Aryabatta Journal of Matematics & Informatics Vol. 5, No., July-ec., 03, ISSN : 0975-739 Journal Impact Factor (0) : 0.93 THE COMPLETE SOLUTION PROCEURE FOR THE FUZZY EOQ INVENTORY MOEL WITH LINEAR AN FIXE BAC ORER COST W. Rita * and S. Rexlin Jeyakumari ** * epartment of Matematics, Holy Cross College (Autonomous), Tricirapalli-60 00, Tamil Nadu, India. ** epartment of Matematics, Holy Cross College (Autonomous), Tricirapalli-60 00, Tamil Nadu, India. ABSTRACT Te nature of te inventory problem consists of repeatedly placing and receiving orders of given sizes at set intervals. In tis paper, we discuss te inventory problem wit fuzzy backorder. Yager s ranking metod for fuzzy numbers is utilized to find te inventory policy in terms of te fuzzy total cost. Finally a numerical example is given to illustrate te model. eywords : Fuzzy number, inventory, backorder, optimization. INTROUCTION Witin te context of traditional inventory models, te pattern of demands is eiter deterministic or uncertain. In practice, te latter corresponds more to te real-world environment. To solve tese inventory problems wit uncertain demands, te classical inventory models usually describe te demands as certain probability distributions and ten solve tem. However, some times, demands may be fuzzy, and more suitably described by linguistic term rater tan probability distributions. If te traditional inventory teories can be extended to fuzzy senses, te traditional inventory models would ave wider applications. Usually, inventory systems are caracterized by several parameters suc as cost coefficients, demands etc. Accordingly, most of te inventory problems under fuzzy environment can be addressed by fuzzying tese parameters. For instance, Park [8] discussed te EOQ model wit fuzzy cost coefficients. Isii and onno [3], Petrovic et. al. [9], and ao and Hsu [] investigated te Newsboy inventory model wit fuzzy cost coefficients and demands respectively. Roy and Maiti [0] developed a fuzzy EOQ model wit a constraint of fuzzy storage capacity. Cang [] construct a fuzzy EOQ model wit fuzzy defective rate and fuzzy demand. Yao and Ciang [3] compare te EOQ model wit fuzzy demand and fuzzy olding cost in different solution metods. ao and Hsu [5] find te lot size-reorder point model wit fuzzy demand. Besides, tere is anoter kind of studies wic fuzzes te decision variables of inventory models. For example: Yao and Lee [5] developed te EOQ model wit fuzzy ordering quantities; Cang and Yao [] investigated te EOQ model wit fuzzy order point; Wen-ai. Hsu and Jun-Wen Cen [] studied Fuzzy EOQ model wit stock out. Madu & eepa [7] developed an EOQ model for deteriorating items aving exponential declining rate of demand under inflation & sortage. un-jen Cung, Leopoldo Eduardo Cárdenas-Barrón [6] compare te complete solution procedure for te EOQ and EPQ inventory models wit linear and fixed backorder costs. Recently W. Rita etal. [] fuzzified EOQ Model wit one time discount offer allowing back. -365-
W. Rita and S. Rexlin Jeyakumari In tis paper, we discuss te inventory problem wit fuzzy back order. Te decision variables are te ordering quantity Q and te back order quantity S. Te approac of tis paper is to find te optimal order quantity Q * wit te minimum cost determined from Yager s ranking metod. Te rest of tis paper is organized as follows; in section, te preliminaries are given. In section 3, te total inventory cost of te problem constructed from te α-cut of te back order. In section, optimal ordering quantity is derived using Yager s ranking metod []. Finally, a numerical example is given to illustrate te model.. PRELIMINARIES.. efinition: Fuzzy Set A fuzzy set A is defined by A = {(x, μ (x) A ) : x X, μ (x) A [0, ]}. In te pair {(x, μ (x) A )}, te first element x belong to te classical set A, te second element μ (x), belong to te interval [0, ], called membersip function or grade of membersip. Te membersip function is also a degree of compatibility or a degree of trut of x in A. A.. α - Cut Te set of elements tat belong to te fuzzy set A at least to te degree α is called te α level set or α - cut. A ) = {x X : μ (x) A α.3. Generalized Fuzzy Number Any fuzzy subset of te real line R, wose membersip function satisfies te following conditions, is a generalized fuzzy number (i) μ (x) A is a continuous mapping from R to te closed interval [0, ]. (ii) μ (x) A = 0, - < x a, (iii) μ (x) A = L(x) is strictly increasing on [a, a ], (iv) μ (x) A =, a x a 3, (v) μ (x) = R(x) is strictly decreasing on [a 3, a ], (vi) A μ (x) = 0, a x <, were a, a, a 3 and a are real numbers. A.. Trapezoidal Fuzzy Number Te fuzzy number A = (a, a, a 3, a ), were a < a < a 3 < a and defined on R is called te trapezoidal fuzzy number, if te membersip function A is given by -366-
μ (x) A = Te complete solution procedure for te fuzzy eoq inventory model wit linear and Fixed back order cost 0 ; x < a or x > a (x - a ) ; a x a (a - a ) ; a x a (x - a ) ; a x a (a 3 - a ) 3 3.5. Yagers Ranking Metod If te cut of any fuzzy number A is [A L ), A g ()], ten its ranking index I( A ) is 3. TOTAL INVENTORY COST: Notations Used: - emand rate in units /unit of time - Ordering cost/ order/set up - Holing cost / unit/unit of time p - Linear back order cost - Fixed back order cost Q - Size of order quantity S - size of te back order quantity Q* - optimal value of Q TC (Q, S) - Total annual cost function of Q and S in $ / year Te total cost function is (Q - S) ps S TC(Q,S) = + + + Q Q Q Q Q S ps S + - S + + + Q Q Q Q Te objective is to find te optimal order quantity wic minimize te total cost Te necessary conditions for minimum TC(Q,S) ps S = 0 - + - S - - Q Q Q Q Q ifferentiate () partially w.r.to S, we obtain, TC(Q,S) S ps + + Q Q Q S 0 - + 0 A ) + A ) dα. L g 0... () -367-
W. Rita and S. Rexlin Jeyakumari Q - we get, S = + p Substitute S in (), Hence te optimal order quantity is * = Q (+p) - p. THE EOQ MOEL WITH BAC ORER AN FUZZY EMANS Let be a normal fuzzy number wit parameters = (l, m, n, u), ten te membersip function of can be defined by a left sape function L(x) and a rigt sape function R(x) as: L(x), l x m μ (x) =, m x n R(x), n x u Te above equation can also be described by te terms of -level cut of as: () = min. μ ), max. μ ) = L ), R ), 0 α First, we discussed te EOQ model wit fuzzy demand, according to te extension principle; te model can be described by terms of α as Tα = TC Q = L ), TC Q = R ), 0 α Were TC (Q) = + Q Q Since te annual cost function T α is a fuzzy number, we can compare T α of different Q by using some ranking metods to find te optimal solution Q* wit a minimal total cost, of wic, not every metod is applicable to rank T α of all possible Q. Te metod proposed by Yager [], does not need to know te explicit form of te membersip functions, and can tus be applied ere. Te Yager s ranking index ranks te fuzzy numbers by an area measurement defined as I L(T) I R(T) I(T) = were I L(T) represents te area bounded by te left sape function of T α, te x axis, te y axis and te orizontal line μt = and I (T) R represents te area bounded by te rigt sape function of T α, te x axis, te y axis and te orizontal line μt =. T α,α tus can be calculated as Te Yager s ranking index of - - - - I(T) = TC Q,S, =L ) L ) dα+ TC Q,S, = R )R ) dα 0 0-368-
Let Te complete solution procedure for te fuzzy eoq inventory model wit linear and Fixed back order cost - - - - ( α,α)= L ) L )dα + R )R )dα 0 0 Now α α (, ) Q I T = + Q wit respect to Q and setting to zero, te necessary condition of optimal Taking te partial derivative of T α solution of T α can be found as α,α Q * = * * - * - T α = TC Q = L ), TC Q = R ), 0 Now if, back order cost is permitted, te model can be described by terms of as - - - - T ) = Q,S, =L ) L ) + Q,S, = R )R ) TC TC 0 Te Yager s index of ten can be derived as T α - - - - I( T ) = TC Q,S, =L ) L ) dα+ TC Q,S, = R )R ) dα 0 0 - - - - Let ( α,α)= L ) L )dα + R )R )dα 0 0 Te necessary conditions for IT ' as attain te minimum on IQ T = 0 wic can be calculated as follows. ( α, α) Q S ( α,α) I(T) Q Q I(T) ( α, α) p S ( α, α) = 0 + S S 0 Q Q Q Q Q Q = ( α,α ) + ( + p) S + S (, ) α α Q = ( α,α ) + ( + p) S + S ( ) α,α - () Te sufficient conditions for te IT " to attain te minimum are T IQ > 0. Because of (, ) (, ) > 0, te sufficient conditions are clearly eld from te above α α > 0 and αα equations. Te optimal solutions Q * tat can be found from equation.(3..3) and te optimal annual cost can be calculated as, = - - - - * * L α L α p R α R α * T) 0 TC Q, S (,) = ( ) ( ), TC Q, S (,) = ( ) ( ), To sow te caracteristics of proposed models a trapezoidal fuzzy demand is employed. Let be te trapezoidal fuzzy demands wit parameters: -369-
W. Rita and S. Rexlin Jeyakumari = [l, m, n, u] It is easy to find tat ) = l + m + n + u For te EOQ model wit, we ave l + m + n + u Q * = If is a symmetrical fuzzy number ten u n = m l Tat is., u + l = m + n Let 0 = m + n, te mean of, ten m + n = Q * = 0 Q * is te conventional EOQ wit crisp demands, m + n. Tis result implies tat no matter wat te spreads of fuzzy demands, as long as te fuzzy demands are symmetric wit te same mean, te Q * will be te same and equal to te conventional EOQ wit te mean of fuzzy demands. Te fuzzy number of annual cost can be calculated as = * * * T) TC Q = l + α(m - l), TC Q = u - α(u - n), 0 Te above equation sows tat te annual cost will also be a trapezoidal fuzzy number and wit parameter as * * * * * l Q m Q n Q u Q TCQ = +, +, +, + * * * * Q Q Q Q * Tis implies tat te sape of membersip function of TCQ is te same as te, but wit a different scale. * Accordingly, te spread of TCQ will vary according to te spread of. Q S ps S TC(Q,S) + - S + + + Q Q Q Q Let,,, p, be fuzzy trapezoidal numbers and tey are defined as follows. [Tat is., tey are described by te -cuts] α = min. μ ), max. μ ) = L ), R ), 0 α α = min. μ ), max. μ ) = L ), R ), 0 α and α = min. μ ), max. μ ) = L ), R ), 0 α p α p = min. μ p p ), max. μ p p ) -370-
Te complete solution procedure for te fuzzy eoq inventory model wit linear and Fixed back order cost = L p p), R p p), 0 α α = min. μ ), max. μ ) = L ), R ), 0 α Yager s ranking index can be derived as (,α) Q ( α) S ( )S 3( αp) S I(T) α α,α = + ( α) - ( α)s + + + Q Q Q Q Were - - - - ( α,α)= L ) L )dα + R )R )dα 0 0 - - ( α)= ( ) dα+ ( ) dα L α R α 0 0 - - 3( αp)= Lp ( αp)d αp+ R p ( αp)dαp 0 0 - - - - ( α,α)= L ) L )dα + R )R )dα 0 0 And te optimal solution can be obtained as Q ( α,α ) ( α ) + ( α ) - ( α ). ( α ) * 3 = p 3 p α,α ( )... (3) Numerical Example: As an illustration, consider a fuzzy inventory problem wit a trapezoidal demand, ordering cost, linear back order, olding cost and fixed back order cost we ave, = (80,90,0,0); = (30,0,60,70); p = (8,9,,); = (.6,.8,.,.); = (,.5,.5, 3) Substituting tese values in (3), we obtain Q * = 63 units. 5. CONCLUSION: Tis paper provides a metodology for constructing te fuzzy total inventory cost wen te linear and fixed back orders are fuzzy. Tis metod is applicable to oter inventory problems like inventory problems wit sortages lost sales and sortages lead time. REFERENCES: [] Cang, H. C. An application of fuzzy teory to te EOQ model imperfect quality items, Computers and Operations Researc, 3, 079-09 (00). -37-
W. Rita and S. Rexlin Jeyakumari [] Cang, S. C. and J. S. Yao, Economic reorder point for fuzzy backorder quantity, European Journal of Operational Researc, 09,83-0 (998). [3] Isii, H and T. onno, A stocastic inventory problem wit sortage cost, European Journal of Operational Researc, 06, 90-9(998). [] ao. C. and W.. Hsu, A Single-period inventory model wit fuzzy demand, Computer and Matematic wit Application,3,8-88(00). [5] ao. C. and W.. Hsu, Lot size-reorder point inventory model wit fuzzy demand, Computer and Matematic wit Application, 3, 9-30(00). [6] un-jen Cung, Leopoldo Eduardo Cárdenas-Barrón, Te complete solution procedure for te EOQ and EPQ inventory models wit linear and fixed backorder costs Matematical and Computer Modeling, 55, 5-56 (0). [7] Madu Jain and eepa Cauan Inventory Model wit eterioration, Inflation and permissible delay in payment. Aryabatta J. of Mats & Info. Vol. () pp 6-7 [00] [8] Park,. S, Fuzzy-set teoretic interpretation of economic order quantity, IEEE Trans. System, Man, Cybernetics, SMC-7, 08-08(987). [9] Petrovic., R. Petrovic and M. Vujosevic, Fuzzy model for te newsboy problem, International Journal of Production Economics,5, 35-(996). [0] Roy, T.., and M. Maiti. A fuzzy EOQ model wit demand-dependent unit cost under limited storage capacity, European Journal of Operational Researc, 99, 5-3(997). [] Wen-ai. Hsu and Jun-Wen Cen, Fuzzy EOQ model wit stock out, epartment of Sipping Transportation & Management, National aosiung Marine University, Taiwan, R.O.C. [] Yager, R.R., A procedure for ordering fuzzy subsets of te unit interval, Information Sciences,, 3-6 (98). [3] Yao, J. S. and J. Ciang, Inventory witout backorder wit fuzzy total cost and fuzzy storing cost defuzzified by centroid and signed distance, European Journal of Operational Researc, 8,0-09(003). [] W. Rita and M. Sumanti Fuzzy EOQ Model wit one time discount offer and Allowed Back order. Aryabatta J. of Matematics & Informatics vol. () pp. 3- [0]. [5] Yao, J. S. and H. M. Lee, Fuzzy inventory wit or witout backorder for fuzzy order quantity wit trapezoid fuzzy number, Fuzzy Sets and Systems, 05, 3-337 (999). -37-