A New Numerical Method and AHP for Mining Method Selection

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Shahriar / Bakhtavar / Akbarpour Shirazi: A New Numerical Method and AHP for Mining Method Selection 289

A New Numerical Method and AHP for Mining Method

Selection

K.Shahriar, E.Bakhtavar & Gh.Saeedi,

Amirkabir University of Technology, IR

M. Akbarpour Shirazi, Khajehnasir University of Technology, IR

ABSTRACT

Mining method selection decision is the most important phase which affects costs of project. In the

study for selecting suitable exploitation method, a new numerical Shahriar and Bakhtavar (Sh&B)

approach and the Analytic Hierarchy Process (AHP) were used. The method is a combined and

modified system of Nicholas, Modified Nicholas and UBC. Expert Choice software based on AHP

was utilized that it also allows users to specify the mining method considering specific relative

importance of each of governing factors. In the decision making process, the input data for the

software and the new system (Sh&B) were caught from information of Third Anomaly Gol-E-Gohar

Iron Ore Deposit located in Kerman province in south eastern of Iran. Finally according to the

achieved results of the software out put data and the new system, as well as comparison with results

of other numerical systems, open pit mining and sublevel caving were selected as the most suitable

mining methods.

ZUSAMMENFASSUNG

Die Auswahl des Abbauverfahrens hat den größten Einfluss auf die Kosten eines Projekts. In dieser

Studie wurden eine neuartiger numerischer Shahriar und Bakhtavar (Sh&B) Ansatz und das

Analytic Hierarchy Process (AHP) Verfahren benutzt. Die Methode ist eine Kombination und

Modifikation der Systeme von Nicholas, Modified Nicholas and UBC. Über auf AHP-gestützte

Expertensysteme können vom Benutzer bevorzugte Abbaumethoden gewichtet betrachtet werden.

Der Vergleich von AHP und des neuen (SH&B)-Systems wurde am Beispiel der Third Anomaly

Gol-E-Gohar Iron Ore Deposit in der Provinz Kerman im Süd-Osten des Irans getestet. Der

Vergleich der von AHP und (SH&B) erzeugten Ergebnisse mit den Ergebnissen anderer

numerischer Verfahren, führte zur Auswahl von Tagebau und Teilsohlen-Pfeilerbruchbau als das zweckmäßigste Verfahren.



Shahriar / Bakhtavar / Akbarpour Shirazi: A New Numerical Method and AHP for Mining Method Selection290

INTRODUCTION

In the past, selection of mining method for a new property was based primarily on operating

experience at similar type deposits and on methods already in use in the districts of the deposit.

Then, the chosen method was modified during the early years of mining as ground conditions and

ore character were better understood. Now a day, however, the large capital investment required to

open a new mine or change an existing mining system make it imperative that the mining methods

examined during the feasibility studies and the method actually selected have a high probability of

attaining the projected production rates [1,2]. Several methods have been developed to evaluate

suitable mining methods for an ore deposit based on physical characteristics of the deposit such as

shape, thickness, plunge, depth, grade distribution, and geo-mechanical properties of the rock. The

Nicholas method (1981) is one such procedure, which applies a numerical approach to rate different

mining methods based on the rankings of specific input parameters. The UBC Mining Method

Selection Algorithm is a modification to the Nicholas approach, which places more emphasis on

stoping methods, thus better representing typical Canadian mining design practices [3]. In 2002 the

MMS system was proposed based on The UBC Algorithm but provides the opportunity to describe

the parameters using fuzzy logic [4].

In the study for selecting suitable exploitation method, a new numerical named Shahriar and

Bakhtavar (Sh&B) approach and the Analytic Hierarchy Process (AHP) were used. The new

quantitative approach is a combined and modified system of Nicolas, Modified Nicolas and UBC

methods. Expert Choice software based on AHP was used that it also allows users to specify the

mining method considering specific relative importance of each of governing factors. In this paper

the input data for the software and the new system (Sh&B) were caught from information of third

anomaly Gol-E-Gohar iron ore deposit. Then according to the achieved results of the new system

and the software out put data, as well as comparison with results of some numerical approaches,

open pit mining and sub level caving were selected.




THE SHAHRIAR&BAKHTAVAR METHOD

Introduction

In order to combine and modify rankings of the previous quantitative approach named Nicholas,Modified Nicholas, and UBC and suggested weighting rates in Modified Nicholas, a new numerical

approach was improved. The new method was named Sh&B (Shahriar and Bakhtavar are names of

authors). The authors believe the new method in mining method selection process for all deposits

can be most effect than other privious methods. Such as the previous numerical methods, a finite

number of methods are considered in the Sh&B approach. It is recognized that certain mining

methods, for example Square Set Stoping, are no longer in common use. No attempt has been made

in this paper to update these mining methods to be considered.

Mining Method Selection in Sh&B Approach

Except the “Grade Quantity” which added, all input parameters of the new approach and UBC are

the same. This parameter was added to the effective input parameters collection because of its

significant in deposit evaluation. The selection process proceeds in the same fashion as modified

Nicolas and UBC. However, most of rankings and numbering system as well as range of input

parameters are different. The Sh&B method uses the input parameters to rate the various mining

methods and arrive at an appropriate mining method (as shown in table 1).

Deposit Geometry Rock Mechanics

Ore Thickness Rock Mass Rating of Ore

Ore Plunge Rock Mass Rating of Hanging Wall

General shape Rock Mass Rating of Foot Wall

Grade Distribution Rock Substance Strength of Ore

Grade Quantity Rock Substance Strength of Hanging Wall

Depth Rock Substance Strength of Foot Wall

Table 1: Input parameters for Sh&B approach




Ore Thickness

In the Sh&B Method the ore thickness category of the UBC approach to recognize that many mines

may work ores less than 10 meters in width was applied. It includes a “Very Narrow” category to

consider ore thickness between 0 and 3 meters. Table 2 shows the categories and input value ranges

for ore thickness.

Description Range in Values

Very Narrow < 3 metres

Narrow 3-10 metres

Intermediate 10-30 metres

Thick 30-100 metres

Very Thick > 100 metres

Table 2: Ore Thickness

Ore plunge

The plunge of a deposit is important parameter to consider as it influences the mining method

directly, the location of development and hence overall mining costs. Furthermore, certain deposit

geometries are more applicable to certain mining methods than others. The Sh&B Method modifiesthe deposit plunge categories of Nicholas and UBC approach to increase accuracy degree in the

mining method selection process. Table 3 summarizes the categories and input value ranges for

deposit plunge.


Flat < 15 degree

Low dip 15-30 degree

Intermediate 30-45 degree

Rarely steep 45-60 degree

steep > 60 degree

Table 3: Ore Plunge




General shape

The Sh&B Method utilizes the same categories as Nicholas and UBC for general shape of a deposit.

These factors are important parameters to consider as they directly influence development

requirements and equipment selection. Furthermore, certain deposit geometries are more applicable

to certain mining methods than are others. The table 4 shows the categories for general shape.

General Shape Description

Massive All dimensions are on the same order of magnitude

Platy-tabular Two dimensions are many times thickness, which does not usually exceed 35m

Irregular Dimensions vary over short distances

Table 4: General Deposit Shape

Grade distribution

The New Method also uses the same categories as the Nicholas and UBC for grade distribution to

influence the selected method (table 5). For example, deposits having an erratic grade distribution,

with ore grade changing over short intervals, are more faverably mined using more expensive, but

more selective techniques, such as Cut&Fill Stoping. However, lower tonnage rates can be expected

utilizing the more selective methods.


UniformThe grade at any point in the deposit does not vary

significantly from the mean grade for that deposit

GradationalGrade values have zonal characteristics, and grades

change gradually from one to another

ErraticGrade values change radically over short distances and

do not exhibit any discernible pattern in their changes

Table 5: Grade Distribution




Deposit Depth

The Nicolas Method does not explicitly account for the depth of mining during development of the

mining method rankings, but uses it to later modify the rankings. The UBC Method utilizes the

depth of mining to eliminate or restrict the use of open pit mining. In some instances, the Nicholas

approach may give erroneous results leading to selection of open pit mining as a preferred method

even for deep deposits. The Sh&B Method modifies (with increasing depth because of systems and

machines development in mining technology especially about open pit) the deposit depth categories

of UBC approach to increase accuracy degree in the mining method selection process. Table 6

summarizes the categories and input value ranges for deposit depth.


Shallow 0-200 metres

Intermediate 200-500 metres

Rarely Deep 500-800 metres

Deep > 800 metres

Table 6: Deposit Depth

Deposit Grade ValueThe Nicolas and UBC Methods do not explicitly account for the deposit grade value of the mining

method rankings. The Sh&B Method utilizes the grade value of the deposit in title of cost

alternative and because of this factor influence on mining method selection process. Table 7 shows

the categories for deposit grade value. It is differ in various minerals and their prices of market, for

example Iron with average deposit grade less than 30% take place in Low Grade category.


Low Grade Depends on kind of mineral and its market price

Medium Depends on kind of mineral and its market price

High Grade Depends on kind of mineral and its market price

Table 7: Average deposit Grade Value




Rock Mass Rating

Parameters of Uni-axial Compressive Strength (UCS) of the intact rock, Rock Quality Designation

(RQD), Discontinuities Spacing, Surface Condition of Discontinuities, and Groundwater Conditions

are used to classify the ore zone, hanging wall and foot wall rock mass rating. The final adjusted

RMR value is rated into one of 5 classes, which describes the relative quality of rock mass (table 8).


Very Poor (0-20)%

Poor (20-40)%

Fair (40-60)%

Good (60-80)%

Very Good (80-100)%

Table 8: Rock Mass Rating

Rock Substance Strength (RSS)

The parameters such as UCS, Vertical Stress various with depth, and Ratio of Horizontal to Vertical

Stress are applied to define the Rock Substance Strength (RSS). With respect to the UBS Method,

RSS is a dimentionless parameter defined as the ratio the UCS of rock mass to the maximum in situ

stress at the depth of mining in the new approach too. Table 9 summarizes the categories and input

value ranges for RSS.


Very Weak < 5

Weak 5-10

Moderate 10-15

Strong > 15

Table 9: Rock Substance Strength




Rating the Mining Methods in Sh&B Method

The mining methods are ranked according to table 10-13. The resulting rankings are summed to

arrive at a rating for each method based on the input parameters. In the tables these abbreviations

were applied: (OP-Open Pit; BC-Block Caving; SLS-Sublevel Stoping; SLC-Sublevel Caving; LW-

Long Wall; RP-Room&Pillar; SHK-Shirinkage; CF-Cut&Fill; TS-Top Sclicing; SS-Square Set)

MM Ore Thickness Ore Plunge

< 3 3-10 10-30 30-100 > 100 < 15 15-30 30-45 45-60 > 60

OP 1 2 3 4 4 4 4 4 4 4

BC -50 -50 0 3 4 -10 0 1 3 4

SLS -10 2 4 3 2 -10 0 2 4 4

SLC -50 -10 3 4 4 -10 0 1 3 4

LW 4 2 -10 -50 -50 4 3 1 -50 -50

RP 4 2 -10 -50 -50 4 2 -10 -50 -50

SHK 4 4 3 -10 -50 -50 -10 1 4 4

CF 3 4 4 1 -10 1 1 3 4 4

TS 1 0 0 2 1 3 4 2 0 0

SS 4 3 2 0 0 1 2 3 4 4

Table 10: Ranking of Thickness and Plunge of Ore

M M General Shape Grade Distribution Grade Value Depth

MA T/P IR U G E L M H SH I RD D

OP 4 3 3 3.8 2.85 1.9 1.6 1.6 1.6 2.4 0.6 -25 -50

BC 4 2 0 2.85 1.9 1.9 1.6 0.8 0.4 0.6 1.2 1.8 2.4

SLS 3 4 1 3.8 3.8 2.85 0.4 1.6 0.8 1.2 1.8 2.4 2.4

SLC 3 4 1 2.85 1.9 1.9 0.4 1.6 0.8 1.2 1.8 1.8 2.4

LW -50 4 -50 3.8 0.95 0 0.4 1.6 0.8 0.6 1.2 1.8 2.4

RP 0 4 2 3.8 2.85 0 0.4 1.6 0.8 1.8 2.4 1.2 0.6

SHK 1 4 2 3.8 1.9 1.9 0 0.8 1.6 1.8 1.8 1.8 0.6

CF 1 4 4 2.85 3.8 3.8 0.4 1.2 1.6 0.6 1.2 2.4 2.4

TS 1 2 0 1.9 0.95 0.95 0.4 1.2 1.2 1.2 1.2 0.6 0.6

SS 0 2 4 0.95 1.9 2.85 0 0.4 4 0.6 0.6 1.2 2.4

Table 11: Ranking of Shape, Grade Distribution and Value, and Depth of Ore




In the table 11: (MA-Massive, T/P-Tabular/Platy, IR-Irregular, U-Uniform, G-Gradational,

E-Erratic, L-Low, M-Medium, H-High, SH-Shallow, I-Intermediate, RD-Rarely Deep, D-Deep)

MM Ore Hanging Wall Foot Wall

< 5 5-10 10-15 > 15 < 5 5-10 10-15 > 15 < 5 5-10 10-15 > 15

OP 2.63 3.5 3.5 3.5 2.1 2.1 2.8 2.8 1.32 1.32 1.76 1.76

BC 3.5 2.63 1.75 0 2.8 2.1 1.4 0 1.76 1.32 0.88 0.44

SLS 0 0.88 3.5 3.5 0 0.7 2.8 3.5 0 0.44 0.88 1.32

SLC 0.88 2.63 2.63 1.75 2.8 2.1 1.4 0.7 0.44 0.88 0.88 0.88

LW 5.25 4.38 1.75 0.88 4.2 3.5 2.1 0.7 0 0.44 0.88 0.88

RP 0 0 2.63 5.25 -7 0 1.4 4.2 0 0 0.44 1.32

SHK 0 0.88 2.63 3.5 0 0.7 2.1 2.8 0 0.88 1.32 1.32

CF 0 0.88 2.63 2.63 2.1 3.5 2.8 1.4 0.44 1.32 0.88 0.88

TS 2.63 1.75 0.88 0 2.1 1.4 1.4 1.4 0.88 0.88 0.44 0.44

SS 3.5 2.63 0.88 0 2.8 2.1 0.7 0 1.32 0.88 0 0

Table 12: Ranking of Rock Substance Strength

MM Ore Hanging Wall Foot Wall

VP P F G VG VP P F G VG VP P F G VG

OP 2.63 2.63 2.63 2.63 2.63 1.4 2.1 2.8 2.8 2.8 0.88 1.32 1.76 1.76 1.76

BC 3.5 2.63 1.75 0 -50 2.1 2.1 2.1 1.4 1.4 1.32 1.32 1.32 0.88 0.88

SLS 0.88 2.63 3.5 3.5 3.5 -50 0 2.1 2.8 2.8 0 0 0.88 1.32 1.32

SLC 2.63 3.5 2.63 0.88 0 2.8 2.8 2.1 1.4 1.4 0.44 1.32 1.32 1.32 1.32

LW 5.25 5.25 3.5 1.75 1.75 4.2 3.5 2.8 2.1 2.1 1.32 1.76 1.32 0.88 0.44

RP -50 0 2.63 4.38 5.25 -50 0 2.1 3.5 4.2 1.32 1.32 0.88 0.44 0

SHK 0 0.88 2.63 2.63 2.63 0 0 1.4 2.8 2.8 0 0 0.88 1.32 1.32

CF 0 0.88 1.75 0.88 2.63 2.1 3.5 2.8 2.1 2.1 1.32 1.32 1.32 0.88 0.88

TS 2.63 1.75 0.88 0 0 0 0 1.4 2.1 2.1 0 0 0.44 0.88 0.88

SS 3.5 3.5 0.88 3.5 0 2.8 2.8 0.7 0 0 1.32 0.44 0 0 0

VP-Very Poor P-Poor F-Fair G-Good VG-Very Good

Table 13: Ranking of Rock Mass Rating




ANALYTIC HIERARCHY PROCESS (AHP)

Overview

Multiple Attribute Decision Making (MADM) deals with the problem of choosing an alternativefrom a set of alternatives which are characterised in terms of their attributes. Usually MADM

consists of a single goal, but this may be of two different types. The first is where the goal is to

select an alternative from a set of scored ones based on the values and importance of the attributes

of each alternative. The second type of goal is to classify alternatives, using a kind of role model or

similar cases. MADM is a qualitative approach due to the existence of criteria subjectivity [5]. The

decision maker might express or define a ranking for the criteria as importance/weights. There are

many forms for expressing these weights, but the most common are: Utility Performance Function,

Analytical Hierarchy Process [6,7], and Fuzzy Version of the classical linear weighted average

[8,9]. AHP is a multi-criteria decision method that uses hierarchical structures to solve complicated,

unstructured decision problems, especially in situations where there are important qualitative

aspects that must be considered in conjunction with various measurable quantitative factors. The

AHP is based on four main axioms [10]:

1) Given any two alternatives (or sub-criterion), the decision-maker is able to provide a pairwise

comparison of these alternatives under any criterion on a ratio scale which is reciprocal.2) When comparing two alternatives, the decision-maker never judges one two be infinitely better

than another under any criterion.

3) One can formulate the decision problem as a hierarchy.

4) All criterion and alternatives which impact a decision-problem are represented in the hierarchy.

The above axioms describe the two basic tasks in the AHP: formulating and solving problem as a

hierarchy, and eliciting judgements in the form of pairwise comparison.

Saaty (1980) has been developed the mathematics necessary to combine the results of the pairwise

comparisons made at different levels in order to final priority value for each of the alternatives at

the bottom of hierarchy [10].




Figure 1: AHP Hierarchy [10]

Expert Choice Software

Expert Choice (EC) software is a multi-objective decision support tool based on the Analytic

Hierarchy Process. A mathematical theory first developed at the Wharton school of the

Pennsylvania University by one of Expert Choice's founders, Thomas L. Saaty. The AHP is a powerful and comprehensive methodology designed to facilitate sound decision making by using

both empirical data as well as subjective judgments of the decision-maker [11].

CASE STUDY

Introduction

The Sh&B Method has been validated utilizing input data of the Third Anomaly of Gol-E-Gohar

Iron Ore Deposit. This deposit located in Kerman province in south eastern of Iran. The length of

the deposit is about 2200 meters (north-south) with an average width of 1800 meters (east-west).

The main Iron ore is Magnetite with hanging and foot wall of Schist and Shale respectively. The

input parameters of the anomaly for Mining Method Selection (MMS) process are given in table 14.

Goal

Criteria 1 Criteria 2 Criteria 3

Alternative A Alternative B Alternative CAlternative A Alternative B Alternative CAlternative A Alternative B Alternative C




Input Parameters Description

Ore Thickness 40 meters

Ore Plunge 20 degrees

General Deposit Shape Platy

Grade Distribution Gradational

Grade Value High

Depth 350 meters

RQD 75%

Joint ConditionFilled with talk strength less than

rock substance strength

Rock Substance Strength 8.7

Rock Mass Rating 63.5

Ore

Zone

Uniaxial Compressive Strength 128 MPa

RQD 38%

Joint Condition Clean joint with a smooth surface


Rock Mass Rating 50

Hinging

Wall


RQD 38%

Joint Condition Clean joint with a smooth surface


Rock Mass Rating 50

Foot

Wall


Table 14: Input parameters of third anomaly of Gol-E-Gohar for MMS Systems




Mining Method Selection

Note that the final outcome will depend on which category is chosen. Using Nicholas, UBC, and

SH&B Methods the total values are shown in table 15.

Method OP BC SLS SLC LW RP SHK CF TS SS

Nicholas 39 32 -29 33 -19 -28 35 37 34 35

UBC 33 26 31 29 -25 -32 -29 29 18 14

Sh&B 30.16 20.11 19.76 22.67 -25.6 -37.59 -44.91 10.02 16.92 21.45

Table 15: Evaluted total values of the Mining Method Selection Systems

According to the evaluated total value of the Nicholas Method, Open Pit, Cut&Fill, and Shirinkagemining methods were selected respectively. But the results of UBC Method show that Open Pit,

Sublevel Stoping, Sublevel Caving and Cut&Fill are most suitable respectively. Based on the

calculated rankings in the Sh&B Method, the most effective and fitness mining methods for the

third anomaly of Gol-E-Gohar Iron Deposit are: 1- Open Pit; 2- Sublevel Caving; 3- Square Set.

For choosing a most suitable mining method from these three alternative ( emphasis on the results

of the Sh&B Mining Method Selection approaches), the Expert Choice Code based on AHP Method

was used.

In first step, it is necessary to create an Expert Choice Model from the problem with Goal, Criteria

(Objectives), and alternatives (figure 2). The number of objectives must be limited to four. In the

next step, judgments/pairwise comparisons with the goal and working down to the alternatives (top-

down) have to started.




Figure 2: AHP Model for Mining Method Selection of Gol-E-Gohar Deposit

In this way, decision alternatives are evaluated by pair wise comparisons, thus allowing more

accurate judgements than the simple weighted product model. Figure 3 and 4 show the weighting

results of pair wise comparison between Geometry and Geotechnical parameters respectively.

Figure 3: Weighting results of pair wise comparison between Geometry parameters

Figure 4: Weighting results of pair wise comparison between Geotechnical parameters

Mining Method Selection

GEOTECHNICAL PARAMETERS GEOMETRY PARAMETERS

Ore Thickness

Ore Plunge

Ore Shape

Grade Distribution

Grade Value

Deposit Depth

RSS of Ore

RSS of Hanging Wall

RSS of Foot Wall

RMR of Ore

RMR of HangingWall

RMR of Foot Wall

Sublevel

Caving

Square

Set

Open Pit

Alternatives




According to Experts judgments, pair wise comparisons of the alternatives based on sub-criteria

were implemented. The alternatives characteristics based on all sub-criteria (for example mining

methods comparison with emphesis on Ore Plunge, Ore Depth, RMR of Hanging Wall, and RSS of

Foot Wall are shown in Figures 5 to 9 respectively) and the total weight of criteria were evaluated.

Figure 5: Mining Methods Comparison with emphesis on Ore PLunge

Figure 6: Mining Methods Comparison with emphesis on Ore Depth

Figure 7: Mining Methods Comparison with emphesis on RMR of Hanging Wall

Figure 8: Mining Methods Comparison with emphesis on RSS of Foot Wall

Figure 9: Importance Degree based on Geometry and Geomechanical Parameters




Finally the weight of each alternative was determined. Table 16 shows the achieved results of the

applied software.

Parameters Weight Open Pit Sublevel Caving Square Set

Geometry 0.454 0.451 0.293 0.257

Geomechanical 0.455 0.416 0.3 0.283

Total Weight 0.435 0.296 0.269

Table 16: Final weight of alternatives

In respect to the out put data of the Expert Choice for the Third Anomaly of Gol-E-Gohar Iron Ore

Deposit, Open Pit was selected as the most suitable mining method [11,12].

CONCLUSION

Generally, systems applied to choose potential mining methods basis on a finite number of defined

input parameters. Due to the evaluated total value of the Nicholas Method, Open Pit, Cut&Fill, and

Shirinkage mining methods were selected respectively. But the results of UBC Method show that

Open Pit, Sublevel Stoping, Sublevel Caving and Cut&Fill are most suitable respectively. Based on

the calculated rankings in the Sh&B Method, the most effective and fitness mining methods for the

Third Anomaly of Gol-E-Gohar Iron Deposit are: Open Pit, Sublevel Caving, and Square Set. Due

to the output results of the Expert Choice Code, Open Pit method was selected for the Third

Anomaly of Gol-E-Gohar Iron Ore Deposit as the most suitable mining method.




LITERATURE

[1] Nicholas, D. E: Method Selection-A Numerical Approach, Design and Operation of Caving

and Sublevel Stoping Mines, 1981, 39-51

[2] Miller-Tait, R. et al: UBC Mining Method Selection, Mine Planning and Equipment

Selection, 1995, 163-168

[3] Husrulid, W. A: Underground Mining Methods Handbook, New York, SME-AIME, 1982.

[4] Clayton, C. et al: A Knowledge-based System for Selecting a Mining Method, IPPM

Conference, Canada, 2002.

[5] Saaty, T. L: Exploration the Interface between Hierarchies, Multiple Objectives and Fuzzy

Sets, Fuzzy Sets and Systems, 1978, 57-68.

[6] Saaty, T. L: Modelling Unstructured Decision Problems-The Theory of Analytical

Hierarchies, Math. Comput. Simulation, 1978, 147-158.

[7] Yager, R. R: Fuzzy Decision Making Including Unequal Objectives, Fuzzy Sets and Systems,

1978, 87-95.

[8] Baas, M. S; Kwakernaak, H: Rating and Ranking of Multiple Aspect Alternatives Using

Fuzzy Sets, Automatiaca, 1977, 47-58.

[9] Baldwin, J. F; Guilg, N. C. F: Comparison of Fuzzy Sets on the Same Decision Space, Fuzzy

Sets and Systems, 1979, 213-232.

[10] Saaty, T. L. et al: The Analytical Hierarchy Process, McGraw Hill, New York, 1980.

[11] Saaty, T. L. et al: Expert Choice Quick Start Guide, Pittsburgh, PA, 2000, 1-14.

[12] Saaty, T. L. et al: Expert Choice Tutorials, Pittsburgh, PA, 2000, 1-40.

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A New Numerical Method and AHP for Mining Method Selection