Intuitionistic fuzzy optimization technique for Pareto optimal solution of manufacturing inventory models with shortages

Accepted Manuscript Production, Manufacturing and Logistics Intuitionistic Fuzzy Optimization Technique for Pareto Optimal Solution of Manufacturing Inventory Models With Shortages Susovan Chakrabortty, Madhumangal Pal, Prasun Kumar Nayak PII: DOI: Reference:

S0377-2217(13)00093-3 http://dx.doi.org/10.1016/j.ejor.2013.01.046 EOR 11496

To appear in:

European Journal of Operational Research

Received Date: Revised Date: Accepted Date:

8 December 2011 14 September 2012 27 January 2013

Please cite this article as: Chakrabortty, S., Pal, M., Nayak, P.K., Intuitionistic Fuzzy Optimization Technique for Pareto Optimal Solution of Manufacturing Inventory Models With Shortages, European Journal of Operational Research (2013), doi: http://dx.doi.org/10.1016/j.ejor.2013.01.046

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Intuitionistic Fuzzy Optimization Technique for Pareto Optimal Solution of Manufacturing Inventory Models With Shortages Susovan Chakrabortty, Madhumangal Pal Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721 102, India

Prasun Kumar Nayak∗ Bankura Christian College, Bankura, 722 101, India,

Abstract This paper discusses a manufacturing inventory model with shortages where carrying cost, shortage cost, setup cost and demand quantity are considered as fuzzy numbers. The fuzzy parameters are transformed into corresponding interval numbers and then the interval objective function has been transformed into a classical multi-objective EPQ (economic production quantity) problem. To minimize the interval objective function, the order relation that represents the decision maker’s preference between interval objective functions has been defined by the right limit, left limit, center and half width of an interval. Finally, the transformed problem has been solved by intuitionistic fuzzy programming technique. The proposed method is illustrated with a numerical example and Pareto optimality test has been applied as well. Keywords: Inventory, Fuzzy Sets, Pareto Optimal Solution, Intuitionistic Fuzzy Optimization Technique, Multi-objective Programming. 1. Introduction Inventory problems are common in manufacturing, maintenance service and business operations in general. Often uncertainties may be associated with demand and various rel∗

Corresponding author Email address: nayak_prasun @rediffmail.com (Prasun Kumar Nayak)

Preprint submitted to European Journal of Operation Research

September 13, 2012

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evant costs like those of carrying, shortage and set-up. In conventional inventory models, uncertainties are treated as randomness and are handled by probability theory. However, in certain situations, uncertainties are due to fuzziness and in such cases the fuzzy set theory, originally introduced by Zadeh (1965), is applicable. Today most of the real-world decision-making problems in economic, technical and environmental ones are multidimensional and multi-objective. It is significant to realize that multiple-objectives are often non commensurable and are at loggerheads with each other in optimization problem. An objective within exact target value is termed as fuzzy goal. So a multi-objective model with fuzzy objectives is more realistic than deterministic model. Usually researchers considered different parameters of an inventory model either as constant or as dependent on time or as probabilistic in nature for the development of the EOQ/EPQ model. But, in real life situations, these parameters may have slight deviations from the exact value which may not follow any probability distribution. In these situations, if they are treated as fuzzy parameters, such a model is more realistic. Recently, the concept of fuzzy parameters has been introduced in the inventory problems by several researchers. In decision making process Bellman and Zadeh (1970), were the first to introduce fuzzy set theory. Tanaka et al. (1974) applied the concept of fuzzy sets to decision-making problems to consider the objectives as fuzzy goals over the α-cuts of fuzzy constraints. Zimmermann (1976, 1978) showed that the classical algorithms can be used in a few inventory models. Li et al. (2002) discussed fuzzy models for single-period inventory problem. Abuo-El-Ata et al. (2003) considered a probabilistic multi-item inventory model with varying order cost. A single-period inventory model with fuzzy demand was analyzed by Kao and Hsu (2002). Fergany and El-Wakeel (2004) considered a probabilistic single-item inventory problem with varying order cost under two linear constraints. A survey of literature on continuously deteriorating inventory models was discussed by Raafat (1991). Hala and EI-Saadani (2006) analyzed a constrained single period stochastic uniform inventory model with continuous distributions of demand and varying holding cost. Some

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inventory problems with fuzzy shortage cost was discussed by Katagiri and Ishii (2000). Moon and Choi (1998) implemented a note on lead time and distributional assumptions in continuous review inventory models. Hariga and Ben-Daya (1999) considered some stochastic inventory models with deterministic variable lead time. A fuzzy EOQ model with demand dependent unit cost under limited storage capacity was implemented by Roy and Maiti (1997). Zheng (1994) discussed optimal control policy for stochastic inventory systems with Markovian discount opportunities. Park (1987), Vujosevic et al. (1996), Chung (2003), Lee and Yao (2004), Lin and Yao (2000), Maiti (2011), Maity and Maity (2006) proposed the EOQ / EPQ model in fuzzy sense where inventory parameters are triangular fuzzy numbers (TFNs). Lai and Hwang (1992, 1994) elaborately discussed fuzzy mathematical programming and fuzzy multiple objective decision making in their two renowned contributions. Ouyang and Chang (2002) analyzed a minimax distribution free procedure for mixed inventory models involving variable lead time with fuzzy lost sales. Teghem et al. (1986) discussed an interactive method for multi-objective linear programming under uncertainty. Mahapatra and Roy (2006) discussed fuzzy multi-objective mathematical programming on reliability optimization model. Deshpande et al. (2011) proposed a bacterial foraging approach to solve fuzzy multi-objective function in inventory management. Ben Abdelaziz (2012) surveyed most solution approaches to the multi-objective stochastic optimization field and pointed out the importance of multi-objective stochastic programming in modeling many practical and complex situations. Based on an order relation of interval number Jiang et al. (2008) suggested a method to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Steuer (1981), Tang (1994), Ishibuchi and Tanaka (1990), Wolfe (2000) applied mathematical programming inexact, fuzzy and interval programming techniques, to deal with the ambiguous coefficients or parameters in an objective function. The programming technique is more flexible and allows one to find out the solutions which are more or less sufficient for the real problem. In fuzzy optimization the degree of acceptance

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of objectives and constraints is considered only. Nowadays, the fuzzy set theory has also been developed in a large area and its modifications and generalized forms have appeared. Intuitionistic fuzzy set (IFS) is one of the generalized forms of the fuzzy set. The concept of an IFS can be seen as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of conventional fuzzy sets. Therefore it is expected that IFS could be used to simulate the human decision-making process and any activity requiring human expertise and knowledge which are inevitably imprecise or not totally reliable. Here the degrees of rejection and satisfaction have been considered so that the sum of both values is always less than one. Atanassov also analyzed intuitionistic fuzzy sets in a more explicit way. Atanassov and Gargov (1989) discussed an open problem in intuitionistic fuzzy sets theory. Nikolova et al. (2002) presented a survey of the research on intuitionistic fuzzy sets. The intuitionistic fuzzy set has received increasingly more attention since its glowing appearance in Wei (2009, 2010) and Wei et al. (2011). Angelov (1997) implemented the optimization in an intuitionistic fuzzy environment. Angelov (1995) also contributed in another important paper, based on intuitionistic fuzzy optimization. Wei (2008) used the maximizing deviation methods to solve the intuitionistic fuzzy multiple attribute decision making problems with incomplete weight information. Pramanik and Roy (2004) solved a vector optimization problem using an intuitionistic fuzzy goal programming. A transportation model was solved by Jana and Roy (2007) using multi-objective intuitionistic fuzzy linear programming. In this paper, we propose an inventory model with fuzzy inventory costs and fuzzy demand rate. The said fuzzy parameters are then converted into appropriate interval numbers following Grzegorzewski (2002). We propose a method to solve the EPQ inventory model using the concept of interval. We have constructed an equivalent multi-objective deterministic model corresponding to the original problem with interval coefficients. To obtain the solution of this equivalent problem, we have used intuitionistic fuzzy program-

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ming technique where the degree of acceptance (satisfaction) is considered as exponential function and rejection of objectives is considered as quadratic function. Then this intuitionistic fuzzy optimization is converted into a crisp one and the resultant solution becomes a (α − β) Pareto optimal solution. The advantage of the intuitionistic fuzzy optimization technique is twofold: It gives the richest apparatus for formulation of optimization problems and the solutions of intuitionistic fuzzy optimization problems can satisfy the objective(s) in greater degree compared to the analogous fuzzy optimization problem and the crisp one. In order to illustrate the solution method, numerical examples are provided. 2. Preliminaries Definition 1. (Atanassov (1986, 1999)) Let X = {x1 , x2 , . . . , xn } be a finite universal set. An Atanassov’s IFS A is a set of ordered triples, A = {hxi , µA (xi ), νA (xi )i : xi ∈ X} where µA (xi ) and νA (xi ) are functions mapping from X into [0, 1]. For each xi ∈ X, µA (xi ) represents the degree of membership and νA (xi ) represents the degree of non-membership of the element xi to the subset A of X. For the functions µA (xi ) and νA (xi ) mapping into [0, 1] the condition 0 ≤ µA (xi ) + νA (xi ) ≤ 1 holds . Definition 2. Let A and B be two Atanassov’s IFSs in the set X. The intersection of A and B is defined as follows : A ∩ B = {hxi , min(µA (xi ), µB (xi )), max(νA (xi ), νB (xi ))i|xi ∈ X}. Definition 3. (Interval number): (Moore (1979)) Let ℜ be the set of all real numbers. An interval may be expressed as a = [aL , aR ] = {x : aL ≤ x ≤ aR , aL ∈ ℜ, aR ∈ ℜ}, where aL and aR are called the lower and upper limits of the interval a, respectively. If aL = aR then a = [aL , aR ] is reduced to a real number a, where a = aL = aR . The set of all interval numbers in ℜ is denoted by I(ℜ). 5

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Basic interval arithmetic : Let a = [aL , aR ] and b = [bL , bR ] ∈ I(ℜ), then a + b = [aL + bL , aR + bR ].

The multiplication of an interval by a real number c 6= 0 is defined as ca = [caL , caR ]; if c ≥ 0 and ca = [caR , caL ]; if c < 0. The difference of these two interval numbers is a − b = [aL − bR , aR − bL ]. The product of these two distinct interval numbers is given by a.b = [min{aL .bL , aL .bR , aR .bL , aR .bR }, max{aL .bL , aL .bR , aR .bL , aR .bR }] . The division of these two interval numbers with 0 6∈ b is given by      aL aL aR aR aL aL aR aR , max . , , , , , , a/b = min bL bR bL bR bL bR bL bR Optimization in interval environment : Now we defined a general nonlinear objective function with coefficients of the decision variables as interval numbers as Pn Qk rj i=1 [aLi , aRi ] j=1 xj Minimize Z(x) = Pl Qn qj j=1 xj i=1 [bLi , bRi ]

(1)

subject to xj > 0, j = 1, 2 . . . n and x ∈ S ⊂ ℜ where S is a feasible region of x, 0 < aLi < aRi , 0 < bLi < bRi and ri , qj are positive numbers. Now we exhibit the formulation of the original problem (1) as a multi-objective non-linear problem. Now Z(x) can be written in the form Z(x) = [ZL (x), ZR (x)] where Qk Pn rj a L i j=1 xj i=1 ZL (x) = Pl Qn qj , i=1 bRi j=1 xj

and

Qk rj j=1 xj i=1 aRi Pl Qn qj i=1 bLi j=1 xj

Pn

(2)

.

(3)

1 ZC (x) = [ZL (x) + ZR (x)]. 2

(4)

ZR (x) = The center of the objective function

Thus the problem (1) is transformed in to Minimize {ZL (x), ZR (x); x ∈ S}

(5)

subject to the non-negativity constraints of (1), where ZL , ZR are defined by (2) and (3). 6

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2.1. Nearest Interval Approximation In this section, according to Grzegorzewski (2002) we determine the interval approxie = (a1 , a2 , a3 ) be an arbitrary triangular fuzzy number mation of a fuzzy number. Let A with α - cuts [AL (α), AR (α)] and with the following membership function  x−a1   ; if a1 ≤ x < a2   a2 −a1 a3 −x µAe(x) = ; if a2 < x < a3 a3 −a2     0 ; otherwise.

(6)

Then by nearest interval approximation method, the lower limit CL and upper limit CR of the interval [CL , CR ] are Z 1 Z 1 a1 + a2 , CL = AL (α)dα = [a1 + (a2 − a1 )α]dα = 2 0 0 Z 1 Z 1 a2 + a3 CR = . AR (α)dα = [a3 − (a3 − a2 )α]dα = 2 0 0 e as a triangular fuzzy number is Therefore, the interval number considering A

a

1 +a2

2

 2 , a3 +a . 2

3. Model formulation The purpose of the EPQ model is to find the optimal order quantity of inventory items at each time such that the sum of the order cost, carrying cost and shortage cost, i.e., total cost is minimal. Notations : For the sake of clarity, the following notations are used throughout the paper. T :

the interval between production cycle ;

t1 :

the time after which the production is stopped ;

e = (d − ∆d1 , d, d + ∆d2 ) ; The demand rate per unit time is imprecise in nature, i.e D

e Q:

fixed lot size per cycle;

Se2 :

the shortage level at the end of t3 ;

Se1 :

e C(Q):

the inventory level at the end of t1 ;

total average cost in the plan period; 7

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The inventory carrying/ holding cost and shortage cost per unit item per unit time and the ordering or setup cost per unit item are imprecise in nature i.e, e1 = (C1 − ∆c11 , C1 , C1 + ∆c12 ), C e2 = (C2 − ∆c21 , C2 , C2 + ∆c22 ) and C e3 = (C3 − C

∆c31 , C3 , C3 + ∆c32 ) ;

Assumptions : The following assumptions are made: (i) Production rate or replenishment rate is finite, say, K units per unit item. (ii) Shortages are allowed and fully backlogged. (iii) Lead time is zero. (iv) The inventory planning horizon is infinite and the inventory system involves only one item and one stocking point. 3.1. Diagrammatic representation A typical behavior of the EPQ model with uniform demand and with shortage is de6

A

K−D 

O

t1

@@ D @@ R @ S1 @ @ @  t3 - t4 B -  t2 -@ E @ @ @ S2  D@ K−D R@ @

-

C

Figure 1: EPQ inventory model with shortage

picted in Figure 1. In each production cycle time T consists of two parts t12 and t34 which are further subdivision into t1 , t2 and t3 , t4 . During the time interval [0, t1 ] the inventory is building up at a constant rate K − D units per unit time and at time t = t1 , the production is stopped and the stock level decreases due to meet up the customer’s demand only up to the time t = t1 + t2 , after that shortages are accumulated at a constant rate of D units per unit time t3 and then shortages are being filled up immediately at a

8

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constant rate K − D units per unit time during the time t4 . In this way the production cycle then repeat itself after the time T where T = t1 + t2 + t3 + t4 . Hence the total cost in the plan period [0, T ] can be expressed as
2 Q − S2 1 C1 K K−D 1 C2 KS22 K X = C3 + + 2 D(K − D) 2 D(K − D)

Therefore total average cost C(Q, S2 ) is given by

Q − S2 C3 D(K − D) + 21 C1 K K−D X K C(Q, S2 ) = = T Q(K − D)


2

+ 21 C2 KS22

(7)

In crisp environment, by using calculus, we optimize C(Q, S2 ) and we get optimum values of Q, T , t1 , t2 , t3 , t4 , S1 , S2 and C. 3.2. Fuzzy EPQ model We assume that the demand rate, holding cost, shortage cost, and set up cost are fuzzy numbers. Then the equation (7) reduces to  2 e1 K K−De Q − S2 + 1 C e KS22 e3 D(K e e + 1C C − D) 2 K 2 2 e C(Q, S2 ) = where 0 ≤ S2 ≤ Q e Q(K − D) 3.2.1. Deterministic representation of the proposed model

(8)

Now, we represent this fuzzy EPQ model to a deterministic form such that it can be easily tackled. Following Grzegorzewski (2002), the fuzzy numbers are transformed into interval numbers as

       f C1 = (C1 − ∆C11 , C1 , C1 + ∆C12 ) ≡ [C1L , C1R ]  f2 = (C2 − ∆C21 , C2 , C2 + ∆C22 ) ≡ [C2L , C2R ]   C     f3 = (C3 − ∆C31 , C3 , C3 + ∆C32 ) ≡ [C3L , C3R ].  C e = (d − ∆d1 , d, d + ∆d2 ) ≡ [dL , dR ] D

(9)

Using (9) the expression (8) becomes

where, and

fL =

e C(Q, S) = [fL , fR ],

R Q − S2 C3L dL (K − dR ) + 12 C1L K K−d K Q(K − dL )


2

L Q − S2 C3R dR (K − dL ) + 21 C1R K K−d K fR = Q(K − dR )

9


2

(10) + 12 C2L KS22 + 12 C2R KS22

(11) (12)

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Addition and other composition rules on interval numbers are used in these equations. Hence, the proposed model can be stated as Minimize {fL (S2 , Q), fR (S2 , Q)}.

(13)

Generally, the multi-optimization problem (13), in the case of minimization problem, is formulated in a conservative sense from (2) as

subject to

Minimize {fC (S2 , Q), fR (S2 , Q)} . fL + fR . 0 ≤ S ≤ Q, where fC = 2

(14)

Here the interval problem (7) is represented as Minimize {fL (S2 , Q), fC (S2 , Q), fR (S2 , Q)} subject to 0 ≤ S ≤ Q

(15)

The formulation (15) gives better approximate solutions than those obtained from (14), Moreover, by the formulation (15) the decision maker (DM) does have the freedom to choose any one of the three functions fL , fC and fR for minimization. 4. Intuitionistic Fuzzy (IF) Programming Technique for Solution To solve multi-objective minimization problem given by (15), we have used the following IF programming technique. For each of the objective functions fL (S2 , Q), fC (S2 , Q), fR (S2 , Q), we first find the lower bounds LL , LC , LR (best values) and the upper bounds UL , UC , UR (worst values), where LL , LC , LR are the aspired level achievement and UL , UC , UR are the highest acceptable level achievement for the objectives fL (S2 , Q), fC (S2 , Q), fR (S2 , Q) respectively and dk = Uk −Lk is the degradation allowance, or leeway, for objective fk (S2 , Q), k = L, C, R. Once the aspiration levels and degradation allowance for each of the objective function has been specified, we formed a fuzzy model and then transform the fuzzy model into a crisp model. The steps of intuitionistic fuzzy programming technique is given below. Step 1: Solve the multi-objective cost function as a single objective cost function using one objective at a time and ignoring all others. 10

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Step 2: From the results of step 1, determine the corresponding values for every objective at each solution derived. Step 3: From step 2, we find for each objective, the best Lk and worst Uk values corresponding to the set of solutions. The initial fuzzy model of (8) can then be stated as, in terms of the aspiration levels for each objective, as follows : Find S2 and Q satisfying e k , k = L, C, R, subject to the non negatively conditions. fk
Step 4 : Define membership function (µfk ; k = L, C, R,) and a non membership

function (νfk ; k = L, C, R,) for each objective function. An exponential membership function is defined as

µ fk =

   1,    e

−w

     0,



fk −Lk Uk −Lk



1−e−w

if fk ≤ Lk −e−w

,

if Lk ≤ fk ≤ Uk

(16)

if fk ≥ Uk .

A quadratic non-membership function is defined as   0, if fk ≤ Lk     2 fk −Lk νfk = , if Lk ≤ fk ≤ Uk Uk −Lk     1, if fk ≥ Uk .

(17)

µfk is strictly monotonic decreasing function with property µf (Lk ) = 1, µf (Uk ) = 0, where as νfk is a parabolic functions with property νf (Lk ) = 0 and νf (Uk ) = 1. These two functions are continuous within [Lk , Uk ]. Therefore, quite naturally the functions meet at a point somewhere in [Lk , Uk ]. Step 5 : After determining the exponential membership and quadratic non-membership function defined in (16) and (17) for each objective function following Nayak and Pal (2011), Angelov (1997) the problem (15) can be formulated an equivalent crisp model on the basis of definition 2 of this paper as max α,

min β

α ≤ µfk (x);

k = L, C, R

β ≥ νfk (x);

k = L, C, R

α ≥ β;

and α + β ≤ 1; α, β ≥ 0 11

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where α denotes the minimal acceptable degree of objective(s) and constraints and β denotes the maximal degree of rejection of objective(s) and constraints. The intuitionistic fuzzy optimization model can be changed into the following crisp (non-fuzzy) optimization model as :        

max(α − β) α ≤ µfk (x); β ≥ νfk (x); which can be written in the form

α≤ α≤

max(α − β) f −LL −w( L UL −LL ) −e−w e 1−e−w e

−w

(

fC −LC UC −LC

) −e−w

1−e−w fR −LR −w UR −LR e

(

(18)

      and α + β ≤ 1; α, β ≥ 0  k = L, C, R

α ≥ β;

α≤

k = L, C, R

) −e−w

1−e−w

α ≥ β, α + β ≤ 1;

;

β≥

;

β≥

;





fL −LL UL −LL fC −LC UC −LC

2

2

            

     fR −LR  β ≥ UR −LR      and α, β ≥ 0, S2 ≤ Q  

2

(19)

Step 6 : Now the above problem can be solved by a non-linear optimization technique and optimal solution of α, (say α∗ ) and β, (say β ∗ ) are obtained. Step 7 : Now after obtaining α∗ and β ∗ , the decision maker (DM) selects the most important objective function from among the objective functions fL , fC and fR . Here fR is selected as DM would like to minimize his/her worst case. Then the problem becomes (for α = α∗ and β = β ∗ )         fL ≤ mL , fC ≤ mC , fR ≤ mR ;    fL ≥ nL , fC ≥ nC , fR ≥ nR ;      0 ≤ S2 ≤ Q; α ≥ β;      and α + β ≤ 1; α, β ≥ 0 min fR

12

(20)

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where mL = LL − mC = LC − mR = LR −

UL −LL w UC −LC w UR −LR w

  [ln{α (1 − e )} + e ] , nL = LL + − LL )     √  ∗ −w −w ∗  [ln{α (1 − e )} + e ] , nC = LC + µ (UC − LC )    √ ∗ −w −w ∗ [ln{α (1 − e )} + e ] , nR = LR + µ (UR − LR )      0 ≤ S2 ≤ Q; α ≥ β;      and α + β ≤ 1; α, β ≥ 0 ∗

−w

−w

√

µ∗ (UL

(21)

Step 8 : Pareto-Optimal Solution

Now after deriving the optimum decision variables, Pareto-optimality test is performed according to Sakawa (1983). Let the decision vector Q∗ , S ∗ and the optimum values fL∗ = fL (Q∗ , S ∗ ), fC∗ = fC (Q∗ , S ∗ ) and fR∗ = fR (Q∗ , S ∗ ) are obtained from (20). With these values, the following problem is solving using a non-linear optimization technique       ∗ ∗ ∗  subject to fL + ǫL = fL , fC + ǫC = fC , fR + ǫR = fR ;    ǫL , ǫC , ǫR ≥ 0, 0 ≤ S2 ≤ Q, α ≥ β      and α + β ≤ 1; α, β ≥ 0 min V = (ǫL + ǫC + ǫR )

(22)

The optimal solution of (22), say Q∗∗ , S2∗∗ , fL∗∗ , fC∗∗ and fR∗∗ are called strong Pareto optimal solution provided V is very small otherwise it is called weak Pareto solution. 4.1. Numerical example We consider an inventory system with the following fuzzy valued parameters: The demand for an item in a company is (1450, 1550, 1650) units per unit time. The company can produce the item at a rate of 4000 units per unit time and the set-up cost of one unit is $(400, 600, 800). The holding cost and shortage cost of one unit per unit time are $(17.5, 22.5, 27.5) and $(0.135, 0.165, 0.195) respectively. Now we use the following methodology to find the optimum manufacturing quantity, optimum shortage quantity and the optimal average cost.

13

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Considering the above parameters as TFN, the nearest interval approximation according e ≡ [dL , dR ] = [1500, 1600], C e1 ≡ [C1L , C1R ] = $[0.15, 0.18], to Grzegorzewski (2002) are D e2 ≡ [C2L , C2R ] = $[20, 25], C e3 ≡ [C3L , C3R ] = $[500, 700], K = 4000. Individual miniC

mum and maximum of objective functions fL , fC , fR are given in Table 1. Table 1: Individual minimum and maximum of objective functions Objective

optimize

optimize

optimize

functions

fL

fC

fR

fL

fL′ = $351.4112 fL′′ = $351.9052 fL′′′ = $352.7971

fC

fC′ = $437.2531 fC′′ = $436.6393 fC′′′ = $436.9186

fR

fR′ = $523.0950 fR′′ = $521.3734 fR′′′ = $521.0401

Now we calculate LL = min(fL′ , fL′′ , fL′′′ ) = $351.4112, UL = max(fL′ , fL′′ , fL′′′ ) = $352.7971, LC = min(fC′ , fC′′ , fC′′′ ) = $436.9186, UC = max(fC′ , fC′′ , fC′′′ ) = $437.2531, LR = min(fR′ , fR′′ , fR′′′ ) = $521.0401, UR = max(fR′ , fR′′ , fR′′′ ) = $523.0950. Using the equation (19), we formulate the following problem as : max z = α − β; −w

(1 − e−w )α ≤ −e−w + e −w

(1 − e−w )α ≤ −e−w + e



−w

(1 − e−w )α ≤ −e−w + e



2 −8785.28Q 18000000+1.08Q2 −3.6QS2 +403S2 34.6475Q



;

2 −3.978QS −10479.3432Q 22640000+1.2215Q2 +445.24S2 2 14.7312Q



2 −4.5QS −12504.9624Q 28000000+1.40625Q2 +503.6S2 2 49.3176Q





;

;

                        

     2 2 2 2  217.0083βQ ≥ (22640000 + 1.2215Q + 445.24S2 − 3.978QS2 − 10479.3432Q) ;      2 2 2 2  2432.2257βQ ≥ (28000000 + 1.40625Q + 503.6S2 − 4.5QS2 − 12504.9624Q) ;        0 ≤ S2 ≤ Q; α ≥ β;      and α + β ≤ 1; α, β ≥ 0 1200.4493βQ2 ≥ (18000000 + 1.08Q2 − 3.6QS2 + 403S22 − 8785.28Q)2;

The solutions obtained from Eq.23 is given in Table 2 and Table 3.

14

(23)

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Table 2: Optimal values of α and β w

Maximum α

Minimum β

0.1

0.7406

0.0624

Table 3: Optimal results when FR is chosen as the most important objective functions. fL∗

fC∗

fR∗

S2∗

Q∗

$351.7575 $436.7971 $521.8368 23.9885 4256.050 Table 4: Pareto-Optimal results. V

fL∗∗

fC∗∗

fR∗∗

S2∗∗

Q∗∗

0.0004 $351.7581 $436.7978 $521.8375 23.9882 4256.050 In Table 4, the value of V is quite small and hence, the optimal results in Table 3 are strong Pareto-optimal solution and can be accepted. 5. Practical implication In developing the EPQ model, most of the earlier researchers have considered demand rate and inventory costs as constant quantities. But in real world applications, the parameters in the inventory problem may not be known precisely due to some uncontrollable factors. Due to the cost of taking proper action to prevent deterioration of items, labour charges in different seasons inventory carrying cost may differ in various seasons. Changes in the price of fuels, mailing and telephonic charges may also make the ordering cost fluctuating. In real life, the period of any product in the market contains three phases: the growth phase, mature phase and decline phase. For example, the demand for soft drinks, ice-creams, refrigerators and air-conditioners generally increases in summer, while rain-coats, umbrellas, etc. are in high demand in the rainy season. Hence approximate solution methodologies have been illustrated for the solution of a class of realistic inventory problems.

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6. Conclusion The present paper proposes a solution procedure for inventory model, where inventory costs and demands are fuzzy numbers. Here shortages are allowed and fully backlogged. The new concept of the optimization problem in an intuitionistic fuzzy environment is introduced in this paper. This concept allows one to define a degree of rejection which may not simply complement the degree of acceptance. Several membership and nonmembership functions have been employed in intuitionistic fuzzy optimization : (i) linear (ii) piece-wise linear (iii) exponential (iv) hyperbolic (v) logistic (vi) parabolic (vii) Sshaped. Linear membership and non-membership functions are most commonly used because they are defined by fixing two points - upper and lower levels of acceptability and rejection. Furthermore, if the membership and non-membership functions are interpreted as the intuitionistic fuzzy utility of decision maker used for describing levels of indifference, preference or aversion towards uncertainty, then non-linear membership and non-membership functions provide a better representation. Each fuzzy member is now approximated to an interval number. After that the problem is converted into a multi-objective inventory problem where the objective functions represented by left limit, right limit and centers of interval functions, are minimized. To obtain the solution to the deterministic multi-objective inventory problem, the intuitionistic fuzzy optimization technique has been used. The advantage of this procedure is that the decision makers can easily minimize the worse cases and maximize the better once. In the process of solution it is desirable that the shape parameters are heuristically and experimentally decided by the DM, because, after obtaining the optimum value in any situation, if the DM /Practitioner is not satisfied with the outputs, he/she may perform the analysis again, re-choosing the membership and non-membership functions fL , fC and fR as others until the strong Pareto-optimum solutions are obtained. Acknowledgements The author would like to thank the Editor, European Journal of Operational Research 16

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21

For finding optimal solution we use Intuitionistic fuzzy optimization technique. Here decision maker easily minimize the worse case and maximize the better case. Non-linear membership and non-membership functions are used for better result. Pareto Optimality test is done for assuring that the result is optimal.