Applying Software Reliability Techniques to Low Retail Demand Estimation

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1 Applyng Sofwar Rlably Tchnqus o Low Ral Dmand Esmaon Ma Lndsy Unvrsy of Norh Txas ITDS Dp P.O. Box Dnon, TX lndsym@un.du Robr Pavur Unvrsy of Norh Txas ITDS Dp P.O. Box Dnon, TX pavur@un.du ABSTRACT Assssng ral dmand s basc o nvnory modlng rsarch and has rcvd much anon n h lraur. Howvr, smang h ra of producs sold whn fw hr ar fw, f any, sals has no rcvd h adqua anon. Ths papr borrows a chnqu usd n assssng sofwar rlably o sma h fuur ra of sals of slow movng producs. An smaor s proposd o forcas h ra of fuur sals for producs ha hav no sold ovr a spcfd nrval of m. Ths smaor uss sascal nformaon from h ra of producs ha hav sold. Th dsrbuon of hs smaor s shown o b posvly skwd for larg m nrvals and somwha ngavly skwd for shor m nrvals. A smpl approach o smang h ra of sals for any spcfd produc s o compu h rao of h numbr of ms sold ovr a m nrval. Ths papr pons ou ha hs sma can b bas f h m nrval slcd s such ha nds only afr a cran numbr of ms hav bn sold. INTRODUCTION Many ral ms sll slowly and smang fuur sals ra may b dffcul afr a fw wks of slow sals. Masrs (1993) ss svral xampls of producs ha hav low dmand, such as spcfc clohng ms, auomobl rpar pars, and spcfc compac dsc ls. Wha yp of sma can a sals managr oban o sma h ra of sals of h spcfc syl shrs ha hav no sold? Wha ar h ncssary assumpons for hs sma o b rlabl? Ths quson wll b addrssd n hs papr. 1

2 Sofwar Rlably Bfor addrssng hs problm, w frs mnon ha hs problm s rlad o a smlar problm n mananng rlabl sofwar. Th nhrn complxy of h sofwar dvlopmn procss s crad by many facors and maks dffcul o manan rlabl sofwar. Esmang h rlably of ndvdual sofwar packags or of sofwar avalabl on a cln srvr nwork may b an lusv ask vn afr dsgn rvws, modul sng, and slf-chckng. Sofwar ngnrs ypcally pu sofwar hrough a sng phas ovr a spcfd prod of m o drmn whn s rady o b rlasd o consumrs, whou ovr sng crang an xcssv m-o-mark (Lyu, 1995). Durng hs prod of m, any bug or faul n h sofwar s usually rmovd. Th rm bug s dfnd n h Rlably, Avalably, and Mananably Dconary by Omdahl (1988) as a program dfc. Th rm bug s usd hr as an quvaln rm for faul., h rmnology of faul and rror ra n sofwar rlably wll b dfnd as n Musa, Iannno, and Okumoo (1987). A faul s dfnd as Aa dfcv, mssng, or xra nsrucon or s of rlad nsrucons ha s h caus of on or mor acual or ponal falur yps@ by Omdahl (1988). A faul wh h sofwar dos no ncssarly caus h sysm o cas opraons. Snc sofwar compans ar ypcally undr prssur o rlas sofwar o b frs on h mark, hr s no nough m o mak hr sofwar complly bug-fr. Thus, sofwar compans ofn rlas sofwar knowng ha bugs (rrors n codng) ar prsn. Th sofwar company amps o sma h ra of fauls occurrng n h sofwar whou havng obsrvd h rrors mad by h rmanng bugs. Assumpons abou h dsrbuon of h numbr of rrors nd o b mad. Schulmyr and McManus (199) suggs rrors ar ofn assumd o occur undr h assumpons of a Posson procss. Applcaon of Sofwar Rlably Logc In ral sllng, dmand n many suaons s also ofn assumd o occur as a Posson procss. Axsar (1993) suggss ha for many consumabls, rparabl or modular ms wh low dmands and a connuous rvw polcy, h appropra dmand dsrbuon follows a Posson procss. Axsar (000) suggss ha for h gnral and praccally mporan problm of mananng spar pars ha dmand can b modld wh a Posson or compound Posson procss, whl ohr assumpons ar br for hghr dmand producs. In nvnory applcaons, h produc can b hough of as a bug and h numbr of ms ha h produc slls pr un of m as h rror ra. For produc ms ha sll, h sals ra can b smad as h numbr of ms ha hav sold pr un of m. For producs ha hav no sold, a zro sals ra s acually oo consrvav, as n sofwar, bcaus may b ha h prod of m ovr whch h sals ar bng smad may no hav bn long nough. In h nx scon, w dscuss how h sma of sals ra should b compud f an unbasd sma s dsrd. Thn, a proposd sma for h sals ra of producs ha hav no sold s prsnd. Th skwnss of h dsrbuon of hs sma s nvsgad. An xampl of an applcaon of h sma s prsnd followd by h concluson scon.

3 ESTIMATING SALES RATE FOR PRODUCTS WITH NONZERO SALES FIGURES Snc sals dmand has ofn bn modld as followng a Posson Procss, w wll mak h assumpon ha sals follows a Posson Procss. Now, smang h sals ra may appar o b a sragh forward procss by smply dvdng h numbr of sals by h m. Praconrs may hnk ha hr of h followng procdurs s accpabl n smang h sals ra, whch w wll b dno by λ. Esmaon Procdur 1. Obsrv h m T ha aks for a fxd k* numbr of rrors o appar n sng and sma λ o b k*/. Esmaon Procdur. Obsrv h numbr of rrors ha occur ovr a fxd m * and sma λ o b k/*. In Esmaon Procdur 1, noc ha k* s fxd and T s a random varabl. Thus, by h assumpons of a Posson procss, T s h sum of k* xponnal random varabls. So T s dsrbud as a Gamma (k*, λ). Snc h lklhood funcon s proporonal o λ k* xp(-λ), h smaor s k*/. No ha E(k*/T) = λ (k*/ (k* - 1)). So, hs smaor wll b basd unlss s adjusd. So h unbasd smaor s (k*-1)/t. Th varanc of hs smaor s Var((k*- 1)/T)=λ (1/ (k*- )). In Esmaon Procdur, noc ha * s fxd and K s a random varabl. Hr, h random varabl K (wh k bng a valu of h vara K) s a Posson random varabl. Snc h lklhood funcon of K s agan proporonal o λ k xp(-λ*), h smaor s k/*. Now, E(K/*) = λ. So, hs smaor s unbasd and s varanc s Var(K/*) = λ/*. Whn s h varanc of h frs smaon procdur approxmaly h sam as h varanc of h scond smaon procdur? If w s Var(K/*) = Var((k*-1)/T) hn λ/* = λ (1/ (k* - )) or λ = (k*-)/*. So f k* and * ar slcd such ha λ s approxmaly qual o k*/* hn h wo procdurs ar consdrd quvaln. Whch smaor s h prfrrd smaor? If k s larg, hn smaon procdur 1 should b ssnally unbasd. For small numbr of occurrncs, hs rror ra may b oo basd o b usful. A usr has o assss h varanc of hs smaors o dcd whch procdur s mos dsrabl. Slcng * o b oo larg may no b praccal and slcng a larg k* may ak much longr han s praccal n obsrvng h numbr of ms sold. Thrfor, praconrs should b awar of h dffrncs n slcng hs smaors o forcas fuur sals ras. ESTIMATING SALES RATE FOR PRODUCTS WITH ZERO SALES FIGURES W agan mak h assumpon ha h sals of ach produc n a larg pool of producs follow a Posson dsrbuon. Th noaon ha w wll us s for h undrlyng unknown ra of produc s unknown ra λ. Whn producs sll, w can us h smaon procdurs n h prvous scon o sma h ra a whch hy wll sll. For producs showng no sals ovr a spcfd prod of m, a zro fuur ra of sals s oo consrvav snc hy may vnually sll. 3

4 Th problm n hs scon s o drmn h fuur sals (or dmand) ra for producs wh no sals. An smaor ha Ross (1985, 1993) proposs for h falur ra of sofwar s h sum of h numbr of bugs ha caus xacly on falur (call hs sum M 1 ()) dvdd by m prod n whch hs falurs occur. Thus h smaor s M 1 ()/ and h oal numbr of bugs n h sysm dos no hav o b known. In addon, s possbl for h falur ra of ach bug can b dffrn. Now, hs smaor could b appld o h suaon n whch h ra of sals s bng obsrvd. Th varabl M 1 () could b usd o rprsn h numbr of producs ha hav sold only on m. Ross (1993) provds an sma of h varanc of hs proposd smaor as (M 1 () and M ())/. Agan, hs varanc smaor has h advanag of no rqurng knowldg of h oal numbr of bugs n h sysm. Th varabl M () could b usd n h sals suaon o rprsn h numbr of producs ha hav sold xacly ms. A dsadvanag o Ross (1993)=s smaor s ha h dsrbuon of M 1 () s no known. A normal approxmaon may b usd, bu h accuracy of hs procdur s dpndn on h m of h nrval and h numbr of producs n h sysm. To formally dvlop h smaor of h fuur sals ra of h poold s of producs ha hav no sold, frs w consdr all producs and us an ndcaor funcon o rprsn whhr a produc has sold any ms ovr m. Each produc has an undrlyng sals ra, λ. L Ψ () = 1 f produc has no sold by m and 0 f has sold. W wsh o sma h valu of h followng random varabl n whch n s h numbr of producs for sal. n Λ( ) = λ Ψ ( ). =1 (1) Th xpcd valu of hs random varabl s h followng. E( Λ( )) = = m m λ E [ Ψ - λ λ ( )] () As mnond arlr, w dfn M 1 () and M () as h numbr of bugs ha wr rsponsbl for causng xacly on falur and wo falurs, rspcvly. W now sa svral rsuls whch can b provd usng h sandard sascal assumpons of a Posson procss. Thus, M 1 ()/ and Λ () hav h sam xpcd valu. To b a Agood@ smaor of Λ(), M 1 ()/ s varanc should b small. 4

5 E (M E (M E (M ( ) / ) = E ( Λ ( ) ) ( ) ( ) ) = ) = (1/ ) E ( ( Λ ( ) - M 1( ) / ) ) = 1 1 λ ( λ ) ( λ = E ( M 1 ( ) + M ( ) ) / n m n - λ - λ - λ + λ - λ / ) (3) No ha f h sals ras λ for ach of h n producs was larg, hn - λ λ would b small. In hs cas, h xpcd valu of M1() would yld a small numbr of producs wh no sals. Ths s conssn wh wha on would xpc M 1 () o b whn producs ar sllng a a fas ra. In addon, no ha f (m for sng) s larg, hn ach of h rms n h xpcd valu of M 1 ()/ wll b small, hus yldng a small sals ra. Th las quaon abov shows ha (M 1 () + M ())/ s an unbasd smaor of h squard dffrnc of Λ() and M 1 ()/ (rror n smang h ru sals ra of producs ha hav no sold). Agan f s larg, hs varanc bcoms small. W drv a bound on h E[(Λ() -M 1 ()/) ] as follows: d - λ f ( λ ) = ( λ + λ / ) f ( λ ) - λ = 0 ==> ( λ +1 / ) + ( λ + λ /) d λ (- ) - λ = 0 (4) Thrfor (λ ) - λ - 1 = 0. Th posv soluon o hs quaon s: λ = (1+ 5 ) / (). (5) Hnc, w can subsu hs valu n h formula for E((Λ() - M 1 ()/) ) and produc h followng bound whch provds h praconr wh an uppr lm for h xpcd squard rror wh knowldg of only n and - (1 + / ( ( Λ () - M 1 ( ) / ) ) (n / ) [ ( (1 + 5 ) / ) + (1 + 5 ) / ] 5 ) = (.83996) n/ (6) E To nvsga h dsrbuon of M 1 ()/, a smulaon was rpad 500 ms of 300 producs wh a Man Tm Bwn Sals of 40 hours (ra = 1/40) ovr dffrn m prods. For a m prod of = 50, noc h skwnss sma of for hs dsrbuon as shown n Tabl 1. Noc ha for abl wh = 70, h skwnss s a small ngav numbr. In Tabl 3, h skwnss s a small posv numbr and hn n Tabl 4 for = 150, h skwnss s approxmaly.3. Thus, h skwnss of hs smaor s somwha ngav for small m prod and vnually bcoms posv for largr m prods. 5

6 Tabl 1: Dsrbuon of mprcal sals ra of producs wh no sals ovr a prod of 40 hours,tabl : Dsrbuon of mprcal sals ra of producs wh no sals ovr a prod of 70 Hours, Tabl 3: Dsrbuon of mprcal sals ra of producs wh no sals ovr a prod of 130 hours, and Tabl 4: Dsrbuon of mprcal sals ra of producs wh no sals ovr a prod of 150 hours ar avalabl upon rqus. Exampl Suppos ha a ral managr rcvd a shpmn of sus. Afr 100 hours of sllng hs sock of sus, assum ha hr wr 16 syls n whch xacly on su sold and n syls n whch xacly sus wr sold. Th sma of h fuur sals ra for hos syls ha hav no sold s M 1 ()/ = 3/100 =.03. If h dsrbuon of M1()/ could b assumd o b normally dsrbud, hn a 95% confdnc nrval on h xpcd fuur rror ra s M 1 () / ± 1.96 (M 1 () + M ( ) ) /. (7) Ths nrval s.03 ± / 100 or o.14. Hnc, h fuur sals ra for h producs, n aggrga, ha hav no sold s approxmad o b bwn 0 o.14. Or n rms of man m bwn sals, bwn 8 hours (1/.14) and nfny (1/0), hus, h ral managr should no xpc o sll mor han on su pr day (8 hours) from h syls ha hav no sold. CONCLUSION Ths sudy appls an approach usd n sofwar rlably o sma h fuur sals ra of producs ha hav no sold ovr a spcfd prod of m. How good of an smaor s h proposd mhod? As mnond n h scon for smang sals for producs ha show no sals, h dsrbuon of h smaor may b mor skwd for small m prods and for larg m prods. Wha drmns larg and small hr? Tha could dpnd on h numbr of producs and h sals ras of hos producs. Fuur rsarch should b dvod o undrsandng h shap of h dsrbuon of hs smaor. Anohr quson of mporanc s h usfulnss of hs approach o h ral managr. Ths may dpnd on h cycl of m n whch a managr has o sll goods. Th proposd approach assums a Posson procss, whch may no b a good modl for h sllng of som producs. In summary, h approach prsnd hr gvs anohr ool o h supply chan managr o oban a hard-o-oban sma on fuur sals ras of producs ha ar slow movrs. REFERENCES Asxar, S., (000). Exac analyss of connuous rvw (R,Q) polcs n wo chlon nvnory Sysms wh compound Posson Dmand., Opraons Rsarch 48(5), 686. Axsar. S. (1993). Connuous rvw polcs for mul lvl nvnory sysms wh sochasc dmand. In S. C. Gravs al. ds. Handbooks n OR &MS. Vol. 4. Norh Holland, Amsrdam. Th Nhrlands, Lyu, M. R. (Ed.), (1995). Sofwar Rlably Engnrng. Slvr Sprng, MD: IEEE Compur Socy Prss. 6

7 Masrs, J. M. (1993) Drmnaon of nar opmal sock lvls for mul-chlon dsrbuon nvnors. Journal of Busnss Logscs, 14(), , Musa, J., Iannno, A., Okumoo, K. (1987), Sofwar Rlably: Masurmn, Prdcon, and Applcaon, McGraw-Hll, Nw York. Omdahl, T. P. (1988), Rlably, Avalably, and Mananably Dconary, ASQC Qualy Prss, Mlwauk. Ross, S. M. (1985), Sascal Esmaon of Sofwar Rlably, IEEE Trans. Sofwar Eng 11, Ross, S. M. (1993), Probably Modls, Acadmc Prss, Nw York. Schulmyr and McMann (199), Handbook of Sofwar Qualy Assuranc, VanNosrand Rnhold, Nw York. 7