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Application of push-relabel and heuristics to open pit mines 1996
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Title | Application of push-relabel and heuristics to open pit mines |
Creator |
Sahni, Jaspreet |
Date Created | 2009-03-06T22:38:58Z |
Date Issued | 2009-03-06T22:38:58Z |
Date | 1996 |
Description | The pit limit problem is crucial to mine planning. The use of computer models to design ultimate open pit limits is becoming increasingly popular. One solution method adopted is to transform the pit limit problem to a maximum flow network. A popular maximum flow technique is push relabel. The purpose of this thesis is twofold. The first is to check if push relabel algorithm performs better than other MF algorithms on real, rather than randomly generated data (as in the past). The second is to develop and test heuristics that can take advantage of the characteristics of the open pit mine network structure to further renhance the push relabel routine. |
Extent | 3410542 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2009-03-06T22:38:58Z |
DOI | 10.14288/1.0087587 |
Degree |
Master of Science - MSc |
Program |
Commerce |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Graduation Date | 1997-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/5693 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0087587/source |
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APPLICATION OF PUSH-RELABEL AND HEURISTICS TO OPEN PIT MINES by JASPREET SAHNI M.B.A., Simon Fraser University, 1992 M.Sc., The University of British Columbia, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Commerce) We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA October 1996 ( C ) J a s p r e e t S a h n i , 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C o ^ i ^ e r t a and- © u s i ^ e s c , A . d m i r t t s ' T r a \ 100 The University of British Columbia Vancouver, Canada Date D e c ^ w x b - e v H , ' °> S fe -6 (2/88) A b s t r a c t The p i t l i m i t problem i s c r u c i a l to mine plan n i n g . The us of computer models to design u l t i m a t e open p i t l i m i t s i s becoming i n c r e a s i n g l y popular. One s o l u t i o n method adopted i s to transform the p i t l i m i t problem to a maximum flow network. A popular maximum flow technique i s push r e l a b e l . The purpose of t h i s t h e s i s i s twofold. The f i r s t i s to check i f push r e l a b e l a l g o r i t h m performs b e t t e r than other MF algorithms on r e a l , r a t h e r than randomly generated data (as i n the p a s t ) . The second i s to develop and t e s t h e u r i s t i c s that can take advantage of the c h a r a c t e r i s t i c s of the open p i t mine network s t r u c t u r e to f u r t h e r enhance the push r e l a b e l r o u t i n e . ii T a b l e o f C o n t e n t s A b s t r a c t i i Table of Contents i i L i s t of Tables i v L i s t of Figures V Acknowledgement v i Chapter One I n t r o d u c t i o n 1 Chapter Two M o d e l l i n g 4 Chapter Three Op t i m i s a t i o n Techniques 6 Graph Theory- 8 Two dimensional dynamic programming 9 Three dimensional dynamic programming 10 Lin e a r programming 12 Network Flow 13 FKS c l o s u r e a l g o r i t h m 15 Johnson's Network flow 17 Push Relabel 19 H e u r i s t i c s 22 Chapter Four Our Implementation 24 Various H e u r i s t i c s 28 I l l u s t r a t i v e example 39 Chapter Five Results and Conclusions 47 Tables and Figures 55 References 70 iii L I S T O F T A B L E S S o l u t i o n Times T o t a l Times FKS D i n i c Push R e l a b e l (FIFO) H e u r i s t i c Push R e l a b e l (HL) H e u r i s t i c Expanded iv L I S T O F F I G U R E S S o l u t i o n time S o l u t i o n time (no FKS) S o l u t i o n time S o l u t i o n time (no FKS) T o t a l time Log S o l u t i o n time vs. Log Nodes A c k n o w l e d g e m e n t I would l i k e to thank a l l the members of my committee f o r t h e i r time and patience. In p a r t i c u l a r my s u p e r v i s o r Dr Tom McCormick deserves s p e c i a l r e c o g n i t i o n f o r h i s time spent i n numerous meetings o r g a n i s i n g my research. I would l i k e to acknowledge the help of Lynx Geosystems, Inc. f o r p r o v i d i n g mine data (blockmod and s a b o 2 ) and the code f o r generating the .spp and .spv f i l e s . I would a l s o l i k e to thank Todd Stewart f o r w r i t i n g the network generator code. F i n a l l y , I thank my e n t i r e f a m i l y , e s p e c i a l l y my parents and wife Meera, f o r t h e i r constant support and encouragement. vi C H A P T E R 1 I n t r o d u c t i o n Mine planning c o n s i s t s of determining a sequence of e x t r a c t i o n over a given time horizon. The optimum open-pit l i m i t of a mine i s defined by Caccetta and Giannini (1990) to be a f e a s i b l e contour whose e x t r a c t i o n r e s u l t s i n maximum t o t a l p r o f i t . F e a s i b i l i t y necessitates the p i t to have safe and workable w a l l slopes which depend on the g e o l o g i c a l structure and the mining equipment. The ultimate p i t l i m i t s a f f e c t the e n t i r e mine layout and provide e s s e n t i a l information f o r the evaluation of the economic p o t e n t i a l of the mineral deposit and i n formulating long, intermediate, and short range mine plans. Consequently, knowledge of the optimum p r o f i t a b l e ultimate p i t l i m i t s during the i n i t i a l stages can avoid c e r t a i n problems such as mining u n p r o f i t a b l e expansions, abandoning p r o f i t a b l e ore, and r e l o c a t i n g surface f a c i l i t i e s . A mine planning s i t u a t i o n requires d i f f e r e n t l e v e l s of p r e c i s i o n at each stage. Johnson (1973) argues that a good estimation of the p i t l i m i t provides a s a t i s f a c t o r y b a s i s f o r the i n i t i a l f e a s i b i l i t y evaluation of a mining p r o j e c t . But knowledge of the true optimum becomes more valuable as mine l i f e progresses and the knowledge of each aspect of the operation increases. The a v a i l a b i l i t y of accurate economic 1 p i t l i m i t s f o r intermediate range planning leads to b e t t e r scheduling r e s u l t i n g i n increased p r o f i t a b i l i t y f o r the project. The ultimate decision to abandon an open-pit mining venture requires that the p i t l i m i t c a l c u l a t i o n be nothing short of optimum. Unfortunately, i n the past the techniques that guarantee optimum so l u t i o n s have been too complex to be understood by the mining community or too c o s t l y to implement i n terms of computer processing times. This r e s u l t s i n the use of simple and f a s t h e u r i s t i c s . A l l stages of mine planning may therefore b e n e f i t from a technique which provides an optimal s o l u t i o n at a low cost and i n an e a s i l y understood manner. One technique used to provide the optimal p i t l i m i t i s to transform the network i n t o a maximum flow problem. The maximum flow problem can then be solved with many standard r o u t i n e s of which the one with the best reputation f o r p r a c t i c a l performance i s push-relabel. This t h e s i s shows that network implementations which i n the past have been regarded to be too slow, are indeed very f a s t now. I t assesses the push r e l a b e l algorithm's performance as compared to other maximum-flow algorithms on ac t u a l c l a s s of graphs (open p i t mining) rather than randomly generated data as has been done i n the past at the DIMACS Challenge (Cherkassky and Goldberg 1994). Also, we attempt to take advantage of the 2 s p e c i a l s t r u c t u r e of an open p i t mine by developing h e u r i s t i c s i n order to enhance the performance of the push- r e l a b e l routine on t h i s c l a s s of problems. 3 C H A P T E R 2 M o d e l l i n g Caccetta and Giannini (1990) define the optimum u l t i m a t e p i t l i m i t of a mine as that contour which i s the r e s u l t of e x t r a c t i n g the volume of m a t e r i a l which provides the t o t a l maximum p r o f i t while s a t i s f y i n g c e r t a i n p r a c t i c a l o p e r a t i o n a l requirements such as safe w a l l slopes. As i n d i c a t e d by Lerchs and Grossmann(1965), the problem can be expressed a n a l y t i c a l l y as f o l l o w s . Let v, c, and m be the three density functions defined at each point (x,y,z) of a three dimensional region containing the ore-body with v ( x , y , z ) : mine value of ore per u n i t volume c ( x , y , z ) : e x t r a c t i o n cost per u n i t volume m(x,y,z): p r o f i t per u n i t volume = v(x,y,z) - c(x,y,z) Let a(x,y, z) be the set of angles s p e c i f y i n g the w a l l slope r e s t r i c t i o n s at the point (x,y, z) with respect to the h o r i z o n t a l plane, f o r a given set of azimuths. This set i s known as the v a r i a b i l i t y of w a l l slope requirement. Define S as the family of surfaces such that at no point does the slope with respect to the h o r i z o n t a l plane exceed the corresponding angle i n a. Denote the family of volumes corresponding to the family S of surfaces by V. The problem then i s to f i n d a volume VQ which maximises the i n t e g r a l of the p r o f i t function m(x,y, z ) . Since, i n p r a c t i c a l 4 situations, there i s no simple analytical representation for the functions v and c, numerical techniques must be used to solve the problem. Generally, the ore body i s divided into blocks. The mineral deposit must somehow be represented as a model to f a c i l i t a t e mine planning. The representation must incorporate both the three dimensional mine structure and the economic value of the deposit. The most popular method i s the regular 3-D fixed block model. This model was f i r s t published i n the early 1960's. In this model the t o t a l volume i s transformed into a three dimensional grid whose each block i s an independent unit with i t s own economic content. Since there are such a large number of blocks i n a typical block model i t i s essential that the optimisation algorithm delivers the solution i n minimum computer time. A fast algorithm could also be run on a personal computer. Because of the costly'ramifications associated with the mine contours generated, i t i s imperative that the accuracy of these contours be high. Further compounding this problem i s the desire for s e n s i t i v i t y information which means that the model must be run many times. 5 C H A P T E R 3 O p t i m i s a t i o n T e c h n i q u e s Optimisation refers to optimisation with respect to certain constraints. Some of these constraints are i m p l i c i t i n the assumptions being used, and nearly a l l techniques make some key assumptions. Kim (1977) states some of these as: (1) The grade and cost of mining each block i s known and accurate. (2) The cost of mining each block does not depend on the sequence of mining. (3) The desired slopes and p i t outlines can be approximated by removed blocks. (4) The objective i s to maximise t o t a l undiscounted p r o f i t , which i s questionable due to the difference i n present values of cash flows. In a l l optimisation techniques, the predominant type of model used i s the regular 3-D fixed block model. The block model i s represented as a weighted digraph with the vertices representing blocks and the arcs representing mining restrictions. A digraph G consists of a set of vertices V(G) and a set of arcs E (G) . There i s an incidence r e l a t i o n which associates with each arc of G an ordered pair of vertices. A (node) weighted digraph i s one i n which each vertex has an assigned weight. Here, the graph contains the arc (x,y) i f 6 the mining of block x requires the removal of block y. The p r o f i t from mining a block i s represented by an appropriate vertex weight. A closure of a weighted digraph i s a set of vertices C such that i f x i s a member of C and (x,y) i s an arc then y must be a member of C. The weight w(C) of the closure C i s the sum of the weights of the vertices of C. Here, a closure represents a feasible p i t contour, with the weight representing the prof i t realised from that p i t contour and conversely the graph theory problem of determining a maximum weight i n a weighted digraph i s equivalent to determining the optimum p i t contour. There have been at least four rigorous optimising techniques presented since 1965: (1) Lerchs Grossmann (2) Dynamic programming (3) Linear programming (4) Network Flow, including (a) FKS (b) Johnson (c) Push-Relabel (d) Dinitz, Ford-Fulkerson, etc. In addition, there are other nearly optimising heuristics such as Floating Cone. 7 G r a p h T h e o r y The Lerchs-Grossmann graphical technique was developed by Helmut Lerchs and Ingo Grossmann (Lerchs and Grossman 1965). The Lerchs-Grossmann algorithm augments the digraph representing the ore body into a rooted tree by adding a root vertex and arcs to a l l other v e r t i c e s . Each set of v e r t i c e s and arcs i s c l a s s i f i e d as e i t h e r strong or weak. A rooted tree whose root i s common to a l l strong arcs i s a normalised tree. The Lerchs-Grossmann algorithm proceeds by c o n s t r u c t i n g a sequence of normalised trees, terminating when the set of strong v e r t i c e s of the normalised tree form a closure of the graph representing the ore body because of the f o l l o w i n g theorem: I f , i n a weighted digraph G, a normalised spanning t r e e T can be constructed such that Y the set of strong v e r t i c e s of T, i s a closure of G, then Y i s a maximum closure. There has been a considerable amount of work done on Lerchs Grossmann to date such as the W h i t t l e 4D commercial software p r o j e c t which costs $15,000 and i s the standard i n the i n d u s t r y now. This i s due to the f a c t that i t provides a true 3-D optimal. 8 Two D i m e n s i o n a l D y n a m i c P r o g r a m m i n g Lerchs and Grossmann f i r s t presented the b a s i c two- dimensional optimal cross section method (Lerchs and Grossman 1965) . The method determines the optimum ultimate p i t l i m i t s i n each v e r t i c a l c r o s s - s e c t i o n of the economic block model. The union of these optimal cross s e c t i o n p i t o u t l i n e s y i e l d s a 3-D p i t contour which u s u a l l y v i o l a t e s the w a l l slope r e s t r i c t i o n s and i s thus i n f e a s i b l e . The advantages of t h i s technique are ease of implementation and use, i t s f l e x i b i l i t y and speed of sol u t i o n . There are two disadvantages of t h i s method. F i r s t , i t optimises only with regard to cr o s s - s e c t i o n and thus r e q u i r e s extensive e f f o r t to smooth out the p i t bottom and end sections. Secondly, the two-dimensional contours do not n e c e s s a r i l y l i n e up from cross s e c t i o n to cross s e c t i o n r e s u l t i n g i n an i n f e a s i b l e s o l u t i o n . 9 T h r e e D i m e n s i o n a l D y n a m i c P r o g r a m m i n g A number of dynamic programming algorithms attempt to extend the bas i c two-dimensional cross s e c t i o n routine presented by Lerchs and Grossmann to handle the three- dimensional problem. One such technique was developed by Johnson and Sharp (1971). This technique eliminates the bottom end smoothing which i s the primary disadvantage of the two- dimensional technique. The algorithm employs r e p e t i t i v e a p p l i c a t i o n of the two-dimensional algorithm to obtain the optimum block configuration on each cr o s s - s e c t i o n , f o r each l e v e l . Then a f i n a l a p p l i c a t i o n of the two-dimensional algorithm on a l o n g i t u d i n a l s e c t i o n where the blocks represent the optimal s e c t i o n contours down to each l e v e l . There are two advantages of the three-dimensional dynamic programming technique. F i r s t l y , ease of understanding and implementation. Secondly, f l e x i b i l i t y . The major disadvantage of t h i s method i s that the s o l u t i o n may not be f e a s i b l e because the method does not require adjacent c r o s s - s e c t i o n a l p i t configurations to l i n e up. Other dynamic programming methods such as those of Koenigsberg (1982) and B r a t i c e v i c (1984) seek to extend the Johnson-Sharp method i n overcoming t h i s d i f f i c u l t y but unfortunately r e s u l t i n s i g n i f i c a n t increases i n computational complexity; s t i l l neither achieves a proper 3-D 10 optimum s o l u t i o n . A f u r t h e r disadvantage i s t h a t the 3-D t e c h n i q u e w i l l not c o n s i d e r waste b l o c k s except those a l o n g the two p r i n c i p a l axes o f the d e p o s i t ( c r o s s - s e c t i o n a l and l o n g i t u d i n a l d i r e c t i o n s ) . The waste b l o c k s i n the c o r n e r a rea o f the p i t are n e g l e c t e d i n d e v e l o p i n g the p i t l i m i t optimum r e s u l t i n g i n s teeper p i t s l o p e s i n these a r e a s . T h i s i s known as the corner e f f e c t . 11 L i n e a r P r o g r a m m i n g I t i s natural, to formulate the open p i t problem as a l i n e a r program since l i n e a r programming i s one of the most popular operations research modelling techniques. The formulation as a l i n e a r program i s very simple but the so l u t i o n of such a model i s not computationally f e a s i b l e f o r lar g e ore bodies. In order to apply l i n e a r programming techniques the problem s i z e must be reduced. 12 N e t w o r k F l o w Picard (1976) proposed a network flow model to solve the maximum closure problem. The network N i s obtained from the digraph D of the orebody by adding two vertices s (source) and t (sink). The source i s joined to a l l vertices having a positive weight by an arc of capacity equal to the weight of the vertex. A l l vertices of negative weight are joined to the sink by an arc having capacity equal to the absolute value of the weight of the vertex. A l l the internal arcs of D are given i n f i n i t e capacity. Picard formulated the problem as a 0-1 program and showed that the maximal closure of D corresponds to a minimal cut of N which can be found by any of the standard maximum flow routines. There are a number of algorithms to solve the maximum flow problem (Ahuja, Magnanti, and Orlin) such as the network simplex method of Dantzig, the augmenting path method of Ford- Fulkerson, the blocking flow method of Dinic, and the push- relabel method of Goldberg and Tarjan. Network flow methods have been regarded by the mining community as too slow and requiring too much memory, but this was before good modern implementations. It i s not theoretical bounds but run time that matters i n practice. The empirical performance of push-relabel and Dinit's algorithm are established i n a paper by Cherkassky and 13 Goldberg (1994). They implemented two versions of push- r e l a b e l (highest l a b e l and FIFO) each combined with g l o b a l and gap r e l a b e l i n g h e u r i s t i c s , and one implementation of D i n i c ' s algorithm. They ran these routines on randomly generated set of problems which were used at the F i r s t DIMACS Challenge (Cherkassky and Goldberg 1994). Their experiments show that both versions of push-relabel are e m p i r i c a l l y f a s t e r than D i n i t z f o r a l l set of problems and that the highest l a b e l version of push-relabel i s e m p i r i c a l l y f a s t e r than the FIFO versio n f o r a l l sets of problems. The performance of these routines on a c t u a l mining data i s i n v e s t i g a t e d i n t h i s t h e s i s . Some network flow techniques presented to date are discussed i n the f o l l o w i n g s e c t i o n s . 14 F K S C l o s u r e A l g o r i t h m This technique was developed by Faaland, Kim, and Schmitt (1990) to solve maximum closure problems. It works en t i r e l y within a compact representation of the weighted digraph. Each vertex i s given a certain weight c a l l e d supply sup(j) and every arc has a pair of capacities which i n i t i a l l y are set to cap(i,j) = i n f i n i t y and cap(j,i) = 0. The unique feature of this algorithm i s a s t a y l i s t which i s the set of nodes which need not be considered i n subsequent iterations of the algorithm. The s t a y l i s t i s i n i t i a l l y empty. The algorithm proceeds as follows: Step 1: I n i t i a l i s e : sup(j) = weight of j . S t a y l i s t = empty. cap(i,j) = i n f i n i t y . cap(j,i) = 0. Step 2: Find a positive supply node i not on the s t a y l i s t and set i t s predecessor label P(i) = 0. Step 3: Find a node j unlabelled i n this i t e r a t i o n that i s not on the s t a y l i s t and with cap(i,j) > 0. If none jump to step 7. Step 4: Send flow from i to j . Label P(j) = i . sup (i) -=flow. cap(i,j)-=flow. sup(j)+=flow. cap(j)+=flow. 15 Step 5: If sup(j) > 0 then i = j go to step 2. Otherwise sup(j) = 0 so backtrack to k=P(j). Step 6: If sup(k) > 0 then erase labels of j and nodes labelled a f t e r j i n this i t e r a t i o n and set i=k and go to step 3. Otherwise sup(k) = 0 so set j=k and go to step 5. Step 7: Backtrack from i to k=P(i). Step 8: If P(i) > 0 then send flow from i to k and reset sup and cap and i=k and go to step 4. Otherwise P(i) =0 so add i and nodes labelled after i to s t a y l i s t . The maximum closure i n the digraph at the end of the algorithm i s i d e n t i f i e d by the nodes i n the s t a y l i s t . Faaland, Kim, and Schmitt's paper tests on small randomly generated two-dimensional version of the open p i t problem with this technique and so there i s no standard to compare the empirical complexity of this algorithm with the others stated above. We implemented this algorithm i n C to provide a basis for the performance of push relabel and our heu r i s t i c . 16 J o h n s o n ' s N e t w o r k F l o w This technique was f i r s t proposed by Johnson i n 1968. It uses the same block pattern representation of allowable mining slopes as the Lerchs-Grossmann method and starts with the same network representation of required allowable mining sequence and converts i t into a b i p a r t i t e network. The algorithm as i l l u s t r a t e d by Johnson (1973) proceeds as follows: Step 1: We divide the blocks into two sets, (a) positive blocks and (b) negative or zero blocks. Step 2: Connect each positive block to a l l negative blocks which are required to be mined to remove the positive block by adding arcs from the positive block to the negative block. Step 3: For each positive block allocate flow from the positive block to i t s restricting negative blocks, mine this block and i t s r e s t r i c t i n g blocks i f the sum i s positive. This process usually requires reallocating flow for shared blocks i n the network. This technique leads to an optimal ultimate p i t l i m i t but i t s disadvantage i s i t s in e f f i c i e n c y caused due to reall o c a t i n g flow. The need to reallocate flow occurs when 17 r e s t r i c t i n g negative blocks are shared by positive blocks. The reason i s that a uneconomic positive block should not be used to support the stripping of an economic block which can support i t s own stripping. This reallocation process has been compared to the normalisation step i n the Lerchs- Grossmann technique though Johnson's i s computationally less expensive. 18 P u s h - R e l a b e l Push-Relabel i s an algorithm for finding a maximum flow through a network. A flow network i s a directed graph G = (V,E ,S/t,u) where V and E are the node and arc set. Nodes s and t are the source and the sink and u i s the nonnegative capacity on the arcs. A flow i s a function that s a t i s f i e s capacity constraints on a l l arcs and conservation constraints on a l l nodes except the source and the sink. The conservation constraint at a node v states that the excess e(v), defined as the difference between the incoming and the outgoing flows, i s equal to zero. A preflow s a t i s f i e s the capacity constraints and the relaxed conservation constraints that only require the excesses to be nonnegative. An arc i s residual i f the flow on i t can be increased without v i o l a t i n g the capacity constraints, and saturated otherwise. The residual capacity of an arc i s the amount by which the arc flow can be increased. A distance l a b e l l i n g d:V-> N s a t i s f i e s the following conditions: d(t) =0 and for every residual arc (v,w), d(v) <= d(w) + 1. A residual arc (v,w) i s admissible i f d(v) = d(w) + 1. A node v i s active i f d(v) < n and e(v) > 0. The push-relabel algorithm maintains a preflow and a distance labelling t i l l there are no active nodes. Then the preflow i s converted to a flow. The push relabel routines of 19 Cherkassky and Goldberg (1994) proceed as follows: Step 1: Find an active node v. An active node i s one with positive excess and distance label less than n. If there are no active nodes then stop. Step 2: Find the next edge (v,w) of v. If none go to step 4. Step 3: If(v,w) i s admissible then send d = (0,min(excess(v) , residual capacity(v, w)) units of flow from v to w and repeat step 3 u n t i l excess(v) equal zero. If (v,w) not admissible go to step 2. Step 4: Replace v's distance label d(v) by min(ViW)d(w) + 1. Go to step 1. The main discharge operation i s applied i t e r a t i v e l y to active nodes as above. A l l that i s l e f t i s to decide the order i n which these active nodes must be processed. There are two alternatives presented by Goldberg and Cherkassky. The f i r s t one i s known as the FIFO algorithm and maintains a l l active nodes i n a queue adding active nodes to the rear of the queue and discharging positive nodes from the front of the queue i n a F i r s t - I n First-Out order. The second one i s the HL algorithm and always selects the node with the highest 20 label to discharge f i r s t . Cherkassky and Goldberg improve the p r a c t i c a l performance of the push relabel algorithms by adding global and gap r e l a b e l l i n g heuristics to the maximal flow code. In their paper (1994) Cherkassky and Goldberg show that both the push relabel routines perform better than Dinitz on randomly generated sets of data from the DIMACS generators. 21 H e u r i s t i c M e t h o d s Although a number of heuristic methods have been proposed, only one of them has been widely accepted, the f l o a t i n g cone. The method's wide acceptance stems from the fact that i t very easily understood and easy to program. Also, there i s no re s t r i c t i o n on wall slope v a r i a b i l i t y . The method as i l l u s t r a t e d by Johnson (1973) proceeds as follows: Step 1: Define a certain potential mining volume by stating a bottom block and the allowable wall slopes. Step 2: Sum the value of a l l the blocks whose centres l i e inside the defined cone. Step 3: Select the cone for mining i f the t o t a l value i s greater than zero. The above process i s continued u n t i l i t i s not possible to find any cones of positive value. The advantages of this technique are that i t i s e a s i l y to understand, allows variable p i t slopes, i s easy to implement, and that variable size blocks present no d i f f i c u l t y . The basic shortcoming of the method i s i t s i n a b i l i t y to consider j o i n t contributions of multiple ore blocks located l a t e r a l l y a distance apart thereby missing the optimal solution. 22 C H A P T E R 4 O u r I m p l e m e n t a t i o n As described earlier, the open p i t l i m i t problem can be converted to a maximum flow problem. The maximum flow problem i s very easy to state: In a capacitated network, we want to send as much flow as possible between two nodes c a l l e d the source s and the sink t, without v i o l a t i n g the capacity r e s t r i c t i o n on any arc. There are a number of algorithms for solving the maximal flow problem. As mentioned i n Chapter 3, the basic methods are: (1) Network simplex method of Dantzig (2) The Augmenting Path method of Ford and Fulkerson (3) The Blocking Flow method of Dinic (4) Push Relabel method of Goldberg and Tarjan Studies by Goldfarb and Grigoriadis (1988) showed that Dinitz algorithm i s i n practice superior to the network simplex and the Ford-Fulkerson methods. Several recent studies such as those by Anderson and Setubal (1993) and Nguyen and Venkateswaran (1993) show that the push relabel method i s superior to Dinic's method i n practice. In particular the paper by Goldberg and Cherkassky (1994) showed that two versions of push relabel perform better than Dinitz for a l l problem sets generated by the First DIMACS challenge; The data used to come to this conclusion are randomly 23 generated by the DIMACS network generators. This thesis attempts to confirm these conclusions on real sets of data for open-pit mining problems. Tom McCormick and Todd Stewart's code generates a bounded network with a minimum possible number of blocks and arcs. I imbed several implementations of heuristics as the network i s generated to check i f we can improve the performance of push relabel by giving i t a "warm sta r t " . Since the push-relabel deals with preflows one can, greedily or by taking advantage of a open p i t mine structure, push as much flow as we want and input these excesses as a preflow into the push-relabel routine thereby giving i t a warm star t . F i r s t l y , the network generation routine establishes the lowest positive value block i n each row column pair of the ore body because there i s no need to consider any blocks beneath these. Next we have to add on those blocks that are outside the ore body that would need to be removed i n order to remove blocks within the ore body. This i s given by the .spp f i l e which i s the cone of slopes of azimuths above any block. The .spp f i l e gives us a l l the blocks that need to be removed i n order to remove a specific block. F i n a l l y , we also have to add on blocks that are necessary to s a t i s f y wall slope r e s t r i c t i o n . This i s given by the .spv f i l e which gives the minimum set of arcs whose tra n s i t i v e closure gives 24 the same cone given by the .spp f i l e . The .spv f i l e gives us the outer boundary blocks of the .spp f i l e . The .spp and .spv f i l e s are generated v i a Lynx's code and examples of these f i l e s are given on pages 47 and 52. We attempt to enhance the speed of the network flow routine by giving i t a warm start. Several heuristics are studied i n order to take advantage of the three dimensional structure of the open p i t mine network that i s solved by a generic maximum flow problem. The network generator proceeds as follows: Step 1: Read i n the 3-D block array with block values. Step 2: Establish a bottom layer for each row-column pair as the lowest positive block layer. Step 3: Extend the bottom array to a new bottom whose size i s expanded as indicated by the .spp f i l e to allow blocks outside the o r i g i n a l orebody to be mined. Step 4: Establish a linked l i s t of a l l positive value bottom layers. Step 5: Run through this linked l i s t and add any new bottom 25 layers that are needed as indicated by the .spp f i l e to s a t i s f y wall slope requirements. Step 6: Number a l l the blocks that are i n the network. Step 7: Run through each block from bottom up generating the minimum possible number of arcs needed to remove that block as given by the .spv f i l e . Attempt to push flow as dictated by the heu r i s t i c . 26 V A R I O U S H E U R I S T I C S We are using a generalised maximum flow routine (push relabel) to solve a sp e c i f i c application which i s the open p i t mine problem. The open p i t mine problem has a three dimensional structure and f i n i t e capacities on source and sink arcs only. The idea then i s to use this special structure to enhance the performance of the push relabel routine. Various heuristics are implemented and incorporated into the network generation routine, and give the push re l a b e l routine a warm start. The performance of each h e u r i s t i c i s judged by the percentage reduction i n solution time of the push relabel routine over three instances of networks chosen to show a range of performance of the he u r i s t i c s . The three networks are generated from the blockmod data which i s a r t i f i c i a l l y generated to represent a real mine. There are two classes of heuri s t i c s . F i r s t l y , those that push flow using arc information only. Secondly those that push flow using node information also. C l a s s 1 M I N I M U M l x We know that i n order to remove blocks on one layer we need to remove blocks from the layers above this layer. The way the network i s generated the ore body generally l i e s i n the central (row,column) area. Therefore we expect the 27 central blocks i n the upper layer to be i n the optimal solution. The f i r s t and most obvious heur i s t i c we consider i s based on the minimum lx distance. Each arc has a t a i l , head pair. The difference between the t a i l ' s row number and the head's row number i s the row offset of t h i s arc. S i m i l a r l y we have a column and a layer o f f s e t . The lx distance i s then defined as the row offset plus the column of f s e t . The l x norm i s the sum of the absolute values of offsets. So a smaller lx distance means an arc that goes up straight for and a smaller l a t e r a l deviation. Therefore we want to push flow on minimum l x distance arcs since they push flow straight up straying the least from the central blocks. This i s accomplished through creating an array of linked l i s t s where the array index equals the l x distance of the arc. Then a l l arcs with the same lx distance are part of one linked l i s t so each time an arc i s generated we push flow f i r s t to the sink i f possible, and then to arcs beginning from minimum l a distance linked l i s t to increasing 1 distance linked l i s t s a l tering the excess on the respective nodes and the flow on the arcs. The mean solution times with and without the h e u r i s t i c are shown below for the representative set of networks. 28 Network a60 b45 c50 Nodes 2813 4277 4097 Heuristic .55 1.12 1.78 Push-Rel . 63 1.23 1.8 Reduction 13 % 9 % 1 % M A X I M U M Z We define the z distance of any arc as z =layer of f s e t . We push flows on arcs i n order of decreasing layer o f f s e t . Arcs connected to higher layers get flow f i r s t with the exception of arcs to the sink getting flow f i r s t . Here we do not care i f flow i s moving away from the central (row, column) blocks as long as i t s going up as fast as possible. The performance of this heuristic was tested on blockmod and a sample follows: Network Nodes Heuristic Push-Rel Reduction a60 2813 .57 .63 10 % b45 4277 1.20 1.23 2 % c50 4097 1.8 1.8 0 % M A X I M U M l x We know that before any block i s i n the optimal solution i t s value must be greater than the value of the head of any of i t s arcs not considering the combined flow effect from other nodes. The central nodes are more l i k e l y to get 29 flow from other nodes than the outer nodes. Therefore an argument can be made for pushing flows to the outer blocks f i r s t so as to reduce reallocation of flows at a l a t t e r stage. In the maximum lx flow i s pushed on arcs i n decreasing order of l x distance with the exception of arcs to sink getting flow f i r s t . The performance was tested on blockmod and the results follow: Network Nodes Heuristic Push-Rel Reduction a60 2813 .62 .63 2 % b45 4277 1.23 1.23 0 % c50 4097 1.8 1.8 0 % M A X I M U M Z - M I N I M U M l x From the preceding results, i t appears that pushing flow on arcs i n the upward direction i s better than pushing flow on arcs away from the centre i n the l a t e r a l direction. From this we infer that we should push flows on arcs with the highest slope. The slope of an arc i s given by z/l1. We would l i k e to push flow on arcs with higher slopes f i r s t , with preference given to flow to sink. The d i f f i c u l t y with t h i s i s that there are arcs with l x distance of zero and therefore i n f i n i t e slope and the above mentioned procedure amounts to the same as the minimum l x h e u r i s t i c . Therefore we pushed flows on various combinations of Max f=(az-bll) 30 where the coefficients a and b give different weights to upward and l a t e r a l movement of flow. The results are as follows for a = 2 and b = 1: Network Nodes Heuristic Push-Rel Reduction a60 2813 .57 .63 10 % b45 4277 1.18 1.23 4 % c50 4097 1.79 1.8 1 % B O U N D E D M I N I M U M l x We know that i t i s preferable to send flow to the top layers. The question then i s that i s there a l i m i t on the flow beyond which i t i s no longer advantageous to push flows to the top layers. The next question i s how do we choose this l i m i t or bound on the amount of flow to send on an arc. As a l l the arcs have i n f i n i t e capacity except those flowing to the sink. The bound chosen depends on the layer number of the head of the arc. For example with a 45 degree dip on each azimuth and waste block values of -W we do not need to push more than 5W to a l a s t layer node. So a flow of max(excess t a i l , 5W) i s pushed i n increasing 11 distance on arcs with head (layer upper bound-1) and so on. The results of this heuristic were promising except i t requires knowledge of waste block values and number of arcs which we do not have u n t i l we see the problem. The results are as follows: 31 Network Nodes Heuristic Push-Rel Reduction a60 2813 .55 .63 13 % b45 4277 1.10 1.23 11 % c50 4097 1.78 1.8 1 % N O FLOWS I N K For each of the heuristic above, we prefer to send flow to the sink f i r s t and then to the arcs indicated by the h e u r i s t i c . It i s only natural to try what happens i f we avoid any arcs flowing to the sink rather than prefer them. The performance with minimum l x h e u r i s t i c was tested on blockmod and the results were as follows: Network Nodes Heuristic Push-Rel Reduction a60 2813 .65 .63 -3 % b45 4277 1.26 1.23 -2 % c50 4097 1.83 1.8 -2 % C L A S S 2 We have been considering just the characteristics of the arc without considering any information on the nodes. The next set of heuristics use information about both arcs and nodes. From now on, we push on arcs with minimum lt distance giving preference to flow to sink since this was the best heuristic but we also apply information from the nodes. 32 TOWARDS P O S I T I V E B L O C K S We prefer positive blocks over negative blocks so we push on arcs whose head i s positive valued. F i r s t preference i s given to arcs connected to the sink. Secondly we push on the arc with minimum lx distance only i f the value [head] i s po s i t i v e otherwise we push on the arc with second lowest lr distance i f i t s head i s positive and so on. Unfortunately the performance of this heuristic was not good when tested on blockmod as shown below: Network Nodes Heuristic Push-Rel Reduction a60 2813 .63 .63 0 % b45 4277 1.22 1.23 1 % c50 4097 1.80 1.8 0 % TOWARDS N E G A T I V E B L O C K S The only logic behind using this h e u r i s t i c i s that since i t s opposite had such a bad performance this should perform very well. But once again i t s performance was very bad as shown below: Network Nodes Heuristic Push-Rel Reduction a60 2813 .65 .63 -3 % b45 4277 1.25 1.23 -2 % c50 4097 1.83 1.8 -2 % 33 TOWARDS V E R Y P O S I T I V E B L O C K S The reasoning behind this was that i t i s possible that the two heuristics above (push to positive, push to negative) don't affect performance because they r e a l l y do not consider how positive or negative the blocks are i.e. a block of -1 i s very different from a block of -12500 yet are treated i n the same manner. Therefore, we push on arcs with increasing 11 distance giving preference to flow to sink but pushing only on arcs whose head have a greater value than the excess carried on the t a i l i . e . very positive blocks. The results from testing on blockmod are as follows: Network Nodes Heuristic Push-Rel Reduction a60 2813 .58 .63 8 % b45 4277 1.15 1.23 7 % c50 4097 1.78 1.8 1 % TOWARDS V E R Y N E G A T I V E B L O C K S For the purpose of completeness we push on arcs with increasing 11 distance giving preference to flow to sink but pushing on arcs whose head have a negative value greater than the excess on t a i l . The results are as follows: Network Nodes Heuristic Push-Rel Reduction a60 2813 .65 .63 -3 % b45 4277 1.26 1.23 -2 % c50 4097 1.80 1.8 -2 % 34 A N A L Y S I S & C O N C L U S I O N The average (over the sample size of three) reduction in solution time for a l l the heuristics i s as follows. Other tests similar to these were done on sab02 confirming these results but s t i l l the sample size i s small and the number of base instances (2) i s also small. (1) Bounded minimum 11 8% (2) Minimum l x 8% (3) Maximum z - minimum l a 5% (4) Very positive 5% (5) Maximum z 4% (6) Maximum lt 1% (7) Positive 0% (8) Negative -2% (9) Very negative -2% (10) No flowsink -2% The heuristics with the greatest percentage reduction in mean solution time are bounded minimum l x and minimum 1}. The extra information required to run bounded minimum lx does not lead to a significant reduction i n solution time. Also, the bounded value for flow i s problem s p e c i f i c and requires knowledge of the waste block values and .spv f i l e s beforehand. The inherent v a r i a b i l i t y i n solution times i s very 35 l i t t l e (i.e. .02 seconds at the most for a l l heuristics) . The difference in solution times of these two heuristics and push relabel are strongly s t a t i s t i c a l l y s i g n i f i c a n t . As mentioned before we do suffer from a lack of more base instances (sab02 and blockmod) but an attempt was made to generate networks that are as different as possible. This i s further discussed in Chapter 5. On the basis of these runs the minimum lx h e u r i s t i c was adopted. The output from the network flow generator (incorporated with the minimum l x heuristic) i s now read into a parser and maximal flow routines. The parser routine must be altered to read an extended format which i s required for reading i n the output of the heuristic i n order to give the maximal flow routine a warm start. The output from the parser i s read into Cherkassky and Goldberg's (1994) push relabel routines with changes made to incorporate the warm start. The networks are run on two maximum flow routines with and without the he u r i s t i c . The same networks are run on Dinic's algorithm without the warm start and our implementation of FKS (chapter 2) to provide a basis for comparison with push relabel. 36 I l l u s t r a t i v e e x a m p l e We consider a small problem to i l l u s t r a t e the steps that the minimum l x heuristic incorporated network generation routine runs through. The problem we consider has three rows, three columns, and three layers. So the o r i g i n a l ore body three dimensional block structure has twenty seven blocks. Let's consider the ore body f i r s t : Layer 1 (Bottom layer) -1 -1 -1 -1 -1 -1 10 -1 10 Layer 2 -1 -1 8 -1 -1 -1 -1 -1 8 Layer 3 -1 -1 -1 -1 -1 -1 -1 -1 -1 The 3*3*3 ore body model i s read into the routine. For example, the value of block row 1, column 3, and layer 2 i s 8. The next step i s to find the lowest positive layer i n each row-column pair and put i t i n a two dimensional array bottom[i] [j] = value. If there i s no positive layer then i t s value i s set to M. 37 bottom[i] [j] M M 2 M M M 1 M 1 The following counters for upper bounds on rows, columns, and layers are set as rub = cub =lub =3. Next we read i n the information from the .spp f i l e which looks l i k e t h i s . NX: maximum number of columns i n spp/spv f i l e NY: maximum number of rows i n spp/spv f i l e NB: number of deepest layer XC: midpoint column number i n spp/spv f i l e YC: midpoint row number i n spp/spv f i l e The azimuth i s a l i s t of angles ( a t o t a l of 360°) at which different wall slopes are permitted. In this example at each of the angles of 0, 90, 180, 270 we must have a minimum wall slope of 45 degrees. By varying these parameters we can vary the spp/spv f i l e s which varies the constraints on the wall slopes at different angles. 38 NX, NY, NB, XC, YC, NP, BX, BY, BH= 7 7 3 4 4 4 10 10 10 AZIMUTH = .0 90.0 180.0 270.0 DIP = 45.0 45.0 45.0 45.0 001:00000003000000 002:00030302030300 003:00030201020300 004:03020101010203 005:00030201020300 006:00030302030300 007:00000003000000 After reading the spp f i l e we create an expanded bottom array bottom[i] [j] to allow for mining of blocks outside the ore body i n order to remove blocks at the edges of the ore body three dimensional blocks. It i s the size of this block structure which represents the number of blocks we would have i n our model i f we did not have a bounding routine which i n this case would be 7*7*3=142. The new bottom array w i l l have dimensions equal to old dimensions + (NX,NY) which means (7,7). The information from the old bottom i s transferred to the new bottom array as (row=row + NY/2) and (column=column + NX/2). new bottom[i] [j] M M M M M M M M M M M M M M M M M M 2 M M M M M M M M M M M 1 M 1 M M M M M M M M M M M M M M M M 39 Next we construct a linked l i s t of each non-M bottom row-column pair for each l e v e l . For example our linked l i s t for layer 1 would contain two elements and their respective row-column pairs and for layer 2 we have one element and i t s row-column pair. We then run through our linked l i s t and apply the information available i n the spp f i l e by putting i t s centre (4,4) over each block i n the linked l i s t to find out what blocks need to be added to s a t i s f y the wall slope r e s t r i c t i o n s . If any blocks are needed then we have to create new bottom blocks for that row-column pair. So we get an updated new_bottom[i][j]. 40 new bottom[i][j] M M M M M M M M M M M 3 M M M M 3 3 2 3 M M 3 2 3 2 3 M 3 2 1 2 1 3 3 M 3 2 3 2 3 M M M 3 M 3 M M The next step i s to number the blocks i s some format. We adopt a simple sequential numbering system where we go along each row numbering each column from i t s bottom layer upward. The block numbers then become as follows: 41 36 31 32 34, 33 35 24 26, 25 27 29, 28 30 10 12, 15, 17, 20, 22, 23 11 14, 13 16 19, 18 21 3 5 4 6 8 7 9 1 2 So our bounding structure uses a t o t a l of 36 blocks as opposed to an o r i g i n a l number of 142 blocks. We can now pri n t out each block and i t s value to the output f i l e . A l l blocks outside the o r i g i n a l ore body can be given any value. In this example they are given a value of -1. 42 We now input information form the .spv f i l e which looks l i k e this NX, NY, NB, XC, YC, NP, BX, BY, BH= 7 7 3 4 4 4 10 10 10 AZIMUTH = .0 90.0 180.0 270.0 DIP = 45.0 45.0 45.0 45.0 001:00000000000000 002:00030000000300 003:00000001000000 004:00000101010000 005:00000001000000 006:00030000000300 007:00000000000000 We use this f i l e to set up the arcs. By placing the centre (4,4) over each block we can set up the arcs that lead out from each block. In general, the order i n which you generate these arcs doesn't matter. In our case we want to set up a heuristic which pushes flow as i t generates the arcs on arcs of increasing l x distance so that we can . Consequently, we must generate these arcs according to ascending 11 distance. We read each nonzero value from the spp f i l e and assign i t a row offset and a column offset number. Then we create an array of linked l i s t s where the array index equals the 12 distance of each nonnegative value i n the spp f i l e . Now we can generate arcs by running through the linked l i s t of each the array i n ascending order of the array index. For example, from the spp f i l e above the entry i n (4,4) has l x distance 0 and so i s the f i r s t member of the linked l i s t of the array element b[0] . Similarly we have four members i n 43 the linked l i s t of array element b[l] with lx distance 1 i . e . (5,4), (4,5), (4,3), (3,4). In order to run the heuristic we need to assign excess on blocks and flows on arcs as we generate the arcs. A l l pos i t i v e value blocks are assigned an excess equal to th e i r value and the rest of the blocks have zero excess to start with. Once the heuristic i s done we output each arc and i t s flow and each block and i t s excess and i t s flow to the sink. The heuristic has pushed excess to the top layers i n order to give the maximal flow routine a warm star t . This output i s now read into the parser and the modified maximal flow routines. We see that our bounding procedure resulted i n decreasing the number of blocks from 142 to 36 and the he u r i s t i c has pushed excess to the upper levels i n order to give the maximal flow routine a warm start. 44 C H A P T E R 5 R e s u l t s The usefulness of any technique cannot be measured u n t i l i t i s applied to an actual problem. We use the data from a Brazilian copper mine (sab02) and data constructed to resemble a real mine (blockmod) to address the following issues: 1) Whether the push relabel routine works well i n practice on this class of graphs. 2) Whether we can take advantage of the structure of an open p i t problem being solved with a generic maximum flow code. 3) To what extent does the network generation procedure reduce the number of blocks. These questions are answered using both a small example and real mine deposit data. The Brazilian copper mine(sab02) has 23 layers, 52 rows, and 80 columns for a t o t a l of 95680 blocks. The small example (blockmod) has 12 layers, 11 rows, and 23 columns for a t o t a l of 3036 blocks. We use these two data sets to generate a number of different networks to do our testing. We were limited to only these two base instances of which r e a l l y only one i s actual data, and the other i s small and a r t i f i c i a l . Ideally i t would be better to have more base instances. Consequently we generated the networks to be as different as possible from each other. The 45 networks generated through different .spp and .spv f i l e s range from 2813 to 27926 nodes. We run our routines on a HP/Apollo 9000, model 730 with 64 MB RAM with a GNU C compiler and the implementations are written i n C. We have tested on 4 different routines which are (1) The FKS closure algorithm which we implemented i n C. (2) Dinic's algorithm as implemented by Goldberg and Cherkassky (1990). (3) The queue push relabel implementation of Goldberg and Cherkassky. (4) The highest label push relabel implementation of Goldberg and Cherkassky. We implemented the minimum 11 h e u r i s t i c which we incorporated into Tom McCormick and Todd Stewart's network generator and implemented a parser routine i n order to incorporate the warm start into both push relabel routines. S o l u t i o n T i m e ( P u s h R e l a b e l ) Goldberg and Cherkassky (1994) show that the empirical performance for both push relabel routines ( highest label and FIFO ) i s better than Dinic's. Their paper also establishes that the highest label push relabel implementation performs better than the FIFO push relabel routine for a l l random problem sets. Goldberg and Cherkassky 46 perform their tests on randomly generated data from the f i r s t DIMACS challenge. The empirical performance of the push relabel on actual data has not been tested as far as we know. Therefore we run a l l five routines on real data (sab02) and generated data (blockmod) to v e r i f y the empirical properties established by Goldberg and Cherkassky. Table One gives the solution times over a l l tested networks respectively for FKS, Dinics, Push Relabel (FIFO), Push Relabel (FIFO) plus minimum l x heuristic, Push Relabel (Highest Label), and Push Relabel (Highest Label) plus minimum 12 h e u r i s t i c . Figure One plots the solution times for FKS, Dinic's, and Push Relabel (FIFO), Push Relabel (FIFO) plus heuristic for both sets of data (sab02 and blockmod) together and separately. Figure Two plots the same solution times from Figure One except for FKS i n order to better scale the graph since FKS times are much larger than the other routines. Figure Three plots the solution times for FKS, Dinic's, Push Relabel (Highest Label) and Push Relabel (Highest Label) plus heuristic for both sets of data (sab02 and blockmod) together and separately. Figure Four plots these solution times i n Figure Three except for FKS i n order to scale the graph. The results from testing on actual data confirm those of Cherkassky and Goldberg (1994) on randomly generated data. The highest label push relabel implementation i s the fastest 47 followed by FIFO push relabel and then Dinic's. Our implementation of FKS was the slowest. S o l u t i o n T i m e ( H e u r i s t i c ) The push relabel incorporated with minimum l x h e u r i s t i c performs better than just the push relabel for both routines. Each routine i s run on the same network many times to check i f the difference i n solution times i s s t a t i s t i c a l l y significant. The solution and to t a l times i n CPU seconds for three networks (A45, B45, and C60) are shown with the t- s t a t i s t i c for difference i n mean solution time. Heuristic Push Relabel A45 FIFO Total Solution Total Solution 1 62.98 19.17 63.65 20.88 2 64.62 19.82 63.48 20.90 3 63.02 19.17 63.42 20.88 4 63.00 19.07 63.18 20.80 5 63.23 19.15 63.67 20.88 6 63.62 19.17 64.03 20.95 t = -14.10 P = .0000 B45 FIFO Total Solution Total Solution 1 89.17 26.43 91.17 28.40 2 88.72 26.43 89.70 28.35 3 89.33 26.40 89.07 28.28 4 88.87 26.45 89.05 28.40 5 89.45 26.43 89.93 28.40 6 89.17 26.43 89.82 28.35 t = -94.36 P = .0000 A45 Highest Label Total Solution Total Solution 1 58.48 14.47 58.50 15.78 2 58.55 14.45 58.68 .15.77 3 58. 67 14.40 58.58 15.73 4 61.10 14.42 65.15 15.88 5 58.55 14.50 58.70 15.82 6 58. 67 14.50 58.73 15.80 t = -50.09 P - .0000 48 C60 FIFO Total Solution Total 92.42 92.95 92.22 94.42 92.75 93.68 Solution 1 88.97 2 90.57 3 92.72 4 88.80 5 91.13 6 89.17 t = -23.18 23.70 23.80 24.90 23.70 23.70 23.78 28.52 28.43 28.47 28.40 28.50 28.48 p = .0000 Tables One and Two show the solution and t o t a l times for a l l networks with and without the he u r i s t i c . These are also plotted in Figures One to Four. These results show that the minimum l x heuristic s t a t i s t i c a l l y s i g n i f i c a n t l y improves the solution time performance of both push relabel routines. E s t i m a t e o f t h e A s y m p t o t i c G r o w t h R a t e f o r S o l u t i o n T i m e We are also interested i n the asymptotic growth rate of an algorithm because in practice the problem size can be very large for certain applications. We assess whether push- re l a b e l does indeed have the best asymptotic growth i n practice. We run the regression of log(solution time) on log(nodes) on the combined sab02 and blockmod data for a l l the routines. Plots of these graphs for Dinitz, FKS, and Push-Relabel are shown on page 79. The asymptotic growth rates are as follows: 49 FKS 1.58 Dinic 1.82 PR(Q) 1.54 PR(Q)+H 1.53 PR(H) 1.51 PR(H)+H 1.50 , The push relabel algorithm has a better asymptotic growth rate for this class of real problems implying that as the problem size increases the performance of push relabel improves even further. This i l l u s t r a t e s how well the push relabel routine performs i n practice on actual data. T o t a l t i m e It i s plausible that we might be concerned about the total time taken to read the network, solve the maximum flow problem and output the maximum closure. We see the t o t a l and solution times for each of the routines in Tables 3, 4, 5, 6, 7, 8 respectively. Also Figure 5 plots the to t a l time for a l l routines. The most s t r i k i n g thing we notice i s that the t o t a l time i s much greater than the solution time for a l l the algorithms except the FKS closure algorithm. This observation raises two points. F i r s t l y , the push relabel algorithm i s so good i n practice that most of the time (see Table 9) i s spent 50 just reading the network and a very l i t t l e time i s spent solving i t . Secondly, most of the reading time of FKS disappears because the parse routine doesn't have to create the overhead of source and sink arcs. That i s the FKS closure algorithm works with the o r i g i n a l digraph and does not need extra arcs. One point to notice i s that while the parser read time i s linear, the solution time i s not so therefore the r e l a t i v e advantage that FKS enjoys due to a shorter parser time decreases as the number of nodes increase. For example with 2813 nodes the t o t a l time for FKS i s 19.86 and for push re l a b e l h e u r i s t i c i s 2.82. For 27926 nodes t o t a l time i s 441.97 for FKS and 95.96 for push relabel h e u r i s t i c . As the problem size has increased the difference i n t o t a l times has increased s i g n i f i c a n t l y from 17.04 to 345.01 i l l u s t r a t i n g that the disadvantage of a large superlinear solution time overrides the advantage of a linear read time for the FKS routine. B o u n d i n g It i s seen that the network generator created by Todd Stewart and Tom McCormick i s very effective at bounding the problem size. The o r i g i n a l number of blocks i n the copper mine are 95,680 while the reduced number of blocks range from 27,926 to 18385 blocks. Caccetta and Giannini(1991) also 51 reduce the problem size by bounding the problem using a three dimensional dynamic programming technique of Johnson and Sharpe. The network generator i s altered and the bounding routine i s disabled to check what happens i f we solve the expanded (not bounded) model. The results for blockmod are shown i n Table Nine. We notice that the expanded model i n f l a t e s the reading time more than the solution time. The solution time increases between 5 to 8 fold while the t o t a l time increases between 8 to 12 fold. This i l l u s t r a t e s how well the push relabel works i n practice on this set of graphs. We t r i e d to run the expanded model on sab02 but ran out of memory space. In fact, the bounded networks generated from sab02 are about the largest models that can run on our machine. Larger instances w i l l run out of memory space. Consequently memory space can be a bigger problem than solution time. The Lerchs-Grossman technique overcomes this problem by running without an ex p l i c i t network, i . e . arcs are generated each time that they are required, trading memory for time. It i s possible that the push-relabel routine could be implemented i n a similar manner trading solution time for memory space. C o n c l u s i o n The results show that the empirical performance of push 52 relabel i s better than Dinic's and FKS both i n terms of solution times and asymptotic solution times. Also, incorporating the minimum 11 heuristic further improves the performance of both push relabel routines. There i s the need to test against Lerchs Grossmann and Floating Cone but a lack of money and time. The results also i l l u s t r a t e that a s i g n i f i c a n t portion of the t o t a l time of the push relabel i s read time and not solution time. This implies that the solution time of push rela b e l routine i s not the constraining factor for using i t on p r a c t i c a l problems but rather the read time. Further research should aim on reducing the read time and not the solution time. A hint can be taken from the FKS routine which has such low read time since i t deals with just the internal network and does not need source and sink arcs. It i s probable that one can implement a push relabel routine which does not need to create source and sink arcs which would significantly reduce the total time of push relabel and thus make i t more suitable for p r a c t i c a l problems. This thesis did not proceed further i n this direction due to lack of time but strongly suggests this research path. In conclusion, the mining community has regarded the network flow technique as being too slow and using too much memory, but now with modern implementations such as push 53 relabel and also push relabel plus heu r i s t i c this b e l i e f should be outdated. The computational complexity of the push relabel which i s improved by the heuristic i s so good that we w i l l run into problems of reading i n the network and memory space much before the network flow solution time becomes excessive. 54 Table 1 (Solution time in sees) Networks Nodes FKS Dinics a60 2813 19.78 0.68 a55 3065 20.33 1.67 b60 3200 20.73 0.87 b55 3430 21 0.98 a50 3509 20.62 1.85 c60 3663 19.35 1.05 b50 3781 21.38 1.97 c55 3838 21.37 1.67 c50 4097 21.4 1.96 a45 4120 18.2 1.19 b45 4277 18.63 1.28 c45 4455 18.97 1.6 a60 18385 379.05 16.28 b60 20281 401.03 28.4 a55 20346 391.12 19.1 b55 21834 409.13 31.53 a50 22644 405.98 31.85 b50 23841 416.18 37.27 c60 25560 417.22 52.5 a45 25668 406.3 43.58 c55 26108 433.63 63.87 b45 26299 428.5 64.82 c50 26935 431.83 74.98 c45 27926 441.87 62.13 Push-Rel Push-Rel Push-Rel Push-Rel FIFO Heuristic H L Heuristic 0.63 0.55 0.47 0.42 1.17 1.03 0.88 0.72 0.97 0.77 0.77 0.65 1.87 1.32 0.85 0.65 1.65 1.52 1.23 1.22 1.33 1.23 1.12 0.93 1.9 1.97 1.45 1.33 1.32 1.33 1.18 1.15 1.8 1.78 1.41 1.38 1.13 0.98 1.03 0.98 1.23 1.12 1.37 1.27 1.47 1.43 1.2 1.11 10.65 9.4 8.27 8.13 16.62 15.12 13.21 12.21 20.73 14.08 13.43 10.47 25.9 22.22 16.07 15.43 20.73 19.55 15.75 13.32 27.5 25.63 19.32 17.82 28.47 23.78 19.9 18 20.88 19.27 15.95 14.15 31.18 28.55 21.93 18.13 28.32 26.38 20.97 19.17 30.43 30.08 26.89 20.98 33.06 27.18 22.95 21.18 55 Table 2 (Total time in sees) Networks Nodes a60 2813 a55 3065 b60 3200 b55 3430 a50 3509 c60 3663 b50 3781 c55 3838 c50 4097 a45 4120 b45 4277 c45 4455 a60 18385 b60 20281 a55 20346 b55 21834 a50 22644 b50 23841 c60 25560 a45 25668 c55 26108 b45 26299 c50 26935 c45 27926 FKS Dinics 19.86 2.97 20.41 5.87 20.85 3.82 21.35 6.43 20.68 9.13 19.51 4.87 21.48 8.92 21.42 6.93 21.61 8.97 18.24 3.94 18.85 4.57 19.62 5.67 379.12 42.62 401.08 73.1 391.17 57.55 409.2 91.8 406.3 87.78 416.25 109.88 417.87 116.67 406.37 86.72 433.7 140.87 428.55 127.05 431.92 153 441.97 136.8 Push-Rel Push-Rel FIFO Heuristic 2.97 2.93 6.17 5.43 3.97 3.88 6.88 6 9.77 7.45 5.55 5.12 8.55 8.62 6.3 6.35 8.6 8.5 3.97 3.85 4.63 4.43 5.6 5.55 36.73 36.68 58.98 58.57 73.72 51.1 85.85 83.17 75.43 74.63 99.43 98.37 92.6 89.08 63.67 63.32 107.72 106.37 89.38 89.2 110.77 108.73 102.95 107.35 Push-Rel Push-Rel H L Heuristic 2.85 2.82 5.53 5.32 3.77 3.67 5.97 5.67 7.27 7.2 5.05 5.05 8.15 8.07 6.32 6.22 9.02 9 3.97 3.73 4.6 4.48 5.57 5.51 33.92 33.35 54.45 54.87 66.5 46.17 75.63 75.78 66.57 65.33 91.08 89.43 82.73 84.52 58.4 57.5 98.1 97.88 81.43 80.67 105.12 100.35 96.95 95.96 56 Table 3 (FKS) Network Nodes Solution Total Difference a60 2813 19.78 19.86 0.08 a55 3065 20.33 20.41 0.08 b60 3200 20.73 20.85 0.12 b55 3430 21 21.35 0.35 a50 3509 20.62 20.68 0.06 c60 3663 19.35 19.51 0.16 b50 3781 21.38 21.48 0.1 c55 3838 21.37 21.42 0.05 c50 4097 21.4 21.61 0.21 a45 4120 18.2 18.24 0.04 b45 4277 18.63 18.85 0.22 c45 4455 18.97 19.62 0.65 a60 18385 379.05 379.12 0.07 b60 20281 401.03 401.08 0.05 a55 20346 391.12 391.17 0.05 b55 21834 409.13 409.2 0.07 a50 22644 405.98 406.3 0.32 b50 23841 416.18 416.25 0.07 c60 25560 417.22 417.87 0.65 a45 25668 406.3 406.37 0.07 c55 26108 433.63 433.7 0.07 b45 26299 428.5 428.55 0.05 c50 26935 431.83 431.92 0.09 c45 27926 441.87 441.97 0.1 Time versus Nodes (Sab02) 450 - p s r 5 = = , , , i - ^ — 370 -| , , i i , 1 , , , !| 18000 19000 20000 21000 22000 23000 24000 25000 26000 27000 28000 N o d e s 57 Table 4 (Dinic) Network Nodes Solution Total Difference a60 2813 0.68 2.97 2.29 a55 3065 1.67 5.87 4.2 b60 3200 0.87 3.82 2.95 b55 3430 0.98 6.43 5.45 a50 3509 1.85 9.13 7.28 c60 3663 1.05 4.87 3.82 b50 3781 1.97 8.92 6.95 c55 3838 1.67 6.93 5.26 c50 4097 1.96 8.97 7.01 a45 4120 1.19 3.94 2.75 b45 4277 1.28 4.57 3.29 c45 4455 1.6 5.67 4.07 a60 18385 16.28 42.62 26.34 b60 20281 28.4 73.1 44.7 a55 20346 19.1 57.55 38.45 b55 21834 31.53 91.8 60.27 a50 22644 31.85 87.78 55.93 b50 23841 37.27 109.88 72.61 c60 25560 52.5 116.67 64.17 a45 25668 43.58 86.72 43.14 c55 26108 63.87 140.87 77 b45 26299 64.82 127.05 62.23 C50 26935 74.98 153 78.02 c45 27926 62.13 136.8 74.67 Time versus Nodes (Sab02) 160 0 -| 1 1 = H 1 1 18000 20000 22000 24000 26000 28000 Nodes 58 Table 5 (Push relabel(FIFO)) Network Nodes Solution Total Difference a60 2813 0.63 2.97 2.34 a55 3065 1.17 6.17 5 b60 3200 0.97 3.97 3 b55 3430 1.87 6.88 5.01 a50 3509 1.65 9.77 8.12 c60 3663 1.33 5.55 4.22 b50 3781 1.9 8.55 6.65 c55 3838 1.32 6.3 4.98 c50 4097 1.8 8.6 6.8 a45 4120 1.13 3.97 2.84 b45 4277 1.23 4.63 3.4 c45 4455 1.47 5.6 4.13 a60 18385 10.65 36.73 26.08 b60 20281 16.62 58.98 42.36 a55 20346 20.73 73.72 52.99 b55 21834 25.9 85.85 59.95 a50 22644 20.73 75.43 54.7 b50 23841 27.5 99.43 71.93 c60 25560 28.47 92.6 64.13 a45 25668 20.88 63.67 42.79 c55 26108 31.18 107.72 76.54 b45 26299 28.32 89.38 61.06 c50 26935 30.43 110.77 80.34 c45 27926 33.06 102.95 69.89 Time versus Nodes (Sab02) 120 -i 18000 20000 22000 24000 26000 28000 Nodes 59 Table6(Heuristic(FIF0)) Network Nodes Solution Total Difference a60 2813 0.55 2.93 2.38 a55 3065 1.03 5.43 4.4 b60 3200 0.77 3.88 3.11 b55 3430 1.32 6 4.68 a50 3509 1.52 7.45 5.93 c60 3663 1.23 5.12 3.89 b50 3781 1.97 8.62 6.65 c55 3838 1.33 6.35 5.02 c50 4097 1.78 8.5 6.72 a45 4120 0.98 3.85 2.87 b45 4277 1.12 4.43 3.31 c45 4455 1.43 5.55 4.12 a60 18385 9.4 36.68 27.28 b60 20281 15.12 58.57 43.45 a55 20346 14.08 51.1 37.02 b55 21834 22.22 83.17 60.95 a50 22644 19.55 74.63 55.08 b50 23841 25.63 98.37 72.74 c60 25560 23.78 89.08 65.3 a45 25668 19.27 63.32 44.05 c55 26108 28.55 106.37 77.82 b45 26299 26.38 89.2 62.82 c50 26935 30.08 108.73 78.65 c45 27926 27.18 107.35 80.17 Time versus Nodes (Sab02) 120 -, 0 -| 1 , , i 1 18000 20000 22000 24000 26000 28000 Nodes 60 Table 7 (Push relabel(HL)) Network Nodes Solution Total Difference a60 2813 0.47 2.85 2.38 a55 3065 0.88 5.53 4.65 b60 3200 0.77 3.77 3 b55 3430 0.85 5.97 5.12 a50 3509 1.23 7.27 6.04 c60 3663 1.12 5.05 3.93 b50 3781 1.45 8.15 6.7 c55 3838 1.18 6.32 5.14 c50 4097 1.41 9.02 7.61 a45 4120 1.03 3.97 2.94 b45 4277 1.37 4.6 3.23 c45 4455 1.2 5.57 4.37 a60 18385 8.27 33.92 25.65 b60 20281 13.21 54.45 41.24 a55 20346 13.43 66.5 53.07 b55 21834 16.07 75.63 59.56 a50 22644 15.75 66.57 50.82 b50 23841 19.32 91.08 71.76 c60 25560 19.9 82.73 62.83 a45 25668 15.95 58.4 42.45 c55 26108 21.93 98.1 76.17 b45 26299 20.97 81.43 60.46 c50 26935 26.89 105.12 78.23 c45 27926 22.95 96.95 74 Time versus Nodes (Sab02) 120 0 -I 1 , 1—1 18000 20000 22000 24000 • 26000 28000 Nodes 61 Table 8 (Heuristic(HL)) Network Nodes Solution Total Difference a60 2813 0.42 2.82 2.4 a55 3065 0.72 5.32 4.6 b60 3200 0.65 3.67 3.02 b55 3430 0.65 5.67 5.02 a50 3509 1.22 7.2 5.98 c60 3663 0.93 5.05 4.12 b50 3781 1.33 8.07 6.74 c55 3838 1.15 6.22 5.07 c50 4097 1.38 9 7.62 a45 4120 0.98 3.73 2.75 b45 4277 1.27 4.48 3.21 c45 4455 1.11 5.51 4.4 a60 18385 8.13 33.35 25.22 b60 20281 12.21 54.87 42.66 a55 20346 10.47 46.17 35.7 b55 21834 15.43 75.78 60.35 a50 22644 13.32 65.33 52.01 b50 23841 17.82 89.43 71.61 c60 25560 18 84.52 66.52 a45 25668 14.15 57.5 43.35 c55 26108 18.13 97.88 79.75 b45 26299 19.17 80.67 61.5 c50 26935 20.98 100.35 79.37 c45 27926 21.18 95.96 74.78 Time versus Nodes (Sab02) 120 - 18000 20000 22000 24000 26000 28000 Nodes 62 Table 9 (Expanded) FIFO Network Nodes Push-Rel Push-Rel Expanded Expanded Solution Total Solution Total a60 2813 0.55 2.93 3.98 22.13 a55 3065 1.03 5.43 11.93 73.87 b60 3200 0.77 3.88 8.03 44.68 b55 3430 1.32 6 14.65 84.2 a50 3509 1.52 7.45 7.77 79.43 c60 3663 1.23 5.12 11.92 66.38 b50 3781 1.97 8.62 19.62 109.65 c55 3838 1.33 6.35 17.08 115.17 c50 4097 1.78 8.5 memory memory a45 4120 0.98 3.85 7.28 35.47 b45 4277 1.12 4.43 9.47 48.02 c45 4455 1.43 5.55 13.62 68 Highest Label Network Nodes Push-Rel Push-Rel Expanded Expanded Solution Total Solution Total a60 2813 0.42 2.82 2.98 26.45 a55 3065 0.72 5.32 8.27 70.02 b60 3200 0.65 3.67 5.45 42.53 b55 3430 0.65 5.67 9.53 78 a50 3509 1.22 7.2 9.58 80.77 c60 3663 0.93 5.05 7.68 61.8 b50 3781 1.33 8.07 12.02 102.45 c55 3838 1.15 6.22 11.07 88.13 c50 4097 1.38 9 memory memory a45 4120 0.98 3.73 4.75 33.55 b45 4277 1.27 4.48 6.25 44.35 c45 4455 1.11 5.51 8.45 62.43 63 Figure 1 (Solution time) Solution time versus Nodes — • — F K S —{§— Dinic — * — Push-Relabel(FIFO) —X—Heuristic 12000 17000 Nodes 22000 27000 25 20 15 I 10 0 4- 2700 Solution time versus Nodes (Blockmod) •FKS - Dinic Push-Relabe(FIFO) -X— Heuristic 3200 3700 Nodes 4200 4700 Solution time versus Nodes (Sab02) «• 300 | 250 "| 200 150 100 50 0 18000 • • — F K S •—Din ic -is— Push-Relabel(FIFO) Heuristic 20000 22000 24000 26000 28000 Nodes 64 Figure 2 (Solution time no FKS) Solution time versus Nodes (Blockmod) •Dinic -Push-Relabel(FIFO) -Heuristic Solution time versus Nodes (Sab02) •Dinic -Push-Relabel(FIFO) Heuristic 1 8 0 0 0 2 0 0 0 0 2 2 0 0 0 2 4 0 0 0 N o d e s 2 6 0 0 0 2 8 0 0 0 65 Figure 3 (Solution time) 25 20 -k 15 10 Solution time versus Nodes (Blockmod) - •—Dinic - » - F K S ~ * Series3 - H — Heuristic I ^ ^ S B 2800 3000 3200 3400 3600 3800 4000 4200 4400 Nodes Solution time versus Nodes (Sab02) 450 400 350 250 17 300 o ~V 200 H 150 18000 20000 22000 24000 Nodes 26000 28000 - • — F K S - • — Dinic ~ i t — Push-Relabel(HL) | -X—Heuristic 66 Figure 4 (Solution time no FKS) Solution time versus nodes (Blockmod) ( A O a E 2800 • Dinic -Push-Relabel(HL) •Heuristic 3300 3800 N o d e s 4300 Solution time versus Nodes (Sab02) -Dinic -Push-Relabel(HL) -Heuristic 18000 20000 22000 24000 26000 28000 N o d e s 67 Figure 5 (Total time) Total time versus Nodes 2800 7800 12800 17800 22800 27800 Nodes Total time versus Nodes (Blockmod) 25 20 -b- 15 1 10 2800 3300 3800 4300 -•—FKS -f§— Dinic ™ir- - Push-Relabel(HL) - X — Heuristic Nodes Total time versus Nodes (Sab02) -•—FKS - *—Dinic - A — Push-Relabel(HL) | -H—Heuristic 18000 20000 22000 24000 Nodes 26000 28000 68 Log(Sol. time) vs Log(No.Nodes) Dinitz o (/} at o •Seriesl 3.4 3.6 3.8 4 4.2 Log Nodes 4.4 4.6 FKS -Seriesl 3.8 4 4.2 Log Nodes 4.6 Push-Relabel o in ui o 2.8 2.3 1.8 1.3 0.8 0.3 -0.2 -Seriesl 3.4 3.6 3.8 4 4.2 Log Nodes 4.4 4.6 69 R E F E R E N C E S A N D B I B L I O G R A P H Y 1. Caccetta, L. And L. M . Giannini (1985), On bounding techniques for the open pit limit problem, Proceedings of the Australasian Institute of Mining and Metallurgy 290, 87-89. 2. Caccetta, L. and L. M . Giannini (1986), Optimisation techniques for the open pit limit problem, Proceedings of the Australasian Institute of Mining and Metallurgy 291, 57-63. 3. Caccetta, L. andL. M . Giannini (1988a), An application of discrete mathematics in the design of an open pit limit, Discrete Applied Mathematics 21, 1-19. 4. Caccetta, L. andL. M . Giannini (1988b), The generation of minimum search patterns in the optimum design of open pit mines, Proceedings of the Australasian Institute of Mining and Metallurgy 293, 57-61. 5. Caccetta, L. andL. M . Giannini (1990), Application of Operations Research techniques in open pit mining, Asian-Pacific Operations Research: APORS'88, 707- 724 6. Chen, T., 1976," 3D Pit Design with variable wall slope capabilities, "14th Applications of Computers in the mineral Industries Symposium, R. V. Ramani, ed., AIME, New York. 7. Cherkassky, B.V. and Goldberg, A.V. (1994), On implementing Push-Relabel Method for the Maximum Flow problem, Technical Report Stanford University. 8. Faaland, B., Kim. K. and Schmitt. T., 1990, A new algorithm for computing the maximal closure of a graph, Management Science, Vol. 36, No.3. 9. Johnson, T.B., 1973, " A Comparative Study of Methods for determining Ultimate Open-Pit Mining Limits," 11th Applications of Computers in the Mineral Industries Symposium, University of Arizona, Tucson, AZ. 10. Johnson, T.B., 1968, " Optimum Open-Pit Mine Production Scheduling, "Operations Research Centre, University of California, Berkeley, 120 pp. 11. Johnson, T.B., and Sharp, W. R., 1971, A Three-Dimensional Dynamic Programming Method for Optimal Open Pit Design, U S B M Report of Investigations 7553. 12. Kim, Y.C. , 1978. "Ultimate Pit Limit Design Methodologies using Computer Models- The State of the Art," Mining Engineering, October 1978, pp. 1454-1459. 70 13. Kim, Y.C. , Cai, W., and Meyer, W.L., Comparison of microcomputer-based optimum pit limit design algorithms, "Society of Mining Engineers. 15. Lemieux, Marc, 1979, "Moving Cone Optimising Algorithm", Computer Methods for the 80's in the Mineral Industry, ed. A. Weiss, Port City Press, Baltimore, M D , pp. 329-345. 16. Lerchs, H. , Grossmann, IF., 1965, Optimum Design of Open-Pit Mines" Transactions, C.I.M., Vol. LXV111, pp. 17-24. 17. Lipkewich, M.P., Borgman, L., 1969, "Two-and-Three Dimensional Pit Design Optimisation Techniques", A Decade of Digital Computing in the Minerals Industry, ed. A. Weiss, SME of ATME, New York, pp. 505-523. 18. Picard, J., 1976, Maximal Closure of a Graph and applications to combinatorial problems. Management Science, Vol. 22, No. 11. 71
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