Pascale Bendotti

The min-up/min-down unit commitment polytope

The min-up/min-down unit commitment problem (MUCP) is to find a minimum-cost production plan on a discrete time horizon for a set of fossil-fuel units for electricity production. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up-time and down-time constraints besides featuring production and start-up costs. A full polyhedral characterization of the MUCP with only one production unit is provided by Rajan and Takriti (Minimum up/down polytopes of the unit commitment problem with start-up costs. IBM Research Report, 2005). In this article, we analyze polyhedral aspects of the MUCP with n production units. We first translate the classical extended cover inequalities of the knapsack polytope to obtain the so-called up-set inequalities for the MUCP polytope. We introduce the interval up-set inequalities as a new class of valid inequalities, which generalizes both up-set inequalities and minimum up-time inequalities. We provide a characterization of the cases when interval up-set inequalities are valid and not dominated by other inequalities. We devise an efficient Branch and Cut algorithm, using up-set and interval up-set inequalities.

Anchored reactive and proactive solutions to the CPM-scheduling problem

In a combinatorial optimization problem under uncertainty, it is never the case that the real instance is exactly the baseline instance that has been solved earlier. The anchorage level is the number of individual decisions with the same value in the solutions of the baseline and the real instances. We consider the case of CPM-scheduling with simple precedence constraints when the job durations of the real instance may be different than those of the baseline instance. We show that, given a solution of the baseline instance, computing a reactive solution of the real instance with a maximum anchorage level is a polynomial problem. This maximum level is called the anchorage strength of the baseline solution with respect to the real instance. We also prove that this latter problem becomes NP-hard when the real schedule must satisfy time windows constraints. We finally consider the problem of finding a proactive solution of the baseline instance whose guaranteed anchorage strength is maximum with respect to a subset of real instances. When each real duration belongs to a known uncertainty interval, we show that such a proactive solution (possibly with a deadline constraint) can be polynomially computed.

Feasibility recovery for the Unit-capacity Constrained Permutation Problem

The Unit-capacity Constrained Permutation Problem with Feasibility Recovery (UCPPFR) is to find a sequence of moves for pieces over a set of locations. From a given location, a piece can be moved to a unit capacity location, i.e. the latter location must be free of its original piece. Each piece has a specific type, and in the end every location must contain a piece of a required type. A piece must be handled using a specific tool, thus incurring a setup cost whenever a tool changeover is required. It could be necessary to use some Steiner locations to find a solution, thus incurring a Steiner cost. The aim of the UCPPFR is to find a sequence of moves with a minimum total setup and Steiner cost. We give a necessary and sufficient condition to check whether an instance admits a solution with no Steiner location. We show that the UCPPFR reduces to finding simultaneously a vertex-disjoint path cover and a shortest common supersequence. Finally, using a compact encoding for solutions and integer linear programming tools, we solve real instances coming from the nuclear power plant fuel renewal problem.

The Unit-capacity Constrained Permutation Problem

Abstract The Unit-capacity Constrained Permutation Problem (UCPP) is to find a sequence of moves for pieces over a set of locations. From a given location, a piece can be moved towards a location with a unit-capacity constraint, i.e. the latter location must be free of its original piece. Each piece has a specific type and at the end every location must contain a piece of a required type. A piece must be handled using a specific tool incurring a setup cost whenever a tool changeover is required. The aim of the UCPP is finding a sequence of moves with a minimum total setup cost. This problem arises in the Nuclear power plant Fuel Renewal Problem (NFRP) where locations correspond to fuel assemblies and pieces to fuel assembly inserts. We first show that the UCPP is NP-hard. We exhibit some symmetry and dominance properties and propose a dynamic programming algorithm. Using this algorithm, we prove that the UCPP is polynomial when two tools and two types are considered. Experimental results showing the efficiency of the algorithm for some instances coming from the NFRP are presented.