Applied Mthemticl Sciences, Vol. 3, 2009, no. 9, 42-428 Deteriorting Inventory Model for Witing Time Prtil Bcklogging Nit H. Shh nd 2 Kunl T. Shukl Deprtment of Mthemtics, Gujrt university, Ahmedbd. 2 JG College Of Computer Appliction, Drive in rod, Ahmedbd Gujrt, Indi Corresponding Author: Dr. Nit H. Shh e mil: nithshh@gmil.com, shhnith@gmil.com. Abstrct. In this study, deterministic inventory model in which items re subject to constnt deteriortion nd shortges re llowed. The unstisfied demnd is bcklogged which is function of time. The optiml order quntity is derived by minimizing the totl cost. The numericl exmple is given to support the result. The convexity of the cost function is shown numericlly. Sensitivity nlysis is crried out to nlyze the effect of criticl prmeters on decision vribles nd the totl cost of n inventory system. Mthemtics Subject Clssifiction: 90B05 Keywords: Deterministic demnd, deteriortion, prtil bcklogging. Introduction: Deteriortion is defined s decy, spoilge, loss of utility of the product. The process of deteriortion is observed in voltile liquids, beverges, medicines, blood components, food stuffs, diry items etc. As result, while determining optiml replenishment order, the loss due to deteriortion cn not be voided. The literture survey by Rft (99), Shh nd Shh (2000) nd Goyl nd Giri (200) cite upto dte review on deteriorting inventory models. Another scenrio of inventory model is bout stock outs. In mrket, it is observed tht becuse of good reputtion of the retiler, some customers re willing to wit for new stocks rrivl, or if the wit will be short, while other my go elsewhere. Abd (996, 200) derived pricing nd ordering policy for vrible rte of deteriortion nd prtilly bcklogging. The prtil bcklogging ws ssumed to be exponentil function of witing time till the next replenishment. The ssumptions of exponentil bcklogging re unrelistic in developing countries. Secondly, Abd s rticle does not include bckorder cost nd lost sle cost in the formultion of the
422 N. H. Shh nd K. T. Shukl objective function which influences service level to customers. Dye et l (2007) took into ccount the bckorder cost nd lost sle. The im of the pper is to mke the pper of Abd (996, 200) more relistic nd pplicble in prctice. It is ssumed tht the bcklogged units re proportionl to the witing time. The rest of the pper is orgnized s follows. In the next section, the ssumptions nd nottions re listed for the foresid concept. In section 3, mthemticl model is derived. Section 4 dels with numericl exmple nd sensitivity nlysis. The concluding remrks re given in section 5. 2. Assumptions nd Nottions: The mthemticl model is bsed on the following nottions nd ssumptions. 2. Nottions: A the ordering cost per order C the purchse cost per unit. h the inventory holding cost per unit per time unit π b the bckordered cost per unit short per time unit. π L the cost of lost sles per unit. t the time t which the inventory level reches zero, t 0 t 2 the length of period during which shortges re llowed, t 2 0 T (= t + t 2) the length of cycle time IM the mximum inventory level during [0, T]. IB the mximum bckordered units during stock out period. Q (= IM + IB) the order quntity during cycle of length T. I (t) the level of positive inventory t time t, 0 t t I 2 (t) the level of negtive inventory t time t, t t t + t 2 K(t, t 2 ) the totl cost per time unit. 2.2 Assumptions:. The inventory system dels with single item. 2. The demnd rte is known nd constnt. 3. The replenishment rte is infinite. 4. The led time is zero or negligible. 5. The plnning horizon is infinite. 6. During the stock out period, the bcklogging rte is vrible nd is dependent on the length of the witing time for the next replenishment. (Ouyng et l (2005)). The proportion of the customers who would like
Deteriorting inventory model 423 to ccept the bcklogging t time t is with the witing time (T t ) for the next replenishment, i.e. for the negtive inventory the bcklogging rte is Bt () = ; > 0 denotes the bcklogging prmeter + ( T t) nd t t T 3. Mthemticl Model: Under bove ssumption, the on hnd inventory level t ny instnt of time is exhibited in figure. Invent. level Q IM t 2 T Time 0 t IB Lost sles Figure Representtion of inventory system During the period [0, t ], the inventory depletes due to the cumultive effects of demnd nd deteriortion. Hence, the inventory level t ny instnt of time during [0, t ] is described by the differentil eqution di() t dt = I() t ; 0 t t (3.) with the boundry condition I (t ) = 0, the solution of differentil eqution (3.) is ( t t I ) () t ( e = ) ; 0 t t (3.2) From time t onwrds shortges occur nd inventory level reches to zero. During the intervl [t, t + t 2 ], the inventory level depends on demnd nd frction of
424 N. H. Shh nd K. T. Shukl the demnd is bcklogged. The stte of inventory during [t, t + t 2 ] cn be represented by the differentil eqution, di2() t = ; t t t+ t2 (3.3) dt + ( t + t t) 2 Using I 2 (t ) = 0, the solution of differentil eqution is I2( t) = (ln( + ( t+ t2 t) ln( + ) (3.4) The mximum positive inventory is t IM = I (0) = ( e ) (3.5) The mximum bckordered units re IB= I2( t+ = ln( + (3.6) Hence, the order size during [0, T] is Q = IM + IB. t Q ( e ) ln( (3.7) The totl cost per cycle consists of following cost components.. Ordering cost per cycle; OC = A 2. Inventory holding cost per cycle; t IHC = h I() t dt 0 h ( t ) 2 e = t 3. Bckordered cost per cycle; t+ t2 BC = π I () 2 t dt b t = πb ( t ln( + ) 4. cost due to lost sles per cycle; t+ t2 LS = π L ( ) dt ( t t2 t) t + + = πl ( t2 ln( + ) 5. Purchse cost per cycle;
Deteriorting inventory model 425 PC = C Q = t C( ( e ) ln( ) + + Therefore, the totl cost per time unit is K ( t, t 2) = [ ] ( ) OC + IHC + t t BC + LS + + PC 2 (3.8) The necessry condition for the totl cost per time unit, to be minimize is t t 2 t K h( e ) + ce A h( + t e ) = t ( t+ ( t+ t ( t2 ln( + )( πb+ πl) c( ( e ) + ln( + ) = 0 (3.9) 2 ( t+ ( t+ 2 t K π bt2 + π L t2 + c A h( + t e ) = t2 ( t+ ( + ( t+ t ( t2 ln( + )( πb+ πl) c( ( e ) + ln( + ) = 0 (3.0) 2 ( t+ ( t+ Provided 2 K K K ( )( ) ( ) > 0 (3.) t t2 t t2 for obtined pir of (t, t 2 ). The equtions (3.9) nd (3.0) re highly non liner. Using mthemticl softwre, for given set of prmetric equtions (3.9) nd (3.0) cn be solved. The obtined vlues of t nd t 2 must stisfy eqution (3.) to minimize the totl cost per time unit of n inventory system. To illustrte nd vlidte the proposed model, let us consider numericl dt in the following section nd crry out sensitivity nlysis with respect to bcklogging prmeter, deteriortion rte nd demnd. 4. Numericl exmple nd Sensitivity Anlysis: units. Consider n inventory system with following prmetric vlues in proper [A, C, h, π b, π L,, ] = [250, 8, 0.50, 2, 5, 25, 2]
426 N. H. Shh nd K. T. Shukl when deteriortion rte is 5 %, time; t t which positive inventory is zero is 4.28 units nd stock out period; t 2 is of length 0.25 units. This dvices retiler to buy 24 units which will minimum cost $ 307.50. The three dimensionl totl cost per time unit grph is shown in Figure 2 by plotting t in the rnge of [4.25, 4.30] nd t 2 in the rnge of [0.20, 0.30]. The grph given in Figure 2 indictes tht totl cost per time unit is strictly convex. Figure 2 Totl cost per time unit Figure 3 (t K with t 2 fixed) Figure 4 (t 2 K with t fixed) Tble Vrition in deteriortion rte t t 2 Q K 0.05 4.28 0.25 24 307.49 0.0 3.37 0.33 07 330.28 0.5 2.82 0.42 95 352.79
Deteriorting inventory model 427 It is observed in tble tht increse in deteriortion rte increses shortges nd totl cost per time unit of n inventory system nd decreses positive inventory time period nd procurement quntity. See figures 3 nd 4. Tble 2 Vrition in bcklogging prmeter t t 2 Q K 5 4.32 0.7 24 308.45 0 4.34 0. 24 309.09 5 4.36 0.06 23 309.65 Increse in bcklogging prmeter decreses positive inventory time period nd totl cost per time unit of n inventory system. Tble 3 Vrition in demnd α t t 2 Q K 50 3.09 0.5 74 550.79 00 2.23 0.09 244 0.94 50.59 0.06 343 898.37 Increse in demnd increses procurement quntity nd totl cost per time unit of n inventory system significntly. 5. Concluding Remrks: In this study, n optiml replenishment schedule is derived under the ssumption of witing time bckordering when units in n inventory re subject to constnt deteriortion. The model exhibits previling relistic mrket. References []. Abd, P.L., Optiml pricing nd lot-sizing under conditions of perish bility nd prtil bckordering. Mngement Science, 42 (996), 093 04. [2]. Abd, P.L., Optiml price nd order-size for reseller under prtil bcklogging. Computers nd Opertion Reserch, 28(200), 53 65. [3]. Chung Yun Dye, Tsu Png Hsieh nd Ling Yuh Ouyng. Determining optiml selling price nd lot size with vrying rte of deteriortion nd exponentil prtil bcklogging. Europen Journl of Opertionl Reserch, 8(2007), 668 678.
428 N. H. Shh nd K. T. Shukl [4]. Goyl, S.K. nd Giri, B.C. Recent trends in modeling of deteriorting inventory. Europen Journl of Opertionl Reserch, 34(200), 6. [5]. Ouyng, L Y., Wu, K S nd Cheng, M C. An inventory model for deteriorting items with exponentilly declining demnd nd prtil bcklogging. Yugoslv Journl of Opertions Reserch, 5(2) (2005), 277 288. [6]. Rft, F., Survey of literture on continuously deteriorting inventory models. Journl of the Opertionl Reserch Society, 40(99), 27 37. [7] Shh, Nit H. nd Shh, Y. K., Literture survey on inventory model for deteriorting items. Economic nnls, 44(2000), 22 237. Receiced: July 7, 2008