**3. Problem Formulation**

Scheduling for the general multi-target and multi-radar model is formulated as the following equations. The parameters and decision variables for the objective function and for the constraints are described in Tables 1 and 2.


**Table 1.** List of parameters.

**Table 2.** List of decision variables.


The objective function is the sum of target–radar–interval assignment *θt*,*r*,*i*, tracking duration *τp <sup>t</sup>*,*r*, and target assignment *xt* minus target–radar assignment *xt*,*<sup>r</sup>* with appropriate weight values for each term in the above formulation. The terms of the objective function have the following roles: the first term identifies the importance of the target–radar pair over time, the second term maximizes the tracking duration of the whole assignment, the third maximizes the number of targets to track, and the last one minimizes the number of handovers between different radars.

Maximize

$$c\_1 \sum\_{\theta \in \Theta} w\_{t,r,i} \theta\_{t,r,i} + c\_2 \sum\_{t \in T, r \in R} w\_t \pi\_{t,r}^p + c\_3 \sum\_{t \in T} w\_t \mathbf{x}\_t - c\_4 \sum\_{t \in T, r \in R} w\_t \mathbf{x}\_{t,r} \tag{4}$$

subject to

$$\mathbf{x}\_t = \max \{ \mathbf{x}\_{t,1}, \dots, \mathbf{x}\_{t,\mathbf{u}^R} \} \quad \forall t \in T \tag{5}$$

$$\mathbf{r}\_{t,r}^p \le \mathbf{M} \mathbf{x}\_{t,r} \tag{6a} \\ \tag{6b} \\ \tag{6a}$$

$$\mathbf{r}^{p,min} \mathbf{x}\_{t,r} \le \mathbf{r}^p\_{t,r} \tag{6} \\ \tag{6}$$

$$
\gamma\_{t,r}^{TW} \le \tau\_{t,r}^{s} \tag{7a}
$$

*τs <sup>t</sup>*,*<sup>r</sup>* <sup>+</sup> *<sup>τ</sup><sup>p</sup> <sup>t</sup>*,*<sup>r</sup>* <sup>≤</sup> *<sup>δ</sup>TW <sup>t</sup>*,*<sup>r</sup>* ∀*t* ∈ *T*,*r* ∈ *R* (7b) (*τ<sup>s</sup>*

$$
\delta\_i^{\text{Int}} - \tau\_{t,r}^s \le M \theta\_{t,r,i}^{\text{start}} \tag{8a}
\\
\qquad \qquad \forall t \in T, r \in R, i \in I \tag{8a}
$$

$$\tau\_{t,r}^s + \tau\_{t,r}^p - \gamma\_i^{Int} \le M \theta\_{t,r,i}^{end} \tag{8b} \\ \tag{8b}$$

$$
\theta\_{t,r,i} \le \mathbf{x}\_{t,r} \tag{8c}
$$

$$
\begin{array}{ccccc}
\theta\_{t,r,i} \le \mathbf{x}\_{t,r} & & \forall t \in T, r \in R, i \in I \\
& \dots & \dots & \dots & \dots \\
\end{array}
\tag{8c}
$$

$$\begin{aligned} \theta\_{t,r,i} \le \theta\_{t,r,i}^{start} \\ \theta\_{t,r,i} \le \theta\_{t,r,i}^{end} \end{aligned} \qquad \begin{aligned} \forall t \in T, r \in R, i \in I \end{aligned} \tag{8d}$$

$$\begin{aligned} \theta\_{t,r,i} \le \theta\_{t,r,i}^{\text{cut}} \quad & \forall t \in T, r \in R, i \in I \end{aligned} \tag{8e}$$
  $\left( \mathbf{x}\_{t,r} + \theta\_{t,r,i}^{\text{start}} + \theta\_{t,r,i}^{\text{end}} - 2 \right) \le \theta\_{t,r,i} \quad \quad \forall t \in T, r \in R, i \in I \tag{8f}$ 

$$\sum\_{r \in R} \theta\_{t,r,i} \le 2 \tag{9}$$

$$\sum\_{r \in R} \alpha\_{r,i} < \dots \tag{9}$$

$$\sum\_{t \in T} \theta\_{t,r,i} \le n^{capa} \tag{10}$$

$$(\tau\_{t,r\_1}^s + \tau\_{t,r\_1}^p) - (\tau\_{t,r\_2}^s + \tau^{HO}) \le My\_{t,r\_1,r\_2}^{\prime} \tag{11a}$$

$$y\_{t,r\_1,r\_2} \le x\_{t,r\_1} \tag{11b}$$

$$y\_{t,r\_1,r\_2} \le x\_{t,r\_2} \tag{11c}$$

$$\left[y\_{t,r\_1,r\_2} \le y'\_{t,r\_1,r\_2}\right] \tag{11d}$$

$$\mathbf{x}\_{t,r\_1} + \mathbf{x}\_{t,r\_2} + y'\_{t,r\_1,r\_2} - \mathbf{2} \le y\_{t,r\_1,r\_2} \tag{11e}$$

$$\begin{aligned} \boldsymbol{\varepsilon}\_{t,r\_2}^s + \boldsymbol{\tau}^{HO} (- (\boldsymbol{\tau}\_{t,r\_1}^s + \boldsymbol{\tau}\_{t,r\_1}^p) &= M(1 - y\_{t,r\_1,r\_2}) \\ \forall t \in T, \quad r\_1, r\_2 \in \mathbb{R}, \quad \boldsymbol{\delta}\_{t,r\_1}^{TW} &< \boldsymbol{\delta}\_{t,r\_2}^{TW} \end{aligned} \tag{111}$$

$$y\_{t,r\_1,r\_2} + x\_{t,r} \le 1 \tag{11g}$$

$$\forall t \in T, \quad r\_1, r, r\_2 \in \mathbb{R}, \quad \gamma\_{t,r\_1}^{TW} < \gamma\_{t,r}^{TW} < \gamma\_{t,r\_2}^{TW}, \quad \delta\_{t,r\_1}^{TW} < \delta\_{t,r}^{TW} < \delta\_{t,r\_2}^{TW} \tag{11g}$$

The constraints in Equation (5) bind target-radar assignment indicates to a single variable with an OR operator.

The constraints in Equation (6) represent the lower bound of the tracking duration for each target–radar pair if the corresponding binary indicator variable *xt*,*<sup>r</sup>* equals 1. *M* in the equations is a very large positive number and used to effectively activate the constraint only when the variables multiplied to this *<sup>M</sup>* take zero [32]. Briefly, *<sup>τ</sup>p*,*minxt*,*<sup>r</sup>* <sup>≤</sup> *<sup>τ</sup><sup>p</sup> <sup>t</sup>*,*<sup>r</sup>* if *xt*,*<sup>r</sup>* <sup>=</sup> 1, otherwise *<sup>τ</sup><sup>p</sup> <sup>t</sup>*,*<sup>r</sup>* becomes 0.

The constraints in Equation (7) ensure the lower and upper bounds of the start and end time for each target–radar pair; the assignment should be inside the corresponding time window.

Let us call the constraints in Equation (8) the occupying constraints. For target–radar pair (*t*,*r*), *θt*,*r*,*<sup>i</sup>* indicates whether an assignment exists or not in Interval *i*. Briefly, *θt*,*r*,*<sup>i</sup>* equals 1 if the following conditions are met simultaneously: *yt*,*<sup>r</sup>* = 1, *τ<sup>s</sup> <sup>t</sup>*,*<sup>r</sup>* ≤ *<sup>δ</sup>Int <sup>i</sup>* , and *<sup>γ</sup>Int <sup>i</sup>* ≤ *<sup>τ</sup><sup>s</sup> <sup>t</sup>*,*<sup>r</sup>* <sup>+</sup> *<sup>τ</sup><sup>p</sup> t*,*r*.

The constraints in Equation (9) limit the maximum number of radars used for tracking a single target to two; two radars are assigned when the handover occurs, otherwise a single radar tracks the target. Simply, the handover of a single target will only occur between two radars. The constraints in Equation (10) limit the capability of simultaneous tracking for each radar.

The constraints in Equation (11) are for the handover. To decide the handover indicator variable *yt*,*r*1,*r*<sup>2</sup> , we define the support variable *y*- *<sup>t</sup>*,*r*1,*r*<sup>2</sup> as in Equation (11a); *y*- *<sup>t</sup>*,*r*1,*r*<sup>2</sup> is 1 if the end time of Radar *r*<sup>1</sup> tracking Target *t* is equal to or higher than the sum of start time and handover duration for Radar *r*2. The reason for using the support variable is that both target–radar pairs (*t*,*r*1) and (*t*,*r*2) must be assigned the schedule simultaneously as well as satisfying the handover time constraint (Equation (11b)–(11f)). The constraints in Equation (11g) ensure that radars with duplicating and smaller time windows are excluded from the assignment.
