# Network Optimization Models

## Presentation on theme: "Network Optimization Models"— Presentation transcript:

Network Optimization Models
Chapter 9: Hillier and Lieberman Dr. Hurley’s AGB 328 Course

Terms to Know Nodes, Arcs, Directed Arc, Undirected Arc, Links, Directed Network, Undirected Network, Path, Directed Path, Undirected Path, Cycle, Connected, Connected Network, Tree, Spanning Tree, Arc Capacity, Supply Node, Demand Node, Transshipment Node, Sink, Source, Residual Network, Residual Capacity, Augmenting Path, Cut, Cut Value, Max-Flow Min-Cut Theorem, Feasible Solutions Property, Integer Solutions Property, Reverse Arc, Basic Arcs, Nonbasic Arcs

Terms to Know Cont. Spanning Tree Solution, Feasible Spanning Tree, Fundamental Theorem for the Network Simplex Method, Program Evaluation and Review Technique (PERT), Critical Path Method (CPM), Immediate Successor, Immediate Predecessor, Project Network, Activity-on-Arc, Activity-on-Node, Project Duration, Critical Path, Crashing an Activity, Crashing the Project, Normal Point, Crash Point, Marginal Cost Analysis

The Shortest Path Problem
A shortest path problem usually has an node known as the origin and a node known as the destination The objective of this problem is to find the shortest path from the origin node to the destination node The shortest path could be measured in time, distance, etc. Since this problem is a special case of the linear programming problem, the simplex method could be used to solve it

Mathematical Model for Seervada Shortest Path Problem
min 𝑤.𝑟.𝑡. 𝑥 𝑂𝐴 , 𝑥 𝑂𝐵 , 𝑥 𝑂𝐶 , 𝑥 𝐴𝐵 , 𝑥 𝐴𝐷 , 𝑥 𝐵𝐷 , 𝑥 𝐵𝐸 , 𝑥 𝐵𝐶 , 𝑥 𝐶𝐸 , 𝑥 𝐸𝐷 , 𝑥 𝐸𝑇 , 𝑥 𝐷𝑇 , 2 𝑥 𝑂𝐴 +4 𝑥 𝑂𝐶 +5 𝑥 𝑂𝐵 +7 𝑥 𝐴𝐷 +2 𝑥 𝐴𝐵 + 𝑥 𝐵𝐶 + 4 𝑥 𝐵𝐷 +4 𝑥 𝐶𝐸 +3 𝑥 𝐵𝐸 +5 𝑥 𝐷𝑇 + 𝑥 𝐷𝐸 + 7𝑥 𝐸𝑇 Subject To: 𝑥 𝑂𝐴 + 𝑥 𝑂𝐶 + 𝑥 𝑂𝐵 = 1 𝑥 𝑂𝐴 − 𝑥 𝐴𝐷 + 𝑥 𝐴𝐵 = 0 𝑥 𝑂𝐵 + 𝑥 𝐴𝐵 + 𝑥 𝐵𝐶 − 𝑥 𝐵𝐷 − 𝑥 𝐵𝐸 = 0 𝑥 𝑂𝐶 − 𝑥 𝐵𝐶 − 𝑥 𝐶𝐸 = 0 𝑥 𝐴𝐷 + 𝑥 𝐵𝐷 + 𝑥 𝐷𝐸 − 𝑥 𝐷𝑇 = 0 𝑥 𝐶𝐸 + 𝑥 𝐵𝐸 − 𝑥 𝐷𝐸 − 𝑥 𝐸𝑇 = 0 𝑥 𝐷𝑇 + 𝑥 𝐸𝑇 = 1 𝑥 𝑂𝐴 , 𝑥 𝑂𝐶 , 𝑥 𝑂𝐵 , 𝑥 𝐴𝐷 , 𝑥 𝐴𝐵 , 𝑥 𝐵𝐶 , 𝑥 𝐵𝐷 , 𝑥 𝐶𝐸 , 𝑥 𝐵𝐸 , 𝑥 𝐷𝑇 , 𝑥 𝐷𝐸 , 𝑥 𝐸𝑇 ∈(0,1)

Examine in Class

Minimum Spanning Tree Problems
The goal of the minimum spanning tree problem is to connect all the nodes either directly or indirectly at the lowest cost The minimum spanning tree problem will have one less link the number of nodes in the optimal solution The solution to the minimum spanning tree problem can be accomplished using the greedy algorithm

Greedy Algorithm Choose any two nodes initially and connect them
Identify the closest unconnected node and then connect it Continue until all nodes have been connected to the tree Ties can be broken arbitrarily

The Maximum Flow Problem
The purpose of the maximum flow problem is to get as much flow through the network based on the capacity constraints of the network This can be measured by the amount leaving the source or by the amount entering the sink It has a source where supply originates from and a sink which absorbs the supply that makes it through the network

Augmenting Path Algorithm for Maximum Flow Problems
Identify an augmenting path that takes flow from the source to the sink in the residual network such that every arc on this path has strictly positive residual If this path does not exist, you have the optimal Identify the residual capacity c* by finding the minimum of the residual capacities of the arcs on the path Increase the flow in this path by c* Decrease the residual capacities by c* for each arc on the augmenting path

Max-Flow Min-Cut Theorem
Another way of figuring out the maximum flow is by using the Max-Flow Min-Cut Theorem The theorem states that if you have a single source and sink, then the maximum flow through the network is equal to the smallest cut value for all the cuts of the network The cut value is found by summing up all the arcs which are directly affected by the cut of a network A cut is defined as a set of directed arcs that separate the source from the sink

Mathematical Model for the Seervada Max Flow Problem
𝑥 𝑂𝐴 − 𝑥 𝐴𝐵 − 𝑥 𝐴𝐷 =0 𝑥 𝑂𝐵 + 𝑥 𝐴𝐵 − 𝑥 𝐵𝐶 − 𝑥 𝐵𝐷 − 𝑥 𝐵𝐸 =0 𝑥 𝑂𝐶 + 𝑥 𝐵𝐶 − 𝑥 𝐶𝐸 =0 𝑥 𝐴𝐷 + 𝑥 𝐵𝐷 − 𝑥 𝐷𝑇 + 𝑥 𝐸𝐷 =0 𝑥 𝐵𝐸 + 𝑥 𝐶𝐸 − 𝑥 𝐸𝐷 − 𝑥 𝐸𝑇 =0 0≤𝑥 𝑂𝐴 ≤5, 0≤𝑥 𝑂𝐵 ≤7, 0≤𝑥 𝑂𝐶 ≤4, 0≤𝑥 𝐴𝐵 ≤1, 0≤𝑥 𝐴𝐷 ≤3, 0≤𝑥 𝐵𝐶 ≤2, 0≤𝑥 𝐵𝐷 ≤4, 0≤𝑥 𝐵𝐸 ≤5, 0≤ 𝑥 𝐶𝐸 ≤4, 0≤𝑥 𝐷𝑇 ≤9, 0≤𝑥 𝐸𝐷 ≤1,0≤ 𝑥 𝐸𝑇 ≤6

Excel Formulation of Seervada Max Flow Problem
Examined in Class

In Class Max Flow Activity (Not Graded)
7 7 B F 8 10 6 7 3 6 A C E G I 6 5 4 7 D H

Minimum Cost Flow Problem Requirements
At least one supply node At least one demand node The network is directed and connected If the node is not a supply or demand node, then it is a transshipment node Flow through an arc is directed There is enough arc capacity to get the total supply to the total demand Costs are proportional to the amount of flow The objective is to minimize cost

Minimum Cost Flow General Mathematical Model
xij = the arc representing the flow from nodes i to j cij = the cost of flow through xij uij = the arc capacity for xij bi = net flow generated at node i Supply node (bi > 0), demand node (bi < 0), transshipment node (bi = 0) 𝑀𝑖𝑛𝑖𝑚𝑖𝑚𝑧𝑒 𝑍= 𝑖=1 𝑛 𝑗=1 𝑛 𝑐 𝑖𝑗 𝑥 𝑖𝑗 Subject to: 𝑗=1 𝑛 𝑥 𝑖𝑗 − 𝑗=1 𝑛 𝑥 𝑗𝑖 = 𝑏 𝑖 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑛𝑜𝑑𝑒 𝑖 0≤ 𝑥 𝑖𝑗 ≤ 𝑢 𝑖𝑗 for each arc

Minimum Cost Flow Problem in Excel
We will examine the Distribution Unlimited Co. in class

What is Project Management
Project management can be defined as the coordination of activities with the potential use of many organizations, both internal and external to the business, in order to conduct a large scale project from beginning to end. There are two management science techniques that are used for project management: Program and Evaluation Review Technique (PERT) Critical Path Method (CPM)

PERT/CPM PERT PERT was designed to examine projects from the standpoint of uncertainty. CPM CPM was designed to examine projects from the standpoint of costs. PERT and CPM techniques have been combined over time. PERT and CPM both rely heavily on the use of networks to help plan and display the coordination of all the activities for a project.

The Reliable Construction Company
Reliable has just secured a contract to construct a new plant for a major manufacturer. The contract is for \$5.4 million to cover all costs and any profits. The plant must be finished in a year. A penalty of \$300,000 will be assessed if Reliable does not complete the project within 47 weeks. A bonus of \$150,000 will be paid to Reliable if the plant is completed within 40 weeks.

Needed Terminology Activity Start Node Finish Node
A distinct task that needs to be performed as part of the project. Start Node This is a node that represents the beginning of the project. Finish Node This node represents the end of the project.

Needed Terminology Cont.
Immediate Predecessor These are activities that must be completed by no later than the start time of the given activity. Immediate Successor Given the immediate predecessor of an activity, this activity becomes the immediate successor of each of these immediate predecessors. If an immediate successor has multiple immediate predecessors, then all must be finished before an activity can begin.

Immediate Predecessors Estimated Duration (Weeks)
Activity List for Reliable Construction Activity Activity Description Immediate Predecessors Estimated Duration (Weeks) A Excavate 2 B Lay the foundation 4 C Put up the rough wall 10 D Put up the roof 6 E Install the exterior plumbing F Install the interior plumbing 5 G Put up the exterior siding 7 H Do the exterior painting E, G 9 I Do the electrical work J Put up the wallboard F, I 8 K Install the flooring L Do the interior painting M Install the exterior fixtures N Install the interior fixtures K, L

How can the project be displayed graphically? How much time is required to finish the project if no delays occur? When is earliest start and finish times for each activity if no delays occur? What activities are critical bottleneck activities where delays must be avoided to finish the project on time?

Questions Needed to be Answered Cont.
For non bottleneck activities, how much can an activity be delayed and yet still keep the project on time? What is the probability of completing the project by the deadline? What is the least amount of money needed to expedite the project to obtain the bonus? How should costs be monitored to keep the project within budget?

Project Network A project network is a network diagram that uses nodes and arcs to represent the progression of the activities is a project from start to finish. Three pieces of information needed: Activity information Precedence relationship Time information

Project Network Cont. Two types of project networks
Activity-on-Arc (AOA) On this diagram, the activity is represented on an arc, while a node is used to separate an activity from its immediate predecessors. Activity-on-Node (AON) On this diagram, the activity is represented by the node, while the arc is used to showed the precedence relationship between the activities.

Scheduling Using PERT/CPM
A path through a project network is a route that follows a set of arcs from the start node to the finish node. The length of a path is defined as the sum of the durations of the activities of the path. What are the paths and their corresponding lengths for Reliable?

Critical Path This is the path that has the longest length through the project. The shortest time that a project can conceivably be finished is the critical path. Why?

More Terminology Earliest start time of an activity (ES)
The time at which an activity will begin if there are no delays in a project. Earliest finish time of an activity (EF) The time at which an activity will finish if there are no delays in a project. Latest start time of an activity (LS) The latest possible time that an activity can start without delaying the project.

More Terminology Cont. Latest finish time of an activity (LF)
The latest possible time that an activity can be completed without delaying the project. Forward pass The process of moving through a project from start to finish to determine the earliest start and finish times for the activities in the project.

More Terminology Cont. Backward pass Slack for an activity
The process of moving through a project from finish to start to determine the latest start and finish times for the activities in the project. Slack for an activity The amount of time that a particular activity can be delayed without delaying the whole project. It is calculated by taking the difference between the latest finish time with the earliest finish time.

More Terminology Cont. Earliest start time rule
The earliest start time for an activity is equal to the largest of the earliest finish times of its immediate predecessors. Latest finish time rule The latest finish time is equal to the smallest of the latest start times of its immediate successors.

Procedure for Obtaining Earliest Times
Step 1: For the activity that starts the project, assign an earliest start time of zero, i.e., ES=0. Step 2: For each activity whose ES has just been obtained, calculate its earliest finish time as ES plus duration of the activity. Step 3: For each new activity whose immediate predecessors have EF values, obtain its ES by using the earliest start time rule.

Procedure for Obtaining Earliest Times Cont.
Step 4: Apply step 2 to calculate EF. Step 5: Repeat step 3 until ES and EF have been obtained for all activities including the finish node.

Procedure for Obtaining Latest Times
Step 1: For each of the activities that together complete the project, set its latest finish time equal to the earliest finish time of the finish node. Step 2: For each activity whose LF value has just been obtained, calculate its latest start time as LS equals LF minus the duration of the activity.

Procedure for Obtaining Latest Times Cont.
Step 3: For each new activity whose immediate successors now have LS values, obtain its LF by applying the latest finish time rule. Step 4: Apply step 2 to calculate its LS. Step 5: Repeat step 3 until LF and LS have been obtained for all activities.

Ways of Finding the Critical Path
Examine all the paths and find the path with the maximum length. Calculate the slack for an activity. If the slack is zero, it is on the critical path. If the slack is positive, it is not on the critical path.

Time-Cost Trade-Offs Reliable had an incentive bonus of \$150,000 to finish the project in 40 weeks. Is it worth while for Reliable to speed-up the project?

Crashing Crashing an activity refers to taking on extra expenditures in order to reduce the duration of an activity below its expected value. Crashing a project refers to crashing a number of activities to reduce the duration of the project.

This is a method concerned with whether it is worthwhile to crash activities to reduce the anticipated duration of the project to a desired value. This assumes that there is a trade-off between time and cost that has an inverse relationship.

More Terminology Normal Point is the time and cost of an activity when it is performed in a normal way. Crash point show the time and cost when the activity is fully crashed.

Graph of Normal and Crash Points

Marginal Cost Analysis
It is a method of using the marginal cost of crashing individual activities on the current critical path to determine the least expensive way of reducing the project duration to an acceptable level. This method requires you to calculate the cost per desired time unit and compare each cost with the other costs.

Maximum Reduction in Time (weeks) Crash Cost per Week Saved
Activity Normal Crash A 2 1 \$180,000 \$280,000 \$100,000 B 4 320,000 420,000 50,000 C 10 7 620,000 860,000 3 80,000 D 6 260,000 340,000 40,000 E 410,000 570,000 160,000 F 5 180,000 G 900,000 1,020,000 H 9 200,000 380,000 60,000 I 210,000 270,000 30,000 J 8 430,000 490,000 K L 250,000 350,000 M 100,000 N 330,000 510,000

Marginal Cost Analysis Cont.
Once the marginal cost for crashing each activity has been conducted, you next want to choose the crashing that has the smallest marginal cost. Next, calculate the effect that the crash has on each path. Note: Crashing can potentially cause another path to become a critical path.

Solving Crashing Problems Using LP
There are three decisions to be made: The start time of each activity The reduction in each activity due to crashing The finish time of the project LP model will be examined in class.