Network Flow Problems

1
Network Flow Problems

• Example of Network Flow problems
• What shipping plan minimizes cost to ship from m
  warehouses to n customers?
• How do you maximize efficiency of a machine shop
  through the assignment of jobs to a group of
  machines?
• What is the maximum flow that can be obtained
  through a series of pipes?
• How do you maximize the flow of traffic through
  a series of one-way streets?
• What is the shortest route if a truck must make
  a milk-run through a series of stops (TSP)?
• How do you schedule a series of project
  activities in order to minimize the total project
  duration time?

 
2
Network Flow Problems Transportation Problem
a warehouse capacity b customer demand
1
b1
c11
1
a1
c12
2
b2
2
a2
3
b3
 
3
a3
4
b4
Warehouse
Customer
3
Network Flow Problems Transportation Problem
Given Supply Vector a a1 a2 am Demand
Vector b b1 b2 bn Transportation cost
matrix c Objective Find shipping plan
that minimizes transportation cost that meets all
customer demands while being constrained by
supply capacities.
4
Network Flow Problems Transportation Problem
Minimize s.t.
Total Cost (supply restriction) i
1m (demand requirement) j 1n
5
Network Flow Problems The Assignment Problem
Consider the problem of assigning n assignees to
n tasks. Only one task can be assigned to an
assignee, and each task must be assigned. There
is also a cost associated with assigning an
assignee i to task j, cij. The objective is to
assign all tasks such that the total cost is
minimized.
 
6
Network Flow Problems The Assignment Problem

• Examples
• Assign people to project assignments
• Assign jobs to machines
• Assign products to plants
• Assign tasks to time slots

 
7
Network Flow Problems The Assignment Problem

• To fit the assignment problem definition, the
  following assumptions must be satisfied
• The number of assignees and the number of tasks
  are the same (denoted by n).
• Each assignee is to be assigned to exactly one
  task.
• Each task is to be assigned to exactly one
  assignee.
• There is a cost cij associated with assignee i
  performing task j.
• The objective is to determine how all n
  assignments should be made to minimize the total
  cost.

 
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Assignment Problem Flow Diagram
a assignee t tasks
a1
c11
1
1
t1
c12
a2
2
2
t2
a3
3
3
t3
 
an
n
n
t4
cnn
assignees
tasks
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Assignment Problem Cost Matrix
Let the following represent the standard
assignment problem cost matrix, c
10
Assignment Problem Conversion to Standard Cost
Matrix
Consider following cost matrix, how do you
convert to satisfy the standard definition of the
assignment problem?
Add big M to avoid incompatible assignments,
and add a dummy assignee (or task) to have equal
assignees and tasks.
11
Assignment Problem Math Formulation
Minimize s.t.
Total Cost
i
j
Does this formulation look familiar? Is this a
Linear Program?
12
Network Flow Problems Maximal Flow Problems
Consider the following flow network
k1n
ks1
1
n
s
k13
k21
k3n
3
ks2
2
k23
 
The objective is to ship the maximum quantity of
a commodity from a source node s to some sink
node n, through a series of arcs while being
constrained by a capacity k on each arc.
13
Maximal Flow Problems

• Examples
• Maximize the flow through a companys
  distribution network from its factories to its
  customers.
• Maximize the flow through a companys supply
  network from its vendors to its factories.
• Maximize the flow of oil through a system of
  pipelines.
• Maximize the flow of water through a system of
  aqueducts.
• Maximize the flow of vehicles through a
  transportation network.

 
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Maximal Flow Problems
Definitions Flow network consists of nodes and
arcs Source node node where flow
originates Sink node node where flow
terminate Transshipment points intermediate
nodes Arc/Link connects two nodes Directed arc
arc with direction of flow indicated Undirected
arc arc where flow can occur in either
direction Capacity(kij) maximum flow possible
for arc (i,j) Flow(f ij) flow in arc (i,j).
Forward arc arcs with flow out of some
node Backward arc arc with flow into some
node Path series of nodes and arcs between some
originating and some terminating node Cycle
path whose beginning and ending nodes are the same
 
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Maximal Flow Problems LP Formulation
f
1
n
f
s
3
2
Objective Maximize Flow (f) Constraints 1) The
flow on each arc, fij, is less than or equal to
the capacity on each arc, kij. 2) Conservation
of flow at each node. Flow in flow out.
 
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Maximal Flow Problems LP Formulation
f
1
n
f
s
3
Max Z f st s) fs1 fs2 f 1) f13 f1n
fs1 f21 2) f21 f23 fs2 3) f3n f13
f23 n) f f3n f1n 0 lt fij lt kij

2

• Objective Maximize Flow (f)
• Constraints
• The flow on each arc, fij, is less than or
• equal to the capacity on each arc, kij.
• Conservation of flow at each node.
• Flow in flow out.

 
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Maximal Flow Problems Conversion to Standard
Form
What if there are multiple sources and/or
multiple sinks?
n1
s1
1
n2
 
3
s2
2
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Maximal Flow Problems Conversion to Standard
Form
Create a supersource and supersink with arcs
from the supersource to the original sources and
from the original sinks to the supersink. What
capacity should we assign to these new arcs?
n1
f
s1
n
1
f
s
n2
 
 
3
s2
2
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Maximal Flow Problems Conversion to Standard
Form
What if there is an undirected arc (flow can
occur in either direction)? See arc (1,2).
f
1
n
f
s
k12
3
 
2
20
Maximal Flow Problems Conversion to Standard
Form
Create two directed arcs with the same capacity.
Upon solving the problem and obtaining flows on
each arc, replace the two directed arcs with a
single arc with flow fij fji , in the
direction of the larger of the two flows.
f
1
n
f
s
k21
k12
3
 
2
21
Project Management - PERT/CPM
Let each node represent a project event/milestone
(node 1 is start of project, node 11 is end of
project). Let each arc represent a project
task/job. Each arc is identified by a job
letter and duration. Note the dummy jobs
indicating precedence that jobs H and I must
complete before K or L begins.
J,2
7
H,5
M,5
D,12
K,8
0
A,5
B,8
C,15
G,11
E,10
11
2
1
3
4
5
6
9
10
0
L,14
N,5
F,8
I,4
8
 
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Project Management - PERT/CPM

• What questions might project managers be
  interested in?
• How long will the project take?
• Can I add manpower or tools to reduce the
  overall project length?
• To which tasks should I add manpower?
• What tasks are on the critical path?
• Is the project on schedule?
• When should materials and personnel be in place
  to begin a task?
• Other?

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Project Management - Examples

• University Convocation Center
• Windsor Engine Plant
• Other major construction projects
• Large defense contracts
• NASA projects (space shuttle)
• Maintenance planning of oil refineries, power
  plants, etc
• other

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Project Management Minimum Completion Time
A,3
C,4
E,5
2
1
4
5
0
D,2
B,1
3
LP Solution Let ti be the time of event
i. Min Z t5 t1 s.t. t2 t1 gt 3
t3 t2 gt 0 t3 t1 gt 1 t4 t2 gt
4 t4 t3 gt 2 t5 t4 gt 5 ti gt
0 for all i
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CPM Critical Path Method

• Can normal task times be reduced?
• Is there an increase in direct costs?
• Additional manpower
• Additional machines
• Overtime, etc
• Can there be a reduction in indirect costs?
• Less overhead costs
• Less daily rental charges
• Bonus for early completion
• Avoid penalties for running late
• Avoid cost of late startup
• CPM addresses these cost trade-offs.

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CPM Critical Path Method
LP Approach Let tij decision variable for
time to complete task connecting
events i and j. kij normal completion
time of task connecting events i and j. lij
minimum completion time of task connecting
events i and j. Cij incremental cost of
reducing task connecting events i and j. Model
I Given project must be complete by some time T,
which tasks should be reduced to minimize the
total cost? Min s.t.
for all jobs (i,j) for all jobs (i,j) for all i
27
CPM Critical Path Method
LP Approach Model II Given an additional
budget of B for crashing tasks, what minimum
project completion time can be obtained while
staying within your budget? Min s.t.
for all jobs (i,j) for all jobs (i,j) for all i