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of the car rails is located to the right of the guardrail,
it is supporting the tripper front right wheels.
Figure 2: Example of tripper car from Brucutu mine – Vale.
The flowchart shown in Figure 3 provides an il-
lustrative example of a bulk mineral processing. The
system’s feed flow comes from a conveyor belt. The
tripper has the role of distributing this material along
the upper opening of the storage silo. The ore is wit-
hdraw by feeders and directed to sieves. In this stage,
the material is split according to its granulometry, one
part returning to the previous process as by-product 1
and another part proceeds further as by-product 2.
The tripper car movement is controlled by au-
tomation systems. Electric motors attached to the
equipment wheels are driven by this systems (Boyer,
2010) in such way it is possible to position the equip-
ment, defining which silo’s bin will receive the ma-
terial loaded by the conveyor belt. Three actions are
possible through the automation operation interface:
move to the right, move to the left, and halt. The ore
flow isn’t interrupted during the moving process.
Model-based predictive control (MPC) systems
are a class of controllers that define their control
action outputs based on future predictions of the pro-
cess states. These predictions are performed by dyna-
mic models that represent the behavior of the system
over time, inside a finite time frame. As an optimizer,
the MPC algorithm uses these informations to build
an optimal control output aiming a desired behavior
for the process. Only the present action of the entire
output forecast window is sent to the plant at each
iteration. In the next run cycle, the entire window
is recalculated and the new control action is formed
(García et al., 1989; Camacho and Bordons, 2007).
This first version of MPC was not able to handle
discrete variables, restricting its application to conti-
nuous process. As an evolution of the initial proposal,
the algorithm was expanded to consider discrete vari-
ables, logics and rules, in addition to the continuous
variables that were already considered in the first ver-
sion. This expanded form is known as hybrid MPC
(Bemporad and Giorgetti, 2006; Borrelli et al., 2005).
This controller is suitable to deal with the problem of
tripper positioning since this process has both conti-
nuous (level, mass flow) and discrete (position) vari-
ables. A framework based on hybrid MPC was used
by (Karelovic et al., 2015) to solve the tripper positi-
oning problem. The criteria used by the authors to set
up the controller objectives were bins level stabiliza-
tion and minimization of the car movement.
The purpose of this work is to develop a solution
to the tripper positioning problem based directly on
mixed integer linear programming. In this case, the
objective of the control system is to minimize the va-
riations of bins levels, ensuring that these variables
remain stable regarding the process constraints. Fi-
nally, a dynamic programming algorithm is proposed
aiming the reduction of the time required to find the
solution of the optimization problem.
2 DEVELOPMENT
The development of the mathematical models that
describe the behavior of the silo-tripper system is the
starting point for the study of the solutions to the pro-
blem of car positioning. The features of the optimi-
zation model, such as instance input data, decision
variables, constraints and objective function, are des-
cribed in detail. This formulation will be solved later
by means of mixed integer linear programming and
dynamic programming.
2.1 Mathematical Model
The Figure 4 presents the moving possibilities of a
n-bins silo. At any position, the equipment can only
move to one of the adjacent places or remain in the
same position over which it is located. For example,
if it’s in p
2
position then the movement options will
be {p
1
, p
2
, p
3
}. Moving the car to a position not ad-
jacent to the current one implies crossing all the in-
termediate positions between the points of departure
and destination. Consequently, tripper directs the ore
flow to the bin whose position is overlapped by cur-
rent equipment displacement. The choice of feeding
position is a task with temporal dependence since the
movement speed of the equipment is finite.
There is a risk that the material may eventually
lacks in one of the bins since tripper may only be in
a particular place at any time. This may occur even
in situations where the mass balance of the system
is balanced, that is, the total mass flow fed is equal to
the total withdrawn. To work around this problem, the
tripper must be moved along the bins openings so that
Proposed Solutions to the Tripper Car Positioning Problem
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