EOQ: How much to order; ROP: When to order

Re-Order Point (ROP) in periodic inventory system is the start of the period. Where is ROP in a perpetual system?

ROP in Perpetual System: Quantity on hand drops to a predetermined amount.

If there were no variations in demand, and demand was really constant; ROP is when inventory on hand is equal to the [average] demand during lead time.

Lead Time is the time interval from placing an order until receiving the order.

When to re-order with EOQ Ordering

The greater the variability of demand during lead time, the greater the need for additional inventory to reduce the risk of shortage (i.e., the greater the safety stock).

When variability exists in demand or lead time, actual demand during lead time may exceed average demand during lead time.

If ROP is equal to average demand during lead time, there is 50% probability of satisfying all the demand.

But we want this probability to be greater than 50%. We prefer 90%, 95%, or 99%.

Safety Stock (SS): serves to reduce the probability of a stockout during the lead time.

ROP in perpetual inventory control is the inventory level equal to the average demand during the lead time + a safety stock (to cover demand variability)

When to re-order with EOQ Ordering

Example 1

Average demand for an inventory item is 200 units per day, lead time is three days, there is no variation in demand and lead time and therefore, safety stock is zero. What is the reorder point?

In a perpetual inventory system, the ROP is the point at which inventory on hand is equal to the [average] demand in lead time plus safety stock.

ROP = 3(200)

Whenever inventory level drops to 600 units we place an order.

We receive the order in 3 days. Because there is no variations in demand, there is no stock out.

Example 2

Average demand for an inventory item is 200 units per day, lead time is three days, and safety stock is 100 units. What is the reorder point

In a perpetual inventory system, the ROP is the point at which inventory on hand is equal to the average demand in lead time plus safety stock.

ROP = 3(200)+100

Whenever inventory level drops to 700 units we place an order.

Since there is variation in demand, during the next three days, we may have a demand of 500, 600, 700, or even more.

Time

Inve

ntor

yDemand During Lead Time

LT

ROP when demand during lead time is fixed

Time

Inve

ntor

yDemand During Lead Time

Demand During Lead Time is Variable

Inventory

Time

Demand During Lead Time is Variable

Average demandduring lead time

A large demandduring lead time

ROP

Time

Qu

an

tity

Safety stock reduces risk ofstockout during lead time

Safety Stock

Safety stock

LT

ROP

Time

Qu

an

tity

Safety Stock

LT

When to re-order (ROP)

Demand during lead time has Normal distribution.

We can accept some risk of being out of stock, but we usually like a risk of less than 50%.

If we order when the inventory on hand is equal to the average demand during the lead time; then there is 50% chance that the demand during lead time is less than our inventory.

However, there is also 50% chance that the demand during lead time is greater than our inventory, and we will be out of stock for a while.We usually do not like 50% probability of stock out

ROP

Risk of astockout

Service level

Probability ofno stockout

Safetystock

0 z

Quantity

z-scale

Safety Stock and ROP

Each Normal variable x is associated with a standard Normal Variable z x is Normal (Average x , Standard Deviation x) z is Normal (0,1)

There is a table for z which tells us Given any probability of not exceeding z. What is the value of z Given any value for z. What is the probability of not exceeding z

Averagedemand

Common z Values

Risk Service level z value0.1 0.9 1.280.05 0.95 1.650.01 0.99 2.33

ROP

Risk of astockout

Service level

Probability ofno stockout

Safetystock

0 z

Quantity

z-scale

Averagedemand

Relationship between z and Normal Variable x

ROP

Risk of astockout

Service level

Probability ofno stockout

Safetystock

0 z

Quantity

z-scale

Averagedemand

z = (x-Average x)/(Standard Deviation of x)x = Average x +z (Standard Deviation of x)μ = Average x σ = Standard Deviation of x

Risk Service z value level0.1 0.9 1.280.05 0.95 1.650.01 0.99 2.33

x = μ +z σ

Relationship between z and Normal Variable ROP

ROP

Risk of astockout

Service level

Probability ofno stockout

Safetystock

0 z

Quantity

z-scale

Averagedemand

LTD = Lead Time DemandROP = Average LTD +z (Standard Deviation of LTD)ROP = Average LTD +ssss = z (Standard Deviation of LTD)

Safety Stock and ROP

ROP

Risk of astockout

Service level

Probability ofno stockout

Safetystock

0 z

Quantity

z-scale

Risk Service level z value0.1 0.9 1.280.05 0.95 1.650.01 0.99 2.33

Averagedemand

ss = z × (standard deviation of demand during lead time)

Demand During Lead Time is Variable N(μ,σ)

Average Demand of sand during lead time is 50 tons.Standard Deviation of demand during lead time is 5 tons.Assume that management is willing to accept a risk no

more that 5%.

What is the service level?Service level = 1-risk of stockout = 1-.05 = 0.95

What is the z value corresponding to this service level?z = 1.65

What is safety stock (ss)? ss = z (standard deviation of demand during lead time)ss = 1.65 (5) = 8.25

What is ROP?ROP = Average demand during lead time + ss

μ and σ of demand during lead time

ROP = 50 + 1.65(5) = 58.25

Average Demand of sand during lead time is 75 tons.Standard deviation of demand during lead time is 10 tons.Assume that the management is willing to accept a risk no

more that 10%.

What is the service level?Service level = 1-risk of stockout = 1-0.1 = 0.9

Risk Service level z value0.1 0.9 1.280.05 0.95 1.650.01 0.99 2.33

Example 2; total demand during lead time is variable

z = 1.28What is safety stock?ss = 1.28(10) = 12.8ROP = Average demand during lead time + ssROP = 75 + 12.8 = 87.8

The general relationship between service level, risk, and safety stock:

Service level increasesRisk decreasesss increases

Example 2; total demand during lead time is variable

: Demandd

: Average Demandd

:Standard deviation of demandd

: Lead timeLT

If demand is variable and Lead time is fixed

Average demand during lead time : ( )LT d

: LTdLT d

:Standard deviation of demand during lead timedLT

μ and σ of demand per period and fixed LT

Average Demand of sand is 50 tons per week.Standard deviation of the weekly demand is 3 tons.Lead time is 2 weeks.Assume that the management is willing to accept a risk

no more that 10%. z = 1.28

: 50 tonsd: 2 weeksLT : 3 tonsd

dROP d LT z LT

50 2 1.28 2 3ROP

100 5.43ROP

μ and σ of demand per period and fixed LT

: Lead timeLT

: Average Lead timeLT

:Standard deviation of Lead TimeLT

If lead time is variable and demand is fixed

Average demand during lead time : ( )LT d

: dLT LTd

: Demandd

μ and σ of Lead Time and fixed demand

Lead Time Variable, Demand fixed

Demand of sand is fixed and is 50 tons per week.The average lead time is 2 weeks.Standard deviation of lead time is 0.5 week.Assuming that the management is willing to accept a risk

no more that 10%. Compute ROP and SS.

Acceptable risk; 10% z = 1.28

: 50 tonsd : 2 weeksLT : .5 weekLT

LTROP d LT zd 50 2 1.28 50 .5ROP

100 32ROP

(a) Average demand per day is 20 units and lead time is 10 days. Assume zero safety stock (ss=0). Compute ROP.

(b) Average demand per day is 20 units and lead time is 10 days. Assume 70 units of safety stock. ss=70. Compute ROP.

(c) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute ss.

(d) If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute ss.

(e) If demand per day is 20 units and lead time is 16 days and standard deviation of lead time is 4 days. Compute ROP at 90% service level. Compute ss.

Assignment 12.d