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					            TM 631 Optimization
                  Fall 2006
          Dr. Frank Joseph Matejcik
13th Session: Ch. 18 Inventory Theory
12/8/08




Frank Matejcik SD School of Mines & Technology   1
                       Activities
•   Review assignments and resources
•   Exam Preview
•   No Assignment this week
•   Chapter 18 alternative




Frank Matejcik SD School of Mines & Technology   2
             Tentative Schedule
          Chapters Assigned                Chapters Assigned
9/01/2008   Holiday             11/17/2008 9 9.3-3, 9.4-1, 9.5-6
9/08/2008   1, 2 ________       11/24/2008 9 9.6-1, 9.8-1
9/15/2008 3 3.1-8,3.2-4,3.6-3 12/01/2008 27 27.4-2, 27.5-3,
9/22/2008 4 4.3-6, 4.4-6, 4.7-6 27.6-2, 27.6-3, & 27.8-2
9/29/2008   6 6.3-1, 6.3-5,     12/08/2008 18 None
            and 6.8-3(abce) 12/15/2008 Final
10/06/2008 Exam 1
10/13/2008 Holiday
10/20/2008 8 8.1-5, 8.1-6, 8.2-6, 8.2-7(ab), 8.2-8
10/27/2008 8 8.4 Answers in Slides
11/03/2008 21
11/10/2008 Exam 2
 Frank Matejcik SD School of Mines & Technology   3
             Web Resources
• Class Web site on the HPCnet system
• http://sdmines.sdsmt.edu/sdsmt/directory/co
  urses/2008fa/tm631M021
• May use D2L, also. It has a password
  protection
• Answers found from the Fall 2006 site
  http://sdmines.sdsmt.edu/sdsmt/directory/co
  urses/2006fa/tm631021 Look at the Answer
  Link and the User guide and password for
  the answer link.

 Frank Matejcik SD School of Mines & Technology   4
       Exam Study Guide Short
• Just problems
• Chapter 9
   – Shortest Path Network
   – Minimum Spanning Tree
   – Maximum Flow
• Chapter 27
   – Last-Value Forecasting Method
   – Averaging Forecasting Method
   – Moving-Average
Frank Matejcik SD School of Mines & Technology   5
       Exam Study Guide Short
• Chapter 27 continued
   – Exponential Smoothing
   – Seasonally Adjusted Time Series
   – Seasonal factors
   – Exponential Smoothing Method for a Linear
     Trend
   – Forecasting Errors MAD, MSE
   – Linear Regression
• Chapter 18 no problems

Frank Matejcik SD School of Mines & Technology   6
                     The EOQ Model



      To a pessimist, the glass is half empty.
      to an optimist, it is half full.

                                                 – Anonymous




Frank Matejcik SD School of Mines & Technology       7
                   EOQ History
   – Introduced in 1913 by Ford W. Harris, “How
     Many Parts to Make at Once”
   – Interest on capital tied up in wages, material
     and overhead sets a maximum limit to the
     quantity of parts which can be profitably
     manufactured at one time; “set-up” costs on
     the job fix the minimum. Experience has
     shown one manager a way to determine the
     economical size of lots.
   – Early application of mathematical modeling to
     Scientific Management
Frank Matejcik SD School of Mines & Technology   8
            MedEquip Example
• Small manufacturer of medical diagnostic
  equipment.
• Purchases standard steel “racks” into which
  components are mounted.
• Metal working shop can produce (and sell) racks
  more cheaply if they are produced in batches due
  to wasted time setting up shop.
• MedEquip doesn’t want to tie up too much
  precious capital in inventory.
Frank Matejcik SD School of Mines & Technology   9
    EOQ Modeling Assumptions
1. Production is instantaneous – there is
   no capacity constraint and the entire lot is
   produced simultaneously.
2. Delivery is immediate – there is no time
   lag between production and availability to
   satisfy demand.
3. Demand is deterministic – there is no
   uncertainty about the quantity or timing of
   demand.

Frank Matejcik SD School of Mines & Technology   10
1.   EOQ Modeling Assumptions
4. Demand is constant over time – in fact, it
   can be represented as a straight line, so that
   if annual demand is 365 units this translates
   into a daily demand of one unit.
5. A production run incurs a fixed setup
   cost – regardless of the size of the lot or the
   status of the factory, the setup cost is
   constant.
6. Products can be analyzed singly – either
   there is only a single product or conditions
   exist that ensure separability of products.
Frank Matejcik SD School of Mines & Technology   11
                         Notation
    D       demand rate (units per year)
    c unit production cost, not counting setup or
      inventory costs (dollars per unit)
    A fixed or setup cost to place an order
      (dollars)
    h holding cost (dollars per year); if the holding
      cost is consists entirely of interest on money
      tied up in inventory, then h = ic where i is an
      annual interest rate.
    Q the unknown size of the order or lot size
          decision variable
Frank Matejcik SD School of Mines & Technology   12
       Inventory vs Time in EOQ
      Inventory
                 Model


                  Q




                      Q/D     2Q/D      3Q/D      4Q/D

                              Time

Frank Matejcik SD School of Mines & Technology   13
                          Costs
 • Holding Cost:          average inventory 
                                               Q
                                               2
                                               hQ
                         annual holding cost 
                                                2
                                               hQ
                           unit holding cost 
                                               2D


 • Setup Costs: A per lot, so                   unit setup cost 
                                                                    A
                                                                    Q



 • Production Cost: c per unit
                                         hQ A
                               Y (Q)       c
 • Cost Function:                        2D Q
Frank Matejcik SD School of Mines & Technology            14
      MedEquip Example Costs

    • D = 1000 racks per year

    • c = $250

    • A = $500 (estimated from supplier’s
       pricing)

    • h = (0.1)($250) + $10 = $35 per unit per
       year
Frank Matejcik SD School of Mines & Technology   15
                Costs in EOQ Model
                           20.00

                           18.00

                           16.00

                           14.00
           Cost ($/unit)




                           12.00

                           10.00                                   Y(Q)
                                                 Q* =169
                            8.00

                            6.00                                          hQ/2D

                            4.00

                            2.00       c                          A/Q

                            0.00
                                   0       100      200     300         400   500

                                                 Order Quantity (Q)
Frank Matejcik SD School of Mines & Technology                                      16
       Economic Order Quantity

              dY (Q)    h   A
                          2 0
               dQ      2D Q


                     2 AD
              Q*                EOQ Square Root Formula
                       h



                     2(500)(1000)
              Q*                  169    MedEquip Solution
                          35


Frank Matejcik SD School of Mines & Technology   17
    EOQ Modeling Assumptions
      1. Production is instantaneous – there is
       no capacity constraint
                                      relax via
       and the entire lot is produced EPL model
       simultaneously.




Frank Matejcik SD School of Mines & Technology   18
          Notation – EPL Model
    D       demand rate (units per year)
    P production rate (units per year), where P>D
    c unit production cost, not counting setup or
      inventory costs (dollars per unit)
    A fixed or setup cost to place an order
      (dollars)
    h holding cost (dollars per year); if the holding
      cost is consists entirely of interest on money
      tied up in inventory, then h = ic where i is an
      annual interest rate.            decision variable
    Q the unknown size of the production lot size
Frank Matejcik SD School of Mines & Technology   19
Inventory vs Time in EPL Model

                          Production run of Q takes Q/P time units

 (P-D)(Q/P)
   Inventory




               -D   P-D
                                                                (P-D)(Q/P)/2




                                   Time




Frank Matejcik SD School of Mines & Technology          20
            Solution to EPL Model
• Annual Cost Function:
                                   AD h(1  D / P )Q
                        Y (Q )                      Dc
                                   Q        2

                                   setup   holding   production


• Solution (by taking derivative and setting
  equal to zero):
                                           • tends to EOQ as P
                     2 AD
           Q* 
                  h(1  P / D)             • otherwise larger than EOQ
                                             because replenishment takes
                                             longer


Frank Matejcik SD School of Mines & Technology         21
        The Key Insight of EOQ
   There is a tradeoff between lot
    size and inventory

• Order Frequency:            F
                                 D
                                 Q



                                      cQ cD
• Inventory Investment:            I    
                                       2   2F

Frank Matejcik SD School of Mines & Technology   22
                                 EOQ Tradeoff Curve
                            50
                            45
     Inventory Investment




                            40
                            35
                            30
                            25
                            20
                            15
                            10
                             5
                             0
                                 0   20   40      60   80   100
                                          Order/Year

Frank Matejcik SD School of Mines & Technology         23
    Sensitivity of EOQ Model to
              Quantity
• Optimal Unit Cost:
                                  hQ* A            We neglect unit cost, c,
                     Y  Y (Q ) 
                       *          *
                                     
                                  2 D Q*           since it does not affect Q*
                               h 2 AD h     A
                                       
                                 2D       2 AD h
                                 2A
                           
                                2 AD h




• Optimal Annual Cost: Multiply Y* by D and
  simplify,    Annual Cost  2 ADh

Frank Matejcik SD School of Mines & Technology         24
    Sensitivity of EOQ Model to
          Quantity (cont.)
• Annual Cost from Using Q':
                                     hQ AD
                          Y (Q)       
                                      2   Q



             Cost(Q) Y (Q) hQ 2  AD Q 1  Q Q* 
                                          * 
• Ratio:     Cost(Q* ) Y (Q* )   2 ADh     2  Q Q 




• Example: If Q' = 2Q*, then the ratio of the actual
  to optimal cost is           (1/2)[2 + (1/2)] = 1.25

Frank Matejcik SD School of Mines & Technology     25
    Sensitivity of EOQ Model to
           Order Interval
• Order Interval: Let T represent time (in years)
  between orders (production runs)
                                               Q
                                            T
                                               D
• Optimal Order Interval:
                         2 AD
                 *
          Q                h           2A
      T *
            
          D               D             hD
Frank Matejcik SD School of Mines & Technology   26
    Sensitivity of EOQ Model to
      Order Interval (cont.)
• Ratio of Actual to Optimal Costs: If we use T'
  instead of T* annual cost under T      1 T T* 
                                           * 
                   annual cost under T *
                                          2 T  T 
• Powers-of-Two Order Intervals: The optimal
  order interval, T* must lie within a multiplicative
  factor of 2 of a “power-of-two.” Hence, the
  maximum error from using the best power-of-two
  is
                     1     1 
                       2  2   1.06
                     2       
Frank Matejcik SD School of Mines & Technology   27
        The “Root-Two” Interval



        2m       T1*      2m 2         T2*            2 m1
              divide by            multiply by
              less than            less than
              2 to get            2 to get
              to 2m                to 2m+1




Frank Matejcik SD School of Mines & Technology   28
                Medequip Example
• Optimum: Q*=169, so T*=Q*/D
  =169/1000 years = 62 days
              hQ * AD 35(169 ) 500 (1000 )
   Y (Q*)                               $5,916
               2    Q*   2        169
• Round to Nearest Power-of-Two: 62 is
  between 32 and 64, but since
  322=45.25, it is “closest” to 64. So,
  round to T’=64 days or Q’= Only 0.07%were lucky
                                because we
                                           error

  T’D=(64/365)1000=175.         and happened to be
              hQ' AD 35(175 ) 500 (1000 )     close to a power-of-two.
   Y (Q ' )                            $5,920
                                              But we can’t do worse
               2   Q'   2        175
                                              than 6%.
Frank Matejcik SD School of Mines & Technology 29
 Powers-of-Two Order Intervals

 Order Interval                   Week
                  0   1   2   3     4    5       6    7   8
       1 2   0




       2  21

       4  22

       8  23


Frank Matejcik SD School of Mines & Technology   30
               EOQ Takeaways
• Batching causes inventory (i.e., larger lot
  sizes translate into more stock).
• Under specific modeling assumptions the
  lot size that optimally balances holding
  and setup costs is given by the square
  root formula: *        2 AD
                      Q 
                            h
• Total cost is relatively insensitive to lot
  size (so rounding for other reasons, like
  coordinating shipping, may be attractive).
Frank Matejcik SD School of Mines & Technology   31
             The Wagner-Whitin Model




    Change is not made without inconvenience,
    even from worse to better.
                                         – Robert Hooker




Frank Matejcik SD School of Mines & Technology   32
             EOQ Assumptions
   1. Instantaneous production.
   2. Immediate delivery.
   3. Deterministic demand.
   4. Constant demand. WW model relaxes this one
   5. Known fixed setup costs.
   6. Single product or separable products.


Frank Matejcik SD School of Mines & Technology   33
    Dynamic Lot Sizing Notation
   t a period (e.g., day, week, month); we will
      consider t = 1, … ,T, where T represents the
      planning horizon.
   Dt        demand in period t (in units)
   ct        unit production cost (in dollars per
      unit), not counting setup or inventory costs in
      period t
   At        fixed or setup cost (in dollars) to place
      an order in period t
   ht        holding cost (in dollars) to carry a unit
      of inventory from period t to period t +1
   Qt        the unknown size of the order or lot
      size in period t     decision variables
Frank Matejcik SD School of Mines & Technology   34
        Wagner-Whitin Example
  • Data
   t     1   2   3   4   5   6   7   8   9   10
   Dt    20 50 10 50 50 10 20 40 20 30
   ct    10 10 10 10 10 10 10 10 10 10
   At   100 100 100 100 100 100 100 100 100 100
   ht    1   1   1   1   1   1   1   1   1   1


  • Lot-for-Lot Solution
   t               1     2     3     4     5     6     7     8     9     10   Total
   Dt              20    50    10    50    50    10    20    40    20    30    300
   Qt              20    50    10    50    50    10    20    40    20    30    300
   It              0     0     0     0     0     0     0     0     0     0      0
   Setup cost     100   100   100   100   100   100   100   100   100   100   1000
   Holding cost    0     0     0     0     0     0     0     0     0     0      0
   Total cost     100   100   100   100   100   100   100   100   100   100   1000
Frank Matejcik SD School of Mines & Technology                    35
 Wagner-Whitin Example (cont.)
• Fixed Order Quantity Solution

    t               1    2    3   4  5    6   7  8    9    10 Total
    Dt              20   50   10 50 50    10 20 40    20   30 300
    Qt             100   0    0 100 0     0 100 0     0    0 300
    It              80   30   20 70 20    10 90 50    30   0   0
    Setup cost     100   0    0 100 0     0 100 0     0    0 300
    Holding cost    80   30   20 70 20    10 90 50    30   0 400
    Total cost     180   30   20 170 20   10 190 50   30   0 700




Frank Matejcik SD School of Mines & Technology        36
        Wagner-Whitin Property
•Under an optimal lot-sizing policy either
the inventory carried to period t+1 from a
previous period will be zero or the
production quantity in period t+1 will be
zero.




Frank Matejcik SD School of Mines & Technology   37
   Basic Idea of Wagner-Whitin
            Algorithm
•By WW Property I, either Qt=0 or
Qt=D1+…+Dk for some k. If jk* = last period of
production in a k period problem then we will
produce exactly Dk+…DT in period jk*.
•We can then consider periods 1, … , jk*-1 as if
they are an independent jk*-1 period problem.




Frank Matejcik SD School of Mines & Technology   38
        Wagner-Whitin Example
 • Step 1: Obviously, just satisfy D1 (note we
   are neglecting production cost, since it is
   fixed). Z *  A  100
               1        1

              j1*  1
 • Step 2: Two choices, either j2* = 1 or j2* = 2.
                            A1  h D , produce in 1
                   Z*
                      min  * 1 2
                            Z1  A2 , produce in 2
                    2


                           100  1(50)  150
                      min 
                            100  100  200
                      150


                   j2  1
                    *


Frank Matejcik SD School of Mines & Technology         39
 Wagner-Whitin Example (cont.)
• Step3: Three choices, j3* = 1, 2, 3.
                    A1  h1 D2  (h1  h2 ) D3 , produce in 1
                    *
         Z 3  min Z1  A2  h2 D3 ,
           *
                                                  produce in 2
                    Z*  A3 ,
                    2                            produce in 3
                   100  1(50)  (1  1)10  170
                   
              min 100  100  (1)10         210
                   150  100
                                             250
              170


          j3  1
           *


Frank Matejcik SD School of Mines & Technology    40
 Wagner-Whitin Example (cont.)

 • Step 4: Four choices, j4* = 1, 2, 3, 4.
                   A1  h1 D2  (h1  h2 ) D3  (h1  h2  h3 ) D4 , produce in 1
                   Z*  A  h D  ( h  h ) D ,
                   1                                                produce in 2
        Z 4  min  *
          *                2    2 3       2    3   4

                   Z 2  A3  h3 D4 ,                               produce in 3
                   Z*  A4 ,
                   3                                                produce in 4
                  100  1(50)  (1  1)10  (1  1  1)50  320
                  100  100  (1)10  (1  1)50              310
                  
             min 
                  150  100  (1)50                          300
                  170  100
                                                             270
             270


         j4  4
          *



Frank Matejcik SD School of Mines & Technology                        41
      Planning Horizon Property
•If jt*=t, then the last period in which
production occurs in an optimal t+1
period policy must be in the set t,
t+1,…t+1.
•In the Example:
     –We produce in period 4 for period 4 of a
      4 period problem.
     –We would never produce in period 3 for
      period 5 in a 5 period problem.
Frank Matejcik SD School of Mines & Technology   42
 Wagner-Whitin Example (cont.)
• Step 5: Only two choices, j5* = 4, 5.
                  Z 3  A4  h4 D5 , produce in 4
                     *
          Z  min  *
           *

                   Z 4  A5 ,
           5
                                      produce in 5
                  170  100  1(50)  320
             min 
                  270  100            370
                320


           j5  4
            *


• Step 6: Three choices, j6* = 4, 5, 6.
   And so on.
Frank Matejcik SD School of Mines & Technology       43
        Wagner-Whitin Example
              Solution
       Last Period                   Planning Horizon (t)
     with Production     1   2   3   4    5    6    7      8                          9       10
            1           100 150 170 320
            2               200 210 310
            3                   250 300
            4                       270 320 340 400 560
            5                            370 380 420 540
            6                                 420 440 520
            7                                      440 480                          520      610
            8                                             500                       520      580
            9                                                                       580      610
           10                                                                                620
            Zt          100 150 170 270 320 340 400 480                             520      580
           jt            1       1      1       4      4      4       4      7     7 or 8      8
                       Produce in period 1      Produce in period 4         Produce in period 8
                       for 1, 2, 3 (20 + 50 +   for 4, 5, 6, 7 (50 + 50 +   for 8, 9, 10 (40 + 20 +
                       10 = 80 units)           10 + 20 = 130 units)        30 = 90 units
Frank Matejcik SD School of Mines & Technology                               44
        Wagner-Whitin Example
           Solution (cont.)
 • Optimal Policy:
   – Produce in period 8 for 8, 9, 10 (40 +
     20 + 30 = 90 units)
   – Produce in period 4 for 4, 5, 6, 7 (50 +
     50 + 10 + 20 = 130 units)
   – Produce in period 1 for 1, 2, 3 (20 +
     50 + 10 = 80 units)
      Note: we produce in 7 for an 8 period problem, but this never
      comes into play in optimal solution.
Frank Matejcik SD School of Mines & Technology          45
        Wagner-Whitin Example
           Solution (cont.)


 t               1    2    3  4  5        6    7  8  9        10 Total
 Dt              20   50   10 50 50       10   20 40 20       30 300
 Qt              80   0    0 130 0        0    0 90 0         0 300
 It              60   10   0 80 30        20   0 50 30        0   0
 Setup cost     100   0    0 100 0        0    0 100 0        0 300
 Holding cost    60   10   0 80 30        20   0 50 30        0 280
 Total cost     160   10   0 180 30       20   0 150 30       0 580

      Note: we produce in 7 for an 8 period problem, but this never
      comes into play in optimal solution.
Frank Matejcik SD School of Mines & Technology          46
  Problems with Wagner-Whitin
   • 1. Fixed setup costs.
   • 2. Deterministic demand and production
     (no uncertainty)
   • 3. Never produce when there is
     inventory (WW Property I).
        –safety stock (don't let inventory fall to
         zero)
        –random yields (can't produce for
         exact no. periods)
Frank Matejcik SD School of Mines & Technology   47
         Statistical Reorder Point Models



       When your pills get down to four,
       Order more.

                    – Anonymous, from Hadley &Whitin




Frank Matejcik SD School of Mines & Technology   48
             EOQ Assumptions
   1. Instantaneous production.
                            EPL model relaxes this one
   2. Immediate delivery.
                      lags can be added to EOQ or other models
   3. Deterministic demand. and (Q,r) relax this one
                         newsvendor
   4. Constant demand. WW model relaxes this one
   5. Known fixed setup costs. can use constraint approach
   6. Single product or separable products.
                                 Chapter 17 extends (Q,r) to
                                 multiple product cases



Frank Matejcik SD School of Mines & Technology   49
      Modeling Philosophies for
       Handling Uncertainty
   1. Use deterministic model – adjust
    solution
      - EOQ to compute order quantity, then add
     safety stock
      - deterministic scheduling algorithm, then
     add safety lead time
   2. Use stochastic model
      - news vendor model
      - base stock and (Q,r) models
      - variance constrained investment models
Frank Matejcik SD School of Mines & Technology   50
     The Newsvendor Approach
• Assumptions:
   1. single period
   2. random demand with known distribution
   3. linear overage/shortage costs
   4. minimum expected cost criterion
• Examples:
   – newspapers or other items with rapid
     obsolescence
   – Christmas trees or other seasonal items
   – capacity for short-life products
Frank Matejcik SD School of Mines & Technology   51
    Newsvendor Model Notation
         X  demand (in units), a random variable.

      G ( x)  P( X  x), cumulative distributi on function of demand
             (assumed continuous.)

                 d
      g ( x)       G ( x)  density function of demand.
                 dx

        co  cost (in dollars) per unit left over after demand is realized.


        c s  cost (in dollars) per unit of shortage.


         Q  production/order quantity (in units); this is the decision variable.

Frank Matejcik SD School of Mines & Technology                       52
              Newsvendor Model
• Cost Function:                                               Note: for any given
                                                               day, we will be either
      Y ( x)  expected overage  expectedshortage cost        over or short, not both.
                                                               But in expectation,
            co E units over  c s E units short           overage and shortage
                                                               can both be positive.
                                               
            co  maxQ  x,0g ( x)dx  c s  maxx  Q,0g ( x)dx
                 0                            0



                 Q                        
            co  (Q  x) g ( x)dx  c s  ( x  Q) g ( x)dx
                 0                        Q




Frank Matejcik SD School of Mines & Technology                   53
      Newsvendor Model (cont.)
• Optimal Solution: taking derivative of Y(Q)
  with respect to Q, setting equal to zero, and
  solving yields:             c
                                      
                     G(Q* )  P X  Q*            s
                                               co  c s
                                                               Critical Ratio is
                                                               probability stock
                                                               covers demand


                     Q *  co
                                           1
                                      cs                                G(x)
• Notes:             Q *  cs       co  c s


                                                          Q*
Frank Matejcik SD School of Mines & Technology                 54
Newsvendor Example – T Shirts
• Scenario:
   – Demand for T-shirts is exponential with mean
     1000 (i.e., G(x) = P(X  x) = 1- e-x/1000). (Note
     - this is an odd demand distribution; Poisson
     or Normal would probably be better modeling
     choices.)
   – Cost of shirts is $10.
   – Selling price is $15.
   – Unsold shirts can be sold off at $8.
• Model Parameters: cs = 15 – 10 = $5
                     co = 10 – 8 = $2
Frank Matejcik SD School of Mines & Technology   55
Newsvendor Example – T Shirts
           (cont.)
• Solution:
                                      Q
                                                 cs      5
                G (Q )  1  e
                     *               1000
                                                             0.714
                                                co  cs 2  5
                    Q*  1,253

• Sensitivity: If co = $10 (i.e., shirts must be
  discarded) then
                                      Q
                                                 cs      5
                G (Q )  1  e
                    *                1000
                                                              0.333
                                                co  cs 10  5
                   Q*  405



Frank Matejcik SD School of Mines & Technology                           56
Newsvendor Model with Normal
         Demand
• Suppose demand is normally distributed
  with mean  and standard deviation .
  Then the critical ratio formula reduces to:
                     Q *  
                                                   3.00



                                cs                                (z)
        G (Q * )          
                       co  c s

       Q *                            cs
                  z where ( z )                                                                                  0              z
                                     co  c s
                                                   0.00
                                                          1   7   13   19   25   31   37   43   49   55   61   67   73   79   85   91   97 103 109 115 121 127 133 139 145 151 157




                            Note: Q* increases in both
              Q*    z
                             and  if z is positive (i.e.,
                            if ratio is greater than 0.5).
Frank Matejcik SD School of Mines & Technology 57
       Multiple Period Problems
• Difficulty: Technically, Newsvendor model
  is for a single period.
• Extensions: But Newsvendor model can be
  applied to multiple period situations,
  provided:
   – demand during each period is iid,
     distributed according to G(x)
   – there is no setup cost associated with
     placing an order
   – stockouts are either lost or backordered
Frank Matejcik SD School of Mines & Technology   58
                       Example

• Scenario:
   – GAP orders a particular clothing item every
     Friday
   – mean weekly demand is 100, std dev is 25
   – wholesale cost is $10, retail is $25
   – holding cost has been set at $0.5 per week (to
     reflect obsolescence, damage, etc.)
• Problem: how should they set order amounts?
Frank Matejcik SD School of Mines & Technology   59
                Example (cont.)
• Newsvendor Parameters:
   c0 = $0.5
   cs = $15
                           15
               G (Q * )          0.9677
                        0.5  15
                 Q  100 
• Solution:    
                 25 
                            0.9677

               Q  100                    Every Friday, they should
                        1.85             order-up-to 146, that is, if
                 25
               Q  100  1.85(25)  146 there are x on hand, then
                                          order 146-x.

Frank Matejcik SD School of Mines & Technology         60
       Newsvendor Takeaways
• Inventory is a hedge against demand
  uncertainty.
• Amount of protection depends on
  “overage” and “shortage” costs, as well as
  distribution of demand.
• If shortage cost exceeds overage cost,
  optimal order quantity generally increases
  in both the mean and standard deviation of
  demand.
Frank Matejcik SD School of Mines & Technology   61
            The (Q,r) Approach
• Assumptions:
   1. Continuous review of inventory.
   2. Demands occur one at a time.
   3. Unfilled demand is backordered.
   4. Replenishment lead times are fixed and
     known.
• Decision Variables:
   – Reorder Point: r – affects likelihood of
     stockout (safety stock).
   – Order Quantity: Q – affects order frequency
     (cycle inventory).
Frank Matejcik SD School of Mines & Technology   62
Inventory vs Time in (Q,r) Model
     Inventory




                     Q

                 r
                         l

                              Time



Frank Matejcik SD School of Mines & Technology   63
 The Single Product (Q,r) Model
• Motivation: Either
  1. Fixed cost associated with
    replenishment orders and cost per
    backorder.
  2. Constraint on number of replenishment
    orders per year and service constraint.
                min fixed setup cost  holding cost  backordercost
                 Q,r

• Objective: Under (1)                  As in EOQ, this makes
                                        batch production attractive.

Frank Matejcik SD School of Mines & Technology            64
        Summary of (Q,r) Model
            Assumptions
1. One-at-a-time demands.
2. Demand is uncertain, but stationary over
   time and distribution is known.
3. Continuous review of inventory level.
4. Fixed replenishment lead time.
5. Constant replenishment batch sizes.
6. Stockouts are backordered.
Frank Matejcik SD School of Mines & Technology   65
                       (Q,r) Notation
       D  expected demand per year
         replenishment lead time (assumed constant)
      X  (random) demand during replenishment lead time
        E[ X ]  expected demand during replenishment lead time
        standard deviation of demand during replenishment lead time
     p(x)  P( X  x)  pmf of demand during lead time
    G ( x)  P( X  x)  cdf of demand during lead time
       A  fixed cost per order
       c  unit cost of an item
       h  annual unit holding cost
       k  cost per stockout
       b  annual unit backorder cost


Frank Matejcik SD School of Mines & Technology             66
             (Q,r) Notation (cont.)
• Decision Variables:
      Q       order quantity
       r      reorder point
       s      r    safety stock implied by r
• Performance Measures:
       F (Q )  average order frequency
     S (Q, r )  average service level (fill rate)
     B (Q, r )  average backorder level
      I (Q, r )  average inventorylevel



Frank Matejcik SD School of Mines & Technology       67
Inventory and Inventory Position
          for Q=4, r=4
           9

           8                                                                                       Inventory Position
                                                                                                   uniformly distributed
           7
                                                                                                   between r+1=5 and
           6                                                                                       r+Q=8
           5
Quantity




           4

           3

           2

           1

           0
                0   2   4   6   8   10   12   14    16    18    20   22   24   26   28   30   32
           -1

           -2

                                                   Time

                                    Inventory Position         Net Inventory

Frank Matejcik SD School of Mines & Technology                                           68
            Costs in (Q,r) Model
  • Fixed Setup Cost: AF(Q)

  • Stockout Cost: kD(1-S(Q,r)), where k is
    cost per stockout

  • Backorder Cost: bB(Q,r)

  • Inventory Carrying Costs: cI(Q,r)
Frank Matejcik SD School of Mines & Technology   69
Fixed Setup Cost in (Q,r) Model
• Observation: since the number of orders
  per year is D/Q,
                               D
                        F(Q) 
                               Q




Frank Matejcik SD School of Mines & Technology   70
     Stockout Cost in (Q,r) Model
• Key Observation: inventory position is
    uniformly distributed between r+1 and r+Q.
    So, service in (Q,r) model is weighted sum
    of service in base stock model.
                     1 r Q
                         
                                          1
        S (Q, r )             G( x  1)  [G(r )      G (r  Q  1)]
                    Q x  r 1            Q
•   Result:             1                     Note: this form is easier to use
        S (Q, r )  1  [ B(r )  B(r  Q)] in spreadsheets because it does
                       Q
                                               not involve a sum.



Frank Matejcik SD School of Mines & Technology           71
  Service Level Approximations
• Type I (base stock):
         S (Q, r )  G(r )             Note: computes number
                                       of stockouts per cycle,
                                       underestimates S(Q,r)



• Type II:                     B(r )   Note: neglects B(r,Q)
               S (Q, r )  1          term, underestimates S(Q,r)
                                Q



Frank Matejcik SD School of Mines & Technology   72
 Backorder Costs in (Q,r) Model
• Key Observation: B(Q,r) can also be
  computed by averaging base stock
  backorder level function over the range
  [r+1,r+Q].
                                1 r Q            1
                     B(Q, r )     
                                Q x  r 1
                                           B( x)  [ B(r  1)      B(r  Q)]
                                                  Q
• Result:
 Notes:
 1. B(Q,r) B(r) is a base stock approximation for backorder level.

 2. If we can compute B(x) (base stock backorder level function),
   then we can compute stockout and backorder costs in (Q,r) model.
Frank Matejcik SD School of Mines & Technology                73
  Inventory Costs in (Q,r) Model
• Approximate Analysis: on average
  inventory declines from Q+s to s+1 so
                           (Q  s )  ( s  1) Q  1      Q 1
             I (Q, r )                             s        r 
                                   2            2          2




• Exact Analysis: this neglects backorders,
  which add to average inventory since on-
                          Q 1
              I (Q, r )        r    B (Q, r )
  hand inventory can       2 never go below zero.
  The corrected version turns out to be
Frank Matejcik SD School of Mines & Technology                  74
 Inventory vs Time in (Q,r) Model
                   Expected Inventory     Actual Inventory
                                                                  Exact I(Q,r) =
             s+Q                                                  Approx I(Q,r)
                                                                   + B(Q,r)
 Inventory




        r                                                         Approx I(Q,r)
s+1=r-+1


                                        Time



 Frank Matejcik SD School of Mines & Technology              75
       Expected Inventory Level for
             Q=4, r=4, =2
                     7


  s+Q 6

                     5
   Inventory Level




                     4


                     3


            s 2

                     1


                     0
                         0   5   10   15          20   25    30   35

                                           Time

Frank Matejcik SD School of Mines & Technology              76
(Q,r) Model with Backorder Cost
• Objective Function:
             D
  Y (Q, r )  A  bB(Q, r )  hI (Q, r )
             Q
• Approximation: B(Q,r) makes optimization
   complicated because it depends on both Q
   and r. To simplify, approximate with base
   stock backorder formula, B(r):
            ~          D                Q 1
Y (Q, r )  Y (Q, r )  A  bB(r )  h(       r    B(r ))
                       Q                 2
Frank Matejcik SD School of Mines & Technology   77
         Results of Approximate
             Optimization
• Assumptions:
   – Q,r can be treated as continuous variables
   – G(x) is a continuous cdf
• Results:
                  2 AD
          Q*               Note: this is just the EOQ formula
                    h
                     b                          Note: this is just the
          G (r*)         r *    z
                   hb                          base stock formula
                         if G is normal(,),
                         where (z)=b/(h+b)

Frank Matejcik SD School of Mines & Technology        78
                  (Q,r) Example
    Stocking Repair Parts:
    D = 14 units per year
    c = $150 per unit
    h = 0.1 × 150 + 10 = $25 per unit
    l = 45 days
    = (14 × 45)/365 = 1.726 units during
      replenishment lead time
    A = $10 b = $40
    Demand during lead time is Poisson
Frank Matejcik SD School of Mines & Technology   79
          Values for Poisson()
              Distribution
           r       p(r)       G(r)       B(r)
           0      0.178      0.178       1.726
           1      0.307      0.485       0.904
           2      0.265      0.750       0.389
           3      0.153      0.903       0.140
           4      0.066      0.969       0.042
           5      0.023      0.991       0.011
           6      0.007      0.998       0.003
           7      0.002      1.000       0.001
           8      0.000      1.000       0.000
           9      0.000      1.000       0.000
          10      0.000      1.000       0.000



                                                      80
Frank Matejcik SD School of Mines & Technology   80
       Calculations for Example
            2 AD        2(10)(14)
    Q*                           4.3  4
              h            15

      b     40
                  0.615
    h  b 25  40

     (0.29)  0.615, so z  0.29


    r*    z  1.726  0.29(1.314)  2.107  2
Frank Matejcik SD School of Mines & Technology   81
      Performance Measures for
              Example
                   D   14
      F (Q*)             3.5
                   Q* 4

                    1                              1
  S(Q * ,r * )  1   [ B (r*)  B (r * Q*)]  1  [ B (2)  B (2  4)]
                   Q*                              Q
                   1
               1  [0.389  0.003]  0.904
                   4

                 1 r * Q *           1
   B (Q*, r*)        
                Q * x  r *1
                              B ( x)  [ B(3)  B(4)  B(5)  B (6)]
                                      Q
                1
               [0.140  0.042  0.011 0.003]  0.049
                4

                   Q * 1                         4 1
    I (Q*, r*)            r *   B (Q*, r*)        2  1.726  0.049  2.823
                     2                              2

Frank Matejcik SD School of Mines & Technology                         82
      Observations on Example

• Orders placed at rate of 3.5 per year

• Fill rate fairly high (90.4%)

• Very few outstanding backorders (0.049
  on average)

• Average on-hand inventory just below 3
  (2.823)
Frank Matejcik SD School of Mines & Technology   83
•
                       Example
         Varying theorder twice as often
    Change: suppose we
    so F=7 per year, then Q=2 and:
                       1                          1
     S (Q, r )  1      [ B(r )  B(r  Q)]  1  [0.389  0.042]  0.826
                       Q                          2
• which may be too low, so increase r from 2
  to 3:   1                 1
     S (Q, r )  1      [ B(r )  B(r  Q)]  1  [0.140  0.011]  0.936
                       Q                          2
• This is better. For this policy (Q=2, r=4) we
  can compute B(2,3)=0.026, I(Q,r)=2.80.
• Conclusion: this has higher service and
  lower inventory than the original policy
  (Q=4, r=2). But the cost of achieving this is
  an extra 3.5 replenishment orders per year.
Frank Matejcik SD School of Mines & Technology             84
  (Q,r) Model with Stockout Cost
• Objective Function:
            D
 Y (Q, r )  A  kD(1  S (Q, r ))  hI (Q, r )
            Q
• Approximation: Assume we can still use
   EOQ to compute Q* but replace S(Q,r) by
   Type II approximation and B(Q,r) by base
   stock approximation:
            ~          D       B(r )      Q 1
Y (Q, r )  Y (Q, r )  A  kD        h(       r    B(r ))
                       Q        Q          2
Frank Matejcik SD School of Mines & Technology   85
         Results of Approximate
             Optimization
• Assumptions:
   – Q,r can be treated as continuous variables
   – G(x) is a continuous cdf
• Results:
                2 AD
         Q*                 Note: this is just the EOQ formula
                  h
                    kD                               Note: another version
         G(r*)              r*    z
                 kD  hQ                             of base stock formula
                         if G is normal(,),        (only z is different)
                         where (z)=kD/(kD+hQ)

Frank Matejcik SD School of Mines & Technology         86
   Backorder vs. Stockout Model
• Backorder Model
   – when real concern is about stockout time
   – because B(Q,r) is proportional to time orders wait for
     backorders
   – useful in multi-level systems
• Stockout Model
   – when concern is about fill rate
   – better approximation of lost sales situations (e.g., retail)
• Note:
   – We can use either model to generate frontier of solutions
   – Keep track of all performance measures regardless of
     model
   – B-model will work best for backorders, S-model for
     stockouts
 Frank Matejcik SD School of Mines & Technology   87
           Lead Time Variability
• Problem: replenishment lead times may be variable,
  which increases variability of lead time demand.
• Notation:
    L = replenishment lead time (days), a random variable
    l   = E[L] = expected replenishment lead time (days)
    L = std dev of replenishment lead time (days)
    Dt = demand on day t, a random variable, assumed
      independent and identically distributed
    d = E[Dt] = expected daily demand
    D= std dev of daily demand (units)
Frank Matejcik SD School of Mines & Technology   88
 Including Lead Time Variability
           in Formulas
• Standard Deviation of Lead Time Demand:
                         if demand is Poisson

             D  d 2 L    d 2 L
                 2        2            2



                                                Inflation term due to
                                                lead time variability
          R    z    z   d 2 L
                                      2

                                                Note:  can be used in any
• Modified Base Stock                           base stock or (Q,r) formula
                                                as before. In general, it will
  Formula (Poisson demand                       inflate safety stock.
  case):
Frank Matejcik SD School of Mines & Technology         89
        Single Product (Q,r) Insights
•   Basic Insights:
     – Safety stock provides a buffer against stockouts.
    – Cycle stock is an alternative to setups/orders.
•   Other Insights:
    1. Increasing D tends to increase optimal order quantity Q.
    2. Increasing  tends to increase the optimal reorder point.
      (Note: either increasing D or l increases .)
    3. Increasing the variability of the demand process tends to
      increase the optimal reorder point (provided z > 0).
    4. Increasing the holding cost tends to decrease the optimal
      order quantity and reorder point.
    Frank Matejcik SD School of Mines & Technology   90

				
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