Optimal control approaches for open pit planning

Abstract

This work tackles the open pit planning problem in an optimal control framework. We study the optimality conditions for the so-called continuous formulation using Pontryagin’s Maximum Principle, and introduce a new, semi-continuous formulation that can handle the optimization of a two-dimensional mine profile. Numerical simulations are provided for several test cases, including global optimization for the one-dimensional final open pit, and first results for the two-dimensional sequential open pit. Theses indicate a good consistency between the different approaches, and with the theoretical optimality conditions.

Appendices

Appendix A: Implementation details for the semi-continuous approach

Time discretization The sequential open pit for the semi-continuous approach described in 2.2 is a multi phase problem. Instead of duplicating the variables for each time-frame, we use here in practice a more compact implementation, by using a time step $\Delta t$ of 1 time-frame, i.e. the time discretization $t_k = 0 \ldots T$ is the sequence of time-frames. This choice makes sense from the operational point of view, since the sequential open pit planning precisely consists in determining the optimal mine profile at each time-frame. It also simplifies a lot the computation of the integrals of the gain and effort functions between two successive mine profiles. We choose an implicit Euler scheme for the time discretization, which gives the trivial discrete dynamics

$$\begin{aligned} P_i^{k+1} = P_i^k + U_i^{k+1} \end{aligned}$$

(20)

that easily gives the next / previous mine profile when needed in the computations.

Gain An additional state variable g is added to represent the gain realized along the time-frames, whose dynamics can be written as

$$\begin{aligned} \dot{g}(t_k) = \frac{1}{(1+\alpha )^{k-1}} \int _{P^{k-1}}^{P^k} G(x,z) dxdz, \quad \forall k=1,\ldots ,T \end{aligned}$$

(21)

The objective is then to maximize g(T). For the 1D case, we approximate the 2-dimensional integral of G by trapezoidal rule over x then along z. In the 2D profile case, the 3D integral of G for the computation of the gain is approximated using a 2D trapezoidal rule along (x, y) then a standard trapezoidal rule along z.

Capacity At each time-frame, the integral of the excavation effort over the domain $\Omega$ can be approximated by

$$\begin{aligned} \int _{P^{k-1}}^{P^k} E(x,z) dxdz \approx \sum _{i=0}^{N} \Delta x \left( \int _{P_i^{k-1}}^{P_i^k} E(x_i,z) dz \right) \end{aligned}$$

(22)

Since $E=1$ and from the discrete dynamics $P_i^{k} = P_i^{k-1} + U_i^{k}$, we can use the following formula

$$\begin{aligned} \int _{P^{k-1}}^{P^k} E(x,z) dxdz \approx \Delta x \sum _{i=0}^{N-1} U_i^k. \end{aligned}$$

(23)

Similarly, for the 2D profile case, the excavation effort at time-frame k is approximated as

$$\begin{aligned} \int _{P^{k-1}}^{P^k} E(x,y,z) dxdydz \approx \Delta x \Delta y \sum _{i=0}^{N-1} \sum _{j=0}^{M-1} U_{i,j}^k. \end{aligned}$$

(24)

B: Additional examples for the final open pit: continuous formulation

1.1 B.1: FOP with infinite capacity and constant slope

Fig. 12

1D profile with infinite capacity—global optimization (HJB method)

We show here the basic example with unconstrained capacity, namely $C_{max} = \infty$. Figure 12 shows the solution obtained by the global method, and Fig. 13 shows the solution from the local method, and we observe that both solutions match. With infinite capacity, the solution, as expected, digs as much as possible with respect to the maximal slope, until it reaches negative gain. This corresponds to the observed Bang-Singular-Bang control structure (neglecting the two very small constrained arcs $P=P_0=0$ at the extremities). As stated in Lemma 2, the singular arc in the middle follows the geodesic $G=0$. The corresponding control also matches the theoretical expression of the singular control (14), despite some oscillations at the junctions with the bang arcs.

Fig. 13

1D profile with infinite capacity—local optimization (direct method) and consistency with PMP optimality condition