Model reformulation for conflict-free routing problems using Petri Net and Deterministic Finite Automaton

Abstract

It is well known that routing problems are recast as a binary integer linear programming in overhead hoist transport (OHT) vehicle systems. Two kinds of discrete-time state space models of the routing problem are proposed in this paper that simplifies the constraints and binary variables. First, a Petri Net (PN) method is used to simplify the constraints. Second, a Deterministic Finite (DF) Automaton method is used for reducing the binary inputs of the state space equation in OHT systems. Finally, some numerical examples are shown to verify the validity of the proposed methods.

Appendix

Suppose that Eq. (2) includes Eq. (1), and Eqs. (5b) and (7) hold. From Eqs. (2) and (8), we get

$$ \hat{x}^{{v_{s} }} \left( {k + 1} \right) = \hat{A} \hat{x}^{{v_{s} }} \left( k \right) + (\hat{B}^{ + } - \hat{B}^{ - } )\widehat{u}^{{v_{s} }} \left( k \right) $$

(25)

for $s = 1, \ldots ,S$ and 0 ≤ k ≤ M − 1. In addition, $B^{ + } \widehat{u}^{{v_{s} }} \left( k \right) \ge {\mathbf{0}}$ holds because of B ⁺ ∊ {0, 1}^n×m and $\widehat{u}^{{v_{s} }} \left( k \right) \in \left\{ {0,1} \right\}^{m}$. By using Eq. (7) and $\hat{A}\text{ := }I_{n}$, we get

$$ \begin{aligned} {\mathbf{0}}& \le \hat{x}^{{v_{s} }} \left( k \right) - \hat{B}^{ - } \widehat{u}^{{v_{s} }} \left( k \right) \\ &\le \hat{A} \hat{x}^{{v_{s} }} \left( k \right) + (\hat{B}^{ + } - \hat{B}^{ - } )\widehat{u}^{{v_{s} }} \left( k \right) \hfill \\ \end{aligned} $$

(26)

for $s = 1, \ldots ,S$ and 0 ≤ k ≤ M - 1. From Eqs. (26) and (8),

$$ \begin{aligned} {\mathbf{0}} &\le \hat{A} \hat{x}^{{v_{s} }} \left( k \right) + \left( {\hat{B}^{ + } - \hat{B}^{ - } } \right)\widehat{u}^{{v_{s} }} \left( k \right) \\ &= \hat{A}_{a} \hat{x}^{v} \left( k \right) + \hat{B}_{a} \widehat{u}^{v} \left( k \right) = \hat{x}^{{v_{s} }} \left( {k + 1} \right). \hfill \\ \end{aligned} $$

(27)

Thus, we get $\hat{x}^{{v_{s} }} \left( {k + 1} \right) \ge {\mathbf{0}}$, that is, Eq. (4a). Also, from $\hat{x}^{{v_{s} }} \left( {k + 1} \right) \ge {\mathbf{0}}$, Eqs. (2) and (5b), we obtain

$$ \begin{aligned} \hat{x}^{{v_{s} }} \left( {k + 1} \right) &= \hat{A} \hat{x}^{{v_{s} }} \left( k \right) + \hat{B}\widehat{u}^{{v_{s} }} \left( k \right) \hfill \\ &\le \mathop \sum \limits_{s = 1}^{S} \left\{ {\hat{A} \hat{x}^{{v_{s} }} \left( k \right) + \hat{x}\widehat{u}^{{v_{s} }} \left( k \right)} \right\} \\ &\le \mathbf{1} \hfill \\ \end{aligned} $$

(28)

for $s = 1, \ldots ,S$ and 0 ≤ k ≤ M − 1. From Eq. (28), we obtain $\hat{x}^{{v_{s} }} \left( {k + 1} \right) \le {\mathbf{1}}$, that is, Eq. (4b). Therefore, Eqs. (27) and (28) prove Eqs. (2), (5b), and (7) are sufficient conditions for Eq. (4).

Second, from Eqs. (26), (27), and (8), we get

$$ \begin{aligned} &{\mathbf{0}} \le \hat{x}^{{v_{s} }} \left( k \right) - \hat{B}^{ - } \widehat{u}^{{v_{s} }} \left( k \right) \hfill \\ &\le \hat{A}\hat{x}^{{v_{s} }} \left( k \right) + ( \hat{B}^{ + } - \hat{B}^{ - } )\widehat{u}^{{v_{s} }} \left( k \right) \le \hfill \\ &\mathop \sum \limits_{s = 1}^{S} \left\{ {\hat{A} \hat{x}^{{v_{s} }} \left( k \right) + \hat{B}^{{v_{s} }} \widehat{u}\left( k \right)} \right\} \hfill \\ \end{aligned} $$

(29)

for $s = 1, \ldots ,S$ and 0 ≤ k ≤ M − 1. From Eq. (29), we get ${\mathbf{0}} \le \mathop \sum \nolimits_{s = 1}^{S} \left\{ {\hat{A} \hat{x}^{{v_{s} }} \left( k \right) + \hat{B}\widehat{u}^{{v_{s} }} \left( k \right)} \right\}$. Thus, this equation proves that Eqs. (2), (5b), and (7) are also sufficient conditions for Eq. (5a).

Finally, consider that Eq. (6b) contributes to the converging and diverging junctions as follows:

$$ \mathop \sum \limits_{s = 1}^{S} \widehat{u}_{c1}^{{v_{s} }} (k) + \widehat{u}_{c2}^{{v_{s} }} (k) \le 1\quad {\text{and}} $$

(30)

$$ \mathop \sum \limits_{s = 1}^{S} \widehat{u}_{d1}^{{v_{s} }} (k) + \widehat{u}_{d2}^{{v_{s} }} (k) \le 1 $$

(31)

for 0 ≤ k ≤ M − 1. $\widehat{u}_{c1}^{{v_{s} }}$ and $\widehat{u}_{c2}^{{v_{s} }}$ are the gates of the converging area. $\widehat{u}_{d1}^{{v_{s} }}$ and $\widehat{u}_{d2}^{{v_{s} }}$ are also the gates of diverging area. Defining $\hat{x}_{c}^{{v_{s} }}$ and $\widehat{u}_{c}^{{v_{s} }}$ as the converging area and its output gate, we see that Eq. (2) includes the following equation:

$$ \hat{x}_{c}^{{v_{s} }} \left( {k + 1} \right) = \hat{x}_{c}^{{v_{s} }} \left( k \right) + \widehat{u}_{c1}^{{v_{s} }} (k) + \widehat{u}_{c2}^{{v_{s} }} (k) - \widehat{u}_{c}^{{v_{s} }} \left( k \right) $$

(32)

for 0 ≤ k ≤ M − 1. From Eqs. (32) and (5b), we obtain

$$ \mathop \sum \limits_{s = 1}^{S} x_{c}^{{v_{s} }} (k + 1) \le 1 $$

$$ \Leftrightarrow \mathop \sum \limits_{s = 1}^{S} \left\{ {\hat{x}_{c}^{{v_{s} }} \left( k \right) + \widehat{u}_{c1}^{{v_{s} }} \left( k \right) + \widehat{u}_{c2}^{{v_{s} }} \left( k \right) - \widehat{u}_{c}^{{v_{s} }} \left( k \right)} \right\} \le 1 $$

(33)

for 0 ≤ k ≤ M − 1. In addition, from Eq. (7), we also obtain

$$ 0 \le \hat{x}_{c}^{{v_{s} }} \left( k \right) - \widehat{u}_{c}^{{v_{s} }} \left( k \right) \le \mathop \sum \limits_{s = 1}^{S} \left\{ { \hat{x}_{c}^{{v_{s} }} \left( k \right) - \widehat{u}_{c}^{{v_{s} }} \left( k \right)} \right\} $$

(34)

For 0 ≤ k ≤ M − 1. Equation (34) proves that $0 \le \mathop \sum \nolimits_{s = 1}^{S} \left\{ { \hat{x}_{c}^{{v_{s} }} \left( k \right) - \widehat{u}_{c}^{{v_{s} }} \left( k \right)} \right\}$ holds, and then, Eq. (33) results in

$$ \begin{aligned} &\mathop \sum \limits_{s = 1}^{S} \left\{ {\widehat{u}_{c1}^{{v_{s} }} \left( k \right) + \widehat{u}_{c2}^{{v_{s} }} \left( k \right)} \right\} \\&\quad \le \mathop \sum \limits_{s = 1}^{S} \left\{ { \hat{x}_{c}^{{v_{s} }} \left( k \right) + \widehat{u}_{c1}^{{v_{s} }} \left( k \right) + \widehat{u}_{c2}^{{v_{s} }} \left( k \right) - \widehat{u}_{c}^{{v_{s} }} \left( k \right)} \right\} \le 1 \end{aligned} $$

(35)

for 0 ≤ k ≤ M − 1. Thus, Eq. (35) proves that Eqs. (2), (5b), and (7) are also sufficient conditions for Eq. (30). Defining $\hat{x}_{d}^{{v_{s} }}$ as the diverging area, we see the following equation from Eq. (7):

$$ \mathop \sum \limits_{s = 1}^{S} \left\{ { \widehat{u}_{d1}^{{v_{s} }} \left( k \right) + \widehat{u}_{d2}^{{v_{s} }} \left( k \right)} \right\} \le \mathop \sum \limits_{s = 1}^{S} \widehat{x}_{d}^{{v_{s} }} \left( k \right) $$

(36)

for 0 ≤ k ≤ M − 1. We obtain Eq. (31) from Eqs. (5b) and (36). This proves that Eqs. (2), (5b), and (7) are also sufficient conditions for Eq. (31).

Therefore, the statement of Lemma 1 holds.