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Inventory Management

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**Lecture Outline Elements of Inventory Management**

Inventory Control Systems Economic Order Quantity Models Quantity Discounts Reorder Point Order Quantity for a Periodic Inventory System

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**What Is Inventory? Stock of items kept to meet future demand**

Purpose of inventory management how many units to order when to order

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**Types of Inventory Raw materials Purchased parts and supplies**

Work-in-process (partially completed) products (WIP) Items being transported Tools and equipment

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**Inventory and Supply Chain Management**

Bullwhip effect demand information is distorted as it moves away from the end-use customer higher safety stock inventories to are stored to compensate Seasonal or cyclical demand Inventory provides independence from vendors Take advantage of price discounts Inventory provides independence between stages and avoids work stop-pages

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**Two Forms of Demand Dependent Independent**

Demand for items used to produce final products Tires stored at a Goodyear plant are an example of a dependent demand item Independent Demand for items used by external customers Cars, appliances, computers, and houses are examples of independent demand inventory

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**Inventory and Quality Management**

Customers usually perceive quality service as availability of goods they want when they want them Inventory must be sufficient to provide high-quality customer service in TQM

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**Inventory Costs Carrying cost Ordering cost Shortage cost**

cost of holding an item in inventory Ordering cost cost of replenishing inventory Shortage cost temporary or permanent loss of sales when demand cannot be met

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**Inventory Control Systems**

Continuous system (fixed-order-quantity) constant amount ordered when inventory declines to predetermined level Periodic system (fixed-time-period) order placed for variable amount after fixed passage of time

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**ABC Classification Class A Class B Class C 5 – 15 % of units**

70 – 80 % of value Class B 30 % of units 15 % of value Class C 50 – 60 % of units 5 – 10 % of value

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**ABC Classification: Example**

1 $ 60 90 PART UNIT COST ANNUAL USAGE

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**ABC Classification: Example (cont.)**

1 $ 60 90 PART UNIT COST ANNUAL USAGE TOTAL % OF TOTAL % OF TOTAL PART VALUE VALUE QUANTITY % CUMMULATIVE 9 8 2 4 4, 3 3, 6 3, 5 3, 10 2, 7 1, $85,400 Example 10.1

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**Economic Order Quantity (EOQ) Models**

optimal order quantity that will minimize total inventory costs Basic EOQ model Production quantity model

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**Assumptions of Basic EOQ Model**

Demand is known with certainty and is constant over time No shortages are allowed Lead time for the receipt of orders is constant Order quantity is received all at once

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**Inventory Order Cycle Demand rate Time Lead time Order placed**

Order receipt Inventory Level Reorder point, R Order quantity, Q

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**EOQ Cost Model Co - cost of placing order D - annual demand**

Cc - annual per-unit carrying cost Q - order quantity Annual ordering cost = Annual carrying cost = Total cost =

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**EOQ Cost Model TC = + CoD Q CcQ 2 = + Q2 Cc TC Q 0 = + C0D Qopt =**

= Q2 Cc TC Q 0 = C0D Qopt = Deriving Qopt Proving equality of costs at optimal point = Q2 = Qopt =

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**EOQ Cost Model (cont.) Order Quantity, Q Annual cost ($) Slope = 0**

Minimum total cost Optimal order Qopt

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**EOQ Example Cc = $0.75 per yard Co = $150 D = 10,000 yards Qopt = 2CoD**

Qopt = yards TCmin = CoD Q CcQ 2 TCmin = TCmin = = Orders per year = D/Qopt = = orders/year Order cycle time = days/(D/Qopt) = = store days

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**Production Quantity Model**

An inventory system in which an order is received gradually, as inventory is simultaneously being depleted AKA non-instantaneous receipt model assumption that Q is received all at once is relaxed p - daily rate at which an order is received over time, a.k.a. production rate d - daily rate at which inventory is demanded

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**Production Quantity Model (cont.)**

Inventory level (1-d/p) Q 2 Time Order receipt period Begin order receipt End Maximum inventory level Average

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**Production Quantity Model (cont.)**

p = production rate d = demand rate Maximum inventory level = Q d = Q 1 - Q p d Average inventory level = 2 TC = CoD CcQ Qopt = 2CoD Cc 1 -

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**Production Quantity Model: Example**

Cc = $0.75 per yard Co = $150 D = 10,000 yards d = 10,000/311 = 32.2 yards per day p = 150 yards per day Qopt = = = yards 2CoD Cc 1 - d p TC = = d p CoD Q CcQ 2 Production run = = = days per order Q p

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**Production Quantity Model: Example (cont.)**

Number of production runs = = = runs/year D Q Maximum inventory level = Q = = yards d p

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**P = per unit price of the item**

Quantity Discounts Price per unit decreases as order quantity increases TC = PD CoD Q CcQ 2 where P = per unit price of the item D = annual demand

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**Quantity Discount Model (cont.)**

Qopt Carrying cost Ordering cost Inventory cost ($) Q(d1 ) = 100 Q(d2 ) = 200 TC (d2 = $6 ) TC (d1 = $8 ) TC = ($10 ) ORDER SIZE PRICE $10 100 – (d1) (d2)

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**Quantity Discount: Example**

QUANTITY PRICE $1,400 ,100 Co = $2,500 Cc = $190 per computer D = 200 Qopt = = = PCs 2CoD Cc TC = PD = CoD Qopt CcQopt 2 For Q = 72.5 TC = PD = CoD Q CcQ 2 For Q = 90

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**Reorder Point Level of inventory at which a new order is placed R = dL**

where d = demand rate per period L = lead time

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**Reorder Point: Example**

Demand = 10,000 yards/year Store open 311 days/year Daily demand = 10,000 / 311 = yards/day Lead time = L = 10 days R = dL = = yards

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**Safety Stocks Safety stock Stockout Service level**

buffer added to on hand inventory during lead time Stockout an inventory shortage Service level probability that the inventory available during lead time will meet demand

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**Variable Demand with a Reorder Point**

point, R Q LT Time Inventory level

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**Reorder Point with a Safety Stock**

point, R Q LT Time Inventory level Safety Stock

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**Reorder Point With Variable Demand**

R = dL + zd L where d = average daily demand L = lead time d = the standard deviation of daily demand z = number of standard deviations corresponding to the service level probability zd L = safety stock

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**Reorder Point for a Service Level**

Probability of meeting demand during lead time = service level a stockout R Safety stock dL Demand zd L

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**Reorder Point for Variable Demand**

The carpet store wants a reorder point with a 95% service level and a 5% stockout probability d = 30 yards per day L = 10 days d = 5 yards per day For a 95% service level, z = R = dL + z d L = = yards Safety stock = z d L = = yards

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**Order Quantity for a Periodic Inventory System**

Q = d(tb + L) + zd tb + L - I where d = average demand rate tb = the fixed time between orders L = lead time sd = standard deviation of demand zd tb + L = safety stock I = inventory level

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**Fixed-Period Model with Variable Demand**

d = 6 bottles per day sd = 1.2 bottles tb = 60 days L = 5 days I = 8 bottles z = 1.65 (for a 95% service level) Q = d(tb + L) + zd tb + L - I = = bottles

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