Modeling and optimization of multi objective non-linear programming problem in intuitionistic fuzzy environment

Accepted Manuscript Modeling and Optimization of Multi Objective Non-linear Programming Problem in Intuitionistic Fuzzy Environment Sujeet Kumar Singh, Shiv Prasad Yadav PII: DOI: Reference:

S0307-904X(15)00268-1 http://dx.doi.org/10.1016/j.apm.2015.03.064 APM 10561

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Appl. Math. Modelling

Please cite this article as: S.K. Singh, S.P. Yadav, Modeling and Optimization of Multi Objective Non-linear Programming Problem in Intuitionistic Fuzzy Environment, Appl. Math. Modelling (2015), doi: http://dx.doi.org/ 10.1016/j.apm.2015.03.064

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Modeling and Optimization of Multi Objective Non-linear Programming Problem in Intuitionistic Fuzzy Environment Sujeet Kumar Singh1,∗, Shiv Prasad Yadav1 1

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, INDIA

Abstract In dealing with real world practical optimization problems, a decision maker usually faces a state of uncertainty as well as hesitation, due to various unpredictable factors. Sometimes it is necessary to optimize several non-linear and conflicting objectives simultaneously. To deal with the uncertain parameters which arise in such situations, intuitionistic fuzzy numbers are utilized. We formulate a multiobjective non-linear programming problem in intuitionistic fuzzy environment. We propose a linear ranking function and utilize it to convert the intuitionistic fuzzy model into a crisp model. After converting the problem into equivalent crisp problem, we propose a non-linear membership function and develop various approaches for solving it by using different operators and fuzzy programming technique. We apply our methodologies for justification to a numerical problem in manufacturing systems.

Keywords: Multiple objective programming; Intuitionistic fuzzy number; Accuracy function; Nonlinear programming

1. Introduction Most of the real world problems are inherently characterized by multiple and conflicting aspects of evaluation. This evaluation is generally judged by optimizing multiple objective functions. Furthermore, when addressing real world problems, frequently the parameters are imprecise numerical quantities due to different uncontrollable factors. Fuzzy quantities are very adequate for modeling these situations. The application of fuzzy set theory to decision making has gained considerable attention by many authors ([2],[5]) after the pioneering work of Bellman and Zadeh ([12]). However, the current research on fuzzy mathematical programming is limited to the range of linear programming and multiobjective linear programming ([1], [5], [8], [11], [13],[16]). Fuzzy non-linear programming (FNLP) is rarely involved. In many practical problems such as in industrial planning there exist many fuzzy and intuitionistic fuzzy nonlinear production, planning and scheduling problems. These problems cannot be modeled and solved by traditional techniques due to presence of imprecise information. So, the research on modeling and optimization for nonlinear programming under Corresponding author Email addresses: [email protected] (Sujeet Kumar Singh ), [email protected] (Shiv Prasad Yadav) ∗

Preprint submitted to Applied Mathematical Modelling

April 22, 2015

fuzzy environment and intuitionistic fuzzy environment is not only important in the fuzzy optimization theory but also has great and wide value in application to the various real world practical problems of conflicting nature. The most common approach for solving fuzzy linear programming problem is to change it into corresponding crisp linear programming problem. Zimmermann ([3]) has introduced fuzzy programming approach to solve crisp multi-objective linear programming problem. Some authors have transformed the fuzzy programming problem into crisp problem by utilizing ranking function ([4],[7]) and then solved it by conventional methods.

In real world situation, generally, we have to deal with uncertainty as well as with hesitation. In such situations intuitionistic fuzzy quantities are more reliable representatives of the imprecise data. Intuitionistic fuzzy set theory introduced by Atanassov ([6]) is a generalization of fuzzy set theory which has very pioneering applications in modeling of physical problems as this theory includes both uncertainty and hesitation in its structure. Thus intuitionistic fuzzy modeling may be more relevant in this situation because of its structure which includes acceptance, non-acceptance and hesitation levels of the quantity for the decision maker (DM). Currently, authors have developed an algorithm to solve non-linear programming problems in manufacturing systems ([15]). In this paper, we propose to model a general multiobjective non-linear programming problem in intuitionistic fuzzy environment where some objectives are to be maximized and some are to be minimized. We have used triangular intuitionistic fuzzy numbers (TIFNs) for representing the parameters of the problem. To the best of our knowledge there is no any method in the literature to solve such problems. So, we developed various approaches to solve such problems with the help of Zimmerman’s fuzzy technique, Gamma-operator and Min-bounded sum operator. After modeling, the problem is converted into a crisp multiple objective non-linear programming problem by using a linear ranking function and then various approaches are developed to solve it. The paper is organized as follows: Section 2 deals with some mathematical preliminaries from literature. In Section 3, we have proposed an ordering of TIFNs using accuracy function. Section 4 deals with intuitionistic fuzzy multiobjective non-linear programming problem (IFMONLPP) with solution techniques. Section 5 gives a summary of the techniques developed in Section 4. In Section 6, We have illustrated the techniques developed by a suitable numerical example followed by conclusion in Section 7.

2. Preliminaries All definitions in this section are taken from literature ([6], [9],[14]). Definition 1. Let X be a universe of discourse. Then an intuitionistic fuzzy set (IFS) A˜I in X is defined by a set of ordered triples A˜I = {< x, µA˜I (x), νA˜I (x) >: x ∈ X}, where µA˜I , νA˜I : X → [0, 1] are functions such that 0 ≤ µA˜I (x) + νA˜I (x) ≤ 1, ∀x ∈ X . The value µA˜I (x) represents the degree of membership and νA˜I (x) represents the degree of nonmembership of the element x ∈ X being in A˜I . h(x) = 1 − µA˜I (x) − νA˜I (x) represents the degree of hesitation for the element x being in A˜I .

2

Definition 2. An IFS A˜I = {< x, µA˜I (x), νA˜I (x) >: x ∈ X} is called an intuitionistic fuzzy number (IFN) if the following hold: • There exists m ∈ R such that µA˜I (m) = 1 and νA˜I (m) = 0 ( m is called the mean value of A˜I ), • µA˜I and νA˜I are piecewise continuous functions from R to the closed interval [0, 1] and 0 ≤ µA˜I (x) + νA˜I (x) ≤ 1, ∀x ∈ R, where  g1 (x), m − a ≤ x < m    1, x=m µA˜I (x) = h (x), m < x≤m+b  1   0, otherwise, and  0 g2 (x), m − a ≤ x < m; 0 ≤ g1 (x) + g2 (x) ≤ 1    0, x=m νA˜I (x) = 0 h (x), m < x ≤ m + b ; 0 ≤ h1 (x) + h2 (x) ≤ 1    2 1, otherwise. Here m is the mean value of A˜I ; a and b are the left and right spreads of membership 0 0 function µA˜I respectively; a and b are the left and right spreads of non- membership function νA˜I respectively; where g1 and h1 are piecewise continuous, strictly increasing and strictly decreasing functions in [m−a, m) and (m, m+b] respectively; g2 and h2 are 0 piecewise continuous, strictly decreasing and strictly increasing functions in [m − a , m] 0 0 0 and [m, m + b ] respectively. The IFN A˜I is represented by A˜I = (m; a, b; a , b ). Definition 3. A TIFN A˜I is an IFN with the membership function µA˜I and non-membership function νA˜I given by  x − a1   , a 1 < x ≤ a2     a2 − a1 1, x = a2 µA˜I (x) = a − x 3   , a2 ≤ x < a3   a3 − a2   0, otherwise, and  a −x 0 2  a 1 < x ≤ a2  0 ,   a − a1   2 0, x = a2 νA˜I (x) = x − a2 0   , a2 ≤ x < a3 0   − a a  2  3 1, otherwise. 3

0 0 0 0 where a1 ≤ a1 ≤ a2 ≤ a3 ≤ a3 . This TIFN is denoted by A˜I = (a1 , a2 , a3 ; a1 , a2 , a3 ). The set of all TIFNs is denoted by IF (R).

Figure 1: Triangular Intuitionistic Fuzzy Number

Definition 4. Arithmetic operations on TIFNs: 0 0 ˜ I = (b1 , b2 , b3 ; b01 , b2 , b03 ) then Let A˜I = (a1 , a2 , a3 ; a1 , a2 , a3 ) and B ˜ I = (a1 + b1 , a2 + b2 , a3 + b3 ; a01 + b0 , a2 + b2 , a0 + b0 ). Addition: A˜I ⊕ B 1 3 3 ˜ I = (a1 − b3 , a2 − b2 , a3 − b1 ; a01 − b0 , a2 − b2 , a0 − b0 ). Subtraction: A˜I B 3 3 1 ˜ I = (l1 , l2 , l3 ; l10 , l2 , l30 ) Multiplication: A˜I ⊗ B where l1 = min{a1 b1 , a1 b3 , a3 b1 , a3 b3 }, l3 = max{a1 b1 , a1 b3 , a3 b1 , a3 b3 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 l1 = min{a1 b1 , a1 b3 , a3 b1 , a3 b3 }, l3 = max{a1 b1 , a1 b3 , a3 b1 , a3 b3 }, l2 = a2 b2 . Scalar multiplication: 0 0 1. k A˜I = (ka1 , ka2 , ka3 ; ka1 , ka2 , ka3 ), k > 0. 0 0 2. k A˜I = (ka3 , ka2 , ka1 ; ka3 , ka2 , ka1 ), k < 0. ˜ I = (0, 2, 4; −1, 2, 5) be TIFNs. Let A˜I = (2, 4, 5; 1, 4, 6) and B ˜ I = (2, 6, 9; 0, 6, 11), A˜I B ˜ I = (−2, 2, 5; −4, 2, 7), A˜I ⊗ B ˜ I = (0, 8, 20; −6, 8, 30) Then A˜I ⊕ B and 2A˜I = (4, 8, 10; 2, 8, 12), −2A˜I = (−10, −8, −4; −12, −8, −2). 3. Accuracy function and Ordering of TIFNs 0 0 Definition 5. ([14]) Let A˜I = (a1 , a2 , a3 ; a1 , a2 , a3 ) be a TIFN. The score function for the membership function µA˜I is denoted by S(µA˜I ) and is defined by S(µA˜I ) = a1 +2a42 +a3 .The score function for the non-membership function νA˜I is denoted by S(νA˜I ) and is defined 0

by S(νA˜I ) = f (A˜I ) =

0

a1 +2a2 +a3 . 4

S(µA˜I )+S(νA˜I ) 2

The accuracy function of A˜I is denoted by f (A˜I ) and defined by 0

=

0

(a1 +2a2 +a3 )+(a1 +2a2 +a3 ) . 8

4

0 0 ˜ I = (b1 , b2 , b3 ; b0 , b2 , b0 ) then Definition 6. ([14]) Let A˜I = (a1 , a2 , a3 ; a1 , a2 , a3 ) and B 1 3 ˜ I ) ⇒ A˜I ≥ B ˜I (i) f (A˜I ) ≥ f (B ˜ I ) ⇒ A˜I ≤ B ˜I (ii) f (A˜I ) ≤ f (B ˜ I ) ⇒ A˜I = B ˜I (iii) f (A˜I ) = f (B ˜ I ) = A˜I , If A˜I ≤ B ˜ I or B ˜ I ≥ A˜I (iv) min(A˜I , B I I I I I ˜ ) = A˜ , If A˜ ≥ B ˜ or B ˜ I ≤ A˜I . (v) max(A˜ , B

Theorem 1. The accuracy function f is a linear function. 0 0 ˜ I = (b1 , b2 , b3 ; b01 , b2 , b03 ) be two TIFNs and Proof. Let A˜I = (a1 , a2 , a3 ; a1 , a2 , a3 ) and B k(> 0) ∈ R. ˜ I ) = f [k(a1 , a2 , a3 ; a01 , a2 , a03 )⊕(b1 , b2 , b3 ; b01 , b2 , b03 )] = f [ka1 +b1 , ka2 +b2 , ka3 + Then f (k A˜I ⊕ B 0 0 0 0 b3 ; ka1 + b1 , ka2 + b2 , ka3 + b3 ] = 0

0

0

0

+

0

(b1 +2b2 +b3 )+(b1 +2b2 +b3 )} 8

0

0

{ka1 +b1 +2(ka2 +b2 )+ka3 +b3 }+{ka1 +b1 +2(ka2 +b2 )+ka3 +b3 } 8

=

0

k{(a1 +2a2 +a3 )+(a1 +2a2 +a3 )} 8

˜ I ). = kf (A˜I ) + f (B

˜ I ) = f [k(a1 , a2 , a3 ; a01 , a2 , a03 ) ⊕ (b1 , b2 , b3 ; b01 , b2 , b03 )] = Next, let (k < 0) ∈ R. Then f (k A˜I ⊕ B 0 0 0 0 f [ka3 + b1 , ka2 + b2 , ka1 + b3 ; ka3 + b1 , ka2 + b2 , ka1 + b3 ] = 0

0

0

0

+

0

(b1 +2b2 +b3 )+(b1 +2b2 +b3 )} 8

0

0

{ka3 +b1 +2(ka2 +b2 )+ka1 +b3 }+{ka3 +b1 +2(ka2 +b2 )+ka1 +b3 } 8

=

0

k{(a3 +2a2 +a1 )+(a3 +2a2 +a1 )} 8

˜ I ). = kf (A˜I ) + f (B

˜ I ) = kf (A˜I ) + f (B ˜ I ). Proved. Therefore in each case, we have f (k A˜I ⊕ B

4. Description of Intuitionistic Fuzzy Multiobjective Non-linear Programming Problem (IFMONLPP) A conventional multiobjective non-linear programming problem is given by M axfk (x), k = 1, 2, ..., K1 , M infk (x), k = K1 + 1, K1 + 2, ..., K, s.t. gj (x) ≤ bj , j = 1, 2, ..., m1 , gj (x) ≥ bj , j = m1 + 1, m1 + 2, , ..., , m2 , gj (x) = bj , j = m2 + 1, m2 + 2, ..., m, x ≥ 0, where fk , k = 1, 2, ...K and gj , j = 1, 2, ..., m are real valued non-linear functions. x is ndimensional decision variable vector x = (x1 , x2 , ..., xn ). Let Ω be the set of all feasible solutions.

5

Definition 7. ([10]) A feasible solution x¯ ∈ Ω is said to be efficient if there is no x∗ ∈ Ω such x), k = K1 + 1, K x) for at least one k; and fk (x∗ ) ≤ fk (¯ x), k = 1, 2, ..., K1 , and fk (x∗ ) > fk (¯ that fk (x∗ ) ≥ fk (¯ ∗ x) for at least one k. and fk (x ) < fk (¯

x) ≥ fk (x), k = 1, 2, ..., K Definition 8. ([10]) A feasible solution x¯ ∈ Ω said to dominate strictly x ∈ Ω if fk (¯ x) ≤ fk (x), k = K1 + 1, K1 + 2, ..., K. and fk (¯

In real world practical problems such as in production, planning, scheduling, agriculture etc. the available quantity of resources as well as the production quantity or the demand quantity or the goal in a period might be uncertain and possess different types of fuzziness due to many factors such as market price, availability of men power, comprehension with the workers, weather, rain, transportation, traffic etc. Also, the ob jectives defined by the decision maker (DM) may be ill defined due to various estimated parameters. Thus intuitionistic fuzzy modeling seems to be more realistic in such situations. In modeling such problems the following points are to be noted.

(1) The available quantity of some types of resources bj , (j = 1, 2, ..., m1 ) has some increment which is acceptable by the DM, i.e., some tolerance is allowed which is governed by the managerial body. Assuming that the planned available quantity of such type of resource j is bj ; (j = 1, 2, ..., m1 ), the largest acceptable increment by DM is upto brj . But the increment 0

after brj is not acceptable at any cost. Then the availability in this case is suitably represented 0 by the right TIFN b˜Ij = (bj , bj , brj ; bj , bj , brj ), j = 1, 2, ..., m1 .

(2) The planned production quantity or demand or the goal or some resources should not be less than bj , j = m1 + 1, m1 + 2, , ..., , m2 . However, a tolerance upto bj l is acceptable by the DM, i.e., the production quantity at least bj l is acceptable. But the production lesser 0 than blj is not acceptable at any cost. Then the quantity in this case is suitably represented 0 by the left TIFN b˜Ij = (blj , bj , bj ; blj , bj , bj ), j = m1 + 1, m1 + 2, , ..., , m2 .

(3) The available quantity of some other type of resources bj , (j = m2 + 1, m2 + 2, ..., m) may be imprecision. Assuming that the available quantity bj , (j = m2 + 1, m2 + 2, ..., m) of such type of resources has estimated error upto bj l and brj on left and right respectively. Also 0

0

it is planned that the value of the resource quantity lesser than blj and greater than brj is not acceptable at any cost. Then such type of imprecision is suitably expressed by the TIFN 0 0 b˜Ij = (blj , bj , brj ; blj , bj , brj ), j = m2 + 1, m2 + 2, ..., m.

Further, assuming that the objective functions f˜kI and the resource constraints functions

g˜jI are nonlinear with estimated coefficient parameters which are in terms of TIFNs.

6

Then IFMONLPP can be modeled as: I M axf˜k (x), k = 1, 2, ..., K1 , I M inf˜k (x), k = K1 + 1, K1 + 2, ..., K, I

g˜j I (x) ≤ b˜j , j = 1, 2, ..., m1 ,

s.t.

I

g˜j I (x) ≥ b˜j , j = m1 + 1, m1 + 2, , ..., , m2 ,

(4.1)

I g˜j I (x) = b˜j , j = m2 + 1, m2 + 2, ..., m, x ≥ 0,

PKi Q n βl Qn αl PRj I I I I where f˜k (x) = rj ik r=1 a˜ i=1 c˜ l=1 xl , j = l=1 xl , k = 1, 2, ..., K and g˜j (x) = I I 1, 2, ..., m. c˜ik and a˜rj are assumed to be TIFNs. x is n-dimensional decision variable vector x = (x1 , x2 , ..., xn ). Therefore (4.1) can be re-written as follows:

P i Q n αl I I M axf˜k (x) = K ik i=1 c˜ l=1 xl , k = 1, 2, ..., K1 P i Qn αl I I M inf˜k (x) = K ik i=1 c˜ l=1 xl , k = K1 + 1, K1 + 2, ..., K, PRj

s.t.

a˜rj I

k=1

PRj

a˜rj I

Qn

PRj

a˜rj I

Qn

r=1

r=1

Qn

l=1

I xβl l ≤ b˜j , j = 1, 2, ..., m1 , I

l=1

l=1

xβl l ≥ b˜j , j = m1 + 1, m1 + 2, ..., m2 ,

(4.2)

I xβl l = b˜j , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n.

Using the accuracy function, the model in (4.2) is transformed to the following crisp MONLPP:

P i 0 Qn αl M axfk (x) = K i=1 cik l=1 xl , k = 1, 2, ..., K1 , P i 0 Qn αl M infk (x) = K i=1 cik l=1 xl , k = K1 + 1, K1 + 2, ..., K, PRj

s.t

r=1

PRj

r=1

PRj

arj

r=1

0

arj

Qn

arj

Qn

l=1

xβl l ≤ bj , j = 1, 2, ..., m1 , 0

xβl l ≥ bj , j = m1 + 1, m1 + 2, , ..., , m2 , 0

l=1

0

0

Qn

l=1

(4.3)

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n, 0

where the accuracy functional values are f (c˜ik I ) = cik , k = 1, 2, ..., K, f (a˜rj I ) = arj , j = 0

I 0 1, 2, ..., m and f (b˜j ) = bj , j = 1, 2, ..., m.

7

0

Theorem 2. An efficient solution for Problem(4.3) is efficient for Problem (4.2). Proof: Let X = (x1 , x2 , ..., xn ) be an efficient solution of Problem (4.3). Then X is feasible for Problem (4.3), i.e., the following hold. PRj

r=1

PRj

arj

PRj

arj

r=1

r=1

0

arj

0

l=1

xβl l ≤ bj , j = 1, 2, ..., m1 , 0

Qn

xβl l ≥ bj , j = m1 + 1, m1 + 2, ..., m2 ,

Qn

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n.

0

l=1

0

Qn

0

l=1

since f is linear,

PRJ

f (a˜rj I )

r=1

PRJ

r=1

f (a˜rj I )

PRJ

r=1

l=1

I xβl l ≤ f (b˜j ), j = 1, 2, ..., m1 , I

Qn

l=1

f (a˜rj I )

Qn

xβl l ≥ f (b˜j ), j = m1 + 1, m1 + 2, , ..., , m2 ,

Qn

l=1

I xβl l = f (b˜j ), j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n.

which imply that PRJ

I rj r=1 a˜

PRJ

r=1

a˜rj I

PRJ

r=1

Qn

a˜rj I

l=1

Qn

Qn

l=1

I xβl l ≤ b˜j , j = 1, 2, ..., m1 ,

I xβl l ≥ b˜j , j = m1 + 1, m1 + 2, , ..., , m2 ,

l=1

I xβl l = b˜j , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n.

Therefore X is a feasible solution of problem (4.2). ¯ = (x¯1 , x¯2 , ..., x¯n ) Next, since X is efficient for Problem (4.3), there does not exist any X ¯ ¯ such that fk (X) ≥ fk (X), k = 1, 2, ..., K1 and fk (X) > fk (X) for at least one k; and ¯ ≤ fk (X), k = K1 + 1, K1 + 2, ..., K and fk (X) ¯ < fk (X) for at least one k. Thus fk (X) P PKi 0 Qn 0 Qn K α i l ¯ such that M ax we have no X ¯l αl , k = 1, 2, ..., K1 i=1 cik l=1 xl ≥ M ax i=1 cik l=1 x P i PKi 0 Qn 0 Qn αl and M ax K ¯l αl for at least one k. i=1 cik l=1 xl > M ax i=1 cik l=1 x PKi P i 0 Qn 0 Qn αl ¯l αl , k = K1 + 1, K1 + 2, ..., K, and Also M in K l=1 x l=1 xl ≤ M in i=1 cik i=1 cik P i PKi 0 Qn 0 Qn αl M in K ¯l αl for at least one k. i=1 cik l=1 xl < M in i=1 cik l=1 x ¯ such that Since P f is linear, no X Qnwe have PKi Qn Ki αl I I ¯l αl , k = 1, 2, ..., K1 and M ax i=1 c˜ik ik l=1 xl ≥ M ax i=1 c˜ l=1 x P i Q n αl PKi Qn I I M ax K ¯l αl for at least one k. ik ik i=1 c˜ l=1 xl > M ax i=1 c˜ l=1 x

8

P i Q n αl PKi Qn I I Also M in K ¯l αl , k = K1 + 1, K1 + 2, ..., K, and ik ik i=1 c˜ l=1 xl ≤ M in i=1 c˜ l=1 x PKi P i Qn Qn αl I I M in K ¯l αl for at least one k. ik ik i=1 c˜ i=1 c˜ l=1 x l=1 xl < M in Hence X is efficient for Problem (4.2). Now to find the optimal solution of a crisp multi-objective non-linear programming problem we proceed as follow. Solve the multi objective nonlinear programming problem by considering one objective function at a time and ignoring all others. Repeat the process K times for K different objective functions. Let the solutions obtained be X1 , X2 , ..., XK respectively. Let X = {Xk , k = 1, 2, ..., K}. Find the values of the objective functions fk , k = 1, 2, ..., K at each point in X. Next we find minimum and maximum values of the ob jective functions. Let Lk be the minimum and Uk be the maximum value of fk , i.e., Lk = min{fk (x), x ∈ X} and Uk = max{fk (x), x ∈ X}.

Since the objectives fk , k = 1, 2, ..., K1 are to be maximized, the level of satisfaction of the DM will increase as the solution tends towards upper bound to each objective and hence the DM will satisfy fully if the objectives reach to their upper bounds. Let µUk denote the degree of attainability of the upper bound Uk of the objective function fk . Then µUk (fk (x)), k = 1, 2, ..., K1 is defined as follows:

µUk (fk (x)) =

    

0, (fk (x))t −Ltk , Ukt −Ltk

1,

fk (x) < Lk , Lk ≤ fk (x) ≤ Uk , fk (x) > Uk ,

where t > 0 is prescribed by the DM. Since the ob jectives fk , k = K1 + 1, K1 + 2, ..., K are to be minimized, the level of satisfaction of the DM will increase as the solution tends towards lower bound to each objective an hence the DM will satisfy fully if the objectives reach to their lower bounds. Let µLk denote the degree of attainability of the lower bound Lk of the objective function fk . Then µLk (fk (x)), k = K1 + 1, K1 + 2, ..., K is defined as follows:

µLk (fk (x)) =

    

1, Ukt −(fk (x))t , Ukt −Ltk

0,

fk (x) < Lk , Lk ≤ fk (x) ≤ Uk , fk (x) > Uk ,

where t > 0 is prescribed by the DM. Now our problem is reduced to increase the satisfaction level of the DM subject to the given constraints. For this we can adopt any one of the following techniques:

9

4.1. Using Zimmerman’s technique Let λ = min{µLk (fk (x)), µUk (fk (x)), k = 1, 2, ..., K}. Then MONLPP can be described as how to make a reasonable plan so that the DM is most satisfied with objectives as well as constraints. That is, there should be the highest degree of balance between objectives and constraints. This can be modeled as follows:

M ax λ µUk (fk (x)) ≥ λ, k = 1, 2, ...K1 , µLk (fk (x)) ≥ λ, k = K1 + 1, K1 + 2, ...K, PRJ 0 0 Qn βl a rj k=1 l=1 xl ≤ bj , j = 1, 2, ..., m1 ,

s.t.

PRJ

arj

PRJ

arj

k=1 k=1

0

Qn

xβl l ≥ bj , j = m1 + 1, m1 + 2, ..., m2 ,

Qn

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n.

0

l=1

0

0

l=1

Thus the problem is reduced to the following single objective NLPP, which can be solved easily by suitable crisp NLPP method. M ax λ s.t. (fk (x))t − Ltk ≥ λ(Ukt − Ltk ), k = 1, 2, ...K1 , Ukt − (fk (x))t ≥ λ(Ukt − Ltk ), k = K1 + 1, K1 + 2, ...K, PRJ 0 Qn 0 βl k=1 arj l=1 xl ≤ bj , j = 1, 2, ..., m1 , PRJ

k=1

arj

PRJ

k=1

0

Qn

arj

0

xβl l ≥ bj , j = m1 + 1, m1 + 2, , ..., , m2 , 0

l=1

Qn

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n. 0

l=1

4.2. Using the γ-operator We define the γ-operator ([10]) with γ ∈ (0, 1) as follows: Q 1 Q 1 (fk (x))t −Ltk QK U t −(fk (x))t 1−γ )] [1 − ( K )( k=K1 +1 kU t −L φ(x) = [( K t k=1 (1 − k=1 U t −Lt k

k

k

k

Q (fk (x))t −Ltk ))( K k=K1 +1 (1 Ukt −Ltk

−

Ukt −(fk (x))t ))]γ . Ukt −Ltk

Then our problem is reduced to the following problem.

PRJ

s.t.

k=1

PRJ

k=1

arj

PRJ

k=1

0

arj

Qn

arj

M ax φ(x) 0 βl l=1 xl ≤ bj , j = 1, 2, ..., m1 ,

Qn

xβl l ≥ bj , j = m1 + 1, m1 + 2, , ..., , m2 , 0

l=1

0

0

Qn

l=1

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n. 0

10

(4.4)

Theorem 3. If x¯ is an optimal solution of Problem (4.4), then x¯ is efficient for Problem(4.1). Proof. Suppose x¯ is optimal for Problem (4.4) and not efficient for Problem(4.1). Then there exists x∗ ∈ X such that fk (¯ x) ≤ fk (x∗ ), k = 1, 2, ..., K1 and fk (¯ x) < fk (x∗ ) for at least one k; and fk (¯ x) ≥ fk (x∗ ), k = K1 + 1, K1 + 2, ..., K and fk (¯ x) > fk (x∗ ) for at least one k. Then we have fk (¯ x) − Lk ≤ fk (x∗ ) − Lk , k = 1, 2, ..., K1 and fk (¯ x) − Lk < fk (x∗ ) − Lk for at least one k; and Uk − fk (¯ x) ≤ Uk − fk (x∗ ), k = K1 + 1, K1 + 2, ..., K and Uk − fk (¯ x) < Uk − fk (x∗ ) for at least one k. ∗ )−L ∗ )−L x)−Lk x)−Lk k k for < fkU(xk −L , k = K1 + 1, K1 + 2, ..., K and fkU(¯ ≤ fkU(xk −L This implies that fkU(¯ k k −Lk k k −Lk at least one k; and∗ ∗ x) Uk −fk (¯ x) k (x ) k (¯ k (x ) for at least one k. < UkU−f , k = K1 + 1, K1 + 2, ..., K and UUk −f ≤ UkU−f Uk −Lk k −Lk k −Lk k −Lk As 0 < γ < 1, we have

(

K1 Y fk (¯ x ) − Lk

Uk − Lk

k=1

1−γ

)

K Y fk (x∗ ) − Lk 1−γ ) and <( Uk − Lk i=1

(4.5)

K Y Uk − fk (x∗ ) 1−γ Uk − fk (¯ x) 1−γ ) . ) <( U U k − Lk k − Lk i=1 +1

K Y

(

k=K1

∗

∗

x)−Lk )−Lk x)−Lk )−Lk for < 1 − fkU(¯ , k = 1, 2, ..., K1 and 1 − fkU(xk −L ≤ 1 − fkU(¯ We also have 1 − fkU(xk −L k −Lk k k −Lk k at least one∗ k; and x) Uk −fk (x∗ ) Uk −fk (¯ x) k (¯ k (x ) for at < 1 − UUk −f , k = K + 1, K + 2, ..., K and 1 − ≤ 1 − 1 − UkU−f 1 1 Uk −Lk U −L −L k −Lk k k k k least one k. This gives

1−(

K1 Y fk (¯ x ) − Lk

Uk − Lk

k=1

1−(

1−γ

)

K Y fk (x∗ ) − Lk γ ) and <1−( Uk − Lk i=1

(4.6)

K Y Uk − fk (x∗ ) γ Uk − fk (¯ x) 1−γ ) . ) <1−( U − L U k k k − Lk i=1 +1

K Y

k=K1

Equations (4.5) and (4.6) together give φ(x∗ ) > φ(¯ x). This contradicts the fact that x¯ is optimal for Problem (4.4). Thus x¯ is efficient for Problem (4.1). 4.3. Using the min-bounded sum operator The min-bounded sum operator ([10]) for γP∈ (0, 1) is given PKby K ψ(x) = [γ min{µLk (x), µUk (x)}+(1−γ)min(1, k=1 µUk (x)+ k=K1 +1 µLk (x))], k = 1, 2, ..., K.

11

Then Problem (4.3) is reduced to PRJ

s.t

k=1

PRJ

k=1

arj

0

arj

Qn

k=1 arj

0

M ax ψ(x) 0 βl l=1 xl ≤ bj , j = 1, 2, ..., m1 ,

Qn

xβl l ≥ bj , j = m1 + 1, m1 + 2, , ..., , m2 , 0

l=1

PRJ

0

Qn

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, xl ≥ 0, l = 1, 2, ..., n. 0

l=1

(4.7)

Theorem 4. x¯ is optimal for Problem (4.7) if and only if (¯ x, η¯, µ ¯) is optimal solution to the following Problem (4.8). max[γ η + (1 − γ)µ] Q n βl 0 k=1 arj l=1 xl ≤ bj , j = 1, 2, ..., m1 ,

PRJ

s.t PRJ

k=1

arj

PRJ

k=1

0

0

Qn

arj

0

xβl l ≥ bj , j = m1 + 1, m1 + 2, , ..., , m2 , 0

l=1

Qn

l=1

xβl l = bj , j = m2 + 1, m2 + 2, ..., m, 0

µ≤

(4.8)

PK

µLk (x)), η ≤ µLk (x) µ ≤ 1, x = (x1 , x2 , ..., xn ) ≥ 0, P 1 PK where η¯ = min(µLk (¯ x), µUk (¯ x)), µ ¯ = min(1, K i=1 µUk (x) + k=K1 +1 µLk (x)). i=1

P 1 Lemma 5. If x¯ is optimal for (4.8) then η¯ = min(µLk (¯ x), µUk (¯ x)) and µ ¯ = min(1, K k=1 µUk (x)+ PK k=K1 +1 µLk (x)). Proof. Suppose x¯ is optimal for Problem (4.8) and η¯ 6= min(µLk (¯ x), µUk (¯ x)). Let η ∗ = min(µLk (¯ x), µUk (¯ x)). Then η ∗ > η¯. Also (¯ x, η ∗ , µ ¯) satisfies the constraints of Problem(4.8) and γ η¯ + (1 − γ)¯ µ < γ η ∗ + (1 − γ)¯ µ. This contradicts the factPthat (¯ x, η¯, µ ¯) isPoptimal for Problem (4.8). In similar way, it can be K 1 proved that µ ¯ = min(1, K k=1 µUk (x) + k=K1 +1 µLk (x)). Proof of the Theorem 4. Necessary part. If x¯ is optimal for Problem(4.7), then

[γ min(µLk (¯ x), µUk (¯ x)) + (1 − γ)min(1,

K1 X

µUk (¯ x) +

≥ [γ min(µLk (x), µUk (x)) + (1 − γ)min(1,

k=1

12

µUk (x) +

µLk (¯ x))]

k=K1 +1

k=1

K1 X

K X

K X

k=K1 +1

µLk (x))], ∀x ∈ Ω.

(4.9)

Suppose (¯ x, η¯, µ ¯) is not optimal for Problem(4.8), where η¯ = min(µLk (¯ x), µUk (¯ x)), µ ¯ = PK P 1 ∗ ∗ ∗ x, η¯, µ ¯) such that min(1, K k=1 µUk (x) + k=K1 +1 µLk (x)). Then there exists (x , η , µ ) 6= (¯ P 1 PK µ∗ ≤ K k=1 µUk (x) + k=K1 +1 µLk (x)), ∗ η ≤ µUk (x), k = 1, 2, ..., K1 , η ∗ ≤ µLk (x), k = K1 + 1, K1 + 2, ..., K, µ∗ ≤ 1 , x∗ ∈ Ω, and [γ η ∗ + (1 − γ)µ∗ ] ≥ [γ η¯ + (1 − γ)¯ µ]. It follows that P 1 PK ∗ ∗ ∗ ∗ γ min(µLk (x∗ ), µUk (x∗ ))+(1−γ)min(1, K k=K1 +1 µLk (x )) ≥ γ η +(1−γ)µ > k=1 µUk (x )+ γ η¯ + (1 − γ)¯ µ P P 1 x) + K x)) x)) + (1 − γ)min(1, K x), µUk (¯ = γ min(µLk (¯ k=1 µUk (¯ k=K1 +1 µLk (¯ which contradicts (4.9). Therefore (¯ x, η¯, µ ¯) is optimal for Problem (4.8 Sufficient Part. Suppose (¯ x, η¯, µ ¯) is optimal Problem(4.8). PKfor PK Then by Lemma 5, we have 1 η¯ = min(µLk (¯ x), µUk (¯ x)) and µ ¯ = min(1, k=1 µUk (x) + k=K1 +1 µLk (x)). ∗ Let x¯ be not optimal for Problem(4.7). P Then there exists PKx such that∗ K1 ∗ ∗ ∗ x))+ x), µUk (¯ γ min(µLk (x ), µUk (x ))+(1−γ)min(1, k=1 µUk (x )+ k=K1 +1 µLk (x )) > γ min(µLk (¯ P P 1 x) + K x)) (1 − γ)min(1, K k=1 µUk (¯ k=K1 +1 µLk (¯ = γ η¯ + (1 − γ)¯ µ. P 1 PK ∗ ∗ ∗ Let η = min(µLk (x∗ ), µUk (x∗ )) and µ∗ = min(1, K µ (x ) + U k k=K1 +1 µLk (x )). k=1 ∗ ∗ ∗ Then (x , η , µ ) satisfies the constraints of Problem(4.7) and γ η¯ + (1 − γ)¯ µ < γ η ∗ + (1 − γ)µ∗ . This contradicts the fact that (¯ x, η¯, µ ¯) is optimal for Problem(4.8). Therefore (¯ x) is optimal solution for Problem (4.7). Theorem 6. If x¯ is unique optimal solution for Problem(4.7), then it is efficient for Problem (4.1). Proof. Suppose x¯ is unique optimal solution for Problem(4.7). Then ψ(¯ x) > ψ(x), ∀x ∈ Ω. ∗ Let x¯ be not efficient for Problem(4.1). Then there exists x ∈ Ω such that fk (¯ x) ≤ ∗ ∗ ∗ x) ≥ fk (x ), k = fk (x ), k = 1, 2, ..., K1 and fk (¯ x) < fk (x ) for at least one k; and fk (¯ K1 + 1, K1 + 2, ..., K and fk (¯ x) > fk (x∗ ) for at least one k. Then we have fk (¯ x) − Lk ≤ fk (x∗ ) − Lk , k = 1, 2, ..., K1 and fk (¯ x) − Lk < fk (x∗ ) − Lk for at least one k;and Uk − fk (¯ x) ≤ Uk − fk (x∗ ), k = K1 + 1, K1 + 2, ..., K and Uk − fk (¯ x) < Uk − fk (x∗ ) for at least one k. fk (x∗ )−Lk fk (¯ x)−Lk fk (x∗ )−Lk x)−Lk for at least one < , k = 1, 2, ..., K and ≤ These imply that fkU(¯ 1 Uk −Lk U −L U −L −L k k k k k k

k; and Uk −fk (¯ x) Uk −Lk

≤

Uk −fk (x∗ ) ,k Uk −Lk

= K1 + 1, K1 + 2, ..., K and

Uk −fk (¯ x) Uk −Lk

<

Uk −fk (x∗ ) Uk −Lk ∗)

∗

x) )−Lk x)−Lk k (x k (¯ ≤ min UkU−f and min UUk −f ≤ min fkU(xk −L Then min fkU(¯ k −Lk k −Lk k k −Lk

13

for at least one k.

As 0 < γ < 1, we have γ min(

We have PK1also Uk −fk (x∗ ) k=1

Uk −Lk

⇒ min(1, This gives

PK1

+

k=1

fk (x∗ ) − Lk Uk − fk (x∗ ) x) fk (¯ x) − Lk Uk − fk (¯ ). , ) ≤ γ min( , Uk − L k Uk − L k Uk − Lk Uk − Lk

PK

k=K1 +1

Uk −fk (x∗ ) Uk −Lk

+

fk (x∗ )−Lk Uk −Lk

<

PK

k=K1 +1

(1 − γ)min(1,

k=1

Uk −fk (¯ x) Uk −Lk

fk (x∗ )−Lk ) Uk −Lk

Uk − Lk

K1 X Uk − fk (¯ x) k=1

+

PK

k=K1 +1

≤ min(1,

K1 X Uk − fk (x∗ ) k=1

≤ (1 − γ)min(1,

PK1

Uk − Lk

+

PK1

k=1

(4.10)

fk (x∗ )−Lk . Uk −Lk

Uk −fk (¯ x) Uk −Lk

+

PK

k=K1 +1

fk (x∗ )−Lk ). Uk −Lk

K X fk (x∗ ) − Lk ) U k − Lk k=K +1 1

K X fk (x∗ ) − Lk ). + U k − Lk k=K +1

(4.11)

1

Equations (4.10) and (4.11) together give ψ(x∗ ) ≥ ψ(¯ x). This contradicts the fact that x¯ is unique optimal solution for Problem (4.7). Thus x¯ is efficient for Problem (4.1). 5. Algorithm for IFMONLPP

The solution technique discussed in Section 4 can be summarized as an algorithm as given below. Step 1. Model the IFMONLPP. Step 2. Using the accuracy function, find corresponding crisp MONLPP. Step 3. Solve the MONLPP by considering one objective function at a time and ignoring all others. Repeat the process K times for K different objective functions. Let the solutions obtained be X1 , X2 , ..., XK respectively. Let X = {Xk , k = 1, 2, ..., K}. Step 4. Find the value of objective functions fk , k = 1, 2, ..., K at each point obtained in Step 3. Step 5. Find minimum and maximum value of the objective functions. Let Lk be the minimum and Uk be the maximum value of fk , i.e., Lk = min{fk (x), x ∈ X} and Uk = max{fk (x), x ∈ X}. Step 6. Utilize any one of the techniques discussed in Section 4 (Zimmerman’s technique, γoperator or minimum bounded sum operator) to transform the MONLPP into single objective NLPP and then solve it.

6. Numerical example A manufacturing factory is going to produce 3 kinds of products A, B and C in a period (say one month). The production of A, B and C require 3 kinds of resources R1 , R2 and R3 . The requirements of each kind of resource to produce each product A are around 2, 4 14

and 3 units respectively. To produce each product B, the respective requirements are 4, 2 and 2 units approximately and that for each unit of C are around 3, 2 and 3 units . The planed available resource of R1 and R2 are around 325 and 360 units respectively. But there are around 30 and 20 units of additional safety store of materials which are administrated by the general manager. For better quality of the products at least 365 units of resource R3 has to be utilized. However, tolerance around 20 units can be allowed by the managerial board. Also the estimated time requirements in producing each unit of products are 4, 5 and 6 hours respectively. Assuming that the planned production quantities of A, B and C are x1 , x2 and x3 respectively. Further, assuming that unit cost and sale’s price of product A, B and ˜3 I ˜2 I ˜1 I respectively, ,U S3 = s1/a , U S2 = s1/a C are U C1 = c˜1 I , U C2 = c˜2 I , U C3 = c˜3 I and U S1 = s1/a 3 2 1 x1

x2

x3

where a1 , a2 and a3 are +ve real numbers. The Manager hopes to maximize total profit and minimize total time requirement. To deal with uncertainties possessing hesitation, let us assume that all parameters of the problem are TIFNs. Then this problem can be modeled as follows: 1−1/a3 1−1/a2 1−1/a1 − c˜3 I x3 , − c˜2 I x2 + s˜3 I x3 − c˜1 I x1 + s˜2 I x2 M ax f˜1 (x)I = s˜1 I x1 M in f˜2 (x)I = ˜4I x1 + ˜5I x2 + ˜6I x3 ,

˜2I x1 + ˜4I x2 + ˜3I x3 ≤ 325 ˜ I,

s.t

˜4I x1 + ˜2I x2 + ˜2I x3 ≤ 360 ˜ I,

(6.1)

˜3I x1 + ˜2I x2 + ˜3I x3 ≥ 365 ˜ I, x = (x1 , x2 , x3 ) ≥ 0, I

˜ = (325, 325, 355; 325, 325, 360) Suppose the estimated parameters taken by the manager are TIFNs. Let 325 ˜ I = (95, 100, 104; 95, 100, 105), ˜ I = (345, 365, 365; 345, 365, 365), s˜1 I = 100 (360, 360, 380; 360, 360, 385), 365 ˜ I = (7.5, 7.5, c˜1 I = 7.5 ˜ I = (118, 120, 122; 115, 120, 124), c˜2 I = 10 ˜ I = (9, 10, 11; 9.5; 6.5, 7.5, 10.5), s˜2 I = 120 I

˜ = (93, 95, 96; 92, 95, 100), c˜3 I = ˜8I = (7.5, 8, 8.5; 7, 8, 9), ˜2I = (1, 2, 3; 9, 10, 12), s˜3 I = 95 0.5, 2, 4), ˜3I = (2, 3, 4; 1.5, 3, 4), ˜4I = (3, 4, 5; 2, 4, 5), ˜5I = (4.5, 5, 6; 4, 5, 6.5), ˜6I = (5.5, 6, 6.5; 5, 6, 6.5), a1 = a2 = 2, a3 = 3. Using accuracy function, Problem (6.1) is transformed to the following crisp MONLPP. 1/2

1/2

1/3

M axf1 (x) = 99.875x1 − 8x1 + 119.875x2 − 10.125x2 + 95.125x3 − 8x3 , M inf2 (x) = 3.875x1 + 5.125x2 + 5.9375x3 , s.t 2.0625x1 + 3.875x2 + 2.9375x3 ≤ 333.125, 3.875x1 + 2.0625x2 + 2.0625x3 ≤ 365.625, 2.9375x1 + 2.0625x2 + 2.9375x3 ≥ 360, x1 , x2 , x3 ≥ 0. 15

(6.2)

Solving the MONLPP (6.2) as a single ob jective NLPP by using LINGO, we have the following solutions. X1 = (57.82, 13.09, 55.53), X2 = (62.26, 0, 60.28), with L1 = 180.72, U1 = 516.70 and L2 = 599.23, U2 = 620.84. The first ob jective is to be maximized. The satisfaction level for the Manager increases if the objective value tends towards upper bound. Thus the membership function or the degree of attainability of upper bound can be defined as follows:

µU1 (f1 (x))

=

  

0, 1/2

(99.875x1

f1 (x) < 180.72,

1/2

1/3

−8x1 +119.875x2 −10.125x2 +95.125x3 (516.70)t −(180.72)t

 

−8x3 )t −(180.72)t

1,

, 180.72 ≤ f1 (x) ≤ 516.70, f1 (x) > 516.70,

The second objective is to be minimized. The satisfaction level increases if the objective value tends towards lower bound. Thus the membership function or the degree of attainability of lower bound can be defined as follows:

µL2 (f2 (x)) =

 

1, 620.84t −(3.875x1 +5.125x2 +5.9375x3 )t , 620.84t −599.23t



0,

f2 (x) < 599.23, 599.23 ≤ f2 (x) ≤ 620.84, f2 (x) > 620.84,

where t > 0 is prescribed by the Manager and x = (x1 , x2 , x3 ). 6.1. Solution by Zimmerman’s approach By Zimmerman’s approach, Problem (6.2) reduces to

M ax λ 1/2 1/3 s.t. − 8x1 + 119.875x2 − 10.125x2 + 95.125x3 − 8x3 )t −(180.72)t ≥ λ(516.70)t − (180.72)t , 620.84t − (3.875x1 + 5.125x2 + 5.9375x3 )t ≥ λ(620.84t − 599.23t ), 2.0625x1 + 3.875x2 + 2.9375x3 ≤ 333.125, 3.875x1 + 2.0625x2 + 2.0625x3 ≤ 365.625, 2.9375x1 + 2.0625x2 + 2.9375x3 ≥ 360, x1 , x2 , x3 ≥ 0. 1/2 (99.875x1

Taking t = 2 and using LINGO, the solution is (x1 , x2 , x3 ) = (60.60, 4.90, 58.5) with satisfaction level λ = 0.62. f1 (x) = 409.70, f2 (x) = 607.28.

16

6.2. Solution using the γ-operator Here the γ operator is 1/2

φ(x) = [(

1/2

1/3

(99.875x1 −8x1 +119.875x2 −10.125x2 +95.125x3 (516.70)t −(180.72)t

t

−8x3 )t −(180.72)t

)

t

t

t

1 +5.125x2 +5.9375x3 ) 1 +5.125x2 +5.9375x3 ) ))(1−( )]1−γ ×[1−(1−( 620.84 −(3.875x ( 620.84 −(3.875x 620.84t −599.23t 620.84t −599.23t

1/2

(99.875x1

Problem (6.2) is transformed to M ax φ(x) s.t. 2.0625x1 + 3.875x2 + 2.9375x3 ≤ 333.125, 3.875x1 + 2.0625x2 + 2.0625x3 ≤ 365.625, 2.9375x1 + 2.0625x2 + 2.9375x3 ≥ 360, x1 , x2 , x3 ≥ 0. Taking t = 2 and using LINGO, the solution is (x1 , x2 , x3 ) = (62.24, 0.76, 60.25), γ = 0.99, φ(x) = 0.96, f1 (x) = 288.86, f2 (x) = 599.64.

6.3. Solution using minimum bounded sum operator Using minimum bounded sum operator, Problem (6.2) is transformed to the following problem: M ax [γη + (1 − γ)µ], s.t.

1/2

1/2

1/3

(99.875x1 − 8x1 + 119.875x2 − 10.125x2 + 95.125x3 − 8x3 )t −(180.72)t ≥ η(516.7)t − (180.72)t , 620.84t − (3.875x1 + 5.125x2 + 5.9375x3 )t ≥ η(620.84t − 599.23t ), 1/2

(99.875x1

1/2

1/3

−8x1 +119.875x2 −10.125x2 +95.125x3 (101.8999)t −(−142.5707)t

−12x3 )t −(−142.5707)t

+

642.5766t −(3.875x1 +5.125x2 +5.9375x3 )t 642.5766t −633.7818t

≥ µ, 2.0625x1 + 3.875x2 + 2.9375x3 ≤ 333.125, 3.875x1 + 2.0625x2 + 2.0625x3 ≤ 365.625, 2.9375x1 + 2.0625x2 + 2.9375x3 ≥ 360, µ ≤ 1, γ > 0, γ < 1, x1 , x2 , x3 ≥ 0. Taking t = 2 and using LINGO, the solution is (x1 , x2 , x3 ) = (60.48, 5.26, 58.37), η = 0.6, µ = 1, f1 (x) = 416.58,f2 (x) = 607.88 for γ ∈ (0, 1). The comparative study of the obtained solutions are given in Table-1. It is clear from Table-1 that the minimum bounded sum operator gives better solution for the first objective, where as γ-operator gives better solution for the second objective. Hence it depends on the DM that which ob jective has higher priority to be achieved. Accordingly, the methods should be applied.

17

1/2

−8x1 +119.875x2 −10. (516.70)t −

f1 f2 Deviations from U1 Deviations from L2

Zimmerman’s Technique 409.7 607.28 107 8.05

γ-operator 288.86 599.64 227.84 0.41(min.)

Min. Bounded sum operator 416.58 607.88 100.12(min.) 8.65

Table 1: Comparison table

7. Conclusion In this paper, we have proposed multi-objective non-linear programming problem in intuitionistic fuzzy environment having mixed type of conflicting objectives. We introduced a linear ranking function for TIFNs. We have also proposed a general non-linear membership function. Using the proposed ranking and membership functions, we developed various approaches to solve the IFMONLPP with the help of γ-operator, min-bounded sum operator and fuzzy programming technique. For illustration, the methodologies are applied to solve a non-linear problem in manufacturing system. To the best of our knowledge so far, there is no method suggested to find the solution of such problems in the literature. So, the proposed modeling as well as methodologies will be very helpful for decision making problems possessing uncertainty and hesitation with multiple ob jectives in production, planning, scheduling and manufacturing systems. Acknowledgements The authors are thankful to the reviewers for fruitful comments which helped us to improve the original manuscript. Further, the first author gratefully acknowledges the financial support given by the Ministry of Human Resource and Development (MHRD), Govt. of India, India.

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