Research Article International Journals of Advanced Research in Computer Science and Software Engineering ISSN: X (Volume-7, Issue-6)

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1 International Journals of Advanced Research in Computer Science and Software Engineering ISSN: 77-8X (Volume-7, Issue-6) Research Article June 07 A Fuzzy Inventory Model having Exponential Demand with Weibull Distribution for Non-Instantaneous Deterioration, Shortages under Partially Backlogging and Time Dependent Holding Cost Sahidul Islam, Abhishek Kanti Biswas Department of Mathematics, University of Kalyani, Nadia, West Bengal, India Department of Mathematics, R. K. M. Vivekananda Centenary College, Kolkata, West Bengal, India DOI: 0.956/ijarcsse/V7I6/08 Abstract: In this paper, we introducea deterministic Economic Order Quantity Inventory Modelwith exponential demand rate, Weibull Distribution for deterioration and time dependent holding cost in crisp and fuzzy environment.also each cycle Shortages are allowed and partially backlogged to suit present day competition in the market objectives. The backlogging rate is variable and as depends on the length of waiting time for next replacement. The costsare assumed as a Triangular Fuzzy Numbers and Graded Mean Integration Representation Methodis used to defuzzify the model. This Model aids in minimizing the total inventory cost by finding the optimal time interval in different environment. A sensitivity analysis is also given to show that the total cost function is extremely influence by the variation of the costs and other parameters. The model is illustrated with numerical example. Keywords and Phrases: Non-Instantaneous Deteriorating Inventory, Exponential demand, Weibull Distribution, Partial Backordering, Time Varying Holding Cost, FuzzyTriangular Numbers, Graded Mean Integration Representation Method. I. INTRODUCTION The inventory system takes an important part of cost controlling in business. The mathematical properties of inventory systems and details of different inventory models have been described in the well-known books ( Hardley and Whitin, 958 [0], Naddor, 96, [] ). For the last few years, researchers in this area have extended investigation into various models with considerations of demand patterns, deterioration, shortage,payment option, order cycles and their combinations. Demand is different for different types of item, it also varies for different period of time. So demand cannot be taken constant for practical use. It can be linear or quadratic function of time. The deteriorating items with shortages have received much attention of several researchers in the recent years because most of the physical goods undergo decay or deterioration over time, example being fruits, vegetables, volatile goods and so on. In some inventory systems such as fashionable commodities, the length of the waiting time for the next replenishment would determine whether the backlogging will be accepted or not. Therefore, the backlogging rate should be variable and depends on the waiting time for the next replenishment. Most of models are based on the economic order quantity (E. O. Q. )/ economic production quantity (E. P. Q.) model developed by Harris [5]. The first attempt to describe the optimal ordering policies for such items was made by Ghare and Schrader [5]. Philip [] developed an generalised inventory model with three parameter Weibull Distribution rate without considering shortages. Many researchers assume that the deterioration of the items in inventory starts from the instant of their arrival in stock. But in real-life most goods would have a span of maintaining quality or original condition, namely electronic goods, food grains etc., during that period, there is no deterioration occurring. K. S. Wu et. al [6]define the phenomenon as non-instantaneous deterioration. In a classical inventory models, the demand rate and holding cost is assumed to be constant. In reality, the demand and holding cost for physical goods may be time dependent. In this connection, see the performed works by Mishra et. al [9] and Abad []. Among the most resent investigations in this field, the studies performed Jong Wuu Wu & Wen Chuan Lee ( [], Giri et. al [5], Shah and Shukla [], Hariga [], Sharma & Chaudhary [], Tripathy and Pradhan [8], Palanivela & Uthayakumara [0] are note-worthy. In the crisp inventory models, all the parameters in the total cost are known and have definite values.but in the practical situation it is not possible, these models provide some general understating of the behaviour of inventory under different assumptions, they are not capable of representing real-life situations. So, applying these models as they are, generally, leads to erroneous decisions. Further, using these models require inventory managers to have some flexibility when deciding on the sizes of the order quantities to reduce the cost of uncertainty.hence fuzzy inventory models fulfil that gap. Using fuzzy set theory to solve inventory problems, instead of the traditional probability theory, produces more accurate results. Different fuzzy inventory models occur due to fuzzy various cost parameters in the total cost. Fuzzy set theory, introduced by Zadeh [7], has been receiving considerable attention from researchers in production and inventory management as well as in other fields. Kacprzyk and Staniewski [] investigated long-term inventory policy-making using fuzzy decision models for a multi-stage inventory planning problem. Park [7] proposed an EOQ model with the ordering and inventory holding costs being trapezoidal fuzzy numbers. Researchers related to this area are: Zimmermann [], Dutta and Kumar [9], Bellman and Zadeh [8], Islam and Roy [0], Lee and Yao [6], Yao and Su [], Maragatham and Laksmidevi [6], Zhao [7], Masatoshi Sakawa [], Bing-Yuan Cao et. al. [], Carlsson and Fuller [9],Mondal and Islam [8] are note-worthy. All Rights Reserved Page

2 ISSN: 77-8X (Volume-7, Issue-6) In the present work, we developed a fuzzy deterministic inventory model for non-instantaneous deteriorating items with exponential demand pattern proposed which the deterioration is a Weibull two parameter distribution and holding cost is expressed as linearly increasing function of time. Shortages are allowed and partially backlogged. The inventory costs are taken as Triangular Fuzzy Number. Graded mean Integration representation method is applied for defuzzification. To the author s best of knowledgesuch type of model has not yet been discussed in the inventory literature. The rest of the paper is organised as follows: In section II, some definition and preliminaries are given. The notations and assumptions are describein section III, used throughout this paper. In section IV, the mathematical model is to minimize the total inventory cost is established in different environment. Numerical example provided in section V to illustrate the theory and solution procedure. This is followed by sensitivity analysis (Section VI) and conclusion (Section VII). II. DEFINITIONS AND FUZZY PRELIMINARIES For Graded Mean Representation Method to defuzzify, we need the following definition: Definition.: Let X denotes a universal set. Then the fuzzy subset A of X is defined by its membership function μ A (x):x [0,] which assigns a real number μ A (x) in the interval [0, ],to each element x X where the value of μ A (x) at x shows the grade of membership of x. Definition. : The α cut of A is defined by A α ={x:μ A (x)=α, α 0}. Definition. : A is normal if there exists x X such that μ A (x)=. Definition. : A fuzzy set Aon R is convex if A(λx (-λ)x ) min[ A x, A(x ) ] for allx, x R and [0,]. Definition.5 : A fuzzy set in the universe of discourse X is called as a fuzzy number in the universe of discourse X Triangular fuzzy number. We consider the situation where fuzzy numbers are represented by triangular membership functions. The fuzzy number Ais said to be triangular fuzzy number if it is fully determined by (a, a, a ) of crisp numbers such that (a < a < a ) whose membership function, representing triangle, can be denoted by L x = x a, a a a x a μ A (x) = R x = a x, a a a x a 0, Oterwise When a = a = a, we have fuzzy point (a, a, a ) = a. The family of all triangular fuzzy number on R, denoted as F N = {(a, a, a ),a < a < a, a, a, a R}. The α cut of A = (a, a, a ) F N, 0 α is A(α) = [A L (α), A R (α)] where A L α = a (a a )α and A R (α) = a (a a )α are the left and right end-point of A(α)..6: The Function Principle :The function principle was introduced by Chen to treat fuzzy arithmetical operations. This principle is used for the operation for Addition, Subtraction, Multiplication anddivision of fuzzy numbers. Suppose A = a, a, a and B = (b, b, b ) are two triangular fuzzy numbers. Then_ (i) Addition: A B = (a b, a b, a b ), where a, a, a ; b, b, b are any real numbers. (ii) Subtraction: A - B = (a - b, a - b, a - b ), where a, a, a ; b, b, b are any real numbers. (iii) Multiplication: A B = (a b, a b, a b ), where a, a, a ; b, b, b are all non-zero positive real numbers. (iv)division: A B = ( a b, a b, a b ), where b, b, b are all non-zero positive real numbers. (v) Scalar Multiplication: For any real number K, KA = ( Ka, Ka, Ka ), Where K 0, KA = ( Ka, Ka, Ka ) Where K< 0, Definition.7: If A = a, a, a is triangular fuzzy number then the graded mean integration representation ofais given as P(A) = W 0 Or, P(A) = A( L h R (h ) )dh W Ahdh 0 ( a a a h a a a h )dh 0 hdh 0, (0 h W A and 0 W A ) Or, P(A) = a a a 6. Definition.8: The Jacobean of the derivatives f x, f x,..., f x n of a function f( x, x,.., x n ) with respect to x, x,.., x n is called the Hessian matrix H of f, i.e., All Rights Reserved Page 5

3 ISSN: 77-8X (Volume-7, Issue-6) As in the case of the Jacobian, the term "Hessian" unfortunately appears to be used both to refer to this matrix and to the determinant of this matrix, In the second derivative test for determining extrema of a function f( x, y, z ), the discriminant D is given by III. NOTATIONS AND ASSUMPTIONS This inventory model is developed on the basis of the following Assumptions and Notations are used throughout this paper in Crisp and Fuzzy Environment. Notations:- I(t) : The inventory level at any time t, t 0. C : The carrying cost per order per unit item. C : The storage cost for backlogged items per unit per unit time. C : The unit cost of lost sales per unit per unit time. C 5 : The purchasing cost per unit. C 6 : The Deteriorating Cost per unit per unit time. TAc: Total average cost per unit. C : Fuzzy carrying cost per order per unit item. C : Fuzzy storage cost for backlogged items per unit per unit time. C : Fuzzy unit cost of lost sales per unit per unit time. C 5 : Fuzzy purchasing cost per unit. C 6 : Fuzzy Deteriorating Cost per unit per unit time. TAc: Fuzzy total average cost per unit. L M : The maximum inventory level. L b : The maximum backordered unit during stock out period. Q ( = L M L b ) : The order quantity during a cycle of length. t : The time at when the inventory level decreases due to demand and deterioration, t > 0. t : The time at when the inventory level reaches to zero, t > 0. T : The length of cycle time. Where, T > 0. Assumptions: The rate of deterioration at any time t > 0 follows the two parameter Weibull Distribution θ(t) = αt - where α ( 0 < α < ) is the scale parameter. When =, θ(t) becomes a constant which is the case of an exponential decay. When <, the rate of deterioration is decreasing with t and when >, it is increasing with t. The demand is taken as exponential, D(t) = ae bt, where a and b < are positive constant. Shortages are allowed and during stock out period, the backlogging rate is variable and as depends on the length of waiting time for next replacement. So that, backlogging rate for negative inventory is γ(t) = δ( T t), where δ is backlogging parameter, 0<δ< and (T t) is waiting time ( t t T ) and the remaining fraction - γ(t) is lost. Holding cost is C (t) = qt, where q is a constant The replenishment takes place at an infinite rate. Lead time is zero or negligible. Deterioration takes place after the life time of items. There is no replenishment or repair of deteriorating items takes place in the given cycle. All Rights Reserved Page 6

4 ISSN: 77-8X (Volume-7, Issue-6) IV. MATHEMATICAL FORMULATION AND SOLUTION OF THE MODEL IN DIFFERENT ENVIRONMENTS Purchasing Inventory Model in Crisp Environment is developed as follow: Let us suppose that the business community purchases a certain amount of items in the inventory system at the beginning of each cycle when t = 0. During the time interval [ 0, t ], the inventory level is decreasing only due to demand rate. The inventory level is dropping to zero owing to demand and deterioration during the time interval [ t, t ]. Finally, shortage occurs due to demand and partial backlogging during the time interval [ t, T]. The behavior of inventory model is demonstrated in Figure. Figure : Graphical Representation of Proposed Inventory System Based on the above description, during the time interval [ 0, t ], the inventory status is given by the following differential equation- di(t) dt = - D (t), ( 0 t t ) () With the boundary condition, I 0 = L m, we get from above equation, L m = a b c, ( 0 t t ) () And I(t) = L m a b a b ebt, ( 0 t t ) () In the time interval [ t, t ], the inventory level decreases due to customer demand and deterioration. In this interval the inventory status is given by the following differential equation- di(t) dt θ t I(t ) = - D (t), (t t t ) () With the boundary condition I(t ) = 0, taking the first two terms of the exponential series and disregarding the higher power ofα and integrating, we get the solution of () is, I(t) = - at - ab t t - aα aαt aα bt at ab t aα t aαt aαb t, (t t t ) (5) Putting t=t in ()and (5), then we find the value of - L m = a b ebt a at b - ab t - aα t aαt aα bt And substituting (6) in (), we get, at ab t aα t aαt aαb t I(t) = a b ebt - a b ebt at - ab t - aα t aαt aα bt at ab t aα t aαt aαb t, ( 0 t t ) (7) In during the third interval ( t, T ), shortages occurs and the demand is partially backlogged. That is the inventory level at T is governed by the following Differential Equation - di(t) = D(t)γ(t), (t dt t T ) (8) With the initial condition I(t ) =0, taking the first two terms of the exponential series and neglecting the higher power of δand integrating we get the solution of (8) is - I(t)= a δ ebt log δ T t log δ T t, (t t T ) (9) The maximum backorder inventory L b is obtained at t =T, i.e. with boundary ConditionI(T) = L b, we get from the above, L b = I(T)= a δ ebt log δ T t (0) (6) All Rights Reserved Page 7

5 ISSN: 77-8X (Volume-7, Issue-6) Thus the order size during total time interval [ 0, T ] is Q = L m L b = a b ebt a at b - ab t - aα t aαt aα bt aαb t aαb t a δ ebt log δ T t () Total number of deteriorated units are, t D = θ(t)i(t)dt t Ignoring the terms containingα or higher degree of it since 0<α<, we get from above equation, D= aα[ t b t Then, t b t t b t All Rights Reserved Page 8 t t b t at ab t aα t aαt t ] (). The Caring cost during the period [ 0, T] : C (). The Inventory Holding cost during the period [ 0, T]: C (t)i(t)dt C 0 (t)i(t)dt t Using equation (7) and equation (5), we get from above, q a b ebt t t a t e bt a ebt at b b ab t aα t aαt aα bt at t abtt aαtt aαtt aαbtt qabq at abt8 aα t ()() t aα t aαb t aαb t a t ab t aα t aα t () () t t aα aα t at ab t aα t () T. The Backordering cost during the period [ 0, T] : C I(t)dt Using equation (9), we get from above, = C e bt ( δ T t t t t aαt t aαb t aαb t t ) a t ab t 8 ) (5). The Opportunity Cost due to lost sales per cycle :C ( t )D(t)dt δ T t Neglecting the higher power of δ and taking the first two terms of the series we get the solution of above equation bellow, = C a(t δt δ T T T T b δb δb t δtt δ t t b t δbt t δb ) (6) 5. The Purchase cost during the period [ 0, T]:C 5 Q Using equation (), we get from above, = C 5 [ a b ebt a at b ab t aα t T aαt aα bt aαt aαb t a δ ebt log δ T t ] (7) 6. The Deteriorating cost during the period [ 0, T]: C 6 D Using equation (), we get from above, = C 6 aα[ t b t t b t t b t t t b t at ab t aα t t ] (8) Therefore the total average cost per cycle is, TAc( t, t, T ) = [Caring cost Inventory Holding cost T Backordering cost Cost due to lost sales Purchase cost Deteriorating cost] at t aα t aαb t = T [C q a b ebt t ab t t aαb t aα t a t e b t b t a ebt at b ab t aα t aαt t aαb t t a t ab t aα t aα t () () t at ab t T T δb δb b T aαt aα bt t C 6 aα[ t b t t aαb t aαb t t aα t aαt t t δtt δ t t b t δbt t δb at ab t aα t aαt aαb t t b t t b t aαt aα bt q a b q( a t ab t 8 aα t ()() a t ab t aα t aα t 8 ) C e bt ( ) C δ T t a(t δt δ T )C5 [ a b ebt a at b ab t aα t a δ ebt log δ T t ] t t b t t ] (9)

6 ISSN: 77-8X (Volume-7, Issue-6) The necessary condition to minimize equation (9) are TA c i ( t,t,t) = 0, where r =,,. (0) t i The solution, which may be called feasible solution of the problem, of the equations(9) and (0) give the optimal solutions of t, t andt which minimize TAc ( t, t, T ) provide they satisfy the sufficient conditions H TAc t, t, T > 0 Where, () We solve the non-linear equation (9) and (0) by using computer software LINGO 6.0 and optimal values of t, t and T can be found respectively. Purchasing Inventory Model in Fuzzy Environment is developed as follow: We consider the model in fuzzy environment due to uncertainly. Since all costs in the inventory model are in fuzzy nature, we represent them in by triangular fuzzy number. Let us assume that, C = (C, C, C ),q = q, q, q ; C = (C, C, C ), C = (C, C, C ),C 5 = (C 5, C 5, C 5 ),C 6 = (C 6, C 6, C 6 )be triangular fuzzy number then the total average cost of the system per unit time in fuzzy sense is given by, TAc ( t, t, T)== [C T q a b ebt t a t e bt a ebt at b b ab t aα t aαt aαbt att abtt aαtt aαtt aαbtt qabq a t ab t aα t aα t aαb t a t ab t aα t aα t aαb t a t 8 ()() () () ab t aα t aα t aαb t at t 8 ab t t aα t t aαt t aαb t t ) C e bt ( ) C δ T t a(t δt δ T T T T b δb δb t δtt δ t t b t δbt δb t )C5 [ a b ebt a at b ab t aα t aαb t t t b t aαt aα bt a δ ebt log δ T t ]C 6 aα[ t b t t at ab t aα t aαt t ] () b t t b t Defuzzification : We defuzzify the fuzzy total cost TAc ( t, t, T ) by graded mean representation method as follows, TAc t, t, T = 6 [TAc t, t, T, TAc t, t, T, TAc t, t, T ] Where,TAc r t, t, T = T [C r q r a b ebt t aα bt aα t at ()() t t ab t aα t t aα t aαb t aαb t a t e bt b t aαt t a ebt at b ab t aα t aαb t t a t ab t aα t aα t ( ) () t t All Rights Reserved Page 9 t aαt q r a b qr ( a t ab t 8 aαb t a t ab t 8 aα aα t at ab t aα t aαt t aαb t ) C r e bt ( ) C r a(t δt δ T T T T b δb δb t δtt δ t t b t δbt t δb r )C5 [ a b ebt a at b ab t aα t a aαt aα bt δ ebt log δ T t ] C r 6 aα[ t b t b t t ]; where r =,, () t at ab t aα t aαt aαb t t b t t b t δ T t t t

7 ISSN: 77-8X (Volume-7, Issue-6) The necessary condition to minimize equations () are TA c r ( t,t,t ) t r = 0, where r =,, () The solution, which may be called feasible solution of the problem, of the equations () and () give the optimal solutions of t, t andt which minimize TAc ( t, t, T )provide they satisfy the sufficient conditions H TAc t, t, T > 0 Now we solve the non-linear equations () and () by using computer software LINGO 6.0 and optimal values of t, t and T can be found respectively. V. NUMERICAL EXAMPLE To illustrate the proposed model, let us consider the following in-put data: For crisp model: Consider an Inventory System with C q C C C 5 C 6 Q a b α γ (/O/C) (/O/C) (/O/C) (/O/C) (/O/C) (/O/C) (U) Rs. 0 Rs. Rs. Rs.0 Rs.00 Rs in their respective units. And we get the optimal out-put values as: t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) For fuzzy model:let us suppose the cost parameters in fuzzy environment as:c = (0, 0, 0 ),q= (0,, ), C = (,, ), C = (00, 0, 0 ), C 5 = (00, 00, 500),C 6 = (00, 80, 60 ). The solution of fuzzy model by graded mean representation is, () When C, q, C, C, C 5,C 6 are all triangular fuzzy numbers then, t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) () When C,q, C, C, C 5 are all triangular fuzzy numbers then, t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) () When C,q, C, C are triangular fuzzy numbers then, t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) () WhenC,C, C are triangular fuzzy numbers then, t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) (5) When C, q are triangular fuzzy numbers then, t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) (6) When C is triangular fuzzy numbers then, t (Y) t (Y) T (Y) TAc( t, t, T) ( Rs.) VI. SENSITIVITY ANALYSIS We now examine the effect of change in the values of the system parameters for sensitivity analysis of the optimal solution of the model. The sensitivity analysis is performed by changing each of the parameters by -0%, -5%, All Rights Reserved Page 0

8 ISSN: 77-8X (Volume-7, Issue-6) 5% and 0% taking one at a time, keeping the other parameters unchangedthe initial data taken from the above numerical example. The results are shown in the bellow table: SENSITIVITY ANALYSIS FOR VARIOUS INVENTORY PARAMETERS Parameters Changed Value *PCPV TAc( t, t, T) (Rs.) t (Y) t (Y) T (Y) ; C = q = C = C = C 5 = C 6 = a = b = Q = α = = δ = *PCPV = Percentage Change in Parameter Values. All Rights Reserved Page

9 ISSN: 77-8X (Volume-7, Issue-6) From the above table we can conclude the following : () The Total Average Cost TAc( t, t, T)increased slowly for increased of C. () The total Average CostTAc( t, t, T)increased and time t, t decreasedslowlyfor increase of q. () The total Average Cost TAc( t, t, T)decrease slightly and time t, t decrease slowly for increase of C. () The total Average CostTAc( t, t, T), time t, t increaserapidly and time Tdecrease slightly for increase of C. (5) The tonal Average CostTAc( t, t, T)increase rapidly, time T increase,t decrease slowlyand time t decreaserapidlyfor increase of C 5. (6) The total Average Cost TAc( t, t, T), time Tdecrease slowlyand time t, t decrease rapidlyfor increase of C 6. (7) The tonal Average Cost TAc( t, t, T),time t increaserapidly,time t increase slowly and time T decrease rapidly for increase of a. (8) The total Average Cost TAc( t, t, T), time t, t increase rapidly alsotime Tdecrease rapidly for increase of b. (9) The total Average Cost TAc( t, t, T),time Tincrease rapidly andtime t, t decrease slowlyfor increase of Q. (0) The Total Average Cost TAc( t, t, T) decrease slightly and time t, t, Tincrease slowly for increase of α. ()The Total Average CostTAc( t, t, T), time t, t,increase slowlyalsotdecrease slightly for increase of. () The Total Average Cost TAc( t, t, T) increase,time t, t decrease rapidly and time T increase slowlyfor increase of δ. VII. CONCLUSIONS In this paper, we have proposed a real life E. O. Q. Inventory Model in a fuzzy environment and presented solution along with sensitivity analysis approach. The inventory model developed with time depended holding cost and shortages arepartially backlogged. Deterioration is non-instantaneous. Here demand rate is considered as Weibull Distribution. This model has been developed for single item. Also, we have developed a crisp model then it transformed to fuzzy model taking triangular fuzzy number and solved by Graded Mean Integration Representation Method. Here decision maker may obtain the optimal results according to his expectation. In future, the other type of membership functions such as piecewise linear hyperbolic, L-R fuzzy number, Trapezoidal Fuzzy Number (TrFN), Parabolic flat Fuzzy Number (PfFN), Parabolic Fuzzy Number (pfn), pentagonal fuzzy number etc. can be considered to construct the membership function and then model can be easily solved by using signed distance method, Werner s Approach, Geometric Programming (GP) technique, Nearest Symmetric Triangular Defuzzification (NSTD) method, etc.. ACKNOWLEDGEMENT The authors are thankful to University of Kalyani for providing financial assistance through DST-PURSE Programme. The authors would like to thank the editor and anonymous reviewers for their valuable and constructive comments and suggestions which have led to a significant improvement in the manuscript. REFERENCES [] Abad, P.L., (996) : Optimal pricing and lot-sizing under conditions of perish ability and partial backlogging. Management Science, (996), [] Haringa, M. ( 995 ) : An EOQ Model for Deteriorating Items with Shortages and Time-Varying Demand. The Journal of the Operational Research Society, Vol. 6, No. (Mar., 995), pp [] Jong Wuu Wu & Wen Chuan Lee (00) An EOQ inventory model for items with Weibull deterioration, shortages and time varying demand, Journal of Information and Optimization Sciences, :, 0-, DOI: 0.080/ [] Kacprzyk, J., and Staniewski, P.(98). Long-term inventory policy-making through fuzzy decision making models. Fuzzy Sets and Systems, 8(),7.doi:0.06/065-0(8) [5] Giri,B.C., Goswami,A. and Chaudhuri,K. S. (996) : An EOQ model for deteriorating items with time varying demand and costs, Journal of the Operational Research Society, vol. 7, no., pp , 996. [6] Lee, H.M., and Yao, J.S. (998). Economic production quantity for fuzzy demand quantity and fuzzy production quantity. European Journal of Operational Research, 09(), 0-.doi:0.06/S077-7(97) [7] Lianxia Zhao ( 0 ) : An Inventory Model under Trapezoidal Type Demand, Weibull-Distributed Deterioration, and Partial Backlogging, Hindawi Publishing Corporation, Journal of Applied Mathematics, Volume 0, Article ID 779, 0 pages, [8] Mandal, W. A. and Islam, S. (05) Fuzzy Inventory Model for Weibull Deteriorating Items, with Time Depended Demand, Shortages, and Partially Backlogging, International Journal of Engineering and Advanced Technology, v., n. 5. [9] Mishra, V. K., Singh, L. S. and Kumar, R. (0) : An inventory model for deteriorating items with timedependent demand and time-varying holding cost under partial backlogging, Journal of Industrial Engineering International 0, 9: [0] Palanivel,M. & Uthayakumara R. (0) : An EOQ Model for Non-Instantaneous deteriorating items with power demand, time dependent holding cost, partial backlogging and permissible delay in payment.international Journal of Mathematics, Computational, Physics and Engineering Vol:8, No: 8, 0 All Rights Reserved Page

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