An EOQ model for non-instantaneous deteriorating items with two levels of storage under trade credit policy

Journal of Industrial Engineering International, Sep 2017

A deterministic inventory model with two levels of storage (own warehouse and rented warehouse) with non-instantaneous deteriorating items is studied. The supplier offers the retailer a trade credit period to settle the amount. Different scenarios based on the deterioration and the trade credit period have been considered. In this article, we have framed two models considering single warehouse (Model-I) and two warehouses (Model-II) for non-instantaneous deteriorating items. The objective of this work is to minimize the total inventory cost and to find the optimal length of replenishment and the optimal order quantity. Mathematical theorems have been developed to determine the existence and the uniqueness of the optimal solution. Computational algorithms for the two different models are designed to find the optimal order quantity and the optimal cycle time. Comparison between the optimal solutions for the two models is also given. Numerical illustrations and managerial insights obtained demonstrate the application and the performance of the proposed theory.

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An EOQ model for non-instantaneous deteriorating items with two levels of storage under trade credit policy

An EOQ model for non-instantaneous deteriorating items with two levels of storage under trade credit policy R. Udayakumar 0 K. V. Geetha 0 0 & K. V. Geetha A deterministic inventory model with two levels of storage (own warehouse and rented warehouse) with non-instantaneous deteriorating items is studied. The supplier offers the retailer a trade credit period to settle the amount. Different scenarios based on the deterioration and the trade credit period have been considered. In this article, we have framed two models considering single warehouse (Model-I) and two warehouses (Model-II) for non-instantaneous deteriorating items. The objective of this work is to minimize the total inventory cost and to find the optimal length of replenishment and the optimal order quantity. Mathematical theorems have been developed to determine the existence and the uniqueness of the optimal solution. Computational algorithms for the two different models are designed to find the optimal order quantity and the optimal cycle time. Comparison between the optimal solutions for the two models is also given. Numerical illustrations and managerial insights obtained demonstrate the application and the performance of the proposed theory. Inventory; Non-instantaneous deterioration; Permissible delay in payment; Two warehouses - Department of Mathematics, R & D Centre, Bharathiar University, Coimbatore, Tamilnadu 624 046, India Department of Mathematics, PSNA College of Engineering and Technology, Dindigul, Tamilnadu 624622, India Introduction Deterioration plays an essential role in many inventory systems. Deterioration is defined as decay, damage, obsolescence, evaporation, spoilage, loss of utility, or loss of marginal value of a commodity which decreases the original quality of the product. Many researchers such as Ghare and Schrader (1963) , Philip (1974) , Goyal and Giri (2001) , Li and Mao (2009) , Geetha and Udayakumar (2015) and Mahata (2015) assume that the deterioration of the items in inventory starts from the instant of their arrival. However, most of the goods such as medicine, volatile liquids, and blood banks, undergo decay or deterioration over time. Wu et al. (2006) defined the term ‘‘non-instantaneous’’ for such deteriorating items. He gave an optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. In this direction, researchers have developed their inventory model for a single warehouse which has unlimited capacity. This assumption is not applicable in real-life situation. When an attractive price discount for bulk purchase is available, the management decides to purchase a huge quantity of items at a time. These goods cannot be stored in the existing storage (the owned warehouse with limited capacity). However, to take advantage, it may be profitable for the retailer to hire another storage facility called the rented warehouse. Units are continuously transferred from rented warehouse to owned and sold from owned warehouse. Usually, the holding cost in rented warehouse is higher than that in owned warehouse, due to the non-availability of better preserving facility which results in higher deterioration rate. Hence to reduce the holding cost, it is more economical to consume the goods of rented warehouse at the earliest. Trade credit is an essential tool for financing growth for many businesses. The number of days for which a credit is given is determined by the company allowing the credit and is agreed on by both the company allowing the credit and the company receiving it. By payment extension date, the company receiving the credit essentially could sell the goods and use the credited amount to pay back the debt. To encourage sales, such a credit is given. During this credit period, the retailer can accumulate and earn interest on the encouraged sales revenue. In case of an extension period, the supplier charges interest on the unpaid balance. Hence, the permissible delay period indirectly reduces the cost of holding cost. In addition, trade credit offered by the supplier encourages the retailer to buy more products. Hence, the trade credit plays a major role in inventory control for both the supplier as well as the retailer. Goyal (1985) developed an EOQ model under the condition of a permissible delay in payments. Aggarwal and Jaggi (1995) then extended Goyal’s model to allow for deteriorating items under permissible delay in payments. Uthayakumar and Geetha (2009) developed a replenishment policy for non-instantaneous deteriorating inventory system with partial backlogging. In this direction, we have formulated a model for noninstantaneous deteriorating items with two levels of storage and the supplier offers the retailer a trade credit period to settle the amount. The rest of this paper is organized as follows. Literature review carried is given in the ‘‘Literature review’’. The assumptions and notations which are used throughout the article are presented in ‘‘Problem description’’. In ‘‘Model formulation’’, mathematical model to minimize the total cost is formulated and the solution methodology comprising some useful theoretical results to find the optimal solution is given. Computational algorithm is designed to obtain the optimal values in the ‘‘Algorithm’’. ‘‘Numerical examples’’ is provided to illustrate the theory and the solution procedure. Following this, sensitivity analysis for the major parameters of the inventory system has been analyzed and the comparison between the two models is studied in ‘‘Comparative study of the results between the two models’’. Managerial implications with respect to the sensitivity analysis were given in ‘‘Managerial implication’’. Finally, we draw a conclusion in ‘‘Conclusion’’. Literature review During the last few decades, a number of research papers in the inventory area for deteriorating items have been published by several researchers. Mukhopadhyay et al. (2004) considered joint pricing and ordering policy for a deteriorating inventory. Malik and Singh (2011) developed an inventory model for deteriorating items with soft-computing techniques and variable demand. Taleizadeh (2014b) developed an economic-order quantity model with partial backordering and advance payments for an evaporating item. Taleizadeh and Nematollahi (2014) established an inventory control problem for deteriorating items with backordering and financial considerations. Taleizadeh (2014a) developed an economic-order quantity model for deteriorating item in a purchasing system with multiple prepayments. Taleizadeh et al. (2015) gave a joint optimization of price, replenishment frequency, replenishment cycle, and production rate in vendor-managed inventory system with deteriorating items. Tavakoli and Taleizadeh (2017) gave a lot sizing model for decaying item with full advance payment from the buyer and conditional discount from the supplier. Ouyang et al. (2006) derived an inventory model for non-instantaneous deteriorating items with permissible delay in payments. Liao (2008) discussed an EOQ model with non-instantaneous receipt and exponentially deteriorating items under two-level trade credits. Maihami and Kamal Abadi (2012 ) gave a joint control of inventory and it is pricing for non-instantaneously deteriorating items under permissible delay in payments and partial backlogging. Soni (2013) established an optimal replenishment policy for non-instantaneous deteriorating items with price and stock-sensitive demand under permissible delay in payment. Tat et al. (2013) developed and EOQ model with non-instantaneous deteriorating items in vendor-managed inventory system. Udayakumar and Geetha (2014) gave an optimal replenishment policy for non-instantaneous deteriorating items with inflation-induced time-dependent demand. Maihami and Karimi (2014) developed pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts. Geetha and Udayakumar (2016) developed an optimal lot sizing policy for non-instantaneous deteriorating items with price and advertisement-dependent demand under partial backlogging. Wu et al. (2014) gave a note on optimal replenishment policies for non-instantaneous deteriorating items with price and stock-sensitive demand under permissible delay in payment. Zia and Taleizadeh (2015) gave a lot sizing model with backordering under hybrid linked to order multiple advance payments and delayed payment. Udayakumar and Geetha (2016) developed an economic-ordering policy for non-instantaneous deteriorating items over finite-time horizon. Taleizadeh et al. (2016) developed an imperfect economic production quantity model with up-stream trade credit periods linked to raw material-order quantity and downstream trade credit periods. Heydari et al. (2017) discussed a two-level day in payments contract for supply chain coordination in the case of credit-dependent demand. In existing literature, Sarma (1987) was the first to develop a deterministic inventory model with two levels of storage and an optimum release rate. Murdeshwar and Sathe (1985) gave some aspects of lot size model with two levels of storage. Pakkala and Achary (1992) developed a deterministic inventory model for deteriorating items with two warehouses and finite-replenishment rates. Goswami and Chaudhuri (1992) established an economic-order quantity model for items with two levels of storage for a linear trend in demand. Benkherouf (1997) established a deterministic-order-level inventory model for deteriorating items with two storage facilities. Bhunia and Maiti (1994 , 1998) gave a two-warehouse inventory model for a linear trend in demand. Ray et al. (1998) developed an inventory model with two levels of storage and stock-dependent demand rate. Lee and Ying (2000) derived an optimal inventory policy for deteriorating items with two warehouses and time-dependent demand. Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon, was derived by Kar et al. (2001) . Yang (2004) gave two-warehouse inventory model for deteriorating items with shortages under inflation. Zhou and Yang (2005) derived the model for two warehouses with stock-level-dependent demand. Yang (2006) developed two-warehouse partial backlogging inventory models for deteriorating items under inflation. Lee (2006) investigated two-warehouse inventory model with deterioration under FIFO dispatching policy. Chung and Huang (2007) derived an optimal retailer’s ordering policies for deteriorating items with limited storage capacity under trade credit financing. Hsieh et al. (2007) determined an optimal lot size for a two-warehouse system with deterioration and shortages using net present value. Rong et al. (2008) gave a two-warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time. Lee and Hsu (2009) gave a two-warehouse production model for deteriorating inventory items with time-dependent demands. Liang and Zhou (2011) developed the two-warehouse inventory model for deteriorating items under conditionally permissible delay in payment. Agrawal et al. (2013) derived the model with ramp-type demand and partially backlogged shortages for a two-warehouse system. Liao et al. (2012 , 2013) developed two-warehouse inventory models under different assumptions. Jaggi et al. (2014) discussed under credit financing in a two-warehouse environment for deteriorating items with price-sensitive demand and fully backlogged shortages. Bhunia and Shaikh (2015) gave an application of PSO in a two-warehouse inventory model for deteriorating item under permissible delay in payment with different inventory policies. Lashgari et al. (2016) considered partial upstream advanced payment and partial up-stream delayed payment in a three-level supply chain. Lashgari and Taleizadeh (2016) developed an inventory control problem for deteriorating items with backordering and financial considerations under two levels of trade credit linked to order quantity. In the literature, the warehouse owned by the retailer is referred to as owned warehouse OW, while the one hired on rent is referred to as rented warehouse RW. The major assumptions used in the previous articles are summarized in Table 1. From Table 1, it is clear that, the two-warehouse system for non-instantaneous deteriorating items under trade credit policy with the assumption of a [ b [ 0 has not been considered previously in the literature, represents several practical real-life situations. A typical example of industries that actually operate under the same set of assumptions is the food industry, vegetable markets, fruits stall, supermarkets, etc., and the product may deteriorate after certain time. With longer storage durations, many processed food items require more sophisticated warehousing facilities. Moreover, in the model developed by Liang and Zhou (2011) , they considered instantaneous deteriorating items under delay in payment. In the present work, we have made an attempt to investigate the above issues together and derive a model that helps the retailer to reduce the total inventory cost of the inventory system, where permissible delay in payment is offered. The parameters of the proposed model are given in Table 2. Problem description To the best of our knowledge, there is no work considering both single-warehouse and two-warehouse models for noninstantaneous deteriorating items with trade credit. To bridge this gap, we have framed two models considering single warehouse (Model-I) and two warehouses (ModelII). Different scenarios based on deterioration time and trade credit period are considered and the theoretical results to find the optimal solution are derived. The main objective of the proposed work is to determine the optimal cycle time and the optimal-order quantity in the above-said situations, such that the total cost is minimized. We consider the different types of storage capacity, so that it will suit to different situations in realistic environment. To develop the mathematical model, the following assumptions are being made. Assumptions i. Demand rate is known and constant. Demand is satisfied initially from goods stored in RW and continues with those in OW once inventory stored at RW is exhausted. This implies that tw\T . The replenishment rate is infinite and the lead time is a ¼ b b [ a 0\a; b\\1 Demand rate Constant Time dependent Time dependent Time dependent Constant Constant Constant Constant Constant Constant Price dependent Time dependent Constant Constant Constant Price dependent Price dependent Deterioration Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Instantaneous Yes Delay in payment No No No No No No No No No No No Yes Yes Yes Yes Yes Permissible shortage Completely backlogged No Completely backlogged Completely backlogged Completely backlogged No Partially backlogged Completely backlogged No Partially backlogged Partially/completely backlogged No No No No Completely backlogged Partially backlogged Objective function Cost Cost Cost Cost Cost Profit Cost Cost Cost Cost Profit Cost Cost Cost Cost Profit Profit iii. iv. The items deteriorate at a fixed rate a in OW and at b in RW, for the rented warehouse offers better facility, so a [ b, and hr ho [ cða bÞ (following Liang and Zhou (2011) ). To guarantee that the optimal solution exists, we assume that aW \D; that is, deteriorating quantity for items in OW is less than the demand rate. When T M, the account is settled at T ¼ M. Beyond the fixed credit period, the retailer begins paying the interest charges on the items in stock at rate Ip. Before the settlement of the replenishment amount, the retailer can use the sales revenue to earn the interest at annual rate Ie, where Ip Ie: When T M, the account is settled at T ¼ M and the retailer does not pay any interest charge. Alternatively, the retailer can accumulate revenue and earn interest until the end of the trade credit period. Notations In addition, the following notations are used throughout this paper: OW RW D k c p hr ho a b M Ip Ie td I0ðtÞ IrðtÞ W Q TCi tw T The owned warehouse The rented warehouse The demand per unit time The replenishment cost per order ($/order) The purchasing cost per unit item ($/unit) The selling price per unit item p [ c The holding cost per unit per unit time in RW The holding cost per unit per unit time in OW The deterioration rate in OW The deterioration rate in RW Permissible delay in settling the accounts The interest charged per dollar in stocks per year The interest earned per dollar per year The length of time in which the product has no deterioration The inventory level in OW at time t The inventory level in RW at time t The storage capacity of OW The retailer’s order quantity (a decision variable) The total relevant costs The time at which the inventory level reaches zero in RW The length of replenishment cycle (a decision variable) Model formulation In this article, we consider two different inventory models, namely, single-warehouse system and two-warehouse system. Based on the values of M, td, and tw, the classification for the two different models is given in Table 3. Model-I (single-warehouse system) Scenario I: td\T In this case, demand becomes constant before the inventory level becomes zero. Thus, inventory level at OW decreases because of the increasing demand in the interval ð0; tdÞ and because of the constant demand and deterioration in the interval ðtd; T Þ. The behavior of the model is given in Fig. 1. Hence, the change in the inventory level in OW at any time t in the interval ð0; T Þ is given by the following differential equations: Based on the assumptions and description of the model, the total annual relevant costs (ordering cost ? holding cost ? interest payable - interest Dho htd2 þ td eaðT tdÞ i 1Þ þ 2 Case 2 (td\M T) 1 TC2ðTÞ ¼ T D k þ a2 ðho þ caÞ eaðT tdÞ Dho htd2 þ td eaðT tdÞ i þ 2 cIpD heaðT MÞ þ a2 aðT MÞ Case 3 (M [ T) 1 TC3ðTÞ ¼ T D k þ a2 ðho þ caÞ eaðT tdÞ þ D2ho htd2 þ td eaðT tdÞ i ð3Þ Since T is the decision variable, the necessary condition to find the optimum value of T to minimize the total cost is ddTCT1 ¼ 0; ddTCT2 ¼ 0; ddTCT3 ¼ 0, which yield dTC1 dT ¼ aðT tdÞ 1 1i aðT tdÞ 1 i 1 i ð1Þ td þ td eaðT tdÞ 2 i 1 Based on the assumptions and description of the model, the total annual relevant costs is given by Since, T is the decision variable, the necessary condition to find the optimum value of T to minimize the total cost is dT ¼ 0 and ddTCT5 ¼ 0, which yield dTC4 dTC4 dT k ¼ T2 þ hoD 2 cIpD þ T ðT cIp T2 T2 2 MÞ þ and dTC5 dT k ¼ T2 þ hoD 2 pIeD T2 ¼ 0; provided that they satisfy the sufficient condition d2TC4ðTÞ [ 0 and d2TC5ðTÞ [ 0. dT2 dT2 Model-II (two-warehouse system) There are certain circumstances, where the owned warehouse of the retailer is insufficient to store the goods. In that situation, the retailer may go for rented warehouse. To suit to this case, we develop an inventory model, where there are two warehouses (owned warehouse OW and rented warehouse RW) (refer Table 3). The inventory system evolves as follows: Q units of items arrive at the inventory system at the beginning of each cycle. Out of which W units are kept in OW and the remaining ðQ WÞ units are stored in RW. The items of OW are consumed only after consuming the goods kept in RW. For the analysis of the inventory system, it is necessary to compare the value of the parameter td and M with the possible values that the decision variables tw and T can take on. This results in the following three scenarios. Scenario I: td\tw\T During the time interval ð0; tdÞ, the inventory level at RW is decreasing only owing to demand rate. The inventory level is dropping to zero due to demand and deterioration during the time interval ðtd; twÞ. The behavior of the inventory system is depicted in Fig. 3. Hence, the change in the inventory level in RW at any time t in the interval ð0; twÞ is given by the following differential equations: dIr1ðtÞ dt dIr2ðtÞ dt ¼ D; 0\t\td ¼ D bIr2ðtÞ; td\t\tw; with the boundary condition Ir2ðtwÞ ¼ 0: The solutions of the above equations are given, respectively, by Ir1ðtÞ ¼ Dðtd D hebðtw tÞ Ir2ðtÞ ¼ b D hebðtw tdÞ tÞ þ b i 1 ; 0\t\td i 1 ; td\t\tw: W and continuity of Furthermore, since Ir1ð0Þ ¼ Q IrðtÞ at t ¼ td, we get D hebðtw tdÞ Q ¼ W þ Dtd þ b i 1 : During the interval ð0; tdÞ, there is no change in the inventory level in OW as demand is met from RW. Hence, at any epoch t, the inventory level at OW is Based on the assumptions and description of the model, the total annual cost which is a function of tw and T is given by 8 TC6ðtw; TÞ; TCðtw; TÞ ¼ :>>< TTCCT78Cððtt9wwðt;;wTT;ÞÞT;; Þ; 0\M td\M tw\M M [ T td tw ; T where TC6ðtw; TÞ ¼ T1 k þ bD2 hðhr þ cbÞ ebðtw tdÞ bðtw tdÞ 1 i þ cIp b22td2 þ ebðtw tdÞðbtd Mb þ 1Þ b2 Mtd þ aD2 h ho þ ca þ cIp eaðT twÞ aðT twÞ 1 i pIeD2M2 ; bðtw MÞ 1 ð15Þ ð17Þ ð18Þ ho þ ca þ cIp eaðT twÞ 1 i oTC6ðtw; TÞ oT 1 1 D h ¼ T TC6ðtw; TÞ þ T a 1 D h ¼ T a i TC6ðtw; TÞ ¼ 0: ð16Þ From Eqs. (15) and (16), we have the following expressions: aDhðhr þ cbÞ ebðtw tdÞ 1 þ cIp ebðtw tdÞðbtd Mb þ 1Þ i þ Wabh ho þ ca þ cIp eaðtw tdÞi ¼ bDh ho þ ca þ cIp eaðT twÞ 1 i; k þ bD2 hðhr þ cbÞ ebðtw tdÞ bðtw tdÞ 1 i þ cIp b22td2 þ ebðtw tdÞðbtd Mb þ 1Þ b2 Mtd M2 2 W h þ a ðho þ caÞ eaðtw tdÞ 1 þ cIp eaðtw tdÞ þ aðtd þ aD2 h ho þ ca þ cIp eaðT twÞ aðT twÞ 1 i To derive the optimal solutions for the proposed model, we need the following lemma. Lemma 1 D hr þ cb þ cIpe btd ebt [ aW ho þ ca þ cIp eat; D hr þ cb þ cIpe bM ebt [ aW ho þ ca þ cIp eat; Dðhr þ cbÞebt [ aWðho þ caÞeat; Proof (See Appendix) Case 1 (0\M td) The necessary conditions for the total annual cost in (11) to be the minimum are oTCo6tðwtw;TÞ ¼ 0 and oTC6oðTtw;TÞ ¼ 0, which give i þ cIphebðtw tdÞðbtd Mb þ 1Þ 1i þWh ho þ ca þ cIp eaðtw tdÞ i Da h ho þ ca þ cIp eaðT twÞ 1 i 1 i 1 i and Theorem 1 If 0\M td, then the total annual cost TC6ðtw; T Þ is convex and reaches its global minimum at the point ðtw6 ; T6 Þ, where ðtw6 ; T6 Þ is the point which satisfies Eqs. (17) and (18). Proof Let tw6 and T6 be the solution of Eqs. (17) and (18) and H1ðtw6 ; T6 Þ be the Hessian matrix of TC6ðtw; T Þ evaluated at tw6 and T6 . It is known that if this matrix is positive definite, then the solution ðtw6 ; T6 Þ is an optimal solution. Taking the second derivative of TC6ðtw; T Þ with respect to tw and T , and then, finding the values of these functions at point ðtw6 ; T6 Þ, we obtain o2TC6ðtw; TÞ otw2 tw6 ;T6 þWah ho þ ca þ cIp eaðtw tdÞ i þDh ho þ ca þ cIp eaðT twÞ io [ T1 Dh ho þ ca þ cIp eaðT twÞ i ¼ T1 nDhðhr þ cbÞ ebðtw tdÞ i þ cIphebðtw tdÞðbtd Mb þ 1Þi Hence, we obtain that o2TC6 o2TC6 otw2 oT 2 o2TC6 o2TC6 otwoT oTotw ðtw6 ;T6 Þ holds, which implies that the matrix H1ðtw6 ; T6 Þ is positive definite and ðtw6 ; T6 Þ is the optimal solution of TC6ðtw; TÞ: Case 2 (td\M tw) The necessary conditions for the total annual cost in Eq. (12) to be the minimum are oTCo7tðwtw;TÞ ¼ 0 and oTC7oðTtw;TÞ ¼ 0, which give o2TC6ðtw; TÞ oT2 tw6 ;T6 1 oTC6 1 T oT þ T ¼ 1 ¼ T [ T1 n o2TC6ðtw; TÞ otwoT ðtw6 ;T6 Þ D ho þ ca þ cIp eaðT twÞ D ho þ ca þ cIp eaðT twÞo D ho þ ca þ cIp eaðT twÞ oTC6 oT 1 n ¼ T ¼ o2TC6ðtw; TÞ oTotw D ho þ ca þ cIp eaðT twÞo tw6 ;T6 ðtw6 ;T6 Þ 2 oTC1 oT ðtw6 ;T6 Þ and aðT twÞ DT h ¼ a ho þ ca þ cIp eaðT twÞ 1 þ cIp eaðtw MÞ 1 i 1 i: 1 i: ð22Þ Theorem 2 If td\M tw, then the total annual cost TC7ðtw; TÞ is convex and reaches its global minimum at the point ðtw7 ; T7 Þ, where ðtw7 ; T7 Þ is the point which satisfies Eqs. (21) and (22). Proof (Similar to the proof of Theorem 1). Case 3 (tw\M T ) The necessary conditions for the total annual cost in Eq. (13) to be the minimum are oTCo8tðwtw;TÞ ¼ 0 and oTC8oðTtw;TÞ ¼ 0, which give oTCo8ðtwtw; T Þ ¼ T1 Db hðhr þ cbÞ ebðtw tdÞ 1 i and Case 4 (M [ T) The necessary conditions for the total annual cost in Eq. (14) to be the minimum are oTCo9tðwtw;TÞ ¼ 0 and oTC9oðTtw;TÞ ¼ 0, which give ð27Þ ð28Þ ð29Þ 1 i oTC9ðtw; T Þ otw 1 ¼ T D h b ðhr þ cbÞ ebðtw tdÞ 1 i and oTC9ðtw; TÞ oT 1 ¼ T TC9ðtw; TÞ þ T1 Da hðho þ caÞ eaðT twÞ 1 þ cIpeaðT MÞ 1i ¼ T1 Da hðho þ caÞ eaðT twÞ 1 þ cIpeaðT MÞ 1i TC9ðtw; TÞ ¼ 0: Equations (27) and (28) can be written as aDhðhr þ cbÞ ebðtw tdÞ 1 i þ Wabhðho þ caÞeaðtw tdÞi ¼ bDhðho þ caÞ eaðT twÞ 1 i; aðT 1 i twÞ 1 i DT h 1Þ ¼ a ðho þ caÞ eaðT twÞ ð30Þ Theorem 4 If M [ T , then the total annual cost TC9ðtw; TÞ is convex and reaches its global minimum at the point ðtw9 ; T9 Þ, where ðtw9 ; T9 Þ is the point which satisfies Eqs. (29) and (30). Proof (Similar to the proof of Theorem 1). Scenario II: tw\td\T In this case, during the time interval ð0; twÞ, the inventory level at RW decreases only owing to demand rate, where tw is the epoch at which the inventory level in RW is zero. The inventory level is dropping to zero due to demand and deterioration during the time interval ðtd; twÞ. This case is demonstrated in Fig. 4. Hence, the change in the inventory level in RW at any time t in the interval ð0; twÞ is given by the following differential equations: with the boundary condition Ir1ðtwÞ ¼ 0 and the solution of the above differential equation is given by Ir1ðtÞ ¼ Dðtw tÞ; 0\t\tw: Again, during the interval ð0; twÞ, demand is met from RW alone, and there is no change in the inventory level in OW. Thus, at any instant t, the inventory level I02ðtÞ at OW is After time tw, demand is met from OW. Hence, the inventory level at OW decreases because of the increasing demand rate during the interval ðtw; tdÞ and then because of the demand rate and deterioration during the interval (td; TÞ. Thus, differential equations governing the inventory level in OW during the interval ðtw; TÞ are dI0d2tðtÞ ¼ D; td\t\tw dI0d3tðtÞ ¼ D aI03ðtÞ; tw\t\T ; with the boundary condition I03ðT Þ ¼ 0, and the solutions of the above equations are given, respectively, by I02ðtÞ ¼ Da eaðT tdÞ 1 ¼ T aðtw tdÞ ðeaðT tdÞ ¼ 0; Case 1 (0\M tw) The necessary conditions for the total annual cost in (31) to be the minimum are oTC1o0tðwtw;TÞ ¼ 0 and oTC1o0Tðtw;TÞ ¼ 0, D hrDtw þ a ho þ cIp i 1Þ þ aW þ cIpDðtw MÞ 1 TC11ðtw; TÞ ¼ T k þ k þ aðT tdÞ 1 þho Wtw þ Da eaðT tdÞ þD td2 2 aðT tdÞ 1 þho Wtw þ Da eaðT tdÞ Proof (Similar to the proof of Theorem 1). Case 2 (tw\M td) The necessary conditions for the total annual cost in (32) to be the minimum are oTC1o1tðwtw;TÞ ¼ 0 and oTC1o1Tðtw;TÞ ¼ 0, which give oTC11ðtw; T Þ otw 1 ¼ T ¼ 0: TC11ðtw; TÞo ¼ 0: ð40Þ Proof (Similar to the proof of Theorem 1). Theorem 6 If td\M tw, then the total annual cost TC11ðtw; T Þ is convex and reaches its global minimum at the point ðt11; T11Þ, where ðt11; T11Þ is the point which satisfies Eqs. (41) and (42). Case 3 (td\M T ) The necessary conditions for the total annual cost in (33) to be the minimum are oTC1o2tðwtw;TÞ ¼ 0 and oTC1o2Tðtw;TÞ ¼ 0, which give oTC12ðtw; TÞ otw 1 ¼ T þ DhoeaðT tdÞðtd twÞ þ Da cIp eaðT MÞ 1 1 ¼ T D h a ðho þ caÞ eaðT tdÞ i 1 þ DhoeaðT tdÞðtd twÞ D cIp eaðT MÞ þ a 1 TC12ðtw; T Þ ¼ 0: From Eqs. (43) and (44), we have the following expressions: aðhrDtw þ hoW Þ ¼ Dho eaðT tdÞ 1 aDðtw tdÞ; k þ Theorem 7 If td\M T, then the total annual cost TC12ðtw; T Þ is convex and reaches its global minimum at the point ðtw12 ; T12Þ, where ðtw12 ; T12Þ is the point which satisfies Eqs. (45) and (46). Proof (Similar to the proof of Theorem 1). Case 4 (M [ T) The necessary conditions for the total annual cost in (34) to be the minimum are oTC1o3tðwtw;TÞ ¼ 0 and oTC1o3Tðtw;TÞ ¼ 0, which give oTC13ðtw; TÞ otw 1 ¼ T DT h ¼ a ðho þ caÞ eaðT tdÞ 1 i þ DhoTeaðT tdÞðtd twÞ Theorem 8 If M [ T , then the total annual cost TC13ðtw; T Þ is convex and reaches its global minimum at the point ðtw13 ; T13Þ, where ðtw13 ; T13Þ is the point which satisfies Eqs. (49) and (50). Proof (Similar to the proof of Theorem 1). Scenario III: td [ T In this case, the inventory levels both in RW as well as in OW become zero before the demand stabilises. Thus, the inventory levels at both the warehouses decrease only because of the increasing demand. The case is depicted in Fig. 5. The inventory level at RW at any epoch t in the time interval ð0; twÞ is given by ð46Þ ¼ 0; ð47Þ ð48Þ ð49Þ pIe2DT : ð50Þ Using the boundary condition I02ðTÞ ¼ 0, the solution of the above equation is given by I02ðtÞ ¼ DðT tÞ; tw\t\T: Based on the assumptions and description of the model, the total annual relevant costs is given by TCðtw; TÞ ¼ where 1 TC14ðtw; TÞ ¼ T pIeDM2 2 ; T2 2 pIeDM2 2 t2 Ttw þ 2w ; Theoretical results which give oTC14ðtw; TÞ otw 1 ¼ T Case 1 (0\M tw) The necessary conditions for the total annual cost in (51) to be the minimum are oTC1o4tðwtw;TÞ ¼ 0 and oTC1o4Tðtw;TÞ ¼ 0, hrDtw þ ho þ cIp Dðtw TÞ þ WÞ þcIpDðtw MÞ ¼ 0; ¼ 1 ¼ T ð54Þ ð55Þ oTC14ðtw; TÞ oT 1 1 T TC14ðtw; TÞ þ T D ho þ cIp ðT twÞ D ho þ cIp ðT twÞ TC14ðtw; TÞ ¼ 0: From Eqs. (54) and (55), we have the following expressions: D ho þ cIp ðtw TÞ þ WÞ þ cIpðtw MÞ ¼ hrDtw; k þ 1 ¼ T fhrDtw þ hoðW þ Dðtw TÞÞg ¼ 0; k þ which give 1 ¼ T fhrDtw þ hoðW þ Dðtw TÞÞg ¼ 0; twÞ pIeDT 2 : Theorem 11 If M [ T, then the total annual cost TC16ðtw; TÞ is convex and reaches its global minimum at the point ðtw16 ; T16Þ, where ðtw16 ; T16Þ is the point which satisfies Eqs. (64) and (65). Proof (Similar to the proof of Theorem 1). Algorithm Based on the above analysis, we state the algorithm which enables us to obtain the overall optimal policy for the singlewarehouse system and two-warehouse inventory system. Algorithm I (single-warehouse system) Step 1: Input all the parameters of the inventory system. Step 2: Compare the values of M and td. If M\td, then go to step 3, and if M [ td, go to step 4. Step 3: (i) (ii) (iii) (iv) Determine T1 ; from Eq. (4). If td\T, let T ¼ T1 and TC ¼ TC1, otherwise go to step (ii). Determine T4 ; from Eq. (9). If T td, let T ¼ T4 and TC ¼ TC4; otherwise, go to step (iii). Determine T5 ; from Eq. (10). If T\M td, let T ¼ T5 and TC ¼ TC5; otherwise, go to step (iv). Let T = arg min TC1; TC4; TC5 , output the optimal T and TC . ð62Þ ð63Þ ð64Þ ð65Þ Step 4: (i) (ii) (iii) Determine T2 ; from Eq. (5). If td\T, let T ¼ T2 and TC ¼ TC2; otherwise, go to step (ii). Determine T3 ; from Eq. (5). If M\T td, let T ¼ T3 and TC ¼ TC3; otherwise, go to step (iii). Let T = arg min TC2; TC3 , output the optimal T and TC . Algorithm II (two-warehouse system) Step 1: Input all the parameters of the inventory system. Step 2: Compare the values of M and td. If M\td, then go to step 3, and if M [ td, go to step 4. Step 3: (i) (ii) (iii) (iv) (v) (vi) (vii) (i) (ii) (iii) Step 4: Determine tw6 and T6 , from Eqs. (15) and (16). If tw6 \T6 , let tw ¼ tw6 , T ¼ T6 , and TC ¼ TC6ðtw6 ; T6 Þ; otherwise, go to step (ii). Determine tw10 and T10, from Eqs. (35) and (36). If M\tw10 td\T10, let tw ¼ tw10 , T ¼ T10, and TC ¼ TC10ðtw10 ; T10Þ; otherwise, go to step (iii). Determine tw11 and T11, from Eqs. (39) and (40). If tw11 \T11, let tw ¼ tw11 , T ¼ T11, and TC ¼ TC11ðtw11 ; T11Þ; otherwise, go to step (iv). Determine tw14 and T14, from Eqs. (54) and (55). If M\tw14 \T14 td, let tw ¼ tw14 , T ¼ T14, and TC ¼ TC14ðtw14 ; T14Þ; otherwise, go to step (v). Determine tw15 and T15, from Eqs. (58) and (59). If tw15 \M\T15 td, let tw ¼ tw15 , T ¼ T15, and TC ¼ TC15ðtw15 ; T15Þ; otherwise, go to step (vi). Determine tw16 and T16, from Eqs. (62) and (63). If tw16 \T16, let tw ¼ tw16 ; T ¼ T16, and TC ¼ TC16ðtw16 ; T16Þ; otherwise, go to step (vii). Let tw; T = arg min n TC6ðtw6 ; T6 Þ; TC10 ðtw10 ; T10Þ; TC11ðtw11 ; T11Þ; TC14ðtw14 ; T14Þ; TC15 ðtw15 ; T15Þ; TC16ðtw16 ; T16Þg, output the optimal tw, T and TC . Determine tw7 and T7 , from Eqs. (19) and (20). If tw7 \T7 , let tw ¼ tw7 , T ¼ T7 , and TC ¼ TC7ðtw7 ; T7 Þ; otherwise, go to step (ii). Determine tw8 and T8 , from Eqs. (23) and (24). If tw8 \M\T8 , let tw ¼ tw8 , T ¼ T8 , and TC ¼ TC8ðtw8 ; T8 Þ; otherwise, go to step (iii). Determine tw9 and T9 , from Eqs. (27) and (28). If tw9 \T9 M, let tw ¼ tw9 , T ¼ T9 , and TC ¼ TC9ðtw9 ; T9 Þ; otherwise, go to step (iv). (iv) (v) The following examples illustrate our solution procedure when single warehouse (Model-I) is considered. Example 1 (M\td) Consider an inventory system with the following data: k ¼ 450, D ¼ 1000; ho ¼ 10, c ¼ 20, p ¼ 25, Ie ¼ 0:2, Ip ¼ 0:5, M ¼ 0:0833, a ¼ 0:08, and td ¼ 0:1045, in appropriate units. In this case, we see that M\td: Therefore, applying algorithm I, we get the optimal solutions, T ¼ 0:5554, the corresponding total cost TC ¼ 5092:42, and the ordering quantity Q ¼ 563:64: Example 2 (M [ td) The data are the same as in Example 1 except: M ¼ 0:0417 and td ¼ 0:0322, in appropriate units. Here, we see that M [ td. Therefore, applying algorithm I, we get the optimal solutions, T ¼ 0:2067, the corresponding total cost TC ¼ 3712:26, and Q ¼ 207:90: Example 3 (td [ T ) The data are the same as in Example 1 except: M ¼ 0:99 and td ¼ 0:9984, in appropriate units. In this case, we see that td [ T . Therefore, applying algorithm I, we get the optimal solutions, T ¼ 1:0440, the corresponding total cost TC ¼ 8206:40, and Q ¼ 1044:10: To illustrate the situations, where two warehouses (Model-II) are considered, we have the following set of examples. Example 4 (M\td) Consider an inventory system with the following data: k ¼ 450, D ¼ 1000; hr ¼ 15; ho ¼ 10, c ¼ 20, p ¼ 25, Ie ¼ 0:2, Ip ¼ 0:5, M ¼ 0:0833, W ¼ 100, a ¼ 0:08, b ¼ 0:02, and td ¼ 0:1045, in appropriate units. Here, we see that M\td: Therefore, applying algorithm II, we get the optimal solutions tw ¼ 0:1179 and T ¼ 0:2429, the corresponding total cost TC ¼ 2714:80, and Q ¼ 251:88: Example 5 (M [ td) The data are the same as in Example 4 except: M ¼ 0:0417, and td ¼ 0:0322, in appropriate units. Here, we see that M [ td. Therefore, applying algorithm II, we get the optimal solutions tw ¼ 0:0888 and T ¼ 0:2502, the corresponding total cost TC ¼ 3505:30, and Q ¼ 252:51: Example 6 (td [ T ) The data are the same as in Example 4 except: M ¼ 0:99 and td ¼ 0:9984, in appropriate units. Here, we see that td [ T . Therefore, applying algorithm II, we get the optimal solutions tw ¼ 0:3548 and T ¼ 0:9874, the corresponding total cost TC ¼ 1379:60, and Q ¼ 458:91: Comparative study of the results between the two models Comparative study with respect to the major parameters for the single and two-warehouse models is done in this section. In this article, we discussed two models. Single warehouse is considered in Model-I and Model-II is framed with two-warehouse system. Different scenarios based on the time in which the product deteriorates is classified. In ‘‘Numerical examples’’, we have given six numerical data sets for obtaining the solution using the computational algorithms. Example 1, Example 2, and Example 3 represent the single-warehouse model (Model-I) for the various scenarios M\td,M [ td, and td [ T , respectively. From Example 3, when td [ T , the total cost of the singlewarehouse inventory system is TC ¼ 8206:40 and Q ¼ 1044:10: From this, we infer that the retailer should avail the permissible delay in payment before the cycle time, so that the total cost of the inventory system can be reduced when compared to the case M\td and M [ td. Similarly, from Example 4 (M\td) and Example 5 ðM [ tdÞ which represent two-warehouse system (Model-II), we see that the total cost of the inventory system in the case M\td is less than the total cost of case M [ td. In addition, Table 4 infers that the total cost of the inventory system is reduced effectively when the retailer avails the rented warehouse facility, that is, when the retailer adopts two-warehouse storage facilities. For example, under scenario M\td, the Table 4 Comparison of the results between the two models Model Scenario tw T Q TC Single warehouse M\td M [ td td [ T Two warehouses M\td M [ td td [ T Table 5 Effect of change in various parameters of the inventory in the two-warehouse model Changing parameter Change in parameter tw Table 6 Optimal solutions for different ordering cost k in Example 4 Table 7 Sensitivity analysis with respect to the parameters hr and ho W 50 75 100 k 450 500 550 450 500 550 450 500 550 D total cost of the system TC ¼ 5092:42 which is effectively reduced to TC ¼ 2714:80 when the retailer avails the rented warehouse facility. Furthermore, consider the case ðM [ tdÞ, the total cost of the integrated system in singlewarehouse model is TC ¼ 3712:26, whereas in twowarehouse model, the total cost is TC ¼ 3505:30 (less = 206.96). In addition when we consider the case td [ T, the difference between the total cost in two models is very much significant (8206:40 - 1379:60 = 6826.80). In all the scenarios, the total cost is effectively reduced in a two-warehouse model comparatively. Furthermore, the comparative study infers that the retailer should order less quantity more frequently in two-warehouse model, but in single-warehouse model, the optimal replenishment policy suggests that more quantity may be ordered less frequently. Therefore, the retailer can gain more profit by improving the storage facility such as warehouses, godowns, and so on to store materials. Table 8 Sensitivity analysis with respect to the parameters a and b Managerial implication In this section, we perform the sensitivity analysis on the key parameters of Model-II, to study their effect on the inventory system. The results are summarized in Tables 5, 6, 7, and 8 and the graphical representation of the sensitivity analysis is shown in Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16. Based on the computational results obtained hr # 22 24 26 28 a # 0.1188 0.2527 2638.60 218.83 0.1180 0.2519 2639.30 218.02 0.1172 0.2511 2640.00 217.20 0.1167 0.2508 2640.60 216.70 0.09 Fig. 6 Effect of change in c on the optimal solution Fig. 7 Effect of change in p on the optimal solution from the sensitivity analysis, the following inferences can be made from managerial view point: When k increases, the optimal cycle time T and the minimum total relevant cost per unit time TC increase simultaneously. For example, when W = 50 and D = 1000, k increases from 450 to 550 units, T increases from 0.2709 to 0.3006, and also TC increases from 2542.20 to 2892.40. This implies that, from managerial view point, if the ordering cost per order is reduced effectively, then the total cost per unit • time could be reduced. The retailer should order more quantity per order when the ordering cost per order is high. When retailer’s warehouse capacity W is increasing, the optimal replenishment cycle time T will decrease, but the relevant total costs TC will increase. For example, when k = 450/order and D = 1000 units, W increases from 50 to 100 units, T decreases from 0.2709 to 0.2429, but TC increases from 2542.20 to Fig. 11 Effect of change in Ip on the optimal solution 2714.80. This implies that the retailer should order less frequently to reduce the total inventory cost when warehouse storage capacity is more. When there is an increase in the value of M, the optimal order quantity Q increases, whereas the optimal total cost TC decreases. This shows that the retailer can minimize the total cost if the retailer obtains a longer permissible delay period from the supplier. When the holding cost increases, the length of the cycle time T decreases and the total cost TC increases. If • the retailer can effectively reduce the holding cost of the item by improving equipment of storehouse, the total cost will be lowered. When the holding cost increases, the ordering quantity Q decreases. From the managerial point of view, when the holding cost for a product is more, the retailer should order less. When the fresh product time increases, the optimal total cost TC decreases and Q increases. Hence, from our model, we suggest that when the fresh product time of a product is more, the retailer should order more quantity. In addition, it shows that the model with nonFig. 14 Effect of change in b on the optimal solution Fig. 15 Effect of change in td on the optimal solution instantaneous deteriorating items always has smaller total annual inventory cost than with instantaneous deteriorating items. If the retailer can extend effectively the length of time, the product has no deterioration for a few days or months, then the total annual cost will be reduced obviously. When the selling price p increases, there is a decrease in the optimal order quantity Q . The larger the value of p, the smaller is the value of the optimal cycle time T . That is, when the unit selling price is increasing, the retailer will order less quantity more frequently. Fig. 16 Effect of change in M on the optimal solution Conclusion The purpose of this article is to frame a model that will help the retailer to determine the optimal replenishment policy for non-instantaneous deteriorating items. The supplier offers a permissible delay in payments with two levels of storage facilities. Our model suits well for the retailer in situations involving unlimited storage space. Thus, the decision maker can easily determine whether it will be financially advantageous to rent a warehouse to hold much more items to obtain a trade credit period. It was assumed that the rented warehouse charges are higher holding cost than the owned warehouse. To reduce the inventory costs, it will be economical to consume the goods of the rented warehouse at the earliest. From the results obtained, we see that the retailer can reduce total annual inventory cost by ordering lower quantity when the supplier provides a permissible delay in payments by improving storage conditions for non-instantaneous deteriorating items. Incorporating more realistic assumptions such as allowable shortages, probabilistic demand, or quantity discounts, this article paves way to extend future research works. Acknowledgements The authors are grateful to the Editor and the anonymous reviewers for their valuable suggestions and constructive comments which have led to a significant improvement of this manuscript. appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Appendix Proof of Lemma 1 Based on the assumptions, 0\b\a\\1; ho þ ca\hr þ cb; D is sufficiently small. we know that aW [ 0 and bM Let f ðtÞ ¼ D hr þ cb þ cIpe btd ebt cb þ cIpÞeat; t [ 0, then we have f ð0Þ ¼ D hr þ cb þ cIpe btd aW hr þ cb þ cIp aW ðhr þ ¼ ðhr þ cbÞðD f 0 ðtÞ ¼ bD hr þ cb þ cIpe btd ebt þ a2W ðhr þ Hence, f ðtÞ is an increasing function and f ðtÞ [ 0 for all t [ 0: As a result, D hr þ cb þ cIpe btd ebt [ aW hr þ cb þ cIp eat [ aW ho þ ca þ cIp eat holds. Let gðtÞ ¼ D hr þ cb þ cIpe bM ebt cbþ cIpÞeat; t [ 0, then we have aW ðhr þ gð0Þ ¼ D hr þ cb þ cIpe bM aW hr þ cb þ cIp ¼ ðhr þ cbÞðD [ ðhr þ cbÞðD aWÞ þ cIpðDe bM aWÞ þ cIpðDð1 aWÞ bMÞ aWÞ [ 0; cb þ cIpÞeat; t [ 0. and g0 ðtÞ ¼ bD hr þ cb þ cIpe bM ebt þ a2W ðhr þ Hence, gðtÞ is an increasing function and gðtÞ [ 0 for all t [ 0: As a result, Dðhr þ cb þ cIpe bM Þebt [ aW hr þ cb þ cIp eat [ aW ðho þ ca þ cIpÞeat holds. 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R. Udayakumar, K. V. Geetha. An EOQ model for non-instantaneous deteriorating items with two levels of storage under trade credit policy, Journal of Industrial Engineering International, 2017, 1-23, DOI: 10.1007/s40092-017-0228-4