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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

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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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Initial Feasible Solution with Vogel’s Approximation Method

The Initial Solution 21

Z = $5,125


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The Stepping Stone Method

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24


25


1A →1B → 3B → 3A + 6 – 8 + 5 – 4 = –$1 26

INITIAL SOLUTION IS NOT OPTIMAL


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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


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34


35


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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 > 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

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