On solving the multi-period location-assignment problem under uncertainty

Transcription

1 On solving the multi-period location-assignment problem under uncertainty María Albareda-Sambola 2 Antonio Alonso-Ayuso Laureano Escudero Elena Fernández 2 Celeste Pizarro Romero. Departamento de Estadística e Investigación Operativa Universidad Rey Juan Carlos, Madrid 2. Departamento de Estadística e Investigación Operativa Universidad Politécnica de Cataluña, Barcelona 3th Combinatorial Optimization Workshop Aussois (France), January 2-7, 2009

2 Outline Introduction 2 Problem description 3 Uncertainty 4 Impulse-Step variables based (DEM) 5 Algorithmic framework 6 Computational comparison 7 Conclusions

3 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.

4 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.

5 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

6 The problem t = t =2 t =3 a a a b 3 b 3 b c c c n =2,p =2 f facilities customers n 2 =4,p 2 = n 3 =5,p 3 =0 Consider a network including a set of facilities and a set of customers

7 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

8 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

9 The problem Costs Assigning a customer to a facility at a given period incurs a assignment cost, cij t, even if the customer does not have a need for service in this period. There is a setup depreciation cost, fi t for the open facilities. the penalty cost, ρ j, for the customers not served in time by the facilities

10 Uncertainty: Modeling via scenario tree scenario scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 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.

11 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 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.

12 Uncertainty in the problem Variables Non-anticipativity principle (Rockafellar and Wets) If two different scenarios s and s are identical until stage t as to as the disponible information in that stage, then the decisions (variables) in both scenarios must be the same too until stage t. scenario scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9

13 Uncertainty in the problem Impulse-Step variables based formulation 0 variables 8, if facility i is open by time period t(g) under >< y g = scenario group g i >: 0, otherwise i I, g G : t(g) T and 8, if customer j is assigned to facility i at time >< x g = period t(g) under scenario group g ij >: 0, otherwise i I, j J, g G. 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.

14 Impulse-Step variables based formulation (DEM) Pure 0 Model Objective function min X i I X X g G j J fi 0 yi 0 + X» X w g g G i I w g ρ j d g j + min X i I» f t(g) i (y g y γ(g) )+ X i i j J X w g (f t(g) i g G:t(g) T c t(g) ij x g ij + X j J f t(g)+ )y g + X i i ρ j d g j ` X i I X w g g G j J c g ij x g ij x g = ij () Note where c g ij = c t(g) ij ρ j d g j γ(g), inmediate scenario group to group g

15 Impulse-Step variables based formulation (DEM) Pure 0 Model Objective function min X i I X X g G j J fi 0 yi 0 + X» X w g g G i I w g ρ j d g j + min X i I» f t(g) i (y g y γ(g) )+ X i i j J X w g (f t(g) i g G:t(g) T c t(g) ij x g ij + X j J f t(g)+ )y g + X i i ρ j d g j ` X i I X w g g G j J c g ij x g ij x g = ij () Note where c g ij = c t(g) ij ρ j d g j γ(g), inmediate scenario group to group g

16 Impulse-Step variables based formulation (DEM) Pure 0 Model Constraints X X i I j J X X i I X (y g i i I x γ(g) ij i I X i I x g n g g G : t(g) < T (2) ij x g j J, g G : t(g) < T (3) ij x g = j J, g G ij T (4) X x g ij i I x g y γk (g) ij i j J, g G : t(g) > (5) i I, j J, g G, where k = min{t(g), τ} (6) y γ(g) ) = p t g G : t(g) T (7) i X i I y γ(g) i y 0 i = p 0 (8) y g i i I, g G : t(g) T (9) x g {0, } i I, j J, g G (0) ij y tg i {0, } i I, g G : t(g) T ()

17 Splitting variables representation For a general problem: min X p ω`c T x ω +q ωt y ω ω Ω s. t. Ax ω = b T ω x ω +Wy ω = h ω ω Ω x ω x ω+ = 0 ω Ω x ω, y ω {0, } ω Ω x y x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 x 6 y 6 A T W I I A T 2 W I I A T 3 W I I A T 4 W I I A T 5 W I I A T 6 W I I = b = h = 0 = b = h 2 = 0 = b = h 3 = 0 = b = h 4 = 0 = b = h 5 = 0 = b = h 6 = 0

18 Branch-and-Fix Coordination Non-anticipativity constraints are relaxed: sc. sc. 2 sc. 3 sc. 4

19 Branch-and-Fix Coordination Non-anticipativity constraints are relaxed: sc. sc. 2 sc. 3 sc. 4

20 Branch-and-Fix Coordination Non-anticipativity constraints are relaxed: s Independent problems Each problem is solved by a Branch and Fix, coordinatting the decisions sc. 2 sc. 2 3 sc. 3 sc

21 Branch-and-Fix Coordination Fix-and-Relax Coordination Fix-and-Relax Introduced by Dillebenger et al. in 994. See also Escudero and Salmerón in It is an heuristic approach to solve multi-level linear integer problems: Initially only variables of the first level are 0- defined. In successive stages, previous level variables are fixed, and current level variables are declared integer. Fix-and-Relax Coordination (FRC) Combine Fix-and-Relax with Branch-and-Fix Coordination: Non-anticipativity conditions are relaxed. Each scenario group is solved by Fix and Relax. Decisions are coordinated via Branch-and-Fix Coordination approach.

22 Branch-and-Fix Coordination Fix-and-Relax Coordination Fix-and-Relax Introduced by Dillebenger et al. in 994. See also Escudero and Salmerón in It is an heuristic approach to solve multi-level linear integer problems: Initially only variables of the first level are 0- defined. In successive stages, previous level variables are fixed, and current level variables are declared integer. Fix-and-Relax Coordination (FRC) Combine Fix-and-Relax with Branch-and-Fix Coordination: Non-anticipativity conditions are relaxed. Each scenario group is solved by Fix and Relax. Decisions are coordinated via Branch-and-Fix Coordination approach.

23 Branch-and-Fix Coordination FRC scheme Fix-and-Relax submodel Given V... V K a partition of K elements of the set of the variables V, problem IP : min cx x X s. t. x j {0, } j V k, k =,..., K, This problem can be approximated by the model It is the so-called FR level k IP k : min x X s. t. x j = x j j V k, k < k, x j {0, } j V k, x j [0, ] j V k, k < k,

24 FRC scheme FR levels t = t = 2 t = 3 t = 4 t = t = 2 t = 3 t = C lu ster C lu ster C lu ster C lu ster (a) Scenario tree (b ) C lu ster stru ctu re

25 FRC scheme t = t = 2 t = 3 t = 4 t = t = 2 t = 3 t = 4 t = t = 2 t = 3 t = 4 Lev el Lev el Lev el Lev el Lev el Lev el Lev el Lev el 8 t = t = 2 t = 3 t = Lev el Lev el 0 Lev el Lev el n n N o des w h o se n o n -a n ticip a tiv ity co n stra in ts a re n o t rela x ed N o des w h o se v a ria b les h a v e b een 0 - fi x ed N o des w h o se v a ria b les h a v e b een 0 - in teg er defi n ed Lev el 3 Lev el n N o des w h o se v a ria b les h a v e b een 0 - co n tin u o u s defi n ed Lev el

26 FRC scheme Branching strategy We have chosen the depth first strategy for the TNF branching selection The criterion for branching consists of choosing the candidate TNF with the smallest Lagrangean Substitution value among the two sons of the last branched TNF. We have chosen two largest small deterioration strategies for the selection of the branching variable.

27 On candidate TNF bounding The bounding of a given candidate TNF, J f, f F, can be obtained by using Lagrangian Decomposition (LD): Z D (µ) = min j J f w j (c j x j + a j y j ) + j J f µ j (x j x j+ ) s. t. Ax j + By j = b j j J f 0 x j, y j 0 j J f, Our aim is to obtain the bound Z D (µ ), where µ = argmax{z D (µ)}. Note The number of Lagrange multipliers depends on the number of non yet branched on/fixed common variables in vector x j and the number of nodes, J f, in the family.

28 On candidate TNF bounding The bounding of a given candidate TNF, J f, f F, can be obtained by using Lagrangian Decomposition (LD): Z D (µ) = min j J f w j (c j x j + a j y j ) + j J f µ j (x j x j+ ) s. t. Ax j + By j = b j j J f 0 x j, y j 0 j J f, Our aim is to obtain the bound Z D (µ ), where µ = argmax{z D (µ)}. Note The number of Lagrange multipliers depends on the number of non yet branched on/fixed common variables in vector x j and the number of nodes, J f, in the family.

29 BFC algorithm ν

30 Why we use this modelization for the problem? A study of the deterministic problem has been done We have done a computational comparison among: Impulse variables based model 2 Step variables based model 3 Impulse-step variables based model

31 Computational Experience Deterministic Version Implemented in experimental C++ code. CPLEX v for solving each instance. Pentium IV,.8Ghz, 52 RAM Microsoft Visual Studio C++ compiler v6.0. J I T instances for each row: Total 60 instances

32 Models dimensions Impulse model Impulse-step model Step model Instance nv nr nel dens dens dens nv nr nel nv nr nel (%) (%) (%) E4-P30-C E8-P30-C E2-P30-C E4-P30-C E8-P30-C E2-P30-C

33 Computational experience

34 Computational experience

35 Graphical comparison of the results obtained with the different models Model M with respect to Model M2 % time dev % o.f. dev.

36 Graphical comparison of the results obtained with the different models Model M3 with respect to Model M % time dev % o.f. dev.

37 Conclusions and Future work A variation of the multi-period incremental service facility location problem has been presented. A variation with uncertainty in the demand and the set of periods when each customer requieres service is presented by using Stochastic Programming. Three 0 equivalent formulations are proposed for the deterministic models, based on the impulse and step variables approaches. An intensive computational experimentation has been performed for the deterministic models. Impulse-Step variables based formulation shows better results. Work-in-progress: Computational experience for the stochastic model