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Network Design Problem. Manufacturing Automation & Integration Lab. 2002. 02. 21 Eoksu Sim([email protected]). Contents. A multi-commodity, multi-plant, capacitated facility location problem: Formulation and efficient heuristic solution(1998)

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Network design problem

Network Design Problem

Manufacturing Automation & Integration Lab.

2002. 02. 21

Eoksu Sim([email protected])


Contents

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


Network design problem

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


Model formulation

WC

PW

Plant

Warehouse

Demand

Throughput limit

Plant capacity

Model formulation

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Solution procedure

Solution procedure

Lagrangian multiplier

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A multiperiod two echelon multicommodity capacitated plant location problem

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


Introduction 1 2

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

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Introduction 2 2

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

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The model 1 5

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

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The model 2 5

The model(2/5)

  • Index, parameters

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The model 3 5

The model(3/5)

  • Decision variables

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The model 4 5

PW

WC

Warehouse

Plant

The model(4/5)

  • Objective function

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The model 5 5

Demand

Capacity limit

Minimum number

The model(5/5)

  • Constraints

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Alternative formulation 1 4

Alternative formulation(1/4)

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Alternative formulation 2 4

Alternative formulation(2/4)

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Alternative formulation 3 4

Alternative formulation(3/4)

사이트 사용/폐쇄 시점의 용량

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Alternative formulation 4 4

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

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Decomposition of the problem lr 1 3

Decomposition of the problem: LR(1/3)

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Decomposition of the problem lr 2 3

Decomposition of the problem: LR(2/3)

  • Analysis of LR

    • We will leave constraints (5a) aside

      • LR1 can be separated into m subproblems

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Decomposition of the problem lr 3 3

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

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Heuristic to construct a feasible solution 1 3

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

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Heuristic to construct a feasible solution 2 3

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)

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Heuristic to construct a feasible solution 3 3

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

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    Computational study 1 6

    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.)

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    Computational study 2 6

    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

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    Computational study 3 6

    Computational study(3/6)

    • Table 2

      • The size of each test problem for the considered planning horizons

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    Computational study 4 6

    Computational study(4/6)

    • The results for the considered planning horizons

      • At least 10 instances have been solved

      • The average results are reported

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    Computational study 5 6

    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

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    Computational study 6 6

    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

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    Conclusions

    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

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