## 6 Numerical Illustrations  
Our proposed mUAQMSTP is demonstrated through an example involving the service network of a recently established bus transportation company. Here is a brief overview of the problem. When planning to launch their service, the bus company focused on seven of the country's most important towns. To that end, twenty-one potential transit routes connecting the cities are being studied. Additionally, in an effort to entice additional customers, the firm provides a discount on the total cost of the bus ticket to customers who use two of their buses in succession during their trip. In order to maximize efficiency and decrease expenses, the company's management plans the bus route for the upcoming quarter in a way that minimizes the overall cost of providing bus services between every town, taking into account any potential discounts. Within this framework, the organization also recognizes multiple variables, such as the cost of gasoline, labor, and bus maintenance and overhaul, that impact the cost of providing the transportation service and fundamentally change with time. Because of the complex social and economic factors at play in decision-making problems, such as the multi-objective adjacent only quadratic minimum spanning tree problem, it is not always feasible to precisely establish or obtain the problem parameters in many real-world contexts. Problem parameters that could vary over time include, but are not limited to, the cost of gasoline, the cost of labor, the cost of refurbishment and maintenance, the cost of the toll tax, the length of time spent in traffic, and the cost of vehicle maintenance. Therefore, when adding such characteristics into decision-making problems, it is vital to use the opinion of domain experts to estimate and evaluate the belief degrees of indeterminacy in them. The uncertainty theory paradigm [PERSON] [18; 34; 35] is a good choice for representing individual beliefs. Therefore, the estimations of the decision makers on the cost of the bus service and the discount amount are considered as uncertain variables. A suitable numerical example is provided in this part while unfolding the feature of the proposed mUAQMSTP. Here, the project managers consider the possible bus services between the cities as an undirected weighted connected network (UWCN) \(G\), \(G=\left(V_{G},E_{G}\right)\) as shown in Figure 2. In Figure 2, each vertex is represented by a city \(v_{i},i=1,2,\ldots,\left|V_{G}\right|\)and an edge \(e_{j},j=1,2,\ldots,|E_{G}|\) represents the possible bus service between the pair of cities to be provided by the company. Each edge of \(E_{G}\) is associated with a linear weight, which determines the possible cost to the company while providing the bus service between a pair of cities. Moreover, the quadratic weight, for any pair of adjacent edges, determines the possible discount on the total bus fare of two consecutive buses of the company when availed by a passenger. These linear and quadratic weights, which are considered as zigzag uncertain variables are reported, respectively, in Tables 12 and 13 (cf. Appendix A). Consequently, the parameters of \(G\) are modelled within the context of uncertainty theory in this investigation. In this case, the uncertain parameter, \(\xi_{i}\), represents the linear cost (weight) associated to an edge, \(e_{i}\), in \(G\). While the indeterminate parameter, \(\xi_{ij}\), represents the quadratic cost (weight) that corresponds to two adjacent edges, \(e_{i}\) and \(e_{j}\). In order to determine an optimal bus route, the project managers determine a multi-objective uncertain adjacent quadratic minimum spanning tree (mUAQMST) of \(G\) using the deterministic Model, (8), of the CCM (as demonstrated in Model (6)) of the proposed mUAQMST (cf. Model (5)). Model (8) is the MOOP and, thus, we used the global criterion method, a classical multi-objective solution technique, on its related compromise Model (9). The model's compromise solution is found using an optimization problem solver, Lingo 11.0.