A formulation for measuring the bullwhip effect with spreadsheets Una formulación para medir el efecto bullwhip con hojas de cálculo

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1 irecció y rgaizació 48 (01) A formulaio for measurig he bullwhip effec wih spreadshees Ua formulació para medir el efeco bullwhip co hojas de cálculo Javier Parra-Pea 1, Josefa Mula ad Fracisco Campuzao-Bolari 3 1 Uiversidad isrial Fracisco José de Caldas, Faculad Tecológica, Bogoá, Colombia. Cero de Ivesigació e Gesió e Igeiería de Producció (CIGIP). Escuela Poliécica Superior de Alcoy, Plaza Ferrádiz y Carboell,, Alcoy - Alicae, Spai. 3 Uiversidad Poliécica de Caragea. Campus Muralla del Mar.30 Caragea -Murcia. Spai. japarpe1@posgrado.upv.es, fmula@cigip.upv.es, fracisco.campuzao@upc.es Fecha de recepció: Fecha de acepació: Absrac: The bullwhip effec refers o he sceario i which orders o suppliers ed o prese larger flucuaios ha sales o buyers, ad he resulig disorio icreasigly amplifies upsream i a supply chai. We propose a rasformaio of he kow formulaio by Che e al. (000) o calculae he bullwhip effec i a supply chai for he purpose of beig easily applied wih spreadshees wihou usig VBA macros. We prese his formulaio modeled by Ms Excel usig a umerical example. Key words: bullwhip effec, measurig, spreadshee. Resume: El efeco bullwhip se refiere al esceario e el que los pedidos al proveedor iede a presear mayores flucuacioes que las veas a los compradores, y la disorsió resulae se amplifica crecieemee aguas arriba e ua cadea de sumiisro. Se propoe ua modificació de la coocida fórmula de Che e al. (000) para calcular el efeco bullwhip e ua cadea de sumiisro co el objeivo de que sea aplicada fácilmee a ravés de hojas de cálculo si usar macros VBA. Se presea esa formulació co Ms Excel usado u ejemplo umérico. Palabras clave: efeco bullwhip, medida, hoja de cálculo. 1. Iroducio A iegraed supply chai icludes purchasig raw maerials, maufacurig wih assembly, or someimes also disassembly, ad he disribuio ad repackagig of produced goods se o ed cusomers. Various operaig sages i he logisic chai (chai odes) ca be represeed by a simple model of some maerial-rasformaios or locaio-chages processig cells ad arcs. I each processig cell, a value is added ad some coss are icurred. There is also supply ad demad, ad boh are ofe sochasic i aure. Iveories are a isurace agais risk of shorage of goods i each cell of a logisic chai. They are limied by o oly he give capaciy of each processig ode, bu also by he rasporaio capabiliy of he ipu ad oupu flows. The bullwhip effec, a well-kow pheomeo, was already iesively sudied by he mid-0h ceury, ad was kow previously uder he ame of demad amplificaio or he Forreser effec (Forreser, 1961). I is ofe cosidered o reduce uceraiy i demad, ad also because of a lead ime sochasic behavior. The Forreser effec is also ecompassed i Serma s bouded raioaliy (Serma, 1989). Such a approach is paricularly welcome whe mahemaical ools are eiher o well developed or o well udersood. Mos research works o demad amplificaio have focused o demosraig is exisece, ideifyig is possible causes, ad developig mehods for miigaig is egaive effec. Lee e al. (1997a) ideify four mai causes of he bullwhip effec: wrog mehods of demad forecasig, supply shorage aicipaio, bach orderig ad price variaio. emad amplificaio occurs mosly because of fiie perurbaios i fial demad ad i lead imes hroughou he supply chai, which is always aicipaed ad ieracs wih oher causes. How he idusry faces his pheomeo has bee broadly sudied by Lee e al. (1997b), where some cosideraios of he bullwhip effec i supply chais are preseed i deail. For a exesio of he

2 30 Javier Parra e al/irecció y rgaizació 48 (01) 9-33 bullwhip effec cocep i a supply chai, we refer readers o Campuzao ad Mula (011). The mai coribuio of his paper is o propose a rasformaio of he formula by Che e al. (000) i order o measure he bullwhip effec i supply chais, for he purpose of usig i easily wih spreadshees ad wihou havig o develop VBA macros. The res of he paper is orgaized as follows: Secio reviews he mai aleraives o measure he bullwhip effec. Secio 3 proposes a rasformaio of he formulaio by Che e al. (000) o measure he bullwhip effec. Secio 4 preses he applicaio of our proposal o spreadshees usig a umerical example. Fially, Secio 5 provides he coclusios ad furher research.. Measurig he bullwhip effec As meioed above, he bullwhip effec refers o he sceario i which orders o suppliers ed o prese larger flucuaios ha sales o buyers, ad disorio propagaes up a supply chai i a amplified maer. As disorio creaes addiioal coss, bullwhip idicaors are assumed o be i coigecy wih coss or added value. rderig goods (ipu flow) i disribuio ceers ca be sudied as a muli-period dyamic problem. The demad durig each period ca be cosidered a sochasic variable. The disribuio of his variable is ofe described wih a cerai probabiliy fucio. I is ideically disribued i our proposal, based o he producio ad iveory corol resuls, especially o he variabiliy rade off sudy as preseed by ejockheere e al. (003). I is also based o he sudy of he impac of iformaio erichme o he bullwhip effec i supply chais (ejockheere e al. 004), where some measures have bee iroduced. Amplificaio upsream of he supply chai ca be measured hrough he variace of demad alog he supply chai. Lee a al. (1997b) proposed chages o he variace i demad upsream as a measure of he bullwhip effec. I is a good measure, bu oly whe uis of flow alog he chai do o chage, which is o he case of may logisics problems. Che e al. (000) sugges ha i order o avoid his problem, he bullwhip effec should be measured by chagig he raio of /µ upsream of he supply chai, where µ is he expeced value of he iesiy of flows. ce agai however, i does o help o avoid he effec of chagig he ui of measure. Che e al. (000) sugges ha a measure of he bullwhip effec could be he raio of hese parameers eiher bewee he ipu ad oupu flows a each aciviy cell i a supply chai upsream whe cosiderig oly oe sage or bewee fial demad ad he firs maufacurig sage whe he oal supply chai is o be evaluaed as i has more sages. Bullwhip / µ / µ (1) Uforuaely /µ is o a measureless raio. Therefore, i is difficul o udersad he raio of hese measures bewee he ipu ad oupu flows, especially where assemblies ad disassemblies chage he ui of measure of µ bewee ipu ad oupu, ad whe we wish o compare his pheomeo i differe supply chais. Thus i his paper, we assume ha he ui of measure does o chage i our supply chai. I he saisical aalysis, he acual variaio described by variace or sadard deviaio is ofe replaced wih he measure of relaive dispersio. If absolue dispersio is measured by sadard deviaio as he roo mea square of he deviaio from he mea, ad if he average is he mea iesiy of supply chai flows µ, he he relaive dispersio ormally used i basic saisics is he coefficie of variaio V /µad he esimaio is geerally expressed as a perceage, which would also be a good idicaor here. Bu followig he approaches of previous auhors, we maiai he bullwhip measure, as suggesed by Che e al. (000), ad we shall o chage he uis upsream of flows. 3. Formulaio proposal We propose he followig rasformaio of Formula (1) provided by Che e al. (000) o measure he bullwhip effec for he purpose of easily usig i wih spreadshees. I order o compue he measure proposed by Che e al. (000), ad cosiderig he imporace of calculaig performace measures o supply chai behavior, we buil a spreadshee based o is measure. The calculus of he bullwhip effec is based o he variace of boh variables: sales ad orders. As he bullwhip effec is a amplificaio of orders i he supply chai as a resul of ushared iformaio a all he supply chai levels, we aalyze hese variables as follows.

3 Javier Parra e al/irecció y rgaizació 48 (01) Firs, he calculus of he variace of sales is doe by uilizig he spreadshee variace formula because i mus be calculaed by usig all he daes available; i.e., he daes associaed wih each period, from zero o he period. Moreover, he variace associaed wih he variable of orders mus be calculaed oly o each period. This preses some differeces bewee sales ad orders. Give he assumpio ha i depeds oly o he curre period (periods i which differeces bewee orders ad sales differ from zero), i ca be ierpreed as follows: he vecor of orders x i o period is a vecor of elemes, for which he firs - 1 elemes are equal o zero ad he -h erm is he value of he orders o period. As a resul of his assumpio, he calculus of he variace of orders is based o he geeral formula of he variace, as show i Equaio (). This formula cosiders -1 elemes because i is a sample variace, geerally for a limied umber of daa. i 1 ( xi x) 1 () As he assumpio ha he vecor of orders is composed of -1 erms which are equal o zero, he variace calculus implies a sum of he erms ha has -1 ideical erms ad ha oe erm is associaed wih he -h period. Give he srucure of he vecor of orders, we ca express he variace formula as show i Equaios (3) ad (4). 1 i 1 ( x x) + ( x x) i 1 (3) The expressio show i Equaio (5) ca be implemeed easily i a spreadshee because i depeds oly o daa quaiy () ad o he las orders iformaio. The, we impleme he expressio of he bullwhip effec measure (BEM) o he period, BEM, as a cumulaive expressio of he effec of idividual period disorios by usig a codiioal expressio preseed i Equaio (6). BEM 1 BEM Var ( rders) BEM 1 + Var ( Sales) if orders sales 0 i oher wise (6) Noe ha he BEM ca be calculaed as a cumulaive value o each period. The formulas i Equaios (5) ad (6) were implemeed by usig a umerical example i order o evaluae heir performace. I he ex secio, we prese a umerical example ad he spreadshees used o calculae he bullwhip effec. 4. Numerical example wih spreadshees The example preseed i Figure 1 shows he BEM, which was calculaed by usig he expressios preseed i Equaios (5) ad (6). emads were geeraed by a discree uiform disribuio bewee x ad y; i.e., hey are ideically disribued. Figure 1 Spreadshee implemeaio of he BEM formula ( 1)( x) + ( x x) 1 (4) Accordig o he assumpio of he vecor of orders, he mea of orders ca be calculaed as he orders i periods divided by. As a resul of he replaceme of he mea of orders i Equaio (4), we ca express he formula of he variace of orders as follows: ( 1) x x + x 1 (5)

4 3 Javier Parra e al/irecció y rgaizació 48 (01) 9-33 The spreadshee formulaio used o calculae he BEM should be carried ou as deailed below. iffereces formula i Cell N8 L8-B8 Variace orders formula i Cell 8 (IF(N8<>0;(A8)*PWER(L8/(A8+1);)+P- WER(L8-(L8/(A8+1)););0))/A8 Variace sales formula i Cell P8 VAR(B$7:B8) Bullwhip Effec Measure formula i Cell Q8 Q7+IF(N80;0;(8/P8)) The las colum is he BEM, ad i is he cumulaive value of he variace of orders divided by he variace of sales. The implemeaio of his measure is preseed i Figure 3. All he expressios ca be exeded o he period i which we eed o calculae BEM. Figure 3 The bullwhip effec measure (BEM); a umerical example The firs expressio correspods o he differece bewee he values of orders ad sales o he period oe. Sales ad orders are preseed i Figure. Figure Sales ad orders versus ime; a umerical example 5. Coclusios I his paper, we prese a rasformaio of he formula by Che e al. (000) o measure he bullwhip effec i supply chais for he purpose of beig easily implemeed wih spreadshees ad wihou havig o develop VBA macros. The, we show i deail he implemeaio of his modified formula o calculae he bullwhip effec i supply chais via spreadshees. The secod expressio shows he form i which he curre relaioship o he variace of orders is calculaed. Here, he formula is oly calculaed whe he differece bewee orders ad sales differs from zero (N8); is he period i which he variace of orders is calculaed, ad i is he value i cell A8 plus 1 (because he period zero is icluded i he daa quaiy); -1 is he value of cell A8; L8 represes he orders i he period, which is used o calculae he mea of orders. The formula of he variace of sales is merely a spreadshee formula of he variace, which is calculaed accordig o he sales from he zero period o he curre period (he period is i he A colum). A forhcomig work is relaed o he rasformaio of he formula by Frasoo ad Wouers (000), which measures he bullwhip effec a a paricular level i a muli-level supply chai as he quoie of he demad variace coefficie geeraed by his level, ad as he coefficie of he demad variace received by his level, i order o also impleme i easily wih spreadshees. Refereces CAMPUZAN, F., MULA, J. (011). Bullwhip effec i supply chais. I: Supply chai simulaio. A sysem dyamics approach for improvig performace. Ed. Spriger. CHEN, F., REZNER, Z., RYAN, J.K., SIMCHI-LEVI,., (000). «Quaifyig he bullwhip effec i a simple supply chai: The impac of forecasig, lead-imes ad iformaio». Maageme Sciece 46 (3),

5 Javier Parra e al/irecció y rgaizació 48 (01) EJNCKHEERE, J., ISNEY, S.M., LAMBRECHT, M.R., T- WILL,.R., (003). «Measurig ad avoidig he bullwhip effec: A corol heoreic approach». Europea Joural of peraioal Research 147, EJNCKHEERE, J., ISNEY, S.M., LAMBRECHT, M.R., T- WILL,.R., (004). «The impac of iformaio erichme o he bullwhip effec i supply chais: A corol egieerig perspecive». Europea Joural of peraioal Research 153, Forreser, J. (1958). «Idusrial yamics. A Major Breakhrough for ecisio Makers». Harvard Busiess Review, July-Augus, pp FRRESTER, J. (1961). Idusrial yamics. MIT Press, Cambridge, MA Frasoo, J.C., Wouers, M.J.F. (000). «Measurig he bullwhip effec i he supply chai». Supply Chai Maageme 5, LEE, H.L., PAMANABHAN, V., WHANG, S., (1997a). «The bullwhip effec i supply chais». Sloa Maageme Review 38 (3), LEE, H.L., PAMANABHAN, P., WHANG, S., (1997b). «Iformaio disorio i a supply chai: The bullwhip effec». Maageme Sciece 43, STERMAN, J.. (1989). «Modellig maagerial behavior: mispercepios of feedback i a dyamic decisio-makig experime». Maageme Sciece 35 (3),