Individual Items with Probabilistic

Demand

TERMINOLOGY•On-hand stock: Physically present on

shelf. Cannot be -ve number.

•Net stock: (on-hand) - (Backorders)

•Inventory position: (on-hand) + (on order)

- (Backorders) - (Committed)

•Safety stock: Average level of net stock

just before a replenishment arrives

Backorder vs. Lost sales

•Complete backordering: Government

organizations, wholesale-retail link of few

distribution systems (exclusive dealership)

•Complete lost sales: Retail-consumer link

mostly for FMCGs

•Combination of the two

•Numerical value of SS depends on the degree

to which backorders or lost sales occur

Three Key Issues

•How often the inventory status should be

determined?

•When a replenishment order should be

placed?

•How large the replenishment order should

be?

4 Questions for Inventory Policies

•How important is the item?

•Can, or should, the stock status be reviewed

continuously or periodically?

•What form should the inventory policy take?

•What specific cost or service objectives

should be set?

IMPORTANCE OF THE ITEM

•A items comprise roughly 20% of the total

number of items, but represent 80% of

the dollar sales volume

•B items comprise roughly 30% of the

items and 15% of the dollar volume

•C items comprise of balance 50% of items

and only 5% of the total dollar volume

CONTINUOUS vs. PERIODIC

•How often should the inventory status be

determined? - Periodic Review

•Transactions Reporting - Continuous review

•COMPARISON

•Items may be produced on the same piece of

equipment, purchased from same supplier or

shipped in the same transportation mode -

Periodic Review

CONTINUOUS Vs. PERIODIC

• COMPARISON

• Periodic review allows a reasonable prediction

of the level of the workload on the staff. While

the replenishment can be done at any moment

in continuous review making it less predictable

• Continuous review is expensive. Especially in

fast moving goods (FMCG). So reviewing errors

and costs are high

CONTINUOUS Vs. PERIODIC

• COMPARISON

• For slow moving goods, periodic review is still

better as it can trace any pilferage or spoilage

during the review period but if it continuous

review it doesn’t detect unless a transaction

happens

• Continuous review, to provide the same level of

customer service, requires less safety stock

than periodic review

Four Types of Control Systems

•Order Point, Order Quantity (s, Q) System

•Order Point, Order-up-to-level (s, S)

System

•Periodic-Review, Order-up-to-level (R, S)

System

•(R, s, S) System

(s, Q) System•Continuous review (R = 0)

•Inventory position and not Net stock is

used to trigger an order.

•Two-bin system: Amount in the second bin

corresponds to order point.

•(s, Q) system is easy to understand even

for the stock clerk

(s, Q) System•One disadvantage is that it cannot accommodate a

large order. For ex., if Q=10 units, D=15 units occurring

when the position is 1 unit just above s. Order in

multiples.

s

s + Q

A BL Time

Invento

ry

Level

(s, S) System•Order quantity is variable enough to raise the

position to order-up-to level ‘S’

•For unit-sized demand, (s,S) is same as (s,Q) system.

s

S

A BL Time

Invento

ry

Level

(s, S) System

•Also called as min-max system as

inventory position, except for momentary

drop is always between s & S.

•Optimal value of (s, S) is difficult to find

and is most of the times arbitrarily fixed.

(R, S) System•Used by those companies not using computer

controls

•Control procedure is that every R units of

time, enough is ordered to raise the inventory

position to S

•Carrying costs are high

•Provides opportunity to adjust the position to

S if demand pattern is changing with time

RL L

S

0

(R, s, S) System

•Combination of (R, S) and (s, S) systems.

•Every R units of time, check the inventory

position, if it is below s, then order to raise

the level to S. If the level is above s, nothing

is done until the next review.

•(s, S) is special case where R=0

•(R, s, S) is a periodic version of (s, S) system

(R, s, S) System

•(R, s, S) provides the lower total cost

comprising replenishment, holding and

shortage costs than any other method but

is difficult to get all 2 parameters

Specific Cost & Service Objectives

•Two types of risks to be balanced:

•If demand is large, stockout may occur

•If demand is lower, then large inventory is

carried

Specific Cost & Service Objectives

•Four methods of modeling these are:

•Safety stocks established through use of

simple approach

•Safety stocks based on minimizing cost

•Safety stocks based on customer service

levels

•Safety stocks based on aggregate

consideration

SINGLE PERIOD MODEL•Christmas trees Cost (C) = $4 per tree and SPrice (S)

= $9, Profit = $10000, so demand was around 2000

units

•Xbar = 2000 and σ = 100 trees

•Let salvage value be V = $1

•Marginal Profit (MP) on a tree = S - C = 9 - 4 = $5

•Marginal Loss (ML) on a tree = C - V = 4 - 1 = $3

•To order Q rather than Q - 1, the expected profit on

marginal tree must be ≥ expected loss

SINGLE PERIOD MODEL

•Let ‘p’ be the probability of selling the marginal tree

•Profit criterion is expected profit ≥ expected loss,

•i.e. p(MP) ≥ (1 - p)ML

•p(MP) ≥ ML - p(ML)

•p(MP+ML) ≥ ML

•Minimum acceptable probability of selling the Qth tree

is given by p ≥ ML / (MP + ML) or SOR ≥ ML / (MP + ML)

•Q = xbar + zSOR σ

MULTI PERIOD MODEL

•Service Level and Safety Stocks: In reality,

sometimes it is difficult to calculate ML (Stockout

costs which has intangibles)

•When stockout costs are not available, surrogate is

the customer service level

•Service level can be classified in two ways: i) Order

Service Level (OSL) and ii) Unit Service Level (USL)

ORDER SERVICE LEVEL (OSL)•Proportion of cycles that customer demand was

satisfied. Order Service Level represents the probability

of not having a stockout during the placement of order.

•Suppose OSL = 0.90 and number of orders/cycles =

D/Q = 20, then customer demand will be satisfied in

OSL * No. Of orders = 0.90 * 20 = 18 orders.

•It also means that during 2 orders, stockout can be

expected. (0.10 * 20 or 20 - 18 = 2)

•OSOR doesn’t tell how many units were short or not

filled during any cycle

UNIT SERVICE LEVEL (USL)•USL indicates the percentage of units of demand

filled during any period of time, whereas the USOR

specifies the quantities of units unfilled or short

during that period.

• Suppose USL = 0.95 and D = 5000 units for an item.

Then 0.95 * 5000 = 4750 units of customer demand

will be filled on an average during the year.

•It also means that 250 units of stockout can be

expected. i.e. If there are 10 orders then approx 25

units per order will be short of.

PERCENT ORDER SERVICE LEVEL

•Suppose the L and MADL are obtained from forecasting

systems, then σL must be calculated i.e. σL = 1.25MAD.

•Suppose the following details are available, L = 200,

MADL = 32, L = 5 days, D = 10000, S = 1500 and h = 30,

then using EOQ we get, Sqrt(2DS / h) = Sqrt

((2*10000*1500)/30) = 1000, so D/Q = 10 orders or cycles

•Suppose the manager is willing to accept a 0.375

probability of stockout on any cycle. Then OSL = 1 - 0.375

= 0.625

•And the ROP = L + Safety Stock = L + Z0.375 * σL

⨱

⨱

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PERCENT UNIT SERVICE LEVEL •Percent Unit Service Level tells us what percentage of

units demanded can be supplied from stock.

•It is usually called as fill rate.

• L

•x - R = σL (z - k)

⨱

s x

0

k z

PERCENT UNIT SERVICE LEVEL•USOR = E(X - R) / Q

•USOR = σL g(k) / Q

•g(k) = (Q * USOR) / σL

•Specify USL, then find g(k), then k, then Safety

Stock, then R

•Suppose USL = 0.99, then USOR = 0.01, then g(k)

= Q*0.01/σL, then go to the table, pick ‘k’ value

corresponding to calculated g(k), then calculate

ROP = xL + kσL

BACKORDER COST•When we are provided with stockout costs and not

the service levels, then how to calculate safety stock

•Increasing the safety stock by one more unit has the

same effect as adding one more unit to reorder level

‘s’

•Raising the reorder point by 1 unit will cost (holding

cost) (Q*h) / D = (1000*30) / 10000 = $3 per cycle

•If we do not add one more unit and suffer the

stockout, then backorder penalty is given by $b / unit

with SOR probability.

BACKORDER COST•Per cycle marginal cost of adding 1 unit to R = per

cycle marginal cost of not adding 1 unit to R

•Qh / D = OSOR*(b)

•So, b = Qh/ (D * OSOR) i.e. Backorder cost ‘b’

•At 99% USL, 14 units were safety stock, so, z σL = 14

where σL = 40.

•z * 40 = 14 leading to z = 0.35 leading to OSOR =

0.363 or 36.3%

•b = (30 * 1000) / (0.363 * 10000) = $8.26