SHIP PRODUCTION COMMITTEEFACILITIES AND ENVIRONMENTAL EFFECTSSURFACE PREPARATION AND COATINGSDESIGN/PRODUCTION INTEGRATIONHUMAN RESOURCE INNOVATIONMARINE INDUSTRY STANDARDSWELDINGINDUSTRIAL ENGINEERINGEDUCATION AND TRAINING

THE NATIONALSHIPBUILDINGRESEARCHPROGRAM

August 1987NSRP 0281

1987 Ship Production Symposium

Paper No. 4: The Use ofComputer Simulation of MergedVariation to Predict Rework Levels on Ship's Hull Blocks

U.S. DEPARTMENT OF THE NAVYCARDEROCK DIVISION,NAVAL SURFACE WARFARE CENTER

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Paper presentedattheNSRP 1987 Ship Production Symposium, Hyatt Regency Hote, New Orleans, Louisiana August 26-28, 1987

The Use of Computer Simulation of Merged No. 4

Variation to Predict Rework Levels on Ship’s Hull BlocksR. L. Storch, Member andP.J. Giesy, Student Member, University of Washington, Seattle, WA

ABSTRACT

In the modular construction ofships, significant productivitylosses can occur during theerection stage, when the modules,or hull blocks, are joinedtogether. Frequently, adjacentblocks do not fit togetherproperly, and rework of one or bothof the mating block interfaces isnecessary to correct the problem.The specific cause of rework is thevariation of plate edges at theblock interface, which is itself acumulative product of numerousmanufacturing variations inherentin hull block construction.Variation in manufacturing isunavoidable, but notuncontrollable. The application ofaccuracy control techniques inshipbuilding has proven that astatistical analysis of variationmakes possible an accurateprediction of its effects. Thisreport presents an examination ofblock interface variation, and thesubsequent development of acomputer simulation method ofpredicting rework levels on thoseblocks.

The complex interaction of allthe edges’ random variations at theblock interface gives rise to aunique rework probabilitydistribution. This probabilitydistribution is evaluated by meansof the computer simulation program,which provides estimates of theaverage rework anticipated, theshape of the probability curve, andother parameters. Similarpredictions are also available forcost and labor of required rework.In addition to predicting reworklevels, the simulation program canbe a useful tool for reducing thoselevels.

1. INTRODUCTION

why Predict Rework?

A shipyard’s need to predictrework is no different from itsneed to be in control of all otheraspects of its operation. Thereare both short term and long termimperatives at work. The shortterm concern is the scheduling ofthe current project. It isnecessary to have accurateforecasts of the time required forevery work package in the project.The construction of a large vesselinvolves the coordination ofthousands of work packages into asingle, interdependent network ofactivities. If the duration of ajob is overestimated, the result isan underutilization of resources.Scheduling inadequate time for aspecific job, however, can disruptthe whole network. In the longterm, a shipyard must directattention to winning futurecontracts. A yard that knows itscosts, including projected reworkcosts, is in the best position tobid realistically, and thereforesuccessfully.

Rework is an intrusion ontraditional construction schedules.Because it is an “unplanned”activity, there has beenproportionally little effortinvested in characterizing therework function, compared to“regular” jobs. But rework can bea significant fraction of the totalproject. Quoting from Michael Wadeof the University of Michigan:

"Regardless of how refined orstandardized a planning systembecomes, there is a highprobability that during the lifecycle of a ship construction

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project, rework ....will befall theproduction schedule with verylittle warning. It is unrealisticto plan vessel withoutconsideration for the impact thesedisruptive factors can have on man-hours and completion dates ....Theability to measure performance atall levels of production will haveadirect effect on a shipyard’sability to bid new workconsistently and confidently." [1](emphasis added.)

2. THE REASONS FOR ERECTION STATEREWORK

Variation at The Block Interface

The cause of rework at theerection stage, neglecting designerror, is variation at the blockinterface. Variation, in itsformal definition, is a deviationfrom design dimensions. In anabsolute sense, there is variationexisting in every dimension ofevery item that has ever beenmanufactured; so long as anattribute can be measured closelyenough, it can be found indeviation from what it is supposedto be. The question of practicalconcern is the magnitude ofvariation.

When two hull blocks are to bejoined at erection, the criticaldimension is the gap between themating edges of the respectiveblocks . A uniform gap between allthe edges at the erection jointallows the welding of the blocks -in many cases, robotic welding - toproceed as scheduled. Excessivevariation of the edges of one orboth of the block interfaces spoilsthis uniform weld gap andinterrupts the erection schedule,as a certain percentage of theinterface must be reworked toachieve a proper fit.

Specifications on weld jointpreparation vary with the differenttypes of welding, but there is ineach case a gap tolerance, an upperlimit and a lower limit on gapwidth, beyond which the quality ofthe weld suffers. As shown in’Figure 1, when the weld gap is toonarrow, or if there is interferencebetween the plates, material mustbe removed by torch cutting fromone or both sides. If the existinggap is too wide, a backing stripmust be welded across the gapbefore the joining weld can bemade.

UPPER GAP TOLERANCE

PROPER GAP-NOREWORK REQUIRED

GAP TOO SMALL-CUITING REQUIRED

GAP TOO WIDE-BACK STRIPWELDINGREQUIRED

Fig.1. Rework Criteria - Cuttingand Backstrip Welding

Of the two types of work,backstrip welding to close a gap ismore expensive than torch cuttingto widen one. Traditionally,shipbuilders, resigned toperforming considerable rework aterection but anxious to minimizebackstrip welding, would add amargin to part dimensions at theblock interface to insure that,whatever the final block variation,a uniform gap could be achieved bycutting away from all the edges theportion of margin remaining. Thepractice is essentially acommitment to rework, andconsidering this, it is no surprisethat erection stage rework levelsat traditional shipyards are quitehigh. The use of margins may havebeen the minimum cost solution ofthe past, before the advent ofstatistical accuracy analysis, buttimes have changed. Theapplication of accuracy controltechniques is now permittingprogressive builders to achievemuch greater accuracy in hull blockconstruction, making it possible tojoin hull blocks with less rework,and without margins.

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Quantifying Variation

These same accuracy controltechniques that make possible thereduction in block variation havean additional use as well. Theycan also be used to help determinehow to deal most effectively withthe variation that can not beeliminated. Through statisticalcharacterization of the interfacevariation associated with aparticular block design, it ispossible to anticipate some of theconsequences of that variation.Specifically, it is possible tomake a prediction, before any steelis cut, on the amount of rework theblock will require at erection.

Consider Figure 2, which showsa simple block interface and thevariation of its edges. The designspecifications of this hypotheticalblock are that the edges of alldecks, bulkheads, and other membersat the interface will lie on asingle plane, as seen in Figure2(a). However, due to variationsof parts and processes in theconstruction of the block, eachedge will exhibit some measurablevariation from the design plane.Each edge’s variation can bemodeled separately as a randomvariable with a normaldistribution. It is possible to

INTERFACE

(c) SIX-SIGMA

VARIATION LIMITS

Fig.2. Longitudinal Variation ofEdges at Block Interface

predict the random variations ofeach of these edges by writing aseries of variation mergingequations. Figure 2(b) representsthe normal probabilitydistributions of longitudinalvariation of all the edges, withrespect to the design plane(transverse and vertical variationcan be evaluated as well, but notwithin the scope of this paper).These probability distributions areeach characterized by a merged mean

some of the distributison curves arecentered a little aft of the designplane and some are centered a-bitforward. This illustrates ascattering of mean variationsvalues above and below a value ofzero.

A necessary precondition to thewriting of variation mergingequations is that all random partand process variations associatedin the block construction be known,and known to vary under a normaldistribution. A full description ofthe process of writing mergingequations can be found in “ThreeDimensional Accuracy ControlVariation Merging Equations,” byR.L. Storch and P. Giesy. A briefdescription of the principle ofmerging equations is provided byL.D. Chirillo:

"If the distribution Of suchvariations for a specific workprocess is Gaussianr that is,normal per a bell-shape curve, theprocess is said to be undercontrol. When work is socontrolled, and verified daily bynominal random sampling, the normaldistribution of a work stage can inaccordance with the Theorem ofVariance, be added to that for asecond work stage in order topredict the distribution for athird work stage.” [2]

It is impossible to predictexactly where a given edge will endup within its probabilitydistribution. That is a randomvariable. Under a normaldistribution, however, it can besaid with fair certainty that theresultant positions will be withinthree standard deviations of themean, within the so-called "six-sigma envelope." Figure 2(c) showsthese six-sigma limits at the blockinterface. If 100 blocks werebuilt from this design, they wouldall be different, but theconfiguration of each blockinterface will fall with certaintysomewhere within that six-sigma

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matrix. Knowing that variation atthe interface is thus constrained,is the first step in thedevelopment of a method forpredicting erection stage reworklevels.

3. DEVELOPING A REWORK STRATEGY

Variation vs. Weld Gap Tolerance

It can be stated then, thaterection stage rework is primarilya function of two opposing factors:the random variations of edgesthroughout the block interface, andthe weld gap tolerance (there is athird factor, of course, called“economics, “ which will beincorporated presently). Thegreater the variation at theinterface, and (or) the smaller theweld gap tolerance - the greaterthe probability that rework will berequired; and expected levels ofrework will be greater as well.Figure 3, which is a continuationof the hull block example startedin Figure 2, illustrates thisrelationship. The two upperdrawings show again the blockinterface and the six-sigmaenvelopes for all the edges. Thediagram of variation limits at thebottom is simply a differentrepresentation of the six-sigmaenvelopes; it emphasizes therelative widths and longitudinalpositions of the edges’ variationlimits. Since the relative lengthsof the edges has been lost in thetransition, that information isgiven in a column beside thediagram.

Note the cross-hatched areaoverlaying the variation limits inFigure 3. This represents the weldgap tolerance. As stated earlier,the weld gap throughout theerection joint must be betweencertain boundary values to avoidthe necessity of reworking one orboth edges of the gap. It does notmatter what the upper and lowertolerance limits are, only thewidth of the tolerance zone isimportant. This visual comparisongives a feel for the probabilitiesof rework being required at theblock interface.

To simplify the rework modelbeing developed, this example willbe presented as a case of one-sidedvariation. Under this constraint,manufacturing variations arepresent only on the block shown.The adjoining block is assumed tobe “perfect, " and therefore not a

factor in determining reworkrequirements. Extension to themare realistic model of two-sided

Fig.3. Diagram of Variation Limits

variation will be dealt with later.Simply stated, the rework criteria(with one-sided variation) is this:when the measured longitudinal spanof plate edges at a block interfaceexceeds the weld gap tolerance,then rework is required. In thecase of Figure 3, it is apparentthat the variation limits are muchwider than the weld gap tolerance.Intuitively, it is clear that theodds are very low of having thesenine edges (effectively nine randomvariables) ending up in a zonesmaller than the width of the weldgap tolerance. This is the same asstating a high probability thatrework will be required at thatinterface.

The Optimum Rework Solution

But how much rework will beneeded? Which edges will likelyrequire cutting or backstripwelding? To answer thesequestions, it is necessary toexamine the decision criteria oferection stage rework. The reworksolution (which edges to cut, whichto backstrip weld) for a specificblock is dependent not only on theresultant longitudinal position ofeach edge after random variationhas taken its toll, but on the

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length of each edge as well. lneach case, the problem becomes oneof finding the optimum solution outof a set of feasible solutions.

To demonstrate this process ofrework optimization, consider thatour hull block from the previousexamples has finally been built.Figure 4 shows the relativelongitudinal positions of the nineedges at the interface.Maintaining the assumption of one-sided variation, the adjoiningblock can be represented as a flatwall, shown on the right. Theshaded region near the wallrepresents the weld gap tolerancezone.

Finding the optimum reworksolution can be viewed as aniterative thought experiment thatis performed by moving the wallthrough the group of edges,stopping at each edge to calculatethe implied rework for that case,and then selecting as the optimumsolution the case requiring theminimum amount of rework. Sincethere are nine edges in ourexample, there are nine possiblerework solutions: A, B, and C,shown in Figure 4, represent threeof these. Solution A would be thefirst one evaluated. The wall ismoved to the left until the firstedge coincides with the minimumweld gap. At this position, thesecond edge is also within thetolerance zone, and so escapesrework. The remaining edges mustbe backstrip welded, for a total114 feet of rework. Solution B isbetter than solution A. With thewall (actually the minimum weldtolerance) at the third edge, thefirst two need cutting and the lastfour need backstrip welding, for atotal of 107 feet. Solution C, at101 feet, is better than A or B.An evaluation of all nine solutionswould confirm that c is in fact theoptimum solution.

This example has represented acase where the unit costs of torchcutting and backstrip welding areequal. In actuality, backstripwelding is a more costly operationthan cutting, and this affects thederivation of the optimum reworksolution. The selection criteriachanges from minimum rework tominimum cost. One would expectthis to result in a shift, on theaverage, to somewhat higher levelsof rework, but with a much smallerpercentage of backstrip welding.

I I

Fig.4. Rework Optimization

4. DETERMINING REWORKPROBABILITIES THROUGH SIMULATION

Estimating the Rework Profile

It has been established thatthe optimum rework solution is afunction of edge variation, edgelength, the weld gap tolerance, andrework costs. The only problemremaining is the one that we beganwith, that of how to predict theamount of rework that a given blockdesign is likely to require. It isa problem that does not lend itselfto an analytical solution. Thoughedge lengths, weld tolerance, andcosts are all constants, and thevariation distribution of each edgeis characterized by a mean and astandard deviation, the complexinteraction of those randomvariations, influenced by all ofthe constants, defies expression.

But analysis is not the onlymethod available. Much can be saidabout rework. Since rework is afunction of random events, it isitself a random variable, and canbe represented as a probabilitydistribution of optimum solutions.It is not a continuousdistribution, since it cannot take

4-5

on a continuous range of values.The values that rework can take areconstrained to the finite ofall possible combinations of sumsof edge lengths.

This type of problem is bestsolved through statisticalmodelling. In other words, usingempirical methods, rather thananalytical. The moststraightforward method would be tosample a large number of hullblocks built from the same design,and generate statistics, such asaverage rework and standard

deviation, to describe the reworkdistribution. Sampling is avaluable statistical tool, whichhas already played an importantrole earlier in this chain ofanalysis: it was sampling that wasused to determine the parameters ofthe specific shipyard processvariations. And the processvariations, of course, are what thedistributions of merged variationof edges at the interface arederived from. Sampling of hullblocks, however, would appear todefeat the”purpose of predictingrework prior to construction -unless a computer was used togenerate the sample. The followingsection describes a computerprogram written for such a purpose.

The Rework Simulation Program

With a rework simulationprogram, it is possible to “build,”and evaluate for rework, many hullblocks at no cost and in verylittle time. And many hull blockswill be needed. If optimum reworkwas known to have a normaldistribution, then a mean andstandard deviation could beinferred from as few as ten ortwenty observations. But since theshape of the rework distribution isnot (yet) well defined, the profilemust be “constructed” as ahistogram of a large number ofobservations. This programestimates the rework profile with ahistogram derived from two hundredsimulated hull blocks.

The-program described here iswritten in Pascal, and runs on anApple Macintosh personalcomputer. The Macintosh hasexcellent graphics capabilities,and the mouse-interface enhancesthe “friendliness” of the program.A complete listing of the programis given in the appendix.

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BUILD A BLOCK INTERFACE:ASSIGN A “MEASURED VARIATION” TO

EACH EDGE USING A RANDOM NUMBERGENERATOR WITH NORMAL DISTRIBUTION

DETERMINE AND SAVE THE OPTIMUMREWORK SOLUTION FOR THAT SAMPLE

NEXT 2

I

FROM SAMPLE POPULATION, CALCULATE:

> AVERAGE REWORK> REWORK RANGE> % SAMPLES REWORKED

SCREEN OUTPUT:

> HISTOGRAM OF POPULATION DISTRIBUTION

> EDGE-SPECIFIC REWORK PROBABILITIES

Fig.5. Simulation Subroutine Flowchart

The mechanics of running asimulation are outlined in Figure5. For each sample hull block, a"reultant variation” is assigned

to each edge at the interface usinga random number generator thatcomplies with the normaldistribution of merged variation ofthat edge. The algorithm for thisis as follows:

First, a random number (N) with a[0,11 normal probability distri-bution (i.e., mean = O, standarddeviation = 1) is generated withthe equation:

N= (-2 logeR1)1/2

where R and R are uniformdistrib tion r ndom numbers from Oto 1.

Then{tthe "resultant variation”edge is:

merged mean variation and standarddeviation, respectively.

A new “N” is generated for eachedge.

After each block interface iscreated in this manner, the programthen determines that block’soptimum rework solution, using apreselected weld gap tolerance andcosts of torch cutting andbackstrip welding. The optimumsolution, chosen on the basis ofminimum cost, is recorded in termsof total linear feet of rework,irrespective of type. At the sametime, a cumulative counter (overthe 200 samples) makes note of thespecific edges that requiredrework, and which type.

This whole procedure isrepeated two hundred times tosimulate the construction andrework of the entire sample of hullblocks . The two hundred optimumrework values become the raw datathat are used to estimate therework distribution. The reworkmean and standard deviation arecalculated from the sample data,and the shape of the distributioncurve is approximated by ahistogram of the data.

A full flowchart of the programis shown in Figure 6. on startup,the user must load a blockvariation table (either by hand, orfrom a file) into the programmemory. This variation table lists

Fig.6. Rework Simulation ProgramFlowchart

the names, merged mean variations,and standard deviations of all theedges at the block interface, andtheir respective lengths. Theprogram then proceeds to the mainmenu, where the user may choose torun a simulation, display or editthe variation table, or end theprogram. After each simulation,the user can call to the screen, orprint, four different graphicalreports: the Rework Distribution,Cost Distributionr LaborDistribution, or Edge SpecificRework Probabilities.

5. A CASE STUDY: THE T-AGOSRRWORK PROFILE

An Introduction to The T-AGOS Case

In 1983, R.L. Storch produced apaper called “Accuracy Control: AGuide to its Application in U.S.Shipyards” [3], which was based onresearch that had been done at theUniversity of Washington and at theTacoma Boatbuilding Co. in Tacoma,Washington. The main purpose ofthat research was to outline theprocedures for determining typicalshipyard process variations andconstructing variation mergingequations. A major project then atTacoma Boat was a Navy contract tobuild a series of twelve T-AGOSclass ocean surveillance vessels.

Three years later, in “ThreeDimensional Accuracy ControlVariation Merging Equations” [4],Storch and Giesy wrote a series ofmerging equations, characterizingthe merged longitudinal variationof all edges at the erectioninterface of a specific hull block:the T-AGOS stern section. A fulllist of the edges at the sternblock interface is given in Figure7.

This complete collection ofblock interface variationparameters provides a realisticdata set to run through thesimulation program. However, it isfirst necessary to explain anadditional complication in the T-AGOS variation table that was notcovered earlier.

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12

34

56

7

8

9

10

11

12

13

14

15

16

Fig.7.

Longitudinal Bulkhead(port)1X6-steeringGearFlatCenterlineFloorfarmeDiagonalFloorfarme(1)DiagonalFloorFrame(2)DiagonalFloor Frame(3)DiagonalFloorFrame(4)Side Shell(sttd)Side shell(port)BilgeStrake(stbd)BilgeStrake(port)

Summary of T-AGOS SternBlock Interface

The T-AGoS variation table isshown in Table 1. Note theappearance of a factor called"Mutual Variation" associated withsome of the edges. This indicatesthe presence of the phenomenon ofRelated Variation, revealed throughthe writing of the variationmerging equations. Edges 3 through8 are a group of edges whose mergedvariations are related; they willbe said to comprise Related Group#1. Likewise, the Main Deck,originally seen as one continuousedge, is more accuratelyrepresented as five shorter edgeswith related variation, making upRelated Group #2. The variation ofan edge in a related group ischaracterized by a randomindependent variation and also arandom mutual variation that iscommon to every edge in that group.The rework simulation program mustbe able to take occurrences ofrelated variation into account torealistically predict rework onhull blocks that contain theserelated groups.

Table 1. T-AGOS Variation Table

T-AGOS -VARIATION TABLEName Length (ft)

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The explanation of Group 1’srelated variation is found in theinternal structure of the T-AGOSstern block. Figure 8 shows apartial exploded view of the block.The location of the block interfaceis at station 96, where the forwardedges of the 13’6” flat, thecenterline frame, and the fourdiagonal frames are seen to lie.The merged variation of these edges(and of all the other edges at theinterface) are calculated withrespect to bulkhead 100. Theexploded view shows the 13.6” eggbox abutting the 15’ egg box, andthe 15’ egg box in turn abuttingbulkhead 100. The forwardtransverse of the 15’ egg box (atstation 96) therefore determinesthe position of the 13’6” egg box.The location of this transverseframe, however, will have variationwith respect to bulkhead 100,variation that will affect equallythe variation of the edges atstation 96. This, then, is themutual variation that is shared byall edges in related Group #1. Theedges! independent variations comefrom process variations that occurforward of station 96.

Fig.8. T-AGOS Stern Section -Exploded View

The reason that the Main Deckwas subdivided into a related groupis because of its assemblysequence. The Main Deck isoriginally assembled from five flatpanels, running fore and aft.There is variation associated withthe construction of these fivepanels that will manifest itselfindependently for each panel.After the panels are joined,however, they constitute the Main

hull block results in additionalvariation that is mutuallyexperienced for each of the fivepreviously separate edges.

When variation tables withrelated groups, such as the T-AGOStable, are loaded into the reworksimulation program, both mutual andindependent variation are randomlygenerated to represent the"construction" of the two hundredhull blocks. The following sectionpresents the program’s estimate ofrework for the T-AGOS stern block,and a sensitivity analysis toevaluate options on improving it.

The T-AGOS Rework Profile

The probability of rework onthe T-AGOS stern section will beassessed in terms of the laborrequired instead of by the actuallinear feet of rework (cutting andbackstrip welding) at theinterface. A focus on rework laborcan be an equally effective methodof monitoring accuracy performance,and projections of laborrequirements are more useful forpurposes of scheduling the buildsequence. The simulation programevaluates rework labor byallocating predetermined man-hourrates (per unit length), forcutting and backstrip welding, tothe optimum rework solutionsgenerated in the simulation.

For the T-AGOS simulation, a

labor rate of 0.25 man-hours perfoot for cutting and 0.58 man-hoursper foot for backstrip welding willbe used. These are hypotheticalvalues, and do not imply standardsof welding performance at TacomaBoat or any other shipyard. Thisrepresents a ratio of labor ratesof about 2.3, and since laborconstitutes the major elementcontributing to total rework costs,a cost ratio of 2.5 will be used todetermine the optimum reworksolutions.

4-9

. .

Figure 9 shows the distributionof rework labor for two separateruns of the simulation program.Both profiles are skewed to theright, though there are differencesin the details. The mode of theupper profile is at approximately28 man-hours, while that for thelower profile lies at around 24

man-hours. The labor averages,however, differ by only about 2%,at 22.7 and 22.2 man-hours,respectively. If a betterapproximation of the truedistribution is needed, it can behad by taking a greater sample sizein the simulation.

F

N

T-AGOS -Distribution of Rework Labor

AVERAGE LABOR:

22.7 man-hrs

STAND DEVIATION:

4.64 man-hrs

SAMPLES REWORKED100%

BackstripLabor:0.58 man-hrs/ft

Gas Cut Labor:

0.25man-hrs/ft

15 21 26 32.1

REWORK LABOR (man-hrs)

(200Samples,025” Gap Tolerance, Strip/Cost Ratio:2.50)

T-AGOS -Distribution of Rework Labor

AVERAGELABOR:

22.5 man-hrs

STAND.DEVIATION

4.86 man-hrs

SAMPLES REWORKED

100%

Backstrip Labor:

0.58 man-hrs/ft

Gas Cut Labor:

9.5 15 21 27 32.3REWORK LABOR(man-hrs)

(200Samples,025" Gap Tolcrance, Strip/cost Ratio:2.5O)

Fig.9. T-AGOS Labor Profiles fromTwo Separate Simulation Runs

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Having obtained an estimate ofthe anticipated rework on the T-AGOS stern block, the next step isto run a few more simulations toobserve how certain design changeswill affect the profile. The firstaxiom of quality control is theimportance of reducing variability.In the T-AGOS case, there areseveral ways of approaching theproblem. Figure 10 is the diagramof variations limits for the edgesat the block interface (theselimits come directly from thevariation table in Table 1). Thefigure shows that the edges inrelated group #1 - the 13’6” Flat,and the Centerline and diagonalframes - exhibit the greatestamount of variation, while theforward edges of the side shellsand bilge strakes have the leastvariation. A reduction in thesevariation limits would certainlyreduce variability. But sincethese are merged variations, thisimplies the need for either a

in Figure 10 in the misalignment ofthe six-sigma variation limits.Lining up the variation limits isaccomplished by normalizing all ofthe mean variations to a singlevalue. A merged mean variation canbe changed bysimply introducing an

"engineering variation”somewherein the build sequence - by, forinstance, telling the N.C. cuttingmachine to cut out a plate that isslightly longer than called for inthe drawing. This would change themean variation at the blockinterface without affecting thestandard deviation.

This strategy was tried out onthe simulation program. The T-AGOSvariation table was edited to bringall of the edges’ mean variationsto zero, and the new tabledesignated “T-AGOS(zero)." Theresults given in Figure 11, show areduction in average rework labor,but not by much. The improvementamounts to something between 2% and

Fig.10. T-AGOS Variation Limits

different assembly sequence or areduction in the process variationsthroughout the shipyard; neitherof which might be immediatelyavailable to the engineering staff.

The case does present, however,an element of variability that canbe very easily dealt with, and thisis that the merged mean variationsof the edges at the interface arenot all the same. This is evident

4% of the original average.Clearly, there is still muchimprovement to be gained through areduction of merged standarddeviations.

To evaluate the effect of ageneral reduction in standarddeviation, two more simulationswere run. The two new variationtables are called T-AGOS(90%) andT-AGOS(80%), reflecting an overall

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T-AGOS(zero) - Distribution of Rework Labor

10.6 16 21 27 32.1REWORKLABOR (man-hrs)

(200 Samples, 025” Gap Toleranoe, Strip/Cost Ratio: 2.50)

Fig.11. T-AGOS (zero ) Labor Profile

AVERAGELABOR:

22.7 man-hrs

STAND. DEVIATION:

4.57 man-hrs

SAMPLES REWORKD

100%

BackstripLabor:0.58man-hrs/ftGasCutLabor:0.25man-hrs/ft

reduction (from the original T-AGOS) of all the edges’ mergedstandard deviations by 10% and 20%,respectively. The results areshown in Figure 12. The 10% and20% reductions in standarddeviation produce around 5% and 9%reductions in average rework labor.

It is difficult, and probablyof little value, to try to comparethese two different approaches toreducing variability. Going from aT-AGOS to a T-AGOS(zero) is verysimple, once the merged variationsare understood, but the benefitsare limited. Getting from a T-AGOSto a T-AGOS(90%) may take manyyears of Accuracy Control work, butultimately there is much morepotential for economic reward alongthat path. Even though it allfalls under the heading of AccuracyControl, it appears that accuracyis relatively easy to achieve -it’s Precision that takes a lot ofwork.

6. STEPS TOWARD PRACTICALAPPLICATION

Sections 1 through 4 have beendevoted to developing a model ofmerged variation at the blockinterface, explaining the decisioncriteria for performing rework onthe interface, and introducing andtesting a simulation programwritten to predict the reworkoutcome on a given hull block,based on the assumptions in themodel. The program is shown to be

capable of producing useful output.Its graphical representations ofthe rework, cost, and labordistributions are easy tointerpret, giving the user a goodgrasp of the probabilitiesassociated with easy case.

Given all this, however, theprogram is still not ready forservice in a real application. Thevariation/rework model presentedhere contains several majorsimplifications, as is appropriatein early stages of research, whichneed to be addressed before theprogram is finally ready for use.This section presents a briefdiscussion on some of theseremaining issues, and sketches outwhat work is left to be done forthe refinement of the model and theimplementation of the simulationprogram

Choosing an Effective Sample Size

At several points in thisreport, the axiom, “the bigger thesample, the better theapproximation,"hashas been used to

acknowledge the topic of samplesize. The sample size of twohundred hull blocks, used in thesesimulations, was chosen fairlyarbitrarily. It is necessary,however, in an industrialapplication, to address morespecifically the questions of “howmuch” versus “how good, “ becausethe decisions have an economicconsequence.

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T-AGOS(90%) - Distribution of Rework Labor

0.15

ENcY

0.05

F 0.10RE

AVERAGELABOR:21.7 man-hrs

STAND. DEVIATION:

4.67 man-hrs

SAMPLES REWORKED

100%

Backstrip Labor:

0.58 man-hrs/ft

Gas Cut Labor:

0.25 man-hrs/ft

72 13 18 24 29.2REWORKLABOR(man-hrs)

(200Samples, 0.25” Gap Tolerance, Strip/Cost Ratio: 2.50)

T-AGOS(80%) - Distribution of Rework Labor

Backs-Labor:0.58Man-hrs/ftGasCutLabor:025man-hrs/ft

6.0 12 18 24 30.3REWORKLABOR(man-hrs)

(200 Samples, 025” Gap Tolrance, Strip/cost Ratio:2.50]

Fig.12. T-AGOS (90%) and T-AGOS (80%) Labor Profiles

How good will a prediction of “ about the sample mean, within which

average rework be for a given it can be stated (at a certain

sample size? Actually, the quality level of confidence) that the

of the prediction depends not only population mean lies. A 95%on sample size, but also on the confidence interval implies a 5%

profile and standard deviation of chance of error, or an "alphathe population. Statistically, the error” of 0.05.

best way to answer this sort ofquestion is in terms of aconfidence interval. A confidenceinterval is an interval, centered

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Assuming that the reworkprofile is a normal distribution(which it isn’t), then it is asimple matter to calculateconfidence intevals. The formulais:

where :p=population meanX=sample meanS=standard deviationn=sample size

=the standard normal valuewith an a/2 probability.

Applying this formula to thefirst T-AGOS simulation, with asample mean of 22.7 man-hours and astandard deviation of 4.45, a 95%confidence interval is calculatedto be: 22.7 k 0.62 man-hours, or

the confidence inteval is about 5%of the value it constrains. Table2 lists 95% confidence intervalsfor the T-AGOS case for samplesizes of 50, 100, 200, and 500.Since the rework function is not anormal distribution, these are onlyrough estimates, but they provideat least a basis for comparing thesize of the simulation with theaccuracy it delivers.

Table 2. Confidence Intervals forVarious Sample Sizes

95% Confidence

sample Size Intervals (man-hours)

50

100

200

500

Characterizing Merged Variation inThree Axes

In this paper, fluctuations inthe erection weld gap have beenattributed to merged variation atthe block interface only in thelongitudinal direction. Obviously,a constructed hull block willexperience some variation along thetransverse and vertical axes aswell, affecting the weld gap, andconsequently rework. This would

seem to imply that three orthogonalsets of variation merging equationsmust be written for each edge atthe interface to fully characterizeits impact on the rework function.A simulation program couldcertainly be written to accommodatethis, though at some point, theadded complexity of thecalculations may render the programunworkable on a mere personalcomputer.

It’s possible, however, thatsuch complete characterization isnot always necessary. An edge’scontribution to the rework functionmight be found to consist of onlytwo factors: its longitudinalvariation, and its perpendicularvariation. For instance, in thecase of a vertical bulkhead, thelongitudinal and transversevariations are the only relevantfactors; any vertical variationencountered will not affect theweld gap. Likewise, for ahorizontal deck, only itslongitudinal and vertical variationmight need be considered. Thevariation of obliquely angled edgeswould have to be characterized inall three directions, but even thiscase can be resolved to justlongitudinal and perpendicularvariation through a rotation ofcoordinate axes. Curved edges,unfortunately, are not amenable toany of this rationalization.

The nature of the erection weldjoint might also have a bearing onhow many axes of variation must beaddressed. This brings theadjoining block into consideration.If a weld joint is edge-to-edge,then the play (or rather, theinterplay) of both longitudinal andperpendicular variation willdetermine the weld gap. Dependingon welding technology, reworkcriteria may either remain in termsof overall weld gap tolerances, ordepend on the interrelated resultof a longitudinal gap and a planergap. On the other hand, if an edgeon the first block is to be weldedto the face of a bulkhead on thesecond, then the edge’sperpendicular variation is not acontributing factor to the qualityof the weld joint (though, granted,it may be of great concern to theAmerican Bureau of Shipping’sstrength requirements).

Considering Two-Sided Variation

Since the “adjoining block" hasentered the discussion again, it isan appropriate time to talk aboutanother shortcoming of our presentvariation/rework model. As it

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stands now, the simulation programassumes a model of one-sidedvariation, that is, variation ononly one of the blocks at theerection joint. But in reality,variation from both of the blockswill actually determine the reworkfunction. There are two ways thatthis can be addressed.

The first method is to revisethe rework program to simulate thevariation at the interface of bothof the blocks. Two variationtables would be loaded into theprogram instead of one, and thesimulation would begin by“building" two hundred blocks ofthe first type and two hundred ofthe second. Determination of theoptimum rework solution of eachcase would in principle be the sameas before, but would necessarilyaccount for the variation on bothsides of the weld joint. Insteadof moving a flat plane through theinterface of the one block, andevaluating in turn each possiblerework solution encountered, oneblock would be moved through theother, with the coincidence of eachpair of mating elementsrepresenting a possible reworksolution. At each out-of-tolerancejoint, it would be immaterial whichof the two edges actually receivedthe rework. The optimum solutionwould still be the one thatincurred the minimum cost. Thesubroutine to perform this taskwould be more complex than the onein current use, but still withinthe scope of a competentprogrammer.

A second method for modelingtwo-sided variation would, asopposed to the first, require norevision of the current program,and should yield an equivalentsolution. The plan involves"merging" the merged variation Ofmating elements at the interface tocreate a “two-sided variation table" that can be processed by thecurrent, one-sided model. This canalso be described as the action of“folding," or transferring, the

variation of the second block ontothe first block, therebymaintaining the model of one-sidedvariation. If an edge on one blockhas a mean variation of 0.25” and astandard deviation of 0.20", andits mating edge on the other blockhas a mean variation of -0.25” witha standard deviation of 0.30", thenthe combined effect wouldcorrespond to a one-sided meanvariation of zero, with a standarddeviation of 0.36”.

Assuming that the secondproposed method is equivalent tothe first, it would accomplish thesame task with much lesscomputational effort. Thereasoning seems intuitively sound,but at this time, a formal proof ofthe equivalence cannot bepresented. The moststraightforward test would be towrite two parallel simulationprograms, one for each method, andcompare the results.

Using Feedback to Improve TheSystem

No matter how complex the modelbecomes, it will always remain justan approximation of real life.Unforeseen factors, or inaccuraterepresentation of chosen factors,can bias the results of thesimulation. This is not to implythat the simulation program cannotbe a valuable tool, but it doessuggest a strategy for furtherimproving the quality of theprogram’s output. Once the systemis in place, recorded rework can becompared to the programrspredictions, to characterize theoverall accuracy of the model. Theconcept is similar to the analysisof residuals in a designedexperiment.

The error of each prediction -that is, the difference between theprojected and actual values - canbe determined for every erectionjoint. If the predicting errorsare normalized to (for instance) apercentage of the actual outcome,then they can all be plottedtogether to detect possible trends.There work prediction for oneinterface might be 10% high; forthe next interface, it might be 6%low. If there is no bias in themodel, then the average error willbe zero. If the model does containbias, then future simulationresults can be amended tocompensate for the averagepercentage error, and achieve amore accurate prediction. Themonitoring of error can also leadto an improvement of the modelitself, if it can point outspecific inaccuracies in thecurrent assumptions. The goal of acontinuously improvingmanufacturing system is facilitatedin part by a continuously improvingcontrol system.

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7. CONCLUSIONS

The purpose of this report hasbeen to show the capabilities ofcomputer simulation in predictingrework on ship’s hull blocks aterection. This simulation of therework function is made possiblebecause of two very powerfulconcepts that have effected greatchanges in shipbuilding technologyover the last few decades. Theseare Group Technology Manufacturingand Statistical Process Analysis.Group technology promotes therational organization of a largeproject into categories of similarwork packages, shifting focus fromthe building of ships to thebuilding of interim products.Statistical process analysis givesthe shipyard a direct understandingof its own manufacturingcapabilities, and at the same time,a Practical framework forcontinuously improving thosecapabilities.

This greater element of controlin shipbuilding technology permitsa characterization of the factorsthat lead to erection-stage rework.Random block variation at theerection interface is modeledthrough the writing of variationmerging equations. Rework for agiven hull block design is thefunction of this random variation,as well as several fixed factors.All of these factors can berepresented in a computersimulation. This reportdemonstrates the use and usefulnessof the author’s simulation programby applying it in the context of acase study. The significantfindings from the variation andrework studies, as well as thesimulation results, are summarizedbelow.

1. Rework on hull blocks isperformed to rectify the effects ofvariation of the edges at the blockinterface. The specific goal ofrework is to create a uniform weldgap at the erection interface bybringing all of the edges into thesame weld tolerance zone. Whenconsidering a given constructedblock, there are many reworksolutions through which theinterface can be made acceptable.The optimum rework solution is theone incurring the minimum cost,based on the four-way interactionbetween the resultant variation ofthe block’s various edges at theinterface, the lengths of theedges, the weld gap tolerance, andthe relative costs of rework.

2. As merged variation at theblock interface occurs randomly,the optimum rework solution isitself a random variable, having aunique probability distributionprofile. The rework simulationprogram, by modelling all of thefactors listed above, can samplefrom the “population” of hullblocks and generate an estimate ofthe rework distribution to anyaccuracy desired. The program alsoproduces estimates of the reworkcost and labor profiles, and therework probabilities of thespecific edges at the interface.

3. The characterization of therework function can be very usefulwhen writing schedules and budgetsfor the erection stage ofconstruction. The forecasts foreach of the ship’s blocks can beassessed during the design phase tolook for blocks with high reworkprobabilities, where design changesmight be needed. The estimate ofedge specific rework probabilitiescan identify when certain edges arecontributing an excessive amount torework levels at the interface.Such early detection of potentialproblems can help the shipyard toavoid costly disruptions in thebuilding schedule.

4. Overall projections ofrework levels for the entire shipcan be obtained by summing theindividual block projections. Themanagement can use overallprojections to evaluate theproducibility of the design, andthe product’s acceptability withrespect to the buyer’sexpectations. Preliminaryprojections may indicate alikelihood of cost or scheduleoverruns, in which case,negotiation can be initiated asearly as possible to reach the mostsatisfactory outcome.

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5. In addition to its value incosting and scheduling, thesimulation program can also be animportant tool for increasingproductivity. The program can beused to assess the impact ofproposed process improvements, suchas greater precision of certainmanufacturing operations, or anincrease in weld gap tolerance.With this information, operationsspending can be prioritized toyield the greatest impact for thedollar.

The program presented here isjust a demonstration model. Everyshipyard that elects to make use ofsuch a program will incorporateinto it the characteristics ofthose fabrication and reworkpractices that are unique to thatyard. It should evolve andimprover in reflection of theshipyard itself, becoming avaluable asset to future productioncapabilities.

8. REFERENCES

1. Wade, M., “Use of StandardTask Blocks to Simplify the ShipProduction Process,” Journal ofShip Production, VO1.2, No.2, May1986, PP.101-109.

2. Chirillo, L.D., “InterimProducts - An Essential Innovationin Shipyards, " Journal Of ShipProduction, Vol.1, No.3, Aug. 1985,pp.170-173.

3. Storchp R.L., “AccuracyControl: A Guide to itsApplication in U.S. Shipyards,"U.S. Department of Transportation,Maritime Administration, Feb. 1983.

4. Storch, R.L. and Giesy,P.J., "Three Dimensional AccuracyControl Variation MergingEquations, " U.S. Department of

Transportation, MaritimeAdministration, Report NO.MA-RD-76086031, Sept. 1986.

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Additional copies of this report can be obtained from theNational Shipbuilding Research and Documentation Center:

http://www.nsnet.com/docctr/

Documentation CenterThe University of MichiganTransportation Research InstituteMarine Systems Division2901 Baxter RoadAnn Arbor, MI 48109-2150

Phone: 734-763-2465Fax: 734-763-4862E-mail: Doc.Center@umich.edu