1. Network Optimization Models
Chapter 10
Network Optimization Models

2. 10.1 Prototype Example The road system for Seervada Park
Location O: park entrance
Location T: a scenic wonder
Trams transport sightseers from park entrance to location T and back

3. Prototype Example Park management faces three problems
Determine the route with the smallest total distance
A shortest-path problem
Determine where telephone lines should be laid
A minimum spanning tree problem
Determine how to route tram to maximize number of trips during peak season
A maximum flow problem

4. 10.2 The Terminology of Networks
Network consists of a set of points and a set of lines connecting points
Node: point (vertex) in the network
Lines: links, arcs, edges, or branches
Labeled by naming the node at each end
From node precedes the to node
Have a flow of some type through them
Directed arcs have unidirectional flow
Undirected arcs (links) allow bidirectional flow

5. The Terminology of Networks
Directed network
Network has only directed arcs
Undirected network
Network has only undirected arcs
Path between two nodes
A sequence of distinct arcs connecting the nodes
Directed path from node i to node j
Sequence of connecting arcs toward node j

6. The Terminology of Networks
Undirected path from node i to node j
Sequence of connecting arcs whose direction can be with toward or away from node j
Connected network
Every pair of nodes in the network has at least one undirected path between them
Tree (spanning tree)
Connected network with no undirected cycles

7. The Terminology of Networks

8. The Terminology of Networks

9. The Terminology of Networks
Arc capacity
Maximum amount of flow that can be carried on a directed arc
Supply node
Flow out exceeds flow in
Demand node
Flow in exceeds flow out
Transshipment node
Flow in equals flow out

10. 10.3 The Shortest-Path Problem
Consider an undirected, connected network
Contains origin and destination nodes
Each link has a nonnegative distance
The problem
Find the shortest path from origin to destination

11. The Shortest-Path Problem
Algorithm
Objective of nth iteration: find the nth nearest node to the origin
Repeat for n = 1, 2… until destination is reached
Input for nth iteration: n − 1 nearest nodes to the origin, including shortest path and distance from the origin
These are called the solved nodes

12. The Shortest-Path Problem
Algorithm (cont’d.)
Candidates for nth nearest node: unsolved node with shortest connecting link to the solved node
Calculation of nth nearest node
For each solved node and its candidate, add the distance between them and the distance of the shortest path from the origin to this solved node
Candidate with smallest total distance is the nth nearest node

14. The Shortest-Path Problem
Shortest path for the Seervada park problem
Looking at last column in Table 10.2, two potential shortest paths exist from the destination to the origin
T→ D → E → B → A → O or
T → D → B → A → O
Total of 13 miles on either path

15. The Shortest-Path Problem
Network simplex method
An alternate option for solving shortest-path problems
Three categories of applications
Minimize total distance traveled
Minimize total cost of a sequence of activities
Minimize total time of a sequence of activities

16. 10.4 The Minimum Spanning Tree Problem
Given: nodes of a network, potential links, and positive length of each link if it is inserted into the network
Design the network by inserting links
A path must exist between every pair of nodes
Problem: minimize total length of links inserted into the network
Network of n nodes requires only n−1 links
Choose the links to form a spanning tree

17. The Minimum Spanning Tree Problem
Applications
Design of telecommunications networks
Design of a lightly-used transportation network to minimize cost of providing links
Design network of power transmission lines
Electrical equipment wiring
Piping systems

18. The Minimum Spanning Tree Problem
Algorithm
Select any node arbitrarily and then add a link to connect it to its nearest node
Identify the unconnected node that is closest to a connected node, and add a link between them
Repeat until all nodes have been connected
Ties may be broken arbitrarily
There may be multiple optimal solutions

19. The Minimum Spanning Tree Problem
Example of graphical approach to implementing the algorithm
Problem: installing telephone lines in Seervada park
See Pages in the text for solution

20. 10.5 The Maximum Flow Problem
General problem description
All flow through a directed, connected network originates at a source, and terminates at a sink
Remaining nodes are transshipment nodes
Flow through an arc is allowed in only one direction (indicated by the arrowhead)
Maximum flow is given by arc capacity
Objective: maximize total flow from source to sink

21. The Maximum Flow Problem
Applications
Maximize flow through company’s distribution network from factories to customers
Maximize flow through company’s supply network from vendors to factories
Maximize oil flow through a system of pipelines
Maximize water flow through aqueducts
Maximize flow of vehicles through a transportation network

22. The Maximum Flow Problem
Algorithms
Simplex method can be used
Augmenting path algorithm is more efficient
Residual network
Remaining arc capacities after some flows have been assigned
Augmenting path
Directed path from source to sink in residual network such that every arc on path has positive residual capacity

23. The Maximum Flow Problem
Algorithm (each iteration follows these steps)
Identify an augmenting path
If none exists, net flows already constitute an optimal flow pattern
Identify the residual capacity, c* of this augmenting path
It will equal the minimum residual capacity of the arcs on this path
Increase the flow in this path by c*

24. The Maximum Flow Problem
Algorithm (cont’d.)
Decrease by c* the residual capacity of each arc on this augmenting path
Increase by c* the residual capacity of each arc in the opposite direction on this augmenting path
Return to the first step
Example: Seervada park transportation problem
See Pages in the text

25. 10.6 The Minimum Cost Flow Problem
General description of the minimum cost flow problem
The network is directed and connected
At least one of the nodes is a supply node, and one of the other nodes is a demand node
All remaining nodes are transshipment nodes
Flow is only allowed in direction of the arrowhead
Arc capacity gives maximum allowable flow

26. The Minimum Cost Flow Problem
General description (cont’d.)
Network has enough arcs with sufficient capacity to enable all flow generated at supply nodes to reach all demand nodes
Cost of flow through each arc is proportional to the amount of flow
Objective: minimize total cost of sending available supply through the network to meet the given demand

27. The Minimum Cost Flow Problem

28. The Minimum Cost Flow Problem
Linear programming problem formulation

29. The Minimum Cost Flow Problem
Feasible solutions property
Integer solutions property
For minimum cost flow problems where every bi and uij have integer values, all the basic variables in every basic feasible solution also have integer values

30. The Minimum Cost Flow Problem
Special cases that fit the minimum cost flow problem
The transportation problem
The assignment problem
The transshipment problem
The shortest-path problem
The maximum flow problem

31. The Minimum Cost Flow Problem
Network simplex method
An alternative method to solving the special cases when the special-purpose algorithms are not available

32. 10.7 The Network Simplex Method
Streamlined version of the simplex method
Same basic steps
Finding the entering basic variable
Determining the leaving basic variable
Solving for the new BF solution
General concepts of the method are covered in the text

33. The Network Simplex Method
Incorporate the upper bound technique:
To deal with the arc capacity constraints
Network representation of BF solutions
Basic arcs: arcs corresponding to basic variables
Key property: they never form undirected cycles
Nonbasic arcs: arcs corresponding to nonbasic variables

34. The Network Simplex Method
BF solutions can be obtained by solving spanning trees
For arcs not in the spanning tree, set the corresponding variables (xij or yij) equal to zero
For arcs in the spanning tree, solve for the corresponding variables (xij or yij) in the system of linear equations provided by the node constraints

35. The Network Simplex Method
Feasible spanning tree
Spanning tree whose solution from the node constraints also satisfies all the other constraints
Fundamental theorem for the network simplex method
Basic solutions are spanning tree solutions (and conversely)
BF solutions are solutions for feasible spanning trees (and conversely)

36. 10.8 A Network Model for Optimizing a Project’s Time-Cost Trade-off
Network based OR techniques developed in the 1950s
PERT (Program Evaluation Review Technique)
CPM (Critical Path Method)
Both are used in project management
Concepts have merged into PERT/CPM
CPM method for time-cost tradeoff
Addresses a project with a specific deadline

37. A Network Model for Optimizing a Project’s Time-Cost Trade-off
CPM method for time-cost trade-off (cont’d.)
Problem: find optimal plan for expediting activities to minimize the total cost of completing the project within the deadline
General approach
Use a network to display the various activities
And the order in which they need to be performed
Form optimization model
Solve using linear programming

38. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Prototype example
The Reliable Construction Co. won the contract to construct a new plant within a time period of 40 weeks
See Table 10.7
Project network options
Activity-on-arc (AOA)
Each activity is represented by an arc
Nodes separate activities from predecessors
Used by original versions of PERT and CPM

40. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Project network options (cont’d.)
Activity-on-node (AON)
Each activity is represented by a node
Arcs show precedence relationships between activities
Has several advantages over AOA
May become the standard format for project networks

41. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Path
One of the routes following the arcs from start to finish
The critical path
Relevant: length of each path through the network
Sum of estimated durations of activities on the path

42. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Estimating the critical path (project duration) for the Reliable Construction Co. example
See Pages in the text
Crashing an activity
Taking special costly measures to reduce an activity’s duration
Crashing the project involves crashing a number of activities

43. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Example problem: determine least expensive way to crash activities to reduce overall duration to 40 weeks
Solution methods
Marginal cost analysis
See Table and Table on Pages 419 and 420 of the text
Linear programming
Follow steps on Pages of the text

44. 10.9 Conclusions Problems addressed with network models
Optimizing an existing network
Designing a new network
Minimum spanning tree problem
CPM method of time-cost trade-offs
Powerful way of applying network optimization to project management