Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, Cotour Approac for Aalysis of Miimum Regios for te Ecoomic Statistical Desig of Xbar Cotrol Carts Satiago-Omar Caballero-Morales Logistics ad Supply Cai Maagemet Uiversidad Popular Autóoma del Estado de Puebla, UPAEP Puebla, México scaballero99@yaoo.com Abdur Raim Faculty of Busiess Admiistratio Uiversity of New Bruswick Fredericto, Caada raim@ub.ca Abstract Visualizatio of te patters or beaviors of complex fuctios is a importat approac to idetify teir solutio spaces. Tis approac ca lead to improve te desig of searc algoritms for optimizatio purposes. tis paper, a D visualizatio metod based o miimum cost scores (MCS) is preseted to idetify te most likely regios to fid optimal values for te ecoomic statistical desig of X-bar cotrol carts. Uiform samplig was cosidered for te aalysis of te cost fuctio of Raim ad Baerjee (99). Elliptical regios were foud to model te miimum regios defied by MCS values estimated for tis cost fuctio troug differet values of sample size (), legt of te samplig iterval () ad coefficiet of te cotrol cart's limits (L). Assessmet of tis metod was performed by fidig te optimal set of ad values witi tese elliptical regios. Optimal values reported i te literature wit Geetic Algoritms (GA) were cosidered for tis case. t was observed tat te optimal values were located witi te boudaries of te elliptical regios ad were associated wit sub-regios wit te igest MCS values. Tis cofirms te suitability of tis approac to obtai te a-priori estimatio of te solutio space. Keywords cotour plots; ecoomic-statistical desig; D visualizatio. NTRODUCTON Visualizatio is a importat approac to uderstad te complex relatiosips betwee variables ad objective fuctios to improve data aalysis [-]. For combiatorial problems, visualizatio ca improve te uderstadig of te static ad dyamic caracteristics of teir solutio spaces [,]. As a cosequece, visualizatio ca be applied to uderstad te searc mecaisms of solvig algoritms ad improve teir performace [-]. tis work, te ecoomic statistical desig (ESD) problem for X-bar cotrol carts is cosidered. A cotrol cart is defied by tree mai parameters: te sample size (), te legt of te samplig iterval () ad te coefficiet of te cart's cotrol limits (L). Te ESD combiatorial problem cosists i fidig te optimal values for, ad L, tat globally miimize costs (modeled by a cost fuctio) cosiderig ecoomic ad statistical restrictios [9]. Previously, full tree-dimesioal (D) visualizatios were performed to uderstad te dyamics of cost fuctio models wit geeral failure distributio [] ad o-ormality [] for te ESD problem. tis work, a geeral cotour approac is proposed to idetify bi-dimesioal (D) miimum regios for te ESD problem. Te estimatio of miimum cost scores (MCS) is defied to obtai te D visualizatio of te miimum regios. t is expected tat tis approac ca lead to improvemets o te searc mecaisms of solvig algoritms to obtai te optimal cart's parameters. cotrast to a D visualizatio [,] te proposed MCS D visualizatio ca be computed faster, improvig its practical use. Te preset paper is structured as follows: i Sectio a overview of te cost fuctio model for te ESD of X-bar cotrol carts is preseted. Tis overview icludes a descriptio of referece data sets for te assessmet of te MCS D visualizatio approac wic is described i Sectio. Te, i Sectio V te cotour-based MCS D visualizatios are preseted ad discussed for eac referece data set. Fially, i Sectio V te coclusios ad future work are preseted. EOM Society teratioal 9
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, COST FUNCTON MODEL. tis work te cost fuctio model of Raim ad Baerjee [9] wit Gamma failure distributio ad uiform samplig was cosidered. Te objective fuctio of tis cost model cosists o miimizig F(,,L) = E(C)/E(T), were E(T) is te Expected Cycle Legt, ad E(C) is te total Expected Cost per Cycle [9,]. Tese costs are defied as follows: α α β β, () β β, () ad β were α Φ Φ δ Φ δ. f α ad represet te maximum ad miimum values for te type ad type error probabilities respectively, te miimizatio of F(,,L) is subject to te followig restrictios: α α, β,, ad. Costat elemets of te cost fuctio idetify time ad ecoomic costs tat are described as follows [9,]: Z is te expected searc time associated to a false alarm (false positive detectio of te "out-of-cotrol" state) wile Z is te expected searc ad repairig time associated to a true alarm. a ad b are te fixed cost per sample ad te cost per uit sample respectively. D is te expected productio cost (per our) of ocoformig items if te process is "i cotrol" state. D is te expected productio cost of ocoformig items if te process is i "out-of-cotrol" state (D > D). W is te expected cost of searcig ad repairig a assigable cause of failure (restorig te process to "i cotrol" state). Y is te expected cost of a false alarm (false positive detectio of te "out-of-cotrol" state). Fially, δ is te detected sift size of te process ad is te scale parameter of te failure distributio wic (i tis case) is Gamma (, ). A. Referece Data Sets As preseted i () ad () te cost model cosiders a total of idepedet variables: Z, Z, D, D, W, Y, a, b, δ, αu, pl, ad. order to obtai a geeral overview of te beavior of te cost fuctio model, sets of value levels for tese variables were cosidered. Tese value levels were selected accordig to te data preseted i [] were represetative combiatios of tese levels were defied by a L ortogoal array. Table presets te value levels defied for eac idepedet variable wile Table presets te L ortogoal array []. TABLE. VALUE LEVELS FOR TE NDEPENDENT VARABLES OF TE COST FUNCTON MODEL Variable Level Level Level Z... Z.. D D 9 9 W Y a b... δ... αu... pl..9.9... EOM Society teratioal 9
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, [] te assessmet of PSO (Particle Swarm Optimizatio) was preseted as a promisig metod for solvig te ESD problem for X-bar cotrol carts uder () ad (). Performace of GA (Geetic Algoritms) was preseted for compariso purposes. Tis was importat for te preset work as te results reported i [] could be used for assessmet of te regios described by te proposed D visualizatio metod. ece, Table icludes te best,, ad L values estimated by te GA metod for eac combiatio of te L ortogoal array []. Tese referece data sets are te basis for te discussio preseted i Sectio V. TABLE. ORTOGONAL ARRAY AND BEST PARAMETERS FOR TE CONTROL CART WT GENETC ALGORTMS (DATA OBTANED FROM []) Trail Z Z D D W Y a b δ αu -β L E(C)/E(T) 9 9...................................................... 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9...................................................................................9.9.9..9.9.9...9.9.9..9.9.9...9.9.9..9.9.9............................ 9 9 9 9...9.......9..9.9 9.9.9..9..9.9.........9..9.9....9.9..9..9.9.9....9..9..9...9.9..9..9 9.... 9..9. 9...9 9. 9...... 9... 9. 9.9. TE MCS D CONTOUR ALGORTM Te followig steps were performed to process te MCS D visualizatio data for te cotour plots of te miimum regios for F(,,L) = E(C)/E(T): ) itialize te rage vectors for te differet values of,, ad L.For tis case, differet values witi te rage {, } ad {, } were cosidered for ad respectively. Tese values were stored i te vectors rag_ ad rag_. For L, differet values were cosidered. Tese were stored i te vector rag_l. ) Estimate E(C)/E(T) for eac set of,, ad L values defied by rag_, rag_ ad rag_l. Tis was performed as follows: a) For eac set of,, ad L values, E(C)/E(T) was estimated. Tis led to create a square matrix of E(C)/E(T) values for a particular L value. b) Eac E(C)/E(T) matrix is stored witi a tree-dimesioal array called costs. Wile te x ad y axes are associated to te values of rag_ ad rag_ respectively, te z-axis is associated to te values of rag_l. ) Normalize te values witi te array costs to pixel values. Tis was performed i two steps: a) Normalize te values of costs for eac L to {,}:. () b) Scale te ormalized values of costs for eac L (VormL) to {, } (pixel values):. () ) Keep oly te pixels tat represet miimum cost values. f a pixel i PixL as a magitude iger ta. te it gets a costat value of ; owever if it as a magitude less or equal to. te it gets a costat value of.. EOM Society teratioal 9
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, ) Estimatio of miimum cost regios troug all L values (estimatio of MCS values). Tis was performed by addig all ormalized (ad scaled) PixL matrices troug all L values (z-axis). Tis led to te creatio of a sigle matrix called miimum_regios were te regios most likely to cotai overall miimum cost values ave iger values of miimum cost scores (MCS) wic are give by te cumulative sum of PixL's. Te miimum_regios matrix presets te followig advatages: (a) it oly cotais iteger MCS values witi te rage {,}, (b) its umerical data (MCS values) ca be used directly by te solvig algoritm to adjust te limit restrictios for te cart's parameters (ad tus, reduce te searc space), (c) it ca be visualized i te D domai wit ay cotour or surface visualizatio metod. Figure (a) presets a example of te miimum_regios matrix i its umerical form (MCS values) wile Figure (b) ad Figure (c) preset two differet cotour visualizatios of te same matrix. (a) Numerical Data (MCS) (b) Cotour Visualizatio (c) Cotour Visualizatio (Surface) Fig.. Examples of umerical data (MCS) ad D cotour visualizatios from a miimum_regios matrix. As preseted i Figure (b) ad Figure (c), miimum regios tat were cosistet troug L ave te igest umerical data (igest MCS values). Tese regios are represeted wit dark red toes. Blue toes represet miimum regios tat were observed just for very few (if ay) values of L. te followig sectio te visualizatio of te miimum regios for all te cases (trails) of Table are preseted ad discussed. V. MCS D CONTOUR PLOTS Tables ad V preset te MCS D visualizatio cotour plots for te miimum_regios matrix of eac trail from te L ortogoal array (see Table ). Te x-y coordiate poits defied by te best - values reported i Table [] are plotted witi te cotour plots. Tese poits are idetified wit te symbol "O". Te cotours of te miimum regios preseted i Tables ad V were observed to be similar to elliptical cotours. Figure presets te te elliptical cotours tat were idetified to cover te miimum regios of te trails. Tese elliptical cotours were labeled from A-to-. Note tat, altoug C* as te same size tat C, C* is located closer to te y-axis (). Table V presets te elliptical cotours tat model te miimum regios of eac trail. B C* C F A 9 D 9 9 9 G E 9 9 Fig.. Elliptical cotours tat cover te MCS D miimum regios of te cost fuctio for eac trail of te L array. EOM Society teratioal 9
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 EOM Society teratioal 9 9 9 9 9 9 9 9 9 9 9 9 9 9 Cotour Plot 9 Cotour Plot 9 9 Cotour Plot MCS D CONTOUR PLOTS FOR TE TRALS OF TE ORTOGONAL ARRAY (PART ) TABLE. 9 9 9 9
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 EOM Society teratioal 9 9 9 9 9 9 9 9 9 9 9 Cotour Plot 9 9 9 Cotour Plot 9 9 Cotour Plot MCS D CONTOUR PLOTS FOR TE TRALS OF TE ORTOGONAL ARRAY (PART ) TABLE V. 9 9 9 99
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, TABLE V. Trail 9 ELLPTCAL CONTOURS FOR EAC TRAL Elliptical Cotour C E A D C* B G Trail Elliptical Cotour D A G C* E Trail 9 Elliptical Cotour D B F E C* f te searc mecaisms of a solvig algoritm is oly focused o te miimum regios wit te igest MCS values, te te size of tese regios (ad of te searc space) ca be reduced furter. As preseted i Tables ad V, for most of te trails, te best - pairs (as preseted i Table ) were located witi te miimum regios wit te igest MCS values. Tis cofirms te advatages ad suitability of te proposed approac to improve te searc mecaisms of solvig algoritms. V. CONCLUSONS AND FUTURE WORK Tis paper preseted a fast estimator of te solutio space for a cost fuctio model for te ecoomic statistical desig (ESD) of X-bar cotrol carts. Tis estimator itroduced te cocept of D visualizatio of miimum cost scores (MCS) to illustrate te patters of te cost fuctio model for te ESD problem. t was observed tat te miimum regios of te cost fuctio for differet referece data sets could be modeled wit elliptical boudaries. Also, it was observed tat sub-regios wit te igest MCS were more likely to cotai te optimal values for te cart's parameters. ece, tis approac ca be itegrated witi a euristic metod to perform a iitial samplig of te solutio space, makig te searc process more specific. Neverteless, more work as to be performed to measure te suitability of tis approac. Te followig poits are cosidered for future work: Explore alteratives to (or improvemets o) te metric of miimum cost scores (MCS). Because tis metric ca be see as a codig metric, oter metrics ca be cosidered for represetatio of miimum values for a cost fuctio model. Tis could make te searc processes of euristic ad exact solvig algoritms faster. Apply te MCS D visualizatio approac to track te searc performace of solvig algoritms. REFERENCES [] [] [] [] [] [] [] [] [9] [] S. Wag, V. terrate ad E. Logmire, "Multi-variate Visualizatio of D Turbulet Flow Data," i Proc. of te Coferece o Visualizatio ad Data Aalysis (VDA, ), pp. -,. A. Goel, C. Baker, C. Saffer, B. Grossma, W. Maso, L. Watso ad R. aftka, "VizCraft: A Problem-Solvig Eviromet for Aircraft Cofiguratio Desig," Computig i Sciece & Egieerig, pp. -,. Q. Wag ad B. Sauders, "teractive D Visualizatio of Matematical Fuctios Usig VRML," Gaitersburg, MD: U.S. Dept. of Commerce Tecology Admiistratio, Natioal stitute of Stadards ad Tecology, Wasigto, D.C.: U.S. Dept. of Eergy, pp. -, 999. S.O. Caballero-Morales, Buildig cost fuctio D becmarks to improve te ecoomic statistical desig of cotrol carts, Matematical Problems i Egieerig, vol., pp. -,. S.O. Caballero-Morales ad M.A. Raim, "Aalyzig te effect of o-ormality o te solutio space for te ecoomic statistical desig of X-bar cotrol carts," i Proc. of te t teratioal Coferece o dustrial Egieerig ad Operatios Maagemet (EOM ), pp. -,. S. alim, R. Yap ad. Lau, "Visualizatio for Aalysig Trajectory-Based Metaeuristic Searc Algoritms," i Proc. of te t Europea Coferece o Artificial telligece (ECA ), pp. -,. S.alim ad R. Yap, "Desigig ad Tuig SLS Troug Aimatio ad Grapics: A Exteded Walk-Troug," i Egieerig Stocastic Local Searc Algoritms: Desigig, mplemetig ad Aalyzig Effective euristics, Lecture Notes i Computer Sciece, Vol., T. Stützle et al. (Eds), Spriger Berli eidelberg, pp. -,. J. Pérez, N. Almaza, M. idalgo, G. Vela, L. Alvarado, M. García, A. Mexicao ad C. Zavala, "A Grapical Visualizatio Tool for Aalyzig te Beavior of Metaeuristic Algoritms," teratioal Joural of Emergig Tecology ad Advaced Egieerig, Vol., No., pp. -,. M.A. Raim ad P.K. Baerjee, "A geeralized ecoomic model for te ecoomic desig of cotrol carts for productio systems wit icreasig failure rate ad early replacemet," Naval Researc Logistics, vol., pp. - 9, 99. M. Ci, L.L. Ye ad F.C. Li, "Particle swarm optimizatio for te ecoomic ad ecoomic statistical desigs of te cotrol cart," Applied Soft Computig, vol., pp. -,. EOM Society teratioal 9
Proceedigs of te teratioal Coferece o dustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Marc -, BOGRAPY Satiago-Omar Caballero-Morales is curretly a professor-researcer i te Departmet of Logistics ad Supply Cai Maagemet at te Uiversidad Popular Autóoma del Estado de Puebla (UPAEP) i Mexico. e olds a B.Sc. degree i Electroics ad Commuicatios Egieerig from te Uiversity of te Americas-Puebla, a M.Sc. degree i dustrial Egieerig from te Uiversity of te AmericasPuebla, ad a P.D. degree i Computer Sciece (Speec Recogitio) from te Uiversity of East Aglia. Sice e as bee a member of e Natioal System of Researcers (SN) i Mexico ad curretly olds te Level- distictio. e as bee a active reviewer for iteratioal jourals as PLOS ONE, Computers & dustrial Egieerig, ad Expert Systems wit Applicatios. is researc iterests are quality cotrol, operatios researc, combiatorial optimizatio, patter recogitio, automatic speec traslatio, aalysis ad simulatio of maufacturig processes, ad uma-robot iteractio. Abdur Raim is a seior lecturer i te Faculty of Busiess Admiistratio at te Uiversity of New Bruswick i Caada. e olds B.Sc. ad M.Sc. degrees i Statistics from te Uiversity of Daka, a M.Sc. degree i Systems Sciece ad Teory from te Uiversity of Ottawa, ad a P.D. degree i dustrial Egieerig from te Uiversity of Widsor. e is a recogized world expert i Total Quality Maagemet ad as taugt a wide rage of courses i uiversities i Caada, Baglades, Malaysia ad Saudi Arabia. e as worked as a cosultat for UNDP (Uited Natios Developmet Program) ad FAO (Food ad Agriculture Orgaizatios of te Uited Natios). e is editor ad member of advisory ad editorial boards of importat iteratioal jourals as Egieerig Optimizatio ad Quality Egieerig. is researc iterests are quality cotrol, ivetory cotrol, reliability, productio ad operatios maagemet, ad as produced over refereed joural articles ad book capters. EOM Society teratioal 9