The STEINER TREE PROBLEM in a weighted graph is to find a least cost sub TREE in the graph such that it contains special vertices called terminal points. The STEINER TREE PROBLEM on graphs is one of the best-known NP-hard PROBLEMs. For this reason, many approximation algorithms for solving STEINER TREE PROBLEM have been proposed. The traditional algorithms achieve near optimal results, but cost more computational time. In this paper we propose an iterative algorithm based on learning automata for solving DYNAMIC STEINER TREE PROBLEM. Simulation results show that our proposed algorithm generates STEINER TREEs of low cost in reasonable running time.

Considering convergent product as an important manufacturing technology for digital products, we integrate functions and sub-functions using a comprehensive fuzzy mathematical optimization process. To form the convergent product, a web-based fuzzy network is considered in which a collection of base functions and sub-functions configure the nodes and each arc in the network is to be a link between two nodes. The aim is to find an optimal TREE of functionalities in the network adding value to the product in the web environment. First, a purification process is performed in the product network to assign the links among bases and sub-functions. Then, numerical values as benefits and costs are determined for the arc and the node, respectively. Also, a fuzzy customers’ value corresponding to the arcs is considered. Next, the STEINER TREE methodology is adapted to a multi-objective model of the network to find the optimal TREE. A fuzzy multi-objective solution methodology is developed for solving the proposed PROBLEM. Finally, an example is worked out to illustrate the proposed approach.

A convergent product is an assembly shape concept integrating functions and sub-functions to form a final product. To conceptualize the convergent product PROBLEM, a web-based network is considered in which a collection of base functions and sub-functions configure the nodes, and each arc in the network is considered to be a link between two nodes. The aim is to find an optimal TREE of functionalities in the network, adding value to the product in the web environment. First, an algorithm is proposed to assign the links among bases and sub-functions. Then, numerical values, as benefits and costs, are determined for arcs and nodes, respectively, using a mathematical approach. Also, customer value corresponding to the benefits is considered. Finally, the STEINER TREE methodology is adapted to a multi-objective model optimized by an augmented e-constraint method. An example is worked out to illustrate the proposed approach.

In this paper we investiagte STEINER loops introduced by N.S. Mendelsohn [Aeq. Math. 6 (1991), 228–230] and provide six (seven) equivalent identities to characterize it. We also prove the power associativity of Bol loops by using closure (Hexagonal) conditions.

The Wiener index W (G) of a connected graph G is defined as W (G) = Su, v Î v (G) dG (u, v) where dG (u, v) is the distance between the vertices u and v of G. For S Í V (G), the STEINER distance d (S) of the vertices of S is the minimum size of a connected subgraph of G whose vertex set is S. The k-th STEINER Wiener index SWk (G) of G is defined as SWk (G) = S SÍV (G)÷s÷=k d(S). We establish expressions for the k-th STEINER Wiener index on the join, corona, cluster, lexicographical product, and Cartesian product of graphs.