Final Report Summary - FLOUE (Facility location optimization under disruption and in an uncertain environment)
Facility location optimisation is an optimisation problem in operations research, particularly important in network design decisions. A 'Facility' is understood here in a general sense as an entity providing any kind of services used by some customers, such as: a hospital, a cash dispenser, a nuclear power centre, etc.
Given some facilities, with known capacities, some candidate location sites, and customer demands, the problem consists in locating facilities and assigning customers to them in such a way to satisfy all demands, without exceeding capacities, and at a minimal fixed and variable costs. Usually, it is assumed that the decision problem occurs in some deterministic and reliable conditions. The customer demands and different costs are predictable, and facilities are so reliable that customers always get the service. One may cite as fundamental problems in this category the well known p-median problem (p-MP) and the capacitated facility location problem (CFLP).
Over the life of a logistic network, the demands, the transportation costs or the facility capacities, may experience disruptions or fluctuations. By which, a network design that looks very good taking into account deterministic evaluation may be quite poor if the situation were to change.
The FLOUE project was focused on providing modelling and resolution techniques by which unreliable versions of facility location problems may be mathematically formulated, theoretically analysed and numerically solved:
i. Median problem with unreliable facilities (MPUF) with independent disruption probabilities : A nonlinear approach
The first part of the project was concerned by MPUF which is the unreliable version of p-MP. It consists in finding the best locations of p facilities, with theoretically infinite capacities, that may be disrupted with known independent probabilities. The problem has been introduced by Berman, Krass and Menezes and up to the beginning of the project no exact method, usable for any type of probabilities, was developed. We have proposed the first pseudo-boolean formulation of this problem able to find the optimal value of short size instances involving any type of probabilities. The contribution corresponding to this part has been submitted for publication in the review OR Spectrum (ORSP-S-12-00233-2).
ii. Median problem with unreliable facilities (MPUFsp) with correlated disruption probabilities: A stochastic programming approach
In a second step of the project, we have considered a more general MPUF version, noticed here MPUFsp, in which the assumption of probability independence has been relaxed. This is MPUFsp disruption probabilities may be dependent and correlation is introduced. Formulating the problem as a two-stage stochastic problem, we have proved that all known asymptotic and submodularity results for MPUF remain valid for MPUFsp. As a practical consequence, the simple 'myopic' (or greedy) heuristic used for the standard p-median problem provides for MPUFsp a good heuristic solution with a guaranteed maximal deviation to the optimal value. Based on the two-stage stochastic programme, we have also proposed a general decomposition methodology allowing to derive a lower bound of the optimal value, and a multistart heuristic solution. Theoretically, our heuristic provides a solution at least better or equal to the solution found by the myopic algorithm. The methodology has been experimented on randomly generated instances containing up to 100 nodes, as well as on a real-life instance dealing with the location of the Toronto hospitals. The paper corresponding to this work has been submitted for publication in Mathematics of Operations Research (ID MOR-2013-012).
iii. Unreliable capacitated facility location problem (UCFLP)
In a third step, a new general problem containing all the previous ones, as well as the unreliable version of CFLP has been introduced. The problem is UCFLP. In UCFLP, a set of facilities of different designs (or types) is considered, each design with an associated fixed cost, limited unreliable capacity, and a budget constraint restricting the number of facilities that may be opened. We have to decide what facilities to choose and where to locate them in order to maximise the expected shipping profit. As for MPUFsp, we formulate the problem as a two-stage stochastic program, and show that the submodular property also holds. Then, the greedy heuristic used for MPUFsp also fulfilled a solution with a garanteed bound. This work has been presented in an invited seminar. It remains some numerical results, before publication, corresponding to the application of the decomposition methodology proposed for MPUFsp.
Notice finally that the scenarios approach used for MPUFsp and UCFLP can be extended similarly to study problems including both unreliable facilities and uncertain demands without changing the theoretical results proved.
Given some facilities, with known capacities, some candidate location sites, and customer demands, the problem consists in locating facilities and assigning customers to them in such a way to satisfy all demands, without exceeding capacities, and at a minimal fixed and variable costs. Usually, it is assumed that the decision problem occurs in some deterministic and reliable conditions. The customer demands and different costs are predictable, and facilities are so reliable that customers always get the service. One may cite as fundamental problems in this category the well known p-median problem (p-MP) and the capacitated facility location problem (CFLP).
Over the life of a logistic network, the demands, the transportation costs or the facility capacities, may experience disruptions or fluctuations. By which, a network design that looks very good taking into account deterministic evaluation may be quite poor if the situation were to change.
The FLOUE project was focused on providing modelling and resolution techniques by which unreliable versions of facility location problems may be mathematically formulated, theoretically analysed and numerically solved:
i. Median problem with unreliable facilities (MPUF) with independent disruption probabilities : A nonlinear approach
The first part of the project was concerned by MPUF which is the unreliable version of p-MP. It consists in finding the best locations of p facilities, with theoretically infinite capacities, that may be disrupted with known independent probabilities. The problem has been introduced by Berman, Krass and Menezes and up to the beginning of the project no exact method, usable for any type of probabilities, was developed. We have proposed the first pseudo-boolean formulation of this problem able to find the optimal value of short size instances involving any type of probabilities. The contribution corresponding to this part has been submitted for publication in the review OR Spectrum (ORSP-S-12-00233-2).
ii. Median problem with unreliable facilities (MPUFsp) with correlated disruption probabilities: A stochastic programming approach
In a second step of the project, we have considered a more general MPUF version, noticed here MPUFsp, in which the assumption of probability independence has been relaxed. This is MPUFsp disruption probabilities may be dependent and correlation is introduced. Formulating the problem as a two-stage stochastic problem, we have proved that all known asymptotic and submodularity results for MPUF remain valid for MPUFsp. As a practical consequence, the simple 'myopic' (or greedy) heuristic used for the standard p-median problem provides for MPUFsp a good heuristic solution with a guaranteed maximal deviation to the optimal value. Based on the two-stage stochastic programme, we have also proposed a general decomposition methodology allowing to derive a lower bound of the optimal value, and a multistart heuristic solution. Theoretically, our heuristic provides a solution at least better or equal to the solution found by the myopic algorithm. The methodology has been experimented on randomly generated instances containing up to 100 nodes, as well as on a real-life instance dealing with the location of the Toronto hospitals. The paper corresponding to this work has been submitted for publication in Mathematics of Operations Research (ID MOR-2013-012).
iii. Unreliable capacitated facility location problem (UCFLP)
In a third step, a new general problem containing all the previous ones, as well as the unreliable version of CFLP has been introduced. The problem is UCFLP. In UCFLP, a set of facilities of different designs (or types) is considered, each design with an associated fixed cost, limited unreliable capacity, and a budget constraint restricting the number of facilities that may be opened. We have to decide what facilities to choose and where to locate them in order to maximise the expected shipping profit. As for MPUFsp, we formulate the problem as a two-stage stochastic program, and show that the submodular property also holds. Then, the greedy heuristic used for MPUFsp also fulfilled a solution with a garanteed bound. This work has been presented in an invited seminar. It remains some numerical results, before publication, corresponding to the application of the decomposition methodology proposed for MPUFsp.
Notice finally that the scenarios approach used for MPUFsp and UCFLP can be extended similarly to study problems including both unreliable facilities and uncertain demands without changing the theoretical results proved.