Developing economic order quantity model for non-instantaneous deteriorating items in vendor-managed inventory (VMI) system

This paper develops an economic order quantity model for non-instantaneous deteriorating items with and without shortages to investigate the performance of the vendor-managed inventory (VMI) system. This model is developed for a two-level supply chain consisting of a single supplier and single retailer with a single non-instantaneous deteriorating item. A numerical example and sensitivity analysis are provided to illustrate how increasing or reducing the related parameters change the optimal values of the decision variables of the two proposed models. The results show that VMI works better and charges lower cost in all conditions.


Introduction and literature review
Most of the existing inventory models in the literature assume that items can be stored limitlessly to satisfy the future demand. However, some types of items either deteriorate or become outdated in the course of time and hence, are unstable. In general, deterioration is defined as the damage, spoilage, dryness, vaporisation, etc. that leads to a decrease in the usefulness of a commodity. Some items will deteriorate significantly, such as certain types of foods, chemicals, electronic components and radioactive substances. Instinctively, deteriorating items may require greater consideration in collaborative actions between suppliers and retailers since a mutual conflict between these two sides can easily induce an over-estimated stock because of deterioration. Thus, in this case the loss will be greater than that for general goods.
Vendor-managed inventory (VMI) has been described as an inventory and supply-chain management system in which the supplier is responsible for making decisions concerning the timing and replenishment quantity. In a VMI system, in addition to the traditional supply-chain information and material flows, the supplier controls the retailer's inventory in order to ensure that desirable customer service levels are met (Disney, Potter, and Gardner 2003). We organise the literature review section for both inventory models in VMI systems and those for deteriorating items.
Evidence has indicated that a VMI system can enhance supply-chain performance by decreasing inventory levels and increasing fill rates (Emigh 1999). An early conceptual framework for VMI was presented by Magee (1958) * Corresponding author. Email: taleizadeh@ut.ac.ir when discussing who is authorised to control the inventories. However, interest in the concept developed during the 1990s. Companies have tried to improve their supply chains as a way of generating a competitive advantage using VMI system. This strategy has been specifically popular in the grocery sector but has also been implemented in different sectors such as steel, books and petrochemicals (Disney et al. 2003). In recent years, various VMI models have been widely studied by researchers. Dong and Xu (2002) demonstrated an analytical model to evaluate the short-term and long-term impacts of VMI system on supply-chain profitability by analysing the inventory systems of the parties which are involved. Yao, Evers, and Dresner (2007), using Dong and Xu (2002)'s assumptions and another additional assumption (the supplier order quantity is likely to be an integer multiple of the buyer's replenishment quantity), examined how VMI system can effect the cost savings. They then explained how the benefits were likely to be distributed between a buyer and a supplier in a supply chain. They concluded that the replenishment quantities between the supplier and the buyer are reduced after implementing VMI. Vlist, Kuik, and Verheijen (2007) compared a two-echelon supply chain with and without VMI policy, consisting of a single buyer and single supplier. They extended Yao et al. (2007)'s research by considering the delivery cost between the supplier and the buyer in their inventory modelling. However, they concluded that the shipment sizes from the supplier to the buyer increase after implementing VMI. Since the conclusions of Vlist et al. (2007) and Yao et al. (2007) seemed to conflict with each other, Wang, Wee, and C 2013 Taylor & Francis Jacob Tsao (2010) revisited both papers and concluded that both of them are valid in their study scopes and assumptions. They summarised the factors that must be expressed clearly to resolve the conflict and to avoid the confusion. Ji, Shen, and Wei (2008) concentrated on VMI's role as a strategy of integrated supply chain. This study helps to provide a more clear comprehension of how ordering costs and carrying charges affect the inventory cost. Hong and Yang (2010) examined the comparison of the profit generated in a supply chain before and after the VMI model is implemented, and also the logistics costs lying in the enterprise supply-chain system and the relevant linkages by determining a two-echelon supply-chain model based on the economic order quantity (EOQ) model under which the logistics costs are subdivided. Cárdenas-Barrón developed a new analytic geometry approach to optimise both EOQ and EPQ models. Pasandideh, Akhavan Niaki, and Roozbeh Nia (2010)'s research is the most related one to our paper in which they considered the retailer-supplier partnership through a VMI system and developed an analytical model to explore the effects of important supply-chain parameters on the cost savings realised from collaborative initiatives. Cárdenas-Barrón, Treviño-Garza, and Wee (2012) presented an alternative heuristic algorithm to solve a multiproduct, multi-constraint VMI system based on EOQ with backorders considering two classical backorder costs: linear and fixed.
To introduce some works about the EOQ model for deteriorating items we can refer to Ghare and Schrader (1963)'s work in which they have presented a classical inventory model with a constant rate of decay. Covert and Philip (1973) developed Ghare and Schrader's constant deterioration rate to a two-parameter Weibull distribution. Then Philip (1974) extended the inventory model with a threeparameter Weibull distribution rate without considering shortages. Deb and Chaudhuri (1986) presented an inventory model with time-dependent deterioration rate. Wee and Yu (1997) extended two models for the exponentially deteriorating items with a temporary price discount for regular and non-regular inventory replenishment time to maximise the total cost saving during the temporary price discount order cycle. A detailed review of deteriorating inventory literatures is given in Raafat (1991) and Goyal and Giri (2001). Law and Wee (2006) developed an integrated productioninventory model from the perspectives of both the manufacturer and the retailer. They considered deterioration, amelioration, multiple deliveries, partial backordering and time discounting in their model. Lo, Wee, and Huang (2007) developed an integrated production-inventory model with a varying rate of deterioration (two-parameter Weibull distribution) under imperfect production processes, partial backordering and inflation. Chung and Wee (2008) extended an integrated deteriorating inventory model with imperfect production process, partial backordering, warranty-period and stock-level-dependent demand for both the manufac-turer and the retailer. Wee, Lo, and Hsu (2009) presented a multi-objective joint replenishment deteriorating items inventory model with fuzzy demand and shortage cost constraint. Widyadana, Cárdenas-Barrón, and Wee (2011) developed an EOQ model for deteriorating items with and without backorders. Their study is one of the first attempts by researchers to solve a deteriorating inventory problem with a simplified approach. They compared optimal solutions with the classical methods for solving a deteriorating inventory model. All the aforementioned articles assume that the deterioration of items in inventory starts from the instant of their arrival. However, many items retain freshness or original condition for a certain period of time. Wu, Ouyang, and Yang (2006) analysed inventory model for non-instantaneous deteriorating items. Following this study, Wu, Ouyang, and Yang (2009) formulated and solved an inventory system with non-instantaneous deteriorating items in which the demand is price-sensitive. Recently, Maihami and Kamalabadi (2012) established joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging. In their study the demand is assumed time and price dependent. Ouyang, Wu, and Yang (2006), Geetha and Uthayakumar (2010) and Musa and Sani (2012) have studied ordering policies for non-instantaneous deteriorating items. In their research delay in payment was permissible. Li, Yu, and Dong (in press) developed a cooperative-game-based profit-sharing method to stabilise VMI partnership in a supply chain.
In the above-cited papers on non-instantaneous deteriorating items, researchers have considered a constant deterioration rate.
Since the combination of these two respective researches is scarce, the aim of this paper is to fill the gap and to propose VMI as one of the new and effective policies for managing inventories in supply chains for deteriorating items.
The rest of this paper is organised as follows. Section 2 presents the structure of models. In Section 3, the models are introduced along with their assumptions and the models are developed for traditional and VMI modes. Numerical examples are presented in Section 4 to analyse the influence of different parameters on the optimal EOQ and the total cost before and after the implementation of VMI system. Finally, conclusions and future research topics are presented in Section 5.

Model structure
In this research, the problem of a single non-instantaneous deteriorating product under the VMI policy is studied. We construct a two-level supply chain consisting of a single supplier and single retailer and examine the inventory management practices before and after the implementation of VMI. We assume that the retailer faces external demand from consumers and investigate the model in two cases including (a) shortage is not permitted and (b) shortage is permitted and will be fully backordered.

The EOQ model with non-instantaneous deteriorating items
The mathematical models are developed based on the following assumptions.
(1) A single-supplier single-buyer supply chain with one non-instantaneous deteriorating item is considered.
(2) Deliveries of orders are assumed to be instantaneous and the lead time is zero. (3) The costumer's demand is deterministic.
(4) The production rate is infinite. (5) During the fixed period, t 1 , the product has no deterioration. After t 1 , the product will deteriorate at a fixed rate. (6) There is no repair or replacement of the deteriorated inventory during the period under consideration.
To model the problem following notations are used too.
Q: the order quantity Q VMI : the order quantity in the VMI policy β: the constant rate of non-instantaneous deterioration A S : the supplier's ordering cost per order A B : the buyer's ordering cost per order C: the deterioration cost per unit D: the buyer's constant demand rate h B : the inventory holding cost held in buyer's store in a period per unit per unit time b: the maximum level of backordering shortage b VMI : the maximum level of backordering shortage in VMI system π : the fixed cost of shortage per unit π : the cost of shortage per unit per unit time T : the time cycle before VMI T VMI : the time cycle after VMI F : the percentage of cycle length in which inventory is positive KB 0i : the buyer's inventory cost before VMI in case i KB 1i : the buyer's inventory cost after VMI in case i KS 0i : the supplier's inventory cost before VMI in case i KS 1i : the supplier's inventory cost after VMI in case i T C: the total cost before VMI T C VMI : the total cost of VMI system the inventory level is decreasing because of the demand rate.

Case 1: shortage is not permitted
But during [t 1 , T ], the inventory level is dropping to zero because of demand and deterioration. So the differential equation shown in Equation (2) shows the changing of the inventory level during [t 1 , T ].
Therefore we have Substituting I max into I 1 (t) yields The total inventory system cost for the cycle time T is made up of the buyer's ordering cost, supplier's ordering cost, and product's holding and deterioration costs. The buyer's holding cost is Moreover the fixed cost of the buyer is A B and the deterioration cost will be

Analysis of inventory costs
So prior to implementing VMI, the total cost of the buyer and the total cost of the supplier are respectively shown in Equations (9) and (10). Then the total cost of the chain is shown in Equation (11).
Using approximation of the Taylor series expansion > 0, the total cost function is convex. So the optimal value of T can be obtained by setting the first derivative of the total cost function with respect to T equal to zero, yielding Therefore But under the VMI policy, since the supplier should pay the buyer's costs, then Equations (9)-(11) will change to Using approximation of the Taylor series expansion, Equations (18) and (19) will change to T 3 VMI > 0, the total inventory cost of the integrated supply chain shown in Equation (21) is convex. So the optimal value of T can be obtained by setting the first derivative of Equation (21) with respect to T equal to zero, yielding And the order quantity in the VMI policy can be determined as below.

Case 2: with shortage
In this section, we develop the previous model presented in subsection 3.1 with an additional assumption that shortage is allowed and completely backlogged (π = 0, π = 0). A pictorial description of the inventory policy with shortage is given in Figure 2.
During the time interval [0, t 1 ] the inventory level is decreasing because of the demand rate. Since tan α = x t 1 = D yields x = t 1 D, we have But during [t 1 , F T ] the inventory level is dropping to zero because of demand and deterioration. So the differential equation shown in Equation (36) shows the changing of the inventory level during [t 1 , F T ].
Therefore we have Furthermore, at time FT, shortage occurs and the inventory level starts dropping below 0. ( Therefore we have Substituting I max into I 1 (t) yields The total system cost for the cycle time T is made up of the buyer's ordering cost, supplier's ordering cost, Buyer's holding cost, deterioration cost and cost of shortage. The buyer's holding cost is Moreover the fixed cost of the buyer is A B and the deterioration cost will be And the shortage cost isπ

Analysis of inventory costs
So prior to implementing VMI, the total cost of the buyer and the total cost of the supplier are respectively shown in Equations (36) and (37). Then the total cost of the chain is shown in Equation (38).

Using approximation of the Taylor series expansion e β(F T −t
The buyer's inventory cost in Equation (39) is a function of T and F. So, the global optimal values of T and F can be obtained by taking the partial derivative of Equation (39) with respect to T and F, then setting them equal to zero (in Appendix we prove that T * and F * give a global optimal solution for the EOQ with non-instantaneous deteriorating items and shortage).
Substituting F * into T * (F ) yields After multiplying both sides by 1 T 2 we get Considering Equation (28), amount of shortage before VMI can be calculated as follows: Considering Equation (32) and using approximation of the Taylor series expansion, the ordering quantity over the cycle before VMI can be determined as But under the VMI policy, since the supplier should pay the buyer's costs, then Equations (36)-(37) will change to KB 12 = 0 (51) Using approximation of the Taylor series expansion, Equations (52) and (53) will change to Once again, since the total inventory cost in Equation (55) is a function of T and F, so the global optimal values of T and F can be obtained by taking the partial derivative of Equation (55) with respect to T and F, then setting them equal to zero (as we mentioned in the previous section, in Appendix we prove that T * and F * give a global optimal solution for the EOQ with non-instantaneous deteriorating items and shortage).
Substituting F * VMI into T * VMI (F ) yields After multiplying both sides by 1 T 2 VMI we get, Amount of shortage after VMI can be calculated as follows: The ordering quantity over the cycle after VMI can be determined as:

Numerical example and sensitivity analysis
In order to illustrate the above solution procedure, let us consider an inventory system with the data presented in Table 1. We consider this example for both models. However, in the second model in which shortage is allowed and completely backlogged we haveπ = 80, π = 0. We solve this example and get the results presented in Table 2.
We now study the effect of the changes in the system parameters A S , A B , h B , D, C and β for both the models andπ just for the second model on the optimal order quantity per cycle Q and the total relevant inventory cost per unit before and after the implementation of the VMI policy. The sensitivity analysis is performed by changing the parameters A S , A B , h B , D and C by + 75%, + 50%, + 25%, −25%, −50% and −75%, and increasing β cumulatively at the rate of 0.05 in the interval [0.005,0.5], taking one parameter at a time and keeping the remaining parameters unchanged. The results are shown in Tables 3 and 4. Regarding the results obtained from Tables 3 and 4, the following analyses are fulfilled. (1) Increasing the supplier's ordering cost causes no effect on the optimal order quantity in both models before the implementation of VMI. But, it leads to a rise in the optimal order quantity after the implementation of VMI. Moreover, increasing the supplier's ordering cost leads to an increase of inventory total costs, having a sharp increase in the traditional supply chain. The inventory total costs before and after VMI are close to each other for lower-order quantities. But in general, the inventory total costs after the implementation of VMI are lower than that in the traditional supply chain and this gap will grow by increasing the supplier's ordering cost.
(2) Increasing the buyer's ordering cost in both models leads to an increase in the optimal order quantity having a greater slope in the traditional supply chain without VMI. In general, the optimal order quantity after VMI is greater than it's quantity before VMI. Moreover, the inventory total cost before VMI has a downward trend, increasing the buyer's ordering cost, making this trend slighter. However, increasing the buyer's ordering cost leads to an increase in the inventory total cost after the VMI implementation. In general, the inventory total costs after the implementation of VMI are lower than that in the traditional supply chain. (3) Increasing the buyer's holding cost in both the models eventuates a decline in the optimal order quantity. But, the optimal order quantity is greater after implementing the VMI policy. Furthermore, increasing the buyer's holding cost leads to an increase in the inventory total cost before and after the VMI policy. It is obvious that the inventory total costs in the traditional supply chain are greater than in a VMI supply chain and this deviation will grow by increasing the buyer's holding cost. (4) Increasing the buyer's demand rate in both before and after the VMI policy leads to an increase in the optimal order quantity, having a greater slope in the VMI supply chain. However, the optimal order quantity after implementing the VMI policy is higher than before it. This distance will grow by increasing the demand rate. Moreover, increasing the buyer's demand rate in both before and after the VMI policy leads to an increase in the inventory total costs. As it is observed in those tables, the inventory total costs after the implementation of VMI are lower than that in the traditional supply chain. (5) Increasing the deterioration cost has a slight effect on the optimal order quantity in the traditional supply chain and the VMI supply chain. As it is obvious in the tables, the optimal order quantity in the VMI supply chain is significantly greater than that in the traditional supply chain. Furthermore, increasing  the deterioration cost in both before and after the VMI policy eventuates an increase in the inventory total costs. The inventory total costs after VMI are much less than before VMI implementation. (6) Increasing the non-instantaneous deterioration rate has a different trend in our models; in the first model, it leads to a decline in the optimal order quantity in both before and after the VMI policy. However, the optimal order quantity after the VMI policy is greater than the quantity before it. Fur-thermore, increasing the non-instantaneous deterioration rate leads to an increase in the inventory total cost before and after the VMI implementation. It is clear that the inventory total costs in the VMI supply chain are lower than in the traditional supply chain and this distance will grow with an increasing deterioration rate.
In the second model considering shortage, by increasing the non-instantaneous deterioration rate, the optimal order quantities before and after the VMI implementation gradually decrease and then increase. However, the optimal order quantity in the VMI supply chain is higher than in the traditional supply chain. Moreover, by increasing the non-instantaneous deterioration rate before the VMI policy, the inventory total cost first increases and then significantly decreases. But in the VMI supply chain, increasing the noninstantaneous deterioration rate leads to an increase in the inventory total cost. In general, the inventory total costs after the implementation of VMI are lower than that in the traditional supply chain.
(7) According to Table 4, increasing the shortage cost in both before and after the VMI policy eventuates a decline in the optimal order quantity. But, the optimal order quantity after the VMI implementation is higher than the one before the VMI implementation. Furthermore, increasing the shortage cost leads to an increase in the inventory total cost before and after the VMI policy. In general, the inventory total costs in the traditional supply chain are higher than that in the VMI supply chain and this distance will grow with increasing shortage cost.

Conclusions and future research
In this paper, the performance of VMI system in a supply chain of a single deteriorating item has been compared with the traditional inventory control system. Two EOQ models for a non-instantaneous deteriorating item, with and without shortage, have been developed to derive the total inventory cost, the optimal order and the shortage quantities as the performance measures. A numerical example and sensitivity analysis have been provided to illustrate the difference in the total cost and optimal decision variables of both systems. It has been demonstrated that the VMI system is more beneficial for the coordination system and delivers lower cost in all conditions. Moreover, optimal order quantity in all conditions in a VMI supply chain is greater than its quantity in a traditional supply chain. For future research, considering more than one buyer can be suitable. Moreover, assuming partial backordering instead of full backordering shortage can enhance the research. It is also suggested to consider and analyse the problem presented in this paper in the three-level mode.