A Fuzzy Inventory Model With Lot Size Dependent Carrying / Holding Cost

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1 IOSR Journl of Mthemtics (IOSR-JM e-issn: ,p-ISSN: 9-765X, Volume 7, Issue 6 (Sep. - Oct. 0, PP A Fuzzy Inventory Model With Lot Size Dependent Crrying / olding Cost P. Prvthi, S. Glkshmi ed & Associte Professor, Deprtment of Mthemtics, uid- E -Millth Government College for Women (Autonomous, Reserch Scholr, Deprtment of Mthemtics, uid- E -Millth Government College for Women (Autonomous, Chenni Abstrct: In the clssicl rris Wilson inventory model l the cost ssocited with the formul ws tken to be constnt nd which does not dependent on ny quntity. In this pper we hve tken Ordering cost, holding cost nd order quntity ll re tringulr fuzzy numbers. Grded men integrtion representtion method is used for defuzzifiction. In this pper we consider n inventory model where the holding cost depends on order quntity. An lgorithm is developed to find the economic order quntity long with numericl exmples. Keywords: Fuzzy inventory model, Tringulr fuzzy numbers, Defuzzifiction I. Introduction: Pricing nd inventory policy re two importnt fctors in success of business of different items. One of the chllenging problems for the reserchers is to study nd nlyze the inventory systems. Those systems cnnot only reduce the costs, but lso reduce stock outs nd improve customer stisfctions. The rris Wilson Formul [5] for determining the optimum lot size is the first nd pioneering model in the inventory systems. The Bsic EO model considers three type of costs like ordering or set up cost, holding cost nd shortge cost. Severl reserchers del with the modifictions of these cost structures. Gupt [0] discussed n inventory model with lot size dependent ordering cost. Gordon [9], Emery [7], Arrow [] hve given so mny worth results in this field. The rris Wilson formul for determining the optimum lot size, the quntity in which n item of inventory should be purchsed or produces is DA ( where D Demnd, A Ordering cost, olding cost. Our Endevour is to introduce more relistic inventory model for which n lgorithm is developed. Our proposed model is illustrted through numericl exmples. This Formul follows tht, ( The optiml quntity minimizes the sum of the nnul set up cost nd holding cost nd ( The Totl cost is DA TC( -----( i. e the costs considered re ssumed to be constnt. This body of reserch ssumes tht the prmeters involved in the EO model, such s the demnd nd the purchsing cost, re crisp vlues or rndom vribles. owever, in relity, the demnd nd the cost of the items often chnge slightly from one cycle to nother. Moreover, it is very hrd to estimte the probbility distribution of these vribles due to lck of historicl dt. Insted, the cost prmeters re often estimted bsed on experience nd subective mngeril udgment. Thus, the fuzzy set theory, rther thn the trditionl probbility theory, is well suited to the inventory. The pper developed by Krbi Dutt Choudhury nd Sumit Sh [6], they considered ll the costs s constnt in their model. To mke it more relistic we hve tken ordering cost, holing cost nd order quntity s tringulr fuzzy numbers. Grded men integrtion representtion method is used for defuzzifiction Pge

2 A Fuzzy Inventory Model With Lot Size Dependent Crrying / olding Cost II. Methodology Fuzzy Numbers Any fuzzy subset of the rel line R, whose membership function µ A stisfy the following conditions, is generlized fuzzy number A. (i µ A is continuous mpping from R to the closed intervl [0, ], (ii µ A = 0, x (iii µ A = L(x is strictly incresing on [, ] (iv µ A = w A, x (v µ A = R(x is strictly decresing on [, 4 ] (vi µ A = 0, 4 x Where 0 < w A nd,, nd 4 re rel numbers. Also this type of generlized fuzzy number is denoted s A (,,, ; ; when w A =, it cn be simplified s A (,,, ;. 4 w A LR 4 w A LR Tringulr fuzzy number The fuzzy set A (,, where < < nd defined on R, is clled the tringulr fuzzy number, The membership function of A is given by x, x x A, x 0, otherwise The Function Principle The function principle ws introduced by Chen [6] to tret fuzzy rithmeticl opertions. This principle is used for the opertion for ddition, subtrction, multipliction nd division of fuzzy numbers. Suppose A (,, nd B ( b, b, re two tringulr fuzzy numbers. Then b (i The ddition of A nd B is A B ( b, b, where,,, b, b, b re ny rel numbers. (ii The multipliction of b A nd B is A B ( c, c, c T ( b, b, b, b, c Where if,,, b, b, b re ll non zero positive rel min T, c b, c mx T numbers, then A B ( b, b, b. (iii B ( b, b, then the subtrction of B from A is A B ( b, b, b where,,, b, b, b re ny rel numbers. b 07 Pge

3 A Fuzzy Inventory Model With Lot Size Dependent Crrying / olding Cost (iv B (,, where b, b, b re ll non zero positive rel numbers, then the division of B b b b nd B A is (,, B b b b KA ( K, K, K ifk 0 (v For ny rel number K, KA ( K, K, K ifk 0 A Grded Men Integrtion Representtion Method Defuzzifiction of A cn be done by Grded Men Integrtion Representtion Method. If A is tringulr fuzzy number nd is fully determined by (,, then defuzzified vlue is defined s ( A 0 h[ h( Model Development nd Anlysis: 0 hdh h( ] dh 4 In this model, we ssume tht the holding cost increses stepwise s the lot size increses. The following nottions re used. D Totl / Annul Demnd A - Fuzzy Ordering cost per order - olding cost for the lot size if q q Where =,,,..m. q 0 = 0 nd Also ssume < < <. m. q For holding cost rris Wilson EO is given by D A ( 6 If does not lie within the intervl [q -, q ] i. e is not order fesible, then the optiml lot size will be determined by q if q (4 q if q (5 With the known vlue of thus obtined from the eqution (, (4, nd (5, T C( cn be clculted t the holding cost by D A T C ( (6 If is order fesible then TC( will be the optiml cost, otherwise the vlue of thus obtined by equtions (4 nd (5, TC( is clculted by (6. Thus mong ll the clculted vlues of T C(, the 08 Pge

4 A Fuzzy Inventory Model With Lot Size Dependent Crrying / olding Cost smllest vlue will be the optiml cost. After defuzzifiction of nd TC( we will get optiml TC ( opt nd the corresponding will be the optiml lot size i. e opt. Algorithm. Input, n = number of lot size. Set =. D A. Clculte 4. If is order fesible, D A 5. Clculte T C (. Go to Step If is not order fesible, then by eqution (4 nd (5, cn be clculted nd hence find TC( by eqution (6. 7. Set = +, until <=n. Go to step 8. Defuzzify nd T C(, then we will get nd TC(. 9. Among ll the clculted TC(, obtin the minimum vlue of TC(, nd set TC( opt = min [ TC ( ]. 0. Thus TC ( opt obtined is the optiml cost nd corresponding is the opt the optiml lot size.. Thus opt nd TC( opt re the required results. Numericl Exmples: Exmple : D = 800 A = (40, 50, 60 (00.99, 07.0,.84 (9.5, 98.44, 0.9 (87.46, 9.65, (80.6, 8.666, (76.45, 79.7, Rnge Fesible 0 0 (00, 0, (0, 0, (40, 50, (70, 80, (90, 00, 0 T C( ( , 77.85, (500.8, , (48.9, 747.7, (8.87, , (466.97, , TC( For the lot size opt = 0 units (pprox the optiml holding cost is TC( opt = Rs Exmple : D = 000 A = (40, 50, Pge

5 A Fuzzy Inventory Model With Lot Size Dependent Crrying / olding Cost (75.77, 79.77, (69.69, 7.8, (65.9, 68., (59.8, 6.6, (56.9, 59.6, 6.56 Rnge Fesible 0 0 (00, 0, (0, 0, (40, 50, (70, 80, (90, 00, 0 T C( (7770.9, , ( , 959.9, (904.6, , 59. (009.04, 4.97, (098.5, 8.6, TC( For the lot size opt = 0 units (pprox the optiml holding cost is TC( opt = Rs III. Conclusion: In this pper the clssicl rris Wilson model hs been extended with fuzzy holding cost depending on lot size. It is observed tht if order quntity lies within the intervl i. e order fesible it will give the optiml costs. And if no such vlue cn be obtined which is order fesible then we cn mke order fesible by eqution (4 nd (5. From ll the clculted vlues of the minimum vlue will give the optiml holding costs. We conclude tht the optiml cost depends on demnd required. The lgorithm hs been tested using numericl exmple. The results show tht the lgorithms described in this pper perform well. We hve considered fuzzy nture of ordering cost, holding cost nd order quntity. Even though the totl cost in this model ppers to be slightly higher thn tht in the clssicl model, this model is more suitble to rel life situtions. A trder cn chieve his gol by dopting fuzzy nture. References: []. Fuzzy systems nd Logics by Zdeh []. Arrow. K. rris, T., nd Mrschk, J. (95, Optiml Inventory Policy, Econometric, 9, []. Bensoussn, A., nd Proth, J. M. (984 Inventory plnning in deterministic environment: concve set up, lrge scle system, 6, [4]. Binghm, P. (986, Economic order quntities for systems with step up function ordering costs, production nd inventory mngement, 7, 9-7. [5]. Emery, J. C. (954, Some theoreticl spect of inventory control mster s thesis in industril mngement, M. I. T. [6]. Golbi, K, (985, Optiml inventory policies when ordering prices re rndom, Opertions Reserch,, [7]. Gorden,, M. J. nd Tylor, W. J. (956, The condition for lot- size production, J, of Americn Sttisticl Assocition, vol. 5, No. 76, [8]. Gupt K. Omprksh, (994, An inventory model with lot size dependent ordering cost Production plnning nd control, 6, [9]. rris, F. W. (95, wht quntity to mke t once. The librry of fctory mngement, 5, Opertions nd Cost (A. W. Shw Compny, Chicgo pp Pge