Competitive location and pricing on a line with metric transportation costs

• Published in: European journal of operational research Vol. 282; no. 1; pp. 188 - 200
• Main Authors: Arbib, Claudio, Pınar, Mustafa Ç., Tonelli, Matteo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-04-2020
• Subjects: Computational complexity; Location; Multi-level programming; Pricing; Sequential games; Multi-level programming; Computational complexity; Sequential games; Pricing; Location
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2019.08.042

•Locate-price problems with metric transportation cost: a Stackelberg game.•2-level pricing on a line difficult to solve even with 2 competing facilities.•3-level locate-price with opening costs (1 competing facility) solved in poly-time.
Consider a three-level non-capacitated location/pricing problem: a firm first decides which facilities to open, out of a finite set of candidate sites, and sets service prices with the aim of revenue maximization; then a second firm makes the same decisions after checking competing offers; finally, customers make individual decisions trying to minimize costs that include both purchase and transportation. A restricted two-level problem can be defined to model an optimal reaction of the second firm to known decision of the first.
For non-metric costs, the two-level problem corresponds to Envy-free Pricing or to a special Network Pricing problem, and is APX-complete even if facilities can be opened at no fixed cost. Our focus is on the metric 1-dimensional case, a model where customers are distributed on a main communication road and transportation cost is proportional to distance. We describe polynomial-time algorithms that solve two- and three-level problems with opening costs and single 1st level facility. Quite surprisingly, however, even the two-level problem with no opening costs becomes NP-hard when two 1st level facilities are considered.