FUZZY INVENTORY MODEL WITH SINGLE ITEM UNDER TIME DEPENDENT DEMAND AND HOLDING COST.

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1 Int J of Intelligent omputing and pplied Sciences 5 FZZY INVENORY MODEL WIH SINGLE IEM NDER IME DEPENDEN DEMND ND HOLDING OS S Barik SKPaikray S Misra K Misra orresponding autor: odma@driemsacin bstract: e objective of tis model is to discuss te inventory model for time varying demand and time dependent olding cost Matematical model as been developed for determining te optimal order quantity te optimal cycle time and optimal total inventory cost in fuzzy environment For defuzzification graded unit preference integration metod is used Numerical examples are given to validate te proposed model Sensitivity analysis as been carried out to analyze te effect of canges in te optimal solution wit respect to te canges in various parameters Keywords: Fuzzy Inventory system time dependent demand olding cost Introduction Stock administration is utilized to minimize te stock conveying cost In conventional EO model te demand rate is tougt to be consistent In real life it is as often as possible watced tat interest for a specific item can be impacted by inner elements suc as price time and availability e adjustment in te interest because of stock or advertising coices relies upon interest flexibility In tis manner wen te interest rate is steady te impact of variability of te olding expense of te total inventory cost functions of suc models as likewise been considered Different models ave been proposed for consistent interest rate wit steady olding cost eng et al [7] built up an EO model on ideal evaluating and requesting arrangement under permissible deferral in installments by expecting tat te offering cost is essentially iger tan te purcase cost ey built up a suitable model for a retailer to establis its ideal cost and lot size simultaneously wen te supplier offered an admissible postponement in installment Mulemann and Valtis-Spanopoulos [] examined te consistent rate EO model but wit variable olding cost expressed as a percentage of te average value of capital investigated in stock Vander Veen [9] exibited an EO stock framework wit te olding cost as a nonlinear function of stock Weiss [] examined conventional EO model wit te olding cost per unit adjusted as a nonlinear function of te lengt over time for wic an item was eld in stock Go [] presented an EO model wit general request and olding cost capacity and interest rate for an item was considered a function of existing stock level and conveying cost per unit was permitted to cange Fuzzy set teory as been applied to inventory problems to andle te uncertainties related to te demand or cost coefficients n extended review of te application of te fuzzy set teory in inventory management can be found in [7] e advantage of using te fuzzy set teory in modeling te inventory problems is its ability to quantify vagueness and imprecision In certain situations uncertainties are due to fuzziness primarily introduced by Zade[] is applicable In 97 Zade et al[7] proposed some strategies for decision making in fuzzy environment Jain[] worked on decision making in te presence of fuzzy variables Kacpryzk et al[] discussed some long-term inventory policy-making troug fuzzy-decision making models Wide applications of fuzzy set teory can be found in Zimmerman[8] and Park[5] In basic EO model we identify te order size tat minimizes te sum of annual costs of inventory olding and fixed setup to place orders In tis model tere are some assumptions: e demand is known fixed and independent uantity discounts are not allowed Inventory replenisment is instantaneous Only variable costs are setup cost and inventory olding cost opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

2 Int J of Intelligent omputing and pplied Sciences 55 No safety stock us EO model serves as a useful approximation to many real life problems In literature tere are many papers on fuzzified problems of EO model rgeletti [8] treated EO model in fuzzy sense and used triangular fuzzy number en and Wang[] used trapezoidal fuzzy number to fuzzify te order cost inventory cost and backorder cost in te total cost of inventory model witout backorder en tey found te estimate of te total cost in te fuzzy sense by functional principle Kao and Hsu [] considered a single-period inventory model wit fuzzy demand Hsie[] analyzed some production inventory models in fuzzy sense and e proposed some optimal strategies Syed & ziz [] used trapezoidal fuzzy number De and Rawat [5] proposed an EO model witout sortage cost by using triangular fuzzy number e total cost as been computed by using signed distance metod Fig : e rapezoidal Fuzzy In te proposed study a fuzzy inventory model as been developed were we consider te demand rate is time varying and olding cost is constant e main objective of tis paper is to obtain minimum total inventory cost order quantity and corresponding order cycle n algoritm tat minimizes te total inventory cost is developed Numerical examples are discussed to illustrate te procedure of solving te model e proposed model is developed in bot te crisp and fuzzy environments In fuzzy environment te related inventory parameters ie te inventory olding cost and ordering cost are fuzzified as te trapezoidal fuzzy numbers and ten apply te Graded Mean Integration Representation metod for defuzzification e objective is to obtain fuzzy optimal solution to minimize te total cost per time unit of an inventory control system based on te fuzzy aritmetical operations under Function Principle Definition and Principles Suppose is a generalized fuzzy number as sown in Figure and is described as any fuzzy subset of te real line R wose membersip function satisfies te following conditions x is a continuous mapping from R to te closed interval [ ] x x a μ is strictly increasing on ] x L x x a x a 5 x R x [ a a is strictly increasing on ] [ a a opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

3 Int J of Intelligent omputing and pplied Sciences 5 x a x a a and a a were are real numbers Fig : Generalised Fuzzy Number In 998 en and Hsie [ ] propose graded mean integration representation for representing generalized fuzzy number Now we describe graded mean integration representation GMIR as follows Suppose L and R are inverse functions of functions L and R respectively and te graded mean -level value of generalized fuzzy number c a b d : w is [ L R ]/ LR as Figure: en te graded mean integration representation of generalized fuzzy number based on te integral value of graded mean -level is w w L R d / P d were is between and w and w fuzzy number are denoted as c a b d : w and c a d : w Generalized trapezoidal fuzzy number and generalized triangular respectively en and Hsie [] already find te general formulae of te representation of generalized trapezoidal fuzzy number or generalized triangular fuzzy number as follows Suppose c a b d : w is a trapezoidal fuzzy number Since x c x d L x w c x a and R x w b x d a c b d en L c a c / w w and L R R c d a c d b / w By formula te graded mean integration representation of is w d d b / w w w P c d a c d b / w d / d e fuzzy aritmetical operations under Function Principle Here we describe some fuzzy aritmetical operations under Function Principle as follows opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

4 Int J of Intelligent omputing and pplied Sciences 57 and B b b b Suppose a a a a e addition of and B is B a b a b a b a b a a and b b a b a b b were are any real numbers e multiplication of and B is B c c c c Were a b a b a b a b } { a b a b a b a } { b c min c min c max c max are two trapezoidal fuzzy numbers en a b a b a b a and b B ab ab ab ab Were B is a trapezoidal fuzzy number B b b b b ten te subtraction of and B is B a b a b a b a lso if are non zero positive real numbers ten b were a b a b a b a and b are any real numbers / B B / b / b / b / b were b b b and b are all positive real numbers If a b a b a b a and b are all positive real numbers ten te division of and B B a / b a / b a / b a / b Let R ten i a a a a ii a a a a Example: Suppose and B are two trapezoidal fuzzy numbers and 5 B 57 B B 5 Ø B 5 B 5755 en is Ø ssumptions and Notations Following assumptions are made for te proposed model: Single inventory will be used Lead time is zero e model is studied wen sortages are not allowed e demand rate Rt is decreasing function of time wit increase of e olding cost is constant Following notations are made for te given model: opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

5 Int J of Intelligent omputing and pplied Sciences 58 I t = On and inventory level at any time t t =e lengt of cycle time = e ordering cost per unit time = e constant annual demand rate t = e time dependent olding cost R t = e time varying demand rate ie R t t Here parameter of te demand curve otal inventory cost per cycle Formulation Ordering quantity is te demand Let I t be te on-and inventory level at any time t e demand rate is assumed to be positive in its entire domain e amount of stock depletes in te period [ ] due to te effect of demand By tis process te stock reaces zero at time Hence te inventory level at any instant of time t is described as follows t time t t te on-and inventory in te interval will be I t t I t d t t Dividing by t and ten taking as t we get di t t dt Wit te condition I ; t e solution of te differential equation is given by I t t 5 Now is te ordering quantity of stock wic is given by were From we obtain 7 Now te average total cost per cycle is given by opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

6 Int J of Intelligent omputing and pplied Sciences 59 opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7 λ dt t λ dt t I Holding t Ordering ost cos 8 sing 7 we obtain λ λ 9 e necessary condition for minimization of is 9 e sufficient condition for minimization of is for Now te function will be maximum if Now from 9 we ave Now on solving implies minimizing total cost to determine optimal given by and ence te optimal cost can be evaluated If ten te optimal and Fuzzy Model and Solution Procedure We consider te model in fuzzy environment Due to fuzziness it is not easy to define

7 Int J of Intelligent omputing and pplied Sciences opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7 all te parameters precisely We use te following variables : fuzzy Ordering cost c : fuzzy carrying cost Suppose a a a a c are nonnegative trapezoidal fuzzy numbers e total average cost per unit time is given by c 5 were Now Now on defuzzifying te fuzzy total average cost we ave P 7 o minimize te average total cost per unit time te optimal value of can be obtained by solving te following equation d P d 8 were λ us minimum value of te total cost denoted by 9 5 omputational lgoritm: Step-: Start Step-: Initialize te value of te variables Step-: Evaluate

8 Int J of Intelligent omputing and pplied Sciences Step-: Evaluate Step-5: Solve te equation Step-: oose te solution from Step-5 Step-7: Evaluate Step-8: If te value of Step-7 is greater tan zero ten tis solution is optimal minimum and go to Step- Step-9: Oterwise go to Step- Step-: End Numerical Examples o illustrate te proposed metod let us consider te following input data risp Model: e values of te parameters in proper units are considered as follows: 5 Optimal Fuzzy Model: We can apply te fuzzy inventory model wit fuzzy order quantity to find te optimal fuzzy total average cost First we represent te case of vague value as te type of trapezoidal fuzzy number Suppose a a a a c 5 Equation 5 can be minimized by using MLB Software to determine optimal & e optimal ordering quantity average cost and time are found to be Sensitivity nalysis BLE : opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

9 Int J of Intelligent omputing and pplied Sciences BLE : BLE : Important points from te table e effect of optimality due to cange of values of different parameters associated in tis model is discussed below & increase wile decreases wit increase in value of te parameter increases wile & decrease wit increase in value of te parameter & increase wit increase in value of te parameter 7 Variation of ime duration ordering uantity and otal cost wrt different parameters 7 Variation of ime duration % -% % % % opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

10 Int J of Intelligent omputing and pplied Sciences 7 Variation of Ordering uantity % -% % % % 7Variation of otal cost 7 5 -% -% % % % 8 onclusion is paper presented a fuzzy inventory control model for time varying demand and time dependent olding cost respectively e proposed model is developed in bot te crisp and fuzzy environments In fuzzy environment all related inventory parameters were assumed to be trapezoidal fuzzy numbers e optimum results of fuzzy model are defuzzified using graded mean = level integration representation metod So te decision maker after analyzing te result can plan for te optimal value for te related parameters e model can furter be studied for sortage state and for multiple items under identical conditions is can also be extended for deterioration conditions and also for discounted cas flow approac opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7

11 Int J of Intelligent omputing and pplied Sciences References [] ang S Fuzzy production inventory for fuzzy product quantity wit triangular fuzzy number Fuzzy Sets and Systems [] en S H Operations on Fuzzy Numbers wit Function Principle amkang Journal of Management Sciences 985 [] enwang Backorder fuzzy inventory model under function principle Information Science [] en S H and Hsie H Optimization of fuzzy simple inventory models999 IEEE International Fuzzy System onference Proceedings 999 Seoul Korea [5] De PK Rawat fuzzy inventory model witout sortages using triangular fuzzy number Fuzzy Information & Engineering 59-8 [] Go M 99 EO model wit general demand and olding cost function European Journal of Operational Researc [7] Guiffrida L Fuzzy inventory models in: Inventory Management: Non-lassical Views apter 8 MY Jaber EdR Press FL Boca Raton pp 7-9 [8] Hadley G Witin M nalysis of inventory systems Prentice-Hall Englewood clipps NJ 9 [9] Harris F Operations and cost W Saw o icago 95 [] Hsie H Optimization of Fuzzy Production Inventory Models Information Sciences - 9- [] Jain R Decision making in te presence of fuzzy variables IIIE ransactions on systems Man and ybernetics [] Kacpryzk J Staniewski P Long-term inventory policy-making troug fuzzy-decision making models Fuzzy Sets and Systems [] Kao K Hsu WK single-period inventory model wit fuzzy demand omputers and Matematics wit pplications 8-88 [] Mulemann P and Valtis-Spanopoulos NP 98 variable olding cost rate EO model European Journal of Operational Researc - 5 [5] Park KS Fuzzy Set eoretical Interpretation of economic order quantity IEEE rans Systems Man ybernet SM [] Syed JK ziz L Fuzzy inventory model witout sortages using signed distance metod pplied Matematics & Information Sciences 7-9 [7] eng J ang and Goyal SK 5 Optimal pricing and ordering policy under permissible delay in payments International Journal of Production Economics 97-9 [8] rgeletti inarelli G Inventory control models and problems European Journal of Operational Researc 98 - [9] Van der Veen B 97 Introduction to te teory of operational Researc Pilips ecnical Library Springer- Verlag New York [] Weiss HJ 98 Economic order quantity models wit non linear olding cost European Journal of Operational Researc 9 5- [] Wilson R scientific routine for stock control Harvard Business Review 9 8 [] Zade L Fuzzy sets Information ontrol [] Zade L Bellman RE Decision Making in a Fuzzy Environment Management Science [] Zimmerman HJ sing fuzzy sets in operational Researc European Journal of Operational Researc 98 - opyrigt DRIEMS ISSN Print: - Vol 5 Issue 7