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Inventory Management

Chapter 12

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Learning Objectives

When you complete this chapter you should be able to:

Conduct an ABC analysis

Explain and use cycle counting

Explain and use the EOQ model

Compute a reorder point

Explain and use the quantity discount model

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Inventory Management

The objective of inventory management is to strike a balance between inventory investment and customer service

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Importance of Inventory

One of the most expensive assets of many companies representing as much as 50% of total invested capital

Less inventory lowers costs but increases chances of running out

More inventory raises costs but always keeps customers happy but it can damage your business!!!

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Functions of Inventory

Inventory can serve several functions that add flexibility to a firm’s operations. The four functions of inventory are:

To provide a selection of goods for anticipated demand and to separate the firm from fluctuations in demand.

To decouple or separate various parts of the production process. For example, if a firm’s supplies fluctuate, extra inventory may be necessary to decouple the production process from suppliers.

To take advantage of quantity discounts

To hedge against inflation and upward price changes

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Types of Inventory

Raw material

Purchased but not processed

Work-in-process (WIP)

Undergone some change but not completed

Finished goods

Completed product awaiting shipment

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The Material Flow Cycle

Figure 12.1

Input Wait for Wait to Move Wait in queue Setup Run Output

inspection be moved time for operator time time

Cycle time

95% 5%

FYI

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Inventory Management at Amazon.com

Amazon.com started as a “virtual” retailer – no inventory, no warehouses, no overhead – just computers taking orders to be filled by others

Growth has forced Amazon.com to become a world leader in warehousing and inventory management

FYI

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Inventory Management at Amazon.com

Each order is assigned by computer to the closest distribution center that has the product(s)

A “flow meister” at each distribution center assigns work crews

Technology helps workers pick the correct items from the shelves with almost no errors

Items are placed in crates on a conveyor, bar code scanners scan each item 15 times to virtually eliminate errors

FYI

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Inventory Management at Amazon.com

Crates arrive at central point where items are boxed and labeled with new bar code

Gift wrapping is done by hand at 30 packages per hour

Completed boxes are packed, taped, weighed and labeled before leaving warehouse in a truck

Order arrives at customer within 1 - 2 days

FYI

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Managing Inventory

How inventory items can be classified and monitored?

ABC analysis

How accurate inventory records can be maintained?

Record accuracy

Cycle counting

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ABC Analysis

ABC analysis is an inventory categorization method which consists in dividing items into three categories (A, B, C) based on annual dollar volume:

Class A - high annual dollar volume

Class B - medium annual dollar volume

Class C - low annual dollar volume

This method aims to draw managers’ attention on the critical few (A-items) not on the trivial many (C-items).

Used to establish policies that focus on the few critical parts and not the many trivial ones (Based on Pareto Principle, 80/20)

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ABC Classes

The ABC approach states that a company should rate items from A to C, basing its ratings on the following rules:

A-items About 20% of the items account for about 70-80% of the dollar usage.

B-items 30% of the items account for about 15-25% of the dollar usage.

C-items About 50% of the items account for about 5% of the dollar usage.

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The ABC analysis

The annual consumption value is calculated with the formula:

(Annual demand) x (item cost per unit)

Through this categorization, the supply manager can identify inventory hot spots, and separate them from the rest of the items, especially those that are numerous but not that profitable.

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4 Steps for the classification of items:

Find out the unit cost and the usage of each material over a given period;

Multiply the unit cost by the estimated annual usage to obtain the net value;

List out all the items and arrange them in the descending value (Annual Value);

Accumulate value and add up number of items and calculate percentage on total inventory in value and in number

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ABC analysis How to Interpret the Data

	Percentage of items	Percentage value of annual dollar volume	Action
Class A items	About 20%	About 70-80%	Close day to day control
Class B items	About 30%	About 15-25% of total value	Regular review
Class C items	About 50%	About 5%	Infrequent review

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ABC Analysis

A Items

B Items

| | | | | | | | | |

10 20 30 40 50 60 70 80 90 100

Percentage of annual dollar usage

80 –

70 –

60 –

50 –

40 –

30 –

20 –

10 –

0 –

Percentage of inventory items

C Items

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ABC Analysis

ABC ANALYSIS FOR A CHIP ANUFACTURER

Silicon Chips, Inc., maker of superfast DRAM chips, wants to categorize its 10 major inventory items

using ABC analysis.

APPROACH ABC analysis organizes the items on an annual dollar-volume basis. Shown below (in columns 1–4) are the 10 items (identified by stock numbers), their annual demands, and unit costs.

SOLUTION Annual dollar volume is computed in column 5, along with the percentage of the total represented by each item in column 6. Column 7 groups the 10 items into A, B, and C categories.

INSIGHT The breakdown into A, B, and C categories is not hard and fast. The objective is to try to separate the “important” from the “unimportant.”

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ABC Analysis

ABC Calculation
(1)	(2)	(3)		(4)		(5)		(6)		(7)
ITEM STOCK NUMBER	PERCENT OF NUMBER OF ITEMS STOCKED	ANNUAL VOLUME (UNITS)	x	UNIT COST	=	Annual $ usage		% $ usage		CLASS
#10286	20%	1,000		$ 90.00			$ 90,000		38.8%	A
#11526		500		154.00			77,000		33.2%	A
#12760		1,550		17.00			26,350		11.3%	B
#10867	30%	350		42.86			15,001		6.4%	B
#10500		1,000		12.50			12,500		5.4%	B
#12572		600		14.17			8,502		3.7%	C
#14075		2,000		.60			1,200		.5%	C
#01036	50%	100		8.50			850		.4%	C
#01307		1,200		.42			504		.2%	C
#10572		250		.60			150		.1%	C
		8,550					$232,057		100.0%

72%

23%

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ABC Analysis

Policies employed may include

More emphasis on supplier development for A items

Tighter physical inventory control for A items and should be stored in more secure area.

The accuracy of inventory records for A items should be verified more frequently

More care in forecasting A items

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Example: classify the following items according to ABC analysis

Item number	Unit cost	Annual demand
101	5	48,000
102	11	2,000
103	15	300
104	8	800
105	7	4,800
106	16	1,200
107	20	18,000
108	4	300
109	9	5,000
110	12	500

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Step 1

Calculate the total spending per year (Annual $ usage)

Item number	Unit cost	Annual demand	Annual $ usage
101	5	48,000	240,000
102	11	2,000	22,000
103	15	300	4,500
104	8	800	6,400
105	7	4,800	33,600
106	16	1,200	19,200
107	20	18,000	360,000
108	4	300	1,200
109	9	5,000	45,000
110	12	500	6,000
Total usage			737,900

Total cost per year: Unit cost * total cost per year

Unit cost x Annual demand

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Step 2

Sort the items by annual usage (Descending Order)

Annual $ usage = Unit Cost x Annual Demand

Item number	Unit cost	Annual demand	Annual $ usage
107	20	18,000	360,000
101	5	48,000	240,000
109	9	5,000	45,000
105	7	4,800	33,600
102	11	2,000	22,000
106	16	1,200	19,200
104	8	800	6,400
110	12	500	6,000
103	15	300	4,500
108	4	300	1,200
Total usage			737,900

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Step 3

Calculate the usage of item in total usage (% $ usage)

X 100

Item number	Unit cost	Annual demand	Annual $ usage	% $ usage
107	20	18,000	360,000	48,8%
101	5	48,000	240,000	32,5%
109	9	5,000	45,000	6,1%
105	7	4,800	33,600	4,6%
102	11	2,000	22,000	3,0%
106	16	1,200	19,200	2,6%
104	8	800	6,400	0,9%
110	12	500	6,000	0,8%
103	15	300	4,500	0,6%
108	4	300	1,200	0,2%
Total usage			737,900

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Step 4

Find the Cumulative % $

Item number	Unit cost	Annual demand	Annual $ usage	% $ usage	Cum % $
107	20	18,000	360,000	48,8%	48,8%
101	5	48,000	240,000	32,5%	81,3%
109	9	5,000	45,000	6,1%	87,4%
105	7	4,800	33,600	4,6%	92%
102	11	2,000	22,000	3,0%	94,9%
106	16	1,200	19,200	2,6%	97,5%
104	8	800	6,400	0,9%	98,4%
110	12	500	6,000	0,8%	99,2%
103	15	300	4,500	0,6%	99,8%
108	4	300	1,200	0,2%	100%
Total usage			737,900

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Step 5

Classify items into classes

Item number	Unit cost	Annual demand	Annual $ usage	% $ usage	CUM% $	Class	% of items	Action
107	20	18,000	360,000	48,8%	48,8%	A	20% of items	Close control
101	5	48,000	240,000	32,5%	81,3%	A
109	9	5,000	45,000	6,1%	87,4%	B	30% of items	Regular review
105	7	4,800	33,600	4,6%	92%	B
102	11	2,000	22,000	3,0%	94,9%	B
106	16	1,200	19,200	2,6%	97,5%	C	50% of items	Infrequent review
104	8	800	6,400	0,9%	98,4%	C
110	12	500	6,000	0,8%	99,2%	C
103	15	300	4,500	0,6%	99,8%	C
108	4	300	1,200	0,2%	100%	C
Total usage			737,900

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Inventory Record Accuracy

Inventory Record Accuracy (IRA) is a measure of how closely official inventory records match the physical inventory.

Accurate records are a critical ingredient in production and inventory systems.

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Record Accuracy

Incoming and outgoing record keeping must be accurate, specially for class A items.

Stockrooms should be secure specially for class A items.

Necessary to make precise decisions about ordering, scheduling, and shipping, specially for class A items.

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Cycle Counting

Cycle Counting: is an inventory auditing procedure, where a small subset of inventory, in a specific location, is counted on a specified day.

Often used with ABC analysis

A items will be counted frequently (once a month)

B items will be counted less frequently (once a quarter)

C items will be counted (one every 6 months)

Has several advantages:

Eliminates shutdowns and interruptions

Eliminates annual inventory adjustment

Allows causes of errors to be identified and corrected faster

Maintains accurate inventory records

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Cycle Counting Example

5,000 items in inventory, classified as: 500 A items, 1,750 B items, 2,750 C items

Policy is to count A items every month (20 working days), B items every quarter (60 days), and C items every six months (120 days)

ITEM CLASS	QUANTITY	CYCLE COUNTING POLICY	NUMBER OF ITEMS COUNTED PER DAY
A	500	Each month	500/20 =	25/day
B	1,750	Each quarter	1,750/60 =	29/day
C	2,750	Every 6 months	2,750/120 =	23/day
				77/day

Each day 77 items are counted, which is more efficient and accurate than conducting a massive inventory count once a year

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Inventory Models

Need to determine when and how much to order

Basic economic order quantity (EOQ) model

Production order quantity model

Quantity discount model

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Basic EOQ Model

Demand is known and constant

Lead time is known and constant

Receipt of inventory is instantaneous and complete

Quantity discounts are not possible

The only variable costs (that are considered) are setup (ordering ) and holding (carrying) cost .

Stock-outs can be completely avoided

Important assumptions

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Dollar Costs Associated with Inventory Models

Setup costs (S) or Ordering Cost: the costs to prepare a machine or process for manufacturing an order OR the costs of placing an order and receiving goods

Holding cost (H) or Carrying Cost: is the cost of holding a unit in inventory for a given time period.

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Holding Costs (H)

Holding cost might be given in terms of cost/unit/year or it might be given in the form of rate%, in that case we need to find the holding cost by:

H = I* p

p = Unit production cost or unit procurement cost, the cost of manufacturing (materials, labor, overhead) or purchasing one unit

I = Percentage based upon rates for borrowing, insuring, investing (opportunity), and so on.

H includes expenses such as:

storage (rent, lease, mortgage, utilities, maintenance, etc.), tracking and monitoring of inventory, damage and pilferage, insurance, interest on money to produce or procure the items in inventory, and the opportunity cost of money tied up in inventory (and not available for use elsewhere).

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Holding Costs

TABLE 12.1	Determining Inventory Holding Costs
CATEGORY		COST (AND RANGE) AS A PERCENT OF INVENTORY VALUE
Housing costs (building rent or depreciation, operating costs, taxes, insurance)			6% (3 - 10%)
Material handling costs (equipment lease or depreciation, power, operating cost)			3% (1 - 3.5%)
Labor cost (receiving, warehousing, security)			3% (3 - 5%)
Investment costs (borrowing costs, taxes, and insurance on inventory)			11% (6 - 24%)
Pilferage, space, and obsolescence (much higher in industries undergoing rapid change like tablets and smart phones)			3% (2 - 5%)
Overall carrying cost			26%

FYI

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Holding Costs

TABLE 12.1	Determining Inventory Holding Costs
CATEGORY		COST (AND RANGE) AS A PERCENT OF INVENTORY VALUE
Housing costs (building rent or depreciation, operating costs, taxes, insurance)			6% (3 - 10%)
Material handling costs (equipment lease or depreciation, power, operating cost)			3% (1 - 3.5%)
Labor cost (receiving, warehousing, security)			3% (3 - 5%)
Investment costs (borrowing costs, taxes, and insurance on inventory)			11% (6 - 24%)
Pilferage, space, and obsolescence (much higher in industries undergoing rapid change like PCs and cell phones)			3% (2 - 5%)
Overall carrying cost			26%

Holding costs vary considerably depending on the business, location, and interest rates. Generally greater than 15%, some high tech and fashion items have holding costs greater than 40%.

FYI

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Inventory Usage Over Time

Figure 12.3

Order quantity = Q (maximum inventory level)

Usage rate

Average inventory on hand

Minimum inventory

Inventory level

Time

Total order received

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Minimizing Costs

Objective is to minimize total costs

Table 12.4(c)

Annual cost

Order quantity

Total cost of holding and setup (order)

Holding cost

Setup (order) cost

Minimum total cost

Optimal order quantity (Q*)

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Example

Annual Demand = 4000 units

Setup/ordering cost = $100

Holding cost = $5/unit per year

Which one of the following quantities will minimize the total cost?

	C1	C2	C3	C4
Quantity (Q)	50	400	2000	4000
Annual Holding Cost (AHC)
Annual Setup Cost (ASC)
Total Cost (TC)

FYI

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WHAT DO YOU OBSERVE ?!

	C1	C2	C3	C4
Quantity (Q)	50 units	400 units	2000 units	4000 units
Annual Holding Cost (AHC)	$125	$1000	$5000	$10000
Annual Setup Cost (ASC)	$8000	$1000	$200	$100
Total Cost (TC)	$8125	$2000	$5200	$10100

FYI

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Minimizing Costs

By minimizing the sum of setup (or ordering) and holding costs, total costs are minimized

Optimal order size Q* will minimize total cost

A reduction in either cost reduces the total cost

Optimal order quantity occurs when holding cost and setup cost are equal

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Minimizing Costs

Q = Number of units per order

Q* = Optimal number of units per order (EOQ)

D = Annual demand in units for the inventory item

S = Setup or ordering cost for each order

H = Holding or carrying cost per unit per year

Annual setup cost = (Number of orders placed per year)

x (Setup or order cost per order)

Annual demand

Number of units in each order

Setup or order cost per order

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Minimizing Costs

Annual setup cost = (Number of orders placed per year)

x (Setup or order cost per order)

Annual demand

Number of units in each order

Setup or order cost per order

Q = Number of pieces per order

Q* = Optimal number of pieces per order (EOQ)

D = Annual demand in units for the inventory item

S = Setup or ordering cost for each order

H = Holding or carrying cost per unit per year

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Q = Number of pieces per order

Q* = Optimal number of pieces per order (EOQ)

D = Annual demand in units for the inventory item

S = Setup or ordering cost for each order

H = Holding or carrying cost per unit per year

Minimizing Costs

Annual holding cost = (Average inventory level)

x (Holding cost per unit per year)

Order quantity

(Holding cost per unit per year)

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Minimizing Costs

Optimal order quantity is found when annual setup cost equals annual holding cost

Solving for Q*

Q = Number of pieces per order

Q* = Optimal number of pieces per order (EOQ)

D = Annual demand in units for the inventory item

S = Setup or ordering cost for each order

H = Holding or carrying cost per unit per year

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D = Annual demand (units)

S = Cost per order ($)

P = Cost per unit ($)

I = Holding cost (%)

H = Holding cost ($) = I x P

Economic Order Quantity (EOQ)

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EOQ Model Equations

Annual total cots = +

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EOQ Example

You’re a buyer for SaveMart.

SaveMart needs 1000 coffee makers per year. The cost of each coffee maker is $50. Ordering cost is $100 per order. Carrying cost is 40% of per unit cost. Lead time is 3 days. SaveMart is open 200 days/yr.

What is the optimal order quantity & ROP?

Find the total cost

Determine expected number of orders

Determine the time between orders

Determine the ROP?

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1. SaveMart EOQ

D = 1000

S = $100

P = $50

I = 40%

H = P x I

H = $20

EOQ = 100 coffeemakers

EOQ = Q* =

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2. SaveMart total cost

D = 1000

S = $100

H = $20

Q* = 100

Annual total cots = +

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3. Number of orders

D = 1000

S = $100

H = $20

Q* = 100

Number of order (N) =

N = = 10 orders

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4. Time between orders

D = 1000

S = $100

H = $20

Q* = 100

No. working days = 200

N = 10 orders

Time between orders (T) =

T= = 20 days

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5. SaveMart ROP

ROP = daily demand x lead time (days)

ROP = d x LT

Daily Demand (d) =

d = = 5 units

ROP = 5 x 3 = 15 units

It means: once you reach 15 units, you have to place an order of 100 units (i.e your EOQ)

D = 1000

No. working days = 200

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An EOQ Example

Determine optimal number of needles to order

D = 1,000 units

S = $10 per order

H = $.50 per unit per year

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An EOQ Example

Determine expected number of orders

D = 1,000 units Q* = 200 units

S = $10 per order

H = $.50 per unit per year

N = = 5 orders per year

1,000

200

= N = =

Expected number of orders

Demand

Order quantity

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An EOQ Example

Determine optimal time between orders

D = 1,000 units Q* = 200 units

S = $10 per order N = 5 orders/year

H = $.50 per unit per year

T = = 50 days between orders

250

= T =

Expected time between orders

Number of working days per year

Expected number of orders

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An EOQ Example

Determine the total annual cost

D = 1,000 units Q* = 200 units

S = $10 per order N = 5 orders/year

H = $.50 per unit per year T = 50 days

Total annual cost = Setup cost + Holding cost

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The EOQ Model

When including actual cost of material P

Total annual cost = Setup cost + Holding cost + Product cost

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The EOQ Model

When including actual cost of material P

Total annual cost = Setup cost + Holding cost + Product cost

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Robust Model

The EOQ model is robust

It works even if all parameters and assumptions are not met

The total cost curve is relatively flat in the area of the EOQ

FYI

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Reorder Points

EOQ answers the “how much” question

The reorder point (ROP) tells “when” to order

Lead time (L) is the time between placing and receiving an order

ROP =

Lead time for a new order in days

Demand per day

ROP = d x L

d =

Number of working days in a year

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Reorder Point Curve

ROP (units)

Inventory level (units)

Time (days)

Figure 12.5

Lead time = L

Slope = units/day = d

Stock is replenished as order arrives

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Reorder Point Example

Demand = 8,000 iPhones per year

250 working day year

Lead time for orders is 3 working days, may take 4

ROP = d x L

d =

Number of working days in a year

= 8,000/250 = 32 units

= 32 units per day x 3 days = 96 units

= 32 units per day x 4 days = 128 units

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Production Order Quantity Model

Used when inventory builds up over a period of time after an order is placed

Used when units are produced and sold simultaneously

Inventory level

Time

Demand part of cycle with no production (only usage takes place)

Part of inventory cycle during which production (and usage) is taking place

Maximum inventory

Figure 12.6

FYI

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Production Order Quantity Model

Q = Number of units per order p = Daily production rate

H = Holding cost per unit per year d = Daily demand/usage rate

t = Length of the production run in days

= (Average inventory level) x

Annual inventory holding cost

Holding cost per unit per year

= (Maximum inventory level)/2

Annual inventory level

= –

Maximum inventory level

Total produced during the production run

Total used during the production run

= pt – dt

FYI

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Production Order Quantity Model

Q = Number of units per order p = Daily production rate

H = Holding cost per unit per year d = Daily demand/usage rate

t = Length of the production run in days

= –

Maximum inventory level

Total produced during the production run

Total used during the production run

= pt – dt

However, Q = total produced = pt ; thus t = Q/p

Maximum inventory level

= p – d = Q 1 –

Holding cost = (H) = 1 – H

Maximum inventory level

FYI

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Production Order Quantity Model

Q = Number of units per order p = Daily production rate

H = Holding cost per unit per year d = Daily demand/usage rate

t = Length of the production run in days

FYI

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Production Order Quantity Example

D = 1,000 units p = 8 units per day

S = $10 d = 4 units per day

H = $0.50 per unit per year

FYI

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Production Order Quantity Model

When annual data are used the equation becomes:

Note:

d = 4 = =

Number of days the plant is in operation

1,000

250

FYI

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Quantity Discount Models

Reduced prices are often available when larger quantities are purchased

Trade-off is between reduced product cost and increased holding cost

TABLE 12.2	A Quantity Discount Schedule
PRICE RANGE		QUANTITY ORDERED	PRICE PER UNIT P
Initial price		0 to 119	$ 100
Discount price 1		200 to 1,499	$ 98
Discount price 2		1,500 and over	$ 96

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Quantity Discount Models

Total annual cost = Setup cost + Holding cost + Product cost

where Q = Quantity ordered P = Price per unit

D = Annual demand in units I = Holding cost per unit per year

S = Ordering or setup cost per order expressed as a percent of price P

Because unit price varies, holding cost is expressed as a percent (I) of unit price (P)

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Quantity Discount Models

Steps in analyzing a quantity discount

Starting with the lowest possible purchase price, calculate Q* until the first feasible EOQ is found. This is a possible best order quantity, along with all price-break quantities for all lower prices.

Calculate the total annual cost for each possible order quantity determined in Step

Select the quantity that gives the lowest total cost.

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Quantity Discount Models

Figure 12.7

520,053 –

517,155 –

Annual Total Cost

Order Quantity

550,000 –

540,000 –

530,000 –

510,000 –

500,000 –

120

1,500

TC for No Discount

TC for Discount 2

TC for Discount 1

Initial Price

Discount Price 1

Discount Price 2

Possible Order Quantities

Not Feasible

Feasible

FYI

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Quantity Discount Models

D = 5200 units S = $200 I = 28%

Because unit price varies, holding cost is expressed as a percent (I) of unit price (P)

H = IP

TABLE 12.2	A Quantity Discount Schedule
PRICE RANGE		QUANTITY ORDERED	PRICE PER UNIT P
Initial price		0 to 119	$ 100
Discount price 1		200 to 1,499	$ 98
Discount price 2		1,500 and over	$ 96

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Quantity Discount Example

Calculate Q* for every discount starting with the lowest price

Q$96* = = 278 drones/order

2(5,200)($200)

(0.28)($96)

Q$98* = = 275 drones/order

2(5,200)($200)

(0.28)($98)

Infeasible – calculate Q* for next-higher price

Feasible

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Quantity Discount Example

TABLE 12.3		Total Cost Computations for Chris Beehner Electronics
ORDER QUANTITY	UNIT PRICE		ANNUAL ORDERING COST	ANNUAL HOLDING COST	ANNUAL PRODUCT COST	TOTAL ANNUAL COST
1,500	$96		$693	$20,160	$499,200	$520,053
275	$98		$3,782	$3,773	$509,600	$517,155

Choose the price and quantity that gives the lowest total cost

Buy 275 drones at $98 per unit

12 - ‹#›

Quantity Discount Variations

All-units discount is the most popular form

Incremental quantity discounts apply only to those units purchased beyond the price break quantity

Fixed fees may encourage larger purchases

Aggregation over items or time

Truckload discounts, buy-one-get-one-free offers, one-time-only sales

FYI

12 - ‹#›

Probabilistic Models and Safety Stock

Used when demand is not constant or certain

Use safety stock to achieve a desired service level and avoid stockouts

ROP = d x L + ss

Annual stockout costs = The sum of the units short for each demand level x The probability of that demand level x The stockout cost/unit x The number of orders per year

FYI

12 - ‹#›

Safety Stock Example

NUMBER OF UNITS	PROBABILITY
30		.2
40		.2
ROP  50		.3
60		.2
70		.1
		1.0

ROP = 50 units Stockout cost = $40 per frame

Orders per year = 6 Carrying cost = $5 per frame per year

FYI

12 - ‹#›

Safety Stock Example

ROP = 50 units Stockout cost = $40 per frame

Orders per year = 6 Carrying cost = $5 per frame per year

SAFETY STOCK	ADDITIONAL HOLDING COST	STOCKOUT COST	TOTAL COST
20	(20)($5) = $100	$0	$100
10	(10)($5) = $ 50	(10)(.1)($40)(6) = $240	$290
0	$ 0	(10)(.2)($40)(6) + (20)(.1)($40)(6) = $960	$960

A safety stock of 20 frames gives the lowest total cost

ROP = 50 + 20 = 70 frames

FYI

12 - ‹#›

Safety stock

16.5 units

ROP 

Place order

Probabilistic Demand

Inventory level

Time

Minimum demand during lead time

Maximum demand during lead time

Mean demand during lead time

Normal distribution probability of demand during lead time

Expected demand during lead time (350 kits)

ROP = 350 + safety stock of 16.5 = 366.5

Receive order

Lead time

Figure 12.8

FYI

12 - ‹#›

Probabilistic Demand

Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined

ROP = demand during lead time + ZsdLT

where Z = Number of standard deviations

sdLT = Standard deviation of demand during lead time

FYI

12 - ‹#›

Probabilistic Demand

Safety stock

Probability of no stockout 95% of the time

Mean demand 350

ROP = ? kits

Quantity

Number of standard deviations

Risk of a stockout (5% of area of normal curve)

FYI

12 - ‹#›

Probabilistic Example

m = Average demand = 350 kits

sdLT = Standard deviation of demand during lead time = 10 kits

Stockout policy = 5% (service level = 95%)

Using Appendix I, for an area under the curve of 95%, the Z = 1.645

Safety stock = ZsdLT = 1.645(10) = 16.5 kits

Reorder point = Expected demand during lead time + Safety stock

= 350 kits + 16.5 kits of safety stock

= 366.5 or 367 kits

FYI

12 - ‹#›

Other Probabilistic Models

When data on demand during lead time is not available, there are other models available

When demand is variable and lead time is constant

When lead time is variable and demand is constant

When both demand and lead time are variable

FYI

12 - ‹#›

Other Probabilistic Models

Demand is variable and lead time is constant

ROP = (Average daily demand x Lead time in days) + ZsdLT

where sdLT = sd Lead time

sd = Standard deviation of demand per day

FYI

12 - ‹#›

Probabilistic Example

Average daily demand (normally distributed) = 15

Lead time in days (constant) = 2

Standard deviation of daily demand = 5

Service level = 90%

Z for 90% = 1.28

From Appendix I

ROP = (15 units x 2 days) + ZsdLT

= 30 + 1.28(5)( 2)

= 30 + 9.02 = 39.02 ≈ 39

Safety stock is about 9 computers

FYI

12 - ‹#›

Other Probabilistic Models

Lead time is variable and demand is constant

ROP = (Daily demand x Average lead time in days) + Z x (Daily demand) x sLT

where sLT = Standard deviation of lead time in days

FYI

12 - ‹#›

Probabilistic Example

Daily demand (constant) = 10

Average lead time = 6 days

Standard deviation of lead time = sLT = 1

Service level = 98%, so Z (from Appendix I) = 2.055

ROP = (10 units x 6 days) + 2.055(10 units)(1)

= 60 + 20.55 = 80.55

Reorder point is about 81 cameras

FYI

12 - ‹#›

Other Probabilistic Models

Both demand and lead time are variable

ROP = (Average daily demand x Average lead time) + ZsdLT

where sd = Standard deviation of demand per day

sLT = Standard deviation of lead time in days

sdLT = (Average lead time x sd2) + (Average daily demand)2s2LT

FYI

12 - ‹#›

Probabilistic Example

Average daily demand (normally distributed) = 150

Standard deviation = sd = 16

Average lead time 5 days (normally distributed)

Standard deviation = sLT = 1 day

Service level = 95%, so Z = 1.645 (from Appendix I)

FYI

12 - ‹#›

Single-Period Model

Only one order is placed for a product

Units have little or no value at the end of the sales period

Cs = Cost of shortage = Sales price/unit – Cost/unit

Co = Cost of overage = Cost/unit – Salvage value

Service level =

Cs + Co

FYI

12 - ‹#›

Single-Period Example

Average demand =  = 120 papers/day

Standard deviation =  = 15 papers

Cs = cost of shortage = $1.25 – $.70 = $.55

Co = cost of overage = $.70 – $.30 = $.40

Service level =

Cs + Co

= = .579

.55

.55 + .40

.55

.95

Service level 57.9%

Optimal stocking level

 = 120

FYI

12 - ‹#›

Single-Period Example

From Appendix I, for the area .579, Z  .20

The optimal stocking level

= 120 copies + (.20)()

= 120 + (.20)(15) = 120 + 3 = 123 papers

The stockout risk = 1 – Service level

= 1 – .579 = .422 = 42.2%

FYI

12 - ‹#›

Fixed-Period (P) Systems

Fixed-quantity models require continuous monitoring using perpetual inventory systems

In fixed-period systems orders placed at the end of a fixed period

Periodic review, P system

FYI

12 - ‹#›

Fixed-Period (P) Systems

Inventory counted only at end of period

Order brings inventory up to target level

Only relevant costs are ordering and holding

Lead times are known and constant

Items are independent of one another

FYI

12 - ‹#›

Fixed-Period (P) Systems

On-hand inventory

Time

Target quantity (T)

Figure 12.9

FYI

12 - ‹#›

Fixed-Period Systems

Inventory is only counted at each review period

May be scheduled at convenient times

Appropriate in routine situations

May result in stockouts between periods

May require increased safety stock

FYI

12 - ‹#›

Annual setup cost = D Q S

Annual setup cost =

Annual holding cost = Q 2 H

Annual holding cost =

= D Q ⎛

⎝ ⎜

⎞

⎠ ⎟S

Annual setup cost = D Q S

Annual setup cost =

= Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟H

D Q

⎛

⎝ ⎜

⎞

⎠ ⎟S =

Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟H

S =

2DS =Q2H

Q2 = 2DS H

Q* = 2DS H

2DS=Q

2DS

EOQ

Optimal Or

der Quanti

Expected N

umber Orde

Expected T

ime Betwee

n Orders

Working Da

Year

Working Da

Year

ROP

EOQ

Q* = 2DS H

2DS

Q* = 2(1,000)(10)

0.50 = 40,000 = 200 units

2(1,000)(10)

0.50

=40,000=200 units

D Q*

TC = D Q S + Q 2 H

= 1,000 200

($10)+ 200 2 ($.50)

= (5)($10)+(100)($.50) =$50+$50=$100

TC=

S+

1,000

200

($10)+

200

($.50)

=(5)($10)+(100)($.50)

=$50+$50=$100

TC = D Q S + Q 2 H +PD

TC=

S+

H+PD

Setup cost = (D /Q)S Holding cost = 1

2 HQ 1− d p( )⎡⎣ ⎤⎦

Setup cost = (D/Q)S

Holding cost =

HQ1-dp

(

)

D Q S = 1

2 HQ 1− d p( )⎡⎣ ⎤⎦

Q2 = 2DS

H 1− d p( )⎡⎣ ⎤⎦

Qp * =

2DS H 1− d p( )⎡⎣ ⎤⎦

HQ1-dp

(

)

2DS

H1-dp

()

2DS

H1-dp

()

Qp * =

2DS H 1− d p( )⎡⎣ ⎤⎦

Qp * =

2(1,000)(10) 0.50 1−(4 8)⎡⎣ ⎤⎦

= 20,000

0.50(1 2) = 80,000

= 282.8 hubcaps, or 283 hubcaps

2DS

H1-dp

()

2(1,000)(10)

0.501-(48)

20,000

0.50(12)

=80,000

=282.8 hubcaps, or 283 hubcaps

Qp * =

2DS

H 1− Annual demand rate Annual production rate

⎛

⎝ ⎜

⎞

⎠ ⎟

2DS

H1-

Annual demand rate

Annual production rate

Q* = 2DS IP

2DS

TC = D Q S+ Q 2 IP+PD

TC=

S+

IP+PD

Q* = 2DS IP

2DS

ROP=(150 packs×5 days)+1.645σ dLT

σ dLT

= 5 days×162( )+ 1502 ×12( ) = 5×256( )+ 22,500×1( ) = 1,280( )+ 22,500( ) = 23,780 ≅154

ROP =(150×5)+1.645(154)≅750+253=1,003 packs

ROP=(150 packs´5 days)+1.645s

dLT

=5 days´16

( )

+150

´1

( )

=5´256

()

+22,500´1

( )

=1,280

()

+22,500

( )

=23,780@154

ROP =(150´5)+1.645(154)@750+253=1,003 packs