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### Network Design Problem

### A multi-commodity, multi-plant, capacitated facility location problem: Formulation and efficient heuristic solution

### A multiperiod two-echelon multicommodity capacitated plant location 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)
- A multiperiod two-echelon multicommodity capacitated plant location problem(2000)
- Network design problem in G7 IMS Project(2002)

MAI-LAB Seminar

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

W location problem: C

PW

Plant

Warehouse

Demand

Throughput limit

Plant capacity

Model formulationMAI-LAB Seminar

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 location problem(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

- Introducing the dynamic aspect into the problem

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Introduction location problem(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

- Assumption
- 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 location problem(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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Alternative formulation location problem(1/4)

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Alternative formulation location problem(2/4)

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Alternative formulation location problem(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 location problem(1/3)

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Decomposition of the problem: LR location problem(2/3)

- Analysis of LR
- We will leave constraints (5a) aside
- LR1 can be separated into m subproblems

- We will leave constraints (5a) aside

MAI-LAB Seminar

Decomposition of the problem: LR location problem(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 location problem(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

- The first step looks for capacities each time period t

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Heuristic to construct a feasible solution location problem(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 location problem(3/3) Step2:

- 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

- 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 location problem(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.)

- The transportation cost

MAI-LAB Seminar

Computational study location problem(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 location problem(3/6)

- Table 2
- The size of each test problem for the considered planning horizons

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Computational study location problem(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 location problem(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

Computational study location problem(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

Conclusions location problem

- 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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