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 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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