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Transportation Model (Powerco)

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Send electric power from power plants to cities where power is needed at minimum ... Transportation between supply and demand points, with the objective of ... – PowerPoint PPT presentation

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Title: Transportation Model (Powerco)


1
Transportation Model (Powerco)
  • Transportation between supply and demand points,
    with the objective of minimizing cost.
  • Send electric power from power plants to cities
    where power is needed at minimum cost
  • Objective Minimize total cost of all shipments
  • There is a unit shipping cost on each shipping
    route
  • This is multiplied by the amount shipped and
    summed over all routes

2
Powerco Contd.
Constraints
  • Cant ship more than is available from each power
    plant (supply point)
  • Must ship at least the amount needed to each city
    (demand point)

Inputs
  • Unit shipping costs along each route
  • Amount of supply at each power plant
  • Demand at each city

Decision Variables
  • The amount to ship along each route
  • There is a route from each supply point to each
    demand point
  • No other routes are allowed

3
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4
Producing Sailboats at Sailco(Inventory
Problem Modeled as Transportation Problem)
  • Produce sailboats over a multiperiod horizon to
    meet known (forecasted) demands on time
  • Regular-time and overtime labor are available
  • Minimize total production and holding costs

Supply
RT
OT
10
10
0
0
0
Inventory
Month
Demand
5
Objective
  • Minimize total costs, which include
  • Regular-time labor costs, Overtime labor costs,
    Inventory holding costs

Inputs
  • Beginning inventory of sailboats
  • Maximum boats that can be produced per month with
    regular-time labor
  • Regular-time and overtime cost per boat
  • Unit holding cost per month in inventory
  • Monthly demands for boats

Decision Variables
  • Number of boats to be supplied for each month
    from possible supplies
  • Supplies indicate the source of the boats
  • Initial inventory
  • Regular-time labor in a particular month
  • Overtime labor in a particular month

6
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7
Job Assignments at MachincoThe Assignment Problem
  • Assign jobs to machines so that each job is
    assigned and each machine does at most one job
  • Minimize total time to do all jobs

8
Job Assignments at MachincoModeling Approach
  • Model as a transportation problem, where all
    supplies and demands are 1
  • Supplies correspond to machines (each with a
    supply of 1)
  • Demands correspond to jobs (each with a demand of
    1)

9
Job Assignments at MachincoObjective
  • Minimize the total time to complete all jobs

10
Job Assignments at MachincoConstraints
  • Each job must be assigned to some machine
  • Each machine can do at most one job

11
Job Assignments at MachincoInputs
  • The time required to do each job on each machine

12
Job Assignments at MachincoDecision Variables
  • Which job-to-machine assignments to make

13
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14
Critical Path ModelBasic Problem
  • Analyze the length of time required to complete a
    project composed of activities with precedence
    relations (some activities cant begin until
    others are completed)
  • See which activities are critical (the total
    project would be delayed if they were delayed)

15
Critical Path ModelObjective
  • Schedule the activities in order to minimize the
    total project time

16
Critical Path ModelConstraints
  • Because of built-in precedence relations,
    activities cant begin until their predecessors
    are completed

17
Critical Path ModelInputs
  • Precedence relations
  • Durations of activities

18
Critical Path ModelDecision Variables
  • The times corresponding to the nodes in the
    project network
  • These are actually the earliest times certain
    activities can begin (e.g., node 2 is the
    earliest activities C and D can begin)

19
Project Network(See Chart1 sheet in Excel)
  • Precedence relations can be summarized in a graph
    called an activity-on-arc network
  • Each node corresponds to a point in time
  • Each arc corresponds to an activity
  • Precedence relations are obtained by joining
    certain nodes with certain arcs
  • Node 1 is a start node (time 0)
  • The last node is a finish node

20
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21
Shipping Food at FoodcoBasic Problem
  • Ship food from production plants to customers at
    least cost
  • Food can be shipped directly to customers or from
    plants to warehouses and then to customers
  • See Chart1 sheet in Excel

22
Shipping Food at FoodcoObjective
  • Minimize the total shipping cost
  • Each shipping cost is proportional to the amount
    shipped along the route

23
Shipping Food at FoodcoConstraints
  • Arc capacities cant be exceeded
  • There must be flow balance at each node
  • There is positive net outflow at each supply
    point (plants)
  • There is zero net outflow at each transshipment
    point (warehouses)
  • There is positive net inflow (negative net
    outflow) at each demand point (customers)

24
Shipping Food at FoodcoInputs
  • Unit shipping costs
  • Arc capacities
  • Supplies at supply points
  • Demands at demand points

25
Shipping Food at FoodcoDecision Variables
  • Flows along all arcs
  • Includes flows into dummy node (which is excess
    capacity not shipped)

26
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27
Maximum Oil Flow at SuncoBasic Problem
  • Ship as much oil (per unit time) from a source
    node to a sink (destination) node as possible
    along a given network of pipelines
  • See Chart1 sheet in Excel

28
Maximum Oil Flow at SuncoObjective
  • Maximize the total flow from source to sink per
    unit of time

29
Maximum Oil Flow at SuncoConstraints
  • Dont exceed arc (pipeline) capacities
  • Achieve flow balance at each node
  • By adding a dummy arc from the sink to the
    source, we can let all net outflows be zero

30
Maximum Oil Flow at SuncoInputs
  • Arc capacities
  • These indicate how much oil can go through a
    given pipeline per unit of time

31
Maximum Oil Flow at SuncoDecision Variables
  • Arc flows
  • These include the flow along the dummy arc (which
    isnt an actual physical flow)

32
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33
Shortest Route Car Replacement Basic Problem
  • Decide on a least-cost purchasing/selling
    strategy for cars, given that a car is needed at
    all times
  • Economic reason for selling cars is that
    maintenance costs increase with age and trade-in
    value decreases with age

34
Shortest Route Car ReplacementSolution Strategy
  • Model as a shortest route problem
  • Origin is year 1
  • Destination is end of planning horizon
  • Any path from node 1 to node 6 represents a
    replacement strategy

35
Shortest Route Car Replacement Objective
  • Minimize the total cost of owning a car during
    the planning horizon, including
  • The cost of purchasing new cars
  • The maintenance cost of owning cars
  • The trade-in value of replaced cars

36
Shortest Route Car Replacement Constraints
  • Flow balance constraints

37
Shortest Route Car Replacement Inputs
  • Length of planning horizon
  • Cost of a new car
  • Maintenance cost per year, which increases with
    the age of the car
  • Trade-in value of car, which decreases with the
    age of the car

38
Shortest Route Purchasing CarsDecision Variables
  • Flows on the arcs

39
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40
Investing at StockcoBasic Problem
  • Choose the investments that stay within a budget
    and maximize the NPV
  • Each investment is an all-or-nothing decision

41
Investing at Stockco Objective
  • Maximize the NPV of the investments chosen

42
Investing at Stockco Constraints
  • Cash spent on investments cant be greater than
    cash available

43
Investing at Stockco Inputs
  • Amount of cash required for each investment
  • Amount of NPV obtained from each investment

44
Investing at Stockco Decision variables
  • Whether to invest or not in each investment
  • This is indicated by a 0-1 changing cell, which
    is 1 for an investment that is chosen, 0 otherwise
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