Abstract
This paper presents an efficient algorithm based on the shortest path algorithm for a directed weighted graph where all arc cots depend on which node is thought of as the source node. This approach can be applied well to the optimal assortment problem for a network good industry with vertical differentiation. The assortment problem was introduced for decision-makers to select what products to be produced or stocked from a large product category when it is not desirable to stock all of them. A network good is a good with network effect—the effect that the consumption of a good by one has an impact on the value of that product to others. Vertical differentiation occurs when the several goods can be ordered according to their objective quality in the sense that a high-quality good is “better” than the low-quality one. It is shown that an optimal assortment plan can be identified in a polynomial time by our algorithms. The theoretical results also provide qualitative managerial insights into the optimal assortment plan for decision-makers.
Highlights
The shortest path problem, in graph theory, involves finding a path between two vertices in a graph such that the sum of the weights of its constituent arcs is minimized
One algorithm is designed to solve the optimal assortment problem with exogenous prices which are determined by the market environment and the seller cannot alter them by himself; and the other aims at solving the optimal assortment problem with endogenous prices which can be decided by the seller
This paper presents a modified shortest path algorithm for a directed weighted graph where all arc cots depend on which node is thought of as the source node, and applies this approach to an optimal assortment problem for a network good industry with vertical differentiation
Summary
The shortest path problem, in graph theory, involves finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent arcs is minimized. Shortest path algorithms are widely applied in practices, such as finding driving directions on web mapping websites like Google Maps, finding the minimal delay path in telecommunication networks, etc. This assortment problem can be modeled as a shortest path problem in a graph. Unlike classic shortest path problem in which all arc costs are priorly given, in the graph modeled by this paper, all arc costs depend on which node is thought of as the source node. A modified approach based on the shortest path algorithm has been developed to solve this problem and to further investigate how network effects influence the assortment decision
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