A subgradient-based branch-and-bound algorithm for the ...

A subgradient-based
branch-and-bound algorithm
for the capacitated facility location problem
Simon Görtz a
a Faculty of Economics – Schumpeter School of Business and Economics,
University of Wuppertal, 42119 Wuppertal, Germany
Andreas Klose b
b Department of Mathematical Sciences,
University of Aarhus, Ny Munkegade, Building 1530, 8000 Aarhus C, Denmark
Abstract
This paper presents a simple branch-and-bound method based on Lagrangean relaxation
and subgradient optimization for solving large instances of the capacitated
facility location problem (CFLP) to optimality. In order to guess a primal solution
to the Lagrangean dual, we average solutions to the Lagrangean subproblem.
Branching decisions are then based on this estimated (fractional) primal solution.
Extensive numerical results reveal that the method is much more faster and robust
than other state-of-the-art methods for solving the CFLP exactly.
Key words: Mixed-integer programming; Lagrangean relaxation; Capacitated
Facility Location; Subgradient optimization; Volume algorithm; Branch-and-bound
1 Introduction
The capacitated facility location problem (CFLP) is a well-known combinatorial optimization
problem with a number of applications in the area of distribution and
production planning. It consists in deciding which facilities to open from a given set
J of potential facility locations and how to assign customers i ∈ I to those facilities.
The objective is to minimize total fixed and shipping costs. Constraints are that
each customer’s demand di ≥ 0 must be satisfied and that each plant cannot supply
more than its capacity sj > 0 if it is open. Denoting the cost of supplying all of
Email addresses: simon.goertz@wiwi.uni-wuppertal.de (Simon Görtz), aklose@imf.au.dk
(Andreas Klose).