Prepaired by: Hashem Al-Sarsak Supervised by: Dr.Sanaa Alsayegh

Inventory theories Inventory is a stock of items kept on hand to meet demand stocks of goods being held for future use or sale, so the owner must replenish inventories soon enough to avoid shortages, These stores could benefit from the kinds of techniques of scientific inventory management that are described in this chapter. Maintaining inventories is necessary for any company dealing with physical products The total value of all inventory—including finished goods, partially finished goods, and raw materials The costs associated with storing (“carrying”) inventory are also very large

How do companies use operations research to improve their inventory policy for when and how much to replenish their inventory? They use scientific inventory management comprising the following steps: 1. Formulate a mathematical model describing the behavior of the inventory system. 2. Seek an optimal inventory policy with respect to this model. 3. Use a computerized information processing system to maintain a record of the current inventory levels. 4. Using this record of current inventory levels, apply the optimal inventory policy to signal when and how much to replenish inventory.

The mathematical inventory models used with this approach can be divided into two broad categories—deterministicقطعي models and stochasticعشوائي The demand for a product in inventory is the number of units that will need to be withdrawn from inventory for some use. There is case of known demand where a deterministicيقيني قطعي inventory model would be used. However, when demand cannot be predicted very well, it becomes necessary to use a stochasticعشوائي inventory model where the demand in any period is a random variable rather than a known constant. Dependent demand items are used internally to produce a final product.

Inventory Costs There are three basic costs associated with inventory: carrying (or holding) costs, ordering costs, and shortage costs. Carrying costs are the costs of holding items in storage. Carrying costs Ordering costs are the costs associated with replenishingتجديد the stock of inventory being held Ordering costs Shortage costs, also referred to as stock-out costs, occur when customer demand cannot be met because of insufficient inventory on hand. The purpose of inventory management is to determine how much and when to order.

Inventory Control Systems here are two basic types of inventory systems: continuous system, an order is placed for the same constant amount whenever the inventory on hand decreases to a certain level,عندما تنخفض لمستوى معين periodic system, an order is placed for a variable amount after an established passage of time. بمرور الزمن

Economic Order Quantity Models EOQ is a continuous inventory system. The function of the EOQ model is to determine the optimal order size that minimizes total inventory costs. In this sections we will describe three model versions: The basic EOQ model. EOQ model with non-instantaneousغير لحظي receipt. EOQ model with shortages

1- The basic EOQ model EOQ is the optimal order quantity that will minimize total inventory costs. The model formula is derived under a set of simplifying and restrictive assumptions, as follows: 1. Demand is known with certainty and is relatively constant over time. 2. No shortages are allowed. 3. Lead time for the receipt of orders is constant. 4. The order quantity is received all at once.

The graph above reflects the basic model assumptions. (inventory order cycle)

1.1- Carrying Cost average inventory =

Annual average inventory

Now that we know that the amount of inventory available on an annual basis is the average inventory, Q/2 We can determine the total annual carrying cost by multiplying the average number of units in inventory by the carrying cost per unit per year, Cc:

1.2- Ordering Cost The total annual ordering cost is computed by multiplying the cost per order (Co ) by the number of orders per year Because annual demand is assumed to be known and constant, the number of orders will be D/Q, where Q is the order size: The only variable in this equation is Q, both Co and D are constant parameters. Thus, the relative magnitude of the ordering cost is dependent upon the order size(Q)

Total Inventory Cost The total annual inventory cost is simply the sum of the ordering and carrying costs:

optimal value of Q The optimal value of Q corresponds to the lowest point on the total cost curve (Fig 2)

The total minimum cost is determined by substitutingالاستعاضة the value for the optimal order size, Qopt, into the total cost equation:

2. The EOQ Model with Non- instantaneous Receipt In this EOQ variation, the order quantity is received gradually over time and the inventory level is depletedمستنفذة at the same time it is being replenishedتجديد

To determine the average inventory level, we define the following parameters that are unique to this model: p = production rate d = inventory is demanded

The total carrying cost, using this function for average inventory, is Thus, the total annual inventory cost is determined according to the following formula:

see Figure.2 Therefore, to find optimal Qopt, we equate total carrying cost with total ordering cost:Figure

3. The EOQ Model with Shortages One of the assumptions of our basic EOQ model is that shortages and back ordering are not allowed. The third model variation that we will describe, the EOQ model with shortages,

S = the shortage level Cs = equals the annual per-unit cost of shortages: Combining these individual cost components results in the total inventory cost formula:

You will notice in Figure that the three cost component curves do not intersect at a common point, as was the case in the basic EOQ model. As a result, the only way to determine the optimal order size and the optimal shortage level, S, is to differentiate the total cost function with respect to Q and S, set the two resulting equations equal to zero, and solve them simultaneously. Doing so results in the following formulas for the optimal order quantity and shortage level:Figure

Example We will use the following example to demonstrate how the optimal value of Q is computed. The I-75 Carpet Discount Store in north Georgia stocks carpet in its warehouse and sells it through an adjoining showroom. The store keeps several brands and styles of carpet in stock; however, its biggest seller is Super Shag carpet. The store wants to determine the optimal order size and total inventory cost for this brand of carpet, given an estimated annual demand of 10,000 yards of carpet, an annual carrying cost of $0.75 per yard, and an ordering cost of $150. The store would also like to know the number of orders that will be made annually and the time between orders (i.e., the order cycle), given that the store is open every day except Sunday, Thanksgiving Day, and Christmas Day (which is not on a Sunday(.

Solution We can summarize the model parameters as follows: Co = 150 $ Cc = 0.75 $ D= 10,000 yd The optimal order size is computed as follows:

Solution cont. The total annual inventory cost is determined by substituting Qopt into the total cost formula, as follows:

Solution cont. The number of orders per year is computed as follows:

Solution cont. Given that the store is open 311 days annually (365 days minus 52 Sundays, plus Thanksgiving and Christmas), the order cycle is determined as follows:

Quiz What is the carrying costs and how to compute?