The coastal seaspace patrol sector design and allocation problem

Abstract

In this paper, we model and solve the problem of designing and allocating coastal seaspace sectors for steady-state patrolling operations by the vessels of a maritime protection agency. The problem addressed involves optimizing a multi-criteria objective function that minimizes a weighted combination of proportional measures of the vessels’ distances between home ports and patrol sectors, the sector workload, and the sector span. We initially assure contiguity of each patrol sector in our mixed-integer programming formulation via an exponential number of subtour elimination constraints, and then propose three alternative solution methods, two of which are based on reformulations that suitably replace the original contiguity representation with a polynomial number of constraints, and a third approach that employs an iterative cut generation procedure based on identifying violated subtour elimination constraints. We further enhance these reformulations with symmetry defeating constraints, either in isolation or in combination with a suitable perturbation of the objective function using weighted functions based on such constraints. Computational comparisons are provided for solving the problem using the original formulation versus either of our three alternative solution approaches for a representative instance. Overall, a model formulation based on Steiner tree problem (STP) constructs and enhanced by the reformulation-linearization technique (RLT) yielded the best performance.

Appendix

Herein, we detail the steps for reformulating P to the equivalent representation P1 using the reformulation-linearization technique (Sherali and Adams 1990, 1994), as summarized in Sect. 3.1.

1.1 Reformulation phase

As in Sherali et al. (1998), we compose additional constraints for each $v \in \mathcal V$ by taking products of the bounding constraints

$$\begin{aligned} \left[\begin{array}{c} \sum \limits _{i\in \delta ^-(j)\cup \{o_v\}} r^v_{ij}=s^v_j, \quad \forall \ j\in N \\ r^v_{ij} \ge 0, \quad \forall \ (i, j)\in A^v\\ r^v_{ij}+r^v_{ji} \le s^v_i, \quad \forall \ (i, j)\in A \\ s^v_j \le 1, \quad \forall \ j\in N \end{array} \right] \begin{array}{c} \text{ with} \\ \text{ appropriate} \\ \text{ elements} \text{ of} \end{array} [ \begin{array}{c} 1\le u^v_j \le n_v, \quad \forall \ j\in N\end{array}], \nonumber \\ \end{aligned}$$

(45)

wherein the lower and upper bounds respectively imposed on the $r^v_{ij}$- and $s^v_j$-variables are induced by Constraints (18) and (19). Although additional RLT constraints can be generated beyond the products delineated in (45) above (these serve to further tighten the model representation, but the foregoing are sufficient to validate it), we restrict our attention to this set in order to avoid a proliferation of new variables and constraints. The resulting so-called RLT constraints are composed as follows:

• (i) Using Constraint (5), we construct the following equalities:

  $$\begin{aligned} \left[ \sum _{i\in \delta ^-(j)\cup \{o_v\}} r^v_{ij}=s^v_j\right]* u^v_j , \quad \forall \ j\in N, v\in \mathcal V . \end{aligned}$$

  (46)

• (ii) Considering the nonnegativity of the $r^v_{ij}$-variables, we have, in particular, that

  $$\begin{aligned} (u^v_i-1) r^v_{ij}&\ge 0, \quad \forall \ (i, j)\in A^v, v\in \mathcal V , \end{aligned}$$

  (47)
  $$\begin{aligned} (n_v-u^v_j) r^v_{ij}&\ge 0, \quad \forall \ (i, j)\in A^v, v\in \mathcal V . \end{aligned}$$

  (48)

  Additionally, regarding the arcs in $A$, Constraint (25) in combination with Constraint (26) indicates that if $r^v_{ij}=1$ then $u^v_j\ge 2$ and $u^v_i\le n_v-1$, validating the following additional inequalities:

  $$\begin{aligned} (u^v_j-2) r^v_{ij}&\ge 0, \quad \forall \ (i, j)\in A^v, v\in \mathcal V , \end{aligned}$$

  (49)
  $$\begin{aligned} (n_v-1-u^v_i) r^v_{ij}&\ge 0, \quad \forall \ (i, j)\in A^v, v\in \mathcal V . \end{aligned}$$

  (50)

• (iii) The respective products of the lower and upper bounding constraints on the $u^v_i$-variables with Constraint (6) yields the inequalities

  $$\begin{aligned} (s^v_i-r^v_{ij}-r^v_{ji}) (u^v_i - 1 )&\ge 0, \quad \forall \ (i, j)\in A^v, v \in \mathcal V , \end{aligned}$$

  (51)
  $$\begin{aligned} (s^v_i-r^v_{ij}-r^v_{ji}) (n_v-u^v_i)&\ge 0, \quad \forall \ (i, j)\in A^v, v \in \mathcal V . \end{aligned}$$

  (52)

• (iv) Likewise, the respective products of $1 \le u^v_j \le n_v$ with $s^v_j \le 1, \ \forall \ j\in N, \ v \in \mathcal V ,$ results in the inequalities

  $$\begin{aligned} (1-s^v_j) (u^v_j - 1)&\ge 0, \quad \forall \ j\in N, v \in \mathcal V , \end{aligned}$$

  (53)
  $$\begin{aligned} (1-s^v_j) (n_v-u^v_j)&\ge 0, \quad \forall \ j\in N, v \in \mathcal V . \end{aligned}$$

  (54)

1.2 Linearization phase

In accordance with Sherali and Adams (1990, 1994), we introduce two new classes of RLT variables, $\rho \equiv (\rho ^v_{ij})_{(i, j) \in A^v,v\in \mathcal V }$ and $\sigma \equiv (\sigma ^v_{j})_{j\in N,v\in \mathcal V }$ to linearize Constraints (46)–(54) by using the following substitutions:

$$\begin{aligned} \rho ^v_{ij}=u^v_{i}r^v_{ij}, \quad \forall \ (i, j)\in A^v, v\in \mathcal V , \text{ and} sigma^v_{j}=u^v_j s^v_j, \quad \forall \ j \in N, v\in \mathcal V .\nonumber \\ \end{aligned}$$

(55)

As a further simplification, we can derive the following identities for linearizing the terms $u^v_{j}r^v_{ij}, \quad \forall \ (i, j)\in A, v\in \mathcal V ,$ using Constraint (25):

$$\begin{aligned} u^v_{j}r^v_{ij}=\rho ^v_{ij}+ r^v_{ij}, \quad \forall \ (i, j)\in A^v, v\in \mathcal V . \end{aligned}$$

(56)

Substituting (55)–(56) into (49)–(50) affects the same linearized constraints as (47)–(48). Hence, we omit (49)–(50) and substitute (55)–(56) into (46)–(48) and (51)–(54), yielding the alternative formulation P1, as presented in Sect. 3.1.