Solving ultimate pit limit problem through graph closure (L-G algorithm) and the fundamental tree algorithm презентация
Содержание
- 2. Contents 1. Introduction 2. Lerchs – Grossmann (Graph closure) 3. Fundamental Tree Algorithm 4. Assignment
- 3. Introduction Two ojectives for open pit optimization Design ultimate pit limits (Moving cone, LG, Net work
- 4. Block model gdg Source: William Hustrulid: Open Pit Mine Planning & Design
- 5. DEPOSIT REPRESENTATION OREBODY MODEL A set of mining blocks of a given size with block value
- 6. DEPOSIT REPRESENTATION OREBODY MODEL Geologic Model, Copper Grades (lb/ton) Economic Model, Value per block ($/ton)
- 7. Ultimate Pit Limit Problems Moving (Floating) Cone Algorithm Dynamic Programming (LG) Graph closure (Lerchs-Grossmann, Minimum Cut
- 8. 2. Graph closure (LG) The optimal pit outline is the 3D pit outline which, if mined
- 9. Lerchs-Grossman Algorithm Mathematical search technique which works from just two sources of information: 1. Economic or
- 10. Arc Relationships Each arc goes from one block (A) to a second block (B) and indicates
- 11. Lerchs-Grossman Algorithm In 3-D each of the blocks is connected to 4, 5 or 9 blocks
- 12. DEFINITIONS Directed graph G(X, A) is defined by a set of nodes X and a set
- 13. BASIC TERMINOLOGY Vertex, xi is the value of block i. Arc, aij=(xi,xj), an arrow drawn from
- 14. Root is the vertex without predecessor A closure is any surface satisfying the slope constraints An
- 15. An m-arc is weak if it is supported by positive mass (mw), strong otherwise (ms). A
- 16. Example of LG -3 -4 +6 +9 X0 p-s m-w p-w p-s Root node
- 17. Steps of the algorithm (1) -2 -2 -3 -4 +6 +9 X0 Step 0: Connect all
- 18. Steps of the algorithm (2) -2 -3 -3 -4 +6 +9 X0 Step 1: Connect p-strong
- 19. Step 3: Normalize the tree. Root (x0) must be common to all strong edges, if not,
- 20. Application of the steps for small orebody model Step 0:
- 21. Application of the steps … Step 1 and 2:
- 22. Application of the steps … Step 1-2 -1 -2 -1 -2 +4 +5 X0 p-w p-w
- 23. Application of the steps … Step 1-2 -1 -2 -1 -2 +4 +5 X0 mw p-w
- 24. Application of the steps … Step 1-2 arc (x0-x2)= 4 -1-2 + 5 -1-2= +3 ?
- 25. Application of the steps … Step 1-2 -1 -2 -1 -2 +4 +5 X0 mw p-w
- 26. Final Pit Limit Mine all the nodes connected to a strong arc. G(X, A) = 3
- 27. Optimal Pit The L-G algorithm will flag each block as being inside or outside the optimal
- 28. Varying Slope Angles It is possible to have different slope angles in different areas of the
- 29. 3. Fundamental Tree Algorithm Long-term production scheduling consists of determining an optimal sequence of extracting the
- 30. The Approach To simplify the MIP we would like to decrease the number of blocks in
- 31. The Approach A Fundamental Tree is defined as a combination of blocks such that: The blocks
- 32. 3.1. Fundamental Tree Algorithm
- 33. Step 1 - Generate a Starting Network Given a our pushback block model represent blocks by
- 34. Example of Generating a Starting Network Example 2-D block model, shows the node numbers and the
- 35. Step 2 - Find the Cone Value The "cone" value of a block (CVi) is the
- 36. Step 3 – Assign cofficients to the Positive Nodes Each positive node will be assigned a
- 37. Example CV6 = +1; CV7 = —3; CV8 = —2 Coefficients Ci are assigned to positive
- 38. Step 4 - Set up the LP Set up a Linear Program to find the first
- 39. Step 5 - Check if we are done If the number of Fundamental Trees found is
- 40. 3.2. Formulating the Linear Program * The objective function: Ci is the coefficient for node (block)
- 41. Example of Step 4
- 42. Example of Step 4 The network representation of the initial solution containing two trees surrounded by
- 43. Example of Step 4 The network representation the solution containing three trees surrounded by dashed lines.
- 44. 3.3. Properties of the fundamental trees Property 1 - All fundamental trees have positive value is
- 45. Property 2 of the fundamental trees All fundamental trees obey the slope constraints if they are
- 46. Property 3 of the fundamental trees A tree found by FTcannot contain a subset of a
- 47. 3.4. Optimization of annual production scheduling using fundamental trees The optimization model is maximization of the
- 48. using fundamental trees Where: DEVpi discounted economic value to be feberated from mining and processing all
- 49. Subject to: ❶. Slope constraints: If j is the index for the fundamental trees that have
- 50. ❷. Sequencing of mining ore and waste If mining the ore tonnages from a given FT,
- 51. ❸. Grade blending constraints Upper bound constraints: The average grade of the material sent to the
- 52. ❹. Processing capacity constraints Upper bound: The total tonnage of ore processed cannot be more than
- 53. ❻. The limit on the stripping ratio in each period Where SRmax is the maximum stripping
- 54. ❼. Mining capacity The total amount of material mined during a year cannot be more than
- 55. 3.5. Case Study: MIP Scheduling Formulation One of the case studies is performed on a large
- 56. ASSIGNMENT 8 Consider the above 2D orebody model: 1. Show steps to determine the ultimate pit
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