Fuzzy relation inequality-based consistency of the wireless communication basic-station system considering the non-working state stations

Abstract

Requirements of the electromagnetic intensities of radio waves in a wireless communication basic-station (WCBS) system could be reduced into a system of fuzzy relation inequalities with max-product composition. In order to check the consistency of a WCBS system with some non-working basic stations, we first introduce the consistency checking method for a WCBS system without any non-working basic station. An index set approach is designed for the consistency checking, with some numerical examples illustrating the effectiveness of our proposed approach. Moreover, in order to reduce the maintenance cost of the basic stations, we further discuss the minimum number of working stations, ensuring that the WCBS system is consistent. The corresponding minimization problem is equivalently converted into a 0-1 integer programming problem, which could be directly solved by some common softwares. The major contribution of this work is to develop an effective approach for obtaining the minimum number of working stations for a given WCBS system. Our obtained results have been formally proved in theory and illustrated by some numerical examples.

Appendix

Proof of Lemma 1

For any $i\in I$ and $j\in J$, we next check $a_{ij}{\hat{x}}_{j}\le d_{i}$ in two cases.

Case 1. If ${\hat{I}}_{j}=\emptyset$, then it follows from (5) and (6) that $a_{ij}\le d_{i}$ and ${\hat{x}}_{j}=1$. So we get

$$\begin{aligned} a_{ij}{\hat{x}}_{j}=a_{ij}\le d_{i}. \end{aligned}$$

(60)

Case 2. If ${\hat{I}}_{j}\ne \emptyset$, then it follows from (6) that ${\hat{x}}_{j}=\bigwedge \limits _{k\in {\hat{I}}_{j}} \frac{d_{k}}{a_{kj}}$. Moreover, when $i\in {\hat{I}}_{j}$, we have

$$\begin{aligned} a_{ij}{\hat{x}}_{j}=a_{ij}\bigwedge \limits _{k\in {\hat{I}}_{j}} \frac{d_{k}}{a_{kj}}\le a_{ij}\frac{d_{i}}{a_{ij}} = d_{i}. \end{aligned}$$

(61)

When $i\notin {\hat{I}}_{j}$, it follows from (5) that $a_{ij}\le d_{i}$. Hence, we have $a_{ij}{\hat{x}}_{j}\le a_{ij}\cdot 1=a_{ij}\le d_{i}$. $\square$

Proof of Lemma 2

$X(A,b,d,J^{nw''})\ne \emptyset$ means that the WCBS system is consistent with the working stations $\{B_{i}|i\in J-J^{nw''}\}$. Since $J^{nw'}\subseteq J^{nw''}$, we have $\{B_{i}|i\in J-J^{nw''}\} \subseteq \{B_{i}|i\in J-J^{nw'}\}$. The working stations $\{B_{i}|i\in J-J^{nw'}\}$ could also make the WCBS system consistent. Thus, we have $X(A,b,d,J^{nw'})\ne \emptyset$. $\square$

Proof of Lemma 3

($\Rightarrow$) Suppose the station $B_{j}$ emits the radio waves with electromagnetic intensity ${\hat{x}}_{j}$, where ${\hat{x}}=({\hat{x}}_{1},{\hat{x}}_{2},\ldots ,{\hat{x}}_{n})$ is the maximum solution of system (1). Then, when the radio waves arrive at $T_{i}$, the electromagnetic intensity is $a_{ij}{\hat{x}}_{j}$. According to (7), $j\in J_{i}$ implies that

$$\begin{aligned} b_{i}\le a_{ij}{\hat{x}}_{j} \le d_{i}. \end{aligned}$$

(62)

This indicates the requirement of $T_{i}$ is satisfied by $B_{j}$.

($\Leftarrow$) Assume that the requirement of $T_{i}$ could be satisfied by $B_{j}$. Then there exists a solution $x=(x_{1},x_{2},\ldots ,x_{n})$ of system (1), such that

$$\begin{aligned} b_{i}\le a_{ij}x_{j} \le d_{i}. \end{aligned}$$

(63)

In system (1), ${\hat{x}}$ is the maximum solution. Thus, ${\hat{x}}\ge x$, i.e., ${\hat{x}}_{j}\ge x_{j}$ for all $j\in J$. By (63), we have

$$\begin{aligned} a_{ij}{\hat{x}}_{j} \ge a_{ij}x_{j} \ge b_{i}. \end{aligned}$$

(64)

It follows from (7) that $j\in J_{i}$. $\square$