QUBO formulation for aircraft load optimization

Abstract

In this article, we tackle the aircraft load optimization problem using classical optimization algorithms and optimization algorithms with QUBO (quadratic unconstrained binary optimization) formulation to run on quantum annealers. The problem is realistic based on plans of a certain aircraft model, the Airbus A330 200F, and can be adapted to other models from other manufacturers. We maximize a characteristic of the combination of containers (unit load device, ULD) to be transported, be it weight, volume, profit, or another, while complying with necessary parameters related to the flight such as the balance of the center of gravity as well as stress in the structure. Finally, examples of the results of different runs on QUBO in the D-Wave simulator are presented.

Appendices

Appendix A ULDs compatible with the model

Given the diverse fuselage radii observed in different cargo aircraft models, despite the standardization of container dimensions and pallet base size, a notable variety of ULDs exists, setting them apart from sea cargo containers. The A330-200F model distinguishes itself by accommodating nearly all existing ULD types due to its modern design. In Appendix A, Tables 2 and 3 enumerate the selected containers and pallets for this model, chosen based on their widespread usage.

Table 2 Selection of ULDs used for main deck

Table 3 Selection of ULDs used for lower deck

Appendix B Shear force calculations

Shear force is caused by the weight of the ULDs and is calculated assuming the aircraft is static and level. The lower decks are modeled to be continuously supported by the same beams, which means that the load distribution is simply summed to the main deck. ULDs are modeled as uniformly distributed loads along their length in the direction of the beam, as shown in Fig. 4.

The discrete model calculates the following equations:

For the front of the wings:

$$\begin{aligned}Q(x) = \int _{a_0}^\textrm{supp} q(x) \,\textrm{d}x \end{aligned}$$

where $a_0$ is the point on the main deck closest to the cabin, x goes from $a_0$ to the point of support, and q(x) is the linear load density.

For the cargo loaded to the back of the wings:

$$\begin{aligned}Q(x^{'}) = \int _{a_1}^\textrm{supp} q(x^{'}) \,\textrm{d}x^{'} \end{aligned}$$

where $a_1$ is the point on the main deck furthest from the cabin, $x^{'}$ goes from $a_1$ to the point of support, and $q(x^{'})$ is the linear load density.

The integrals are calculated as a Riemann sum with partitions of one inch over the length of the beams:

$$\begin{aligned}Q(x_{i}) = \sum _{i=0}^{i_\textrm{supp}} q(x_{i}) \cdot 1 \text { in} \end{aligned}$$

for the front beam, and

$$\begin{aligned}Q(x^{'}_{i}) = \sum _{i=0}^{i_\textrm{supp}} q(x^{'}_{i}) \cdot 1 \text { in} \end{aligned}$$

for the rear beam.

When ULDs are uniformly distributed loads, the sum gives exact results and will continue to do so if the load distribution is any step function with steps in whole inches.