reclaimer schedules.

Graphical representation of a feasible schedule
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The schedule $H_k$ of reclaimer $R_k$ ($k=0,1$) with makespan $C_k$ can be described by a piecewise linear function representing the position of the reclaimer on the rail as a function of time. Such a function can be represented by an ordered list of breakpoints $$B_k=\left(\left(t_i^{(k)},x^{(k)}_i\right)\in {\mathbb{R}}^{+}\times [0,L]\ :\ i=0,1,\ldots,q_k \right)$$ in the time-space Cartesian plane, where $0=t_0^{(k)}<t_1^{(k)}<\cdots<t_{q_k}^{(k)}=C_k$, $x_0^{(k)}=x_{q_k}^{(k)}=kL$. For $t\in[t_i^{(k)},t^{(k)}_{i+1}]$ we have $$H_k(t)=x_i^{(k)}+\frac{x^{(k)}_{i+1}-x^{(k)}_{i}}{t^{(k)}_{i+1}-t^{(k)}_{i}}\left({t-t^{(k)}_{i}}\right),$$ and the slope between consecutive points $(t^{(k)}_i,x^{(k)}_i)$ and $(t^{(k)}_{i+1},x^{(k)}_{i+1})$ is either:

-   $0$, the reclaimer is idle;

-   $+s$, the reclaimer is moving to the right without processing any stockpile;

-   $-s$, the reclaimer is moving to left without processing any stockpile;

-   $+1$, the reclaimer is moving to right while processing a stockpile on either one of the two pads; and

-   $-1$, the reclaimer is moving to left while processing a stockpile on either one of the two pads.

This is illustrated in Figure \[fig:example\].

(0.0,0.0) – (15.0,0.0) node\[anchor =west\] [[time]{}]{} ; (0.0,0.0) – (0.0,13.0) node\[anchor = south\] [[space]{}]{} ; in [0,...,6]{} [(-.1,2\*)–(.1,2\*) node\[left, xshift = -0.02\] [[$\y$]{}]{};]{} in [0,...,7]{} [(2\*, 0.1) – (2\*, -0.1) node\[below, yshift = -0.02\] [[$\x$]{}]{};; ]{} in [0,...,12]{} (0.0,) – (15,) 