Diapositivo 1

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EURO Working Group on Locational AnalysisEWGLA
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A JOINT REPLENISHMENT INVENTORY-LOCATION
MODEL FRANCISCO SILVA CEEAplA University of
the Azores Ponta Delgada, Portugal fsilva_at_notes.ua
c.pt LUCIA GAO UMass Boston College of
Management, Boston, USA lucia.silva-gao_at_umb.edu
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EURO Working Group on Locational AnalysisEWGLA
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A JOINT REPLENISHMENT INVENTORY-LOCATION
MODEL We introduce a distribution center
location model that incorporates joint
replenishment inventory costs at the distribution
centers. The model is formulated as a Facility
Location Problem (FLP) which objectively
considers not only location specific costs but
also inventory replenishment costs. We propose a
Greedy Randomized Adaptive Search Procedure
(GRASP) to solve the problem. For larger problems
we propose a Heuristics Concentration algorithm

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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• Abstract

• The problem

• Joint Replenishment

The model we propose results from an integrated
approach between location and inventory
decisions. We assume that the distribution
centers are to be jointly replenished. The model
may be more suitable in cases where the
distribution centers share the same
transportation facility and where a coordinated
joint replenishment policy may lead to important
savings in transportation costs.

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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NOTATION
A Major replenishment cost (independent of the
DCs included in the order). aj - Variable
cost of including DC j in the joint order, with j
1, 2,... , n. hj Holding cost (maintenance)
of a unit of the item in DC j. dj Demand at
DC j by unit of time, constant and known. T
Joint replenishment cycle time, i.e., period of
time elapsing between each revision of the stocks
(Basic Cycle Time). kj Frequency of
replenishment at DC j, assuming discrete values
which are multiples of T. K Vector composed of
the elements kj (k1, k2, ... , kn). Ctr
Relevant average total costs of the joint
replenishment system by unit of time.

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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Joint Replenishment Problem (JRP)
Facility Location Problem (FLP)

• Abstract

• The problem

Number of DC (n)

• Joint Replenishment

Basic Cycle Time (T)

• Location Problem

Location of the DCs (Y)

• GRASP

Frequency of Replenishment (k)

• Numerical Examples

Allocation of DC to the retail outlets (X)

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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ASSUMPTIONS OF THE JRP

• Abstract

• the demand is known and constant
• the stock replenishment admits non integer
  quantities of the items
• the costs of output and/or the prices of
  acquisition do not depend on the quantities
  ordered
• the stock replenishment is immediate, assuming an
  infinite quantity available of each item
• stock shortage is not admitted
• the waiting time of supply is zero
• there is no limitation for the space of storage
• the operation of the system of storage admits an
  infinite time horizon
• the joint replenishment of the orders, requires
  that, at least, one of the items to be always
  ordered, i.e., to have T as periodicity of order
  (restricted cycle policy).

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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RELEVANT COSTS
The relevant costs associated to the problem of
joint replenishment of stocks in accordance with
the model assumptions are classified as ordering
costs and holding costs (hj). The ordering
costs are subdivided into a fixed component A
(major set-up cost), that is incurred whenever an
order occurs, independently of the number of DCs
in replenishment, and in a variable costs
component (aj), that is related with each DC
integrated in the order (minor set-up cost).

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

Grouping all DCs object of a joint replenishment
we will be able to identify the equation of the
medium relevant total costs by unit of time (see
as an example Viswanathan, 1996)

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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SOLVING THE JRP

• Abstract

Goyal (1974)

• The problem

• Joint Replenishment

For each cycle time T, there is one and only one
vector K (integer, positive), which replenishment
frequencies minize the Ctr for every item j .

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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• Abstract

For every item j variable relevant costs by unit
of time are given by

• The problem

• Joint Replenishment

• Location Problem

• GRASP

minimum value for t

• Numerical Examples

• HC

• Simulation results

lower bound for T

• Conclusions

upper bound for T ki 1 j 1, ... , n (all
items are included in the order of replenishment)
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Viswanathan (1996) offers a new algorithm based
in Goyal (1974) and Van Eijs (1993), proposing
new bounds for the value of T

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

Ctr(T) and Ctr (K) are monotonically decreasing
except for values close to the optimum T.
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• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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A- Construction phase A.1 Randomly choose p
locations from the candidate nodes from the
RCL. A.2 Allocate the demand nodes to a facility
location. A.2.1 Initial Solution Allocate the
demand nodes to their closest facility
location. A.2.2 Local Search Starting with the
first demand point in your list change its actual
allocation by all other possible allocations, one
at a time and compute the value for the
objective. If the solution improves (lower
objective) keep the new allocation, otherwise
restore the initial allocation. Repeat the
procedure for all demand points in the list.
A.3 Compute location costs considering the
locations/allocations obtained in A.1 and
A.2. A.4 Use Viswanathans algorithm to find the
optimal joint replenishment policy and compute
the replenishment costs objective. A.5 Sum the
location costs and replenishment costs
objectives in order to find the total costs
objective.

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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A- Construction phase A.1 Randomly choose p
locations from the candidate nodes from the
RCL. A.2 Allocate the demand nodes to a facility
location. A.2.1 Initial Solution Allocate the
demand nodes to their closest facility
location. A.2.2 Local Search Starting with the
first demand point in your list change its actual
allocation by all other possible allocations, one
at a time and compute the value for the
objective. If the solution improves (lower
objective) keep the new allocation, otherwise
restore the initial allocation. Repeat the
procedure for all demand points in the list.
A.3 Compute location costs considering the
locations/allocations obtained in A.1 and
A.2. A.4 Use Viswanathans algorithm to find the
optimal joint replenishment policy and compute
the replenishment costs objective. A.5 Sum the
location costs and replenishment costs
objectives in order to find the total costs
objective.

• Abstract

• The problem

• Joint Replenishment

Greedy function

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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B- Local Search phase B For each distribution
center, de-allocate the demands that were
allocated to it and move the distribution center
location to all possible unused potential
locations, repeating the following steps B.1
Allocate the demand nodes to the facility
locations. B.1.1 Initial Solution Allocate the
demand nodes to their closest facility
location. B.1.2 Local Search Starting with the
first demand point in your list change its actual
allocation by all other possible allocations, one
at a time and compute the value for the
objective. If the solution improves (lower
objective) keep the new allocation, otherwise
restore the initial allocation. Repeat the
procedure for all demand points in the list.
B.2 Compute the location costs objective
corresponding to the locations/allocations
obtained in B.1.1. B.3 Use Viswanathans
algorithm to find the optimal joint replenishment
policy and compute the replenishment costs
objective. B.4 Sum the location costs and
replenishment costs objectives in order to find
the total costs objective. B.5 If the objective
improves, keep the new locations and
replenishment frequencies as the solution for the
problem. Otherwise, keep the old locations and
replenishment frequencies as the solution for the
problem.

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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Numerical examples with location costs

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

Numerical examples with location costs and
replenishment costs and

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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Numerical examples A 45

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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Heuristics Concentration

• Stage 1
• Find p random initial locations for the
  distribution centers
• Allocate each demand node to its closest
  distribution center location.
• For every demand point, change its actual
  allocation by all other possible allocations, one
  at a time and compute the value for the
  objective. If the solution improves (lower
  objective) keep the new allocation, otherwise
  restore the initial allocation.

• Abstract

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Stage 2
• Use all final locations obtained from all
  starting solutions or use the final locations
  from the best k out of the multiple starting
  solutions in Stage 1 to form the new, reduced set
  of potential locations (the concentration set -
  CS).
• Apply the GRASP heuristic proposed in the
  previous section considering as potential
  locations for the distribution centers the ones
  in the CS.

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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Construction set (CS) 1000 iterations

• Abstract

Local search in CS 100 iterations

• The problem

• Joint Replenishment

• Location Problem

Percentage of optimal solutions

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions

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EURO Working Group on Locational AnalysisEWGLA
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• Abstract

The conclusions point out to the fact that, when
considering replenishment costs and when
replenishment costs are not irrelevant in
comparison to location costs, the locations of
the distribution centers will be different from
those obtained with the original FLP. From the
model, we may obtain the solution for the
strategic problem of the location of the
distribution centers, as well as the solution for
the inventory problem corresponding to the
located distribution centers.

• The problem

• Joint Replenishment

• Location Problem

• GRASP

• Numerical Examples

• HC

• Simulation results

• Conclusions