Network Optimization Models - Agribusiness De Optimization Models Chapter 10: Hillier and Lieberman Chapter 8: Decision Tools for Agribusiness Dr. Hurley’s AGB 328 Course

  • Published on
    24-May-2018

  • View
    214

  • Download
    2

Embed Size (px)

Transcript

  • Network Optimization ModelsChapter 10: Hillier and Lieberman

    Chapter 8: Decision Tools for Agribusiness

    Dr. Hurleys 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, NonbasicArcs

  • 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 a 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... ,,,,,,,,

    ,,,,,

    2 + 5 + 4 + 2 + 7 + 1 + 4

    + 3 + 1 + 4 + 1 + 1 + 5 + 7Subject To:

    + + = 1 = 0 + + = 0 + = 0 + + = 0 + + = 0 + = 1, , , , , , , , , , , , , (0,1)

  • Seervada Spreadsheet Model

    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 than 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

    max... ,,,,,,

    ,,,,,

    + +

    = 0 + = 0

    + = 0 + + = 0 + = 0

    0 5, 0 7, 0 4, 0 1,0 3, 0 2, 0 4, 0 5,0 4, 0 9, 0 1,0 6

    Note: This is based on Figure 10.11 in the text.

  • Excel Formulation of Seervada Max

    Flow Problem Examined in Class

  • In Class Max Flow Activity (Not

    Graded)

    A

    D

    C

    B

    E

    H

    G

    F

    I

    8

    7

    6

    7

    106 3

    5

    7

    6

    47

    0

    0

    0

    0

    00

    0

    0

    0

    0

    0

    0

  • 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)

    = =1 =1

    Subject to:

    =1 =1

    =

    0 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

    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.

  • Activity List for Reliable Construction

    Activity Activity Description

    Immediate

    Predecessors

    Estimated

    Duration (Weeks)

    A Excavate 2

    B Lay the foundation A 4

    C Put up the rough wall B 10

    D Put up the roof C 6

    E Install the exterior plumbing C 4

    F Install the interior plumbing E 5

    G Put up the exterior siding D 7

    H Do the exterior painting E, G 9

    I Do the electrical work C 7

    J Put up the wallboard F, I 8

    K Install the flooring J 4

    L Do the interior painting J 5

    M Install the exterior fixtures H 2

    N Install the interior fixtures K, L 6

  • Questions Needed to be Answered

    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 for 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

Recommended

View more >