𝑋𝑘 ) 𝑓1′

𝑓3 𝑓3 (𝑋1 ) 𝑓3 (𝑋2 ) 𝑓3∗

𝑓2′

311

... ... ... ..

𝑓𝑘 𝑓𝑘 (𝑋1 ) 𝑓𝑘 (𝑋2 ) 𝑓𝑘 (𝑋3 )

...

𝑓𝑘∗ 𝑓𝑘′

𝑓3′

Max 𝑓 … Table 1.1. Positive Ideal Solution

X 𝑋1 𝑋2 𝑋3 : :

𝑋𝑘

Step 6 From PIS, obtain the lower bounds and upper bounds for each objective functions, where 𝑓𝑘∗ and 𝑓𝑘′ 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 𝜇 𝑈𝑘 = max(𝑍𝑘 (𝑋𝑟 )) 𝜇 𝐿𝑘 = min(𝑍𝑘 (𝑋𝑟 )), 0≤𝑟≤𝐾 For non -membership functions: 𝜇 𝜇 𝜇 𝜇 𝑈𝑘𝜈 = 𝑈𝑘 − 𝜆(𝑈𝑘 − 𝐿𝑘 ), 𝑈𝑘𝜈 = 𝐿𝑘 , 0 < 𝜆 < 1. Step 8 Consider the membership functions 𝜇𝑘 (𝑓𝑘 (𝑥)) and non-membership functions 𝜈𝑘 (𝑓𝑘 (𝑥)) As following linear functions: 𝜇 1, 𝑓𝑘 (𝑥) ≤ 𝐿𝑘 𝜇

𝑈𝑘 −𝑓𝑘 (𝑥)

𝜇𝑘 (𝑓𝑘 (𝑥)) =

𝜇

𝜇

𝑈𝑘 −𝐿𝑘

𝜇

,

𝜇

𝐿𝑘 ≤ 𝑓𝑘 (𝑥) ≤ 𝑈𝑘

(9)

𝐿𝜈𝑘 ≤ 𝑓𝑘 (𝑥) ≤ 𝑈𝑘𝜈

(10)

𝜇

{0, 𝑓𝑘 (𝑥) ≥ 𝑈𝑘 1, 𝑓𝑘 (𝑥) ≥ 𝑈𝑘𝜈 𝜈𝑘 (𝑓𝑘 (𝑥)) = {

𝑓𝑘 (𝑥)−𝐿𝜈𝑘 𝑈𝑘𝜈 −𝐿𝜈𝑘

,

0, 𝑓𝑘 (𝑥) ≤ 𝐿𝜈𝑘

312


Step 9 An intuitionistic fuzzy optimization technique for TS-MOLP problem with such membership and non-membership functions can be written as: Maximize 𝛼 − 𝛽 Such that 𝜇𝑘 (𝑓𝑘 (𝑥)) ≥ 𝛼, 𝜈𝑘 (𝑓𝑘 (𝑥)) ≤ 𝛽, 𝛼 + 𝛽 ≤ 1, 𝛼 ≥ 𝛽, 𝛽 ≥ 0, 𝑛

∑ 𝑥𝑖𝑗 = 𝑎𝑖 , 𝑖 = 1,2, … , 𝑚, 𝑗=1 𝑚

∑ 𝑥𝑖𝑗 = 𝑏𝑗 , 𝑗 = 1,2, … , 𝑛, 𝑖=1

𝑥𝑖𝑗 ≥ 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 𝑍𝑡𝑜𝑡𝑎𝑙 = 𝑍𝑠𝑡𝑎𝑔𝑒−𝐼 + 𝑍𝑠𝑡𝑎𝑔𝑒−𝐼𝐼 . Stage 12 Calculate acceptance and rejection degrees for 𝑍𝑡𝑜𝑡𝑎𝑙 by using Zadeh extension principle. 4. ILLUSTRATION OF PROPOSED SOLUTION METHOD Consider the intuitionistic fuzzy transportation problem Min 𝑧1 = 1̃𝑥11 + 2̃𝑥12 + 7̃𝑥13 + 7̃𝑥14 + 1̃𝑥21 + 9̃𝑥22 + 3̃𝑥23 + 4̃𝑥24 + 8̃𝑥31 + 9̃𝑥32 + 4̃𝑥33 + 6̃𝑥34 ̃ 𝑥24 + 6̃𝑥31 + Min 𝑧2 = 4̃𝑥11 + 4̃𝑥12 + 3̃𝑥13 + 4̃𝑥14 + 5̃𝑥21 + 8̃𝑥22 + 9̃𝑥23 + 10 2̃𝑥32 + 5̃𝑥33 + 1̃𝑥34 Such that 𝑥11 + 𝑥12 + 𝑥13 + 𝑥14 = 8 𝑥21 + 𝑥22 + 𝑥23 + 𝑥24 = 19 𝑥31 + 𝑥32 + 𝑥33 + 𝑥34 = 17 𝑥11 + 𝑥21 + 𝑥31 = 11 𝑥12 + 𝑥22 + 𝑥32 = 3 𝑥13 + 𝑥23 + 𝑥33 = 14


313

𝑥14 + 𝑥24 + 𝑥34 = 16 𝑥𝑖𝑗 ≥ 0 where, 1̃ = {(0.1,1,1.5), (0.5,1,1.2)}, 2̃ = {(1,2,2.5), (1.5,2,2)} 7̃ = {(3,7,9), (5,7,8)}, 7̃ = {(4,7,9), (6,7,8)} 1̃ = {(0.1,1,1.6), (0.6,1,1.5)}, 9̃ = {(7,9,11), (8,9,10)} 3̃ = {(1,3,5), (2,3,4)}, 4̃ = {(2,4,6), (3,4,5)} 8̃ = {(6,8,10), (7,8,9)}, 9̃ = {(7,9,12), (8,9,11)} 4̃ = {(2,4,7), (3,4,6)}, 6̃ = {(4,6,9), (5,6,8)} 4̃ = {(2,4,6), (3,4,5)}, 4̃ = {(3,4,7), (3,4,5)} 3̃ = {(1,3,5), (2,3,4)}, 4̃ = {(2,4,6), (3,4,5)} 4̃ = {(2,4,6), (3,4,5)}, 5̃ = {(3,5,7), (4,5,6)} 8̃ = {(5,8,10), (5.5,8,11)}, 9̃ = {(8,9,10), (8.5,9,9.5)} ̃ = {(8,10,12), (9,10,10.5)}, 6̃ = {5,6,6.5), (5.5,6,6.1)} 10 2̃ = {(1,2,3), (1.2,2,2.8)}, 5̃ = {(4,5,6), (4.5,5,5.5)}. First stage optimal solutions are: 𝑥11 = 0.5, 𝑥12 = 1, 𝑥13 = 0, 𝑥14 = 0, 𝑥21 = 2, 𝑥22 = 0, 𝑥23 = 0.32, 𝑥24 = 0.17, 𝑥31 = 0, 𝑥32 = 0, 𝑥33 = 0.17, 𝑥34 = 1.82, Min 𝑧1 = 28.64, Min 𝑧2 = 34.43, 𝛼 = 0.78, 𝛽 = 0. Second stage optimal solutions are: 𝑥11 = 4.5, 𝑥12 = 2, 𝑥13 = 0, 𝑥14 = 0, 𝑥21 = 4, 𝑥22 = 0, 𝑥23 = 12.06, 𝑥24 = 0.43, 𝑥31 = 0, 𝑥32 = 0, 𝑥33 = 1.43, 𝑥34 = 13.56, Min 𝑧1 = 223.41, Min 𝑧2 = 302.74, 𝛼 = 0.64, 𝛽 = 0. Now using definition [3] we get total minimum values are given as: Min 𝑍1 = 252.05, Min 𝑍2 = 337.17 with degree of acceptance 𝛼 = 0.78 and degree of acceptance 𝛽 = 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.

314


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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