Rudarski radovi br 4 2011 - Institut za rudarstvo i metalurgiju Bor

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In the course of construction project<br />realization, in addition to the assessment<br />of its duration as well as the probability of<br />duration of individual activities and the<br />completion of the project entirely or<br />someof its stages, it is also important to<br />determine the possibility of the contractor<br />to complete the foreseen activities, parts of<br />the project or the entire project within the<br />set or contract deadline. The notion of possibility<br />is connected to the capability and<br />readiness of a subject (project contractor)<br />to complete under the set conditions and<br />within the foreseen time the obligations or<br />tasks undertaken and it is different from<br />the notion of probability which is connected<br />to statistical data [3].<br />The possibility distribution function πi<br />is further defined, for which the following<br />applies:<br />π i ( ti ) = Poss{<br />Ti<br />= ti}<br />μ ( ti<br />)<br />and it expresses the possibility that the<br />element Ti ∈ Ti<br />.<br />For the project consisting of m activities<br />the ti durations of which are the elements<br />of a Ti fuzzy set, the membership<br />function μi represents the possibility of a<br />contractor to complete this activity within<br />some time limit ti. The function πi represents<br />a degree of possibility of a contractor<br />and when πi(ti)=0, it is not possible to<br />complete the activity within a time limit. If<br />πi(ti)=1, then the possibility is maximum.<br />Within the deterministic procedure of<br />the CPM method, the value of this function<br />is:<br />⎧1<br />za ti<br />= ai<br />⎫<br />π i ( ti<br />) = ⎨ ⎬<br />⎩0<br />za t ≠ a ⎭<br />i<br />i<br />where ai is a set or determined value of i<br />activity duration.<br />The duration of activity ti as a fuzzy<br />variable and the degree of completion<br />possibility πi are estimated based on experience.<br />CALCULATION OF PROJECT<br />DURATION WHEN ACTIVITY<br />DURATIONS ARE FUZZY<br />NUMBERS<br />Let the given fuzzy numbers be<br />{ , μ ( )<br />A }<br />{ μ<br />A }<br />A = x x x∈ R and<br />B = y, ( y) y∈ R . The membership<br />functions of the considered fuzzy numbers<br />are continuous and their values belong to<br />the interval (0, 1).<br />Let ∗ denote the operation over fuzzy<br />numbers. Then A * B is also a fuzzy number<br />denoted as C = A* B,<br />so that<br />{ , μ ( ) }<br />c<br />C = z z z∈R where:<br />z=x*y and the membership function is<br />calculated according to the expansion<br />principle, i.e.<br />μ ( z) = supmin( μ ( x), μ ( y)).<br />C z= x+ y A B<br />In a special case when fuzzy numbers<br />are linear, in other words when the membership<br />functions are triangular in shape,<br />then the expressions by which we calculate<br />the sum, the difference, the product<br />and the quotient of fuzzy numbers are<br />considerably simpler.<br />No 4, 2011. 159<br />MINING ENGINEERING
For every α intersection the fuzzy<br />numbers are represented by :<br />α α α α<br />A = ⎡<br />⎣xL, x ⎤ D ⎦ and B = ⎡<br />⎣ yL , y ⎤ D ⎦<br />Then the operations with triangular<br />fuzzy numbers are defined by the following<br />expressions [5].<br />{ , μ ( ) , }<br />C<br />C = A+ B= z z z∈ R<br />⎡ α α α α⎤<br />z= ⎢xL + yL , xR + yR⎥; μ ( z)<br />= α; α = 0,1<br />⎣ ⎦ C<br />{ , μ ( ) , }<br />C<br />C = A− B= z z z∈R [ ]<br />⎡ α α α α⎤ z= ⎢xL − yR, xR − yL⎥; μ ( z)<br />= α; α = 0,1<br />⎣ ⎦ C<br />{ , μ ( ) , }<br />C<br />C = A⋅ B= z z z∈R [ ]<br />⎡ α α α α⎤<br />z= ⎢xL ⋅yL , xR ⋅ yR⎥; μ ( z)<br />= α; α = 0,1<br />⎣ ⎦ C<br />{ μ }<br />C<br />C = A: B= z, ( z) z∈ R,<br />[ ]<br />⎡ α α α α⎤<br />z= ⎢xL / yR, xR / yL⎥; μ ( z)<br />= α; α = 0,1<br />⎣ ⎦ C<br />Example:<br />[ ]<br />There are strictly positive fuzzy numbers<br />given A = x, μ ( x) x∈<br />[ 6,10]<br />and<br />{ }<br />A<br />{ , μ ( ) [ 8,10 ] } .<br />A<br />B = y y y ∈<br />μ ( x)<br />=<br />A<br />⎧ 1<br />⎫<br />x −3, 6 ≤ x ≤8 ⎪ 2<br />⎪<br />⎨ ⎬<br />⎪ 1<br />− x + 5, 8 ≤ x ≤10⎪<br />⎩⎪ 2<br />⎭⎪<br />( x)<br />=<br />⎧ y − 8, 8 ≤ y ≤ 9 ⎫<br />⎨ ⎬<br />⎩− y + 10, 9 ≤ x ≤ 10 ⎭<br />No 4, 2011. 160<br />MINING ENGINEERING<br />μ<br />B<br />Determine: C = A+ B<br />In order to make the marked operations<br />over the given triangular fuzzy numbers,<br />it is necessary to determine the left<br />and right confidence limit for every confidence<br />level α ∈[<br />0,<br />1]<br />. In other words, it is<br />necessary to determine the extreme values<br />in α intersection of A and B fuzzy numbers.<br />The left limit value of A fuzzy number<br />for the α ∈[<br />0,<br />1]<br />confidence level α x L<br />is obtained according to the expression:<br />1 α<br />α<br />xL<br />− 3 = α → xL<br />= 6 + 2α<br />2<br />The right limit value of A fuzzy number<br />for the α ∈[<br />0,<br />1]<br />confidence level α x R is<br />obtained according to the expression:<br />1<br />− xα<br />5 α α<br />R + = → xR<br />= 10 − 2α<br />2<br />α intersection of a fuzzy number is:<br />[ α , α ] = [ 6 + 2α<br />, 10 − 2α<br />]<br />x<br />1<br />α = L R<br />2<br />x A<br />Analogue to the previous expression<br />we obtain for α intersection of B fuzzy<br />number:<br />[ α α<br />y ] = [ 8 + α,<br />−α<br />]<br />B α = L , yR<br />10<br />Let us calculate the sum of A and B<br />fuzzy numbers which is marked as C . C<br />fuzzy number is formally written down as<br />{ , μ ( ) } .<br />C<br />C = z z<br />According to the rule of addition of<br />triangular fuzzy numbers, it follows that:<br />[ (6 2 ) (8 ),(10 - ) (10 2 ]<br />[ 14 3 α,203α] C = + α + + α α + − α =<br />= + −
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INSTITUT ZA RUDARSTVO I METALURGIJU
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SADR@AJ CONTENS M. Bugari
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vršeno je sve do 1987. god. a kona
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MINING AND METALLURGY INSTITUTE BOR
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Based on realized geological drilli
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istočno od đerdapske navlake lež
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GEOLOŠKE KARAKTERISTIKE LEŽI
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Sl. 4. Teren u domenu ležišta gli
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Ispitivanje granulometrijskog sasta
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literature data, semi-detailed polu
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GEOLOGICAL CHARACTERISTICS OF
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Fig. 4. Field in the domain of clay
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Testing of grain-size distribution
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Ova ankerna smeša će biti koriš
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This anchor mixture will be used in
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Tabela 2. Rezultati ispitivanja kam
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Sl. 4. Određivanje odnosa vlažnos
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Table 2. Test results of stone aggr
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Structural characteristics Det
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305 -100 Sl. 1. Izgled za
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Nakon zapunjavanja preostalog prost
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10 7 BRANA 1 LEGENDA
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305 -100 Fig. 1. View of
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Fig. 3. Review of the present appea
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4. CONCLUSION Fig. 4. Disposit
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Dobivanje uglja iznad raskršća ot
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matematičkom modelu, a međusobne<
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4. ZAKLJUČAK Proračunom dobi
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this stage is carried out alternate
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Rearranging the expression for the<
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4. CONCLUSION Calculation of t
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Tabela 1. Pregled polaznih tehno ek
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5. ZAKLJUČAK Bilansne rezerve
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Table 1. Summary of initial techno-
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5. CONCLUSION Balance reserves
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odmah iza valjka, kome je jedan deo
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Sl.3. Tehnološka šema laboratorij
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further testing, the laboratory pla
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Fig. 3. Technological scheme of lab
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Međutim, na početku ćemo pojasni
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Page 118 and 119: Tabela 3.[8] Vjerovatnost Ovim
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Page 126 and 127: usiness policy of the company and o
Page 128 and 129: Table 1. NATURAL (workpla
Page 130 and 131: will occur from exposure to hazard
Page 132 and 133: ing to the overall economy. However
Page 134 and 135: and measures of their control can h
Page 136 and 137: INSTITUT ZA RUDARSTVO I METALURGIJU
Page 138 and 139: Tabela 1. Rudne rezerve u ležišti
Page 140 and 141: Tabela 3. Proizvodnja pratećih pro
Page 142 and 143: Tabela 6. Prosečne neto zarade<br
Page 146 and 147: Table 1. Ore reserves in the active
Page 148 and 149: Table 3. Production of accompanying
Page 150 and 151: The average net wages have a tenden
Page 154 and 155: i izražava mogućnost da element t
Page 156 and 157: i p ( RZ t ) RZ = ma
Page 158 and 159: 8. Iskop jama za temeije samce 1 2
Page 160 and 161: Pokazani postupak zatim analogno If it is not possible t
Page 170 and 171: Rz3 = Rz2+ t3 [ ] Rz3 = 3
Page 172 and 173: The difference in computation betwe
Page 176 and 177: Tabela 4. Zastupljenost objekata na
Page 178 and 179: 8. TROŠKOVI SAOBRAĆAJNIH NEZ
Page 180 and 181: 11. KRITERIJUMI VREDNOVANJA I 
Page 182 and 183: Tabela 15. Vrednosti kriterijumskih
Page 186 and 187: Table 3. Slopes of level Varia
Page 188 and 189: 7. EXPLOITATION COSTS Exploita
Page 190 and 191: • changes in natural look of terr
Page 192 and 193: Fig. 2. Distribution of responses f
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Page 199: JP PEU RESAVICA ЈП ПЕУ Р
Page 202: INSTRUCTIONS FOR THE AUTHORS M
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Rudarski radovi br 4 2011 - Institut za rudarstvo i metalurgiju Bor

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