Modeling Service networks

Modeling service networks
Laura Albert
The Industrial & Systems Engineering Department
University of Wisconsin-Madison
laura@engr.wisc.edu
punkrockOR.wordpress.com
@lauraalbertphd
This work was in part supported by the National Science Foundation under Award No. CMMI 1361448, 1444219, 1541165

Service networks for public sector OR
Simple definition: Public sector operations research (OR)
is a problem whose outputs are subject to public
scrutiny.
Public sector OR could include problems in these areas:
• Public health and safety: police, fire, emergency
services, public health
• Community development: planning, transportation
• Human services: public assistance, welfare,
drug/alcohol treatment, homeless services
• Nonprofit management: management of community-
oriented service providers
23/26/2018 Laura A. Albert, UW-Madison

Service networks in public sector OR
• Public health and safety
• Fire fighters, police officers, paramedics, post-disaster
response and recovery
• Location models for locating vehicles and designing
response districts
• Community development
• School buses, library loan networks
• Vehicle routing problem for bus schedules
• Human services
• Public assistance, meals on wheels
• Staffing models to schedule employees and volunteer shifts
• Nonprofit management
• Humanitarian logistics, volunteers, blood bank operations
• Multicommodity flow models to deliver relief aid

The origins of public sector OR
44
The President’s
Commission on Law
Enforcement and the
Administration of Justice
(1965)
Al Blumstein chaired the
Commission’s Science
and Technology Task
Force (CMU)
1972
The research was put
into practice
The research was
influential
The research led to many
applied papers published in
the best journals
The research won the top
awards in the field

Early public sector OR models for public safety
service networks
5
Set cover / maximum cover models
How can we “cover” the maximum
number of locations with
ambulances?
Church, R., & ReVelle, C. (1974). The maximal covering
location problem. Papers in regional science, 32(1),
101-118.
Markov models
How many fire engines should we send?
Swersey, A. J. (1982). A Markovian decision model for deciding how
many fire companies to dispatch. Management Science, 28(4), 352-
365.
Data analytics
How far will a fire
engine travel to a call?
Kolesar, P., & Blum, E. H.
(1973). Square root laws
for fire engine response
distances. Management
Science, 19(12), 1368-1378.
Hypercube queuing models
What is the probability that our first choice
ambulance is unavailable for this call?
Larson, R. C. (1974). A hypercube queuing model for facility location
and redistricting in urban emergency services. Computers &
Operations Research, 1(1), 67-95.

Motivating example:
Locating ambulances at stations
• We want to locate 𝑠𝑠 idental ambulances at stations in a
geographic region to “cover” the most calls in 9 minutes
• Our initial assumptions:
1. Locate 𝑠𝑠 ambulances at stations
2. Call volumes to vary by location
3. Deterministic travel times (a call is
“covered” by an ambulance or not)
4. Consider the closest ambulance to each call
(ignore backup coverage for now)

Anatomy of a 911 call
Goal: Response times
Service provider:
Emergency 911 call
Unit
dispatched
Unit is en
route
Unit arrives
at scene
Service/care
provided
Unit leaves
scene
Unit arrives
at hospital
Patient
transferred
Unit returns
to service

Objective functions
• All EMS departments evaluate service according to
response time threshold (RTT)
• Most common RTT: nine minutes for 80% of calls
• A call with response time of 8:59 is covered
• A call with response time of 9:00 is not covered
• Yields a coverage objective function
Why RTTs?
• Easy to measure
• Intuitive
• Unambiguous

Facility location to model
service networks

Location models overview
10
• Decide where to locate facilities to serve customers
• (stations / warehouses / shelters / hubs / etc.)
• In order to achieve some balance between
1. Service (coverage, distance)
2. Cost (number of facilities)
Usually two decisions
to make:
1. Where to locate?
2. Which customers are
assigned/allocated to
which facilities?
• Sometimes referred to as
“location–allocation models”

Applications of Facility Location Models
• Widely applied in public and private sectors:
• Emergency medical services (EMS) / fire stations
• Airline hubs
• Blood banks
• Hazardous waste disposal sites
• Hurricane shelters
• Fast-food restaurants
• Public swimming pools
• Schools
• Vehicle inspection stations
• Bus stops
• etc.

Classical Models
12
1. P-median problem: minimize demand-weighted distance (DWD)
s.t. locate ≤ P facilities
2. Uncapacitated fixed-charge location problem (UFLP):
minimize fixed cost + DWD
3. P-center problem: minimize maximum distance
4. Maximum covering location problem (MCLP):
maximize covered demands
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10

Model formulations

Notation
14
• Sets
• 𝐼𝐼 = {customers}
• 𝐽𝐽 = {potential facility sites}
• Parameters
• ℎ𝑖𝑖= annual demand of customer 𝑖𝑖 ∈ 𝐼𝐼
• 𝑐𝑐𝑖𝑖𝑖𝑖 = cost to transport one unit from 𝑗𝑗 ∈ 𝐽𝐽 to 𝑖𝑖 ∈ 𝐼𝐼(distance)
• 𝑓𝑓𝑗𝑗 = fixed (annual) cost to open a facility at site 𝑗𝑗 ∈ 𝐽𝐽
• 𝑃𝑃 = number of facilities
• 𝑉𝑉𝑖𝑖= set of facilities that can cover customer 𝑖𝑖 with 𝑉𝑉𝑖𝑖 = {𝑗𝑗 ∈ 𝐽𝐽: 𝑐𝑐𝑖𝑖𝑖𝑖 ≤
𝑅𝑅} and 𝑅𝑅 is the coverage radius
• Decision variables
• 𝑥𝑥𝑗𝑗 = 1 if facility 𝑗𝑗 ∈ 𝐽𝐽 is opened, 0 otherwise
• 𝑦𝑦𝑖𝑖𝑖𝑖 = 1 if facility 𝑗𝑗 ∈ 𝐽𝐽 serves customer 𝑖𝑖 ∈ 𝐼𝐼, 0 otherwise

Uncapacitated fixed-charge location problem
formulation
15
jiy
jx
jixy
iy
ij
j
jij
Jj
ij
,}1,0{
}1,0{
,
1s.t.
∀∈
∀∈
∀≤
∀=∑∈
∑∑∑ ∈ ∈∈
+
Ii Jj
ijiji
Jj
jj ychxfmin Min fixed + transportation cost
Satisfy all demands
Don’t assign customer to closed facility
Integrality
NOTE: It is always optimal to assign each customer solely to its nearest open facility
Therefore, we think of yij as binary since there is an optimal solution with yij ∈{0,1} for all i, j
Talk about “the facility to which customer i is assigned”
Rather than “the amount of i’s demand served”

P-median Formulation
16
jiy
jx
Px
jixy
iy
ij
j
Jj
j
jij
Jj
ij
,}1,0{
}1,0{
,
1s.t.
∀∈
∀∈
=
∀≤
∀=
∑
∑
∈
∈
∑∑∈ ∈Ii Jj
ijiji ychmin Min demand-weighted distance
(transportation cost)
Satisfy all demands
Integrality
Locate P facilities

P-median solution
17
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10

P-Center Formulation
18
jiy
jx
Px
jixy
iy
ij
j
Jj
j
jij
Jj
ij
,}1,0{
}1,0{
,
1s.t.
∀∈
∀∈
=
∀≤
∀=
∑
∑
∈
∈
Max radius is the largest customer distance
(transportation cost)
Satisfy all demands
Integrality
Locate P facilities
min 𝑟𝑟
s.t. 𝑐𝑐𝑖𝑖𝑖𝑖 𝑦𝑦𝑖𝑖𝑖𝑖 ≤ 𝑟𝑟 ∀𝑖𝑖
Minimize max radius
r = maximum coverage radius

P-Center Solution
19
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10

Maximal Covering Formulation
20
iz
jx
ixz
Px
i
j
Vj
ji
Jj
j
i
∀∈
∀∈
∀≤
=
∑
∑
∈
∈
}1,0{
}1,0{
s.t.
∑∈Ii
ii zhmax Maximize covered demand
Locate P facilities
Definition of coverage
Integrality
where Vi = set of facilities that can cover customer i with 𝑉𝑉𝑖𝑖 = {𝑗𝑗 ∈ 𝐽𝐽: 𝑐𝑐𝑖𝑖𝑖𝑖 ≤ 𝑠𝑠}
zi = 1 if customer i is covered, 0 otherwise
How can we adjust this basic model to address ambulances not always being
available?

Maximal Covering Solution
21
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10

Tradeoffs
22
All models achieve some balance between
1. Service  distance, cost or coverage
2. Cost  often the number of facilities to locate (𝑃𝑃)
Capacity
• Most models also have a capacitated version
• Facilities have fixed throughput capacity (an input)
• Balance workload among service providers
• Sometimes capacity is a decision variable
• Discrete choices (50,000 sq ft / 100,000 sq ft / 200,000 sq ft)
• Continuous variable (cost is a function of capacity)

P-Median model
23
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Uncapacitated model Capacitated model
Too much demand assigned
to two stations
New stations selected
Not all demand assigned to closest
open facility

Let’s revisit our motivating example
• We want to locate 𝑠𝑠 ambulances at stations in a geographic
region to “cover” the most calls in 9 minutes
• We extend the base models to include:
1. Ambulances that are not always available (backup coverage
is important)
2. Each ambulance responds to roughly the same number of
calls (hint: use capacitated model variations)
3. Non-deterministic travel times leading to non-binary
coverage
4. Different types of ambulances

Lift assumption that every vehicle/facility is always available
Deterministic covering models
with backup coverage

Maximal Covering Extension
26
Previous models match each customer with one facility
• UFCL, P-median, P-center
Or count calls as “covered” if they are covered at least once
• 𝑧𝑧𝑖𝑖 = 1 if customer 𝑖𝑖 ∈ 𝐼𝐼 is covered at least once, and 0 otherwise.
Ambulances are unavailable for new calls when they are
serving a customer
• Backup service is important!
Models must give credit for backup service.
• Examine through coverage model extensions
• Introduce k-coverage (coverage by k ambulances)
• 𝑧𝑧𝑖𝑖 𝑖𝑖 = 1 if customer 𝑖𝑖 is covered at least 𝑘𝑘 times, 0 otherwise

Maximal Covering Extensions
27
How could you extend the maximal covering model?
𝑧𝑧𝑖𝑖 𝑖𝑖 = 1 if customer 𝑖𝑖 is covered at least 𝑘𝑘 times, 0 otherwise
Consider 𝑘𝑘 = 2
If location 𝑖𝑖 is covered once: 𝑧𝑧𝑖𝑖 𝑖 = 1, 𝑧𝑧𝑖𝑖 𝑖 = 0
If location 𝑖𝑖 is covered twice: 𝑧𝑧𝑖𝑖 𝑖 = 1, 𝑧𝑧𝑖𝑖 𝑖 = 1
Three cases:
1. We maximize double coverage (𝑧𝑧𝑖𝑖 𝑖)
2. We weigh single and double coverage
𝜃𝜃 = weight associated with single coverage (≥ 1/2)
(1 − 𝜃𝜃 = weight associated with double coverage)
3. We weigh single and double coverage by considering the
proportion of time ambulances are available for service

Maximal Double Covering Formulation
Hogan and ReVelle 1986
28
Maximize double covered demand
Locate P facilities
Definition of double coverage
Integrality
Note: we are only looking at double
coverage here!
max �
𝑖𝑖∈𝐼𝐼
ℎ𝑖𝑖 𝑧𝑧𝑖𝑖 𝑖
𝑠𝑠. 𝑡𝑡. �
𝑗𝑗∈𝐽𝐽
𝑥𝑥𝑗𝑗 = 𝑃𝑃
1 + 𝑧𝑧𝑖𝑖 𝑖 ≤ �
𝑗𝑗∈𝑉𝑉𝑖𝑖
𝑥𝑥𝑗𝑗 , 𝑖𝑖 ∈ 𝐼𝐼
𝑥𝑥𝑗𝑗 ∈ 0,1 , 𝑗𝑗 ∈ 𝐽𝐽
𝑧𝑧𝑖𝑖 𝑖 ∈ 0,1 , 𝑖𝑖 ∈ 𝐼𝐼
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10

Maximal Multiobjective Covering Formulation
Hogan and ReVelle 1986
29
Weighted average of single and
double covered demand
Locate P facilities
Integrality
max 𝜃𝜃 �
𝑖𝑖∈𝐼𝐼
ℎ𝑖𝑖 𝑧𝑧𝑖𝑖 𝑖 + (1 − 𝜃𝜃) �
𝑖𝑖∈𝐼𝐼
ℎ𝑖𝑖 𝑧𝑧𝑖𝑖2
𝑗𝑗∈𝐽𝐽
𝑧𝑧𝑖𝑖 𝑖 + 𝑧𝑧𝑖𝑖 𝑖 ≤ �
𝑧𝑧𝑖𝑖2 ≤ 𝑧𝑧𝑖𝑖1, 𝑖𝑖 ∈ 𝐼𝐼
𝑧𝑧𝑖𝑖1, 𝑧𝑧𝑖𝑖 𝑖 ∈ 0,1 , 𝑖𝑖 ∈ 𝐼𝐼
Hierarchical coverage
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Full coverage for double coverage
Partial credit for single coverage

Maximal Expected Covering Location Model
Daskin 1983
30
Let’s introduce ambulance busy probability 𝑞𝑞 and assume it’s the same for
all ambulances
zik = 1 if customer i is covered at least k times, 0 otherwise
If location 𝑖𝑖 is covered 𝑘𝑘 times, then the expected covered demand is:
𝐸𝐸𝑘𝑘 = ℎ𝑖𝑖(1 − 𝑃𝑃 not covered by 𝑘𝑘 ambulances )
𝐸𝐸𝑘𝑘 = ℎ𝑖𝑖(1 − 𝑃𝑃(all 𝑘𝑘 ambulances busy)
𝐸𝐸𝑘𝑘 = ℎ𝑖𝑖(1 − 𝑞𝑞𝑘𝑘
)
The marginal contribution of the kth ambulance to this expected value is:
𝐸𝐸𝑘𝑘 − 𝐸𝐸𝑘𝑘−1 = ℎ𝑖𝑖 1 − 𝑞𝑞 𝑞𝑞𝑘𝑘−1 = ℎ𝑖𝑖 𝜃𝜃𝑘𝑘
We can then use this in an integer programming model

Maximal Expected Covering Location Model
Daskin 1983
Let’s introduce ambulance busy probability 𝑞𝑞 and assume it’s the same for
all ambulances
zik = 1 if customer i is covered at least k times, 0 otherwise
Weight for k-coverage: 𝜃𝜃𝑘𝑘 = 1 − 𝑞𝑞 𝑞𝑞𝑘𝑘−1
31
max �
𝑖𝑖∈𝐼𝐼
�
𝑘𝑘=1
𝑃𝑃
ℎ𝑖𝑖 1 − 𝑞𝑞 𝑞𝑞𝑘𝑘−1
𝑧𝑧𝑖𝑖 𝑖𝑖
𝑗𝑗∈𝐽𝐽
�
𝑘𝑘
𝑧𝑧𝑖𝑖 𝑖𝑖 ≤ �
𝑧𝑧𝑖𝑖 𝑖𝑖 ∈ 0,1 , 𝑖𝑖 ∈ 𝐼𝐼, 𝑘𝑘 = 1, … , 𝑝𝑝
Maximize covered demand
Locate P facilities
Definition of coverage
Integrality

You survived your crash course into
location models to support the
modeling of service networks!
3/26/2018 Laura A. Albert, UW-Madison 32

References
• Church, Richard, and Charles R. Velle. "The maximal covering location
problem." Papers in regional science 32, no. 1 (1974): 101-118.
• Schilling, David, D. Jack Elzinga, Jared Cohon, Richard Church, and Charles
ReVelle. "The TEAM/FLEET models for simultaneous facility and equipment
siting." Transportation Science 13, no. 2 (1979): 163-175.
• Hogan, Kathleen, and Charles ReVelle. "Concepts and applications of backup
coverage." Management science 32, no. 11 (1986): 1434-1444.
• Daskin, Mark S. "A maximum expected covering location model: formulation,
properties and heuristic solution." Transportation science 17, no. 1 (1983): 48-
70.
• Ansari, Sardar, Laura Albert McLay, and Maria E. Mayorga. "A maximum
expected covering problem for district design." Transportation Science 51, no. 1
(2015): 376-390.
• Goldberg, Jeffrey B. "Operations research models for the deployment of
emergency services vehicles." EMS management Journal 1, no. 1 (2004): 20-39.
• McLay, Laura Albert. "Discrete Optimization Models for Homeland Security and
Disaster Management." Tutorials in Operations Research (2015).

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