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Sheet 1: Introduction
| Simulating Inventory Control with Orders that Cross during Lead Time | |||||||||
| sS.xls Version 1.4 03/22/02 | |||||||||
| 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. | |||||||||
| 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. | ||||||||
| 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) | ||||||
| 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 | |||||||||
| 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 + mD2 VarL for continuous review if orders do not cross. | ||||||||
| (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 + mD2 VarL | ||||||||
| 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 + mD2 VarL | ||||||||
| 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, | |||||||||
| 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. | |||||||||
| 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. | |||||||||
| Simulation of a Min-Max (s,S) Inventory System in Continuous Time: Discrete or Continuous Review | |||||||||||||
| Current Simulation Design: | Periodic Review. Q=30, D=10, varD=10 | ||||||||||||
| 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. | |||||||||||||
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Theoretical Values | Mean | Var | StDev | |||||||
| Periodic Review? | 1 | Gamma IDT? | 1 | Gamma LT? | 0 | LTD | 30.00 | 230.00 | 15.166 | ||||
| LT+1 Dem | 40.00 | 240.00 | 15.492 | ||||||||||
| Periodic Review |
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LT | 3.00 | 2.00 | 1.414 | |||||||
| mean | 0.1 | mean | 3 | Inter-Demand Time | 0.1 | 0.01 | 0.100 | ||||||
| StDev | 0.1 | StDev | 1.4142135623731 | Demand/period | 10.00 | 10.00 | 3.162 | Variance is an approx? | |||||
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| Minimum Q = S-s | 30 | IDT | f(IDT) | LT | f(LT) | Conversion Formulas: | |||||||
| 0 | 0.5 | 0 | Mean | Var | StDev | ||||||||
| 0.033333333333333 | 1 | 0.2 | Demand per Period: | 10 | 10 | 3.162 | |||||||
| 0.066666666666667 | 2 | 0.2 | Inter-demand Time: | 0.1 | 0.01 | 0.100 | |||||||
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0.1 | 3 | 0.2 | ||||||||||
| Runin Periods |
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0.133333333333333 | 4 | 0.2 | Inter-Demand Time | 0.1 | 0.01 | 0.100 | |||||
| Run Periods |
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0.166666666666667 | 5 | 0.2 | Demand/period | 10 | 10 | 3.162 | |||||
| RNSeed |
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0.2 | 0.5 | 6 | 0 | ||||||||
| mean | 0.1 | mean | 3 | ||||||||||
| StDev | 0.1 | StDev | 1.4142135623731 | ||||||||||
| Click here to view simulation results. | |||||||||||||
| Click here to view graphs of the distributions | |||||||||||||
| Periodic Review, Q=50. D=10 (Var=10). Discrete LT: Mu= 3, Std=1.41 | View Performance Summary | ¬ More at these links. |
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| Gamma InterDemandTime: Mu= 0.1, Std=0.1 Crossings/Delivery=0 | View Graphs of Distributions | ||||||||||||||
| Links ® | View Simulation Data | ||||||||||||||
| EOQ = | 32 | Number of columns of data available: | 8 | Go To Simulation Design | |||||||||||
| Min order Qty, Q = S-s = | 50 | Use data in column number: |
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¬Choose your data set here | |||||||||||
| Order-up-to level, S = | 80 | Reorder trigger level (to vary): s = |
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¬Set the Reorder Level here | |||||||||||
| Performance Statistics for s=30, S=80 |
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Using Normal Distribution to approximate: |
LTDem | Shortfall @Deliv. | LT+1 Dem | Shortfall | |||||
| Mean = | 29.980 | 35.014 | 41.017 | 57.046 | Mean = | 29.980 | 35.014 | 41.017 | 57.046 | ||||||
| Variance = | 230.033 | 242.520 | 239.770 | 489.283 | Variance = | 230.033 | 242.520 | 239.770 | 489.283 | ||||||
| E[Inventory] = | 6.476 | 4.293 | 2.138 | 24.736 | $1.00 | Inventory Cost per unit time | E[Inventory] = | 6.061 | 4.025 | 2.170 | 24.668 | ||||
| E[Backorders] = | 6.456 | 9.307 | 12.156 | 1.783 | $9.00 | Backorder Cost per unit time | E[Backorders] = | 6.041 | 9.039 | 13.187 | 1.714 | ||||
| P[Backorders>0] = | 0.478 | 0.577 | 0.697 | 0.158 | $- | Cost whenever Backorders > 0 | P[Backorders>0] = | 0.486 | 0.614 | 0.752 | 0.145 | ||||
| Backorder Rate = |
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$- | Cost per unit Backordered | Backorder Rate = | 0.089 | 0.112 | 0.137 | 1.550 | ||||
| Demands=500784, Orders=9100, Periods=50000, Orders/Period= | 0.182 | $50.00 | Fixed Cost of Ordering | ||||||||||||
| For s=30, S=80, Total Cost per Unit Time = | $49.88 | ||||||||||||||
| Crossings=0 | Crossings per Delivery = | 0.000 | |||||||||||||
| Analysis of Service Level Accuracy. | Target Probability of No Backorders: |
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¬This target's meaning: | ||||||||||||
| Using simulated distribution of: |
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For "Shortfall", | Using Normal Distribution to approximate: |
LTDem | Shortfall @Deliv. | LT+1 Dem | Shortfall | |||||
| it means "% of time during which | |||||||||||||||
| To achieve target probability, s= |
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44 | there are some backorders." | To achieve target probability, s= | 55 | 61 | 65 | 43 | |||||
| S=s+Q: | 105 | 111 | 116 |
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For "Shorfall@Delivery", | S=s+Q: | 105 | 111 | 115 | 93 | |||||
| Actual P(no BO@Deliv): | 0.895 | 0.956 | 0.984 | 0.696 | it means "% of orders for | Actual P(no BO@Deliv): | 0.895 | 0.956 | 0.979 | 1.000 | |||||
| Actual P(no BO, time av.): | 0.989 | 0.997 | 0.999 | 0.952 | which some backorders exist | Actual P(no BO, time av.): | 0.989 | 0.997 | 0.999 | 0.946 | |||||
| Predicted by LTD | 0.956 | 0.988 | 0.996 | 0.792 | when the order arrives." | Predicted by LTD | 0.956 | 0.988 | 0.996 | 1.000 | |||||
| Predicted by LT+1 Dem | 0.795 | 0.895 | 0.949 | 0.580 | Predicted by LT+1 Dem | 0.795 | 0.895 | 0.941 | 1.000 | ||||||
| Search for Minimum Cost Reorder Level (s) holding constant S-s=50 | |||||||||||||||
| Make sure "Calculation" is set to "Automatic". If it is not, then press F9 to recalculate the cost curve whenever you change anything. | |||||||||||||||
| Table for Optimizing s | |||||||||||||||
| s | Cost | ||||||||||||||
| Graph of Cost vs s, starting at | 30 | $49.88 | |||||||||||||
| First value of s for the graph: |
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¬this value, and incrementing by | 35 | $47.96 | |||||||||||
| Interval between values: |
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¬this amount. | 36 | $47.85 | |||||||||||
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at s =37, S =87 |
37 | $47.83 | |||||||||||||
| 38 | $47.89 | ||||||||||||||
| 39 | $48.02 | ||||||||||||||
| 40 | $48.22 | ||||||||||||||
| Graphs of Distributions for simulation run number 8 | 41 | $48.50 | |||||||||||||
| 42 | $48.84 | ||||||||||||||
| Change which simulation | 43 | $49.24 | |||||||||||||
| is graphed by changing the | 44 | $49.70 | |||||||||||||
| column number in cell G5. | 45 | $50.22 | |||||||||||||
| 46 | $50.78 | ||||||||||||||
| 47 | $51.40 | ||||||||||||||
| 48 | $52.06 | ||||||||||||||
| Minimum Cost = $ | $47.83 | ||||||||||||||
| at s = | 37 | ||||||||||||||
| , S = | 87 | ||||||||||||||
| Page Down for cumulative distributions ¯ |
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| Description of Simulation Data being Viewed: | |||||||||||||||
| Inputs: | Outputs: | ||||||||||||||
| Runin Periods | 5,000 | SimTime | 50,000 | ||||||||||||
| Run Periods | 50,000 | Demands | 500,784 | ||||||||||||
| Q = S-s | 50 | Orders | 9,100 | ||||||||||||
| Exp. Demand | 10 | Deliveries | 9,100 | ||||||||||||
| Var. Dmd (approx) | 10 | Crossings | 0 | ||||||||||||
| RNSeed | 5235 | Cross/Deliv | 0 | ||||||||||||
| Periodic Review? | 1 | ||||||||||||||
| Gamma LT? | 0 | ||||||||||||||
| Gamma IDT? | 1 | ||||||||||||||
| Exp. Inter-Demand Time | 0.1 | Shortfall, Avg. | 57.0462538231744 | ||||||||||||
| StDev Inter-Demand Time | 0.1 | Shortfall, var. | 489.282705097935 | ||||||||||||
| Exp. LT | 3 | Shortfall@Deliv. Avg. | 35.013956043956 | ||||||||||||
| StDev LT | 1.4142135623731 | Shortfall@Deliv. var. | 242.520354679386 | ||||||||||||
| Exp LTD | 30 | LTD Avg. | 29.9803296703297 | ||||||||||||
| Var LTD | 230 | LTD var. | 230.032909781428 | ||||||||||||
| Exp D(LT+1) | 40 | LT+1 Dem Avg | 41.0173626373626 | ||||||||||||
| Var D(LT+1) | 240 | LT+1 Dem Var | 239.769808428935 | ||||||||||||
| 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? | 0 | ||||||||||||||
| 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? | 0 | ||||||||||||||
| x | LTDem | Shortfall @Deliv. | LT+1 Dem | Shortfall | Cumul LTDem | Cumul Shortfall @Deliv. | Cumul LT+1 Dem | Cumul Shortfall | |||||||
| 0 | 0.00% | 0.01% | 0.00% | 0.00% | 0.00% | 0.01% | 0.00% | 0.00% | |||||||
| 1 | 0.03% | 0.01% | 0.00% | 0.00% | 0.03% | 0.02% | 0.00% | 0.00% | |||||||
| 2 | 0.01% | 0.00% | 0.00% | 0.00% | 0.04% | 0.02% | 0.00% | 0.00% | |||||||
| 3 | 0.12% | 0.01% | 0.00% | 0.01% | 0.16% | 0.03% | 0.00% | 0.01% | |||||||
| 4 | 0.41% | 0.07% | 0.00% | 0.02% | 0.57% | 0.10% | 0.00% | 0.03% | |||||||
| 5 | 0.68% | 0.13% | 0.00% | 0.03% | 1.25% | 0.23% | 0.00% | 0.06% | |||||||
| 6 | 1.11% | 0.19% | 0.00% | 0.05% | 2.36% | 0.42% | 0.00% | 0.11% | |||||||
| 7 | 1.93% | 0.43% | 0.00% | 0.09% | 4.30% | 0.85% | 0.00% | 0.20% | |||||||
| 8 | 2.12% | 0.59% | 0.02% | 0.12% | 6.42% | 1.44% | 0.02% | 0.32% | |||||||
| 9 | 2.43% | 0.87% | 0.01% | 0.15% | 8.85% | 2.31% | 0.03% | 0.46% | |||||||
| 10 | 2.81% | 1.23% | 0.02% | 0.22% | 11.66% | 3.54% | 0.05% | 0.68% | |||||||
| 11 | 2.64% | 1.34% | 0.10% | 0.26% | 14.30% | 4.88% | 0.15% | 0.94% | |||||||
| 12 | 2.21% | 1.62% | 0.18% | 0.31% | 16.51% | 6.49% | 0.33% | 1.25% | |||||||
| 13 | 2.09% | 1.55% | 0.25% | 0.35% | 18.59% | 8.04% | 0.58% | 1.60% | |||||||
| 14 | 1.85% | 1.98% | 0.53% | 0.38% | 20.44% | 10.02% | 1.11% | 1.98% | |||||||
| 15 | 1.78% | 1.76% | 0.71% | 0.40% | 22.22% | 11.78% | 1.82% | 2.38% | |||||||
| 16 | 1.99% | 1.95% | 1.07% | 0.45% | 24.21% | 13.73% | 2.89% | 2.83% | |||||||
| 17 | 1.60% | 2.18% | 1.32% | 0.46% | 25.81% | 15.90% | 4.21% | 3.29% | |||||||
| 18 | 1.99% | 2.07% | 1.48% | 0.52% | 27.80% | 17.97% | 5.69% | 3.81% | |||||||
| 19 | 2.12% | 1.90% | 1.81% | 0.55% | 29.92% | 19.87% | 7.51% | 4.36% | |||||||
| 20 | 2.24% | 2.00% | 2.23% | 0.58% | 32.16% | 21.87% | 9.74% | 4.94% | |||||||
| 21 | 2.13% | 2.26% | 1.91% | 0.63% | 34.30% | 24.13% | 11.65% | 5.57% | |||||||
| 22 | 1.93% | 2.12% | 2.14% | 0.67% | 36.23% | 26.25% | 13.79% | 6.23% | |||||||
| 23 | 1.89% | 1.90% | 2.08% | 0.69% | 38.12% | 28.15% | 15.87% | 6.93% | |||||||
| 24 | 1.89% | 2.05% | 2.12% | 0.72% | 40.01% | 30.21% | 17.99% | 7.65% | |||||||
| 25 | 2.40% | 1.92% | 2.13% | 0.76% | 42.41% | 32.13% | 20.12% | 8.41% | |||||||
| 26 | 1.90% | 2.05% | 1.96% | 0.82% | 44.31% | 34.19% | 22.08% | 9.23% | |||||||
| 27 | 1.91% | 2.05% | 1.95% | 0.84% | 46.22% | 36.24% | 24.02% | 10.07% | |||||||
| 28 | 1.95% | 1.93% | 2.05% | 0.88% | 48.16% | 38.18% | 26.08% | 10.95% | |||||||
| 29 | 2.07% | 2.11% | 2.01% | 0.90% | 50.23% | 40.29% | 28.09% | 11.85% | |||||||
| 30 | 1.98% | 2.04% | 2.21% | 0.94% | 52.21% | 42.33% | 30.30% | 12.79% | |||||||
| 31 | 1.99% | 1.77% | 2.00% | 0.99% | 54.20% | 44.10% | 32.30% | 13.78% | |||||||
| 32 | 2.01% | 1.96% | 2.00% | 1.01% | 56.21% | 46.05% | 34.30% | 14.79% | |||||||
| 33 | 2.13% | 2.13% | 2.01% | 1.06% | 58.34% | 48.19% | 36.31% | 15.86% | |||||||
| 34 | 2.01% | 1.84% | 1.93% | 1.11% | 60.35% | 50.02% | 38.24% | 16.97% | |||||||
| 35 | 1.99% | 1.78% | 2.09% | 1.13% | 62.34% | 51.80% | 40.33% | 18.10% | |||||||
| 36 | 1.97% | 1.95% | 2.02% | 1.16% | 64.31% | 53.75% | 42.35% | 19.27% | |||||||
| 37 | 1.96% | 2.16% | 1.96% | 1.19% | 66.26% | 55.91% | 44.31% | 20.46% | |||||||
| 38 | 1.73% | 2.07% | 1.89% | 1.24% | 67.99% | 57.98% | 46.20% | 21.70% | |||||||
| 39 | 1.95% | 1.93% | 1.56% | 1.25% | 69.93% | 59.91% | 47.76% | 22.95% | |||||||
| 40 | 1.99% | 1.90% | 2.22% | 1.33% | 71.92% | 61.81% | 49.98% | 24.28% | |||||||
| 41 | 1.98% | 2.04% | 1.92% | 1.34% | 73.90% | 63.86% | 51.90% | 25.62% | |||||||
| 42 | 1.73% | 2.01% | 1.84% | 1.38% | 75.63% | 65.87% | 53.74% | 27.00% | |||||||
| 43 | 1.81% | 1.84% | 2.41% | 1.44% | 77.44% | 67.70% | 56.14% | 28.44% | |||||||
| 44 | 1.73% | 1.91% | 1.85% | 1.44% | 79.16% | 69.62% | 57.99% | 29.88% | |||||||
| 45 | 1.90% | 1.99% | 2.02% | 1.46% | 81.07% | 71.60% | 60.01% | 31.34% | |||||||
| 46 | 1.85% | 1.78% | 1.81% | 1.50% | 82.91% | 73.38% | 61.82% | 32.85% | |||||||
| 47 | 1.87% | 2.16% | 2.05% | 1.54% | 84.78% | 75.55% | 63.88% | 34.38% | |||||||
| 48 | 1.85% | 1.76% | 1.87% | 1.55% | 86.63% | 77.31% | 65.75% | 35.94% | |||||||
| 49 | 1.77% | 1.89% | 2.07% | 1.61% | 88.40% | 79.20% | 67.81% | 37.54% | |||||||
| 50 | 1.63% | 2.01% | 2.15% | 1.63% | 90.02% | 81.21% | 69.97% | 39.18% | |||||||
| 51 | 1.45% | 1.87% | 1.92% | 1.65% | 91.47% | 83.08% | 71.89% | 40.82% | |||||||
| 52 | 1.34% | 1.87% | 2.08% | 1.73% | 92.81% | 84.95% | 73.97% | 42.55% | |||||||
| 53 | 1.09% | 1.43% | 1.70% | 1.68% | 93.90% | 86.37% | 75.67% | 44.23% | |||||||
| 54 | 0.92% | 1.49% | 1.81% | 1.74% | 94.82% | 87.87% | 77.48% | 45.97% | |||||||
| 55 | 0.75% | 1.59% | 2.01% | 1.73% | 95.57% | 89.46% | 79.49% | 47.70% | |||||||
| 56 | 0.70% | 1.24% | 1.67% | 1.73% | 96.27% | 90.70% | 81.16% | 49.43% | |||||||
| 57 | 0.64% | 1.21% | 1.79% | 1.76% | 96.91% | 91.91% | 82.96% | 51.19% | |||||||
| 58 | 0.45% | 1.12% | 1.55% | 1.75% | 97.36% | 93.03% | 84.51% | 52.95% | |||||||
| 59 | 0.52% | 1.04% | 1.86% | 1.71% | 97.88% | 94.08% | 86.36% | 54.65% | |||||||
| 60 | 0.51% | 0.84% | 1.54% | 1.72% | 98.38% | 94.91% | 87.90% | 56.37% | |||||||
| 61 | 0.41% | 0.68% | 1.59% | 1.72% | 98.79% | 95.59% | 89.49% | 58.10% | |||||||
| 62 | 0.21% | 0.71% | 1.33% | 1.66% | 99.00% | 96.31% | 90.82% | 59.75% | |||||||
| 63 | 0.24% | 0.65% | 1.27% | 1.66% | 99.24% | 96.96% | 92.10% | 61.41% | |||||||
| 64 | 0.21% | 0.54% | 1.09% | 1.60% | 99.45% | 97.49% | 93.19% | 63.02% | |||||||
| 65 | 0.13% | 0.45% | 0.92% | 1.59% | 99.58% | 97.95% | 94.11% | 64.60% | |||||||
| 66 | 0.05% | 0.44% | 0.82% | 1.58% | 99.64% | 98.38% | 94.93% | 66.18% | |||||||
| 67 | 0.10% | 0.34% | 0.76% | 1.51% | 99.74% | 98.73% | 95.69% | 67.69% | |||||||
| 68 | 0.09% | 0.24% | 0.60% | 1.48% | 99.82% | 98.97% | 96.30% | 69.18% | |||||||
| 69 | 0.04% | 0.20% | 0.60% | 1.42% | 99.87% | 99.16% | 96.90% | 70.60% | |||||||
| 70 | 0.07% | 0.11% | 0.66% | 1.41% | 99.93% | 99.27% | 97.56% | 72.01% | |||||||
| 71 | 0.03% | 0.15% | 0.46% | 1.38% | 99.97% | 99.43% | 98.02% | 73.39% | |||||||
| 72 | 0.00% | 0.14% | 0.37% | 1.33% | 99.97% | 99.57% | 98.40% | 74.72% | |||||||
| 73 | 0.02% | 0.07% | 0.27% | 1.32% | 99.99% | 99.64% | 98.67% | 76.03% | |||||||
| 74 | 0.00% | 0.10% | 0.30% | 1.30% | 99.99% | 99.74% | 98.97% | 77.33% | |||||||
| 75 | 0.00% | 0.07% | 0.18% | 1.24% | 99.99% | 99.80% | 99.14% | 78.57% | |||||||
| 76 | 0.01% | 0.07% | 0.18% | 1.21% | 100.00% | 99.87% | 99.32% | 79.78% | |||||||
| 77 | 0.00% | 0.07% | 0.14% | 1.15% | 100.00% | 99.93% | 99.46% | 80.94% | |||||||
| 78 | 0.00% | 0.03% | 0.18% | 1.11% | 100.00% | 99.97% | 99.64% | 82.04% | |||||||
| 79 | 0.00% | 0.01% | 0.05% | 1.11% | 100.00% | 99.98% | 99.69% | 83.16% | |||||||
| 80 | 0.00% | 0.02% | 0.10% | 1.05% | 100.00% | 100.00% | 99.79% | 84.21% | |||||||
| 81 | 0.00% | 0.00% | 0.04% | 1.00% | 100.00% | 100.00% | 99.84% | 85.22% | |||||||
| 82 | 0.00% | 0.00% | 0.03% | 0.99% | 100.00% | 100.00% | 99.87% | 86.20% | |||||||
| 83 | 0.00% | 0.00% | 0.03% | 0.91% | 100.00% | 100.00% | 99.90% | 87.12% | |||||||
| 84 | 0.00% | 0.00% | 0.05% | 0.93% | 100.00% | 100.00% | 99.96% | 88.05% | |||||||
| 85 | 0.00% | 0.01% | 0.00% | 0.88% | 100.00% | 100.01% | 99.96% | 88.93% | |||||||
| 86 | 0.00% | 0.00% | 0.02% | 0.84% | 100.00% | 100.01% | 99.98% | 89.77% | |||||||
| 87 | 0.00% | 0.00% | 0.01% | 0.80% | 100.00% | 100.01% | 99.99% | 90.57% | |||||||
| 88 | 0.00% | 0.00% | 0.00% | 0.76% | 100.00% | 100.01% | 99.99% | 91.33% | |||||||
| 89 | 0.00% | 0.00% | 0.01% | 0.71% | 100.00% | 100.01% | 100.00% | 92.04% | |||||||
| 90 | 0.00% | 0.00% | 0.00% | 0.69% | 100.00% | 100.01% | 100.00% | 92.74% | |||||||
| 91 | 0.00% | 0.00% | 0.00% | 0.66% | 100.00% | 100.01% | 100.00% | 93.40% | |||||||
| 92 | 0.00% | 0.00% | 0.00% | 0.62% | 100.00% | 100.01% | 100.00% | 94.02% | |||||||
| 93 | 0.00% | 0.00% | 0.00% | 0.58% | 100.00% | 100.01% | 100.00% | 94.60% | |||||||
| 94 | 0.00% | 0.00% | 0.00% | 0.56% | 100.00% | 100.01% | 100.00% | 95.15% | |||||||
| 95 | 0.00% | 0.00% | 0.00% | 0.53% | 100.00% | 100.01% | 100.00% | 95.68% | |||||||
| 96 | 0.00% | 0.00% | 0.00% | 0.47% | 100.00% | 100.01% | 100.00% | 96.16% | |||||||
| 97 | 0.00% | 0.00% | 0.00% | 0.45% | 100.00% | 100.01% | 100.00% | 96.60% | |||||||
| 98 | 0.00% | 0.00% | 0.00% | 0.41% | 100.00% | 100.01% | 100.00% | 97.01% | |||||||
| 99 | 0.00% | 0.00% | 0.00% | 0.37% | 100.00% | 100.01% | 100.00% | 97.38% | |||||||
| 100 | 0.00% | 0.00% | 0.00% | 0.33% | 100.00% | 100.01% | 100.00% | 97.70% | |||||||
| 101 | 0.00% | 0.00% | 0.00% | 0.31% | 100.00% | 100.01% | 100.00% | 98.01% | |||||||
| 102 | 0.00% | 0.00% | 0.00% | 0.27% | 100.00% | 100.01% | 100.00% | 98.29% | |||||||
| 103 | 0.00% | 0.00% | 0.00% | 0.24% | 100.00% | 100.01% | 100.00% | 98.53% | |||||||
| 104 | 0.00% | 0.00% | 0.00% | 0.21% | 100.00% | 100.01% | 100.00% | 98.75% | |||||||
| 105 | 0.00% | 0.00% | 0.00% | 0.19% | 100.00% | 100.01% | 100.00% | 98.94% | |||||||
| 106 | 0.00% | 0.00% | 0.00% | 0.16% | 100.00% | 100.01% | 100.00% | 99.10% | |||||||
| 107 | 0.00% | 0.00% | 0.00% | 0.15% | 100.00% | 100.01% | 100.00% | 99.25% | |||||||
| 108 | 0.00% | 0.00% | 0.00% | 0.13% | 100.00% | 100.01% | 100.00% | 99.38% | |||||||
| 109 | 0.00% | 0.00% | 0.00% | 0.11% | 100.00% | 100.01% | 100.00% | 99.50% | |||||||
| 110 | 0.00% | 0.00% | 0.00% | 0.09% | 100.00% | 100.01% | 100.00% | 99.58% | |||||||
| 111 | 0.00% | 0.00% | 0.00% | 0.09% | 100.00% | 100.01% | 100.00% | 99.67% | |||||||
| 112 | 0.00% | 0.00% | 0.00% | 0.07% | 100.00% | 100.01% | 100.00% | 99.74% | |||||||
| 113 | 0.00% | 0.00% | 0.00% | 0.05% | 100.00% | 100.01% | 100.00% | 99.79% | |||||||
| 114 | 0.00% | 0.00% | 0.00% | 0.05% | 100.00% | 100.01% | 100.00% | 99.84% | |||||||
| 115 | 0.00% | 0.00% | 0.00% | 0.03% | 100.00% | 100.01% | 100.00% | 99.87% | |||||||
| 116 | 0.00% | 0.00% | 0.00% | 0.02% | 100.00% | 100.01% | 100.00% | 99.90% | |||||||
| 117 | 0.00% | 0.00% | 0.00% | 0.02% | 100.00% | 100.01% | 100.00% | 99.92% | |||||||
| 118 | 0.00% | 0.00% | 0.00% | 0.02% | 100.00% | 100.01% | 100.00% | 99.94% | |||||||
| 119 | 0.00% | 0.00% | 0.00% | 0.02% | 100.00% | 100.01% | 100.00% | 99.95% | |||||||
| 120 | 0.00% | 0.00% | 0.00% | 0.01% | 100.00% | 100.01% | 100.00% | 99.97% | |||||||
| 121 | 0.00% | 0.00% | 0.00% | 0.01% | 100.00% | 100.01% | 100.00% | 99.97% | |||||||
| 122 | 0.00% | 0.00% | 0.00% | 0.01% | 100.00% | 100.01% | 100.00% | 99.98% | |||||||
| 123 | 0.00% | 0.00% | 0.00% | 0.01% | 100.00% | 100.01% | 100.00% | 99.98% | |||||||
| 124 | 0.00% | 0.00% | 0.00% | 0.01% | 100.00% | 100.01% | 100.00% | 99.99% | |||||||
| 125 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 99.99% | |||||||
| 126 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 99.99% | |||||||
| 127 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 128 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 129 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 130 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 131 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 132 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 133 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 134 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 135 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 136 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 137 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 138 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 139 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 140 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 141 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 142 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 143 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 144 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 145 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 146 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 147 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 148 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 149 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 150 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 151 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 152 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 153 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 154 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 155 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 156 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 157 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 158 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 159 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 160 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 161 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 162 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 163 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 164 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 165 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 166 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 167 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 168 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 169 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 170 | 0.00% | 0.00% | 0.00% | 0.00% | 100.00% | 100.01% | 100.00% | 100.00% | |||||||
| 100% | 100% | 100% | 100% | ||||||||||||
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