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**Topic 6. INVENTORY MANAGEMENT**

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**I. Introduction What is inventory? Types of Inventories:**

stored resource used to satisfy current or future demand Types of Inventories: Raw Materials/Components In-Process Goods (WIP) Finished Goods Supplies

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**Introduction Inventory Related Costs:**

Holding Cost -- cost to carry a unit in inventory for a length of time (annual), includes interest opportunity cost, insurance, taxes, depreciation, obsolescence, deterioration, May be expressed as a percentage of unit price or as a dollar amount per unit

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**Introduction Inventory Related Costs (continued):**

Order Cost -- Cost of ordering and receiving inventory, Include determining how much is needed, preparing invoices, shipping costs, inspecting goods upon receipt for quantity and quality, Generally expressed as a fixed dollar amount, regardless of order size Inventory may also influence purchasing cost Inventory is costly

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**Introduction Inventory Related Costs (continued):**

Shortage Cost-- result when demand exceeds the inventory on hand, Include the opportunity cost of not making a sales, loss of customer goodwill, late charges, and in the case of internal customers, the cost of lost production or downtime, difficult to measure, thus may have be subjectively estimated

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**Introduction Why Hold Inventories? Meet anticipated demand**

Lead time – the time period between place an order until receive the order Average lead time demand is considered as anticipate demand Protect against stock-out Safety stock – more than average lead time demand inventory

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**Introduction Why Hold Inventories (continued)?**

De-couple successive operations - separate production from distribution Wine production and inventory Smooth production process Snowmobile production and inventory Buy/Produce in economic lot sizes - take advantage of quantity discounts Hedge against price increases

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Introduction JIT Inventory – minimum inventory needed to keep a system running, small lot sizes Advantages lower inventory costs easy to identify problems and potential problems Disadvantages requires accurate timing and cooperation breakdowns stop everything

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**Introduction Inventory Classification A Identify important**

Annual $ volume of items A B C High Low Few Many Number of Items Inventory Classification Identify important Items and more inventory control on important items Measure of importance: ABC analysis: A = 70-80% of total inventory value, but only 15% of items B = 15-25% of total inventory value, but 30% of items C = 5% of total inventory value, but 55% of items

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**Introduction Monitor Inventory**

As important as demand forecast for decision making Universal Product Code - Bar code printed on a label that has information about the item to which it is attached Cycle counting: taking physical counts of items and reconciling with records on a continual rotating basis, regular inventory audits, ABC approach

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**Introduction Inventory Systems**

Objective: minimize annual total inventory cost and maintain satisfied service level. service level: probability of no shortage Total Inventory Cost is not Inventory Cost Annual total inventory cost (TC) = annual product cost + annual inventory cost Annual product cost = annual demand * unit price Annual inventory cost = annual holding cost + annual setup (order) cost + annual shortage cost

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**Introduction Possible performance measures customer satisfaction**

number of backorders/lost sales number of customer complaints inventory turnover ratio of annual cost of goods sold to average inventory investment days of inventory expected number of days of sales that can be supplied from existing inventory

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**Introduction Requirements for Effective Inventory Management :**

A system to keep track of the inventory on hand and on order A classification system for inventory items A reliable forecast of demand that includes an measure of forecast error Reasonable estimates of inventory holding costs, ordering costs, and shortage costs Knowledge of lead times and lead time variability

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Introduction 1. Continuous (Perpetual) Review System: (event-triggered) Monitor the inventory level all the time, order a fixed quantity (Q) when the inventory level drops to the reorder point (ROP) Calculate: Q and ROP Re-Order Point (ROP) – an inventory level when actual inventory drops to it will trigger an activity of re-order.

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**Introduction 2. Periodic Review System: (time-triggered)**

Place an order every fixed period T. Each time bring the current inventory to a target level M Calculate: T and M 3. Advantages and Disadvantages?

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**Introduction Dependent and Independent Demand:**

Dependent demand: derived demand, lumpy (subassemblies and components) cars Independent demand: from customer side, smooth (end items and finished goods) tires

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**II. Inventory Models On Order Quantity**

Model Basics (consider as annual) Total Cost (TC) = Product Cost + Inventory Cost Inventory Cost = Holding Cost + Setup (Order) Cost + Shortage Cost TC = Product Cost + Holding Cost

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**Inventory Models On Order Quantity**

Product Cost = Annual Demand * Unit Price Holding Cost = average inventory level * Holding Cost per unit per year Ordering Cost = # of orders * Setup Cost per order # of orders = annual demand / order quantity Shortage Cost = Shortage Cost per unit * average # of shortage per year Best Order Quantity = a quantity that minimizes TC

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**Inventory Models On Order Quantity**

EOQ Model (Economic Order Quantity), Fixed-Order-Quantity Model Assumptions There is one product type Demand is known and constant Lead time is known and constant Receipt of inventory is instantaneous (one batch, same time) Shortage is not allowed

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**EOQ Model (continued) Q Lead time Reorder point Place order Receive**

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**EOQ Model (continued) Notation and Terminology**

Q = order quantity(# of pieces per order) Q0 = Economic Order Quantity (EOQ) D = demand for the time period considered (units per year) S = setup/order cost ($ per order) H = holding cost per unit per year ($ per unit per year) in general proportional to the price, H = I*P

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**EOQ Model (continued) Notation and Terminology (continued)**

I = Interest rate (expanded) (% per year) P = unit price ($ per unit) IC = inventory cost = setup cost + holding cost TC = IC + product cost Find Out EOQ

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**EOQ Model (continued) Average Inventory Level = Holding Cost =**

Number of orders per year = Setup (Order) Cost = Shortage Cost = 0, why?

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**EOQ Model (continued) Product Cost = IC = Total Cost (TC) =**

Minimize TC Minimize IC, why?

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EOQ Model (continued) Observation: at the best order quantity EOQ (Q0), holding cost = setup cost Solve Q0, we have

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**EOQ Model (continued) The Inventory Cost Curve is U-Shaped Annual Cost**

Carrying Costs Annual Ordering Costs QO (EOQ) Order Quantity (Q)

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**EOQ Model (continued) Example:**

Annual demand = 10,000 unit/year, ordering cost = $50/order, unit cost (price) = $4/unit, expanded interest rate = 25%/year. EOQ? TC at EOQ?

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**EOQ Model (continued) Sensitivity of IC with related to Q**

-- Example (continued) Avg. Inventory Holding Cost # of orders per year Order Cost IC Q (Q/2) (Q/2)*H (D/Q) (D/Q)*S +(D/Q)*S 500 250 $250 20 $1,000 $1,250 1000 $500 10 1500 750 $750 6.667 $333 $1,083

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**EOQ Model (continued) Conclusion: Thinking Challenge:**

1. Inventory cost curve is flat around EOQ 2. Flatter when Q increases than when Q decreases from EOQ Thinking Challenge: If the order quantity Q = 2*EOQ, by how much IC will increase?

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**EOQ Model (continued) Sensitivity of EOQ with related to D, H, S, P, I**

1. Insensitive to parameter change 2. Directions?

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EPQ Model EPQ (Economic Production Quantity) Model: Fixed Order Quantity Model with Incremental Replenishment Problem description: Assumptions There is one product type Demand is known and constant Receipt of inventory is gradual and at a constant replenishment (production) rate Shortage is not allowed

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**EPQ Model (continued) Q Production rate - usage rate Usage rate**

Quantity on hand Usage rate Reorder point Time Start to produce Finish production Start to produce Production run length

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**EPQ Model (continued) Notation and Terminology**

Qp = production quantity(# of pieces/production run) Qp0 = Best production quantity (EPQ) p = daily production rate (units per day) d = daily demand rate (units per day) D = demand rate (units per year) S = production setup (order) cost($ per setup) H = holding cost per unit per year (again H = I*P in general) T = production run length = Q/p

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**EPQ Model (continued) Maximum Inventory Level =**

Average Inventory Level = Annual Holding Cost =

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**EPQ Model (continued) Number of production runs per year =**

Order Cost = IC = TC = Minimize TC Minimize IC, why?

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**EPQ Model (continued) Observation: at EPO, holding cost = setup cost**

Best Production Quantity (EPQ) formula:

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**EPQ Model (continued) Remarks: EPQ > EOQ (why?)**

Example: D=2000 unit/year, S=$5/setup, H=$0.4/unit/year, p=100 unit/day, 200 working days/year. Find the best production batch size and the # of production runs/year.

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**EOQ with discount EOQ with Discount Model:**

Assumptions: same as with EOQ, plus discount on all units Terminology Price breaks: the smallest order quantity to receive a discount price Feasibility: the order quantity matching the claimed price is feasible, otherwise infeasible.

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**EOQ with discount (continued)**

Example: Order Price $2.1/unit $2.0 Great equal 700 $1.9 Idea is to compare TC curves under different prices - why TC?

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**EOQ with discount (continued)**

Order Quantity Total Cost Curve for Price 1 Total Cost Curve for Price 2 $ cost Total Cost Curve for Price 3 400 700

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**EOQ with discount (continued)**

Order Quantity Total Cost Curve for Price 1 Total Cost Curve for Price 2 $ cost Total Cost Curve for Price 3 400 700

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**EOQ with discount (continued)**

Observations: EOQ with a lower price, if feasible, is better than any order quantity with the same or higher price. Potential best order quantity: cheapest feasible EOQ, price breaks associated with lower prices.

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**EOQ with discount (continued)**

Solution Procedure: 1. Find the feasible EOQ with cheapest possible price. 2. Calculate TCs of the EOQ (from Step 1) and price breaks above EOQ. 3. Pick the order quantity with lowest TC

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**EOQ with discount (continued)**

Example (continued) Annual demand = 10,000 unit/year, order cost = $5.5/order. Assuming holding costs are proportional to unit prices and annual interest rate = 20%. Find the best order quantity.

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**III. Models on Reorder Points - When to Order?**

Find ROP (Re-Order Point) ROP depends on: Lead Time: time between placing and receiving an order Demand Distribution: how uncertain Desired Service Level: probability of no shortage = 1-P(s), where P(s) = probability of shortage

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**Models on Reorder Points - When to Order ? (continued)**

Constant Demand Rate: Constant daily demand rate = d, Lead time = L days ROP = d * L = Lead time demand Remark: no uncertainty in demand service level = 100% safety stock = 0

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**Models on Reorder Points - When to Order ? (continued)**

Variable Demand with Stable Average Rate How continuous review system works? Lead time demand: demand during the lead time ROP Lead time demand ==> ROP < Lead time demand ==> ROP = Average lead time demand + Safety Stock = m + SS

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**Models on Reorder Points - When to Order ? (continued)**

Remarks: Higher the desired service level ---> More uncertain the demand ---> Two methods to determine the SS

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**Models on Reorder Points - When to Order ? (continued)**

1. Determine SS and ROP based on shortage cost inf. (if available) SS increases Holding cost ? Shortage cost ? Best SS minimizes total inventory cost

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**Models on Reorder Points - When to Order ? (continued)**

1. Determine SS and ROP based on shortage cost inf. (continued) -- Example: Consider a light switch carried by Litely. Litely sells 1,350 of these switches per year, and places order for 300 of these switches at a time. The carrying cost per unit per year is calculated as $5 while the stock out cost is estimated at $6 ($3 lost profit per switch and another $3 lost in goodwill, or future sales loss). Find the best SS level and ROP for Litely.

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**Models on Reorder Points - When to Order ? (continued)**

Determine SS and ROP based on shortage cost inf. (continued) 1. Determine SS and ROP based on demand inf. during each lead time period: Lead Time Demand 5 10 15 20 25 30 Probability 0.1 0.15 0.2

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**Models on Reorder Points - When to Order ? (continued)**

Determine SS and ROP based on shortage cost inf. (continued) If SS = 0, ROP = m = 15 switches

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**Models on Reorder Points - When to Order ? (continued)**

Determine SS and ROP based on shortage cost inf. (continued) # of orders per year = For no safety stock, Litely has the following shortage table. Why? Shortage Level no shortage 5 10 15 Probability 0.55 0.2 0.15 0.1

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**Models on Reorder Points - When to Order ? (continued)**

Determine SS and ROP based on shortage cost inf. (continued) Determine the best SS in following table Safety stock Add. Holding cost Avg. shortage (per order) Annual shortage cost Total cost 5 10 15

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**Models on Reorder Points - When to Order ? (continued)**

2. Determine ROP and SS based on lead time demand distribution and desired service level:

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**Models on Reorder Points - When to Order ? (continued)**

Case 1. Empirical Lead time demand distribution -- Example: Lead Time Demand Frequency Probability ROP Service Level 3 2 4 5 6 7 8

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**Models on Reorder Points - When to Order ? (continued)**

Find R and SS to achieve the service level of 85% and 95%, respectively.

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**Models on Reorder Points - When to Order ? (continued)**

Case 2. Lead time demand is Normally distributed with (m, ) SS = , ROP = m + SS, z = single tail normal score of desired service level. ( is the standard deviation) Example: Lead time demand is Normally distributed with mean = 4 and standard deviation = 3. Find ROP and SS to achieve the service level of 85% and 95%, respectively.

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**IV. Single Period Model and Marginal Analysis (Newsvendor Problem)**

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**Homework (Additional problems)**

Problem 1: A toy manufacturer uses approximately 36,000 silicon chips annually. The chips are used at a steady rate during the 240 days the plant operates. Annual holding cost is 50 cents per chip, and ordering cost (per order) is $25/order. Assume that each of their orders comes in one batch. Determine: a. .the best order quantity b. demonstrate that your order quantity is optimal by showing that annual ordering costs = annual holding costs c. the average inventory level d. the number of orders per year e. the number of working days between orders (Hint: days between orders = # days in a year / # of orders per year. Why?)

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**Homework (Additional problems)**

Problem 2. The Dine Corporation is both a producer and a user of brass couplings. The firm operates 200 days a year and uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 150 per day. Inventory holding cost is estimated at $5 per unit per year. Machine setup costs are $40 per production run. Determine: a. the best production run size b. demonstrate that your production run size is optimal by showing that annual set up costs = annual holding costs (Hint: find the formula of holding and setup cost for EPQ model in my lecture note.) c. the maximum inventory level (Hint: find the formula in the derivation of EPQ) d. the number of production runs per year e. the cycle time and the production time within each cycle (Hint: cycle time is given by Q/d and production time is given by Q/p. Why? Think before using the formula)

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**Homework (Additional problems)**

A small manufacturing firm used roughly 3,400 pounds of chemical dye each year. Currently the firm purchases 300 pounds per order and pays $3 per pound. The supplier has just announced that orders of 1,000 pounds or more will be filled at a price of $2.5 per pound. The manufacturing firm incurs a cost of $100 each time it submits an order and assigns an annual holding cost of 20% of the purchase price per pound. a. determine the best order size that minimizes the total cost b. if the supplier offered the discount at 2,500 pounds instead of at 1,000 pounds, what order size would minimize total cost?

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**Homework (Additional problems)**

Problem 4: A product is ordered four times every year. Inventory carrying cost is $20 per unit per year, and the cost of shortage for each unit is $40. Given the following demand probabilities during the reorder period Lead Time Demand 40 80 120 160 Probability 0.1 0.25 0.3

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**Homework (Additional problems)**

Problem 4 (continued) a) What is the average lead time demand? b) What would be the reorder point without safety stock? c) What would be the probabilities of the following shortage levels if the company uses the reorder point without safety stock?

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**Homework (Additional problems)**

Problem 4 (continued) d) Follow the Litely example in my lecture to find out the best safety stock level to minimize the total cost. e) What is the reorder point to achieve the 95% service level? What is the associated safety stock? (Hint: you need to follow the example in my lecture note under Case 1) Shortage Level 40 80 Probability

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