A fuzzified Industrial warehouse inventory model for deteriorating items with decreasing demand and various fuzzy cost parameters

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1 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 A fuzzified Industrial warehouse inventory model for deteriorating items with dereasing demand and various fuzzy ost parameters Parag Dutta,M.S Mathematis(Gold Medalist), Purai De Choudhury,M.S Mathematis Department of Mathematis, Assam University, Silhar, parag0611@gmail.om,spurai91@gmail.om Astrat: In this researh study, an attempt has een made to develop an industrial warehouse inventory model onsidering demand as exponentially dereasing funtion of time. Shortages are not allowed in the model and fuzziness is introdued in the system y assuming the various ost omponents(holding ost,ordering ost,purhase ost,deterioration ost).in the fuzzy environment ost parameters are taken to e triangular fuzzy numers.the purpose is to minimize the total ost assoiated with the inventory system.a numerial example is given to illustrate the model approximately. Keywords: Inventory model, Deterioration, Dereasing demand, Triangular fuzzy numer, Holding ost, Defuzzifiation. Introdution: Demand has always een the prime issue in dealing an inventory system.sometimes the demand of the items may e proailisti in nature, sometimes it may e stati i.e, onstant for eah time period.further it may follow the pattern of inreasing and dereasing type or onstant type.so, due to the hange in the market senario demand varies from time to time.seasonal fator is one of the major things on whih the demand depends.for an example when winter approahes Room Heater,Woolen lothes et are of high demand and at the end of winter season demand of these items dereases.in this ontext oth inreasing and dereasing demands are oming in to the piture.but in an inventory model not only demand ut also different ost fators play a vital role as ost parameters(holding ost,ordering ost et) are known and have definite value with amiguity.some of the usiness situation fit suh onditions ut in most of the ases due to the hange in market senario these parameters are impreise.this unertainty onept an e defined as fuzziness or vagueness.the industrial authority have to deide the quantity to e manufatured.also deterioration fator must e taken in to onsideration as so many physial goods are there whih deteriorate during the stok in periods due to different fators like dryness,rusting of iron,damage,spoilage and vaporization.so onsidering all the fators an inventory model has to e prepared so that the total ost assoiated with the system is minimum and profit is maximum. During the last few deades innumerale numers of inventory model have een prepared. Ghare and Shrader(1963) developed for the first time an inventory model for deteriorating items. Convert and Philip(1973) extended their work. Hartely (1976) first proposed a prolem in his ook Operations Researh A Managerial Emphasis. Dave (1988) disussed the two-warehouse inventory models for finite and infinite rate of replenishment.donaldson (1977) developed an optimal algorithm for solving lassial no shortage inventory model.benkherouf (1997) presented a two-warehouse model for deteriorating items with the general form of time dependent demand under ontinuous release fashion. Lee and Ma (2000) developed a no-shortage inventory model for perishale items with free form of time dependent demand and fixed planning horizon. In their model, some yles are of single warehouse system and the remaining is of two-warehouse system.on the other hand, onsidering two-storage failities, Yang (2004) developed two inventory models for deteriorating items with uniform demand rate and ompletely aklogged shortages under inflation. Reently, Yang (2006) extended the models introdued in Yang (2004) y inorporating the partially aklogged shortages.de Choudhury.P and Dutta.P (2015) developed a two warehouse inventory model onsidering demand as ui funtion of time. Also De Choudhury.P and Dutta.P(2015) have fuzzified the same model. The onept of fuzzy logi was first proposed y Zadeh(1965).Bellam and Zadeh(1970) disussed the differene etween randomness and fuzziness. Zimmermann(1985) gave a review on appliations of fuzzy set theory. Park (1987) disussed the EOQ model in whih trapezoidal fuzzy numers are used. Yao and Lee(1999) presented a fuzzy inventory model with and without akorder for fuzzy order quantity with trapezoidal fuzzy numer. As fuzziness means vagueness,ertain parameters like various ost parameters are not always measuraly properly,so we have assumed these parameters as fuzzy numer in fuzzy system. A omparative study etween risp and fuzzy system has een highlighted properly.in the urrent researh study, an inventory model has een prepared onsidering demand as dereasing funtion of time.fuzzifiation is allowed in the system onsidering the ost parameters as triangular fuzzy numers.signed distane method has een used for defuzzifiation. 16

2 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 PRELIMINARIES: FUZZY SET A fuzzy set A in a universe of disourse X is defined as the set of pairs A = {(x, μ A (x)): x X}, where μ A (x):x [0,1] is a mapping and μ A (x) is alled memership funtion of A or grade of memership of x in A. CONVEX FUZZY SET A fuzzy set A in a universe of disourse is alled onvex if for all x 1, x 2 X, μ A (δx 1 + (1 δ)x 2 min{μ A (x 1 ), μ A (x 2 )}, where δ [0,1]. NORMAL FUZZY SET A fuzzy set A is alled normal fuzzy set if there exists at least one x X suh that μ A (x 1 ) = 1. FUZZY NUMBER A fuzzy numer is a speial ase of a fuzzy set. Different definitions and properties of fuzzy numers are enountered in the literature.but it atually represents the notation of a set of real numers loser to a where a is the numer eing fuzzified. A fuzzy numer is a fuzzy set whih is oth onvex and normal. TRIANGULAR FUZZY NUMBER (TFN) A triangular fuzzy numer A is represented y the triplet (a 1, a 2, a 3 ) and is defined y its ontinuous memership funtion where μ A (x):x [0,1] is given y x a 1 if a a 2 a 1 x a if x = a μ A (x) = f(x) = 2 a 3 x if a a 3 a 2 x a 3 2 { 0 otherwise Assumptions and Notation: The following assumptions have een used in developing the model: Replenishment rate is infinite. Lead time is onstant. Shortages are not allowed in the system. Costs are onsidered as triangular fuzzy numers. Model is formulated oth in Crisp and Fuzzy system. 17

3 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 A single item is onsidered over the presried period T units of time, whih is sujet to variale demand rate. Model is onsidered for imperfet items. Deterioration rate is onstant. Demand is assumed as D = Ae t, dereases with time. A > 0, > 0. t (Demand Vs Time graph, demand is of dereasing type) t The following notations have een used in the present model : T=total yle length. I(t)=Inventory level in the Industrial warehouse. h=holding ost in Industrial warehouse per unit per unit time. =Deterioration rate. C d =Deterioration ost per unit. O =Ordering ost per unit. C p =Purhase ost per unit. Q =Ordering quantity. t 1 =time at whih the inventory level falls to zero. C h =Inventory holding ost. 18

4 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 Mathematial Model: Let Q e the total amount of inventory purhased at the eginning of eah period. Now the demand may e a dereasing funtion of time ut still the inventory level is falling during [0, t 1 ]due to the demand an deterioration and vanishes ompletely at t = t 1 Let I(t) e the on-hand inventory level at time t. The differential equations assoiated with the system is The solution of the equation is given y, I(t) = Ae t. Inventory Holding ost, C h = i.e, C h = t 1 0 Ah ( ) hi(t)dt e t1 Deterioration ost= C d Aet1 t1.e t et 1 t1 t 1 0 = C dah I(t)dt e t1 di dt + I = Ae t, I(t 1 ) = 0 (1) et 1 t (2) + 1 ]... (3) + 1 ]... (4) Ordering ost=o Purhase ost= C p Aet 1 t1. ] (5) Total ost of the system= 1 T [C h + C d + O + Purhase Cost] = 1 { Ah e t1 et 1 t1 T ( ) The total ost per unit time is minimum if + 1 ] + C dah 19 e t1 et 1 t1 C p Aet 1 t1. ]}... (6) TC t 1 = ] + O +

5 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 Fuzzy model: By using signed distane method we have solved the model in fuzzy environment. We have used triangular fuzzy numer for holding osts, deterioration ost, ordering ost, purhase ost. (i) h [h 1, h + 2 ], 0 < 1 < h, 0 < 1 2 (ii) C d [C d 3, C d + 4 ], 0 < 3 < C d, 0 < 3 4 (iii) O [O 5, O + 6 ], 0 < 5 < O, 0 < 5 6 (iv) C p [C p 7, C p + 8 ], 0 < 7 < C p, 0 < 7 8 The signed distane method of the aove fuzzy numers are as (i) d(h, 0) = h + 1 ( ) (ii) d(c d, 0) = C d + 1 ( ) (iii) d(o, 0) = O + 1 ( ) (iv) d(c p, 0) = C p + 1 ( ) Now, TC = (TC 1, TC 2, TC 3 ) TC 1 = 1 T {A(h 1) ( ) e t1 et 1 t 1 + (C p 7 ) Aet 1 t 1. ]} + 1 ] + (C d 3 )A(h 1 ) e t 1 et 1 t ] + (O 5 ) TC 2 = TC TC 3 = 1 T {A(h + 2) ( ) e t1 et 1 t 1 + (C p + 8 ) Aet 1 t 1. ]} + 1 ] + (C d + 4 )A(h + 2 ) e t 1 et 1 t ] + (O + 6 ) The total inventory ost per unit time y signed distane method is d(tc ) = TC + 1 T {A( 2 1 ) e t 1 et 1 t1 ( ) 7 ) Aet 1 t1. ]}...(7) + 1 ] + ( 4 3 )A( 2 1 ) e t 1 et 1 t1 + 1 ] + ( 6 5 ) + ( 8 Numerial Example: In order to illustrate the aove system of equations onneting total ost of the system, onsider an inventory system with the following data and ompute in oth risp and fuzzy system. 20

6 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 Crisp Model: Consider = 0.05, h = 0.4, C d = 0.5, C p = 0.6, O = 2000 then t 1 = 1.2 and TC = Fuzzy Model: Consider = 0.05, h = (0.3, 0.4, 0.6), C d = (0.4,0.5,0.7), C p = (0.5,0.6,0.8), O = (1900,2000,2200) then t 1 = 1.15 and TC = Conlusion: We have developed an industrial warehouse inventory model for deteriorating items having time varying exponentially dereasing demand.the model has een formulated with the pratial assumption of demand rate for seasonal produts.more preisely during the end of partiular seasons like winter the demand rate of room heaters,woolen lothes, et dereases so taking in to aount these quantities the system has een solved.also deterioration of produts is a natural phenomenon as almost all the produts undergo deay during the ourse of time so deterioration fator has played a vital role in the inventory model.the model we have solved is highlighted in oth risp and fuzzy system.as ost parameters are impreise and sometimes it is not possile to get the suitale result so fuzziness is used.well known triangular memership funtion is used for all the fuzzy numers.signed distane defuzzifiation method is also used to formulate the resulting equations in fuzzy system. Aknowledgement: The authors are thankful to eah other.they would also like to thank other researhers regarding this topi aross the gloe. REFERENCES: [1] Benkherouf, L A. (1997), A deterministi order level inventory model for deteriorating items with two storage failities, International Journal of Prodution Eonomis, 48, [2] Bellman, E, and Zadeh, L.A (1970),Deesion making in a fuzzy environment.management Siene17 (4):B141-B164. [3] Covert,R.P and Philip,G.P(1973), An EOQ model for items with Weiull distriution deterioration,aiie Trans,5(4), [4] De Choudhury.P and Dutta.P(2015), A Two Warehouse Inventory Model for Deteriorating Items with Cui Demand, Quadrati Holding Cost and Variale Baklogging Rate, IJAENT,Volume-2 Issue-10, Septemer [5] Dave, U. (1988), On the EOQ models with two levels of storage, Opsearh, 25. [6] Donaldson W.A. (1977), Inventory replenishment poliy for a linear trend in demand-an analytial solution, Operational Researh Quarterly, 28, [7] Dutta P, De Choudhury P(2015): A Fuzzy ased Two Warehouse Inventory model for deteriorating items with ui demand and different fuzzy ost parameters, International Journal of Engineering Researh and General Siene,Volume 3, Issue 5, Septemer-Otoer, [8] Ghare, P.M and Shrader,G.P (1963) A model for exponentially deaying inventory, Journal of Industrial Engineering(J.I.E),14, [9] Hartely, R. V. (1976), Operations Researh-a managerial emphasis, Goodyear pulishing Company, [10] Lee, C. C. and Ma, C. Y. (2000), Optimal inventory poliy for deteriorating items with two warehouse and time dependent demands, Prodution Planning And Control, 7, [11] Park, K.S (1987),Fuzzy set theoreti interpretation of eonomi order quantity, IEEE Transations on systems, Man and Cyernetis,17(6), [12] Wu, K.S, Ouyang, L.Y.and Yang, C.T. (2006), An optimal Replenishment poliy for non-instantaneous deteriorating items with stok dependent demand and partial aklogging, I.J.P.E, [13] Yang, H.L (2004). Two-warehouse inventory models for deteriorating items with shortages under inflation, European Journal of Operational Researh, 157, [14] Yang, H.L (2006). Two-warehouse partial aklogging inventory models for deteriorating items under inflation, International Journal of Prodution Eonomis, 103,

7 International Journal of Engineering Researh and General Siene Volume 5, Issue 2, Marh-April, 2017 [15] Yao J.S and Lee H.M,(1999) Fuzzy inventory with or without akorder for fuzzy order quantity with trapezoidal fuzzy numer. Fuzzy sets and Systems,105, [16] Zadeh,L.A(1965).Fuzzy sets,information and ontrol,8, [17] Zimmermann, H.J (1991).Fuzzy set theory and its appliations,kluwer Aademi Press: Dordreht 22