Quality Insights: Measurement and quality rationing: an analytical approach. Adedeji B. Badiru* and Anna E. Maloney

Transcription

1 Int. J. Qualty Engneerng and Technology, Vol. X, No. Y, xxxx Qualty Insghts: Measurement and qualty ratonng: an analytcal approach Adedej B. Badru* and Anna E. Maloney Ar Force Insttute of Technology, 2950 Hobson Way, Wrght Patterson AFB, OH , USA Emal: Emal: *Correspondng author Abstract: Measurement s as mportant n qualty management as t s n many aspects of human endeavours. Take, for example, a busness that s attemptng to formulate a yearly budget. Ths busness cannot create a budget out of thn ar; t must use qualtatve and quanttatve assessments based on nterpretatons and equatons. The exact methods busnesses use to formulate ther budgets may vary, but they should all nclude statstcal analyss n order to acheve accuracy and precson. Budget management and qualty management are analogous wth respect to the applcaton of analytcal technques to make decsons. Ths paper captalses on that relatonshp to apply budget allocaton and ratonng technques to qualty. Ths paper presents an analytcal approach to measurement and qualty ratonng n order to meet qualty goals. Qualty on a scale of measurement s the premse of ths paper. Keywords: measurement; qualty management; qualty control; captal ratonng; budgetng; qualty modellng. Reference to ths paper should be made as follows: Badru, A.B. and Maloney, A.E. (xxxx) Qualty nsghts: Measurement and qualty ratonng: an analytcal approach, Int. J. Qualty Engneerng and Technology, Vol. X, No. Y, pp Bographcal notes: Adedej B. Badru s a Professor of Systems Engneerng at the Ar Force Insttute of Technology (AFIT). He s a regstered professonal engneer (PE), a certfed Project Management Professonal (PMP), and a fellow of the Insttute of Industral Engneers. He receved hs BS n Industral Engneerng, MS n Mathematcs and MS n Industral Engneerng from Tennessee Technologcal Unversty, and PhD n Industral Engneerng from the Unversty of Central Florda. He s the author of several books and journal artcles. Anna E. Maloney s an ntern at the Graduate School of Engneerng and Management at the Ar Force Insttute of Technology. She s currently attendng The Oho State Unversty, where she s pursung degrees n nternatonal securty and ntellgence and Russan. Copyrght 20XX Inderscence Enterprses Ltd.

2 2 A.B. Badru and A.E. Maloney Introducton Statstcs, whether n qualtatve or quanttatve form, s the foundaton for accuracy and precson of measurements. Ths appendx presents a collecton of useful statstcal defntons, explanatons, nterpretatons, llustratons, examples, formulatons, formulas, and equatons. Usng the tools and technques mentoned n ths paper, readers can: dentfy and quantfy the sources of measurement varablty assess the effect of the measurement system varablty on process varablty dscover opportuntes for measurement system and total process varablty mprovement. Measurement pervades everythng we do. Ths apples to techncal, management, and socal actvtes and requrements. Even n ordnary stuatons, such as human lesure, the mportance of measurement comes to the surface. How much, how far, how good, how fast, and how long are typcal connotatons of measurement. Throughout hstory, humans have strved to come up wth better tools, technques, and nstruments for measurement. From the very ancent tmes to the present fast-paced socety, our search for more precse, more convenent, and more accessble measurng devces has led to new developments over the years. Ths pursut of better measurements s partcularly amenable to the management of qualty n products and servces. Because t s not cost-effectve to overdesgn qualty nto everythng for the sake of provdng a safeguard, we must fnd analytcal ways to determne when qualty s enough. In other words, what level of qualty s achevable, practcal, and acceptable for the msson at hand? Ths paper uses the analytcal approach of captal ratonng to suggest how qualty ratonng can be acheved n products and servces. Any resource (.e., captal) needed to mpact ncremental qualty onto a product s the same resource that s needed n other aspects of the busness enterprse. Thus, the analogy of captal ratonng of qualty s applcable. 2 Measurement scales for qualty Every decson requres data collecton, measurement, and analyss. In practce, we encounter dfferent types of measurement scales dependng on the partcular tems of nterest. Data may need to be collected on decson factors, costs, performance levels, outputs, and so on. The dfferent types of data measurement scales that are applcable for qualty assessment nclude nomnal scale, ordnal scale, nterval scale, and rato scale. Nomnal scale s the lowest level of measurement scales. It classfes tems nto categores. The categores are mutually exclusve and collectvely exhaustve. That s, the categores do not overlap and they cover all possble categores of the characterstcs beng observed. For example, n the analyss of the crtcal path n a project network, each job s classfed as ether crtcal or not crtcal. Gender, type of ndustry, job classfcaton, and colour are examples of measurements on a nomnal scale.

3 Qualty Insghts: Measurement and qualty ratonng 3 Ordnal scale s dstngushed from a nomnal scale by the property of order among the categores. An example s the process of prortsng project tasks for resource allocaton. We know that frst s above second, but we do not know how far above. Smlarly, we know that better s preferred to good, but we do not know by how much. In qualty control, the ABC classfcaton of tems based on the Pareto dstrbuton s an example of a measurement on an ordnal scale. Interval scale s dstngushed from an ordnal scale by havng equal ntervals between the unts of measurement. The assgnment of prorty ratngs to project objectves on a scale of 0 to 0 s an example of a measurement on an nterval scale. Even though an objectve may have a prorty ratng of zero, t does not mean that the objectve has absolutely no sgnfcance to the project team. Smlarly, the scorng of zero on an examnaton does not mply that a student knows absolutely nothng about the materals covered by the examnaton. Temperature s a good example of an tem that s measured on an nterval scale. Even though there s a zero pont on the temperature scale, t s an arbtrary relatve measure. Other examples of nterval scale are IQ measurements and apttude ratngs. Rato scale has the same propertes of an nterval scale, but wth a true zero pont. For example, an estmate of zero tme unt for the duraton of a task s a rato scale measurement. Other examples of tems measured on a rato scale are cost, tme, volume, length, heght, weght, and nventory level. Many of the tems measured n engneerng systems wll be on a rato scale. An mportant aspect of measurement nvolves the classfcaton scheme used. Most systems wll have both quanttatve and qualtatve data. Quanttatve data requre that we descrbe the characterstcs of the tems beng studed numercally. Qualtatve data, on the other hand, are assocated wth attrbutes that are not measured numercally. Most tems measured on the nomnal and ordnal scales wll normally be classfed nto the qualtatve data category whle those measured on the nterval and rato scales wll normally be classfed nto the quanttatve data category. The mplcaton for engneerng system control s that qualtatve data can lead to bas n the control mechansm because qualtatve data are subject to the personal vews and nterpretatons of the person usng the data. As much as possble, data for qualty management and control should be based on a quanttatve measurement. Fgure shows the basc measurement characterstcs for qualty assessment. Precson, accuracy, correlaton, stablty, and lnearty are essental for the purpose of determnng when qualty s adequately algned and suffcent for the specfc purpose of nterest. Fgure 2 llustrates a measure of accuracy bas, n whch the followng equatons apply: μ = μ + μ σ σ σ total product measurement system 2 total = 2 product + 2 measurement Bas s the dfference between the observed average value of measurements and the true value. The true value s an accepted, traceable reference standard. The accuracy of the measurement system s determned by conductng a bas study. Fgure 3 shows graphcal representaton of lnearty. Lnearty s a measure of the dfference n accuracy or precson over the range of measurement nstrument capablty.

4 4 A.B. Badru and A.E. Maloney Fgure Basc measurement characterstcs for qualty assessment Precson True Values Measured Values Tme Tme 2 Accuracy True Values Measured Values Stablty Bas Measurement offset 0 Correlaton Method 2 No Offset Method Measurement Lnearty Fgure 2 Measure of accuracy bas Master Value (Standard Reference) Average Value Fgure 3 Assessment of lnearty (see onlne verson for colours) Gage : Lnearty s an ssue here Bas Gage 2: Lnearty may be an ssue here Bas Bas Gage 3: Lnearty s NOT an ssue here Measurement Measurement Measurement

5 Qualty Insghts: Measurement and qualty ratonng 5 3 Qualty captal allocaton For the purpose of ths secton, consder qualty desgn as a budgetng exercse. Budgetng nvolves sharng lmted resources among several project groups or functons contaned n a project. A budget analyss can serve as any of the followng: a plan for resources expendture a product selecton crteron a projecton of qualty polcy a bass for qualty control a performance measure for the organsaton a standardsaton of resource allocaton an ncentve for qualty mprovement. Top-down budgetng nvolves collectng data from upper-level sources such as top and mddle managers. The fgures suppled by the managers may come from ther personal judgment, past experence, or past data on smlar project actvtes. The cost estmates are passed to lower-level managers, who then break the estmates down nto specfc work components wthn the project. These estmates may, n turn, be gven to lne managers, supervsors, and lead workers to contnue the process untl ndvdual actvty costs are obtaned. Top management provdes the global budget, whle the functonal-level worker provdes specfc budget requrements for project tems. In ths method, elemental actvtes and ther schedules, descrptons, and labour skll requrements are used to construct detaled budget requests. Lne workers famlar wth specfc actvtes are asked to provde cost estmates. Estmates are made for each actvty n terms of labour tme, materals, and machne tme. The estmates are then converted to an approprate cost bass. The dollar estmates are combned nto composte budgets at each successve level up the budgetng herarchy. If estmate dscrepances develop, they can be resolved through the nterventon of senor management, mddle management, functonal managers, project manager, accountants, or standard cost consultants. Elemental budgets may be developed on the bass of the tmed progress of each part of the project. When all the ndvdual estmates are gathered, a composte budget can be developed. Such analytcal tools as learnng curve analyss, work samplng, and statstcal estmaton may be employed n the cost estmaton and budgetng processes. 4 Mathematcal formulaton of qualty captal allocaton Captal ratonng nvolves selectng a combnaton of projects that wll optmse the on nvestment (Badru et al., 202; Seger et al., 2000). A mathematcal formulaton of the captal (qualty) budgetng problem s presented below:

6 6 A.B. Badru and A.E. Maloney Maxmse z = Subject to n = n = cx v x x = 0, ; =, K, n B where n number of product characterstcs v measure of performance for the product characterstc c cost of product characterstcs x ndcator varable for product characterstc B budget (qualty) avalablty level. A soluton of the above model wll ndcate what product characterstc levels should be selected n combnaton wth other product characterstcs. The example that follows llustrates a qualty ratonng problem. 5 Qualty ratonng model Plannng a portfolo of products s essental n resource-lmted producton system. The captal-ratonng example presented here demonstrates how to determne the optmal combnaton of product qualty nvestments so as to maxmse total on nvestment (.e., product qualty value). Let us thnk of each qualty level opton as a specfc project opton. Suppose a statstcal analyst s gven N projects, X, X 2, X 3,,X N, wth the requrement to determne the level of nvestment n each project so that total nvestment s maxmsed subject to a specfed lmt on avalable budget. The projects are not mutually exclusve. The nvestment n each project starts at a base level b ( =, 2,,N) and ncreases by varable ncrements k j (j =, 2, 3,,K ), where K s the number of ncrements used for project. Consequently, the level of nvestment n project X s defned as follows: K j x = b + k j= where x 0, For most cases, the base nvestment wll be zero. In those cases, we wll have b = 0. In the modellng procedure used for ths problem, we have X f the nvestment n project s greater than zero = 0 otherwse

7 Qualty Insghts: Measurement and qualty ratonng 7 and th f j ncrement of alternatve s used Yj = 0 otherwse The varable x s the actual level of nvestment n project, whle X s an ndcator varable ndcatng whether or not project s one of the projects selected for nvestment. Smlarly, k j s the actual magntude of the j th ncrement whle Y j s an ndcator varable that ndcates whether or not the j th ncrement s used for project. The maxmum possble nvestment n each project s defned as M, such that b x M. There s a specfed lmt, B, on the total budget avalable to nvest, such that x B. There s a known relatonshp between the level of nvestment, x, n each project and the expected, R(x ). Ths relatonshp wll be referred to as the qualty utlty functon, f(.), for the project. The utlty functon may be developed through hstorcal data, regresson analyss, and forecastng models. For a gven project, the utlty functon s used to determne the expected, R(x ), for a specfed level of nvestment n that project. That s, R( x) = f ( x) K = ry j j j= where r j s the ncremental obtaned when the nvestment n project s ncreased by k j. If the ncremental decreases as the level of nvestment ncreases, the utlty functon wll be concave. In that case, we wll have the followng relatonshp: r r or r r 0. j j+ j j+ Thus, Y Y or Y Y 0. j j+ j j+ So that only the frst n ncrements (j =, 2,,n) that produce the hghest s are used for project. Fgure 4 shows an example of a concave nvestment utlty functon. If the ncremental s do not defne a concave functon, f(x ), then one has to ntroduce the nequalty constrants presented above nto the optmsaton model. Otherwse, the nequalty constrants may be left out of the model, snce the frst nequalty, Y j Y j+, s always mplctly satsfed for concave functons. Our objectve s to maxmse the total. That s, Maxmse Z ry j j = j

8 8 A.B. Badru and A.E. Maloney Subject to the followng constrants: x = b + k Y j j j b x M Yj Yj+, j x B j x 0 Y = 0or, j Now, suppose we are gven four projects (.e., N = 4) and a qualty budget lmt of $0 mllon. The respectve nvestments and s are shown n Table, Table 2, Table 3, and Table 4. Fgure 4 Utlty curve for nvestment yeld R(x) curve h4 Return h3 y y2 y3 y4 Table Investment data for Project for captal ratonng Stage (j) y j x r j R(x ) nvestment Level of nvestment Total

9 Qualty Insghts: Measurement and qualty ratonng 9 Table 2 Investment data for Project 2 for captal ratonng Stage (j) y 2j x 2 R 2j R(x 2 ) nvestment Level of nvestment Total Table 3 Investment data for Project 3 for captal ratonng Stage (j) y 3j x 3 r 3j R(x 3 ) nvestment Level of nvestment Total Table 4 Investment data for Project 4 for captal ratonng Stage (j) y 4j x 4 r 4j R(x 4 ) nvestment Level of nvestment Total

10 0 A.B. Badru and A.E. Maloney All of the values are n mllons of dollars. For example, n Table, f an ncremental nvestment of $0.20 mllon from stage 2 to stage 3 s made n Project, the expected ncremental from the project wll be $0.30 mllon. Thus, a total nvestment of $.20 mllon n Project wll yeld a total of $.90 mllon. The queston addressed by the optmsaton model s to determne how many nvestment ncrements should be used for each project. That s, when should we stop ncreasng the nvestments n a gven project? Obvously, for a sngle project, we would contnue to nvest as long as the ncremental s are larger than the ncremental nvestments. However, for multple projects, nvestment nteractons complcate the decson so that nvestment n one project cannot be ndependent of the other projects. The LP model of the captal-ratonng example was solved wth LINDO software. The soluton ndcates the followng values for Y j. Project : Y =, Y2 =, Y3 =, Y4 = 0, Y5 = 0 Thus, the nvestment n Project s X = $.20 mllon. The correspondng s $.90 mllon. Project 2: Y2 =, Y22 =, Y23 =, Y24 =, Y25 = 0, Y26 = 0, Y27 = 0 Thus, the nvestment n Project 2 s X2 = $3.80 mllon. The correspondng s $6.80 mllon. Project 3: Y3 =, Y32 =, Y33 =, Y34 =, Y35 = 0, Y36 = 0, Y37 = 0 Thus, the nvestment n Project 3 s X3 = $2.60 mllon. The correspondng s $5.90 mllon. Project 4: Y4 =, Y42 =, Y43 = Thus, the nvestment n Project 4 s X4 = $2.35 mllon. The correspondng s $3.70 mllon. The total nvestment n all four projects s $9,950,000. Thus, the optmal soluton ndcates that not all of the $0,000,000 avalable should be nvested. The expected from the total nvestment s $8,300,000. Ths translates nto an 83.92% on nvestment. The optmal soluton ndcates an unusually large on total nvestment. In a practcal settng, expectatons may need to be scaled down to ft the realtes of the project envronment. Not all optmsaton results wll be drectly applcable to real stuatons. Possble extensons of the above model of captal ratonng nclude the ncorporaton of rsk and tme value of money nto the soluton procedure. Rsk analyss would be relevant, partcularly for cases where the levels of s for the varous levels of nvestment are not known wth certanty. The ncorporaton of tme value of money would be useful f the nvestment analyss s to be performed for a gven plannng

11 Qualty Insghts: Measurement and qualty ratonng horzon. For example, we mght need to make nvestment decsons to cover the next fve years rather than just the current tme. 6 Conclusons As mentoned at the begnnng of ths paper, budgetng and captal ratonng processes convey mportant messages about the nvestment potental of a project. The technques presented here offer addtonal tools for assessng how and where lmted resources should be drected. The outputs of the computatonal analyss can serve as a plan for resource expendture on qualty pursut, a qualty targetng crteron, a projecton of qualty polcy, a bass for qualty control, a performance measure, a standardsaton of qualty goal, and an ncentve for mprovement of organsatonal practces. References Badru, A.B., Oye, I-O. and Ayen, B.J. (202) Industral Control Systems: Mathematcal and Statstcal Models and Technques, Taylor & Francs CRC Press, Boca Raton, FL. Seger, D., Badru, A.B. and Mlatovc, M. (2000) A metrc for aglty measurement n product development, IIE Transactons, Vol. 32, No. 4, pp