On solving the multi-period location-assignment problem under uncertainty ıa Albareda-Sambola 2 Antonio Alonso-Ayuso 1 Laureano Escudero 1 Mar´ andez 2 Celeste Pizarro Romero 1 Elena Fern´ 1. Departamento de Estad´ ıstica e Investigaci´ on Operativa Universidad Rey Juan Carlos, Madrid 2. Departamento de Estad´ ıstica e Investigaci´ on Operativa Universidad Polit´ ecnica de Catalu˜ na, Barcelona 13th Combinatorial Optimization Workshop Aussois (France), January 12-17, 2009
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Outline Introduction 1 Problem description 2 Uncertainty 3 Impulse-Step variables based (DEM) 4 Algorithmic framework 5 Computational comparison 6 Conclusions 7
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational The MISFLP problem Given a time horizon, a set of customers and a set of facilities (e.g., production plants), Multi-period Incremental Service Facility Location Problem (MISFLP) is concerned with: locating the facilities within a given discrete set of potential sites and assigning the customers to the facilities along given periods in a time horizon.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational The MISFLP problem Assumptions Ensuring at each single period t the service of a minimum number of customers, say n t . The allocation of any customer to the servers might change in different periods. Once a customer is served in a time period it must be served at any subsequent period. Once a facility is opened it remains open until the end of the time horizon.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Variation of the MISFLP problem We present a variation of the MISFLP where: Each customer needs to be serviced only in a subset of the periods of the time horizon we assume that this set of periods is known for each customer There is uncertainty in some parameters of the problem
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational The problem t = 1 t = 2 t = 3 1 1 1 a a a 2 2 2 b 3 b 3 b 3 4 4 4 c c c 5 5 5 n 1 = 2 , p 1 = 2 n 2 = 4 , p 2 = 1 n 3 = 5 , p 3 = 0 facilities customers f 1 Consider a network including a set of facilities and a set of customers
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational The problem For the costumers: The allocation of a customer to the facilities might change at different time periods but Once a customer is assigned in a given time period, he must continue to be assigned to one facility. A costumer cannot be assigned to more than one facility at each period All customers must be found to have been assigned at the end of the time horizon At each single period exactly p t facilities are opened at least n t new customers are covered
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational The problem For the costumers: The allocation of a customer to the facilities might change at different time periods but Once a customer is assigned in a given time period, he must continue to be assigned to one facility. A costumer cannot be assigned to more than one facility at each period All customers must be found to have been assigned at the end of the time horizon At each single period exactly p t facilities are opened at least n t new customers are covered
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational The problem Costs Assigning a customer to a facility at a given period incurs a assignment cost, c t ij , even if the customer does not have a need for service in this period. There is a setup depreciation cost, f t for the open facilities. i the penalty cost, ρ j , for the customers not served in time by the facilities
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Uncertainty: Modeling via scenario tree scenario 1 scenario 1 scenario 2 scenario 2 scenario 3 scenario 3 scenario 4 scenario 4 scenario 5 scenario 5 scenario 6 scenario 6 scenario 7 scenario 7 scenario 8 scenario 8 scenario 9 scenario 9 Two stages Multi stages Scenario is an execution of uncertain and deterministic parameters along different stages of the temporal horizon. Scenario group for a given stage is the set of scenarios with the same realization of the uncertain parameters up to the stage. Scenario tree scheme is a technique used to model and interpret the uncertainty.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Parameters Uncertainty in the problem The most important uncertainty that we find in this problem is the number of costumers that will need to be serviced in each time period Parameters d g j , coefficient that takes the value 1 or 0 depending on whether or not customer j is available for being serviced at time period t ( g ) under scenario group g , ∀ j ∈ J . n g , minimum number of customers to be serviced in time period t ( g ) under scenario group g .
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Uncertainty in the problem Variables Non-anticipativity principle ( Rockafellar and Wets ) ′ are identical until stage t as to as the If two different scenarios s and s disponible information in that stage, then the decisions (variables) in both scenarios must be the same too until stage t . scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Uncertainty in the problem Impulse-Step variables based formulation 0–1 variables 8 1 , if facility i is open by time period t ( g ) under > > < y g scenario group g ∀ i ∈ I , g ∈ G : t ( g ) ∈ T ∗ i = > > 0 , otherwise : and 8 1 , if customer j is assigned to facility i at time > > < x g period t ( g ) under scenario group g ∀ i ∈ I , j ∈ J , g ∈ G − . ij = > > 0 , otherwise : where T ∗ = { t ∈ T : t ≤ |T | − τ } and G − ≡ G \ { 0 } Note The x –variables still are impulse variables, but the y –variables are step variables.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Impulse-Step variables based formulation (DEM) Pure 0–1 Model Objective function »X ´– “ f t ( g ) ( y g i − y γ ( g ) c t ( g ) x g ” ρ j d g x g X f 0 i y 0 X w g X X X ` min i + )+ + 1 − = i i ij ij j ij i ∈I g ∈G − i ∈I j ∈J j ∈J i ∈I » – w g ρ j d g w g ( f t ( g ) − f t ( g )+ 1 ) y g c g ij x g X X X X X w g X j + min i + i i ij g ∈G : t ( g ) ∈T ∗ g ∈G j ∈J i ∈I g ∈G − j ∈J (1) Note where c g ij = c t ( g ) − ρ j d g ij j γ ( g ) , inmediate scenario group to group g
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Impulse-Step variables based formulation (DEM) Pure 0–1 Model Objective function »X ´– “ f t ( g ) ( y g i − y γ ( g ) c t ( g ) x g ” ρ j d g x g X f 0 i y 0 X w g X X X ` min i + )+ + 1 − = i i ij ij j ij i ∈I g ∈G − i ∈I j ∈J j ∈J i ∈I » – w g ρ j d g w g ( f t ( g ) − f t ( g )+ 1 ) y g c g ij x g X X X X X w g X j + min i + i i ij g ∈G : t ( g ) ∈T ∗ g ∈G j ∈J i ∈I g ∈G − j ∈J (1) Note where c g ij = c t ( g ) − ρ j d g ij j γ ( g ) , inmediate scenario group to group g
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Impulse-Step variables based formulation (DEM) Pure 0–1 Model Constraints ∀ g ∈ G − : t ( g ) < |T | x g ij ≥ n g X X (2) i ∈I j ∈J ∀ j ∈ J , g ∈ G − : t ( g ) < |T | x g X ij ≤ 1 (3) i ∈I x g X ij = 1 ∀ j ∈ J , g ∈ G |T | (4) i ∈I x γ ( g ) x g X X ≤ ∀ j ∈ J , g ∈ G : t ( g ) > 1 (5) ij ij i ∈I i ∈I ij ≤ y γ k ( g ) x g ∀ i ∈ I , j ∈ J , g ∈ G − , where k = min { t ( g ) , τ } (6) i ( y g i − y γ ( g ) ) = p t ∀ g ∈ G − : t ( g ) ∈ T ∗ X (7) i i ∈I y 0 = p 0 X (8) i i ∈I y γ ( g ) ≤ y g ∀ i ∈ I , g ∈ G − : t ( g ) ∈ T ∗ (9) i i x g ∀ i ∈ I , j ∈ J , g ∈ G − ij ∈ { 0 , 1 } (10) y tg ∀ i ∈ I , g ∈ G : t ( g ) ∈ T ∗ ∈ { 0 , 1 } (11) i
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