Susan Cholette DS855 Fall 2006

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Susan CholetteDS855 Fall 2006
Managing Uncertainty in theSupply Chain Safety
Inventory
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Outline

• The role of safety inventory in a supply chain
• Determining the appropriate level of safety
  inventory
• Impact of supply uncertainty on safety inventory
• Impact of aggregation on safety inventory
• Impact of replenishment policies on safety
  inventory
• Estimating and managing safety inventory in
  practice
• Managing safety inventory in a multi-echelon
  supply chain will implicitly be covered in
  Chapter 16 SC-coordination

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The Role of Safety Inventory in a Supply Chain

• Forecasts are never completely accurate
• If average demand is 1000 units per week, every
  once in a while actual demand is 1000. But about
  half the time actual demand will be greater than
  1000, and about half the time actual demand will
  be less than 1000
• If you kept only enough inventory in stock to
  satisfy average demand, half the time you would
  run out
• Safety inventory Inventory carried for the
  purpose of satisfying demand that exceeds the
  amount forecasted in a given period

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Role of Safety Inventory

• Average inventory is cycle inventory safety
  inventory
• The fundamental tradeoff
• Raising the level of safety inventory provides
  higher levels of product availability and
  customer service
• Raising the level of safety inventory also raises
  the level of average inventory and therefore
  increases holding costs
• Very important in high-tech or other industries
  where obsolescence is a significant risk (where
  the value of inventory, such as PCs, can drop in
  value). i.e. Compaq vs. Dell in PCs
• As cycle inventory had a cost of hCQ/2, a 100 lot
  order of 10 wine (at 20 cost of capital ) had
  an annual holding cost of ___? What would the
  safety stock cost be to hold safety stock of 100
  bottles?

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Determining the AppropriateLevel of Safety
Inventory

• Two questions that we need to ask
• What is the appropriate level of safety inventory
  to carry?
• What actions can be taken to improve product
  availability while reducing safety inventory?
• We will discuss the following
• Demand uncertainty
• Product availability
• Replenishment policies
• Cycle service level and fill rate
• Determining safety level given desired cycle
  service level or fill rate
• Determining the impact of required product
  availability and uncertainty on safety inventory

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Measuring Demand Uncertainty

• Appropriate level of safety inventory determined
  by
• supply or demand uncertainty
• desired level of product availability
• Demand has a systematic component and a random
  component
• The estimate of the random part is the measure of
  demand uncertainty and is usually measured by the
  standard deviation of demand
• Notation
• D or m Average demand per period (day or week
  most common)
• sD standard deviation of demand per period
• L lead time time between when an order is
  placed and received
• Coefficient of variation is the size of
  uncertainty relative to the demand
• cv sD / m std_dev-of_demand/ mean_demand
• You can ignore the covariance equation, r, in the
  textbook, as for all lectures, homeworks and
  quizzes/final we will assume demands are
  independent between regions/stores/days and thus
  will have no measurable correlation effects

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Measuring Product Availability Terms

• Product availability a firms ability to fill a
  customers order out of the available inventory
• Not Rainchecks or Well Fed-Ex it to you free of
  S/H
• Out-of-stock (OOS) the product is no longer
  available, we run out
• not a problem per se if no customer demand the
  product before our next order comes in
• Fill Rate (fr) fraction of demand that is
  satisfied from inventory
• Can relate to product or orders (multiple
  products)
• We will focus on customer demand for a single
  item in 855
• Cycle service level (CSL or just SL) the
  fraction of replenishment cycles that end with
  all customer demand met

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Replenishment Policies

• Replenishment policy decisions regarding when to
  reorder and how much to reorder
• Continuous review inventory is continuously
  monitored and an order of size Q is placed when
  the inventory level reaches the reorder point ROP
• Periodic review inventory is checked at regular
  (periodic) intervals and an order is placed to
  raise the inventory to a specified threshold (the
  order-up-to level) (a.k.a. Fixed Order
  Intervals)
• We will first discuss Continuous Review, and then
  briefly cover Periodic Review towards the end

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Continuous Review Policy Safety Inventory and
Cycle Service Level

• L Lead time for replenishment- if it remains
  invariant
• D Average demand per unit time (sometimes m)
• ?D Standard deviation of demand per period
• DL Mean demand during lead time
• ?L Standard deviation of demand during lead time
• CSL Cycle service level (also denoted SL or SL)
• ss Safety stock
• ROP Reorder point

Average Inventory Q/2 ss
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Review Using Standard Normal Distributions

• Recall from BUS786 (and statistics- DS512)
• z (D-m)/s
• Once you know z, you can determine SL (and vice
  versa)
• How?
• Option 1 The Standard Normal can be referenced
  in Excel,
• F(z)NORMSDIST(z) gives SL i.e.
  NORMSDIST(1.65) .95
• F-1(SL) NORMSINV(SL) gives the z value
  corresponding to the SL, i.e. NORMSINV(.99)
  2.33
• You can use the regular normal distribution
  shown in the book, but it is easier to calculate
  the z value and just use the Standard Normal.
• See next slide for Option 2 Table-Lookups
• On any test or quiz you will be provided sample
  values or a table

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Option 2 Table Look-ups for Standard Normal

• If we discover z 1.32, our SL 90.66
• What z does an 80 SL correspond to?

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Examples 11.111.2 Estimating Safety Inventory
(Continuous Review Policy)

• Example Weekly demand for PalmPCs averages
  2,500 with a standard deviation of 500 units.
  We place an order of 10,000 units when we drop to
  6000 units, and the order takes 2 weeks to
  arrive.
• What is our average inventory?
• What is the average time a unit spends on the
  shelf?
• What is our chance of running out of stock before
  the order arrives?
• 1. DL DL (2500)(2) 5000
• ?L sqrt(LT) ?L 1.41500 707
• ss ROP - DL 6000 - 5000 1000
• Cycle inventory Q/2 10000/2 5000
• Average Inventory cycle inventory ss 5000
  1000 6000
• 2. Average Flow Time Avg inventory / throughput
  6000/2500 2.4 weeks
• 3. SL NORMSDIST (ss/?L) NORMSDIST(1000/707)
• 92 (This value can also be determined
  from a probability distribution table)
• So we have an 8 chance of running out

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Estimating Unmet Demand Fill Rate

• Fill Rate Proportion of customer demand
  satisfied from stock
• Stock-out occur when demand during lead time
  exceeds the reorder point
• ESC is the expected shortage per cycle (average
  demand in excess of reorder point in each
  replenishment cycle)
• ss is the safety inventory
• Q is the order quantity, which is the average
  demand, D, and so can be used interchangeably

ESC -ss1-NORMDIST(ss/?L, 0, 1, 1) ?L
NORMDIST(ss/??L, 0, 1, 0)
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Example 11.3 Evaluating Fill Rate

• This example can also be performed in Excel
• Examples on sheets 1 and 2 in Ch11_ss_inv.xls
• Given ss 1,000, Q 10,000, sL 707, Fill
  Rate (fr) ?
• ESC -ss1-NORMDIST(ss/?L, 0, 1, 1)
• ??L NORMDIST(ss/?L, 0, 1, 0)
• -1,0001-NORMDIST(1,000/707, 0, 1, 1)
• 707 NORMDIST(1,000/707, 0, 1, 0)
• 25.13
• For every order cycle, we expect to be short
  about 25 units
• fr 1- ESC/Q 1- (25.13)/10,000 0.9975
• So only .25 of demand is unmet (yet have a mere
  92 CSL!)
• Second (easier!!) option for calculation
• Look up E(z), given z or SL on Unit Normal Loss
  Table
• I will provide you a copy of this Table for
  quizzes and tests
• ESC E(z) ?L
• Overall Fill Rate 1- ESC/Q

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Service Level and Fill Rate

• Fill Rate and Service Level are not the same!
• The Fill Rate increases as Service Level
  increases, but is affected by other factors such
  as
• Standard Deviation of Demand
• Lead Time
• Order Size
• Stock-outs themselves (hence CSL) are not the
  problem- if we run out of inventory, but have no
  customers until the next order comes in, we have
  no lost sales- so no problem!
• For most real-life situations, Fill Rates usually
  turn out to be much higher than Service Levels

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Example 11.4 EvaluatingSafety Inventory Given
CSL

• Demand for LegosTM D 2,500/week ?D 500/week
• L 2 weeks Q 10,000 CSL 0.90
• Calculations show
• DL 5000, ?L 707 (from earlier example)
• ss FS-1(CSL)?L NORMSINV(0.90)(707) 906
• this value can also be determined from a Normal
  probability distribution table
• ROP DL ss 5000 906 5906

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Evaluating Safety InventoryGiven Desired Fill
Rate

• D 2500/wk, sD 500/wk, Q 10000, LT 2wks
• If desired fill rate is 97.5, what safety
  inventory should be held?
• ESC (1 - fr)Q 250
• We arent going to attempt to take the inverse of
  the ESC function(!), so we have two options See
  sheet 2 of Ch11_ss_inv.xls
• Option 1) Using Excel, plug different values of
  SS in- the larger the SS, the lower the ESC.
• Option 2) Solve for E(z), given ESC E(z) ?L
  Then look up closest z on the lookup table.
  E(z) 250/707 .35 -gt z .1 (or a CSL of
  54)
• Discussion how can CSL be so low for a high Fill
  Rate?
• BTW, it is possible to have negative values for
  z. This is when you order less than you expect
  to be able to sell.
• Get SS 67 units
• What happens when we increase our desired fill
  rate?

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Determine Safety Inventory for a Desired Fill
Rate (try different values of ss)
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Impact of Supply Uncertainty

• Everything weve done so far assumes that our
  suppliers will deliver the product within the
  specified LT. But what if that is not the case
  and LT is variable? (Assume normal distribution)
• D Average demand per period
• ?D Standard deviation of demand per period
• L Average lead time
• ?sL Standard deviation of lead time

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Example Impact of Supply Uncertainty

• Daily Demand for Computers D 2,500/day ?D
  500/day
• But now Lead time is variable L 7 days sL
  7 days
• Our order and SL policies Q 10,000 CSL
  0.90
• DL DL (2500)(7) 17500
• So ss F-1s(CSL)sL NORMSINV(0.90) x 17550
• 22,491 computers
• Open example on sheet 3 of ch11-ss-inv.xls

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Impact of Supply Uncertainty

• Given demand averages 2500/day with ?D 500/day
  and that average LT 7 days
• Safety inventory when sL 0 days is 1,695
• Safety inventory when sL 1 is 3,625
• Safety inventory when sL 2 is 6,628
• Safety inventory when sL 3 is 9,760
• Safety inventory when sL 4 is 12,927
• Safety inventory when sL 5 is 16,109
• Also, compare to LT 14 days, with sL 0 is 2398

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Impact of Required Product Availability and
Uncertainty on Safety Inventory

• As desired product availability (as measured by
  service level or fill rate) increases, required
  safety inventory increases
• Demand uncertainty (sL) increases, required
  safety inventory increases
• Managerial levers to reduce safety inventory
  without reducing product availability include
• reducing supplier lead time, L or reduce
  variability in lead time (better relationships
  with suppliers)
• reducing uncertainty in demand, sL (better
  forecasts, better information collection and use)
• 9/2005 CSCMP Forum Market conditions
• Ghiradellis clients 1 concern
• On-time delivery, neither late or early

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Impact of Using Periodic Review Instead of
Continuous Review Policies

• To date weve assumed that we can re-order when
  stock drops to a ROP. But what if we can order
  only at fixed, pre-determined intervals?
• Instead of setting Q, now use an
  Order-up-to-level (OUL) that we place every T
  periods, where OUL D(LT) ss A
• A on-hand inventory, where, generally, wed
  expect A ss DL
• We can determine safety stock, ss z ?TL
  where
• D Average demand per period
• ?D Standard deviation of demand per period
• L Average lead time
• T Review Interval

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Example Periodic Review Policy

• Take the demand distribution from the Legos
  example and assume that Lead time is constant at
  1 week, but that we are only allowed to place an
  order every 4 weeks. How does our Safety stock
  differ from using ROP policy?
• See Sheet 5 in ch11_ss_inv.xls
• D 2,500/wk ?D 500/wk
• L 2 weeks T 4 weeks, CSL 0.90
• DLT D(LT) (2500)(24) 15,000
• Every 4 weeks we order up to the level of 16750
  units (order size adjusted downward by existing
  inventory)
• Our safety stock is 1570
• If we could order with ROP, our Safety stock
  would be 906 boxes, or 58 of what is required
  now. If annual H is only .1/box, the
  difference in costs is 66.

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Cycle and Safety Stock Inventory Periodic Review
Policy

• What is our average cycle inventory? Not in book
• Cycle stock .5 DT, same as with ROP
• Given SS needs are higher,
• What are reasons we might use Periodic Review?

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Impact of Aggregationon Safety Inventory

• Aggregation is a potentially powerful way to
  reduce safety inventory and, thus, costs, without
  impacting Service Level
• It is also called consolidation or risk-pooling
• Some of the possible methods to achieve it
• Aggregation through consolidation
• Information centralization
• Specialization
• Product substitution
• Component commonality
• Postponement

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Formulae for Impact of Aggregation
Will not use covariance formulae
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Impact of Aggregation(Example 11.7)

• Car Dealer 4 dealership locations
  (disaggregated)
• D 25 cars sD 5 cars L 2 weeks desired
  CSL0.90
• What would the effect be on safety stock if the 4
  outlets are consolidated into 1 large
  (aggregated) location?
• At each disaggregated outlet
• For L 2 weeks, sL 7.07 cars
• ss Fs-1(CSL) x sL (z1.28) x 7.07 9.06
• Each outlet must carry 9 cars as safety stock, so
  safety inventory for the 4 outlets in total is
  49 36 cars

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Impact of Aggregation, cont.

• One outlet (aggregated option)
• DC D1 D2 D3 D4 25252525 100
  cars/wk
• sRC Sqrt(52 52 52 52) 10
• sLC sDC Sqrt(L) (10)Sqrt(2) (10)(1.414)
  14.14
• ss Fs-1(CSL.9) x sLC (z1.28) x 14.14 18.12
  or about 18 cars
• What is the factor of improvement in Safety Stock
  with aggregation?
• Caveat If covariance, r does not equal 0 (demand
  is not completely independent), the impact of
  aggregation is not as great
• What are some situations where covariance is very
  likely to be present and cannot be ignored?
• In this class, we will assume covariance is
  negligible

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Generalization Consolidating n Identical
Facilities

• The optimal order quantity (EOQ) increases by a
  factor of
• The average inventory decreases by a factor of 1/
• True of both cycle and safety stock inventory
• The total number of setups decreases by a factor
  of 1/
• This translates to a proportional decrease in
  setup/order costs
• The total cost decreases by a factor of 1/
• - Where total costs carrying costs of cycle
  stock, carrying costs of safty stock order
  costs

Note that the cycle stock at the combined
facility is larger by a factor of than the
cycle stock at a single pre-consolidation
facility. But, because there would were n of
these pre-consolidation cycle stocks, the total
inventory is smaller after consolidation.
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Impact of Aggregation

• If number of independent stocking locations
  decreases by n, the expected level of safety
  inventory will be reduced by square root of n
  (square root law)
• E-commerce retailers can attempt to take
  advantage of aggregation (Amazon) more easily
  compared to bricks and mortar retailers (Borders)
• Aggregation has two major disadvantages
• Increase in response time to customer order
• Increase in transportation cost to customer
• Some e-commerce firms (such as Amazon) have
  reduced aggregation to mitigate these
  disadvantages
• Open Question How might we get some of the same
  benefits of aggregation without the disadvantages?

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Information Centralization

• Virtual aggregation
• Information system that allows access to current
  inventory records in all warehouses from each
  warehouse
• Most orders are filled from closest warehouse
• In case of a stock-out, another warehouse can
  fill the order
• Better responsiveness, lower transportation cost,
  higher product availability, but reduced safety
  inventory

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Specialization

• Stock all items in each location or stock
  different items at different locations?
• Different products may have different demands in
  different locations (e.g., snow shovels)
• There can be benefits from aggregation
• E.g. Barnes and Noble- use of kiosks for
  low-volume items
• Benefits of aggregation can be affected by
• coefficient of variation of demand (higher cv
  yields greater reduction in safety inventory from
  centralization)
• value of item (high value items provide more
  benefits from centralization)

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Value of Aggregation at Grainger (Table 11.4)
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Product Substitution

• Use of one product to satisfy another products
  demand
• Manufacturer-driven one-way substitution
• Ship a 120Gig HD instead of 100Gig HD
• Customer-driven two-way substitution
• Buy 180 tablet bottle of Advil instead of 90
  tablet bottle, or buy store brand
• Analysis and proper product placement are
  necessary for substitution to be fully effective
• Clothing retailers Design collection so several
  tops match several pants (Zara)
• Caveats (not in text)
• Substitution is not as prevalent as grocers
  would like (H.Dunn, Inventory Management Expert
  and 855 guest lecturer, 9/30/2003)
•   There are certain items which a grocery store
  simply must have on its shelf.   We've seen
  someone push a nearly-full cart down the
  detergent aisle, see the empty slot where Tide
  was, and walk out of the store leaving the cart
  by the empty Tide slot. The moral is people
  expect certain things when it comes to service,
  and one of those is a standard item or brand no
  one wants to be the one responsible for letting
  the store run out of Tide. (Robert Knedlik, 855
  Student who worked in Albersons IT Dept.)   

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Component Commonality

• Using common components in a variety of different
  products
• Can be an effective approach to exploit
  aggregation and reduce component inventories
• Can be an effective approach to reduce component
  inventories
• Used extensively in electronics (Dell) and
  automotive (Toyota)
• Clothing manufacturers Sports Obermeyers
  zippers (remove unnecessary differentiation)
• The cost savings from expanding usage from 2 to 3
  products is much higher than expanding from 4 to
  5 products
• See example on sheet 4 of Ch11-ss-inv.xls

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Postponement

• The ability of a supply chain to delay product
  differentiation or customization until closer to
  the time the product is sold
• Goal is to have common components in the supply
  chain for most of the push phase and move product
  differentiation as close to the pull phase as
  possible
• An analysis of the potential cost savings from
  postponement is errr postponed until Chapter 12
• Examples
• Dell in electronics
• Benetton and Mango both use gray fabric for
  garment dyeing

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Estimating and ManagingSafety Inventory in
Practice

• Account for the fact that supply chain demand is
  lumpy
• Adjust inventory policies if demand is seasonal
• Use simulation to test inventory policies first.
• Simulation is essential to evaluate complex
  policies and is useful to examine implications of
  simple ones (will see examples in Ch12_
• Why use Simulation? see Dr. Savages Flaw of
  Averages http//www.stanford.edu/savage/flaw/Art
  icle.htm
• Then start with a limited pilot before rolling
  out company-wide!
• Monitor service levels
• Focus on reducing safety inventories (but dont
  forget 5!)

• Dr. Savage is the Dave Barry of Decision
  Science. If you are studying accounting, or want
  to read a humorous but disturbingly relevant
  article on FASB http//www.stanford.edu/dept/MSan
  dE/faculty/savage/AccountingRemarks.pdf

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Summary of Learning Objectives

• What is the role of safety inventory in a supply
  chain?
• What are the factors that influence the required
  level of safety inventory?
• What are the different measures of product
  availability?
• What managerial levers are available to lower
  safety inventory and improve product availability?