Multi Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions.

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1 It. Joural of Math. Aalyss, Vol. 8, 204, o. 4, HIKARI Ltd, Mult Objectve Fuzzy Ivetory Model wth Demad Depedet Ut Cost ad Lead Tme Costrats A Karush Kuh Tucker Codtos Approach S. Ragaayak ad 2 C.V. Seshaah Dept. of Mathematcs, Sr Ramakrsha Egeerg College Combatore,TN, Ida 2 Dept. of Mathematcs Sr Ramakrsha Egeerg College, Combatore, TN, Ida Copyrght 204 S. Ragaayak ad C.V. Seshaah. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract I ths paper a mult-objectve vetory model wth demad depedet ut cost ad leadg tme has bee formulated wth umber of orders ad producto cost as costrats. I most of the real world stuatos the cost parameters, the objectve fucto ad costrats of the decso makers are mprecse ature. A demad depedet ut cost s assumed ad solved usg Karush Kuh Tucker codtos. Here the ut producto cost s cosdered uder fuzzy evromet. The model has bee solved wth demad, lot sze ad leadg tme as decso varables. Keywords: vetory, membershp fucto, Karush-Kuh-Tucker codto, demad depedet, leadg tme

2 88 S. Ragaayak ad C.V. Seshaah. Itroducto I geeral the classcal vetory problems are desged by cosderg that the demad rate of a tem s costat ad determstc ad that the ut prce of a tem s cosdered to be costat ad depedet ature. But practcal stuato, ut prce ad demad rate of a tem may be related to each other. Whe the demad of a tem s hgh, a tem s produced large umbers ad fxed cost of producto are spread over a large umber of tems. Hece the ut cost of the tem decreases. (.e) the ut prce of a tem versely relates to the demad of that tem. So demad rate of a tem may be cosdered as a decso varable. A demad depedet ut cost had bee treated by some researchers the problem of EOQ model. Chag[4] studed a EOQ model wth demad depedet ut cost of sgle tem. Be-Daya ad Abdul Raouf [3] descrbed the problem of vetory models volvg lead tme as a decso varable. Abou-et- Ata ad Kotb [] developed a crsp vetory model uder two restrctos. Also Teg & Yag [0] examed determstc vetory lot sze model wth tme varyg demad ad cost uder geeralzed holdg costs. Other related studes were wrtte by Jag & Kle [6].The cocept of fuzzy set theory was frst troduced by Zadeh []. Later o Bellma ad Zadeh [2] used the fuzzy set theory to the decso makg problems. Hece Fuzzy set theory has made o etry to the vetory cotrol system. May researchers solved fuzzy mult tem mult objectve vetory model especally usg Geometrc programmg method. Here we solve the model usg Karush -Kuh-Tucker Codtos wth ut producto cost uder fuzzy evromet. 2. Notatos & Assumptos To costruct the model, we defe the followg otatos D = Aual demad rate (decso varable) p = ut purchase(producto) cost S = orderg cost H = ut holdg (Ivetory varyg) cost per ut tem Ss= Kσ L = Safety stock

3 Mult objectve fuzzy vetory model 89 = umber of dfferet tems (carred vetory) L = Leadg rate tme (decso varable) Q = Producto (order) quatty take (decso varable) TC( D, Q, L ) = Average aual total cost for the th tem B = Total vestmet cost for repleshmet t = umber of orders Assumptos The followg basc assumptos are made the model () Tme horzo s fte (2) No shortages are allowed (3) Ut producto cost p = AD β, =,2,3, s versely related to the demad rate where A > 0, β. (4) Lead tme crashg cost s related to the lead tme by a fucto of the b form R(L )= α, =,2,..., α >0, 0 < b 0.5 L where α, b are real costats, selected to provde the best ft of the estmated cost fucto. (5) Objectve s to mmze the aual relevat total cost. 3. Mathematcal formulato The aual relevat total cost [sum of producto, order, vetory carryg ad lead tme crashg costs] whch accordg to the basc assumptos of the EOQ model s: SD Q D TC( D, Q, L) = pd + + Kσ L H R( L) = Q Q () Substtutg p ad R (L ) () gves

4 90 S. Ragaayak ad C.V. Seshaah TC( D, Q, L ) AD SD Q K L H D L β b = + + σ α = Q Q To derve the optmal total cost a vetory problems, there are some restrctos o avalable resources. () Ivestmet amout o total producto cost caot be fte, t may have a upper lmt o the maxmum vestmet (.e) (or) = = p Q B β AD Q B () A upper lmt o the umber of orders that ca be made a tme D cycle o the system (.e) t Q = 4. Fuzzfcato of Cost Parameter I ths paper the ut producto cost p s defed uder fuzzy evromet. The membershp fucto for the fuzzy varable p s defed as follows., p L L UL p μ p ( X) =, L L p UL UL L L 0, p U L Here U L ad L L are upper lmt ad lower lmt of p respectvely 5. Numercal example The decso varables amely the optmal order quatty Q, optmal demad rate D ad optmal lead tme L whose values determe the mmum aual relevat total cost are computed for dfferet values of β.

5 Mult objectve fuzzy vetory model 9 The parameters of the model are show Table - Assume the stadard devato σ =6 ut/year & K=2, 6 p 8 The optmal values of the producto batch Q,demad rate D, lead tme L ad mmum total cost are gve Table 2 Table S A H α B t 200$ 5 0.8$ 200 Table 2 The optmal soluto of Q, D & L as a fucto of β α β b D Q L M TC p µ p x x x x The optmal soluto s D =.342, Q = , L = 8.26x0-6 ad M TC = $ whch correspods to maxmum membershp fucto It has bee see that as β value creases, the lot sze Q also creases. But coversely the demad D, lead tme L ad the mmum total cost decreases.

6 92 S. Ragaayak ad C.V. Seshaah 7. Cocluso I ths paper we have proposed a cocept of the optmal soluto of the vetory problem wth fuzzy cost prce per ut tem. A vetory model wth demad depedet ut cost ad lead tme depedet o leadg tme crashg cost wth lmted lot sze ad vestmet s solved usg Karush -Kuh-Tucker codtos. Here the optmal soluto s calculated wth fuzzy ut prce per tem. The result reveals the mmum expected aual total cost of the vetory model. The values of the varables amely, the demad D, lot sze Q ad the lead tme L ca be calculated for varous values of α & b. Also the problem ca be exteded for more tha oe tem wth other costrats lke lmted warehouse ad setup cost. Refereces [] M. O. Abou-El-Ata ad K. A. M. Kotb, Mult-Item EO Q Ivetory model wth varyg Holdg cost uder two Restrctos: A Geometrc Programmg Approach, Pro-ducto Plag & Cotrol, Vol. 8, No. 6, 997, pp do:0.080/ [2] R.E.Bellma, L.A.Zadeh, Decso makg a fuzzy evromet,maagemet Scece 7(4)(970) B4-B64.New York, 93 [3] M.Be-Daya ad A.Raouf, Ivetory models volvg lead tme as a decso varable, Joural of the operatoal Research socety, Vol.45,No.5.994,pp [4] T.C.E. Cheg A Ecoomc order quatty wth Demad-Depedet Ut Cost, Europea Joural of Operatoal Research, Vol. 40, No. 2, 989, pp do:0.06/ (89) [5] P.K.Gupta, Ma Moha, Problems operatos Research (Methods & Solutos), S. Chad Co., (2003) [6] H. Jug ad C. M. Kle, Optmal Ivetory Polces uder Decreasg Cost Fuctos va Geometrc Programmg, Europea Joural of Operatoal Research, Vol. 32, No. 3, 200, pp do:0.06/s (00)0068-5

7 Mult objectve fuzzy vetory model 93 [7] K. A. M. Kotb ad H. A. Fergay, Mult-Item EOQ Model wth Varyg Holdg cost :A Geometrc Programmg Approach, Iteratoal Mathematcal Fo-rum, Vol. 6, No. 23, 20, pp [8] Kotb A.M. Kotb, Hala A.Fergacy Mult Item EOQ Model wth Both Demad-Depedet Ut Cost ad varyg Leadtme va Geomrtrc programmg Appled mathematcs,20,2, [9] N. K. Madal, T. K. Roy ad M. Mat, Ivetory Model of Deterorated Items wth a Costrats: A Geometrc Programmg Approach, Europea Joural of Opera-toal Research, Vol. 73, No., 2006, pp do:0.06/j.ejor [0] J. T. Teg ad H. L. Yag, Determstc Ivetory Lot-Sze Models wth Tme-Varyg Demad ad Cost uder Geeralzed Holdg Costs, Iformato ad Ma-agemet Sceces, Vol. 8, No. 2, 2007, pp [] L.A. Zadeh Fuzzy sets, Iform. ad Cotrol (965) Receved: October 5, 203