Deterministic Inventory Management

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					Inventory Management

 Operations Management
  Dr. Ron Tibben-Lembke
          Purposes of Inventory

   Meet anticipated demand
     Demand variability
     Supply variability

   Decouple production & distribution
       permits constant production quantities
   Take advantage of quantity discounts
   Hedge against price increases
   Protect against shortages
2006   13.81   1857   24.0% 446   801   58 1305   9.9
                           US Inventory, GDP ($B)
14,000


12,000


10,000


 8,000


 6,000


 4,000


 2,000


   -
         84


               86


                     88


                           90


                                  92


                                          94


                                                 96


                                                        98


                                                              00


                                                                    02


                                                                          04
       19


              19


                    19


                          19


                                 19


                                        19


                                                19


                                                       19


                                                             20


                                                                   20


                                                                         20
                                Business Inventories    US GDP
                               US Inventories as % of GDP
           25.0%



           20.0%



           15.0%
% of GDP




           10.0%



           5.0%



           0.0%
               84


                     86


                           88


                                   90


                                         92


                                               94


                                                          96


                                                                98


                                                                      00


                                                                            02


                                                                                  04
              19


                    19


                          19


                                 19


                                        19


                                              19


                                                     19


                                                               19


                                                                     20


                                                                           20


                                                                                 20
                                                   Year


Source: CSCMP, Bureau of Economic Analysis
            Two Questions

Two main Inventory Questions:
   How much to buy?
   When is it time to buy?
Also:
Which products to buy?
From whom?
         Types of Inventory

   Raw Materials
   Subcomponents
   Work in progress (WIP)
   Finished products
   Defectives
   Returns
        Inventory Costs

What costs do we experience because
 we carry inventory?
           Inventory Costs

Costs associated with inventory:
   Cost of the products
   Cost of ordering
   Cost of hanging onto it
   Cost of having too much / disposal
   Cost of not having enough (shortage)
               Shrinkage Costs

   How much is stolen?
       2% for discount, dept. stores, hardware,
        convenience, sporting goods
       3% for toys & hobbies
       1.5% for all else
   Where does the missing stuff go?
       Employees: 44.5%
       Shoplifters: 32.7%
       Administrative / paperwork error: 17.5%
       Vendor fraud: 5.1%
   Inventory Holding Costs

Category                 % of Value
 Housing (building) cost          4%
 Material handling                3%
 Labor cost                       3%
 Opportunity/investment           9%
 Pilferage/scrap/obsolescence     2%
Total Holding Cost               21%
            Inventory Models

   Fixed order quantity models
     How much always same, when changes
     Economic order quantity
     Production order quantity

     Quantity discount

   Fixed order period models
       How much changes, when always same
        Economic Order Quantity

Assumptions
   Demand rate is known and constant
   No order lead time
   Shortages are not allowed
   Costs:
       S - setup cost per order
       H - holding cost per unit time
                    EOQ
   Inventory
   Level


Q*             Decrease Due to
Optimal        Constant Demand
Order
Quantity




                                 Time
               EOQ
   Inventory
   Level


Q*             Instantaneous
Optimal        Receipt of Optimal
Order          Order Quantity
Quantity




                                    Time
                     EOQ

      Inventory
      Level
 Q*



Reorder
Point
(ROP)

                           Time

             Lead Time
                     EOQ

      Inventory
      Level
 Q*
                           Average
                           Inventory Q/2
Reorder
Point
(ROP)

                              Time

             Lead Time
               Total Costs

   Average Inventory = Q/2
   Annual Holding costs = H * Q/2
   # Orders per year = D / Q
   Annual Ordering Costs = S * D/Q
   Annual Total Costs = Holding + Ordering
                   Q      D
       TC (Q)  H *  S *
                   2      Q
         How Much to Order?

Annual Cost




                          Holding Cost
                          = H * Q/2



                       Order Quantity
         How Much to Order?

Annual Cost

       Ordering Cost
       = S * D/Q



                          Holding Cost
                          = H * Q/2



                       Order Quantity
         How Much to Order?
                     Total Cost
Annual Cost          = Holding + Ordering




                         Order Quantity
         How Much to Order?
                          Total Cost
Annual Cost               = Holding + Ordering




              Optimal Q       Order Quantity
                Optimal Quantity

                        Q    D
Total Costs =         H* S*
                        2    Q
Take derivative       H      D       Set equal
with respect to Q =      S * 2 0   to zero
                      2      Q
Solve for Q:

 H DS                2 DS       2 DS
   2            Q 
                  2
                             Q
 2 Q                  H          H
            Adding Lead Time

   Use same order size           2 DS
                              Q
                                   H
   Order before inventory depleted
   R = d * L where:
     d = demand rate (per day)

     L = lead time (in days)
       both in same time period (wks, months,
        etc.)
              A Question:

   If the EOQ is based on so many
    horrible assumptions that are never
    really true, why is it the most
    commonly used ordering policy?
            Benefits of EOQ

   Profit function is very shallow
   Even if conditions don’t hold
    perfectly, profits are close to optimal
   Estimated parameters will not throw
    you off very far
                 Sensitivity

   Suppose we do not order optimal Q*, but
    order Q instead.
   Percentage profit loss given by:
        TC(Q) 1  Q * Q 
                      
        TC(Q*) 2  Q Q * 
   Should order 100, order 150 (50% over):
    0.5*(1.5 + 0.66) =1.08 an 8%cost increase
          Quantity Discounts

   How does this all change if price
    changes depending on order size?
   Holding cost as function of cost:
       H=I*C
   Explicitly consider price:
                     2DS
                  Q
                     I C
       Discount Example

D = 10,000   S = $20     I = 20%

    Price    Quantity     EOQ
c = 5.00     Q < 500      633
    4.50     501-999      666
    3.90     Q >= 1000    716
                 Discount Pricing

Total Cost

       Price 1      Price 2           Price 3


                   X 633
                    X 666
                       X 716

             500              1,000      Order Size
                 Discount Pricing

Total Cost

       Price 1      Price 2           Price 3


                   X 633
                    X 666
                       X 716

             500              1,000      Order Size
        Discount Example

Order 666 at a time:
Hold 666/2 * 4.50 * 0.2= $299.70
Order 10,000/666 * 20 = $300.00
Mat’l 10,000*4.50 = $45,000.00 45,599.70

Order 1,000 at a time:
Hold 1,000/2 * 3.90 * 0.2=$390.00
Order 10,000/1,000 * 20 = $200.00
Mat’l 10,000*3.90 = $39,000.00 39,590.00
            Discount Model

1. Compute EOQ for next cheapest price
2. Is EOQ feasible? (is EOQ in range?)
     If EOQ is too small, use lowest possible
     Q to get price.
3. Compute total cost for this quantity
4.   Repeat until EOQ is feasible or too big.
5.   Select quantity/price with lowest total
     cost.
Inventory Management
  -- Random Demand
        Master of the Obvious?

   If you focus on the things the
    customers are buying it’s a little
    easier to stay in stock
    •   James Adamson CEO, Kmart Corp.
        3/12/02
    Fired Jan, 2003
          Random Demand

   Don’t know how many we will sell
   Sales will differ by period
   Average always remains the same
   Standard deviation remains constant
    Impact of Random Demand

How would our policies change?
   How would our order quantity change?
   How would our reorder point change?
            Mac’s Decision

   How many papers to buy?
   Average = 90, st dev = 10
   Cost = 0.20, Sales Price = 0.50
   Salvage = 0.00
   Overage:     CO = 0.20 - 0.00 = 0.20
   Underage: CU = 0.50 - 0.20 = 0.30
            Optimal Policy

F(x) = Probability demand <= x
Optimal quantity:
                            Cu
           P(unit sold) 
                          Co  Cu
Mac: F(Q) = 0.3 / (0.2 + 0.3) = 0.6
From standard normal table, z = 0.253
 
=Normsinv(0.6) = 0.253
Q* = avg + zs = 90+ 2.53*10 = 90 +2.53 = 93
            Optimal Policy

   Model is called “newsboy problem,”
    newspaper purchasing decision
   If units are discrete, when in doubt,
    round up
   If u units are on hand, order Q - u
    units
              Multiple Periods

   For multiple periods,
       salvage = cost - holding cost
   Solve like a regular newsboy
          Random Demand
   If we want to satisfy all of the
    demand 95% of the time, how many
    standard deviations above the mean
    should the inventory level be?
         Probabilistic Models
Safety stock = x m
                       x m
From statistics, z 
                           s
                 Safety stock
Therefore, z =                  & Safety stock = zs
                       s
From normal table z.95 = 1.65
Safety 
       stock = zs= 1.65*10 = 16.5
ROP = m + Safety Stock
     =350+16.5 = 366.5 ≈ 367
              Random Example
   What should our reorder point be?
       demand over the lead time is 50 units,
       with standard deviation of 20
       want to satisfy all demand 90% of the time
   To satisfy 90% of the demand, z = 1.28
     Safety stock = zs = 1.28 * 20 = 25.6
   R = 50 + 25.6 = 75.6
        St Dev Over Lead Time

   What if we only know the average daily
    demand, and the standard deviation of
    daily demand?
     Lead time = 4 days,
     daily demand = 10,
     standard deviation = 5,

   What should our reorder point be, if z =
    3?
            St Dev Over LT
   If the average each day is 10, and the
    lead time is 4 days, then the average
    demand over the lead time must be
    40.
   What is the standard deviation of
    demand over the lead time?
   Std. Dev. ≠ 5 * 4
       St Dev Over Lead Time
   Standard deviation of demand =
         Ls
      4  5  10
   R = 40 + 3 * 10 = 70
          Service Level Criteria

   Type I: specify probability that you
    do not run out during the lead time
       Chance that 100% of customers go
        home happy
   Type II: proportion of demands met
    from stock
       100% chance that this many go home
        happy, on average
   Service levels easier to estimate
        Two Types of Service

Cycle   Demand   Stock-Outs
   1      180         0       Type I:
   2       75         0       8 of 10 periods
   3      235        45       80% service
   4      140         0
   5      180         0       Type II:
   6      200        10       1,395 / 1,450 =
   7      150         0              96%
   8       90         0
   9      160         0
  10       40         0
Sum     1,450        55
              Type I Service

   a = desired service level
   We want F(R) = a
   R=m+s*z
Example: a = 0.98, so z = 2.05
if m = 100, and s = 25, then
R = 100 + 2.05 * 25 = 151
         Type II Service

 b = desired service level
 Number of mad cust. = (1- b) EOQ

 L(z) = EOQ (1- b) / s

Example: EOQ = 100, b = 0.98
L(z) = 100 * 0.2 / 25 = 0.8
P. 835: z = 1.02
R = 126 -- A very different answer
     Inventory Recordkeeping

Two ways to order inventory:
 Keep track of how many delivered, sold

 Go out and count it every so often

If keeping records, still need to double-
   check
 Annual physical inventory, or

 Cycle Counting
               Cycle Counting
   Physically counting a sample of total
    inventory on a regular basis
   Used often with ABC classification
       A items counted most often (e.g., daily)
   Advantages
     Eliminates annual shut-down for
      physical inventory count
     Improves inventory accuracy
     Allows causes of errors to be identified
           Fixed-Period Model
   Answers how much to order
   Orders placed at fixed intervals
     Inventory brought up to target amount
     Amount ordered varies

   No continuous inventory count
       Possibility of stockout between intervals
   Useful when vendors visit routinely
       Example: P&G rep. calls every 2 weeks
         Fixed-Period Model:
           When to Order?

Inventory Level       Target maximum




       Period                  Time
         Fixed-Period Model:
           When to Order?

Inventory Level        Target maximum




       Period Period           Time
         Fixed-Period Model:
           When to Order?

Inventory Level        Target maximum




       Period Period           Time
         Fixed-Period Model:
           When to Order?

Inventory Level           Target maximum




       Period Period Period       Time
         Fixed-Period Model:
           When to Order?

Inventory Level           Target maximum




       Period Period Period       Time
         Fixed-Period Model:
           When to Order?

Inventory Level           Target maximum




       Period Period Period       Time
          Fixed Order Period
   Standard deviation of demand over T+L =


     s T  L  T  Ls
   T = Review period length (in days)
   σ = std dev per day
   Order quantity (12.11) =

       q  d (T  L)  zs T  L  I
                ABC Analysis
   Divides on-hand inventory into 3 classes
       A class, B class, C class
   Basis is usually annual $ volume
       $ volume = Annual demand x Unit cost
   Policies based on ABC analysis
     Develop class A suppliers more
     Give tighter physical control of A items

     Forecast A items more carefully
                Classifying Items
                     as ABC
% Annual $ Usage
  100
   80
   60
   40
            A
   20              B               C
   0
        0           50          100      150
                  % of Inventory Items
  ABC Classification Solution

Stock #   Vol.     Cost    $ Vol.     %   ABC
 206      26,000   $ 36 $936,000
 105         200    600 120,000
 019       2,000     55 110,000
 144      20,000      4   80,000
 207       7,000     10   70,000
 Total                    1,316,000
  ABC Classification Solution

Stock #   Vol.     Cost    $ Vol.    %      ABC
 206      26,000   $ 36 $936,000     71.1    A
 105         200   600 120,000        9.1    A
 019       2,000     55 110,000       8.4    B
 144      20,000      4   80,000      6.1    B
 207       7,000     10   70,000      5.3    C
 Total                    1,316,000 100.0