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In a Vendor-Management-Inventory (VMI) system, the distributor is authorized to coordinate the inventories at retailers and replenish the customer’s stock. The advantage of VMI is that the stock-out situations can be reduced. This provides a framework for synchronizing transportation decisions and hence reduced the transportation cost significantly. Here a stochastic model for VMI system is used to present an analytic model for quantity-time-based dispatching policy from the quantity-based dispatching policy and time-based dispatching policy. The model takes into the account of the inventory cost, the transportation cost, the dispatching cost of the model is obtained by using the renewal theory, since a new inventory cycle begins whenever there is a dispatching of products.



Inventory management is very important to most industries as well as commercial sectors because it helps companies respond quickly to customer’s
demand, which is an important element of competitive markets. An inventory system is a collection of people, equipment, and procedures that function to keep account of the quantity of items in inventory and to determine which item to purchase or what quantity to produce and at what time (Allen, 2006). In order to maintain a high customer service level, the demand from customers should be satisfied within a stipulated period of time. In view of this, the inventory level will be such that probability of stock-out should be low and at the same time holding cost of inventory should be low.

According to Allen (2006), other limitations, such as warehouse and its locations, plants and its locations and replenishment lead-time may increase the complexity in inventory management. Usually, in inventory analysis, the following costs are considered, setup cost, holding cost, item cost and penalty cost. But in this study we will not consider penalty cost because backlogging will not be considered. By balancing these costs, useful marketing strategies can be provided. A well-maintained inventory system can lead to remarkable reduction in cost (Allen, 2006).
Different strategies are used in inventory management but in this work we are going to consider vendor-managed-inventory (VMI).
1.2 Vendor – Managed Inventory (VMI) Vendor-managed-inventory is a Supply chain practice where the inventory is monitored, planned, and managed by the vendor of the consuming organization, based on the expected demand and on previously agreed minimum and maximum inventory level (Waller et.al, 1999).

VMI is based on the belief that supplying parties are in a better position to manage inventory production capacities and lead times. Also, it is believed that vendor-managed inventory reduces bottleneck in the supply chain, and increases stock monitory, (Eri Goa, 1988). It is also referred to as a program of supplier managed inventory or direct replenishment inventory which emerged in the late 1980’s as a partnership to coordinate replenishment decisions in a supply Chain while maintaining the independence of Chain member, (Mehmet, 2006). In this relationship between a vendor and a customer, it is the vendor that decides when and what quantity the customer’s stock is replenished.
VMI is one of the most widely discussed partnering initiatives for improving multi-firm supply chain efficiency. It started in the retail business and grew to
Efficient Consumer Response (ECR), where consumer satisfaction or rather consumer expectation of stock availability is an important way to have a competitive edge over others, (Eri Goa, 1988). Furthermore, it is known as continuous replenishment inventory or supplier-managed inventory. This was
popularized or successfully implemented in the late 1980s by Wal-Mart and its supplier Procter & Gamble, (Waller et al, 1999).


Numerous firms including Campbell soup, Johnson and Johnson Company in United States of America, (Clark, 1994), Barrila, the pasta manufacturer in Europe, (Haminond, 1994), applied this type of inventory system in their business transaction. In a full VMI partnership, the supplier, usually the manufacturer or a distributor, makes the main inventory replenishment decisions for the consuming organization. This means that the vendor monitors the buyer’s inventory levels (physically or otherwise) and makes periodic re-supply decisions regarding order quantity, shipping, and timing, Waller et al (1999). Transactions customarily initiated by the buyer (such as purchase orders) are initiated by the supplier.
Indeed, the  purchase order acknowledgement from the vendor may be the first indication that a transaction is taking place: an advance shipping notice informs the customer of materials in transit.
Matt et al (2001) stated that it is the retailers that furnish the manufacturer with accurate and up-to-date information about their current demand. This accurate information enables manufacturer to plan production more efficiently. The retailer benefits by lowering administrative costs and achieving a better ability to respond to demand and avoid stock-out. Full VMI is a concept that is based on the cooperation of supplier and customer regarding inventory management and the connected delivery concept.

(Konstocks et al, 2008) stated that despite the range of full VMI relationships, the question is whether full VMI is beneficial to all parties. Barke, (1996), claimed that vendors are unwillingly forced into a full VMI agreement by powerful customers. Saccomano, (1997), argued that full VMI is just a way to transfer the risks involved in inventory management from customers to vendor.
In full VMI, the supplier has to solve and integrated inventory-routing or production-inventory-routing problem. The supplier can reduce the level of
inventories while maintaining the same level of service or can increase the level of service and reduce the transportation cost by a more uniform utilization of the transportation capacity, (Claudia et al, 2005).
Full VMI have a random demand with a known distribution function for each period in the planning horizon. The distributor manages the stock for the inventory cost incurred on the locations. The distributor has two types of costs related to these points: the holding cost (the cost of an item held in stock) and a stock-out cost (cost per unit not sold due to stock unavailability), (Rita et al, 2003).

1.3 Statement of the problem

Many researchers have worked on vendor-managed inventory (VMI), where the supplier is authorized to manage inventories of agreed-upon stock-keeping retail locations. The vendor realized a sequence of random demands from a group of retailers located in a given geographical area and the demands are shipped immediately. The vendor has the autonomy of holding small orders until an agreeable dispatching time with the expectation that an economical consolidated dispatch quantity accumulates (Cetinkaya and Lee, 2000).
In this work, we simultaneously compute a quantity-based policy and a replenishment quantity. We balanced between pure quantity-based and time-based policies. Under quantity-and-time policy, a dispatch decision is taken at min (Tq, T) where Tq denotes the arrival time of qth demand and T is the normal time.

1.4 Assumptions of the Model


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