probabilistic demand copy
TRANSCRIPT
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Individual Items with Probabilistic
Demand
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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
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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
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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?
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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?
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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
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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
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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
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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
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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
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(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
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(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
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(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
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(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.
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(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
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RL L
S
0
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(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
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(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
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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
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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
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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
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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
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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)
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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
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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.
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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
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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
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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.
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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