Transportation Model (Powerco)

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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

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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

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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
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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

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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

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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)

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Job Assignments at MachincoObjective

• Minimize the total time to complete all jobs

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Job Assignments at MachincoConstraints

• Each job must be assigned to some machine
• Each machine can do at most one job

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Job Assignments at MachincoInputs

• The time required to do each job on each machine

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Job Assignments at MachincoDecision Variables

• Which job-to-machine assignments to make

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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)

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Critical Path ModelObjective

• Schedule the activities in order to minimize the
  total project time

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Critical Path ModelConstraints

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

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Critical Path ModelInputs

• Precedence relations
• Durations of activities

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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)

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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

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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

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Shipping Food at FoodcoObjective

• Minimize the total shipping cost
• Each shipping cost is proportional to the amount
  shipped along the route

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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)

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Shipping Food at FoodcoInputs

• Unit shipping costs
• Arc capacities
• Supplies at supply points
• Demands at demand points

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Shipping Food at FoodcoDecision Variables

• Flows along all arcs
• Includes flows into dummy node (which is excess
  capacity not shipped)

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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

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Maximum Oil Flow at SuncoObjective

• Maximize the total flow from source to sink per
  unit of time

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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

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Maximum Oil Flow at SuncoInputs

• Arc capacities
• These indicate how much oil can go through a
  given pipeline per unit of time

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Maximum Oil Flow at SuncoDecision Variables

• Arc flows
• These include the flow along the dummy arc (which
  isnt an actual physical flow)

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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

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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

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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

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Shortest Route Car Replacement Constraints

• Flow balance constraints

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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

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Shortest Route Purchasing CarsDecision Variables

• Flows on the arcs

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Investing at StockcoBasic Problem

• Choose the investments that stay within a budget
  and maximize the NPV
• Each investment is an all-or-nothing decision

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Investing at Stockco Objective

• Maximize the NPV of the investments chosen

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Investing at Stockco Constraints

• Cash spent on investments cant be greater than
  cash available

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Investing at Stockco Inputs

• Amount of cash required for each investment
• Amount of NPV obtained from each investment

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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