Transportation and Assignment Problems Operations Research
Which Factory should supply to which Warehouse and how much? Factory 2 Warehouse 1 Factory 1 Warehouse 2 Warehouse 3 Factory 3
What is a transportation problem?
Transportation as a Linear Programming Problem
Transportation Problem - Matrix
Transportation Problem - MatrixTransportation Cost ofroute AB (from Factory Ato Store B) Is Total Supply = Total Demand?
Transportation Problem - Types
How to solve a transportation problem?1. Formulate the problem and set up in matrix form2. 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 NWCMWhat 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 celland 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 theclosed 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 itselfwas 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 - Setting up a new problem• Introduce artificial small quantity d that doesn’t otherwise impact supply-demand constraints
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 outall other zeros in that column. If all zeros have either been markedIdentify columns with exactly one with square or crossed out –zero. Draw a square on that zero.Cross out all other zeros in that row. If there is at least one and only one square in each row, problem has been solved.
Draw minimum number of lines tocover 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