Inventory II

8/9/2019 Inventory II

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Inventory DecisionMaking

Economic Order Quantity Models Two basic approaches to the reorder decisions are the Q-

system and the P-system. With certainty demand and leadtimes, they yield the same policies. With uncertain demandand lead times, there are significant differences. In practice,these systems are often modified. We will concentrate onlyon the Q and P systems.

1. Q-System (Fixed Order Quantity) when stock falls to apredetermined level (reorder point), an order is placed for a

fixed quantity of the good. This Q-system entails highermonitoring costs than the P-system but often lowers thecarrying costs.

on hand inventory

timeReorder PointApril 1st April 10th April 17th

Q is fixed


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2. P-System (Fixed order Interval, variable orderquantity)

a. Inventory reviewed at preset times (say, once amonth) and an order is placed for the differencebetween a predetermined maximum inventory level andthe actual amount on hand and on order from previousreviews.

b. The P-System has higher carrying and stockout costsbut lower monitoring costs than fixed order quantity Q-system.

c. Allows coordination of multiple purchases to takeadvantage of quantity discounts and better schedulingand work patterns in warehouse.

April 1st May 1st June 1st July 1st

time

Fixed order-quantity models

1. Economic order quantity(EOQ)

2. Production order quantity(POQ)

3. Quantity discount Probabilistic models

Fixed order-period models

How much and

when to order?

Inventory Models


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Known and constant demand

Known and constant lead time

Instantaneous receipt of material

No quantity discounts

Only order cost and holding cost

No stockouts

EOQ Assumptions

More units must be stored if more are ordered

Purchase Order Purchase Order

Description Qty.

Microwave 1000

Order quantity

Why Holding Costs Increase

Description Qty.Microwave 2

Order quantity


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Cost is spread over more units

Example: You need 1000 microwave ovens

Purchase Order

Description Qty.

Microwave 1

Purchase Order

Description Qty.

Microwave 1

Purchase Order

Description Qty.

Microwave 1

Purchase Order

Description Qty.Microwave 2

1 Order (Postage 0.34) 500 Orders (Postage 170)

Order quantity

Purchase Order

Description Qty .Microwave 1000

Why Order Costs Decrease

Deriving an EOQ

Develop an expression for total costs Total cost = order cost + holding cost

Find order quantity that gives minimumtotal cost (use calculus) minimum is when slope is flat

slope = derivative set derivative of total cost equal to 0 and

solve for best order quantity


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Expected Number of Orders per year = =N D

Q

D = Demand per year (known and relatively constant)

S = Order cos t per order

H = Holding (carrying) cost per unit per year

d = Demand per day

L = Lead time in days (known and relatively constant)

Q = order size (number of pieces or it ems per order)

EOQ Model Equations

Order Cost per year = S D

Q

Holding Cost per year = (average inventory level) H

EOQ Model - average inventory level

Average

Inventory

(Q/2)

Time

Inventory Level

Order

Quantity

(Q)

0

Maximum inventory = Q

Minimum inventory = 0


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Inventory Carrying Cost

Order or Setup Cost


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Inventory Costs

Order Quantity

Annual Cost

H o l d i n g

C o s t = ( Q

/ 2 ) H T o t a

l C o s t C u r

v e = ( D / Q

) S + ( Q / 2 ) H

Order Cost Curve = (D/Q)S

Optimal

Order Quant ity (EOQ=Q*)

EOQ Model - How Much toOrder?


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= × ×EOQ = Q* D S H

2

EOQ Total Cost Optimization

Total Cost =D

Q S +

Q

2 H

Take derivative of total cost w ith respect to Q and set

equal to zero:

Solve for Q to get optimal order size:

D

Q 2 S +

1

2 H = 0

Optimal Order Quantity = =× ×

Q* D S

H

Expected Number of Orders = =N D

Q *

Expected Time Between OrdersWorking Days / Year

= =T N

2

D = Demand per year

S = Order cost per order

H = Holding (carrying) cost

d = Demand per day

L = Lead time in days

EOQ Model Equations


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Working Days / Year =

= ×

d D

ROP d L


d = Demand per day (known and relatively constant)

L = Lead time in days (known and relatively constant)

ROP = reorder point (number of pieces or items remaining when

order is to be placed)

EOQ Model - When to order?

Suppose demand is 10 per day and

lead time is (always) 4 days.

When should you order?

When 40 are left!

EOQ Model - When To Order

Time

Inventory Level

Q*

Reorder

Point

(ROP)

2nd order 3rd order 4th order 1st order

placed

1st order

received

Lead Time = time between placi ng

and receiving an order


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2 ×1200 ×50

EOQ ExampleDemand = 1200/year

Order cos t = 50/order Holding cost = 5 per year per item

260 working days per year

=Q* 5

= 154.92 uni ts/order ; so order 155 each time

Expected Number of Orders = N = 1200/year

155 = 7.74/year

Expected Time Between Orders = T =260 days/year

7.74 = 33.6 days

Total Cost = 1200 155

50 + 155 2

5 = 387.10 + 387.50 = 774.60/year

EOQ is RobustDemand = 1200/year

Order cost = 50/order

Holding cost = 5 per year per item


Q = 155 units/order TC = 774.60/year

Q* = 154.92 units/order TC = 774.60/year = 387.30 + 387.30

Suppose we must order in multiples of 20:

Q = 140 units/order TC = 778.57/year (+0.5%)Q = 160 units/order TC = 775.00/year (+0.05%)

Suppose we wish to order 6 times per year (every 2 months):

Q = 1200/6 = 200 units/order TC = 800.00/year = 300.00 + 500.00

(200 units/order is 29% above Q* - but cost is only 3.3% above optimal )


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Order Quantity

Annual Cost

T o t a l C o s

t C u r v e

154.92

EOQ Model is Robust

Small

variation

in cost

Large variation

in order size

EOQ amount can be adjusted to facilitatebusiness practices.

If order size is reasonably near optimal (+ or- 20%), then cost will be very near optimal

(within a few percent)

If parameters (order cost, holding cost,demand) are not known with certainty, thenEOQ is still very useful.

Robustness


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260 days/year =d

1200/year

ROP = 4.615 units/day 5 days = 23.07 uni ts

-> Place an order whenever inventory falls t o (or below) 23 units

EOQ Model - When to order?

Demand = 1200/year Order cos t = 50/order



Lead time = 5 days

= 4.615/day

Sawtooth Models


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Total Costs for Various EOQ Amounts

Graphical Representation of the EOQExample


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Assume material is not receivedinstantaneously

for example, it is produced in-house

Other EOQ assumptions apply

Model provides production lot size (likeEOQ amount) for one product

Similar to EOQ with setup cost ratherthan order cost

Production Order Quantity Model

Consider one product at a time.

Produce Q units in a production run; thenswitch and produce other products.

Later produce Q more units in 2nd

production run (Q units of product ofinterest).

Later produce Q more units in 3rdproduction run, etc.

Production Order Quantity Model


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POQ Model Inventory Levels

Inventory Level

Time

Production

Begins

ProductionRun Ends

Production portion of cycle

Demand portion of cyc le with no

production (of this product)


Inventory Level

Time

Production

Begins

Production

Run Ends

Production rate = p = 20/day

Demand rate = d = 7/day

Slope = -d = -7/day

Slope = p-d = 13/day

Note: Not all of produc tion goes into

inventory


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Inventory Level

Time

Production

Begins

ProductionRun Ends

Production rate = p = 20/day

Demand rate = d = 7/day

Slope = -d = -7/day

Inventory decreases by 7/day

after producing

Slope = p-d = 13/day

Inventory increases by 13 each day

while producing

Note: 1-(d/p) = fraction of production

that goes into inventory

Number of Production Runs per year = D Q


S = Setup cost per order


d = Demand per day

p = Production rate per day (known and relatively constant)

Q = Production run size (number of pieces or items per production

run)

POQ Model Equations

Setup Cost per year = S D

Q

Holding Cost per year = (average inventory level) H


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Time

Inventory Level

Production

Portion of Cycle

Maximum Inventory

= Q(1-(d/p))

Demand portion of cyclewith no supply

Number of Production Runs per year =

D

Q


S = Setup cost per setup


d = Demand per day

p = Production rate per day (known and relatively constant)

Q = Production run size (number of pieces or items per production run)

POQ Model Equations

Setup Cost per year

= H [1-(d/p)] Q

2 Holding Cost per year = (ave. inventory level) H

=D

Q S


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Optimal Production Run Size = =× ×

Q p * D S

H[1-(d/p)]

Maximum inventory level = Q p [1- (d/p)]

2

D = Demand per year

S = Setup cost per setup


d = Demand per day

p = Production rate per day

POQ Model Equations

Total Cost =D

Q S +

Q

2 H [1-(d/p)]

Production Run length (time) = Q p /p

POQ Model Equations - cont.

Time

I n v e n t o r y

L e v e l

Cycle length (time) = Q p /d


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POQ Example

Demand = 1000/year (of product A)Setup cos t = 100/setup


Production rate = 10/day


2 ×1200 ×50 =Q p * 5 ×[1-(2.74/10)]

= 117.36 units/run

Total Cost =1000

117.36

100 + 117.36

2

20 [1-(2.74/10)]

Maximum inventory level = 117.36 [1- (2.74/10)] = 85.2 units

Demand rate = d = 1000/365

= 2.74/day

= 852.08 + 852.03 = 1704.11/year

POQ ExampleDemand = 1000units/year Production rate = 10 units/day

Q p * = 117.36 units per run

Demand rate = d = 1000/365

= 2.74/day

Number of Production Runs per year = 1000/117.36 = 8.52

Cycle length = 117.36/(2.74/day) = 42.8 days

Production Run length = 117.36/(10/day) = 11.74 days

11.74

42.8


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POQ is robust (like EOQ): Can adjust production run size.

Useful even when parameters are uncertain.

Consider last example: Set production runlength to 14 days (2 weeks) rather than11.74 days (as was optimal). Q = 140 units per run = 10/day*14 days (19%

over optimal Q )

Total cost = 1730.68 (1.6% over optimal Q )

Robustness of POQ

POQ computes a production run size for asingle product

For multiple products made on the sameequipment: Compute POQ, run time, and cycle time for each

product.

Combine into a production schedule using acommon cycle time.

May require adjusting run size, run time and cycletime to a common value.

POQ & Multiple Products


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Example: Company makes 3 products: A, B, C A: optimal run time = 3 days; optimal cycletime = 10 daysB: optimal run time = 8 days; optimal cycletime = 18 daysC: optimal run time = 10 days; optimal cycletime = 33 days

Multiple Products Example

3 6 9 6 33 A B C B A A

Use 30 days as a common cycle; adjust run & cycletimes:

A: run time = 3 days; cycle time = 10 days(3 runs/30 days)

B: run time = 6 days; cycle time = 15 days(2 runs/30 days)

C: run time = 9 days; cycle time = 30 days(1 run/30 days)


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Variation of EOQ (not POQ)

Allows quantity discounts

Reduced price for purchasing larger quantities

Other EOQ assumptions apply

Trade-off lower price to purchase item &increased holding cost from more items

Total cost must include annual purchase cost

Total Cost = Order cost + Holding cost + Purchase cost

Quantity Discount Model

Holding cost

depends on price

usually expressed as a % of price per unit time

20% of price per year, 2% of price per month, etc.

I = holding cost percent of price per year

P = price per unit

H = Holding cost = IP

Quantity Discount Model - Holding Cost


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Order Quantity = =× ×

Q* D S

IP

2

D = Demand per year

S = Order cos t per order

H = Holding (carrying) cost = IP

I = Inventory holding cost % per year

P = Price per unit

Quantity Discount Equations

Total Cost = D Q

S + Q 2

IP + PD


D = 1000/year

S = 100/order

I = 20% per year

Q P IP


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Quantity Discount Example

D = 1000/year

S = 100/order

I = 20% per year

Q P IP


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Order

Quantity

TotalCost

Quantity toearnDiscount 2

Discount 2

Quantity toearnDiscount 1

Discount 1

T C f o r N o

D i s c o u n t

Initial Price

T C f o r D

i s c o u n t

1

T C f o r

D i s c o u

n t 2

Lowest cost not in

discount range

Best order quantity

in range


OrderQuantity

TotalCost

Quantity toearn

Discount 2

Discount 2

Quantity toearn

Discount 1

Discount 1

T C f o r N o

D i s c o u n t

Initial Price

T C f o r D

i s c o u n t

1

T C f o r

D i s c o u

n t 2

Lowest Cost


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Fixed Order Quantity Approach(Condition of Certainty)

Summary and Evaluation of theFixed Order Quantity Approach: EOQ is a popular inventory model.

EOQ doesn’t handle multiple locations as well as asingle location.

EOQ doesn’t do well when demand is not constant.

Minor adjustments can be made to the basic model.

Newer techniques will ultimately take the place of EOQ.

Fixed Order Quantity Approach(Condition of Uncertainty)

Uncertainty is a more normal condition.

Demand is often affected by exogenousfactors---weather, forgetfulness, etc.

Lead times often vary regardless of carrierintentions.

Note the variability in lead times anddemand.


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Fixed Order Quantity Model

under Conditions of Uncertainty


Reorder Point – A Special Note

With uncertainty of demand, the reorderpoint becomes the average daily demandduring lead time plus the safety stock.


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Uncertainty of Demand Affects SimpleEOQ Model Assumptions: a constant and known replenishment time.

constant cost/price, independent of orderquantity or time.

no inventory in transit costs.

one item and no interaction amongthe inventory items.

infinite planning horizon. no limit on capital availability.

Probability Distribution ofDemand during Lead Time

ProbabilityDemand

0.01160

0.06150

0.241400.38130

0.24120

0.06110

0.01100 units


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Possible Units of Inventory Short or in Excessduring Lead Time with Various Reorder Points

0-10-20-30-40-50-60160

100-10-20-30-40-50150

20100-10-20-30-40140

3020100-10-20-30130

403020100-10-20120

50403020100-10110

6050403020100100

160150140130120110100

Reorder Points ActualDeman

d

Possible Units of Inventory Short or in Excessduring Lead Time with Various Reorder Points

0.01

0.06

0.24

0.38

0.24

0.06

0.01

Proba-bility

0-0.1-0.2-0.3-0.4-0.5-0.6160

0.60-0.6-1.2-1.8-2.4-3.0150

4.82.40-2.4-4.8-7.2-9.6140

11.4

7.63.80-3.8-7.6-11.4

130

9.67.24.82.40-2.4-4.8120

3.02.41.81.20.60-0.6110

0.60.50.40.30.20.10.0100

160150140130120110100

Reorder Points ActualDeman

d


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160150140130120110100Demnd

$750

$517.50

$390

$682.50

$1640

$3018

$4500

TAC

$0$15

$12

0$585

$162

0

$301

5

$45

00GR/Q

$0$1$8$39$108$201$300

G=gw

0.00.10.83.910.820.130(g)

$750

$502.50

$270

$97.50$20$2.500(VW)

30.020.110.83.90.80.10.0(e)

Calculation of Lowest-Cost Reorder Point

Fixed Order Quantity Approach(Condition of Certainty): ExpandedEOQ Model

Where R = 3600 units V = $100; W = 25%; A = $200 per order; G = 8

Q = √ 2 R(A + G) VW

√ 2 * 3600 * ($200 + 8)$100 * 25%

Q = approximately 242 units


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Fixed Order Quantity Approach(Condition of Certainty): ExpandedEOQ Model

Where R = 3600 units V = $100; W = 25%; A = $200 per order; G = 8; Q = 242; e = 10.8

TAC = QVW + AR + eVW + GR

2 Q QTAC = (242*$100*25%) + (200*3600) + (10.8*$100*25%) + (8*3600)

2 242 242

TAC = $3025 + $2975 + $270 + $119

TAC = $6389 (New value for TAC when uncertainty introduced)

Fixed Order Quantity Approach(Condition of Uncertainty):Conclusions

Following costs will rise to cover the uncertainty:

Stockout costs.

Inventory carrying costs of safety stock

Results may or may not be significant.

In text example, TAC rose $389 or approximately

6.5%. The greater the dispersion of the probability

distribution, the greater the cost disparity.


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Area under the Normal Curve

Reorder Point Alternatives andStockout Possibilities


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Fixed Order Interval Approach A second basic approach

Involves ordering at fixed intervals andvarying Q depending upon the remainingstock at the time the order is placed.

Less monitoring than the basic model

Examine Figure 7-11.

Amount ordered over each five weeks in the

example varies each week.

Fixed Order Interval Model(with Safety Stock)


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Summary and Evaluation of EOQ Approaches to Inventory

Management Four basic inventory models:

Fixed quantity/fixed interval

Fixed quantity/irregular interval

Irregular quantity/fixed interval

Irregular quantity/irregular interval

Where demand and lead time are known,basic EOQ or fixed order interval model best.

If demand or lead time varies, then safety

stock model should be used

Summary and Evaluation of EOQ Approaches to InventoryManagement

Relationship to ABC analysis “A” items suited to a fixed quantity/irregular

interval approach.

“C” items best suited to a irregularquantity/fixed interval approach.

Importance of trade-offs Familiarity with EOQ approaches assists the

manager in trade-offs inherent in inventorymanagement.


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Summary and Evaluation of EOQ Approaches to Inventory

Management New concepts

JIT, MRP, MRPII, DRP, QR, and ECR also takeinto account a knowledge and understandingof applicable logistics trade-offs.

Number of DCs The issue of inventory at multiple locations in a

logistics network raises some interesting

questions concerning the number of DCs, theSKUs at each, and their strategic positioning.

Additional Approaches toInventory Management

Three approaches to inventorymanagement that have special relevanceto supply chain management:

JIT (Just in Time)

MRP (Materials Requirements into Planning) DRP (Distribution Resource Planning)


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Sawtooth Model Modified for

Inventory in Transit

EOQ Costs Considering VolumeTransportation Rate


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Annual Savings, Annual Cost, andNet Savings by Various Quantities

Using Incentive Rates

Net Savings Function for IncentiveRate

Documents

Inventory II