d08540000120114004session 4_inventory control systems

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

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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


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.


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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