Chapter 15
Deterministic EOQ Inventory Models
to accompany
Operations Research: Applications and Algorithms
4th edition
by Wayne L. Winston
Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.
Description
We now begin our formal study of inventory modeling.
We begin by discussing some important concepts used to
describe inventory models and then develop versions of the
famous economic order quantity (EOQ) model that can be
used to make optimal inventory decisions when demand is
deterministic.
2
15.1 Introduction to Basic Inventory
Models
 The purpose of inventory theory is to
determine rules that management can use to
minimize the costs associated with maintaining
inventory and meeting customer demand.
 Inventory models answer the following
questions
1. When should an order be placed for a product?
(產品何時下定單)
1. How large should each order be?
每次定單需下多少量
3
Costs Involved in Inventory Models
 The inventory models that we will discuss
involve some or all of the following costs:
 Ordering and Setup Cost
 These costs do not depend on the size of the order.
They typically include things like paperwork, billing
or machine setup time if the product is made
internally.
 Unit Purchasing Cost
 This cost is simply the variable cost associated with
purchasing a single unit. Typically, the unit
purchasing cost includes the variable labor cost,
variable overhead cost, and raw material cost.
4
 Holding or Carrying Cost
 This is the cost of carrying one unit of inventory for
one time period. The holding costs usually includes
storage cost, insurance cost, taxes on inventory and
others. Usually, however, the most significant
components of holding cost if the opportunity cost
incurred by tying up capital in inventory.
 Stockout(缺貨) or Shortage Cost(短缺)
 When a customer demands a product and the
demand is not met on time, a stockout, or shortage,
is said to occur. If they will accept delivery at a later
date, we say the demands are back-ordered. This
case is often referred to as the backlogged
demand case. If they will not accept late delivery,
we are in the lost sales case. These costs are often
harder to measure than other costs.
5
Assumptions of EOQ Models
 Repetitive Ordering
 The ordering decision is repetitive in the sense that it
is repeated in a regular fashion.
 Constant Demand
 Demand is assumed to occur at a known, constant
rate.
 Constant Lead Time(前置時間)
 The lead time for each order is a known constant, L.
By the lead time we mean the length of time
between the instant when an order is placed and the
instant at which the order arrives.
6
 Continuous Ordering
 An order may be placed at any time. Inventory
models that allow this are called continuous review
models. If the amount of on-hand inventory is
reviewed periodically and orders may be placed only
periodically, we are dealing with a periodic review
model.
7
15.2 Basic Economic Order Quantity
Model

For the basic EOQ model to hold, certain
assumptions are required:
1. Demand is deterministic and occurs at a constant
rate.
2. In an order of any size (say q units is placed, an
ordering and setup cost K is incurred.
3. The lead time for each order is zero.
4. No shortages are allowed.
5. The cost per unit-year of holding inventory is h.
8
Assumptions of the Basic EOQ Model
 We define D to be the number of units
demanded per year.
 The setup cost K of assumption 2 is in addition
to a cost pq of purchasing or producing the q
unites ordered. We assume p (the purchasing
cost) does not depend on the size of the order.
 Assumption 3 assumes each order arrives on
time.
 Assumption 4 implies that all demand is met
on time.
9
 Assumption 5 implies that if I units are held for
T years, a holding cost of ITh is incurred.
 Given these assumptions, the EOQ model
determines an ordering policy that minimizes
the yearly sum of ordering cost, purchasing
cost, and holding cost.
10
Derivation of Basic EOQ Model
 We begin by making some simple observations.
 We should never place an order when I, the inventory
level, is greater than zero; if we place an order then
the we are incurring an unnecessary holding cost.
 If I=0, we must place an order to prevent a shortage.
 Each time an order is placed (when I=0) we should
order the same quantity, q.
 TC(q)= annual cost of placing orders + annual
purchasing cost + annual holding cost
 To determine the annual holding cost, we need
to examine the behavior of I over time.
11
 A key concept in the study of EOQ models is
the idea of a cycle.
 Definition: Any interval of time that begins
with the arrival of an order and ends the
instant before the next order is received is
called a cycle.
 In the figure the average inventory during any
cycle is simply half of the maximum inventory
level attained during the cycle.
 This result will hold in any model for which
demand occurs at a constant rate.
12
 The economic order quantity, or EOQ,
 2 KD 
q*  

 h 
1
2
minimizes TC(q).
 Thus, q* does indeed minimize total annual
cost.
 The figure on the next page confirms the fact
that at q*, the annual holding and ordering
costs are the same.
13
Example 1
 Braneast Airlines uses 500 taillights per year.
 Each time an order for taillights is placed, an
ordering cost of $5 is incurred.
 Each light cost 40¢, and the holding cost is
8¢/light/year.
 Assume that demand occurs at a constant rate
and shortages are not allowed.
 What is the EOQ? How many orders will be
placed each year? How much time will elapse
between the placement of orders?
14
Example 1 Solution
 We are given that K = $5, h = $0.08/light/year,
and D = 500 lights/year.
 The EOQ is
 Hence, the airline should place an order for
250 taillights each time that inventory reaches
zero.
15
 The time between placement (or arrival) of
orders is simply the length of a cycle.
 Since the length of each cycle is , q*/D, the
time between orders will be
q * 250 1

 year
D 500 2
16
Sensitivity of Total Cost to Small
Variations in the Order Quantity
 In most situations, a light deviation from the
EOQ will result in only a slight increase in costs.
 For Example 1, let’s see how derivations from
the EOQ change the total annual cost.
 We focus our attention on how the annual
holding cost and ordering costs are affected by
the changes in order quantity.
 Let
HC(q) = annual holding cost if the order quantity is q
OC(q) = annual ordering cost if the order quantity is q
17
 We find that HC(q)=0.04q and OC(q)=2500/q.
 The figure shows that HC(q) + OC(q) is very
flat near q*. The flatness of the HC(q) +
OC(q) is very important, because it is often
difficult to estimate h and K.
 Inaccurate estimation of h and K may result
in a value of q that differs slightly from the
actual EOQ.
 The flatness of the HC(q) + OC(q) curve
indicates that even a moderate error in the
determination of the EOQ will only increase
costs by a slight amount.
18
Determination of EOQ When Holding Cost is
Expressed in Terms of Dollar Value of Inventory
 Often the annual holding cost is expressed in
terms of the cost of holding one dollar’s worth
of inventory for one year.
 Suppose that hd = cost of holding one dollar in
inventory for one year.
 Then the cost of holding one unit of inventory
for one year will be phd and may be written as
 2 KD 

q*  
 phd 
1
2
19
The Effect of Nonzero Lead Time
 We now allow lead time L to be greater than
zero.
 The introduction of a nonzero lead time leaves
the annual holding and ordering costs
unchanged. Hence, the EOQ still minimizes
total costs.
 To prevent shortages from occurring and to
minimize holding cost, each order must be
placed at an inventory level that ensures that
when each order arrives, the inventory level
will equal zero.
 Definition: The inventory level at which an
order should be placed is the reorder point.
20
 To determine the reorder point for the basic
EOQ model, two cases must be considered:
 Case 1: Demand during the lead time does not
exceed the EOQ. In this case the reorder point occurs
when the inventory level equals LD. The order will
arrive L time units later, and upon arrival of the order,
the inventory level will equal LD - LD=0.
 Case 2: Demand during the lead time exceeds EOQ.
In this case the reorder point does not equal LD. In
general it can be shown that the reorder point equals
the remainder when LD is divided by the EOQ.
21
 The determination of the reorder point
becomes extremely important when demand is
random and stockouts can occur.
22
Spreadsheet Template for the Basic
EOQ Model
 The user inputs the values of K, h, lead time
(L), and D.
 In A8 the EOQ is determined by the formula
(2*K*D/H)^.5.
 In B8 we computer annual holding costs by the
formula .5*A8*H.
 In D5 we computer orders per year for the
EOQ by the formula D/A8.
 In C8 we compute annual holding costs for the
EOQ from the formula K*D5.
23
 In D8 we computer total annual cost for the
EOQ by the formula B8+C8.
 In B11 we computer the reorder point by the
formula =MOD(L*D,A8)
 This yields the remainder obtained when L*D is
divided by the EOQ.
24
Power of Two Ordering Policies
 Suppose a company orders three products and
the EOQ’s for each product yield time between
orders of 3.5 days, 5.6 days and 9.2 days. It
would rarely be the case that they all arrive on
the same days.
 If we could synchronize our reorder intervals
so that orders for different products often
arrived on the same day we often can greatly
reduce our fixed costs.
25
 Roundy (1986) devised an elegant and simple
method called Power of Two Ordering
Policies to ensure that the placement of
orders for multiple products are well
synchronized.
 Let q* = EOQ. Then the optimal reorder
interval for a product is t* = q*/D.
 The virtue of a Power of Two policy is that
different products will frequently arrive at the
same time. This will greatly reduce fixed costs.
26
15.3 Computing the Optimal Order Quantity
When Quantity Discounts Are Allowed
 Up to now, we have assumed that the annual
purchase cost does not depend on the order
size.
 In real life, however, suppliers often reduce the
unit purchasing price for large orders. Such
price reductions are referred to as quantity
discounts.
 The approach used previously is no longer valid
and we need a new approach to find the
optimal quantity.
27
 If we let q be the quantity ordered each time
an order is placed, the general quantity
discount model analyzed in this section may
be described as follows:
 If q<b1, each item cost p1 dollars.
 If b1≤q<b2, each item costs p2 dollars.
 If bk-2≤q<bk-1, each item cost pk-1 dollars.
 If bk-1≤q<bk=∞, each item costs pk dollars.
 Since b1, b2, …., bk-1 are points where a price
change occurs, we refer to them as price
break points.
28
Example 4
 A local accounting firm in Smalltown orders
boxes of floppy disks from a store in
Megalopolis.
 The per-box price charged by the store
depends on the number of boxes purchased.
 The accounting firm uses 10,000 disks per year.
 The cost of placing an order is assumed to be
$100.
29
Example 4
 The only holding cost is the opportunity cost
of capital, which is assumed to be 20% per
year.
 For this example,
 b1 = 100
 b2 = 300
 p1 =$50.00
 p2 = $49.00
 p3 = $48.50
30

Before explaining how to find the order quantity
minimizing total annual costs, we need the following
definitions:

TCi(q) = total annual cost if each order is q units at a
price pi

EOQi = quantity that minimizes total annual cost if, for
any order quantity, the purchasing cost of the item is pi

EOQi is admissible if bi-1≤EOQi<bi.

TC(q) = actual annual cost of q items are ordered each
time an order is placed.

These definitions are illustrated on the next slide.

Our goal is to find a value of q minimizing TC(q).
31

In general, the value of q minimizing TC(q)
can be either a break point (shown in second
figure) or some EOQi (shown in first figure).

The following observations are helpful in
determining the point that minimizes TC(q).
1. For any value of q,
TCk(q) < TCk-1(q) < … <TC2(q) <TC1(q)
2. If EOQi is admissible, then minimum cost for bi1≤q<bi occurs for q=EOQi (see the first figure on the
next slide). If EOQi<bi-1, the minimum cost for bi1≤q<bi occurs for q=bi-1 (see second figure on next
slide).
32
3. If EOQi is admissible, then TC(q) cannot be
minimized at an order quantity for which the
purchasing price per item exceeds pi. Thus if EOQi is
admissible, the optimal order quantity must occur for
either price pi, pi+1,…, or pk.

These observations allow us to use the
following method to determine the optimal
order quantity when quantity discounts are
allowed.

Beginning with the lowest price, determine for
each price the order quantity that minimizes
total annual cost for bi-1 ≤ q <bi.
33
 Continue determining q*k, q*k-1…. Until one of
the q*i’s is admissible; from observation 2, this
will mean that q*i = EOQi.
 The optimal order quantity will be the member
of {q*k, Q*k-1,…q*i} with the smallest value of
TC(q).
34
Example 4 continued
 Each time an order is placed for disks, how
many boxes of disks should be ordered?
 How many orders will be placed annually?
 What is the total annual cost of meeting the
accounting firm’s disk needs?
35
Example 4 Solution
 Note the K=$100 and D = 1000 boxes per year.
 We first determine the best order quantity for
p3 = $48.50 and 300 ≤ q. Then
1
2
 2(100)1000 
  143.59
EOQ3  
 0.2(48.50) 
 Since EOQ3 < 300, EOQ3 is not admissible.
Therefore q ≥ 300, TC3(q) is minimized by q*3
= 300.
36
Example 4 Solution
 We next consider p2=$49.00 and 100 ≤ q <
300. Then
1
2
 2(100)(1000) 
EOQ2  
  142.86
9.8


 Since 100 ≤ EOQ2 < 300, EOQ2 is admissible,
and for a price p2 = $49.00, the best we can
do is to choose q*2 = 142.86.
 Since q*2 is admissible, p1 =%50.00 and 0 ≤
q < 100 cannot yield the order quantity
minimizing TC(q). Thus, either q*2 or q*3 will
minimize TC(q).
37
Example 4 Solution
 To determine which of these order quantities
minimizes TC(q), we must find the smaller of
TC3(300) and TC2(142.86).
 For q*3 the annual holding cost is $9.70. Thus,
for q*3,
Annual ordering cost = 100(1000/300) = $333.33
Annual purchasing cost = 1000(48.50) = $48,500
Annual holding cost = (½)(300)(9.7) = $1455
TC3(300) = $50, 288.33
38
Example 4 Solution
 For q*2 the annual holding cost is $9.80. Thus,
for q*2,
Annual ordering cost = 100(1000/142.86) = $699.99
Annual purchasing cost = 1000(49) = $49,000
Annual holding cost = (½)(142.86)(9.8) = $700.01
TC3(142.86) = $50, 400
 Thus q*3 = 300 will minimize TC(q).
 Our analysis shows that each time an order is
placed, 300 boxes of disks should be ordered.
Then 3.33 orders are placed each year.
39
Spreadsheet Template for Quantity
Discounts
 The table illustrates how inventory problems
with a quantity discount can be solved on a
spreadsheet.
 In cell C2 we enter D, the annual demand.
 In cell D2(HD) we enter the annual cost of
holding $1 of goods in inventory for one year.
 In the cell range A6:C8 we enter the left-hand
endpoint, right-hand endpoint, and price for
each interval.
40
Spreadsheet Template for Quantity
Discounts
 In D6 we computer the EOQ for the interval b0
= 0 ≤ order quantity < 100=b1 by entering
the formula (2*$K*$D/(($HD*C6))^.5).
 This statement computes the order quantity in
the first interval that in E6 we enter the
formula
=IF(AND(D6>=A6,D6<B^),D6,IF(D6<A6,A6,B^-1)
41
15.4 The Continuous Rate EOQ Model
 Many goods are produced internally rather than
purchased from an outside supplier.
 In this situation, the EOQ assumption that each
order arrives at the same instant seems
unrealistic; it isn’t possible to produce, say,
10,000 cars at the drop of a hard hat.
 If a company meets demand by making its own
products, the continuous rate EOQ model will
be more realistic than the traditional EOQ
model.
42
 The continuous rate EOQ model assumes that
a firm can produce a good at a rate of r units
per time period.
 This means that during any time period of
length, t, the firm can produce rt units.
 We define
 q = number of units produced during each
production run
 K = cost of setting up a production run
 H = cost of holding one unit in inventory for one
year
 D = annual demand for the product
 The variation of inventory over time is shown.
43
 r 
Optimal run size  EOQ

r D
1
2
As r increases, production occurs at a more rapid
rate. Hence, for large r, the rate model should
approach the instantaneous delivery situation
of the EOQ model.
44
Spreadsheet Template for the
Continuous Rate EOQ Model
 The table in the book illustrates a template for
the continuous rate EOQ model.
 In cell A6 the user inputs K; in B6, h; in C6, D;
and in D6, the production rate r.
 In A8 the formula (2*K*D/H)^.5*(R/(R-D))^.5
computes the optimal size.
 In B8 the formulas D/Q computers the number
of runs per year.
 In C8 we computer the annual cost with the
formula (H*Q*(R-D)/(2*R))+K*D/Q.
45
15.5 & 15.6 The EOQ Model with Back
Orders Allowed & When to Use EOQ Models
 In many real-life situations, demand is not met
on time, and shortages occur. When a shortage
occurs, costs are incurred.
 Let s by the cost of being short one unit for
one year.
 The variables K, D, and h have their usual
meanings.
 In most situations s is very difficult to measure.
 We assume all demand is backlogged and no
sales are lost.
46
 To determine the order policy that minimizes
annual costs, we define
 q = order quantity
 q - M = maximum shortage that occurs under an
ordering policy
 Equivalently, the firm will be q – M units short
each time an order is placed.
 We assume lead time for each order is zero.
 The firm’s maximum inventory level will be
M – q + q = M.
47
 TC(q,M) is minimized by q* and M*:
1
2
 2 KD(h  s) 
hs
q*  

EOQ



hs
s




1
2
 2 KDs 
 s 
M*  

EOQ



h
(
h

s
)
h

s




1
2
1
2
 Maximum shortage = q* - M*
 As s approaches infinity, q* and M* both
approach the EOQ, and the maximum
shortage approaches zero.
48
Spreadsheet Template for the EOA
Model with Back Orders
 A spreadsheet template for the EOQ model
with back orders.
 In cells A6, B6, C6, and D6 we enter the values
of K, D, h, and s, respectively.
 In A8 we computer the optimal order quantity
with the formula (2*K*D(H+S)/(H*S))^.5
 In B8 we computer the optimal value of M with
the formula (2*K*D*S/(H*(H+S)))^.5
 In C8 we compute the maximum shortage with
the formula =Q-M.
49
 In D8 we compute the annual total cost
TC(q,M) with the formula
(M^2*H)/(2*Q))+((Q-M)^2S/2*Q))+(K*D/Q).
50
15.7 Multiple Product Economic Order
Quantity Models
 Supposed a company orders several products.
 Each time an order is received shipments of
some of the products arrive.
 Each time an order arrives there is a fixed cost
associated with the order and there is another
fixed cost associated with each product
included in the order.
 How can we minimize the sum of annual
holding and fixed costs?
51
 Chopra and Meindl(2001) devised a method to
find a near optimal solution to this type of
problem.
 To begin we find the product that is most
frequently ordered.
 We assume this product will be included in
each order.
52
 We then setup a Solver model that determines
the following Changing Cells:
 Number of orders received per year.
 For all products other than the chosen product the
number of orders that need to be received before an
order of the product is received.
 We can easily determine the total fixed cost
and the total holding cost for each product.
 The sum of these costs will be our target cell
for Solver. Our model is highly nonlinear and it
is necessary to use the Evolutionary Solver to
find the optimal solution.
53