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Selection of the Longwall Face Crew with Respect to Stochastic Character of the Production Process – Part 1 – Procedural Description / Wyznaczanie Obsady Przodka Ścianowego Z Uwzględnieniem Stochastycznego Charakteru Procesu Produkcyjnego. Cz. 1 – Opis Metody

Selection of the Longwall Face Crew with Respect to Stochastic Character of the Production... Arch. Min. Sci., Vol. 57 (2012), No 4, p. 1071­1088 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.2478/v10267-012-0071-9 RYSZARD SNOPKOWSKI*, MARTA SUKIENNIK* SELECTION OF THE LONGWALL FACE CREW TH RESPECT TO STOCHASTIC CHARACTER OF THE PRODUCTION PROCESS ­ PART 1 ­ PROCEDURAL DESCRIPTION WYZNACZANIE OBSADY PRZODKA CIANOWEGO Z UWZGLDNIENIEM STOCHASTYCZNEGO CHARAKTERU PROCESU PRODUKCYJNEGO. CZ. 1 ­ OPIS METODY A proposal of the method aimed at the longwall face crew selection th respect to stochastic character of the production process has been described in this study. Modules, which can be isolated from the production cycle, as well as methods of determination of the probability function density describing duration of individual action realized in production process, have been described in the first part of the study. Procedure of crew selection of individual modules, including optional crew selection, has been described in next chapters. Statement of action, which should be executed in order to apply the proposed method, including final conclusions, is discussed in the last chapter. Keywords: longwall face crew selection, probability function density, production cycle, longwall face Zagadnienie wyznaczania obsady przodka cianowego jest przedmiotem bada i analiz praktycznie od momentu rozpoczcia stosowania systemu cianowego w kopalniach wgla kamiennego. Metoda opisana w niniejszej pracy uwzgldnia jednak czynnik dotychczas nie uwzgldniany w opracowaniach z tego zakresu, a mianocie stochastyczny charakter realizowanego w przodku procesu. Pocztki prac z zakresu analizy funkcjonowania przodków cianowych z uwzgldnieniem stochastycznego charakteru procesu produkcyjnego sigaj lat 90 ­ tych, kiedy zaczto wykorzystywa metod symulacji stochastycznej jako metod badawcz. Pierwszym krokiem w proponowanej metodzie jest podzial procesu produkcyjnego na moduly. Kryterium podzialu stano sposób realizacji poszczególnych czynnoci lub operacji w danym module. Zaproponowano cztery rodzaje modulów i oznaczono odpoednio literami od A do D. Moduly typu A to moduly z czynnociami wykonywanymi w sposób równolegly, wród których wystpuje tzw. czynno odca. Czynno odca jest to taka czynno, której realizacja nie ponna by wstrzymywana z powodu zbyt wolnego wykonywania pozostalych czynnoci wystpujcych w tym module. Moduly typu B to takie, w których czynnoci lub operacje wykonywane s w sposób równolegly, ale wród niech nie wystpuje czynno odca. Czynnoci wykonywane w sposób szeregowy * AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MINING AND GEOENGINEERING, DEPARTMENT OF ECONOMICS AND MANAGEMENT IN INDUSTRY, A. MICKIECZA 30 AVE., 30-059 KRAKOW, POLAND charakteryzuj moduly typu C. W modulach tych moe by wykonywana dowolna ilo czynnoci w ukladzie szeregowym, dodatkowo czynno pojedyncz traktuje si jak szeregow. Moduly typu A, B i C wyodrbnione s z cyklu produkcyjnego na rysunku 1. Cech charakterystyczn modulów typu D jest wystpowanie czynnoci lub operacji zarówno w ukladzie równoleglym, jak i szeregowym. Na rysunku 2 zamieszczono przyklad takiego modulu. Kolejnym krokiem w metodzie wyznaczania obsady przodka cianowego jest wyznaczenie funkcji gstoci prawdopodobiestwa, opisujcych czas realizacji poszczególnych czynnoci w ramach wyodrbnionych modulów. Schemat wyznaczania funkcji opisujcych czas trwania czynnoci lub operacji w ramach modulów zamieszczono na rysunku 3. Przestaony schemat zaklada zebranie danych pomiarowych a nastpnie przeprowadzenie analizy statystycznej, która polega na wyznaczeniu funkcji aproksymujcych f1,i,j, majcych wlasnoci funkcji gstoci prawdopodobiestwa. Funkcje te opisuj czas realizacji czynnoci lub operacji ,,i"-tej wykonywanej w ramach danego modulu ,,j "-tego, na odcinku jednego metra. Nastpnie wyznacza si splot otrzymanych funkcji w celu wyznaczenia funkcji splotowych fi,j , które opisuj czas realizacji czynnoci lub operacji ,,i"-tej w danym module ,,j"-tym. Otrzymane funkcje splotowe maj wlasnoci funkcji gstoci prawdopodobiestwa. Mona je wyznaczy dema metodami: metod analityczn lub metod symulacyjn. W metodzie analitycznej wykorzystuje si definicj splotu funkcji, natomiast w metodzie symulacyjnej schemat postpowania, który zamieszczono na rysunku 4. Jeeli w module znajduj si czynnoci lub operacje, które mog by wykonywane przez rón liczb pracowników (obsad), wówczas funkcja fi,j wyznaczana jest dla kadego wariantu obsady z osobna. Symbolem ,,k " oznaczono obsad, dla której funkcja fi,j zostala wyznaczona. Po wyznaczeniu funkcji gstoci prawdopodobiestwa, opisujcych czas realizacji poszczególnych czynnoci, nastpuje wyznaczenie obsady w ramach poszczególnych modulów. W zzku z wydzieleniem trzech typów modulów przedstaono algorytmy wyznaczania obsady uwzgldniajce to zrónicowanie. Algorytm wyznaczania obsady dla modulów z czynnociami lub operacjami równoleglymi i odcymi przedstaony jest w rozdziale 4.1. Na rysunku 5 zamieszczono przyklad modulu, w którym wystpuje czynno odca a nastpnie z wykorzystaniem wzorów od 1 do 4 opisano procedur postpowania przy wyznaczaniu obsady w modulach typu A. Algorytm wyznaczania obsady dla modulów z czynnociami lub operacjami równoleglymi bez odcych opisano w rozdziale 4.2. Na rysunku 6 zmieszczono przykladowy modul z dema czynnociami równoleglymi, z których adna nie jest odc. Wzorami od 5 do 10 opisano procedur wyznaczania obsady w modulach typu B. Moduly typu C oraz schemat wyznaczania obsady opisane s w rozdziale 4.3. Rysunek 7 prezentuje przykladowy modul z dema czynnociami szeregowymi, a wzory od 11 do 15 przedstaaj proces wyznaczania obsady w modulach tego typu. Algorytm wyznaczania obsady dla modulów z czynnociami lub operacjami wykonywanymi szeregowo i równolegle zaprezentowany jest w rozdziale 4.4. Na rysunku 8 zamieszczono przykladowy modul, a wzory od 16 do 26 prezentuj procedur wyznaczania obsady w modulach typu D. Zaproponowana metoda zaklada wykorzystanie funkcji gstoci prawdopodobiestwa czasów trwania czynnoci do wyznaczania obsady przodka wydobywczego. W metodzie wykorzystano odmienne od deterministycznego podejcie, polegajce na traktowaniu czasów realizacji czynnoci jako zmiennych losowych. Zastosowanie opracowanej metody wymaga realizacji szeregu czynnoci, z których najwaniejsze to: ­ identyfikacja kluczowych czynnoci w procesie produkcyjnym, ­ podzial procesu produkcyjnego na charakterystyczne moduly, ze wzgldu na jednoczesno realizacji czynnoci, ­ identyfikacja funkcji gstoci czasów trwania czynnoci w wydzielonych modulach, ­ przyjcie wstpnych wariantów obsady dla poszczególnych modulów ­ optymalizacja obsady w modulach poprzez uwzgldnienie prawdopodobiestw realizacji czynnoci przy zaloonej obsadzie z uwzgldnieniem charakteru modulów. Mona take zauway, ze: 1. Kady proces produkcyjny mona podzieli na skoczon liczb modulów rónicych si jednoczesnoci realizacji czynnoci. 2. Wyodrbnianie z procesu produkcyjnego modulów, pozwala na latejsz analiz procesu produkcyjnego, a co za tym idzie ulata dobór obsady. 3. Uyte w metodzie kryterium prawdopodobiestwa osignicia zaloonego czasu trwania realizacji modulu, pozwala na racjonalny dobór obsady, gdy realizacja modulu jako caloci ma wyszy priorytet ni realizacja poszczególnych czynnoci. Slowa kluczowe: obsada przodka cianowego, funkcje gstoci prawdopodobiestwa, cykl produkcyjny, przodek cianowy 1. Introduction Studies aimed at the analysis of longwall face operations th respect to stochastic character of the production process were started in the nineties of the last century (Snopkowski, 1990, 1994). Studies aimed at the analysis of production of coal excavated from longwall faces th respect to chosen probability distributions were continued in the next decades (Snopkowski, 2000a, 2000b, 2002). Stochastic simulation method was used as the research method (Snopkowski, 2005, 2007a, 2007b, 2009) and (Snopkowski & Napieraj, 2012). Thus the problem of the longwall face crew selection was started at the same time as longwall exploitation system was applied in hard coal mines. However, the method described in the present study takes under consideration not considered so far factor, i.e. stochastic character of the process realized in the longwall face (Sukiennik, 2011). Division of the production process into modules is the first procedural step of the method of question. Manner of the realization of individual activities or operations in given module is considered as a criterion of this division. 2. Division of the production process into modules In the production process realised thin the longwall face of hard coal mines we can isolate its smaller fragments, which are determined as "modules" in the present study. Such partition is aimed at development of the crew selection method, which takes under consideration different character of these modules. The modules differ from each other th manner of realization of individual activities or operations. Detail information related th procedure of defining particular modules are described in next chapters of this study. 2.1. Modules comprising activities or operations executed in parallel To this group of the production process are prescribed activities or operations, which are executed in parallel. Modules th co called leading activities and modules thout such activities can be distinguished. Modules th leading activities or operations are characterized th two features. The first feature comprises parallelism of the executed activities or operations and the second one is characterized th presence of so called leading activity. Example of the module of this type marked th symbol "A" is shown in Fig. 1. Module marked th symbol ,,A" comprises fragment of the production cycle, in which shearer cuts coal rock body and is followed by support and conveyor shifting, which are moved toward the longwall face. Shearer cutting is considered as leading activity in this module. This activity determinates two other activities. In practice it means that the shearer shouldn't wait for the support or conveyor shift. Thus shearer standstill on the section (L ­Xp) shouldn't take place because it could result in daily output decrease. Module of type ,,A" occur anywhere where fragments of the parallel activities or operations are present, in which single activity is considered as leading activity. Fig. 1. Scheme of the production cycle for two way shearer mining th isolated modules. Source: (Snopkowski, 1997) th modifications Parallelism and lack of leading activity or operations is a characteristic feature of module thout leading activities or operations. Example of the module of this type marked as "B" s shown in Fig. 1. Operation of the drive unit shift when the shearer is stopped in a distance (x1 + x2 + x3) from the driver unit occurs in this module. Shearer is exposed to short maintenance th eventual replace of cutting picks. It should be noted that in case of other mining processes we can also isolate modules, which in Fig. 1 are marked as ,,A" and ,,B". 2.2. Modules th activities and operations executed in series Lack of activities and operations executed in parallel is the most important feature of the module of this type, which can be isolated fro production process. Example of the module of this type marked as ,,C" is shown in Fig. 1. This is shearer advance, during which no other activities or operations are executed. In case of several operations executed in series, the operations are classified as the operations of the same module. 2.3. Modules and operations executed in series and in parallel Occurrence of activities or operations both in parallel and in series system is a characteristic feature of the module of this type. Example of such module is shown in Fig. 2. activity 1 (operation) activity 2 (operation) activity 3 (operation) acitivity 4 (operation) module ,,j" Fig. 2. Example of module of the type D th activities or operations executed In series and in parallel. Source: Authors materials In this module occur both activities (operations) executed in the in series, i.e. marked th sequent digits 1,2 and 4, and also 1 and 3, and parallel system comprising activities (operations) 2,3 and 4. 3. Developing the functions describing activity or operation duration executed thin the modules Scheme of the development of functions describing duration of activities and operations thin individual modules is shown in Fig. 3. Realization of the scheme presented in Fig. 3 is preceded by collection of suitable data in conditions of concrete longwall face. For example, for a module marked as "A" in Fig. 1, the data comprise shearer operational speed, support advance rate and conveyor shifting rate. Then the data are exposed to statistical analysis comprising determination of approximation functions f1,i, j, having properties characteristic for probability density function. These functions describe time of realization of "i" activity or operation in scope of given "j" module on a distance of 1 meter. Next procedural step shown in the scheme comprises development of obtained convolution in order to determine convolution functions fi, j, which describe time of realization of "i" activity or operation thin "j " module. Obtained convolution functions have properties characteristic for probability density functions. They can be determined th use of two methods: analytical method or simulation method. Definition of convolution function is used in analytical method and the procedural scheme is shown in Fig. 4. measurement data {t1, t2,...,tN} statistical analysis made in order to determine func tions f1,i,j describing duration of activities or operations on a distance if 1 m determining of convolu tion function f1,i,j by analytic method determining of convolu tion function f1,i,j by simulation method h d d functions fi,j describing duration of activity or opera tions thin module Fig. 3 Scheme of determining functions describing duration of activities or operations in the module. Source: Authors materials Range of activity realized in given module is marked th symbol "S" in the scheme. If this range is for example 15 m, S = 15. Assumed number of simulations is marked th symbol "LS ". If the module comprises activities and operations, which can be executed by various number of workers (crew) the function fi, j is determined for each crew selection variant separately. Marking fi,kj, where symbol "k" comprises crew, for which function fi,j was determined has been introduced in order to differentiate the mentioned functions. Procedure of selection the crew thin the k modules th use of the function fi,j. is characterized in next parts of this study. ^d Zd ^hD ^hD ^hD < ^ < < ^hD > >^ > > >^ Fig. 4. Scheme of determining the function th use of simulation method. Source: Authors materials 4. Selection of crew for individual module of the production process Crew of the whole production process realized in the longwall face is determined for each module of the production process. In order to select four module types, algorithms of crew selection taking into account this differentiation is described in next parts of this study. 4.1. Algorithm for the crew selection for modules th parallel and leading operations According to mentioned description, leading activity or operations is defined as activity or operation, which realization shouldn't be stopped thin given module (it shouldn`t "wait" for execution of other activities or operations). Operation of the shearer in longwall face is an ex ample of such activity in production process. The shearer shouldn't wait until the mechanized support of conveyor is shifted. Each stoppage of the shearer advance is inconvenient from the production efficiency point of view. Thus in the module, in which leading activity occurs, the crew selection procedure takes under consideration the mentioned conditions. Example of module, in which leading activity occurs is shown in Fig. 5. Fig. 5. Example of the module th leading activity (operations). Source: Authors materials Symbols used in Fig. 5: f1, j f2, j f2, j f3, j k =2 k =2 k =4 ­ function density, describing duration of activity (operation) 1 in module "j" for crew k = 2, ­ density function, describing duration of activity (operation) 2 in module "j" for crew k = 2, ­ density function, describing duration of activity (operation) 2 in module "j" for crew k = 4, ­ density function describing duration (operation) 3 in module "j " for crew k = 3. Module "j " can be realized for two crew variants: ­ ­ Variant I ­ 7 persons crew, activities (operations) realized according to function k f1,kj=2 , f2,kj=2 , f3,j=3, ­ 9 persons crew, activities (operations) realized according to function k =2 k =4 f1, j , f2, j , f3,j . Scheme of the crew selection in module "j" is divided into the follong stages: Stage I Calculation of the value t0 for complex probability p according to formula: t0 f1,k j= 2 (t) dt = p (1) where: p -- probability of realization of the leading activity in time shorter than t0 (level 0,95 is recommended), t0 -- determined boundary value of the leading activity (operations) in module "j", for which probability of exceeding amounts for 1 ­ p (for p = 0,95, this probability amounts for only 0,05) Stage II Calculation of the probability of activities (operations) realization, which are executed in time sorter than t0 for all variants of the crew selection module "j". Variant I In the first variant, the crew selection is realized by 7 persons and activities (operations) k =2 k =2 are characterized by functions f1,j , f2,j , f3,j . The follong integrals marked th symbols 2 and 3 should be calculated in order to calculate the probabilities: t0 f k =2 (t) dt 2, j t0 f (t ) dt 3, j k = p2,= 2 j (2) (3) = p3, j In the second variant the crew is realized by 9 persons and the activities (operations) are k =2 k =4 characterized by functions f1, j , f2, j , f3, j . The searched probabilities are calculated from fomulas: f k = 4 (t )dt 2, j whereas p3, j from formula 3. t0 k = p2,=j 4 (4) Stage III Procedure of the variant of the crew selection on the basis of values of calculated probabilities. Three options are distinguished in this procedure: k =2 k =4 Option 1. All calculated probabilities are greater than p, thus p2, j > p; p3, j > p; p2, j > p (for module "j"). In practice it means that in no crew selection variant occur, which can stop leading activities (operations). In this case, variant of the crew selection th the minimal number of workers should be selected. In the cited example it is variant I, for which the crew consists of 7 persons. Option 2. Only in one variant the calculated probabilities are greater than p. That means that the only one variant of the crew selection assures execution of activities (operations) in the module in such manner that the realization of the leading activity is not stopped. In this case, crew selection assumed in this variant is taken as the module crew. Option 3. All calculated probabilities are smaller than p. For the ex ample value it means k =2 k =4 that: p2, j < p; p3, j < p; p2, j < p In practice it means that none crew variant assures execution of the activity (operation) in sorter time than the leading activity, i.e. the leading activity ll be stopped. In such situation possibility of crew size should be considered. If there is no such possibility, variant for which calculated probabilities are closest to value p are considered. Based on the dada from example the follong quotients should be calculated: ­ for variant I: ­ for : k k p2,=2 + p3,=3 j j k k p2,= 4 + p3,=3 j j , . Variant, for which the quotient is larger, should be considered as the variant of the crew selection of module "j ". Analogical calculations can be executed for selection of optimal variant of the crew selection if only a part of calculated probabilities satisfies assumed criterion (are smaller than p). In practice it means that activities (operations), which can stop execution of the leading activity can occur in each variant of the crew selection procedure. However, follong the mentioned calculation procedure we are able to select such crew, which assures minimized stoppages of activity (operation), which is considered as leading activity. 4.2. Algorithm of crew selection for a module th parallel activities or operations thout leading operations Example of the model th two parallel operations, where none of them is leading. The follong symbols were used in Fig: f1, j f1, j ­ density function, describing time of the activity (operation) 1 realization in module "j" for the crew k = 1, ­ density function, describing time of the activity (operation) 1 realization in module "j" for the crew k = 2, k =2 Fig. 6. Example of the module th activities (operations) executed simultaneously thout leading activity Source: Authors materials f2,kj=2 ­ density function, describing time of the activity (operation) 2 realization in module "j" for the crew k = 2. Module "j" can be realized in two options: ­ ­ Variant I ­ 3 person crew, activities (operations) realized according to function k =2 f1, j , f2, j , ­ 4 person crew, activities (operations) realized according to function k =2 k =2 f1, j , f2, j . Procedure of the crew selection in module th paralel activities (operations) thout leading operations comprises the follong stages: Stage I Value of the boundary module realization time, which probability indicates that the limit ll not bee exceeded is p, is calculated for each crew variant. Thus further calculations executed th respect to assumed probability level p. value of p = 0.95 are recommended, (like confidence level), although specific value depends on person performing the calculations. The calculations are as follow: Variant I k t1,=1 j k t 2,=j2 f1, j (t) dt = p f 2, j (t) dt = p k =2 (5) (6) k k t0 I = max t1,=1 ; t2,=j2 j W (7) where: W t0 I -- boundary module realization time for variant I, not exceeding probability amounting for p. k t1, =2 j f1, j (t )dt = p k =2 (8) t 2, j k =2 -- calculated from formula 6, k k t0 II = max t1, =2 ; t2,=2 j j W (9) where: t0 II -- boundary module realization time for , not exceeding probability amounting for p. Stage II Calculation of this variant of the crew selection , assures the shortest realization time of this module, thus: (10) For ex ample, if the time is the shortest t0 II (for ), the crew, which should realize activities (operations) in module "j" comprises 4 persons, whereas the activity (operation) ,,2" should be realized also by two persons. The calculations were executed for assumed probability level p. 4.3. Algorithm of the crew selection for module th activities (operations) executed in series Example of module th two activities executed in series is shown in Fig. 7. The follong symbols were used in Fig. 7: f1, j f1, j f2, j ­ density function, describing time of activity (operation) 1 realization in module "j" for crew k = 1, ­ density function, describing time of activity (operations) realization 1 in module "j" for crew k = 2, ­ density function, describing time of activity (operations) realization 2 in module "j" for crew k = 3. k =2 Fig. 7 Example of the module th activities (operations) executed in series Source: Authors materials Module "j" can be realized in two variants: ­ Variant I ­ 4 person crew, activities (operations) realized according to function f1, j , f2, j , ­ ­ 5 person crew, activities (operations) realized according to function f1, j , f2, j . Scheme of the module crew selection th activities (operations) executed in series comprises the follong stages Stage I Function describing realization time of module "j" is determined for each crew variant. These functions are determined th use of convolution operations and it has the follong form: Variant I k =2 where: k k = f1, j=1 Ä f 2, =3 j (11) -- density function of random variable time of realization of module "j" for variant I. where: k = f1,kj=2 Ä f 2, =3 j (12) -- density function of random variable time of realization of module "j" for . Stage II Boundary value of time of realization of module "j", not exceeding probability amounting for p is determined for each crew selection variant. The calculations are as follow: Variant I W t0 I f j I (13) where: t0 I -- boundary realization time for variant I, not exceeding probability p. t0 f j II (t )dt = p (14) where: t0 II -- boundary realization time for , not exceeding probability p. Stage III Determination of the crew selection variant , which assures the shortest realization time of given module: (15) For example, if the time t0 II is the shortest (for ), the crew which should realize the activities (operations) in module "j" amounts for 5 persons, whereas realization time of this W module can not exceed value t0 II th probability p, if the first activity (operation) ll be executed by two persons and the second activity (operation) is executed by three persons. 4.4. Algorithm of the crew selection for module th activities or operations executed both in parallel and in series Example of module th activities executed in series is shown in Fig. 8. The second activity can be executed in two crew variants. Symbols used in Fig. are as follow: f1, j f2, j f2, j f4, j k =2 ­ Density function describing time of realization of activity (operation) 1 in module "j" for crew k = 2, ­ Density function describing time of realization of activity (operation) 2 in module "j" for crew k = 2, ­ Density function describing time of realization of activity (operation) 3 in module "j" for crew k = 2, ­ Density function describing time of realization of activity (operation) 4 in module "j" for crew k = 2. k =2 k =2 k =2 Fig. 8. Example of the module th activities (operations) executed in series and in parallel. Source: Authors materials Module "j" can be realized in two crew variants: ­ Variant I ­ 8 persons crew, activities (operations) realized according to function f1, j , f2, j , f3,j , f4,j , ­ ­ 10 persons crew, activities (operations) realized according to function f1, j , f2, j , f3,j , f4,j . Scheme of crew selection in the module th activities (operations) executed both in series and in parallel, comprises the follong stages: Stage I Calculation of the module duration th use of complex probability level p, for each crew variant. Concept of full paths was introduced into calculations. Full paths are defined as a set of successive activities (operations), from which the first activity is started in the initial point of the module and the last one is finished in final point of the module. Two full paths marked th symbols S1 and S2 can by distinguished for the module presented in Fig. 8. Path S1 comprises activities (operations) 1,2 and 4, whereas path S2 comprises activities (operations) 1 and 3. Functions describing duration of full paths are developed for all crew variants, as convolution of suitable functions in the follong manner: Variant I k =2 k =4 k =2 k =2 k =2 k =2 k =2 k =2 S , k k k = f1, j= 2 Ä f 2, =2 Ä f 4, =2 j j (16) (17) S2 , k = f1,kj=2 Ä f 3, = 2 j where: S1, -- density function describing duration of path S1 for crew variant in module "j", S2, -- density function describing duration of path S2 for crew variant in module "j". S , k k k = f1, j=2 Ä f 2, = 4 Ä f 4, =2 j j k k = f1, j= 2 Ä f3, =2 j (18) (19) where: S1, S2, S 2 , -- density function describing duration of path S1 for crew variant in module "j", -- density function describing duration of path S2 for crew variant in module "j". Stage II Duration of module "j" th not exceeding probability p is determined for each crew variant. Calculations for each variants are as follow: Variant I S S The follong integrals should be calculated in order to determine time t0 1, and t0 2,: S t0 1, S t 0 2, 1 S , (20) S 2 , (21) where: t0 1, -- duration of path S1 in crew variant , not exceeding probability p, t0 2, -- duration of path S2 in crew variant , not exceeding probability p. Duration of module "j" for crew variant , not exceeding probability p, is calculated from formula: S S t0 I = max t0 1 where: t0 { S ,W ; t0S ,W } I 2 I (22) -- duration of module "j" for crew variant , not exceeding probability p. S ,W S ,W In order to determine time t0 1 II and t0 2 II we should calculate: t0S 2 , t0S2 , S , (23) S2 , (24) where: t0 1 S S , -- duration of path S1 in crew variant , not exceeding probability p, t0 2, -- duration of path S2 in crew variant , not exceeding probability p. Duration of module "j" for crew variant , not exceeding probability p, is calculated from formula: t0 II = max t0 1 where: t0 { S ,W ; t0S ,W } II 2 II (25) -- duration of module "j" for crew variant , not exceeding probability p. Stage III For handling the activity (operation) of module "j" should be assumed such crew variant , which assures the shortest time of the module realization, thus: {W (26) where: W t0 i -- the shortest time of realization of module "j" for crew variant . For example, if time t0 II (for ) is the shortest, the crew, which should realize activities (operations) in module "j" amounts for 10 persons, whereas the first activity (operation) should be executed by two persons, the second activity (operation) ­ four persons, the third activity (operation) by two persons and the forth activity (operation) also by two persons. The calculations were executed th assumed probability p. 5. Final conclusions Observations of durations of activities (operations) executed in production process confirm the thesis that these times are exposed to some fluctuations resulting from mining and geological conditions. The method proposed in this study assumes application of probability density function of durations of the executed activities for the longwall face selection. Different from the deterministic approach, i.e. considering the activity realization times as random variables was applied in the proposed method. Application of the developed method requires execution of some activities, from which the most important are: ­ identification of key activities of the production process, ­ division of the production process into characteristic modules, th respect to simultaneity of the activities realization, ­ identification of the density functions of the durations of executed activities thin separated module, ­ assuming preliminary crew variants for individual module, ­ optimization the crew selection in module via taking under consideration probabilities of the activities realization for assumed crew, according to the module character. It should be also noted that: 1. Each production process can be divided into finite number of modules differing th simultaneity of the activity realization. 2. Isolation of the modules from production process allows easier analysis of the production process and in consequence proper crew selection. 3. The applied criterion of the probability of achieving assumed module realization duration allows rational crew selection because module realization as a whole has higher priority than realization of individual activities. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archives of Mining Sciences de Gruyter

Selection of the Longwall Face Crew with Respect to Stochastic Character of the Production Process – Part 1 – Procedural Description / Wyznaczanie Obsady Przodka Ścianowego Z Uwzględnieniem Stochastycznego Charakteru Procesu Produkcyjnego. Cz. 1 – Opis Metody

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10.2478/v10267-012-0071-9
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Abstract

Arch. Min. Sci., Vol. 57 (2012), No 4, p. 1071­1088 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.2478/v10267-012-0071-9 RYSZARD SNOPKOWSKI*, MARTA SUKIENNIK* SELECTION OF THE LONGWALL FACE CREW TH RESPECT TO STOCHASTIC CHARACTER OF THE PRODUCTION PROCESS ­ PART 1 ­ PROCEDURAL DESCRIPTION WYZNACZANIE OBSADY PRZODKA CIANOWEGO Z UWZGLDNIENIEM STOCHASTYCZNEGO CHARAKTERU PROCESU PRODUKCYJNEGO. CZ. 1 ­ OPIS METODY A proposal of the method aimed at the longwall face crew selection th respect to stochastic character of the production process has been described in this study. Modules, which can be isolated from the production cycle, as well as methods of determination of the probability function density describing duration of individual action realized in production process, have been described in the first part of the study. Procedure of crew selection of individual modules, including optional crew selection, has been described in next chapters. Statement of action, which should be executed in order to apply the proposed method, including final conclusions, is discussed in the last chapter. Keywords: longwall face crew selection, probability function density, production cycle, longwall face Zagadnienie wyznaczania obsady przodka cianowego jest przedmiotem bada i analiz praktycznie od momentu rozpoczcia stosowania systemu cianowego w kopalniach wgla kamiennego. Metoda opisana w niniejszej pracy uwzgldnia jednak czynnik dotychczas nie uwzgldniany w opracowaniach z tego zakresu, a mianocie stochastyczny charakter realizowanego w przodku procesu. Pocztki prac z zakresu analizy funkcjonowania przodków cianowych z uwzgldnieniem stochastycznego charakteru procesu produkcyjnego sigaj lat 90 ­ tych, kiedy zaczto wykorzystywa metod symulacji stochastycznej jako metod badawcz. Pierwszym krokiem w proponowanej metodzie jest podzial procesu produkcyjnego na moduly. Kryterium podzialu stano sposób realizacji poszczególnych czynnoci lub operacji w danym module. Zaproponowano cztery rodzaje modulów i oznaczono odpoednio literami od A do D. Moduly typu A to moduly z czynnociami wykonywanymi w sposób równolegly, wród których wystpuje tzw. czynno odca. Czynno odca jest to taka czynno, której realizacja nie ponna by wstrzymywana z powodu zbyt wolnego wykonywania pozostalych czynnoci wystpujcych w tym module. Moduly typu B to takie, w których czynnoci lub operacje wykonywane s w sposób równolegly, ale wród niech nie wystpuje czynno odca. Czynnoci wykonywane w sposób szeregowy * AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY, FACULTY OF MINING AND GEOENGINEERING, DEPARTMENT OF ECONOMICS AND MANAGEMENT IN INDUSTRY, A. MICKIECZA 30 AVE., 30-059 KRAKOW, POLAND charakteryzuj moduly typu C. W modulach tych moe by wykonywana dowolna ilo czynnoci w ukladzie szeregowym, dodatkowo czynno pojedyncz traktuje si jak szeregow. Moduly typu A, B i C wyodrbnione s z cyklu produkcyjnego na rysunku 1. Cech charakterystyczn modulów typu D jest wystpowanie czynnoci lub operacji zarówno w ukladzie równoleglym, jak i szeregowym. Na rysunku 2 zamieszczono przyklad takiego modulu. Kolejnym krokiem w metodzie wyznaczania obsady przodka cianowego jest wyznaczenie funkcji gstoci prawdopodobiestwa, opisujcych czas realizacji poszczególnych czynnoci w ramach wyodrbnionych modulów. Schemat wyznaczania funkcji opisujcych czas trwania czynnoci lub operacji w ramach modulów zamieszczono na rysunku 3. Przestaony schemat zaklada zebranie danych pomiarowych a nastpnie przeprowadzenie analizy statystycznej, która polega na wyznaczeniu funkcji aproksymujcych f1,i,j, majcych wlasnoci funkcji gstoci prawdopodobiestwa. Funkcje te opisuj czas realizacji czynnoci lub operacji ,,i"-tej wykonywanej w ramach danego modulu ,,j "-tego, na odcinku jednego metra. Nastpnie wyznacza si splot otrzymanych funkcji w celu wyznaczenia funkcji splotowych fi,j , które opisuj czas realizacji czynnoci lub operacji ,,i"-tej w danym module ,,j"-tym. Otrzymane funkcje splotowe maj wlasnoci funkcji gstoci prawdopodobiestwa. Mona je wyznaczy dema metodami: metod analityczn lub metod symulacyjn. W metodzie analitycznej wykorzystuje si definicj splotu funkcji, natomiast w metodzie symulacyjnej schemat postpowania, który zamieszczono na rysunku 4. Jeeli w module znajduj si czynnoci lub operacje, które mog by wykonywane przez rón liczb pracowników (obsad), wówczas funkcja fi,j wyznaczana jest dla kadego wariantu obsady z osobna. Symbolem ,,k " oznaczono obsad, dla której funkcja fi,j zostala wyznaczona. Po wyznaczeniu funkcji gstoci prawdopodobiestwa, opisujcych czas realizacji poszczególnych czynnoci, nastpuje wyznaczenie obsady w ramach poszczególnych modulów. W zzku z wydzieleniem trzech typów modulów przedstaono algorytmy wyznaczania obsady uwzgldniajce to zrónicowanie. Algorytm wyznaczania obsady dla modulów z czynnociami lub operacjami równoleglymi i odcymi przedstaony jest w rozdziale 4.1. Na rysunku 5 zamieszczono przyklad modulu, w którym wystpuje czynno odca a nastpnie z wykorzystaniem wzorów od 1 do 4 opisano procedur postpowania przy wyznaczaniu obsady w modulach typu A. Algorytm wyznaczania obsady dla modulów z czynnociami lub operacjami równoleglymi bez odcych opisano w rozdziale 4.2. Na rysunku 6 zmieszczono przykladowy modul z dema czynnociami równoleglymi, z których adna nie jest odc. Wzorami od 5 do 10 opisano procedur wyznaczania obsady w modulach typu B. Moduly typu C oraz schemat wyznaczania obsady opisane s w rozdziale 4.3. Rysunek 7 prezentuje przykladowy modul z dema czynnociami szeregowymi, a wzory od 11 do 15 przedstaaj proces wyznaczania obsady w modulach tego typu. Algorytm wyznaczania obsady dla modulów z czynnociami lub operacjami wykonywanymi szeregowo i równolegle zaprezentowany jest w rozdziale 4.4. Na rysunku 8 zamieszczono przykladowy modul, a wzory od 16 do 26 prezentuj procedur wyznaczania obsady w modulach typu D. Zaproponowana metoda zaklada wykorzystanie funkcji gstoci prawdopodobiestwa czasów trwania czynnoci do wyznaczania obsady przodka wydobywczego. W metodzie wykorzystano odmienne od deterministycznego podejcie, polegajce na traktowaniu czasów realizacji czynnoci jako zmiennych losowych. Zastosowanie opracowanej metody wymaga realizacji szeregu czynnoci, z których najwaniejsze to: ­ identyfikacja kluczowych czynnoci w procesie produkcyjnym, ­ podzial procesu produkcyjnego na charakterystyczne moduly, ze wzgldu na jednoczesno realizacji czynnoci, ­ identyfikacja funkcji gstoci czasów trwania czynnoci w wydzielonych modulach, ­ przyjcie wstpnych wariantów obsady dla poszczególnych modulów ­ optymalizacja obsady w modulach poprzez uwzgldnienie prawdopodobiestw realizacji czynnoci przy zaloonej obsadzie z uwzgldnieniem charakteru modulów. Mona take zauway, ze: 1. Kady proces produkcyjny mona podzieli na skoczon liczb modulów rónicych si jednoczesnoci realizacji czynnoci. 2. Wyodrbnianie z procesu produkcyjnego modulów, pozwala na latejsz analiz procesu produkcyjnego, a co za tym idzie ulata dobór obsady. 3. Uyte w metodzie kryterium prawdopodobiestwa osignicia zaloonego czasu trwania realizacji modulu, pozwala na racjonalny dobór obsady, gdy realizacja modulu jako caloci ma wyszy priorytet ni realizacja poszczególnych czynnoci. Slowa kluczowe: obsada przodka cianowego, funkcje gstoci prawdopodobiestwa, cykl produkcyjny, przodek cianowy 1. Introduction Studies aimed at the analysis of longwall face operations th respect to stochastic character of the production process were started in the nineties of the last century (Snopkowski, 1990, 1994). Studies aimed at the analysis of production of coal excavated from longwall faces th respect to chosen probability distributions were continued in the next decades (Snopkowski, 2000a, 2000b, 2002). Stochastic simulation method was used as the research method (Snopkowski, 2005, 2007a, 2007b, 2009) and (Snopkowski & Napieraj, 2012). Thus the problem of the longwall face crew selection was started at the same time as longwall exploitation system was applied in hard coal mines. However, the method described in the present study takes under consideration not considered so far factor, i.e. stochastic character of the process realized in the longwall face (Sukiennik, 2011). Division of the production process into modules is the first procedural step of the method of question. Manner of the realization of individual activities or operations in given module is considered as a criterion of this division. 2. Division of the production process into modules In the production process realised thin the longwall face of hard coal mines we can isolate its smaller fragments, which are determined as "modules" in the present study. Such partition is aimed at development of the crew selection method, which takes under consideration different character of these modules. The modules differ from each other th manner of realization of individual activities or operations. Detail information related th procedure of defining particular modules are described in next chapters of this study. 2.1. Modules comprising activities or operations executed in parallel To this group of the production process are prescribed activities or operations, which are executed in parallel. Modules th co called leading activities and modules thout such activities can be distinguished. Modules th leading activities or operations are characterized th two features. The first feature comprises parallelism of the executed activities or operations and the second one is characterized th presence of so called leading activity. Example of the module of this type marked th symbol "A" is shown in Fig. 1. Module marked th symbol ,,A" comprises fragment of the production cycle, in which shearer cuts coal rock body and is followed by support and conveyor shifting, which are moved toward the longwall face. Shearer cutting is considered as leading activity in this module. This activity determinates two other activities. In practice it means that the shearer shouldn't wait for the support or conveyor shift. Thus shearer standstill on the section (L ­Xp) shouldn't take place because it could result in daily output decrease. Module of type ,,A" occur anywhere where fragments of the parallel activities or operations are present, in which single activity is considered as leading activity. Fig. 1. Scheme of the production cycle for two way shearer mining th isolated modules. Source: (Snopkowski, 1997) th modifications Parallelism and lack of leading activity or operations is a characteristic feature of module thout leading activities or operations. Example of the module of this type marked as "B" s shown in Fig. 1. Operation of the drive unit shift when the shearer is stopped in a distance (x1 + x2 + x3) from the driver unit occurs in this module. Shearer is exposed to short maintenance th eventual replace of cutting picks. It should be noted that in case of other mining processes we can also isolate modules, which in Fig. 1 are marked as ,,A" and ,,B". 2.2. Modules th activities and operations executed in series Lack of activities and operations executed in parallel is the most important feature of the module of this type, which can be isolated fro production process. Example of the module of this type marked as ,,C" is shown in Fig. 1. This is shearer advance, during which no other activities or operations are executed. In case of several operations executed in series, the operations are classified as the operations of the same module. 2.3. Modules and operations executed in series and in parallel Occurrence of activities or operations both in parallel and in series system is a characteristic feature of the module of this type. Example of such module is shown in Fig. 2. activity 1 (operation) activity 2 (operation) activity 3 (operation) acitivity 4 (operation) module ,,j" Fig. 2. Example of module of the type D th activities or operations executed In series and in parallel. Source: Authors materials In this module occur both activities (operations) executed in the in series, i.e. marked th sequent digits 1,2 and 4, and also 1 and 3, and parallel system comprising activities (operations) 2,3 and 4. 3. Developing the functions describing activity or operation duration executed thin the modules Scheme of the development of functions describing duration of activities and operations thin individual modules is shown in Fig. 3. Realization of the scheme presented in Fig. 3 is preceded by collection of suitable data in conditions of concrete longwall face. For example, for a module marked as "A" in Fig. 1, the data comprise shearer operational speed, support advance rate and conveyor shifting rate. Then the data are exposed to statistical analysis comprising determination of approximation functions f1,i, j, having properties characteristic for probability density function. These functions describe time of realization of "i" activity or operation in scope of given "j" module on a distance of 1 meter. Next procedural step shown in the scheme comprises development of obtained convolution in order to determine convolution functions fi, j, which describe time of realization of "i" activity or operation thin "j " module. Obtained convolution functions have properties characteristic for probability density functions. They can be determined th use of two methods: analytical method or simulation method. Definition of convolution function is used in analytical method and the procedural scheme is shown in Fig. 4. measurement data {t1, t2,...,tN} statistical analysis made in order to determine func tions f1,i,j describing duration of activities or operations on a distance if 1 m determining of convolu tion function f1,i,j by analytic method determining of convolu tion function f1,i,j by simulation method h d d functions fi,j describing duration of activity or opera tions thin module Fig. 3 Scheme of determining functions describing duration of activities or operations in the module. Source: Authors materials Range of activity realized in given module is marked th symbol "S" in the scheme. If this range is for example 15 m, S = 15. Assumed number of simulations is marked th symbol "LS ". If the module comprises activities and operations, which can be executed by various number of workers (crew) the function fi, j is determined for each crew selection variant separately. Marking fi,kj, where symbol "k" comprises crew, for which function fi,j was determined has been introduced in order to differentiate the mentioned functions. Procedure of selection the crew thin the k modules th use of the function fi,j. is characterized in next parts of this study. ^d Zd ^hD ^hD ^hD < ^ < < ^hD > >^ > > >^ Fig. 4. Scheme of determining the function th use of simulation method. Source: Authors materials 4. Selection of crew for individual module of the production process Crew of the whole production process realized in the longwall face is determined for each module of the production process. In order to select four module types, algorithms of crew selection taking into account this differentiation is described in next parts of this study. 4.1. Algorithm for the crew selection for modules th parallel and leading operations According to mentioned description, leading activity or operations is defined as activity or operation, which realization shouldn't be stopped thin given module (it shouldn`t "wait" for execution of other activities or operations). Operation of the shearer in longwall face is an ex ample of such activity in production process. The shearer shouldn't wait until the mechanized support of conveyor is shifted. Each stoppage of the shearer advance is inconvenient from the production efficiency point of view. Thus in the module, in which leading activity occurs, the crew selection procedure takes under consideration the mentioned conditions. Example of module, in which leading activity occurs is shown in Fig. 5. Fig. 5. Example of the module th leading activity (operations). Source: Authors materials Symbols used in Fig. 5: f1, j f2, j f2, j f3, j k =2 k =2 k =4 ­ function density, describing duration of activity (operation) 1 in module "j" for crew k = 2, ­ density function, describing duration of activity (operation) 2 in module "j" for crew k = 2, ­ density function, describing duration of activity (operation) 2 in module "j" for crew k = 4, ­ density function describing duration (operation) 3 in module "j " for crew k = 3. Module "j " can be realized for two crew variants: ­ ­ Variant I ­ 7 persons crew, activities (operations) realized according to function k f1,kj=2 , f2,kj=2 , f3,j=3, ­ 9 persons crew, activities (operations) realized according to function k =2 k =4 f1, j , f2, j , f3,j . Scheme of the crew selection in module "j" is divided into the follong stages: Stage I Calculation of the value t0 for complex probability p according to formula: t0 f1,k j= 2 (t) dt = p (1) where: p -- probability of realization of the leading activity in time shorter than t0 (level 0,95 is recommended), t0 -- determined boundary value of the leading activity (operations) in module "j", for which probability of exceeding amounts for 1 ­ p (for p = 0,95, this probability amounts for only 0,05) Stage II Calculation of the probability of activities (operations) realization, which are executed in time sorter than t0 for all variants of the crew selection module "j". Variant I In the first variant, the crew selection is realized by 7 persons and activities (operations) k =2 k =2 are characterized by functions f1,j , f2,j , f3,j . The follong integrals marked th symbols 2 and 3 should be calculated in order to calculate the probabilities: t0 f k =2 (t) dt 2, j t0 f (t ) dt 3, j k = p2,= 2 j (2) (3) = p3, j In the second variant the crew is realized by 9 persons and the activities (operations) are k =2 k =4 characterized by functions f1, j , f2, j , f3, j . The searched probabilities are calculated from fomulas: f k = 4 (t )dt 2, j whereas p3, j from formula 3. t0 k = p2,=j 4 (4) Stage III Procedure of the variant of the crew selection on the basis of values of calculated probabilities. Three options are distinguished in this procedure: k =2 k =4 Option 1. All calculated probabilities are greater than p, thus p2, j > p; p3, j > p; p2, j > p (for module "j"). In practice it means that in no crew selection variant occur, which can stop leading activities (operations). In this case, variant of the crew selection th the minimal number of workers should be selected. In the cited example it is variant I, for which the crew consists of 7 persons. Option 2. Only in one variant the calculated probabilities are greater than p. That means that the only one variant of the crew selection assures execution of activities (operations) in the module in such manner that the realization of the leading activity is not stopped. In this case, crew selection assumed in this variant is taken as the module crew. Option 3. All calculated probabilities are smaller than p. For the ex ample value it means k =2 k =4 that: p2, j < p; p3, j < p; p2, j < p In practice it means that none crew variant assures execution of the activity (operation) in sorter time than the leading activity, i.e. the leading activity ll be stopped. In such situation possibility of crew size should be considered. If there is no such possibility, variant for which calculated probabilities are closest to value p are considered. Based on the dada from example the follong quotients should be calculated: ­ for variant I: ­ for : k k p2,=2 + p3,=3 j j k k p2,= 4 + p3,=3 j j , . Variant, for which the quotient is larger, should be considered as the variant of the crew selection of module "j ". Analogical calculations can be executed for selection of optimal variant of the crew selection if only a part of calculated probabilities satisfies assumed criterion (are smaller than p). In practice it means that activities (operations), which can stop execution of the leading activity can occur in each variant of the crew selection procedure. However, follong the mentioned calculation procedure we are able to select such crew, which assures minimized stoppages of activity (operation), which is considered as leading activity. 4.2. Algorithm of crew selection for a module th parallel activities or operations thout leading operations Example of the model th two parallel operations, where none of them is leading. The follong symbols were used in Fig: f1, j f1, j ­ density function, describing time of the activity (operation) 1 realization in module "j" for the crew k = 1, ­ density function, describing time of the activity (operation) 1 realization in module "j" for the crew k = 2, k =2 Fig. 6. Example of the module th activities (operations) executed simultaneously thout leading activity Source: Authors materials f2,kj=2 ­ density function, describing time of the activity (operation) 2 realization in module "j" for the crew k = 2. Module "j" can be realized in two options: ­ ­ Variant I ­ 3 person crew, activities (operations) realized according to function k =2 f1, j , f2, j , ­ 4 person crew, activities (operations) realized according to function k =2 k =2 f1, j , f2, j . Procedure of the crew selection in module th paralel activities (operations) thout leading operations comprises the follong stages: Stage I Value of the boundary module realization time, which probability indicates that the limit ll not bee exceeded is p, is calculated for each crew variant. Thus further calculations executed th respect to assumed probability level p. value of p = 0.95 are recommended, (like confidence level), although specific value depends on person performing the calculations. The calculations are as follow: Variant I k t1,=1 j k t 2,=j2 f1, j (t) dt = p f 2, j (t) dt = p k =2 (5) (6) k k t0 I = max t1,=1 ; t2,=j2 j W (7) where: W t0 I -- boundary module realization time for variant I, not exceeding probability amounting for p. k t1, =2 j f1, j (t )dt = p k =2 (8) t 2, j k =2 -- calculated from formula 6, k k t0 II = max t1, =2 ; t2,=2 j j W (9) where: t0 II -- boundary module realization time for , not exceeding probability amounting for p. Stage II Calculation of this variant of the crew selection , assures the shortest realization time of this module, thus: (10) For ex ample, if the time is the shortest t0 II (for ), the crew, which should realize activities (operations) in module "j" comprises 4 persons, whereas the activity (operation) ,,2" should be realized also by two persons. The calculations were executed for assumed probability level p. 4.3. Algorithm of the crew selection for module th activities (operations) executed in series Example of module th two activities executed in series is shown in Fig. 7. The follong symbols were used in Fig. 7: f1, j f1, j f2, j ­ density function, describing time of activity (operation) 1 realization in module "j" for crew k = 1, ­ density function, describing time of activity (operations) realization 1 in module "j" for crew k = 2, ­ density function, describing time of activity (operations) realization 2 in module "j" for crew k = 3. k =2 Fig. 7 Example of the module th activities (operations) executed in series Source: Authors materials Module "j" can be realized in two variants: ­ Variant I ­ 4 person crew, activities (operations) realized according to function f1, j , f2, j , ­ ­ 5 person crew, activities (operations) realized according to function f1, j , f2, j . Scheme of the module crew selection th activities (operations) executed in series comprises the follong stages Stage I Function describing realization time of module "j" is determined for each crew variant. These functions are determined th use of convolution operations and it has the follong form: Variant I k =2 where: k k = f1, j=1 Ä f 2, =3 j (11) -- density function of random variable time of realization of module "j" for variant I. where: k = f1,kj=2 Ä f 2, =3 j (12) -- density function of random variable time of realization of module "j" for . Stage II Boundary value of time of realization of module "j", not exceeding probability amounting for p is determined for each crew selection variant. The calculations are as follow: Variant I W t0 I f j I (13) where: t0 I -- boundary realization time for variant I, not exceeding probability p. t0 f j II (t )dt = p (14) where: t0 II -- boundary realization time for , not exceeding probability p. Stage III Determination of the crew selection variant , which assures the shortest realization time of given module: (15) For example, if the time t0 II is the shortest (for ), the crew which should realize the activities (operations) in module "j" amounts for 5 persons, whereas realization time of this W module can not exceed value t0 II th probability p, if the first activity (operation) ll be executed by two persons and the second activity (operation) is executed by three persons. 4.4. Algorithm of the crew selection for module th activities or operations executed both in parallel and in series Example of module th activities executed in series is shown in Fig. 8. The second activity can be executed in two crew variants. Symbols used in Fig. are as follow: f1, j f2, j f2, j f4, j k =2 ­ Density function describing time of realization of activity (operation) 1 in module "j" for crew k = 2, ­ Density function describing time of realization of activity (operation) 2 in module "j" for crew k = 2, ­ Density function describing time of realization of activity (operation) 3 in module "j" for crew k = 2, ­ Density function describing time of realization of activity (operation) 4 in module "j" for crew k = 2. k =2 k =2 k =2 Fig. 8. Example of the module th activities (operations) executed in series and in parallel. Source: Authors materials Module "j" can be realized in two crew variants: ­ Variant I ­ 8 persons crew, activities (operations) realized according to function f1, j , f2, j , f3,j , f4,j , ­ ­ 10 persons crew, activities (operations) realized according to function f1, j , f2, j , f3,j , f4,j . Scheme of crew selection in the module th activities (operations) executed both in series and in parallel, comprises the follong stages: Stage I Calculation of the module duration th use of complex probability level p, for each crew variant. Concept of full paths was introduced into calculations. Full paths are defined as a set of successive activities (operations), from which the first activity is started in the initial point of the module and the last one is finished in final point of the module. Two full paths marked th symbols S1 and S2 can by distinguished for the module presented in Fig. 8. Path S1 comprises activities (operations) 1,2 and 4, whereas path S2 comprises activities (operations) 1 and 3. Functions describing duration of full paths are developed for all crew variants, as convolution of suitable functions in the follong manner: Variant I k =2 k =4 k =2 k =2 k =2 k =2 k =2 k =2 S , k k k = f1, j= 2 Ä f 2, =2 Ä f 4, =2 j j (16) (17) S2 , k = f1,kj=2 Ä f 3, = 2 j where: S1, -- density function describing duration of path S1 for crew variant in module "j", S2, -- density function describing duration of path S2 for crew variant in module "j". S , k k k = f1, j=2 Ä f 2, = 4 Ä f 4, =2 j j k k = f1, j= 2 Ä f3, =2 j (18) (19) where: S1, S2, S 2 , -- density function describing duration of path S1 for crew variant in module "j", -- density function describing duration of path S2 for crew variant in module "j". Stage II Duration of module "j" th not exceeding probability p is determined for each crew variant. Calculations for each variants are as follow: Variant I S S The follong integrals should be calculated in order to determine time t0 1, and t0 2,: S t0 1, S t 0 2, 1 S , (20) S 2 , (21) where: t0 1, -- duration of path S1 in crew variant , not exceeding probability p, t0 2, -- duration of path S2 in crew variant , not exceeding probability p. Duration of module "j" for crew variant , not exceeding probability p, is calculated from formula: S S t0 I = max t0 1 where: t0 { S ,W ; t0S ,W } I 2 I (22) -- duration of module "j" for crew variant , not exceeding probability p. S ,W S ,W In order to determine time t0 1 II and t0 2 II we should calculate: t0S 2 , t0S2 , S , (23) S2 , (24) where: t0 1 S S , -- duration of path S1 in crew variant , not exceeding probability p, t0 2, -- duration of path S2 in crew variant , not exceeding probability p. Duration of module "j" for crew variant , not exceeding probability p, is calculated from formula: t0 II = max t0 1 where: t0 { S ,W ; t0S ,W } II 2 II (25) -- duration of module "j" for crew variant , not exceeding probability p. Stage III For handling the activity (operation) of module "j" should be assumed such crew variant , which assures the shortest time of the module realization, thus: {W (26) where: W t0 i -- the shortest time of realization of module "j" for crew variant . For example, if time t0 II (for ) is the shortest, the crew, which should realize activities (operations) in module "j" amounts for 10 persons, whereas the first activity (operation) should be executed by two persons, the second activity (operation) ­ four persons, the third activity (operation) by two persons and the forth activity (operation) also by two persons. The calculations were executed th assumed probability p. 5. Final conclusions Observations of durations of activities (operations) executed in production process confirm the thesis that these times are exposed to some fluctuations resulting from mining and geological conditions. The method proposed in this study assumes application of probability density function of durations of the executed activities for the longwall face selection. Different from the deterministic approach, i.e. considering the activity realization times as random variables was applied in the proposed method. Application of the developed method requires execution of some activities, from which the most important are: ­ identification of key activities of the production process, ­ division of the production process into characteristic modules, th respect to simultaneity of the activities realization, ­ identification of the density functions of the durations of executed activities thin separated module, ­ assuming preliminary crew variants for individual module, ­ optimization the crew selection in module via taking under consideration probabilities of the activities realization for assumed crew, according to the module character. It should be also noted that: 1. Each production process can be divided into finite number of modules differing th simultaneity of the activity realization. 2. Isolation of the modules from production process allows easier analysis of the production process and in consequence proper crew selection. 3. The applied criterion of the probability of achieving assumed module realization duration allows rational crew selection because module realization as a whole has higher priority than realization of individual activities.

Journal

Archives of Mining Sciencesde Gruyter

Published: Dec 1, 2012

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