In practice, some special LR fuzzy numbers, like the triangular fuzzy number, the Gaussian fuzzy number and the Cauchy fuzzy number, are widely used in many areas to deal with various vague information. With regard to these special LR fuzzy numbers, called regular LR fuzzy numbers in this paper, an operational law is proposed for fuzzy arithmetic, providing a novel approach to analytically and exactly calculating the inverse credibility distribution of some specific arithmetical operations based on the credibility measure. As an application of the operational law, an equivalent form of the expected value operator as well as a theorem for computing the expected value of strictly monotone functions is suggested. Finally, we utilize the operational law to construct a solution framework of fuzzy programming with parameters of regular LR fuzzy numbers, and such type of fuzzy programming problems can be handled by the operational law as the classic deterministic programming without any particular solving techniques.

Online-to-offline (O2O) food delivery service refers to an emerging modern business model that enables customers to order foods from local restaurants via an online platform, and then receive and enjoy them offline after the delivery, offered by couriers. Such service, discussed in this article, specifies that a customer can order from multiple restaurants in a single order and choose for them to be delivered together or separately, whereas the commonly discussed mode only permits placing an order in one restaurant at once. In this service, one crucial issue is how to dispatch these orders to couriers for offline delivery. For this, we propose a new three-stage order dispatch scheme, namely, pseudo-assign first, re-route second, and courier selection last, aiming to deliver the orders in time and balance the couriers’ workload. Due to the dynamism and uncertainty inherently involved in this issue, we also take responsiveness to future demands and robustness into consideration when making the dispatch. Compared with existing approaches, the new one significantly balances the couriers’ workload and, meanwhile, keeps good performance in delay rate, making the decisions more practical. Furthermore, this study analyzes the influence of customers’ preference for order splitting and number of couriers on the efficiency of the distribution system, thereby interesting managerial insights for O2O food delivery are revealed.

In this paper, the minimum spanning tree problem is investigated on the graph with fuzzy edge weights. The notion of fuzzy $\mathit{\alpha }$-minimum spanning tree is presented based on the credibility measure, and then the solutions of the fuzzy $\mathit{\alpha }$-minimum spanning tree problem are discussed under different assumptions. First, we respectively, assume that all the edge weights are triangular fuzzy numbers and trapezoidal fuzzy numbers and prove that the fuzzy $\mathit{\alpha }$-minimum spanning tree problem can be transformed to a classical problem on a crisp graph in these two cases, which can be solved by classical algorithms such as the Kruskal algorithm and the Prim algorithm in polynomial time. Subsequently, as for the case that the edge weights are general fuzzy numbers, a fuzzy simulation-based genetic algorithm using Prüfer number representation is designed for solving the fuzzy $\mathit{\alpha }$-minimum spanning tree problem. Some numerical examples are also provided for illustrating the effectiveness of the proposed solutions.