Transportation Model

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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…


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 doesn’t hold?
Integer Solutions Property: Unimodularity
Unimodularity relates to the properties of the A matrix
(determinants of the submatrices, beyond scope)
Transportation problems are unimodular, so we get the integers solutions property:

For transportation problems, when every si and dj have an integer value, every BFS is integer valued.

Most network problems also have this property.
Transportation Simplex Method
Since any transportation problem can be formulated as an LP, we can use the simplex method to find an optimal solution Because of the special structure of a transportation LP, the iterations of the simplex method have a very special form The transportation simplex method is nothing but the original simplex method, but it streamlines the iterations given this special form Transportation Simplex Method

The Transportation Simplex Tableau
Prototype Problem
Holiday shipments of iPods to distribution centers
Production at 3 facilities,
A, supply 200k
B, supply 350k
C, supply 150k
Distribute to 4 centers,
N, demand 100k
S, demand 140k
E, demand 300k
W, demand 250k
Total demand vs. total supply
Prototype Problem
Finding an Initial BFS
The transportation simplex starts with an initial basic feasible solution (as does regular simplex) There are alternative ways to find an initial BFS, most common are The Northwest corner rule
Vogel’s method
Russell’s method (beyond scope)
The Northwest Corner Rule
Begin by selecting x11, let x11 = min{ s1, d1 }
Thereafter, if xij was the last basic variable selected,
Select xi(j+1) if source i has any supply left
Otherwise, select x(i+1)j
The Northwest Corner Rule
Vogel’s Method
For each row and column, calculate its difference:
= (Second smallest cij in row/col) - (Smallest cij in row/col) For the row/col with the largest difference, select entry with minimum cij as basic Eliminate any row/col with no supply/demand left from further steps Repeat until BFS found

Vogel’s Method (1): calculate...
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