Fuzzy Inventory Model for Deteriorating Items with Time Dependent Demand and Partial Backlogging

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1 Applied Mathematics, 05, 6, Published Online March 05 in SciRes. Fuzzy Inventory Model for Deteriorating Items with ime Dependent Demand and Partial Backlogging Sushil Kumar *, U. S. Rajput Department of Mathematics & Astronomy, University of Lucknow, Lucknow, India * sushilmath4444@gmail.com Received 0 February 05; accepted 0 March 05; published March 05 opyright 05 by authors and Scientific Research Publishing Inc. his work is licensed under the reative ommons Attribution International License ( BY. Abstract In this paper we developed a fuzzy inventory model for deteriorating items with time dependent demand rate. Shortages are allowed and completely backlogged. he backlogging rate of unsatisfied demand is assumed to be a decreasing exponential function of waiting time. he demand rate, deterioration rate and backlogging rate are assumed as a triangular fuzzy numbers. he purpose of our study is to defuzzify the total profit function by signed distance method and centroid method. Further a numerical example is also given to demonstrate the developed crisp and fuzzy models. A sensitivity analysis is also given to show the effect of change of the parameters. Keywords Inventory, Deterioration, Shortages, riangular Fuzzy Number, Signed Distance Method and entroid Method. Introduction In many inventory models uncertainty is due to fuzziness and fuzziness is the closed possible approach to reality. In recent years some researchers gave their attention towards a time dependent rate because the demand of newly launched products such as fashionable garments, electronic items, mobiles etc. increases with time and later it becomes constant. Deterioration is defined as damage, decay or spoilage of the items that are stored for future use always loose part of their value with passage of time, so deterioration cannot be avoided in any business scenarios. F. Harris (95 [] developed first inventory model. Lotfi A. Zadeh (965 [] introduced the concept of fuzzy set theory in inventory modeling. L. A. Zadeh [] and R. E. Bellman (970 considered an inven- * orresponding author. How to cite this paper: Kumar, S. and Rajput, U.S. (05 Fuzzy Inventory Model for Deteriorating Items with ime Dependent Demand and Partial Backlogging. Applied Mathematics, 6,

2 tory model on decision making in fuzzy environment. R. Jain (976 [4] developed a fuzzy inventory model on decision making in the presence of fuzzy variables. D. Dubois and H. Prade (978 [5] defined some operations on fuzzy numbers. J. Kacpryzk and P. Staniewski (98 [6] developed an inventory model for long term inventory policy making through fuzzy decisions. H. J. Zimmerman (98 [7] tried to use fuzzy sets in operational research. G. Urgeletti inarelli (98 [8] considered the inventory control models and problems. K. S. Park (987 [9] define the fuzzy set theoretical interpretation of an EOQ problem. M. Vujosevic, D. Petrovic and R. Petrovic (996 [0] developed an EOQ formula by assuming inventory cost as a fuzzy number. J. S. Yao and H. M. Lee (999 [] developed a fuzzy inventory model by considering backorder as a trapezoidal fuzzy number.. K. Kao and W. K. Hsu (00 [] developed a single period inventory model with fuzzy demand.. H. Hsieh (00 [] developed an inventory model and give an approach of optimization of fuzzy production. J. S. Yao and J. hiang (00 [4] developed an inventory model without backorders and defuzzified the fuzzy holding cost by signed distance and centroid methods. Sujit D. Kumar, P. K. Kund and A. Goswami (00 [5] developed an economic production quantity model with fuzzy demand and deterioration rate. J. K. Syed and L. A. Aziz (007 [6] consider the signed distance method for a fuzzy inventory model without shortages. P. K. De and A. Rawat (0 [7] developed a fuzzy inventory model without shortages by using triangular fuzzy number.. K. Jaggi, S. Pareek, A. Sharma and Nidhi (0 [8] developed a fuzzy inventory model for deteriorating items with time varying demand and shortages. Sumana saha and ripti hakrabarty (0 [9] developed a fuzzy EOQ model with time varying demand and shortages. D. Dutta and Pawan Kumar (0 [0] considered a fuzzy inventory model without shortages using a trapezoidal fuzzy number. D. Dutta and Pawan Kumar (0 [] [] considered an optimal replenishment policy for an inventory model without shortages by assuming fuzziness in demand, holding cost and ordering cost. Dipak Kumar Jana, Barun Das and apan Kumar Roy (0 [] give a fuzzy generic algorithm approach for an inventory model for deteriorating items with backorders under fuzzy inflation and discounting over random planning horizon. In this paper we consider an inventory model for deteriorating items with time dependent demand rate and partial backlogging. Shortages are allowed and completely backlogged for the next replenishment cycle. he demand rate, deterioration rate and backlogging rate are assumed as triangular fuzzy numbers. he purpose of our study is to defuzzify the total profit function by signed distance method and centroid method and comparing the results of these two methods with the crisp model. Figure shows the developed model and Figure and Figure show the graphs of total profit function with respect to deterioration and backlogging rates.. Definitions and Preliminaries When we are considering the fuzzy inventory model then the following definitions are needed. ( A fuzzy set A on the given universal set X is denoted and defined by A = x, λ x : x X where, λ : X [ 0,] A {( A }, is called the membership function, Figure. With respect to described model. 497

3 P P Figure. With respect σ P P And, ( x Figure. With respect θ. λ = A degree of x in A. ( A fuzzy number A is a fuzzy set on the real line R, if its membership function λa has the following properties λ x is upper semi continuous. A λ ( x = 0 A, outside some interval [ a, a 4]. real numbers a and a, a a a a4 such that λ A ( x is increasing on [ a, a ] [ a, a 4] and λ ( x = A, for each x in [ a, a ]. ( A triangular fuzzy number is specified by the triplet ( a, a, a where a a a continuous membership function λ : X [ 0,] as follows A λ A x a, a x a; a a a x, ; 0, otherwise. x = a x a a a (4 Let A be a fuzzy set defined on R, then the signed distance of A is defined as d( A,0 = AL( α AR dα where, Aα = AL( α AR = a ( b a α, d ( d c α [ (5 If A= ( abc,, 0, decreasing on and defined by its α is an α cut of a fuzzy set A. is a triangular fuzzy number then the signed distance of A is defined as d( A,0 = ( a b c

4 is a triangular fuzzy number then the centroid method on A is defined as (6 If A= ( abc,, ( A = ( a b c.. Assumptions and Notations We consider the following assumptions and notations. he demand rate is R( t = αt where α is a positive constant, for a increasing demand >, and for a decreasing demand <.. θ is the deterioration parameter.. σ is the backlogging parameter.. A is the ordering cost per order. 4. h c is the holding cost per unit per unit time. 5. d c is the deterioration cost per unit per unit time. 6. s c is the shortages cost per unit per unit time. 7. p c is the purchase cost per unit. 8. s p is the selling price per unit, where sp p. 9. o p is the opportunity cost per unit due to lost sales. 0. is the length of order cycle.. θ is the fuzzy deterioration parameter.. σ is the fuzzy backlogging parameter.. α is the fuzzy demand parameter. P is the total fuzzy profit per unit time. 4., 5. is the time at which shortage starts. P is the total profit per unit time. 6. (, 7. I( t is the inventory level at any time in [ ] 8. he inventory system consists only one item. 9. he time horizon is infinite. 0. he lead time is zero.. he replenishment rate is infinite... Mathematical Formulation 0,. Suppose an inventory system consists Q units of the product in the beginning of each cycle. Due to demand and deterioration the inventory level decreases in [ 0, ] and becomes zero at t =. he interval [, ] is the shortages interval. During the shortages interval the unsatisfied demand is backlogged at a rate of e σ t, where t is the waiting time. he instantaneous inventory level at any time t in [ 0, ] are governed by the following differential equations with boundary condition I( 0 di dt = Q with boundary condition I( = 0 I t θi = αt, 0 t ( di σt = α t e, t ( dt θt αt αθt αθ t = Qe ( ( ( 499

5 he solution of Equation ( is I t he solution of Equation ( is using I( = 0, in Equation ( ( = θ t α αθ αθ αt αθt αθ t e ( ( ( ( ασ α I t t t Q he ordering cost per cycle is he holding cost per cycle is 0 H = h I t d; t H = ( ( θ α αθ αθ e = O ( ( = A (7 α αθ αθ = h. ( ( ( he deterioration cost per cycle is he shortage cost per cycle is d ; D = d Q αt t 0 D αθ αθ = d. (4 (5 (6 (8 (9 S = s I t d; t ( ασ α S = s. he purchase cost per cycle in [ 0, ] is P = p Q; he purchase cost per cycle in [, ] is α αθ αθ P = p σt ( α P = p t e d; t p α p α P =. Due to lost sales the opportunity cost per cycle in [, ] is. (0 ( ( 500

6 p σt ( α α O = co t t e d; t p copασ Op = he sales revenue cost per cycle in [ 0, ] is herefore the total profit per unit time is α ασ ασ R ( σt SR = sr αt dt ( αt e d t ; 0 S R α ασ ασ = sr. P (, = [ SR O H D S P P OP] ; P(, = s A. α αθ αθ h ( ( ( (5 αθ αθ ασ ( α d s ( α αθ αθ pα p α p co p ασ (. For a Ist order approximation of e θt ( ( ( α ασ ασ α αθ αθd P(, = [ sr A h ( ασ α s α αθ pα p α p copασ (, ( ( he necessary condition for P (, to be maximum is that P(, = 0 and ( (4 (6 P, = 0, and solving these equations we find the optimum values of and say and for which profit is maximum and the sufficient condition is (, (, (, P P P 0 and (, P 0. 50

7 (, αθ ( 4 ( P d αθ = p ασ h α { θ ( } α s σα ( ( p α ( θ p α p α co ασ, p ( P(, ασ α = s α( σ s ( p α ( α( p co σ P (,, P p (7 (8.. Fuzzy Model Let =, = and Let us consider the inventory model in fuzzy environment due to uncertainty in parameters let us assume that the parameters θ, α and σ may change within some limits. θ θ, θ, θ α α, α, α σ = σ, σ, σ are triangular fuzzy numbers then the total profit per unit time in fuzzy sense is ( α ασ ασ α αθ αθ d P(, = sr A h ( ασ α s α αθ p co α p α pασ p ( ( ( Now we defuzzify the total profit P in two cases.,, (9... Signed Distance Method By signed distance method the total profit per unit time is where, 4 P (, = P, P, P(, α ασ ασ α αθ P (, = sr A h 4 ( ( ασ α s α αθ p α p α p ( ( αθ co d pασ (, (0 50

8 α ασ ασ α αθ P (, = sr A h 4 ( ( ασ α s α αθ p α p α p ( ( αθ co d pασ (, α ασ ασ α αθ P (, = sr A h 4 ( ( ασ α s α αθ p α p α p ( ( αθ co d pασ (, From Equation (0 we have P (, ( α α α ( ασ ασ ασ ( ασ ασ ασ = sr 4A 4 ( α α α ( αθ αθ αθ ( h ( ασ ασ ασ ( ( α α α s ( α α α ( αθ αθ αθ p ( α α α p ( α α α ( αθ αθ αθ p ( ( ασ ασ ασ ( d. co p ( he necessary condition for P (, to be maximum is that P (, = 0 and P (, = 0, and solving these equations we find the optimum values of and say and for which profit is maximum and the sufficient condition is P, P, P, 0 and P (, 0. 50

9 P (, = p 4 ( ασ ασ ασ P ( ( ( 4 αθ αθ αθ ( α α α ( ( αθ αθ αθ ( θ θ θ ( h d { } s ( ασ ασ ασ ( α α α ( α α α ( θ ( α α α ( α α α p p p co, ( ασ ασ ασ p, = sp 4 ( α α α( σ ( ( ασ ασ ασ α α α s ( ( α α α ( α α α ( σ σ σ p p co p (, P,... entroid Method By entroid method the total profit per unit time is P (, P (, = P, P, P(, ( α α α ( ασ ασ ασ ( ασ ασ ασ = sr A ( α α α ( αθ αθ αθ h ( ( ασ ασ ασ ( ( α α α s ( ( α α α ( αθ αθ αθ p p ( α α α p ( α α α ( ( αθ αθ αθ d cop ( ασ ασ ασ (, ( ( (4 (5 504

10 he necessary condition for P (, to be maximum is that P (, = 0 and P (, = 0, and solving these equations we find the optimum values of and say and for which profit is maximum and the sufficient condition is P ( P, P, P, 0 and P (, 0 ( αθ αθ αθ ( 4 (, = p ( ασ ασ ασ h ( α α α d ( αθ αθ αθ { ( θ θ θ( } s ( ασ ασ ασ ( α α α ( α α α ( θ ( α α α ( α α α p p p ( ασ ασ ασ cop, (6 P (, = sp ( α α α( σ ( ( ασ ασ ασ α α α s ( ( α α α ( α α α ( σ σ σ p p cop (7.. Numerical Example P (,, Let us consider an inventory system with the following parameters in appropriate units as A = Rs00/order s = Rs0/unit/year, co = Rs6/unit/year, σ = Rs/unit/year, p α = 8 unit/year, = unit/year, θ = 0.0/year. able shows that as we increase deterioration parameter θ then the total profit increases. able shows that as we increase backlogging parameter σ then the total profit increases. able shows that as we increase demand parameter α then the total profit increases.... Fuzzy Model 5,0, 0 Let α =, σ = (,8,5 and ( 0.00, 0.005, 0.50 p θ = are triangular fuzzy numbers. he solution of the fuzzy inventory model can be determined by the following two methods.... Signed Distance Method When α, σ and θ are triangular fuzzy numbers, then able 4 shows the value of total profit. When α and σ are triangular fuzzy numbers, then able 5 shows the value of total profit. When α and θ are triangular fuzzy numbers, then able 6 shows the value of total profit. 505

11 able. Variation in total profit with respect to θ. θ P able. Variation in total profit with respect to σ. σ P able. Variation in total profit with respect to α. α P able 4. Variation in total profit with fuzzy numbers, α, σ and θ. P able 5. Variation in total profit with fuzzy numbers, α and σ. P entroid Method When α, σ and θ are triangular fuzzy numbers, then able 7 shows the value of total profit. When α and σ are triangular fuzzy numbers, then able 8 shows the value of total profit. When α and θ are triangular fuzzy numbers, then able 9 shows the value of total profit. 506

12 able 6. Variation in total profit with fuzzy numbers, α and θ. P able 7. Variation in total profit with fuzzy numbers, α, σ and θ. P able 8. Variation in total profit with fuzzy numbers, α and σ. P able 9. Variation in total profit with fuzzy numbers, α and θ. P Sensitivity Analysis From able, we see that as we increase the deterioration parameter θ then the optimal time period, the optimal cycle time and total profit increases. From able, we see that as we increase the backlogging parameter σ then the optimal time period, the optimal cycle time decreases and total profit increases. From able, we see that as we increase the demand rate parameter α then the optimal time period decreases and the optimal cycle time and total profit increases. In the case of crisp model we see that the backlogging parameter σ is more sensitive than the deterioration parameter θ and the demand rate parameter α. From the tables for signed distance method and centroid method we see that the fuzzy variables θ and α are more sensitive than the fuzzy variable σ. As we increase the fuzzy variables θ and α in the signed distance method and centroid method than the total profit increases rapidly in centroid method. herefore in the sense of fuzziness the centroid method is better one than the signed distance method. 5. onclusion In this paper we studied a fuzzy inventory model for deteriorating items with time dependent demand rate and partial backlogging. Shortages are allowed and completely backlogged. As we increase the parameters α, σ and θ in the crisp model then the total profit increases and due to the uncertainties in the demand rate, deterioration rate and backlogging rate the parameters α, σ and θ are consider as triangular fuzzy numbers. For defuzzification by signed distance method and centroid method it has been observed that the total profit decreases as the optimal cycle time decreases and the profit given by the signed distance method is minimum as compared to the centroid method. Further this model can be generalized by considering time dependent deterioration rate, holding cost, shortage cost and so many types. 507

13 Acknowledgements he author would like to thank anonymous referees for their valuable comments and suggestions for the improvement of this paper. References [] Harris, F. (95 Operations and ost. AW Shaw O., hicago. [] Zadeh, L.A. (965 Fuzzy Set. Information ontrol, 8, [] Zadeh, L.A. and Bellman, R.E. (970 Decision Making in a Fuzzy Environment. Management Science, 7, [4] Jain, R. (976 Decision Making in the Presence of Fuzzy Variables. IIIE ransactions on Systems, Man and ybernetics, 7, [5] Dubois, D. and Prade, H. (978 Operations on Fuzzy Numbers. International Journal of System Science, 9, [6] Kacpryzk, J. and Staniewski, P. (98 Long erm Inventory Policy Making through Fuzzy Decision Making Methods. Fuzzy Sets and System, 8, [7] Zimmerman, H.J. (98 Using Fuzzy Sets in Operational Research. European Journal of Operation Research,, [8] Urgeletti inarelli, G. (98 Inventory ontrol Models and Problems. European Journal of Operation Research, 4, -. [9] Park, K.S. (987 Fuzzy Set heoretical Interpretation of Economic Order Quantity. IEEE ransactions on Systems, Man and ybernetics, 7, [0] Vujosevic, M. and Petrovic, D. (996 EOQ Formula When Inventory ost Is Fuzzy. International Journal of Production Economics, 45, [] Yao, J.S. and Lee, H.M. (999 Fuzzy Inventory with or without Backorder for Fuzzy Order Quantity with rapezoidal Fuzzy Number. Fuzzy Sets and Systems, 05, [] Yao, J.S. and Lee, H.M. (999 Economic Order Quantity in Fuzzy Sense for Inventory without Backorder Model. Fuzzy Sets and Systems, 05, -. [] Kao,.K. and Hsu, W.K. (00 A Single Period Inventory Model with Fuzzy Demand. omputers and Mathematics with Applications, 4, [4] Hsieh,.H. (00 Optimization of Fuzzy Production Inventory Models. Information Sciences, 46, [5] Yao, J.S. and hiang, J. (00 Inventory without Backorder with Fuzzy otal ost and Fuzzy Storing ost Defuzzified by entroid and Signed Distance. European Journal of Operational Research, 48, [6] Kumar, Sujit De, Kundu, P.K. and Goswami, A. (007 An Economic Production Quantity Inventory Model Involving Fuzzy Demand Rate and Fuzzy Deterioration Rate. Journal of Applied Mathematics and omputing,, [7] Syed, J.K. and Aziz, L.A. (007 Fuzzy Inventory Model without Shortages by Using Signed Distance Method. Applied Mathematics and Information Sciences,, [8] De, P.K. and Rawat, A. (0 A Fuzzy Inventory Model without Shortages Using riangular Fuzzy Number. Fuzzy Information and Engineering,, [9] Jaggi,.K., Pareek, S., Sharma, A. and Nidhi (0 Fuzzy Inventory Model for Deteriorating Items with ime Varying Demand and Shortages. American Journal of Operational Research,, [0] Saha, S. and hakrabarti,. (0 Fuzzy EOQ Model with ime Dependent Demand and Deterioration with Shortages. IOSR Journal of Mathematics,, [] Dutta, D. and Kumar, P. (0 Fuzzy Inventory Model without Shortage Using rapezoidal Fuzzy Number with Sensitivity Analysis. IOSR Journal of Mathematics, 4, -7. [] Dutta, D. and Kumar, P. (0 Optimal Ordering Policy for an Inventory Model for Deteriorating Items without Shortages by onsidering Fuzziness in Demand Rate, Ordering ost and Holding ost. International Journal of Advanced Innovation and Research,,

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