Transportation and Assignment Problems Chapter 5
The Transportation Model
1.
A product is transported from a number of sources to a number of destinations at the minimum possible cost.
2.
Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product.
2
Example DEMAND (TONS) İSTANBUL
SUPPLY (TONS) SAMSUN
150
200
175
100 İZMİR
300 ANTALYA
3
ERZURUM
275 GAZİANTEP
Supply – Demand
Grain Elevator
Supply
Mill
Demand
1. Samsun 2. Erzurum 3. Gaziantep
150 175 275
A. İstanbul B. İzmir C. Antalya
200 100 300
Total
600 tons
Total
600 tons
4
Transportation Costs
Transportation Cost Relations ($/ton)
Mill
Grain Elevator 1. Samsun 2. Erzurum 3. Gaziantep
5
A. Ä°stanbul
B. Ä°zmir
C. Antalya
$6 $7 $4
$8 $ 11 $5
$ 10 $ 11 $ 12
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
11
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 15
Z = $4,550
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Vogel’s Approximation Method
2
4
1
2 Penalty Cost is the difference between the lowest and the next lowest cell cost in a row (or 17column).
3
1 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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19
20
Initial Feasible Solution with Vogel’s Approximation Method
The Initial Solution 21
Z = $5,125
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The Stepping Stone Method
23
24
25
1A →1B → 3B → 3A + 6 – 8 + 5 – 4 = –$1 26
INITIAL SOLUTION IS NOT OPTIMAL
27
28
-
+
+
-
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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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32
33
34
35
36
The Alternative Optimal Solution
37
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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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46
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The Unbalanced Transportation Model An Unbalanced Model (Demand > Supply)
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The Unbalanced Transportation Model An Unbalanced Model (Demand < 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
35
60
61
62
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