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A Periodic Review, Order-up-to Inventory Model

Outline  A Periodic Review, Order-up-to Inventory Model

 System description

 Replenishment policy

 Performance measures

 In-stock Probability

 Total Annual Policy-related Cost

 Choosing policy parameters

 Comparison of continuous-review and periodic-review

models

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Related Readings  Operations Management (10th Edition, Prentice Hall): pages 325-329

of Chapter 9

 Matching Supply with Demand (3rd Edition, McGraw-Hill): Sections

14.1-14.5 (before Expected Back Order) and 14.6, pages 287-301 and

304

 Operations Management (13th Edition, Pearson): pages 514-515 of

Chapter 12

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Timing in the Periodic Review, Order-up-to Model  Time is divided into periods of equal length, e.g., one hour, one month.

 During a period the following sequence of events occurs:

 A replenishment order can be submitted.

 Inventory is received.

 Random demand occurs.

 Lead times:

 An order is received after a fixed number of periods, called the lead time.

 Let L represent the length of the lead time.

Order Receive period 0

order

Demand occurs

Order Receive period 1

order

Demand occurs

Order Receive period 2

order

Demand occurs

Time

Period 1 Period 2 Period 3

An example with L = 1

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

What is a Periodic Review, Order-up-to Inventory Model?  Periodic review (P) system: A system in which an item’s inventory

position is reviewed periodically rather than continuously

 Order-up-to level, T  the maximum inventory position we allow.  sometimes called the base stock level.

 Implementation: Review the inventory status at the end of each review interval with

length P. If the inventory position is below some target order-up-to

level T, then place an order to bring the inventory position to the target

order-up-to level T.

 Two parameters (P, T) in a periodic review policy Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Periodic Review, Order-up-to System with Uncertain Demand

P P Time

O n

-h a n

d i n

v e n

to ry

Order placed

Order received

Order placed

Order received

L L

IP1

IP3

IP2

IP IPIP

Note: T is the amount of inventory we will use to satisfy demand during (P+L) periods (i.e., the protection interval).

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Order-up-to Model Implementation  A period’s order quantity = T – Inventory position

 Suppose T = 4.

 If a period begins with an inventory position = 1, then three units are ordered.

(4 – 1 = 3 )

 If a period begins with an inventory position = -3, then seven units are ordered

(4 – (-3) = 7)

 A period’s order quantity = the previous P period’s demand:

 The order up-to model is a pull system because inventory is

ordered in response to demand.

 The order up-to model is sometimes referred to as a 1-for-1

ordering policy.

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Push vs. Pull

Push: Inputs availability triggers production

supplier process customer

inputs outputs

Pull: Outputs need triggers production

supplier process customer

inputs outputs

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Determining How Much to Order Example 12.6: A distribution center has a backorder for five 46” color TV

sets. No inventory is currently on hand, and now is the time to review.

How many should be reordered to achieve an inventory position level of

T = 400 if there are no scheduled receipts?

Inventory Position = On-hand inventory + Pipeline Inventory – Backorders

= 0 + 0 – 5 = –5 sets

Qt = T – IPt = 400 – (–5) = 405 sets

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

What Determines the Inventory Level?  Short answer:

 Inventory level at the end of a period = T - demand over L + P periods.

 Explanation via an example with T = 6, L = 3, P =1 and 2 units on-

hand at the start of period 1

D1 D2 D3 D4 ?

Period 1

Time

Inventory level at the end of period four = 6 - D1 – D2 – D3 – D4

Keep in mind:

At the start of a period the Inventory level + On-order equals T.

All inventory on-order at the start of period 1 arrives before the end of period 4

Nothing ordered in periods 2-4 arrives by the end of period 4

All demand is satisfied so there are no lost sales.

Period 2 Period 3 Period 4

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Stock-out and In-stock Probabilities, On- order Inventory  The stock-out probability is the probability at least one unit is backordered in a

period:

 The in-stock probability is the probability all demand is filled in a period:

 Expected on-order inventory = Expected demand over one period x lead time

 This comes from Little’s Law. Note that it equals the expected demand over

L periods, not L + P periods.

     T periods PLover DemandProb1

T periods PLover DemandProb y probabilitStockout

 

  T periods PLover DemandProb yprobabilitStockout -1 y probabilitstock -In

 

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

(P,T) Policy Calculation  Two parameters specify the policy: P and T.  The value of P can be determined as follows:

 Use EOQ model to calculate Q (the average order quantity in each

review)

 P = Q/D (this is actually the TBO, time-between-orders)

 The value of T can be determined by

 If demand is certain, then

 If demand is uncertain, then

T = Demand during (P + L) periods

T = Mean demand during (P + L) periods + SAFETY STOCK

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Use In-stock Probability to Determine T  In-stock Probability (also called Cycle service level, CSL) measures the

likelihood of not running into a stockout by the end of the P+L periods.

CSL = Prob {demand during P + L periods ≤ T}

 Using statistics (for the case of normal distributions):

z = NORMSINV(CSL),

NORMSINV is the inverse function of the standard normal distribution

function.

 Safety stock = zP+L

 z = The number of standard deviations needed for a given cycle

service level.

 P+L = The standard deviation of demand during P+L periods. Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

A Normal Probability Distribution for an 85% In-stock Probability

Average demand

during lead time

Average demand

during P + L periods

Cycle-service level = 85%

Probability of stockout (1.0 – 0.85 = 0.15)

zP+L

Probability distribution of demand during P+L

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Distributions for the P+L Demand  Suppose the demand distribution in each period has a mean  and

standard deviation .

 The lead time is L periods and the review interval is P periods. The

demand distributions are independent and identical across period.

 Then the distribution for the demand during the P+L periods has a

mean P+L = (P+L) and a standard deviation LPLP  

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Calculating Total Annual Policy-related Cost in A Periodic Review Model  Total annual policy-related cost for the periodic review system is the

sum of three cost components:

= Annual cycle inventory holding cost + annual ordering cost

+ annual safety stock holding cost

where P: Average order quantity for each review D: Mean annual demand K: Fixed ordering cost H: Annual inventory carrying cost

LPHzK P

D H

P  

 )()(

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Example 12.7: Finding T and Total Costs Bird feeder demand is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units, operating 52 weeks a year. Lead time (L) is 2 weeks and EOQ is 75 units with a cycle-service level of 90%. Annual demand (D) is 936 units. Annual holding cost rate (H) is $15 per unit per year, and fixed ordering cost (K) is $45 per order.

What is the optimal review interval (in weeks, rounded to the nearest integer)?

P = (52) = (52) = 4.2 or 4 weeks EOQ

D 75

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

What is the optimal target order-up-to level (rounded to the nearest integer)?

P = (52) = (52) = 4.2 or 4 weeks EOQ

D 75

P+L =  P + L = 5 6 = 12 units Standard deviation of demand over the protection period

T = Average demand during the protection interval + Safety stock

=  (P + L) + zP + L = (18 units/week)(6 weeks) + 1.29(12 units) = 123 units

z value for a 90% cycle-service level

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

What is the optimal total annual policy-related cost (Rounded to the nearest integer)?

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Example 12.7: Cont’d

D = (18 units/week)(52 weeks) = 936 units Safety Stock during P = 15 Holding Costs = $15/unit Ordering Costs = $45

 = 18 units L = 2 weeks Cycle service level = 90% EOQ = 75 units

The time between reviews (P) = 4 weeks T = 123 units

C = ($15) + ($45) + 15($15) 4(18)

2 936

4(18)

C = $540 + $585 + $225 = $1350

The total P-system cost for the bird feeders is:

The P system requires 15 units in safety stock. Recall from Example 12.4 (see Class 16 notes) that the Q system needs 9 units for safety stock. If cost were the only criterion, the Q system would be the better choice ($1259.10 vs. $1350).

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Calculating Performance Measures for a (P,T) Policy

 Suppose we use a (P,T) inventory replenishment policy, with given

parameters P and T.

 What is the CSL (i.e., Instock Probability) this policy achieves?

 

  

  



LPTNORMSDISTCSL  

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Example: Finding (P, T) and the CSL A regional warehouse purchases hand tools from various suppliers and then distributes on demand to retailers in the region. The warehouse operates 5 days a week, 52 weeks per year. The following data are estimated for 3/8-inch hand drills with double insulation and variable speeds:

Average daily demand  = 100 drills Standard deviation of daily demand  = 30 drills Lead time L = 3 days

Holding cost H = $9.40 per unit per year

Ordering cost K = $35 per order

Cycle-service level = 92%

Suppose a periodic (P) system is used at the warehouse.

(a) Calculate the P (in workdays, rounded to the nearest integer) that gives about the same number of orders per year as the EOQ.

P = (EOQ/D) years = 260(EOQ/D) days = 4.4  4 days.

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Average daily demand  = 100 drills Standard deviation of daily demand  = 30 drills Lead time L = 3 days

Holding cost H = $9.40 per unit per year

Ordering cost K = $35 per order

Cycle-service level = 92%

Suppose a periodic (P) system is used at the warehouse.

(b) What is the value of the target inventory position, T (rounded to the nearest integer)?

At 92% CSL, we have z = 1.41 (from Normal distribution Table). The standard deviation for the demand over protection interval is

Therefore, safety stock = zP+L = 1.41 * 79.37  112 drills and then T = 100(4+3) + 112 = 812 drills.

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

drillsLPLP 37.793430  

Example: Finding (P, T) and the CSL A regional warehouse purchases hand tools from various suppliers and then

distributes on demand to retailers in the region. The warehouse operates 5 days a week, 52 weeks per year. The following data are estimated for 3/8-inch hand drills with double insulation and variable speeds:

Average daily demand  = 100 drills Standard deviation of daily demand  = 30 drills Lead time L = 3 days

Holding cost H = $9.40 per unit per year

Ordering cost K = $35 per order

Cycle-service level = 92%

Suppose a periodic (P) system is used at the warehouse.

The comparison of the Q and P systems are as follows: Continuous review Periodic review

z 1.41 z 1.41

Safety stock 73 Safety stock 112

Reorder point R 373 Target inventory position T 812

Annual policy-related costs $4822.38 Annual policy-related costs $5207.80

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Average daily demand  = 100 drills Standard deviation of daily demand  = 30 drills Lead time L = 3 days

Holding cost H = $9.40 per unit per year

Ordering cost K = $35 per order

Cycle-service level = 92%

Suppose a periodic (P) system is used at the warehouse.

(d) Suppose the manager has chosen T = 800. What will the CSL be? Keep three digits after the decimal point.

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

 

  

  



LPTNORMSDISTCSL  

.896.0)260.1( 37.79

700800 

 

   

 NORMSDISTNORMSDIST

Comparison of Two Systems

 Continuous review (Q) system

 Review frequencies can be tailored to each item

 Lower, less-expensive safety stocks

 Periodic review (P) system

 Convenient to administer (Inventory Position only required at

review)

 Orders for multiple items from the same supplier may be

combined

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Summary of the Inventory Models

 Continuous review systems (Q system)

 Periodic review systems (P system)

 EOQ is a special case of the above two types of

systems

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Summary of the Inventory Models Notations

 IP = inventory position = On-hand Inventory + Pipeline Inventory - Backorders

 Q = order quantity

 mean of per period demand

  = standard deviation of per period demand

 L = supply lead time

 D = mean of annual demand

 H = annual holding cost per unit

 K = ordering cost

 R = reorder point in continuous review

 T = target inventory position in periodic review

 P = length of review cycle in periodic review

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Key Equations  EOQ model

 Cycle inventory = Q/2 EOQ =

 Total annual policy-related cost =

 Total annual cost =

 Continuous review models (R, Q)  Order quantity = EOQ Reorder point R = L + zL  Total annual policy-related costs =

 Total annual cost =

 Periodic review models (P, T)

 P = EOQ/D Target inventory position T = (P+L) + zP+L  Total annual policy-related cost =

 Total annual cost =

HDK /2

)()( 2

K Q

D H

Q 

LHzK Q

D H

Q  )()(

cDHzK Q

D H

Q L  )()(

cDK Q

D H

Q  )()(

LPHzK P

D H

P  

 )()(

cDHzK P

D H

P LP  

 )()(

2 Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

True or False?  For a periodic review, order-up-to inventory system:

 Demand is a random variable, so total demand between reviews varies.

True

 The actual order quantity Q remains the same from one order to the next.

False

 The order quantity is equal to the demand in the last review interval (of P

periods).

True

 The EOQ model with parameters (R, Q) can also be implemented as a

periodic review model, where P = Q/D year and T = R + Q.

True

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang

Summary  The periodic review, order up-to model is appropriate for products with

random demand but many replenishment opportunities over fixed review

periods .

 Expected inventory and service are controlled via the order up-to level:

 The higher the order up-to level the greater the expected inventory and

the better the service (in-stock probability).

 The key factors that determine the amount of inventory needed are…

 The length of the replenishment lead time.

 The desired service level (in-stock probability).

 Demand uncertainty.

Classes 17-19 Periodic Review Inventory Model MGT 303 Prof. Yang