Optimal design of container inspection strategies considering multiple objectives via an evolutionary approach

Abstract

The size and complexity of containerized shipping across the globe has increased the vulnerability of seaports to the attack of terrorist networks and contraband smuggling. As a result, the creation of inspection strategies to check incoming containers at ports-of-entry has been necessary to enable the detection of containers carrying prohibited items. However, since costs and tardiness considerations related to the inspection process prevent all cargo to be manually checked, including different non-intrusive screening technologies as part of the inspection strategies is essential to optimize inspection needs. In this paper, inspection strategies are represented as decision-tree structures where each node illustrates a screening device, and links represent the two possible classifications a screened container can get (i.e. suspicious or unsuspicious). Based on such classification, one of three actions is taken: to continue screening, release or physically check the container. The contribution of this paper is a mathematical framework that provides an approximation to the Pareto optimal solutions (i.e. inspection strategies) that enable decision-makers to: (1) identify tradeoffs among vulnerability, inspection cost, and tardiness for different inspection strategies, and based on this (2) find the strategy that best suits current inspection needs. The mathematical framework includes: (1) a multi-objective optimization model that concurrently minimizes vulnerability, cost, and tardiness while determining screening device allocation and threshold settings, as well as, (2) an evolutionary approach used to solve the optimization model.

Appendix

According to the binary decision-tree structure presented in this paper, the sensor assigned to the first node (i=1), is always used to screen all containers. At subsequent nodes (i>1), the fraction of “suspicious” and “unsuspicious” containers—$Q_{i}^{1}$ and $Q_{i}^{0}$ respectively—screened, released or physically inspected, changes based on the results obtained at the preceding nodes. These fractions are calculated using (A.1), (A.2) and (A.3). At the first node, both, $Q_{i}^{1}$ and $Q_{i}^{0}$ have a value of 1 since 100% of containers are inspected at such node.

$$Q_{i}^{z} = Q_{\Omega (i)}^{z} \bigl[I_{i}^{0}\Phi_{L_{\Omega (i)}}^{z}(t_{L_{\Omega (i)}}) +\bigl( 1 - I_{i}^{0} \bigr) \bigl( 1 - \Phi_{L_{\Omega (i)}}^{z}(t_{L_{\Omega (i)}})\bigr) \bigr] $$

(A.1)

where:

$$I_{i}^{0} =\begin{cases}0& \text{if the link between }L_{i}\text{ and }L_{\Omega (i)}\text{ is of type ``unsuspicious''} \\1&\text{otherwise}\end{cases}$$

Ω(i):

represents the node immediately preceding node i

$\Phi_{j}^{z}(t_{j})$ :

is the cumulative distribution function for sensor readings r _j of sensor type j for both, “suspicious and “unsuspicious” containers considering the corresponding threshold value t _j and assuming such readings are normally distributed $N(\mu_{j}^{z},\sigma_{j}^{z} )$

$t_{L_{\Omega (i)}}$ :

is the threshold value of the sensor type located at node Ω(i)

Equation (A.1) indicates that the fraction of “unsuspicious” (z=0) or “suspicious” (z=1) containers at each node, $Q_{i}^{z}$, is a function of: (1) the fraction of “unsuspicious” or “suspicious” containers, respectively, screened at the immediate preceding node Ω(i), (2) the reliability of the sensor located at such preceding node $\Phi_{L_{\Omega (i)}}^{z}(t_{L_{\Omega (i)}})$ and (3) its corresponding threshold value $t_{L_{\Omega (i)}}$.

However, when at a given node containers are either released or physically inspected, the fraction of “suspicious” and “unsuspicious” containers at that particular node is obtained differently and it depends on the decision taken at the preceding node. Equations (A.2) and (A.3) are used to obtain $Q_{i}^{z}$ in such cases.

(A.2)

(A.3)

where $\alpha_{i}^{h} = \frac{i}{2^{h - 1}}$ and h represents the echelon number (Ramirez-Marquez 2008).

Equation (A.2) ensures that same sensor types are not allocated at subsequent positions to avoid having containers inspected twice with the same type of device. Equation (A.3) guarantees that once a container is released or physically inspected, no further action is taken with respect to the inspection strategy.

The utilization of each sensor type is calculated following equation (A.4). It represents the fraction of “unsuspicious” containers unnecessarily screened by sensor type j. The false-suspicious rate is obtained using (A.5) and it accounts for the expected fraction of “unsuspicious” containers that unnecessarily go through a physical inspection when using strategy L.

(A.4)

(A.5)

where:

$$\mathbf{L} = ( L_{1},\ldots,L_{i},L_{i + 1},\ldots,L{}_{2^{n + 1} - 1} )$$

L _i represents a sensor type j, a release (“OK”) or a physical inspection (“CHK”) allocated at node i. For further details on the development of these equations refer to (Ramirez-Marquez 2008).