Topic 6. INVENTORY MANAGEMENT

Topic 6. INVENTORY MANAGEMENT

I. Introduction

• What is inventory? – stored resource used to satisfy current or

future demand

• Types of Inventories:– Raw Materials/Components– In-Process Goods (WIP)– Finished Goods– Supplies

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

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

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

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

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

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

Introduction

• Inventory Classification– Identify importantItems 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

Annual $ volume of items

AA

BB

CC

High

Low

Few ManyNumber of Items

Introduction

• Monitor Inventory– As important as demand forecast for decision making– Universal Product Code - Bar code

printed on a label that hasinformation 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

0

214800 232087768

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

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

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

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.

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?

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

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

+ Setup (Order) Cost + Shortage Cost

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

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

EOQ Model (continued)

Placeorder

Receive order

Lead time

Receive order

Placeorder

Receive order

Reorderpoint

Q


• 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


• 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


– Average Inventory Level =

– Holding Cost =

– Number of orders per year =

– Setup (Order) Cost =

– Shortage Cost = 0, why?

2

Q

HQ

2

Q

D

SQ

D


– Product Cost =

– IC =

– Total Cost (TC) =

– Minimize TC Minimize IC, why?

SQ

DH

QDP

2

DP

SQ

DH

Q

2


• Observation: at the best order quantity EOQ (Q0),

holding cost = setup cost

Solve Q0, we have

SQ

DH

Q

0

0

2

PI

SD

H

SDQEOQ

22

0


SQ

DH

QIC

2

The Inventory Cost Curve is U-Shaped

An

nu

al C

os

t

(EOQ)QO Order Quantity (Q)

AnnualCarrying Costs

AnnualOrdering Costs


• 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?


• Sensitivity of IC with related to Q

-- Example (continued)

Avg.Inventory

Holding Cost

# of ordersper year

Order Cost IC

Q (Q/2) (Q/2)*H (D/Q) (D/Q)*S(Q/2)*H+(D/Q)*S

500 250 $250 20 $1,000 $1,250

1000 500 $500 10 $500 $1,000

1500 750 $750 6.667 $333 $1,083


• Conclusion: – 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?


• Sensitivity of EOQ with related to D, H, S, P, I– 1. Insensitive to parameter change– 2. Directions?

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

EPQ Model (continued)

Finishproduction

Start toproduce

Reorderpoint

Quantityon hand

Q Production rate- usage rate

Usage rate

Start toproduce

Production run length

Time


• 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


• Maximum Inventory Level =

• Average Inventory Level =

• Annual Holding Cost =

p

p Qp

dp

p

Qdp

pQ

p

dp

2

1

HQ

p

dpp

2

1


• Number of production runs per year =

• Order Cost =

• IC =

• TC =

• Minimize TC Minimize IC, why?

pQ

D

SQ

D

p

S

Q

DQ

p

dp

pp

2

1

S

Q

DQ

p

dpDP

pp

2

1


• Observation: at EPO,

holding cost = setup cost

• Best Production Quantity (EPQ) formula:

dp

p

H

SDQEPQ p

2

0

S

Q

DQ

p

dp

pp

002

1


• 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.

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.

EOQ with discount (continued)

• Example:

Order Price

0-399 $2.1/unit

400-699 $2.0

Great equal 700 $1.9

• Idea is to compare TC curves under different prices - why TC?

SQ

DH

QDPTC

2


Order Quantity

Total Cost Curvefor Price 1



$ co

st

400 700


Order Quantity




$ co

st

400 700


• 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.


• 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


• 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.

III. Models on Reorder Points - When to Order?

• 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

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


• 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

yprobabilit itsdemand possibleeach m


• Remarks:– Higher the desired service level --->– More uncertain the demand --->

• Two methods to determine the SS


• 1. Determine SS and ROP based on shortage cost inf. (if available) – SS increases Holding cost ? Shortage

cost ?– Best SS minimizes total inventory cost


• 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.


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

0 5 10 15 20 25 30

Probability 0.1 0.15 0.2 0.1 0.2 0.15 0.1


• Determine SS and ROP based on shortage cost inf. (continued)

• If SS = 0, ROP = m = 15 switches

151.03015.0252.020

1.0152.01015.051.00

yprobabilit itsdemand possibleeach

m


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


• Determine SS and ROP based on shortage cost inf. (continued)

• Determine the best SS in following table

Safety stock

Add. Holdingcost

Avg. shortage (per order)

Annual shortage cost

Total cost

0

5

10

15


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


• Case 1. Empirical Lead time demand distribution • -- Example:

Lead Time Demand

Frequency Probability ROP Service Level

3 2

4 3

5 5

6 5

7 3

8 2


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


• 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.

2z

IV. Single Period Model and Marginal Analysis (Newsvendor

Problem)

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?)

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)

Homework (Additional problems) • Problem 3• 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?


• 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

0 40 80 120 160

Probability 0.1 0.25 0.3 0.25 0.1


• 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?


• 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 0 40 80

Probability

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Topic 6. INVENTORY MANAGEMENT