I have come across the paper that deals with spacial positioning of mobile sensors to optimally detect sound source, or position mobile cellphone towers to maximize the coverage.
The region $Q$ is partitioned into mutually exclusive $n$ voronoi polytopes $W=\{ W_1,..,W_n \}$. A function $\phi :Q \to \mathbb{R}_+$ assigns probability density that a certain event (here sound source) has happened over $Q$. There are $n$ sensors to be distributed $P=(p_1,..p_n)$ over the each voronoi partition that satisfies the following equation:
$$H(P,W)=\textrm{minimize}\sum_{i=1}^n \int_{W_i} f(\| q-p_i\|)d \phi(q)$$
The quality of observation at point $q$ from sensor $p_i$ is the distance of the point $q$ from the sensor $f(\| q-p_i\|)$ within sensor boundary region $W_i$ of sensor $p_i$ which makes sense. Then we sum the same for all the sensors.
Can anyone explain why are we multiplying $(q)$ after $d\phi$ in the objective formula?
This is just notation: they are not multiplying by $q$. By $d\phi(q)$, the authors are just emphasizing that $\phi$ depends on $q$ which is the variable of integration.
The authors sort of alternate between two notations using $d\phi(q)$ in sections II.A and II.B where $\phi$ is the distribution density function and then switching in section II.C to $\rho(q)\,dq$. The only difference here is that $\phi$ is assumed to be a probability distribution (and integrates to $1$) while $\rho$ is just a positive density function. But they could write $\phi(q)\,dq$ in the earlier sections, too.