programacion lineal N 1.pdf

8/11/2019 programacion lineal N 1.pdf

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IMPLEMENT TION OF LINE R PROGR MMING MODEL

FOR OPTIMUM OPEN PIT PRODUCTION SCHEDULING PROBLEM

By

Edgar Urbaez

and Kadri

Dagdelen

Colorado School of Mines

Golden, Colorado 80401

ABSTRACT

Production scheduling

is one

of

the most important factors affecting mine

planning. A program -

Mine Scheduler

- that aids mining engineers with the

setup of

an

optimum open pit production scheduling algorithm has been

developed. A mathematical model for open pit scheduling that optimizes the Net

Present Value of the cash flows was formulated. A simulated data is formulated

as

a Mixed Integer Linear Programming MILP) problem. Wha.t s

Best

by Lindo

Systems is used as the MILP solver in

Microsoft Excel

The implementation of

the algorithm focuses on problem setup, allowing users to easily evaluate,

simplify, and develop the model that best fits their production scheduling.

INTRODUCTION

The objective of production scheduling is to try to best answer the

questions of whether a portion of a deposit should be mined or not, and if it is to

be mined when it should be mined and how it should be processed. Many

approaches have been taken in this area. However, due to the complexity of the

problem, in which sequencing of the pushbacks is one of the critical issues, it has

been difficult to come up with an algorithm that can truly produce an optimum

production schedule. Also as complex as the algorithm is the manipulation of

the data. Mines nowadays are getting larger. The number of metallurgical

processes involved in a single operation has been increasing lately. The

optimum open pit production scheduling algorithm needs predefined pushbacks

before it can be used. As a consequence, in order to obtain the best results

possible from its implementation, the right open pit pushback design algorithm

has to be used. Due to the formulation of the algorithm, optimum cutoff grade

per period of time is achieved and different costs and prices

can

be included for

the life of a project.


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PREVIUOS

WORK

Different methods to solve the production scheduling problem have been

studied. Asarco's needs for a production scheduler (Williams, 1974) resulted

in

the contracting of Systems Control, Inc. SCI) to develop a computer program

that would solve the production scheduling problem one period

at

a time. Then,

through dynamic programming,

the

periods were combined to create the final

production schedule. Later, a linear programming formulation by Gershon

(Gershon, 1982) introduced the concept

of

long/short term interfaces. These

interfaces consisted

of

coming

up

with a short term schedule while keeping an

optimal long term plan. A new algorithm, based on the lagragian concepts of

mathematics, was then developed by Dagdelen and Johnson (Dagdelen and

Johnson, 1986) as an attempt to create a production schedule that was optimum.

The concepts

of

sequencing and multiple periods scheduling were successfully

introduced in this approach.

Since the mathematical programming approaches developed to address

the scheduling problem were difficult to implement and understand, Gershon's

heuristic approach seemed

to

yield a nearly optimal solution within a reasonable

period of time by utilizing the concept of a ranked positional weight

to

determine

whether a block should be

mined

or not (Gershon, 1987). This approach lacked

rigorous mathematical proof of

the

optimality of the solutions. A dynamic

programming algorithm was then formulated by Seymour (Seymour, 1994) that

finds the mining and cutoff grade sequence that maximizes the Net Present

Value. Since the algorithm tried to

find

a solution by exhaustive searching

techniques, it was prone to combinatorial explosion. Given that price

of

metal is

not constant, Wang and Sevim (Wang

and

Sevim, 1995) utilized Gershon's

downward cone concept to derive a method that is not a function of price but of

maximum-metal content as an alternative to parametrization.

In

more recent works, techniques

such

as

Gershon's heuristic, parametric

analysis, Wang and Sevim's heuristic, MILP, and exhaustive search via dynamic

programming, among others,

have been

developed. The

main

problem facing

each of these techniques is to come up with an ultimate pit limit and the

production schedule required to achieve it in one formulation; not one

independent

of

the other.

Also, the

time it takes to solve the problem is critical

for a given technique to be useful and effective for the mining engineers doing

the mine planning.

PROBLEM FORMULATION

Definition of the Value Coefficient

The Value Coefficient determines the economic value of a material type in

a given increment

and

period of time. It is used by the algorithm to find the most

profitable solution based on

Net

Present

Value.

It is expressed

in

/ton

and

calculated by using the following formula:


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1

0

k.o * k o * k o k.o *

Ci,j,l,m ( /ton) =

Hgi j / rj,j'/

P- S ] =

l7l

j

,j,l,m

- Pi.j.I,m} l+d)m

Where.

c t t l m

is the discounted value of material coming from mine

k.

sequence '. material type o. increment i. to be processed by j in

time period m

As

such the superscripts

k,o and

subscripts

;,j,

used

in

the variables refer

to material type 0

in

mine k from sequence '. increment ; to be processed by j in

a given time period m. The other variables are:

k.o

gj,j,1

rk,o

i,j,1

p

S

k.o .

l7l

i

,j,I,m

k o

Pi,j /.m

d

Grade (units/ton) associated with material described

by the superscripts k.

0

and subscripts i.

j, ,

Recovery ( ) for the process j

Commodity Price ( /units)

Marketing cost ( /units)

Mining cost

( Iton)

Processing cost ( Iton)

Discount rate or accepted rate of return that could be

realized

on

similar alternative investments of

equivalent risk (Stermole and Stermole .1993)

Mathematical formulation of scheduling problem

The scheduling problem is formulated

as

a mixed integer programming

problem by Dagdelen (Dagdelen. 1996) and is presented

as

follows:

The objective function is to maximize the Net Present Value by varying the

flow of tonnages from source to destination over multiple time periods. The

mathematical formulation attempts to solve the different source-destination

combinations found

in

real life problems as represented in Figure 1. The

mathematical formulation is:

K I

J

L M N 0

Max Z -

( C ~ ~

t k ~

+ z ~ , o s t ~ o A ~ . i t ~ . )

- .l-.i J..i ,J,I,m ,j.I ,m ,I,m,n ,I,m,n I,J,m,n ,j,m,n

=1

1=1 j=1 =1 m=1

n=1

0=1

Let define the individual summations as:

K,l,J

,L,M ,N ,0

Max Z =

( C ~ ~ t ~ ~ + z ~ , o s t ~ o

+ i t ~ .

)

. .

I,j,/,m 1t},I,m

',[,m,n

,,/,m,n

I,},m,n l.j,m,n

k,I,j,

,nt,n,0=1


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Where,

... ,,

~ P l t 1 ) ~

~ ~ ~ ~ Nil


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Let simplify the summations from 1 to K,I,J,L,M,N,O as:

K f J .f-.M .N.O

k i , j , ~ n O _ l

=

k . i , j ~ . n , o

Therefore, the objective function can be written

as:

MaxZ=

~ C

k

?

t k , ~ Zk.o st

k

o

+Ao it? )

.

,j.I,m l,j.I,m

1.I,m.n 1.I.m.n I.j .m.n I.j,m,n

k ... m,n,o

Tonnage from mine

k

and material type

0

in increment

i

from a sequence

I

can go to any destination

j

in any given time period

as

shown in Figure

2.

This

tonnage is represented by

the variable

tU I ,m

The objective function is subject to

the following constraints:

~ E Q l

I1l a

t

'

IIU I

I I t t , Q

Figure 2 Variable Definition

1.

Deposit reserves: ensures that material mined is what

is

available in the

geological reserves. Otherwise, the algorithm could allocate a tonnage of

material to a given value coefficient that exceeds what physically can be

found in the actual deposit, so that it can increase the resulting Net

Present Value. T i ~ o is the tonnage available for material 0

in

mine k from

sequence increment i. For k =

1 K; 0

=

1 ;

i =

1

I;

I

= 1 L;

n =

1

N.

~ (t

k

.

o

+ stk O ) ; Tk,o

i,j,l,m i,l,m,n i,l

j,m

2.

Source mine production capacity: sets the mining rate.

In

other words, it

defines the minimum and maximum tonnage that

can

be mined from mine

k

in time period

m.

For k =1

K;

m =1

M.


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k ko ko

M

k

M mmm ~ fi.J /.m

+

sti.i.m.n max

m

'.J n.o

3. Destination processing capacity: sets the processing rate per destination.

It

defines the minimum and maximum tonnage that can be sent to

destination j in time period m. For

j

=1, J; m =1,

M.

C min j.m

l tU,.m +

i t i ~ j . m . n ) C max

j.m

k ,fJ.:..o

4.

Commodity unit production by destination and time period: sets the

commodity production target

in

measured units, such as ounces and

pounds among others.

It

defines the minimum and maximum commodity

production units that can be sent to process j in time period m. G is

recovered grade by k,o,i j,l. SG is assumed recovered grade for each

stockpile. For

j

=

1,

J;

m =

1, M.

P

min

i.m l

[ tU, *

CU,) + i t i ~ i . n

*

sctj .n)] P

max

j.m

k .fJ.:..o

5.

Destination average percentage minimum two sets): sets the lower head

grade limit per destination. It defines,

in

percentage, the minimum head

grade requirement to be processed by j

in

time period m. For j

= 1,

J; m =

1, M.

[

tk .o

*

G

k

o

_

G .

0

. 0

*

SG

k

.

o _

G .

0

]

...

0

i.j.l,m i,1 l ln

j m

+ lti,j,m,n n l ln

j m c:.

k.,

n,o

6.

Destination average percentage maximum two sets): sets the upper head

grade limit per destination. It defines, in percentage, the maximum head

grade requirement to be processed by

j in

time period

m.

For

j

= 1, J; m =

1, M.

1

[tUI m

* G i ~ i o - Gmaxj,m) +

i t ~ j / 1 l , n

* ( S G ~ o - Gmaxj,m)] 0

k /;: ; ,0

7.

Stockpile inventory: determines the availability of material in the stockpile.

It defines the tonnage available that can be sent to any destination from

the stockpile by keeping track of the current stockpiled material and the

new stockpiled material in a given time period. It::: n is the stockpile

inventory consisting of material 0 from mine k increment i,

in

time period

m in

stockpile n. For k

=1, K; 0 =

1 0; i

=1,

I; m

=1,

M; n

=1, N.


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/k .o = / k.o

+

l s t ~ o

_

itO )

.m.n

,.

m I . n ,.I.m.n I.j.m.n

j

8. Integer check: ensures that only one integer has a value of

one.

If

an

integer has a value of one, a sequence

can

start mining. If a sequence

can start mining, any previous sequence

can

be partially mined

or

completely mined

out. Y

m

is

one if cumulative tons from mine

k

sequence

I are depleted in period m or before. Y/

m

is zero otherwise. For k = 1,

K;

I =

1,

L

9.

Integer conditioning: activates the mining of a sequence in a given period

of time. T is the total tonnage in sequence Ifrom mine k. For k = 1,

K;

1=

1, L-1; mm

= 1, M.

M M

E E

tUI.mm - [

E Y/

mm

) *

T ]

~

0

mm l

l . j O mm I

10. Sequence enforcing: ensures that a sequence has to be completely mined

out before mining material from a third sequence. This constraint allows

for partial mining, which means that two sequences

can

be mined at the

same time, but a third sequence can not start mining until the first

sequence is all mined out. For k

=

1, K; I

= 1,

L; m

= 1,

M.

k.o (Y

k *

T

k ) 0

/ J ti.j,/+I.m

-

I.m 1+1::::

. j.O

SIMULATED MODEL

Setting up the scheduling problem

The simulated data

is

designed

to

convey the output information in a way

that is easy to access by the

user.

The assumptions are realistic

and

the data

represents a real life gold deposit. The simulated model consists of 1 source, 3

processes

1

mill, 1 heap pile, 1 dump site , 2 material types, 3 pushbacks, 5

material increments, 3 time periods

and

1 stockpile resulting in 370 numeric

variables as shown in Figure 3.

Cost parameters are setup in Microsoft Excel by Mine Scheduler as

shown in Figure 4. The light blue cells represent the input fields. As can be

seen, cost can incrementally increase as the pit gets deeper. Also, none of the


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parameters has to

be

assumed constant throughout the mine life. Net Present

Value formula is included

in

this model. Discounting starts at the

end

of a year.

Costs for stockpiling and rehandling are also included.

ECONOMIC PARAMETERS

Discount Rate

Gold Price ( /oz)

Sales Cost

( /oz)

Stockpile Cost

( /ton)

Rehandle Cost

( /ton)

Process Cost :

Mill - Oxide

( /ton)

- Sulfide( /ton)

Leaching - Oxide

( /ton)

' ~ ~ ~ i I ~ & ~ ~

- Sulfide( /ton)

14

Mining Cost :

PushBack 1

( Iton)

Push Back 2

( /ton)

PushBack 3

( /ton)

Waste Cost

Figure 4 Cost parameters table

This model contains 2 different material types: oxide and refractory. Figure

5 shows the reserve table for the refractory

and

oxide material types. Both

material types

can

be defined in terms of grade increments as well as tonnage

and average grade within these increments. Refractory material

can

be further

defined in terms of sulfur/sulfide (SS) and carbon/carbonate (CC) content. The

average grade is determined by calculating the weighted average of the grade

increment.

Constraints are setup for the different destinations

as

shown in Figure 6.

Large numbers are inputted as maximum limits for Stockpile

and

Waste Dump to

simulate unlimited capacities. Stockpile constraints are removed from the last

year since the objective is to handle and process

all

the material available by the

end of the mine life. There is a minimum annual production of 20,000,000 tons

from the mine.

Also

the mill has to have an annual supply of at least 7,000,000

tons with a minimum average grade of 0.04 gold ounces per ton.


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P

Figure 5 Resource table

Mine Tons max

Tons min

Mill Tons max

Leaching

Waste Dump

Stockpile

Tons min

Grade max

Grade min

SS

content m a x l l ~ ~ 2 2 ~ ~ t J I I L ~ ~

Tons max

Tons min

Grade max

Grade min

SS

content max

H + + ' ~ ~ + + + ~ H . ~ ~ R

SS

content min

content

1 t i ; : ; j c t j : : t ~ ~ t : i t 7 ~ ~

Tons min

Tons max

Tons min

Figure 6 Constraint table

Recovery tables are created for each material type Figure 7 shows the

recovery tables for both material types Recovery values are inputted for each

process These values do not need to be constant They can vary by pushback

and also by grade increment within a pushback This approach results

in

a more

powerful and flexible tool as more detailed information coming from a grade-

recovery curve can be inputted


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OXI E

PushBack 3

Figure 7 Recovery table

OXIDE

1998

PushBack 1

PushBack 2 - - : ~ ~ _ ~ ~ _ . . . . . . . . : ~

Figure 8 Year 1998 input solution table

Variables are created to represent the tonnage that is mined from a given

year material type pushback and material increment. Figure 8 shows a solution

table for the year 1998. A similar table is created for each period of time and for

each material type. The resulting output has the same table format as the input


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Figure 10 Output solution tables

Figure 10 also shows that no refractory material is mined in 1998 thus the

total tonnage mined from pushback 1 is 15 millions. In 1999 another 15 millions

tons are mined from pushback 1 but 446 058 tons of refractory material are sent

to waste. Since pushback 1 has not been depleted by 1999 mining of pushback


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2 and pushback 3

is

not possible. Furthermore, 3,578,120 tons of oxide material

are sent to the stockpile

in

1999.

CONCLUSION

A mathematical model for open pit scheduling problem that optimizes the

Net Present Value of the cash flows has been formulated as a Mixed Integer

Linear Programming problem. This work focuses on the implementation of the

open pit scheduling problem formulation. A program,

Mine Scheduler

is

developed to setup the studied formulation. Mine Scheduler provides a fast and

reliable way to setup open pit production scheduling problems. It eliminates the

risks of accidentally typing the wrong formula in a cell. Its ability to relatively

quickly setup

an

open pit production scheduling algorithm makes for a powerful

tool

as

more scenarios can be evaluated per period of time compared to

traditional methods. Also, its

Microsoft Windows

interface makes it extremely

easy to use requiring no special training. Just a quick tutorial, taking place in few

minutes, is enough to start using

Mine Scheduler.

Some of the advantages of using

Mine Scheduler

are the dynamic

determination of cutoff grades per destination and per time period, the non

constant yearly price and cost input permitting forecasting, and its flexibility of

use of the different constraint sets. Another advantage is the manipulation of the

final results, which can

be

exported in any text format and read by a variety of

text editors. Results

can

also be expressed in 2-D and 3-D graphics by using the

existing tools in Microsoft Excel

REFERENCES

Dagdelen,

K.

1985 Optimum Multi Period Open Pit Mine Production

Scheduling , Ph.D. Dissertation, Colorado School of Mines, Golden, Colorado

Dagdelen, K. and Johnson, T. 1986, Optimum Open Pit Mine Production

Scheduling by Lagrangian Parametrization , 19

T

APCOM SYMPOSIUM, pp.

127-142

Dagdelen, K. 1996, Formulation of Open Pit Scheduling Problem Including

Sequencing

as

MILP , Internal Report, Mining Engineering Department, Colorado

School of Mines, Golden, Colorado

Davis,

R

and Williams,

C.

1972, Optimization Procedures for Open Pit Mine

Scheduling , pp. C1-C18

Gershon, M. 1982, A Linear Programming Approach to Mine Scheduling

Optimization ,

17TH

APCOM SYMPOSIUM, pp. 483-499


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