2.1 SURVEY OF INVENTORY MODELS

Transcription

1 CHAPTER-II LITERATURE REVIEW Supply chain models are very much important in inventory control situations. An effective supply chain network requires a cooperative relationship between the manufacturers, the distributors and the retailers. 2.1 SURVEY OF INVENTORY MODELS Review of literature and survey of the developed inventory models are being treated for consideration in this section of the chapter two. Although there is an abundance of the literature of general inventory systems, however, an attempt has been made to cover some of them related to the problem. The aim of this section is to present a complete and updated picture of inventory theory related to the problem Inventory Models With Inflation In the classical inventory it is assumed that all the costs associated with the inventory system remains constant over time. Since most decision makers think that the inflation does not have significant influence on the inventory policy and most of the inventory models developed so far does not include inflation and time value of money as parameters of the system. But due to large scale of inflation the monetary situation in almost all the countries has changed to an extent during the last thirty years. Nowadays inflation has become a permanent feature in the inventory system. Inflation enters in the picture of inventory only because it may have an impact on the present value of the future inventory cost. Thus the inflation plays a vital role in the inventory system and production management though the decision makers may face difficulties in arriving at answers related to decision making. At present, it is impossible to ignore the effects of inflation and it is necessary to consider the effects of inflation on the inventory system. Buzacott (1975) was the first author to include the concept of inflation in inventory modeling. He showed that the effect of inflation results in cost increases. The effect of inflation was considered in the main cost components, and the optimal order quantity expression was developed. He developed deterministic inventory models under inflation for two cases, first

2 when the price is subject to the same inflation rate as costs and in the second case when the price is dependent on ordering policy assuming constant rate of inflation which means that cost C(t) at time t becomes C(t + δt) at time (t + δt) through inflation, where, c(t+δt)=c(t)+kc(t)δt dc(t) kt =kc(t) which has solution C(t)=C 0e, where C 0 is cost dt as δt at time zero. After the pioneer work of Buzacott (1975), it was found that inflation plays a vital role in the inventory system and production management though the decision makers may face difficulties in arriving at answers related decision making. At present, it is impossible to ignore the effects of inflation. Between 1975 to 1985 many researchers have been addressing the inflationary effect on an inventory policy. Misra (1975) investigated inventory systems under the effects of inflation. Ray and Chaudhuri (1977) presented an EOQ model under inflation and time discounting allowing shortages. Biermans and Thomas (1977) suggested the inventory decision policy under inflationary conditions. Donaldson (1977) was the first one to have introduced the concept of variable demand rate to the research world. But this paper did not incorporate the concept of inflation. An expression for the EOQ was derived by minimizing the average annual cost. Misra (1975-a, 1979) investigated inventory systems under the effects of inflation. Bierman and Thomas (1977) suggested the inventory decision policy under inflationary conditions. Datta et al. (1991) made an attempt to investigate the effects of inflation and time-value of money on an inventory model with linear time-dependent demand rate without shortages. The model developed in this article is a finite-horizon (T, Si) policy inventory model. Hariga (1994) studied the effects of inflation and time value of money on the replenishment policies of items with time continuous non-stationary demand over a finite planning horizon. He developed dynamic programming models for three commonly used replenishment policies in the inventory lot-sizing literature with time varying demand and shortages. Economic analysis of dynamic inventory models with non-stationary costs and demand was presented by Hariga (1994). The effect of inflation was also considered in this analysis. An economic order quantity inventory model for deteriorating items was developed by Bose et al. (1995). Authors developed inventory model with linear trend in demand allowing inventory shortages and backlogging. The effects of inflation and time-value of money were

3 incorporated into the model. The inventory policy was discussed over a finite time-horizon with several reorder points. It was assumed that the goods in the inventory deteriorate over time at a constant rate. The results were discussed with a numerical example and sensitivity analysis of the optimal solution with respect to the parameters of the system was carried out. Several particular cases of the model were discussed in brief. Effects of inflation and time-value of money on an inventory model was discussed by Hariga (1995) with linearly increasing demand rate and shortages. Hariga and Ben-Daya (1996) then discussed the inventory replenishment problem over a fixed planning horizon for items with linearly time-varying demand under inflationary conditions. Ray and Chaudhuri (1997) developed a finite time-horizon deterministic economic order quantity inventory model with shortages, where the demand rate at any instant depends on the on-hand inventory at that instant. The effects of inflation and time value of money were taken into account. A generalized dynamic programming model for inventory items with Weibull distribution deterioration was proposed by Chen (1998). The demand was assumed to be time-proportional, and the effects of inflation and time-value of money were taken into consideration. Shortages were allowed and partially backordered. The effects of inflation and time-value of money on an economic order quantity model have been discussed by Moon and Lee (2000). Authors have considered the normal distribution as a product life cycle in addition to the exponential distribution. The two-warehouse inventory models for deteriorating items with constant demand rate under inflation were developed by Yang (2004). The shortages were allowed and fully backlogged in the models. Some numerical examples for illustration were provided. Chang (2004) proposed an inventory model under a situation in which the supplier has provided a permissible delay in payments to the purchaser if the ordering quantity is greater than or equal to a predetermined quantity. Shortage was not allowed and the effect of the inflation rate, deterioration rate and delay in payments were discussed as well. An inventory model for deteriorating items with stock-dependent consumption rate with shortages was produced by Hou (2006). Model was developed under the effects of inflation and time discounting over a finite planning horizon. The results were discussed with a numerical example and particular cases of the model were discussed in brief. Sensitivity analysis of the

4 optimal solution with respect to the parameters of the system was carried out. Models for ameliorating / deteriorating items with time-varying demand pattern over a finite planning horizon were proposed by Moon et al.(2005). The effects of inflation and time value of money were also taken into account. Jolai et al. (2006) presented an optimization framework to derive optimal production over a fixed planning horizon for items with a stock-dependent demand rate under inflationary conditions. Deterioration rate was taken as two parameter Weibull distribution function of time. Shortages in inventory were allowed with a constant backlogging rate. Two-warehouse partial backlogging inventory models for deteriorating items were discussed by Yang (2006). The inflationary effect was considered in the models. Deterioration rates in both the warehouses were taken as constant. Some numerical examples for illustration were provided and sensitivity analysis on some parameters was made. Jaggi et al. (2007) presented the optimal inventory replenishment policy for deteriorating items under inflationary conditions using a discounted cash flow (DCF) approach over a finite time horizon. Shortages in inventory were allowed and completely backlogged and demand rate was assumed to be a function of inflation. Optimal solution for the proposed model was derived and the comprehensive sensitivity analysis has also been performed to observe the effects of deterioration and inflation on the optimal inventory replenishment policies. Two stage inventory problem over finite time horizon under inflation and time value of money was discussed by Dey et al. (2008) Inventory Models With Permissible Delay In Payments Inventory policy with trade credit financing was formulated by Haley and Higgins (1973). Retailer s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments was discussed by Hwang and Shinn (1997).The effect of payment rules on ordering and stock holding in purchasing was suggested by Kingsman (1983). Different inventory policies with trade credit was developed by Chapman et al. (1984) and Chapman and Ward (1988). Goyal (1985) was the first to develop the economic order quantity under conditions of permissible delay in payments. Author had assumed that the unit selling price and the purchase price were equal. In practice, the unit selling price should be

5 greater than the unit purchasing price. Dave (1985) corrected Goyal s model by assuming the fact that the selling price is necessarily higher than its purchase price. Aggarwal and Jaggi (1995) developed ordering policies of deteriorating items under permissible delay in payments. The demand and deterioration were considered as constant. An ordering policy for deteriorating items under permissible delay in payments was proposed by Jamal et al. (1997). In this policy shortage was allowed. Jamal et al. (2000) presented optimal payment time for a retailer under permitted delay of payment by the wholesaler. The wholesaler allowed a permissible credit period to pay the dues without paying any interest for the retailer. In the study, a retailer model was considered with a constant rate of deterioration. The inventory replenishment policy for deteriorating items under permissible delay in payments was presented by Chung (2000). The problem was illustrated numerically. An inventory model for initial-stock-dependent consumption rate was suggested by Liao et al. (2000). Shortages were not allowed. The effects of inflation rate, deterioration rate, initial stock-dependent consumption rate and delay in payments were discussed. Chang and Dye (2001) developed and inventory model for deteriorating items with partial backlogging and permissible delay in payments. Chung et al. (2002) have discussed the inventory decision for EOQ inventory model under permissible delay in payments. Dye (2002) developed a deteriorating inventory model with stock-dependent demand and partial backlogging. The conditions of permissible delay in payments were also taken into consideration. The shortages were neither completely backlogged nor completely lost assuming the backlogging rate to be inversely proportional to the waiting time for the next replenishment. Numerical examples were also presented to illustrate the model numerically. An EOQ model for deteriorating items with credit policy was discussed by Chang et al. (2003). Lot sizing decision policy was presented by Chung and Liao (2004). In this policy trade credit was depending on the ordering quantity. The effect of trade credit policy with limited storage capacity within the economic order quantity was proposed by Chung and Huang (2004). Numerical examples were given to illustrate all the theorems obtained in the study. The optimal cycle time for deteriorating items

6 under permissible delay in payments was proposed by Chung and Huang (2005). Teng et al. (2005) presented an optimal pricing and ordering policy under permissible delay in payments. Deterioration rate was taken as constant and shortages were not allowed in the study. The optimal ordering policy in a DCF analysis for deteriorating items was suggested by Chung and Liao (2006). In this policy trade credit was also depending on the ordering quantity. Huang (2007) developed optimal retailer s decisions under two levels of trade credit policy with in the economic production quantity (EPQ) framework. Author has assumed that the supplier would offer the retailer a delay period and the retailer also adopted the trade credit policy to stimulate his/her customer demand to develop the retailer s replenishment model. Replenishment rate was taken as finite. Chen and Kang (2007) developed the integrated models with the permissible delay in payments for determining the optimal replenishment time interval and replenishment frequency. Sana and Chaudhuri (2008) formulated the retailer s profit maximizing strategy. Increasing deterministic demands were discussed in the environment of permissible delay in payment and discount offer to the retailer. An inventory model under two levels of trade credit policy was proposed by Jaggi et al. (2008) with credit-linked demand. A mathematical was developed by Soni and Shah (2008) to formulate optimal ordering policies for retailer when demand was partially constant and partially dependent on the stock. Progressive credit period was offered to settle the account by the supplier Inventory Models For Deteriorating Items Commodities such as fruits, vegetables and foodstuffs suffer from depletion by direct spoilage while kept in store. Highly volatile liquids such as alcohol, gasoline, etc. undergo physical depletion over time through the process of evaporation. Electronic goods, photographic film, grain, chemicals, pharmaceuticals etc. deteriorate through a gradual loss of potential or utility with the passage of time. Thus deterioration of physical goods in stock is very realistic feature. As a result, while determining the optimal inventory policy of that type of products, the loss due to deterioration cannot be ignored. In the literature of inventory theory, the deteriorating inventory models have been continually modified so as to accumulate more practical features of the real inventory systems. The analysis of deteriorating inventory began with Ghare and Schrader (1963), who established

7 the classical no-shortage inventory model with a constant rate of decay. However, it has been empirically observed that failure and life expectancy of many items can be expressed in terms of Weibull distribution. This empirical observation has prompted researchers to represent the time to deterioration of a product by a Weibull distribution. Covert and Philip (1973) extended Ghare and Schrader s (1963) model to obtain an economic order quantity model for a variable rate of deterioration by assuming a two-parameter Weibull distribution. Philip (1974) presented an EOQ model for items with Weibull distribution deterioration rate. Pierskalla and Roach (1972) suggested optimal issuing policies for perishable inventory. Misra (1975-b) presented a production lot size model for an inventory system with deteriorating items with variable rate of deterioration while rate of production was finite. An order level inventory model for a system with constant rate of deterioration was presented by Shah and Jaiswal (1977). Roychowdhury and Chaudhuri (1983) formulated an order level inventory model for deteriorating items with finite rate of replenishment. Dave and Patel (1981) suggested an inventory model for deteriorating items with time proportional demand. Aggarwal (1978) and Aggarwal and Jaggi (1989) presented inventory ordering policies for deteriorating items. Hollier and Mak (1983) developed inventory replenishment policies for deteriorating item with demand rate decreases negative exponentially and constant rate of deterioration. The authors developed two models, in the first model it was assumed that replenishment orders were placed at an equal interval. Goyal (1986) and Gupta and Vrat (1986) suggested inventory models with variable rates of demand. Dave (1986) presented an order level inventory model for deteriorating items. Model was continuous in units but allowed discrete opportunities for replenishment. In this model, demand kept on changing from time unit to time unit and occurs instantaneously at the beginning of each time unit. Deterioration rate was assumed to be constant and lead-time was zero. A deterministic inventory system with stock dependent demand rate was formulated by Baker and Urban (1988). Dutta and Pal (1988) investigated an order level inventory model with power demand pattern with a special form of Weibull function for deterioration rate, considering deterministic demand as well as probabilistic demand. Some particular cases for demand pattern have also been discussed in the model.an inventory model concerning a single item was suggested by

8 Mandal and Phaujdar (1989) for deteriorating items with a variable rate of deterioration. Model was developed with stock-dependent consumption rate and uniform production rate. Shortage was allowed and the excess demand was backlogged as well. The rate of deterioration was first assumed as constant and then variable. Dutta and Pal (1990) presented a note on an inventory level dependent demand rate. An EOQ model for deteriorating items with a linear trend in demand was formulated by Goswami and Chaudhuri (1991). Survey of literature on continuously deteriorating inventory models was presented by Raafat (1991). Inventory model for deteriorating items with stock dependent demand rate was proposed by Pal et al. (1993). Inventory models for perishable items with stock dependent selling rate were suggested by Padmanabhan and Vrat (1995). The selling rate was assumed to be a function of current inventory level and rate of deterioration was taken to be constant with complete, partial backlogging and without backlogging. Sachan (1984) and Hill (1995) formulated inventory policies for deteriorating items with time varying demand. An inventory model with exponential demand and constant rate of deterioration was proposed by Kishan and Mishra (1995). Shortages were allowed in the model. Wee (1995) presented inventory policies for deteriorating items with declining market. Balkhi and Benkherouf (1996) presented a production lot size model for deteriorating items and arbitrary production and demand rates. In this model demand and production were allowed to vary with time in an arbitrary way and shortages were allowed. The rate of deterioration was taken as constant and the method was illustrated with the help of numerical example. Giri et al. (1996) discussed an inventory model with an inventory-level-dependent demand rate followed by a constant demand rate for items deteriorating at a constant rate. The sensitivity of the decision variables to change in the parameter values was examined and the effects of these changes on the optimal policy were discussed in brief. Mandal and Maiti (1997) presented an inventory model for damagable items with fully backlogged shortages for both linear and non-linear damage and demand functions. The authors considered production rate as finite and solved the model with profit maximization criteria together with sensitive analysis. An economic order quantity model for perishable products with nonlinear holding cost was proposed by Giri and Chaudhuri (1998). The demand rate was taken as function of onhand inventory. Su et al. (1999) formulated a deterministic production inventory model for

9 deteriorating items with an exponential declining demand over a fix time horizon. The production rate at any instant depends on demand at that time. The item deteriorates at a constant rate and shortages were allowed in the model. Approximate expressions were obtained for the production scheduling period, maximum inventory level, unfilled order backlog and the total average cost. Some special cases and a numerical example were also discussed in the concluding phase of the model. An EOQ inventory model with ramp-type demand rate for items with Weibull deterioration was formulated by Wu et al. (1999). Giri et al. (2000) presented a note on a lot sizing heuristic for deteriorating items with time varying demand and shortages. An order level inventory model for deteriorating items was proposed by Gupta and Agarwal (2000). The demand was taken as a linear function of time and production rate was taken as demand dependent. A production inventory model for deteriorating items with exponential declining demand was discussed by Kumar and Sharma (2000). Time horizon was fixed and the production rate at any instant was taken as the linear combination of on hand inventory and demand. Approximate expressions were obtained for production scheduling period, maximum inventory level, unfilled order backlog and total average cost. Aggarwal and Jain (2001) presented an inventory model for exponentially increasing demand rate with time. The items were deteriorating at a constant rate and shortages were allowed. The authors solved the model by two methods. The first method was efficient for getting accurate results, although, lengthy and complicated while the second method was faster than the first method and thus easy to apply. Model for deteriorating items over a finite planning horizon was considered by Balkhi (2001). The demand, production and deterioration rate were assumed to be known and continuous functions of time. Shortages were allowed and completely backlogged. Chung and Lin (2001) proposed the discounted cash flow approach to investigate inventory replenishment problem for deteriorating items taking account of time value of money over a fixed planning horizon. The constant rate of deterioration was applied to on-hand inventory. Two models were analyzed; in the first model shortage was not allowed and in the second model shortage was allowed and it was fully backlogged. Goyal and Giri (2001-a) presented a review of the advances of deteriorating inventory literature since early 1990s. The models available in the relevant literature have been suitably

10 classified by shelf-life characteristic of the inventoried goods. A comment on Chang and Dye (1999) was presented by Goyal and Giri (2001-b). In this comment an EOQ model for deteriorating items with time varying demand and partial backlogging was proposed. A singlevender and multiple-buyers production-inventory policy for a deteriorating item was formulated by Yang and Wee (2002). Production and demand rates were taken to be constant. A mathematical model incorporating the costs of both the vender and the buyers was developed. Goyal and Giri (2003) considered the production-inventory problem in which the demand, production and deterioration rates of a product were assumed to vary with time. Shortages of a cycle were allowed to be partially backlogged. Two models were developed for the problem by employing different modeling approaches over an infinite planning horizon. Solution procedures were derived for determining the optimal replenishment policies. An order level inventory model for deteriorating items with an exponential increasing demand was proposed by Sharma and Singh (2003). Huang (2003) formulated the deterministic inventory models with shortage and defective units. An order-level inventory problem for a deteriorating item with time dependent quadratic demand was presented by Khanra and Chaudhuri (2003). The inventory was assumed to deteriorate at a constant rate and shortages was not allowed. The model was solved analytically for an infinite time-horizon. Economic order quantity model with Weibull distribution deterioration, shortage and ramp type demand was formulated by Giri et al. (2003). A production-inventory model for a deteriorating item over a finite planning horizon was presented by Sana et al. (2004). Deterioration rate was taken as constant fraction of the on-hand inventory. The demand was taken to be linear time-varying function. An order-level inventory model for a deteriorating item with Weibull distribution deterioration, time-quadratic demand and shortages was suggested by Ghosh and Chaudhuri (2004). An economic production quantity model for deteriorating items was discussed by Teng and Chang (2005). Demand rate was taken as dependent on the display stock level and the selling price per unit. Authors have provided the necessary conditions to determine an optimal solution for maximization of profit for the EPQ model. Sensitivity analysis was applied on the parameter effects of the optimal price and the production run time.

11 A deteriorating multi-item inventory model with limited storage capacity was suggested by Mandal et al. (2006). The demand rate for the items was finite and deterioration rate was taken to be constant. The replenishment rate was taken as constant. The problem was supported by numerical examples. An order level inventory system for deteriorating items has been discussed by Manna and Chaudhuri (2006). The demand rate was taken as ramp type function of time and the finite production rate was proportional to the demand rate at any instant. The deterioration rate was time proportional and the unit production cost was inversely proportional to the demand rate. Results were illustrated with two numerical examples along with its sensitivity. Order level inventory systems with ramp type demand rate for deteriorating items were discussed by Mandal and Pal (1998) and Panda et al. (2007). A note on the inventory models for deteriorating items with ramp type demand was developed by Deng et al. (2007). They have proposed an extended inventory model with ramp type demand rate and its optimal feasible solution Two Warehouse Inventory Models Sarma (1983) proposed a two warehouse inventory model by assuming the cost of transporting K-unit from RW to OW as constant and called it as K-release rule (KRR). The rate of replenishment was assumed as infinite. Murdeshwar and Sathe (1985) formulated some aspects of lot size models with two level of storage and derived complete solution for optimum lot size under finite production rates. The authors assumed while deriving the K-release rule that K units were transferred n-times from OW to RW during production stage with constant transportation cost. Sarma (1987) developed a deterministic inventory model for a single deteriorating item which was stored in two different warehouses of non deteriorating product. The preserving facilities were better in rented warehouse than own warehouse resulting in a lower rate of deterioration. Goswami and Chaudhuri (1992) developed an economic order quantity model for items with two levels of storage for a linear trend in demand. In the first phase of the paper, a deterministic model without shortage was developed with numerical examples and in the second phase deterministic model with shortage was considered with numerical example and sensitivity analysis. In this model, the time interval between two successive shipments was constant while

12 the numbers of units transported in a shipment was different and increased with the number of shipment. However, the above assumption is doubtful in many cases like consumer goods type, where the customers are attracted with the easy availability or abundance supply of the items. An inventory model for deteriorating items with two warehouses was formulated by Pakkala and Achary (1992). Pakkala and Achary (1994) proposed an inventory model for deteriorating products when two separate warehouses were used. In developing the model, the demand was assumed as uniform with finite replenishment rate and shortages were allowed in the model. An analytical study of the model was presented and the model for the case of nondeteriorating items was deduced. A two warehouse inventory model for a linear trend in demand was proposed by Bhunia and Maiti (1994) for single item with infinite rate of replenishment and linear increasing trend in demand. Shortage was fully backlogged. The transportation cost was assumed to be dependent on the quantity transported from RW to OW. The optimal values of the number of shipment made, quantity transported in each shipment from RW to OW and highest storage level with cost minimization criteria were obtained, sensitivity analysis was shown with graphs at the end of the paper together with concluding remarks. A deterministic order level inventory model for deteriorating items with two storage facilities was discussed by Benkherouf (1997). Bhunia and Maiti (1998) developed a deterministic inventory model with two warehouses for deteriorating items taking linearly increasing demand with time, shortages were allowed and excess demand was backlogged as well. Stock was transferred from RW to OW under a continuous release pattern taking transportation cost into account. The deterioration rate was different in both the warehouses and scheduling period was taken as variable. Optimal inventory policy for deteriorating items with two warehouse and time dependent demand was produced by Lee and Ma (2000). Yang (2004) developed the two-warehouse inventory models for deteriorating items with constant demand rate under inflation. An inventory model with two warehouses and stock-dependent demand rate was proposed by Zhou and Yang (2005). Shortages were not allowed in the model and the transportation cost for transferring items from RW to OW was taken to be dependent on the transported amount. A computational procedure was proposed to obtain the optimal replenishment quantity, the optimal replenishment cycle and the optimal shipment schedule so that the average total profit of the system should be

13 minimum. Two warehouse inventory models with LIFO and FIFO dispatching policies were developed by Lee (2006). In this study, Pakkala and Achary s (1992) model was first modified and then a FIFO dispatching two-warehouse model with deterioration was propped. It has been observed by the comparison of these two models that the FIFO model is less expensive to operate than LIFO, if the mixed effects of deterioration and holding cost in RW were less than that of OW. Niu and Xie (2008) presented a note on Lee s (2006) model. This note points out that the conclusion of Lee s model was incorrect and misleading. Authors have provided a new sufficient condition such that the modified LIFO model always has lower cost than Pakkala and Achary s (1992) model. Besides, authors have compared Pakkala and Achary s original LIFO model with Lee s FIFO model for the special case where the two warehouses have the same deterioration rates. Hsieh et al. (2008) suggested a deterministic inventory model for deteriorating items with two warehouses by minimizing the net present value of the total cost. Deterioration rates in both warehouses were taken as different. Shortage in inventory was allowed and fully backlogged. A two warehouse inventory model with shortages for a deteriorating item has been formulated by Rong et al. (2008) and an inventory policy was proposed for maximum profit Supply Chain Inventory Models It is only when these requirements are seen together as part of a large picture that ways can be found to effectively balance their different demands. Customer service at its most basic level means consistently high order fill rates, high ontime delivery rates and a very low rate of products returned by customers for whatever reason. Internal efficiency for organizations in a supply chain means that these organizations get an attractive rate of return on their investments in inventory and other assets. Manufacturers procure raw material and process them in to finished goods, and sell the finished goods to distributors, then to retailer and/or customer. When an item moves through more than one stage before reaching the final customer, it forms a multi-echelon inventory system. A large amount of researches on multi-echelon inventory control has appeared in the literature during the last decades. Clark and Scarf (1960) were the first to study the two-echelon inventory model. They proved the optimality of a base stock policy for the pure serial inventory system and developed

14 an efficient decomposing method to compute the optimal base stock ordering policy. Sherbrooke (1968) considered an ordering policy of two-echelon model for warehouse and retailer. It is assumed that stockouts at the retailers are completely backlogged. Diks and De Kok (1998) determined a cost optimal replenishment policy for a divergent multi-echelon inventory system. Van der Heijden et al. (1997) presented stock allocation policies in general single-item and N-echelon distribution systems, where it is allowed to hold stock at all levels in the network. A joint replenishment policy for multi-echelon inventory control was proposed by Axsater and Zhang (1999). A multi-echelon inventory model for a deteriorating item was developed by Rau et al. (2003). An optimal joint total cost has been derived from an integrated perspective among the supplier, the producer and the buyer. A deteriorating item inventory model in a supply chain was proposed by Wu and Wee (2005). Shortages in inventory were allowed and fully backlogged. Two-echelon inventory model with lost sales was proposed by Hill et al. (2007) Partial Backlogging Inventory Models The inventory involving backlogging and lost sales is called mixture inventory. Mixture inventory means mixture of backorders and lost sales. Therefore in the third case, it is assumed that a fixed fraction of the Zangwill (1966) developed a production multi period production scheduling model with backlogging. Inventory models with a mixture of backorders and lost sales were formulated by Montgomery et al. (1973). Rosenberg (1979) presented the analysis of a lot-size model with partial backlogging. Inventory models with partial backorders were proposed by Park (1982). A dynamic production scheduling model with lost sales was proposed by Sung and Rhee (1987). Mak (1987) proposed optimal production-inventory control policies for an inventory system. Shortages in inventory were allowed and partially backlogged. Economic production lot size model for deteriorating items with partial backordering was suggested by Wee (1993). Abad (1996) formulated a generalized model of dynamic pricing and lot-sizing for perishable items. Shortage was allowed and the demand was partially backlogged. A comparison of two replenishment strategies for the lost sales inventory model was presented by Donselaar et al. (1996). Time-limited free back-orders EOQ model was formulated by Abbound and Sfairy (1997). The effects of lost sales on composite lot sizing were discussed by Sharma and

15 Sadiwala (1997). Deteriorating inventory model with quantity discount, pricing and partial backordering was developed by Wee (1999). In this study, demand was assumed to decrease with the increment in price for the product. The numerical example was also given to illustrate the model. An EOQ model for deteriorating items with time varying demand was discussed by Chang and Dye (1999). Shortages in inventory were allowed and partially backlogged. A note on Chang and Dye (1999) was given by Wang (2001). Lead time and ordering cost reduction in continuous review inventory systems with partial backorders was proposed by Ouyang et al. (1999).Optimal price and order size inventory policy for a reseller under partial backlogging was proposed by Abad (2000-a). Shortage in inventory was allowed with partial backordering and lost sale. Author has taken a problem of determining the production lot size of a perishable product that decays at an exponential rate. An EOQ inventory model over a finite planning horizon, with constant deterioration rate and time varying demand was suggested by Papachristos and Skouri (2000). Shortages in inventory were allowed with time-dependent backlogging rate. Wu (2000) developed inventory model for items with Weibull distribution deterioration rate, time dependent demand and partial backlogging. An EOQ inventory model for items with Weibull distribution deterioration rate and ramp type demand was formulated by Wu (2001). Shortages in inventory were allowed and assumed to be partially backlogged. The problem of determining the optimal price and lot size for a reseller was considered by Abad (2001). Selling price was taken as constant in the inventory cycle. Analysis was also presented for the reselling situation in which a nonperishable product was sold. A two-echelon inventory model with lost sales was suggested by Andersson and Melchiors (2001). A type of the capacitated lot-sizing problem was formulated by Cheng et al. (2001). Authors have presented a general model for the lot sizing problem with backorder. The objective was to determine lot sizes that will minimize the sum of setup costs, holding cost, backorder cost, regular time production cost and over time production costs subject to resource constraints. A two reorder level inventory system with renewal demands and partial backlogging was suggested by Longo and Arivarignan (2002). Ouyang and Chang (2002) presented stochastic continuous review inventory model with variable lead time and partial backorders to capture the

16 reality of uncertain backorders. Perumal and Arivarignan (2002) considered a situation with two different rates of production. It was assumed that production started at one rate and after some time it may be switched over to another rate. A continuous review inventory model for perishable items with time-varying demand and shortages was studied by Skouri and Papachristos (2002). The backlogging rate was an exponentially decreasing, time dependent function specified by a parameter. Teng et al. (2002) presented an optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging. The inventory problem for deteriorating items with time-varying demand and shortages over a finite planning horizon was presented by Wang (2002). Author has defined an appropriate time dependent partial backlogging rate and introduced the opportunity cost due to lost sales. An EOQ model for deteriorating items with time-varying demand and partial backlogging was suggested by Teng et al. (2003). Cost comparison of four inventory models for deteriorating items with variable demand was discussed by Skouri and Papachristos (2003). The backlogging rate was considered as to be a function of the waiting time up to the beginning of the next cycle, that is a function of time between the moment that demand occurs and the next production starting time.papachristos and Skouri (2003) considered a model where the demand rate was a convex decreasing function of the selling price and backlogging rate was a time dependent function. The time to deterioration of the item was distributed as two parameter Weibull distribution function of time. Abad (2003) presented the pricing and lot sizing problem for a perishable good under finite production, exponential decay, partial backordering and lost sale. The backlogging phenomenon in the literature was often modeled using backordering and lost sale costs. Aksen et al. (2003) considered the single item lot sizing problem with immediate lost sales. Zhou et al. (2003) considered a variable production scheduling strategy for deteriorating items with variable demand and partial lost sale. In this policy each cycle of a schedule starts with a period of shortages, and then followed by continuous replenishment. A general time-varying demand inventory lot-sizing model with waiting-time-dependent backlogging and a lot-size-dependent replenishment cost was developed by Zhou et al. (2004). Some convenient mathematical properties of the cost function were identified, with which an effective numerical solution procedure was developed for determining the optimal replenishment

17 policy. Chu and Chung (2004) discussed the sensitivity of the inventory model with partial backorders. Deterministic economic order quantity models with partial backlogging were formulated by Teng and Yang (2004). Authors have extended the economic order quantity model to allow for not only time-varying demand but also fluctuating unit cost. The backlogging rate of unsatisfied demand was a decreasing function of the waiting time. A note on inventory model with a mixture of back orders and lost sales was proposed by Chu et al. (2004). The minimum total cost was evaluated under the diversity cycle time and illustrations were applied to explain the calculation process. An EOQ model with time varying deterioration and linear time varying demand over finite time horizon was proposed by Ghosh and Chaudhuri (2005). Shortages in inventory were allowed and partially backlogged with waiting time dependent backlogging rate. An analytic solution of the model was discussed and it was illustrated with the help of a numerical example. An economic manufacturing quantity problem for an unreliable manufacturing system was presented by Giri and Yun (2005) where machine was subject to random failure and at most two failures can occur in a production cycle. The shortages, if occur due to longer repair time, were backlogged partially by resuming the production run after machine repair. Some characteristics of the model with exponential failure and exponential/constant repair times were studied. A single-item production-inventory system over an infinite time horizon with increasing demand and finite production rate was discussed by Giri et al. (2005). The demand during the stock-out period was partially captive with a known fraction. The proposed model was also shown to be suitable for a prescribed time horizon. Numerical example was also taken to illustrate the solution procedure of the developed model. An EOQ model for deteriorating items with linear demand was produced by Lin and Lin (2005). Shortage in inventory was allowed and the backlogging rate was dependent on the waiting time up to the arrival of next lot. Ouyang et al. (2005) discussed an EOQ inventory mathematical model for deteriorating items with exponentially decreasing demand. The shortages were allowed and partially backordered. The backlogging rate was variable and dependent on the waiting time for the next replenishment. The sensitivity analysis of the major parameters was performed. A comparison among various partial backlogging inventory lot-size models for deteriorating items was given by Yang (2005). Demand function was assumed to be positive and fluctuating with time and the backlogging rate was a differentiable and decreasing

18 function of time. Author has used maximizing profit as the objective to find the optimal replenishment policy. A production inventory model for a deteriorating item with imperfect quality was formulated by Yu et al. (2005). Shortages due to imperfect quality were partially backordered. Dye and Ouyang (2005) proposed an EOQ model for perishable items under stock dependent selling rate and time dependent partial backlogging and then established the unique optimal solution to the problem when building up inventory is not profitable. However, they did not provide the optimal solution to the problem when building up inventory is profitable. Chang et al. (2006-b) then established an appropriate model in which building up inventory is profitable, and then provided an algorithm to find the optimal solution to the problem. A numerical was also used to illustrate the model numerically. Pal et al. (2006) have developed an inventory model for single deteriorating item by considering the impact of marketing strategies such as pricing and advertising as well as the displayed stock level on the demand rate of the system. Shortages were allowed and the backlogging rate was dependent on the duration of waiting time up to the arrival of next lot. The storage capacity of the showroom/shop was finite. The effects of change in demand, deterioration, backlogging parameters and mark-up rate on the initial on hand inventory level, shortage level, frequency of advertisement per cycle along with the maximum average profit were presented numerically. Optimal ordering policy for deteriorating items with partial backlogging was formulated by Ouyang et al. (2006) when delay in payment was permissible. An economic production lot size model for deteriorating items with stock-dependent demand was produced by Jolai et al. (2006) under the effects of inflation. Shortages were allowed and partially backlogged. Computation algorithm for inventory model with a service level constraint and lead time demand was discussed by Lee et al. (2006). The backorder rate was dependent on the length of lead time through the amount of shortages. Authors have developed two computational algorithms to find the optimal order quantity and the optimal lead time. The two numerical examples were also given to illustrate the results. A deteriorating inventory model with time-varying demand was discussed by Dye et al. (2006). Shortages were allowed and partially backlogged with a shortages dependent backlogging rate. Authors have used couple of numerical examples to illustrate the results. San

19 Jose et al. (2006) presented an inventory model with backlogging, where the fraction of backlogged demand was a negative exponential function of the waiting time. Numerical examples were used to illustrate how the approach works. An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand was proposed by Wu et al. (2006). Shortages were allowed and the backlogging rate was variable and dependent on the waiting time for the next replenishment. The necessary and sufficient conditions of the existence and uniqueness of the optimal solution were shown. Sensitivity analysis of the optimal solution with respect to the major parameters was carried out. A deterministic inventory model for deteriorating items with price dependent demand was developed by Dye et al. (2007-b). The demand and deterioration rates were continuous and differentiable function of price and time, respectively. Shortages for unsatisfied demand were allowed and backlogging rate was taken as negative exponential function of the waiting time. Numerical example was also used to study the problem numerically. Dye (2007) proposed joint pricing and ordering policy for deteriorating inventory. Shortages in inventory were allowed and partial backlogged. A note on sensitivity analysis of inventory model with partial backorders was presented by Yang (2007). Teng et al. (2007) extended Abad s (2003) pricing and lot-sizing model by adding not only the shortage cost for backlogged items but also the cost of lost goodwill due to lost sales into the objective. Authors established a modeling approach as in Goyal and Giri s (2003) model to the same pricing and lot sizing problem and then proved that both the models provide the same profit if all the parameters are constant. If any single parameter varied with time, then performances of both models varied. They have obtained some theoretical results that show the conditions under which one model has more net profit per unit time than the other. An inventory lot-size model for deteriorating items with partial backlogging was formulated by Chern et al. (2008). Authors have taken time value of money in to consideration. The demand was assumed to fluctuating function of time and the backlogging rate of unsatisfied demand was a decreasing function of the waiting time. The effects of inflation and time value of money were also considered in the model. Thangam and Uthayakumar (2008) presented a twolevel supply chain model with partial backordering and approximated Poisson demand.