Sheet: Introduction File: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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 ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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 120130140 150160170
Sheet: Graphs File: ec629f86-2113-4eec-8f4b-d97a56309cb2.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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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 -10
5020
5025
4995
5000
5005
5010
5015
5030
5035
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: ec629f86-2113-4eec-8f4b-d97a56309cb2.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
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Sheet: Trace File: ec629f86-2113-4eec-8f4b-d97a56309cb2.xls Page 13 of 41
Arriving Inventory Order Lead Delivery - Leadtime Inv. before orders Order Pipeline, Arrival Times
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Sheet: Trace File: ec629f86-2113-4eec-8f4b-d97a56309cb2.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
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5024.357531 1 -35 30 -5
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5030.221076 1 -56 30 -26
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Sheet: Trace File: ec629f86-2113-4eec-8f4b-d97a56309cb2.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
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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
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5032.499536 1 -55 30 -25
5032.536676 1 -56 30 -26
5032.562803 1 -57 30 -27
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5034.649256 1 -59 30 -29
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5035.085558 1 -63 60 -3
5035.179073 1 -64 60 -4
5035.528264 1 -65 60 -5
5035.54545 1 -66 60 -6
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Sheet: Trace File: ec629f86-2113-4eec-8f4b-d97a56309cb2.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
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5038.283992 1 -30 0 -30
5038.283992 -30 30 0 30 3 5041.284 0 5041
5038.443712 1 -31 30 -1
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5038.665112 1 -33 30 -3
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5038.846401 1 -35 30 -5
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Sheet: Trace File: ec629f86-2113-4eec-8f4b-d97a56309cb2.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
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10 10
11 11
12 12
13 13
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142 142
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162 162
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165 165
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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
ec629f86-2113-4eec-8f4b-d97a56309cb2.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
ec629f86-2113-4eec-8f4b-d97a56309cb2.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.
ec629f86-2113-4eec-8f4b-d97a56309cb2.xls, Warning, p. 41 of 41