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**Chapter 15 Deterministic EOQ Inventory Models**

to accompany Operations Research: Applications and Algorithms 4th edition by Wayne L. Winston Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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**Description We 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.

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**15.1 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 questions When should an order be placed for a product? (產品何時下定單) How large should each order be? 每次定單需下多少量

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**Costs 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.

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**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(缺貨) or Shortage Cost(短缺) 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.

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**Assumptions 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(前置時間) 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.

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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.

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**15.2 Basic Economic Order Quantity Model**

For the basic EOQ model to hold, certain assumptions are required: Demand is deterministic and occurs at a constant rate. In an order of any size (say q units is placed, an ordering and setup cost K is incurred. The lead time for each order is zero. No shortages are allowed. The cost per unit-year of holding inventory is h.

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**Assumptions 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.

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**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.

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**Derivation 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.

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**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.

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**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.

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**Example 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 8¢/light/year. Assume that demand occurs at a constant rate and shortages are not allowed. What is the EOQ? How many orders will be placed each year? How much time will elapse between the placement of orders?

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Example 1 Solution We are given that K = $5, h = $0.08/light/year, and D = 500 lights/year. The EOQ is Hence, the airline should place an order for 250 taillights each time that inventory reaches zero.

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**The time between placement (or arrival) of orders is simply the length of a cycle.**

Since the length of each cycle is , q*/D, the time between orders will be

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**Sensitivity of Total Cost to Small Variations in the Order Quantity**

In most situations, a light deviation from the EOQ will result in only a slight increase in costs. For Example 1, let’s see how derivations from the EOQ change the total annual cost. We focus our attention on how the annual holding cost and ordering costs are affected by the changes in order quantity. Let HC(q) = annual holding cost if the order quantity is q OC(q) = annual ordering cost if the order quantity is q

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**We find that HC(q)=0.04q and OC(q)=2500/q.**

The figure shows that HC(q) + OC(q) is very flat near q*. The flatness of the HC(q) + OC(q) is very important, because it is often difficult to estimate h and K. Inaccurate estimation of h and K may result in a value of q that differs slightly from the actual EOQ. The flatness of the HC(q) + OC(q) curve indicates that even a moderate error in the determination of the EOQ will only increase costs by a slight amount.

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**Determination of EOQ When Holding Cost is Expressed in Terms of Dollar Value of Inventory**

Often the annual holding cost is expressed in terms of the cost of holding one dollar’s worth of inventory for one year. Suppose that hd = cost of holding one dollar in inventory for one year. Then the cost of holding one unit of inventory for one year will be phd and may be written as

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**The Effect of Nonzero Lead Time**

We now allow lead time L to be greater than zero. The introduction of a nonzero lead time leaves the annual holding and ordering costs unchanged. Hence, the EOQ still minimizes total costs. To prevent shortages from occurring and to minimize holding cost, each order must be placed at an inventory level that ensures that when each order arrives, the inventory level will equal zero. Definition: The inventory level at which an order should be placed is the reorder point.

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**To determine the reorder point for the basic EOQ model, two cases must be considered:**

Case 1: Demand during the lead time does not exceed the EOQ. In this case the reorder point occurs when the inventory level equals LD. The order will arrive L time units later, and upon arrival of the order, the inventory level will equal LD - LD=0. Case 2: Demand during the lead time exceeds EOQ. In this case the reorder point does not equal LD. In general it can be shown that the reorder point equals the remainder when LD is divided by the EOQ.

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The determination of the reorder point becomes extremely important when demand is random and stockouts can occur.

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**Spreadsheet Template for the Basic EOQ Model**

The user inputs the values of K, h, lead time (L), and D. In A8 the EOQ is determined by the formula (2*K*D/H)^.5. In B8 we computer annual holding costs by the formula .5*A8*H. In D5 we computer orders per year for the EOQ by the formula D/A8. In C8 we compute annual holding costs for the EOQ from the formula K*D5.

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**In D8 we computer total annual cost for the EOQ by the formula B8+C8.**

In B11 we computer the reorder point by the formula =MOD(L*D,A8) This yields the remainder obtained when L*D is divided by the EOQ.

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**Power of Two Ordering Policies**

Suppose a company orders three products and the EOQ’s for each product yield time between orders of 3.5 days, 5.6 days and 9.2 days. It would rarely be the case that they all arrive on the same days. If we could synchronize our reorder intervals so that orders for different products often arrived on the same day we often can greatly reduce our fixed costs.

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Roundy (1986) devised an elegant and simple method called Power of Two Ordering Policies to ensure that the placement of orders for multiple products are well synchronized. Let q* = EOQ. Then the optimal reorder interval for a product is t* = q*/D. The virtue of a Power of Two policy is that different products will frequently arrive at the same time. This will greatly reduce fixed costs.

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**15.3 Computing the Optimal Order Quantity When Quantity Discounts Are Allowed**

Up to now, we have assumed that the annual purchase cost does not depend on the order size. In real life, however, suppliers often reduce the unit purchasing price for large orders. Such price reductions are referred to as quantity discounts. The approach used previously is no longer valid and we need a new approach to find the optimal quantity.

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If we let q be the quantity ordered each time an order is placed, the general quantity discount model analyzed in this section may be described as follows: If q<b1, each item cost p1 dollars. If b1≤q<b2, each item costs p2 dollars. If bk-2≤q<bk-1, each item cost pk-1 dollars. If bk-1≤q<bk=∞, each item costs pk dollars. Since b1, b2, …., bk-1 are points where a price change occurs, we refer to them as price break points.

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Example 4 A local accounting firm in Smalltown orders boxes of floppy disks from a store in Megalopolis. The per-box price charged by the store depends on the number of boxes purchased. The accounting firm uses 10,000 disks per year. The cost of placing an order is assumed to be $100.

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Example 4 The only holding cost is the opportunity cost of capital, which is assumed to be 20% per year. For this example, b1 = 100 b2 = 300 p1 =$50.00 p2 = $49.00 p3 = $48.50

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**These definitions are illustrated on the next slide. **

Before explaining how to find the order quantity minimizing total annual costs, we need the following definitions: TCi(q) = total annual cost if each order is q units at a price pi EOQi = quantity that minimizes total annual cost if, for any order quantity, the purchasing cost of the item is pi EOQi is admissible if bi-1≤EOQi<bi. TC(q) = actual annual cost of q items are ordered each time an order is placed. These definitions are illustrated on the next slide. Our goal is to find a value of q minimizing TC(q).

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In general, the value of q minimizing TC(q) can be either a break point (shown in second figure) or some EOQi (shown in first figure). The following observations are helpful in determining the point that minimizes TC(q). For any value of q, TCk(q) < TCk-1(q) < … <TC2(q) <TC1(q) If EOQi is admissible, then minimum cost for bi-1≤q<bi occurs for q=EOQi (see the first figure on the next slide). If EOQi<bi-1, the minimum cost for bi-1≤q<bi occurs for q=bi-1 (see second figure on next slide).

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If EOQi is admissible, then TC(q) cannot be minimized at an order quantity for which the purchasing price per item exceeds pi. Thus if EOQi is admissible, the optimal order quantity must occur for either price pi, pi+1,…, or pk. These observations allow us to use the following method to determine the optimal order quantity when quantity discounts are allowed. Beginning with the lowest price, determine for each price the order quantity that minimizes total annual cost for bi-1 ≤ q <bi.

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**Continue determining q. k, q. k-1…. Until one of the q**

Continue determining q*k, q*k-1…. Until one of the q*i’s is admissible; from observation 2, this will mean that q*i = EOQi. The optimal order quantity will be the member of {q*k, Q*k-1,…q*i} with the smallest value of TC(q).

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Example 4 continued Each time an order is placed for disks, how many boxes of disks should be ordered? How many orders will be placed annually? What is the total annual cost of meeting the accounting firm’s disk needs?

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**Example 4 Solution Note the K=$100 and D = 1000 boxes per year.**

We first determine the best order quantity for p3 = $48.50 and 300 ≤ q. Then Since EOQ3 < 300, EOQ3 is not admissible. Therefore q ≥ 300, TC3(q) is minimized by q*3 = 300.

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Example 4 Solution We next consider p2=$49.00 and 100 ≤ q < 300. Then Since 100 ≤ EOQ2 < 300, EOQ2 is admissible, and for a price p2 = $49.00, the best we can do is to choose q*2 = Since q*2 is admissible, p1 =%50.00 and 0 ≤ q < 100 cannot yield the order quantity minimizing TC(q). Thus, either q*2 or q*3 will minimize TC(q).

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Example 4 Solution To determine which of these order quantities minimizes TC(q), we must find the smaller of TC3(300) and TC2(142.86). For q*3 the annual holding cost is $9.70. Thus, for q*3, Annual ordering cost = 100(1000/300) = $333.33 Annual purchasing cost = 1000(48.50) = $48,500 Annual holding cost = (½)(300)(9.7) = $1455 TC3(300) = $50,

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Example 4 Solution For q*2 the annual holding cost is $9.80. Thus, for q*2, Annual ordering cost = 100(1000/142.86) = $699.99 Annual purchasing cost = 1000(49) = $49,000 Annual holding cost = (½)(142.86)(9.8) = $700.01 TC3(142.86) = $50, 400 Thus q*3 = 300 will minimize TC(q). Our analysis shows that each time an order is placed, 300 boxes of disks should be ordered. Then 3.33 orders are placed each year.

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**Spreadsheet Template for Quantity Discounts**

The table illustrates how inventory problems with a quantity discount can be solved on a spreadsheet. In cell C2 we enter D, the annual demand. In cell D2(HD) we enter the annual cost of holding $1 of goods in inventory for one year. In the cell range A6:C8 we enter the left-hand endpoint, right-hand endpoint, and price for each interval.

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**Spreadsheet Template for Quantity Discounts**

In D6 we computer the EOQ for the interval b0 = 0 ≤ order quantity < 100=b1 by entering the formula (2*$K*$D/(($HD*C6))^.5). This statement computes the order quantity in the first interval that in E6 we enter the formula =IF(AND(D6>=A6,D6<B^),D6,IF(D6<A6,A6,B^-1)

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**15.4 The Continuous Rate EOQ Model**

Many goods are produced internally rather than purchased from an outside supplier. In this situation, the EOQ assumption that each order arrives at the same instant seems unrealistic; it isn’t possible to produce, say, 10,000 cars at the drop of a hard hat. If a company meets demand by making its own products, the continuous rate EOQ model will be more realistic than the traditional EOQ model.

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**The variation of inventory over time is shown.**

The continuous rate EOQ model assumes that a firm can produce a good at a rate of r units per time period. This means that during any time period of length, t, the firm can produce rt units. We define q = number of units produced during each production run K = cost of setting up a production run H = cost of holding one unit in inventory for one year D = annual demand for the product The variation of inventory over time is shown.

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**As r increases, production occurs at a more rapid rate**

As r increases, production occurs at a more rapid rate. Hence, for large r, the rate model should approach the instantaneous delivery situation of the EOQ model.

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**Spreadsheet Template for the Continuous Rate EOQ Model**

The table in the book illustrates a template for the continuous rate EOQ model. In cell A6 the user inputs K; in B6, h; in C6, D; and in D6, the production rate r. In A8 the formula (2*K*D/H)^.5*(R/(R-D))^.5 computes the optimal size. In B8 the formulas D/Q computers the number of runs per year. In C8 we computer the annual cost with the formula (H*Q*(R-D)/(2*R))+K*D/Q.

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**15.5 & 15.6 The EOQ Model with Back Orders Allowed & When to Use EOQ Models**

In many real-life situations, demand is not met on time, and shortages occur. When a shortage occurs, costs are incurred. Let s by the cost of being short one unit for one year. The variables K, D, and h have their usual meanings. In most situations s is very difficult to measure. We assume all demand is backlogged and no sales are lost.

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**To determine the order policy that minimizes annual costs, we define**

q = order quantity q - M = maximum shortage that occurs under an ordering policy Equivalently, the firm will be q – M units short each time an order is placed. We assume lead time for each order is zero. The firm’s maximum inventory level will be M – q + q = M.

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**TC(q,M) is minimized by q* and M*:**

Maximum shortage = q* - M* As s approaches infinity, q* and M* both approach the EOQ, and the maximum shortage approaches zero.

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**Spreadsheet Template for the EOA Model with Back Orders**

A spreadsheet template for the EOQ model with back orders. In cells A6, B6, C6, and D6 we enter the values of K, D, h, and s, respectively. In A8 we computer the optimal order quantity with the formula (2*K*D(H+S)/(H*S))^.5 In B8 we computer the optimal value of M with the formula (2*K*D*S/(H*(H+S)))^.5 In C8 we compute the maximum shortage with the formula =Q-M.

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**In D8 we compute the annual total cost TC(q,M) with the formula (M^2**

In D8 we compute the annual total cost TC(q,M) with the formula (M^2*H)/(2*Q))+((Q-M)^2S/2*Q))+(K*D/Q).

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**15.7 Multiple Product Economic Order Quantity Models**

Supposed a company orders several products. Each time an order is received shipments of some of the products arrive. Each time an order arrives there is a fixed cost associated with the order and there is another fixed cost associated with each product included in the order. How can we minimize the sum of annual holding and fixed costs?

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**Chopra and Meindl(2001) devised a method to find a near optimal solution to this type of problem.**

To begin we find the product that is most frequently ordered. We assume this product will be included in each order.

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**We then setup a Solver model that determines the following Changing Cells:**

Number of orders received per year. For all products other than the chosen product the number of orders that need to be received before an order of the product is received. We can easily determine the total fixed cost and the total holding cost for each product. The sum of these costs will be our target cell for Solver. Our model is highly nonlinear and it is necessary to use the Evolutionary Solver to find the optimal solution.

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