1. Network Optimization Models
Chapter 10
Network Optimization Models

2. Network Optimization Models

3. 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

4. 10.1 Prototype Example

5. Prototype Example Summary: 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

6. 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

7. The Terminology of Networks

8. 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

9. 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 (note a directed path also satisfies the definition of an undirected path, but not vice versa). Example: in Fig. 10.2, A->B->C->E is a directed path and B->C->A->D is an undirected path
Cycle: A path begins and ends at the same node; DE- ED is a directed cycle; AB-BC-CA is an undirected cycle
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; every spanning tree has n – 1 arcs (the minimum number of arcs needed to have a connected network and the maximum number of arcs possible without having undirected cycles).

10. The Terminology of Networks

11. The Terminology of Networks
There
are
several
alternative
choices
of adding
an arc at
each stage
of the
process.

12. The Terminology of Networks
Arc capacity
Maximum amount of flow that can be carried on a directed arc
Supply node (or source node)
Flow out exceeds flow in
Demand node (sink node)
Flow in exceeds flow out
Transshipment node (intermediate node)
Flow in equals flow out

13. 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

14. The Shortest-Path Problem
This is a special type of linear programming problem (strictly speaking it is an integer programming problem, but due to the special structure of the model, the integer restriction on the decision variables can be relaxed). Each node can be thought of as having a unit flow passing through if it is on the selected path, no flow otherwise. Thus, O is the source node, T is the demand node, and other nodes are transshipment nodes.

15. The Shortest-Path Problem
Minimize Z = 2xOA + 5xOB + 4XOC + 2XAB + 7xAD + xBC + 4xBD + 3xBE + xCB + 4xCE + xDE + 5xDT + xED + 7xET s.t. (net flow constraints) xOA + xOB + xOC = 1 (source node O) -xDT – xET = -1 (sink node T) -xOA + xAB + xAD = 0 (node A) xBC + xBE + xBD – xOB – xAB – xCB = 0 (node B) xCB + xCE – xOC – xBC = 0 (node C) xDE + xDT – xAD – xBD – xED = 0 (node D) xED + xET – xDE – xBE – xCE = 0 (node E) xij ≥ 0 The optimal solution by Excel Solver (see course website) is xOA = xAB = xBE = xED = xDT = 1 (and other xij = 0) with Z = 13.

16. The Shortest-Path Problem
The Excel Solver uses the general simplex method. However, using the shortest path algorithm to solve the problem is much more efficient.

17. The Shortest-Path Problem
Algorithm (using a table or directly applying the algorithm on the graph)
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 (these nodes are called solved nodes and other nodes are called unsolved nodes), including shortest path and distance from the origin

18. The Shortest-Path Problem
Algorithm (cont’d.)
Candidates for nth nearest node: for each n – 1 solved node, find an unsolved node (if any) with shortest connecting link to the solved node and make this unsolved node a candidate node (ties provide additional candidates)
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

20. 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

21. 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
Other shortest path problems can be solved by slightly modify the algorithm
For example:
- shortest directed path (only directed arcs should be considered)
- shortest path from the origin to all other nodes (stop the algorithm until all nodes are solved nodes)

22. The Shortest-Path Problem

23. 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

24. The Minimum Spanning Tree Problem

25. 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

26. 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

27. The Minimum Spanning Tree Problem
Example of graphical approach to implementing the algorithm
Problem: installing telephone lines in Seervada park

28. The Minimum Spanning Tree Problem
The optimal
solution does not
depend on the
initial node chosen.
Try the algorithm
again using a
different initial
node.

29. 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

30. 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

31. 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
For example, if 5 units flow was sent from O to B with capacity 7, then the residual capacity of arc from O to B is 2. Note that the residual capacity of arc from B to O is 5 (for cancelling some previously assigned flow from O to B).

32. The Maximum Flow Problem
Augmenting path
Directed path from source to sink in residual network such that every arc on path has positive residual capacity. The minimum of these residual capacities is called the residual capacity of the augmenting path.

33. The Maximum Flow Problem

34. 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*

35. 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

36. The Maximum Flow Problem

37. The Maximum Flow Problem

38. The Maximum Flow Problem

39. The Maximum Flow Problem

40. The Maximum Flow Problem
Use Excel Solver to solve the problem (see the course website)

41. The Maximum Flow Problem

42. The Maximum Flow Problem

43. 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

44. 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

45. The Minimum Cost Flow Problem

46. The Minimum Cost Flow Problem
Linear programming problem formulation

47. The Minimum Cost Flow Problem
Linear programming problem formulation

48. 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

49. The Minimum Cost Flow Problem
Example in Section 3.4 (Distribution Unlimited Co.)
One of the constraints
is redundant due to
summation of bi’s = 0.

50. The Minimum Cost Flow Problem

51. 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
All the above problems can be modeled as the minimum cost flow problem by slightly adjusting the problems.

52. The Minimum Cost Flow Problem
The transportation problem: there are no transshipment nodes and no capacity constraints on xij (i.e. uij = ∞)
The assignment problem: same as the transportation problem and bi = 1 for each supply node and bi = -1 for each demand node.
The shortest path problem: one supply node with supply 1 unit at origin and one demand node with demand 1 unit at the destination; replace each link by directed arcs except for the links with one end at the origin or destination; the distance between i and j becomes cij or cji and uij = ∞.

53. The Minimum Cost Flow Problem

54. The Minimum Cost Flow Problem
The maximum flow problem: source becomes supply node and sink becomes demand node; let cij = 0 for all existing arcs; select a quantity F which is a safe upper bound on the maximum feasible flow through the network and assign F to the supply node and –F to the demand node; add an arc going directly from the supply node to the demand node with cij = M (big number) and uij = ∞ (since cij = M, the minimum cost flow problem will send F from the supply node to the demand node through other paths as much as possible).

55. The Minimum Cost Flow Problem

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

57. The Minimum Cost Flow Problem

58. 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
Incorporate the upper bound technique:
To deal with the arc capacity constraints like xij ≤ uij which are handled like nonnegativity constraints. For example, for arc A->B with uAB = 10 and cAB = 2 and bA = 50 and bB = 40, suppose after some iterations xAB has become the leaving basic variable by reaching it upper bound 10. Then xAB = 10 is replaced by xAB = 10 – yAB, and yAB becomes the new nonbasic variable. Meanwhile, we replace arc A->B by arc B->A (reverse arc) with yAB as its flow quantity and assign this new arc a capacity of 10 and unit cost -2, i.e. uBA = 10, cBA = -2.

59. The Network Simplex Method
Also, to take xAB = 10 into account, decrease bA from 50 to 40 and increase bB from 40 to 50, i.e. bA = 40 and bB = 50.

60. The Network Simplex Method
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

61. The Network Simplex Method

62. 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

63. 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)

64. The Network Simplex Method

65. The Network Simplex Method

66. The Network Simplex Method
Iterations: Iteration I (1) selecting entering basic: candidate arcs are BA, ED and AC.
Consider arc AC: add AC to the initial feasible spanning tree with flow Ө. This will create an undirected cycle. Within the cycle, arcs having the same direction as arc AC increase their net flow by Ө, and arcs having opposite direction as arc AC decrease their flow by Ө. Arcs not in the cycle their net flow is unchanged. As a result, we compute the change in Z: ΔZ = cAB*θ + cCE*θ + cDE*(-θ) + cAD*(-θ) = 4θ + θ -3θ - 9θ = -7θ

67. The Network Simplex Method
ΔZ= -2θ+9θ+3θ
-θ-3θ = 6θ

68. The Network Simplex Method
ΔZ = 2θ + 3θ = 5θ
Since it’s a minimizing problem, adding arc AC decreases Z value. Thus,
xAC is the entering basic variable.
(2) Selecting the leaving basic variable: By adding arc AC, consider the
upper bound of the arcs which have the same direction as AC (since
the net flow increased by θ) and the lower bound of the arcs which have
the opposite direction as arc AC (the net flow decreased by θ).

69. The Network Simplex Method

70. The Network Simplex Method

71. The Network Simplex Method
Iteration II

72. The Network Simplex Method

73. The Network Simplex Method
Iteration III:

74. The Network Simplex Method

75. The Network Simplex Method
Iteration IV:
No improvement in Z, thus the current BF solution is optimal.

76. The Network Simplex Method

77. 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

78. 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

79. 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 (a network used to represent a project) 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

81. 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

82. A Network Model for Optimizing a Project’s Time-Cost Trade-off

83. 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 (the length of the longest path through the project network)
Equals the estimated project duration
Ties provide more critical paths

84. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Crashing an activity
Taking special costly measures to reduce an activity’s duration (by special costly measures such as using overtime, hiring additional temporary help and etc.)
Crashing the project involves crashing a number of activities (normal point: no crashing activity; crash point: activities are fully crashed)

85. 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 and the company will be paid $5.4 million (normal point: 44 weeks and cost $4.55 million and crash point: 28 weeks and cost $6.15 million)

86. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Solution methods
Marginal cost analysis
In each iteration, crash an activity (which is allowed to be crashed) on the critical path with the smallest cost to reduce its duration by a week. If there is a tie, crash the activity on all critical paths that costs the least.

87. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Total crashing cost is $140,000. Figure shows the
resulting project network, where the darker arrows show the critical
paths.
If we need go further (for example, the deadline is 39 weeks),
the next step would require looking at the activities on all
three critical paths to find the least expensive way of
shortening all three critical paths by a week, and that’d be B).

88. A Network Model for Optimizing a Project’s Time-Cost Trade-off

89. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Solution methods cont’d
Linear programming

90. A Network Model for Optimizing a Project’s Time-Cost Trade-off
For example, consider activity J

91. A Network Model for Optimizing a Project’s Time-Cost Trade-off

92. A Network Model for Optimizing a Project’s Time-Cost Trade-off
Excel Solver solution (also see on the course website)

93. 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