Chapter 15 Deterministic EOQ Inventory ModelsDescription15.1 Introduction to Basic Inventory ModelsCosts Involved in Inventory ModelsAssumptions of EOQ ModelsSlide 715.2 Basic Economic Order Quantity ModelAssumptions of the Basic EOQ ModelSlide 10Derivation of Basic EOQ ModelSlide 12Slide 13Example 1Example 1 SolutionSlide 16Sensitivity of Total Cost to Small Variations in the Order QuantitySlide 18Determination of EOQ When Holding Cost is Expressed in Terms of Dollar Value of InventoryThe Effect of Nonzero Lead TimeSlide 21Slide 22Spreadsheet Template for the Basic EOQ ModelSlide 24Power of Two Ordering PoliciesSlide 2615.3 Computing the Optimal Order Quantity When Quantity Discounts Are AllowedSlide 28Example 4Slide 30Slide 31Slide 32Slide 33Slide 34Example 4 continuedExample 4 SolutionSlide 37Slide 38Slide 39Spreadsheet Template for Quantity DiscountsSlide 4115.4 The Continuous Rate EOQ ModelSlide 43Slide 44Spreadsheet Template for the Continuous Rate EOQ Model15.5 & 15.6 The EOQ Model with Back Orders Allowed & When to Use EOQ ModelsSlide 47Slide 48Spreadsheet Template for the EOA Model with Back OrdersSlide 5015.7 Multiple Product Economic Order Quantity ModelsSlide 52Slide 53Chapter 15Deterministic EOQ Inventory Modelsto accompanyOperations Research: Applications and Algorithms 4th editionby Wayne L. WinstonCopyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.2DescriptionWe now begin our formal study of inventory modeling.We begin by discussing some important concepts used to describe inventory models and then develop versions of the famous economic order quantity (EOQ) model that can be used to make optimal inventory decisions when demand is deterministic.315.1 Introduction to Basic Inventory ModelsThe purpose of inventory theory is to determine rules that management can use to minimize the costs associated with maintaining inventory and meeting customer demand.Inventory models answer the following questions1. When should an order be placed for a product?2. How large should each order be?4Costs Involved in Inventory ModelsThe inventory models that we will discuss involve some or all of the following costs:Ordering and Setup CostThese costs do not depend on the size of the order. They typically include things like paperwork, billing or machine setup time if the product is made internally.Unit Purchasing CostThis cost is simply the variable cost associated with purchasing a single unit. Typically, the unit purchasing cost includes the variable labor cost, variable overhead cost, and raw material cost.5 Holding or Carrying CostThis is the cost of carrying one unit of inventory for one time period. The holding costs usually includes storage cost, insurance cost, taxes on inventory and others. Usually, however, the most significant components of holding cost if the opportunity cost incurred by tying up capital in inventory.Stockout or Shortage CostWhen a customer demands a product and the demand is not met on time, a stockout, or shortage, is said to occur. If they will accept delivery at a later date, we say the demands are back-ordered. This case is often referred to as the backlogged demand case. If they will not accept late delivery, we are in the lost sales case. These costs are often harder to measure than other costs.6Assumptions of EOQ ModelsRepetitive OrderingThe ordering decision is repetitive in the sense that it is repeated in a regular fashion. Constant DemandDemand is assumed to occur at a known, constant rate.Constant Lead TimeThe lead time for each order is a known constant, L. By the lead time we mean the length of time between the instant when an order is placed and the instant at which the order arrives.7 Continuous OrderingAn order may be placed at any time. Inventory models that allow this are called continuous review models. If the amount of on-hand inventory is reviewed periodically and orders may be placed only periodically, we are dealing with a periodic review model.815.2 Basic Economic Order Quantity ModelFor the basic EOQ model to hold, certain assumptions are required:1. Demand is deterministic and occurs at a constant rate.2. In an order of any size (say q units is placed, an ordering and setup cost K is incurred.3. The lead time for each order is zero.4. No shortages are allowed.5. The cost per unit-year of holding inventory is h.9Assumptions of the Basic EOQ ModelWe define D to be the number of units demanded per year.The setup cost K of assumption 2 is in addition to a cost pq of purchasing or producing the q unites ordered. We assume p (the purchasing cost) does not depend on the size of the order.Assumption 3 assumes each order arrives on time.Assumption 4 implies that all demand is met on time.10 Assumption 5 implies that if I units are held for T years, a holding cost of ITh is incurred.Given these assumptions, the EOQ model determines an ordering policy that minimizes the yearly sum of ordering cost, purchasing cost, and holding cost.11Derivation of Basic EOQ ModelWe begin by making some simple observations.We should never place an order when I, the inventory level, is greater than zero; if we place an order then the we are incurring an unnecessary holding cost.If I=0, we must place an order to prevent a shortage.Each time an order is placed (when I=0) we should order the same quantity, q.TC(q)= annual cost of placing orders + annual purchasing cost + annual holding costTo determine the annual holding cost, we need to examine the behavior of I over time.12 A key concept in the study of EOQ models is the idea of a cycle.Definition: Any interval of time that begins with the arrival of an order and ends the instant before the next order is received is called a cycle.In the figure the average inventory during any cycle is simply half of the maximum inventory level attained during the cycle.This result will hold in any model for which demand occurs at a constant rate.13 The economic order quantity, or EOQ,minimizes TC(q).Thus, q* does indeed minimize total annual cost.The figure on the next page confirms the fact that at q*, the annual holding and ordering costs are the same.212*hKDq14Example 1Braneast Airlines uses 500 taillights per year. Each time an order for taillights is placed, an ordering cost of $5 is incurred.Each light cost 40¢, and the holding cost is