Predictio of returs i WEEE reverse logistics based o spatial correlatio Ju LV (JLV@dbm.ecu.edu.c) School of Busiess, East Chia Normal Uiversity 8, Dogchua Road, Shaghai 41, Chia. Jia-pig XIE School of Iteratioal Busiess, Shaghai Uiversity of Fiace & Ecoomics, 777, Guodig Road, Shaghai 433, Chia Abstract Uder the stress of eviromet legislatio ad resource lack, logistics maagemet turs from liear mode ito closed mode. It is difficult for academic circle ad idustry to predict the returs of recyclig WEEE products due to the ucertaity of the umber of returs. From a ew perspective, we cosider spatial correlatio of retured WEEE, itroduce Krigig method of spatial statistics ad build Krigig spatial model of reverse logistics retur predictio.four mai properties about the retur volume of recycle ceters are foud. Simulatio experimets based o Mote Carlo are coducted to validate the effectiveess of the developed model. Keywords: Reverse logistics, Krigig method, Spatial correlatio Itroductio Huma is facig some problems about sustaiability of the eviromet such as the icreasig waste ad lack of resources, etc. However, it is difficult for academic circle ad idustry to predict the returs of recyclig WEEE products due to the ucertaity of the umber of returs. 1
I the field of retur product predictio, some articles have emerged that provided predictio modelig methods. Kelle ad Silver(1989) assumed that the retur product quatity is the fuctio of past sale ad retur, ad built the retur predictio model based o sale ad retur history data. Yag ad Williams (9) built a logic regressio model to predict the retur tedecy of computers. Polyomial regressio aalysis does well i fittig the polyomial tedecy of data series, but has higher modelig error, because the highest power of polyomial is radomly chose by researcher. Gómez et al. () developed a predictio model based o back propagatio eural etwork to predict the retur quatity of WEEE. However, the etwork structure ad etwork ode umber are decided ot by mathematics aalysis, but oly by user s experiece. Toktay et al. () assumed the retur quatity follows biomial probability distributio, ad used Bayes ifereces ad expectatio maximizatio algorithm to fid the total probability ad delay probability parameter of reverse logistics, ad the fid the retur quatity ad time. The above articles cosider the retur quatity of etwork odes as time series data, ad the time correlatio is the key cosideratio, ad uder give coditios these methods have good results to predict retur quatity. However, i the WEEE reverse logistics etwork, the retur quatity ot oly has the time correlatio, but also has the spatial correlatio. Mi et al. (6) built oliear model closed-loop supply chai etwork based o both space ad time, ad researched o solutio based o geetic algorithm. Graczyk ad Witkowski (11) poited out that oly if both space ad time are cosidered, better result ca be realized i reverse logistics. Three Japaese scholars from atioal graduate istitute for policy studies of Japa foud that from the space perspective, the solid waste has stroger iterdepedecy (Daisuke et al., 1). They used the 6-1 data from miistry of the eviromet of Japa, classified solid waste ito two types: household solid waste ad busiess solid wastes. Ad the spatial correlatio is aalyzed i 47 areas by the method of data evelopmet aalysis, ad the result showed that the solid waste quatities i 47 areas are highly spatial correlatio. I what follows, we develop ad aalyze Krigig models to provide isights to iform policy maagers ad WEEE compaies regardig spatial correlatio betwee the two retur ceters. I, we descript the problems ad secod order radomess. I 3, we provide a aalysis of spatial structure ad variogram fuctio of WEEE retur ad develop Krigig spatial model. I 4, the simulatio experimet ad results aalysis are coducted. We ed with a summary ad coclusio i 5. Problem Descriptios ad Secod Order Radomess Hypothesis Problem Descriptio We assume that oe electroic compay has q retur ceters located withi a two-dimesio fiite regio D. The retur ceters are s 1 ( x, y ), s ( x, y ),..., (, ) s x y, x ad y represet the retur ceter geographical locatio of the abscissa ad ordiate i this regio, the distace q
betwee ay two poits is h, ad the vector forms of all poits ares ( s1, s, s q ), ad all retur ceters i D are s ( x, y) D. The retur quatity of each ceter is deoted by 1 Z( s ), Z( s ), Z( s ). The vector form of the retur quatity of all ceters i this regio is q Z ( ( ), ( ), ( )), which is cosidered as a stochastic variable. Z s1 Z s Z s q It is assumed that i oe period ad uder radom coditio, the retur quatities of ceters poits are Z( s1), Z( s), Z( s ), two questios exist: How to quatitatively describe the spatial structure correlatio betwee the kow retur quatity of ay ceters ad the rest ukow retur quatity of q- ceters? Based o the quatitative descriptio of their spatial structure correlatio, how to predict the ukow retur quatity usig kow retur quatity ad to predict retur quatity i the whole regio? Tryig to aswer the above two questios, we eed to aalyze the spatial structure of the retur quatity ad build spatial correlatio stochastic model uder the coditio of spatial correlatio ad ucertaity. Secod Order Radomess Krigig method is oe of the most importat spatial predictio methods of stochastic variable i spatial statistics. This method is first used by South Africa miig egieer D.R.Krige to search gold i 1951, ad is amed by him. Ad the, the Krigig method was systematically proposed by Frech mathematicia Mathero, ad became the importat part of spatial statistics. Radomess idicates that variables are arbitrary value i oe period, ad secod order radomess idicates that variables keep o beig arbitrary, ad the covariace cov si( x, y), s j( x, y) at ay various locatios depeds oly o distace betwee two retur locatios h si s j. Secod order radomess is expressed as: E[ Z( s)] m,( sd) (1) Cov[ Z( s h), Z( s)] C( h),( s, s hd ) () where E() is the mathematic expectatio ad Cov() is the covariace of the icremet betwee two retur ceters, m is a costat, deotes the total retur quatity i oe period ad i oe area, h is the distace betwee ay two retur ceters, Z( s h) deotes the retur quatity of oe ceter with a give distace h from the other ceter s. 3
Spatial Predictio Mathematics Model of WEEE Retur Quatity Spatial Structure Aalysis of WEEE Retur Quatity The variogram fuctio of spatial statistics is used to aalyze spatial variatio structure of retur quatity variable. For ay distace h, the variable of [ Z( s h) - Z( s)] has a defiite variace, which depeds o distace h ad for all s ad h satisfies (Mathero, 1973) : Var[ Z( s h) - Z( s)] E{{[ Z( s h) - Z( s)] E[ Z( s h) - Z( s)]} } ( h) (3) where ( h) is called variogram i spatial statistics, deotes the degree of spatial correlatio i a spatial statioary process at a give distace h, which will be calculated i 3.3. Propositio 1: I the whole regio D, spatial variogram fuctio, covariace of retur quatity betwee various ceters ad variace of retur quatity of oe ceter ca be writte as: ( h) C( h) (4) Var[ Z( s)] E{[ Z( s) m] } C( h) Cov[ Z( s h), Z( s)] E[ Z( s h) m][ Z( s) m] E[ Z( s h) Z( s)] m Propositio : I the whole regio D, spatial variogram fuctio, variace of retur quatity of oe ceter ad spatial correlatio coefficiet ca be writte as: ( h) [1 ( h)] (5) where ( h) is spatial correlatio coefficiet ad is defied as: ( h) C( h) / WEEE Retur Quatity Variogram Fuctio Retur quatity variogram fuctio ( h) explores spatial variatio degree ad correlatio structure of the retur quatity variables of various spatial positios i regio D for oe WEEE compay, ad ca also describes the variables radomess ad structure. Figure 1 shows WEEE retur quatity variogram fuctio. The three mai parameters i figure 1 are explaied as follows: 4
Variogram fuctio ( h) Rage( a ) Sill ( C C 1 ) Nugget C Distace Figure 1 WEEE retur quatity variogram fuctio Sill ( C C 1 ): i figure 1, as distace h gets large, the correlatio betwee the two ceters which is separated with the distace h becomes egligible ad the value of variogram teds to be statioary. The value of sill is the maximum height of the variogram curve. Nugget ( C ): the variace at zero distace due to measuremet or samplig error.correlatio rage ( a ): the distace at which the variace achieves the plateau. Whe h a, pairs of ceters have spatial correlatio; whe h a, the spatial correlatio is egligible. I spatial statistics, popular theoretical fittig variogram fuctios iclude spherical, expoetial, Gauss fuctios, which are empirically proved that ca effectively aalyze spatial variatio structure: 1) Spherical variogram fuctio h C C h a h a h a (6) 3 ( ) 1[1.5( / ) -.5( / ) ], ( h) C C, h a (7) 1 The covariace fuctio of spherical variogram fuctio is obtaied: C C C h () 1, C h C h a h a h a 3 ( ) 1[1-1.5( / ).5( / ) ], C( h), h a ) Expoetial variogram fuctio ( h) C C (1 e 3 h/ a ) (9) 1 The covariace fuctio of expoetial variogram fuctio is obtaied: (8) 5
C() C C ; C( h) C e 3 ha / 1 1 (1) 3) Gauss variogram fuctio 1 3( ha / ) ( h) C C (1 e ) The covariace fuctio of Gauss variogram fuctio is obtaied: 3( ha / ) 1 1 C() C C ; C( h) C e (11) (1) Propositio 1: whe the value of WEEE retur quatity variogram fuctio ( h) is icreasig with the distace h till a statioary costat (Sill, C C ), the costat reflects the 1 biggest variatio rage or total variatio degree of WEEE retur system. Propositio : whe distace h is zero, () equals C (Nugget, C ), this value reflects the size of radomess of retur quatity variable i oe ceter. Propositio 3: Let total variatio degree ( C C1) be divided by variatio rage ( C 1 ) C1 betwee various ceters, ad the we ca obtai structure variace ratio C C 1. The ratio characterizes the degree of spatial heterogeeity, which meas the spatial variability caused by the autocorrelatio occupies the proportio of the total variatio of the system. It reflects the spatial correlatio. The larger the ratio is, the stroger the spatial correlatio is. It is geerally believed that i WEEE reverse logistics whe the ratio value is less tha 5%, the variables are with weaker spatial correlatio; ad whe 5% to 75%, the variables have a moderate spatial correlatio; ad whe greater tha 75%, the variables have a stroger spatial correlatio. Propositio 4: I the whole regio D, there exists a distace a, whe pairs of ceters less tha this distace have spatial correlatio; otherwise, pairs of ceters are egligibly correlated. Krigig Spatial Model of the WEEE Returs Based o the secod order radomess ad spatial structure of WEEE retur quatity, we ca build a WEEE Krigig spatial predictio model. The retur quatity Zs ˆ( ) at the ukow ceter locatio s ca be determied by a liear combiatio of kow retur quatity Zs ( i ), see equatio (13): Zˆ( s ) m [ Z ( s ) m ] i1 i i (13) where Zs ˆ( ) is the retur quatity i ukow ceter s to be predicted, Zs ( i ) is the kow 6
retur quatity i ceter s i, i are weights characterized by the impacts of kow ceters o the value of the ukow ceter s, ad are the umber of kow ceters, ad the weights i are as well determied variables, obtaied from the coditio that the estimatio is ubiased ad variace error is miimized. 1) Ubiasedess of predictio E[ Zˆ ( s ) Z( s )] (14) E[ Zˆ ( s ) Z( s )] m { E[ Z( s )] m} E[ Z( s )] m { m m} m i i i i1 i1 (15) Whatever the weights are, the retur predictio of Krigig method is ubiased, ad it is also demostrated that Krigig method is effective. ) Miimum variace of predictio i i i j i j i1 i1 j1 (16) Var[ Zˆ ( s ) Z( s )] C( s s ) C( s s ) where deotes the variace of error betwee predictio value ad real value. To miimize the Krigig error variace, the weight coefficiets are calculated by: 1 1 C11 C1 C1 C C C 1 (17) Cij Cov( si s j ) are covariace betwee si ad s j, C ii are variace value. Proof: Whe the miimum variace of Krigig method is satisfied, the weighted Krigig liear equatio is writte as: C( s s ) C( s s ) C( s s )] i1 1 1 [ C( s s ) C( s s ) C( s s )] 1 1 1 1 1 1 [ C( s s ) C( s s ) C( s s )] 1 1 [ C( s s ) C( s s ) C( s s )] 1 1 Liear equatio system is obtaied as: C( s s ) C( s s ), j 1,, i i j j (18) (19) 7
C11 C1 1 C1 C C C 1 Krigig weights ca be calculated by equatio (), ad the to estimate the retur quatity of ukow ceters. Ad error variace ca be calculated as well: ˆ SK Var Z s Z s ic s si i1 [ ( ) ( )] ( ) (1) Accordig to equatios (13) to (1), we ca predict the retur quatity of ay ukow retur ceter i regio D, ad fially predict the total retur quatity i the whole regio. () Simulatio ad results aalysis Mote Carlo Simulatio Data Geeratio We use Mote Carlo method to geerate four samples with spatial correlatio (see table 1). The umbers of retur ceters are respectively 1, 5, ad 1, basically coverig the possible umber rage of retur ceters i practice. For sample, 3 ad 4, the simulatio is repeated for 5 times. For sample 1, sice there are 1 ceters ad too much computatio, the simulatio is repeated for 5 times. So the simulatio is totally repeated for 15 times, ad detailed statistics descriptio is show i table 1. 8% of each sample is as kow retur ceter, that is, for four samples, the umber of kow retur ceters are respectively 8, 4, 16 ad 8. The rest % of each sample, as ukow ceters, is used to be tested. Table 1 Table of Mote Carlo Simulatio Data Sample Order Number of Simulatio Miimum Maximum Stadard Media Number Ceters Repetitio times Value Value Deviatio 1 1 5 9543 541.6 5 5 8945 458.7 3 5 1 511.5 4 1 5 7849 3459.6 Spatial Structure Aalysis ad Variogram Fuctio Calculatio I spatial statistics, popular theoretical fittig variogram fuctios iclude spherical, expoetial ad Gauss fuctios. Accordig to sectio 3. ad 3.3, three variogram fuctios are fially fitted after multiple repeated simulatios. Simulatio result 1: Fittig result shows that for three variogram fuctio, all the correlatio leghth h is 9 kilometer, which meas as h becomes larger tha 9 kilometer, the spatial correlatio disappears. The comparative aalyze results of three variogram fuctios are show i table 3. The ratio 8
Number of Ceters 1 5 1 of the structural variace is the ratio of ugget to sill ad sill, C1 / ( C C1 ), reflects spatial variatio degree. The larger the ratio is, the stroger the spatial correlatio is (see sectio 3.). Simulatio result : Predictio result usig expoetial fuctio is best ad the error is smallest. From table, we ca fid that the structural variace ratio i expoetial mode, compared with spherical model ad Gauss model, has the largest value, which meas that the spatial correlatio betwee samples fitted by expoetial model is strogest. Table Quatity aalyze results of three variogram fuctio Variogram fuctio Sill(1 5 ) C+C1 Nugget(1 5 ) C Ratio of structural variace (%) predictio error percetage stadard deviatio of predictio error Spherical fuctio.8778.1488 81.64% 1.8%.15657 Expoetial fuctio.8664.163 87.8%.87%.166 Gauss fuctio.8173.44 7.17% 1.71%.1183 Spherical fuctio.8451.14717 8.15% 1.4%.1449 Expoetial fuctio.84158.891 89.41%.84%.1343 Gauss fuctio.817.185 77.8% 1.76%.14894 Spherical fuctio.91.1414 84.33%.84%.14158 Expoetial fuctio.9157.986 9.14%.48%.1841 Gauss fuctio.918.164 8.17% 1.4%.1951 Spherical fuctio.941.11194 88.1% 1.7%.16518 Expoetial fuctio.9485.55578 94.15%.87%.1438 Gauss fuctio.9349.1733 76.47%.49%.15571 WEEE Retur Predictio Result Usig Krigig Model We use two idexes of predictio error percetage ad stadard deviatio of predictio error to evaluate the accuracy of Krigig method. The predictio error percetage is that the predictio value is subtracted from actual value ad the divided by actual value, ad stadard deviatio of predictio error is the stadard deviato of Krigig predictio erro of multiple smaples. Table shows the WEEE retur predictio results of krigig model. Simulatio result 3: The total predictio error percetage is 1.1%, stadard error is.153. The result shows that the Krigig model ca accurately predicate WEEE retur quatity, ad the method of Krigig is feasible. Simulatio result 4: As for variogram fuctio, the error percetage of expoetial fuctio is.77%, lower tha spherical fuctio (1.1%) ad Gauss fuctio (1.85%). Therefore, the result of expoetial fuctio is best ad the error is smallest i sectio 4. is validated. Simulatio result 5: From perspective of the umber of ceters affect o accuracy of predictio model, the predictio error percetages o the ceters with umber of 1,5, ad 1 are 1.%, 1.1%,.91% ad 1.48% respectively, big predictio error does ot exist. It is demostrated that retur Krigig model ca be used for the WEEE retur system with differet umbers of ceters. 9
Coclusios Uder the stress of eviromet legislatio ad resource lack, logistics maagemet turs from liear mode ito closed mode. It is difficult for academic circle ad idustry to predict the returs of recyclig WEEE products due to the ucertaity of the umber of returs. From a ew perspective, we cosider spatial correlatio of retured WEEE, itroduce Krigig method of spatial statistics ad build Krigig spatial model of reverse logistics retur predictio. The coclusios ca be reached that: 1) Based o Mote Carlo simulatio, the WEEE retur spatial model preseted i this paper ca predict WEEE retur accurately, ad the total predictio error percetage is 1.1%. ) Three variogram fuctios ca be used to aalyze the spatial structure of WEEE retur. The predictio error percetage of expoetial fuctio, spherical fuctio ad Gauss fuctio is.77%, 1.1% ad 1.85% respectively, ad expoetial fuctio gets a better predictio result. Therefore, whe WEEE retur predictio based o spatial correlatio is used, expoetial fuctio is recommeded. 3) WEEE retur system ofte has differet umber of ceters. I simulatio study, the umber of ceters rages from 1 to 1, which covers rage of the umber of ceters i actual coditio. The predictio error for Krigig model is comparatively stable. 4) I Mote Carlo simulatio, WEEE retur system exists a distace a at which the variace achieves the plateau, ad whe h a, ay pairs of ceters have spatial correlatio, whe h a, the spatial correlatio is egligible. Referece: Daisuke I., Masashi Y., Yuichiro Y. 1. Ecoomic geography ad productive efficiecy of solid-waste logistics i Japa's prefectures: measuremets via the data evelopmet aalysis, GRIPS Discussio. Gómez J. M., Rautestrauch C., Nürberger A., Kruse R.. Neuro-fuzzy approach to forecast returs of scrapped products to recyclig ad remaufacturig. Kowledge-Based Systems 15(1):119-18. Hokey Mi, Chag Seog Ko, Hyu Jeug Ko. 6. The spatial ad temporal cosolidatio of retured products i a closed-loop supply chai etwork. Computers ad Idustrial Egieerig 51(1): 39-3. Kelle P, Silver E.A. 1989. Forecastig the returs of reusable cotaiers. Joural of Operatios Maagemet 8(1): 17-35. Magdalea G., Krzysztof W. 11. Reverse logistics processes i plastics supply chais. Total Logistic Maagemet 4:43-55. Mathero, G. 1973. The itrisic radom fuctios, ad their applicatios, Advaces i Applied Probability, 5: 439-468 Tokay L.B, Weu L.M, Zeios S.A.. Ivetory maagemet of remaufacturable products. Maagemet Sciece, 46(11): 141-146. Ya Y., Eric W. 9. Logistic model-based forecast of sales ad geeratio of obsolete computers i the U.S. Techological Forecastig ad Social Chage 76: 115-1114. Yig L., Nig Y. 5. Spatial data: Its ature, effects ad aalysis. Advaces i Earth Sciece, (1):49-56. 1