Transportation and Assignment Problems

Operations Research

Which Factory should supply to which Warehouse and how much?

Factory 1

Factory 3

Factory 2

Warehouse 3

Warehouse 1

Warehouse 2

What is a transportation problem?

Transportation as a Linear Programming Problem

Transportation Problem - Matrix

Transportation Problem - Matrix

Is Total Supply = Total Demand?

Transportation Cost of route AB (from Factory A to Store B)

Transportation Problem - Types

How to solve a transportation problem?

1. Formulate the problem and set up in matrix form

2. Obtain initial basic feasible solution3. Test the solution for optimality4. If yes, Stop5. If no, determine new optimal solution6. Go to step 3

Methods for finding initial solution

• North West Corner Method• Minimum Matrix Method• Vogel’s Corner Method

North West Corner Method

Initial Solution using NWCM

What is the number of positive allocations? -------- (6)

What is (number of rows + number of columns -1) -------- (3+4-1 = 6)

Testing for optimality

• Is there any alternative route (empty cell) which is better than existing routes?

• i.e. If I shift one unit from current route to any other route, does overall cost increase or decrease?

• Which out of alternative routes is best (which one reduces cost by maximum amount)?

Stepping Stones MethodTo evaluate each empty cell, draw a closed path starting at empty cell and returning to empty cell through at least 3 occupied cells.Add +1 (one unit) to the empty cell. Correspondingly subtract/ add one unit to each occupied cell on the closed path so that row and column sums remain balanced.

Increase in transportation cost = +4-6+5-3 = 0. There is no benefit to be gained by shifting units to route AD.

Initial solution itself was optimal in this case!

Special Cases

• Multiple optimum solution – A scenario where multiple routes have same overall cost.

• Unbalanced transportation problem - If total supply not equal to total demand

• Degeneracy – number of positive allocations < (number of rows + number of columns -1)

• Maximization

Unbalanced transportation problem

– If supply is more add a dummy demand column– If demand is more add a dummy supply column– Dummy cells have transportation cost zero

Which one is greater, demand or supply?

What should we add, dummy row or column?

Now solve using regular approach

Degeneracy

Degeneracy - Setting up a new problem

• Introduce artificial small quantity d that doesn’t otherwise impact supply-demand constraints

Maximization Problem

Convert to minimization problem

Assignment Problem• Special case of transportation problem• Here each source can supply to only one

destination– Number of sources equal to number of destinations– Only one unit supplied from source to destination

• Assigning jobs to workers• Assigning teachers to classes• Can be solved using simple enumeration of

combinations, regular transportation method or simplex method

Hungarian Method - Kuhn

Identify rows with exactly one zero. Draw a square on that zero. Cross out all other zeros in that column.

Identify columns with exactly one zero. Draw a square on that zero. Cross out all other zeros in that row.

If all zeros have either been marked with square or crossed out –

If there is at least one and only one square in each row, problem has been solved.

Draw minimum number of lines to cover all zeros.

Special cases

• Unbalanced – Sources and Destinations not equal.– Add a dummy source or destination with 0 cost.

• Maximization– Convert to minimization problem using

opportunity cost