programacion lineal n 1.pdf
TRANSCRIPT
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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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