The Single Period Coverage Facility Location Problem

1. The Single Period Coverage Facility Location Problem María Albareda-Sambola Elena Fernández Yolanda Hinojosa Justo Puerto Dpt Stat & OR Dpt Stat & ORDpt. App. Economy I Dpt Stat & OR U. Politécnica de Cataluña U. Politécnica de Cataluña Univ. Sevilla Univ. Sevilla

2. Index • Multi-period facility location problems • The single period coverage facility location problem Lagrangean Relaxation I Lagrangean relaxation II • Set partitional formulation • Column Generation • Computational experiments • Conclusions and future research

3.  Multiperiod discrete location problems What are multiperiod discrete location problems ? Discrete location problems where the decision to make is: what to at each time period of given planning horizon T T = {t1, t2, …, t|T|}. tr Planning horizon

4.  Multiperiod discrete location problems What are multiperiod discrete location problems ? Discrete location problems where the decision to make is: what to at each time period of given planning horizon T T = {t1, t2, …, t|T|}. tr+1 Planning horizon

5. State of the art • Ballou, 1968. Dynamic warehouse location analysis. J. of Marketing Research. • Scott, 1971. Dynamic location-allocation systems. Environment and Planning. • Warszawski, 1973. Multi-dimensional location problems. Operations Research Quarterly. • Wesolowsky, 1973. Dynamic facility location. Management Sci.. • Wesolowsky, Truscott, WG, 1975. The multiper. fac. loc-alloc prob with reloc of fac. Management Sci. • Khumawala, Whybark, 1976. Solving the dynamic warehouse location problem. Intl.J. Prod.n Res. • van Roy, Erlenkotter, 1982. A dual-based procedure for dynamic facility location. Management Sci.. • Kelly, Maruckeck, 1984. Planning horizon results for the dynamic warehouse loc. prob. J.Op.Man.. • Galvão, Santibañez-González, 1992, A Lagrangean heur for the p-k-median dyn. loc. Prob., EJOR. • Chardaire, Sutter, Costa, 1996. Solving the dynamic facility location problem. Networks. • Current, Tatick, ReVelle, 1997. Dyn. facility loc. when the total number of facilities is uncertain. EJOR • Saldanha da Gama, Captivo, 1998, A heur. approach for the discrete dynamic loc. problem., Loc. Sci. • Hinojosa, Puerto, F.R. Fernández, 2000. A multiper. two-echelon multicom cap. plant loc. prob. EJOR • Dias, Captivo, Climaco, 2006, Cap.dyn. loc. prob. opening, closure & reop of fac.IMA J. Man Maths. • Melo, Nickel, Saldanha da Gama, 2006, Dynamic multi-commodity capac. fac. Loc.: a math. modeling framework for strategic supply chain planning. C&OR. • Hinojosa, Kalcsics, Nickel, Puerto, Velten,2008, Dynamic supply chain with inventory. C&OR.

6. To the best of our knowledge … • Most of the existing work the assignment variable are continuous. • All the customers must be served at all periods. • The multi-period sequential coverage facility location problem • Albareda-Sambola, Fernández, Hinojosa, Puerto C&OR 08 • Albareda-Sambola, Alonso-Ayuso, Escudero, Fernández, Hinojosa, Pizarro, GOM 08 • The multi-period facility concentrator problem • Saldanha da Gama 08

7. Single Period Coverage Facility Location Problem:Multiperiod Discrete location problem • Facilities are opened from scratch at each period. • A facility only gives service during the period it is opened. (it is possible to open it in later periods). • Customers are only serviced once during the planning horizon. • Health campaigns.

8. The Single Period Coverage Facility Location Problem (SPCFLP) Location-allocation problem to minimize the total costs throughout a finite time planing horizon. t=1 t=2 t=3 • At each time period at least pt facilities are opened. • There is a maximum number (H) of customers that can be served by each open facility. • A facility operates during the time period when it is opened. (it is possible to open it again in subsequent periods). • Each customer is served in one single period throughout the planning horizon.

9. A model for the problem I: set of customers, indexed by i I. J: set of possible locations for plants, indexed by j  J. T: set of time periods, indexed by t  T. ctij. cost of allocating customer i to plant j at time period t. f tj : cost of opening plant j at time period t. pt: minimum number of plants to be opened at period t H: maximum capacity of plants.

10. |T| = 1, pt=0, Capacitated Plant Location Problem. • It is NP-hard (SPLP particular case) • Penalty for not servicing a customer ri:

11. The allocation subproblem If the set of open facilities in period t, Ĵ t , is known, the optimal allocation for the customers can be obtained by solving the following linear program: (AS) Coefficient matrix totally unimodular (It can be transformed into a transportation problem )

12. Lagrangean Relaxation I (capacity constraints) • L1(u) can be descomposed in two independent problems. • L1(u) has the integrality property. • Proposition: d1=max u0 L1(u) coincides with the value of the LP relaxation of M1

13. Lagrangean heuristic • How to obtain a feasible solution? • For a given u, let (x(u),y(u)) denote the optimal solution to L1(u). • tT, define J t (u) = { jJ: ytj(u) =1 } • Solve the allocation subproblem (AS) with J t = J t (u)  tT,

14. Lagrangean Relaxation II (allocation constraints)

15. Lagrangean Relaxation II (allocation constraints) • L2(v) can be solved by inspection: •  tT,which is the cost of opening jJ ? • Order the index set of customers by increasing values of cijt-vi. • s* = min { H , argmax{cirjt - vi }<0 } •  tT do • Order the index set of plants by increasing values of wjt • j(t)* = max {pt, argmax{ wjrt<0}} • Do yjrt =1, r=1,…,j(t)* (and the corresponding assignment)

16. Lagrangean heuristic • d2=max v € R L2(v) is solved with subgradient optimization • How to obtain a feasible solution? • For a given v, let (x(v),y(v)) denote the optimal solution to L2(v). • tT, define J t (v) = { jJ: ytj(v) =1 } • Solve the allocation subproblem (AS) with J t = J t (v)  tT,

17. t j=1 j j=|J|           i              ztkj             A Set Partitioning Formulation for the SPCFLP Sk I: subset of customers. If | Sk | ≤ H it is possible to assign the customers indexed in Sk to a facility j at a given period. zkjt : Binary decision variable that takes value 1 if at period t facility j is opened and customers indexed in Sk are assigned to it. pkjt : fjt + iSkcijt . (cost associated with zkjt ) aik : Coefficient that takes value 1 if customer i is indexed in Sk and 0 otherwise

18. Column Generation for SP Solve SP without generating explicitely all columns. (optimal solution uses very few columns) Reduced cost of zkjt Pricing problem p g r Feasible columns kKiI aik H • Order coefficients cijt-pi by increasing values. {i1, i2,…, i|H|, …, i|I|} • s*= min { H, argmax{cirjt-pi<0} } • âir=1, r=1,…,s*, âi=0, otherwise. If vjt < -fjt + gjt + rt , then rkjt < 0

19. Column Generation Algorithm for SP • Generate an initial set of feasible columns B  K. • While not end do • Solve LSP(B). Let p,g,r denote the optimal dual multipliers. • Forall jJ, tT do • Solve the pricing problem • If vjt < -fjt + gjt + rtgenerate the corresponding column and the variable zkjt • If no column has been generated, end Remark: LSP(B) is not a valid lower bound until the end of the process. Proposition: is a valid lower bound

20. Relationship between Lagrangean Relaxation  Column Generation Proposición: The value of the Lagrangean Dual D2, coincides with the value of the LP relaxation of M2

21. Some numerical results |I|=100, 150, 200, 500, |J|=10, 15, 30, |T|=4, 8, 12: H different values up to 30. 10 instances of each considered combination

22. Percent gaps

23. cpu times (in secs.) Number of instances in the group for which the optimality of the best solution found could not be proved

24. Larger instances with LR2

25. Some concluding comments • The Single Period Coverage Facility Location Problem • Comparison of formulations and possible solution techniques • Exploit the properties and the structure of the problems • Good Computational results with LR2 • Improve the convergence of column Generation (stabilization metods) • More sophisticated problems (general capacity constraints, …)

26. Thank you for your attention