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 a2 2 3 b3 a3 3 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 = 1…m
(demand requirement)
j = 1…n

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.

8. 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
cnn t4
assignees
tasks

9. 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 company’s distribution network from its factories to its customers.
Maximize the flow through a company’s 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.

14. 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

15. Maximal Flow Problems – LP Formulation1 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.

16. Maximal Flow Problems – LP Formulation1 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 <= fij <= 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.

17. Maximal Flow Problems – Conversion to Standard Form
What if there are multiple sources and/or multiple sinks? n1 s1 1 n2 3 s2 2

18. 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

19. 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
D,12
H,5 7
M,5
K,8
A,5
B,8
C,15
G,11
E,10 1 2 3 4 5 6 9 10 11
L,14
N,5
F,8
I,4 8

22. 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?…

23. 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…

24. Project Management – Minimum Completion Time1 2 4 5
B,1
D,2 3
LP Solution: Let ti be the time of event i.
Min Z = t5 – t1
s.t. t2 – t1 >= 3
t3 – t2 >= 0
t3 – t1 >= 1
t4 – t2 >= 4
t4 – t3 >= 2
t5 – t4 >= 5
ti >= 0 for all i

25. 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.

26. 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 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 i