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Network Design Problem

Network Design Problem. Manufacturing Automation & Integration Lab. 2002. 02. 21 Eoksu Sim(ses@ultra.snu.ac.kr). Contents. A multi-commodity, multi-plant, capacitated facility location problem: Formulation and efficient heuristic solution(1998)

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Network Design Problem

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  1. Network Design Problem Manufacturing Automation & Integration Lab. 2002. 02. 21 Eoksu Sim(ses@ultra.snu.ac.kr)

  2. Contents • A multi-commodity, multi-plant, capacitated facility location problem: Formulation and efficient heuristic solution(1998) • A multiperiod two-echelon multicommodity capacitated plant location problem(2000) • Network design problem in G7 IMS Project(2002) MAI-LAB Seminar

  3. A multi-commodity, multi-plant, capacitated facility location problem: Formulation and efficient heuristic solution Computers & Operational Research, Vol. 25, No. 10, pp. 869-878, 1998 Hasan Pirkul* and Vaidyanathan Jayaraman * * * School of Management, University of Texas, Dallas, Richarddon, TX 830688, USA ** College of Business and Economics, Washington State Unicersity, Vancouver, WA 98686, USA

  4. WC PW Plant Warehouse Demand Throughput limit Plant capacity Model formulation MAI-LAB Seminar

  5. Solution procedure Lagrangian multiplier MAI-LAB Seminar

  6. A multiperiod two-echelon multicommodity capacitated plant location problem EJOR, 123 (2000) 271-291 Y. Hinojosa*, J. Puerto**, F.R. Fernandez** *Dept. Economia Aplicada I. Fac. De Ciencias Econom. Y Empresar., Universidad de Sevilla, 41018, Sevilla, Spain **Dept. Estadistica e IO, Universidad de Sevilla, Sevilla, Spain

  7. Introduction(1/2) • A discrete plant location problem • Uncapacitated plant location problems(UPLP) • Capacitated plant location problems(CPLP) • MIP formulation but NP-hard • Focus of this paper • Introducing the dynamic aspect into the problem • Not only the transportation plan but also the time-staged establishment of the facilities • The assumption of a certain structure in thetransportation pattern • The transportation follows a two-step path MAI-LAB Seminar

  8. Introduction(2/2) • A multiperiod two-echelon multi commodity capacitated location problem • Assumption • The capacities of plants and warehouses, as well as demands and transportation costs change over T time periods • We do not consider holding decisions • The formulation permits both the opening of new facilities and the closing of existing ones • A very large MIP • 50 customers, 20 warehouses, 20 plants, 2 products, 4 time period  11,360 variables and 764 constraints • An alternative approach • A Lagrangean relaxation scheme incorporating a dual ascent method together with a heuristic construction phase method MAI-LAB Seminar

  9. The model(1/5) • The objective • Minimizing the total cost for meeting demands of the different products specified over time at various customer locations • Hypotheses • No holding decisions • The set of customers and products, together with the feasible locations • Sites • Once closed they cannot be reopened • If they were open they would not be closed • A minimum number of plants and warehouses must be open at the first and last time period • A minimum coverage of the demand at the beginning and after the time horizon • Sites’ limited capacity which depends on the time period MAI-LAB Seminar

  10. The model(2/5) • Index, parameters MAI-LAB Seminar

  11. The model(3/5) • Decision variables MAI-LAB Seminar

  12. PW WC Warehouse Plant The model(4/5) • Objective function MAI-LAB Seminar

  13. Demand Capacity limit Minimum number The model(5/5) • Constraints MAI-LAB Seminar

  14. Alternative formulation(1/4) MAI-LAB Seminar

  15. Alternative formulation(2/4) MAI-LAB Seminar

  16. Alternative formulation(3/4) 사이트 사용/폐쇄 시점의 용량 MAI-LAB Seminar

  17. Alternative formulation(4/4) • Problem P’ • A MIP problem which includes as a particular instance the UPLP • NP-hard • Cannot expect to solve exactly large sizes of problem P’ in polynomial time • A heuristic method to solve P’ for those instances • (1) using a Lagrangean relaxation • (2) using an “ad hoc” procedure obtaining a feasible solution from the solutions of the relaxed problems MAI-LAB Seminar

  18. Decomposition of the problem: LR(1/3) MAI-LAB Seminar

  19. Decomposition of the problem: LR(2/3) • Analysis of LR • We will leave constraints (5a) aside • LR1 can be separated into m subproblems MAI-LAB Seminar

  20. Decomposition of the problem: LR(3/3) • The subgradient method • To get a lower bound for v(P’) • The selection of the initial set of multipliers is crucial • The quality of the first solution depends very much on this choice • The following set of initial multipliers MAI-LAB Seminar

  21. Heuristic to construct a feasible solution(1/3) • The following scheme that consists of two different steps • The first step looks for capacities each time period t • Both for plants and warehouses • Once these capacities have been established for meeting the demand  second step • The second step looks for the best transportation plan between plants and warehouses and between warehouses and customers MAI-LAB Seminar

  22. Heuristic to construct a feasible solution(2/3) • Compute the total capacity of all the open warehouses as well as the total demand in t. • Ct : the difference b/w the demand and the capacity in this time period • Arrange in nonincreasing sequence with respect to Ct all those time periods • Where the capacity of the warehouses is not enough to cover the demand • Compute I(j,to) MAI-LAB Seminar

  23. Heuristic to construct a feasible solution(3/3) • The greater Ct, the larger the number of warehouses that have to be opened and this affects the remaining time periods • The process consists of opening those warehouses in nondecreasing order of the index I(j,to) • Until the demand in that time period is fulfilled • The same procedure has to be applied to the opening of plants • Step2: • Replace the values of these binary variables in the formulation of P’. • P’ is a continuous linear program that can be easily solved MAI-LAB Seminar

  24. Computational study(1/6) • Experiment • A subcomplex(virtual machine) with six processors and 2Gb of RAM of a machine HP Exemplar SPP-1000 Series • C++, Subroutines of IMSL to solve linear programs • CPLEX 6.0 • The data(randomly) • The transportation cost • Being proportional to the Euclidean distance among the location of final customers and warehouses, and plants and warehouses respectively. • The locations of all the facilities • Uniformly distributed in the square [1,15]×[1,15] • All these costs • An increment b/w 10% and 25% in each time period(inflation rate, etc.) MAI-LAB Seminar

  25. Computational study(2/6) • The minimum number of plants and warehouses open at the first and the last time period • Depends on the difference b/w the total demand requested in each time period and the average of the capacity of warehouses in that time period • Table 1 • The test problems that have been solved MAI-LAB Seminar

  26. Computational study(3/6) • Table 2 • The size of each test problem for the considered planning horizons MAI-LAB Seminar

  27. Computational study(4/6) • The results for the considered planning horizons • At least 10 instances have been solved • The average results are reported MAI-LAB Seminar

  28. Computational study(5/6) • H-Gap : the percentage gap b/w the feasible solution obtained applying the heuristic and the greatest lower bound obtained in each instance b/w the continuous and the Lagrangean relaxation of P’ • Worst-H : the worst result used to compute the average H-Gap. • N : the number of iterations needed by the heuristic algorithm • CPU-H : the average time in seconds used for these iterations • E-Gap : the percentage gap with respect to the exact solution of the problem obtained using CPLEX • Worst-E : the worst result used to compute the average E-Gap • CPU-E : the average time in seconds used by CPLEX to solve the problems MAI-LAB Seminar

  29. Computational study(6/6) • The reason for the missing values • To obtain the exact solutions CPLEX solver needs prohibitive computational times • The heuristic method • Provides solutions whose gap(H-Gap) range b/w 0.24% and 5%. • It is worth noting that these gaps are computed with respect to lower bounds of the optimal values MAI-LAB Seminar

  30. Conclusions • A heuristic method to solve problem • Based on a Lagrangean relaxation which provides solutions(possibly infeasible for the original problem) but verifying the integrality constraints • Computational results • Show the gaps b/w the solutions proposed and lower bounds of the optimal solutions and exact solutions MAI-LAB Seminar

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