Supply Chain Management Notes For Distribution Planning

L#4-6: Multi Period Inventory Modelling SCM 2015, IIM Lucknow

Dr. Sushil 1

• Multi-Period Inventory Systems• Multi-Period Inventory Systems

Inventory ManagementInventory Management

Sushil Kumar, PhD

IIM Lucknow

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Three Levels of Inventory Decisions

• Supply Chain Decisions (strategic)

• What are the potential alternatives to inventory?

• How should the product be designed?

• Deployment Decisions (strategic)

• What items should be carried as inventory?

• In what form should they be maintained?

• How much of each should be held and where?

• Replenishment Decisions (tactical/operational)

• How often should inventory status be determined?

• When should a replenishment decision be made?

• How large should the replenishment be?


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What factors influence inventory

replenishment models?

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Multi-Period Inventory Systems

These systems ensure the material availability on ongoing basis

throughout the year.

• Items are ordered multiple times throughout the year

• System logic dictates:

• Actual quantity ordered

• Timing of the order


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• Fixed-order quantity models are Event Triggered

when inventory drops to a certain level (R)

• Occur at any time depending on the demand rate

• Continuous monitoring is needed and also known as

Perpetual system

• Fixed-time period models are Time triggered in which

Time to reorder is predetermined

• Orders at fixed times quantity depending on demand

rate and therefore monitoring is required periodically.

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Multi-Period Inventory Models

• Fixed-Order Quantity Models

• Event triggered (Example: running out of stock)

• Economic Order Quantity (EOQ Model)

• Q-Model

• Fixed-Time Period Models

• Time triggered (Example: Monthly sales call by sales

representative)

• Periodic system

• Periodic review system

• Fixed order interval system

• P-Model


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Multi-Period Inventory Systems:

Some Differences

• Fixed-order quantity model favours more expensive items

• Fixed-order quantity model more appropriate for important

items

• Fixed-order quantity model require more time to maintain:

each activity is logged

• Fixed-time period model has a larger average inventory

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Multi-Period Inventory Systems:

Some DifferencesFeature Q-Model P-Model

Order Quantity Q— constant q— variable

When to place order R T

Recordkeeping Each addition/

withdrawal

At review period

Inventory size Smaller Larger

Time to maintain Higher

Type of items A, V, X, H


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Fixed-Order Quantity System

Idle stateWaiting for demand

Demand occursUnits withdrawnfrom inventory or backordered

Compute inventory positionPosition = on hand

+ on order - backorder

Issue an order for exactly Q units

Is position ≤

reorder point?

Yes

No

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Fixed-Time Period Reordering System

Idle stateWaiting for demand Demand occurs

Units withdrawnfrom inventory or backordered

Compute inventory positionPosition = on hand

+ on order - backorder

Issue an order for theNumber of units needed

Has review

time arrived?Yes

No

Compute order quantityto bring inventory

up to required level


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Fixed-Order Quantity Model:

Model Assumptions

• Demand for the product is constant and uniform throughout the period

• Lead time (time from ordering to receipt) is constant

• Price per unit of product is constant

• Inventory holding cost is based on average inventory

• Ordering or setup costs are constant

• All demands for the product will be satisfied (No back orders are allowed)

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Inventory Terminology

TC = Total annual cost

D = Demand

C = Cost per unit

Q = Order quantity

S = Cost of placing an order or setup cost

H = Annual holding and storage cost per unit of inventory

R = Reorder point

L = Lead time


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Basic Fixed-Order Quantity Model and

Reorder Point Behavior

R = Reorder point

Q = Economic order quantity

L = Lead time

L L

Q QQ

R

Time

Numberof unitson hand

1. You receive an order quantity Q.

2. You 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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Inventory Costs

• Costs associated with ordering too much (represented by

carrying costs)

• Costs associated with ordering too little (represented by

ordering costs)

• These costs are opposing costs, i.e., as one increases the

other decreases

TC = Total annual cost D = Demand

C = Cost per unit Q = Order quantity

S = Cost of placing an order or setup cost

H = Annual holding and storage cost per unit of inventory

R = Reorder point

L = Lead time


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Basic EOQ Model• Typical assumptions made

• annual demand (D), carrying cost (H) and ordering cost (S) can be

estimated

• average inventory level is the fixed order quantity (Q) divided by 2

which implies

• no safety stock;

• orders are received all at once

• demand occurs at a uniform rate

• no inventory when an order arrives

• Stockout, customer responsiveness, and other costs are

inconsequential

• acquisition cost is fixed, i.e., no quantity discounts

• Annual carrying cost = (average inventory) x (carrying cost) = (Q/2)H

• Annual ordering cost = (average number of orders per year) x

(ordering cost) = (D/Q)S

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Cost Minimization Goal

Ordering Costs

Holding

Costs

Order Quantity (Q)

Annual Cost of

Items (DC)

Total Cost

QOPT

By adding the item, holding, and ordering costs together, we

determine the total cost curve, which in turn is used to find

the Qopt inventory order point that minimizes total costs

CO

ST


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Basic Fixed-Order Quantity (EOQ)

Model Formula

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual cost

D =Demand

C =Cost per unit

Q =Order quantity

S =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

Common point on both cost curves would indicate an optimal point

Using calculus, we take the first derivative of the total cost

function with respect to Q, and set the derivative (slope)

equal to zero, solving for the optimized (cost minimized)

value of Qopt


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Other EOQ related Parameters

We also need a

reorder point to

tell us when to place an order

• Optimal cycle length

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EOQ Example (1)

Annual Demand = 1,000 units

Days per year considered in average daily demand = 365

Cost to place an order = $10 Holding cost per unit per year = $2.50

Lead time = 7 days Cost per unit = $15

Given the information below, what are the EOQ and reorder

point?

In summary, you place an optimal order of 90 units. In the

course of using the units to meet demand, when you only

have 20 units left, place the next order of 90 units.


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Example: Basic EOQ

Zartex Co. produces fertilizer to sell to wholesalers.

One raw material – calcium nitrate – is purchased from a

nearby supplier at $22.50 per ton. Zartex estimates it will

need 5,750,000 tons of calcium nitrate next year.

The annual carrying cost for this material is 40% of

the acquisition cost, and the ordering cost is $595.

a) What is the most economical order quantity?

b) How many orders will be placed per year?

c) How much time will elapse between orders?

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Example: Basic EOQ

• Economical Order Quantity (EOQ)

D = 5,750,000 tons/year

H = .40(22.50) = $9.00/ton/year

S = $595/order

= 27,573.135 tons per order


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Example: Basic EOQ

• Total Annual Stocking Cost (TSC)

TSC = (Q/2)H + (D/Q)S

= (27,573.135/2)(9.00) + (5,750,000/27,573.135)(595)

= 124,079.11 + 124,079.11

= $248,158.22

• Number of Orders Per Year = D/Q

= 5,750,000/27,573.135 = 208.5 orders/year

• Time Between Orders = Q/D

= 1/208.5 = .004796 years/order

= .004796(365 days/year) = 1.75 days/order

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EOQ Limitations

When batch set-up costs are high, EOQ suggests very large

batches.

• Complicates production scheduling

• Give longer lead times to customers

• Needs excess inventory storage

• Too much capital in stocks

• Solution? Artificially high value on holding cost

• EOQ suggests fractional value

• Suppliers unwilling to split standard package sizes

• Deliveries by vehicles with fixed capacities

• More convenient to round order size

• If you shift from EOQ, what happens to the cost?


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An Example

D = 6000 C = 30 S = 125 H = 7

EOQ = Qo = = 462.91

VCo = H x Q = 3,240.37

At 450? VC = DS/Q + QH/2 = 3,241.67

At 500? VC = DS/Q + QH/2 = 3,250.00

For 450 Batch 2.8% below optimal Cost 0.04% High

For 500 Batch 8% above optimal Cost 0.3% High

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Impact on cost with changes in order quantity near EOQ

How Robust is the EOQ Model

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

QO

An

nu

al

Co

st

(optimal order quantity)


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Example

Mae Chow Min works in her bakery for 6 days a week for 49weeks a year. Flour is delivered directly with a charge of $7.5for each delivery. Chow Min uses an average of 10 sacks ofwhole-grain flour a day, for which she pays $12 a sack. She hasan overdraft at the bank which costs 12 per cent a year, withspillage, storage, loss and insurance costing 6.75 per cent a year.

A. What size of delivery should Chow Min use and what arethe resulting costs?

B. How much should she order if the flour has a shelf life of2 weeks?

C. How much should she order if the bank imposes amaximum order value of $1,500?

D. If the mill only delivers on Mondays, how much sheorder and how often?

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Uncertainty in Demand

• Assumption was demand in known

• Suppose we have erred by E percent

• Then demand would be D(1+E)

VC/VCo = ½ [Qo/Q + Q/Qo]


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Uncertainty in Costs

• Assumption was costs are known

• Ordering cost and holding cost

• Suppose we have taken extra E1 and E2 respectively

• Calculations would then have been for

• Ordering Cost = S(1+E1)

• Holding Cost = H(1+E2)

VC/VCo = ½ [Qo/Q + Q/Qo]

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• Production done in batches or lots

• Capacity to produce a part exceeds the part’s usage or demand rate

• Assumptions of EPQ are similar to EOQ except orders are received

incrementally during production

Economic Production Quantity (EPQ)

UsageUsage

Pro

duct

ion

& U

sage

Pro

duct

ion

& U

sage


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• Only one item is involved

• Annual demand is known

• Usage rate is constant

• Usage occurs continually

• Production rate is constant

• Lead time does not vary

• No quantity discounts

Economic Production Quantity Assumptions

Economic Run Size

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Example for Economic Production Quantity

• A toy manufacturer uses 48000 rubber-wheels per year for its popular dump truck series. The firm makes its own wheels, which it can produce at a rate of 800 per day. The toy trucks are assembled uniformly over the entire year. Holding cost is $1 per wheel a year. Setup cost for a production run of wheels is $45. the firm operates 240 days per year. Determine the a. Optimal run size

b. Minimum total annual cost for carrying and setup

c. Cycle time for the optimal run size

d. Run time


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Procedure to Find Best Order Size

Start

Finish

Take the next lowest unit cost

1 Find the lowest point using Qo

2

Calculate the cost at this valid minimum

5

Find the lowest cost and corresponding order size

6Compare the costs of all the points considered

7

Calculate the cost at the break point on the left of the valid range

4 3Is this point valid?

Yes

No

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Quantity DiscountsAll-Units Discount Order Cost Function


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Quantity Discounts

Incremental Discount Order Cost Function

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All-Units Quantity Discounts:An Example for Constant Holding Cost

• The maintenance department of Malamara hospitaluses about 816 cases of liquid cleanser annually.Ordering costs are $12, holding costs are $4 per case ayear, and the new price schedule indicates that ordersare less than 50 cases will cost $20 per case, 50 to 79cases will cost $18 per case, 80 to 99 cases will cost$17 per case, and larger orders will cost $16 per case.Determine the optimal order quantity and the total cost.


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Quantity Discounts:An Example for Proportionate Holding Cost

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Quantity Discounts:An Example for Proportionate Holding Cost

• Surge Electric uses 4,000 toggle switches a year.

Switches are priced as follows: 1 to 499, 90 cents each;

500 to 999, 85 cents each; and 1000 or more, 80 cents

each. It costs approximately $30 to prepare an order

and receive it, and holding costs 40% of purchase price

per unit on an annual basis. Determine the optimal

order quantity and the total annual cost.


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All-Units Discount Quantity :An Example for Proportionate Holding Cost

• The Weighty Trash Bag Company has the following price

schedule for its large trash can liners.

For orders up to 500 bags, the company charges 30 cents

per bag; for orders of more than 500 but 1,000 or less bags,

it charges 29 cents per bag; and for orders more than 1,000,

it charges 28 cents per bag.

• The company considering what standing order to place

with Weighty uses trash bags at a fairly constant rate of 600

per year. The accounting department estimates that the

fixed cost of placing an order is $8, and holding costs are

based on a 20 percent annual interest rate.

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Average Annual Cost Function for Incremental Discount Schedule


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Planned Shortages with Back Orders

Shortages are when demand is not met from stock.

Useful when:

• Shortages are not expensive

• Planned shortages are beneficial, e.g., Car dealer,

furniture shop

• It is more likely when:

• Unit cost is high

• Wide range of items

• Extreme case is ‘Make-to-Order’

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Planned Shortages with Back Orders

Optimal Order size

Optimal amount to be back-ordered


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Example for Shortages with Back Orders

Demand for an item is constant at 100 units a month.

Unit cost is 50, reorder cost is 50, holding cost is 25%

of value a year, shortage cost for back orders is 40% of

value a year. Find an optimal inventory policy for the

item.

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Order-Point Determination


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Stock-Out Occurrence

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L Time

Expected demand

during lead time

Maximum probable demand

during lead time

R

Qu

an

tity

Safety stockSafety stock reduces risk of

stockout during lead time

Safety Stock


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Inventory Level with Safety Stock

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• Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered

• Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.

• Service Level - Probability that demand will not exceed supply during lead time.

When to Reorder with EOQ Ordering


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Determinants of the Reorder Point

• The rate of demand

• The lead time

• Demand and/or lead time variability

• Stockout risk (safety stock)

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ROP

Risk ofa stockout

Service level

Probability of

no stockout

Expecteddemand Safety

stock

0 z

Quantity

z-scale

The ROP based on a normal

Distribution of lead time demand

Reorder Point


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Basis for Setting the Reorder Point

• The reorder point is set based on

• the demand during lead time (DDLT) and

• the desired customer service level

• Reorder point (R) = Expected demand during lead

time (EDDLT) + Safety stock (SS)

• The amount of safety stock needed is based on the

degree of uncertainty in the DDLT and the customer

service level desired

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DDLT Distributions

• If there is variability in the DDLT, the DDLT is

expressed as a distribution

• Discrete (Integer values)

• Continuous (Valid for high demand)


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Reorder Point for a Discrete DDLT Distribution

• Assume a probability distribution of actual DDLTs is

given or can be developed from a frequency

distribution

• Starting with the lowest DDLT, accumulate the

probabilities. These are the service levels for DDLTs

• Select the DDLT that will provide the desired customer

level as the reorder point

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Example for Setting Reorder Point

One of Sharp Retailer’s inventory items is now being

analyzed to determine an appropriate level of safety

stock. The manager wants an 80% service level

during lead time. The item’s historical DDLT is:

DDLT (cases) Occurrences

3 8

4 6

5 4

6 2


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Construct a Cumulative DDLT Distribution

Probability Probability of

DDLT (cases) of DDLT DDLT or Less

2 0 0

3 .4 .4

4 .3 .7

5 .2 .9

6 .1 1.0

To provide 80% service level, R = 5 cases

.8


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Safety Stock (SS)

R = EDDLT + SS

SS = R - EDDLT

EDDLT = .4(3) + .3(4) + .2(5) + .1(6) = 4.0

SS = 5 – 4 = 1


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Continuous DDLT Distribution

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Z Percentageof Cycles

with shortages

Cycle service Level (%)

0.00

0.84

1.00

1.04

1.28

1.48

1.64

1.88

2.00

2.33

2.58

3.00

50.0

20.0

15.9

15.0

10.0

7.0

5.0

3.0

2.3

1.0

0.5

0.1

50.0

80.0

84.1

85.0

90.0

93.0

95.0

97.0

97.7

99.0

99.5

99.9


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Reorder Point for aContinuous DDLT Distribution

average demand during lead time

standard deviation of demand during lead time

Safety Stock =

Reorder point =

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Reorder Point for a


• The resulting DDLT distribution is a normal distribution with the following parameters:

Standard deviation of a series of independent occurrences is equal to the square root of the sum of the variances


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Setting Order Point

for a Continuous DDLT Distribution

• The customer service level is converted into a Z value using the normal distribution table

(Or NORSINV function in excel)

• The safety stock is computed by multiplying the Z value by σDDLT.

• The order point is set using R = EDDLT + SS, or by substitution

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Auto Zone sells auto parts and supplies including a

popular multi-grade motor oil. When the stock of this oil

drops to 20 units, a replenishment order is placed. The store

manager is concerned that sales are being lost due to stockouts

while waiting for an order. It has been determined that lead

time demand is normally distributed with a mean of 15 units

and a standard deviation of 6 units.

The manager would like to know the probability of a

stockout during lead time.



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• EDDLT = 15 units

• σDDLT = 6 units

R = EDDLT + Z(σDDLT )

20 = 15 + Z(6)

5 = Z(6)

Z = 5/6

Z = .833

0 .833

Area = .2967

Area = .5

Area = .2033

z

Standard Normal Distribution

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• The Standard Normal table shows an area of .2967

for the region between the z = 0 line and the z = .833

line. The shaded tail area is .5 - .2967 = .2033.

• The probability of a stockout during lead time is

.2033


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Rules of Thumb in Setting Reorder Point

• Set safety stock level at a percentage of EDDLT

where j is a factor between 0 and 3.

• Set safety stock level at square root of EDDLT

OP = EDDLT + j (EDDLT)Class Description j

1

2

3

4

5

6

Uncritical

Uncertain-uncritical

Critical

Uncertain-critical

Supercritical

Uncertain-supercritical

0.1

0.2

0.3

0.5

1.0

3.0

When stockouts are not particularly undesirable

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Also known as Fixed Order Interval Model

• Orders are placed at fixed time intervals

• Order quantity for next interval?

• Suppliers might encourage fixed intervals

• May require only periodic checks of inventory

levels

• Risk of stockout

Fixed Time Period Models


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• Tight control of inventory items

• Items from same supplier may yield savings in:

• Ordering

• Packing

• Shipping costs

• May be practical when inventories cannot be closely monitored

Fixed Time Period Benefits

• Requires a larger safety stock

• Increases holding/carrying cost

• Costs of periodic reviews

Fixed Time Period Disadvantages

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Behavior of Fixed Time Period Systems

• As demand for the inventoried item occurs, the inventory level drops

• When a prescribed period of time has elapsed, the ordering process

is triggered, i.e., the time between orders is fixed or constant

• At that time the order quantity is determined by finding out the

average demand during the vulnerable period plus some safety stock

and subtracting current inventory level on hand plus on order if any.

• After the lead time elapses, the ordered quantity is received , and the

inventory level increases

• The upper inventory level may be determined by the amount of

space allocated to an item

• This system is used where it is desirable to physically count

inventory each time an order is placed


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Fixed-Time Period Model

with Safety Stock Formula

q = Average demand + Safety stock – Inventory currently on hand

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Determining Quantity in Fixed period Model

• Using an approach similar to that used to derive EOQ, the

optimal value of the fixed time between orders is derived to be

Determining the Value of σT+L

The standard deviation of a sequence of random events equalsthe square root of the sum of the variances.


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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 Solution

So, to satisfy 96 percent of the demand, you shouldplace an order of 645 units at this review period

“z” is found by using the Excel NORMSINV function.For a probability 0.96, z = 1.75


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Order-Up-to-Level in a

Periodic Review System

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Period of Time an Order Must Cover


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Hybrid Inventory Models

• Optional replenishment model

• Similar to the fixed order period model

• Unless inventory has dropped below a prescribed

level when the order period has elapsed, no order

is placed

• Protects against placing very small orders

• Attractive when review and ordering costs are

large

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Hybrid Inventory Models:

Optional Replenishment SystemMaximum Inventory Level, M

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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Base Stock Model

• Start with a certain inventory level

• Whenever a withdrawal is made, an order of equal

size is placed

• Ensures that inventory maintained at an

approximately constant level

• Appropriate for very expensive items with small

ordering costs

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Single Bin System

Order Enough to

Refill Bin

Essentially a P system

• Target inventory level and current inventory

position IP are established


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Two-Bin System

Full Empty

Order One Bin of

Inventory

Essentially a Q system

• When the first bin is empty, it triggers the

replenishment order

• The second bin contains an amount equal to

safety stock,

or the average demand during the lead time

plus the safety stock.

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Inventory Accuracy and Cycle Counting

• Inventory accuracy refers to how well the

inventory records agree with physical count

• Cycle Counting is a physical inventory-taking

technique in which inventory is counted on a

frequent basis rather than once or twice a year

Supply Chain Management Notes For Distribution Planning

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