Simulating Inventory Control with Orders that Cross during

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					Sheet: Introduction          File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                         Page 1 of 41

         Simulating Inventory Control with Orders that Cross during Lead Time
                                      sS.xls Version 1.5 03/18/2010
                                        John.O.McClain@cornell.edu
                                  Johnson Graduate School of Management
                                             Cornell University
                                              Ithaca NY 14853
This workbook is intended for teaching or research. You are welcome to use it in any manner,
and change it as you see fit. It comes without any guarantee whatsoever, and is distributed
free of charge. Changes are frequent, so check back frequently for a new version.

Most inventory control systems use a formula for lead-time demand to set safety stock levels.
Research has uncovered situations where that method leads to large and expensive errors.*
In particular, if replenishment orders might not arrive in the same order in which they are placed, then
the above method will leave you with too much inventory if your objective is a high level of protection,
and too little inventory if you are aiming to run out of stock frequently. That is, the variance of the
inventory level is smaller than the variance of lead-time demand.
     *See Robinson, L.R, J.R. Bradley and L.J. Thomas, "Consequences of Order Crossover under
     Order-up-to Inventory Policies." M&SOM Manufacturing and Service Operations Management ,
     Volume 3, No. 3 (2001), pp.175-188.

This workbook contains a macro, written in Visual Basic, that allows you to simulate the inventory control
system known variously as the Min-Max system, the (s,S) system, or the reorder-level, order-up-to system.
Two versions are available in the simulation:
  > Periodic review: orders may be placed only at specific points of time, such as daily or weekly.
  > Continuous review: orders are placed instantly, as soon as inventory reaches the reorder level.
In both cases, the state of the system is tracked at all times, so that accurate costs may be calculated.

The word "order" refers to an action taken to replenish the supply of an item that is stocked in inventory
and sold to customers. The word "demand" refers to a customer wanting to buy one unit of the item.
A "backorder" is an unsatisfied demand for which the customer will take delivery at a later time.
The simulation assumes that all customers are willing to wait if their demand is backordered.

The simulation allows orders to cross. It assumes that the lead time of one order is independent of
that for any other order, and therefore crossing occurs whenever the lead time for an order is longer
than the interval between orders plus the lead time for the next order.

Please note that the independence assumption is not true in some real circumstances. For example,
if both orders are shipped by rail, and if one freight car cannot pass another, the orders cannot cross.
However, in that case lead times are also not independent, but rather are positively correlated, so
models that assume independence (i.e. most inventory models) are also incorrect.
Sheet: Introduction         File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                     Page 2 of 41

Contents: These are the sheets in this workbook.
  Introduction (this sheet), with the following sections:
          1. Measuring Inventory and Shortages at Time of Delivery
               (a) Shortfall below Reorder Level at Delivery: Shortfall@Deliv
               (b) Distribution of Shortfall@Delivery if Orders do not Cross
                           (i) Continuous Review
                           (ii) Periodic Review
          2. Inventory and Shortages at Any Time
               (a) Shortfall below Order-up-to Level
               (b) Average Inventory and Shortages
          3. Cost of Inventory, Backorders and Ordering:
          4. Optimization
   Simulate where you set up the model and run the simulation.
               Here is where you input your data, as follows.
               In Cell C8, check the box if you want periodic review. Uncheck for continuous review.
               In Cell C15, enter Q, which is the minimum order size. It is the gap between s and S
               In cell F8, if you want Gamma IDT, check the box. Uncheck for discrete IDT.
                           IDT = Inter-Demand Time, the time between occurrences of demand.
                           Gamma will give a smooth, continuous distribution. Discrete allows only a few values.
                  If you use Gamma IDT, put the mean and standard deviation of IDT in cells F11 and F12.
                           These statistics determine the average rate of demand. To convert demand rates into
                           IDT rates, use the "conversion formulas" by entering demand data in cells L17 and N17.
                           The resulting IDT stats are in L18 and N18: enter them in F11 and F12.
                  If you use Discrete IDT, put the possible IDT values in cells E16 to E22,
                           and the corresponding probabilities in cells F16 to F21. NOTE: cell F22 is
                           computed automatically so that the probabilities add to 1.0.
                           The mean and standard deviation of IDT will then show in cells F23 and f24.
               In cell I8, if you want Gamma IDT, check the box. Uncheck for discrete IDT.
                           LT = Lead Time, the time between placing an order and receiving the goods.
                           Gamma will give a smooth, continuous distribution. Discrete allows only a few values.
                  If you use Gamma LT, put the mean and standard deviation of LT in cells I11 and I12.
                  If you use Discrete LT, put the possible LT values in cells H16 to H22,
                           and the corresponding probabilities in cells I16 to I21. NOTE: cell I22 is
                           computed automatically so that the probabilities add to 1.0.
                           The mean and standard deviation of IDT will then show in cells I23 and I24.
               This completes the Data Input. However, before you make a run, you must specify
               Runin Periods in cell C20 (how long to run the model before beginning data collection), and
               Run Periods in cell C21 (how long to continue the model while collecting data).
   Graphs      where simulation results are displayed in detail for any run stored on the Data sheet.
   Trace       where the first part of the most recent simulation run is shown in a table and a graph.
   Data        where the results of all simulation runs are stored, until you erase them.
   Other sheets in this book, if any, may contain data and graphs from previous simulation experiments.

1. Measuring Inventory and Shortages at Time of Delivery:
        s reorder level, or Min                   Inv On-hand inventory
        S order-up-to level, or Max               BO Number of units backordered to customers
        Q S-s                                  NetInv = Inv - BO (can be positive or negative)
Sheet: Introduction          File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                        Page 3 of 41

       L    A value of lead time         InvPosition NetInv + Outstanding Orders
       D    A value of one-period demand          DL Demand that occurs during lead time
 mL, VarL   Average & Variance of L       mDL, VarDL Average & Variance of DL
 mD, VarD   Average & Variance of D

(a) Shortfall below Reorder Level at Delivery: Shortfall@Deliv
     Protection against shortages focuses attention on inventory at the time a replenishment order
     arrives. Safety stock governs the likelihood that backorders will exist at that instant.
     In the simulation, @Deliv refers to events that happen just before replenishments occur.
     However, rather than tracking inventory, which can be positive or negative, the simulation monitors
     "Shortfall below s at delivery," defined as
         1) Shortfall@Deliv = s - NetInv@Deliv at the time (just before) replenishment occurs.
     This is a non-negative variable since net inventory is at or below the reorder level, s , whenever any
     order is outstanding (i.e. not yet received.)
     From Shortfall@Deliv we may compute certain performance measures:
         2) NetInv@Deliv = s - Shortfall@Deliv
         3) Inv@Deliv = MAX(0, s - Shortfall@Deliv )
         4) BO@Deliv = MAX(0, Shortfall@Deliv - s ) = Inv@Deliv - s + Shortfall@Deliv
         5) P(BO@Deliv >0) = P( Shortfall@Deliv > s )
     The latter may also be expressed as a rate, although the meaning is a little confusing. It is NOT the
     rate at which backorders occur, but rather "occurrences per unit time" of the joint event
     "replenishment arrives, backorders exist," or "replenishment arrives too late to prevent backorders."
     That event is denoted "BO@Deliv>0 " and its occurrence rate is
         6) Rate(BO@Deliv>0 ) = P( Shortfall@Deliv > s )×(Replenishment Orders Per Unit Time )

(b) Distribution of Shortfall@Delivery if Orders do not Cross
    The following gives the classical argument for the distribution of shortfall, assuming that orders do
    not cross, and also assuming that lead times are independent, two assumptions which are convenient
    but contradictory. These numbers may be compared to the actual values from the simulation to see
    how much is lost if the classical rules are used.
        With no order crossing, when an order arrives, all prior replenishment orders have already arrived,
    and no subsequent ones have. At the time that order was placed, Inventory Position included the prior
    orders, so to compute inventory at delivery, we only have to account for the demand that occurs in the
    lead time (or lag time) between placing and receiving the order. That is,
        7) NetInv@Deliv = InvPosition@Ordering - DL and
        8) Shortfall@Deliv = s - InvPosition@Ordering + DL if orders do not cross (substitute 7 into 1).

    (i) Continuous Review
    Under continuous review, an order is placed the instant that inventory position reaches the reorder
    level. That is,
        9) InvPosition@Ordering = s for continuous review, so
       10) Shortfall@Deliv = DL for continuous review if orders do not cross            (substitute 9 into 8).
 >> Shortfall@Deliv equals lead-time demand for continuous review, if orders do not cross.
    The probability that backorders occur before an order arrives is
       11) P(BO@Deliv >0) = P( DL > s )                                                (substitute 10 into 5).
    The following formulas assume that Lead Times are independent and identically distributed, and that
Sheet: Introduction            File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                            Page 4 of 41

    the same is true for Demands, and that Lead Times are independent of Demands. They are, in fact, the
    well-known formulas for the mean and variance of lead-time demand.
      12) E[Shortfall@Deliv ] = mD mL and
       13) Var[Shortfall@Deliv ] = mL VarD + mD VarL for continuous review if orders do not cross.
                                               2



    (ii) Periodic Review
    Under periodic review, inventory position can reach the reorder point at a time t that is before the end
    of the period, so inventory position will be at or below the reorder point when the order is placed.
            If t is at the end of the period, the order is placed at the instant that the reorder level is reached.
            If t is just after the beginning of the period, a one-period demand occurs before ordering.
    This leads to the following inequality:
       14) s - D ≤ InvPosition@Ordering ≤ s
    Substituting 14 into 8,
       15) DL ≤ Shortfall@Deliv ≤ DL + D = DL+1 for periodic review if orders do not cross.
 >> Shortfall@Deliv is between the demand during lead time and the demand during one period longer
    than lead time, if orders do not cross. Also, because the probability above s is a nonincreasing
    function of s ,
       16) P(DL > s ) ≤ P(Shortfall@Deliv >s ) ≤ P( DL + D' - 1> s ) , and so
     17) P(DL > s ) ≤ P(BO@Deliv >0) ≤ P( DL + D' - 1> s )                               (substitute 16 into 5).
    The expected value of 15 yields
     18) mD mL ≤ Shortfall@Deliv ≤ mD (1+mL)
    The arguments leading to equation 15 also yield a lower limit for the variance:
     19) Var[Shortfall@Deliv ] ≤ mL VarD + mD VarL
                                                   2

    The upper limit in equation 15 also yields a variance estimate, but it is not necessarily an upper limit:
     20) Var[Shortfall@Deliv ] ≈ (1 + mL) VarD + mD VarL
                                                        2




2. Inventory and Shortages at Any Time
    The simulation also measures the inventory level after every event. Inventory is constant between
    events (by definition, since an event is defined as a change of state), so the distribution is
    tabulated by accumulating the time that each state persists.

(a) Shortfall below Order-up-to Level
      "Shortfall below S" is defined at every time in the simulation as
       20) Shortfall = S - NetInv.
     Notice that Shortfall uses a different reference point than Shortfall@Delivery, namely S rather
     than s . This is necessary to avoid negative values.
     From Shortfall , we may compute more performance measures:
       21) NetInv = S - Shortfall.
       22) Inv = MAX(0, S - Shortfall )
       23) BO = MAX(0, Shortfall - S ) = Inv - S + Shortfall
       24) P(BO >0) = P( Shortfall > S )
     Since a demand is backordered if it arrives when inventory is zero, the average number of demands
     backordered per unit time is
       25) Rate(BO ) = P( Shortfall ≥ S )×(Demand Rate )
     We can also calculate the average time that a backorder endures which, according to Little's Law,
Sheet: Introduction         File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                       Page 5 of 41

    is proportional to the average number of backorders waiting.
           Average duration of a Backorder = (Average # Backordered)(Rate of Backorders Occuring)
       26) Av(Wait per BO ) = Av(BO ){Av(DemandRate ) × P{Shortfall ≥S )}
    If you want to include in this average the fact that many customers have zero backorder time, then
       27) Av(Wait per Demand ) = Av(BO )Av(DemandRate ) (includes zero-length backorders.)

(b) Average Inventory and Shortages
    Average inventory is greater when computed over time than when computed just before a delivery.
    Inventory just before delivery can never be above the reorder point, whereas it can at other times.
    The average inventory over time will include the "sawtooth pattern" commonly seen in textbooks,
    caused by cycle stock represented by the order quantity. Therefore the exact theoretical expression
    for average inventory and backorders is elusive, and I will not try to include it here. However the
    simulation results yield averages from the distribution of Shortfall, using equations 22 through 25.
Sheet: Introduction         File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                        Page 6 of 41

3. Cost of Inventory, Backorders and Ordering:
    The simplest model has linear inventory and backorder costs. However, what constitutes backorder
    cost? There may be a cost per unit time for backorders, and a fixed cost whenever a backorder
    occurs. There also might be a fixed cost per unit time that accrues as long as there are any
    backorders. If the gap between s and S is changed, the number of orders placed will change, which
    changes the cost of ordering. A model that covers all of these costs is
      28) Average Cost per Period = C1 × Av(Inv ) + C2 × Av(BO ) + C3 × mD × P(BO ≥ 0)
                                         + C4 × P(BO >0) + C5 × Av(OrderRate)

4. Optimization
    The value of Q (the gap between s and S ) is held constant during a simulation. (In fact, it operates as
    if the order-up-to level were S =0 with reorder level s = -Q .) However, the output may be used
    to represent any (s,S ) system that has S-s=Q . You can find the optimal value of s among all
    systems that have the same Q as the one in your simulation, and then set S=s+Q .
        On the Graphs sheet, an Excel Table calculates costs for a range of values of s . A graph shows the
    results. You may input the first value of s and the interval between points. To home in on the
    optimum, adjust the first value until the graph is U-shaped, and then lower the interval to 1.
        However, the result is only optimal for the value of Q that you simulated. To find an overall
    optimum, you must repeat the simulation for a series of values of Q , and use the table to find the
    best reorder level for each Q . Record those values and select the one with lowest cost.
Sheet: Simulate                                  File 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                       Page 7 of 41

          Simulation of a Min-Max (s,S) Inventory System in Continuous Time: Discrete or Continuous Review

                                 Continuous Review. Q=30, D=10, varD=10
      Current Simulation Design:
                                 Gamma InterDemandTime: Mu= 0.1, Std=0.1. Discrete LT: Mu= 3, Std=1.41
Change the design by entering numbers in the yellow boxes, and by checking or unchecking the selection boxes.

          Type of Review            Inter-demand time (IDT)             Lead Time (LT)           Theoretical Values   Mean       Var StDev
                                                                                                               LTD    30.00   230.00 15.166
  Periodic Review?     FALSE       Gamma IDT?         TRUE        Gamma LT?         FALSE
                                                                                                             LTDem    30.00   230.00 15.166
       Continuous Review                   Gamma IDT               Gamma LT (not in use)                         LT    3.00     2.00 1.414
                                           mean     0.1                mean        3             Inter-Demand Time      0.1     0.01 0.100
                                           StDev   0.100              StDev 1.41421356               Demand/period    10.00    10.00 3.162
          Order Quantity            Discrete IDT (not in use)            Discrete LT
             Q = S-s     30                  IDT    f(IDT)                  LT     f(LT)         Conversion Formulas:
                                         0            0.5               0                                            Mean     Var       StDev
                                   0.033333333                          1            0.2         Demand per Period:   10       10       3.1623
                                   0.066666667                          2            0.2          Inter-demand Time:  0.1     0.01       0.100
          Run Controls                  0.1                             3            0.2
        Runin Periods   5,000      0.133333333                          4            0.2          Inter-Demand Time   0.1     0.01        0.1
         Run Periods   50,000      0.166666667                          5            0.2              Demand/period   10       10        3.162
             RNSeed     5235            0.2           0.5               6             0
                                           mean       0.1                 mean        3
                                           StDev      0.1                StDev 1.41421356




        Simulate                   Click here to view simulation results.

    Delete Old Data                Click here to view graphs of the distributions
Sheet: Graphs                                       File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                      Page 8 of 41

                  Continuous Review, Q=30. D=10 (Var=10). Discrete LT: Mu= 3, Std=1.41                            View Performance Summary
                  Gamma InterDemandTime: Mu= 0.1, Std=0.1 Crossings/Delivery=0.084                                View Graphs of Distributions
                                                                                                    Links            View Simulation Data
                               EOQ = 32                  Number of columns of data available:           2           Go To Simulation Design
              Min order Qty, Q = S-s = 30                        Use data in column number:             2        Choose your data set here
                Order-up-to level, S = 60                  Reorder trigger level (to vary): s =        30        Set the Reorder Level here

  Performance Statistics for                  Shortfall
                                  LTDem                    LTDem        Shortfall Unit Costs
         s=30, S=60                           @Deliv.
                      Mean =      29.914       29.914       29.914       44.359
                    Variance = 228.594        192.355      228.594      268.229
                 E[Inventory] =    6.481        5.896        6.481       17.186    $    1.00 Inventory Cost per unit time
               E[Backorders] =     6.395        5.809        5.395        1.545    $    9.00 Backorder Cost per unit time
            P[Backorders>0] =      0.477        0.482        0.477        0.176    $      -   Cost whenever Backorders > 0
             Backorder Rate =      0.159        0.160        0.159        1.924    $      -   Cost per unit Backordered
     Demands=499094, Orders=16637, Periods=50000, Orders/Period=          0.333    $ 50.00 Fixed Cost of Ordering
                                        For s=30, S=60, Total Cost per Unit Time = $ 47.73
                                Crossings=1392           Crossings per Delivery =       0.084

Analysis of Service Level Accuracy.                       Target Probability of No Backorders:        0.95       This target's meaning:
                                                         Shortfall                                               For "Shortfall",
 Using simulated distribution of:           LTDem                     LTDem         Shortfall
                                                         @Deliv.                                                 it means "% of time during which
 To achieve target probability, s=            55            53           55            42                        there are some backorders."
                        S=s+Q:                85            83           85            72                        For "Shorfall@Delivery",
        Actual P(no BO@Deliv):              0.971         0.956        0.971         0.798                       it means "% of orders for
      Actual P(no BO, time av.):            0.997         0.994        0.997         0.956                       which some backorders exist
               Predicted by LTD             0.956         0.935        0.956         0.762                       when the order arrives."
         Predicted by LT+1 Dem              0.956         0.935        0.956         0.762

Search for Minimum Cost Reorder Level (s) holding constant S-s=30
                                                       Make sure "Calculation" is set to "Automatic". If it is
         52                                            not, then press F9 to recalculate the cost curve
         50                                            whenever you change anything.
                                                                                                                 Graph of Cost vs s , starting at
  Cost




         48
         46
                                                                  First value of s for the graph:      35        this value, and incrementing by
                                                                       Interval between values:         1        this amount.
         44
              0         20        40          60                     Minimum Cost = $45.26
                       Reorder Level, s                                  at s =36, S =66
Sheet: Graphs                                           File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                    Page 9 of 41

              Continuous Review, Q=30. D=10 (Var=10). Discrete LT: Mu= 3, Std=1.41                                  View Performance Summary
              Gamma InterDemandTime: Mu= 0.1, Std=0.1 Crossings/Delivery=0.084                                      View Graphs of Distributions
                                                                                                         Links        View Simulation Data
                           Graphs of Distributions for simulation run number 2                                        Go To Simulation Design
  Continuous Review, Q=30. D=10 (Var=10). Discrete LT: Mu= 3, Std=1.41
  Gamma InterDemandTime: Mu= 0.1, Std=0.1 Crossings/Delivery=0.084                                                 Change which simulation
                                                                                             LTDem
                                                                                                                   is graphed by changing the
          0.03                                                                               Shortfall @Deliv.     column number in cell G5.
         0.025
          0.02
         0.015
          0.01
         0.005
             0
                       0    10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170




  0.03
                                                                                         LTDem
 0.025
  0.02                                                                                   Shortfall                               Page Down for
 0.015                                                                                                                             cumulative
  0.01                                                                                                                            distributions
                                                                                                                                        
 0.005
     0
         0        10       20   30   40       50   60   70   80    90 100 110 120 130 140 150 160 170

             LT: Mean = 3, CV = 0.471, Discrete                         IDT: Mean = 0.1, CV = 1, Gamma




    0         1            2     3        4        5     6        0.01 0.055   0.1   0.145 0.19 0.235 0.28

                                                                                             LTDem

                                                                                             Shortfall @Deliv.

Gamma InterDemandTime: Mu= 0.1, Std=0.1 Crossings/Delivery=0.084
                                                                                             Shortfall
              1.00
              0.80
              0.60
              0.40
              0.20
              0.00                              Cumulative Distributions
                           0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170
Sheet: Graphs                           File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                             Page 10 of 41

          Continuous Review, Q=30. D=10 (Var=10). Discrete LT: Mu= 3, Std=1.41                  View Performance Summary
          Gamma InterDemandTime: Mu= 0.1, Std=0.1 Crossings/Delivery=0.084                      View Graphs of Distributions
                                                                                    Links         View Simulation Data
Description of Simulation Data being Viewed:                                                      Go To Simulation Design
                                     Inputs:                            Outputs:
                               Runin Periods       5,000                SimTime        50,000
                                Run Periods       50,000               Demands       499,094
                                     Q = S-s           30                 Orders       16,637
                               Exp. Demand             10              Deliveries      16,636
                          Var. Dmd (approx)            10              Crossings        1,392
                                    RNSeed          5235              Cross/Deliv   0.083674
                           Periodic Review?       FALSE
                                Gamma LT?         FALSE
                               Gamma IDT?          TRUE
                    Exp. Inter-Demand Time            0.1           Shortfall, Avg. 44.359368
                  StDev Inter-Demand Time             0.1             Shortfall, var. 268.22924
                                     Exp. LT            3    Shortfall@Deliv. Avg. 29.913561
                                   StDev LT 1.41421356       Shortfall@Deliv. var. 192.35463
                                    Exp LTD            30                LTD Avg. 29.913561
                                    Var LTD          230                  LTD var. 228.59412
                                Exp D(LT+1)            40                Not Used 30.913561
                                Var D(LT+1)          240                 Not Used 228.59412
                                Input Distributions: Below             Output Distributions: Farther Below

Inter-Demand Time Distribution: Ignore Discrete. Gamma used.                                    Distribution Actually Used:
                                                Gamma Parameters: 1.000       0.100        IDT: Mean = 0.1, CV = 1, Gamma
                  Discrete IDT      F(IDT)        f(IDT)          Gamma IDT         f(IDT)          Gamma IDT         f(IDT)
                         0.000       0.500        0.500                 0.010       9.048                  0.010      9.048
                         0.033       0.500        0.000                 0.055       5.769                  0.055      5.769
                         0.067       0.500        0.000                 0.100       3.679                  0.100      3.679
                         0.100       0.500        0.000                 0.145       2.346                  0.145      2.346
                         0.133       0.500        0.000                 0.190       1.496                  0.190      1.496
                         0.167       0.500        0.000                 0.235       0.954                  0.235      0.954
                         0.200       1.000        0.500                 0.280       0.608                  0.280      0.608
                                                         Will Gamma overflow? FALSE

Lead Time Distribution: Ignore Gamma. Discrete used.                                              Distribution Actually Used:
                                                  Gamma Parameters: 4.500       0.667       LT: Mean = 3, CV = 0.471, Discrete
                    Discrete LT
                              Discrete F(LT)Discrete f(LT)           Gamma LT Gamma f(LT)              Discrete LT      f(IDT)
                          0.000        0.000        0.000                 0.300       0.005                  0.000      0.000
                          1.000        0.200        0.200                 1.650       0.259                  1.000      0.200
                          2.000        0.400        0.200                 3.000       0.277                  2.000      0.200
                          3.000        0.600        0.200                 4.350       0.134                  3.000      0.200
                          4.000        0.800        0.200                 5.700       0.046                  4.000      0.200
                          5.000        1.000        0.200                 7.050       0.013                  5.000      0.200
                          6.000        1.000        0.000                 8.400       0.003                  6.000      0.000
                                                           Will Gamma overflow? FALSE
Sheet: Trace                                                    File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                            Page 11 of 41

Order Crossing Simulation                                                      Partial Simulation Results                      Theory
(s, S) with S=0                                    demands      442                              Shortfall Leadtime Leadtime
Continuous time!                              orders placed     15                    Shortfall @ Delivery Demand    Demand
            Q = S-s         30                    deliveries    14             Mean      41.83       56.87     26.87    30.00
      Exp. Demand           10          Number of Crossings      1           Variance   254.65     183.41     221.98   230.00
 Var. Dmd (approx)          10
           RNSeed        5235
  Periodic Review?                                   Continuous Review. Q=30: D=10, var=10: L= 3, Std=1.41
                      FALSE Continuous Review. Q=30: D=10, var=10: L= 3, Std=1.41
       Gamma LT?      FALSE
      Gamma IDT?       TRUE             0
           Exp. IDT        0.1




                                                                                                                  5020




                                                                                                                               5025
                                                4995




                                                         5000




                                                                      5005




                                                                                     5010




                                                                                                      5015




                                                                                                                                            5030




                                                                                                                                                              5035
                                      -10
         StDev IDT         0.1        -20
            Exp. LT          3        -30
                                      -40
          StDev LT    1.41421         -50
                                          -60
                                          -70
                                          -80
        Current Time 55.556


                                 Arriving       Inventory          Order                    Lead      Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period          Dmd       Orders On-Hand On-Order Position Quantity                  Time      Period      Demand Delivery crossed        1       2      3      4     5
               5000                         -48        30     -18                                                                               5000
        5000.173046          1              -49        30     -19
        5000.205754          1              -50        30     -20
        5000.310213          1              -51        30     -21
        5000.382774                    30   -21         0     -21                                                        -21          -51
        5000.467588          1              -22         0     -22
        5000.623709          1              -23         0     -23
        5000.908757          1              -24         0     -24
        5001.043778          1              -25         0     -25
         5001.11675          1              -26         0     -26
        5001.309805          1              -27         0     -27
        5001.342144          1              -28         0     -28
        5001.393685          1              -29         0     -29
        5001.549392          1              -30         0     -30
        5001.549392                         -30        30        0      30                         1 5002.5494                                     0   5003
         5001.62275          1              -31        30       -1
        5001.763145          1              -32        30       -2
        5001.844754          1              -33        30       -3
        5002.349471          1              -34        30       -4
        5002.437536          1              -35        30       -5
        5002.549392                    30    -5         0       -5                                                        -5          -35
        5002.598439          1               -6         0       -6
        5002.611633          1               -7         0       -7
        5002.669455          1               -8         0       -8
        5002.676757          1               -9         0       -9
        5002.690741          1              -10         0     -10
        5002.693341          1              -11         0     -11
        5002.729228          1              -12         0     -12
        5002.731997          1              -13         0     -13
        5002.941784          1              -14         0     -14
        5002.961997          1              -15         0     -15
        5003.179469          1              -16         0     -16
        5003.222885          1              -17         0     -17
        5003.468923          1              -18         0     -18
        5003.502495          1              -19         0     -19
        5003.523262          1              -20         0     -20
         5003.73177          1              -21         0     -21
        5003.769321          1              -22         0     -22
        5003.826027          1              -23         0     -23
        5004.162968          1              -24         0     -24
         5004.17674          1              -25         0     -25
        5004.181105          1              -26         0     -26
        5004.183976          1              -27         0     -27
        5004.191863          1              -28         0     -28
        5004.236135          1              -29         0     -29
        5004.496639          1              -30         0     -30
        5004.496639                         -30        30        0      30                         1 5005.4966                                     0   5005
        5004.744864          1              -31        30       -1
        5004.901383          1              -32        30       -2
        5005.085572          1              -33        30       -3
        5005.234753          1              -34        30       -4
        5005.471668          1              -35        30       -5
        5005.488336          1              -36        30       -6
        5005.496639                    30    -6         0       -6                                                        -6          -36
        5005.507679          1               -7         0       -7
        5005.648534          1               -8         0       -8
        5005.679202          1               -9         0       -9
        5005.735994          1              -10         0     -10
        5005.811116          1              -11         0     -11
        5005.857882          1              -12         0     -12
        5005.958634          1              -13         0     -13
Sheet: Trace                                               File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                    Page 12 of 41

                                Arriving       Inventory          Order       Lead       Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period         Dmd       Orders On-Hand On-Order Position Quantity     Time       Period      Demand Delivery crossed        1       2      3      4     5
        5006.185391         1              -14         0     -14
        5006.294004         1              -15         0     -15
        5006.398898         1              -16         0     -16
        5006.400871         1              -17         0     -17
        5006.444422         1              -18         0     -18
        5006.568489         1              -19         0     -19
        5006.637815         1              -20         0     -20
        5006.769622         1              -21         0     -21
        5006.881362         1              -22         0     -22
        5006.979882         1              -23         0     -23
        5007.027178         1              -24         0     -24
        5007.092416         1              -25         0     -25
        5007.172431         1              -26         0     -26
        5007.391184         1              -27         0     -27
        5007.408501         1              -28         0     -28
        5007.524233         1              -29         0     -29
        5007.598126         1              -30         0     -30
        5007.598126                        -30        30        0      30            2 5009.5981                               0    5010
        5007.657855         1              -31        30       -1
        5007.665675         1              -32        30       -2
        5007.691803         1              -33        30       -3
        5007.988156         1              -34        30       -4
        5008.008356         1              -35        30       -5
        5008.077129         1              -36        30       -6
        5008.354601         1              -37        30       -7
        5008.371281         1              -38        30       -8
         5008.54965         1              -39        30       -9
        5008.550189         1              -40        30     -10
        5008.585484         1              -41        30     -11
        5008.697039         1              -42        30     -12
        5008.828862         1              -43        30     -13
         5008.84935         1              -44        30     -14
        5008.894675         1              -45        30     -15
        5009.125983         1              -46        30     -16
        5009.126611         1              -47        30     -17
        5009.258672         1              -48        30     -18
        5009.457427         1              -49        30     -19
        5009.483553         1              -50        30     -20
        5009.598126                   30   -20         0     -20                                           -20        -50
        5009.642632         1              -21         0     -21
        5009.675938         1              -22         0     -22
        5009.796912         1              -23         0     -23
        5009.865261         1              -24         0     -24
        5010.101792         1              -25         0     -25
        5010.168576         1              -26         0     -26
        5010.303152         1              -27         0     -27
         5010.41784         1              -28         0     -28
        5010.535388         1              -29         0     -29
         5010.57704         1              -30         0     -30
         5010.57704                        -30        30        0      30            5   5015.577                              0    5016
        5010.738756         1              -31        30       -1
        5010.761057         1              -32        30       -2
        5010.904805         1              -33        30       -3
        5011.192288         1              -34        30       -4
        5011.257925         1              -35        30       -5
        5011.340638         1              -36        30       -6
        5011.395519         1              -37        30       -7
        5011.743065         1              -38        30       -8
        5011.845079         1              -39        30       -9
        5011.945671         1              -40        30     -10
        5012.116681         1              -41        30     -11
        5012.138242         1              -42        30     -12
        5012.485316         1              -43        30     -13
         5012.70029         1              -44        30     -14
        5012.794175         1              -45        30     -15
        5012.856272         1              -46        30     -16
        5012.925432         1              -47        30     -17
        5013.001783         1              -48        30     -18
        5013.022007         1              -49        30     -19
        5013.075306         1              -50        30     -20
        5013.285833         1              -51        30     -21
        5013.595849         1              -52        30     -22
        5013.664477         1              -53        30     -23
        5014.168396         1              -54        30     -24
          5014.3426         1              -55        30     -25
        5014.378521         1              -56        30     -26
        5014.394282         1              -57        30     -27
        5014.419987         1              -58        30     -28
        5014.598754         1              -59        30     -29
        5014.650928         1              -60        30     -30
        5014.650928                        -60        60        0      30            2 5016.6509                               0    5016 5016.7
Sheet: Trace                                               File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                   Page 13 of 41

                                Arriving       Inventory          Order       Lead      Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period         Dmd       Orders On-Hand On-Order Position Quantity     Time      Period      Demand Delivery crossed        1       2      3      4     5
        5014.770354         1              -61        60       -1
        5014.783437         1              -62        60       -2
        5015.034334         1              -63        60       -3
        5015.065044         1              -64        60       -4
        5015.127242         1              -65        60       -5
        5015.314618         1              -66        60       -6
        5015.433581         1              -67        60       -7
        5015.474816         1              -68        60       -8
        5015.507533         1              -69        60       -9
         5015.57704                   30   -39        30       -9                                         -39        -69           5016
        5015.585094         1              -40        30     -10
        5015.643736         1              -41        30     -11
         5015.84156         1              -42        30     -12
         5015.94776         1              -43        30     -13
        5016.121705         1              -44        30     -14
        5016.215372         1              -45        30     -15
        5016.297956         1              -46        30     -16
         5016.32307         1              -47        30     -17
        5016.368212         1              -48        30     -18
        5016.440383         1              -49        30     -19
        5016.515035         1              -50        30     -20
         5016.59158         1              -51        30     -21
        5016.597965         1              -52        30     -22
        5016.650928                   30   -22         0     -22                                          -22        -52
        5016.739632         1              -23         0     -23
        5016.742518         1              -24         0     -24
         5016.82915         1              -25         0     -25
        5016.902957         1              -26         0     -26
          5017.0895         1              -27         0     -27
           5017.184         1              -28         0     -28
        5017.262315         1              -29         0     -29
        5017.398921         1              -30         0     -30
        5017.398921                        -30        30        0      30            3 5020.3989                              0    5020
        5017.407893         1              -31        30       -1
        5017.584132         1              -32        30       -2
        5017.592956         1              -33        30       -3
        5017.724044         1              -34        30       -4
        5017.857725         1              -35        30       -5
        5017.896266         1              -36        30       -6
        5017.956079         1              -37        30       -7
        5018.001187         1              -38        30       -8
        5018.231636         1              -39        30       -9
        5018.460737         1              -40        30     -10
        5018.603558         1              -41        30     -11
        5018.681515         1              -42        30     -12
        5018.715656         1              -43        30     -13
        5018.739164         1              -44        30     -14
        5018.842714         1              -45        30     -15
        5018.958991         1              -46        30     -16
        5019.079769         1              -47        30     -17
        5019.191921         1              -48        30     -18
        5019.242912         1              -49        30     -19
        5019.268767         1              -50        30     -20
        5019.518807         1              -51        30     -21
        5019.661261         1              -52        30     -22
        5019.681984         1              -53        30     -23
          5020.1023         1              -54        30     -24
        5020.178911         1              -55        30     -25
        5020.246428         1              -56        30     -26
        5020.314307         1              -57        30     -27
        5020.398921                   30   -27         0     -27                                          -27        -57
         5020.48442         1              -28         0     -28
        5020.488499         1              -29         0     -29
        5020.513344         1              -30         0     -30
        5020.513344                        -30        30        0      30            3 5023.5133                              0    5024
        5020.687838         1              -31        30       -1
         5020.70282         1              -32        30       -2
        5020.804579         1              -33        30       -3
        5020.915913         1              -34        30       -4
        5021.164149         1              -35        30       -5
        5021.205309         1              -36        30       -6
        5021.209908         1              -37        30       -7
        5021.329855         1              -38        30       -8
        5021.372338         1              -39        30       -9
        5021.517346         1              -40        30     -10
        5021.713743         1              -41        30     -11
        5021.737031         1              -42        30     -12
        5021.760438         1              -43        30     -13
        5021.790284         1              -44        30     -14
        5021.839872         1              -45        30     -15
        5022.021183         1              -46        30     -16
Sheet: Trace                                               File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                   Page 14 of 41

                                Arriving       Inventory          Order       Lead      Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period         Dmd       Orders On-Hand On-Order Position Quantity     Time      Period      Demand Delivery crossed        1       2      3      4     5
        5022.254165         1              -47        30     -17
        5022.368792         1              -48        30     -18
        5022.558656         1              -49        30     -19
         5022.63081         1              -50        30     -20
        5022.944227         1              -51        30     -21
        5023.016635         1              -52        30     -22
        5023.112848         1              -53        30     -23
        5023.220841         1              -54        30     -24
        5023.275128         1              -55        30     -25
        5023.386915         1              -56        30     -26
        5023.513344                   30   -26         0     -26                                          -26        -56
         5023.52923         1              -27         0     -27
        5023.632372         1              -28         0     -28
        5023.701703         1              -29         0     -29
        5023.790374         1              -30         0     -30
        5023.790374                        -30        30        0      30            3 5026.7904                              0    5027
        5023.945869         1              -31        30       -1
        5024.007807         1              -32        30       -2
        5024.084874         1              -33        30       -3
        5024.201414         1              -34        30       -4
        5024.357531         1              -35        30       -5
        5024.499417         1              -36        30       -6
        5024.524385         1              -37        30       -7
        5024.542984         1              -38        30       -8
        5025.172501         1              -39        30       -9
        5025.228033         1              -40        30     -10
        5025.317178         1              -41        30     -11
        5025.338606         1              -42        30     -12
        5025.493862         1              -43        30     -13
        5025.623628         1              -44        30     -14
        5025.881797         1              -45        30     -15
        5026.012697         1              -46        30     -16
        5026.176865         1              -47        30     -17
        5026.180452         1              -48        30     -18
        5026.198067         1              -49        30     -19
        5026.350103         1              -50        30     -20
        5026.445077         1              -51        30     -21
        5026.790374                   30   -21         0     -21                                          -21        -51
        5026.842714         1              -22         0     -22
        5026.906976         1              -23         0     -23
        5027.009626         1              -24         0     -24
        5027.228326         1              -25         0     -25
        5027.411797         1              -26         0     -26
        5027.422136         1              -27         0     -27
        5027.475767         1              -28         0     -28
        5027.688143         1              -29         0     -29
         5027.70727         1              -30         0     -30
         5027.70727                        -30        30        0      30            4 5031.7073                              0    5032
        5027.760315         1              -31        30       -1
        5027.772632         1              -32        30       -2
        5027.900233         1              -33        30       -3
        5027.930596         1              -34        30       -4
        5027.931462         1              -35        30       -5
        5027.973992         1              -36        30       -6
         5028.05085         1              -37        30       -7
        5028.129417         1              -38        30       -8
        5028.154958         1              -39        30       -9
        5028.295583         1              -40        30     -10
        5028.344078         1              -41        30     -11
        5028.347114         1              -42        30     -12
        5028.612496         1              -43        30     -13
        5028.664946         1              -44        30     -14
        5028.712273         1              -45        30     -15
        5028.771048         1              -46        30     -16
        5028.912755         1              -47        30     -17
        5029.233837         1              -48        30     -18
        5029.435017         1              -49        30     -19
        5029.593682         1              -50        30     -20
        5029.662773         1              -51        30     -21
        5029.755202         1              -52        30     -22
         5029.84277         1              -53        30     -23
        5030.143891         1              -54        30     -24
        5030.188847         1              -55        30     -25
        5030.221076         1              -56        30     -26
         5030.22418         1              -57        30     -27
        5030.287107         1              -58        30     -28
        5030.354508         1              -59        30     -29
        5030.366136         1              -60        30     -30
        5030.366136                        -60        60        0      30            4 5034.3661                              0    5032 5034.4
        5030.379011         1              -61        60       -1
        5030.508819         1              -62        60       -2
Sheet: Trace                                               File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                    Page 15 of 41

                                Arriving       Inventory          Order       Lead       Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period         Dmd       Orders On-Hand On-Order Position Quantity     Time       Period      Demand Delivery crossed        1       2      3      4     5
        5030.523726         1              -63        60       -3
        5030.530885         1              -64        60       -4
        5030.878612         1              -65        60       -5
         5030.97832         1              -66        60       -6
        5031.006654         1              -67        60       -7
        5031.082726         1              -68        60       -8
         5031.14497         1              -69        60       -9
        5031.280931         1              -70        60     -10
        5031.327971         1              -71        60     -11
        5031.328157         1              -72        60     -12
        5031.463481         1              -73        60     -13
        5031.474148         1              -74        60     -14
         5031.70727                   30   -44        30     -14                                           -44        -74           5032
        5031.837424         1              -45        30     -15
        5031.895509         1              -46        30     -16
         5032.26247         1              -47        30     -17
        5032.262992         1              -48        30     -18
        5032.384426         1              -49        30     -19
        5032.403908         1              -50        30     -20
        5032.431359         1              -51        30     -21
        5032.434821         1              -52        30     -22
        5032.484306         1              -53        30     -23
        5032.488831         1              -54        30     -24
        5032.499536         1              -55        30     -25
        5032.536676         1              -56        30     -26
        5032.562803         1              -57        30     -27
         5032.71013         1              -58        30     -28
        5032.767342         1              -59        30     -29
        5032.768958         1              -60        30     -30
        5032.768958                        -60        60        0      30            4   5036.769                              0    5034 5036.8
        5032.796112         1              -61        60       -1
        5032.867628         1              -62        60       -2
        5032.884919         1              -63        60       -3
        5033.193461         1              -64        60       -4
        5033.227385         1              -65        60       -5
        5033.265257         1              -66        60       -6
         5033.35362         1              -67        60       -7
        5033.404252         1              -68        60       -8
         5033.40712         1              -69        60       -9
        5033.421594         1              -70        60     -10
        5033.450962         1              -71        60     -11
        5033.481796         1              -72        60     -12
        5033.487996         1              -73        60     -13
        5033.536005         1              -74        60     -14
         5033.67025         1              -75        60     -15
        5033.719806         1              -76        60     -16
        5033.741958         1              -77        60     -17
        5033.853085         1              -78        60     -18
        5033.853492         1              -79        60     -19
        5033.911984         1              -80        60     -20
         5034.05012         1              -81        60     -21
         5034.06658         1              -82        60     -22
        5034.120716         1              -83        60     -23
        5034.333459         1              -84        60     -24
        5034.366136                   30   -54        30     -24                                           -54        -84           5034
        5034.437008         1              -55        30     -25
        5034.445628         1              -56        30     -26
        5034.447384         1              -57        30     -27
        5034.464492         1              -58        30     -28
        5034.649256         1              -59        30     -29
        5034.757263         1              -60        30     -30
        5034.757263                        -60        60        0      30            1 5035.7573                               1    5036 5036.8
         5034.75963         1              -61        60       -1
        5035.075138         1              -62        60       -2
        5035.085558         1              -63        60       -3
        5035.179073         1              -64        60       -4
        5035.528264         1              -65        60       -5
         5035.54545         1              -66        60       -6
        5035.757263                   30   -36        30       -6                                           -6        -66           5036
        5035.859144         1              -37        30       -7
        5036.096552         1              -38        30       -8
        5036.218403         1              -39        30       -9
        5036.254702         1              -40        30     -10
        5036.380081         1              -41        30     -11
        5036.414385         1              -42        30     -12
        5036.592434         1              -43        30     -13
        5036.634031         1              -44        30     -14
        5036.726962         1              -45        30     -15
        5036.768958                   30   -15         0     -15                                           -45        -45
        5036.933896         1              -16         0     -16
        5037.017377         1              -17         0     -17
Sheet: Trace                                               File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                    Page 16 of 41

                                Arriving       Inventory          Order       Lead       Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period         Dmd       Orders On-Hand On-Order Position Quantity     Time       Period      Demand Delivery crossed        1       2      3      4     5
        5037.287158         1              -18         0     -18
        5037.404183         1              -19         0     -19
        5037.437786         1              -20         0     -20
         5037.52578         1              -21         0     -21
        5037.570839         1              -22         0     -22
        5037.591512         1              -23         0     -23
        5037.706143         1              -24         0     -24
        5038.102444         1              -25         0     -25
        5038.212345         1              -26         0     -26
          5038.2175         1              -27         0     -27
        5038.218997         1              -28         0     -28
        5038.281305         1              -29         0     -29
        5038.283992         1              -30         0     -30
        5038.283992                        -30        30        0      30            3   5041.284                              0    5041
        5038.443712         1              -31        30       -1
          5038.6124         1              -32        30       -2
        5038.665112         1              -33        30       -3
        5038.753703         1              -34        30       -4
        5038.846401         1              -35        30       -5
        5038.935692         1              -36        30       -6
        5038.997491         1              -37        30       -7
        5039.077191         1              -38        30       -8
        5039.174697         1              -39        30       -9
        5039.182398         1              -40        30     -10
        5039.285203         1              -41        30     -11
        5039.329692         1              -42        30     -12
        5039.377851         1              -43        30     -13
        5039.443212         1              -44        30     -14
        5039.540938         1              -45        30     -15
        5039.548147         1              -46        30     -16
          5039.6252         1              -47        30     -17
        5039.677224         1              -48        30     -18
        5039.934501         1              -49        30     -19
        5039.955243         1              -50        30     -20
        5040.003119         1              -51        30     -21
        5040.079987         1              -52        30     -22
        5040.241041         1              -53        30     -23
        5040.241171         1              -54        30     -24
        5040.436163         1              -55        30     -25
        5040.527045         1              -56        30     -26
        5040.576647         1              -57        30     -27
        5040.594166         1              -58        30     -28
        5040.773648         1              -59        30     -29
        5040.854747         1              -60        30     -30
        5040.854747                        -60        60        0      30            3 5043.8547                               0    5041 5043.9
        5040.865775         1              -61        60       -1
        5040.959478         1              -62        60       -2
        5040.967434         1              -63        60       -3
        5040.997509         1              -64        60       -4
        5041.011809         1              -65        60       -5
        5041.125872         1              -66        60       -6
        5041.151166         1              -67        60       -7
        5041.283992                   30   -37        30       -7                                          -37        -67           5041
        5041.533536         1              -38        30       -8
        5041.672056         1              -39        30       -9
        5041.817197         1              -40        30     -10
        5041.870454         1              -41        30     -11
        5041.909493         1              -42        30     -12
        5041.934927         1              -43        30     -13
         5041.94124         1              -44        30     -14
        5042.037145         1              -45        30     -15
        5042.154729         1              -46        30     -16
        5042.211057         1              -47        30     -17
        5042.353996         1              -48        30     -18
        5042.523147         1              -49        30     -19
        5042.583484         1              -50        30     -20
        5042.622043         1              -51        30     -21
        5042.793783         1              -52        30     -22
        5042.915065         1              -53        30     -23
        5043.099852         1              -54        30     -24
        5043.128838         1              -55        30     -25
        5043.501251         1              -56        30     -26
        5043.602072         1              -57        30     -27
         5043.61249         1              -58        30     -28
        5043.696944         1              -59        30     -29
        5043.697507         1              -60        30     -30
        5043.697507                        -60        60        0      30            5 5048.6975                               0    5044 5048.7
        5043.854747                   30   -30        30        0                                          -30        -60           5044
        5043.880203         1              -31        30       -1
        5043.937087         1              -32        30       -2
        5043.965608         1              -33        30       -3
Sheet: Trace                                               File: 39493bd0-6846-4c8a-8b12-199fc7d77b32.xls                                                  Page 17 of 41

                                Arriving       Inventory          Order       Lead     Delivery   - Leadtime Inv. before orders Order Pipeline, Arrival Times
       Period         Dmd       Orders On-Hand On-Order Position Quantity     Time     Period      Demand Delivery crossed        1       2      3      4     5
        5044.163469         1              -34        30       -4
        5044.318931         1              -35        30       -5
        5044.541464         1              -36        30       -6
        5044.587209         1              -37        30       -7
        5044.619318         1              -38        30       -8
        5044.686789         1              -39        30       -9
        5044.710875         1              -40        30     -10
              Runin Periods         5,000         5,000
                Run Periods        50,000        50,000
                     Q = S-s            20            30
               Exp. Demand              10            10
         Var. Dmd (approx)              10            10
                    RNSeed           5235          5235
          Periodic Review?         FALSE         FALSE
                Gamma LT?          FALSE         FALSE
               Gamma IDT?           TRUE          TRUE
   Exp. Inter-Demand Time              0.1           0.1
  StDev Inter-Demand Time              0.1           0.1
                     Exp. LT             3             3
                   StDev LT    1.4142136     1.4142136
                    Exp LTD             30            30
                    Var LTD           230           230
                Exp D(LT+1)             40            40
                Var D(LT+1)           240           240
                   SimTime         50,000        50,000
                  Demands         499,303       499,094
                      Orders       24,966        16,637
                  Deliveries       24,966        16,636
                  Crossings         5,150          1392
                 Cross/Deliv   0.2062805      0.083674
                   LTD Avg.    29.949852     29.913561
                    LTD var.   228.45531     228.59412
"LT+1 Dem" or "LTD", Avg.      30.949852     30.913561
"LT+1 Dem" or "LTD", Var.      228.45531     228.59412
      S-Inv. (Time Av) Avg.    39.434485     44.359368
       S-Inv. (Time Av) var.   185.21499     268.22924
         s-Inv @Deliv. Avg.    29.949852     29.913561
          s-Inv @Deliv. var.   151.85353     192.35463
                           x             x             x
                                         0             0
                                         1             1
                                         2             2
                                         3             3
                                         4             4
                                         5             5
                                         6             6
                                         7             7
                                         8             8
                                         9             9
                                        10            10
                                        11            11
12   12
13   13
14   14
15   15
16   16
17   17
18   18
19   19
20   20
21   21
22   22
23   23
24   24
25   25
26   26
27   27
28   28
29   29
30   30
31   31
32   32
33   33
34   34
35   35
36   36
37   37
38   38
39   39
40   40
41   41
42   42
43   43
44   44
45   45
46   46
47   47
48   48
49   49
50   50
51   51
52   52
53   53
54   54
55   55
56   56
57   57
58   58
59   59
60   60
61   61
62   62
63   63
64   64
65   65
66   66
67   67
68   68
69   69
70   70
71   71
72   72
73   73
74   74
75   75
76   76
77   77
78   78
79   79
80   80
81   81
82   82
83   83
84   84
85   85
86   86
87   87
88   88
89   89
90   90
91   91
92   92
93   93
94   94
95   95
96   96
97   97
98   98
99   99
100   100
101   101
102   102
103   103
104   104
105   105
106   106
107   107
108   108
109   109
110   110
111   111
112   112
113   113
114   114
115   115
116   116
117   117
118   118
119   119
120   120
121   121
122   122
123   123
124   124
125   125
126   126
127   127
128   128
129   129
130   130
131   131
132   132
133   133
134   134
135   135
136   136
137   137
138   138
139   139
140   140
141   141
142   142
143   143
             144          144
             145          145
             146          146
             147          147
             148          148
             149          149
             150          150
             151          151
             152          152
             153          153
             154          154
             155          155
             156          156
             157          157
             158          158
             159          159
             160          160
             161          161
             162          162
             163          163
             164          164
             165          165
             166          166
             167          167
             168          168
             169          169
             170          170
-Inv         -Inv         -Inv
                0            0
                0   6.128E-06
       2.646E-05    2.059E-05
       5.552E-05    7.142E-05
       0.0001775    0.0001567
       0.0003714    0.0003441
       0.0007612    0.0007457
       0.0011762    0.0011426
       0.0017154    0.0016919
       0.0023825    0.0022837
       0.0030543    0.0030199
       0.0038026    0.0036593
       0.0046104    0.0043444
       0.0053155    0.0048052
       0.0063485    0.0058364
       0.0067831    0.0062321
0.0077791   0.0069116
0.0087434   0.0076005
0.0094145   0.0083077
0.0105391   0.0088757
0.0116726   0.0096537
0.0128322   0.0104885
0.0140328   0.0112809
0.0152131   0.0118038
0.0170631   0.0128571
 0.017772   0.0139093
0.0190689   0.0145269
0.0202201   0.0151656
0.0213202   0.0159298
0.0221948   0.0170096
0.0233565   0.0169454
0.0244317   0.0178661
0.0251406   0.0182014
0.0255983   0.0193204
0.0264334   0.0200977
0.0268699   0.0203536
0.0274836   0.0213798
0.0280486   0.0216118
0.0277169   0.0219001
0.0278298   0.0217289
0.0274234   0.0220579
0.0273628   0.0222683
0.0280595   0.0222535
0.0272542   0.0225058
0.0270159   0.0223755
0.0263981   0.0222811
0.0253343   0.0223969
0.0247088   0.0220006
0.0233546      0.02142
0.0220016   0.0211239
0.0206707   0.0211361
0.0192302   0.0207808
0.0184046    0.019995
0.0168853   0.0200727
0.0161094   0.0191118
0.0153081   0.0187268
0.0140193    0.017764
0.0132491   0.0178156
0.0123487   0.0168225
0.0112462   0.0166022
 0.010224   0.0159633
0.0092215   0.0150082
0.0080537   0.0144814
0.0072478    0.013665
0.0063961   0.0130406
0.0057738   0.0120749
 0.004956   0.0116935
0.0042871    0.011041
0.0036665   0.0100038
0.0031287   0.0092399
0.0027424   0.0081784
0.0021001   0.0074553
0.0018782   0.0066811
0.0014675   0.0060476
0.0011844   0.0052099
0.0009117   0.0047862
0.0007069   0.0043379
0.0005931   0.0036787
0.0004493    0.003314
0.0003196   0.0029566
0.0002431   0.0025305
0.0001964   0.0020789
 0.000143   0.0017672
0.0001246   0.0014978
7.078E-05   0.0013071
4.077E-05   0.0009767
4.799E-05   0.0007483
3.064E-05   0.0006091
2.048E-05      0.00049
1.475E-05   0.0004188
8.048E-06   0.0003286
1.637E-06   0.0002336
8.232E-06    0.000159
1.198E-06   0.0001117
3.456E-08   9.543E-05
        0    5.48E-05
        0   4.862E-05
        0   2.977E-05
        0   2.727E-05
        0   9.165E-06
        0   1.387E-05
        0   2.871E-06
        0    4.76E-06
        0   3.268E-06
0   1.123E-06
0   1.239E-06
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
              0           0
LTD         LTD         LTD
              0           0
      8.011E-05   0.0002404
      0.0006409   0.0003006
      0.0012417   0.0016831
       0.003645   0.0030055
      0.0072098   0.0072133
      0.0138588   0.0119019
      0.0173035    0.018995
      0.0224706   0.0238038
      0.0260354   0.0253066
      0.0258752   0.0268694
      0.0258752   0.0253066
      0.0204278   0.0234431
      0.0193063   0.0175523
      0.0197469   0.0189348
      0.0174638   0.0183938
      0.0185052   0.0183938
      0.0174237   0.0186343
      0.0197869   0.0183337
      0.0209485   0.0188146
0.0218697       0.02176
0.0218697     0.020678
0.0193063    0.0192955
0.0209485    0.0223611
0.0197469     0.020678
0.0196667    0.0218803
0.0197469     0.017913
0.0217496    0.0204376
0.0197068    0.0186944
0.0192662    0.0214595
0.0205079    0.0209185
0.0194264    0.0220005
0.0185452    0.0193556
0.0197869    0.0216398
0.0189858    0.0199567
0.0207883    0.0186343
0.0185452    0.0177326
0.0212689    0.0201371
0.0195065    0.0202573
 0.021309    0.0182736
0.0212289    0.0213994
0.0196267    0.0209185
0.0199071    0.0180933
0.0191861    0.0194157
0.0196667    0.0179731
0.0184651    0.0171916
0.0172635    0.0180933
0.0171033    0.0170714
0.0150605    0.0151479
0.0156613    0.0159894
 0.016182    0.0149675
0.0134583    0.0122025
0.0138989    0.0123828
0.0103741    0.0128036
0.0100937    0.0119019
0.0086518    0.0088363
   0.00733   0.0079947
0.0071698    0.0077543
 0.005848    0.0050493
0.0045662    0.0051695
0.0037651    0.0044482
0.0034046    0.0033662
0.0030842     0.003246
0.0019226    0.0018634
0.0016022   0.0013224
0.0014019   0.0012022
0.0010414   0.0009017
 0.000721   0.0006011
0.0005207    0.000541
0.0004406   0.0002404
0.0002403   0.0001803
0.0001602   0.0002404
0.0003605   6.011E-05
8.011E-05   6.011E-05
8.011E-05           0
4.005E-05   0.0001202
        0   6.011E-05
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
                        0            0
s-Inv @Deliv.s-Inv @Deliv.s-Inv @Deliv.
                        0            0
                        0 0.0001803
                0.0002003 0.0001803
                0.0005608 0.0012022
                0.0018826 0.0020438
                0.0038452 0.0053498
                0.0068493 0.0088363
                0.0084915 0.0138855
                 0.011015 0.0180933
                0.0140191 0.0198365
                0.0143395 0.0202573
                0.0167428     0.021219
                0.0137787 0.0203775
                0.0149804 0.0159894
                0.0157414     0.018454
                0.0160618 0.0175523
                0.0172234 0.0198966
                0.0178643 0.0207382
                0.0197869 0.0209786
                0.0219899 0.0216999
                0.0224706 0.0242246
                0.0236321 0.0224213
                0.0221101 0.0220606
                0.0237523 0.0248858
0.0247136   0.0241645
0.0266362   0.0239841
0.0304414   0.0209185
0.0348874   0.0230825
 0.034527   0.0200769
0.0354482   0.0233229
0.0368902   0.0223611
0.0334054   0.0229623
0.0307618   0.0207983
0.0283986    0.022842
0.0275975   0.0213393
0.0263558   0.0209185
0.0232316   0.0212792
0.0247136   0.0254268
0.0225907   0.0261481
0.0229112   0.0238639
0.0227509   0.0279514
0.0200673   0.0251262
0.0195466   0.0211589
0.0182648   0.0209786
0.0175038    0.018454
0.0159016   0.0180933
0.0143796   0.0166506
 0.013218   0.0149675
0.0119362    0.012503
 0.011776   0.0131041
0.0114956    0.012503
0.0087719   0.0104592
0.0085717   0.0096177
0.0061684   0.0102789
0.0055275   0.0084155
0.0045262   0.0067324
0.0039253   0.0049892
0.0032845   0.0051094
0.0027638   0.0036667
0.0019226   0.0033061
0.0016422   0.0030055
0.0012817   0.0024645
0.0010815   0.0019235
 0.000721   0.0011421
0.0005608   0.0009017
0.0003605   0.0007814
0.0002403   0.0004208
0.0002804   0.0004208
0.0002804   0.0003607
0.0001602   0.0001202
8.011E-05   0.0002404
        0   0.0001202
0.0001202   6.011E-05
        0   6.011E-05
4.005E-05           0
        0   6.011E-05
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
                   0            0
LT+1 Dem      LTDem        LTDem
                   0            0
           8.011E-05   0.0002404
           0.0006409   0.0003006
           0.0012417   0.0016831
            0.003645   0.0030055
           0.0072098   0.0072133
           0.0138588   0.0119019
           0.0173035    0.018995
           0.0224706   0.0238038
           0.0260354   0.0253066
           0.0258752   0.0268694
           0.0258752   0.0253066
           0.0204278   0.0234431
           0.0193063   0.0175523
           0.0197469   0.0189348
           0.0174638   0.0183938
           0.0185052   0.0183938
           0.0174237   0.0186343
           0.0197869   0.0183337
           0.0209485   0.0188146
           0.0218697      0.02176
           0.0218697    0.020678
           0.0193063   0.0192955
           0.0209485   0.0223611
           0.0197469    0.020678
           0.0196667   0.0218803
           0.0197469    0.017913
           0.0217496   0.0204376
0.0197068    0.0186944
0.0192662    0.0214595
0.0205079    0.0209185
0.0194264    0.0220005
0.0185452    0.0193556
0.0197869    0.0216398
0.0189858    0.0199567
0.0207883    0.0186343
0.0185452    0.0177326
0.0212689    0.0201371
0.0195065    0.0202573
 0.021309    0.0182736
0.0212289    0.0213994
0.0196267    0.0209185
0.0199071    0.0180933
0.0191861    0.0194157
0.0196667    0.0179731
0.0184651    0.0171916
0.0172635    0.0180933
0.0171033    0.0170714
0.0150605    0.0151479
0.0156613    0.0159894
 0.016182    0.0149675
0.0134583    0.0122025
0.0138989    0.0123828
0.0103741    0.0128036
0.0100937    0.0119019
0.0086518    0.0088363
   0.00733   0.0079947
0.0071698    0.0077543
 0.005848    0.0050493
0.0045662    0.0051695
0.0037651    0.0044482
0.0034046    0.0033662
0.0030842     0.003246
0.0019226    0.0018634
0.0016022    0.0013224
0.0014019    0.0012022
0.0010414    0.0009017
 0.000721    0.0006011
0.0005207     0.000541
0.0004406    0.0002404
0.0002403    0.0001803
0.0001602    0.0002404
0.0003605   6.011E-05
8.011E-05   6.011E-05
8.011E-05           0
4.005E-05   0.0001202
        0   6.011E-05
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
        0           0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
0   0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
                            0            0
   Discrete IDT Discrete IDT Discrete IDT
                            0            0
                  0.0333333 0.0333333
                  0.0666667 0.0666667
                          0.1          0.1
                  0.1333333 0.1333333
                  0.1666667 0.1666667
                          0.2          0.2
              Discrete F(IDT)
Discrete F(IDT)            Discrete F(IDT)
                          0.5          0.5
                          0.5          0.5
                          0.5          0.5
                          0.5          0.5
                          0.5          0.5
                          0.5          0.5
                            1            1
    Discrete LT Discrete LT Discrete LT
                            0            0
                            1            1
                            2            2
                            3            3
                            4            4
                            5            5
                            6            6
 Discrete F(LT)             Discrete F(LT)
               Discrete F(LT)
                            0            0
                          0.2          0.2
                          0.4          0.4
                          0.6          0.6
                          0.8          0.8
                            1            1
                            1            1
                                     39493bd0-6846-4c8a-8b12-199fc7d77b32.xls, Warning, p. 39 of 41

                     WARNING: YOU NEED TO ENABLE THE MACROS IN THIS FILE
Try these steps. If step 1 does not work, then go to step 2.
STEP 1: Try to enable the macros
 For Excel 2007,
  A. If you see the Security Warning in your menu bar, proceed with step B.
        If the Security Warning is not there, go to STEP 2.
  B. In the Security Warning, Click Options.
  C. In the Alert Window that appears, click Enable This Content and click OK.




 For Excel 2003,
  A. Close this file. Then open it again.
  B. In the window that appears, click Enable Macros.
      If a window like this one
         does NOT appear, then go to STEP 2.




STEP 2: If Step 1 does not work
  If you DID NOT get a Security Message, then your security setting is too high.
  Here is what you should do:

 For Excel 2007,
  A. Click the Microsoft Office Button at the top-left of the screen:
  B. Click Excel Options.
  C. Click Trust Center, then Trust Center Settings, and then Macro Settings.
  D. Click Disable all macros with notification
                                     39493bd0-6846-4c8a-8b12-199fc7d77b32.xls, Warning, p. 40 of 41
  E. Exit from Excel. Closing the file is not enough. On the menu bar, select File, and then Exit.
  F. Open this file again and follow the instructions in STEP 1 to enable the macros.
 For Excel 2003,
  A. On the menu bar at the top of this page, select Tools, then Macro, then Security.
  B. On the Security Level tab, select Medium and click OK.
  C. Then exit from Excel. Closing the file is not enough. On the menu bar, select File, and then Exit.
  D. Open this file again and follow the instructions in STEP 1 to enable the macros.
39493bd0-6846-4c8a-8b12-199fc7d77b32.xls, Warning, p. 41 of 41

				
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