Transportation and Assignment Problems

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Transportation and Assignment Problems
Lecture 5
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The Transportation Model

• A product is transported from a number of sources
  to a number of destinations at the minimum
  possible cost.
• Each source is able to supply a fixed number of
  units of the product, and each destination has a
  fixed demand for the product.

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Example
SUPPLY (TONS)
DEMAND (TONS)
150
200
175
100
275
300
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Supply Demand
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Transportation Costs
Transportation Cost Relations (/ton)
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Linear Programming Model
In a balanced transportation model where supply
equals demand, all constraints are equalities
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The Transportation Tableau
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Methods for Solution

• SS Stepping-Stone Method
• MODI Modified Distribution Method

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Initial Feasible Solution

• Northwest Corner Method
• Minimum Cell Cost Method
• Vogel's Approximation Method - VAM

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The Northwest Corner Method
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Initial Feasible Solution with Northwest Corner
Method
The Initial Solution
Z 5,925
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The Minimum Cell Cost Method
The Initial Minimum Cell Cost Allocation
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The Second Minimum Cell Cost Allocation
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Initial Feasible Solution with Minimum Cell Cost
Method
The Initial Solution
Z 4,550
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Vogels Approximation Method
Penalty Cost is the difference between the lowest
and the next lowest cell cost in a row (or
column).
VAM allocates as much as possible to the minimum
cost cell in the row or column with the largest
penalty cost.
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Initial Feasible Solution with Vogels
Approximation Method
The Initial Solution
Z 5,125
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The Stepping Stone Method
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1A ?1B ? 3B ? 3A 6 8 5 4 1
INITIAL SOLUTION IS NOT OPTIMAL
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-

-
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The Stepping-Stone Path for Cell 1A
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The Second Iteration of the Stepping-Stone Method
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The Alternative Optimal Solution
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The Modified Distribution Method
The Minimum Cell Cost Initial Solution
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Cells with Allocations
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The Initial Solution with all ui and uj Values
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Empty Cells
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The Second Iteration of the MODI Solution Method
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The New ui and uj Values for the Second Iteration
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The Unbalanced Transportation Model An Unbalanced
Model (Demand gt Supply)
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The Unbalanced Transportation Model An Unbalanced
Model (Demand lt Supply)
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Degeneracy
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The Minimum Cell Cost Initial Solution
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The Initial Solution
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The Second Stepping-Stone Iteration
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The Assignment Model
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Example
The Travel Distances to Each Game for Each Team
of Officials
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The Assignment Tableau with Row Reductions
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The Assignment Tableau with Column Reductions
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The Opportunity Cost Table with the Line Test
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The Second Iteration
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