### METRIC

RELIABILITY OF REPAIRABLE SYSTEM, SPARE AND SERVICE PARTS INVENTORY, METRIC AND EXCEL

OBJECTIVE: Development of METRIC (Multi-Echelon Technique for Recoverable Item Control), a non-METRIC, MOD-METRIC and VARI-METRIC models with Microsoft EXCEL.

Disciplines: Operations research, dynamic programming, maximization and minimization of objectives, backorder.

MOTIVATION: the work with Algorithm of KETTELLE (see here) helped the comprehension of the METRIC and construction of this similar web page.

 Models summary table : MODEL CHARACTERISTIC DISTRIBUTION A.1) Multi-item A) KETTELLE's algorithm A.2) Single-level Poisson A.3) Single-indenture B.1) Multi-item B) METRIC B.2) Multi-level Poisson B.3) Single-indenture C.1) Multi-item C) MOD-METRIC C.2) Multi-level Poisson C.3) Multi-indenture D.1) Multi-item D) VARI-METRIC D.2) Multi-level Negative binomial D.3) Multi-indenture

PART I "METRIC" The theory is known from text of Craig C. Sherbrooke and its essence (concepts of probability, backorder, expected number and variance of backorder, etc) is summarized on this table. The numerical example used is from paper by Wagner de Souza Borges of Sao Paulo University.

Basically this case describes a repair system with two levels (one depot and two bases) operating two repairable parts or equipment.

The purpose is to determine two sets of parts : one of type 1 N1 (N10, N11, N12) at three maintenance sites (Depot and two Bases) and other of type 2 N2 (N20, N21; N22)  also at three maintenance sites, all sets restricted by budget and minimizing the total Expected Back Order.

Possible extension to be done to written Excel program quantitatively and/or qualitatively is modification of start hypothesis that be one or combination of items below listed.

·         inventory control by batches, here (S-1, S) that is order and repair demands are one-per-one basis

·         greater number of depot, bases and equipment

·         limited repair capacity (here considered unlimited)

·         addition of pool of spares, scrap rate, cannibalization

·         support between same repair level (the flow of equipment between bases not taken into account)

·         addition of more repair level (insertion of intermediate level between base and depot)

·         breakdown of equipment (repairable part composed by other parts)

·         use of other reliability function than Poisson (demand is always exactly one unit) for parts fail

·         use of other parameter than EBO (expected back order) to maximize or minimize. Fill rates measure the consumption of stocked spares while back orders indicate lack of them. The purpose of named RBS Readiness Based Sparing is to consider Operational Availability Ao that is the relationship between active operational and total numbers of the system and related to system usage that is more significant and understandable than fill rates or numbers of back order both related to parts usage . So RBS method maximizes Ao under fixed budget or alternatively minimizes budget for targeted operational availability.

Ao = MTBF/(MTBF+MTTR+ACWT) where MTBF represents the reliability, MTTR the maintainability and ACWT (EBO=ACWT x Removal Rate or EBO=ACWT/MTBF with one removal to one fail) Average Customer Wait Time the material supportability. ACWT should be another parameter to be minimized instead of EBO.

EBO is considered as the average number of items not filled for which customer has to wait for supply, that is the number of failures that rise from system operation while customer waits for spares during supply delay. According to Little's law number = rate x time, see above EBO=ACWTxRR .......

·         non stationary condition, lost sales, production, ... in place of steady state, backorder, inventory, ...

The reasons that EXCEL was used here is that this MICROSOFT spreadsheet is useful for calculation of mathematical expressions defined by the algorithm, simplify its understanding. The factors are followings:

A.     The input values can be changed to see different outputs once the program is completed, inside limitation of final application and EXCEL itself.

B.     During application development EXCEL allows to check the partial results step by step up to the final one.

C.     EXCEL automatically converts numerical tables to graphical format to see what the algorithm does.

The present computer program with VBA is experimental and does not have any other purpose than to show the METRIC (Multi-Echelon Technique for Recoverable Item Control) example and the codification with VBA is not optimized. This EXCEL project is protected to hide the code and it does not allow internal modification. However the author is not responsible directly or indirectly of any result from its use by anyone.(See here the XLS file)

Symbols and mathematical expressions used are:

λ = mean demand rate

ν = mean time to repair

ρ = repair probability

τ = mean time between shipment and receipt

μ = mean number of part under repair or resupply (pipeline)

p = unit price

B = budget

EBO = expected back order

i = part type (i = 1, 2, ..., k)

j = repair level (j = 0, 1, 2, ..., m)

Spares OnHand = [ InitialStock - ( InRepair + UnderTransportation ) ]+

Spares in BackOrder = [ ( InRepair + UnderTransortation ) - InitialStock ]+

Pipeline = PL = Spares InRepair + Spares UnderTransportation

If InitialStock = 0 then E(BO) = E(PL) = E(IR) + E(UT) and Var(BO) = V(BO) = V(PL) = V(IR) + V(UT).

If IR (repair process) and UT (transportation process) have poisson distribution then PL and BO also follow same distribution.

An elementary simulation of the material flux with Excel is here.

METRIC as expressed below does not think about variance.

EXCEL book herein presented has 5 worksheets:

1.      The 1st worksheet contains input data that the user can change to get different results after running application.

 Part 1i=1 Part 2i=k=2 Unitprice \$5 \$3 Budget \$40 λ1j ν1j ρ1j τ1j λ2j ν2j ρ2j τ2j Depot j=0 0.02531 0.01782 Base 1 j=1 23.2 0.01000 0.5 0.01 35.2 0.00700 0.7 0.01 Base 2 j=m=2 20.1 0.01500 0.6 0.02 30.2 0.01000 0.6 0.02

Here is the schematic description of model and numerical example.

2.      2nd sheet shows calculated values for μi0 and λi0.

3.   3rd sheet has the generated undominating sequence for part type1.

See here graphical illustration.

4.   the 4th page is similar to previous one for part type 2.

See here graphical presentation.

5.   5th page presents final result that is the quantity required of each part type by repair level for total cost or budget.

 N10 N11 N12 N20 N21 N22 C EBO 0 0 0 0 0 0 \$0.00 3.075646 0 0 0 1 0 0 \$3.00 2.411676 1 0 0 0 0 0 \$5.00 2.292245 0 0 0 1 0 1 \$6.00 2.042374 1 0 0 1 0 0 \$8.00 1.628275 1 0 0 1 0 1 \$11.00 1.258972 1 0 0 1 1 1 \$14.00 0.991385 1 0 1 1 0 1 \$16.00 0.939530 1 0 0 2 1 1 \$17.00 0.909466 1 0 1 1 1 1 \$19.00 0.671943 1 0 1 2 1 1 \$22.00 0.590024 1 1 1 1 1 1 \$24.00 0.417037 1 1 1 2 1 1 \$27.00 0.335118 2 1 1 1 1 1 \$29.00 0.304102 1 1 1 2 1 2 \$30.00 0.266018 2 1 1 2 1 1 \$32.00 0.222183 2 1 1 2 1 2 \$35.00 0.153083 2 1 1 2 2 2 \$38.00 0.120005 2 1 2 2 1 2 \$40.00 0.104837

See graphical format here of sequence for parts type 1 and 2.

PART II "NON-METRIC" MODEL As the method becomes closer to real situation the mathematical expressions to describe the data relationship are too complex. However Rustenburg et al present another simple solution (with greedy algorithm, for not repairable items and limited budget) to "Single-period, constrained, multi item spare parts management", see MS-Excel file here and graphical result here.

PART III "MOD-METRIC" THE DEVELOPMENT OF MULTI-INDENTURE MOD-METRIC WITH MS-EXCEL IS SHOWN HERE.

THIS FILE CONTAINS IN:

PAGE 1 THE INPUT DATA FOR TWO ASSEMBLIES, ONE DEPOT AND FIVE BASES.

PAGE 2 THE ARRIVAL RATES EXPRESSION.

PAGE 3 THE PIPELINE EXPRESSION.

PAGE 4 ARRIVAL RATES ALLOCATION.

PAGE 5 EXAMPLE OF PIPELINES AND EBO'S CALCULATION FOR A STOCK LEVEL (QUANTITIES) WITH POISSON DISTRIBUTION.

THE START IS PIPELINES DETERMINATION FOR SUBASSEMBLIES (LOWEST INDENTURE) AT DEPOT (HIGHEST MAINTENANCE LEVEL). THE END IS MINIMIZATION OF ASSEMBLIES (HIGHEST INDENTURE) TOTAL EBO AT BASES (LOWEST MAINTENANCE LEVEL) AS ONLY EBO'S OF ASSEMBLIES (AND NOT OF SUBASSEMBLIES) AFFECT THE SYSTEM AVAILABILITY.

THE MS-EXCEL MOD-METRIC WITH VBA IS AN APPLICATION OF GREEDY HEURISTIC, MARGINAL ANALYSIS ALGORITHM OR BIGGEST BANG PER/FOR BUCK PROCESS AND WE HAVE THE RESULT SETS (BEARING, SEAL, HOUSING, PUMP, ROTOR, STATOR, MOTOR) FOR STOCK QUANTITIES AS (12; 16; 8; 8; 11; 21; 5) FOR DEPOT AND 5 x (0; 0; 0; 6; 1; 1; 4) FOR BASES SUMMING A BUDGET OF \$2,310.20 AND EBO OF 1.89706, AN AVAILABILITY OF 62%, STARTING FROM A KIT OR SET WITH QUANTITIES ZERO FOR ALL ITEMS AND SITES; PLEASE SEE SET OF NUMBER 141 ON SHEETS "MMR" AND "RES" OF THE SPREADSHEET.

IF THE START SETS COMPRISE QUANTITIES ZERO FOR DEPOT AND ONE FOR FIVE BASES, OR ONE FOR ALL, THE RESULTS ARE (12; 16; 8; 8; 12; 21; 5) FOR DEPOT, 5 x (1; 1; 1; 6; 1; 1; 4) FOR BASES, \$2,372.70 FOR BUDGET AND 1.80253 FOR TOTAL EBO OF BASES.

IN THE FIRST SHEET NAMED "MMR" OF EXCEL ONE CAN SEE THE INPUT DATA (REFERENCE ABOVE FILE PAGE 1), ARRIVAL RATES ALLOCATION (REFERENCE FILE PAGE 4 ), THE FOUR COLUMNS WITH VALUES OF PIPELINE (REFERENCE FILE PAGE 3 ) AND OTHER THREE COLUMNS FOR GREEDY ALGORITHM FOR VBA MACRO. THE RESULT FOR EACH STEP OR SET IS SHOWN ON THE REST OF THIS SHEET.

THE SECOND SHEET "RES" BASICALLY SHOWS TOTAL EXPECTED BACKORDER AND COST/PRICE OF EACH SET AS WELL AS THE GRAPHICAL VARIATION OF TOTAL EBO WITH BUDGET.

THIRD SHEET HAS EXCEL FORMULAS RELATING DEMAND RATES, PIPELINES AND EBO OF NUMERICAL EXAMPLE, IT WAS USED TO CHECK INTERMEDIATE VALUES FROM ALGORITHM DURING DEVELOPMENT TO VERIFY AND VALIDATE THE VBA PROGRAM.

THIS EXCEL MOD-METRIC APPLICATION CAN CALCULATE 220 SUCCESSIVE SETS STARTING FROM STOCK QUANTITIES AS ALL ZEROS (OR ZEROS AND ONES) BUT MORE ITERATIONS ARE POSSIBLE CONSIDERING AS NEW START SET THE FINAL ONE OF PREVIOUS RUNNING AND APPLYING THE MACRO ONCE AGAIN. THE LIMITATION OF THIS PROCEDURE IS THE ERROR GIVEN BY EXCEL NATIVE POISSON FUNCTION AROUND A QUANTITY OF 130 UNITS.

PART IV "VARI-METRIC" These above three practical applications should be the bases for implementation with Excel and VBA of a general multi-item, multi-indenture, multi-echelon inventory system called Vari-Metric. It is observed that the Poisson often is not a good fit for experimental results as distribution mean is not constant and varies continuously with trials. The Negative Binomial series used in this method is an extension (as a summation of several processes ) of Poisson but the function has an added work that requires two parameters to be determined that are mean and variance for pipeline quantity calculation. A real sample of spares demand (reference B Brown) and examples of Poisson and Negative Binomial distribution adjusted to it are shown here .

Compare examples of Metric ( with marginal analysis ) and Vari-metric. The expected values of backorder and its variances at depot, the expected backorders at the bases can be checked below.

Here is an example of general vari-metric method described.

1st step - Allocate failure rates (FR's).
Go to assemblies of depot.
Then go to subassemblies of bases.
Finally determine failure rates for subassemblies of depots.
2nd step - Calculate pipeline values E(PL) (=V(PL)).
3rd step - Then calculate EBO and VBO for subassemblies of depot.
4th step - Determine E(PL) and V(PL) for assemblies of depot.
Note that EBO, VBO, E(PL) and V(PL) follow inverse sequence relevant to FR's allocation.
5th step - Determine EBO and VBO for assemblies of depot.
Observe that E(PL) = E[BO(S=0)] and V(PL) = V[BO(S=0)].
Check the values using this MS-Excel file with EBO and VBO functions.
The VBA codes for UDF (user defined function) are here.
6th step - Determine E(PL) and V(PL) for subassemblies of bases.
7th step - Determine EBO and VBO for subassemblies of bases.
8th step - Determine E(PL) and V(PL) for assemblies of bases.
9th step - Determine EBO (and VBO) for assemblies of bases.
10th step-Find out system availability (here not shown).

As conclusion see here the summary of recursion formulas, MS-Excel file with user defined functions for EBO and VBO developed to avoid summation to infinite, visual basic codes used to write these functions.

PART V "TIME DEPENDENT FAILURE" The development of Poisson process applying MS Excel spreadsheet with not constant failure rate and dynamic Palm's theorem is presented here .

Failure rates are modeled by straight line bathtub form function and pipeline and expected backorder values are calculated in three different phases, infant mortality, random failure and wearout in this table .

To see fill rates on table and graphical formats click here .

PART VI "METRIC LIKE MODELS" The following eleven examples detail the development of theoretical expressions with spreadsheet.

A) METRIC (poisson random variable for both depot and bases)
See the simulation item (I) below.

1st step - calculation of EBOs for depot (poisson distribution)

2nd step - determination of pipeline quantities for bases

3rd step - finding of EBOs at bases (poisson).

4th step - presentation of total EBO (depot and two bases) and availability

B) QUEUE-METRIC (Metric with queue at depot)
See the simulation item (K) below.

1st step - introduction of queue M/M/4 to limit repair capacity of depot and calculation of probability distribution per queueing theory.
For PBO and EBO computation see detail in (F) below.

2nd step and on - similar to (A) Metric above.

C) VARI-METRIC (bases distribution approximated by negative binomial)
See the simulation item (I) below.

1st step - calculation of EBOs for depot (poisson distribution)

2nd step - variances for depot by recursive formula

3rd step - pipeline quantities for bases (same as METRIC)

4th step - variance for bases by recursive expression

5th step - EBOs of bases applying negative binomial random variable (see table from MS-Excel).

Note: EBOs of bases for depot quantity equals to zero are got from poisson distribution (same as METRIC).

D) VARI-METRIC (exact or better precise procedure)

1st step - determination of backorder probability, EBO and variance all for depot (see table of MS-Excel spreadsheet)

2nd step - disaggregation of depot backorder probability across bases (see table of spreadsheet)

3rd step - convolution of result of previous step with probability (poissonian) of spares under transportation from depot to base (see table of spreadsheet)

or

n = 0, 1, 2, ..., M and ( n - m ) >= 0 that is

etc, etc.

This operation is performed by an UDF (user defined function) and it requires recalculation each time values change pressing keys combination "Control + Alt + F9".

4th step - finding of EBOs at bases from probability of backorder at bases resulted by convolution (see table of spreadsheet)

Note: this process directly finds all EBOs (of depot and bases). See the numerical table of spreadsheet here.

See here comparisons METRIC to BINOMIAL DISAGGREGATION and VARI-METRIC to BINOMIAL DISAGGREGATION.

E) VARI-METRIC WITH REPAIR QUEUE AT DEPOT (disaggregation method)

1st step - introduction of queue M/M/4 to limit repair capacity of depot and calculation of probability distribution per queueing theory.
For PBO, EBO and VBO computation see detail in (F) below.

2nd step and on - similar to (D) Vari-metric above. The numerical table of the MS-Excel and functions or procedures generated with VBA are contained in these sheets.

F) VARI-METRIC WITH REPAIR QUEUE AT DEPOT (bases EBO approximated by Neg Bin Dist)

1st step - Depot EBO and VBO similar to (C) above. For PBO, EBO and VBO computation see detail clicking here.

2nd step - Pipeline quantities and variances similar to (C) above.

3rd step - Bases EBO per Negative Binomial Distribution.

See the simulation item (J) below.

The theory is explained here. Note that EBO (NBD), VBO (NBD) and VBO (precise) refer to vari-
metric model.

1st step - calculation of subassemblies expected backorder and variances.

2nd step - spreadsheet for backorder probabilities distribution of subassemblies by quantity.

3rd step - convolution of PBO's of subassemblies.

4th step - probability distribution of assembly (poisson).

5th step - convolution of probabilities from steps 3 and 4.

6th step - determination of assembly expected and variance values of backorder.

H) MOD-METRIC with  fill rate and availability

Example similar to previous one comparing:
Fill Rate and EBO that are measures of spare or service parts supply support performance and
Availability that is a measure of system (or here assembly) operational performance.
Note that P[BO=0], Fill Rate and Availability curves start convex and become concave while EBO curve is convex.

I) METRIC (Excel simulation)

1) Depot stock quantity of cell A6 is an input, change it to see spreadsheet calculation.
2) Base stock quantity of cell G6 is another intput, change its value.
3) Press combination keys "Control"+"Alt"+F9 for recalculation to get different outputs.
4) Compare outputs mean (cells C2, D2, H2, I2, J2, K2) and Expected Backorder (E2, L2) calculated value with table contents of example (A) above.
5) Compare outputs variance (cells C4, D4, H4, I4, J4, K4) and Variance Backorder (E4, L4) calculated value with table contents of example (C) above.
6) For difference between workseets 1 and 2 see (6) below.

Worksheets description
1) Cell A6, see above.
2) Column B with 10,000 time period (for instance, weeks, days, ...).
3) Column C, generation of Poisson numbers with mean = 2.4 by UDF (user defined function).
4) Cell G6, see above.
5) Column H, quantity under transportation and generation of Poisson numbers by UDF.
6) Column I shows quantity of one base under repair at depot. The first procedure is the use of Poisson UDF with half of mean given in cell E2, this easy way causes incongruous numbers between columns E and I. The second one is to allocate half of quantities of column E to I, but odd numbers cause some difficulties as column H must contain integer numbers.
7) Column J with summation of columns H and I.
8) Column L for backorder quantity of one of two bases considered.
9) Columns for on hand quantities in depot D and one base K.
10) See here extracts from simulation.

J) MOD-METRIC (Excel simulation)

1) Press "Control"+"Alt"+F9 combination keys for recalculation.
2) 10,000 sample times are simulated.
3) See item (I) above for other comments.
4) Stock quantities cells A6, F6 and K6 are inputs.
5) EBO's and VBO's of cells D2, D4, I2, I4, O2 and O4 are outputs.
6) Each time assembly has a fail it is caused by only one of two subassemblies. When we cannibalize an assembly removing operating subassembly to repair another assembly the final EBO and VBO have different result than shown.
7) Here is extract from simulation, see table of item (G) above for comparison.

K) METRIC WITH QUEUE (Excel simulation)

Simulation similar to item (I) above with queue ditribution probability.
See extract here and compare with items (B) and (F).

Several values of EBO at bases found during the development of this example are shown here.

Note that all MS-Excel applications and files herein have only purpose to demonstrate and explain the methods and the author is not anyway direct or indirectly responsible for any result from its use by anyone.

Jorge Fukuda jfukuda@zipmail.com.br The author earned Mechanical Aeronautical Engineer degree in 1979 by Brazilian Air Force College at Sao Jose dos Campos, Sao Paulo.

From 1980 to 1990 worked for Brazilian Aircraft Manufacturer EMBRAER at Sao Jose dos Campos in Sao Paulo as Engineer in Spare Parts Division.

From 1991 to 1994 worked for JAMCO Co (JAL group company) in Tokyo Japan as Engineer with Technical Publication (for Galley and Lavatory for Boeing & Douglas Aircraft) .

From 1994 to 2019 worked for GAMESA AERONAUTICA (now AERNNOVA Aerospace) at Vitoria Spain as Spare Parts Engineer (for Airbus, Sikorsky, Bombardier, Embraer, Eurocopter and NHI Aircraft Structural Segment).

Retired from July 2019, living in spain.

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