Match factor extensions

Transcription

1 Chapter 3 Match factor extensons For the mnng ndustry, the match factor rato s an mportanndcator wth a dual purpose: durng the equpment selecton phase, t can be used to determne an approprate fleet sze such that the truck fleet productvty matches that of the loader fleet; when a flees selected, the match factor can be used to estmate the relatve effcency of the fleet. Thus far ths rato has been restrcted to homogeneous fleets - however, heterogeneous fleets are common n large scale mnes. We present several extensons to the match factor rato to allow consderaton of heterogeneous fleets. The results of ths chapter have been publshed n the Internatonal Journal of Mnng, Reclamaton and Envronment (Burt & Caccetta 2007). 3. Introducton The mnng and constructon ndustres have long held nterests n determnng the productvty or effcency of a selected fleet of trucks and loaders. One way to study the effcency of a flees to wegh the effcences of the truck fleet and loader fleet aganst one another. servce rates. The match factor s the rato of truck arrval tmes to loader The aforementoned ndustres have used the match factor for many decades as an ndcator of productvty performance. As defned by Morgan & Peterson (968), the match factor rato, MF,, for trucks of type workng wth loaders of type s gven as MF, =, t X, (3.) where s the number of trucks of type, s the number of loaders of type,, s the tme taken to load truck type wth loader type, and t X s the average cycle tme for all trucks. We provde a summary of the notaton n secton

2 Ths rato has htherto reled on the assumpton that the truck and loader fleets are homogeneous. Thas, all the trucks are of the same type, and all the loaders are of the same type. In realty, mxed fleets are common. Heterogeneous fleets can occur when equpment types are dscontnued, equpmens superseded, or smply when a mxed flees cheaper than a homogeneous fleet. Heterogeneous fleets can occur when new equpmens purchased to work alongsde pre-exstng equpment. Is also possble that a heterogeneous fleet can represent a mnmum cost equpment selecton soluton. In ths chapter we propose new ways for defnng match factor for heterogeneous fleets. In partcular, we: Present two ways to defne match factor when heterogeneous truck fleets are present [Secton 3.2]; Present a new method to defne match factor when heterogeneous loadng fleets are operatng [Secton 3.3]; and Present a new method to defne match factor when both truck and loader fleets are heterogeneous [Secton 3.4]. The am of ths chapter s to provde extensons to the productvty and effcency measures currently avalable n the lterature. Ths wll enable greater consderaton of heterogeneous fleets. 3.2 Heterogeneous truck fleets The fleet most lkely to be heterogeneous s the truckng fleet. Ths s due to the large number of trucks requred to meet producton requrements, compared to a relatvely small number of loaders. Also, although there may often be dfferent types of loaders operatng n a mne, they are often n dfferent locatons and so can t be consdered as separate fleets. We begn by consderng the truck arrval rate n the case of a heterogeneous truck fleet wth a homogeneous loadng fleet. Defnton 3.2. The truck arrval rate, A, for a heterogeneous truck fleet wth homogeneous loadng flees the rato of the number of trucks to the truck cycle tme: A = t X, where s the number of trucks of type X (where X s the set of all truck types), and t X s the average cycle tme for all truck types. 37

3 At ths stage ths rate s unaffected by the number of truck types, as we use an average truck cycle tme. The loader servce rate s the number of trucks that are served per second. The loader cycle tme may vary between dfferent truck types. Defnton The loader servce rate, D, for loader type s gven by D = x,, where s the number of trucks of type, s the number of loaders of type X, and, s the tme requred to load truck type from loader type. As the match factor s the rato of truck arrval rate to loader servce tme, the match factor for heterogeneous truck fleets s easly derved from Defntons 3.2. and 3.2.2: MF = A D (3.2) = = x X,, t X (3.3) (3.4) = MF,. (3.5) Is clear thaf only one truck type s operatng n the fleet, then equaton (3.4) wll produce the same result as equaton (3.). Another way to thnk of ths match factor for heterogeneous truck fleets s to add the ndvdual match factors from each of the homogeneous sub-fleets. Note that the alternatve method s only approprate for the case of homogeneous loader fleets workng wth heterogeneous truckng fleets. Sometmes we would lke to use unque truck cycle tmes for dfferent truck types n the fleet. Ths can occur when trucks have dfferent routes. For example, consder a case where larger equpmens used to haul waste whle smaller trucks are used to haul ore: the waste and ore may be sent to dfferent locatons, wth sgnfcantly dfferent cycle lengths. When ndvdual truck cycle tmes are used, the tmes must be weght averaged to produce an accurate match factor. Equaton (3.4) can be easly extended to account for unque truck cycle tmes. 38

4 Defnton The average cycle tme, t X, s gven by: t X =, where s the cycle tme for truck type X and s the number of trucks of type. Now, substtutng ths new truck cycle tme nto equaton (3.4), we have the followng lemma: Lemma 3.2. For heterogeneous truck fleets wth ndvdual truck cycle tmes, the match factor for homogeneous loader fleets of type X can be represented by MF = ( ), x t. (3.6) 3.3 Heterogeneous loader fleets Ths secton consders the case of mxed loaders n the fleet, whle the trucks reman unform n type. The tme requred to load a truck may be dfferent for varous types of loaders. The loader servce rate s the number of trucks served n a defned tme perod. In a heterogeneous fleet, the tme taken to serve a truck may dffer between the varyng loader types. Lemma 3.3. The loader servce rate for heterogeneous loader fleets workng wth truck type X s gven by D =,. (3.7) Proof: The loader servce rate s the rato of the total number of trucks to the tme requred to serve them. We have, for several loader types and one truck type. The number of trucks type served by loader type n, tme s:,. Thus the total number of trucks served by all loader types n a unt of tme s D =,, as requred. 39

5 Recall that the match factor s the rato of truck arrval rate to loader servce rate. The truck arrval rate s: A =, whch gves the followng theorem: Theorem For heterogeneous loader fleets, the match factor for a homogeneous truck fleet of type X s MF =,. (3.8) When only one type of loader operates n the fleet, equaton (3.8) reduces to equaton (3.). In the case of multple dump locatons or routes, equaton (3.8) can be expanded to account for dfferng truck cycle tmes. Frst, we represent the average truck cycle tme by: t X = h,h,h (3.9) where,h s the cycle tme for truck type on route h, and,h s the number of trucks of type workng on route h. Ths gves the followng corollary: Corollary The match factor for heterogeneous loader fleets workng wth truck type (wth ndvdual truck cycle tmes for trucks on route h) can be represented by MF =, ( ) 2 ( h,h,h ). (3.0) 3.3. Example The followng example calculates the match factor of a heterogeneous loader fleet. Table 3. outlnes the equpment set. Equpment Capacty (Tonnes) Cycle Tme (seconds) 22 Truck type A Loader type B Loader type C Table 3.: Example data for a heterogeneous loader fleet wth common truck cycle tme. The cycle tme for the loader s the tme taken for one full swng of the bucket. Some trucks may need several buckets to fll ther trays. The frst step s to determne 40

6 the unque loadng tme for each truck. If the truck capacty s not a round multple of the loader capacty, then we take another scoop f the capacty lefs more than one thrd of the loader bucket sze. Ths s because t takes almost the same amount of tme to move a porton of a scoop as t does to move a full scoop (Gransberg 996). Truck type A and loader type B: Truck type A and loader type C: =2.5 3 swngs 3 35 = 05 seconds =3.6 4 swngs 4 35 = 40 seconds Ths gves the match factor: MF A = = x A x X t A, ( ) =0.88. Ths shows that the flees under-trucked. When a mnmum cost flees desred, one would reasonably expect that under-truckng would provde better solutons than perfectly matchng the fleets wth a match factor of. 3.4 Heterogeneous truck and loader fleets For heterogeneous truck and loader fleets, we consder the tme requred for each loader to serve the avalable truck fleet. Ths s equal to the sum of the number of trucks of type multpled by the tme requred to serve that truck type. We call ths the loadng tmes,, for each loader type. Defnton 3.4. The tme,, requred for a loader of type to serve the entre fleet of trucks s =,. (3.) So the tme taken for one loader to serve one truck s: x. (3.2) Thus we obtan the followng corollary: Corollary 3.4. The loader servce rate for heterogeneous trucks and loaders s 4

7 gven by D =. (3.3) As n Secton 3.3, the truck cycle tme s assumed to be constant for the entre truck fleet for that perod. Theorem The match factor for both heterogeneous truck and loader fleets can be represented by 3.2.: MF = t X. (3.4) Proof: We now consder the truck arrval rate for the entre fleet, gven n defnton MF = A D = t X = t X, as requred. If we have unque truck cycle tmes, equaton (3.4) can be easly extended to: MF = I (,I,I ) (3.5) where,i s the unque truck cycle tme for trucks n subset I X. We fnd an expresson thas equvalent to equaton (3.4) but may be smpler to mplemenn a spreadsheet by frst observng that the loader servce rate can be represented by: D = x x h= (,h ), (,h ). We take the truck arrval rate for the entre fleet from defnton 3.2. to obtan the followng theorem. 42

8 Theorem The match factor for both heterogeneous truck and loader fleets can be represented by MF =,, h= (,h ) t X. (3.6) When only one type of truck and one type of loader operate n the fleet, equatons (3.4)-(3.5) reduce to equaton (3.), as expected Example Ths example determnes the match factor of a heterogeneous truck and loader fleet. Table 3.2 presents the data set. Equpment Capacty (Tonnes) Cycle Tme (seconds) 5 Truck type A Truck type B Loader type C Loader type D Table 3.2: Example data for a heterogeneous truck and loader fleet wth common truck cycle tme. The unque loadng tmes for each truck are determned by the rule of thumb descrbed n Secton 3.3. Truck type A and loader type C: =2.5 3 swngs 3 35 = 05 seconds Truck type A and loader type D: Truck type B and loader type C: Truck type B and loader type D: =3.9 4 swngs 4 30 = 20 seconds =3.8 4 swngs 4 35 = 40 seconds =6. 6 swngs 6 30 = 80 seconds We calculate the loadng tmes,, for each loader of type. t C = = 2555 t D = = 3060 MF = =

9 Ths soluton s close to the theoretcal perfect match of.0. Ths s a good result n terms of overall effcency and productvty of the fleet. However, one should be aware that costng has not been consdered n determnng the match factor and so s possble that the fleet would be cheaper to operate even f the match factor was lower. 3.5 Dscusson For the mnng ndustry, the match factor rato s an mportanndcator whch we have extended for several lkely crcumstances, ncludng heterogeneous truck and loader fleets wth multple routes. The match factor can be used to optmse the truck cycle tme n order to gan maxmal effcency from the selected fleet. Alternatvely, project managers may use the match factor formula to determne the deal number of trucks n the fleet. The formulae presented n ths chapter are less restrcted n ther choce of equpment, select mxed fleets to sut the productvty requrements and mnmse materals handlng expense. Is nterestng to note that the Morgan & Peterson (968) match factor rato excludes watng and queung tmes for trucks and loaders. Ths may be because the watng tme for a truck flees dffcult to estmate wthout frst knowng the sze of the truck fleet. However, f we use the match factor rato as an ndex of overall fleet effcency, then s acceptable to nclude watng tmes that have been estmated by other methods. The formulae presented n ths chapter provde a sensble extenson to the orgnal equaton and brng greater accuracy to the cases where mxed fleets operate together. All of these formulae can be mplemented easly n spreadsheet software such as Mcrosoft Excel. Throughout the rest of ths thess, we employ the heterogeneous match factor ratos to ndcate the overall effcency of the selected fleets. 44

10 3.6 Summary of notaton X the set of all avalable truck types. X the set of all avalable loader types. the truck equpment type ndex, X. the loader equpment type ndex, X. MF, the match factor for homogeneous truck and loader fleets of types and respectvely. MF the match factor for heterogeneous truck fleets workng wth loader type. MF the match factor for heterogeneous loader fleets workng wth a homogeneous truck fleet of type. MF the match factor for heterogeneous truck and loader fleets. the number of trucks of type. the number of loaders of type. the cycle tme for truck type. the tme requred for a loader of type to serve the fleet of trucks., the the tme taken to load truck type wth loader type. t X the average cycle tme for all the trucks. A the truck arrval rate. A the truck arrval rate for truck type. D the loader servce rate for heterogeneous truck fleets workng wth loader type. D the loader servce rate for heterogeneous loader fleets workng wth truck type. D the loader servce rate for heterogeneous loaders workng wth a heterogeneous truck fleet. 45