INDU 342: Logistics Network Models
Design of a logistics system
Claudio Contardo
Mechanical, Industrial and Aerospace Engineering
Concordia University
Lecture 5
Logistics network design
Logistics network design
Logistics network design involves the planning of long-term operations,
by an integrated consideration of strategic planning decisions (location
and size of facilities, and links between them) and tactical/operational
decisions (flows on multiple periods of time)
Logistics network design
Factors triggering a re-design of an existing network
Cost changes
Market changes
Technological changes
Economic changes
Political changes
Organizational changes
Logistics network design
Figure: Example of a two-echelon logistics system
Classification of network design problems
Network design problems can be classified according to the following
attributes
Social-oriented vs profit-oriented
Discrete vs continuous locations
Single- versus multi-commodity flows
Shipping between facilities
Hierarchy of flows
Number of echelons (single-, two-, multiple-echelon)
Binary vs continuous flows
Granularity operational planning modeling
Determining the number of facilities
Locating new facilities (or closing existing ones) can have an impact on
Transportation
Figure: Effect on transportation from adding a consolidation cross-dock
Determining the number of facilities
Inventory costs
Square root law
If order sizes are optimized according to the economic order quantity
formula (we will see this later), then inventory levels on the new network
will follow the formula r
nf
If = I0
n0
Determining the number of facilities
Facility costs (other than transportation + inventory): fixed costs +
operational costs. They tend to increase as the number of facilities
grow due to a loss in economies of scale
Determining the number of facilities
Figure: Number of facilities that minimizes total cost
Determining the number of facilities
Service level. Often, a purely cost-driven analysis is not able to
capture service level (which tends to improve as the number of
facilities increases). We often see that planners tend to grow beyond
the optimal value p∗
Weighted scoring method
Assign weights wj for criteria j ∈ J
Assign scores rij for every candidate location i, criterion j
Select the facility i∗ that maximizes j ri∗ j
P
Weighted scoring method
JetMarket
Jet Market has to decide where to locate a retail outlet in Berne,
Switzerland. External consultants have selected seven criteria which are
considered the most important for the location decision. Each criterion
has a weight from 0 to 1 (see the leftmost Table below). Three possible
commercial areas (indicated as 1, 2, and 3) are evaluated by applying the
weighted scoring procedure. Scores assigned to the location criteria for
the three alternative locations vary between 0 and 10 and are reported in
the right-most Table below
Weighted scoring method
JetMarket
The weighted scores for each candidate facility are 4.55, 5.30 and 4.50.
Therefore, the second facility is preferred over the other two
Continuous facility location problems
Locating a single facility in the Euclidean plane
Input
A finite set of locations V , coordinates (xv , yv ), v ∈ V , demands
dv , v ∈ V
A constant unit transportation cost c
Output: Find the optimal location (x∗ , y ∗ ) that solves
X q
2 2
Minimizex,y f (x, y) = cdv (xv − x) + (yv − y)
v∈V
Continuous facility location problems
Locating a single facility in the Euclidean plane
Since the objective is convex, the optimal location (x∗ , y ∗ ) satisfies
∇f (x∗ , y ∗ ) = 0
or analytically
P
√ dv xv
v∈V (xv −x∗ )2 +(yv −y ∗ )2
∗
x = dv
√
P
v∈V (xv −x∗ )2 +(yv −y ∗ )2
P
√ dv yv
v∈V (xv −x∗ )2 +(yv −y ∗ )2
∗
y = dv
.
√
P
v∈V (xv −x∗ )2 +(yv −y ∗ )2
Continuous facility location problems
We can solve this using a fixed-point method (Weiszfeld method):
P P
v dv xv v dv yv
x0 = P , y0 = P
v dv v dv
For every t ≥ 1:
P
√ d v xv
v∈V(xv −xt−1 )2 +(yv −y t−1 )2
t
x = dv
√
P
v∈V (xv −xt−1 )2 +(yv −y t−1 )2
P
√ dv yv
v∈V(xv −xt−1 )2 +(yv −y t−1 )2
t
y = dv
.
√
P
v∈V (xv −xt−1 )2 +(yv −y t−1 )2
Stopping criterion: |f (xt , y t ) − f (xt−1 , y t−1 )| ≤ ϵ
Continuous facility location problems
Locating multiple facilities
For locating multiple facilities (say p ≥ 2), we can proceed as follows:
Partition the customer set V into p sets V 1 . . . V p
For each subset V k execute Weiszfeld’s method to find the optimal
location (xk , y k ) to serve the customers in V k
Assign each customer node v ∈ V to its closest of the p facilities
and re-construct the subsets V 1 . . . V p . Repeat the previous step
unless the subsets remained the same
Continuous facility location problems
Locating a cross-dock in the Euclidean plane
Figure: Adding a cross-dock
Continuous facility location problems
Locating a cross-dock in the Euclidean plane
By properly selecting the demands, this is equivalent to the
single-echelon problem
Continuous facility location problems
Figure: Location of a cross-dock in the Euclidean plane
Discrete facility location problems
Discrete facility location problem
Input:
A set V1 of candidate locations, warehousing costs Fi (ui ), caps qi
A set V2 of customers, demands dj
Transportation costs Cij (·) ≥ 0 for every (i, j) ∈ V1 × V2
Output:
Select a subset of facilities I ⊆ V1
Assign every customer to the closest chosen facilities
At minimum cost (fixed + transportation)
Discrete facility location problems
X X
Minimize Z= Fi (ui ) + Cij (sij )
i∈V1 i∈V1 ,j∈V2
subject to
X
sij = ui i ∈ V1
j∈V2
X
sij = dj j ∈ V2
i∈V1
ui ≤ qi i ∈ V1
s, u ≥ 0.
Discrete facility location problems
It is often assumed that:
Transportation costs are linear, equal to cij sij for some cost matrix c
Open facilities are subject to a fixed charge cost + a warehousing
cost, equal to fi + qi ui , for ui > 0 (else 0)
By properly selecting the fixed + transportation costs, one can write
the resulting problem as follows
Discrete facility location problems
X X
Minimize Z = fi yi + cij sij
i∈V1 i∈V1 ,j∈V2
subject to
X
sij ≤ qi yi i ∈ V1
j∈V2
X
sij = dj j ∈ V2
i∈V1
s≥0
y binary.
