Elevator dispatching problem: a mixed integer linear programming formulation and polyhedral results

In the static elevator dispatching problem the aim is to design a route for each capacitated elevator to satisfy a set of transportation requests such that a cost function is minimized while satisfying a number of constraints. This problem is a crucial part in the control of an elevator group. So far, the problem has been formulated in various algorithmic-dependent forms, where part of the constraints have been given only verbally. In this paper we present a mixed-integer linear programming formulation of the problem where all constraints are given in explicit mathematical form. This allows, e.g., polyhedral analysis of the problem. We also present some new valid inequalities to strengthen the formulation. Furthermore, we study the polyhedral structure of the problem in a generic case arising in the down-peak traffic pattern. In particular, we show which equalities define a minimal equality system for the polytope of the problem, which is defined as the convex hull of the feasible solutions. In addition, we provide the dimension of the polytope and analyze which valid inequalities derived are facet inducing.