International Journal of Mining and Geological Engineering, 1990, 8, 163-177

Fuzzy evaluation of powered support suitability to coal seams G U O W E N Z H A N G and X I N J I N G M I N Beijino Graduate School, China University of Minin9 and Technology, Beijin9 100083,People'sRepublic of China Received 24 November 1989

Summary The selection of powered supports involves many geological factors where adequate data is lacking and in which subjective judgement is inevitable. This paper proposes one approach, a fuzzy multicriteria decision-making model, to transfer non-quantitativejudgements into quantitative ones that computers can process. A detailed example is included to demonstrate how the model applied to evaluating the suitability of powered supports to coal seams. Keywords: Fuzzy sets; multicriteria decision-making; powered support; subjective judgement; coal mining; longwall mining.

Introduction In mining equipment selection problems, solutions can generally take one of three forms: no solution, single solution and multiple solutions. A single solution is the most satisfactory. The problem of no solution is the most difficult to address. The problem of multiple solutions needs further economical and technical comparisons. Here, attempts are made to deal with the problem of multiple solutions from a technical point of view. In general, the selection of powered support needs takes into account five major groups of geological factors: (1) (2) (3) (4) (5)

roof formation; seam thickness; seam gradient; geological disturbances; seam floor conditions.

When evaluating the influence of each on technical selection of powered supports, one has to define how important they are (or how much impact they have). This definition is subjective. When technically assessing the result of a selection with respect to the geological factors, linguistic terms are commonly used, e.g. excellent, favourable, poor, etc. These descriptive terms are ambiguous and imprecise. The question that is raised immediately is according to what criterion (or scale), does 'excellent' or 'favourable' apply? 0269-0136/90 $03.00+.12 © 1990 Chapman & Hall Ltd.

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164 Basic description of fuzzy sets

Since the notion of fuzzy sets was first introduced in 1965 by Zadeh, the theory of fuzzy sets has been developing at an increasing rate. Its potential usage is being constantly explored and documentation of successful applications is limited (Wenzhang, 1989; Denby and Atkinson, 1988; Brown and Singh, 1987; Nguyen, 1985; Bandopadhyay, 1987). The application of this approach to subjective decision-making in the mining industry is slowly being investigated. The key concept throughout fuzzy set theory is the definition of a fuzzy subset. That is (see Kaufman, 1975; Dubois and Prade, 1980; Chen, 1984). Let tt be a mapping of a reference set U #: U---q-0, 1] then # determines a fuzzy subset, denoted by A_, in U. tt is called the membership function, sometimes indicated by #A" #A(u) is called the grade of membership of It in A_.If #A(U) is either 0 or 1, the fuzzy set A_will be understood as an 'ordinary subset'. There are a number of schemes for notation of fuzzy set A_. For instance, pair representation:

A = {(u, ~A(u)), u~U} A more convenient notation was proposed by Zadeh. When u is a finite set {ul, u2,. • • Un}, a fuzzy set on U is expressed as A

m ~t~U1~

¢-'"-1

/21

#A(Un) _ Un

~A(Ui) i= 1

Ui

When U is not a finite set, then

A_= ~ ~A(u) Ju U In correspondence with an ordinary set, a fuzzy set still has three basic operations: (a) Union ( v ) uBU, tt AvB(u)= max[#(u), #_B(u)]. (b) Intersection ( ^ ) u~U, #A ^ B(U) = min[tt(u), #B (u)]. Where #AvB(U) and #A^B(U) are the membership functions of A_vB_and A_^B_respectively. (c) Complement if) [ u s U , ~.A c] (U)~--- 1 - - # A ( U ) .

The complement Ac of A is defined by the membership function.

Muiticriteria decision making model The problems ofmulticriteria decision making here refer to all the feasible alternatives can be evaluated to a number of criteria and certainly some criteria are subjective rather than

Fuzzy evaluation of powered support suitability to coal seams

165

conventionally objective (Kickert, 1978; Chen, 1984). The versatility of the practical cases has led to the birth of numerous operational research techniques including decision theory, mathematical programming and sensitivity analysis. Fuzziness exists extensively in various kinds of decision problems although it is often ignored. After the concept of fuzzy sets emerged, they have been spreading rapidly into the existing decision-making techniques. Fuzzy mathematical programming, fuzzy relation decision-making and fuzzy integration decision-making are typical examples of the products. In this paper, fuzzy relation decisionmaking will be employed and therefore introduced. Fuzzy relation decision-making has a few slightly different versions (Kickert, 1978), the one that has been particularly useful in this work is 'fuzzy rating and ranking of multicriteria alternatives', to which in fact, many other authors often refer multicriteria decision-making modelling (Dubois and Prade, 1980; Chen, 1984; Hong et al., 1983). The basic idea of this decision making technique is as follows. Given a set of alternatives A = {al • • • am} and a set of n criteria, the merit of alternative a i according to the criterion j is denoted by the rating rij. The relative importance of each criterion is denoted by a weight wj. Then alternative a i receives the weighted average rating. This average rating induces an ordering of the alternatives a i . .. %. In order to apply this technique to the assessment of equipment suitability to coal seams, modifications and enhancements must be made, considering the features and characteristics of the problem stated before and based on the procedures discussed in the literature (Dubois and Prade, 1980; Chen, 1984; Kickert, 1978; Hong et al., 1983), the authors suggest the following consecutive steps. (1) List the alternatives (objects) set X = {xl, • • •, x,}; (2) Work out the factors set u = {u 1. . . . . u,,}, determine an impact ruler and assign a weight number to u~(i= 1, 2 . . . . . m); (3) Define the membership functions of all the basic factors and compute the values rij~[O, 1]. There are two models available to evaluate the basic factors according to the different given conditions:

Model A:

Ug~,Ugconsists

V(u.) =

of m discrete point values.

{v.l, vg2,...,

is the subjective assessment result subset, the quantified value of assessment vgi is (#v,x) (%), i = 1, 2 . . . . , m. Assume the weighted coefficient vector

A(V(uo))= (a 1 , a z . . . . .

a,,),

where ith element a i corresponds to ith assessment %~, i = 1, 2 . . . . .

m. Then uo's grade is:

% = ~ ai~ai(uo) k=l

Model B: ifuvo is a continuous function in [a, b] and u o is a continuous subset [a, fl]e[a, b], then

1F1

ra - fll --~1

~ lava(uo) dug

(4) Evaluate the compound factors. Let a compound factor ul consist of uil, ug2. . . . . uim,

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vii, vi2 . . . . . v~,,and a~a, a~z. . . . . a~mare grades of membership and weighted numbers of u~l, U~z, . . . , u~m respectively. Three models can be used. Model I: ri(u~)= ~ aikVik k=l

1 m

Model II: r i = Z-*V (aikVik) U i k ==

1

m

Model III: r,= T. A

(aikl)ik)

[d,i1 k . =

(5) Form the fuzzy evaluation matrix RIxi - (ril, ri2 . . . . . rirn)9.[0, 1] m, (6) Choose the objective evaluation model f: [0, 1 ] " ~ R (all the real numbers), noted in: D =f(zl, z2 . . . . . zm) (7) Aggregate the evaluation characteristics:

D(xi) = f (ril, ri2 . . . . . r,,,) (i < = m). Thus a dominance relation matrix can be constructed, and the final decision can be made intuitively.

Model application

Problem description As a hypothetical case study, assume that a seam with area of 5 x 400 000 m 2 is 2.0 metres thick and dipping at 10° on average. Based on stratigraphic sequence classification (Peng and Chiang, 1979; China Mining Institute, 1979), the roof type is type 2 (moderately weak roof, immediate roof is about 2-4 times as thick as the coal seam), and the immediate roof is kind 1 (unstable) as a whole. The type of supports should thereby be chock-shield series in nature. Among all the available chock-shield supports, it is supposed that three specific models: support AX, support BY and support CZ, with difference in technical parameters, are feasible in terms of supporting heights, supporting capacity and applicable gradient. The favourable conditions for each one are summarized in Table 1. The detailed seam conditions are as follows: Roof condition: strong and competent roof with high periodic pressure, moderate strong roof, fairly stable, slight periodic pressure, weak immediate roof, negligible periodic pressure,

5 %; 20%; 75%;

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Table 1. Descriptions of supports Roof formation

Seam thickness

Seam gradient

Support AX

Strong and fairly stable roof, slight periodic pressure

Not very flexible, best range is 1.7 ~ 2.3 m

Best gradient is under 10°, with special measures against sliding, workable gradient can be up to 16°; when over 16°, the operation is very difficult

Support BY

Weak immediate roof, negligible periodic pressure

Very flexible, applicable range is 1.4~2.7 m

Best gradient is under 12°, with special measures against sliding, workable gradient can reach 18°

Support CZ

Broken immediate roof, negligible periodic pressure

Moderate flexible, suitable range is

Best gradient is under 15°; if gradient is more than 15°, special measures are required to prevent supports from sliding

1.6 ~ 2.4 m

Seam thickness: washout 1.0 ~ 1.4 1.4~ 1.6 1.6~2.4 2.4~2.7 2.7,-~ 3.0

area ( < 1.0 m), m thick, m thick, m thick, m thick, m thick,

1%; 5 %; 8%; 70%; 10%; 6%.

Seam gradient: < = 8 °, 8 o,~ 12 o, 12°~15 °, 15 ° ~ 19 °,

60%; 25%; 10%; 5%.

The above factors do not have any inherent relationship with one another and none are mutually transferable. Each factor has a different impact on support selection. In fact, the impact of each factor on support further differs as the support alters, and this impact even varies when the factor's values falls in different ranges. H o w does one justify the suitability of each support and decide the one that best matches the conditions? It is obvious that this problem is complex when the five groups of factors are taken into account.

Detailed evaluation models The problem of equipment suitability evaluation described above is graphically represented in Fig. la, where u 0 represents suitability; ul, coal seam gradient; u2, seam thickaness; u3,

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roof formation; u31 , periodic pressure and u3z, roof stability. The basic factors are u~, u 2, us~, u3z, us is a compound factor of u31 and u32. (1) Object set X = {Support AX, Support BY, Support CZ}. (2) The basic factor set u = {ut, uz, us1, U3z}, a compound factor set is u = {u3}. The impact ruler is scaled in Fig. 1b. Although weight number assignment is in most cases made the same for every element of event set X; different assignment schemes of weight number are established in consideration of differences in the supports characteristics. Tables 2, 3 and 4 are weight number distributions of Support AX, Support BY and Support CZ. (3) Different criteria are used for different basic factors. Let v~ = {suitable} be the general assessment m a r k of factor u 1. The definition of membership functions is the most important of all. If the definition is not adequate, the final evaluation result cannot be expected to be satisfactory. However, the definition is very subjective.

Seam gradient u 1 Membership functions of objects AX, BY and CZ to u 1 are defined as:

• /

1

u1<_10 °

20_ul 10

10° < ul _< 16 °

Support AX #~1(ul)= J 30-u 35

16 ° < u I _<30 °

0

u 1> 3 0 °

"1

u1_<12 °

22-u~ 10

12°
30-u 1 30

18°
0

U1 > 30 °

iN

"1---15°

Support BY # v l ( u l ) = .

Support CZ #vl(ul)=

35-ul

15°
u 1 > 35 ° They are also graphically shown in Figs 2a,b,c.

Seam thickness u2 When seam thickness is beyond the upper limit of a particular support, coal has to be left to ensure that the supports contact the roof so that roof convergence is controlled, thus a lower

Fuzzy evaluation of powered support suitability to coal seams

u0

169

112 "

u

3

<

u31 u 32

Ca)

Extremely Important

Very Important

Important

Unimportant

I

I

I

I

I

I

1

0.8

0.6

0.4

0.2

0

Very Unimportant

Extremely Unimportant

(b)

Fig. 1. Evaluation models: (a) Factor tree and (b) Impact ruler

Table 2. Support AX weight number distribution Assessment mark Scale value Standardized weight number

U1

U2

U3

Important 0.6 0.3

Important 0.6 0.3

Very important 0.8 0.4

Table 3. Support BY weight number distribution Assessment mark Scale value Standardized weight number

Ul

/'12

U3

Very important 0.8 0.44

Unimportant 0.4 0.23

Important 0.6 0.33

recovery rate is brought about; otherwise the roof condition will be damaged owing to loss of control of the roof convergence. Conversely, if seam thickness is less than the lower limit, a certain thickness of floor must be cut to provide room for the supports. In between the limits there will be an 'optimum' thickness where the supports operate 'best'.

Wenzhang and Jingmin

170 Table 4. Support CZ weight number distribution

Assessment mark Scale value Standardized weight number

U1

122

123

Important 0.6 0.3

Very important 0.8 0.4

Important 0.6 0.3

bl-v 1 1.00 ~ 0.750.507 U1

0.25'

I

1

r

5° 10° 15° 20 ° 25 ° 30°

(a)

~v 1 1.00 0.75 0.500.25"

u1

I 5°

10° 15° 20 ° 25 ° 30°

Cb) ~Lv 1 1.00 0.75 0.50Ul

0.25 -

I 5°

I

I

I

10° 15° 20° 25 ° 30° 35 °

(c) Fig. 2. Membership grades graphically represented: (a) of support CZ to u l , (b) of support AX to ul and (c) of support BY to u x

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171

Assuming that the above factors are considered,/)2 of u2 is supposed to be {suitable} as well, then the membership functions are defined as: 1

u 2 < 1.3

u2-1.3 0.4

1.3 < u 2 < 1.7

Support AX i~(u2)=.~ 1

1.7
3 --U 2

0.7 0

u 2 > 3.0

0

u2~1.0

u2-L0 J Support BY pv2(u2)=-)

2.3
0.4

1.0Nu2
1

1.4Nu2<2.7

3.5-u 2 0.8

2.7Nu2<3.5

0

u2>3.5 u2N1.4

!

0.4

Support CZ/zv2(u2)= 4 1

1.4Nuz<1.8 1.8Nu2<2.4

!

/30-u

2.4Nuz<3.0

L 0 0.6 u2>3.0 They are graphically shown in Figs 3a,b,c.

Period pressure u31 and roof stability u32. Basic factors u 31 and u32 are normally assessed in linguistic terms. It is in fact very difficult to establish continuous mathematical functions. Using fuzzy set theory to describe them accurately is out of the scope of the study. Here u 31 and u 32 are simply divided into acceptable states and accordingly assigned grades of membership as in Table 5. From our hypothetical data, the basic factors Ul, u 2 and u31 , u32 can be computed.

~v,(ul): The conditions of seam gradient have to be adjusted to be compatible with the membership functions. Take Support AX as an example, the procedure is as follows: (a) Range, 8 ° ~ 12 ° for example, needs to be divided into 8°~ 10 ° and 10° ~ 12 °. Using interpolation methods, 8 ° ~ 10 ° takes (10 - 8)/(12 - 8) = 0.5 = 50% of 8 ° ~ 12°. Since 8° .,. 12 °

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}iv 2 1 ~00 0.75

f f ~ l

0.500.25 -

U2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 (a)

~t v2 1.00! 0.75. 0.500.25 I I I I I ~-0.5 1.0 1.5 2.0 2.5 3.0 3.5 Cb)

1.00 !

}iv 2

0.75. 0.500.25 0.51.01.52.02.53.03.5

(c) Fig. 3. Membership grades graphically represented: (a) of support AX to u 2, (b) of support BY to u2 and (c) of support CZ to u 2 takes 25% of the whole area, 8°~ 10° accounts 50% x 25% = 12.5% of the whole area. Applying this method to the rest of the area, the re-organized gradient distribution is: u l < 1 0 °, 1 0 ° < u i < 1 6 °, 1 6 ° < u 1 < 1 9 °,

72.5% ; 23.75%; 3.75%.

(b) calculating grade of membership in each subrange. ul - 10°, #v(ul) = 1,

Fuzzy evaluation of powered support suitability to coal seams

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Table 5. Grades of membership assigned to factors u31, u32 for supports AX, BY, CZ Support AXi u31 ~1(u31 )

Extremely high 0.2

U32

Very stable 1

~lv3z(U32)

Very high 0.3 Stable 0.9

High 0.4

Moderate stable 0.8

Moderate 0.7 Broken roof 0.5

Low 0.9

Slight 1

Easily caving 0.2

Support BY:

U31

'v~l(u31)

Extremely high 0.1

u32 pv~2(U3z)

Very stable 0.8

Very high 0.2 Stable 0.85

High 0.3

Moderate stable 1

Moderate 0.4 Broken roof 0.9

Low 0.8

Slight 1

Easily caving 0.9

Support CZ: 1/31 #v31(u31)

Extremely high 0.1

ua2 ktv32(us2)

Very stable 0.7

Very high 0.2 Stable 0.85

High 0.3

Moderate stable 0.9

Moderate 0.4 Broken roof 1

Low 0.8

Slight 1

Easily caving 0.95

1 1o(2o_u1 10° < ul < 16°, #v(ul) - 16_ 10

o \

10 J dul=0"8

1 f119(30--Ul~

16°
6 \

35

} du1=0"36

(c) The weighted mean value #v(ul) for support AX: /~v(ul) = 1 x 72.5% +0.8 x 23.75% +0.36 x 3.75% =0.9285 Applying this method to computing/~v(Ul) for Support BY and Support CZ, obtain: Support AX: #~(u 1) = 0.9285, Support BY: #~(ul) = 0.9515, Support CZ: #~(ul)=0.995.

~o2(u2): Like computing #v(ul), the values of/%(u2) can be got: Support AX: #v~(u2)=0.73875, Support BY: #~2(u2)=0.95375, Support CZ: #~2(u2) = 0.7325. #~31(u31) and #v3~(u32) Grades of membership of Supports AX, BY and CZ to u31 and u32 are listed in Table 6.

Wenzhan9and Jinomin

174 Table 6. Grades of membership to u31, u32 High

Moderate

Slight

#v31(U31)

Support AX Support BY Support CZ

0.4 x 5% 0.3 x 5% 0.3 x 5%

0.7 x 20% 0.4 × 20% 0.4 x 20%

1 x 75% 1 × 75% I x 75%

0.91 0.845 0.845

Support AX Support BY Support CZ

Very stable 1 x 5% 0.8 x 5% 0.7 x 5%

Moderate 0.8 x 20% 1 x 20% 0.9 x 20%

Broken roof 0.5 x 75% 0.9 x 75% 1 x 75%

/~32(uaz) 0.585 0.915 0.965

(4) The basic factors u31 and 1232 constitute a c o m p o u n d factor u a. Only after the c o m p o u n d factor u 3 is evaluated, can the objective evaluation be made. Using the three models, their values are calculated, take S u p p o r t AX as an example: 2

Model I:

/zv3=

a3kl)3 k k=l

= 0 . 4 × 0.91 + 0 . 6 x 0.585 =0.715

Model H:

+ # ~ = ~1 k=,

(a3kV3k)

1

max(0.4.0.91, 0.6.0.585) 1

= _ _a*

(0.4.0.91)

Therefore #vs = 0.91. 1

Model III:

2

#v3=~/k k=l

(a3kVsk)

1 a* min(0.4.0.91, 0.6.0.585) 1 = ~ (0.6.0.585) Therefore #v3 = 0.585. Similarly, the calculations can be made to S u p p o r t BY and S u p p o r t CZ. T h e values for three supports are all listed in Table 7. (5) In correspondence with the c o m p o u n d factor u 3, there are three evaluation matrices:

Fuzzy evaluation of powered support suitability to coal seams

175

Table 7. Values of u 3 of three supports Model I 0.91 0.915 0.965

Support AX Support BY Support CZ

Model II 0.585 0.845 0.845

Model III 0.715 0.887 0.917

[

-#,~(AX) #~2(AX)#~3(AX)]

R~= #~(BY) #o:(BY) #o~(BY)

_~,~l(cz) ~t~(cz) #o~(cz)

0.9285 F = [0.9515

0.7388

0.9100]

0.9538 0.9150

/

[_0.9950 0.7325 0.9650

R2=

0.9285 F|0.9515

0.7388 0.5850 1 0.9538

0.8450

L0.9950 0.7325 0.8450 0.9285 F R3= |0.9515

0.7388 0.7150 1 0.9538 0.8870

/

[-0.9950 0.7325

0.9170

(6) The three models used in evaluating the compound factor u 3 are again employed to compute the evaluation characteristics of the objective (suitability): dAx ~= 0.3 x 0.9285 + 0.3 x 0.7388 + 0.4 x 0.91 = 0.8642,

dAx2=max(0.3"0.9285, 0.3"0.7388, 0.4-0.91)=0.91, dAx~----min(0.3.0.9285, 0.3-0.7388, 0.4.0.91) = 0.7388. dByt=0.940, dBv2=0.9515, dBys=0.9538, dcz ~=0.881, dcz=0.995, dcz=0.965. Thus a new evaluation matrix R~ is obtained: dAx~

dAx 2

dAx 3

0.8642 0.9100 0.7388

dBy1 dByz dBy3 =

0.9400 0.9515 0.9538

dcz,

0.8810 0.9950 0.9650

dcz2

dcz ~

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176

Similarly, new matrixes R~ and R~ on Models I and II are built: 0.7342

0.9285

0.7388

R~= 0.9169

0.9515

0.9538

0.8450

0.9950

0.8450

0.7862

0.9285

0.7388

R~= 0.9307

0.9515

0.9538

0.8666

0.9950

0.9170

(7) D(xi) is calculated in:

O(xi) = ~,

ak d x k

k=l

where ak is standardized weight number. The calculation for Support AX and Model I for example is: D(AX1) = 0.3 × 0.864 + 0.3 x 0.9100 + 0.4 x 0.7388 = 0.8278 The values of D(xi) are listed in Table 8. This table can be viewed as a dominance matrix. It is clear that Support BY is the best of the three. Table 8. Values of D(xi) Support AX Support BY Support CZ

Model I

Model II

Model III

0.8278 0.9472 0.9518

0.7927 0.9370 0.9050

0.8099 0.9431 0.9329

Discussion and conclusion Using fuzzy set theory to aid in the evaluation of equipment suitability is a relatively new approach. Fuzzy set ideas will be used in expert systems concerned with many aspects of mining engineering. In this work, the factors under consideration in most cases lack adequate quantitative data, and their influences are often judged in linguistic terms, instead of mathematical language. The theory of fuzzy sets can provide both rational and systematic approaches to the transfer of linguistic, non-quantitative terms into quantitative expressions so that more extensive applications of computers are possible. The basic model proposed in this paper has practical applications and the following points are worth stressing. (1) Determination of influential factors. Obviously, the more numerous the factors, the more thoroughly the objective system is depicted. However the main factors will be more easily underestimated. Therefore identifying only the crucial factors will not only simplify the model and save time but also enhance the model. (2) Weight number assignment. This is still a subjective matter. Irrational assignment of

Fuzzy evaluation of powered support suitability to coal seams

177

weight number will very likely cause an incorrect result no matter how realistically the membership functions are constructed. (3) Membership function construction. This is another subjective matter. Careful study of each factor is essential to describe accurately the relation between the factor and the equipment suitability. (4) Consistency of results from different models. This is often caused by improper assignment of weight numbers. Fuzzy set theory is a developing area of applied mathematics. As its practical significance in many aspects of society is being realized, its potential applications are being explored extensively and progressively. An expert system currently being carried out by the authors for self-advancing support selection will further employ the fuzzy rank-ordering technique to deal with the problem of no solution and fuzzy pattern recognition to aid in roof classification.

References

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