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**Managerial Decision Modeling with Spreadsheets**

Chapter 12 Inventory Control Models

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**Learning Objectives Understand importance of inventory control.**

Use Economic Order Quantity (EOQ) model to determine how much to order. Compute reorder point (ROP) in determining when to order more inventory. Use EOQ with non-instantaneous receipt model to determine how much to order or produce. Handle EOQ problems that allow quantity discounts. Understand use of safety stock with known and unknown stockout costs. Understand importance of ABC inventory analysis.

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12.1 Introduction Inventory is one of most expensive and important assets to many companies. Managers have long recognized good inventory control is crucial.

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**12.2 Importance Of Inventory Control**

Inventory control serves several important functions and adds flexibility to firm’s operations. Five main uses of inventory are: Decoupling function. 2. Storing resources. 3. Irregular supply and demand. 4. Quantity discounts. 5. Avoiding stockouts and shortages.

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**12.3 Inventory Control Decisions**

Fundamental decisions made when controlling inventory: 1. How much to order 2. When to order Inventory model is utilized to determine how much to order and when to order.

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**12.3 Inventory Control Decisions**

Major objective in controlling inventory is to minimize total inventory costs: 1. Cost of items. 2. Cost of ordering. 3. Cost of carrying, or holding, inventory. 4. Cost of stockouts. 5. Cost of safety stock, additional inventory held to help avoid stockouts.

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**Inventory Cost Factors**

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**12.4 Economic Order Quantity (EOQ): Determining How Much To Order**

Economic Order Quantity Model Assumptions: Demand is known and constant. Lead time - time between placement of and receipt of the order is known and constant. Receipt of inventory is instantaneous. Quantity discounts are not possible. Only variable costs are cost of placing an order, ordering cost, and cost of holding or storing inventory over time, holding or carrying cost. If orders are placed at right time, stockouts or shortages can be avoided completely.

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**Inventory Usage Over Time**

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**Ordering and Inventory Costs**

Objective of inventory models is to minimize total costs. With assumptions given, significant costs are ordering cost and carrying cost.

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**Ordering and Inventory Costs**

Total Cost as Function of Order Quantity

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**Ordering and Inventory Costs**

Average inventory on hand is: Average inventory level = ( 0 + Q ) / 2 = Q / 2 Other inventory parameters are: Q* = Optimal order quantity (i.e., EOQ). D = Annual demand in units for inventory item. Co = Ordering cost per order. Ch = Holding or carrying cost per unit per year. P = Purchase cost per unit of inventory item. Holding cost could be constant or calculated as cost of capital: Ch = I x P

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**Inventory Costs and EOQ**

Total ordering cost = ( D / Q ) x Co Total carrying cost = ( Q / 2 ) x Ch Total cost = Total ordering cost + Total carrying cost + Total purchase cost = ( D / Q ) x Co + ( Q / 2 ) x Ch + P x D Economic Order Quantity is:

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**Plot of Costs Versus Order Quantity**

Sumco Pump Company

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**12.5 Reorder Point: Determining When To Order**

Second inventory question is when to order. Time between placing and receipt of an order, called lead time or delivery time, is often few days or few weeks. When to order decision is usually expressed in terms of reorder point (ROP), inventory level at which an order should be placed. Reorder point, ROP, is given as: ROP = (demand per day) x (lead time in days) = d x L Demand, d, expressed in units demanded per day and lead time, L, expressed in days.

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Reorder Point Curve

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**Reorder Point Recall EOQ = 200 and total cost of $5,100.**

Sumco Pump Company Recall EOQ = 200 and total cost of $5,100. Calculations based on annual demand of 1,000 units, ordering cost of $10 per order, annual carrying cost of $0.50 per unit, and purchase cost of $5 per pump housing. Assume lead time of 3 business days between time firm places an order and time order is received. Assume there are 250 business days in year. To calculate reorder point, first determine daily demand rate, d. Since there are 250 business days in year and annual demand is 1,000, daily demand rate is 4 (= 1,000 / 250) pump housings.

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**12.6 EOQ With Non-instantaneous Receipt**

Firm may build up inventory gradually over period of time. Example, firm may receive shipments from suppliers uniformly over period of time. Or firm may be producing at rate of p per day and simultaneously selling at rate of d per day (where p > d). Average inventory level = [ 0 + Q ( 1 - d / p )] / 2 = = Q ( 1 - d / p ) / 2

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**12.6 EOQ With Non-instantaneous Receipt**

Or firm may be producing at rate of p per day and simultaneously selling at rate of d per day (where p > d). Avg. inventory level = Q (1 - d / p) / 2

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**Finding Economic Production Quantity**

Parameters are: Q* = Optimal order or production quantity (EPQ) Cs = Setup cost per setup For given order quantity Q: Total setup cost = ( D / Q ) x Cs Total carrying cost = [ Q ( 1- d / p) / 2 ] x Ch Total cost = Total setup cost + Total carrying cost + Total production cost = (D / Q) x Cs + [Q (1- d / p) / 2] x Ch + P x D Calculate EPQ as:

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**Brown Manufacturing Example**

Produces mini-sized refrigeration packs in batches. Estimated demand for year is 10,000 units. Operates for 167 business days each year. Annual demand translates to daily demand rate of 60 units per day. It costs about $100 to set up manufacturing process, and carrying cost is $0.50 per unit per year. When production process has been set up, 80 refrigeration packs can be manufactured daily. Each pack costs $5 to produce. How many packs should Brown produce in each batch?

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EPQ Model Brown Manufacturing Calculates and reports EPQ as well as following output measures: maximum inventory (= Q*[1- d /p]) average inventory (= Q*[1- d /p] / 2) number of setups (= D / Q*) total holding cost (= Ch x Q*[1- d /p] / 2) total setup cost (= Cs x D / Q*) total purchase cost (= P x D) total cost (= Ch x Q*[1- d /p] / 2 + Cs x D /Q* + P x D)

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**EPQ Model Total setup cost = total carrying cost ($250 each).**

Brown Manufacturing Total setup cost = total carrying cost ($250 each). EPQ: Q* = 4,000 units. Total cost, including production cost of $50,000, is $50,500.

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Inventory Costs Plot Brown Manufacturing

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**12.7 Quantity Discount Models**

To increase sales, companies offer quantity discounts to customers. Quantity discount is simply reduced cost for item when purchased in larger quantities. It is common to have discount schedule with several discounts for large orders. Total cost = Total ordering cost + Total carrying cost Total purchase cost = (D/Q) x Co + (Q/2) x Ch + P x D Find EOQ that incorporates cost with discount to minimize total cost.

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**12.7 Quantity Discount Models**

Find EOQ that incorporates cost with discount to minimize total cost.

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**Four Steps to Analyze Quantity Discount Models**

For each discount price, calculate a Q* value using EOQ formula. For any discount level, if Q* computed in Step 1 is too low to qualify for discount, adjust Q* upward to lowest quantity that qualifies for discount. Using total cost equation,compute total cost for every Q* determined in steps 1 and 2. If Q* had to be adjusted upward because it was below allowable quantity range, be sure to use adjusted Q* value. Select Q* with lowest cost as computed in Step 3. It will be order quantity to minimize total cost.

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**Total Cost Curve for Quantity Discount Model**

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**Brass Department Store Example**

Stocks toy cars. Store given quantity discount schedule for cars as shown in Table 12.2. Normal cost for cars is $5. For orders between 1,000 and 1,999 units, unit cost is $4.80, and for orders of 2,000 or more units, unit cost is $4.75. Ordering cost is $49 per order, annual demand is 5,000 race cars, and inventory carrying charge as percentage of cost, I, is 20% or 0.2. What order quantity will minimize total cost?

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**Plot of Total Cost Versus Order Quantity**

Brass Department Store

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12.8 Use Of Safety Stock

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12.9 ABC Analysis Recognizes fact some inventory items are more important than others. Purpose of analysis is to divide all of company's inventory items into three groups: A, B, and C. Depending on group, decide how inventory levels should be controlled.

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**Silicon Chips, Inc. Example**

Maker of super-fast DRAM chips, has organized its 10 inventory items on an annual dollar-volume basis. Parts are identified by item number, part number, annual demands, and unit costs. How should company classify items into groups A, B, and C?

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**Silicon Chips, Inc. Example**

How should company classify items into groups A, B, and C?

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Summary Focus was to answer two questions in inventory planning: (1) how much to order, and (2) when to order. EOQ makes a number of assumptions: (1) known and constant demand and lead times. (2) instantaneous receipt of inventory. (3) no quantity discounts. (4) no stockouts or shortages. (5) only variable costs are ordering and carrying costs.

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**Summary If assumptions do not hold, more complex models are needed:**

(1) economic production quantity. (2) quantity discount models. Discussed computation of safety stocks when demand during lead time was unknown for two cases: (1) cost of stockout is known. (2) cost of stockout is unknown. Presented ABC analysis to determine how inventory items should be classified based on their importance and value.

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