Transportation Model
Operations Research
Modeling Toolset
Network Problems
Linear programming has a wide variety of applications
Network problems
Special types of linear programs
Particular structure involving networks
Ultimately, a network problem can be represented as a linear programming model
However the resulting A matrix is very sparse, and involves only zeroes and ones
This structure of the A matrix led to the development of specialized algorithms to solve network problems
Types of Network Problems
Shortest Path
Special case: Project Management with PERT/CPM
Minimum Spanning Tree
Maximum Flow/Minimum Cut
Minimum Cost Flow
Special case: Transportation and Assignment Problems
Set Covering/Partitioning
Traveling Salesperson
Facility Location and many more
The Transportation Problem
The Transportation Problem
The problem of finding the minimum-cost distribution of a given commodity from a group of supply centers (sources) i=1,…,m to a group of receiving centers (destinations) j=1,…,n
Each source has a certain supply (si)
Each destination has a certain demand (dj)
The cost of shipping from a source to a destination is directly proportional to the number of units shipped
Simple Network Representation
Example: P&T Co.
Produces canned peas at three canneries
Bellingham, WA, Eugene, OR, and Albert Lea, MN
Ships by truck to four warehouses
Sacramento, CA, Salt Lake City, UT, Rapid City, SD, and Albuquerque, NM
Estimates of shipping costs, production capacities and demands for the upcoming season is given
The management needs to make a plan on the least costly shipments to meet demand
Example: P&T Co. Map
Example: P&T Co. Data
Example: P&T Co.
Network representation
Example: P&T Co.
Linear programming formulation
Let xij denote…
Minimize
subject to
General LP Formulation for Transportation Problems
Feasible Solutions
A transportation problem will have feasible solutions if and only if
How to deal with cases when the equation
Modeling Toolset
Network Problems
Linear programming has a wide variety of applications
Network problems
Special types of linear programs
Particular structure involving networks
Ultimately, a network problem can be represented as a linear programming model
However the resulting A matrix is very sparse, and involves only zeroes and ones
This structure of the A matrix led to the development of specialized algorithms to solve network problems
Types of Network Problems
Shortest Path
Special case: Project Management with PERT/CPM
Minimum Spanning Tree
Maximum Flow/Minimum Cut
Minimum Cost Flow
Special case: Transportation and Assignment Problems
Set Covering/Partitioning
Traveling Salesperson
Facility Location and many more
The Transportation Problem
The Transportation Problem
The problem of finding the minimum-cost distribution of a given commodity from a group of supply centers (sources) i=1,…,m to a group of receiving centers (destinations) j=1,…,n
Each source has a certain supply (si)
Each destination has a certain demand (dj)
The cost of shipping from a source to a destination is directly proportional to the number of units shipped
Simple Network Representation
Example: P&T Co.
Produces canned peas at three canneries
Bellingham, WA, Eugene, OR, and Albert Lea, MN
Ships by truck to four warehouses
Sacramento, CA, Salt Lake City, UT, Rapid City, SD, and Albuquerque, NM
Estimates of shipping costs, production capacities and demands for the upcoming season is given
The management needs to make a plan on the least costly shipments to meet demand
Example: P&T Co. Map
Example: P&T Co. Data
Example: P&T Co.
Network representation
Example: P&T Co.
Linear programming formulation
Let xij denote…
Minimize
subject to
General LP Formulation for Transportation Problems
Feasible Solutions
A transportation problem will have feasible solutions if and only if
How to deal with cases when the equation