Optimal Ordering Policy for Deteriorating Items with Price and Credit Period Sensitive Demand under Default Risk 1

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1 Optimal Ordering Policy for Deteriorating Items with Price and Credit Period Sensitie Demand under Default Risk Dhir Singh & Amit Kumar Dept. of Mathematics, Shaheed Heera Singh Got. Degree College, Dhanapur (Chandauli), India Dept. of Mathematics, Got. Degree College, Nadhabhood(Sahaswan),Badaun, India Abstract In this paper, an inentory model for deteriorating items with selling price and credit period sensitie demand is deeloped. he default risk associated with sales reenue of the retailer is also taken into consideration. Here, shortages are allowed and partially backlogged. o make the study close to reality, the holding cost is considered as a time dependent function. his study proides a procedure to deelop the total retailer s profit function per unit time of the system and optimal ordering quantity per cycle for the retailer. Finally, the model is illustrated with a numerical example and to study the effect of changes of different system s on the total retailer s profit per unit time of the system, sensitiity analysis is performed by changing one at a time and presering the other s at their original alues. Keywords: Inentory, deterioration, price and credit period sensitie demand, and default risk.. Introduction Most of the existing inentory models under trade credit financing are assuming that buyer pays instantly purchasing cost of the items as soon as the items are receied. Howeer, such assumption is not necessarily what happens real world. In practice, the retailer offers a delay period known as the trade credit period to her customers to pay for more purchasing. And this practice may be proed to be an excellent strategy in today s competitie scenario. But at the same time, there may be a chance of default risk for the retailer if buyer can t pay his dues. Obiously, the longer delay period may increase the default risk. During the past few years, many articles dealing with a range of inentory models under trade credit financing hae appeared in arious national and international journals. At the earliest, Goyal [8] established an inentory model for a single item under permissible delay in payments when selling price equal to the purchase cost. Aggarwal and Jaggi [3] extended Goyal [8] model for the deteriorating items. Jamal et al. [9] further generalized the Goyal [8] model to allow the shortages. eng [4] extended Goyal [8] model by considering the difference between selling price and purchasing cost. Abad [] deeloped an optimal pricing and lot sizing inentory model for a reseller considering selling price dependent demand. Abad [] formulated optimal lot sizing policies for perishable goods in a finite production inentory model with partial backlogging and lost sales. Dye et al. [6] determined optimal selling price and lot size with a ariable rate of deterioration and exponential partial backlogging. Kumari et al. [0] presented two warehouse inentory model for deteriorating items with partial backlogging under the conditions of permissible delay in payments. Chang et al.[4] inestigated a partial backlogging inentory model for non instantaneous deteriorating items. hey assumed that the demand of the items are stock dependent, and proposed a mathematical model to find the minimum total releant cost. Liao et al. [] inestigated a distribution free newsendor model with balking and lost sales penalty. eng and Lou [5] proposed the demand rate is an increasing function of the trade credit period. Lou and Wang [] studied optimal trade credit and order quantity by considering trade credit with a positie correlation of market sales, but are negatiely correlated with credit risk. Wu et al. [6] explored optimal credit period and lot size by considering delayed payment time dependent demand under default risk for deteriorating items with expiration dates. Dye and Yang [7] discussed the sustainable trade credit and replenishment policy with credit-linked demand and credit risk considering the carbon emission constraints. Chen and eng [5] extended eng and Lou s model [5] to consider time arying deteriorating items and default risk rates under two leels of trade credit. Wu and hao [7] discussed two retailer supplier supply chain models with default risk under the trade credit policy. Singh and Singh [3] recently deeloped an optimal inentory policy for deteriorating items with stock leel and selling price dependent demand under the permissible delay in payments. Howeer, none of the paper discusses the optimal ordering policy by considering the demand of the products to a credit period sensitie and selling price dependent inoling default risk. herefore, we deelop an economic order quantity model for the retailer under the following scenario: () the retailer ISSN: Page 37

2 proides a trade credit period to his customer, () the retailer s trade credit period to the buyer not only increases sales and reenue but also increases the default risk, (3) the demand aries simultaneously with price and the length of credit period that is offered to the customers, (4) shortages are allowed and partially backlogged, and (5) o make the study close to reality, the holding cost is considered as a linear function of time. Finally, the model is illustrated with a numerical example and to study the effect of changes of different system s on the order quantity and the total retailer s profit per unit time of the system, sensitiity analysis is performed.. Assumptions he following assumptions are used in deeloping the mathematical inentory model.. he product considered in this model is deteriorating in nature and there is no repair or replacement of the deteriorated units during the complete cycle.. he demand D s,n of the products is a linear function of the selling price and the credit period. For simplicity, the demand rate D s,n may be gien by Ds,n a bs cn, where a, b, and c are non-negatie s. Also, s is the selling price of the product and n is the credit period offered by the retailer. 3. Notations he following notations are used in deeloping the mathematical inentory model. i. a, b, c : Demand s ii. h, h : Holding cost s iii. α, β : Deterioration s i. δ : Backlogging. λ : Default risk i. p : Purchasing price per unit ii. s : Selling price per unit iii. k : Shortage cost per unit ix. l : Lost sales cost per unit x. d : Deterioration cost per unit 4. Mathematical Modelling Here, the retailer receies the stock at the initial time t 0. During the time interal 0,, the inentory leel is depleted as a result of cumulatie effects of demand and deterioration until it reduces to zero. At t, the inentory leel becomes zero, and thereafter shortages occur. During the time interal, shortages are accumulated due to demand,, until it reaches to a maximum shortage leel Q 3. Shortages are allowed and backlogged partially. he backlogging rate depends on the length of the waiting time for the next replenishment. For simplicity, the backlogging rate of negatie δ inentory is gien by B t e, Where δ is known as backlogging with 0 δ and is the waiting time up to the next replenishment. θ t of the product is defined as a two- Weibull function β θ t αβt, where0 α,β, α is 4. he deterioration rate known as the scale and β is the shape. 5. Replenishment is instantaneous and lead time is zero. 6. he default risk to the retailer depends on the credit period offered by his/her to their customers. For simplicity, the rate of the default risk with respect to the credit period offered by the retailer is taken as f n n, where is non-negatie default. 7. he planning horizon is infinite and the inentory system inoles only one product. 8. Holding cost h per unit per unit time is assumed to be an increasing function of time and expressed as h h ht where h,h 0 xi. O : Ordering cost per order xii. I : Inentory leel at any time t Q : Initial inentory leel at t 0 Q xiii. xi. : Maximum backordered quantity during stock out period x. Q : Order quantity per cycle xi. : he time at which inentory leel becomes zero xii. : Cycle time xiii. n : Allowable trade credit period offered by the retailer to his customers xix. : otal profit of the retailer per unit time which is partially backlogged. At t, the retailer receies the new stock to clear the preious backlog and to continue the process. he behaior of the inentory leel oer time during a gien cycle is shown in Fig.. herefore, the inentory leel at any instant of time t is described by the following differential equations: I' θ I Ds,n, 0 t () I' B Ds,n, t () With the boundary condition I 0 ISSN: Page 37

3 Inentory leel Q 0 ime Q Lost Sales Fig.: he graphical representation of behaior of the inentory leel oer time he solutions of the aboe differential equations are gien by I I α t a bs β β β cn β t β βt where 0 t t a bs cn t δ t where t (3) (4) With the help of equations (3) and (4), one can get the initial inentory leel α I0 a bs cn β Q (5) β and the maximum backordered quantity is Q I a bs cn δ (6) Hence, the order quantity per cycle is gien by Q Q Q α β β Q a bs cn (7) δ he total profit of the retailer per unit time of the system compromises the following components: Sales Reenue: Since Q is the total backordered quantity of the system, the retailer s sales reenue considering default risk per cycle is gien by n Ds,n SR s dt Q 0 δ λ SR sa bs cn n (8) Ordering Cost: he ordering cost per cycle OC O (9) Purchasing Cost: he purchasing cost per cycle is gien by PC pq α β a bs β PC p (0) cn δ Deterioration Cost: Since Q is the total order quantity per cycle, the deterioration cost per cycle is DC dq Ds,ndt 0 dαa bs cn β DC () β Holding Cost: he cost associated with the holding of the stock is calculated as HC 0 h h t I dt αβ h β a bs cn 6 h β αβ β β β β β 3 () Shortage Cost: he shortage cost of the retailer is SC k I dt δ SC ka bs cn 3 (3) 3 Lost Sales Cost: During the stock-out period, shortages are accumulated due to demand, but all customers do not wait up to the next arrial lot ISSN: Page 373

4 and some of them make their purchasing from other retailers. So, the cost associated with the lost sales per cycle can be calculated as LSC l B td s,ndt a bs cn δ l LSC (4) Hence the total profit of the retailer per unit time of the system is gien by SR OC PC DC (5) HC SC LSC he goal is to maximize the total profit of the retailer per unit time with respect to the critical time and the cycle time. he nonlinearity of the objectie functions in (5) does not allow us to obtain the closed form solution. We analyze the model with numerical alues for the inentory s next section. 5. Numerical Example o illustrate the model deeloped here, consider the following alues in appropriate units: a 00,b 0.,c 3, h, h 0.6,α 0.0, β, n 3,δ 0.0, λ 7, p 00, s 60, k 6,l 7, d 4, and O Sensitiity Analysis and its Graphical Representation able : Sensitiity Analysis with respect to the trade credit period Credit % period change in credit period n With the help of Mathematica software and aboe data, we hae the following optimal solutions:.349,.9689, Q 77.4, and he Concaity of the Retailer profit function with respect to and is also shown in figure Fig. able : Sensitiity Analysis with respect to the holding cost s % change h h Fig.3: /s n % -0% 0% 0% 0% % Change credit period n Fig.4: /s h &h -0% -0% 0% 0% 0% % Change s h h ISSN: Page 374

5 able 3: Sensitiity Analysis with respect to the deterioration s % change α β Fig.5: /s α & β able 5: Sensitiity Analysis with respect to the default risk % change λ Fig.7: /s λ -0% -0% 0% 0% 0% % Change λ % -0% 0% 0% 0% % Change s α β able 6: Sensitiity Analysis with respect to the demand s % change able 4: Sensitiity Analysis with respect to the backlogging % change δ Fig.6: /s δ -0% -0% 0% 0% 0% % Change δ a b c Fig.8: /s a, b, & c -0% -0% 0% 0% 0% % Change s a b c ISSN: Page 375

6 7. Conclusion In this paper, an inentory model for deteriorating items with selling price and credit period sensitie demand has been deeloped. he default risk associated with sales reenue of the retailer is also taken into consideration. Here, shortages are allowed and partially backlogged. o make the study close to reality, the holding cost is considered as a time dependent function. Finally, the model has been illustrated with a numerical example and to study the effect of changes of different system s on the total retailer s profit per unit time of the system, sensitiity analysis has been performed by changing one at a time and presering the other s at their original alues. As a future scope of research this model can be studied under more realistic enironments such as inflationary enironment and fuzzy enironment. o make it more releant to today s scenario certain constraints on storage space and budget can be implemented. References [] Abad, P. L. (00), Optimal price and order size for a reseller under partial backordering, Computers and Operations Research, 8 (), [] Abad, P. L. (003), Optimal pricing and lot sizing under conditions of perishability, finite production and partial backordering and lost sales, European Journal of Operational Research, 44 (3), [3] Aggarwal, S. P., & Jaggi, C. K. (995), Ordering policy for deteriorating items under permissible delay in payments, Journal of the Operational Research Society, 46 (5), [4] Chang, C.., eng, J.., & Goyal, S. K. (00)., Optimal replenishment policies for non instantaneous deteriorating items with stock-dependent demand, International Journal of Production Economics, 3, [5] Chen, S. C., & eng, J.. (05), Inentory and credit decisions for time-arying deteriorating items with up-stream and down-stream trade credit financing by discounted cash flow analysis, European Journal of Operational Research, 43(), [6] Dye, C. Y., Hsieh,. P., & Ouyang, L. Y. (007), Determining optimal selling price and lot size with a arying rate of deterioration and exponential partial backlogging, European Journal of Operational Research, 8, [7] Dye, C. Y., & Yang, C.. (05), Sustainable trade credit and replenishment decisions with credit-linked demand under carbon emission constraints, European Journal of Operational Research, 44(), [8] Goyal, S. K. (985), Economic order quantity under conditions of permissible delay in payments., Journal of Operational Research Society, 36 (4), [9] Jamal, A. M., Sarker, B. R., & Wang, S. (997), An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of Operational Research Society, 48(8), [0] Kumari, R., Singh, S.R., & Kumar, N. (008), wo warehouse inentory model for deteriorating items with partial backlogging under the conditions of permissible delay in payment, International ransactions in Mathematical Sciences and computer, (), [] Liao, Y., Banerjee, A., and Yan, C. (0), A distribution free newsendor model with balking and lost sales penalty, International Journal of Production Economics, 33 (), 4-7. [] Lou, K. R., Wang, W. C. (0), Optimal trade credit and order quantity when trade credit impacts on both demand rate and default risk, Journal of the Operational Research Society,, -6. [3] Singh, S. R., & Singh, D. (07), Deelopment of an optimal inentory policy for deteriorating items with stock leel and selling price dependent demand under the permissible delay in payments and partial backlogging, Global Journal of Pure and Applied Mathematics, 3, (9), [4] eng, J.. (00), On the economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 53 (8), [5] eng, J.., & Lou, K. R. (0), Seller s optimal credit period and replenishment time in a supply chain with upstream and down-stream trade credit, Journal of Global Optimization, 53(3), [6] Wu, J., Ouyang, L. Y., Barron, L., & Goyal, S. (04), Optimal credit period and lot size for deteriorating items with expiration dates under two leel trade credit financing, European Journal of Operational Research, 37(), [7] Wu, C., & hao, Q. (06), wo retailer supplier supply chain models with default risk under trade credit policy, Springerplus 5(), 78. ISSN: Page 376