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Optimal mining of a stratif0rm phosphate field

A b r a h a m M E H R E Z , Uri R E G E V and Alan S T U L M A N Faculty of Engineering Sciences, Ben Gurion University of the Negev, Beer Sheva 84120, Israel

scribed. Then, an optimal control model will be formulated. Finally the solution and an example will be presented.

Received February 1982 Revised March 1983 2. Problem description

An optimal control model is developed to determine opti- mal surface mining operations of a phosphate field. This field contains both a bottom layer and a top layer of phosphates. The deeper the digging,, the more phosphates will be exploited but the higher will be the cost per ton. If the bottom layer is not exploited, it will be lost to future production. The methods of Optimal Control Theory are applied and produce expres- sions which can be evaluated for the optimal scheduling around the phosphate field.

1. Introduction

The "Zin" factory of Negev Phosphates, Ltd., was established in 1976. It is located in the Zin Valley in the south of Israel. This factory mines the phosphates of the area, which are then en- riched for the purpose of producing phosphoric acid. While operating the factory and the mine, a problem concerning the depth of mining and its scheduling arose.

The operations research literature that ad- dresses itself to mining operations problems gener- ally deals with such types of problems as sched- uling and planning. The models suggested are generally discrete linear and non-linear pro- gramming models. For example, see Allais [1], Bishko [4], Babayev [3], Aronofsky [2], Devine [5], and Reddy [7], who discussed the scheduling of a phosphate mine. Babayev [3] pointed out that the problem he treated was of a continuous nature requiring optimal control techniques which are difficult to implement. Our problem of optimizing mining operations will be solved by these optimal control techniques.

In the next section the problem will be de-

North-Holland European Journal of Operational Research 16 (1984) 314- 318

A cross-cut of a phosphate mining field would reveal that the phosphates are distributed in 4 main phosphate layers. The three top layers, in- cluding the unimportant material between them, end at a depth of about 11 meters under the surface, while the bottom layer, which is about one meter in depth, starts at a depth of 18 meters. In order to use the bottom layer it is necessary to remove about six meters of unimportant material, which is done by exploding.

The mining is done in parallel strips. The unim- portant material of each strip is placed on the next strip, where mining has already been completed. Because of this method, it is impossible to mine the top layers in the near future and postpone the mining of the bottom layer to a further date, since during the mining of the top layers the bottom layer is covered by a large quantity of unimportant material. This situation will either prevent mining the bottom layer or significantly increase the cost.

The mining company can at any point dig deep in the field or can dig shallow to recover only the top layer: the second layer then becomes unre- coverable. Shallow digging will produce less phos- phates per unit area than deep digging will. How- ever, the cost of shallow digging is much less than the cost of deep digging and thus the profit per ton of phosphates is greater for shallow digging than for deep digging. The mining company is faced with the problem of where and when they should engage in deep digging and where and when they should use shallow digging in order to maximize the discounted present net worth of the total future profits received by operating the mine.

In this paper we treat the quantity of raw material per unit area as being one of two con- stants which depends on whether the area is minded shallowly or deeply. This assumption was considered reasonable by the company. The claim is that small local variations in the homogeneity of

0377-2217/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)

A. Mehrez et al. / Optima~phosphate mining 3 1 5

the field are made meaningless by the (lack of) precision a bulldozer is capable of. Thus while different fields show variation between themselves, within a field the homogeneity may be considered sufficient for planning purposes.

If indeed the quantity of raw material was not homogeneous throughout the entire field, our sub- sequent model will still be correct, only the subse- quent 'analytical' solution would be complicated considerably. A numerical solution is always possi- ble.

An additional consideration is that the phos- phates must be transported from the site from which they are extracted to a processing facility. The company incurs a transportation cost propor- tional to the distance from the site to the facility.

In the following model we develop an ap- proximation of the actual physical mining opera- tion. Namely, the mining is done in arcs equidis- tant from the facility and transportation is always along a straight line from the mining point to the facility.

3. The model

Let us now define the following parameters, variables and functions:

p

c , =

C =

m s =

m d ~---

r -~-

= y =

Y m

/ ( y ) =

us(y ) =

revenue per ton of phosphates, cost per ton of shallow digging, cost per ton of deep digging, transportation cost per ton per unit dis- tance, tons produced per unit area by shallow digging. tons produced per unit area by deep digging. the cost of money or the discount rate, the capacity of the processing facility, the distance to the processing facility, the furthest distance to the processing facility in the mining field, the function defining the length of the arc in the mining field of equal distance y from the facility, a decision function defining how much phosphates will be mined by shallow mining along the arc of distance y from the facility. Thus 0 ~< u~(y)~< m, l ( y ) ,

Ud(y ) =

A ( y ) =

vs(y ) =

o d ( y ) =

the amount of phosphates mined by deep mining along the arc of distance y from the facility. Thus

ud(y ) = m d / ( y ) 1 m s l ( y ) ,

the area of the field of distance y or less from the facility, or fdl(a) da, the total quantity of phosphates pro- duced by shallow mining within the dis- tance y from the facility, or fdus(a) da, the total quantity of phosphates pro- duced by deep mining within the dis- tance of y from the facility, or

fo" U d ( a ) da = mdA( .v ) - rn---~v,( v).

At this point, it is worth noting that we make no assumptions whatsoever about the shape of the area to be mined. Indeed, the area need not even be continuous (i.e. we do not object to a little mountain rising in the midst of the phosphate field and where mining cannot be done). We will only require that the function I (y ) be completely known. That is, that the field be completely surveyed.

In the real life problem examined, shallow min- ing at y,, is still less expensive per ton than deep mining anywhere else in the field. That is, the large transportation costs of shallow mining at Ym are not sufficient to overcome the difference between C d and C s.

This being so, the optimal scheduling of the mining operation is clear. The objective of maxi- mizing the present worth of the profits is clearly served by incurring the largest profits as soon as possible, or postponing the largest costs as long as possible. Thus for any decision function Us(y) the optimal scheduling of the mining will be to do all the shallow mining first in the order of the closest proximity to the facility, followed by all the deep mining in the order of proximity to the facilitv.

Note that this assumption does in no way sug- gest that there will be only one area deep mined and one area shallow mined. Only that if there are many areas of each kind, the shallow mined areas will be mined chronologically before the deep mined areas.

If Ym is so large that deep digging in close area of the field is actually less expensive per ton than

316 A. Mehrez et a t / Optimal phosphate mining

shallow digging at Ym (due to increased transpor- tation costs) then an equally predictable optimal scheduling scheme can be determined. It will how- ever make the model to determine the depth of mining of each equidistant arc from the facility, that we will develop, inappropriate.

Since the capacity of the processing facility is C 0, it is clear that the rate of mining will always be C 0. To mine at a faster rate will speed up the cost without speeding up the revenues. To mine at a slower rate will just postpone profits and diminish their present worth.

The amount of time that will elapse from the onset of operations till shallow mining at a dis- tance y from the facility is then given by

V s ( Y ) / G .

Then the present worth to the mining company of its shallow digging operations at a distance y from the facility is given by

e ..... ( y , /Co(p_ C. - Cy )us (y ) .

The amount of time that will elapse from the onset of operations till deep mining at a distance y from the facility is then given by

vs(Ym) + vd(y ) /Cp

since all the shallow mining must be completed before any deep mining is begun. Then the present worth to the mining company of its deep digging operations at distance y from the facility is given by

e-,(,,,o.~+v,¢y))/c,( p _ C d - Cy )ud( y ).

The total present worth to the mining company of both types of operations over all distances from the facility is then given by

j _foY'e .~ (y,/C. { p _ Cs _ Cy } us(y) d v

foY"e., ( , ) + , . ( , , / c . { p _ Cd _ Cy } 4- - 'L .'m ' .'

X u d ( y ) dy .

Our objective then is to find the function us(y) (and Ud(Y) by its relationship to us(y)) that will maximize J, subject to the condition that

v ' ( y ) = u s ( y ) ,

(and similarly v' a (y) = Ud (y), but this relationship can be derived from the previous one and the

definitions). By integrating J by write our problem as

maxJ = ? [ P - Cs + K,( P - Ce) u,(.v )

where

parts we then

- K , ( P - C s - Cym)

-K , ( e - q - CY.,)e -''°''''jC']

_ [foY'e .,.o,/c,+ K,e-.,,,,y,/C, dy]

K 1 = e-rt',()'ml/(-'p

and such that

v ~ ( y ) = u s ( Y )

where

O ~ u s ( y ) < ~ m ) ( Y ) .

4. Developing the model

Our problem of finding the us(y) to maximize J can be thought of as an optimal control problem where us(y) is the control function and Vs(y) is the state function. We shall now apply Pontryagin's Maximum Principle to our problem.

Let us define the Hamiltonian function as

H ( v s ( y ) , u s ( Y ) , X ( Y ) }

= _ c [ e - r . . , v , J c ° + K , e - " , ' , ' J c 0 l r

+ ~ ( y ) u ~ ( y ) ,

where h (y ) is the costate or Lagrange Multiplier function. The Maximum Principle states that

H(v ,*(y ) , u*~(y), X*(y)}

>/H{ v : ( y ) , us(y) , X*(y)},

(where the star denotes the optimum solution), for all admissible functions us(y) and for all possible values of y. Observing the form of H we can clearly see that this will only be the case if us(y) is made as small as possible wherever X(y) < 0 and as large as possible whenever h ( y ) > 0. Whenever h ( y ) = 0 , us(y) can assume any possible value what-so-ever. Such a solution is often refered to as a bang-bang solution. In our particular situation

A. Mehrez et at / Optimal phosphate mining 317

this means that on the optimal trajectory

h ( y ) > 0 ~ us(y ) = m s i ( y ) and u d ( y ) = 0,

X ( y ) < 0 =, u~(y) = 0 and Ud(Y ) = m d l ( y ).

Thus the optimal mining of the field will be to dig each arc of equidistance from the facility either deep or shallow but not as combination of both techniques. We can then view our model as com- prised of successive strips of deep digging followed by shallow digging, the strips being placed at various distances from the facility. We must still determine the number of such strips and their placement.

Lemma. The number of transitions from deep dig- ging to shallow digging or, vice-versa, is at most equal to one.

Proof. We will use another relationship of the Maximum Principle, namely that

O X ' (y ) ~v , (y ) H { v * ( y ) , u : ( y ) , ) ~ * ( y ) } .

In our case this becomes

[ ,, , cq X' (v ) = C K l m d e -'L"(~~/c~ -- e . " ms

The second derivative of X(y) is then given by

X"(.v)= C r [ ~,~(y)e-'"""~co

m d - ] - - K 1 -~-'~ Ud ( . l" )e - r ` ' J ( v ) / ( ` P .

It is clear from our previous results that

X ( y ) > 0 = u d ( y ) = 0 =, X" (y ) > 0

and

) , ( y ) < 0 = u , ( y ) = 0 = X"(.v) < O.

We then have that ) t (y )X" (y )> /0 . Now if we consider the function X(y)X ' (y ) we

have

d x d y ( Y ) X ' ( Y ) = X ( Y ) X " ( v ) + X ' ( Y ) 2

which is by our previous results clearly positive. Thus X(y))~'(y) is increasing and there can be no more than one value of.v such that X(y ) - - 0. This

in turn implies that h ( y ) changes sign at most one time or that there is at most one transition from one digging method to the other.

5. Conclusions

We have seen that the optimal mining alloca- tions of a phosphate field is to either use deep mining in the area close to the facility and then perhaps from some distance switch to shallow mining or vice-versa. The procedure for finding the solution is then to solve the two separate problems and choose the maximum of the two solutions.

The first problem will assume that the area nearest to the facility is to be deep mined. Let X denote the radius at which switching is made, then our objective function becomes

max j = Cp [ p - Cs + K , ( P - Ca ) r

- K , ( P - C ~ - C y ~ )

- K , ( P - C d - Cym)e . . . . A,(x)/c.]

_c~ [ ffOe . . . . A:t,.,/C. d y

+ foXK,e-r"~A,( , ,/C, d.v],

where

rm.A~( Vm)/C r K l = e -

A,(y) = [ ' l(a) d(,,). >'~ X. Jo

and

A2(>') =fxl(a) da. >.~ x.

The second problem assume that the area nearest the facility is to be shallow mined. Our previous objective function then becomes

max j = Cp [ P - C , + KI( P - Ca) r

- K , ( p - c~ - C.vm)

- K , ( e - Cd - C,,m ) e - ' ' ° A " , . , j c ~ ]

318 A. Mehrez et al. /Opttmalphosphate mining

- C-~ [ foXe ..... A,¢ , ~/C, d 3,

+ . xf>'Kte . . . . oA:~ , ~/C. d y] ,

where

g 1 = e-rrn.AdX)/Cp

and

A t ( y ), A2(y ) as before.

Both problems can be solved by the regular methods of calculus and present no difficulties.

C = $0 .40 / ton /km, md/rn ~ = 60/56, r = 0.1, Cp = 9000 ton a day, )'m = 6 km, l (y) = approximately 1 km for all values of

y.

Computer analysis of these parameters indicated that the boundry solution, to dig deep throughout the entire field, is the optimal policy for the form. Such a solution would avoid unemployment prob- lems created by changes in the rate of mining.

6. Application References

The model discussed in this paper originated in a problem faced by an Israeli phosphate manufac- turing firm. This firm, which is one of the largest phosphate producers in the world, was facing a problem of optimal exploitation of one of their phosphate producing fields. Specifically, the firm had to decide on the scheduling of the work in different parts of the field and on the depth of the mining in each part to the firm's best economic advantage. The firm was in the situation where the capacity was fixed at a level below the demand but where the price was fixed.

The actual parameters for this problem were:

P = $ 4 5 / t o n , C~ = $10/ton, Cp -- $15/ton,

[ll M. Aliais, Method of appraising economic prospects of mining exploration over large lerritories: Algerian Sahara case study, Management Sci. 3 (1975) 285-347.

[2] J.S. Aronofsky and A.C. Williams, The use of linear pro- gramming and mathematical models in underground oil production, Management Sci. 8 (1962) 394-407.

[3] D.A. Babayev, Mathematical models for optimal timing of drilling in multilayer oil and gas fields. Management Sci. 21 (1975) B1361-1369.

[4] D. Bishko and W.A. Wallace, A planning model for con- struction minerals, Management Sci. 18 (1972) B502-518.

[5] M.D. Devine, A model for minimizing the cost of drilling dual completion oil wells, Management Sci. 20 (1973) 532-535.

[6] G.L. Lilien, A note on offshore oil field development problems and suggested solutions. Management Sci. 20 (1973) 536-539.

[7] J.M. Reddy. A model to schedule sales optimally blended from scarce resources. Interface 6 (1) Part 2 (1975) 97-107.

[8] Sage and White. Optimum Systems Control (Prentice-Hall, Englewood Cliffs. NJ, 1977).