Lerchs Grossman

Hel m ut lerchs"' Ma nager of Scientific Services, Mantreal Datacentre, lnternational Business Machin es Ca . Ltd., lngo F. Grossma nn Manager , Management Science Applications, lnternatianal Business Machines Co. Ltd . Optimum Design of Open-Pit Mines Joint C.O.R.S. ond O.R.S.A. Conference, Montreal , Moy 27-29, 1964 Tronsoct ions, C. l .M .,_ Vo l ume LXV I I I , 1965, pp. 17-24 ABSTRACT An open-pit mining operation can be .vieed ª:' a process by which the 01>en surface of a mme is contmu ously deformed. The planning of a minin g program in rnlves the design of the final shape of this open surface. The approach de,·eloped in this paper is b:ised. on he follow ing assum ptions: l. the ty 1> of material, it nune ,·aJue and its extraction cost is g1ven for each pomt; 2. restrictions on the geometry of the pit are specified (sur face bound aries and maximu m allowable wa ll slopes) ; 3. the objective is to maxim ize total profit - total mine value of material extracted minus total extraction cost. Two numeric methods are proposed : .A sim ple dynamic programming algorithm for the two-dimensional pit (or a single vertical section of a m ine), and. a moe elab?rate graph algorith m for the general threed1mens1onal p1t. lntrod uction SURFACE mi ning prog ram is a complex opera tion that may extend over many years, and in Avolve h uge capital expend i tu res and risk. Before un dertaking such an operation, it must be known what ore there is to be mi ned ( types, grades, quantities and spatial d istri bution ) ancl how much of the ore shou ld be m ined to make the operation profitable. The reserves of ore and i ts spatial distribution are estimated by geological i nterpretation of the informa tion obtai ned from d rill cores. The object of pit de sign then is to determine the amou nt of ore to be mi ned. Assumi ng that the concentration of ores and im purities is known at each point, the problem is to de cide what the ultimate contour of the pit will be and in what stages this contou r is to be reached . Let us note that if, with respect to the global objectives of a mining program, an optimum pit contour exists, and if the mi ning operation is to be optimized, then this contour must be known, if only to minimize the total cost of mining. •Now Senior Resea1·ch Mathematician, General Motors Research Laboratory, Wanen, Mich. Bulletin for January, 1965. Montreal y. and it is meani ngless to consider any one component of pla nni ng separately. This en vironment is d escri bed by the mine value of ali ores present and the extraction cost of ores and waste materials. with respect to a fixed horizontal plane. we can express the problem as follows: Let v. The sote restrictions con cern the geom etry of the pi t . Let V be the fam ily of vol umes corresponding to the family. e and m be three density functions def ined at each point of a three-dimensional space. as a function of time. S. given a real or a hypothetical economical environment (market si tuation. y. z) -c(x. plant configu ration. There is an in ti mate relationsh ip between al! the above points. z) = pro'it per unit \ olume. y. The problem is to find. planning such as: may bear on questions what ma rket to select . The model proposed in this paper will serve to explore alternatives in pit design. z) = mine \"alue of ore per unit volume c(x.m(x. one that maximizes the integral f. z) dx dy dz 47 . z) v(x. A mathema tical model taki ng i n to account al! possible alterna tives simu ltaneously wou ld. y.what mi ni ng m ethocls to use. what u pgrad ing plants to install .what quanti ties to extract. . what transportation facili ties to provide. y. of surfaces.extraction cost per unit volume m (x. z) . The obj ective then is to design the contour of a pit so as to maximize the d ifference between the total mine val ue of ore extrncted and the total extrac tion cost of ore and waste. . exceed a. v (x. etc.Open-Pit Model Besides pit design. Analytically. Let a (x. be of formidable size ancl its solu t ion wou ld be beyond the means of present knowhow. y.). z) define an angle at each point and ! et S be the family of surfaces such that at no point does their slope. the wall slopes of the pit m ust not exceed certai n given angles that may vary with the depth of the pit or with the material. y. V. however. among ali volum es. J ... = k=i mk.L . there is no simple analytical representa tion for the functions v and e. at slopes a (see Figu re 1). Construct a new tableau (Fig ure 2) with the quantities :VI..!JiE.¡+ mkax (P. . It follows that if the element ( i.. column by column starting with column l: P. The conf igu ration of lines yielding the best re sults is then selected..!_=-..Two-Dimensional Pit Let us select the units u.!. In a final tableau. j ) to (i + k.. j ) at its base. represents the profit realized in extracti ng a single column with element (i._-. JC =c ]r= =3c== 3 4 5 ó 7 8. can be traced by fol lowrng the arrows starting from element ( i. rep resenting the bottorn of the pit and two walls.. then this con tou-. . J .-Dynamic Programming. faster and more accu rate.: _¡ !_:. and ui of a rectangular grid system such that Uj = tan a For each unit rectangle (i. Indicate the maximum by an arrow going from (i. The technique used to determine the contour of a section consists in moving three straight Jines. If the maximum value of P in the f irst row is positive.!l:=. k=I Po¡= O F=F 2 = . then there exists no contou r with positive profit. add a row zero and compute the following qu antities: Figure l. and to consider each section as a two-dimensional pit.+ k. consequently. . The traditional approach is to divide the whole pit into parallel vertical sections.. then the optimum contou r is obtained by following the arrows from and to the \ef t of this element.L. is as follows: P11 is the maximum possible contribu tion of columns 1 to j to any feasible pi t that contains the element (i. j-1). a i s taken to be constant over the entire pit ) . = V¡. The interpretation of the P. The following dynamic programming technique is simpler. M.. '--J . j ) is part of the optimum contour.:::. = M. 48 The Canadian Mining and Metallurgical . If ali elements of the f irst row are negative.. Poi = O then.:. j ) ..C. j ) determine the quan tity m 1i = v. numeric methods must be used. j ) on its contour. j ) .Cil 1: mk.. Oi'?E -4- a = 45° ffi¡j = M.L_J Figure 2. Now.L. and in evaluating the ore and the extraction cost of materials limited by the three lines. to the lef t of element ( i.._ ... any f easible pit contour must contain at least one ele ment of the f irst row. (Here. .1) with k = -1' O' 1 Generally._-=:. When a is constant. Associ ate to each volume element V. where F is the resu lting force of all blocks that par ticipate in the movement. sorne of the blocks will be lif ted.. Any f easible contour of the pit is represented by a closure of G. X¡) if V¡ is adjacen t to V. with proper selection of un its on the axis. Draw an are (x1. and if M r is the total mass of a set of vertices Y. are the mi ne value and the extraction cost of element V. Let each element V. where v.. and C. a set of vertices Y such that if a vertex x1 belongs to Y and if the are (x1. V.. Búlletin for January. The resulting force in a block is indicated with an arrow. a mass ffi¡ = \"¡ - C. y. This problem can be viewed as an extreme case of the time-cost optimization problem in project net works. will clepend on the structure of the pit itself and on the function a (x. there are obvious computa tional advantages to be gained from a direct ap proach . A) with a set of vertices X and a set of ares A. The total work done in this movement is F X 1= F". of a graph. The three-d imensional pit model can be illustrated by a physical analogue. T2. each block has a grid point at its center through which there is an upward force (the val ue of the ore in the block) and a downward force (the cost of removing the ore) . Figure 4. • • • T" following simple rnles until no further transformation is possible. We thus obtain a directed three-dimensional graph G = ( X. It can also be transformed into a network flow problem. If a -3 -\ \ \ --- - z. and if the mining of vol ume V. 4. i t can be of help i n "smoothing" the pit. In Figure 5. Hence. The maxi mum closure of G is then given by the vertices of a set of well-iden tif ied branches of the final tree. if such exist. 1965. I 1 - 3 3 1 -1 ' mass m. then the problem of optimum pit design comes to find ing in a graph G a closure Y with maximum mass or. Instead. If the system is Ief t to move freely one unit along a vertical axis. that is. as well as next best solu tions.----:::: Three-Dimensional Pit When the optimum contou rs of ali the vertical sec tions are assembled. Montreal Figure 5. F is the maximum resulting force over any set of blocks that can freely move u p ward in this system. retu rning to ou r model. Tbe decomposition of the pit into elementary vol umes V. a graph al gorithm can be applied. these aclvantages become important when the graphs considerecl contain a very large number of elements. Tº is then transformed into successive trees T'. the movement of any free mechanical system is such as to maximize the work done. as is the case in most instances. shortly. to which severa) solutions have been proposed (2. x1) exists in A then the vertex xi must also belong to Y. The procedure starts with the construction of a tree Tº in G. one of the grid sys tems shown in Figure 4 can be taken. The dynamic programming approach becomes im practical in three dimensions. ( See Figure 3) . it invariably tu rns out that they do not fit together because the wall slopes in a vertical section and at right angles with the sections that were optimized exceed the permissible angle ex. 5) . that is. is not permissible unless volume Vi is also mined. However. However. as may be the case for an open-pit model. and V¡have at Jeast one point in common. The walls and the bottom of the pit are then "smoothed out. 1 • Figure 3. bu t may also be obtained by taking for V.11 alter nate optima." This takes a great amount of effort and the resu lting pit contour may be far from optimum. This division can be qu ite arbitrary. z) . a maxi mum closure of G. An eff ective algorithm to find the maximum clos u re of a graph is developed in the Appendix . the unit volumes defined by a three-dimensional grid.-. 49 . -I -1 -2 1 1- 4 -t _. be represented by a vertex x. and. The model is derived as fol Jows : Let the entire pi t be divided into a set of vol u me elements V. 3. Let us note that because the dynamic programming approach yields not only the optimum contour but also ?. the blocks will separate along the optimum pit contom. is associated to each vertex x1. We thus obtain a directed three-dimensional graph G = (X._ -I -\ 1 \ _ . Figure 4. shortly. the unit \'Olumes defined by a three-dimensional grid. one of the grid sys tems shown in Figure 4 can be taken.. An eff ective algorithm to f ind the maximum clos ure of a graph is developed in the Appendix. and. that is. y. where F is the resulting force of ali blocks that par ticipate in the movement." This takes a great amount of effort and the resulting pit contour may be far from optimum . Let each element V. and if M. z) . X ) exists in A then the vertex Xi must also belong to Y.roach yields not only the optimum contour but also al! alter nate optima. The proced ure starts with the construction of a tree Tº in G. Tº is then transformed into successive trees T'. If a 3 1. and c. a mass where v.. The dynamic programming approach becomes im practical in three dimensions. In Figure 5. a maxi mum closure of G. It can also be transformed i nto a network flow problem.. a graph al gorithm can be applied. Let us note that because the dynamic programming ap¡. However. will depend on the structure of the pit itself and on the function a (x. each block has a grid point at its center through which there is an upward force (the value of the ore in the block ) and a downward force (the cost of removing the ore) . it can be of help in "smoothing" the pit. This division can be qu ite arbitrary. it invariably turns out that they do not fit together because the wall slopes in a vertical section and at right angles with the sections that were optimized exceed the permissib le angle a. If the system is lef t to move freely one unit along a vertical axis. Associ ate to each volume element V. The model is derived as fol lows : Let the entire pi t be divided into a set of vol ume elements V. returning to ou r model. is the total mass of a set of vertices Y.. as is the case in most instances.Three-Dimensional Pit When the optimum contours of ali the vertical sec tions are assembled. belongs to Y and if the are (x1. of a graph. then the problem of optimum pit design comes to finding in a graph G a closure Y with maximum mass or. xi) if Vi is adjacent to V. as well as next best solu tions. and if the mining of volume V1 is not permissible unless volume Vi is also mined. that is. T2. 3. Draw an are (x. be represented by a vertex x. 5) . A) with a set of vertices X and a set of ares A. F is the maximum resulting force over any set of blocks that can freely move up ward in this system. . a set of vertices Y such that if a vertex x.. Vi and Vj have at least one point in common.. sorne of the blocks will be lif ted. Hence. The three-dimensional pit model can be illustrated by a physical analogue. The decomposition of the pit into elementary vol umes V. The resulting force in a block is indicated with an arrow. there are obvious computa tional advantages to be gained from a direct ap proach . as may be the case for an open-pit model. However. Any feasi ble contour of the pit is represented by a closu1e of G. The total work done in this movement is F X 1= F'. the movement of any free mechanical system is such as to maximize the work done. these advantages become important when the graphs considered contain a very large number of elements. This problem can be viewed as an extreme case of the time-cost optimi zation problem in project net works. The maximum closure of G is then given by the vertices of a set of well-iden tified branches of the final tree. The walls and the bottom of the pit are then "smoothed out. ( See Figure 3) . ' L \ 1 -1 4 l 1- 3 l 2 -1 3 -1 mass m 1 is associated to each vertex x. but may also be obtained by taking for V. with proper selection of u nits on the axis. When a is constant. Instead. 4. are the mine value and the extraction cost of element V. • • • Tn following simple rules until no further transformation is possible. if such exist. the blocks will separate along the optimum pit contour. to which severa! solutions have been proposed (2. --. ¡ 1 1 t i í Figure 3.LU . Montreal Figure 5.- .. Búlletin for January. 1965. 49 . . \J¡ -.--f._I 1 1 Vi.·.... virtual\y will highlight sorne properties of the pit model. For . profit in the f i rst year of operations and that our rnining capacity is limited to a total volume V. bu t for suff iciently small incre stay ments of . The following analysis final contour of a pit. This problern can be transforrned into the basic problem by substituti ng each elernenta ry rnass by a new mass m'. ¡... and unlimited nurnbers of ways of reaching a final con the results rnay provide a basis for the selection of tour. interrnediate contours.. 1 111 -.. however.-.-----------. 1 M / f' .. Let us add a restriction to our pit rnodel. What is the optimum contour now ? To answer this question we shall consider the function P= M.. each way having a different cash flow pattern. 1 1 1 1 o 50 V V...An optirnurn digging pattern rnight be one in which the integral of the cash f\ow curve is maximum. 1 1 1 o Mo _ _ _ _ _ _ _ _ ___ M .:_ . J!<..-Steps (left and right..-.\ = O we obtain ou r ol d solution .\ i n- Porometric Anolysis '--.... p v.t. V the volu me of the closu re and . The Canadian Mining and Metallurgical ..--- 1 .I_----.\ a positi ve scalar.\ the optimum contour and V will constan t. when . Supposing that we want to maxirnize the Figure 6 illustrates sorne of the possible cash flows.-.-Cash Flow Patterns T creases.-._ . aboYe) involvcd in determ ining the shape (low er Jef t) of Curve M --= M (V). There are. Figure 6..-1.:::::./..W where M is the mass of a closu re.-/ 1 1 ! Figure 7.... P decrea ses. Instead of maxi mizing M as we did in ou r basic rnodel. .. The problem of designing intermediate pit contours can The established algorithm provides solutions to the become extremely complex..c. = m. we now want to rnaximize P... \2 < Mz + (V¡ . = X1. we draw.) is piecewise linear and con vex . To obtain M1. An edge. completely encloses the con tou r C. 11iass of x.). if we go back to the cu rve V (. V2). of G withO. B) defined by a set of ares Be A and contain ing ali the vertices of G. A tree is a connected and directed graph T = (X.) defined by a set of vertiees of Y e X and containing ali the ares that connect vertiees of Y in G. representi ng the optimum contour for Vi. Between any two of its characteristic points (M2. . In deed. maximum In other words. • • • • • Tº following given rules. and Cb eorresponding to two volumes V. obtained by suppressing an are a. The function V = V (. To each point of this curve corres ponds a contour that is opti mum if the volume mined is exactly V..EY ---+ rx. from B. . A circuit i s a path in whieh the initial vertex co incides with the terminal vertex. and take its intersection wi th OP.). the orebody can be depleted along the cu rve M (V). Y = q. For each Jine segment of P (.\) is a step function.\. is also a closure of G. which implies an orientation.. and branches of a branch are ealled twigs. any volume element eontained in C. is also contained in c•. e.. The graph G also defines a function r mapping X into X and such that (x. y) € A or (y./. A) is defined by a set of elements X called the vertices of G. must be situated below the cu rve P (. A1) which does not contain the root of the tree is ealled a bmnch of T. is a set of two elements sueh that (x. the null set. A partial graph G ( B) of G is a graph (X. x1) . indeed.) sueh that the termina l vertex of each are eorresponds to the initial vertex of the suceeeding are. P is linear with . P = P (A.. The Algorithm The graph G is first augmented with a dummy node x. e. In summary. for V" < Vb.\2. A chain is a sequence of edges [e.\)./. A subgraph G (Y) of G is a graph (Y.x1 f ind a closuvalue re Y m. The algorithm starts with the eonstruction of a tree Tº in G. a2 . (M.\) . y). T2..2V¡ < Mz . u ntil no further transforma tion is possible. M = P + AV Hence. = [x. the intermedi ate volume V. It follows that if no other restrictions are imposed. The root of the branch is the vertex of the branch that is adjaeent to the are a1.and dummy ares (x. Montreal Substituti ng for M¡ . .2V2 = M1 ). A path is a sequenee of ares (a. y] of G. def i nes the value . that is. = (x. By definition. The trees eonstructed during the iterative process are characterized by a given number of properties. An interesting f eature of the curve M (V) is that given two optimum contours C. the root. A) is a set of vertiees Y e X such that x(Y r xfY. (M.\). We can now write M. These points correspond to optimum contours for given vol umes V.. called the ares of G. The proposed graph algorithm can be easily extended to permit such para metric studies. as long as V is constant.With a suf ficiently large . V2). A) and for each vertex a numeric called themass. we have established the shape of the curve M (V) and shown how its characteristic points can be obtained.V2) (2) Frorn (1) and (2). The graph Bulletin for January. the cu rve M (V) is convex.l in which each edge has one vertex in eommon with the sueceeding edge. in a rooted tree T has two eomponents. The total curve M = M (V) cannot be gen erated by this process. the contour will jump to a smaller volume. The component T. together with a set A of ordered pairs of elements a. The maximum closu re is then given by the vertiees of a set of well identified branehes of the final tree.. has a value M'1 = M z + (V. a segment of slope V. The point B on the surface P. A.. V. A branch is a tree itself.. As V jumps to a smaller value so does the slope. it results that point C must in deed be situated below point D. C) eontaining no cycles. finds a set of elements Y e X such that >< X. the value of M corresponding to a volume V is given by the intersection of the segment of slope V and the axis OP. This rnining pattern will maximize the integral of eash flow with respeet to total volume mined [M (V) indeed is a cash flow]. • • • a.2V1 From the equali ty Maximum Closure of a Graph Def initions A directed graph G = (X.eY and Mv = m¡is maximum X. The Problem Given a directed graph G = ( X. Tº is then transformed into sueeessive trees T'. the contour C. A rooted tree is a tree with one disti ngu ished vertex. A closu1·e of a directed graph G = (X. we can obtain a point of the curve M =M (V) .€Y A closure wi th maximu m mass is also called a maxi rnuni closure. then G(Y) is a closed subgmph of G.. y) E A y er X.V2) : = t! 2 (1) But a point D on the segment (M2. To highlight these properties and to avoid unneces sary repetitions we shall next develop sorne additional terminology. but its shape is shown in Fig u re 7. V. x) € A..\. This concept diff ers from that of an are. e. . 1965.. and Vb then.\ and the slope of the line is V. . A cycle is a chain i n which the initial and final vertices co i ncide. If Y is a closure of G. SI . . . Proof : We shal! show that if a vertex. Be eause x. The mass M. also belong to z.is ealled an m-edge (minus-edge) and Tk an 1n-branch. A p-edge (braneh) is strong if it sup ports a mass that is strietly positive. Let ( Figure 3) T(Z) and T ( X-Z) be the subgraphs of T defined by the vertices of Z and X-Z. In a tree T with root x. Similarly.as one of their extremities. joining x. Figure 2. (are a. x. respee tively. then e. A vertex X... The graph G considered in the seque! will be an augmented graph obtained by adding to the original 52 Propert y 1 If a vertex x. We sha!l also disti nguish between strong and weak edges (branehes) .to every vertex x. ali strong edges of a normalized tree will also be p-edges. The Canadian Mining and Metallurgical . x. Note that as ali dummy edges are p-edges.) and we say that the edge e. of a branch T. Finally. the are (x. a tree is norrnal ized if the root x.. These properties will lead us to a basic theorem on maxim um closu res of a direeted graph. T.. if the ter minal vertex of ª• is part of the braneh T•. x. Th is mass is assoeiated with the edge e. with re speet to x. does not belong to Z..) <loes not affect the problem. x.) of a strong p-edge with a dummy are (x. Eaeh edge ek (are a.of the braneh T.) of a tree T defines a braneh. then all the vertiees X.. is ealled a p-edge (plus-edge ) if the are ak points toward the braneh T. to the root x•. x. It is also eonvenient to write x. ali twigs of a branch can be divided into two elasses: p-twigs and m-twigs. We shall next establish properties of normalized trees.). Any tree T of a graph G can be normalized by replaeing the are (x.. say x./ I 1 \ \ \ \ \ \ \ Figu re l. (are a ) is said to support the braneh Tk.. then is ealled a p-branch. Edges (branehes ) that are not strong are said to be weak. (braneh Tk) is eharaeterized by the orientation of the are a. (are a.) of a strong m-edge with a dummy are (x. The tree in Figure 2 has been obtained by normal izing the tree in Figure l.t:X -2 T(2) T(X-e) Figure 3. then Z is not a maxi mum closure. eannot be part of any maximum elosure of G. and assume X1:€Z. x1) . of the branch T. the introduction of du mmy ares (x. an edge e. Vertices that are not strong are said to be weak. Defi11ition s gl'aph a du mmy vertex x. x.for the root of the braneh T•. that is.. The vertex x.€Xs. points away from branch T.is eommon to ali strong edges. an m-edge (braneh ) is strong if it supports a mass that is null or negative. e.) and repeating the proeess until ali strong edges have x.. The edge e... x. noted as Tk = (X. will be the root of ali trees considered. with negative mass and dummy ares (x. is said to be strong if there exists at Ieast one strong edge on the ehain of T joining X. If are a. Ak) . belongs to the maximum closure Z of a normalized tree T.) supports a mass M•. is the sum of the masses of ali vertiees of T. . x. x . .] be the chain of T• linki ng Xm to x•.. however... of T. x.] have changed thei r statu s : a p-edge i n T' becomes an m-edge in T8 and vice versa .] be this m-edge with x. If. ali the edges e1 on the chain [xm. Xm) of T' with the are (x..' M. . ... Theorem I If.-Determine x. and x1 e X-Y'. Go to step l.. This yield s T' '. because of the above relation. On the chain [x. If the tree has no strong vertex. Steps of the Algorit hrn Constru ct a norma lized tree Tº in G and enter the iterative process. This graph is a tree and it is. . 4. T. . must go over x. .. Constrnction of Tº Tº can be obtained by constru cting an arbitrary tree in G and then normalizing this tree as ou tlined earlier. . x. .. otherwise th ere wou ld be a cycle i n T. n.. then obviously Figure 4.J closure of T....' be the component of T (X-Z) containing the vertex x•. in more detai 1. Xm ) supports in T< a branch T. the mass of T'. then the maximum closure . is negative or null. Because T is a partial graph of G and becau se (property 2 ) Y is a maximum The tree T" is obtai ned from T' by replacing the are (x. the mass of any p-twig of T. . .) .. x. x. (a) Construction of T" : If then we find a closure Y of G and a partial graph S of G for which Y is a maximum closu re then. Hence.. normalized. x1) i n G such th at Y.cX-Z (possibly x. The are (x. x. . that is. Let T.. x. then go to step 2. Y must also be a maximu m closure of G. x 0 ... then the maximum closu re is the empty set Z . on the ch ain [xm. .• (4) (5) (6) .. Each tree T' = (X.. The process ter minates when Y is a closu re of G. x. The f irst such edge between Xk and x. Ao) where Ao is the set of ali dummy ares (x.... x. is an m edge.. Go to {l ) For the edge [x. the root of the strong branch con taining x. ·2. x.. A5) is a partial graph of G and if Z and Y are maximum closu res of S and G. . . and thus contain at least one of the edges of A·.. the theorem follows immediately. x-y'' = + Propert y 2 The maximu m closu re of a norma l ized tree T is the set Z of its strong vertices . .] of T• we h ave the followi ng transformation of masses : For an edge e.Ali ares N:· of T that jo in vertices of X-Z with vertices of Z have thei r terminal vertex in Z. the mass of T. = x. Because T is normalized. are all p-edges in T.. .Mv ized tree of G is empty. If we note that any p-branch of T is a closed sub graph of T. .)... in a directed graph G.c¡... x.. A') is characterized by its set of ares A' and its set of strong vertices Y'. is strictly positive. the status of an edge of T• and the mass su pported by the edge are u nchanged by this trans formation.. we obtain the following distribution of masses in T•: m-edge edge e¡ on [xm• . x. lteration t + l transforms a nor malized tree T' into a new normaliz ed tree T'+1. the set of strong vertices of a normal M. . X•] l\IJ¡• édge Mk• = Mm' edge ei on [x1. a norma l ized tree T can be constructed such that the set Y of strong vertices of T is a closure of G..] in T'... A mu ch simpler procedu re. On the chains [x. the transformations taking place in steps 2 and 3 of the algorithm. .] l\lh• = Mm• í2 ) For an edge eJ on the chain [x. all pedges support zero or negative masses and all medges support strictly positive masses as T' is normal ized. x. Hence T. This completes the proof ..) w ith the are (x.. x. Except for this chain.-Norma lize T".' is a branch obtained by re moving p-twigs from the m-branch T. T ransf ormations The steps ou tlined above do not indicate the amoun t of cilcu lation i nvolved in each iterati on nor do they establish that the process will termínate in a f in ite n umber of steps. P1·oof : We shall use the following argument : If S = ( X. in particu lar. x. .' has larg er mass) . Y' is a maximum closu re of G. . ." > Mm' Xo] P-edge < lVI. ..] Mi• = Mm' + M.. • .].. ..of G is the empty set Y = c¡..) . no strong edge. but that an m-branch is not a closed subgraph of T.. The edges of A* connecting vertices of T. .) .. is to constru ct the graph (X. this property follows directly from property l. respectively. is an m-branch of T. as Z is a closure of T.' with mass M.'>0. ...' to vertices of Z. M. is strictly positive and Z is not a maximum closure ( th e closure Z X.. . At least one of the edges of A* is an medge because the chain joining x. ( See Figu re 4) . x..' (3) In addi tion. 3.• = Mm' M.-If there exists an are (x. .cZ and x... then Y is a maxim um closure -0f G.) and [x. Xk and/or x.. to x.. Iteration t + l contain s the following steps: 1.e Y'. Let (Figu re 4 ) [x. Otherw ise go to step 4. Hence.. Let eq = [x.-Terminate. To clarify these points we sha 11 analyze. wi th the exception of ( x. Construct the tree T' by replacing the are (x. . of course.. Montreal It results from these relation s. :Bulletin for January.step 3. 1965. 53 . Vol..E. ." Wiley. "A Network Flow Tenth Annual Minerals Symposium M ay 7-9. Anyone desiring more information may write to Robert G. 2. in the normalization process. . x. H. .with a strong dummy edge (x. and thus ew must be situated on the chain [x1.. .. Grand Junction. Beverly.I. . we now have (7) (8) In any case.. ali strong edges must be on the chain [xm . xb] .M.) . Th is completes the proof. Computation for Project ·Cost Curves. E. Tn will differ either in their masses My or in their sets of strong vertices. . Box 28. we have analyzed them separately to ·establish the following: trees in the sequence Tº. 1965.I. R. x. No.]. January. 1965 T HE lOth annual Minerals Symposium will be held in Grand Junction.. however. 1963. Jr. . Reynold s. Symposiu m secretary treasu rer. May 4-5.S. say e..]. No..t The latter case can on ly occu r if the equ ality ap plies i n (6). under the spon sorship of the Colorado Plateau Section of the American Institute of Mining.. Social events and field trips are also being planned. We now search for the next strong p-edge on th e chain [x.. 7.. . 1962.' and the set y•+i is larger than the set Y'. x. Berge. T'. In addition to Moab. Wyoming.]. 3. 8 and 9. . F. . (2) Fulkerson. l. x. The name was changed two years ago to Miner als Symposium and the scope broadened to inclu de other miner als such as oil.. sections.S.J. Grand Ju nction." M anagement Science. Ottawa. . (4 Kelly. . "Critica! Path Planning and ) Schedul ing: Mathematical Basis. Committees are now at work on the 1965 program. . the Colorado Plateau Section of the A. We replace e. T'. . .. This edge. is general chair man for the event. . Colorado. the set Xw' con tains X..] of T'. Because. ( MayJune)." Manag ement Science. a maxi mum closure of G is obtained in a finite number of steps. 9. x. . = [x. metall urgy and geology.. Be ca use of property 3. . We shall show that either M r decreases du r ing an iteration or else My stays constant but the set Y increases.). Philip Don nerstag. 1961. must be a p-edge ( beca use of property 3. Project Cost Polygons. (b) Normalization of T•: As T' was normalized. x. ( 1961) . and at Riv erton. . Jet us see how My and Y transform during an iteration. . In practice. Indeed. Utah. x. property 3 will remain valid on the chain [xb. Mw• If Mw• If Mw• Mm' because of (4) and (6) < Mrut then Mv'+1 < M.. is chai rman of the host group. . and Lerchs. Colorado. . the Symposi um has been held at Grants..J .E. (3)Grossmann. References Theorem JI In followi ng the steps of the algorithm... of Continental Materials Corp.. Thus. we l·emove a p-twig from the branch T/ and must subtract its mass from all the edges of the chain Lx. New Mexico... At the 1965 Grand Junction meeting. D. . P. ." Op et·ations Resecirch (U. . Proceedings of the Third Annual Conference of the Canadian -Operation a l Research Society. Let ew be this edge and Xwª the vertices of the branch T. transformations (a) and (b) can be carried out simultaneously .t = Mm' then Myt+ 1 = M. Grand Junction. 1961. ." originated ten years ago at Moab. T".. speak ers will cover technical and non techn ical subjects relating to m in ing. Each normalized tree is characterized by its set Y of strong vertices and the mass My of this set. Vol. H. potash and oil shale. C.. x 9.] then the mass M"• is strictly positive and larger than any mass suported by a p-edge that precedes ea on the chain [xm. The Minerals Symposium..Property 3 If. we never generate a strong m-edge it is clear that the last p-edge removed from the chain [xm. . . we on ly have to show that no tree can repeat itself in the sequence Tº. Metallurgical and Pe troleum Engineers.. .] and repeat the process u ntil the last strong p-edge has been re moved from the chain. April. which now rep resents a broad coverage of the mi ning and processing industries of the Western states.] is the p-edge that supports the Jargest positive mass in T•. sponsored by A. all m-edges are weak) . Proof : As the number of trees in a finite graph is finite.]. As a result of steps (a) and (b). Then.O. . x.. "The Theory of Graphs and its Applica tions. on May 7..M. of American Metal Cli max.. "An Algorithm for Directed Grapbs witb Application to tbe Project Cost Curve and In-Process Inventory.. William. form erly known as the "Uranium Sym posium.. Severa] hu ndred people attend this annual event. so that any two (1) C. . Advance programs will be mailed out. . We remove strong edges one by one starting from the first strong edge en countered on the chain [x.x. "A Structural Method of ·Computing (5) Prager. x. e" i s an m -edge on the chain [xm.. . . in T•. 54 The Canadian Mining and Metallurgical . 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