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Presented by:
Phillip Leonard Petate
OPERATIONS RESEARCH:
INVENTORY MODELS
San Beda College
Graduate School of Business
INVENTORY MODELS
I. Basic EOQ Models
II. Quantity Discounts
III. Economic Lot Size (ELS)
2
Inventory
Inventory is any stored resource that is used to
satisfy a current or future need.
Raw materials, work-in-process, and finished
goods are examples of inventory. Inventory
levels for finished goods, such as clothes
dryers, are a direct function of market
demand.
3
Inventory
By using this demand information, it is possible
to determine how much raw materials (e.g.,
sheet metal, paint, and electric motors in the
case of clothes dryers) and work-in-process
are needed to produce the finished product.
4
Types of Inventory
 Raw Materials
Purchased but not processed
 Work-In-Process
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
5
The Material Flow Cycle
6
Importance of Inventory Control
Inventory control serves several important functions and
adds a great deal of flexibility to the operation of a firm.
Five main uses of Inventory are as follows:
1. Decoupling Function
2. Storing Resources
3. Irregular Supply and Demand
4. Quantity Discounts
5. Avoiding Stockouts and Shortages
7
Inventory Control Decisions
8
Even though there are literally millions of different types
of products manufactured in our society, there are only
two fundamental decisions that you have to make when
controlling inventory:
1. How Much to Order
2. When to Order
Purpose of Inventory Models
9
The purpose of all inventory models is to determine
how much to order and when to order. As we know,
inventory fulfills many important functions in an
organization. But as the inventory levels go up to
provide these functions, the cost of storing and
holding inventory also increases. Thus, we must reach
a fine balance in establishing inventory levels.
A major objective in controlling inventory is to
minimize total inventory costs.
Components of Total Cost
10
Some of the most significant inventory costs are as follows:
1. Cost of the items
2. Cost of ordering
3. Cost of carrying, or holding, inventory
4. Cost of stockouts
5. Cost of safety stock, the additional inventory that may be
held to help avoid stockouts
Holding, Ordering, and Setup Costs
 Holding Cost
The cost of holding or “carrying” inventory over time.
 Ordering Cost
The cost of placing an order and receiving goods.
 Setup Cost
The cost to prepare a machine or process for
manufacturing an order.
11
Inventory Cost Factors
12
Independent vs Dependent Demand
 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.
13
INVENTORY MODELS
I. Basic EOQ Models
14
Basic EOQ Model
The Economic Order Quantity (EOQ) model is one of
the oldest and most commonly known inventory
control techniques.
Research on its use dates back to a 1915
publication by Ford W. Harris. This model is still used
by a large number of organizations today.
This technique is relatively easy to use, but it makes
a number of assumptions.
15
Basic EOQ Assumptions
 Demand is known and constant.
 The lead time - that is, the time between the
placement of the order and the receipt of the
order - is known and constant.
 The receipt of inventory is instantaneous. In
other words, the inventory from an order
arrives in one batch, at one point in time.
16
 Quantity discounts are not possible.
 The only variable costs are the cost of placing an
order, ordering cost, and the cost of holding or
storing inventory over time, carrying, or holding,
cost.
 If orders are placed at the right time, stockouts
and shortages can be avoided completely.
17
Basic EOQ Assumptions
Basic EOQ Assumptions Summary
Important Assumptions:
1. Demand is known, constant, and independent
2. Lead time is known and constant
3. Receipt of inventory is instantaneous and complete
4. Quantity discounts are not possible
5. Only variable costs are setup and holding
6. Stockouts can be completely avoided
18
19
Inventory Usage Over Time
[Refer to Graph in Slide 19]
With these assumptions, inventory usage has a
sawtooth shape. In the graph, Q represents the
amount that is ordered. If this amount is 500 units, all
500 units arrive at one time when an order is
received. Thus, the inventory level jumps from 0 to 500
units. In general, the inventory level increases from 0
to Q units when an order arrives.
20
Inventory Usage Over Time
[Refer to Graph in Slide 19]
Because demand is constant over time, inventory drops
at a uniform rate over time. Another order is placed
such that when the inventory level reaches 0, the new
order is received and the inventory level again jumps
to Q units, represented by the vertical lines. This
process continues indefinitely over time.
21
Inventory Usage Over Time
22
Objective of Basic EOQ Model
The objective of most inventory models is to minimize
the total cost. With the assumptions just given, the
significant costs are the ordering cost and the
inventory carrying cost. All other costs, such as the cost
of the inventory itself, are constant. Thus, if we
minimize the sum of the ordering and carrying costs,
we also minimize the total cost.
23
Total Cost as a Function of Order Quantity
24
Objective of Basic EOQ Model
[Refer to Graph in Slide 23]
To help visualize this, Slide 23 graphs total cost as a
function of the order quantity, Q. As the value of Q
increases, the total number of orders placed per year
decreases. Hence, the total ordering cost decreases.
However, as the value of Q increases, the carrying
cost increases because the firm has to maintain larger
average inventories.
25
Objective of Basic EOQ Model
[Refer to Graph in Slide 23]
The optimal order size, Q*, is the quantity that
minimizes the total cost. Note in Slide 23 that Q*
occurs at the point where the ordering cost curve and
the carrying cost curve intersect. This is not by chance.
With this particular type of cost function, the optimal
quantity always occurs at a point where the ordering
cost is equal to the carrying cost.
26
Average Inventory Level
Now that we have a better understanding of inventory
costs, let us see how we can determine the value of Q*
that minimizes the total cost. In determining the annual
carrying cost, it is convenient to use the average inventory.
Referring to Slide 19, we see that the on-hand inventory
ranges from a high of Q units to a low of zero units, with
a uniform rate of decrease between these levels. Thus, the
average inventory can be calculated as the average of
the minimum and maximum inventory levels.
27
Average Inventory Level
That is:
Average Inventory Level = (0 + Q2)/2 = Q/2
We multiply this average inventory by a factor called
the Annual Inventory Carrying Cost Per Unit to
determine the annual inventory cost.
Q = Number of Units in each Order
Q* = Optimal Number of Pieces per Order (EOQ)
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
The EOQ Model
28
Notes:
The unit holding/carrying cost, H, is usually expressed in one of two ways:
1. As a fixed cost. For example, H is $0.50 per unit per year.
2. As a percentage (typically denoted by I) of the item’s unit purchase cost or
price. For example, H is 20% of the item’s unit cost. In general,
H = I x P
I = Annual Carrying Cost as a Percentage of the Unit Cost of the Item
P = Purchase Cost per Unit of the Inventory Item
Q = Number of Units in each Order
Q* = Optimal Number of Pieces per Order (EOQ)
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per 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/Ordering
Cost per Order
=
= (S)
D
Q
EOQ - Annual Setup Cost
29
Annual Holding Cost =(Average Inventory Level)
x (Holding Cost per Unit per Year)
Order Quantity
2
= (Holding Cost per Unit per Year)
= (H)Q
2
30
EOQ - Annual Holding Cost
Q = Number of Units in each Order
Q* = Optimal Number of Pieces per Order (EOQ)
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
Optimal Order Quantity is found when Annual Setup Cost Equals
Annual Holding Cost
D
Q
S = HQ
2
Solving for Q* (EOQ) 2DS = Q2H
Q2 = 2DS/H
Q* = 2DS/H
The EOQ ModelAnnual Setup Cost = S
D
Q
Annual Holding Cost = H
Q
2
31
Q = Number of Units in each Order
Q* = Optimal Number of Pieces per Order (EOQ)
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
EOQ Example
32
Let us now apply these formulas to the case of SBC, a
company that buys alarm clocks from a manufacturer
and distributes to retailers. SBC would like to reduce its
inventory cost by determining the optimal number of
alarm clocks to obtain per order. The annual demand is
1,000 units, the ordering cost is $10 per order, and the
carrying cost is $0.50 per unit per year. Each alarm
clock has a purchase cost of $5. How many clocks
should SBC order each time?
Determine the Optimal Number of Alarm Clocks to Order
D = 1,000 Units
S = $10 per Order
H = $0.50 per Unit per Year
Q* =
2DS
H
Q* =
2(1,000)(10)
0.50
= 40,000 = 200 units
EOQ Example
33
D = 1,000 units Q* = 200 units
S = $10 per order
H = $0.50 per unit per year
= N = =Expected
Number of
Orders
Demand
EOQ
D
Q*
N = = 5 Orders per Year1,000
200
EOQ - Expected Number of Orders
34
= T =Expected Time
Between Orders
Number of Working
Days per Year
N
T = = 50 Days Between Orders
250
5
EOQ - Expected Time Between Orders
35
D = 1,000 units Q* = 200 units
S = $10 per order N = 5 Orders per Year
H = $0.50 per unit per year
250 Working Days per Year
TBOEOQ = (Working Days per Year)
EOQ
D
Q* = Optimal Number of Pieces per Order (EOQ)
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost for per Order
H = Holding or Carrying Cost per Unit per Year
P = Purchase Cost per Unit of the Inventory Item
EOQ - Total Cost
36
Total Cost = Total Setup or Ordering Cost
+ Total Holding or Carrying Cost
+ Total Purchase Cost
= (D/Q* x S) + (Q*/2 x H) + (P x D)
EOQ - Total Cost
37
Total Cost = Total Setup/Ordering Cost (D/Q* x S)
+ Total Holding/Carrying Cost (Q*/2 x H)
+ Total Purchase Cost (P x D)
Observe that the total purchase cost (P x D) does not depend
on the value of Q. This is so because regardless of how many
orders we place each year, or how many units we order each
time, we will still incur the same annual total purchase cost.
D = 1,000 Units Q* = 200 units
S = $10 per Order N = 5 orders per year
H = $0.50 per Unit per Year T = 50 days
P = $5
Total Annual Cost
= Setup Cost + Holding Cost + Purchase Cost
TC = S + H + (P x D)
D
Q*
Q*
2
TC = ($10) + ($.50) + $5 x 10001,000
200
200
2
TC = $50 + $50 + $5,000
EOQ - Total Cost
38
TC = $5,100
39
Purchase Cost of Inventory Items
It is often useful to know the value of the average
inventory level in monetary terms. We know that the
average inventory level is Q/2, where Q is the order
quantity. If we order Q* (the EOQ) units each time, the
value of the average inventory can be computed by
multiplying the average inventory by the unit purchase
cost, P. That is:
Average Monetary Value of Inventory
= P x (Q*/2)
40
Average Monetary Value of Inventory
= P x (Q*/2)
= $5 x (200/2)
= $500
D = 1,000 Units Q* = 200 units
S = $10 per Order N = 5 orders per year
H = $0.50 per Unit per Year T = 50 days
P = $5
Purchase Cost of Inventory Items
In using these formulas, we assumed that the values of the
Ordering Cost (S) and Carrying Cost (H) are known
constants.
S = Q* ² x H/(2D)
H = 2DS/Q* ²
41
Calculating the Ordering Costs (S) and
Carrying Cost (H) for a Given Value of EOQ
EOQ Formula:
S = Q* ² x H/(2D)
= 200² x ($0.50/ 2 x 1,000)
= 40,000 x 0.00025
= $10 per Order
42
Calculating the Ordering Costs (S) and
Carrying Cost (H) for a Given Value of EOQ
D = 1,000 Units Q* = 200 units
S = $10 per Order N = 5 orders per year
H = $0.50 per Unit per Year T = 50 days
P = $5
H = 2DS/Q* ²
= 2 x (1,000 x 10) / 200²
= 20,000 / 40,000
= $0.50 per Unit per Year
Reorder Point (ROP)
Now that we have decided how much to order, we
look at the second inventory question: when to order.
In most simple inventory models, it is assumed that we
have instantaneous inventory receipt. That is, we
assume that a firm waits until its inventory level for a
particular item reaches zero, places an order, and
receives the items in stock immediately.
43
Reorder Point (ROP)
In many cases, however, the time between the placing
and receipt of an order, called the Lead Time, or
Delivery Time, is often a few days or even a few
weeks. Thus, the when to order decision is usually
expressed in terms of a reorder point (ROP), the
inventory level at which an order should be placed.
44
 EOQ answers the “How Much” question.
 The Reorder Point (ROP) tells “When” to order.
ROP =
Lead Time for a New
Order in Days
Demand
per Day
= d x L
Where d = D
Number of Working Days in a Year
45
Reorder Point (ROP)
Q*
ROP
(units)
InventoryLevel(Units)
Time (Days)
Lead time = L
Slope = Units/Day = d
Reorder Point Curve
46
Demand = 1,000 Alarm Clocks per Year
250 Working Days in the Year
Lead Time for Orders is 3 Working Days
ROP = d x L
d =
D
Number of Working Days in a Year
= 1,000/250
= 4 Units per Day x 3 Days
= 12 Units
Reorder Point Example
47
= 4 Units per Day
INVENTORY MODELS
II. Quantity Discounts
48
To increase sales, many companies offer quantity discounts to
their customers. A quantity discount is simply a decreased
unit cost for an item when it is purchased in larger quantities.
It is not uncommon to have a discount schedule with several
discounts for large orders. See example below:
Quantity Discount Models
49
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
As can be seen in Slide 49 Table, the normal cost for
the item in this example is $5. When 1,000 to 1,999
units are ordered at one time, the cost per unit drops
to $4.80, and when the quantity ordered at one time
is 2,000 units or more, the cost is $4.75 per unit. As
always, management must decide when and how much
to order. But with quantity discounts, how does a
manager make these decisions?
Quantity Discount Models
50
As with previous inventory models discussed so far, the
overall objective is to minimize the total cost. Because the
unit cost for the third discount in Slide 49 Table is lowest,
we might be tempted to order 2,000 units or more to
take advantage of this discount. Placing an order for that
many units, however, might not minimize the total inventory
cost. As the discount quantity goes up, the item cost goes
down, but the carrying cost increases because the order
sizes are large. Thus, the major trade-off when
considering quantity discounts is between the reduced
item cost and the increased carrying cost.
Quantity Discount Models
51
Total Annual Cost
= Setup Cost + Holding Cost + Purchase Cost
= S + H + (P x D)
D
Q*
Q*
2
Quantity Discount Models
52
Recall that we computed the total cost (including the total
purchase cost) for the EOQ model as follows:
Next, we illustrate the four-step process to determine the
quantity that minimizes the total cost.
1. For each discount price, calculate a Q* value, using the
EOQ formula. In quantity discount EOQ models, the unit
carrying cost, H, is typically expressed as a percentage (I)
of the unit purchase cost (P). That is, H = I x P. As a result,
the value of Q* will be different for each discounted
price.
2. For any discount level, if the Q* computed in step 1 is too
low to qualify for the discount, adjust Q* upward to the
lowest quantity that qualifies for the discount. For
example, if Q* for discount 2 in Slide 49 Table turns out
to be 500 units, adjust this value up to 1,000 units.
4 Steps to Analyze Quantity Discount Models
53
Total Cost Curve for the Quantity
Discount Model
54
As seen in Slide 54, the total cost curve for the discounts shown in
Slide 49 is broken into three different curves. There are separate
cost curves for the first (0 ≤ Q ≤ 999), second (1,000 ≤ Q ≤
1,999), and third (Q ≥ 2,000) discounts. Look at the total cost
curve for discount 2. The Q* for discount 2 is less than the
allowable discount range of 1,000 to 1,999 units. However, the
total cost at 1,000 units (which is the minimum quantity needed to
get this discount) is still less than the lowest total cost for discount 1.
Thus, step 2 is needed to ensure that we do not discard any
discount level that may indeed produce the minimum total cost.
Note that an order quantity compute in step 1 that is greater than
the range that would qualify it for a discount may be discarded.
Total Cost Curve for the Quantity
Discount Model55
4 Steps to Analyze Quantity Discount Models
56
3. Using the Total Cost Equation, compute a total cost for
every Q* determined in steps 1 and 2. If a Q* had to be
adjusted upward because it was below the allowable
quantity range, be sure to use the adjusted Q* value.
4. Select the Q* that has the lowest total cost, as computed in
step 3. It will be the order quantity that minimizes the total
cost.
1. For each discount, calculate Q*
2. If Q* for a discount doesn’t qualify, choose the
smallest possible order size to get the discount
3. Compute the total cost for each Q* or adjusted
value from Step 2
4. Select the Q* that gives the lowest total cost
Analyzing a Quantity Discount Summary
57
GSB Department Store stocks toy cars. Recently, the
store was given a quantity discount schedule for the
cars, as shown in Slide 49. Thus, the normal cost for the
cars is $5.00. For orders between 1,000 and 1,999
units, the unit cost is $4.80, and for orders of 2,000 or
more units, the unit cost is $4.75. Furthermore, the
ordering cost is $49 per order, the annual demand is
5,000 race cars, and the inventory carrying charge as a
percentage of cost, I, is 20%, or 0.2. What order
quantity will minimize the total cost?
Quantity Discount Example
58
Quantity Discount Example
59
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
D = 5,000 Units
S = $49 per Order
I = 20% of Cost
H = I x P (Cost)
Q* =
2DS
H
Calculate Q* for every Discount Q* =
2DS
IP
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)
Quantity Discount Example
60
In the GSB Department Store example, observe that
the Q* values for discounts 2 and 3 are too low to be
eligible for the discounted prices (Slide 49 Table).
They are, therefore, adjusted upward to 1,000 and
2,000, respectively.
With these adjusted Q* values, we find that the lowest
total cost of $24,725 results when we use an order
quantity of 1,000 units [See Slide 61 and Slide 62].
Quantity Discount Example
61
Calculate Q* for every Discount Q* =
2DS
IP
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
Quantity Discount Example
62
Discount
Number
Unit
Price
Order
Quantity
Annual
Product
Cost
Annual
Ordering
Cost
Annual
Holding
Cost Total Cost
1 $5.00 700 $25,000 $350.00 $350 $25,700.00
2 $4.80 1,000 $24,000 $245.00 $480 $24,725.00
3 $4.75 2,000 $23,750 $122.50 $950 $24,822.50
Choose the Price and Quantity that gives the Lowest Total Cost
Buy 1,000 Units at $4.80 per Unit
Quantity Discount Example
63
INVENTORY MODELS
III. Economic Lot Size (ELS)
64
ELS is the quantity of material or units of a
manufactured good that can be produced or
purchased within the lowest unit cost range. It is
determined by reconciling the decreasing unit
cost of larger quantities with the associated
increasing unit cost of handling, storage,
insurance, interest, etc.
(www.businessdictionary.com)
Economic Lot Size (ELS)
65
Economic Lot Size (ELS)
A manufacturer must determine the production lot size that will
result in minimum production and storage cost.
Economic Order Quantity (EOQ)
A purchaser must decide what quantity of an item to order
that will result in minimum reordering and storage cost.
Economic Lot Size (ELS)
66
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
p = production rate
d = daily demand
67
ELS Model
dp
p
H
2DS
ELS


68
ELS - Total Annual Cost
   S
ELS
D
H
p
dp
2
ELS
C 




 

D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
p = production rate
d = daily demand
69
ELS - Time Between Orders (TBO)
 Days/YearWork
D
ELS
TBOELS 
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
p = production rate
d = daily demand
70
ELS – Production Time per Lot
p
ELS
D = Annual Demand in Units for the Inventory Item
S = Setup or Ordering Cost per Order
H = Holding or Carrying Cost per Unit per Year
p = production rate
d = daily demand
ELS - Example
71
A plant manager of a chemical plant must determine the lot size
for a particular chemical that has a steady demand of 30 barrels
per day. The production rate is 190 barrels per day, annual
demand is 10,500 barrels, setup cost is $200, annual holding cost
is $0.21 per barrel, and the plant operates 350 days per year.
a. Determine the Economic Production Lot Size (ELS)
b. Determine the Total Annual Setup and Inventory Holding Cost for this
item (Total Annual Cost)
c. Determine the time between orders (TBO), or cycle length, for the ELS
d. Determine the Production Time per Lot
72
   barrels4,873.4
30190
190
21.0$
200$500,102



b. The total annual cost with the ELS is
   S
ELS
D
H
p
dpELS
C 




 

2
   200$
4.873,4
500,10
21.0$
190
30190
2
4.873,4





 

82.861$91.430$91.430$ 
a. Solving first for the ELS, we get
dp
p
H
DS


2
ELS
ELS - Example
73
c. Applying the TBO formula to the ELS, we get
  Days/YearWork
ELS
TBOELS
D
days162or162.4
d. The production time during each cycle is the lot size
divided by the production rate:

p
ELS
 350
500,10
4.873,4
days26or25.6
190
4.873,4

ELS - Example
Thank You!
74
References
Inventory Control Models
2013 Pearson Education, Inc. publishing as Prentice Hall
Special Inventory Models
2010 Pearson Education, Inc. publishing as Prentice Hall
6.3 Further Business Applications: Economic Lot Size
Dr. Grethe Hystad, Mathematics Department
The University of Arizona (www.math.arizona.edu)
Business Dictionary (www.businessdictionary.com)
75

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Basic EOQ Model, Quantity Discount, Economic Lot Size

  • 1. Presented by: Phillip Leonard Petate OPERATIONS RESEARCH: INVENTORY MODELS San Beda College Graduate School of Business
  • 2. INVENTORY MODELS I. Basic EOQ Models II. Quantity Discounts III. Economic Lot Size (ELS) 2
  • 3. Inventory Inventory is any stored resource that is used to satisfy a current or future need. Raw materials, work-in-process, and finished goods are examples of inventory. Inventory levels for finished goods, such as clothes dryers, are a direct function of market demand. 3
  • 4. Inventory By using this demand information, it is possible to determine how much raw materials (e.g., sheet metal, paint, and electric motors in the case of clothes dryers) and work-in-process are needed to produce the finished product. 4
  • 5. Types of Inventory  Raw Materials Purchased but not processed  Work-In-Process 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 5
  • 7. Importance of Inventory Control Inventory control serves several important functions and adds a great deal of flexibility to the operation of a firm. Five main uses of Inventory are as follows: 1. Decoupling Function 2. Storing Resources 3. Irregular Supply and Demand 4. Quantity Discounts 5. Avoiding Stockouts and Shortages 7
  • 8. Inventory Control Decisions 8 Even though there are literally millions of different types of products manufactured in our society, there are only two fundamental decisions that you have to make when controlling inventory: 1. How Much to Order 2. When to Order
  • 9. Purpose of Inventory Models 9 The purpose of all inventory models is to determine how much to order and when to order. As we know, inventory fulfills many important functions in an organization. But as the inventory levels go up to provide these functions, the cost of storing and holding inventory also increases. Thus, we must reach a fine balance in establishing inventory levels. A major objective in controlling inventory is to minimize total inventory costs.
  • 10. Components of Total Cost 10 Some of the most significant inventory costs are as follows: 1. Cost of the items 2. Cost of ordering 3. Cost of carrying, or holding, inventory 4. Cost of stockouts 5. Cost of safety stock, the additional inventory that may be held to help avoid stockouts
  • 11. Holding, Ordering, and Setup Costs  Holding Cost The cost of holding or “carrying” inventory over time.  Ordering Cost The cost of placing an order and receiving goods.  Setup Cost The cost to prepare a machine or process for manufacturing an order. 11
  • 13. Independent vs Dependent Demand  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. 13
  • 14. INVENTORY MODELS I. Basic EOQ Models 14
  • 15. Basic EOQ Model The Economic Order Quantity (EOQ) model is one of the oldest and most commonly known inventory control techniques. Research on its use dates back to a 1915 publication by Ford W. Harris. This model is still used by a large number of organizations today. This technique is relatively easy to use, but it makes a number of assumptions. 15
  • 16. Basic EOQ Assumptions  Demand is known and constant.  The lead time - that is, the time between the placement of the order and the receipt of the order - is known and constant.  The receipt of inventory is instantaneous. In other words, the inventory from an order arrives in one batch, at one point in time. 16
  • 17.  Quantity discounts are not possible.  The only variable costs are the cost of placing an order, ordering cost, and the cost of holding or storing inventory over time, carrying, or holding, cost.  If orders are placed at the right time, stockouts and shortages can be avoided completely. 17 Basic EOQ Assumptions
  • 18. Basic EOQ Assumptions Summary Important Assumptions: 1. Demand is known, constant, and independent 2. Lead time is known and constant 3. Receipt of inventory is instantaneous and complete 4. Quantity discounts are not possible 5. Only variable costs are setup and holding 6. Stockouts can be completely avoided 18
  • 20. [Refer to Graph in Slide 19] With these assumptions, inventory usage has a sawtooth shape. In the graph, Q represents the amount that is ordered. If this amount is 500 units, all 500 units arrive at one time when an order is received. Thus, the inventory level jumps from 0 to 500 units. In general, the inventory level increases from 0 to Q units when an order arrives. 20 Inventory Usage Over Time
  • 21. [Refer to Graph in Slide 19] Because demand is constant over time, inventory drops at a uniform rate over time. Another order is placed such that when the inventory level reaches 0, the new order is received and the inventory level again jumps to Q units, represented by the vertical lines. This process continues indefinitely over time. 21 Inventory Usage Over Time
  • 22. 22 Objective of Basic EOQ Model The objective of most inventory models is to minimize the total cost. With the assumptions just given, the significant costs are the ordering cost and the inventory carrying cost. All other costs, such as the cost of the inventory itself, are constant. Thus, if we minimize the sum of the ordering and carrying costs, we also minimize the total cost.
  • 23. 23 Total Cost as a Function of Order Quantity
  • 24. 24 Objective of Basic EOQ Model [Refer to Graph in Slide 23] To help visualize this, Slide 23 graphs total cost as a function of the order quantity, Q. As the value of Q increases, the total number of orders placed per year decreases. Hence, the total ordering cost decreases. However, as the value of Q increases, the carrying cost increases because the firm has to maintain larger average inventories.
  • 25. 25 Objective of Basic EOQ Model [Refer to Graph in Slide 23] The optimal order size, Q*, is the quantity that minimizes the total cost. Note in Slide 23 that Q* occurs at the point where the ordering cost curve and the carrying cost curve intersect. This is not by chance. With this particular type of cost function, the optimal quantity always occurs at a point where the ordering cost is equal to the carrying cost.
  • 26. 26 Average Inventory Level Now that we have a better understanding of inventory costs, let us see how we can determine the value of Q* that minimizes the total cost. In determining the annual carrying cost, it is convenient to use the average inventory. Referring to Slide 19, we see that the on-hand inventory ranges from a high of Q units to a low of zero units, with a uniform rate of decrease between these levels. Thus, the average inventory can be calculated as the average of the minimum and maximum inventory levels.
  • 27. 27 Average Inventory Level That is: Average Inventory Level = (0 + Q2)/2 = Q/2 We multiply this average inventory by a factor called the Annual Inventory Carrying Cost Per Unit to determine the annual inventory cost.
  • 28. Q = Number of Units in each Order Q* = Optimal Number of Pieces per Order (EOQ) D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year The EOQ Model 28 Notes: The unit holding/carrying cost, H, is usually expressed in one of two ways: 1. As a fixed cost. For example, H is $0.50 per unit per year. 2. As a percentage (typically denoted by I) of the item’s unit purchase cost or price. For example, H is 20% of the item’s unit cost. In general, H = I x P I = Annual Carrying Cost as a Percentage of the Unit Cost of the Item P = Purchase Cost per Unit of the Inventory Item
  • 29. Q = Number of Units in each Order Q* = Optimal Number of Pieces per Order (EOQ) D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per 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/Ordering Cost per Order = = (S) D Q EOQ - Annual Setup Cost 29
  • 30. Annual Holding Cost =(Average Inventory Level) x (Holding Cost per Unit per Year) Order Quantity 2 = (Holding Cost per Unit per Year) = (H)Q 2 30 EOQ - Annual Holding Cost Q = Number of Units in each Order Q* = Optimal Number of Pieces per Order (EOQ) D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year
  • 31. Optimal Order Quantity is found when Annual Setup Cost Equals Annual Holding Cost D Q S = HQ 2 Solving for Q* (EOQ) 2DS = Q2H Q2 = 2DS/H Q* = 2DS/H The EOQ ModelAnnual Setup Cost = S D Q Annual Holding Cost = H Q 2 31 Q = Number of Units in each Order Q* = Optimal Number of Pieces per Order (EOQ) D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year
  • 32. EOQ Example 32 Let us now apply these formulas to the case of SBC, a company that buys alarm clocks from a manufacturer and distributes to retailers. SBC would like to reduce its inventory cost by determining the optimal number of alarm clocks to obtain per order. The annual demand is 1,000 units, the ordering cost is $10 per order, and the carrying cost is $0.50 per unit per year. Each alarm clock has a purchase cost of $5. How many clocks should SBC order each time?
  • 33. Determine the Optimal Number of Alarm Clocks to Order D = 1,000 Units S = $10 per Order H = $0.50 per Unit per Year Q* = 2DS H Q* = 2(1,000)(10) 0.50 = 40,000 = 200 units EOQ Example 33
  • 34. D = 1,000 units Q* = 200 units S = $10 per order H = $0.50 per unit per year = N = =Expected Number of Orders Demand EOQ D Q* N = = 5 Orders per Year1,000 200 EOQ - Expected Number of Orders 34
  • 35. = T =Expected Time Between Orders Number of Working Days per Year N T = = 50 Days Between Orders 250 5 EOQ - Expected Time Between Orders 35 D = 1,000 units Q* = 200 units S = $10 per order N = 5 Orders per Year H = $0.50 per unit per year 250 Working Days per Year TBOEOQ = (Working Days per Year) EOQ D
  • 36. Q* = Optimal Number of Pieces per Order (EOQ) D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost for per Order H = Holding or Carrying Cost per Unit per Year P = Purchase Cost per Unit of the Inventory Item EOQ - Total Cost 36 Total Cost = Total Setup or Ordering Cost + Total Holding or Carrying Cost + Total Purchase Cost = (D/Q* x S) + (Q*/2 x H) + (P x D)
  • 37. EOQ - Total Cost 37 Total Cost = Total Setup/Ordering Cost (D/Q* x S) + Total Holding/Carrying Cost (Q*/2 x H) + Total Purchase Cost (P x D) Observe that the total purchase cost (P x D) does not depend on the value of Q. This is so because regardless of how many orders we place each year, or how many units we order each time, we will still incur the same annual total purchase cost.
  • 38. D = 1,000 Units Q* = 200 units S = $10 per Order N = 5 orders per year H = $0.50 per Unit per Year T = 50 days P = $5 Total Annual Cost = Setup Cost + Holding Cost + Purchase Cost TC = S + H + (P x D) D Q* Q* 2 TC = ($10) + ($.50) + $5 x 10001,000 200 200 2 TC = $50 + $50 + $5,000 EOQ - Total Cost 38 TC = $5,100
  • 39. 39 Purchase Cost of Inventory Items It is often useful to know the value of the average inventory level in monetary terms. We know that the average inventory level is Q/2, where Q is the order quantity. If we order Q* (the EOQ) units each time, the value of the average inventory can be computed by multiplying the average inventory by the unit purchase cost, P. That is: Average Monetary Value of Inventory = P x (Q*/2)
  • 40. 40 Average Monetary Value of Inventory = P x (Q*/2) = $5 x (200/2) = $500 D = 1,000 Units Q* = 200 units S = $10 per Order N = 5 orders per year H = $0.50 per Unit per Year T = 50 days P = $5 Purchase Cost of Inventory Items
  • 41. In using these formulas, we assumed that the values of the Ordering Cost (S) and Carrying Cost (H) are known constants. S = Q* ² x H/(2D) H = 2DS/Q* ² 41 Calculating the Ordering Costs (S) and Carrying Cost (H) for a Given Value of EOQ EOQ Formula:
  • 42. S = Q* ² x H/(2D) = 200² x ($0.50/ 2 x 1,000) = 40,000 x 0.00025 = $10 per Order 42 Calculating the Ordering Costs (S) and Carrying Cost (H) for a Given Value of EOQ D = 1,000 Units Q* = 200 units S = $10 per Order N = 5 orders per year H = $0.50 per Unit per Year T = 50 days P = $5 H = 2DS/Q* ² = 2 x (1,000 x 10) / 200² = 20,000 / 40,000 = $0.50 per Unit per Year
  • 43. Reorder Point (ROP) Now that we have decided how much to order, we look at the second inventory question: when to order. In most simple inventory models, it is assumed that we have instantaneous inventory receipt. That is, we assume that a firm waits until its inventory level for a particular item reaches zero, places an order, and receives the items in stock immediately. 43
  • 44. Reorder Point (ROP) In many cases, however, the time between the placing and receipt of an order, called the Lead Time, or Delivery Time, is often a few days or even a few weeks. Thus, the when to order decision is usually expressed in terms of a reorder point (ROP), the inventory level at which an order should be placed. 44
  • 45.  EOQ answers the “How Much” question.  The Reorder Point (ROP) tells “When” to order. ROP = Lead Time for a New Order in Days Demand per Day = d x L Where d = D Number of Working Days in a Year 45 Reorder Point (ROP)
  • 46. Q* ROP (units) InventoryLevel(Units) Time (Days) Lead time = L Slope = Units/Day = d Reorder Point Curve 46
  • 47. Demand = 1,000 Alarm Clocks per Year 250 Working Days in the Year Lead Time for Orders is 3 Working Days ROP = d x L d = D Number of Working Days in a Year = 1,000/250 = 4 Units per Day x 3 Days = 12 Units Reorder Point Example 47 = 4 Units per Day
  • 49. To increase sales, many companies offer quantity discounts to their customers. A quantity discount is simply a decreased unit cost for an item when it is purchased in larger quantities. It is not uncommon to have a discount schedule with several discounts for large orders. See example below: Quantity Discount Models 49 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
  • 50. As can be seen in Slide 49 Table, the normal cost for the item in this example is $5. When 1,000 to 1,999 units are ordered at one time, the cost per unit drops to $4.80, and when the quantity ordered at one time is 2,000 units or more, the cost is $4.75 per unit. As always, management must decide when and how much to order. But with quantity discounts, how does a manager make these decisions? Quantity Discount Models 50
  • 51. As with previous inventory models discussed so far, the overall objective is to minimize the total cost. Because the unit cost for the third discount in Slide 49 Table is lowest, we might be tempted to order 2,000 units or more to take advantage of this discount. Placing an order for that many units, however, might not minimize the total inventory cost. As the discount quantity goes up, the item cost goes down, but the carrying cost increases because the order sizes are large. Thus, the major trade-off when considering quantity discounts is between the reduced item cost and the increased carrying cost. Quantity Discount Models 51
  • 52. Total Annual Cost = Setup Cost + Holding Cost + Purchase Cost = S + H + (P x D) D Q* Q* 2 Quantity Discount Models 52 Recall that we computed the total cost (including the total purchase cost) for the EOQ model as follows: Next, we illustrate the four-step process to determine the quantity that minimizes the total cost.
  • 53. 1. For each discount price, calculate a Q* value, using the EOQ formula. In quantity discount EOQ models, the unit carrying cost, H, is typically expressed as a percentage (I) of the unit purchase cost (P). That is, H = I x P. As a result, the value of Q* will be different for each discounted price. 2. For any discount level, if the Q* computed in step 1 is too low to qualify for the discount, adjust Q* upward to the lowest quantity that qualifies for the discount. For example, if Q* for discount 2 in Slide 49 Table turns out to be 500 units, adjust this value up to 1,000 units. 4 Steps to Analyze Quantity Discount Models 53
  • 54. Total Cost Curve for the Quantity Discount Model 54
  • 55. As seen in Slide 54, the total cost curve for the discounts shown in Slide 49 is broken into three different curves. There are separate cost curves for the first (0 ≤ Q ≤ 999), second (1,000 ≤ Q ≤ 1,999), and third (Q ≥ 2,000) discounts. Look at the total cost curve for discount 2. The Q* for discount 2 is less than the allowable discount range of 1,000 to 1,999 units. However, the total cost at 1,000 units (which is the minimum quantity needed to get this discount) is still less than the lowest total cost for discount 1. Thus, step 2 is needed to ensure that we do not discard any discount level that may indeed produce the minimum total cost. Note that an order quantity compute in step 1 that is greater than the range that would qualify it for a discount may be discarded. Total Cost Curve for the Quantity Discount Model55
  • 56. 4 Steps to Analyze Quantity Discount Models 56 3. Using the Total Cost Equation, compute a total cost for every Q* determined in steps 1 and 2. If a Q* had to be adjusted upward because it was below the allowable quantity range, be sure to use the adjusted Q* value. 4. Select the Q* that has the lowest total cost, as computed in step 3. It will be the order quantity that minimizes the total cost.
  • 57. 1. For each discount, calculate Q* 2. If Q* for a discount doesn’t qualify, choose the smallest possible order size to get the discount 3. Compute the total cost for each Q* or adjusted value from Step 2 4. Select the Q* that gives the lowest total cost Analyzing a Quantity Discount Summary 57
  • 58. GSB Department Store stocks toy cars. Recently, the store was given a quantity discount schedule for the cars, as shown in Slide 49. Thus, the normal cost for the cars is $5.00. For orders between 1,000 and 1,999 units, the unit cost is $4.80, and for orders of 2,000 or more units, the unit cost is $4.75. Furthermore, the ordering cost is $49 per order, the annual demand is 5,000 race cars, and the inventory carrying charge as a percentage of cost, I, is 20%, or 0.2. What order quantity will minimize the total cost? Quantity Discount Example 58
  • 59. Quantity Discount Example 59 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 D = 5,000 Units S = $49 per Order I = 20% of Cost H = I x P (Cost) Q* = 2DS H
  • 60. Calculate Q* for every Discount Q* = 2DS IP 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) Quantity Discount Example 60
  • 61. In the GSB Department Store example, observe that the Q* values for discounts 2 and 3 are too low to be eligible for the discounted prices (Slide 49 Table). They are, therefore, adjusted upward to 1,000 and 2,000, respectively. With these adjusted Q* values, we find that the lowest total cost of $24,725 results when we use an order quantity of 1,000 units [See Slide 61 and Slide 62]. Quantity Discount Example 61
  • 62. Calculate Q* for every Discount Q* = 2DS IP 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 Quantity Discount Example 62
  • 63. Discount Number Unit Price Order Quantity Annual Product Cost Annual Ordering Cost Annual Holding Cost Total Cost 1 $5.00 700 $25,000 $350.00 $350 $25,700.00 2 $4.80 1,000 $24,000 $245.00 $480 $24,725.00 3 $4.75 2,000 $23,750 $122.50 $950 $24,822.50 Choose the Price and Quantity that gives the Lowest Total Cost Buy 1,000 Units at $4.80 per Unit Quantity Discount Example 63
  • 64. INVENTORY MODELS III. Economic Lot Size (ELS) 64
  • 65. ELS is the quantity of material or units of a manufactured good that can be produced or purchased within the lowest unit cost range. It is determined by reconciling the decreasing unit cost of larger quantities with the associated increasing unit cost of handling, storage, insurance, interest, etc. (www.businessdictionary.com) Economic Lot Size (ELS) 65
  • 66. Economic Lot Size (ELS) A manufacturer must determine the production lot size that will result in minimum production and storage cost. Economic Order Quantity (EOQ) A purchaser must decide what quantity of an item to order that will result in minimum reordering and storage cost. Economic Lot Size (ELS) 66
  • 67. D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year p = production rate d = daily demand 67 ELS Model dp p H 2DS ELS  
  • 68. 68 ELS - Total Annual Cost    S ELS D H p dp 2 ELS C         D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year p = production rate d = daily demand
  • 69. 69 ELS - Time Between Orders (TBO)  Days/YearWork D ELS TBOELS  D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year p = production rate d = daily demand
  • 70. 70 ELS – Production Time per Lot p ELS D = Annual Demand in Units for the Inventory Item S = Setup or Ordering Cost per Order H = Holding or Carrying Cost per Unit per Year p = production rate d = daily demand
  • 71. ELS - Example 71 A plant manager of a chemical plant must determine the lot size for a particular chemical that has a steady demand of 30 barrels per day. The production rate is 190 barrels per day, annual demand is 10,500 barrels, setup cost is $200, annual holding cost is $0.21 per barrel, and the plant operates 350 days per year. a. Determine the Economic Production Lot Size (ELS) b. Determine the Total Annual Setup and Inventory Holding Cost for this item (Total Annual Cost) c. Determine the time between orders (TBO), or cycle length, for the ELS d. Determine the Production Time per Lot
  • 72. 72    barrels4,873.4 30190 190 21.0$ 200$500,102    b. The total annual cost with the ELS is    S ELS D H p dpELS C         2    200$ 4.873,4 500,10 21.0$ 190 30190 2 4.873,4         82.861$91.430$91.430$  a. Solving first for the ELS, we get dp p H DS   2 ELS ELS - Example
  • 73. 73 c. Applying the TBO formula to the ELS, we get   Days/YearWork ELS TBOELS D days162or162.4 d. The production time during each cycle is the lot size divided by the production rate:  p ELS  350 500,10 4.873,4 days26or25.6 190 4.873,4  ELS - Example
  • 75. References Inventory Control Models 2013 Pearson Education, Inc. publishing as Prentice Hall Special Inventory Models 2010 Pearson Education, Inc. publishing as Prentice Hall 6.3 Further Business Applications: Economic Lot Size Dr. Grethe Hystad, Mathematics Department The University of Arizona (www.math.arizona.edu) Business Dictionary (www.businessdictionary.com) 75