Mixed-integer linear methods for layout-optimization of screening systems in recovered paper production

Abstract

The industrial treatment of waste paper in order to regain valuable fibers from which recovered paper can be produced, involves several steps of preparation. One important step is the separation of stickies that are normally attached to the paper. If not properly separated, remaining stickies reduce the quality of the recovered paper or even disrupt the production process. For the mechanical separation process of fibers from stickies a separator screen is used. This machine has one input feed and two output streams, called the accept and the reject. In the accept the fibers are concentrated, whereas the reject has a higher concentration of stickies. The machine can be controlled by setting its reject rate. But even when the reject rate is set properly, after just a single screening step, the accept still has too many stickies, or the reject too many fibers. To get a better separation, several separators have to be assembled into a network. From a mathematical point of view this problem can be seen as a multi-commodity network flow design problem with a nonlinear, controllable distribution function at each node. We present a nonlinear mixed-integer programming model for the simultaneous selection of the network’s topology and the optimal setting of each separator. Numerical results are obtained via different types of linearization of the nonlinearities and the use of mixed-integer linear solvers, and compared with state-of-the-art global optimization software.

Appendices

Appendix 1: Linear approximation

Before any nonlinear function can be used in a MILP formulation, it has to be approximated piecewise linearly. The number of binary variables involved in the resulting MILP directly corresponds to the number of intervals, or triangles, respectively, used for the approximation. Therefore, we aim to find a sufficiently good approximation with a small number of intervals (or triangles).

1.1 Linearization of univariate functions

Several different approaches concerning approximation by piecewise linear functions in one dimension are discussed in the literature, e.g., finding the best approximation for a given number of fixed nodes (de Boor 2001; Stone 1961) or free nodes (de Boor 2001; Cantoni 1971; Gavrilovic 1975; Stone 1961; Vandewalle 1975), or finding an approximation with the minimum number of line segments necessary, such that the error is smaller than a specified tolerance (Cameron 1966; Hamman and Chen 1994; IBM ILOG CPLEX 1986; Manis et al. 1977; Tomek 1974).

A piecewise linear function that coincides with a continuous function ${g : [a,b] \rightarrow \mathbb{R}}$ in its breakpoints is called its interpolant and denoted by I _g . Note that the absolute approximation error

$$ \max_{x\in[a,b]} \vert I_g(x)-g(x) \vert $$

(41)

can at best be halved by going over from the interpolation to the best possible approximation by broken lines (de Boor 2001). Halving the error is especially possible for convex and concave functions. In these cases the piecewise linear interpolant completely lies above (below) g, cf. Fig. 7. Therefore, the error can be reduced by moving the interpolant downwards (upwards).

Fig. 7

Shifting an interpolant to a convex function downwards by $\epsilon /2$

Let g be convex or concave and

$$ \varepsilon := \max_{x\in [a,b]}\vert g - I_g\vert. $$

(42)

Then

$$ \max_{x\in [a,b]}\vert g- \left(I_g \mp \frac{\varepsilon}{2}\right) \vert \leq \frac{\varepsilon}{2}. $$

(43)

For finding a good interpolant (de Boor 2001) gives a useful formula for node placement in the following theorem.

Theorem 7.1 (de Boor 2001, p. 36) Let $g\in C^{2}(a,b)$ and $\vert g''\vert$ be monotone near a and b, and $\int_{a}^{b}\vert g''(x)^{1/2} \vert dx < \infty.$ Then , if $a_{2},\ldots,a_{n-1}$ are chosen such that

$$ \int_{a}^{a_{i}}\vert g''(x)\vert^{1/2} dx = \frac{i-1}{n-1} \int_{a}^{b}\vert g''(x)\vert^{1/2} dx, \quad \hbox{for}\; i = 2, \ldots, n-1, $$

(44)

then

$$ \Vert g-I_{g}\Vert = O(n^{-2}), $$

where $\Vert f\Vert := \max_{x\in[a,b]} \vert f\vert$ denotes the uniform norm of $f \in C[a,b].$

Amongst others, (Cameron 1966; Manis et al. 1977) treat the task of finding a broken line that approximates a given function within a given error tolerance with as few line segments as possible. Their algorithms define a tunnel of radius $\varepsilon$ around the original curve (or around sample points of the curve) and find the farthest point visible through this tunnel. The solution received by the algorithms of Cameron (1966) and Manis et al. (1977) is optimal with respect to the number of line segment. But there is no guarantee that the solution yielding the minimal error is chosen among those solutions with minimal number of line segments. In contrast, generally, the constructed solution is not optimal with respect to the error.

We implemented an algorithm based on the ideas of Cameron (1966) in Schönberger (2007). Examples in Schönberger (2007) show that for approximating the logarithm, de Boor’s formula (44) combined with shifting the piecewise linear function upwards by half of the interpolation error results in the same number of intervals needed for a given error tolerance as by using the above techniques. Since the first approach is less time consuming, we used this method to obtain the piecewise linear functions for our further computational results.

1.2 Linearization of bivariate functions

For determining an approximation of the bivariate function given by the application, we used an algorithm developed by (Gürth 2007). Given an error tolerance it determines an L ₂-approximation, or an interpolation, respectively, of functions in two variables over triangulations using $\sqrt{3}$-subdivision (Gürth 2007) as refinement algorithm for triangulations. For calculating the L ₂-approximation hat functions and a discretized scalar product induced by a weighted Sobolev norm are used. A basic implementation of the algorithm in Matlab was made available to us by Gürth (2007). Computational experiments in Schönberger (2007) suggest that this approach leads to less triangles than for example the use of Delaunay triangulations always refining at the point with the currently largest error, when taken into account the absolute error. The algorithm was adapted to fit to our error definition.

Appendix 2: Modeling piecewise linear functions

Consider a continuous piecewise linear function ${h:D \subseteq \mathbb{R}^n \rightarrow \mathbb{R}: x \mapsto h(x)},$ defined on a triangulation $\mathcal{T}$ of D, where the vertices of the triangulation correspond to the breakpoints of the piecewise linear function (see Fig. 8). Let $V(\mathcal{T})$ denote the set of vertices of $\mathcal{T}$, and accordingly V(T) the set of vertices of a simplex $T\in \mathcal{T}$.

Fig. 8

Piecewise linear function h(x)

2.1 Disaggregated convex combination models

2.1.1 Basic model

$$ \begin{aligned} \sum_{T\in {\mathcal{T}}} \sum_{v\in V(T)} \lambda_{T,v} v = x, & \sum_{T\in {\mathcal{T}}} \sum_{v\in V(T)} \lambda_{T,v} h(v) = h(x) \\ & \lambda_{T,v} \geq 0, \; \hbox{for} \; T\in{\mathcal{T}}, v \in V(T)\\ \sum_{v\in V(T)} \lambda_{T,v} = y_T,\quad \sum_{T\in {\mathcal{T}}} y_T = 1, &\quad y_T \in \{0,1\}, \; \hbox{for}\; T \in {\mathcal{T}}. \end{aligned} $$

We refer to this model as disaggregated convex combination model and denote it by DCC. Inter alia, it has been studied by Lowe (1984).

2.1.2 Logarithmic model

The idea is to reduce the number of binary variables and constraints by introducing a unique binary code for every simplex and just binary variables for every digit in this binary code.

$$ \begin{aligned} \begin{array}{ll} \mathop\sum\limits_{T\in {\mathcal{T}}} \mathop\sum\limits_{v\in V(T)} \lambda_{T,v} v = x, & \mathop\sum\limits_{T\in {\mathcal{T}}} \mathop\sum\limits_{v\in V(T)} \lambda_{T,v} h(v) = h(x),\\ \mathop\sum\limits_{T\in {\mathcal{T}}} \mathop\sum\limits_{v\in V(T)} \lambda_{T,v} = 1, & \lambda_{T,v} \geq 0,\; \hbox{for} \; T\in{\mathcal{T}}, v \in V(T),\\ \mathop\sum\limits_{T\in {\mathcal{T}}^0(B,l)} \mathop\sum\limits_{v\in V(T)} \lambda_{T,v} \leq y_l, & \mathop\sum\limits_{T\in {\mathcal{T}}^+(B,l)} \mathop\sum\limits_{v\in V(T)} \lambda_{T,v} \leq 1-y_l,\\ & y_l \in \{0,1\}, \; \hbox{for}\; l \in L({\mathcal{T}}), \end{array} \end{aligned} $$

where $B:\mathcal{T}\rightarrow \{0,1\}^{\lceil \log_2\vert \mathcal{T}\vert \rceil}$ is any injective function, $\mathcal{T}^0(B,l):=\{ T\in \mathcal{T}: B(T)_l = 0 \},\;\mathcal{T}^+(B,l):=\{ T\in \mathcal{T}: B(T)_l = 1 \}$ and $L(\mathcal{T}) = \{ 1,\ldots, \log_2(\lceil \vert \mathcal{T} \vert \rceil)\}.$ We refer to this model as logarithmic disaggregated convex combination model and denote it by Dlog. This formulation has been introduced by Ahmed et al. (2009) and Nemhauser and Vielma (2008).

2.2 Convex combination models

Again a point (x, h(x)) is represented as convex combination of its neighbored grid-points. In contrast to above the number of continuous variables is reduced by aggregating the variables associated with the same vertex (belonging to more than one simplex) in the triangulation.

2.2.1 Basic model

$$ \begin{aligned} \begin{array}{ll} \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_v v = x, & \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_v h(v) =h(x),\\ \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_v = 1, \; \lambda_v \geq 0, & \lambda_v \leq \mathop\sum\limits_{T: v\in V(T)} y_T, \; \hbox{for}\; v\in V({\mathcal{T}}),\\ \mathop\sum\limits_{T\in {\mathcal{T}}} y_T = 1, & y_T \in \{0,1\}, \; \hbox{for} \; T \in {\mathcal{T}}. \end{array} \end{aligned} $$

This approach is known as the convex combination or lambda method (Dantzig 1963; Keha et al. 2004; Lee and Wilson 2001; Lowe 1984; Moritz 2006; Nemhauser and Wolsey 1988; Padberg 2000; Wilson 1998). We refer to it as convex combination model and denote it by CC.

2.2.2 Logarithmic model

Similar to Dlog, the number of binary variables of CC can be reduced by identifying each simplex with a binary code through an injective function $B: \mathcal{T}\rightarrow \{ 0,1 \}^{\lceil \log_2(\vert \mathcal{ T }\vert) \rceil}.$ But this time B has to own special properties in order to ensure the adjacency condition, i.e., the positive λ _v ’s have to correspond to the vertices of the same simplex. Generally speaking, a binary branching scheme complying with the adjacency condition is a family of dichotomies $\{ L_s, R_s \}_s\in S$ with S finite and $L_s, R_s \subset V(\mathcal{T})$ such that for every $T\in \mathcal{T}$ we have $V(T)=\bigcap_{s\in S}(V(\mathcal{T})\setminus A_s),$ where A _s  = L _s or A _s  = R _s for each $s\in S$ (Ahmed et al. 2009). Given such a branching scheme, the piecewise linear function may be modeled as follows.

$$ \begin{aligned} \begin{array}{ll} \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_{v} v = x, & \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_{v} h(v) =h(x),\\ \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_{v} = 1, & \lambda_{v} \geq 0, \ \rm{ for}\; {v \in V({\mathcal{T}}}),\\ \mathop\sum\limits_{v\in L_s} \lambda_{v} \leq y_s, & \mathop\sum\limits_{v\in R_s} \lambda_{v} \leq 1-y_s, \\ & y_s \in \{0,1\},\;\hbox{for}\; s \in S. \end{array} \end{aligned} $$

It is possible to construct such a branching scheme with a logarithmic number of dichotomies for every so-called Union-Jack-triangulation in ${\mathbb{R}^2}$ (Ahmed et al. 2009). For a general triangulation this is not always possible.

For a triangulation in ${\mathbb{R}}$, however, such a branching scheme inducing a logarithmic number of variables can always be constructed using the so-called grey code, i.e., an injective function $B:\{1,\ldots,r\}\rightarrow \{0,1\}^{\lceil \log_2(r) \rceil}$, where $r = \vert V(\mathcal{T})\vert$ is the number of vertices, such that for all $i\in\{2,\ldots,r\}\;B(i-1)$ and B(i) differ in exactly one digit (Nemhauser and Vielma 2008). Then the dichotomies are given by

$$ L_s = \{i\in \{2,\ldots,r\}: B(i)_s = B(i-1)_s = 0\}, $$

$$ R_s = \{1\}\cup\{i\in \{2,\ldots,r\}: B(i)_s = B(i-1)_s = 1\}. $$

We refer to this formulation as logarithmic convex combination model and denote it by Log.

SOS-approach

$$ \begin{aligned} &{\begin{array}{ll} \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_{v} v = x,& \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_{v} h(v) =h(x),\\ \mathop\sum\limits_{v\in V({\mathcal{T}})} \lambda_{v} = 1,& \lambda_{v} \geq 0, \ \rm{ for}\; {v \in V({\mathcal{T}}})\\ \end{array}}\\ &\lambda \; \hbox{satisfies the adjacency condition}. \end{aligned} $$

Instead of introducing binary variables to ensure the adjacency condition, the constraint may also be indirectly implemented by integrating it in a branch-and-bound framework (Moritz 2006). For a triangulation in ${\mathbb{R}}$ the adjacency condition complies with the the so-called Special Ordered Set of Type 2 Condition (SOS2 condition), i.e., at most two of the variables corresponding to the set of vertices are positive and if they are positive they are neighbored. The integration into a branch-and-bound framework may also be possible for higher dimensions, but one has to define its own branching scheme relying on the triangulation (Moritz 2006).

For the one-dimensional case, we refer to this approach as SOS2-approach and denote it by SOS2.

2.3 Incremental model

This formulation requires the vertices of the triangulation to be ordered in a special way (Wilson 1998).

• T _i ∩ T _i-1 ≠ ∅, for $i = 2,\ldots, \vert \mathcal{T} \vert,$

• for each simplex T _i , we can label its k _i vertices as $v_i^0, v_i^1, \ldots, v_i^{k_i}$ so that $v_{i-1}^{k_{i-1}} = v_i^0$, for $i = 2,\ldots, \vert \mathcal{T} \vert$.

Notice that for univariate functions this assumption is always fulfilled by the natural order in ${\mathbb{R}}$.

Wilson (1998) shows, that the ordering assumption holds for any triangulation of a domain D that is a topological disk in ${\mathbb{R}^2}.$ (Bartholdi and Goldsman 2001) give an algorithm to compute such an order fast. (Geißler et al. 2010) pose an algorithm to order triangulations in higher dimensions.

Given a triangulation ordered in this way, we may describe a point x by filling-up all simplices prior to T _i , where $x\in T_i,$ and then presenting x as v ⁰ _i plus a conical combination of the rays v ^j _i  − v ⁰ _i , with $j=1,\ldots, k_i,$ compare Fig. 9.

$$ \begin{aligned} \begin{array}{ll} v_1^0 + \sum\limits_{i=1}^{\vert {\mathcal{T}} \vert} \sum\limits_{j=1}^{k_i} \delta_i^j (v_i^j-v_i^0) = x, & h(v_1^0) + \sum\limits_{i=1}^{\vert {\mathcal{T}} \vert} \sum\limits_{j=1}^{k_i} \delta_i^j (h(v_i^j)-h(v_i^0))=h(x),\\ \sum\limits_{j=1}^{k_i} \delta_i^j \leq 1, \delta_i^j \geq 0 & \hbox{for}\; i = 1, \ldots, \vert {\mathcal{T}} \vert \; \hbox{and} \; j = 1,\dots, k_i,\\ w_i \leq \delta_i^{k_i}, \quad \sum\limits_{j=1}^{k_i} \delta_{i+1}^{j} \leq w_i, & w_i \in \{0,1\}, \; \hbox{for}\; i= 1,\; \ldots, \; \vert {\mathcal{T}} \vert -1. \end{array} \end{aligned} $$

We refer to this formulation as incremental model and denote it by Inc. In literature this model is sometimes also referred to as delta formulation. This formulation has been studied in (Dantzig 1963; Keha et al. 2004; Markowitz and Manne 1957; Padberg 2000; Wilson 1998).

Fig. 9

The point x written as v ⁰₁ plus a sum of vectors

Note that for the (disaggregated) convex combination models a triangulation is not necessary. These approaches may directly be generalized to a finite family of general polytopes (Ahmed 2009), although the number of continuous variables may increase and ambiguities will occur.