© 2008 prentice-hall, inc. chapter 6 to accompany quantitative analysis for management, tenth...
TRANSCRIPT
© 2008 Prentice-Hall, Inc.
Chapter 6
To accompanyQuantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna Power Point slides created by Jeff Heyl
Inventory Control Models
© 2009 Prentice-Hall, Inc.
© 2009 Prentice-Hall, Inc. 6 – 2
Introduction
Inventory is an expensive and important asset to many companies
Lower inventory levels can reduce costs Low inventory levels may result in stockouts
and dissatisfied customers Most companies try to balance high and low
inventory levels with cost minimization as a goal
Inventory is any stored resource used to satisfy a current or future need
Common examples are raw materials, work-in-process, and finished goods
© 2009 Prentice-Hall, Inc. 6 – 3
Introduction
Basic components of inventory planning Planning what inventory is to be
stocked and how it is to be acquired (purchased or manufactured)
This information is used in forecasting demand for the inventory and in controlling inventory levels
Feedback provides a means to revise the plan and forecast based on experiences and observations
© 2009 Prentice-Hall, Inc. 6 – 4
Introduction
Inventory may account for 50% of the total invested capital of an organization and 70% of the cost of goods sold
Inventory Costs
Labor Costs
Energy Costs
Capital Costs
© 2009 Prentice-Hall, Inc. 6 – 5
Introduction
All organizations have some type of inventory control system
Inventory planning helps determine what goods and/or services need to be produced
Inventory planning helps determine whether the organization produces the goods or services or whether they are purchased from another organization
Inventory planning also involves demand forecasting
© 2009 Prentice-Hall, Inc. 6 – 6
The Inventory Process
Suppliers Customers
Finished Goods
Raw Materials
Work in Process
Fabrication/ Assembly
Inventory Storage
Inventory Processing
© 2009 Prentice-Hall, Inc. 6 – 7
Controlling Inventory
Levels
Introduction
Forecasting Parts/Product
Demand
Planning on What Inventory to Stock
and How to Acquire It
Feedback Measurements to Revise Plans and
Forecasts
Figure 6.1
Inventory planning and control
© 2009 Prentice-Hall, Inc. 6 – 8
Importance of Inventory Control
Five uses of inventory The decoupling function Storing resources Irregular supply and demand Quantity discounts Avoiding stockouts and shortages
The decoupling function Used as a buffer between stages in a
manufacturing process Reduces delays and improves efficiency
© 2009 Prentice-Hall, Inc. 6 – 9
Importance of Inventory Control
Storing resources Seasonal products may be stored to satisfy
off-season demand Materials can be stored as raw materials,
work-in-process, or finished goods Labor can be stored as a component of
partially completed subassemblies Irregular supply and demand
Demand and supply may not be constant over time
Inventory can be used to buffer the variability
© 2009 Prentice-Hall, Inc. 6 – 10
Importance of Inventory Control
Quantity discounts Lower prices may be available for larger orders Cost of item is reduced but storage and insurance
costs increase, as well as the chances for more spoilage, damage and theft.
Investing in inventory reduces the available funds for other projects
Avoiding stockouts and shortages Stockouts may result in lost sales Dissatisfied customers may choose to buy from
another supplier
© 2009 Prentice-Hall, Inc. 6 – 11
Inventory Decisions
There are only two fundamental decisions in controlling inventory How much to order When to order
The major objective is to minimize total inventory costs
Common inventory costs are Cost of the items (purchase or material cost) Cost of ordering Cost of carrying, or holding, inventory Cost of stockouts
© 2009 Prentice-Hall, Inc. 6 – 12
Inventory Cost Factors
ORDERING COST FACTORS CARRYING COST FACTORS
Developing and sending purchase orders Cost of capital
Processing and inspecting incoming inventory Taxes
Bill paying Insurance
Inventory inquiries Spoilage
Utilities, phone bills, and so on, for the purchasing department Theft
Salaries and wages for the purchasing department employees Obsolescence
Supplies such as forms and paper for the purchasing department
Salaries and wages for warehouse employees
Utilities and building costs for the warehouse
Supplies such as forms and paper for the warehouse
Table 6.1
© 2009 Prentice-Hall, Inc. 6 – 13
Inventory Cost Factors
Ordering costs are generally independent of order quantity Many involve personnel time The amount of work is the same no matter the
size of the order Carrying costs generally varies with the
amount of inventory, or the order size The labor, space, and other costs increase as
the order size increases Of course, the actual cost of items
purchased varies with the quantity purchased
© 2009 Prentice-Hall, Inc. 6 – 14
Economic Order Quantity
The economic order quantityeconomic order quantity (EOQEOQ) model is one of the oldest and most commonly known inventory control techniques
It dates from a 1915 publication by Ford W. Harris
It is still used by a large number of organizations today
It is easy to use but has a number of important assumptions
© 2009 Prentice-Hall, Inc. 6 – 15
Economic Order Quantity
Assumptions1. Demand is known and constant2. Lead time (the time between the placement and
receipt of an order) is known and constant3. Receipt of inventory is instantaneous
Inventory from an order arrives in one batch, at one point in time
4. Purchase cost per unit is constant throughout the year; no quantity discounts
5. The only variable costs are the placing an order, ordering costordering cost, and holding or storing inventory over time, holdingholding or carrying costcarrying cost, and these are constant throughout the year
6. Orders are placed so that stockouts or shortages are avoided completely
© 2009 Prentice-Hall, Inc. 6 – 16
Inventory Usage Over Time
Time
Inventory Level
Minimum Inventory
0
Order Quantity = Q = Maximum Inventory Level
Figure 6.2
Inventory usage has a sawtooth shape Inventory jumps from 0 to the maximum when the shipment arrives Because demand is constant over time, inventory drops at a uniform
rate over time
© 2009 Prentice-Hall, Inc. 6 – 17
EOQ Inventory Costs The objective is to minimize total costs
The relevant costs are the ordering and carrying/holding costs, all other costs are constant. Thus, by minimizing the sum of the ordering and carrying costs, we are also minimizing the total costs
The annual ordering cost is the number of orders per year times the cost of placing each order
As the inventory level changes daily, use the average inventory level to determine annual holding or carrying cost The annual carrying cost equals the average inventory
times the inventory carrying cost per unit per year The maximum inventory is Q and the average inventory
is Q/2.
© 2009 Prentice-Hall, Inc. 6 – 18
Inventory Costs in the EOQ Situation
Objective is generally to minimize total cost Relevant costs are ordering costs and carrying
costs
2level inventory Average
Q
INVENTORY LEVEL
DAY BEGINNING ENDING AVERAGE
April 1 (order received) 10 8 9
April 2 8 6 7
April 3 6 4 5
April 4 4 2 3
April 5 2 0 1
Maximum level April 1 = 10 unitsTotal of daily averages = 9 + 7 + 5 + 3 + 1 = 25Number of days = 5Average inventory level = 25/5 = 5 units Table 6.2
© 2009 Prentice-Hall, Inc. 6 – 19
Inventory Costs in the EOQ Situation
Develop an expression for the ordering cost.
Develop and expression for the carrying cost.
Set the ordering cost equal to the carrying cost.
Solve this equation for the optimal order quantity, Q*Q*.
© 2009 Prentice-Hall, Inc. 6 – 20
Inventory Costs in the EOQ Situation
Mathematical equations can be developed using
Q = number of pieces to orderEOQ = Q* = optimal number of pieces to order
D = annual demand in units for the inventory itemCo = ordering cost of each orderCh = holding or carrying cost per unit per year
Annual ordering cost Number of
orders placed per year
Ordering cost per
order
oCQD
orderper Cost orderper units ofNumber
Demand Annual
© 2009 Prentice-Hall, Inc. 6 – 21
Inventory Costs in the EOQ Situation
Mathematical equations can be developed using
Q = number of pieces to orderEOQ = Q* = optimal number of pieces to order
D = annual demand in units for the inventory itemCo = ordering cost of each orderCh = holding or carrying cost per unit per year
hCQ2
Annual holding cost Average
inventory
Carrying cost per unit
per year
Total Inventory Cost = ho CQ
CQ
D
2
© 2009 Prentice-Hall, Inc. 6 – 22
Inventory Costs in the EOQ Situation
Minimum Total Cost
Optimal Order Quantity
Curve of Total Cost Curve of Total Cost of Carrying of Carrying and Orderingand Ordering
Carrying Cost CurveCarrying Cost Curve
Ordering Cost CurveOrdering Cost Curve
Cost
Order Quantity
Figure 6.3
Optimal Order Quantity is when the Total Cost curve is at its lowest . This occurs when the Ordering Cost = Carrying Cost
© 2009 Prentice-Hall, Inc. 6 – 23
Finding the EOQ
When the EOQ assumptions are met, total cost is minimized when Annual ordering cost = Annual holding cost
ho CQ
CQD
2
Solving for Q
ho CQDC 22
22Q
C
DC
h
o
*EOQ QQC
DC
h
o 2
© 2009 Prentice-Hall, Inc. 6 – 24
Economic Order Quantity (EOQ) Model
hCQ2
cost holding Annual
oCQD
cost ordering Annual
h
o
C
DCQ
2 *EOQ
Summary of equations
© 2009 Prentice-Hall, Inc. 6 – 25
Sumco Pump Company Example
Company sells pump housings to other companies
Would like to reduce inventory costs by finding optimal order quantity Annual demand = 1,000 units Ordering cost = $10 per order Average carrying cost per unit per year = $0.50
units 20000040500
10000122 ,
.))(,(*
h
o
C
DCQ
© 2009 Prentice-Hall, Inc. 6 – 26
Sumco Pump Company Example
Total annual cost = Order cost + Holding cost
ho CQ
CQD
TC2
).()(,
502
20010
2000001
1005050 $$$
© 2009 Prentice-Hall, Inc. 6 – 29
Purchase Cost of Inventory Items
Total inventory cost can be written to include the cost of purchased items
Given the EOQ assumptions, the annual purchase cost is constant at D C no matter the order policy C is the purchase cost per unit D is the annual demand in units
It may be useful to know the average dollar level of inventory
2level dollar Average
)(CQ
© 2009 Prentice-Hall, Inc. 6 – 30
Purchase Cost of Inventory Items
Inventory carrying cost is often expressed as an annual percentage of the unit cost or price of the inventory
This requires a new variable
Annual inventory holding charge as a percentage of unit price or costI
The cost of storing inventory for one year is then
ICCh
thus,IC
DCQ o2
*
© 2009 Prentice-Hall, Inc. 6 – 31
Sensitivity Analysis with the EOQ Model
The EOQ model assumes all values are know and fixed over time
Generally, however, the values are estimated or may change
Determining the effects of these changes is called sensitivity analysissensitivity analysis
Because of the square root in the formula, changes in the inputs result in relatively small changes in the order quantity
h
o
C
DC2EOQ
© 2009 Prentice-Hall, Inc. 6 – 32
Sensitivity Analysis with the EOQ Model
In the Sumco example
units 200500
1000012
.))(,(
EOQ
If the ordering cost were increased four times from $10 to $40, the order quantity would only double
units 400500
4000012
.))(,(
EOQ
In general, the EOQ changes by the square root of a change to any of the inputs
© 2009 Prentice-Hall, Inc. 6 – 33
Reorder Point:Determining When To Order
Once the order quantity is determined, the next decision is when to orderwhen to order
The time between placing an order and its receipt is called the lead timelead time (LL) or delivery delivery timetime Inventory must be available during this period to
met the demand When to order is generally expressed as a
reorder pointreorder point (ROPROP) – the inventory level at which an order should be placed
Demand per day
Lead time for a new order in daysROP
d L
© 2009 Prentice-Hall, Inc. 6 – 34
Determining the Reorder Point
The slope of the graph is the daily inventory usage Expressed in units demanded per day, d
If an order is placed when the inventory level reaches the ROP, the new inventory arrives at the same instant the inventory is reaching 0
© 2009 Prentice-Hall, Inc. 6 – 35
Procomp’s Computer Chip Example
Demand for the computer chip is 8,000 per year Daily demand is 40 units Delivery takes three working days
ROP d L 40 units per day 3 days 120 units
An order is placed when the inventory reaches 120 units
The order arrives 3 days later just as the inventory is depleted
© 2009 Prentice-Hall, Inc. 6 – 36
The Reorder Point (ROP) Curve In
vent
ory
Lev
el (
Uni
ts)
Q*
ROP(Units)
Slope = Units/Day = d
Lead Time (Days) L
© 2009 Prentice-Hall, Inc. 6 – 37
EOQ Without The Instantaneous Receipt Assumption
When inventory accumulates over time, the instantaneous receiptinstantaneous receipt assumption does not apply
Daily demand rate must be taken into account The revised model is often called the production production
run modelrun model
Inventory Level
Time
Part of Inventory Cycle Part of Inventory Cycle During Which Production is During Which Production is Taking PlaceTaking Place
There is No Production There is No Production During This Part of the During This Part of the Inventory CycleInventory Cycle
t
Maximum Inventory
Figure 6.5
© 2009 Prentice-Hall, Inc. 6 – 38
EOQ Without The Instantaneous Receipt Assumption
Instead of an ordering cost, there will be a setup cost – the cost of setting up the production facility to manufacture the desired product Includes the salaries and wages of employees
who are responsible for setting up the equipment, engineering and design costs of making the setup, paperwork, supplies, utilities, etc.
The optimal production quantity is derived by setting setup costs equal to holding or carrying costs and solving for the order quantity
© 2009 Prentice-Hall, Inc. 6 – 39
Annual Carrying Cost for Production Run Model
In production runs, setup costsetup cost replaces ordering cost
The model uses the following variables
Q number of pieces per order, or production run
Cs setup cost
Ch holding or carrying cost per unit per yearp daily production rated daily demand ratet length of production run in days
© 2009 Prentice-Hall, Inc. 6 – 40
Annual Carrying Cost for Production Run Model
Maximum inventory level (Total produced during the production run) – (Total used during the production run)
(Daily production rate)(Number of days production)– (Daily demand)(Number of days production)
(pt) – (dt)
since Total produced Q pt
we knowpQ
t
Maximum inventory
level
pd
QpQ
dpQ
pdtpt 1
© 2009 Prentice-Hall, Inc. 6 – 41
Annual Carrying Cost for Production Run Model
Since the average inventory is one-half the maximum
pdQ
12
inventory Average
and
hCpdQ
1
2cost holding Annual
© 2009 Prentice-Hall, Inc. 6 – 42
Annual Setup Cost for Production Run Model
sCQD
cost setup Annual
Setup cost replaces ordering cost when a product is produced over time (independent of the size of the order and the size of the production run)
and
oCQD
cost ordering Annual
© 2009 Prentice-Hall, Inc. 6 – 43
Determining the Optimal Production Quantity
By setting setup costs equal to holding costs, we can solve for the optimal order quantity
Annual holding cost Annual setup cost
sh CQD
CpdQ
1
2
Solving for Q, we get
pd
C
DCQ
h
s
1
2*
© 2009 Prentice-Hall, Inc. 6 – 44
Production Run Model
Summary of equations
pd
C
DCQ
h
s
1
2 quantity production Optimal *
sCQD
cost setup Annual
hCpdQ
1
2cost holding Annual
If the situation does not involve production but receipt of inventory over a period of time, use the same model but replace CCss with CCoo
© 2009 Prentice-Hall, Inc. 6 – 45
Brown Manufacturing Example
Brown Manufacturing produces commercial refrigeration units in batches
Annual demand D 10,000 units
Setup cost Cs $100
Carrying cost Ch $0.50 per unit per yearDaily production rate p 80 units daily
Daily demand rate d 60 units daily
How many refrigeration units should Brown produce in each batch?How long should the production cycle last?
© 2009 Prentice-Hall, Inc. 6 – 46
Brown Manufacturing Example
pd
C
DCQ
h
s
1
2*1.
2.
8060
150
100000102
.
,*Q
00000016
4150
0000002,,
.
,,
units 4,000
days 50800004
,
pQ
cycle Production
© 2009 Prentice-Hall, Inc. 6 – 49
Quantity Discount Models
Quantity discounts are commonly available The basic EOQ model is adjusted by adding in the
purchase or materials cost
Total cost Material cost + Ordering cost + Holding cost
ho CQ
CQD
DC2
cost Total
where
D annual demand in units
Cs ordering cost of each orderC cost per unit
Ch holding or carrying cost per unit per year
© 2009 Prentice-Hall, Inc. 6 – 50
Quantity Discount Models
Quantity discounts are commonly available The basic EOQ model is adjusted by adding in the
purchase or materials cost
Total cost Material cost + Ordering cost + Holding cost
ho CQ
CQD
DC2
cost Total
where
D annual demand in units
Cs ordering cost of each orderC cost per unit
Ch holding or carrying cost per unit per year
Holding cost Ch IC
I holding cost as a percentage of the unit cost (C)
Because unit cost is now variable
© 2009 Prentice-Hall, Inc. 6 – 51
Quantity Discount Models
A typical quantity discount schedule
DISCOUNT NUMBER
DISCOUNT QUANTITY DISCOUNT (%)
DISCOUNT COST ($)
1 0 to 999 0 5.00
2 1,000 to 1,999 4 4.80
3 2,000 and over 5 4.75
Table 6.3
Buying at the lowest unit cost is not always the best choice
© 2009 Prentice-Hall, Inc. 6 – 52
Quantity Discount Models
Total cost curve for the quantity discount model
Figure 6.6
TC Curve for Discount 1
TC Curve for Discount 3Total Cost
$
Order Quantity
0 1,000 2,000
TC Curve for Discount 2
EOQ for Discount 2
© 2009 Prentice-Hall, Inc. 6 – 53
Brass Department Store Example
Brass Department Store stocks toy race cars Their supplier has given them the quantity
discount schedule shown in Table 6.3 Annual demand is 5,000 cars, ordering cost is $49, and
holding cost is 20% of the cost of the car The first step is to compute EOQ values for each
discount
order per cars 70000520
49000521
).)(.())(,)((
EOQ
order per cars 71480420
49000522
).)(.())(,)((
EOQ
order per cars 71875420
49000523
).)(.())(,)((
EOQ
© 2009 Prentice-Hall, Inc. 6 – 54
Brass Department Store Example
The second step is adjust quantities below the allowable discount range
The EOQ for discount 1 is allowable The EOQs for discounts 2 and 3 are outside the
allowable range and have to be adjusted to the smallest quantity possible to purchase and receive the discount
Q1 700
Q2 1,000
Q3 2,000
© 2009 Prentice-Hall, Inc. 6 – 55
Brass Department Store Example
The third step is to compute the total cost for each quantity
DISCOUNT NUMBER
UNIT PRICE
(C)
ORDER QUANTITY
(Q)
ANNUAL MATERIAL COST ($)
= DC
ANNUAL ORDERING COST ($) = (D/Q)Co
ANNUAL CARRYING COST ($) = (Q/2)Ch TOTAL ($)
1 $5.00 700 25,000 350.00 350.00 25,700.00
2 4.80 1,000 24,000 245.00 480.00 24,725.00
3 4.75 2,000 23,750 122.50 950.00 24,822.50
The fourth step is to choose the alternative with the lowest total cost
Table 6.4
© 2009 Prentice-Hall, Inc. 6 – 58
Use of Safety Stock
If demand or the lead time are uncertain, the exact ROP will not be known with certainty
To prevent stockoutsstockouts, it is necessary to carry extra inventory called safety stocksafety stock
Safety stock can prevent stockouts when demand is unusually high
Safety stock can be implemented by adjusting the ROP
© 2009 Prentice-Hall, Inc. 6 – 59
Use of Safety Stock
The basic ROP equation is
ROP d Ld daily demand (or average daily demand)L order lead time or the number of working days it takes to deliver an order (or average lead time)
A safety stock variable is added to the equation to accommodate uncertain demand during lead time
ROP d L + SSwhere
SS safety stock
© 2009 Prentice-Hall, Inc. 6 – 60
Use of Safety Stock
StockoutStockoutFigure 6.7(a)
Inventory on Hand
Time