Facility Location Problems

1. Facility Location Problems 2000. 1. 19 Eoksu Sim(ses@ultra.snu.ac.kr)

2. Contents • Overview • Facility Location Decisions • Business Logistics Management(3th Edition) Chapter 11 • Single Facility Location • Multiple Facility Location • Dynamic Warehouse Location • Detailed Modeling • A Local Search Heuristics for the Two-Stage Capacitated Facility Location Problem • Advances in Distribution Logistics • Local search heuristic for the cost matrix

3. Facility Location Decisions Chapter 11 in “Business Logistics Management(3th Edition)” Ronald H. Ballou Case Western Reserve University 1992 by Prentice Hall, Inc.

4. Single Facility Location(1/3) • Exact center-of-gravity approach, the grid method, centroid method, p-median method • Transportation cost is the only locational factor, static continuous location model

5. Single Facility Location(2/3) • Determine the X,Y coordinate points for each source and demand point, along with point volumes and linear transportation rates • Approximate the initial location from the center-of-gravity formulas by omitting the distance term di as follows • Calculate di • di를 이용하여 revised 계산 • 변화가 별로 없을 때까지, 또는 이익이 없을 때까지 계산 • 비용 계산

6. Single Facility Location(3/3) • Some of the simplifying assumptions • Demand volumes are frequently assumed to be concentrated at one point(demand cluster) • The basis of variable costs • Total transportation costs usually are assumed to increase proportionately with distance • Straight-line routes are commonly assumed b/w the facility and other network points • Not dynamic

7. Multiple Facility Location • Basic planning question • How many warehouses should there be in the logistics network? How large should they be, and where should they be located? • Which customers should be assigned to which warehouses? Which warehouses should be assigned to which plants, vendors, and ports? • Which products should be stocked in which warehouses? Which products should be shipped directly from plants/vendors/ports to customers? • Solution Methods • Exact Methods • Multiple Center-of-Gravity Approach - min. transportation cost • Mixed-Integer Linear Programming • Simulation Methods • Heuristic Methods

8. Comparison with cost averaged Current best configuration Current best configuration Projected best configuration for some years Next year’s best configuration Final configuration Final configuration Optimal configuration path Dynamic Warehouse Location dynamic over time

9. A Local Search Heuristics for the Two-Stage Capacitated Facility Location Problem Advances in Distribution Logistics Arno Bruns Universitate St. Gallen, 9000 St. Gallen, Switzerland

10. Introduction • The design of the distribution network determines • The flow of goods from the supply to the demand points • Two Stage Capacitated Facility Location Problem(TSCFLP) is finding • The locations of depots from a set of potential depot sites to satisfy given customer demands • The assignment of customers to the selected depots • The product flow from the plants to the depots • Such that the total system cost is minimized • The new heuristic procedure presented in this paper • Render good solutions for such type of problems with short computational time

11. Problem Formulation Assumptions • Solving the TSCFLP • Depots, Customers, Product amounts, Total system cost • Assumptions • Every customer can only be served by exactly one depot • Previously known • The locations of the plants, the potential depot sites and the customers • The supply capacities of the plants, the transshipment capacities of the depot sites and the demand of the customers • Sum(customer demands) <= Sum(plant capacities) • Sum(throughput capacities) >= total demand

12. Problem Formulation Cost & Figure • The cost components • The manufacturing costs at the plants • The transportation cost b/w plants and depot sites • The fixed costs of operating a depot • The throughput cost per unit for every depot site • The costs of satisfying the demand of the customers from the potential depot sites

13. Assignment of customers Fixed operating cost First transportation Depot Capacity Demand Plant Capacity Flow Conservation Problem Formulation Mathematical Formulation

14. Problem Formulation

15. The Local Search Heuristic • The TSCFLP is reduced to a CFLP with single sourcing by eliminating the first distribution stage

16. First transportation cost Rule of thumb by testing The Local Search Heuristic Initialization Phase • - the total cost of serving customer i from a not specified plant through depot site j • customer i’s share of the fixed cost of depot j • For each depot, the min and the max transportation cost • by adding correction term

17. The Local Search Heuristic Construction Phase(1/4) • The aim : to obtain an estimate of the number of depots used in good solutions of the CFLP and the TSCFLP • Determining the Drop Order(nonincreasing values of ) • The average cost of supplying one unit to any customer from depot j • The average cost of shipping one unit from any plant to a customer through depot j

18. The Local Search Heuristic Construction Phase(2/4) • Constructing a Solution for the CFLP • Assign the customers at least cost to one of the depots of M • The sum of the demands assigned to depot j • Determine the set of depots whose capacity is violated • The minimal additional cost to reassign customer i • Reassignment according to nondecreasing values of regardless of the capacity constraints

19. The Local Search Heuristic Construction Phase(3/4) • Reducing Infeasibility • Infeasibility measure u 1. Calculate the infeasibility u of the solution. Stop if the infeasibility u is zero or no exchange of customers can reduce the infeasibility 2. Take a combination of two open depots j1 and j2 and select subset I1 and I2 of the customers assigned to each depot 3. Compute the change uI1,I2 in the infeasibility defined as: 4. If the infeasibility is reduced by exchanging the subsets, uI1,I2<=0, then the next step is performed. Otherwise, go to step 2.

20. The Local Search Heuristic Construction Phase(4/4) 5. Calculate the change in cost caused by such an exchange 6. Determine the ratio 7. Select the exchange, which has the minimum ratio rI1,I2 for a given combination of depots • Estimating the total cost of the TSCFLP • If the cost improves the best upper bound found so far, the solution is saved as the new best one

21. The Local Search Heuristic Improvement Phase • The transportation problem of the first stage (14) to (17) is solved and the present estimate • The Local Search • To explore the neighborhood of the best solution of the CFLP found so far • “One Out”, “Two Out-One In”, “One Out-One In”, “One In”, “One Out-Two In”

22. Final Results • Combining Results • Calculate the exact cost of the solution found

23. Test Problems(1/2) • Generating problems using 100 largest cities of Switzerland • Six problem classes - the number of plants, depots, customers • Eight problem instance in each class - depot capacity, throughput costs, fixed operating cost

24. Test Problems(2/2) • Demand of customers • Cost of serving customer i • Transportation cost • Plant capacities • Depot capacities • Fixed operating cost • Throughput cost

25. Numerical Results(1/2) Comparison of the objective value • DIH algorithm • SunSoft PASCAL 4.0 on a SUN ULTRA workstation(167 MHz) • The lower bounds and the optimal solutions • LP/MIP Solver CPLEX 3.0 (limit of three hours of CPU-time) • 2.37% in average over all problem classes and instances • 1.82% higher on average with optimal value

26. Numerical Results(2/2) Time Fraction • Comparison with other heuristic • Lagrangean heuristic by Klose(1997) • DIH’s value is 1.53% higher • Time fraction

27. Conclusions and Outlook • A typical heuristic is presented • It is capable of solving medium-sized problems • A further research area • The performance of the heuristic on problem instances of realistic size or on real problems of enterprises • The neighborhood of a specific solution • simulated annealing • tabu search

28. References • Advances in Distribution Logistics • Bernhard Fleischmann, Jo A. E. E. van Nunen, M. Grazia Speranza, Paul Stahly(Eds.), Springer-Verlag, 1998 • Business Logistics Management • Ronald H. Ballou, Prentice Hall, 1992 • Strategic Logistic Planning by Means of Simple Plant Location: A Case Study • Ulrich Tushaus, Stefan Wittmann, Advances in Distribution Logistics, 1998 • Obtaining Sharp Lower and Upper Bounds for Two-Stage Capacitated Facility Location Problems • Andreas Klose, Advances in Distribution Logistics, 1998