[231005] - 02. Basic EOQ & EPQ.docx

IE3215
SISTEM PERSEDIAAN
(2 sks)
TAUFIQ RAHMAN, S.T., M.T
Institut Teknologi Batam Industrial Engineering

Economic Order Quantity (EOQ)
The EOQ calculation is the most important analysis of
inventory control, and arguably one of the most important
results derived in any area of operations management. The
ﬁrst reference to the work is by Harris (1915), but the
calculation is often credited to Wilson (1934) who
independently duplicated the work and marketed the results.

Basic EOQ model assumptions:
֍ 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.
A number of other assumptions are implicit in the model,
including:
֍ 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.

Variables used in Basic EOQ Model
 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,
so a holding cost might be, say, £10 a unit a year.
 Order quantity (Q) which is the ﬁxed order size that we always use. The
purpose of this analysis is to ﬁnd an optimal value for this order quantity.
 Cycle time (T) which is the time between two consecutive
replenishments. This depends on the order quantity, with larger orders
leading to longer cycle times.
 Demand (D) which sets the number of units to be supplied from stock in a
given time period (for example, ten units a week). Here, we assume that
the demand is continuous and constant.

Derivation of the economic order quantity
This derivation uses a standard approach that is suitable for many stock control models.
It has three steps, as follows:
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.

Optimal Order Size
amount entering stock in cycle = amount leaving stock in cycle
𝑸 = 𝑫 𝐱 𝑻
1) Total Cost per Cycle
total cost
per cycle
=
unit cost
component
+
reoredr cost
component
+
holding cost
component
unit cost component = unit cost (UC) × number of units ordered (Q)
= 𝑼𝑪 𝐱 𝑸
reorder cost component = reorder cost (RC) × number of orders placed (1)
= 𝑹𝑪
holding cost component = holding cost (HC) × average stock level (Q/2) x time held (T)
𝑯𝑪 𝐱 𝑸 𝐱 𝑻
=
𝟐
𝒕𝒐𝒕𝒂𝒍 𝒄𝒐𝒔𝒕
= 𝑼𝑪 𝐱 𝑸 + 𝑹𝑪 +
𝒑𝒆𝒓 𝒄𝒚𝒄𝒍𝒆
𝟐

2) Total Cost per Unit Time
𝟐
3) Minimize Cost per Unit Time
The third step of our analysis ﬁnds the minimum cost per unit time.
For this we differentiate the equation for TC with respect to Q and
set the result equal to zero:
The second step divides this cost by the cycle length, T,
to give a total cost per unit time, TC: 𝑻𝑪 = 𝑼𝑪 𝐱 𝑫 +
𝑹𝑪 𝐱 𝑫
𝑸
+
𝑯𝑪 𝐱 𝑸
𝟐
𝑻𝑪 =
𝑼𝑪 𝐱 𝑸
𝑻
+
𝑹𝑪
𝑻
+
𝟐 𝑻
𝒅(𝑻𝑪)
𝒅𝑸
= −
𝑹𝑪 𝐱 𝑫
𝑸𝟐 +
𝑯𝑪
𝟐
= 𝟎
Since, 𝑸 = 𝑫 𝐱 𝑻 𝑸𝟐 =
𝟐 𝐱 𝑹𝑪 𝐱 𝑫
𝑯𝑪
𝑼𝑪 𝐱 𝑫 𝐱 𝑻
𝑻𝑪 =
𝑹𝑪 𝐱 𝑫 𝑯𝑪 𝐱 𝑸
+ + Economic Order Quantity (Qo) Optimal Cycle Length (To)
𝑻 𝑸 𝟐
𝑸 = 𝑻 =
𝑸
𝒐
𝟐 𝐱 𝑹𝑪
=
𝑹𝑪 𝐱 𝑫
𝑻𝑪 = 𝑼𝑪 𝐱 𝑫 +
𝑸
𝑯𝑪 𝐱 𝑸
+
𝟐
𝒐 𝑯𝑪 𝟎
𝑫 𝑫 𝐱 𝑯𝑪
Since, 𝑸 = 𝑫 𝐱 𝑻

We can also find the optimal cost per unit time, TCo,
by substituting the value for Qo. We know that:
If you compare this to the economic order quantity,
you can see that:
𝑹𝑪 𝐱 𝑫
𝑸
𝑯𝑪 𝐱 𝑸
+
𝟐
𝑽𝑪𝒐 = 𝟐 𝐱 𝑹𝑪 𝐱 𝑫 𝐱 𝑯𝑪 𝑸
𝒐
=
𝑯𝑪
The unit cost component is fixed, so we can concentrate
on the last two terms which form the variable cost (VC).
Optimal Variable Cost per Unit Time
Then:
𝑽𝑪 =
𝑹𝑪 𝐱 𝑫
𝑸
+
𝑯𝑪 𝐱 𝑸
𝟐
𝑽𝑪𝒐 = 𝑯𝑪 𝐱 𝑸𝒐
Substituting for Qo to give the optimal value VCo: Then the optimal total cost per unit time is the sum
of this variable cost and the fixed cost:
𝑽𝑪𝒐 = 𝑹𝑪 𝐱 𝑫 𝐱
𝑯𝑪
+
𝑯𝑪
𝟐
𝐱
𝑯𝑪 Optimal Cost per Unit Time
𝑽𝑪𝒐 =
𝑹𝑪 𝐱 𝑫 𝐱 𝑯𝑪
𝟐
+
𝑹𝑪 𝐱 𝑫 𝐱 𝑯
𝑪
𝟐
T𝑪𝒐 = 𝑼𝑪 𝐱 𝑫 𝐱 𝑽𝑪𝒐
An interesting point in the equation which clearly shows that for
𝑽𝑪𝒐 = 𝟐 𝐱 𝑹𝑪 𝐱 𝑫 𝐱 𝑯𝑪 the economic order quantity the reorder cost component equals the
holding cost component (both have the value√RC × HC × D/2)

𝑹𝑪 𝐱 𝑫 𝑯𝑪 𝐱 𝑸
𝑸
+
𝟐
= 𝟑𝟎 𝐱 𝟔, 𝟎𝟎𝟎 +
𝟏𝟐𝟓 𝐱 𝟔, 𝟎𝟎𝟎 𝟔 𝐱 𝟓𝟎𝟎
𝟓𝟎𝟎
+
𝟐
= 𝟏𝟖𝟑, 𝟎𝟎𝟎 𝒂 𝒚𝒆𝒂𝒓
Jaydeep (Trading) 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?
Solution:
Listing the values we know in consistent units: The optimal time between orders is:
Demand = D = 6,000 units a year
Unit cost = UC = $30 a unit
Reorder cost = RC = $125 an order
𝑻𝟎 =
𝑸𝒐
=
𝑫
500
6,000
=
0.083 years
(1 month)
Holding cost = HC = $6 a unit a year
Substituting these ﬁgures into the economic
order quantity equation gives:
The associated variable cost is:
𝑽𝑪𝒐 = 𝑯𝑪 𝐱 𝑸𝒐
= 𝟔 𝐱 𝟓𝟎𝟎 = $𝟑, 𝟎𝟎𝟎 𝐚 𝐲𝐞𝐚𝐫
This gives a total cost of:
𝑸𝒐 =
𝑸𝒐 =
𝑯𝑪
𝟐 𝐱 𝟏𝟐𝟓 𝐱 𝟔, 𝟎𝟎𝟎
𝟔
T𝑪𝒐 = 𝑼𝑪 𝐱 𝑫 𝐱 𝑽𝑪𝒐
= 𝟑𝟎 𝐱 𝟔, 𝟎𝟎𝟎 𝐱 𝟑, 𝟎𝟎𝟎
= 𝟏𝟖𝟑, 𝟎𝟎𝟎 𝐚 𝐲𝐞𝐚𝐫
𝑸𝒐 = 𝟓𝟎𝟎 𝐮𝐧𝐢𝐭𝐬

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′ −
That is:
𝑹𝑪 𝐱 𝑫
𝑸′ +
𝑯𝑪 𝐱 𝑸′ 𝑹𝑪 𝐱 𝑫 𝑯𝑪 𝐱 (𝑸′ − 𝟏)
≤ +
𝟐 (𝑸′−𝟏) 𝟐
𝟏 units. We can simplify by multiplying both sides by 𝟐 𝐱 𝑸′ 𝐱 (𝑸′−𝟏)
𝟐 𝐱 𝑸′ 𝐱 (𝑸′−𝟏)
𝑹𝑪 𝐱 𝑫
𝑸′
𝑯𝑪 𝐱 𝑸′
+
𝟐
≤ 𝟐 𝐱 𝑸′ 𝐱 (𝑸′−𝟏)
𝑹𝑪 𝐱 𝑫
(𝑸′−𝟏)
𝑯𝑪 𝐱 (𝑸′ − 𝟏)
+
𝟐
𝑸′𝟐 𝐱 (𝑸′−𝟏) 𝐱 𝑯𝑪 − 𝑸′𝐱 (𝑸′−𝟏)𝟐 𝐱 𝑯𝑪 ≤ 𝟐 𝐱 𝑸′ 𝐱 𝑹𝑪 𝐱 𝑫 − 𝟐 𝐱 𝑸′ 𝐱 𝑹𝑪 𝐱 𝑫 𝐱 (𝑸′−𝟏)
(𝑸′𝟑−𝑸′𝟐) 𝐱 𝐇𝐂 − (𝑸′𝟑 − 𝟐𝑸′𝟐 + 𝑸′) 𝐱 𝐇𝐂 ≤ 𝟐 𝐱 𝑸′ 𝐱 𝑹𝑪 𝐱 𝑫 − 𝟐 𝐱 𝑸′ 𝐱 𝑹𝑪 𝐱 𝑫 + 𝟐 𝐱 𝑹𝑪 𝐱 𝑫
This suggests a procedure for checking whether it is better to
round up or round down discrete order quantities:
1. Calculate the EOQ, Qo.
2. Find the integers 𝑸′ − 𝟏 and 𝑸′ that surround Qo.
3. If 𝑸′ 𝐱 (𝑸′−𝟏) is less than or equal to Qo2 , order 𝑸′.
4. If 𝑸′ 𝐱 (𝑸′−𝟏) is greater than Qo2 , order 𝑸′ − 𝟏 .
𝑸′𝟐 𝐱 𝐇𝐂 − 𝑸′ 𝐱 𝐇𝐂 ≤ 𝟐 𝐱 𝑹𝑪 𝐱 𝑫
𝐇𝐂 𝐱 (𝑸′ 𝑸′ − 𝟏 ) ≤ 𝟐 𝐱 𝑹𝑪 𝐱 𝑫
𝑸′ 𝐱 𝑸′ − 𝟏 ≤
𝑯𝑪
𝑸′ 𝐱 𝑸′ − 𝟏 ≤ 𝑸𝒐𝟐
Since,
𝑸𝒐 =
𝑯𝑪

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?
Solution:
We can start by substituting the values to ﬁnd an EOQ:
The company obviously cannot order 12.31 motors, so its alternatives
are to order 12 or 13. Here 𝑸′ equals 13 and the rule we developed
above suggests ordering 13 when:
𝑸′ 𝐱 𝑸′ − 𝟏 ≤ 𝑸𝒐𝟐
𝟏𝟑 𝐱 𝟏𝟐 ≤ 𝟏𝟐. 𝟑𝟏𝟐 ⇒ 𝟏𝟓𝟔 ≤ 151.54
This is not true so the best policy is to order motors in batches of 12.
We can check this decision by calculating the variable cost of ordering
in bathces of 12 or 13:
Order size 12 motors
𝑸𝒐 =
𝑯𝑪
𝑽𝒄 =
𝑹𝑪 𝐱 𝑫
𝑸
𝑯𝑪 𝐱 𝑸
+
𝟐
𝟓𝟎 𝒙 𝟏, 𝟎
𝟎
𝟎
=
𝟏𝟐
𝟔𝟔𝟎 𝒙 𝟏
𝟐
+
𝟐
= £ 8,126.67
Order size 13 motors
𝑸𝒐
𝟐 𝐱 𝟓𝟎 𝐱 𝟐𝟎 𝐱 𝟓𝟎
=
𝟔𝟔𝟎
𝑹𝑪 𝐱 𝑫
𝑽𝒄 =
𝑸
𝑯𝑪 𝐱 𝑸
+
𝟐
𝟓𝟎 𝒙 𝟏, 𝟎
𝟎
𝟎
=
𝟏𝟑
𝟔𝟔𝟎 𝒙 𝟏
𝟑
+
𝟐
= £ 8,136.15
𝑸𝒐 = 𝟏𝟐. 𝟑𝟏 𝐮𝐧𝐢𝐭𝐬

Economic Production/Manufacturing Quantity
Q
Economic Production Quantity (EPQ) / Economic
Manufacturing Quantity (EMQ) / Optimal Batch Size
is an inventory model that replenish the stock from
production. The EPQ model was presented by Taft on
May 1918. The EPQ model is fairly similar to the
EOQ model, but it is applied to manufacturing. This
model is used when: B
0
 inventory builds up over a period of time after an
order is placed;
 units are produced and sold simultaneously.
tp L
Time
t1
Quantity
(Q)

Basic EPQ model assumptions:
The assumptions of the EPQ model are similar to those of the EOQ
model, except that instead of orders received in a single delivery,
units are received incrementally during production. The assumptions
are:
֍ Only one product is involved
֍ Annual demand is known
֍ The usage rate is constant
֍ Usage occurs continually, but production occurs periodically
֍ The production rate is constant when production is occurring
֍ Lead time is known and constant
֍ There are no quantity discounts

Parameters:
Q = Number of Units made (Unit)
A = Maximum stock level (Unit)
P = Production rate (Unit/Time)
D = Demand rate (Unit/ Time)
PT = Production time (Time)
DT = Demand Time (Time)
T = Holding Time (Time)
RC = Reorder Cost ($)
HC = Holding Cost ($)
UC = Unit Cost ($)

Looking at the productive part of
the cycle we have:
We also know that total
production during the period is:
Substituting for PT into the
equation for A gives:
𝑨 = 𝑷 − 𝑫 𝐱 𝑷𝑻 𝑸 = 𝑷 𝐱 𝑷𝑻 𝒐𝒓 𝑷𝑻 =
𝑸
𝑷 𝑨 = 𝑸 𝐱
(𝑷 − 𝑫)
𝑷
1) Total Cost per Cycle
total cost
per cycle
=
unit cost
component
+
reoredr cost
component
+
holding cost
component
unit cost component = unit cost (UC) × number of units made (Q)
= 𝑼𝑪 𝐱 𝑸
reorder cost component = reorder cost (RC) × number of production set-ups (1)
= 𝑹𝑪
holding cost component = holding cost (HC) × average stock level (A/2) x time held (T)
𝑯𝑪 𝐱 𝑨 𝐱 𝑻
=
𝟐
=
𝟐
𝐱
(𝑷 − 𝑫)
𝑷
𝟐
𝐱
(𝑷 − 𝑫)
𝑷

2) Total Cost per Unit Time
𝟐
𝐱
(𝑷 − 𝑫)
𝑷
3) Minimize Cost per Unit Time
The third step of our analysis ﬁnds the minimum cost per unit time.
For this we differentiate the equation for TC with respect to Q and
set the result equal to zero:
The second step divides this cost by the cycle length, T,
to give a total cost per unit time, TC:
𝑹𝑪 𝐱 𝑫
𝑸
𝑯𝑪 𝐱 𝑸
+
𝟐
(𝑷 − 𝑫)
𝐱
𝑷
𝑻𝑪 =
𝑼𝑪 𝐱 𝑸
𝑻
+
𝑹𝑪
𝑻
+
𝟐 𝑻
𝐱
(𝑷 − 𝑫)
𝑷
𝒅(𝑻𝑪)
𝒅𝑸
= −
𝑹𝑪 𝐱 𝑫
𝑸𝟐 +
𝑯𝑪
𝟐
𝐱
𝑷 − 𝑫
𝑷
= 𝟎
Since, 𝑸 = 𝑫 𝐱 𝑻 𝑸𝟐 =
𝑯𝑪
𝐱
𝑷 − 𝑫
𝑷
𝑻𝑪 =
𝑼𝑪 𝐱 𝑫 𝐱 𝑻
𝑻
+
𝑹𝑪 𝐱 𝑫
𝑸
+
𝑯𝑪 𝐱 𝑸
𝟐
𝐱
(𝑷 − 𝑫)
𝑷 Optimal Batch Size (Qo)
𝑹𝑪 𝐱 𝑫
𝑸
𝑯𝑪 𝐱 𝑸
+
𝟐
(𝑷 − 𝑫)
𝐱
𝑷
𝑸𝒐 =
𝑯𝑪
𝐱
𝑷
𝑷 − 𝑫

Comparison of ﬁnite and instantaneous replenishment
Finite replenishment rate
Instantaneous replenishment
(for the economic order quantity)
Order Quantity
𝑸
𝒐
=
𝑯𝑪
𝐱
𝑷
𝑷−𝑫
𝑸𝒐 =
𝑯𝑪
Cycle Length
𝑻
𝒐
=
𝟐 𝐱 𝑹𝑪
𝑯𝑪 𝐱 𝑫
𝐱
𝑷
𝑷−𝑫
𝑻𝒐 =
𝟐 𝐱 𝑹𝑪
𝑯𝑪 𝐱 𝑫
Variable Cost
𝑽𝑪𝒐
= 𝟐 𝐱 𝑹𝑪 𝐱 𝐇𝑪 𝐱 𝑫 𝐱
𝑷−𝑫
𝑷
𝑽𝑪𝒐 = 𝟐 𝐱 𝑹𝑪 𝐱 𝐇𝑪 𝐱 𝑫
Total Cost 𝑻𝑪𝒐 = 𝑼𝑪 𝐱 𝑫 + 𝑽𝑪𝒐 𝑻𝑪𝒐 = 𝑼𝑪 𝐱 𝑫 + 𝑽𝑪𝒐
Production Time 𝑷𝑻𝒐 =
𝑸𝒐
𝑷

Demand for an item is constant at 1,800 units a year. The item can be made at a constant rate of 3,500 units a year. Unit cost is £50, batch set-
up cost is £650, and holding cost is 30 per cent of value a year. What is the optimal batch size for the item?
Solution:
Listing the variables we know: The optimal production time, Pto, is: The optimal total cost per unit time, TCo, is
D = 1,800 units a year 𝑸𝒐 566.7 0.16 years
P = 3,500 units a year
UC = £50 a unit
𝑷𝑻𝟎 = 𝑷
=
3,500
=
(8.4 weeks) 𝑻𝑪𝒐 = 𝑼𝑪 𝐱 𝑫 + 𝑽𝑪𝒐
RC = £650 a batch The Optimal Cycle Length, To, is: 𝑻𝑪𝒐 = 𝟓𝟎 𝐱 𝟏, 𝟖𝟎𝟎 + 𝟒, 𝟏𝟐𝟗
HC = 0.3 × 50 = £15 a unit a year
Substituting these values gives an optimal batch
size, Qo, of:
𝑻𝒐 =
𝟐 𝐱 𝑹𝑪
𝐱
𝑯𝑪 𝐱 𝑫
𝟐 𝐱 𝟔
𝟓
𝟎
𝑷
𝑷 − 𝑫
𝟑, 𝟓𝟎𝟎
𝑻𝑪𝒐 = £94,129 a year
0.31 years
𝑸𝒐 =
𝐱
𝑯𝑪
𝑷
𝑷 − 𝑫
𝑻𝒐 = 𝐱
𝟏𝟓 𝐱 𝟏, 𝟖𝟎𝟎 𝟑,𝟓𝟎𝟎 − 𝟏, 𝟖𝟎𝟎
=
(16.4 weeks)
The Optimal variable Cost, VCo, is:
𝑸𝒐
𝟐 𝐱 𝟔𝟓𝟎 𝐱 𝟏, 𝟖
𝟎
𝟎
=
𝟏𝟓
𝟑, 𝟓𝟎𝟎
𝐱
𝟑, 𝟓𝟎𝟎 − 𝟏, 𝟖𝟎𝟎
𝑽𝑪𝒐
𝑷 − 𝑫
= 𝟐 𝐱 𝑹𝑪 𝐱 𝐇𝑪 𝐱 𝑫 𝐱
𝑷
𝟑, 𝟓𝟎𝟎 − 𝟏, 𝟖𝟎𝟎
𝑸𝒐 = 𝟓𝟔𝟔. 𝟕 𝐮𝐧𝐢𝐭𝐬 𝑽𝑪𝒐 = 𝟐 𝒙 𝟔𝟓𝟎 𝒙 𝟏𝟓 𝒙 𝟏, 𝟖𝟎𝟎 𝒙 = £4,129 a year
𝟑, 𝟓𝟎𝟎

Excercise
[1] ITEBA Corporation has got a demand for particular part at
12,000 units per year. The cost per unit is $ 4 and it costs $ 36 to
place an order and to process the delivery. The inventory carrying
cost is estimated at 9 percent of average inventory investment.
Determine
(i) Economic order quantity.
(ii) Optimum number of orders to be placed per annum.
(iii) Minimum total cost of inventory per annum.
[2] ITEBA Company produces a cable at the rate of 4000 metres
per hour. The cable is used at the rate of 2500 metres/hour. The
cost of the cable is $ 6 per metre. The inventory carrying cost is 25
percent and set-up costs are $ 4100 per set-up. Determine the
optimal ‘number of cycles required in a year for the manufacture
of this cable.
[3] An automobile manufacturing company is purchasing an
item from outside suppliers. Demand is 10,000 units per annum.
Cost of the item is $ 8 per unit and procurement cost is estimated
to be $ 120 per order. Cost of carrying inventory is 25 percent. If
the consumption rate is constant determine EOQ.
In the above problem, if the company decides to manufacture the
above item with an equipment which produce 50 units per day.
The cost of units thus produced is $ 3.5 per unit setup cost is
$ 150. How your answer is changed in the second case.

Excercise
[4] ITEBA Corporation currently practices the following system
for the procurement of an item.
No. of orders placed in a year = 8, Ordering cost = $ 800/order
Each time order quantity = 300, Carrying cost = 40 percent
Comment on the ordering policy of the company and estimate the
loss to the company in not practising scientific inventory policy
[5] A contractor undertakes to supply diesel engines to a truck
manufacturer at the rate of 30 per day. He finds that the cost of
holding a completed engine in stock is $ 18 per month. Production
of engines is in batches and each time a new batch is started, there
are set-up costs of $ 10,000. How frequently should the batches be
started and what will be the minimum average inventory cost and
production time if production rate is 40 engines/day. Assume 300
working days in a year.
[6] A company consumes 14000 units of a particular item.
The company has a production capacity of 75 units/day. The
cost of each unit produced by the company is $ 8. The setup
and tooling up cost is $ 96 per setup. The carrying charges are 20
percent of cost per unit. Assume 300 working days per annum.
Determine
(i) Economic quantity to be manufactured in each batch.
(ii) How frequently should the production runs be made?
(iii) Determine the production period.
If Company consider to purchase the item, with Ordering Cost
$ 85 per order. Which decision will company choose? Purchase or
manufacture?

[231005] - 02. Basic EOQ & EPQ.docx

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