What is Inventory?

 Definition--The stock of any item or resource used in an organization  Raw materials  Finished products  Component parts  Supplies  Work in process

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Inventory System Purpose

 The set of policies and controls that determine what inventory levels should be maintained, when stock should be replenished, and how large orders should be

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Purposes of Inventory

1. To maintain independence of operations 2. To meet variation in product demand 3. To allow flexibility in production scheduling 4. To provide a safeguard for variation in raw material delivery time 5. To take advantage of economic purchase order size

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Inventory Costs

 Holding (or carrying) costs  Setup (or production change) costs  Ordering costs  Shortage (or backlog) costs

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Independent vs. Dependent Demand

Independent Demand (Demand not related to other items) Dependent Demand (Derived/Calculated) 5

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Classifying Inventory Models

 Fixed-Order Quantity Models    Event triggered Make exactly the same amount Use re-order point to determine timing  Fixed-Time Period Models  Time triggered  Count the number needed to re-order

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Inventory Control Inventory Inventory Models Fixed Order Quantity Models Constant Demand Simple EOQ EOQ w/usage EOQ w/ Quantity Discounts Find the EOQ and R Determine p and d Calculate Total costs Find the EOQ and R Select Q and find R Uncertainty in Demand Fixed Time Period Models Single Period Models

Fixed-Order Quantity Models

Assumptions

      Demand for the product is constant and uniform throughout the period Lead time (time from ordering to receipt) is constant Price per unit of product is constant Inventory holding cost is based on average inventory Ordering or setup costs are constant All demands for the product will be satisfied (No back orders are allowed)

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EOQ Model--Basic Fixed-Order Quantity Model

Inventory Level Q R Q L R = Reorder point Q = Economic order quantity L = Lead time

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Time L Q 9

Basic Fixed-Order Quantity Model

Total Annual Cost = Annual Purchase Cost + Annual Ordering Cost + Annual Holding Cost Derive the Total annual Cost Equation, where:

TC - Total annual cost D - Annual demand (and d-bar = average daily demand = D/365) C - Cost per unit Q - Order quantity S - Cost of placing an order or setup cost R - Reorder point L - Lead time H - Annual holding and storage cost per unit of inventory

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Cost Minimization Goal

C O S T

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Q OPT Order Quantity (Q) Total Cost Holding Costs Annual Cost of Items (DC) Ordering Costs 11

Deriving the EOQ

 Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero Q OPT = 2DS = H 2(Annual Demand)(Or der or Setup Cost) Annual Holding Cost

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Reorder Point, R = dL

_ d = average demand per time unit L = Lead time (constant)

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EOQ Example

Annual Demand (D) = 1,000 units Days per year considered in average daily demand = 365 Cost to place an order (S) = $10 Holding cost per unit per year (H) = $2.40

Lead time (L) = 7 days Cost per unit (C) = $15

Determine the economic order quantity and the reorder point.

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Solution

Q OPT = 2DS = H 2(1,000 )(10) = 91.287

units 2.40

91 or 92 units???

d = 1,000 units / year 365 days / year = 2.74 units / day

Why do we round up?

_ R eorder point, R = d L = 2.74units / day (7days) = 19.18 or

20 units

When the inventory level reaches 20, order 91 units.

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Problem

 Retailer of Satellite Dishes D = 1000 units S = $ 25 H = $ 100 How much should we order?

What are the Total Annual Stocking Costs?

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EOQ with Quantity Discounts

  What if we get a price break for buying a larger quantity?

To find the lowest cost order quantity:  Since “C” changes for each price-break, H=iC  Where, i = percentage of unit cost attributed to carrying inventory and , C = cost (or price) per unit  Find the EOQ at each price break.

 Identify relevant and feasible order quantities.  Compare total annual costs  The lowest cost wins.

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EOQ with Quantity Discounts Example

Copper may be purchased for $ .82 per pound for up to 2,499 pounds $ .81 per pound for 2,500 to 5,000 pounds $ .80 per pound for orders greater than 5,000 pounds Demand (D) = 50,000 pounds per year Holding costs (H) are 20% of the purchase price per unit Ordering costs (S) = $30

How much should the company order to minimize total costs?

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44 43 42 41 40 0 Problem 28 Feasible <2500 <2500 - 4999 >5000 20 40 60 (Order Quantity 100's of units) 80 100

Inventory Inventory Control Inventory Models Constant Demand Fixed Order Quantity Models Uncertainty in Demand Find the  L Find Z Safety Stock Find the EOQ and R Fixed Time Period Models Single Period Models

What if demand is not Certain?

    Use safety stock to cover uncertainty in demand.

Given: service probability which is the probability demand will

NOT

exceed some amount.

The safety stock level is set by increasing the reorder point by the amount of safety stock.

The safety stock equals z•  L where,  L = the standard deviation of demand during the lead time.

 For example for a 5% chance of running out z  1.65

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Problem

Annual Demand = 25,750 or 515/wk @ 50 wks/year Annual Holding costs = 33% of item cost ($10/unit) Ordering costs are $250.00

 d = 25 per week Leadtime = 1 week Service Probability = 95% Find: a.) the EOQ and R b.) annual holding costs and annual setup costs c.) Would you accept a price break of $50 per order for lot sizes that are larger than 2000?

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Inventory Inventory Control Fixed Order Quantity Models Inventory Models Fixed Time Period Models Current Inventory Find the  T+L Find Z Find order quantity (q) Single Period Models

Fixed-Time Period Models

 Check the inventory every review period and then order a quantity that is large enough to cover demand until the next order will come in.

 The model assumes uncertainty in demand with safety stock added to the order quantity.  More exposure to variability than fixed-order models

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Fixed-Time Period Model with Safety Stock Formula

q = Average demand + Safety stock - Inventory currently on hand q = d (T + L) + Z  T + L I Where : q = quantity t o be ordered T = the number of days between reviews L = lead time in days d = forecast average daily demand z = the number of standard deviations  T + L = standard deviation for a specified service probabilit of demand over the review and lead time y I = current inventory level (includes items on order)

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Determining the Value of

 T+L  T + L = T i +   1 L   d i 2 Since each day is independen t and  d is constant,  T + L = (T + L)  d 2  The standard deviation of a sequence of random events equals the square root of the sum of the variances.

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Example of the Fixed-Time Period Model

Given the information below, how many units should be ordered?

Average daily demand for a product is 20 units.

The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The daily demand standard deviation is 4 units.

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Example of the Fixed-Time Period Model: Solution

 T+ L = (T + L)  d 2 = q = d (T + L) + Z  T + L I  2 = 25.298

q = 20(30 + 10) + (1.75)(25.

298) 200 q = 800  44.272

200 = 644.272, or

645 units

So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period.

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Problem

    A pharmacy orders antibiotics every two weeks (14 days).

the daily demand equals 2000 the daily standard deviation of demand = 800 lead time is 5 days  service level is 99 %  present inventory level is 25,000 units What is the correct quantity to order to minimize costs?

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Inventory Inventory Control Inventory Models Constant Demand Fixed Order Quantity Models Uncertainty in Demand Fixed Time Period Models Single Period Models

Single – Period Model for items w/obsolescence

(newsboy problem)

For a single purchase

  Amount to order is when marginal profit (MP) is equal to marginal loss (ML).

Adding probabilities (P = probability of that unit being sold) for the last unit ordered we want P(MP)  (1-P)ML or P  ML /(MP+ML) Increase order quantity as long as this holds.

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Single–Period Model

(Text Prob. #21)

Famous Albert’s Cookie King

Demand (dozen) 1,800 2,000 2,200 2,400 2,600 2,800 3,000

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Probability of Demand 0.05

0.10

0.20

0.30

0.20

0.10

0.05

Each dozen sells for $0.69

and costs $0.49 with a salvage value of $0.29.

How many cookies should he bake?

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

ABC Classification System

Items kept in inventory are not of equal importance in terms of: 60  dollars invested % of $ Value 30 A  profit potential 0 B  sales or usage volume % of Use 30 60 C  stock-out penalties So, identify inventory items based on percentage of total dollar value, where “A” items are roughly top 15 %, “B” items as next 35 %, and the lower 50% are the “C” items.

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Inventory Accuracy and Cycle Counting

 Inventory accuracy  Do inventory records agree with physical count?

 Cycle Counting   Frequent counts When? (zero balance, backorder, specified level of activity, level of important item, etc.)

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