2. Assumptions and Notation
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1 Volume 8 o. 08, ISS: (on-line version) url: ijpam.eu A IVETORY MODEL FOR DETERIORATIG ITEMS WITHI FIITE PLAIG HORIZO UDER THE EFFECT OF PERMISSIBLE DELAY AD PRESERVATIO TECHOLOGY Himanshu Rathore Department of Mathematics and Statistics Manipal University Jaipur, Jaipur, Rajasthan, India rathorehimanshu003@gmail.com, himanshu.rathore@jaipur.manipal.edu Abstract: In present article we have studied an inventory system for deteriorating items over a finite planning horizon. To make a control on deteriorating nature of items we have incorporated the concept of preservation technologies. The customer s demands are received with exponential rate which directly depends on the time. In stock out situation exponentially partial backordering is put in to consideration. The whole study is carried out under the effect of permissible delay in payment in an inflationary environment. There are two cases according to the length of the credit period, in first case the credit period is less than the stock out time period; whereas in second case, the credit period is greater than the time at which total stock is finished. In the end we have illustrated the study by using a numerical illustration and to sensitize the model, a sensitivity analysis is also performed. Keywords: Demand, Deterioration, Permissible delay, Inflation, Preservation.. Introduction owadays, in inventory theory the concept of deterioration becomes very important to handle. The perfect inventory system is directly related to the better control of deteriorating nature of the items. So, we can say that the concept of preservation technology cannot be avoidable. To enhance the better research one have a deep study of literature available related to this field. Hsu, Wee and Teng [00] have established an inventory model for deteriorating item under the effect of preservation technologies. Huang et al. [0] have developed an inventory model for deteriorating items in which retailer is permitted to invest in preservation technology with incorporation of retailer s unit time profit. Hsieh and Dye [03] have studied a production inventory system for deteriorating item with the effect of preservation technology and time dependent demand rate. Gilding [04] has developed an optimal payment policy over the finite planning horizon in an inflationary environment. Muniappan, Uthayakumar and Ganesh [05] have developed an EOQ model for deteriorating items with time dependent deterioration. They have also focused on permissible delay in payment with time dependent demand and completely backlogged shortages. The whole study was carried out under the effect of inflation and time value effect of money over the finite planning horizon. Palanivel and Uthayakumar [05] have established an EOQ model for deteriorating item with non-instantaneous deterioration rate. They have focused on price and advertisement dependent demand and partial backlogging over the finite planning horizon. The whole study was done under the effect of inflation. Yang, Dye, and Ding [05] have established an optimal payment policy under the effect of preservation technology for deteriorating item. Lashgari, Taleizadeh and Sana [06] have studied the two level credit linked with order quantity for deteriorating item. They have also considered the concept of backordering. Mishra [06] has studied the revenue sharing theory in case of deteriorating item. To put a command on deterioration rate he has considered the concept of preservation technology. In this study he has considered price and stock level dependent demand rate. Singh, Khurana and Tayal [06] have established an EOQ formula for deteriorating products with stock dependent demand with the concept of permissible delay in payment. To put a control on deteriorating nature the item, they have incorporated the concept of preservation technologies. In this study we have established an inventory model for deteriorating item over a finite planning horizon. To put a command on constant rate of deterioration we have considered the effect of preservation technologies. We have taken the exponential function of time for the demand rate; in stock out situation the concept of exponential partial backordering is considered. To enhance the sale we have considered the effect of permissible delay in payment in an inflationary environment. We have discussed two types of cases according to the length of credit period. In the end the model is numerically illustrated and to sensitize the 77
2 model, a sensitivity analysis is also performed. In next section, the important assumptions and notation are detailed which are used in mathematical model formulation.. Assumptions and otation The assumptions and notation, which are used in mathematical model generation, are mentioned below. Assumptions The demand rate D is constant; The partially backlogged shortages are permitted with backlogging rate e δt, where 0< δ<. Replenishment is instantaneous; Finite time horizon; Lead time is zero; Preservation technology is used to reduce deterioration; Deterioration rate is constant; Effect of time value of money and inflation is considered. The suppliers permit delay (M) in payment to a purchaser, during this period, purchaser deposits his revenue in an interest bearing account. At the end of delay period, he has two choices that are: he can pay the amount at the end of delay period or after the delay period (payment time between M and T (shortages started at time T ). Supplier charges high interest for unsold items when purchaser chooses the payment time between M and T. otation A: The ordering cost, $ per order; p: The unit selling price at time t=0, $ per item; C: The unit purchase cost per unit at time t = 0; C t: The unit purchase cost per unit at time t, C t=c e -Rt ; C p(j): The purchase cost of the j th cycle; C h: The holding cost, $ per unit per time unit excluding interest charges; C b: The unit backorder cost; C L: The unit lost sale cost; δ: The backlogging parameter where 0< δ< ; T : The time at which shortages occurs; I (t): The inventory level during time period [0, T ]; I (t): The inventory level during time period [T, T]; Q: IM+IB, the order quantity during cycle length [0, T]; IA: The average inventory level during positive inventory cycle; IM: The maximum inventory level during cycle time [0, T ]; B: The backordered inventory during shortages; IB: The maximum shortage quantity during shortage period [T,T]; L c: The lost sales inventory; θ: The constant deterioration rate 0 < θ < ; m(ξ): The reduced deterioration rate under the effect of preservation technology, as follows: m (ξ)= θ (- e -aξ ) where a>0; ξ: The preservation technology (PT) cost to preserve the product where ξ >0; τ θ: The resultant deterioration rate, τ θ = (θ - m(ξ)); R: The constant represents the difference between time discounting and inflation rate, where 0 R <; I e: The interest earned, $ per unit per year; I p: The interest paid by purchaser, $ per unit in stock per year, which is charged by supplier; M: The permissible delay in payment and 0 < M < (i. e. trade credit for purchaser to settle the account); L: The length of finite planning horizon T: The length of replenishment cycle; I (t): Inventory level at any time t; Q: The order quantity per replenishment cycle; t j: The total time, which elapsed upto and including the j th replenishment cycle where t 0 = 0, t = T, t = L; T j: At this time inventory level, in the j th replenishment cycle is zero; F: The fraction of replenishment cycle, at which net stock level is positive; : The number of replenishment during the planning horizon = (L / T); TC (, F, ξ): The present worth of total relevant cost per time unit, when M T ; where, F and ξ are decision variables; TC (, F, ξ): The present worth of total relevant cost per time unit, when T < M; where, F and ξ are decision variables; ote: The notation with superscript (*) represents the optimal values of respective parameters. The present worth of total relevant cost includes following cost parameters. OC: The order cost; PC: The purchase cost; HC: The present worth of holding cost excluding interest charges; BC: The present worth of backordered cost; LC: The present worth of lost sales cost; IP: The total interest paid for unsold items at initial time or after the permissible delay M; TIP: The present worth of total interest paid for unsold items at initial time or after the permissible delay M; 78
3 IE and IE : The total interest earned from sales revenue, during permissible delay in payment in both cases when T < M and M T respectively. TIE and TIE : The present worth of total interest earned from sales revenue, during permissible delay in payment in both cases when T < M and M T respectively. C p (j) = C (j) [D (T + (θ m(ξ)) ( T ))] C (j+)t [D ((T T) δ ( T T )) ] ow the present worth of total purchase cost is as follows: PC = C p (j) j=0 PC = C [[D (T + (θ m(ξ)) ( T ))] 3. Mathematical Model Formulation Initially, the inventory level is maximum (IM) at t = 0 and it gradually depleted, mainly due to demand and partially due to reduced deterioration during time [0, T ]. At time t= T the inventory level is zero and during [T, T], shortages occur and are partially backlogged. Inventory depletion, during cycle length [0, T], is represented by following differential equations: di (t) dt di (t) dt di (t) dt + τ θ I (t) = D; 0 t T () + (θ m(ξ))i (t) = D; 0 t T () = De δt ; T t T (3) Using boundary conditions I (t=0) =IM, I (t=t ) =0 and I (t=t ) =0. ow solving () and (3), we get I (t) = [D ((T t) + (θ m(ξ)) ( T t )) ( (θ m(ξ))t)] (4) I (t) = [D ((T t) δ ( T t )) ] (5) At time t=0 the inventory level is maximum. Hence IM= I (t=0) therefore from (4), we have IM = [D (T + (θ m(ξ)) ( T ))] (6) The maximum inventory shortage level is IB= -I (t=t), therefore from (5), we have IB = [D ((T T) δ ( T T )) ] (7) The present worth of total relevant cost consists of following cost parameters i. The present worth of ordering cost (OC) ow the total replenishments are, therefore the present worth of order cost is as follows: OC = j=0 Ae jrt = A [ e (+)RT ] ; e RT where T = (L/) OC = A [ e ((+)/)RL ] (8) e RL/ ii. The present worth of purchase cost (PC) The purchase cost for j th cycle is C p (j) = C (j) IM + C (j+)t IB e RT [D ((T T) δ ( T T )) ]] [ e RT PC = [D ( FL + (θ m(ξ)) (F L ))] C e RL/ [D ( [ L ( F ) δ ( F L L ) ) ]] e RT ] [ e RL e RL/ ] (9) iii. The present worth of holding cost(hc) Here is the holding cost for the average inventory; therefore the average inventory is as follows: T IA = I (t)dt = D [ T + (θ m(ξ)) T 3 0 (θ 6 m(ξ)) T 4 ] (0) 8 ow the present worth of holding cost is as follows: HC = j=0 C h e jrt IA = C h D [ T m(ξ)) T 3 6 (θ m(ξ)) T HC == C h D [( F L m(ξ)) ( F4 L (θ ] [e RT ] 8 e RT ) + (θ m(ξ)) (F3 L ) (θ 8 4 )] [ e RL e RL/ ] () iv. The present worth of backordered cost (BC) The backordered inventory, which as follows: T B = I (t)dt T B = D [(T T T T ) δ (T T T 3 T3 3 6 )] () Therefore the present worth of backorder cost over the finite planning horizon is as follows: BC = j=0 C b e jrt B = C b D [(T T T T ) δ ( T 3 T T T3 )] [e RT ] 3 6 e RT BC = C b D [(( FL ) (F L ) ( L )) δ (( F L 3 3 ) (F3 L ) (F3 L 3 ))] [ e RL ] (3) 6 3 e RL/ 79
4 Inventory Level International Journal of Pure and Applied Mathematics v. The present worth of lost sales cost (LC) The average lost sales stock amount is as follows: T L c = ( e δt )Ddt = [Dδ ( T T )] T (4) ow the present worth of total lost sales cost is as follows: LC = j=0 C L e jrt = C L D [δ ( T T )] [ e RT e RT ] LC = C L D [δ ( F ) ( L )] [ e RL e RL/ ] (5) Q T M T Time ow we will find interest paid and interest earned by purchaser, for this there are two cases (i) T < M (ii) M T. These two cases are graphically represented in Figure and Figure. Case : T < M The permissible delay period M is greater than the total inventory depletion period i.e. T. Therefore there is no interest paid by purchaser to the supplier for the items. However purchaser will use the sales revenue to earn interest at the rate of I e during time period [0, T ] and interest from cash invested during period [T, M]. Therefore the total value of interest earned under the effect of inflation is M IE = I e p[ibm + (M T )DT + DT dt] IE = I e p [ MD (δ (T T ) (T T)) +DT (M T ) MD (δ ((F )L ) L (F )) IE = I e p [ FL (M + DFL 0 ] ) ] (6) ow the present worth of total interest earned is as follows: TIE = j=0 IE (j) = j=0 IE e jrt TIE = I e p [ δ ((F )L ) MD ( ) L (F ) + DFL FL (M ) ] [ e RL e RL/ ] (7) Figure. (Case : T < M) Inventory system with permissible delay and shortages. Therefore total relevant cost under the effect of inflation in first case is TC (, F, ξ) = [OC + PC+ HC+BC+LC- TIE ] (8) To minimize total relevant cost, we differentiate TC (, F, ξ) w. r. t to, F and ξ and for optimal value necessary conditions are TC (,F,ξ) Where TC F = 0, TC (,F,ξ) F = 0, TC (,F,ξ) ξ = 0 = { C LDFL 3 + C d D( L FL FL 3 3 δ + 3 F L 3 3 δ) I e p( dfl + DL(M FL ) DM( L + FL δ)) + CD( L D( LR )(L FL δ) + afl θξ) + C h D(FL + af L 3 3 θξ a F 3 L 4 4 θ ξ )} TC 0 = {A + 3 C LD( F )L + C b D( FL 3 + L F L 3 F L 3 δ + 3 F3 L 3 δ) + C b D( FL L F3 L 3 3 δ) I e p( DF L 3 F L F L 3 3 δ + DFL(M FL ) + 730
5 Inventory Level International Journal of Pure and Applied Mathematics DM( ( +F)L + ( + F )L δ)) I e p( DFL(M FL ) DM( ( +F)L + ( + F )L δ)) + CD( FL L D( LR DLR( ( +F)L F L δ) ( +F)L )( F L δ) + af L θξ) + CD( FL L D( LR )(( +F)L F L δ) + af L θξ) + C h D(F L + af3 L 3 θξ a F 4 L 4 3 θ ξ ) + C h D( F L + 6 af3 L 3 3 θξ 8 a F 4 L 4 4 θ ξ )} + IP = I p CD [ FL m(ξ)) (( (FL) (FL ) (FL M) (((FL) ) M ) + (θ M) (((FL)3 ) M3 ))] (9) 3 6 ow the present worth of total interest paid is as follows: TIP = j=0 IP(j) = j=0 IPe jrt TIP = I p CD [ FL (FL M) (((FL) ) M ) + (θ ( (FL) ) (FL M) e m(ξ)) ( )] [ RL ] (0) (( (FL)3 ) M3 ) e RL/ 3 6 TC ξ = acdf L 3 θ + C h D( 6 af3 L 3 3 θ 4 a F 4 L 4 4 θ ξ) Provided the determinant of principal minor of hessian matrix are positive definite, i.e. det(h)>0, det(h)>0, det(h3)>0 where H, H, H3 is the principal minor of the Hessian-matrix. Hessian Matrix of the total cost function is as follows: Q [ TC (,F,ξ ) TC (,F,ξ ) F TC (,F,ξ ) F TC (,F,ξ ) ξ TC (,F,ξ ) ξ TC (,F,ξ ) TC (,F,ξ ) F F ξ TC (,F,ξ ) TC (,F,ξ ) ξ F ξ ] 0 M T T Time Case : M T In this case the permissible delay period M expires before the total inventory depletion period T ; hence purchaser will have to pay interest charged on unsold items during (M, T ). Therefore Present worth of interest paid by purchaser is IP = I p C[ T M [I (t)]dt IP = I p CD [T (T M) ( T M ) + (θ m(ξ)) (( T ) (T M) ( T 3 M 3 ))] 6 Figure. (Case : M T ) Inventory system with permissible delay and shortages. ow the present worth of interest earned during positive inventory and interest from invested cost is (This expression has been used by Muniappan, Uthayakumar and Ganesh (05)). IE = I e p [IBM + M 0 Dtdt] IE = I e p [MD (δ ( T T ) (T T)) + DM IE = I e p [MD ( (F )δl ] (F )L ) + DM ] () 73
6 ow the present worth of total interest earned is as follows: TIE = j=0 IE (j) = j=0 IE e jrt TIE = I e p [MD ( (F )δl DM (F )L ) + ] [ e RL e RL/ ] () Hence the present worth of total relevant cost is TC (, F, ξ) =[OC+PC+HC+BC+LC+TIP TIE ] (3) To minimize total relevant cost, we differentiate TC (, F, ξ) w. r. t to, F and ξ and for optimal value necessary conditions are TC (,F,ξ) TC = 0, (,F,ξ) = 0, TC (,F,ξ) = 0 F ξ Where TC F = { C LDFL + CDI pl( M + FL ) + C bd( L FL FL3 δ) + δ( F L F L 3 δ 3 ) DI emp( L + FL δ ) + a( F L FL ( M + FL ) )θξ + CD( L d( LR )(L FL δ ) + afl θξ ) + C h D( FL + af L 3 θξ 3 a F 3 L 4 θ ξ 4 )} TC = {A C LD( F )L + C bd( L 3 FL 3 + F L 3 ) CDFI pl( M + FL ) C bd( L + FL F L F L 3 δ) 3 + δ( F3 L 3 4 F3 L 3 δ 4 ) ( + F)L DI e Mp( ( + F )L δ 3 ) I e p( dm ( + F)L + DM( + ( + F )L δ )) + a( F3 L 3 4 ) F L ( M + FL 3 )θξ + CD( L 3 FL d( LR ( + F)L )( + F L δ 3 ) DLR( ( +F)L F L δ ) af L θξ 3 ) + CD( L + FL d( LR + F)L )(( F L δ ) + af L θξ ) + C h D( F L 3 af3 L 3 θξ 4 + a F 4 L 4 θ ξ 5 ) + C h D( F L + af3 L 3 θξ 6 3 a F 4 L 4 θ ξ 8 4 )} 73
7 TC M3 = {a( ξ 6 F3 L F L ( M + FL ) )θ + acdf L θ + C h D( af3 L 3 θ 6 3 a F 4 L 4 θ ξ 4 4 )} Provided the determinant of principal minor of hessian matrix are positive definite, i.e. det(h)>0, det(h)>0, det(h3)>0 where H, H, H3 is the principal minor of the Hessian-matrix. Hessian Matrix of the total cost function is as follows: [ TC (,F,ξ ) TC (,F,ξ ) F TC (,F,ξ ) F TC (,F,ξ ) ξ TC (,F,ξ ) ξ TC (,F,ξ ) TC (,F,ξ ) F F ξ TC (,F,ξ ) TC (,F,ξ ) ξ F ξ ] Preservation parameter a Backordering parameter δ Delay period M Inflation parameter R umerical Illustration For the numerical illustration, we have used following values of different parameters of the model. A = 000, D = 500, R = 0.05, L = 0.48, θ = 0.05, C h =., C = 8, I p = 0.06, I e = 0.05, p = 0, M = 0.037, C b =, C L = 3, δ = 0.0, a =. By using mathematical software Mathematica7, the optimal values are as follows: ξ = 3.86, F = 0.677, = , TC = Sensitivity Analysis The convexity of the total cost function is shown in Fig. 3, with respect to the decision variables F and. The sensitivity analysis is given in the table, in which we have mentioned the slight variation in optimal values, with respect to different inventory parameters. Chan ge in Para meter ξ* F* * TC *X 0 Demand parameter D Deterioration parameter ϴ Figure 3. Convexity of optimal total cost (TC *) w. r. t. optimal period with positive stock (F*) and optimal replenishment cycles (*) 6. Conclusion In this paper we have studied a finite horizon inventory model for deteriorating items. The preservation technology is used to put the deterioration under control. To enhance the sale the concept of permissible delay has been incorporated. We have calculated the optimal values of different parameter, which keep the total cost function at optimal point. In the end a numerical illustration is presented with sensitivity analysis and convexity of the total cost function. Further study can be extended by incorporating other parameters of inventory control theory. 733
8 References [] Hsu, P. H.; Wee, H. M. and Teng, H. M. Preservation technology investment for deteriorating inventory, International Journal of Production Economics, 4 (), pp (00). [] Huang, Y. H., Wang, C. -C., Huang, C. J. and Dye, C. Y. Comments on preservation technology investment for deteriorating inventory, African Journal of Business Management, 5(), pp (0). [9] Mishra, V. K. Inventory model for deteriorating items with revenue sharing on preservation technology investment under price sensitive stockdependent demand, International Journal of Mathematical Modelling & Computations,6( (WITER)), pp (06). [0] Singh, S., Khurana, D. and Tayal, S. An economic order quantity model for deteriorating products having stock dependent demand with trade credit period and preservation technology, Uncertain Supply Chain Management, 4(), pp. 9-4 (06). [3] Hsieh, T. P. and Dye, C.Y. A productioninventory model incorporating the effect of preservation technology investment when demand is fluctuating with time, Journal of Computational and Applied Mathematics, 39, pp (03). [4] Gilding, B. H. Inflation and the optimal inventory replenishment schedule within a finite planning horizon, European Journal of Operaional Research, 34, pp (04). [5] Muniappan, P., Uthayakumar, R. and Ganesh, S. An EOQ model for deteriorating items with inflation and time value of money considering time-dependent deteriorating rate and delay payments, Systems Science & Control Engineering: An open Access Journal, 3, pp (05). [6] Palanivel, M., and Uthayakumar, R. Finite horizon EOQ model for non-instantaneous deteriorating items with price and advertisement dependent demand and partial backlogging under inflation, International Journal of Systems Science, 46(0), pp (05). [7] Yang, C. T., Dye, C. Y. and Ding, J. F. Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers & Industrial Engineering, 87, pp (05). [8] Lashgari, M., Taleizadeh, A. A. and Sana, S. S. An inventory control problem for deteriorating items with back-ordering and financial consideration under two levels of trade credit linked to order quantity, Journal of Industrial and management optimization, (3), pp (06). 734
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