PARAMETERS OF DISPERSION FOR ON-TIME PERFORMANCE OF POSTAL ITEMS WITHIN TRANSIT TIMES MEASUREMENT SYSTEM FOR POSTAL SERVICES

Transcription

1 PARAMETERS OF DISPERSION FOR ON-TIME PERFORMANCE OF POSTAL ITEMS WITHIN TRANSIT TIMES MEASUREMENT SYSTEM FOR POSTAL SERVICES Daniel Salava Kateřina Pojkarová Libor Švadlenka Abtract The paper i focued on uage of method for on-time performance evaluation of tet potal item flow ued for meaurement of the tranit time of end-to-end potal ervice for ingle piece priority mail. The reult come out from modelling of meaurement ytem for potal tranport network efficiency in proportion to real potal traffic flow with particular dicriminant characteritic. The reult of on-time performance i repreented by etimate of on-time probability, which can be defined a probability of cae, when tranit time of potal item doe not exceed jut one day. On-time probability i interpreted by variable called weighted etimate of on-time probability, which alo include ditribution of mail flow within geographical tratification according to dijunctive et of defined potal area. The rate of variability caued by ued ample deign i expreed by parameter of ample diperion in variou form and by deign factor regarding particular ample. Aement of thee calculated key parameter i the main objective of thi paper, becaue they evaluate meaurement ytem efficiency and could erve a bae for next modelling of thi ytem. Calculation of thee parameter preume fixed potal tranportation network and dometic mail flow for variou ample ize. Key word: tranit time, priority mail, ample diperion, meaurement ytem JEL Code: C02, C13 Introduction Thi paper i focued on uage of method for aement of tet potal item flow ued for meaurement of the tranit time of end-to-end tranportation ervice for ingle piece priority mail (SPPM). The reult come out from modelling of meaurement of potal tranport ytem efficiency on the bai of real potal traffic flow with particular dicriminant 1607

2 characteritic. Meaurement of tranit time i realized by flow of tet potal item in proportion to real potal traffic flow. Becaue meauring of tranit time i realized by repreentative ample of tet letter, the reult of on-time performance i repreented by etimate of on-time probability. On-time probability for tet letter can be defined a probability of cae, when tranit time of tet letter flow doe not exceed jut one day. On-time probability i interpreted by variable called value of etimate for on-time probability, caely weighted etimate of on-time probability including ditribution of letter flow within geographical tratification according to dijunctive et of defined potal area. The rate of variation caued by ued ample deign i expreed by deign factor from poible viewpoint of untratified end-to-end meauring ytem, tratified random ample and tratified end-to-end ample (ČSN EN 13850, 2013). It i related to ample deign and on-time probability etimate. Significant parameter of diperion decribing exploitation of meauring ytem are repreented by ample variance. Diperion ha very ignificant role to reflect variability feature of ample or meauring data. It can be expreed by variou tool or parameter like tandard deviation, ample variance and o on. They generally have own contruction and poibilitie of uage. Diperion meauring alo may be interpreted by variou term uch variability, cale, pread and o on. Lat mentioned pread can include more general point of view (Wilcox, 2005). Diperion can have narrower conception of variability related to given fix point (Bickel and Lehmann, 1976). Relation of ample variance and mean value can erve a example. Some known meaure of diperion conider one-dimenional data, but it ha not to be ufficient approach for contemporary need of data proceing and aement. Analyi of multidimenional data naturally give poibility to meaure diperion eparately for each variable, but poible relation between variable remain urevealed. Poible way to expre that i covariance matrix with variance of all ingle variable and covariance for all pair of variable. Thi way which lead to multidimenional ample with reearche of ome dimenion with matrix of number equal quare of thi dimenion (Kolacz and Grzegorzewki, 2016). Diperion meauring can apply alo the aggregation theory with other reflection of pread meauring with one-dimenional data (Gagolewki, 2015). It i inpiring, becaue thi theory i focued particularly on meaure of central tendency (Beliakov, Pradera and Calvo, 2007), alo known a meaure of location or centrality of obervation ample quantile a median, max,min, and then mode, arithmetic mean and o on. Thi theory i trongly 1608

3 developed (Calvo and Beliakov, 2010) and definition of aggregation function ha wide equipment of pecific mean (Grabich, Marichal, Meiar and Pap, 2011). For intance, the ample tandard deviation can be applied to evaluate coherence of managerial deciion making (Huang, Chang and Lin, 2013). There exit ome other tudie on the extended pread meaure (Calvo and Mayor, 1999), tudie connected with multi-argument ditance or relationhip between multidimenional diperion meaure and multiditance (Martín and Mayor, 2011) or tudie on method of ample quantile calculation providing nonparametric etimator by implementation in tatitical oftware package (Hyndman and Fan, 1996). Parameter of diperion for aement of potal meaurement ytem efficiency have pecific form reflecting feature and purpoe of potal operation. Thee parameter are included and characterized in next part of thi paper. The main objective of thi paper i aement of meaurement ytem efficiency by calculation of key parameter of diperion for aement of meaurement reult from viewpoint of next modelling. The reult of on-time probability etimate accuracy are related to input aumption, i.e. epecially geographical coverage of potal ervice and geographical tratification on dijunctive et of potal area. Modelling method conider one- Operator field of tudy with et potal tranportation network and conidering dometic mail flow for variou ample ize. 1 Meauring ytem characteritic and key parameter of diperion The SPPM i collected, proceed and delivered by potal operator and meauring proce ue repreentative ample of end-to-end ervice for addreed mail with et level of ervice tranit time. Deign of meauring ytem include election and allocation of tet item. Thee item are poted and received by elected panellit. Sample deign include pecification of panellit and tet item, which mut be repreentative in conideration of deign bai. Deign bai i the mot appropriate tructural information available for characterization of real mail ditributed in particular field of tudy. The reult of on-time performance mut be expreed a percentage of potal item delivered in tranit time jut one day after day of poting. Deign of meauring ytem hould enure repreentative ample of SPPM tet item in the field of tudy. The mot common way to reach repreentative ample would be deign of imple random ample (SRS) compriing real mail and then monitoring of it tranit time (ČSN EN 13850, 2013). Realization of uch deign with high accuracy of meauring would 1609

4 be unrealitic. Thu deign work with prepared tet item ent and received by elected panellit to enure high quality meauring. Thi approach to deign require tet item to integrate into exiting real mail flow with no tructural difference. Related tructure are created by all characteritic of item with ignificant influence on reult of tranit time. Sample deign thu hould be tratified according to et of dicriminant mail characteritic (DMC). Stratification mean decompoition of baic tatitical et to dijunctive and comprehenive ubet (called trata) conidered more homogenou regarding obervated characteritic than the whole baic tatitical et. Poible characteritic are repreented by tructure related to ender, receiver, potal logitic network and the whole tet item. Potal logitic network can be decribed by geographical term and geographical tructure hould reflect potal network. Characteritic of tet item can be baed on panellit, induction and delivery point and the whole tet item, and mut be reflected in meauring tructure of panellit, related point and prepared tet item. The mot frequent dicriminant characteritic (with impact on tranit time reult) are geographical area, type of payment, type of induction, time of poting, format, weight degree and method of addreing. Each characteritic ha it own poible mode important for reult of meauring proce and tranit time reult. Reult for trata mut be weighted to achieve undeviated reult. Weighted etimation for on-time probability defined a follow (ČSN EN 13850, 2013): p weighted and next related parameter are p weighted N N (1) where i value of on-time probability etimate in tratum and N i volume of real mail in geographical tratum (relation between type of potal area of induction/delivery, N i volume of real mail). Sample variance of tratified random ample (StrRS) i calculated a: vâr StrRS N pweighted: vârsrs N 2 2 N N n 1 1 (2) where v âr SRS i ample variance of imple random ample SRS and n (number of tet item in tratum ) mut be higher than 1 for all trata. Then x i number of on-time tet item in tratum and w i weight of tratum. 1610

5 All trata with any or only one valid tet item mut be left out from calculation. Weighted reult are obtained by mean value p weighted conidering real mail trata weight (RSW) in form of ratio N /N, while ample variance i gained by RSW quare (N /N) 2. StrRS 2 n 1 2 n x x weighted w vârsrs w 2 1 n n n 1 n v âr (3) Sample variance of imple random ample (SRS) v âr SRS p ˆ i calculated a follow: 1 1 n x v ârsrs 1 (4) 2 n 1 n n 1 n where i value of on-time probability etimator, n i volume of tet item and x i figure of on-time tet item. Sample variance of tratified end-to-end ample (StrEtE) i formed a: where vâr vârete StrRS weighted vârstrete weighted : (5) vâr p meaurement ytem (EtE). SRS v âr ˆ EtE i ample variance of on-time probability etimate in end-to-end Meauring ytem work with parameter of deign factor to reflect covariance and weighing impact more tranparently. In cae of SRS minimum ample ize i calculated directly on the bai of expected or required deign accuracy. Sample deign hould be more ophiticated and complex with conideration of certain principle, when impact of induction and delivery point on meaured variation (covariance), tratification by weighting ytem and appropriate probability ditribution for accuracy calculation hould be integrated into ontime probability calculation. More complex ample then ha in mot cae higher level of variance. Calculation of accuracy for untratified end-to-end ample how lo of accuracy due to correlation impact among tet item. Thee corellation would exit among item ent in the ame point of induction, received in the ame point of delivery or ditributed by relation of the ame point of induction-delivery. The rate for added variance i expreed by deign factor df, which meaure lo of accuracy. It i defined a ratio of ample variance of on-time probability etimator in et 1611

6 ample deign to ample variance of on-time probability etimator in SRS of the ame ize. Deign factor i alway related to et ample deign and probability etimator (ČSN EN 13850, 2013). Accuracy of any end-to-end ample deign increae not only by higher volume of item, but alo by higher number of ender and/or receiver. Deign factor particularly depend on number of ued point of induction and delivery. Lower level of deign factor i reached by higher number of thee point. Thi fact create compromie ituation between panel ize and ample ize. Larger panel enable to ue more point of induction and delivery and it lead to lower level of deign factor and ample ize. Deign factor thu relate to minimum ample ize, which i directly proportional to deign factor. It i appropriate to apply mall number of tet item belonging to concrete point of induction/delivery to reduce corellation impact. Deign factor parameter are defined by following form (ČSN EN 13850, 2013). Deign factor for end-to-end untratified meauring ytem df EtE and for tratified random ample df StrRS are defined a follow: df df vâr EtE EtE: (6) vârsrs vâr weighted n 1 1 StrRS StrRS : (7) weighted weighted Deign factor for end-to-end tratified meauring ytem df StrEtE following form: i defined in df vâr vârstrrs weighted n 1 1 EtE StrEtE: df EtE df StrRS (8) vârsrs weighted weighted 2 Modelling of tet item ample and reult of parameter Modelling of tet item ample i baed on parameter of geographical coverage by potal ervice and tratification of meauring ample for two period of Modelling preume one-operator field of tudy with dometic SPPM for variou ample ize in proportion of deign bai. Ued ample ize are neceary to cover all potal area with concrete flow of tet item, which mut fulfil requirement of proportionality with deign bai of real SPPM flow. The firt modelling ample conider one-month period with preumed number of

7 panellit, the econd one conider two-month period with preumed number of 144 panellit. Meauring ytem modelling for both period i baed on relevant indicator figure. Thee indicator n, x, p and âr p ˆ EtE weighted v compried in Table 1 are neceary data for key diperion parameter calculation. Applying thee input variable, calculated reult of diperion parameter uing above-mentioned equation follow: Tab. 1: Input indicator and diperion parameter reult of modelling Parameter 1 t period 2 nd period n x p weighted 0, , p v âr ˆ 1,16486E-04 5,82863E-05 EtE p v âr ˆ 6,2168E-05 2,716958E-05 SRS v âr StrRS p 6,0953E-05 2,72279E-05 weighted v âr 1,14209E-04 5,84115E-05 StrEtE weighted df EtE 1, , df StrRS 0, , df StrEtE 1, , Source: author Calculated reult of key parameter of diperion give atifactory value. Sample variance have very mall value and deign factor do not reach high value a well. Reult of 1 t and 2 nd period are naturally influenced by utilization of et meauring ytem and ditribution of tet item within panel among other. Deign factor depend on number of tet item belonging to point of induction and delivery and alo on ditribution of tet item in exiting relation between point of induction and delivery. The number of induction point and ditribution are ignificant point of variation. Stabilization of deign factor require extended deign not only with more tet item, but alo with more induction point. The number of induction point could be increaed to reduce exiting correlation of induction point and to minimize deign factor. It lead to larger ize of ender panel. Due to incorporation of new ender, plan for aignment of tet item, which aign letter to 1613

8 individual relation between point of induction and delivery, could be improved. Thi improvement could be realized by reduction of exiting acummulation of tet item in the ame relation between point od induction and delivery. It lead to higher ditribution of tet item in exiting relation. Thi model deign require approximately twice a much tet item to reach the ame accuracy of SRS, which ue each ender and receiver only one time (df EtE ). Similar reult i given by aumption of SRS by calculation of ample ize, when concrete deign would require in thi cae alo approximately twice a much tet item to reach the ame level of accuracy (df StrEtE ). Uing Z-multiple of induction point the number of exiting relation between point of induction and delivery increae approximately by Z 2 -multiple. It follow from thi fact, that tabilization factor of panel expanion will lie in range from (df StrEtE 2) 1/2 the ame volume of letter per relation between point of induction and delivery to df StrEtE 2 (the ame volume of letter per induction point). Concluion Defined requirement on potal meauring ytem within quality of ervice management, beide main aim of meauring (etimator of tranit time of end-to-end ervice for SPPM), have to meet aim related to accuracy and proportionality to real mail flow including parameter of diperion. Received figure of parameter by meauring ytem modelling are ufficiently atifactory. Value of ample variation are very low and deign factor bring relatively good reult a well. Mentioned increae of panellit number with correponding additional number of tet item i poible olution, but panel etting and tet item ditribution to panellit i in practice compromie between expenivene of meauring ytem and acceptable reult of meaurement. Conidering applied ytem of modelled tet letter flow, next modelling baed on proportional parameter hould give reliable reult a well. Acknowledgment Contractual reearch Audit of meaurement of the tranit time of potal item according to norm ČSN EN ( ) for Czech Telecommunication Office 1614

9 Reference Beliakov, G., Pradera, A., & Calvo, T. (2007). Aggregation Function: A Guide for Practitioner. Springer. Bickel, P. J., & Lehmann, E. L. (1976). Decriptive Statitic for Nonparametric Model (III. Diperion). Annal of Statitic, 4(6), Calvo, T., & Beliakov, G. (2010). Aggregation Function Baed on Penaltie. Fuzzy Set and Sytem, 161(10), Calvo, T., & Mayor, G. (1999). Remark on Two Type of Extended Aggregation Function. Tatra Mountain Mathematical Publication, 16, Gagolewki, M. (2015). Spread Meaure and their Relation to Aggregation Function. European Journal of Operational Reearch, 241(2), Grabich, M., Marichal, J.-L., Meiar, R., & Pap, E. (2011). Aggregation Function: Mean. Information Science, 181(1), Huang, Y.-S., Chang, W.-C., & Lin, Z.-L. (2013). Aggregation of Utility-baed Individual Preference for Group Deciion-making. European Journal of Operational Reearch, 229(2), Hyndman, R. J., & Fan, Y. N. (1996). Sample Quantile in Statitical Package. The American Statitician, 50(4), Kolacz, A., & Grzegorzewki, P. (2016). Meaure of Diperion for Multidimenional Data. European Journal of Operational Reearch, 251(3), Martín, J., & Mayor, G. (2011). Multi-argument Ditance. Fuzzy Set and Sytem, 167(1), Wilcox, R. R. (2005). Introduction to Robut Etimation and Hypothei Teting. Elevier. ČSN EN 13850: Poštovní lužby - Kvalita lužby - Měření přepravní doby lužeb mezi koncovými body pro jednotlivé prioritní záilky a záilky první třídy (2013). Úřad pro technickou normalizaci, metrologii a tátní zkušebnictví. Contact Ing. Daniel Salava, Ph.D. Univerity of Pardubice Jan Perner Tranport Faculty Studentká 95, Pardubice daniel.alava@upce.cz 1615

10 Ing. Kateřina Pojkarová, Ph.D., doc. Ing. Libor Švadlenka, Ph.D. Univerity of Pardubice Jan Perner Tranport Faculty Studentká 95, Pardubice 1616