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7/28/2019 D08540000120114004Session 4_Inventory Control Systems
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Session 03
Inventory Control SystemsEconomic Order Quantity and its variation.
D 0 8 5 4Supply Chain : Manufacturing and Warehousing
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Bina Nusantara University
2
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INVENTORY CONTROL SYSTEMS
The fundamental inventory problem can be succinctlydescribed by two questions :1. When should an order be placed ?
2. How much should be ordered ?
The complexity of the resulting model depends upon theassumptions one makes about the various parametersof the system.
The major distinction is between
a. Inventory Control Subject to Known Demand
b. Inventory Control Subject to Unknown Demand
Bina Nusantara University
3Source : Production and Operations Analysis 4th
Edition, Steven NahmiasMcGraw Hill International Edition
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Econom ic Order Quant i ty and i ts var iat ion
The EOQ Model ( Economic Order Quantity Model )
is the simplest and most fundamental of all inventorymodels.
It describes the most important trade-off between
Fixed Order Costs and Holding Costs.
And is the basis for the analysis of more complex systems.
Bina Nusantara University
4Source : Production and Operations Analysis 4th
Edition, Steven NahmiasMcGraw Hill International Edition
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Order Quantity
Annual Cost
Order (Setup) Cost Curve
Optimal
Order Quantity (Q*)Bina Nusantara University
5
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Notation
D= Demand rate (in units per year).
c= Unit production cost, not counting setup or inventory
costs (in dollars per unit). A = Constant setup (ordering) cost to produce
(purchase) a lot (in dollars).
h= Holding cost
Q= Lot size (in units); this is the decision variable
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The model
Inventory versus time in the EOQ model
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The model
Average inventory level:
The holding cost per unit:
The setup costper unit:
The production cost per unit:
2
Q
D
hQ
D
hQ
2
2
Q
A
c
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Economic order quantity
)(2
02
)(2
quantityordereconomichADQ
conditionorderf irstQ
A
D
h
dQ
QdY
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What-if
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What-if
EOQ EOQ
Annual demand 12,000 12,000
Cost per unit $6.75 $6.75Interest rate to hold 20% 20%
Ordering cost $28.00 $28.00
Quantity each order 461 =INT(C5/C10)
Number of orders 26 26
Unit holding cost $1.35 =C6*C7
Annual holding cost $311 =C9*C11/2Annual ordering cost $728 =C10*C8
Combined cost $1,039 =C12+C13
Annual purchase cost $81,000 =C5*C6
Total cost $82,039 =C14+C15
What-If AnalysisThe minimum costobtained by using theeconomic orderquantity is $952.50, soincreasing the orderquantity by 10% leadsa total cost increase ofonly $4.30. Changingthe order quantity by a
small amount has verylittle effect on the cost,because EOQ formulagives robust solutions.
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Inventory Systems
Single-Period Inventory Model
One time purchasing decision (Example: vendor sellingt-shirts at a football game)
Seeks to balance the costs of inventory overstock andunder stock
Multi-Period Inventory Models
Fixed-Order Quantity Models
Event triggered (Example: running out of stock) Fixed-Time Period Models
Time triggered (Example: Monthly sales call by salesrepresentative)
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Single-Period Inventory Model
uo
u
CC
CP
soldbeunit willy that theProbabilit
estimatedunderdemandofunitperCostC
estimatedoverdemandofunitperCostC
:Where
u
o
P
This model states that we should
continue to increase the size of the
inventory so long as the probability
of selling the last unit added is
equal to or greater than the ratio
of: Cu/Co+Cu
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Single Period Model Example
Our college basketball team is playing in a tournamentgame this weekend. Based on our past experience wesell on average 2,400 shirts with a standard deviation of
350. We make $10 on every shirt we sell at the game,but lose $5 on every shirt not sold. How many shirtsshould we make for the game?
Cu=$10 and Co= $5; P $10 / ($10 + $5) = .667
Z.667 = .432 (use NORMSDIST(.667)
therefore we need 2,400 + .432(350) = 2,551 shirts
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Multi-Period Models:Fixed-Order Quantity Model Model Assumptions (Part 1)
Demand for the product is constant and uniformthroughout the period
Lead time (time from ordering to receipt) is
constant
Price per unit of product is constant
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Multi-Period Models:Fixed-Order Quantity Model Model Assumptions (Part 2)
Inventory holding cost is based on average inventory
Ordering or setup costs are constant
All demands for the product will be satisfied (No backorders are allowed)
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Basic Fixed-Order Quantity Model and Reorder Point Behavior
R = Reorder pointQ = Economic order quantity
L = Lead time
L L
Q QQ
R
Time
Numberof unitson hand
1. You receive an order quantity Q.
2. Your start using
them up over time. 3. When you reach down to
a level of inventory of R,
you place your next Q
sized order.
4. The cycle then repeats.
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Basic Fixed-Order Quantity (EOQ) Model Formula
H2
Q+SQ
D+DC=TC
TotalAnnual =Cost
AnnualPurchase
Cost
AnnualOrdering
Cost
AnnualHolding
Cost+ +
TC=Total annual
cost
D =Demand
C =Cost per unit
Q =Order quantityS =Cost of placing
an order or setup
cost
R =Reorder point
L =Lead time
H=Annual holding
and storage cost
per unit of inventory
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Deriving the EOQ
Using calculus, we take the first derivative of the total cost function withrespect to Q, and set the derivative (slope) equal to zero, solving for theoptimized (cost minimized) value of Qopt
Q =2DS
H=
2(Annual Demand)(Order or Setup Cost)
Annual Holding CostOPT
R eorder point, R = d L_
d = average daily demand (constant)
L = Lead time (constant)
_
We also need areorder point to tell uswhen to place anorder
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EOQ Example (1) Problem Data
Annual Demand = 1,000 unitsDays per year considered in average
daily demand = 365
Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15
Given the information below, what are the EOQ andreorder point?
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EOQ Example (1) Solution
Q =2D S
H=
2(1,000 )(10)
2.50= 89.443 units or OPT 90 u nits
d =1,000 units / year
365 days / year= 2.74 units / day
R eo rder p oin t, R = d L = 2 .7 4u nits / d ay (7d ays) = 1 9.1 8 or _
20 un its
In summary, you place an optimal order of 90 units. In thecourse of using the units to meet demand, when you only have20 units left, place the next order of 90 units.
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EOQ Example (2) Solution
Q =2D S
H=
2(10,000 ) (10)
1.50= 3 6 5 .1 4 8 un its , o r O P T 366 u ni ts
d =10,000 units / year
365 days / year= 27.397 units / day
R = d L = 27.397 units / day (10 days) = 273.97 or
_
274 u nits
Place an order for 366 units. When in the course of using theinventory you are left with only 274 units, place the next order of 366units.
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Multi-Period Models: Fixed-Time Period Model:Determining the Value ofT+L
T+L di 1
T+L
d
T+L d
2
=
Since each day is independent and is constant,
= (T + L)
i
2
The standard deviation of a sequence of randomevents equals the square root of the sum of thevariances
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Example of the Fixed-Time Period Model
Average daily demand for a product is 20
units. The review period is 30 days, and lead
time is 10 days. Management has set a policy
of satisfying 96 percent of demand from items
in stock. At the beginning of the review period
there are 200 units in inventory. The daily
demand standard deviation is 4 units.
Given the information below, how many units should be
ordered?
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Example of the Fixed-Time Period Model: Solution (Part 1)
T+ L d2 2= (T + L) = 30 + 10 4 = 25.298
The value for z is found by using the Excel NORMSINV
function, or as we will do here, using Appendix D. Byadding 0.5 to all the values in Appendix D and finding the
value in the table that comes closest to the service
probability, the z value can be read by adding the column
heading label to the row label.So, by adding 0.5 to the value from Appendix D of 0.4599,
we have a probability of 0.9599, which is given by a z = 1.75
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Example of the Fixed-Time Period Model: Solution (Part 2)
or644.272,=200-44.272800=q
200-298)(1.75)(25.+10)+20(30=q
I-Z+L)+(Td=q L+T
units645
So, to satisfy 96 percent of the demand, you should
place an order of 645 units at this review period
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Price-Break Model Formula
CostHoldingAnnual
Cost)SetuporderDemand)(Or2(Annual
=iC
2DS
=QOPT
Based on the same assumptions as the EOQ model, the price-break model has a similar Qopt formula:
i = percentage of unit cost attributed to carrying inventoryC = cost per unit
Since C changes for each price-break, the formula above willhave to be used with each price-break cost value
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Price-Break Example Problem Data(Part 1)
A company has a chance to reduce their inventory ordering costs by placing
larger quantity orders using the price-break order quantity schedule below.What should their optimal order quantity be if this company purchases this
single inventory item with an e-mail ordering cost of $4, a carrying cost rate
of 2% of the inventory cost of the item, and an annual demand of 10,000
units?
Order Quantity(units) Price/unit($)0 to 2,499 $1.202,500 to 3,999 1.00
4,000 or more .98
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Price-Break Example Solution (Part 2)
units1,826=0.02(1.20)
4)2(10,000)(=
iC
2DS=QOPT
Annual Demand (D)= 10,000 unitsCost to place an order (S)= $4
First, plug data into formula for each price-break value of C
units2,000=0.02(1.00)
4)2(10,000)(=iC
2DS=QOP T
units2,020=0.02(0.98)
4)2(10,000)(=
iC
2DS=QOP T
Carrying cost % of total cost (i)= 2%Cost per unit (C) = $1.20, $1.00, $0.98
Interval from 0 to 2499, theQopt value is feasible
Interval from 2500-3999, theQopt value is not feasible
Interval from 4000 & more, theQopt value is not feasible
Next, determine if the computed Qopt
values are feasible or not
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Price-Break Example Solution (Part 3)
Since the feasible solution occurred in the first price-break, itmeans that all the other true Qopt values occur at the beginnings ofeach price-break interval. Why?
0 1826 2500 4000 Order Quantity
Totalannualcosts So the candidates
for the price-
breaks are 1826,2500, and 4000units
Because the total annual cost function isa u shaped function
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Maximum Inventory Level, M
Miscellaneous Systems:
Optional Replenishment System
MActual Inventory Level, I
q = M - I
I
Q = minimum acceptable order quantity
If q > Q, order q, otherwise do not order any.
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Miscellaneous Systems:
Bin SystemsTwo-Bin System
Full Empty
Order One Bin of
Inventory
One-Bin System
Periodic Check
Order Enough toRefill Bin
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ABC Classification System Items kept in inventory are not of equal importance in
terms of:
dollars invested
profit potential
sales or usage volume
stock-out penalties
0
30
60
30
60
A BC
% of$ Value
% ofUse
So, identify inventory items based on percentage of total dollar value,where A items are roughly top 15 %, B items as next 35 %, and
the lower 65% are the C items
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Inventory Accuracy and Cycle Counting
Inventory accuracy refers to how well theinventory records agree with physical count
Cycle Counting is a physical inventory-takingtechnique in which inventory is counted on afrequent basis rather than once or twice a year
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