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Institutional Archive of the Naval Postgraduate School Calhoun: The NPS Institutional Archive DSpace Repository Theses and Dissertations 1. Thesis and Dissertation Collection, all items 1988-09 The effect of the covariance factor on the Procurement Problem Variance of net leadtime demand Adams, Keith T. http://ndl.handle.net/10945/23190 Downloaded from NPS Archive: Calhoun Calhoun is the Naval Postgraduate School's public access digital repository for ' (8 D U DLEY research materials and institutional publications created by the NPS community. : Calhoun is named for Professor of Mathematics Guy K. 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Adams september 1988 Thesis Advisor: Alan W. McMasters Approved for public release; distribution is unlimited 1238664 Boary GEASSIFIGATION OF THIS PAGE REPORT DOCUMENTATION PAGE _ REPORT SECURITY CLASSIF:CATION 1b RESTRICTIVE MARKINGS Inclassified 7 SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION / AVAILABILITY OF REPORT : eemeveamron Ppublvegmelease; | DECLASSIFICATION / DOWNGRADING SCHEDULE Cuca tblteaon 1S Unlimited iz ORGANIZATION REPORT NUMBER(S) 5 MONITORING ORGANIZATION REPORT NUMBER(S) | NAME OF PERFORMING ORGANIZATION 6d OFFICE SYMBOL | 7a. NAME OF MONITORING ORGANIZATION (if applicable) Naval Postgraduate School 515) Naval Postgraduate School t ADDRESS (City, State, and ZIP Code) 70. ADDRESS (City, State, and ZIP Code) Monterey, California 93943-5000 Monterey, California 93943-5000 i | NAME OF FUNDINGs SPONSORING Go. OFFICES yRISOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (if applicable) | ADDRESS (City, State, and Z/P Code) 10 SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK ELEMENT NO. NO. NO |. TITLE (include Security Classification) THE EFFECT OF THE COVARIANCE FACTOR ON THE PROCUREMENT PROBLEM VARIANCE OF NET LEADTIME DEMAND WORK UNIT ACCESSION NO. |. PERSONAL AUTHOR(S) | ADAMS, Keith T. Ja. TYPE OF REPORT) 13b TIME COVERED 14. DATE OF_REPORT (Year, Month, Day) [15 PAGE COUNT Master's Thesis FROM TO 1988 September 44 .. SUPPLEMENTARY NOTATION The views expressed in this thesis are those of the autho Bnd do not reflect the official policy or position of the Department of Defense: or the U.S. Government. | COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number) FIELD Covariance, Variance, Standard deviation, : fT tC CLeadtime, Demand, Repairables, inventory models, |. AP ICP's, Inventory Control Points, Wholesale . ABSTRACT (Continue on reverse if necessary and identify by block number) | An analysis is made of the formulae used by the Navy's Inventory Control Points in calculating the variance of Net Leadtime Demand of repairable items. A new formula is then derived, which makes use of actual Icalculations of covariance between regenerations and demands. The resulting variance values derived from the new formula are compared with the variance Ivalues resident on the Navy's Ships Parts Control Center data base and are Ishown to produce lower variances. The new formula is also compared to the Joption path formula to determine which formula produces the smallest Ivariance. The comparison suggests an under-estimation of variance results when the option path with its estimate of the covariance is used. The thesis concludes with recommendations for implementation of the new formula. 0. DISTRIBUTION / AVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION Ge unctassieiepvunuimiteoD = () same as RPT = C)otic users | Unclassified 2a. NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (Include Area Code) | 22c. OFFICE SYMBOL Prof. Alan W. McMasters 408-646-2678 54Mg | A | D FORM 1473, 84 MAR 83 APR edition may be used until exhausted SECURITY CLASSIFICATION OF THIS PAGE All other editions are obsolete | BL UNCLASCLEFYED™” Office. 1986—606-24. Approved for public release; distribution is unlimited The Effect of the Covariance Factor on the Procurement Problem Variance of Net Leadtime Demand by Keith T. Adams Lieutenant Commander, Supply Corps, United States Navy B.S., Purdue University, 1973 M.Ed., University of Missouri, 1975 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN OPERATIONS RESEARCH from the NAVAL POSTGRADUATE SCHOOL September 1988 ABSTRACT An analysis is made of the formulae used by the Navy’s Inventory Control Points in calculating the variance of Net Leadtime Demand of repairable items. A new formula is then derived, which makes use of actual calculations of covariance between regenerations and demands. The resulting variance values derived from the new formula are compared with the variance values resident on the Navy’s Ships Parts Control Center data base and are shown to produce lower variances. The new formula is also compared to the option path formula to determine which formula produces the smallest variance. The comparison suggests an under-estimation of variance results when the option path with its estimate of the covariance is used. The thesis concludes with recommendations for implementation of the new formula. 11 THESIS DISCLAIMER The views and judgements presented in this thesis are those solely of the author. They do not necessarily reflect official positions held by the Naval Postgraduate School, the Department of the Navy, the Department of Defense, or any other US government agency or organization. No citation of this work may include references or attributions to any official US government source. 1V TABLE OF CONTENTS MUS GPTG UTC TON occ ccccccscccccesccccc ccs ccacecoucccccccuccovceonccavscossesuecsseccessssscsscevsvsvsenve 1 POMOC OUND) ices cccsecoscorcstlepiatsscossccceosscccccccecsssssesssesesvees 1 MOURNS cig cccieyscecccveacccccesecscccosecseccssectsccsecsscessecssesssecsseessens ene 3 DICT, 3 Be cc eeccicceccecsassccsccssssessssessecssecssesseessesssessonssosssessees 3 I. FORMULA DEVELOPMENT.......cccccsscccsscccsscsssccsccseccsseessessecssesssessessreessesssesseess 4 PUMA IONAL, CAVEAT: ..n.c.ccccscccccccccscccsecoccccescssccosecssccsssssssssessssesseen 4 B. PROCUREMENT PROBLEM VARIABLE...ee.ccsscsssssssesssesosessessseeosee. 4 C. UICP VARIANCE FORMULA .o..cocccssccssessssssesssessesstesstsssesssesstessessece: 5 BAU VOR MMOL As cscscccscvcsccecssccccecoseccccsssecsscsuccscsssseccncsssccsucsseccsesees 7 E. FORMULA COMPARISONS. .eecccscsssscsscsssessecsssessesssessecsvesseesensevesseesees 9 II]. FORMULA COMPARISON METHODOLOGY .0.....::-ssssscscesssssssecessssssessessssvcce 12 AA, YATES AVCOTCMSIER (GIN. eae 12 B. FORMULA COMPARISON PROCEDURES. ..-.cccccccssssssessecssesssecesene 13 ©y AMAT FORMATE, c.cccccc.-ccccccsccsccecssecsscoscsasscssecssecssecssessseessevsee 15 po SB GUUIEINS) a rc 16 V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS. .....scccsocssscssseees 21 (CUI CS Se 21 BO ONC WU STON GS, cic iccccsscccssccccsscccssecscscsvecssecssecseccoscsseccsscssscaneceeseen 21 C. RECOMMENDATIONS .ovcceccssscssssssscosecssessscssecssessesssessscarsessessessessseese 22 D. RELATED FURTHER STUDY .oeccccccccscsssccosssssssssscssecessesesssessstsnecsasce: 23 MMMSNO)E REFERENCES, ......cccc ccc cccccccceccccssscssescsocesscsessessssousecsecsssssussssessestsensevsres. 24 V APPENDIX A - DEN/NOMECLATURE Of DATA ELEMENTS 20 APPENDIN B - FORTRAN PROGRAM FOR VARIANCE CALCULATIONS.... 26 APPENDIX CG - DEVAILET) OUD PU als CIN ye 34 INITIAL DISTRIBUTION Ores iiss ccessecs ce 35 V1 TABLE OF ABBREVIATIONS Variables Z - Procurement Problem Variable (PPV) Z - procurement problem random variable V - Procurement Problem Variance D - mean demand per quarter d - quarterly demand random variable Var(d)- variance of quarterly demand L - mean procurement leadtime ] - procurement leadtime random variable L, - net acquisition time Var(])- variance of leadtime B - mean regenerations per quarter (CR)(SR) b - quarterly regeneration random variable T - mean procurement problem turn around time (or mean repair cycle time) t - repair cycle time random variable Var(t)- variance of repair cycle time Abbreviations A/O - Application/Operation ASO - Aviation Supply Office, Philadelphia, PA COG - Cognizant Activity DEN - Data Element Number 1G - Inventory Control Point ICPDAT - Inventory Control Point Data IHF - Inventory History File FMSO - Fleet Material Support Office, Mechanicsburg, PA MAD - Mean Absolute Deviation NICN - Navy Identification Code Number NIIN - National Item Identification Number NPS - Naval Postgraduate School, Monterey, CA NSF - Navy Stock Fund OPTION - Option path formula for variance calculations at ICPs PVAR - Mathematically correct formula for variance calculation SAS - Statistical Analysis System SIG - Selective Item Generator SPCC - Ships Parts Control Center, Mechanicsburg, PA UICP - Uniform Inventory Control Point Vil I. INTRODUCTION A. BACKGROUND In the U.S. Navy there are approximately 228,800 items classified as repairables. The responsibility for managing these items is shared between the Navy’s two inventory control points (ICPs), the Aviation Supply Office (ASO) in Philadelphia, PA., and Ships Parts Control Center (SPCC) in Mechanicsburg, PA. The tota] dollar value of these items is in excess of $28 billion with an annual Navy Stock Fund (NSF) budget for procurement of just under $2 billion [Ref 1]. To manage the inventories of these high dollar value items, the ICPs use a complex mathematical model which incorporates formulae for the calculation of means and variances of attrition demand over a net leadtime of procurement for specific items. The mean net leadtime demand calculated is called the Procurement Problem Variable (Z) and the variance of that demand is called the Procurement Problem Variance (V). These two parameters are key elements in determining the procurement quantity that 1s necessary to maintain a repairable item inventory at prescribed protection levels. Specifically, the mean is the quantity that should be available to meet the average demand over the net leadtime. Additionally, a percentage of the square root of the variance (standard deviation) could be purchased to meet any additional demand that may be experienced. This is essentially a safety level [Ref 2]. The sum of the mean and safety level is the procurement réorder point used by the ICPs. If, in the calculation of the variance, an error is made resulting in too large a value, more safety stock than necessary may be held. This would tie up money in unnecessary stock and prevent it from being used elsewhere. If the variance calculation was too small, not enough material would be available, resulting in the chance of a ‘stock-out’ being higher than desired. In the late seventies, the ICPs recognized that the variance model being used, generally calculated variance values that were too high. Two attempts to correct this situation were then incorporated into the model. One was a result of a iL study completed by Fleet Material Support Office (FMSQO) in 1977 [Ref 3]. This study hypothesized that the large variances were a result of ignoring a dependent relationship between the quarterly demand for an item and the quarterly regeneration of carcasses that were returned for repair. The dependent relationship manifests itself as a covariance between these two random variables. This was ignored in the original model when calculating the variance of the net leadtime demand. As a result of this study, an estimate of the covariance between regeneration and demand rates was incorporated into the computerized Levels program (UICP A/O D01) by the ICPs. This estimate was provided as an option path in the Levels program [Ref 4]. The second attempt to reduce variance was done by SPCC in a study completed in the same year [Ref 5]. To prevent excessively large safety levels from being created, a "patch" was added to the Levels program which performs a variance to mean ratio check for each item. If this ratio exceeds an ICP selected parameter, it modifies the program to recompute the variances of the net leadtime demand using a power rule formula [Ref 6]. The variance to mean ratio check, the power rule formula and the estimate of covariance are included in DO1, but the use of the covariance term is only an option. This option path is currently not being used at SPCC [Ref 7]. The only definitive reason for not using it was that the ICPs felt that the variance values that were obtained did not provide sufficient safety stock (.e., too small a variance). Thus, the large variances (that are not recalculated by the power rule because they do not exceed the ICP parameter) which precipitated the initial studies, appear to remain on file at SPCC. This thesis will look at possible reasons for the large variances mentioned above and will attempt to offer a method for estimating the value of the variances more accurately. B. OBJECTIVES There are two main objectives of this thesis. The first is to develop a theoretically correct variance formula for the net leadtime demand which will use the expected values of demand and regeneration rates to calculate the covariance. The second is to compare the theoretically correct formula with the actual variance values on file from SPCC'’s data base and the option path variance formula of DO1l. By the comparison with the latter, the degree to which the estimate of the covariance agrees with the theoretically correct formula for covariance can also be obtained. C. SCOPE The comparisons made to satisfy the second objective were limited to using a 5% sample of items resident on SPCC's files. No ASO data was examined. No attempt was made to actually calculate safety level or determine actual changes in costs of stock which would result from different variance calculations. However, it follows that any reduction in variance, with all other factors remaining constant, would reduce the amount of safety stock required to provide a given level of protection. D. PREVIEW In Chapter II, the two alternatives to be used in this thesis for computing the procurement problem variance will be presented. In particular, the theoretically correct variance formula will be derived and the difference between it and the option path formula will be discussed. Chapter III contains a short discussion on how the data was acquired and the procedures used in the comparison of the three alternatives. In Chapter IV, the results of the comparisons are shown and discussed. Chapter V summarizes the previous chapters, presents conclusions from the analysis, and makes recommendations for further testing and implementation. IH. FORMULA DEVELOPMENT This chapter begins with a notational caveat and then discusses the concept of the procurement problem variable as the mean demand for an item over a net leadtime. It continues with an explanation of the variance formula used by the ICPs which includes the covariance estimate and variance to mean ratio check that is used to reduce the variance values. The fourth section presents the derivation of a theoretically correct variance formula which will be called "PVAR’. The chapter concludes with comparisons of the correct formula with the formulae that are currently being used at SPCC. A. NOTATIONAL CAVEAT Capita] letters are used to denote the mean values of the variables that they represent. Occassionally, there will be a need to distinguish between these mean values and the distributed random variable from whence they came. This will be accomplished by adopting the expediancy of using the lower case version of the symbol to represent the random variable. Alj time is measured in quarters. B. PROCUREMENT PROBLEM VARIABLE The Procurement Problem Variable (known as PPV and denoted by Z) is the expected demand over an "average acquisition time". The term "variable", in this case, is a misnomer. It is a mean of the distribution of the procurement problem random variable, not a random variable itself. However, the term has been accepted by convention, to represent the expected demand over a net leadtime. To develop this net leadtime, let B represent the average number of items regenerated per quarter and let D be the average number of items demanded per quarter. The ratio of B/D then represents the average proportion of demands that are satisfied by regenerations and 1 - B/D is the average proportion of demands that are not, and thus have to be procured. Next, let L represent the mean procurement leadtime and T represent the mean repair turn-around time. The average of the net acquisition time, L,, can then be represented by the following formula: (1) L, = (1 - B/D)L + (B/D)T. Multiplying this formula by the average quarterly demand, D, will produce the average demand over L.,. (2) itve= Ol).- BlL+ BT =e Equation (2) is the formula used by the ICPs for computing Z, the mean of the net leadtime demand [Ref 4]. C. UICP VARIANCE FORMULA The variance formula that was used in the middle 1970's was: (3) V =(L- T)[Varid) + Var(b)] + TVar(d) + D2Var(t) + (D - BP[Var(]) + Vari(t)]. The above equation was pieced together from a Fleet Material Support Office (FMSO) Working Memorandum [Ref 3] and the current computerized Levels program documentation (UICP A/O DO1) [Ref 4]. The memorandum, which was a summary of a study completed in 1977, suggested changes to the above equation (3) that would reduce the variance of net leadtime demand of repairables. The problem of observed large variances at the Inventory Control Points (ICP) in the mid seventies was important to them because of increasing funding restrictions and budgetary. hmitations that were being imposed upon the supply system at that time. They recognized that a reduction in variance values would reduce the amount of money needed to fund safety stock. To accomplish this reduction, the ICPs incorporated the changes that were recommended by the study. The major change that was incorporated was an estimate of the covariance between the demand rate and regeneration rate of a repairable item. From the FMSO study the estimate had the form of: (4) Var(d)B/D The ICPs programmed the above covariance estimate into the variance equation as an option path. The option path has the following form: (5) OPTION = (L - T)[Var(d) + Var(b) -2Var(d)B/D] + TVar(d) + D?Var(t) + (D - BP{Var(]) + Var(t)}. The above equation (5) is the same formula that is documented in the current Levels program. However, the option path, according to SPCC’s Operations Analysis Division [Ref 7], is not being used. The only variance reduction technique that is currently being used is a variance to mean ratio check and subsequent power rule recalculation of variance. The variance to mean ratio check and the power rule were implemented as a result of a study completed by SPCC in 1977 [Ref 5] which was also motivated by the excessively large variances of net leadtime demand that were on file. To prevent large safety levels from occurring, a "patch" was added to the Levels program which compared the variance of net leadtime demand, calculated from equation (3), with the mean of net leadtime demand, calculated from equation (2). If this ratio exceeded a preset ICP parameter (SPCC = 150, ASO = 450), the variance calculated by equation (3) was recalculated using the following formula (power rule): (6) VieewalZie where a and b are preset parameters. The above parameters (a,b) are currently set at SPCC as 4.849 and 1.502, respectively, and at ASO as 27.458 and 1.559, respectively [Ref 10]. These parameters are reviewed approximately every three years by FMSO. In summary, the current variance calculations at the ICPs are obtained by using equation (3) and the variance to mean ratio check with the power rule. The actual variance values on file at SPCC will be referred to as "V" throughout the rest of this paper. Note that even though equation (3) and equation (5) are calculations for the variance of net leadtime demand, V, to prevent confusion, the results of equation (5) will be referred to as "OPTION’. OPTION, equation (5), is only programmed as an option path and, as previously mentioned, is not being used. D. PVAR FORMULA The procurement problem variable, as shown in formula (2) can be derived in another way as follows. Let | be the number of quarters required for procurement of a new item. Let t be the repair turn-around time necessary to repair a carcass of the same item. The mean net number of items to buy to meet demand over | can be described by the regression function [Ref 9] as follows: (8) Elz ll] = 1D - (1-t)B This equation has the following interpretation. The first term, ID = IlE[d], is the expected number of items demanded given the procurement leadtime, 1. This value must be offset by the mean number of carcasses expected to be returned to inventory in ‘ready for issue” condition (RFI) over |. For the first t periods of the given | periods a number of carcasses are being repaired. The number of such carcasses is the consequence of the number of items returned to supply for repair prior to our time origin. After t such items can be used to fill demands. The term (1 - t)B = (1-t)E(b) represent a conditional expectation of those regenerations after our time origin. This is the reason for the negative term in (8). Using the basic rule of iterated expectations, (9) Elz] = ElE|z |1.t]]. It then follows: (10) 7 PL = 1B. which can be rewritten to show that it is identical to formula (2): Z= DL - BL + BT. To develop the variance of net leadtime demand, the regression function (8) can be used. Rewriting the regression function of net demand (z) on leadtime (1) and repair cycle time (t) provides the following: (11) Elz {1.t] = d - t)(D - B) + tD, and the conditional variance of z given | and t is: (12) Variz]1t) = ( - t)Var(d - b) + tVar(d), because we are summing (] - t) independent observations of (d - b) and adding it back to independent observations of d. Using the Lemma stated and proved by FMSO [Ref 10] (i.e., the unconditional variance is the mean of the conditional variance plus the variance of the regression function) results in: (13) Var(z) = (L - T)Vari(d - b) + TVar(d) + Var((D - B) + tB), = (L - T)Var(d - b) + TVar(d) + (D - B)?Var(1) + B?Var(t), because procurement leadtime and repair cycle time are independent variables. Since current repairables inventory management procedures [Ref 11] require a return of a carcass concurrently with a requisition for another unit of the repairable (i.e., a one for one exchange), this creates a dependent relationship between the number of carcasses returned to supply for repair and the demand for the same item. Accounting for this dependent relationship, twice the covariance between demand and regeneration (because each is dependent on the other) is subtracted from [Var(d) + Var(b)]. The following formula results: (14) Variz) = (L - T)/Var(d) + Var(b) - 2Cov(d,b)] + TVar(d) + B?Var(t) + [(D - B)?Var(1)]. The covariance term from the above equation (14) can be derived using expectations [Ref 14]: (15) Covid,b) = Ef(d - D)(b - B)], E[db] - DB. When (15) is inserted in (14) the resulting equation, which will be called PVAR, for calculating the variance of demand over a net acquisition leadtime is: (16) PVAR = (L - T)iVari(d) + Var(b) - 2(E[db] - DB)] + TVar(d) + B2Var(t) + [(D - B)?Var(1)]. E. FORMULA COMPARISONS If PVAR, equation (16), is subtracted from V, equation (3), the difference is: (17) V - PVAR = 2DVar(t)(D-B) + (L - T)2Covid,b). Adding PVAR to both sides and expanding terms results in an expression relating V and PVAR: (18) V = PVAR +(D?2 - B2)Varit) + (D - B)*Var(t) + (L - T)2Cov(d,b). Collecting terms and simplifying: (19) V = PVAR + 2DVar(t)(D-B) + (L - T)2Cov(d,b). It is interesting to note when PVAR would equal V. If we assume that L > T, then V = PVAR when the following is true: (20) DVar(t)D - B) = -(L - T)Cov(d,b). A special case of the above would occur when both terms are zero. That results from any one term (on both sides) being zero. This is not an uncommon event (i.e., Cov(d,b) and Var(t) equal to zero) as will be shown in the following chapters. Also note that if the covariance term was negative (1.e., E[db] > DB) and any term on the left side of equation (20) was zero (1.e., Var(t) = 0), then PVAR would be greater than V. Mathematically it is possible for the covariance term to be negative, but conceptually it is not since a probability of a regeneration will exists when a demand occurs and the regeneration rate can never be negative. The negative covariance term is not an uncommon event when working with the data and may suggest problems with the data on file. This investigation is left for further study. The same procedures as above can be used to compare PVAR and OPTION. For simplicity, let the estimate of covariance, equation (4), be represented by Cov’ and let the calculation of covariance, equation (15) be represented by Cov. This comparison results in: (20) OPTION = PVAR - 2[Cov’(d.b) - Cov(d,b)] + (D2? - B2)Var(t) +(D - B?Varit). As discussed above, if Var(t) = 0, then the difference between OPTION and PVAR reduces to: (21) OPTION = PVAR -2[Cov(d.b) - Cov(d,b)]. Then PVAR and OPTION will be equal when: 10 (22) Cov(d,b) = Covid,b), and PVAR will be less than OPTION when: (23) Cov(d,b) > Covid,b). This last situation, equation (23), will be discussed in depth in Chapter IV. tl Il. FORMULA COMPARISON METHODOLOGY This chapter begins with an explanation of how the data was obtained from the files of SPCC and loaded to the Naval Postgraduate School’s (NPS) mainframe computer. It then explains the process used to compare the variance, V, on file at SPCC, with the option path formula for variance, OPTION, and the theoretically correct variance formula, PVAR. A. DATA ACQUISITION The data used to compare the three models was taken from SPCC’s data files on the Univac 494 computer. The data consisted of all repairable items with a cognizant activity code (COG) of 7H, 7I, and 7G. These COGs indicate that the items are specically managed by SPCC. The data elements necessary to calculate the variances were downloaded to tape via the ICPDAT (inventory control point data) network using the computer resources of the Operations Research Department (Code 93) at FMSO. The specific Data Element Number (DEN) and nomenclature of each data element are presented in Appendix A. It was necessary to access two different files to obtain all the data elements. The SIG (selective item generator) file was used for most of the data and the IHF (inventory history file) was accessed for specific data necessary to calculate expected values (for Covid,b)). Once the data was acquired, it was translated into IBM format for storage in National Item Identification Number (NIIN) sequence on the new IBM 3090 mainframe at SPCC. A mainframe data analysis package, SAS, was used to eliminate any NIIN which had blanks or data missing from any DEN. An example would be a NIIN that had data on the SIG file but no IHF entries and vice versa. For the purpose of this data selection, zero was considered a valid data entry, but blanks were not. Finally, a tape was obtained of the remaining data. This tape was taken to the Naval Postgraduate School (NPS) where it was uploaded on the IBM 370/3033AP mainframe and stored in a batch data file. Due to the size of the data (in excess of 47,000 line items or 12 NIINs), a 5% sample was taken from the batched data set and loaded to a private disk (B-disk). The private disk allowed interactive programming, which was not available if kept on the batch file. The 5% size was the largest sample size that could be loaded and stored on a private disk (1672K bytes of disk space). The resulting sample had a total sample size of 2,345 observations. Each observation consisted of a NIIN and all data elements pertaining to that NIIN that were needed for computing the variances being compared. Since the batch file was arranged in NIIN sequence, the sequential sampling technique [Ref 12] was used to ensure a continuous, representative sample across all NIINs. To obtain the 5% sample, the data was sequentially subdivided into blocks of 20 items. A number between 1 and 20 was selected at random to determine which item from each block would be sampled. The 5% sample, therefore, consisted of one item from each block. B. FORMULA COMPARISON PROCEDURES The V, PVAR and OPTION formulae were programmed on the NPS mainframe computer using FORTRAN. The actual code is presented in Appendix B. The resulting variances from each of these equations were compared to the corresponding variance obtained directly off SPCC’s file, V1. The file variance value is denoted by V1 to distinguish it from the programmed UICP variance formula, V. Vl was used as the comparison value because it is the actual variance used in the calculation of inventory levels. V was used only to compare it to V1 to see if the variance on file could be duplicated by a simple formula. If Vi could not be duplicated then some method other than direct calculation of the variance was used by SPCC. It is assumed that the power rules were used to estimate the variances of the components within the variance formula. A recent study by FMSO [Ref 8] indicates that the mean absolute deviations (MAD) that are used to compute the variances of several of the variables in the calculation of the variance are estimated by power rules similar to the one discussed above. The affect of the power rules and the resulting variance values is. left to further study. 13 As discussed under SCOPE, no direct comparison of OPTION and PVAR will be done with V. Thus, the comparison of variance values will be done between V1 (the values on file at SPCC) and V (the UICP variance formula), and between Vi and OPTION (the UICP option formula for calculating variance) and PVAR (the theoretically correct variance formula). A series of data checks were built into the program to remove any item with data that resulted in calculations of a negative Z, a leadtime demand of zero or less or D (mean quarterly demands) that were equal to zero. The last check was done to prevent division by zero when using the OPTION equation. The values of the three variances were tabulated in a series of output files. The output files were then divided into specific categories of demand for several reasons. It was important to reduce the size of the comparison groups to make data analysis easier in GRAFSTAT. When the data set is too large, the graphic output exceeds its capacity. Another reason is that the ICPs use certain mean quarterly demand values as a criteria for determining underlying probability distributions for demand during net leadtime. It was also considered important to separate the high demand items from the lower demand items since they are managed more intensely. A series of demand groupings were therefore defined. Costs associated with a stock-out are higher if the safety levels for these high demand items are inaccurate. The "Low Low Demand" items had mean quarterly demands of less than one unit. The "Medium Low Demand” items had mean quarterly demands equal to or greater than 1 but less than 2 units. Items with mean demands equal to or greater than 2 but less than 5 were grouped into the "High Low Demand’ category. The "Medium Demand’ category contained items with mean demands equal to or greater than 5 but less than 20 and the "High Demand’ items were those with mean quarterly demands of 20 units or more. In addition to V1, PVAR, OPTION, and V, the output files contained an identification number for a specific NIIN (I), the PPV (Z) value and various other data elements. Finally, the standard deviation or square root of each variance (except V because this was not in the comparison) and the ratio (V/Z) were 14 included. This ratio was used to look at how many of the samples exceeded the variance to mean ratio parameter at SPCC of 150. The output files were input to an NPS mainframe statistical analysis package, GRAFSTAT, for graphical analysis. The output from this package did not integrate well into a microcomputer word processor and thus was used only to find trends between the variance calculations. Once trends were observed, the original FORTRAN program was modified to produce summary data of the results. These results were then fed into a microcomputer. Using the microcomputer and Harvard Graphics, graphs of the summary data were then prepared and imported to WordPerfect 5.0 for use in this thesis. C. DATA FILE OUTPUT A total of 1,261 items (53.8%) passed through all the data checks. A cursory look at the items not passing the check showed that most of the items had mean demands that were less than one per quarter and many of the data elements had zero values. A large majority of these items were identified as new items because they were coded with Navy Item Code Numbers (temporary NICN’s appeared instead of NIIN’s) for which little or no historical data was available. Most of these items should have been screened from the data set during initial download at SPCC, but were not because of the presence of zeros in the data fields instead of blanks. It could not be determined why the zeros were entered in the DENs. However, zeros allowed them to pass through the initial screening but then caused them to fail the final data checks built into the calculation programs. In addition, some of these items were identified as having gone through a Cognizant Activity change (i.e., COG migration) which is a change of activity responsible for the supply management of the particular item or reclassification from an item having been identified by a NICN to an item which is now identified by a NIIN. This would cause a "disconnect" between data on the IHF (Inventory History File) which was associated with a NICN and the same item on the SIG (Selective Item Generator) File which is now identified by a NIIN. This normally would have produced blanks and would have been screened out initially but the presence of unexplainable zeros prevented it. 15 IV. RESULTS The data was run through the different variance calculations and the results were divided into demand groups as mentioned in Chapter III. Figure 1 shows the distribution of the items among the different demand groups. Number of Items per Category 800 600 400 200 Low Low Med Low High Low Medium High Demand Category : MM No.per dem. category Figure 1 - Distribution of the 1261 items by quarterly demand category. A sample of the detailed output file for high demand items are presented in Appendix C. 16 As can be seen from Figure 1, most of the items were in the low low demand group. Those items that the ICP consider for intense management are in the medium and high demand group. Even though they are only a-.small percent of the total items in the sample. they reflect the relative percentages for the entire population. Figure 2 shows the percentage of items in each demand category that had a reduction in variance values (over V) as a result of the PVAR calculations and OPTION calculations. % Decrease in Variance _— ce, ee ips ZEEE: Demand Group Low Low MedLow HighLlow Medium Ma PVAR Formula OPTION Formula High Total Figure 2 - Percent of items within each demand group that showed a reduction of variance values by PVAR amma, OPTION. Note that in every demand category, the OPTION formula reduced the variance by a larger percentage than did the PVAR formula. The main reason for this is that a large number of items, when using the PVAR formula, had demand- regeneration covariances equal to zero. ‘This was caused by regeneration data 17 equal to zero. This did not occur when using the OPTION formula because it used the mean regeneration value that was on file while PVAR used the raw data to calculate mean regeneration. This suggests that mean regeneration values are being calculated at SPCC by some other method and not from data on file. The investigation of this point is left for further study. As can be seen in Figure 2, for the high demand items, PVAR reduced the variance for only 57.7% of the items. This was the lowest improvement shown by PVAR. Those items that did not have their variances reduced, fell into two categories. They were either items that had covariances equal to zero (in the PVAR formula) because of regeneration data equal to zero or the variance to mean ratios (as shown by V/Z) were greater than the variance to mean ratio check parameter of 150. In the latter case, V1 was computed using the power rule while PVAR was calculated as programmed (the use of the power rule in calculating V1 was verified by hand). Table I shows typical items in these categories. Item number 313 had a zero covariance when PVAR was used to calculate variance. Item number 1287 had V1 recomputed using the power rule. Finally, item number 560 fell into both categories. TABLE I ITEM Z Vi PVAR V ratio BLS 96.75 2671.48 a) lea Dial 59.40 1287 93.43 AAD? 21 15911.4 15922.9 170.43 560 99.08 4824.57 15793.5 15793.5 159.40 A small quantity of the items (7 items with high demand and 14 total) from the output file had variance to mean ratios greater than 150 (V/Z > 150). Of these 14 items 4 had PVAR values that would not have passed the variance to mean ratio check. This suggests, in this particular case, the cut-off parameter of 150 may be too severe. If this situation is true then not enough safety stock is being held to meet the required protection level. 18 For the rest of the demand categomes, Figure 2 shows that PVAR is only slightly less effective than OPTION, in reducing the variances of the sample. The main reason given that SPCC has not used the OPTION formula is that it calculates variances which have been shown to be too small to provide sufficient safety stock. If PVAR were implemented, then quite possibly the same would hold true. However, the discussions so far have been limited to the number of items for which variances were reduced, not the degree of reduction. To determine the degree of reduction, the differences in standard deviations (square root of the variances) were plotted for all items where V1 (the variance on file) was less than PVAR and PVAR was less than OPTION. From the plots, summary data was gathered and is shown in histogram form in Figure 3. This figure accounts for 94% of the items sampled. The other 6% of the sample that is not included are items where the PVAR formula calculated a zero covariance or where the variance to mean ratio exceeded 150. These items were discussed above. | In Figure 3 the data is grouped by the difference in number of items. Option shows a decrease in standard deviation over PVAR by a median value of 3. PVAR shows a decrease in standard deviation over V1 by a median value of 1. From the difference between V1 and PVAR it appears that for the same level of protection, ‘on the average’, less safety stock would be required if the PVAR formula was used. From the differences between OPTION and PVAR, the OPTION formula, “on the average", provides even less safety stock than PVAR. According to SPCC, the OPTION formula is not used because it reduces the variance of net leadtime demand too much and thus does not provide enough safety stock. On the other hand the variances on file (V1) are apparently too large and have been the object of a number of studies and program modifications to reduce their values. The PVAR formula, as presented in this thesis, reduces the variance, as compared to V1. However, PVAR does not reduce it to the level of OPTION. Thus, PVAR might be the solution to this dilemma. 19 % of total in each group Median of 8.d.PVAR-s.d.OPTION = 3 Median of s.d Vi- 8.d.PVAR = 1 difference in std. dev. [ s.d.PVAR-s.d.Option s.d.V1-s.d.PVAR a Figure 3 - Difference in standard deviations between Vaiiancecewermul ac. V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS A. SUMMARY This thesis has compared different formulae that are or could be used to calculate the variances of the net leadtime demand for SPCC managed repairable items. PVAR, the theoretically correct variance formula, was derived directly calculate the covariance between quarterly demands and quarterly regenerations. The differences between the OPTION formula (documented in UICP A/O DOI), the PVAR formula (derived in Chapter II), and the variance formula used to compute the current values listed on SPCC’s data files were discussed. The variance values listed on SPCC’s data files were then compared with the variance values calculated ‘by both the OPTION formula and the PVAR formula. Finally, an analysis of the results from the comparisons of the different variance formula was presented. B. CONCLUSIONS It is a well known fact that a large variance in net leadtime demand resident on SPCC’s file can result in unusually large safety stock. In the past, various changes to the UICP programs have been implemented which reduce the variance to acceptable levels to prevent large sums of money from being tied up in possibly unused and unnecessary safety stock. The current procedure is to make a variance to mean ratio check and to recalculate the variance of net leadtime demand if it exceeds a predetermined threshold. An alternative available estimates the covariance factor and uses the option path for computing the variance. This approach was designed to reduce the variance to acceptable levels by accounting for covariance between the dependent variables of demand and regeneration. The option path, if it were used, apparently underestimates the variance of net leadtime demand and would excessively reduce the amount of safety stock required. While this would reduce, considerably, the amount of dollars necessary to procure and maintain the safety level, it could also reduce the levels of operational availability of various weapon systems by not providing enough safety stock. The PVAR model, when used with complete and current data, reduces the variance of over 95% of the repairable items sampled. It also does not estimate the covariance of regeneration and demand, but calculates it directly and thus gives a more theoretically correct variance output. In addition, it does not reduce the variances to the levels calculated by the OPTION formula. By using the PVAR model, SPCC could reduce the amount of money tied up in unnecessary safety stock for those items which had large variances on file and redistribute some of the money to items which may require, for what ever reason, an increase in protection level G.e., more safety stock). This would possibly allow an increase in operational availability of weapon systems and at the same time could reduce the amount of money necessary for spares support. It would allow the ICP to do its job cheaper and smarter. C. RECOMMENDATIONS The results indicated here, should not imply that the PVAR model is a panacea for a restrictive funding environment. The model should be thoroughly tested and verified through simulation and under actual operating conditions prior to any consideration being given to incorporating it into levels setting. In particular, PVAR should continue to be tested using data obtained from ASO to see if similar results (as obtained in this study) apply to aviation material. In addition, simulation and actual field testing of PVAR should be done to see if the variance values that are calculated by PVAR provide for enough safety stock. FMSO has recently completed a new Functional Description (PD-80) [Ref 13] for a UICP program system design to forecast leadtime and repair turn-around time. The documentation and program are to be incorporated into the software modifications being made as part of the ICP modernization project. The procedures described in PD-80 include many significant improvements over D0O1, but the basic formula for calculating the variance of demand over net leadtime is 22 still similar to equation (5) in Chapter Il. When the PVAR model passes testing, corrections can easily be made to PD-80 and then implemented without delay. By correcting only the variance formula, and maintaining the other significant improvements of PD-80, the ICPs would not only operate more economically but also provide the necessary spares support for the fleet. D. RELATED FURTHER STUDY Further study should be directed toward the policy governing the use of the power rules for estimating the mean absolute deviation of the components of the variance formula. If the reason for estimating these MADs is due to lack of data, then this lack of data needs to be investigated as well. Blank data fields were screened out of this study. These blanks will affect the new Levels program (PD- 80) that does not use MADs but instead calculates directly the variances of the individual components of the formula for variance of net leadtime demand. 23 LIST OF REFERENCES a ogo ne Naval Supply Systems Command, Command Presentation for 2. Tersine, Richard J. Princtples of Inveutory and Matertals Managenient, 2d ed., pg 126, North-Holland, 1982. 3. Navy Fleet. Materia! Sapa Office, ALRAND Worlung Memorandum 292, Calculation of Procurement Problem Variance, 2 March 1977. 4. Navy Fleet Material Support Office, System Design Documentation (FD- DO1), Levels, appendix O, by Richard S. Jackson, 31 March 1984. Oo. Ships Parts Control Center letter 790C/EE/140_ to Commander, Naval Sung Systeins Command, Subj: “Lnplementation of Variance-to-Mean Ratio Check in UICP A/O DO1", 9 February 1977. 6. Navy Fleet Material Support Office, ACLRAND Working Memorandum 357, Update to Power Rule Parameters, 30: May 1980. 7. — Interview with Mr. John Boyarski, Operations Analysis Division, SPCC, Mechaniesbure, PA, 16 November 1987. 8. Navy Fleet Material Support Ofitcee, AL RAND Working Memorandum 535, Update to Porweer Rule Parameters, 15 November 1987. OF oe Morris UL. Probability aud Statistics, 2™ ed., pg 604, Addison- Wesley, 1986 10. Navy Fleet) Material Support Office, ALRAND Report 50B, Stattstical framing Manual, Vol Lil, pe V22 by 13. 110 Bigs miei te cnet It. Navy Repairables Management Manual ONAVMATINST 4400.14B), pg V-12, 17 Peluso . 12.) Duncan, Acheson J. Qualtty Control and Industrial Statistics, Sth ed., pg 201, Irwin, LOXG. 13. Navy Fleet Material Ce a Office, Uniform Inventory Control Prograni Systent Desien Documentation Forecastug LUEITAT Requtrements Model, pg M-43, December 1987. APPENDIA A DEN/Nomenclature of Data Elements AQ19 - Observed Quarterly Demand MAD BO11B - Procurement Leadtime AO19B - Observed Quarterly Careass Return MAD BOILA - Procurement Leadtinie Forecast . BOI2ZF - Average Procurement Turn-Around Time for repair BO12B - Average Carcass Return Rate BOLSA SVariince om Pile at SPCC BOZ3C - Demand over Procurement Leadtime | BO23I]5 - Regenerations over Procurement Leadtime. BO23G - Regenerations during repair turn-around time BO74 - Average Quarterly Svsten: Demand Forecast BO32C - Observed Leadtune Demanc 3074 A . Neuen RET Regenerations C001 /CO02 - National Item Identification Number C003 - Cognizant Activity C005 - Unit of Issue | F020 - FO20G - Depot completions reported for the last 8 quarters OO - Repair Survival Rate FOO9A - Repair Survival Rate MAD HOM HOMA HOLIC - HO2Z1, HOZIA ,HOZLC- Total quarterly demand reported for the last 8 quarters APPENDIX B FORTRAN PROGRAM FOR VARIANCE CALCULATION tedededededetedesetedetoletedste dv tedtedetetetsieseheteteteioteteiedciohdicekiceiddssdcedicichiciviciciciggihkecledvivietvdetcdlRdict *THIS PROGRAM READS THE DATA FROM THE DATA FILE "STGIHF" AND PUTS IT %* “IN COLUMN VECTORS FOR FURTHER ANALYSIS. THE DATA SET IS IN CHARACTER * “FORMAT WITH A LRCL = 236. THE OUTPUT IS “RATIODAT LISTING A’ ¥ *COMPILE THE PROGRAM USING FORTVS AND USE THESIS EXEC TO RUN ¥ *THE VARIABLES ARE: % CNIIN - NATIONAL ITEM IDENTIFICATION NUMBER (CO01E/C002) ¥ * COG - COGNIZANT ACTIVITY (C003) * * DEMMAD - OBSERVED DEMAND MAD (A019) ve * PLTMAD - PROCUREMENT LEADTIME MAD (BO11B) ¥ * PLTFC - PROCUREMENT LEADTIME FORCAST (BO0114) ve * DEM - AVERAGE QUARTERLY SYSTEM DEMAND FORCAST (B074) ve * LTDEM - OBSERVED LEADTIME DEMAND (BO023C) ve ** RATIO1 - VARIANCE TO MEAN RATIO FROM FILE (V1/PPV) 7% * RATIO2 - NEW VARIANCE TO MEAN RATIO CALCULATED (PVAR/PPV) ¥ * RATIO3 - VARIANCE TO MEAN RATIO CALC FROM (OPTION/PPV) ¥ ** RATIOS - VARIANCE TO MEAN RATIO WITHOUT COVARIANCE DO1 (V/PPV) ve RATDIF - DIFFERENCE BETWEEN CALCULATED VAR/MEAN AND FILE VAR/MEAN * ** CRMAD - OBSERVED CARCASS RETURN MAD (A019B) ¥ * PTAT - AVERAGE PROCUREMENT TURN-AROUND TIME FOR REPAIR (B012F) ve * NTTMAD - NAVY (NON-REPROTING) REPAIR TURN-AROUND TIME (B012B) ve ** AVGCR - AVERAGE CARCASS RETURN RATE (BO22B) ve * LREGEN - RFI REGENERATIONS DURING LEADTIME (BO23E) ¥ ** TREGEN - RFI REGENERATIONS DURING PTAT (B023G) ‘ ** QREGEN - QUARTERLY RFI REGENERATIONS (BO74A) zs * RSRMAD - REPAIR SURVIVAL RATE MAD (FOO9A) * * QTRIRP THRU QTR8RP - DEPOT COMPLETIONS REPORTED FOR THE Gas? S Gis 2 (PO20 THRU BO ZveT x * RSR - REPAIR SURVIVAL RATE (F009) we % CUl= Nee ISSUE Cele * * QTRIDM THRU QTR8DM - TOTAL QTRLY DEMAND REPORTED FOR THE LAST 8 QTR * = (HO14+HO14A+HO14C THRU HO21+HO21A+H021C) * “© OPTION - CALCULATED VARIANCE BY THIS PROGRAHN WITH COVARIANCE COV1 " * V1 - VARIANCE OF PPV ON SPCC'S FILE(BO19A) * ve 6 Vo = VARIANCE FROM DO1 WITH OUT COVARIANCE cf * COV1 - EST OF COVARIANCE FACTOR USED AT THE ICPS % ** =COV - COVARIANCE FACTOR CALCULATED BY EXPECTED VALUES % * PPV - PROCUREMENT PROBLEH YARIABLE (BO23¢G=202 02 5G) ¥ * PVAR - CALCULATED PROCUREMENT PROBLEM VARIANCE WITH COVARIANCE COV * * BDATA - COUNTER FOR BAD DATA WHICH WILL NOT BE USED IN ANALYSIS * %* GDATA - COUNTER FOR GOOD DATA WHICH WILL BE USED IN ANALYSIS ¥ * POSDIF - COUNTER FOR POSITIVE IMPROVENENT IN VARIANCE WITH PROGRAM * * NEGDIF - COUNTER FOR NEGATIVE IMPROVEMENT IN VARIANCE WITH PROGRAM * *% UNCHNG - TOTAL QTY OF NIINS WITH VARIANCE UNCHANGED BY PROGRAM * * VDIF - DIFFERENCE BETWEEN VARIANCE ON FILE AND CALCULATED VARIANCE * * COUNT1-5 - COUNTER FOR VAR EXCEEDING SPCC PARAMETER FOR RATIO * * DELTA - DIFFERENCE IN STANDARD DEVIATION % * NUM - NUMBER OF ITEMS USED 1Os@Ake Ui ice: % *% J - SETS THE NUMBER OF DATA LINES (NIINS) TO BE READ/USED ¥ 26 eee - oelG PRESET PARAMETER * § - CONSTANT FOR THE ESTIMATION OF VARIANCE FROM THE MAD low -SIPeiive Cr Vi pee vARSD - STD DEV OF PVAR feb) = STD DEV OF OPTION Pees. = DIFFERENCE eee ee AND OPTION S.D. ( PVARSD- ON SIe Seseseveves esksestrledevestcvesesevestedevestestevesistc ve * DECLARE VARIABLES , PARAMETER (J=2345, s\e stes'e 77 ves'e ses tose slcs'es tesles's \s svsevesicsles sevesicsiesicsles'c sesvvesesveseseseseveses ‘SET PARAMETERS, DIMENSION ARRAYS = eZee 150 ) Poe deed yy ORMMADC J), DEM(J), LIDEM(J), V1iCJ), CRMAD(CJ), CPTATI( J) ,NITMAD( J), AVGCR(J), LREGEN( J), TREGEN( J), QREGEN(J), CRSRMAD(J), RSR(J), CRATIO2(J), RATIO3(J), PYani@ee, COV (J), AB TUE (rs es PPV(J), RATDIF(J), COV1(J), NCJ), RATION gy RATDEL(J) C,RATIO4(J), OPTION(J), DELTA(J), TOTDEL,V1SD(J) ,PVARSD(J), COPSD( J) ,DIFF(J) INTEGER PLTMAD(J), QTRIRP(J), QTR2RP(J), NEGDIF, POSDIF, NEGDEL Ceeeeowrt, UNDEL, COUNTI, COUNT2, COUNT3, COUNT4, COUNTS, OUirerro?), QPRakP( J), OTRSRP( J), QTRORP(J), QTR7RP(J), UNCHNG, OtEZUM J) , CQTR5DM( J), QTR6DM(J), QTR/DM(C J), QTR8DM( J), BDATA, GDATA, COTR8RP(J), OQTRIDN(J), erOrr 1 , Pee Lis NTE i, NDELI1, OTR). OTRSDMCT) , ee Nee. POE 2 NDELZ , POLE. eee eee, NDELS, PDIf4, NDIF4, PDEL4, NDEL4, PDIF5, NDIF5, Pee. NUE, NUM ,DEL,ONEF ,ONEL, IWOF ,TWOL, THREEF , THREEL, evi! VPI VEL, TEN ,TENL,GIEN?F ,GIENL CHARACTER*9 CNIINCJ) CHARACTER*2 UIC J) ,COG(J) BDATA=0 GDATA=0 NEGDIF= POSDIF= PDIF1=0 0 0 TOTDEL=0. 0 27 9’ ? a ¢ ay ils: NUM=0 DEL=0 ONEF=0 ONEL=0 TWOF=0 TWOL=0 THREEF=0 THREEL=0 FIVEF=0 FIVEL=0 TENF=0 TENL=0 GTENF=0 GTENL=0 WRITECGR 3 — | ae' V1 ',' PVAR ',* OPTION Ce V es Z i) V1 SD ',' PVAR SD ', C’ OPTION sie Vi READ DATA FILE AND CREATE DATA VECTORS DO 10) ieee READ (1,15) CNIIN(I), COG(I), DEMMAD(I), PLTMAD(I), PLTFC(I), DEM(I), LTDEM(I), V1(1), CRMAD(I), PTAT(I), NTTMAD(I), AVGCR(1), LREGEN(I), TREGEN(I), QREGEN(1I), RSRMAD(I), QTRIRP(I), QTR2RP(I), QTR3RP(1), QTR4RP(I), QTRSRP(I), QTR6RP(I), QTR7RP(I), QTR8RP(I), RSR(I), UI(I), QTRIDM(I), QTR2DM(I), QTR3DM(I), QTR4DM(I), QTRSDM(I), QTR6DM( I), QTR7DM(I), QTR8DM(I) FORMAT (A9, A2, F10.4, 13, 2(F9.2), F10.2, 2(F10.4), F4.2, F3,1, Fl10.2, 2(F9.1), F9.2, F3.2, 8(15), F3.2, AZ. S@leue Be 5 Fl SP 1 a Jy oa 1 ea a og * CALCULATE COV, COV1, V, PPV AND PVAR COV(I)= (((QTRIDM(1I)*QTR1IRP( 1) )+(QTR2DM( I )*QTR2RP(I)) C+( QTR3DM(1)*QTR3RP( 1) )+( QTR4DM( 1)*QTR4RP( 1) )+( QTRSDM( 1) C*QTR5RP( 1) )+(QTR6DM( I )**QTR6RP( I) )+( QTR7DM( I )*QTR7RP(I)) C+( QTR8DM( I )**QTR8RP( 1) ))/8)-C( (QTRIDM( I )+QTR2DM( 1)+QTR3DM( I) C+QTR4DM( 1)+QTR5DM(1)+QTR6ODM(1)+QTR7DM(I)+QTR8DM(1))/8) C**( (QTRIRP( I)+QTR2RP( 1)+QTR3RP( 1)+QTR4RP( 1)+QTRSRP(1)+ CQTR6RP( I)+QTR7RP(1)+QTR8RP(1))/8)) IF(DEM( 1). LE.0) THEN BDATA=BDATA + 1 GOTO 10 END COVICTI)= CCRSRCI))*CAVGCRCI))*( (S*DEMMAD( 1) )**2)) /DEM(C I) VCI)= (CPLTFCC(I)-PTATCI))*( CC S*DEMMAD(I))**2) + CRSRCI)**2)* CCC S*CRMAD( 1) )**2)+( AVGCR(I)**2)*((S*RSRMAD( I) )**2) + CCC S*CRMADC I) )**2)*( (S*RSRMAD(I))**2)) + (PTATCI)*((S*DEMMAD( 1) )**2 C))+ CCDEMCI)**2)%*( (S*NTIMADCI))**2)) + ((DEMCI) -QREGEN( I) )**2)* CCC CS*PLTMAD( I) )**2)+¢C CS*NTTMAD( I) )****2)) OPTION(I)= ( PLTFC(1)-PTAT(1))*(((S*DEMMAD(1))**2) + (RSR(1)**2)* 28 ees Oh lee 2+ AVGCR( 1)*-2 )*( (S*RSRMAD( 1) )**2) - 2*COVICL) + Ses enh (ie 2 )*((S*RSRMAD(1))**2)) + (PTATCI)*((S*DEMMAD( I) )**2 Cy) CODEM( 1) *2)*( (S*NITMADC I) )**2)) + (CDEM( I) -QREGEN( 1) )**2)* CGC Ge een) =? )+( (S*NTIMAD( 1) )**2)) PVAR(I)= (PLTFC(1)-PTAT( I) )*( ((S*DEMMAD(I))**2) + (RSR(1)%*2)* 6(( S*#CRMAD( 1) )*"2)+( AVGCR( 1)7*2)e*( (S*RSRMAD(1))**2) - 2*COV(I) + C(( S*CRMAD( 1) )2**2)**( (S*RSRMAD(1))**2)) + (PTAT(1)*((S*DEMMAD(1))**2 C))+ ((QREGEN( I )***2)*( (S*NTTMAD(1))***2)) + (((DEM(1)-QREGEN( I) )** C2)*((S*PLTMAD( 1) )***2)) PPV( 1I)=LTDEM( 1) -LREGEN( I)+TREGEN( I) * DATA CHECK AND SCRUB FOR BAD OR ERRONEOUS DATA ELEMENTS PCE COE Gia me be. 0}, THEN BDATA = BDATA + 1 GO TO 10 tpi VAR( 1). bi. 07 OR, PPV(1). LT. 0.OR. V(1).LT.0) THEN BDATA = BDATA + 1 ce 10 10 ELSE IF (V1(1). LT. 0. OR. OPTION(1I).LT.0) THEN BDATA = BDATA + 1 GO TO 10 END IF GDATA = GDATA + 1 * CALCULATE VARIANCE TO MEAN RATIOS RATIOI(1)=V1(1)/PPV(1) RATIO2( 1)=PVAR(1)/PPV(I) RATIO3(1)=OPTION(1)/PPV(1) RATIO4(1)=V(1)/PPV(I) RATDEL(I) = RATIOI(I) - RATIO3(1) RAMDIE( 1) = RATIO1(1) - RATIO2(1) IF (RATDIF(1).LT.0.) THEN NEGDIF = NEGDIF + 1 ELSE IF(RATDIF(1).GT. 0.) THEN POSDIF = POSDIF + 1 END IF Wee RSeOEn( 1). LT. 0.) THEN NEGDEL = NEGDEL + 1 Pion ir CRATDEL( 1).GT. 0.) THEN POSDEL = POSBEL + i ENDe IF PaGe sib avr) ). LT. 0) THEN DEL=DEL+1 END ar * CALCULATE STANDARD DEVIATION AG. V1ISD(I)=V1(1)**. 5 PVARSD( I1)=PVAR(1)***. 5 OPSD( 1)=OPTION(1)***. 5 * REPORT WRITER AND DATA OUTPUT * DATA OUTPUT FOR IMPROVEMENT CALCULATION IF(PVAR( 1). GT. V1(1). OR. OPTION(I).GT. PVAR(I)) THEN GO TO 100 END IF DIFF(I)=PVARSD(1)-OPSD(T) DELTA(I)=(V1(1)**. 5)-( PVARCI)**. 5) TOTDEL=TOTDEL + DELTA(T) NUM=NUM+1 WRITE(3,95) I, DELTA(1),DIFF(I) 95 FORMAT(' -' ,I15,2X, ‘DELTA = ‘,F10.3,2X, DIFF = ‘,F10.3) IF(DIFF(I). LE. 1) THEN ONEF=ONEF+1 ELSE IF(DIFF(1). LE. 2. AND. DIFF(I).GT. 1) THEN TWOF=TWOF+1 ELSE IF(DIFF(I). LE. 3. AND. DIFF(I). GT. 2) THEN THREEF=THREEF+1 ELSE IF(DIFF(I). LE. 5. AND. DIFF(I). GT. 3) THEN FIVEF=FIVEF+1 ELSE IF(DIFF(i). LE. 10. AND. DIFF(I).GT.5) THEN TENF=TENF+1 ELSE IF(DIFF(1I).GT.10) THEN GTENF=GTENF+1 END IF IF(DELTA(I). LE. 1) THEN ONEL=ONEL+1 ELSE IF(DELTA(1). LE. 2. AND. DELTA( I). GT. 1) THEN TWOL=TWOL+1 ELSE IF(DELTA(1). LE. 3. AND. DELTA( I). GT. 2) THEN THREEL=THREEL+1 ELSE IF(DELTA(I). LE. 5. AND. DELTA( I). GT. 3) THEN FIVEL=FIVEL+1 ELSE IF(DELTA(I). LE. 10. AND. DELTA(1).GT.5) THEN TENL=TENL+1 ELSE IF(DELTA(I).GT.10) THEN GTENL=GTENL+1 END IF * SPLIT DATA INTO LLOW, MLOW, HLOW, MED AND HIGH DEM ITEMS FOR ANALYSIS 100 LES CDEMC HT) Li. 1) THEN GO TO 114 ELSE IF (DEM(1). LT. 2. AND. DEMC1 ). GE=d) Sie GO TO 154 ELSE IF (DEM(1). LT. 5. AND) DEMC1! ) Giz ina GO TO 164 ELSE IF (DEM(1). LT. 20. AND. DEM( 1). GE. 5) aaa GO TO 124 ELSE IP (CBEMC 1). GE. 20) SEN GO TO 134 END IF 51) *LOW LOW DEMAND OUTPUT 114 Pownce LoL. 0. ) THEN NDIF1 = NDIF1 + 1 ELSE IF(RATDIF(I).GE. 0.) THEN PUTRI = POIF1 + 1 END IF PoeCRet DE bCi). Ll. 0. jimainiN NDEL1 = NDEL1 + 1 EUSE DTFCRATDEL( 1). GE. 0. ) THEN PDEL1 = PDEL1 + 1 BND IF IF(RATIO4(1).GE.P) THEN COUNT1 = COUNT1 + 1 END IF WRITE(10,115) I, V1(1), PVARCI), OPTION(I), V(I), PPV(I), CDIFF(I),V1SD(I), PVARSD(I), OPSD(I), RATIO4(I) 115 FORMAT ('-' ,15,9(F10. 3),F10. 3) So) “dol ae *MED LOW DEMAND FILE OUTPUT 154 IF (RATDIF(1I). LT. 0.) THEN NE? = NDIFZ +t bok LPCRATDIE( 1).GE. 0.) THEN Pte 2 = RbIk2 +1 RIND eke Pe GRADER LC Toh, 0.) THEN NDEL2 = NDEL2 + 1 Por UCR TDEL( 1),CGE.0. ) THEN POEL2 = PDELZ s+ 1 END ear PiCRAT LOA jeGE. P) THEN COUNT2 = COUNT2 + 1 END IF WRITE(11,155) I, V1(1), PVAR(I), OPTION(1I), V(1I), PPV(I), CDIFF(I),V1SD(I), PVARSD(I), OPSD(I), RATIO4(1) 155 FORMAT ('-' ,15,9(F10. 3),F10. 3) GO TO 10 *HIGH LOW DEMAND OUTPUT 164 IF (RATDIF(1I).LT.0.) THEN NPWESe— NDIF3 +1 Poobelra@rcsdpih(1).GE.0.) THEN Polror— PDIFS + 1 END IF Balk IF (RAIDER 1). LT.O. ) THEN NDEL3 = NDEL3 + 1 ELSE IFCRATDEL(I).GE.0. ) THEN PDEL3 = PDEL3 + 1 ENDeah IFCRATIOS(1).GE.P) THEN COUNT3 = COUNT3 + 1 oN) Ble he WRITE( 7,165) I, V1C1), PVARC1), OPTIONC Iie eee Cle CDIFF(1I),V1SD(1), PVARSD(I), OPSD(I), RATIO4(I) 165 FORMAT (=o, 9( F10.s) ems) GO TO 10 *MED DEMAND OUTPUT 124 IF CRAQDIF( 1). LT. 0. ) GaEN NDIF4 = NDIF4 + 1 ELSE IFCRATDIF(1).GE.0. ) THEN PDIF4 = PDIF4 + 1 END ek IF (RATDEL(1 ). LT. GepeiEN NDEL4 = NDEL4 + 1 ELSE TFCRATDER 1) Che” ji THEN PDEL4 = PDEL4 + 1 END IF IFCRATIO4(1).GE.P) THEN COUNTS = COUNT4 + 1 ENE WRITE(8,125) I, V1(1), PVARC1), OPTIONGD Sev Gl Sere Gn CDIFF(1),V1iSD(I), PVARSD(I), OPSD(I), RATIO4(1) 125 FORMAT ( = 915, 9¢P1G.c er teen GOERS *HIGH DEMAND OUTPUT 134 IF (RATDIF(1).LT.0.) THEN NDIS = NOUPSs ee ELSE JFCRATDIFC] GE, OC”) Hien EDIEe = Dis + 1 END IF IF (RATDEL(I).LT.0.) THEN NDELS = NDELS + 1 ELSE IF(RATDEL(1).GT. 0.) THEN PDEL5 = PDELS + 1 END IF IFCRATIO4(1).GE.P) THEN COUNTS = COUNTS + 1 32 END IF WRITE(9,135) I, V1(1I), PVAR(I), OPTIONCI), VC1I), PPV(I), CDIFF(I),V1SDC(I), PVARSD(I), OPSD(I),RATIO4(1) ie FORMAT ( = ,15,9(F10.3),F10. 3) 10 CONTINUE UNCHNG = I - (POSDIF + NEGDIF) Nites 1 = (POSDEL + NEGDEL) *TOTAL SUMMARY DATA OUTPUT WRITE (3,145) BDATA, GDATA, NEGDIF, POSDIF, UNCHNG, NEGDEL, POSDEL C, UNDEL, TOTDEL, NUM, DEL 145 FORMAT ('-'/'0O BDATA = ‘',1I5/'0 GDATA = ',I5/'ONEGDIF = ',15/ C'OPOSDIF = ',15/'OTOTAL VARIANCE UNCHANGED = ',15/'ONEGDEL = ',7 CI5/'OPOSDEL = ',I5/'OTOTAL VARIANCE UNCHANGED = ',15/ G'OTOTAL DELTA OF S.D. = ',F10.3/'ONUMBER OF ITEMS = ',I5/ C'ONUMBER OF ITEMS WHEN V<V1 = ',F10. 3) WRITE(3,300) ONEF,TWOF , THREEF ,FIVEF,TENF ,GTEN,ONEL,TWOL, THREEL, CFIVEL,TENL,GTENL ogee FORMAT ('-'/ 12(15)) “SUMMARY DATA OUTPUT BY DEMAND WRITE (3,215) NDIF1, PDIF1, NDEL1, PDEL1, COUNT1 WRITE (10,215) NDIF1, PDIF1, NDEL1, PDEL1, COUNT1 215 FORMAT ('-'/'OLOW DEMAND SAMPLES'/'ONDIF1 = ',I5/'OPDIF1 = ',15/ oer = .15/ OPDEL1 = ,15/'OCOUNT1 = ‘,I5) WRITE (3,255) NDIF2, PDIF2, NDEL2, PDEL2, COUNT2 Meme (11,255) NDIF2, PDIF2, NDEL2, PDEL2, COUNT2 255 FORMAT ('-'/'OMED LOW DEMAND SAMPLE'/'ONDIF2 = ',15/'OPDIF2 = ',I5 own? = —§15/ OPDEL2 = ,15/'OCOUNT2 = ',I5) WRITE (3,265) NDIF3, PDIF3, NDEL3, PDEL3,COUNT3 WRITE (7,265) NDIF3, PDIF3, NDEL3, PDEL3, COUNT3 265 FORMAT ('-'/'OHIGH LOW DEM SAMPLES'/'ONDIF3 = ',15/'OPDIF3 = ',IS/ eee) = 9l5/ OPDELS = ',15/ OCOUNT3 = ',I5) WRITE (3,225) NDIF4, PDIF4, NDEL4, PDEL4, COUNT4 WRITE (8,225) NDIF4, PDIF4, NDEL4, PDEL4,COUNT4 225 FORMAT (‘'-'/'OMEDIUM DEMAND SAMPLES'/'ONDIF4 = ',15/'OPDIF4 = ',I5 me Ege —) .15/ OPDEL4 = ,15/' OCOUNT4 = ' ,15) WRITE (3,235) NDIFS, PDIF5, NDELS, PDELS, COUNTS WRITE (9,235) NDIFS5, PDIF5, NDELS, PDELS, COUNTS 235 FORMAT ( -'/'OHIGH DEMAND SAMPLES'/'ONDIF5 = ',I5/'OPDIFS = ',15/ © eee —) 9157 OPDELS = ,15/ OCOUNTS = ',15) STOP END 52 APPENDIX C DETAILED OUTPUT LISTING I - Item Number PVAR -. Vets colcul eee - Variance calculate OPTION - Variance calculated i OPTION - Variance calculated by UICP formula - Mean net leadtime demand (PPV) 4 sd - difference in between PVAR_ standard deviation and OPTION standard deviation (PVAR s.d. - OPTION s.d.) V1 sd - Square Root of V1 (standard deviation) PVAR sd - Square Root of PVAR (standard deviation) 1 Square Root_of OPTION (standard deviation) OPTION sd V/Z Variance to Mean ratio 1 ] V1 PVAR OPTION Vv Z 4 3D V15D PVAR SD OPTION SD V/Z 36 5023.379 1482.285 254.502 1482.285 105.620 ce25G7 70.876 38.500 15.9 so 14.034 97 13199.719 18595.754 2413.862 18595.756 193.650 0.000 114.890 136.366 49.131 96.028 113° 5271.066 2987.308 274.486 3106.028 64.920 38.089 72.602 54.656 16.568 47.844 241 2942.332 6507, 905 465.541 650.705 717250 a. 7 36 54.245 2oeol 5 cl. 576 9.135 290 4100.773 2518.112 566.498 2589.786 51.450 26.380 64.037 50.181 23.801 50.336 292 6695.910 1497.289 728.998 1997-269 123.220 11.704 81.829 38.695 26.991 12.15) S13 2671.982 S797. 102 SIS7 628657 aie 96077 50 0.000 51.686 75.810 “5. S20 59.402 387 29102.812 11955-1176 260779955 11455 e176 188.350 55.961 lp. col 107.029 a), 06's 60.819 560 482%. 566 157935.531 25783528 515795555) 99.080 0.000 69.459 125. Ga2 S07 779 159.402 879 - 369670 2I eco 2.097 926.285 350k? 82.890 26.950 60.795 37 wo 20.455 39.942 9120 -27E4.769 122469.891 1604.28) 1°351.262 68.710 0.000 Sear) 110.679 40.053 179.760 L123 Sesle.9357 S959 .59e 198) 2 e567 See eee eo 103.700 COL cNG ao 140.759 182.7 1286 11429.637 23712.079 2579.768 22867.633 Ty S710 0.000 106.910 153.987 SOR 179.996 WB? 44072, 21) Pool e76 (1715-97 Wo%ce. 93.4930 0.000 66.500 126.140 41.394 ° 170.426 134) 2603/7 3.969.300 bes0 26c6209e) 5497 2erU 306.980 7.146 162.401 58.381 by ere Lt ll .o3 1365 «67726.999 17683.113 193525038 21a2 ee eA 0.000 87.903 132.978 44.182 158 454 1366 19959. 168 601073.) 36787 055039976, 000 274, 750 0.000 141.277 Zoo. s0 196.949 392.170 1370 87616 .5002319494.875 3624.6%8269876 .062 729.660 0.000 296.001 481.087 euler [1 2) 369.865 1646592673 .500467730.000467309.250467730.000 6558.996 0.308 769.853 683.908 683.600 7h 1663 4712.086 12041.215 2381.762 12041.215 97 2 2eU 0.000 68.645 109.732 “8.805 123.474 1730 15199.637 22515255) S779 28 Fee 2751 Ss. 55) Clee eu 0.000 123.287 1G9 5377 69.137 104.896 1791)» =93207.099 1352.95] 373.564 1998.95] 60.%60 17.455 56.631 36.782 19.328 23.965 1890 15284.000 2287.788 587.166 2326.068 169.690 23.599 123.628 47.851 249 22 5c 13.708 189] 3$5834.844 30795.516 174955.398 37147 .898 438.280 43.368 189.501 175.487 132.119 84.758 1959 5019.367 4016.4]? 375.707 4163.137 65.600 a4 70.847 63.375 19.3835 63.462 2320 10893.074 436%.207 1129.218 4364.207 191.110 52.496 104.370 66.062 33.604 22.856 q1 10. et. Pere eer 1 OUTION LIST No. of Copies Defense Technical Infonmation Center Z Cameron Station — | Alexandria, Virginia 22314 pumech Defense Logistics Studies Information Exchange U. 8. Anny Logistics Management Center Fort Lee, Virginia 23501 Library, Code 0142 Z Naval Postgraduate School Monterey, Calfornia 93943-5002 Professor Rohert R. Read Code 55Re, Department of Operations Research Naval Posteraduate School Monterey, California 93943-5000 Professor Alan W. McMasters _ D Code 54Mg, Departinent of Adiinistrative Sciences Nava] Postgraduate School Monterey, California 93943-5000 Commanding Officer an Navy Fleet: Material Support Office Attn: Code 93 (LCDR Ik. Adams) Mechanicsburg, Pennsylvania 17055-0787 Commanding Officer 3 Naval Supply Svstems Conimmand Attn: SUP 042 (CDR M. Mitchell Washington, 1D. C. 20376-5000 Commanding Officer 3 Navy Ships Parts Control Center Attn: Code 0-412 Mechanicsburg, Pennsylvamia = 17055 Commanding Officer _ 3 Navy Aviation Supply Ofce Attn: Code SDB4-A Philadelphia, Pennsylvania 19111 Professor Peter Purdue ] Departinent. of Operations Research Naval Postgraduate Schoo! Monterey, California 93943-5000 Chief of Naval Operations 1 Navy Hepartment Attn: OP31 Washington, DC 20350-2000 Oe @eh bt "8 7 = BUA Le ee “ t a j - Seth Mp Tile ae 1 jetes wt entate ae - aterts a . thesA232501 wi LN aA TaN RRND hk Ea The eff . . : oe S-Rafat bs t eae pee am ' tidied ter Qhoh hofer 7 Can a Gh tah th, sdiaagiticgtaatatat tes tists Sd smecel ¢ elect of the covariance factor ont [i : 08008 ot some OEE NOE Pc nt he of Lanse! Bhueagg re en ] . FF ott te toh! Ob # tpeytnse | 6 AA Gar dar os a Ne! 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