# OPEN PIT MINE PLANNING AND DESIGN PDF

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Get this from a library! Open pit mine planning & design. [W A Hustrulid; Mark Kuchta; Randall K Martin]. PDF | Definition of Open pit Mining Parameters:Open pit Mining method; Bench; Open Pit Bench Terminology; Bench height; Cutoff grade;. PDF | 95+ minutes read | The objective of this study is to integrate the Open pit mine planning and design is a decision-making process that leads to a realistic.

The step-by-step approach to mine design and planning offers a fast-path approach to the material for both undergraduate and graduate students. Hustrulid Published by A. Balkema in Rotterdam,Brookfield, VT. He recognised that the optimal pit design changes if. MacNeil and Dimitrakopoulos formulated the open pit to underground transition as a stochastic optimisation problem.

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Download site: Customer reviews: Open Pit Mine This review of the first edition is a must have for anyone who is related with open pit mine planning. It is also a great book for mining engineering students. The concepts are clearly presented, and very well organized.

The complete process of mine planning from the early stages to Surface mining planning and design of open pit mining Surface mining planning and design of open pit mining 1. This material is intended for use in lectures, presentations and as handouts to students, and is provided in Power point format so as to allow customization for the individual needs of course instructors.

Permission of the author and publisher is required for any other usage. Surpac mining software free download Windows Surpac mining software free download.

The type and number of shovels are selected during the planning process and their productivity determines the mining rate of the benches, pushbacks and the mine. The space available for loading is part of the pushback design. Condition: New. This is the approach used by Lerchs-Grossman implementation of Whittle [8]. A series of heuristically discounted pits is produced in a greedy fashion until it is no longer profitable to consider any further pits.

The final pit is used as the ultimate pit limits. A simple example of when this would happen is if there was a large section of ore beneath a large amount of waste.

It would not be feasible to mine anything until the scaling factor reaches some threshold value Fig. Producing a series of pits in the fashion described above also suffers from the problem that the pit produced for a given factor may be disconnected. Single pushbacks may have multiple sections that are physically far from each other, making them impractical. Ideally, a pushback should be one connected piece and not fragmented.

A further problem with the technique described is that other geometric limitations open pit pushbacks must adhere too are not considered. This can include requiring the pit base be a convex shape, possibly of a minimum width.

The mine designer typically has to manage these issues by hand. Existing algorithms for pushback and open pit optimization are typically designed to only consider one fixed orebody model. As a rule, optimization in mine design and planning has two major flaws: —inputs are assumed certain while they are not, thus uncertainty from geological, mining and market sources is not accounted; —conventional mathematical models cannot handle input uncertainty models, in distinction from stochastically described inputs.

The difference arises from significant departures in expected cash flows between the traditional single- point ore body estimates and stochastic models, and demonstrates potentially misleading results from combining traditional ore body models with complex optimization algorithms. Furthermore, this and other examples [4, 9] highlight the conceptual and computational inadequacy, and technological limits of mine design and production scheduling technologies currently used, when optimizing under uncertainty.

With advances in stochastic simulation techniques, new algorithms are needed to handle multiple equally likely ore body model realizations. The techniques should provide a robust optimization over all ore body models and not just perform well in expectation.

Lerchs—Grossman Algorithm The most well established procedure for producing ultimate pit limits is the Lerchs—Grossman L—G algorithm [10] and the nested pit L—G implementation for pushback design by Whittle [8] where heuristic techniques are used to incorporate discounting and pushback design. Zhao and Kim [11] developed a 3D algorithm based on the L—G approach. It was the first algorithm that produced the ultimate pit limits for a large sized mine in a reasonable amount of time.

For an introduction to the terminology and notation of graph theory see Bondy and Murty [12]. One can assign the weight ci to node xi where ci is the economic value of the block that xi represents.

Now the problem of finding an ultimate pit is equivalent to finding what is known as a maximum graph closure in G. A depiction of a graph closure, the xi, labeled notes of the graph represent blocks in an ore body model. Graph G with dummy node x0 and arcs added from the dummy node x 0 to all other nodes.

It is clear from the definition of our graph G that a graph closure in G represents a physically feasible pit, if not, then a block not in our closure violating the slope requirements would have an edge from a node in the closure directed towards it, a contradiction to the definition of a closure.

For any feasible pit, the set of nodes within the pit limits clearly defines a graph closure. It follows that there is a one-to-one mapping between feasible pit limit designs and graph closures.

## 13 editions of this work

The maximum weight graph closure corresponds to the ultimate pit limits. The L—G algorithm begins by adding a dummy root node x0 to the graph G with arcs directed from x0 to every node in G Fig.

Some definitions of basic terms will be necessary to proceed. A connected forest is referred to as a tree. A spanning tree T of G is a subgraph of G on the node set V G such that the edges of the spanning tree are a subset of the edges in G and T is a tree. When referring to a tree or spanning tree of our graph, edges are considered as undirected.

A branch bv is defined to be the subtree of the spanning tree rooted at the child v of x0. The mass of a branch is the sum of the weights of the nodes in the branch. The branch is referred to as strong if it's mass is positive and weak otherwise. A node is referred to as weak if it is a member of a weak branch and a node as strong if it is a member of a strong branch.

Denote plus arcs that supports a branch with positive total mass as ps, otherwise use pw. Similarly, use mw to denote a minus-arc that supports a strong branch and ms for a minus-arc that supports a branch having negative total mass.

## Open-Pit-Mine-Planning-and-Design-VolHustrulid.pdf

To find the maximum closure, the L-G algorithm produces a series of normalized trees until the set of strong branches for a normalized tree corresponds to a graph closure.

Beginning with the normalized tree T , taking all positive branches will clearly have weight greater then the weight of the ultimate pit, since initially this is equivalent to choosing all positive weight blocks.

Clearly, such a pit would violate slope constraints. The approach of the L-G algorithm is to produce normalized trees of lesser and lesser value until the slope constraints are satisfied, and the positive mass branches correspond to ultimate pit of our ore body model. Once such a situation is identified, the two branches are merged and the algorithm produces a new normalized tree. If rs is the root of the strong branch, merging the two branches consists of removing the arc x0 , rs from T and adding the arc xs , xw.

After merging the branches, the new branch is traversed to update the mass of the nodes in the combined branch. If a strong arc a, b is created that isn't adjacent to the root x0 , remove arc a, b and add an arc x0 , a from the root to node a if a is diconnected from the root when arc a, b is removed or add the arc x0 , b if b is disconnected from the root when a, b is removed. This process is called renormalizing. When no arc xs , xw exists in G such that xs is a member of a strong branch and xw overlies xs and is a member of a weak branch, the algorithm terminates.

The set of strong branches form a graph closure, and it can also be shown that this graph closure is in fact of maximum value. A small example of the algorithm is presented. After adding all the nodes from the the dummy node x0 , begin with the normalized tree in Fig. An arc is then identified where the arc's tail node is a member of a strong branch and it's head is a member of a weak branch.

The two branches are merged and the normalized tree is updated. Figure 5 shows merging of x1 and x6 and the tree normalizing.

## Open Pit Mine - Planning and Design-3rd Edition

After recognizing the overlying weak branches above node x6 where x6 has value 4 and renormalizing the tree in Fig. When the algorithm terminates, the strong branches connected to the dummy node form the maximum closure. Hence, there is no clear way of alter the L—G algorithm to address the gap problems directly by producing graph closures of a given size. The initial normalized tree; ps arcs are plus-arcs supporting a strong branch and pw arcs are plus- arcs supporting a weak branch.

This image depicts merging branches x 4 and x 6 ; the dashes arc is removed; mw denotes a minus-arc, supporting a strong arc. The image of the tree after all weak branches above x 6 have been merged. The strong branches connected to the dummy node represent the final graph closure. Open pit parametrization produces maximum valued pits as a function of another parameter where this parameter is defined for each block in our orebody model.

Seymour chooses pit volume as the parameter. If one was to plot the economic pit value vs. The algorithm follows the approach of the L—G method with the addition of the parametrized variables and the added ability to notice when a subtree can be regarded as a small pit. A threshold value is used to determine if a branch is strong or weak. By altering the threshold value, a series of nested pits can be produced. The upper convex hull of pit value versus pit size.

When comparing two branches, the branch with greater strength is called stronger, and the branch with less strength is called weaker. Begin by initializing every node as an independent branch.

Then, for every node xs in every strong branch test to see if there is an overlying node that is a member of a weaker or equal strength branch. When a node xw of a weaker or equal strength branch is found overlying a member of a stronger branch an arc is allocated from the stronger branch node to the weaker branch node following the rules for arc allocation.

Convert the strong branch node into the root of its branch. This is done by reversing the arcs on the path between the strong node xs and the root of the strong nodes branch. Adjust the cumulative masses and cumulative value accordingly. During arc reversal switching branch roots , arcs are pruned or deleted if they are down arcs from a node whose cumulative strength is greater than the branch strength.

This splits the branch into two branches and prevents a stronger node from supporting non-overlying weaker nodes. When this occurs the stronger branch has been weekend and must be tested to see if it still qualifies for the arc allocation.

The stronger branch root arc is directed to the overlying weaker branch member, creating one branch out of the two.

## 3rd Edition

The combined branch strength is updated. This is done by traversing the weaker branch from the node at the join to the weaker branches root and reversing arcs and updating cumulative mass and strength along the way. During this weak branch update, arcs are deleted if they point down from a node whose cumulative strength is greater than the branches strength. After each arc allocation, a sweep is made where each node is converted to the branch root to trigger any applicable arc deletions.

If no arc was allocated no node of a weaker branch was found overlying a node of a stronger branch the algorithm terminates. No node has a weaker overlying branch. Otherwise, the procedure is repeated for allocating an arc between the pair of nodes found, until no such pair exists. The branches are sorted in order of decreasing strength b1, This family of pits represents the maximum valued convex hull for a single variable pit limit parametrization.

The plot of the upper convex hull of the pit value vs. Heuristic techniques of combining adjoining blocks and averaging their values to decrease the size of the block model are typically used to produce acceptable run times, at the expense of producing a truly optimal pushback design.

Furthermore, if the pits that lie on upper convex hull are far apart in terms of size then gap problems will continue to persist, since the algorithm will not return pits of the desired size Fig.

Network Flow Approaches Following the success of modeling the ultimate pit problem under the context of a graph closure, Picard [14] showed how to find the maximum closure of a graph by using maximum network flow algorithms.

## Determining the Most Effective Factors on Open Pit Mine Plans and Their Interactions

This allows one to use known efficient algorithms for maximum flow to find the ultimate pit. The maximum flow problem is one where: a directed graph G is given, with capacities on the edges, a source node s , and a sink node t ; one wants to know the maximum amount of flow that can travel from the source node s to the sink t without violating the capacity constraints on the edges.

An arc i, j with a capacity of ci , j can send at most ci , j units of flow from node i to node j. Since any flow going from s to t is constrained to be at most the capacity of a minimum cut, it follows that the maximum s — t flow is at most the size of a minimum cut.

Add arcs from s to every node that has positive weight in G and add arcs from every negative weight node to t.

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Give the edges of the form s, v a capacity c s ,v equal to the weight of v in G and give arcs of the form v, t a capacity c v ,t equal to the absolute value of the weight of v in G.

Consider the small example of a vertical cross-section of an orebody model in Fig. Figure 11 depicts the construction of the network from the orebody model in Fig. The unlabeled arcs have infinite capacity. Vertical cross-section of an orebody model with economical value of blocks. Network constructed from the orebody model in Fig. The minimum cut of the network shown in Fig. In the context of an orebody model, one can think of a minimum cut consisting of arcs directed to the ore that is left in the ground and arcs from the waste that is left inside the pit limits.

The infinite capacity arcs ensure that slope constraints are maintained. Since the block model is finite, minimizing the value of ore left outside the pit plus the cost of the waste left inside the pit is equivalent to maximizing the ore inside the pit minus the waste inside the pit.

Figure 12 shows the minimum cut in our example, the dashed arcs correspond to the arcs in the minimum cut. One can formulate the minimum cut problem as a linear program LP in such a way that the constraint matrix is totally unimodular. This implies that one can get an integral solution by solving the LP, which can be done efficiently in practice for large sized networks. Hochbaum and Chen [15, 16] showed that the L—G algorithm can be used as a network flow algorithm. From the series of normalized trees they showed how one could obtain an optimal network flow.

They also analyzed the runtime of the L—G algorithm and improved it by scaling techniques different from those used to generate pushback designs to show that L—G can be implemented to run in O mnlog n time, where m and n are the number of arcs and nodes in the constructed graph respectively.

The network flow algorithm they developed is known as the pseudoflow algorithm. Muir [17] implemented the pseudoflow algorithm and found it more efficient than the L—G algorithm in practice. Gallo et al [18] developed the way to use a network flow algorithm to produce a series of parameterized minimum cuts. Dagdelen—Johnson Lagrangian Parametrization In [19] Dagdelen and Johnson formalized the process of parametrization under the context of Lagrangian relaxation. The process of Lagrangian relaxation is one where a troublesome constraint is removed from the LP and placed in the objective.

In the context of an open pit optimization problem, the technique applied to the problem of finding a pit of a fixed tonnage is shown. This IP is totally unimodular once again so by relaxing the integrality on the xi 's one can solve it efficiently.

One can therefore view the procedure of finding nested pits by Dagdelen and Johnson's Lagrangian Parametrization as an equivalent procedure to that of scaling the orebody model value and running the L-G algorithm to get a series of nested pits.

It therefore suffers from the same gap problems as those discussed in the review of existibg methods in an earlier section. IP Formulations Due to technical and engineering limits there are many constraints that should be considered that intrinsically limit the size of a pushback based on its period of extraction [20].

Two such constraints are milling constraints and extraction capacity constraints.

The mill should typically be fed a certain minimum and maximum quantity of ore. Since efficient algorithms exist to find optimal pits without such a cardinality constraint on the size, connectivity, and geometric constraints, one would like to know if an efficient algorithm exists with these restrictions. If one considers the problem of finding an optimal pit with only the restriction that the pit must be connected one single entity it can be shown that this problem becomes NP-hard, meaning it is unlikely that an efficient polynomial time algorithm exists.

The complexity class of the problem could however change if one considers the convexity and certain milling constraint as well. Related is the work by Tachefine and Soumis [21] who use Lagrangian relaxation in the presence of any number of cardinality constraints. One approach to solve these large IP systems is to aggregate blocks together [22, 23] to decrease the number of variables in the IP.

Doing this in a naive fashion can alter the shape of the ultimate pit that is produced.

Taking the average of a set of blocks tends to increase the small values and decrease the large values of the blocks in the orebody model which leads to dilution, missclassification and erroneous assessments. This can have a dramatic effect on the feasibility study of a mine and pit optimization, and has the same effect as what is known in mining literature as the effect of selectivity [24, 1]. Related to the above work is that of Akaike [25] who looks at open pit design optimization with multiple destinations such as mill, waste, or stockpile and the destination of a specific block is determined during the optimization process defined in his IP formulation.

Fundamental Tree Algorithm An approach for combining blocks together known as the fundamental tree algorithm was introduced by Ramazan [26]. The fundamental tree method combines blocks in such a way that the ultimate pit produced on the combined blocks is the same as that produced if the blocks were not combined together.

The approach decreases the number of blocks, which in some cases makes solving integer programs feasible for a class of mines of larger volume. Since the number of variables has decreased in the IP formulation, one can put more constraints into the IP and still have efficient run times. The fundamental tree algorithm combines together blocks using the LP model such that the combined blocks would have certain characteristics in order to generate feasible production schedules through applications of mixed integer programming MIP formulations for a given orebody model.

A fundamental tree is defined as any combination of blocks such that: 1 blocks can be profitably mined, 2 blocks obey the slope constraints, 3 there is no proper subset of the chosen blocks that meets 1 and 2. The following definitions and illustrations are adopted from Ramazan [26] in order to follow the description of the algorithm.

If there is no flow going through the arc, the arc is not activated setting xi , j parameter to 0.

The pushbacks that are designed during mine planning must obey the maximum allowable pit slope constraints. In copying the network flow approaches, edges are added between nodes to overlying nodes that must be removed prior to it's removal. The cone value of node i — CVi , is defined as the total value of all the blocks inside the slope constraint cone whose apex is set on positive node i.

The coefficient, or rank Ci is obtained using the cone value of ore block i as discussed below. This coefficient assignment is to force the LP model to start arc and flow settings from the highest cone value block. This procedure cooperates with other constraints discussed below to result in trees that have all the defined properties of fundamental trees.

The coefficient M is used as a large number big M , which is larger than the flow in the network. Mutual support refers to the support of a waste block by more than one ore block. Steps of the algorithm: 1. The first step of the algorithm is to find the ultimate pit limits of the block model.The pushbacks that are designed during mine planning must obey the maximum allowable pit slope constraints.

Limited Google Scholar 5. Dimitrakopoulos, D. A branch bv is defined to be the subtree of the spanning tree rooted at the child v of x0. Farrelly, C.

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