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Oct 23, 2014 - 19 Dipti Dubey and Aparna Mehra (2011).Linear Programming with triangular intutionistic fuzzy numbers,EUSFLAT-LFA 2011,Aix-les-. BainsΒ ...
Applied Mathematical Sciences, vol. 8, 2014, no. 149, 7443 - 7452 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.44233

An Intuitionistic Fuzzy Multi-Objective Vendor Selection Problem Prabjot Kaur* Applied Mathematics Birla Institute of Technology Mesra, Ranchi, Jharkhand, India *Corresponding Author Copyright Β© 2014 Prabjot Kaur. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract A vendor selection problem is a multiobjective decision making problem involving the optimization of cost, quality and delivery performance on the basis of evaluation of criteria like cost, quality, service etc. The problem of vendor selection involves its selection and allocation of order. The selections of appropriate vendors enable firms to improve performance related to customer's needs, requirements and satisfaction. In real world most of the input information related to criteria is not known precisely due to several conflicting factors. Imprecision is best handled by fuzzy set theory. A more advanced form of fuzzy set theory with an additional degree of freedom is intuitionistic fuzzy sets. We choose triangular intuitionistic fuzzy number to represent the criteria because of its ability to handle uncertainty in data and its simple arithmetic operations to solve any linear programming problem. This paper develops an intuitionistic fuzzy multiobjective linear model (IFMOLM) to vendor selection problem with the coefficient of the objectives as triangular intuitionistic fuzzy numbers and rest of the data in the constraints as crisp. The multiple objectives are minimization of net price, maximization of quality and maximization of on time delivery for vendors. Specifically the paper incorporates the LPP is formulated in intuitionistic fuzzy form and a ranking function converts the IFMOLM to equivalent crisp linear form (Dubey and Mehra(2011)). A numerical example illustrates the application of the methodology.

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The results of the numerical example show that the results obtained by this methods is better than fuzzy approach. Keywords: Vendor selection, Triangular Intuitionistic Fuzzy Number (TIFN), Multicriteria decision Making (MCDM)

1.0 Introduction Vendor selection problem is a multiobjective decision making problem where decisions may be driven by more than one objective.The objectives may be of low cost or good quality and good service facility.The vendor selection problem is a selection and allocation of order to various vendors selected. The first study of vendor selection problem was done by Dickson in the 1960's. Since then various multi-criteria decision making approaches have been proposed for vendor selection, such as the analytic hierarchy process(Narasimhan (1983)), case-based reasoning(Choy and Lee (2002)), data envelopment analysis (Baker and Talluri (1997)), fuzzy set theory (Chen et al. (2006),Sarkar and Mohapatra (2006)), genetic algorithm (Ding et al. (2005)), mathematical programming(Weber & Current(1993), Zydiak, & Chaudhry, 1995) and multiple objective programming (Buffa & Jackson, 1983; Feng; Ghoudsypour & O’Brien, 1998; Weber & Ellram,1992). Significant analytical methods are lacking in VSP as discussed by Weber (1991) where he listed only ten articles (Gaballa (1974), Anthony and Buffa (1977), Pan (1989) etc) corresponding to mathematical programming. The disadvantage of such models is that they are application specific or not readily generalizable or limited in terms of the scope of assumptions. The influence of the criteria and its impact on alternatives provided by decision makers are difficult to exactly express by crisp data in the selection of vendors. Fuzzy sets was introduced by Zadeh (1965) to express impreciseness or vagueness in data.A further improvement in fuzzy sets was the concept of intuitionistic fuzzy set (IFS) introduced by Atanassov (1986). It is defined by two functions expressing the degree of membership and the degree of nonmembership, respectively. Accordingly, IFS is an appropriate tool to describe the impreciseness or uncertainity in data with an additional degree of freedom. IFS has found application in various areas and one area being the supply chain. Guo,Qi and Zhao(2010) gave an approach based on intuitionistic fuzzy topsis to deal with supplier selection problem. Boran et al (2009) gave a multicriteria intutionistic fuzzy group decision making for supplier selection with Topsis method. Shahrokhi et al. (2011) gave an integrated method using intuitionistic fuzzy set and linear programming for supplier selection problem.In this paper we set a multiobjective optimization problem in an intuitionistic fuzzy environment and results obtained by intuitionistic approach are better compared to fuzzy approach.

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The organization of the paper is as follows: Section 2 we explain about basics of intuitionistic fuzzy sets. In section 3 model formulation in an intuitionistic fuzzy environment and its solution is discussed. Section 4 takes a numerical example for illustration of the methodology.In section 5 we conclude the results.

2.0 Preliminaries Definition 1 : Given a fixed set X= {x1,x2,…..,xn} , an intuitionistic fuzzy set (IFS) is defined as 𝐴̿= (/xi οƒŽ X ) which assigns to each element xi , a membership degree tA(xi) and a non-membership degree fA(xi) under the condition 0 ο‚£ t A ( xi )  f A ( xi ) ο‚£ 1, for all xi οƒŽ X . Definition 2: A (TIFN) 𝐴̿ = (π‘Ž1 , π‘Ž2 , π‘Ž3 ; π‘€π‘Ž )(π‘Ž1 β€² , π‘Ž2 , π‘Ž3 β€² ; π‘’π‘Ž )} is an IFS in R with the following membership function πœ‡π΄ΜΏ (x) and non- membership πœ—π΄ΜΏ (x). (π‘₯βˆ’π‘Ž1 )π‘€π‘Ž π‘Ž2 βˆ’π‘Ž1

πœ‡π΄ΜΏ (x)

=

π‘€π‘Ž (π‘Ž3βˆ’π‘₯ )π‘€π‘Ž π‘Ž3 βˆ’π‘Ž2 {

0,

π‘Ž2 βˆ’ π‘₯ + π‘’π‘Ž ( π‘₯βˆ’π‘Ž1 β€² ) , π‘Ž1 β€² π‘Ž2 βˆ’ π‘Ž 1 β€²

π‘’π‘Ž , π‘Ž3 β€² βˆ’π‘Ž2 β€²

1,

π‘₯ = π‘Ž2

and

πœ—π΄ΜΏ (π‘₯) =

, π‘Ž2 ≀ π‘₯ ≀ π‘Ž 3 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’ ≀ π‘₯ ≀ π‘Ž2 π‘₯ = π‘Ž2

π‘₯βˆ’π‘Ž2 +π‘’π‘Ž (π‘Ž3 β€² βˆ’π‘₯)

{

, π‘Ž1 ≀ π‘₯ ≀ π‘Ž2

, π‘Ž2 ≀ π‘₯ ≀ π‘Ž3 β€² π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’

The values π‘€π‘Ž and π‘’π‘Ž respectively represent the maximum degree of membership and the non-membership such that 0 ≀ wa ≀1,0≀ua ≀ 1.+ Definition 3: Arithmetic Operations of TIFN is given by: If 𝐴̿ = (π‘Ž1 , π‘Ž2 , π‘Ž3 ; π‘€π‘Ž )(π‘Ž1 β€² , π‘Ž2 , π‘Ž3 β€² ; π‘’π‘Ž )} and 𝐡̿ = (𝑏1 , 𝑏2 , 𝑏3 ; 𝑀𝑏 )(𝑏1 β€² , 𝑏2 , 𝑏3 β€² ; 𝑒𝑏 )} are two TIFNs, and k be a real number then 𝐴̿ + 𝐡̿ = {(π‘Ž1 + 𝑏1 ,π‘Ž2 + 𝑏2 , π‘Ž3 + 𝑏3 ; min{π‘€π‘Ž , 𝑀𝑏 )(π‘Ž1 β€² + 𝑏1 β€² , π‘Ž2 + 𝑏2 , π‘Ž3 β€² + 𝑏3 β€² ; max{π‘’π‘Ž , 𝑒𝑏} )} is also a TIFN. π‘˜π΄ΜΏ =is a TIFN {(ka1, ka2, ka3; wa)(ka1',ka2,ka3 '; ua)}.k>0.

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Definition 4: Let 𝐴̿ = (π‘Ž1 , π‘Ž2 , π‘Ž3 ; π‘€π‘Ž )(π‘Ž1 β€² , π‘Ž2 , π‘Ž3 β€² ; π‘’π‘Ž )} be a TIFN. Then value and the ambiguity of 𝐴̿ is given as follows: 1. The value of the membership function of 𝐴̿ is (π‘Ž +4βˆ—π‘Ž2 +π‘Ž3 )π‘€π‘Ž π‘‰πœ‡ (𝐴̿) = 1 6

… (1)

2. The value of the non-membership function is π‘‰πœ— (𝐴̿) =

(π‘Ž1 β€² +4βˆ—π‘Ž2 +π‘Ž3 β€² )(1βˆ’π‘’π‘Ž ) 6

… (2)

3. The ambiguity of the membership function of 𝐴̿ is (π‘Ž βˆ’π‘Ž )𝑀 π΄πœ‡ (𝐴̿) = 3 61 π‘Ž

… (3)

4.The ambiguity of non-membership function 𝐴̿ is (π‘Ž β€² βˆ’π‘Ž β€² )(1βˆ’π‘’π‘Ž ) π΄πœ— (𝐴̿) = 3 1

… (4)

6

Also π΄πœ‡ (𝐴̿) ≀ π΄πœ— (𝐴̿). Definition 5: Let 𝐴̿ = (π‘Ž1 , π‘Ž2 , π‘Ž3 ; π‘€π‘Ž )(π‘Ž1 β€² , π‘Ž2 , π‘Ž3 β€² ; π‘’π‘Ž )} be a TIFN.Then the value index and ambiguity index of 𝐴̿ is defined as follows: V (𝐴̿, πœ†) = π‘‰πœ‡ (𝐴̿) + πœ†(π‘‰πœˆ (𝐴̿) βˆ’ π‘‰πœ‡ (𝐴̿)) ΜΏ πœ†) = 𝐴𝜈 (𝐴̿) βˆ’ πœ†(𝐴𝜈 (𝐴̿) βˆ’ π΄πœ‡ (𝐴̿ )) and A(𝐴,

… (5) …(6)

where Ξ» Ι›[0,1] is a weight which represents the decision maker's preference information. ΜΏ, Ξ») = V (A ΜΏ, Ξ») –A (A ΜΏ, Ξ») Definition 6: Ranking relation F (A

(7)

3.0 Methodology 3.1 Crisp Multi-Objective linear model The classical linear programming problem is to find an optimum value of a linear function subject to constraints represented by linear inequalities or equations. The formulation of linear model can be expressed as:

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Min Z= cx Subject to Ax≀ b xβ‰₯0

… (8)

where x=[x1,x2,..xn]T is a vector of decision variables and z is the objective function. 3.2. Intuitionistic Fuzzy Multiobjective linear model Consider the intuitionistic linear programming problem (IFMOLM) in which coefficient of the objective function are considered as intutionistic fuzzy numbers. Min z = βˆ‘π‘›π‘—=1 𝑐̃𝑗 π‘₯𝑗

… (9)

Subject to: βˆ‘π‘›π‘—=1 π‘Žπ‘–π‘— π‘₯𝑗 ≀ 𝑏𝑖 , i=1,2,…,m xjβ‰₯0, j=1, 2…,n, where 𝑐 = {(𝑐1 , 𝑐2 , 𝑐3 ; 𝑀𝑐 )(𝑐1 β€² , 𝑐2 , 𝑐3 β€² ; 𝑒𝑐 )} are TIFN's.

3.3 Solution to Intuitionistic Fuzzy Multiobjective linear model The elements in the objective function are intuitionistic fuzzy numbers and the constraints are in the crisp form. To convert the intuitionistic fuzzy objection function to crisp form using the ranking function f, for predefined Ξ» Ι› [0,1], IFLP is equivalent to the following crisp optimization problem. Max F (βˆ‘π‘›π‘—=1 𝑐̃𝑗 , πœ†) … (10)

Subject to βˆ‘π‘›π‘—=1 π‘Žπ‘–π‘— π‘₯𝑗 ≀ 𝑏𝑖 , i=1,2,…,m xj β‰₯ 0, j=1,2…,n. The linear equivalent is as follows: max(1 βˆ’ πœ†) π‘šπ‘–π‘› ⏟ {𝑀𝑐̃𝑗 } βˆ‘π‘›π‘—=1 𝑗

π‘‰πœ‡ (𝑐𝑗 ) 𝑀𝑐̃𝑗

π‘₯𝑗 βˆ’ (1 βˆ’ Ξ») min ⏟ {1 βˆ’ 𝑒𝑐̃𝑗 } βˆ‘π‘›π‘—=1 j

𝑉 (𝑐̃ )

πœ† π‘šπ‘–π‘› ⏟ {1 βˆ’ 𝑒𝑐̃𝑗 } βˆ‘π‘›π‘—=1 1βˆ’πœˆ 𝑒𝑗 π‘₯𝑗 βˆ’ πœ† π‘šπ‘–π‘› ⏟ {𝑀𝑐̃𝑗 } βˆ‘π‘›π‘—=1 𝑗

𝑐̃𝑗

𝑗

𝐴𝜐 (𝑐̃𝑗 )π‘₯𝑗 𝑀𝑐̃𝑗

𝐴𝜐 (𝑐̃𝑗 )π‘₯𝑗 1βˆ’π‘’π‘Μƒπ‘—

+

… (11)

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Prabjot Kaur Subject to:

βˆ‘π‘›π‘—=1 π‘Žπ‘–π‘— π‘₯𝑗 ≀ 𝑏𝑖 ,i=1,2,…,m. xj β‰₯ 0, j=1,2…,n.

For optimistic attitude Ξ»=1, equation (*) reduces to max min ⏟ {1 βˆ’ 𝑒𝑐̃𝑗 } βˆ‘π‘›π‘—=1

π‘‰πœ (𝑐̃𝑗 )π‘₯𝑗

j

1βˆ’π‘’π‘Μƒπ‘—

βˆ’ π‘šπ‘–π‘› ⏟ {𝑀𝑐̃𝑗 } βˆ‘π‘›π‘—=1 𝑗

𝐴¡ (𝑐̃𝑗 )π‘₯𝑗 𝑀𝑐̃𝑗

…(12)

Subject to: βˆ‘π‘›π‘—=1 π‘Žπ‘–π‘— π‘₯𝑗 ≀ 𝑏𝑖 ,i=1,2,…,m

xj β‰₯0, j=1,2…,n For pessimistic Ξ»=0 (*) reduces to 𝑛

𝑛

𝑗=1

𝑗=1

π‘‰πœ‡ (𝑐𝑗 ) 𝐴𝜐 (𝑐̃)π‘₯ 𝑗 𝑗 π‘šπ‘Žπ‘₯ π‘šπ‘–π‘› ⏟ {𝑀𝑐̃𝑗 } βˆ‘ π‘₯𝑗 βˆ’ (1 βˆ’ Ξ») min ⏟ {1 βˆ’ 𝑒𝑐̃𝑗 } βˆ‘ + 𝑀𝑐̃𝑗 1 βˆ’ 𝑒𝑐̃𝑗 j 𝑗 ⏟ {𝑀𝑐̃𝑗 } βˆ‘π‘›π‘—=1 π‘šπ‘–π‘› 𝑗

𝑉¡ (𝑐̃𝑗 ) 𝑀𝑐̃𝑗

π‘₯𝑗 βˆ’ π‘šπ‘–π‘› ⏟ {1 βˆ’ 𝑒𝑐̃𝑗 } βˆ‘π‘›π‘—=1 𝑗

𝐴𝜐 (𝑐̃𝑗 )π‘₯𝑗 1βˆ’ 𝑒𝑐̃𝑗

…(13)

Subject to: βˆ‘π‘›π‘—=1 π‘Žπ‘–π‘— π‘₯𝑗 ≀ 𝑏𝑖 ,i=1,2,…,m xj β‰₯ 0, j=1,2…,n

The above crisp LPP is solved by optimization software Tora 2.0.We obtain solution of MOLPP for the various objectives.

4.0Numerical Example A textile company desires to select suitable suppliers to purchase yarn for a new product (Yucel and Guneri (2011)). A committee of decision makers, D1, D2 and D3 has been constituted and then committee selected net price, quality and ontime delivery as selection criteria and demand as deterministic in the constraint. The data for vendor selection problem is given in Table1.In Table 2 data as TIFNS is presented.

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Table 1: Supplier's quantitative information. Supplier A1 A2 A3

Net Price 5 7 4

Quality(%) 80 90 85

Delivery(%) 90 80 85

Capacity 400 450 450

Table 2:TIFN data for Supplier's quantitative information. Supplier

Net Price

Quality(%)

Delivery(%)

Capacity

A1

(4,5,6;.75)(3,5,7,.25)

(.75,.80,.90;.75))(.75,.80,95;.25)

(.85,.90,.95;1)(.65,.85,.95;0)

400

A2

(6,7,8;1)(5,7,9;0)

(.85,.90,.95;1)(.65,.85,.95;0)

(.75,.80,.90;.75))(.75,.80,95;.25)

450

A3

(3,4,5;.75)(4,5,6;.25)

(.70,.85,.90;.75)(.65,.85,.95;.25)

(.70,.85,.90;.75)(.65,.85,.95;.25)

450

Using data of Table 2 in equations 10, 11, 12, 1nd 13,an IMOLPP is formulated. For Ξ»=0, the equivalent crisp formulation for the three objectives is as follows: Objective function 1: Min 2.75 x1+4.25x2+2.5x3 Subject to: x1+x2+x3=800 x1≀400 x2≀450 x3≀450 x1, x2, x3 β‰₯0 Objective function 2: Max .56x1+.66x2+.55x3 Subject to: x1+x2+x3=800 x1≀400 x2≀450 x3≀450 x1, x2, x3 β‰₯0 Objective function 3: Max .66x1+.55x2+.55x3

Subject to: x1+x2+x3=800 x1≀400 x2≀450 x3≀450 x1 , x2, x3 β‰₯0

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5.0 Results: Using the optimization software Tora 2.0 for solution of the multiobjective problem, obtain the optimal solution for the model as follows: Z1=2662.50, x1=350, x2 = 0, x3 = 450; Z2=493, x1=350, x2=450, x3=0; Z3=488, x1=400, x2=400, x3=0.

The allocation of order for the three objectives varies according to situations. The first objective was minimize the cost so maximum allocation went to vendor 3 whose cost was lowest and vendor 2 got no allocation of order because of its high cost. The second objective was maximize quality ,so allocation went to first and second vendor according to percentage quality .Though maximum allocation went to vendor 2 which had high quality and high price too. The third objective was minimizing delivery time so equal allocation of order to vendor 1 and vendor 2.Though delivery of goods was 90% and 80% for vendors 1 and 2.Other factors were also taken into consideration for order allocation.

6.0 Conclusions In this study based on MOLPP in an intuitionistic environment for selection of vendor and order allocation to vendor is done. Compared to the results of Yucel and Guneri (2011) , the results obtained by TIFN approach are better .In this study we use intuitionistic fuzzy numbers because of its ease of use and its ability to represent vagueness in data with an additional degree of freedom(non-membership function). The future scope of the work includes representing the whole MOLPP in terms of intutionistic fuzzy sets.

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R.A.D. Carvalho, H.G. Costa, Application of an integrated decision support process for supplier selection, Enterprise Information Systems 1 (2007)197-216.

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4. G. Dickson, An analysis of vendor selection: systems and decisions, Journal of Purchasing 2 (1966) 5-17. 5. Narsimhan,R. (1983) An analytical approach to supplier selection, J. of Purchasing and Materials Management, winter, 27–32. 6. Choy, K.L., Lee, W.B., Lo, V., 2002. Development of a case based intelligent customer Supplier relationship management system. Expert Systems with Applications, 23 (3), 281–297. 7. Baker, R.C., Talluri, S., 1997. A closer look at the use of DEA for technology selection. Computers and Industrial Engineering 32 (1), 101– 108. 8. Chen, C.T., Lin, C.T., Huang, S.F., 2006. A fuzzy approach for supplier evaluation and selection in supply chain management. International Journal of Production Economics (102), 289–301. 9 Sarkar, A., Mohapatra, P.K.J., 2006. Evaluation of supplier capability and performance: A method for supply base reduction. Journal of Purchasing and Supply Management 12 (3), 148–163. 10 Ding, H., Benyoucef, L., Xie, X., 2005. A simulation optimization methodology for supplier selection problem. International Journal Computer Integrated Manufacturing 18 (2–3), 210–224. 11 Weber, C.A. and Current, J.R. (1993) A multiobjective approach to vendor selection, European J. of Operational Research, vol 68, 173-184. 12 Ghodsypour,S.H. and Brien,C.O. (1998) A DSS for supplier selection using an integrated AHP and Linear Programming, International J. of Production Economics, vol 56-57, 199-212. 13 Buffa, F.P. and Jackson, W.M. (1983) A goal programming model for purchase planning, Journal of Purchasing and Materials Management, 2734. 14 Weber, C.A. and Ellram, L.M. (1992) Supplier selection using multiobjective programming: A decision support systems approach, International J. of Physical Distribution and Logistics Management, vol 23 (2), 3-14.

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15 Zixue Guo,Merian Qi and Xin Zhao,"A new approach based on intutionistic fuzzy set for selection of suppliers",2010 sixth international conference on Natural Computation(ICNC 2010). 16 Fatih Emre Boran,Serkan Gene,Mustafa Kurt and Diyar Akay, "A multicriteria intutionistic fuzzy group decision making for supplier selection with Topsis method",Expert system with applications ,vol.36,pp.11363-11368,2009. 17 Atanassov,K.(1986).Intuitionistic fuzzy sets.Fuzzy Sets and Systems,vol.79,pp.403-405. 18 M.Shahrokhi,A.Bernard and H.Shidpour ,"An integrated method using intuitionistic fuzzy set and linear programming for supplier selection problem",18th IFAC World Congress, Milano(Italy)August28-September 2,2011. 19 Dipti Dubey and Aparna Mehra (2011).Linear Programming with triangular intutionistic fuzzy numbers,EUSFLAT-LFA 2011,Aix-lesBains, France. 20 Atakan Yucel and Ali Fuat Guneri (2011).A weighted additive fuzzy programming approach for multi-criteria supplier selection,Expert Systems with Applications ,38,6281-6286. Received: March 10, 2014; Published: October 23, 2014