Two variations of the minimum

Given a set S of starting vertices and a set T of terminating vertices in a graph G = (V,E) with non-negative weights on edges, the minimum Steiner network problem is to find a subgraph of G with the minimum total edge weight. In such a subgraph, we require that for each vertex S $${\in}$$ S and T $${\in}$$ T, there is a path from S to a terminating vertex as well as a path from a starting vertex to T. This problem can easily be proven NP-hard. For solving the minimum Steiner network problem, we first present an algorithm that runs in time and space that both are polynomial in n with constant degrees, but exponential in |S|+|T|, where n is the number of vertices in G. Then we present an algorithm that uses space that is quadratic in n and runs in time that is polynomial in n with a degree O(max {max {|S|,|T|}−2,min {|S|,|T|}−1}). In spite of this degree, we prove that the number of Steiner vertices in our solution can be as large as |S|+|T|−2. Our algorithm can enumerate all possible optimal solutions. The input graph G can either be undirected or directed acyclic. We also give a linear time algorithm for the special case when min {|S|,|T|} = 1 and max {|S|,|T|} = 2. The minimum union paths problem is similar to the minimum Steiner network problem except that we are given a set H of hitting vertices in G in addition to the sets of starting and terminating vertices. We want to find a subgraph of G with the minimum total edge weight such that the conditions required by the minimum Steiner network problem are satisfied as well as the condition that every hitting vertex is on a path from a starting vertex to a terminating vertex. Furthermore, G must be directed acyclic. For solving the minimum union paths problem, we also present algorithms that have a time and space tradeoff similar to algorithms for the minimum Steiner network problem. We also give a linear time algorithm for the special case when |S| = 1, |T| = 1 and |H| = 2

An integrated AHP-QFD-based compromise ranking model for sustainable supplier selection

Sustainability in industrial organizations has become a predominant concept in the 21st century due to environmental regulations, economic importance, and social obligations. In this context, sustainable supplier selection plays an epochal role for taking strategic business decisions. So, a systematic approach is required to deal with the sustainable factors. By integrating the three pillars of sustainability, namely economic, environmental, and social factors, this chapter presents an integrated model for supplier selection from a sustainability perspective. The proposed framework combines analytic hierarchy process (AHP) and quality function deployment (QFD) methods to deal with the sustainable criteria. Finally, suppliers are ranked using VIse Kriterijumska Optimizacija kompromisno Resenja (VIKOR) method. The proposed framework is used to analyze a case study of a dairy company, but it can also be implemented for sustainable supplier selection in any industries. The study brings an artefact for managers to effectively analyze suppliers with the integrated decision-making model