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Part II
Methods for
Independent Demand
Chapter 3
Economic Order Quantity
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Introduction
• This chapter introduces some quantitative models for
inventory control.
• The first model takes an idealized stock and finds the fixed
order size that minimizes costs.
• This is the economic order quantity (EOQ), which is the
basis of most independent demand methods.
Defining the EOQ:
Background
• EOQ was first referenced to the work by Harris (1915), but
the calculation is often credited to Wilson (1934) who
independently duplicated the work and marketed the
results.
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Typical Pattern of Stock Level
Stock
level
Stock
out
Place
order
Place
order
Replenishment
Replenishment
Time
Replenishment
Assumptions of Basic EOQ Model
• The demand is known exactly, is continuous and is
constant over time;
• All costs are known exactly and do not vary;
• No shortages are allowed;
• Lead time is zero – so a delivery is made as soon as the
order is placed.
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Assumptions of Basic EOQ Model
Implicit Assumptions:
• We can consider a single item in isolation, so we cannot
save money by substituting other items or grouping several
items into a single order;
• Purchase price and reorder costs do not vary with the
quantity ordered;
• A single delivery is made for each order;
• Replenishment is instantaneous, so that all of an order
arrives in stock at the same time and can be used
immediately.
Constant Demand Assumption
Demand
Time
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Assumptions of Basic EOQ Model
The assumptions might seem unrealistic,
• All models are simplifications of reality and their aim is to
give useful results rather than be exact representations of
actual circumstances. The EOQ is widely used, and we can
infer that it is accurate enough for many purposes. The
results may not be optimal in the strict mathematical sense,
but they are good approximations and do, at worst, give
useful guidelines.
• This is a basic model that we can extend in many ways.
Stock Level with Fixed Order Size
Stock
level
Time
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Variables Used in the Analysis
•
•
•
•
Unit cost (UC) is the price charged by the suppliers for one unit of the
item, or the total cost to the organization of acquiring one unit.
Reorder cost (RC) is the cost of placing a routine order for the item and
might include allowances for drawing-up an order, correspondence,
telephone costs, receiving, use of equipment, expediting, delivery,
quality checks, and so on. If the item is made internally, this might be a
set-up cost.
Holding cost (HC) is the cost of holding one unit of the item in stock for
one period of time. The usual period for calculating stock costs is a
year.
Shortage cost (SC) is the cost of having a shortage and not being able
to meet demand from stock. In this analysis we have said that no
shortages are allowed, so SC does not appear (it is effectively so large
that any shortage would be prohibitively expensive).
Decision Variables
• Order quantity (Q) which is the fixed order size that
we always use.
• Cycle time (T) which is the time between two
consecutive replenishments.
Other Variables
• Demand (D) which sets the number of units to be
supplied from stock in a given time period.
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Derivation of the EOQ
Standard approach
1. Find the total cost of one stock cycle.
2. Divide this total cost by the cycle length to get a
cost per unit time.
3. Minimize this cost per unit time.
Stock Level with Fixed Order Size
Stock
level
Optimal
order size Q
D
Average
stock
level
Time
𝑄 =𝐷×𝑇
Place order &
receive delivery
T
Place order &
receive delivery
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Derivation of the EOQ
1. Find the total cost of one stock cycle.
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑐𝑦𝑐𝑙𝑒
= 𝑢𝑛𝑖𝑡 𝑐𝑜𝑠𝑡 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 + 𝑟𝑒𝑜𝑟𝑑𝑒𝑟 𝑐𝑜𝑠𝑡 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡
+ ℎ𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡
HC × Q × T
= UC × Q + RC +
2
Derivation of the EOQ
2. Divide this total cost by the cycle length to get a
cost per unit time.
UC × Q 𝑅𝐶 HC × Q
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 time =
+
+
𝑇
𝑇
2
𝑆𝑖𝑛𝑐𝑒 𝑄 = 𝐷 × 𝑇
𝑇𝐶 = 𝑈𝐶 × 𝐷 +
𝑅𝐶 × 𝐷 HC × Q
+
𝑄
2
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Derivation of the EOQ
For the economic order
quantity, the reorder cost
component equals the
holding
cost component
Derivation of the EOQ
3. Minimize this cost per unit time.
𝑑(𝑇𝐶)
𝑅𝐶 × 𝐷 HC
=−
+
=0
𝑑(𝑄)
𝑄2
2
Solve for 𝑄𝑜 (EOQ),
𝑄𝑜 =
2 × 𝑅𝐶 × 𝐷
𝐻𝐶
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Derivation of the EOQ
Optimal cycle length
𝑇𝑜 =
𝑄𝑜
=
𝐷
2 × 𝑅𝐶
𝐻𝐶 × 𝐷
Optimal cost per unit time
𝑇𝐶𝑜 = 𝑈𝐶 × 𝐷 +
𝑅𝐶 × 𝐷
HC ×
2 × 𝑅𝐶 × 𝐷
𝐻𝐶
2
+
2 × 𝑅𝐶 × 𝐷
𝐻𝐶
= 𝑈𝐶 × 𝐷 + 2 × 𝑅𝐶 × 𝐻𝐶 × 𝐷 = 𝑈𝐶 × 𝐷 + 𝐻𝐶 × 𝑄𝑜
Worked Example 1:
EOQ
Jaydeep Company buys 6,000 units of an item every year with a unit
cost of $30. It costs $125 to process an order and arrange delivery,
while interest and storage costs amount to $6 a year for each unit
held. What is the best ordering policy for the item?
𝐷 = 6000 units/yr 𝑅𝐶= $125 an order
𝑈𝐶= $30 a unit
𝐻𝐶= $6/yr for a unit
2 × 𝑅𝐶 × 𝐷
2 × 125 × 6000
=
= 500
𝐻𝐶
6
𝑄𝑜
500
1
𝑇𝑜 =
=
=
𝑦𝑒𝑎𝑟 = 1 𝑚𝑜.
𝐷
6000 12
𝑇𝐶𝑜 = 𝑈𝐶 × 𝐷 + 𝐻𝐶 × 𝑄𝑜 = 30 × 6000 + 6 × 500
= $183000
𝑄𝑜 =
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Worked Example 2:
EOQ
Sarah Brown works for a manufacturer that makes parts for marine
engines. The parts are made in batches, and every time a new
batch is started it costs £1,640 for disruption and lost production and
£280 in wages for the fitters. One item has an annual demand of
1,250 units with a selling price of £300, 60% of which is direct
material and production costs. If the company looks for a return of
20 per cent a year on capital, what is the optimal batch size for
the item and the associated costs?
𝐷 = 1250 units/yr
𝑈𝐶= (£300)(60%) = £180 a unit
𝐻𝐶= (£180)(20%) = £36/yr for a unit
𝑅𝐶= £1640 + 280 = £1920 per setup
Worked Example 2:
EOQ
𝐷 = 1250 units/yr
𝑈𝐶= (£300)(60%) = £180 a unit
𝐻𝐶= (£180)(20%) = £36/yr for a unit
𝑅𝐶= £1640 + 280 = £1920 per setup
2 × 𝑅𝐶 × 𝐷
2 × 1920 × 1250
=
= 365
𝐻𝐶
36
𝑄𝑜
365
𝑇𝑜 =
=
= 0.29 𝑦𝑒𝑎𝑟 = 15 𝑤𝑒𝑒𝑘𝑠
𝐷
1250
𝑇𝐶𝑜 = 𝑈𝐶 × 𝐷 + 𝐻𝐶 × 𝑄𝑜 = 180 × 1250 + 36 × 365
= £238140
𝑄𝑜 =
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Adjusting the EOQ
Problems with EOQ Model:
1. When their batch set-up costs are high, the EOQ can
suggest very large batches
2. The EOQ suggests fractional value for things which come
in discrete units
3. Suppliers are unwilling to split standard package sizes
4. Deliveries are made by vehicles with fixed capacities
5. It is simply more convenient to round order sizes to a
convenient number.
Adjusting the EOQ
𝑅𝐶×𝐷
𝑆𝑖𝑛𝑐𝑒 𝑉𝐶𝑜 = 𝐻𝐶 × 𝑄𝑜 and 𝑉𝐶 = 𝑄 +
𝑉𝐶
𝑅𝐶 × 𝐷
𝐻𝐶 × 𝑄
=
+
𝑉𝐶𝑜 𝑄 × 𝐻𝐶 × 𝑄𝑜 2 × 𝐻𝐶 × 𝑄𝑜
Substituting 𝑄𝑜 =
HC× Q
2
2×𝑅𝐶×𝐷
𝐻𝐶
𝑉𝐶
1 𝑄0 𝑄
1 1 𝑘
𝑄
=
+
=
+
where k =
𝑉𝐶𝑜 2 𝑄 𝑄𝑜
2 𝑘 1
𝑄0
Quadratic formula 𝑥 =
−𝑏± 𝑏 2 −4𝑎𝑐
2𝑎
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Adjusting the EOQ
Worked Example 3:
Adjusting EOQ
Each unit of an item costs a company £40 with annual holding costs
of 18% of unit cost for interest charges, 1% for insurance, 2% per
cent allowance for obsolescence, £2 for building overheads, £1.50
for damage and loss and £4 miscellaneous costs. If the annual
demand for the item is constant at 1,000 units and each order costs
£100 to place, calculate the economic order quantity and the total
cost of stocking the item. If the supplier will only deliver batches of
250 units, how does this affect the costs?
𝐷 = 1000 units/yr
𝑈𝐶= £40
𝐻𝐶= (£40 )(21%)+ £7.50 = £15.90 /yr for a unit
𝑅𝐶= £100 per order
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Worked Example 3:
Adjusting EOQ
𝐷 = 1000 units/yr
𝑈𝐶= £40
𝐻𝐶= (£40 )(21%)+ £7.50 = £15.90 /yr for a unit
𝑅𝐶= £100 per order
𝑄𝑜 =
2 × 𝑅𝐶 × 𝐷
=
𝐻𝐶
2 × 100 × 1000
= 112.15
15.90
𝑉𝐶𝑜 = 𝐻𝐶 × 𝑄𝑜 = 15.90 × 112.15 = £1783
𝑉𝐶
1 𝑄0 𝑄
=
+
𝑉𝐶𝑜 2 𝑄 𝑄𝑜
𝑉𝐶 =
𝑉𝐶𝑜 𝑄0 𝑄
1783 112.15
250
+
=
+
= £2388 a year
2 𝑄 𝑄𝑜
2
250
112.15
Worked Example 4:
Adjusting EOQ
Jessica Choi works in her bakery for 6 days a week for 49 weeks a
year. Flour is delivered directly with a charge of £7.50 for each
delivery. Jessica uses an average of 10 sacks of whole-grain flour a
day, for which she pays £12 a sack. She has an overdraft at the
bank which costs 12% a year, with spillage, storage, loss and
insurance costing 6.75% a year.
a) What size of delivery should Jessica use and what are the
resulting costs?
b) How much should she order if the flour has a shelf-life of 2
weeks?
c) How much should she order if the bank imposes a maximum
order value of £1,500?
d) If the mill only delivers on Mondays, how much Jessica order
and how often?
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Worked Example 4:
Adjusting EOQ
𝐷 = 10(6)(49) = 2940 sacks/yr
𝑈𝐶= £12
𝐻𝐶= (£12 )(12%+6.75%) = £2.25 /yr for a sack
𝑅𝐶= £7.50 per order
a) 𝑄𝑜 =
2×𝑅𝐶×𝐷
𝐻𝐶
=
2×7.50×2940
2.25
= 140
𝑉𝐶𝑜 = 𝐻𝐶 × 𝑄𝑜 = 2.25 × 140 = £315
b) Shelf-life = 2 weeks; Q  10(6)(2) = 120
𝑉𝐶 =
𝑉𝐶𝑜 𝑄0 𝑄
315 140 120
+
=
+
= £318.75 a year
2 𝑄 𝑄𝑜
2 120 140
Worked Example 4:
Adjusting EOQ
c) Max. order = £ 1500; Q  125
𝑉𝐶 =
𝑉𝐶𝑜 𝑄0 𝑄
315 140 125
+
=
+
= £317.03 a year
2 𝑄 𝑄𝑜
2 125 140
d) Weekly delivery; T = 1, 2, 3,… weeks
𝑉𝐶𝑜 𝑄0 𝑄
315
+
=
2
𝑄 𝑄𝑜
2
𝑉𝐶𝑜 𝑄0 𝑄
315
𝑇 = 3; 𝑉𝐶 =
+
=
2
𝑄 𝑄𝑜
2
𝑇 = 2; 𝑉𝐶 =
140 120
+
= £318.75 a year
120 140
140 180
+
= £325.00 a year
180 140
e) Consider all conditions simultaneously
Shelf-life = 2 weeks; Q  10(6)(2) = 120
Max. order = £ 1500; Q  125
Weekly delivery; Q = 60, 120, 180, …
Q = 120
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Orders for Discrete Items
• Suppose we calculate the optimal order size as Qo, which is
between the integers Q’−1 and Q’. We should round up the
order size if the variable cost of ordering Q’ units is less
than the variable cost of ordering Q’−1 units.
Orders for Discrete Items
For 𝑉𝐶(𝑄’) ≤ 𝑉𝐶(𝑄 ′ − 1),
𝑅𝐶 × 𝐷 HC × Q′
𝑅𝐶 × 𝐷 HC × (Q′ − 1)
+
≤ ′
+
𝑄′
2
𝑄 −1
2
multiplying both sides by 2×Q’×(Q’−1)
𝐻𝐶(𝑄 ′ )(𝑄 ′ − 1) ≤ 2(𝑅𝐶)(𝐷)
(𝑄 ′ )(𝑄′ − 1) ≤ 𝑄𝑜 2
1.
2.
3.
4.
Calculate the EOQ, 𝑄𝑜 .
Find the integers Q’ and Q’−1 that surround 𝑄𝑜 .
If Q’× (Q’− 1) is less than or equal to 𝑄𝑜 2 , order Q’.
If Q’× (Q’− 1) is greater than 𝑄𝑜 2 , order Q’− 1.
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Worked Example 5:
Orders for Discrete Items
Schlessinger Aeronautic work a 50-week year and stock an electric
motor with the following characteristics:
D = 20 a week
UC = £2,500 a unit
RC = £50
HC = £660 a unit a year
What is the optimal order quantity? Would it make much difference
if this number were rounded up or down to the nearest integer?
Worked Example 5:
Orders for Discrete Items
D = 20 a week
UC = £2,500 a unit
RC = £50
HC = £660 a unit a year
a) 𝑄𝑜 =
2×𝑅𝐶×𝐷
𝐻𝐶
=
2×50×(20×50)
660
= 12.31
Check if (𝑄′ )(𝑄′ − 1) ≤ 𝑄𝑜 2
12 13 > 12.312 ∴ 𝑄 = 12
b) VC 12 =
VC 13 =
𝑅𝐶×𝐷
𝑄
+
HC× Q
2
=
50×(20×50)
660×12
+ 2
12
= £8126.67
𝑅𝐶 × 𝐷 HC × Q′
50 × (20 × 50) 660 × 13
+
=
+
= £8136.15
𝑄
2
13
2
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Uncertainty in Demand and Costs:
Error in Demand
• Suppose that actual demand for an item is D, but there is a
proportional error in the forecasts, E. Then the forecast is
𝐷 × (1 + 𝐸) and instead of using the correct EOQ:
𝑄𝑜 =
• Since
2×𝑅𝐶×𝐷
, we used
𝐻𝐶
𝑉𝐶
1 𝑄
𝑄
= 2 𝑄0 + 𝑄
𝑉𝐶𝑜
𝑜
𝑉𝐶
1
=
𝑉𝐶𝑜 2
𝑄=
1
1+𝐸
2×𝑅𝐶×𝐷×(1+𝐸)
𝐻𝐶
+
1+𝐸
1
Uncertainty in Demand and Costs:
Error in Demand
Error in Cost
160.00%
140.00%
120.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
-150%
-100%
-50%
0%
50%
100%
150%
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Uncertainty in Demand and Costs:
Error in Costs
• Suppose, for example, that we approximate an actual
reorder cost of RC by RC× (1+ E1), and an actual holding
cost of HC by HC × (1+ E2):
𝑄=
𝑉𝐶
1 𝑄0
𝑄
• Since 𝑉𝐶 = 2
𝑜
2 × 𝑅𝐶 × (1 + 𝐸1 ) × 𝐷
𝐻𝐶 × (1 + 𝐸2 )
𝑄
+𝑄
𝑜
𝑉𝐶
1
=
𝑉𝐶𝑜 2
1 + 𝐸2
1 + 𝐸1
+
1 + 𝐸1
1 + 𝐸2
Reverse Calculation:
Estimating Implied Costs
𝑄0 =
2×𝑅𝐶×𝐷
𝐻𝐶
, 𝑅𝐶 =
𝑄0 2 ×𝐻𝐶
2×𝐷
• If this calculation is repeated, it might be possible to get a
reasonable overall estimate for the reorder cost.
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Worked Example 6:
Estimating implied costs
A company has a standing order of 40 units of an item every
month. What can you infer about the costs? If the reorder cost
is actually €160, what is the implied holding cost?
D = 40 a month
RC = €160
𝐻𝐶 =
2 × 𝑅𝐶 × 𝐷
𝑄0 2
=
2 × €160 × 40/𝑚𝑜
= €8.00/mo
402
Adjusting the Order Quantity
𝑄=𝑘×
𝑄=
2 × 𝑅𝐶 × 𝐷
𝐻𝐶
2 × 𝑅𝐶 × 𝐷
𝐻𝐶 × 𝑘
• Factor 𝑘 is introduced to make adjustments to the order
quantity.
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Adding a Finite Lead Time
Causes of lead time:
• Time for order preparation
• Time to get the order to the right place in suppliers
• Time at the supplier
• Time to get materials delivered from suppliers
• Time to process the delivery
Adding a Finite Lead Time:
Reorder Level
• When demand is constant, there is no benefit in carrying
stock from one cycle to the next, so each order should be
timed to arrive just as existing stock runs out.
• To achieve this, we have to place an order a time LT before
the delivery is needed.
• The easiest way of arranging this is to define a reorder level.
• The EOQ does not depend on lead time and remains
unchanged.
• As both demand and lead time are constant, the amount of
stock needed to cover the lead time is also constant at:
lead time × demand per unit time
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Adding a Finite Lead Time:
Reorder Level
Stock
level
Optimal
order size Q
Reorder
level
LT
𝑅𝑂𝐿 = 𝐷 × 𝐿𝑇
Place
order
Time
Receive
delivery
Worked Example 7:
Reorder Level w/Finite Lead Time
Carl Smith uses radiators at the rate of 100 a week, and he
has calculated an EOQ of 250 units. What is his best ordering
policy if lead time is: (a) one week? or (b) two weeks?
D = 100 a week
EOQ = 250
a) LT = 1wk; ROL = 𝐿𝑇 × 𝐷 = 1 × 100 = 100
b) LT = 2 wks; ROL = 𝐿𝑇 × 𝐷 = 2 × 100 = 200
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Adding a Finite Lead Time:
Longer Lead Time
• When the lead time is particularly long, there can be several
orders outstanding at any time.
• In particular, when the lead time is between n and n+1 cycle
lengths, giving:
𝑛 × 𝑇 < 𝐿𝑇 < 𝑛 + 1 𝑇
• There are n orders outstanding when it is time to place
another. Then we subtract 𝑛 × 𝑄𝑜 from the lead time
demand to get the reorder level:
𝑅𝑂𝐿 = 𝐿𝑇 × 𝐷 − 𝑛 × 𝑄𝑜
Stock Level with Longer Lead Time
Stock
level
Time
Place
order B
Place
order C
Receive
delivery B
Receive
delivery C
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Worked Example 8:
Longer Lead Time
Demand for an item is steady at 1,200 units a year with an ordering
cost of £16 and holding cost of £0.24 a unit a year. Describe an
appropriate ordering policy if the lead time is constant at (a) 3
months; (b) 9 months; or (c) 18 months.
𝐷 = 1200 units/yr; 𝐻𝐶= £0.24 /yr for a unit; 𝑅𝐶= £16 per order
2 × 𝑅𝐶 × 𝐷
2 × 16 × 1200
=
= 400
𝐻𝐶
0.24
𝑄𝑜
400
𝑇𝑜 =
=
= 0.3333 𝑦𝑟. = 4 𝑚𝑜.
𝐷
1200
𝑄𝑜 =
𝐿𝑇
𝑇𝑜
𝐿𝑇
9 mo.; 𝑛 = 𝑟𝑜𝑢𝑛𝑑𝑑𝑜𝑤𝑛
𝑇𝑜
𝐿𝑇
18 mo.; 𝑛 = 𝑟𝑜𝑢𝑛𝑑𝑑𝑜𝑤𝑛
𝑇𝑜
a)
LT = 3 mo.; 𝑛 = 𝑟𝑜𝑢𝑛𝑑𝑑𝑜𝑤𝑛
= 0; 𝑅𝑂𝐿 = 𝐿𝑇 × 𝐷 − 𝑛 × 𝑄0 = 300
b)
LT =
= 2; 𝑅𝑂𝐿 = 𝐿𝑇 × 𝐷 − 𝑛 × 𝑄0 = 100
c)
LT =
= 4; 𝑅𝑂𝐿 = 𝐿𝑇 × 𝐷 − 𝑛 × 𝑄0 = 200
Practical Points
• Two-bin system: Bin B contains an amount equal to the
reorder level, and all remaining stock in Bin A. Stock is used
from Bin A until it is empty – Time to place an order.
• Three-bin system: Third bin holds a reserve only for
emergency.
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