Title: Transportation Model (Powerco)
1Transportation 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
2Powerco 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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4Producing 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
5Objective
- 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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7Job 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
8Job 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)
9Job Assignments at MachincoObjective
- Minimize the total time to complete all jobs
10Job Assignments at MachincoConstraints
- Each job must be assigned to some machine
- Each machine can do at most one job
11Job Assignments at MachincoInputs
- The time required to do each job on each machine
12Job Assignments at MachincoDecision Variables
- Which job-to-machine assignments to make
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14Critical 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)
15Critical Path ModelObjective
- Schedule the activities in order to minimize the
total project time
16Critical Path ModelConstraints
- Because of built-in precedence relations,
activities cant begin until their predecessors
are completed
17Critical Path ModelInputs
- Precedence relations
- Durations of activities
18Critical 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)
19Project 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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21Shipping 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
22Shipping Food at FoodcoObjective
- Minimize the total shipping cost
- Each shipping cost is proportional to the amount
shipped along the route
23Shipping 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)
24Shipping Food at FoodcoInputs
- Unit shipping costs
- Arc capacities
- Supplies at supply points
- Demands at demand points
25Shipping Food at FoodcoDecision Variables
- Flows along all arcs
- Includes flows into dummy node (which is excess
capacity not shipped)
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27Maximum 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
28Maximum Oil Flow at SuncoObjective
- Maximize the total flow from source to sink per
unit of time
29Maximum 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
30Maximum Oil Flow at SuncoInputs
- Arc capacities
- These indicate how much oil can go through a
given pipeline per unit of time
31Maximum 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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33Shortest 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
34Shortest 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
35Shortest 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
36Shortest Route Car Replacement Constraints
37Shortest 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
38Shortest Route Purchasing CarsDecision Variables
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40Investing at StockcoBasic Problem
- Choose the investments that stay within a budget
and maximize the NPV - Each investment is an all-or-nothing decision
41Investing at Stockco Objective
- Maximize the NPV of the investments chosen
42Investing at Stockco Constraints
- Cash spent on investments cant be greater than
cash available
43Investing at Stockco Inputs
- Amount of cash required for each investment
- Amount of NPV obtained from each investment
44Investing 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