Susan Cholette DS855 � Fall 2006 by 4ouO90


									 Managing Uncertainty in the
Supply Chain: Safety Inventory
        Susan Cholette
       DS855 – Fall 2006
• 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

       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

            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?

         Determining the Appropriate
          Level of Safety Inventory
•   Two questions that we need to ask
    1. What is the appropriate level of safety inventory to carry?
    2. 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

       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           11-6
  Measuring Product Availability: Terms
• Product availability: a firm’s ability to fill a customer’s order out
  of the available inventory
   • Not Rainchecks or “We’ll Fed-Ex it to you free of S/H”

• Out-of-stock: (OOS) the product is no longer available, we “run
   • 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

           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

     Continuous Review Policy: Safety
     Inventory and Cycle Service Level
L: Lead time for replenishment-
   if it remains invariant
D: Average demand per unit        D L  DL
   time (sometimes m)
sD: Standard deviation of
   demand per period
                                  sL  LsD
DL: Mean demand during lead       ss  F 1 (CSL)  s L
sL: Standard deviation of         ROP  D L  ss
   demand during lead time
CSL: Cycle service level (also    CSL  F ( ROP, D L , s L )
   denoted %SL or SL)
ss:Safety stock               Average Inventory = Q/2 + ss
ROP: Reorder point                                             11-9
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)

• 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

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?

    Examples 11.1+11.2: 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.
1. What is our average inventory?
2. What is the average time a unit spends on the shelf?
3. What is our chance of running out of stock before the order arrives?
•    1.    DL = DL = (2500)*(2) = 5000
            sL = sqrt(LT)* sL = 1.41*500 = 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/sL) = NORMSDIST(1000/707)
         = 92% (This value can also be determined from a probability distribution table)
      • So we have an 8% chance of running out

   Estimating Unmet Demand: Fill Rate
• Fill Rate: Proportion of customer
  demand satisfied from stock
• Stock-out occur when demand during       fr  1 
  lead time exceeds the reorder point                Q
• ESC is the expected shortage per cycle                          ss 
  (average demand in excess of reorder     ESC   ss * (1  F S     )
                                                                 s L 
  point in each replenishment cycle)
                                                                     
• ss is the safety inventory
                                                                ss 
• Q is the order quantity, which is the           s L    f        
                                                               s L 
                                                              S    
  average demand, D, and so can be used

ESC = -ss{1-NORMDIST(ss/sL, 0, 1, 1)} + sL NORMDIST(ss/sL, 0, 1, 0)
    Example 11.3: Evaluating Fill Rate
1. 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 = -ss{1-NORMDIST(ss/sL, 0, 1, 1)} +
                  sL NORMDIST(ss/sL, 0, 1, 0)
            = -1,000{1-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!)
2. 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) sL
•    Overall Fill Rate = 1- ESC/Q                                         11-14
      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

        Example 11.4: Evaluating
       Safety Inventory Given CSL
Demand for LegosTM: D = 2,500/week; sD = 500/week
  L = 2 weeks; Q = 10,000; CSL = 0.90
Calculations show:
  DL = 5000, sL = 707 (from earlier example)

  ss = FS-1(CSL)sL = [NORMSINV(0.90)](707) = 906
   •    this value can also be determined from a Normal
       probability distribution table

        ROP = DL + ss = 5000 + 906 = 5906

       Evaluating Safety Inventory
         Given 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 aren’t 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) sL Then look up closest z on
        the lookup table. E(z) = 250/707 = .35 -> 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.
    2. Get SS = 67 units

•   What happens when we increase our desired fill rate?
Determine Safety Inventory for a
       Desired Fill Rate
     (try different values of ss)
  Desired Fill Rate Necessary Safety Inventory

      97.5%                    67
      98.0%                    183
      98.5%                    321
      99.0%                    499
      99.5%                    767
       Impact of Supply Uncertainty
• Everything we’ve 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
   •   sD: Standard deviation of demand per period
   •   L: Average lead time
   •   sL: Standard deviation of lead time

             D L  DL
             s L  Ls  D s     2
                                         2 2
Example: Impact of Supply Uncertainty
 Daily Demand for Computers: D = 2,500/day; sD = 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

sL  L s 2  D 2 sL

  (7) 5002  (2500)2 (7)2  17550

 So ss = F-1s(CSL)sL = NORMSINV(0.90) x 17550
       = 22,491 computers

 Open example on sheet 3 of ch11-ss-inv.xls                11-20
  Impact of Supply Uncertainty
• Given demand averages 2500/day with sD =
  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

  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
• 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
   • Ghiradelli’s clients’ #1 concern:
      On-time delivery, neither late or early

  Impact of Using Periodic Review Instead
      of Continuous Review Policies
• To date we’ve assumed that we can re-order when stock drops to
  a ROP. But what if we can order only at fixed, pre-determined
• Instead of setting Q, now use an Order-up-to-level (OUL) that
  we place every T periods, where OUL = : D(L+T) +ss – A
   • A = on-hand inventory, where, generally, we’d expect: A = ss + D*L

• We can determine safety stock, ss = z* sT+L where:
   •   D: Average demand per period
   •   sD: Standard deviation of demand per period
   •   L: Average lead time
   •   T: Review Interval

       DT  L  D(T  L)
       s T L  s D T  L
       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; sD = 500/wk
    L = 2 weeks T = 4 weeks, CSL = 0.90;
    DL+T = D(L+T) = (2500)(2+4) =15,000
    s T  L  s D T  L  500 * 4  2  1225
    SS  s D T  L * normsinv(CSL)  *1.28  1570
• 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.
     Cycle and Safety Stock Inventory:
          Periodic Review Policy

• What is our average cycle inventory? Not in book
     Cycle stock = .5* D*T, same as with ROP
• Given SS needs are higher,
       What are reasons we might use Periodic Review?
            Impact of Aggregation
             on 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:
    1.   Aggregation through consolidation
    2.   Information centralization
    3.   Specialization
    4.   Product substitution
    5.   Component commonality
    6.   Postponement

Formulae for Impact of Aggregation
          D   Di

                   i 1
          s C
             D       s i 1

          s  Ls

          ss  F s (CSL)  s L
                 1          C

         Will not use covariance formulae
               Impact of Aggregation
                  (Example 11.7)
Car Dealer : 4 dealership locations (disaggregated)
D = 25 cars; sD = 5 cars; L = 2 weeks; desired CSL=0.90
• 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 = (z=1.28) x 7.07 = 9.06
• Each outlet must carry 9 cars as safety stock, so safety inventory for the 4
  outlets in total is 4*9 = 36 cars

   Impact of Aggregation, cont.
One outlet (aggregated option):
DC = D1 + D2 + D3 + D4 = 25+25+25+25 = 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 = (z=1.28) 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

  Generalization: Consolidating n
        Identical Facilities
• The optimal order quantity (EOQ) increases by a factor of                   n
• The average inventory decreases by a factor of 1/ n
   • True of both cycle and safety stock inventory
• The total number of setups decreases by a factor of 1/ n
   • This translates to a proportional decrease in setup/order costs
• The total cost decreases by a factor of 1/ n
   - 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 n 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.
            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?
        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

• 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

Value of Aggregation at Grainger
                (Table 11.4)
                 Motors         Cleaner
 Mean demand 20                 1,000
 SD of demand 40                100
 Disaggregate cv 2              0.1
 Value/Unit      $500           $30
 Disaggregate ss $105,600,000   $15,792,000
 Aggregate cv    0.05           0.0025
 Aggregate ss    $2,632,000     $394,770
 Holding Cost    $25,742,000    $3,849,308
 Saving / Unit   $7.74          $0.046
                    Product Substitution
• Use of one product to satisfy another product’s 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 Alberson’s IT Dept.)

      Component Commonality
• Using common components in a variety of different
• Can be an effective approach to exploit aggregation and
  reduce component inventories
• Can be an effective approach to reduce component
   • 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
• See example on sheet 4 of Ch11-ss-inv.xls
• 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

                    Estimating and Managing
                   Safety Inventory in Practice
      1. Account for the fact that supply chain demand is lumpy
      2. Adjust inventory policies if demand is seasonal
      3. 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. Savage’s* “Flaw of Averages”

      4. Then start with a limited pilot before rolling out company-
      5. Monitor service levels
      6. Focus on reducing safety inventories (but don’t 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:
   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
• What are the different measures of product availability?
• What managerial levers are available to lower safety inventory
  and improve product availability?


To top