0-AL67 17 PERCENT ERRORS IN THE ESTIMATION OF DEMIND FOR 1/1SECONDARY ITEMS(U) ARMY ARMAMENT MUNITIONS AD CHEMICALCOMMAND ROCK ISLRND IL R.. G 9 ROBINSON NOV 95

UNCLASSIFIED RMSMC-RDR-TN-8502 SBI-RD-E708 024 F/ 15/5 ML

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RDA-TN-850200

_ s-LECTAPR 15 W

JD..

PERCENT ERRORSIN THE

ESTIMATION OF DEMANDFOR SECONDARY ITEMS

George RobinsonAMSMC- SAL

November 1985

U14L4, LA bi r L r. I 4SECURITY CLASSIFICATION OF THIS PAGE (When Dots Entore *

-

REPORT DOCUMENTATION PAGE REAI NSTRUCTIONS13EORE COMPLETING FORM1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIEHT'S CATALOG NUMBER

RDA-TN-8502 1/l4vAI/."7 /7 ___.._._-"_

4. TITLE (mid Subtitle) S. TYPE OF REPORT & PERIOD COVEqEO

Percent Errors in the Estimation of Demand for

Secondary Items Technical NoteS. PERFORMING ORG. REPORT NUMBER

RDA-TN-85027. AUTHOR(m) S. CONTRACT OR GRANT NUMBER(e)

George B. Robinson

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

US Army Armament Munitions and Chemical Command AREAS WORK UNIT NUMBERS

ATTN: AMSMC-SALRock Island, IL 61299-6000

I1. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

November 198513. NUMBER OF PAGES

58

14. MONITORING AGENCY NAME & ADDRESS(I dllfermt from Contrallind Ollc) IS. SECURITY CLASS. (of this report)

Unclassified

ISa. DECL ASSIFICATION/DOWNGRADINGSCHEDULE

I6. DISTRIBUTION STATEMENT (of this Report)

* Approved for Public Release; Distribution Unlimited

17. DISTRIBUTION STATEMENT (of the ebetract entered In Block 20, If different from Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side i lnecessa, aid Identify by block number) bPercent Errors Average Monthly Demand Spares

Estimation Wholesale Level Commodity Command Standard SystemForecasting Logistics (CCSS)

Supply Secondary Items

Demand Spare Parts

20. ABSTRACT (Continue an rever e side If neceeesay md Identify by block number)

AMC subordinate commands use the Commodity Command Standard System (CCSS) tohelp manage secondary items. Percent errors in the estimation of demand are

used by CCSS to compute economic order quantities and variable safety levels.

Because the percent errors had not been verified for AMCCOM, this sLudycomputed percent errors from historical AMCCOM demand and return data. The

percent errors found were roughly ten percent higher than the ones in use.

it was recommended tbit the current percent errors be retained and thata limited number of serviceable returns be deducted from demand estimates.

U 1JA°f 141 EDITION OFI NOV 6S IS OBSOLETE UNCLASSIfiED

SECURITY CLASSIFICATION OF THIS PAGE (When ODet Entered)

ie

I? 6A

AC*QNOWLDGENi'b

signiicarit yjuiuance was receiveo from Mr. Goruy becgum oi.

tne AM~C(Jjh- Niateriei Manac-emeriL LirecctoratetMessrs. t owiand Alan haplan ot tne Inventory kesearcli Oitice.

AcCesiofi Fo

NTIS CRAMIOTIC TAB 03

Unannou.ifce'iJustiicatro. .... - ----- ----

Availability CodesAvail and (or

A-1-

p~p..i NSFICTEO

3'

LOW~ ~ C19 W

A. '

SUMMA RY

Percent errors in the estimation of demand are used in theCommodity Command Standard System (CCSS) Economi c OrderOuantity/Variable Safety Level (EOO/VSL) computations. Thepercent error is the average unsigned percentage differencebetween the estimated item demand and the actual demand. CCSSuses different percent errors for different demand levels anddifferent item annual costs. These percent errors hae never hepnverified at AMCCOM since the inception of CCSS durinq the 1970s.

Two related CCSS parameters also had never been verifiedl:

1. The percentage of serviceable returned items used tooffset demand, currently 20 percent.

2. The annual value of item demand used to classifyitems as low value or high value, currently $200 peryear.

The following data were used to develop the percent errors inthe estimation of demand:

1. Demand Return Disposal (DRD) transactions for thev(ars 1979-1983.

2. National Item Identification Number (NTrN) changes.

3. The NTTNs and other data for managed, non-ammunition "eitems in CCSS.

The following method was used:

1. For each yearly file of DRD transactions thetransaction count, quantity, and dollar value for each

* item were accumulated. Disposals, unserviceable returns,and the following demands were omitted:

a. Non-recurring.

b. One-time.

c. Foreign military sales.

d. Depot overhaul.

e. Zero quantitv.

1. After item NIINs were uprIated, the onlv itemsretained were those which met all six of the followingcriteria:

Ji

. _-A

a. Managed.

b. Non- ammun i t i on.

c. Non-provi sioninq.

d. Either Procurement Appropriation Secondary (PAS)or Army Stock Fund (ASF).

e. Normally stocked, stored, and issued.f. Had nonzero demand in the first Vear of thestudy.

3. For each item, two years of demand history were usedto predict the percent error for the following year.Percent errors were computed by comparing 1981-82 averagedemand level with 1983 net demand, i.e. demand lessserviceable returns. Similarly 1980-81 was used topredict 1982, and 1971-80 to predict 1981. "

4. Percent errors were accumulatecl bv number ofrequisitions and yearly dollar value, and were adjustedto nine-month percent errors to accomodate CCSS.

Results and conclusions are as follows:

1. The computed percent errors were roughly ten percenthigher than those in CCSS.

2. The current CCSS percent errors fit the observed 4.

aistributions more closely than the computed percenterrors did.

3. The expected number of requisitions qererallyexceeded the number observed in the past.

4. When serviceab]e returns were used to offset demands,the percent errors became larger.

5. When the low/high value breakpoint was increased from$200 to $500 or to $1,000, the percent errors change""very little, but the items became somewhat more equallydivided between the low and high value classes.

6. By adjusting the means and the percent errors, better

fits to the observed distributions were obtained.

Tt is recommended:

1. That the CCSS percent errors not be changed atp:present.

IW,

2. That serviceable returns n ot be used to offsetdemands.

3. That CCSS be revised to adjust tlve means before usingthe percent errors to compute the demand distribution.

If it appears feasible to revise CCSS to adjust the means,then it is recommended: 5-

1. That the proposed breakpoint between low and highdollar value items be increased from $200 to some valuebetween $500 and $1,000 per year.

2. That percent errors and factors to adjust the meansbe recomputed using the new breakpoint and the programchange factors.

3. That the costs and benefits of adjusting the meansand changing the percent errors be evaluated.

* TABLE OF WN~TENTS

Summary i

Intxoduction I

Dat a 2

Method

Results b

Conclusions 8

Recommendat ions 9

Distribution List 10

Appendix A -- Percent Errors Lor All Years A-I

Appenuix b -- Using Serviceable Returns to Otiset b-IDemands-Preli minary

Appendix C -- Percent Errors for Different breakpoints C-I

Appendix D -- Items With Percent Errors Exceeding lOU0 D-iPercent

Appendix E -- Histograms of Percent Errors E-I

Appendix F -- The Negative Binomial Distribution F-I

Appendix G -- Expected and Actual Mean Q-.

Appendix H -- Expectea and Actual Stanoard Deviation h-I

Appendix I -- Obtaining Closer Fitito the Demand L-1Di stri but ions

Appendix J -- Using Serviceable Returns to Oftset J-IDemands-Final

ivi

................... *.* ~~... .... *..*.

*~*I~ *~**'%.*~*.*.~ * .. *2~~~*-'- ".\*'-- *

*.

iNT RODUCTI' I

The Commodity Command Standard System (CCS S) is acomputerized system that AMCCOM and other US Army commands usefor materiel management. CCbS processes requisitions, keepstrack ot current items, records demand, return and disposaltransactions, and determines reorder points, reorder quantities,and whether items should be stocked or ordered as needed.

The percent error is defined as the average unsigneu .

.. percentage difference between the estimated item aemana for anine-month time span and the actual item demand tor the nine

• months. For example, if the estimated demand is IOU and theactual demand is either 180 or 20, the percent error is 80percent. Percent errors are provided by CCSS for seven levels ofitem annual demand and two levels of item annual total uollarvalue, <= $200.00 and $20O.UO. This methodology was developedby the US Army Inventory Research Office (IRO).

Percent errors in the estimation at demand are used in C(SSEconomic Order Quantity/Variable Safety Level (EOQ/VSL)computations. The economic order quantity is the order quantity

* which is the least expensive in the long run. The variablesafety level is the excess inventory carried to meet demand inexcess of the estimates. CCSS EOQ/VSL computations use two yearsof demand history, the percent errors, and the program changefactor (PCF) to predict item demana during the procurement

*' leadtime (PROLT) ior the item. The PCF accounts for the expected*- change in usage caused by changes in the type and number of

weapon systems that use the item.

When this study was performed, CCSS did the EOU/VSLcomputations only for nonprovisioning (old) items. (Currently,CCSS does the EOQ/VSL computations for provisioning items aswell.) Provisioning items are denoted by a "4" in the thirdposition of the Financial Inventory and Accounting Code (FIA-CD).Items usually change from provisioning to nonprovisioning statusafter two years in CCSS, but this change can be overridden.

V'

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7.7.

DATA

The Lollowing .ata were used to develop the percent errors inthe estimation ot demand:

1. Demand Return Disposal (DRD) transactions for the..

years 1979-1983. The DRD transactions since 1976 are available Von, magnetic tape for all items, including ammunition items, _-2nonmanage(A items, and items which are no longer in CCSS.

2. National Item Identification NJumber (NIIN) changes -

were extracted from CCSS on 31 January 1984. -

3. NIIN information for managed were non-ammunitionitems in CCSS, extracted 27 January 1984. The informationincluded, for each item:

a. The date that the item entered CCSS (DT-ENTMY).

b. The Financial Inventory and Accounting Code(FIA-CD ).

c. The Inventory Management and Processing Code(IMpC).

2 . -

...............

METTHOD A

For each yearly tile of DRD transactions the transactioncount, quantity, and dollar value for each item were accumulated.The following transactions were omitted: 2a C io

4 •. Disposals: Record Identification (REC-ID) = 3.

2. Unserviceable returns: XEC-ID = 2 and Condition Code..

(COND-CL)) not "A" through "E".

3. Transactions with zero quantity: Final Quantity(FNL-T Y) = U.

4. Nonrecurring or one-time demands: Demand Code(DMD-CD) = "N" or "oil

5. Foreign military sales: the first character otDocument Number (DOC-NO) = "B", and the sixth characterof DOC-NO = "3" through "8" or "e.

6. Depot overhaul demand: Project Code (PROJ-CD) ="Cz "

Aiter item NIlNs were updated, only items which met all thefollowing criteria were retained:

I. AMCCOM managed and non-ammunition: the firstposition of FIA-CD ="M"

2. Non-provisioning: the tirst two digits ot DT-ENTRYwere O or less. Although the FIA-CD icientified items asprovisioning or nonprovisioning, because the 1982 FIA-CLDswere not available, the DT-ENTRY was used instead.

3. Either i rocurement Appropriation Secondary (PAS) orArmy Stock Fund (ASF). Note: All ASF items aresecondary items.

a. A PAS item has the second digit of the FIA-CDequal to "X" through "Z" .

b. An ASF item has the second digit ot the FIA-CD

equal to "2".

4. Stocked: IMPC = "IB" or "IC".

5. Demanded during the first year ot t he study. .Elaccording to the D transactions. Note: This stepinsured that customer demand for the item had not just .obegun.

3

-7- .

, P . lr. , .: . -. . , • - . . . -

. . . . . . . .

Demands could be oitset by serviceable returns in either tilepredicting ("old") or predicted ("new") years. New year demandwas offset by all of the new year serviceable returns, so thatthe accuracy of predicting the net new year demand would befound. Old year demand was offset by returns on a one-tor-onebasis, up to a given percentage of demand. Several maximumoffsets were tested: U percent, 5 percent, 10 percent, 1.1.percent, 20 percent, 50 percent, and 100 percent ot old yeardemand.

The deduction from predicted demand of a percentage ofserviceable returns needs some explanation: Serviceable returnsare items which were requisitioned and distributed to the field,not used, and subsequently returned. Therefore, serviceablereturns indicate past demand, but there is no way to tell whenthat past demand occurred or what type of demand it was. It may"--have been demanded over two years ago or it may nave been a

non-recurring demana. Since CC b only retains two years ofdemand history and because of the uncertainty of when past demandoccurred or what type of demand it was, an adjustment to therecurring demand is mande. Until recently a 5% offset was used.This offset has now been changed to 20% and may be revised again.

Percent errors were computed for each item and accumulated bydemanu Level ana item "extended price," that is, whether theaverage annual dollar value of the item was (= 200.U0 or

- > $200.U0 for the two predicting years. Percent errors were alsoaccumulated for other extended price breakpoints, namely $1UU,$50O, $1, 000, $5,000, and $25, 0U. Old year demand was notoffset by serviceable returns.

Because these computations found an average error for a* 12-month time spa&L but CCSS bases its percent errors on a 9-month

span, each average percent error was then multiplied by thesquare root of 12/9, approximately 1.155. This method is the oneused by CL S, as documented in the CCSS Operating Instructions(CCSSOI) 18-710-102, Volume 4, Appendix C, as corrected by

Mr. W. Karl Kruse of IRO.

Subsequently, percent errors were computed using demandduring 1979-81 and 1960-82.

histograms of the percent errors were computed so that tieobserved distributions could be compared with negative binomialdistributions. For each item the computed ratio of actual demanato expected demand was tallied in the corresponding nistogramcell. For each three year period two histograms were tallied,one for items with total annual value less than the 2u-breakpoint and one for items exceeding the breakpoint. Thesehistograms were then converted to show, for each level of annualdemand, the percent of items in each class.

Moreover, the means and standard deviations were also

'4

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computed tor items in each class of total annual value andaverage annual aemand. The means and standard deviations wereused to compute negative binomial distributions for comparisonwi th:- ,

i. The negative binomial distribution that CCSS would

generate from the current ("old") percent errors.

2. The uistribution that CCSS would generate from thecomputed ("new") percent errors.

3. The observed distribution. These comparisons weredone cell by cell in the histograms by adding theunsigned differences between a negative binomialdistribution and the observed distribution. because theCCSS computation of safety levels involved only caseswhere the actual demand was higher than the predicteddemand, only those cells were used in comparing thedistributions.

In performing Chi-Square tests to compare distributions, allof the cells were used. However, if the expected value of a cellwas five or less, that cell was combined with another cell. Ineach case the test was performed by first computing theChi-Square value between two percent type histograms and thenmultiplying by the ratio of the number of items in the classdivided by 10. Consequently the Chi-Square values are onlyapproximate.

In order to obtain closer fits between the negative binomialand the observed demand distributions, ditierent values of themean and the standard deviation were tried. Closeness of tit wasdefined as the sum of the unsigned differences between thehistogram cells where the demand exceeded the estimate. The sumsof squares of those deviations were also computed.

In response to a suggestion by Mr. Ed Gotwals of IRO, animproved method of considering serviceable returns was developed.For this purpose, the percent error was computed from the ratio:

prediction error

old year demand

. . . . ,

V RESULTS

The percent errors for all years are tabulated in Appendix A.These percent errors were roughly ten percent higher than thosein CCSS. The computed percent errors and those currently in CCSS

are as follows:

Currently in CCSS Comput edAve rage

Annual Number Annual Dollar Value Annual Dollar Valueof Requisitions <= $200. > $200. = $200. > $200.

-------- ----- ----- - ----- --------

2.5 - 4 1.701 1.286 1.969 1.3244.5 - 8 1.262 1.019 1.329 1.1728.5 - 16 1.024 0.792 1.020 0.958

16.5 - 32 0.910 0.656 0.933 0.78832.5 - 62 0.575 0.575 0.664 0.643 :.

62.5 -122 0.469 0.469 0.578 0.558122.5 and over 0.409 0.409 0.460 0.456

The percent errors for 1981-93 for different percentageoffsets of old year serviceable returns are tabulated in AppendixR. The more that serviceable returns were used to help predict,the worse the predictions became. Consequently, serviceable

* returns were subsequently not used to predict future demands.

The percent errors for 1981-83 for different item annual costbreakpoints are tabulated in Appendix C. Because the results forother breakpoints were relatively close, the $200 breakpointcurrently used in CCSS was chosen for use in the rest of thestudy. However, a $500 to $1,000 per year breakpoint would havegiven a more even split between the low and high dollar valueitems.

For general information, items with percent errors exceeding1,000 percent are listed in Appendix D.

Percent error histograms are presented in Appendix E. Byinspection, these histograms look somewhat like histograms for

-. negative binomial distributions.

In appendix F, several negative binomial distributions arecompared with each observed distribution. The column marked "SumHigh Diffs" contains this sum for the region where thecomputation of safety levels would be affected, i.e., where theactual was hiqher than the predicted. In this region, the old

.5+

] ! • *. . ... - . • " . .. . .. S ","S "-°

. . ..-. .. -o'N R . .. . . - ".. - -o- •",'

• ..... S5. . . .. . . . . . .. ..

CCSS percent errors predicted markedly better than the new CC(Spercent errors, ana slightly better than direct computation forthe known mean and standard deviation. In each case the negativebinomial distributions resembled the observed one, even though

* Chi-Square tests usually showed that the observed distributionswere not negative binomial.

Generally, CCSS underestimated the means and standarddeviations o demand, as shown in Appendices G and H. by varyingthe means and standard deviations, closer tits to the observeddemand distributions were obtained, as shown in' an example inAppendix I.

Results ot the improved method for considering serviceabiereturns are shown in Appendix J. The percent errors fellslightly, roughly three percent, as more serviceable returns wereconsidered. The percent errors were generally the least when upto 20 to 50 percent of demand was of tset with serviceable -returns.

7

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The principal conclusions from the above results are:-V

1. The computed percent errors were roughly ten percenthigher than those in C(SS.

2. The current CCSS percent errors fit the observeddistributions more closely than the computed percenterrors did.

3. The expected number of requisitions differed from thenumber observed in the past.

4. When serviceable returns were used to offset demands,the accuracy of prediction improved.

5. When the low/high value breakpoint was increased from$200 to $500 and $1,00, the percent errors changed verylittle, and the items became somewhat more evenlydistributed between the low ana high value classes.

b. by adjusting the means and percent errors, betterfits to the observed distributions were obtained.

Reservations include the following:

1. There is an uncertainty caused by not knowing theProgram Chance Factor (PCF), which could be anywhere from0.01 to 2.0 tor non-provisioning items. This factoradjusts past demand to reflect changes in the number andLocation of weapon systems using the item. Since the PCFwas not available, this adjustment was not made.

2. The percent errors found may be too low because somevalid items without demands in the first year of threewere excludeo.

b1-7

.. .

IT. :,

V7..V7

RE C MMMIAT IOINS

It is recommended:

1. That the C(SS percent errors not be changea atpresent.

2. That CCSS use serviceable returns to offset up to 2Uto 50 percent of demands.

3. That CCSS be revised to adjust the means before usingthe percent errors to compute the demand distribution.

It it appears feasible to revise CCSS to adjust the means,then it is recommended:

1. That the proposed breakpoint between low and highdiollar value items be increased from $200 to sane valuebetween $500 and $1,000 per year.

2. That percent errors and factors to adjust the meansbe recomputed using the new breakpoint and the programchange factors.

3. That the costs and benefits of adjusting the meansand changing the percent errors be evaluated.

9

7-7-7

DISTRIBUfION LIST,

NO. OF %

Commander, US Army Materiel Command, 5001 Eisenhower, Ave.,Alexandria, VA 22333-0001

2 AMCSM-%RS

Commander, US Army Armament Munitions and Chemical Command, ARock Island, IL 61299-6000

1 AMSMC-MM5 AMSMC-MMP-SS -

1 AMSMC-$A5 AM MC-SAL',I AMSMC-SAS

Commander, US Army Armament Munitions and Cnemical Command,Rock Island, IL 61299-7300

4 SMCKR-ESP-L

Commander, US Army Communications - ElectronicsCommand, Fort Monmouth, NJ U7703

: 1 AMSEL-MM. AMSEL-PL-SA

Commander US Army Troop Support Command,4300 Goodfellow Blvd., St. Louis, MO b3120-158-"

" I Al. TS-S h'.

I AMTS-CCCI AM!TS-SPSS (Risch)

Commander, US Army Aviation Systems Command,430U Goodtellow Blvd., St. Louis, MO o3120-1584

I AMSAV-SI A&SAV-CCC .

Commander, US Army Tank-Automotive Command,Warren, M 48397-5000

I AMS TA-FI AMSTA-V* i AMSTA-SA

• I Commandant, US Army Logistics Management Center,Fort Lee, VA 23801

* 1 Defense Logistics Agency, DLA-LO, Cameron Station,Alexandria, VA 22314

I Logistics Studies OtLice, AMXMC-LSO, ALMC,Fort Lee, VA 23801

1 US Army Inventory Research Ottice, ATTN: Mr. Gotwals,US Army Materiel Systems Analysis Activity,800 Custom House, 2d I Chestnut Streets,Philadelphia, PA 19106-2976

-.

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1* *.IAPPENDIX A

PERXNT ERRO1~ FOR ALL ~1&Ai~

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*) *J.

*444

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4-%

.. 4 . 4 . 4.... **4'* '.4.'. . 4.-

PERCENT ERRORS FOR ALL YEARS

_ $200 Per Year ._.

1-2-3

Annual Currently WeightedRequisitions in CCSS 1979-81 1980-82 1981-83 Average

2.5-4 1.701 1.682 1.984 2.055 1.9694.5-8 1.262 1.297 1.238 1.400* 1.3298.5-16 1.024 1.105 .991 1.010 1.02016.5-32 .910 .901 .900* .962 .931

32.5-62 .575 .703 .718 .616 .66462.5-122 .469 .596* 580* .570* .578

122.5 & Over .409 .517* .474* .431* .460

*Smoothed

> $200 Per Year

1-2-3Annual Currently Weighted 'L...

Requisitions in CCSS 1979-81 1980-82 1981-83 Average

2.5-4 1.286 1.194 1.281 1.396 1.3244.5-8 1.019 1.018 1.214 1.195 1.172

8.5-16 .792 .891 .960 .979 .95816.5-32 .656 .753 .818 .779 .78832.5-62 .575 .628 .641 .650* .643

62.5-122 .469 .549 .550* .566 .558122.5 & Over .409 .497 .474 .431 .456

*Smoothed

A-]

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p APPENDIX B

USING SERVI~EA~LE ~IURNS TO OFFSET L~~NUS--PRELlNINARY

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9. 9 5 .!g~ .*.f .. 9-.~ .49 9 . . . . . . .. -9

~ **.**** -

* £..d... * ~ ~ .. ~. ~ ~ ~ *.1.*~

USING SERVICEABLE RETURNS TO OFFSET DEMANDS--PRELIMINARY

Combined Percent Errors (Both $200 and > $200)

Average Annual OffsetNumber of Requisitions

in 1981-82 0% 5% 10% 15% 20% 50% 100%

2.5-4 1.741 1.779 1.820 1.861 1.904 2.259 2.5504.5-8 1.614 1.645 1.676 1.706 1.738 1.987 2.3398.5-16 .987 1.007 1.026 1.045 1.066 1.214 1.47016.5-32 .805 .821 .836 .852 .869 1.003 1.27032.5-62 .753 .773 .793 .814 .838 1.025 .899

-62.5-122 .570 .574 .579 .585 .594 .668 .924122.5 & More .430 .427 .427 .429 .433 .456 .471

B-1

APPENDIX C

PERCENT ERRORS FOR DIFFERENT BREAKPOINTS

r p.

'.

PERCENT ERRORS FOR DIFFERENT BREAKPOINTS.4.

Percent Errors for Annual Value BreakpointME

Average Annual Breakpoint* Number of Requisitions

in 1981-82 $100 $200 $500 $1000 $5000 $25000

2.5-4 2.295 2.055 1.867 1.799 1.740 1.7464.5-8 2.694 2.276 1.967 1.833 1.680 1.6308.5-16 1.068 1.010 1.001 1.023 1.009 .99016.5-32 .997 .962 .847 .829 .813 .80632.5-62 .573 .616 .582 .565 .816 .77662.5-122 .834 .647 .605 .558 .595 .587

122.5 & More .636 .423 .433 .348 .378 .421 -

Percent Errors for Annual Value > Breakpoint

Average Annual Breakpoint* Number of Requisitions

in 1981-82 $100 $200 $500 $1000 $5000 $25000

2.5-4 1.383 1.396 1.448 1.525 1.748 1.4424.5-8 1.225 1.195 1.143 1.113 1.138 1.2248.5-16 .973 .979 .976 .942 .888 .93316.5-32 .789 .779 .788 .788 .787 .79732.5-62 .763 .768 .796 .844 .644 .59862.5-122 .566 .566 .564 .572 .538 .483

122.5 & More .430 .431 .430 .436 .451 .446

c-1

4..

.* * ~ * ..4 . . . . . . . . . . . ..4 . . . . .. .

7V1.7. ..

NUMBER OF ITEMS BY CATEGORY FOR DIFFERENT BREAKPOINTS

FOR 1981-83

V. Breakpoint

$100 $200 $500 $1000 $5000 $25000

Annual Value

Breakpoint 1744 2583 3970 4959 7033 8340

Annual Value> Breakpoint 7378 6539 5152 4163 2089 782

C-2

% - -- -da .

4 M

4

p

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4.

APPENDIX D

ITEMS WITH PERCENT ERRORS EXCEEDING 1000 PERCENT

- -.44..

- -

*J.

4.

I

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D-

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APPENDIX E

.. '. %

HISTOG1W~S OF PERCENT ERRORSesI.~.,

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APPNDI F1*

THE EGATVE BNOMIL DITRIBTIO

a7%

'te

..

The Negative Binomial Distribution.i%

The negative binomial distribution is used by CCSS to estimate the qu ntityof demands for an item. This distribution is discrete and unimodal, and as thevariance approaches the mean it becomes Poisson.

This distribution can be computed from its mean V and standard deviation aas follows:

P= -asa

q= 1-p

r= IJp

q

Let p denote the probability density at each nonnegative integer k. Thenr

P0=P

and for k > 0,

Pk = Pk-1 x q x (k + r - 1)k

For computational purposes we assume that all requisitions for an item are for thesame quantity of items.

Source: Alan Kaplan of the US Army Inventory Research Office (IRO), by tele-phone on 9 September 1983.

In Tables F-I and F-2, theoretical and observed distributions are tabulatedtogether for comparison. The distributions are of four types:

1. Old CCSS -- figured from the current CCSS percent errors, as CCSS would:the mean was the average demand during the past two years, and the standarddeviation was computed from the mean and the percent errors.

2. New CCSS -- figured from the computed percent errors as CCSS would.

3. Direct -- figured directly from the observed mean and standard deviation -

for 1981-1983.

4. Observed -- observed for 1981-1983.

Because the distributions are rounded to the nearest percent, they sometimes do

not add up to 100 percent. Because of outliers in the data, the mean and standarddeviation for two distributions were not readily available. Two other distributions '

were omitted because they had only 6 and 28 data points respectively.

F-1

The numerical column headings need some further explanation, since only on-

limit for each column is given. For example, the column titled "> 50%" includes

only items with percent errors > 50 percent and < 100 percent, and the column

titled "< 1000%" includes only items with percent errors > 400 percent and -< 1000 ypercent.

The "Sum High Diffs" column compares the theoretical distributions with theobserved ones. Because we are primarily concerned with safety level, the distri-butions were compared only for the cases when they exceeded their means. For these

cases for each theoretical distribution, the absolute values of the deviations fromthe observed percentages were added and placed in the Sum High Diffs column.Consequently, the smaller the Sum High Diffs value, the better the fit of thetheoretical distribution to the observed one.

When the Old and New CCSS distributions were compared, the old one fit moreclosely. In only twelve cases were there sufficient items for a valid comparison.In seven cases, the Old CCSS distribution fit more closely; in one case, the New CCSSdistribution fit more closely; and in four cases, the two distributions fit equallyclosely.

In Tables F-3 and F-4 the results of chi-squared tests comparing the variousdistributions are presented. The columns labeled "Reject 10%" and "Reject 1%"give the values of the chi-squared statistic above which we assert with 90 percentand 99 percent confidence, respectively, that the data was not distributedaccording to the test distribution.

IF.

F-2°

................................................

. . . . . . . . . . . . . . . . . . . . . .* - . . . .

. . . . . . . . . . . . ... -

Table F-i *

Distributions for Annual Value _ $200 "

Annual Ty"e W I0d:Ka Actual Was High

Ty Acta SumLo ~t..IDetri- -. .. --_i__h_

Itq on b bution 100 150 % 0 50Z 0% -50% .1 2 2002 '400 -ClOoo "O00 .i t

Od CCSS 56 11. 6 4 3 5 4 5 4 1 25

2.5 -'

New .CCS .64 92 5 ..3.11 ..2.L 3 4 4, 2 28

Direct 53 9 5 4 3 4 4 5 27

Observed 21 17 13 211 8 10 10 6

Olid CCSS 34 20 15 3 7 5 6 3 0 . 214.5

Now CCSS 39 19 13 3 6 4 6 5 4 0 23to -- . --

SDirect I- - .,- t

observed 14 19 18 1 12 9 10 a. 6 3

Old CCSS 14 31 20 2 11 7 a 5 2 0 14

Now OCSS 13 31 20 2 11 7 8 _5 2 0 1to

36 Direct 7 26 21 3 13 9 11 e 2 0 l"

Observed 8 21 24 0 16 13 10 6 2 0- -o _ -___ 2 = =-Old CCSS 6 36 22 1 12 8 8 0 1 0 i-?

16 C5S 8 36 21 1 12 8 S 5 1 0 01

to - - - -7

32 Direct 8 31 16 1 11 8 10 8 4 0

Iobserved 7 15 26 0 25 9 9 6, i

Old CCSS 0 26 32 1 20 11' 7 21 0 0 i-i

ew CCSS 0 28 31 1 19 11 8 2 0 0 1)to ' 8 0 0I0t"

62 Direct 0 23 32 1 21 12 8 2 0 0 10

Observed 17 30 01 0 0 1

Old CCSS 0 19 38 125 11 5 l1 _ 0 0 .2. o 0

Now" CCSS 0 26 33 1 21 7 2 0 0to

Direct .--.

Observed 0 11 43 0 25 14 0 7, 0 0 1 t .m-

'125Old CCSS 0 14 42 0 28 11 4 O0 0 I01- - iiiji o____

122.5

N o CCSS 0 16 41 0 27 11 15 0 0 0 O',

Over Direct __,.__6__ _"

Observed 0 1 0 33 50 0 0 0 0

F- 3

............................................................ - . . . . . . . ." ""** *

Table F-2

Distributions for Annual Value > $200

Annual Type of Actual Was Lo Okay Actual Was High SUm

Requl- Deri- OZ - -~or10-~00 HIhet-on bu ion 2 >O : i5OZ p 6200% W00 OZ O0 1000 Di

OldCCSS 43 15 9 6 5 7 6 5 3 o 142.5.

t New CCSS 47 141 8 6 4 6 6 5 4 0 15to - i - - -'--"- -

4 Direct 36 14 9 7 5 8 7 7 5 1 9

Observed 27 17 I 15 2 11 8 9 7 5 0

Old CCSS 20 23 20 5 11 7 8 5 2 0 6

t No C 31 211 16 4 8 5 7 6 3 0 1to __ _ _ - - - - - - - -

Direct 34 19

1 3 3 7 5 6 6 5 1 15

Observed 20 24 17 1 14 8 7 6 210

Old CCSS 6 29 25 3 15 9 8 4 1 0 5

8.5 New CCSS 12131120 2 12 7 8 5 1 0 9

to

16 Direct 19 30j 16 2 9 6 9 6 3 0 15

Observed 15 26 22 1 16 8 7 3 2 1 0

Old CCSS 1 28 30 2 18 In 8 3 0 0 6

lb.5 NO CCSS 3 2 15 9 8 4 0 C , 7

32 Direct 4 36 25 2 14 8 7 3 0 0 5

Observed 12 25 2b 1 17 B 7 2 1 0= -, -325 Old CCSS 0 26 32 1 20 11 7 2 0 0

ew CCSS 0 30 29 1 18 10 8 2 0 ,0 8

to - - ', .

62 Direct

Observed 6 22 33 1 22 9 5 2 0 0

Old CCSS 0 19 38 1 25 11 5 1 0 0 10

.5 Nev CCSS 0 26 33 1 21 11 7 2 0 0 9to _ _ . - ~ ---

122 Direct 0 33 32 1 18 9 0 0 4

Observed 6 20 42 0 17 10 4 1 0 0

Old CCSS 0 1 142 0 28 11 4 0 0 0 18122.5 .----

and New CCSS 0 16 141 0 27 11 5 0 0 0 18

Ove Direct 0 131 1 37 0 19 8 4 0 0 0 10

Observed 5 1 21 4 1 1 0 0

4. . . -. .

• . ' • . . : ., ? '. . .. - . . . .'- . 'm-

TABLE F-3 a

CHI-SQUARE TEST;SP

*TOTAL VALUE !$2OO/YEAR

Approximate

Annual Type of Degrees of Chi-Square Reject Reject

Reqns Distribution Freedom Statistic 10% 1%

*2.5 New CCSS 9 1273 14.68 21.67

*to 4 Direct 7 865 12.02 18.48

4.5 New CCSS 8 464 13.36 20.09

to 8 Direct

-8.5 New CCSS 8 68 13.36 20.09

to 16 Direct 6 32 10.64 16.81

16.5 New CCSS 6 49 10.64 16.81

to 32 Direct 5 54 9.236 15.09

32.5 New CCSS 4 4.8 7.779 13.28

to 62 Direct 2 1.8 4.605 9.210

*62.5 New CCSS 3 33 6.251 11.34

to 122 Direct--

122.5 New CCSS

and Over Direct--

F-5

TABLE F-4

CHI-SQUARE TESTS

TOTAL VALUE >$200/YEAR

Approximate

Annual Type of Degrees of Chi-Square Reject Reject

Reqns Distribution Freedom Statistic 10% 1%

2.5 New CCSS 8 375 13.36 20.09

to 4 Direct 6 148 10.64 16.81

44.5 New CCSS 8 169 13.36 20.09

to 8 Direct 6 271 10.64 16.81

8.5 New CCSS 7 54 12.02 18.48

to 16 Direct 6 168 10.64 16.81

16.5 New CCSS 7 321 12.02 18.48

to 32 Direct 5 219 9.236 15.09

32.5 New CCSS 5 20 9.236 15.09

to 62 Direct

* 62.5 New CCSS 5 24 9.236 15.09

to 122 Direct 2 25 4.605 9.210

122.5 New CCSS 4 57 7.779 13.28

and Over Direct 2 103 4.605 9.210

* F-6

.,.% ..~

**~~1

* 4't

b *~- -~

~FV~

J.. *~*%'-S -,-.. .~*-

* a,.

4-

APPENDIX C

EXPECTED AND ACTUAL MEAN

bi

A..

p

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.5,.-5.''.v -. ~.

V.-.

1%

USN~

* '-- ~'

R -L

EXPECTED AND ACTUAL MEAN

Item Annual Cost < $200 per year

Average AnnualNumber of Ratio of Actual Mean to Expected Mean

Requisitions 1979-81 1980-82 1981-83

2.5 - 4 1.514 1.972 2.044

4.5 - 8 1.407 1.302 2.295

8.5 - 16 1.347 1.217 1.280

16.5 - 32 1.186 1.548 1.34732.5 - 62 1.207 1.327 1.05162.5- 122 1.640 1.406 1.202

122.5 and more 2.127 1.255 1.299

Item Annual Cost > $200 per year

.. Average AnnualNumber of Ratio of Actual Mean to Expected Mean

Requisitions 1979-81 1980-82 1981-83

2.5 - 4 1.008 1.185 1.338

4.5 - 8 .948 1.205 1.1778.5 - 16 .906 1.050 1.063

16.5 - 32 .849 1.022 .91732.5 - 62 .947 .997 1.057

62.5 - 122 .989 1.151 .896122.5 and more 1.196 1.183 .853

G-1

-- A *.

p

* APPENDIX H

EXPECTED AND ACTUAL STANDARD DEVIATION

L- 7.7-1.- 1- I;-).

EXPECTED AND ACTUAL STANDARD DEVIATION

Item Annual Cost < $200 per year* Average Annual

Number of Ratio of Actual to Expected Standard DeviationRequisitions 1979-81 1980-82 1981-83

2.5 - 4 1.671 1.830 1.8064.5 - 8 1.143 1.233 2.9008.5 - 16 1.393 1.211 1.11016.5 -32 .983 2.116 1.43532.5 -62 1.769 1.223 .95962.5 -122 2.289 .882 1.350122.5 and more 1.920 .931 .686

Item Annual Cost > $200 per yearAverage Annual

Number of Ratio of Actual to Expected Standard DeviationRequisitions 1979-81 1980-82 1981-83

2.5 - 4 1.098 1.081 1.1574.5 - 8 1.295 1.509 1.2988.5 - 16 1.102 1.976 1.28016.5 - 32 1.004 1.807 .97632.5 - 62 .992 1.081 4.776 -3.

62.5 - 122 .955 1.723 .986122.5 and more .966 1.004 1.080

H-I

COMPUTING THE STANDARD DEVIATION

Glossary: o

PCER is the percent error in demand for a nine month period.

PCER-LT is percent error in the procurement lead time.PROLT is procurement lead time in months.PROLT-DMD is the demand during the PROLT.AMD is average monthly demand.STD-DEV is the standard deviation.AYD is average yearly demand.

The CCSS Computation:

1. Source: CCSS Operating Instructions 18-710-102, Volume 4, Appendix C, as corrected

by Karl Kruse of the Inventory Research Office."" ~**0.5 .,

7" 2. Compute PCER-LT PCER/ (PROLT/9) **,.5

3. Compute PROLT-DMD AMD x PROLT

4. If PROLT-DMD > 20 use GAMMA 1.41; otherwise use GAMMA from this table:

PCER-LT GAMMA0 to .5 1.27

Over .5 to .8 1.33Over .8 to 1.0 1.42

Over 1.0 1.52

5. Compute STD-DEV PROLT-DMD x GAMMA x PCER-LT

Our Computation:

1. Compute PCER-LT as in the CCSS computation. To generate a distributioncomparable to the twelve-month histograms in Appendix E, PROLT - 12 months was

used.

2. Compute the AYD as follows.

a. For each interval of annual requisitions, choose a representative to

generate a typical distribution for the interval. The numbers chosen were 3,6, 12, 22, 45, 85, and 175 requisitions per year.

. b. For each interval, compute AYD - Annual Requisitions x Average QuantityRequisitioned, using 2 as the Average Quantity Requisitioned.

3. Compute PROLT-DHD - AYD x PROLT in years. PROLT in years - 1.0.

4. Compute GAMMA as in the CCSS computation. For 3 and 6 annual requisitiunb,• PROLT-DMD = 6 and 12 respectively; since PCER-LT > 1.0, GAMMA = 1.52. For 12

and more annual requisitions, PROLT-DMD > 20, so GAMMA = 1.41.

5. Compute STD-DEV as in the CCSS computation..

H-2

- .- - . . .. . .' ,. . .. . . . - ........ ... . .-.... ... *.. . . . .. .-... .-.-. : . ,. .. ... .. .,..,... ... . . -, : .. _ . . -l',

- -.. - -' - -= --- ~ .... - .... - .-..- - - - . - . -

r4

a

~. J.a a

'.*' .. '

4...

''-V

APPENDIX I

OBTAINING CLOSER FITS TO THE DEMAND DISTRIBUTIONS

.4..

E'm~.

'4.

*4.'~

~ a,

a.- a~* . . . .

- 'a-6 ' S - . **a. t~ A *. *. * *,* ,* *. a. *P**'

In Table I-I closer fits to an observed demand distribution were obtained

by varying the mean and standard deviation of negative binomial distributions.This case involved 2.5 to 4 annual requisitions with an annual value of $2tO orless.

The columns of Table I-I are as follows:

1. The distribution used.

2. Actual Low or Equal: The percent of cases with the observed valueless than or equal to the mean.

3. Actual was High: The percent of cases with the observed value greaterthat the mean, further subdivided into a histogram as follows:

a. < 50% denotes cases with the observed value up to 1.5 times themean.

b. 3 100% denotes cases with the observed value between 1.5 and 2times the mean.

c. Similarly, < 200%, < 400%, and < 1000% denote observed values from

2 to 3 times, from 3 to 5 times, and from 5 to 11 times the mean.

d. > 1000% denotes that demand was more than 11 times the mean.

4. Sum Pos Diffs: For cases when the actual exceeded the mean, the sum ofthe percent differences between the observed and test distributions.

2

5. Sum Pos Diffs The sum of the squares of those differences.

... i

In '

I--'

JIIN

" -.'""'." -" " ".- '. '- -.. - . .""""- . """- " " . """ . """" - "," J '"- ,"." ",". " '' . '.-." ''2 ''" - %I-1 """" ""

The rows of Table I-I are as follows:

1. Observed--The histogram observed when using 1981-82 to predict 1983.

2. Old CCSS--Using the current CCSS percent error with mean f 3 and standarddeviation - 6.717.

3. New CCSS--Using the computed CCSS percent error with mean = 3 and standard

deviation - 8.117.

* 4. Direct--Using the computed mean and computed standard deviation.

5. Variations of old and new CCSS with P=3--One of these distributions couldbe used without revising CCSS but by changing only the percent errors now iL CCSS.

6. Variations of direct with v f 6.132--Possibilities by using the expected mean

and various standard deviations. Use of one of these distributions would requirea revision of CCSS.

7. Variations of direct--Possibilities by using various means and standard devitions.Use of one of these distributions would also require a revision of CCSS.

'1

.1.

S.....

....... ....... .T .~

'.4 .. *. 4 . - . 4 . 4-4

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TABLE I-1

Obtaining Closer Fits to the Demand Distributions

Total Value < $200/yr. Actual Low Actual was H"ighhum Su

2.5 to 4 Annual Reqns. or equal 50% = 100% < 200% 400% = 1000% ) 1000% Diffs Diffs

Observed 53 11 8 10 10 6 2

Old CCSSU =3, a = 6.717 78 3 5 4 5 4 1 25 139

New CCSS=3, a = 8.117 81 2 4 3 4 4 2 28 186

Direct=6.132, a =14.655 71 3 4 4 5 7 5 27 151 ".

Variations of old and

new CCSS with p = 3

o 2 65 15 15 5 0 0 0 34 230a= 3 66 9 12 8 4 0 0 22 1M)Oa= 4 70 7 9 7 5 2 0 19 71a 5 73 5 7 6 6 3 0 20 82a= 6 76 4 5 5 5 4 0 24 116a 8 81 3 4 4 4 4 2 26 156

a 10 84 2 3 3 3 4 2 30 208

Variations of directwith = 6.132

= 6 42 8 14 14 14 7 0 20 82a = 8 52 6 10 10 11 10 1 13 47a - 9 56 6 8 9 10 9 2 9 35a i 10 59 5 7 8 9 9 3 14 52

a = 12 65 4 6 6 7 8 4 20 86a - 16 73 3 4 4 5 7 5 27 151a - 20 79 2 3 3 3 5 5 32 214

Variations of direct

4.5, a - 4.5 53 9 14 12 9 3 0 16 58

a = 6 60 7 10 9 9 5 0 11 27a = 7.5 65 5 7 7 8 6 1 13 51

pu=8, o = 8 35 7 12 14 17 13 1 27 147a = 10 44 6 10 11 13 13 3 19 89a - 12 51 5 8 9 11 12 5 17 83

Note:p = meang a f standard deviation

Also note: The Sum High Dif s and Sum High Diffs columnS include only cases where the W"actual was high.-".

1-3

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APPENDIX J

USING SERVICEABLE RETURNS TO OFFSET DEMANDS -- FINAL

USIN4G SERVICEABLE RETURNS TO OFFSET DEMANDS -- FiNAL

Moaifiei Percent Errors (both <=$200 and > 2UU) ~ 4

(Average Ratios of Error to Old Year Demand)

Average Annual OffsetNumber of Requisitons

in 1981-82 0%4 5% 10% 15% 2U% 50% 1 01

2.5-4 1.741 1.736 1.732 1.728 1.725 1.724 1.7474.5-8 1.614 1.6U7 1.601 1.59b 1.592 1.585 1.6078.5-16 .987 .977 .969 .964 .960 .9b0 .9W)lb.5-32 .805 .791 .781 .773 .7b9 .765 .77832.5-62 .753 .742 .734 .729 .727 .728 .74362.5-122 .570 .553 .542 .535 .531 .530 .544

122.5 &More .430 .412 .402 .398 .39b .:s95 .394

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