Table 1.
The summary of quantum and classical algorithms that support the proposed mathematical model.
Table 2.
Notation for all variables used in the model.
Fig 1.
An illustration of an urban map where the facility j (red dot) is surrounded by its neighbouring demand nodes (coloured urban blocks).
The blue circle determines the extent of the facility’s neighbourhood Aj. The demand nodes are coloured according to their population density.
Fig 2.
Graph networks depicting relations between 49 facilities under three different objective functions.
Each vertex j ∈ {1, …, 49} represents the facility j, and each edge (i, j) represent competition between the facilities i and j. (Left) Inter-facility relations under the unconstrained quadratic objective (1). (Center) Objective function with the p-constraint added as a penalty (16). The resulting graph is dense because each facility must be aware of all other facilities. (Right) The objective function with the partitioned p-constraint as a penalty (24). The resulting graph is sparse because each facility must be aware of only facilities in the subset Sk to which it belongs and its neighbouring competitors.
Table 3.
A user constructed three-facility matrix for TOPSIS algorithm.
We prioritize a facility’s demand attribute by setting Wd = 0.45. If the main goal was to retain a well-connected allocation then Wc would have the largest weight.
Fig 3.
A visual schematic of the weight estimation procedure with the TOPSIS algorithm.
First, the alternatives matrix is constructed and supplied to the TOPSIS algorithm. TOPSIS embeds the attributes into the Euclidean space. The three facility attributes are mapped on three axes: demand, connectedness and landmarks. From the radar plot, it is clear that facility C has overall better characteristics. Hence, it gets a higher ranking. The output is the facility weights.
Fig 4.
A simplified schematic of a single iteration cycle of three branches of solvers.
The SA and TS solvers perform the classical hill-climbing optimization and provide redundancy in the case of decoherence errors in the quantum annealer. The quantum annealer extends the hill-climbing search with quantum tunnelling through optimization landscape barriers in the direction of the global minimum.
Fig 5.
Eigenvalues of for |F| = 49 and different p values.
The objective function with is nonconvex. The eigenvalues of
for p = 10, p = 14, p = 20 are drawn with dotted lines to highlight the loss of positive definiteness and hence convexity.
Fig 6.
Vancouver northbound B20 route with dissemination areas; unoptimized (Left), optimized (Right).
The black dots (bus stops) have different sizes according to their weight wj. The plot on the right demonstrates the optimized route where the retained stops are marked in red.
Fig 7.
The objective function maximum for p retained stops and corresponding theoretical upper bound.
Fig 8.
Unoptimized route (Left). Optimized route (Right). The best solution for p = 14 does not cover all the dissemination areas, resulting in an accessibility decrease.
Fig 9.
Min, max and a median number of stops in dissemination area neighbourhoods for different values of p.
Table 4.
p-values from the Wilcoxon signed-rank test on the 30 synthetic instances with the significance level α = 0.05.
Table 5.
The summary of facility allocation models.