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Inventory Management

PowerPoint presentation to accompany

Heizer and Render

Operations Management, Eleventh Edition

Principles of Operations Management, Ninth Edition

PowerPoint slides by Jeff Heyl

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Outline

Global Company Profile:
Amazon.com

The Importance of Inventory
Managing Inventory
Inventory Models
Inventory Models for Independent Demand

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Outline - Continued

Probabilistic Models and Safety Stock

Single-Period Model

Fixed-Period (P) Systems

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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 for independent inventory demand
Compute a reorder point and safety stock

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Learning Objectives

When you complete this chapter you should be able to:

Apply the production order quantity model
Explain and use the quantity discount model
Understand service levels and probabilistic inventory models

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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

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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

Lights indicate products that are to be picked and the light is reset

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

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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

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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
Operations managers must balance inventory investment and customer service

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Functions of Inventory

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
To take advantage of quantity discounts
To hedge against inflation

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Types of Inventory

Raw material
Purchased but not processed
Work-in-process (WIP)
Undergone some change but not completed
A function of cycle time for a product
Maintenance/repair/operating (MRO)
Necessary to keep machinery and processes productive
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%

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Managing Inventory

How inventory items can be classified (ABC analysis)
How accurate inventory records can be maintained

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ABC Analysis

Divides inventory into three classes based on annual dollar volume
Class A - high annual dollar volume
Class B - medium annual dollar volume
Class C - low annual dollar volume
Used to establish policies that focus on the few critical parts and not the many trivial ones

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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 DOLLAR VOLUME	PERCENT OF ANNUAL DOLLAR VOLUME	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

Figure 12.2

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

Other criteria than annual dollar volume may be used
High shortage or holding cost
Anticipated engineering changes
Delivery problems
Quality problems

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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

More care in forecasting A items

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Record Accuracy

Accurate records are a critical ingredient in production and inventory systems
Periodic systems require regular checks of inventory
Two-bin system
Perpetual inventory tracks receipts and subtractions on a continuing basis
May be semi-automated

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Record Accuracy

Incoming and outgoing
record keeping must be
accurate
Stockrooms should be secure
Necessary to make precise decisions about ordering, scheduling, and shipping

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Cycle Counting

Items are counted and records updated on a periodic basis
Often used with ABC analysis
Has several advantages

Eliminates shutdowns and interruptions

Eliminates annual inventory adjustment

Trained personnel audit inventory accuracy

Allows causes of errors to be identified and corrected

Maintains accurate inventory records

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Cycle Counting Example

5,000 items in inventory, 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

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Control of Service Inventories

Can be a critical component
of profitability
Losses may come from
shrinkage or pilferage
Applicable techniques include

Good personnel selection, training, and discipline

Tight control of incoming shipments

Effective control of all goods leaving facility

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Inventory Models

Independent demand - the demand for item is independent of the demand for any other item in inventory
Dependent demand - the demand for item is dependent upon the demand for some other item in the inventory

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Inventory Models

Holding costs - the costs of holding or “carrying” inventory over time
Ordering costs - the costs of placing an order and receiving goods
Setup costs - cost to prepare a machine or process for manufacturing an order
May be highly correlated with setup time

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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%

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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%.

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Inventory Models for Independent Demand

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, constant, and independent
Lead time is known and constant
Receipt of inventory is instantaneous and complete
Quantity discounts are not possible
Only variable costs are setup (or ordering) and holding
Stockouts can be completely avoided

Important assumptions

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Inventory Usage Over Time

Figure 12.3

Order quantity = Q (maximum inventory level)

Usage rate

Average inventory on hand

Inventory level

Time

Minimum inventory

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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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 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

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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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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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

1,000

200

N = = 5 orders per year

Demand

Order quantity

= N = =

Expected number of orders

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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

250

T = = 50 days between orders

Number of working days per year

Expected number of orders

= T =

Expected time between 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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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

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An EOQ Example

Determine optimal number of needles to order

D = 1,000 units Q* = 200 units

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

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

Only 2% less than the total cost of $125 when the order quantity was 200

1,500 units

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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

= d x L

Lead time for a new order in days

Demand per day

ROP =

Number of working days in a year

d =

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Reorder Point Curve

Figure 12.5

Resupply takes place as order arrives

ROP (units)

Inventory level (units)

Time (days)

Lead time = L

Slope = units/day = d

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Reorder Point Example

Demand = 8,000 iPods per year

250 working day year

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

ROP = d x L

= 8,000/250 = 32 units

= 32 units per day x 3 days = 96 units

= 32 units per day x 4 days = 128 units

Number of working days in a year

d =

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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

Figure 12.6

Inventory level

Time

Demand part of cycle with no production (only usage)

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

Maximum inventory

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Production Order Quantity Model

Q = Number of pieces 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

Annual inventory holding cost

Holding cost
per unit per year

= (Average inventory level) x

Annual inventory level

= (Maximum inventory level)/2

Maximum inventory level

Total produced during the production run

Total used during the production run

= –

= pt – dt

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Production Order Quantity Model

Q = Number of pieces 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

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

Maximum inventory level

Total produced during the production run

Total used during the production run

= –

= pt – dt

Maximum inventory level

= p – d = Q 1 –

Maximum inventory level

Holding cost = (H) = 1 – H

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Production Order Quantity Model

Q = Number of pieces 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

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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

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Production Order Quantity Model

When annual data are used the equation becomes

Note:

Number of days the plant is in operation

1,000

250

d = 4 = =

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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
DISCOUNT NUMBER	DISCOUNT QUANTITY	DISCOUNT (%)	DISCOUNT PRICE (P)
1	0 to 999	no discount	$5.00
2	1,000 to 1,999	4	$4.80
3	2,000 and over	5	$4.75

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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 H = Holding cost per unit per year

S = Ordering or setup cost per order

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

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Quantity Discount Models

Steps in analyzing a quantity discount

For each discount, calculate Q*
If Q* for a discount doesn’t qualify, choose the lowest possible quantity to get the discount
Compute the total cost for each Q* or adjusted value from Step 2
Select the Q* that gives the lowest total cost

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Quantity Discount Models

Figure 12.7

1,000

2,000

Total cost $

Order quantity

Q* for discount 2 is below the allowable range at point a and must be adjusted upward to 1,000 units at point b

1st price break

2nd price break

Total cost curve for discount 1

Total cost curve for discount 2

Total cost curve for discount 3

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Quantity Discount Example

Calculate Q* for every discount

Q1* = = 700 cars/order

2(5,000)(49)

(.2)(5.00)

Q2* = = 714 cars/order

2(5,000)(49)

(.2)(4.80)

Q3* = = 718 cars/order

2(5,000)(49)

(.2)(4.75)

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Quantity Discount Example

Calculate Q* for every discount

Q1* = = 700 cars/order

2(5,000)(49)

(.2)(5.00)

Q2* = = 714 cars/order

2(5,000)(49)

(.2)(4.80)

Q3* = = 718 cars/order

2(5,000)(49)

(.2)(4.75)

1,000 — adjusted

2,000 — adjusted

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Quantity Discount Example

Choose the price and quantity that gives the lowest total cost

Buy 1,000 units at $4.80 per unit

TABLE 12.3	Total Cost Computations for Wohl’s Discount Store
DISCOUNT NUMBER	UNIT PRICE	ORDER QUANTITY	ANNUAL PRODUCT COST	ANNUAL ORDERING COST	ANNUAL HOLDING COST	TOTAL
1	$5.00	700	$25,000	$350	$350	$25,700
2	$4.80	1,000	$24,000	$245	$480	$24,725
3	$4.75	2,000	$23.750	$122.50	$950	$24,822.50

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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 x the probability x the stockout cost/unit
x the number of orders per year

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Safety Stock Example

ROP = 50 units Stockout cost = $40 per frame

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

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

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Safety Stock Example

ROP = 50 units Stockout cost = $40 per frame

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

A safety stock of 20 frames gives the lowest total cost

ROP = 50 + 20 = 70 frames

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

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Probabilistic Demand

Figure 12.8

Safety stock

16.5 units

ROP 

Place order

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

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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

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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)

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Probabilistic Example

m = Average demand = 350 kits

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

Z = 5% stockout policy (service level = 95%)

Using Appendix I, for an area under the curve of 95%, the Z = 1.65

Safety stock = ZsdLT = 1.65(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

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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

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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

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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

Safety stock is about 9 computers

ROP = (15 units x 2 days) + ZsdLT

= 30 + 1.28(5)( 2)

= 30 + 9.02 = 39.02 ≈ 39

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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

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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

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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

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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 = 1.65 (from Appendix I)

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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

Cs + Co

Service level =

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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

.55

.55 + .40

.55

.95

= = .579

Service level 57.9%

Optimal stocking level

 = 120

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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%

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Fixed-Period (P) Systems

Orders placed at the end of a fixed period
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

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Fixed-Period (P) Systems

Figure 12.9

On-hand inventory

Time

Target quantity (T)

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

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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

Printed in the United States of America.

= D Q ⎛

⎝ ⎜

⎞

⎠ ⎟S

Annual setup cost = D Q S

Annual setup cost =

= Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟H

Annual setup cost = D Q S

Annual setup cost =

Annual holding cost = Q 2 H

Annual holding cost =

D Q S =

Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟H

S =

2DS =Q2H

Q2 = 2DS H

Q* = 2DS H

2DS=Q

2DS

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

TC = D Q S + Q 2 H

= 1,500 200

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

=$75+$50=$125

TC=

S+

1,500

200

($10)+

200

($.50)

=$75+$50=$125

= 1,500 244.9

($10)+ 244.9 2

($.50)

=6.125($10)+122.45($.50) =$61.25+$61.22=$122.47

1,500

244.9

($10)+

244.9

($.50)

=6.125($10)+122.45($.50)

=$61.25+$61.22=$122.47

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

Q* = 2DS IP

2DS

ROP = (150 packs×5 days)+1.65σdLT

σdLT = 5 days×16 2( )+ 1502 ×12( ) = 5× 256( )+ 22,500×1( )

= 1,280( )+ 22,500( ) = 23,780 ≅154 ROP = (150×5)+1.65(154)≅ 750+ 254 =1,004 packs

ROP=(150 packs´5 days)+1.65s

dLT

=5 days´16

( )

+150

´1

( )

=5´256

( )

+22,500´1

( )

=1,280

()

+22,500

( )

=23,780@154

ROP =(150´5)+1.65(154)@750+254=1,004 packs