Fuzzy Inventory Optimization in a Two Stage Supply Chain with Full Information Sharing and two Backorder Costs

Transcription

1 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: Fuzz Inentor Optimization in a wo Stage Suppl Cain wit Full Information Saring and two Backorder Cot S. Gajalakmi.arati Head & ociate rofeorepartment of MatematicQuaid- E -Millat Goernment College for Women utonomoucennai Reearc Scolarepartment of Matematic Quaid- E MillatGoernment College for Women utonomoucennai tract Nowada a new pirit of cooperation require etween te uer and te endor in uppl cain enironment. We conider te cae of two tage uppl cain tem wic inole a ingle endor wo upplie a ingle product to a ingle uer. In order to coordinate te repleniment policie and optimizing te operational cot te two uppl cain partner full are teir information. For ti purpoe we deelop te propoed model auming linear and fied ack order cot. In addition et up cot and olding cot are taken a fuzz triangular numer. Signed ditance metod i ued for defuzzification. Numerical eample illutrate te procedure of te propoed model. eword:wo tage uppl cain two ackorder cot fuzz inentor model riangular fuzz numer defuzzification.. Introduction: e inetigation [] on effect in uppl cain reported tat lack of information aring can lead eceie inentor poor cutomer erice lotreenueand unplanned capacitie. e paper deeloped Seliaman a two tageuppl cain inentor model under two tpe of ackorder cot. i model i te etenion of [] wic conidered a ingle endor and a ingle uer. i propoed model we deelop an inentor model uing triangular fuzz numer for olding cot ordering cot and etup cot. Signed ditance metod i ued for defuzzification. ue to irregularitie or pical propertie of te material all te time we cannot take parameter a ariale. For ti ituation we appl fuzz concept.we introduce two ackorder cot and minimizing te total cot. eremainder of ti paper i organized a Metodolog e ignificant adance in information and communication tecnologie facilitated te proiion and aring of te uine information necear for efficienc improement. Goal and Szendroit [] preented a contant lot ize model were te lot i produced troug a fied equence of manufacturing tage wit a ingle et up and witout interruption. one endor multi-uer integrated inentor model wa deeloped in [] wit te ojectie of minimizing te endor total annual cot uject to te maimum cot tat te uer ma e prepared to incur. model. Notation and umption Model deelopment Fuzz ene Numerical eample illutrate te propoed 05 IRJE ISO 900:008 Certified Journal age 9

2 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: Metodolog.. Fuzz Numer n fuzz uet of te real line R woe memerip function µ atified te following condition i a generalized fuzz numer. i µ i a continuou mapping from R to te cloed interal [0 ] a a a a a a a a a a 0 oterwie ii µ = 0 a iii µ = L i trictl increaing on [a a ] a a i µ = w µ = R i trictl decreaing on [a a ] i µ = 0 a Were 0 <w and a a a and a are real numer. lo ti tpe of generalized fuzz numer e denoted a a a a a; w LR; wen w = it can e a a a a ; w LR implified a... riangular fuzz numer were a < a < a e fuzz et a a a and defined on R i called te triangular fuzz numer if te memerip function of i gien.. e Function rinciple e function principle wa introduced Cen [6] to treat fuzz aritmetical operation. i principle i ued for te operation for addition utraction multiplication and diiion of fuzz numer. and B Suppoe a a a two triangular fuzz numer. en i e addition of and B i are B a a a were a a a are an real numer. ii e multiplication of and B i B c c c Were a a a a c min c a c if a a a are all non zero poitie real numer ten B a a a. iii B ten te utraction of B from i B a a a were a a a are an real numer. ma 05 IRJE ISO 900:008 Certified Journal age 9

3 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: i B were are all non B zero poitie real numer ten te diiion of B a i a a and B For an real numer a a a if 0 a a a if 0.. Signed itance Metod endor uer -Fuzz ordering cot of te uer -Fuzz inentor olding cot for te -Fuzz inentor olding cot for te -ackordering cot per unit per unit time linear for te uer -ackorder cot per unit fied for te uer -fuzz ackordering cot per unit per unit time linear for te uer efuzzification of can e found igned ditance metod. If i a triangular fuzz numer and i full determined a a a te igned ditance from to 0 i defined a uer -fuzz ackorder cot per unit fied for te C otal cot for te period [0 ] C - Fuzz total cot for te period [0 ] d 0 d 0. Notation L R a 0 d a -aic ccle time ccle time at te end uer -roduction rate of te endor -integer multiplier at endor -Setup cot of te endor -Ordering cot of te uer -emand rate for te uer a. umption a e model deal wit a ingle endor and a ingle uer for a ingle product. Repleniment i intantaneou c roduction rate and demand rate are determinitic and uniform d Complete information aring polic i adopted e ere are two tpeof ackorder cot f Holding cot ordering cot and etup cot are fuzz in nature endor uer -inentor olding cot per unit time for te -inentor olding cot per unit time for te -te tock- out time at te uer -fuzz etup cot of te endor 05 IRJE ISO 900:008 Certified Journal age 95

4 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: IRJE ISO 900:008 Certified Journal age 96 Figure : te uer inentor leel eaior 5. Model eelopment We conider a two tage uppl cain coniting of a ingle endor and a ingle uer. i uppl cain model i formulated for te integer multiplier inentor repleniment coordination metod were te ccle time at te endor i an integer multiple of te inentor repleniment ccle time ued te uer. e total annual cot for te uer i gien C e total cot for te endor i C B adding te total cot for te uer and total cot for te endor we get te total cot for te entire uppl cain. i.e. C = C + C C 0 C See appendi Second order deriatie i poitie. So total cot function i cone in. Equating te firt order deriatie to 0 we get te alue of Sutituting te aoe in we get te total cot te equation C 5

5 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: IRJE ISO 900:008 Certified Journal age Inentor model in Fuzz ene We conider te model in fuzz enironment. Since te ordering cot and olding cot are fuzz in nature we repreent tem triangular fuzz numer. e fuzz total cot i gien C 6 were O rdering cot et up cot and olding cot for ot uer and endor e triangular fuzz numer Sutituting tee triangular fuzz numer to te total cot equation we get C 7 B uing aritmetic function principle in 7 and implifing we get C =[a c ] a 8 Finding left and rigt α-cut for te fuzz total cot equation we get Left α-cut L = a + a

6 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: IRJE ISO 900:008 Certified Journal age 98 9 Rigt α-cut R = c - c 0 efuzzifing te fuzz total cot C uing igned ditance metod we get 0 ] [ 0 d H C d R L

7 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: IRJE ISO 900:008 Certified Journal age 99 0 Integrating wit repect to α and implifing we get defuzzified alue for total cot. i.e. C Finding firt and econd order differentiation we get 0 C wic i potie See appendi B otal cot C wit repect to i cone. B equating firt order deriatie equal to 0 we get te optimum time we get 8

8 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: Sutituting in total cot equation we get C 7. lgoritm Step.Input in Equation. Step.Find. Step.Sutitute in equation 5 we get otal Cot for crip alue. Step.Input fuzz alue for ordering cot et up cot and olding cot in equation Step 5.Find. Step 6.Sutitute in equation we get total cot for fuzz alue. Step 7.Find difference etween ccle time and total cot found in tep 6 and tep. Step 8.Repeat tee tep canging different demand rate to gie enitiit anali for te propoed model 8. Concluion wo tage uppl cain wit endor uer coordination and cooperation i conidered in ti model. Information aring etween endor and uer i necear for te moot running of a trade and i uitale for te real life ituation. Here olding cot for ot uer and endor et up cot for te endor and ordering cot for te uer are taken a triangular fuzz numer. B taking ti conideration we minimize te total cot of te uppl cain muc etter tan in our ae paper. i i illutrated te numerical eample. 9.Numerical eample: Numerical eample are gien to mention te workailit of te propoed model. Computational work are carried out uing MLB 7.0. For gien =500; =.5; = 0.; =9; = 00; = ; = 0.; = 0.; = 0; en and otal cot are own in ale. For gien =500; =.5 ; = ; =8 9 0; = ; = ; = 0.; = 0.; = 0; ; 505 en defuzzified and otal cot are own in ale. From tale we ee tat a demand increae uer ccle time decreae ut te total cot increae. Since te ccle time decreae uer place more order frequentl and ence uer get more profit. If uer get more profit oioul endor alo endor alo get more profit. Een toug te total increae wit demand te uer can improe i uine placing more order to te endor frequentl. Conequentl te endor alo can deelop i trade and ence te uppl cain will work trongl. i ituation i te ame for ariou olding cot of te uer. More oer in fuzz enironment te total cot i decreaed eril for te ame cange in demand and alo for olding cot of te uer. So we can oere tat fuzz enironment i natural and eneficial for ot endor and te uer. Nutell of te paper: a. Wen demand increae graduall ten te ccle time for te proce decreae.. Wen demand increae and te ccle time decreae ten te total cot increae. c. Wen demand increae difference etween te total cot for crip and fuzz input alo increae. d. Increae of olding cot for uer lead to increae of total cot. e. ecreae of olding cot for uer lead to decreae of total cot. 05 IRJE ISO 900:008 Certified Journal age 90

9 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: ale : Crip model wit different demand rate emand C ale : Fuzz model wit different demand rateutituting triangular fuzz numer of ordering cot olding cot etup cot ackorder cot fied & linear emand C IRJE ISO 900:008 Certified Journal age 9

10 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: ale a: Fuzz model wit different demand rate utituting defuzzified alue of ordering cot olding cot etup cot ackorder cot fied & linear emand C ale : ifference etween crip and fuzz model emand C IRJE ISO 900:008 Certified Journal age 9

11 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: ale : ifferent olding cot for uer crip emand Setup cot Oedering cot Holding cot uer Holding cot endor C ale 5: different olding cot uer fuzz emand Setup cot Ordering cot Holding cot uer Holding cot endor C IRJE ISO 900:008 Certified Journal age 9

12 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: ale 6: ifference etween crip and fuzz model aring olding cot for uer emand C Figure: Figure : crip input Figure : Input defuzzified alue Figure : Input fuzz alue and after defuzzification 05 IRJE ISO 900:008 Certified Journal age 9

13 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: IRJE ISO 900:008 Certified Journal age 95

14 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: IRJE ISO 900:008 Certified Journal age 96 ppendi C C C 0 C ppendi B C C C 0 C

15 International Reearc Journal of Engineering and ecnolog IRJE e-issn: Volume: 0 Iue: 06 Sep-05 p-issn: Reference:. S.. Goal and. Z. Szendroit contant lot ize model wit equal and unequal ized atc ipment etween production tage Engineering cot and production economic ol 0 pp L. Lu one endor multi uer integrated inentor model European journal of Operation Reearc pp H. Lee V. admanaan and S. Wang e ullwip effect in uppl cain MI Sloan Management Reiew pp. 9-0 Spring M. E. Seliaman and. mad generalized algeraic model for optimizing inentor deciion in a multi tage comple uppl cain ranportation Reearc E pp S. S Sana production inentor model of imperfect qualit product in a tree laer uppl cain eciion upport tem pp Moamed E. Seliaman Optimizing te two tage uppl cain inentor model wit full information aring and two ackorder cot uing rid geometric algeraic metod Hindawi uliing corporation. 05 IRJE ISO 900:008 Certified Journal age 97