A COST OPTIl\fiSED PROCESS MEAN SET POINT FOR TWO SIDED SPECIFICATIONS

This paper describesthe derivationof cost minimisingexpressions to optimallyset theprocess meanof a manufacturing processrestrictedby a double-sidedspecification. Two scenariosare considered. For the first scenario,multiplereworkingiterationsare possible,while in the second,only one rework opportunityis allowed. A numericalexampleis also presented. Resultswere obtainedby numerical solutionand are presentedin graphicalformat. OPSOMMING Hierdieartikelbeskryfdie afleidingvan koste-minimerende uitdrukkings wat die proses-gemiddelde van 'n vervaardigingsproses wat deur 'n twee-kantige spesifikasie beperk word,optimaaldaar stel. Twee gevalleword oorweeg. In die eerstegeval is verskeieherwerk-iterasies moontlik,terwyldie tweede gevalslegs een herwerk-geleentheid aanvaar. 'n Numeriesevoorbeeldwordook bespreek. Resultate is deur numerieseanaliseverkryen word grafiesvoorgestel. 87 http://sajie.journals.ac.za


INTRODUCTION
Many manufacturing processes,when under statistical control, exhibitfixed probabilities connectedto the incidence of both scrap and rework.In a machiningprocess,scrap is usually associatedwith too much materialhaving been removed,and reworkis associatedwith not enough machininghavingbeen done.The presenceof finishing defectsis also related to rework.
Definitecostsare associatedwith the occurrenceof both scrap and rework and these costs are normally not the same.For a given processdispersionand technicalspecification, it remains withinthe control of the operations managerto set the processmean to such a value that the total cost of scrap and reworkwill; be a minimum.The need and importanceof this are widelyappreciated, e.g.Grant [5].
The aboveproblemis addressedin this paper and two production scenariosare considered.

LIST OF SYMBOLS USED
Standarddeviationof processoutputfor a processassumedto be under statistical control

DESCRIPTION OF THE TWO PRODUCTION SCENARIOS CONSIDERED
The probabilitiesfor the occurrenceof scrap and rework in a manufacturingprocess that is fairlywell under control, would remain largely constantover time.For such a manufacturingprocess, let the probabilityof a part being scrapped at a particularstage of manufacturebe P s , and let the probabilityof it requiringrework at that stage, be Pro Also, set the sum of the material cost and the cost of productionup to and includingthat point, at C s , and that of rework at that specific stage in the manufacturingprocess,to an average of C,..A reworked item is essentiallyrecycled in the manufacturingprocess.In the first instance considered (scenarioA), multiplereworking iterationsare allowed.In the interest of simplicity,it is assumed that the number of times an item has been reworked previously, has no effect on the probability of it requiring reworkingat the end of the next cycle or productionstage.In the second instance(scenarioB), only one rework opportunityis allowed.The transition-state diagrams of the two scenariosare illustratedin figures (la) and (lb) respectively.

Figure la
When consideringscenario A (refer to figure la), the number of good items Nbeing produced in a run of Iunits, is: The populationI of units requiredto be manufacturedin order to produce N good units is by rearrangement of the above terms: (la) A: I =N.( l-P r J l-P r -P s On the other hand, when consideringscenarioB, the number of good items Nbeing produced in a run of I units,is: for r,« 1 The populationI of units requiredto be manufacturedin order to produce N good units is by rearrangement of the above terms:

TOTAL PRODUCTION COSTS FOR THE TWO SCENARIOS (2a)
The expressions derivedin the previous sectioncan be used to calculatethe total productioncost, TPC, to produceN good units for scenariosA and B respectively. A: where I .(~J= the total number of rework cycles necessaryto produce N good items. 1-Pi-'. B:

TPC =I.C s +I,Pr,C r
In terms of equations(la) and (lb), the above expressionscan be written as: By dividingthe former equationsby N, the productioncost per good unit is obtained as: A: Hence, the problem is to arrive at a set point for the process mean relativeto the nominal specification that would minimisethe TPCPU.Before proceedingto minimisethe total unit cost given by equations B: (2a) and (2b), it is useful to simplify these equations by dividing the TPCPUby C r (divisionor multiplicationby a constant will not in any way affect the value of the dependent variable for which TPCPU will be a minimum) and setting the cost ratio to:

(~:) ~C
This simplificationleads to the following two expressionsthat represent relative measures of the total productioncost per unit.
A: TRPCPU = C-Pr ,C+Pr WithP, and P; arbitrarilychosen as illustratedin figure (2), with scrap being associated with low values of the quality characteristicand rework with high values thereof, it is evident that: The position of the process mean relative to the nominal specification(i.e. the midpoint of the design specification)is given by: with F x( x; u ) representingthe cumulativedistributionfunction(c.df) of the process outputx.The offset Y = p-N .S will therefore be negative when the process mean is set below the nominal specificationand positive otherwise.x Also, the position of the USL and the LSL relative to the process mean may be stated as: A: B: where u> process mean, W= specificationwidth and Y= differencebetween the processmean and the nominal specification.
In order to simplifyand standardisethe relationshipsthat follow, it is useful to express the mean-offset , Y, and specification width, W, as multiplesof the process dispersion, 0; by defining: The special instance of the normal distributionleads to the following four expressionsthat apply to both scenarios: (10) Finally, substitutingequations (10) into equations (8a) and (8b), the following expressions are obtained from which the optimum process location,y, can be determined iterativelyfor the two scenarios: A: Figures (3a) and (3b) are graphs of the offset,y, vs. the specificationwidth, w, for the cost ratio, C, equal to the values 5, 10, 20, 50 and 100 respectively.The values for the different cost ratios indicatethat the cost of scrappingthe part is equal to five times the rework cost, ten times the rework cost, twenty times the rework cost, and finally one hundred times the rework cost.The first instance applies early in the manufacturingprocess and the latter towards the end after numerous stages.
The C p index that measurespotential or inherent capabilityof the production process (assuming a stable process) is also shown for both scenarios in figures (3a) and (3b).This index is defined as: --!-r-, "'-c-r---:....!-'---r=b ----=...." ""' " ...::: '----.. -- From figures (3a) and (3b) it is evident that the narrower the specificationwidth and the greater the cost ratio C = CS (i.e. the lower the relative cost of rework), the greater the offset is required to be in C r absolute and especiallyin relative terms, i.e. in comparisonwith the specificationwidth.This is the situationafter numerousproduction stages, when the total value added to the part becomes high.In other words, C, is a cumulativecost that adds up as the item progresses through the various manufacturing stages or operations.On the other hand, Cr,is the average rework cost per rework operation.Therefore, the cost ratio, C =~, will increase as the item proceeds through its various manufacturing operations."""'b1

A SPECIAL SIMPLIFlED CASE
The theoryabovemay be simplifiedfor both scenarioswhen appliedto a processthat normallygenerates smallpercentages of scrap and rework.To illustrate, equations(3a) and (3b) may be rewrittenas follows: A: B:

f{P-Y-: ;p}
For the specialcase of the normal distribution the optimalprocessmean offsetis given by: Equation( 13) allowsy (and J:? to be calculatedexplicitly, but representsan approximation valid onlyfor smallprobabilities for work being out of specification.The practicalvalue of this simplification is questionable andprobablylimited,becausefor smallP, andP, (a processwith good capability), there is littleneed to adjustthe processmean for optimality.

NUMERICAL EXAMPLE Problem statement:
A steelshaftthat formspart of the armatureassemblyof an electricalmotor requiresvarioussuccessive machining operations before attainingits finalform and dimensions.Duringone of the operations where the shaftis machinedon a centrelathe, the outsidediameteris machinedaccordingto the following tolerance specification: 10.00mID ± 0.05 mID (with the process outputbeing normallydistributed).
Historical data for this operationhas indicateda poor capabilityindex Cp of only 0.70,partly due to the age and physicalconditionof the lathe..Shaftswith diametersbelow 9.95 mID must be scrapped,while thosewith diameters above 10.05mID must be reworkedto bring them withinthe specifiedtolerance range.
The raw material from which the shaft is made, costsR 30.00, and the labour and overheadcosts,up to and includingthis particularoperation,amountsto R 60.00.The average cost to rework a shaft on the centre latheis estimatedat R 10.00.

Solution:
From the problem statement,the value of the cost componentC,=R 30 + R 60 =R 90.
It shouldbe notedthat in the mathematical formulation of the problem,scrap was assumedto be items with smallervaluesof the qualitycharacteristic than rework.Under this assumption, the optimalprocess setpointwill alwayshave positiveoffsets(positive y-values).However,ifscrap is generatedat high

x
Location(constantmean) of processwhich is assumedto be under statistical control Cumulativedistribution function(c.d..f) ofa standardnormal random variablex Ratio of materialcost plus originalmanufacturing cost to that of rework Sum of materialand productioncost per unit up to the currentmanufacturing operation Index that measurespotentialor inherentcapabilityof the productionprocessassuminga stableprocess Reworkcost per unit for the currentmanufacturing operation Probability densityfunction(p.d.f.)of processoutputwith mean set at a value of Il Cumulativedistribution function (c.df) pfprocessoutput with mean set at a valueof Il Size of a productionlot that would ensure-the productionofN good items Lower limit of design specificat ion Numberof good items out of a productionlot of size I Nominalspecification i.e. midpointof design specification Probability of scrap at a givenmachinesetting Probability of rework at a given machinesetting Total productioncost to makeN good items Total productioncost per good item that is manufactured Totalrelevantproductioncost per good item that is manufactured Upper limit of designspecification Specification width (in relativeterms):expressedas a number of standarddeviationsof processoutput Specification width (in absoluteterms) Valueofa qualitycharacteristic ofa manufactured unit http://sajie.journals.ac.zaX Random variablethat describesthe value of a quality characteristic of a manufactureditem y Offset of process mean relative to nominal specification(in relative terms): expressed as number of standard deviationsof process output Y Offset of process mean relative to nominal specification(in absolute terms): i.e.Ji-NS.

5 .
Figure(2) depicts the case where the output of a manufacturingprocess follows any general probability density function I x(x; p) with process mean u.Acceptableproduction units are considered those units that would fall within the upper and lower specificationlimits denoted by USL and LSL respectively.