Inventory Management

MNG221- Management Science Inventory Management

1Lecture OutlineTypes of inventoryReasons for holding inventoryStock costsObjective of inventory managementPareto analysisDeterministic and stochastic models2Inventory ManagementStock may be classified into:Raw materialsWork-in-progressFinished goodsResources, Labor, Cash The classification depends on the nature of the firm.3Inventory ManagementThe main purpose of inventory is simply to meet customer demand.It often represents a significant cost to a business firm, (including insurance, obsolescence, depreciation, interest, opportunity costs, storage costs, etc.) Therefore inventory related costs can be controlled, through the management of inventory levels. 4Elements of Inventory ManagementInventory Management

5Elements of Inventory ManagementInventory is defined as a stock of items kept on hand by an organization to use to meet customer demand.

The Role of Inventory The main reasons for holding inventory are:To satisfy demand immediatelyTo meet seasonal or cyclical demand6Elements of Inventory ManagementThe Role of InventoryTo allow for unimpeded production and provide independence between operations. To take advantage of bulk purchasing price discounts.To absorb seasonal fluctuations.A necessary part of the production process.

7Elements of Inventory ManagementThe Role of Inventory (continued)Inventory may also accumulate because of poor control methods, obsolesce and suboptimal decisions.

8Elements of Inventory ManagementDemandA crucial component and the basic starting point for the management of inventory is customer demand, because it exists for the purpose of meeting the demand of customers.Customers can be Internal (machine operator) or External (Individual purchasing goods from stores)

9Elements of Inventory ManagementDemand (Continued)An essential determinant of effective inventory management is an accurate forecast of demand. The demand for items in inventory is classified as dependent or independentDependent Demand items are used internally to produce a final productIndependent Demand items are final products demanded by an external customer.10Elements of Inventory ManagementInventory CostsThere are three basic costs associated with inventory:Carrying Costs - are the costs of holding items in storage.Ordering Costs - are the costs associated with replenishing the stock of inventory being held.11Elements of Inventory ManagementInventory CostsShortage costs - also referred to as stockout costs, occur when customer demand cannot be met because of insufficient inventory on hand.

12Elements of Inventory ManagementInventory CostsThe objective of inventory management is to employ an inventory control system that will indicate how much should be ordered and when orders should take place to minimize the sum of the above three inventory costs 13Inventory Control SystemsInventory Management

14Inventory Control SystemsAn Inventory System is a structure for controlling the level of inventory by determining how much to order (the level of replenishment) and when to order.There are two basic types of inventory systems: a continuous (or fixed order quantity) system and a periodic (or fixed time period) system.15Inventory Control SystemsThe primary difference in the two systems is that in a:Continuous system - an order for the same amount is placed whenever the inventory decreases to a certain level. Periodic system - order is placed for a variable after an established passage of time.16Inventory Control SystemsContinuous Inventory SystemIn a continuous inventory system, (alternatively referred to as a perpetual system or a fixed order quantity system) a constant amount is ordered when inventory declines to a predetermined level, referred to as the reorder point.This fixed order quantity is called the economic order quantity17Inventory Control SystemsContinuous Inventory SystemThe inventory level is closely and continuously monitored so that management always knows the inventory status.However, the cost of maintaining a continual record of the amount of inventory on hand can also be a disadvantage of this type of system.18Inventory Control SystemsPeriodic Inventory SystemIn a periodic inventory system, (also referred to as a fixed time period system or periodic review system) an order is placed for a variable amount after a fixed passage of time.19Inventory Control SystemsPeriodic Inventory SystemThe inventory level is not monitored at all during the time interval between orders.It has the advantage of requiring little or no record keepingIt has the disadvantage of less direct control20Economic Order Quantity ModelsBasic ModelInventory Management

21Economic Order Quantity ModelsThe most widely used and traditional means for determining how much to order in a continuous system is the Economic Order Quantity (EOQ) model, also referred to as the Economic Lot Size Model.The function of the EOQ model is to determine the optimal order size that minimizes total inventory costs.22Economic Order Quantity ModelsThe Basic EOQ ModelIt is essentially a single formula for determining the optimal order size that minimizes the sum of carrying costs and ordering costs. 23Economic Order Quantity ModelsThe Basic EOQ ModelAssumptionsDemand is known with certainty and is relatively constant over time.No shortages are allowed.Lead time for the receipt of orders is constant.The order quantity is received all at once.24Economic Order Quantity ModelsThe Basic EOQ Model

The Inventory Order Cycle

25Economic Order Quantity ModelsThe Basic EOQ ModelQ is the point at which ordering and carrying costs react inversely to each other in response to an increase in the order size.R is the point at which a new order is placed with enough lead time for the reordering of stock.26Economic Order Quantity ModelsThe Basic EOQ Model Carrying CostsCarrying cost is usually expressed on a per-unit basis for some period of time on an annual basis (i.e., per year), and sometimes as a percentage of average inventory.Average Inventory = Q or Q points over period, t 2 number of points27Economic Order Quantity ModelsThe Basic EOQ Model Carrying Costs

28Economic Order Quantity ModelsThe Basic EOQ Model Carrying Costs Thus, Carrying cost is Ordering Costs is Total Inventory Cost is

The EOQ cost model

29Economic Order Quantity ModelsThe Basic EOQ ModelThe Optimal Value Of Q corresponds to the lowest point on the total cost curve or the point where the ordering cost curve intersects with the carrying cost curve.

30Economic Order Quantity ModelsThe Basic EOQ ModelThus The Optimal Value Of Q by equating the two cost functions and solving for Q, as follows:

31Economic Order Quantity ModelsThe Basic EOQ ModelAlternatively, the optimal value of Q can be determined by differentiating the total cost curve with respect to Q

32Economic Order Quantity ModelsThe Basic EOQ ModelThe total minimum cost

33Economic Order Quantity ModelsThe Basic EOQ Model - ExampleThe I-75 Carpet Discount Store wants to determine the optimal order size and total inventory cost 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). 34Economic Order Quantity ModelsThe Basic EOQ Model ExampleThe model parameters as follows:

35Economic Order Quantity ModelsThe Basic EOQ Model ExampleThe optimal order size is computed as follows:

36Economic Order Quantity ModelsThe Basic EOQ Model ExampleThe total annual inventory cost is determined by substituting Qopt into the total cost formula, as follows:

37Economic Order Quantity ModelsThe Basic EOQ Model ExampleThe number of orders per year is computed as follows:

38Economic Order Quantity ModelsThe Basic EOQ Model ExampleGiven that the store is open 311 days annually (365 days minus 52 Sundays, plus Thanksgiving and Christmas), the order cycle is determined as follows:

39Economic Order Quantity ModelsThe Basic EOQ Model The optimal order quantity determined in general, is an approximate value, because it is based on estimates of carrying and ordering costs as well as uncertain demand.

This in practice it is acceptable to round off the Q values to the nearest whole number.

However, the EOQ model is robust; because Q is a square root, errors in the estimation of D, Cc, and Co are dampened.

40Economic Order Quantity ModelsNon-instantaneous ModelInventory Management

AssumptionsDemand is known with certainty and is relatively constant over time.No shortages are allowed.Lead time for the receipt of orders is constant.The order quantity is received all at once.41Economic Order Quantity ModelsNon-instantaneous Receipt ModelA variation of the basic EOQ model is achieved when the assumption that orders are received all at once is relaxed.It is also referred to as the Gradual Usage, or Production Lot Size, model.In this EOQ variation, the order quantity is received gradually over time and the inventory level is depleted at the same time it is being replenished.42Economic Order Quantity ModelsNon-instantaneous Receipt ModelThis is a situation most commonly found when the Inventory user is also the producerWhen orders are delivered gradually over time When retailer and producer of a product are one and the same. 43Economic Order Quantity ModelsNon-instantaneous Receipt Model

The EOQ model with Non-instantaneous Order Receipt

44Economic Order Quantity ModelsNon-instantaneous Receipt ModelThe ordering cost component of the basic EOQ model does not change.

However, the carrying cost component is not the same for this model variation because average inventory is different.

The maximum inventory level is not simply Q; it is an amount somewhat lower than Q,45Economic Order Quantity ModelsNon-instantaneous Receipt ModelUnique parameters of this model:p = daily rate at which the order is received over time, also known as the production rated = the daily rate at which inventory is demanded 46Economic Order Quantity ModelsNon-instantaneous Receipt ModelAs such, the maximum amount of inventory that is on hand is computed as follows:

47Economic Order Quantity ModelsNon-instantaneous Receipt ModelGiven the maximum inventory level, the average inventory level is determined by dividing this amount by 2, as follows:

48Economic Order Quantity ModelsNon-instantaneous Receipt ModelThe total carrying cost, using this function for average inventory, is:

49Economic Order Quantity ModelsNon-instantaneous Receipt ModelThus, the total annual inventory cost is determined according to the following formula:

50Economic Order Quantity ModelsNon-instantaneous Receipt ModelTherefore, to find optimal Qopt, we equate total carrying cost with total ordering cost:

51Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleAssume that the I-75 Carpet Discount Store has its own manufacturing facilityfurther assume that the ordering cost, Co, is the cost of setting up the production process Recall that Cc = $0.75 per yard and D = 10,000 yards per year.The manufacturing facility operates 311 days and produces 150 yards of the carpet per day.52Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleThus the parameters are:

53Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleThe optimal order size is determined as follows:

54Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleThis value is substituted into the following formula to determine total minimum annual inventory cost:

55Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleThe length of time to receive an order or production run is computed as follows:

56Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleThe number of orders per year is actually the number of production runs that will be made, computed as follows:

57Economic Order Quantity ModelsNon-instantaneous Receipt Model ExampleFinally, the maximum inventory level is computed as follows:

58Economic Order Quantity ModelsShortages ModelInventory Management

59Economic Order Quantity ModelsShortages ModelThe assumptions of our basic EOQ model is that shortages and back ordering are not allowedThe EOQ model with shortages relaxes the assumption that shortages cannot exist.However, it will be assumed that all demand not met because of inventory shortage can be back ordered and delivered to the customer later.

60Economic Order Quantity ModelsShortages Model

The EOQ model with Shortages

61Economic Order Quantity ModelsShortages ModelBecause back-ordered demand, or shortages, (S), are filled when inventory is replenished, the maximum inventory level does not reach Q, but instead a level equal to Q - S.Therefore, the cost associated with shortages has an inverse relationship to carrying costs.As the order size, Q, increases, the carrying cost increases and the shortage cost declines.62Economic Order Quantity ModelsShortages Model

Cost model with shortages

63Economic Order Quantity ModelsShortages ModelThe individual cost functions are provided as follows, where S equals the shortage level and Cs equals the annual per-unit cost of shortages:

64Economic Order Quantity ModelsShortages ModelCombining these individual cost components results in the total inventory cost formula:

65Economic Order Quantity ModelsShortages ModelThe three cost component curves do not intersect at a common point, as was the case in the basic EOQ model.As such, 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. 66Economic Order Quantity ModelsShortages ModelSet 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:

67Economic Order Quantity ModelsShortages Model ExampleAssume that the I-75 Carpet Discount Store allows shortages and the shortage cost, Cs, is $2 per yard per year. All other costs and demand remain the same

68Economic Order Quantity ModelsShortages ModelSeveral additional parameters of the EOQ model with shortages can be computed for this example, as follows:

The time between orders, identified as t in is computed as follows:

69Economic Order Quantity ModelsShortages ModelThe time during which inventory is on hand, t1, and the time during which there is a shortage, t2, during each order cycle can be computed using the following formulas:

70