Modi

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Transportation Problems
Transportation is considered as a “special case” of LP

Reasons?
it can be formulated using LP technique so is its solution

Here, we attempt to firstly define what are them and then studying their solution methods:

Transportation Problem
We have seen a sample of transportation problem on slide 29 in lecture 2

Here, we study its alternative solution method

Consider the following transportation tableau
Review of Transportation Problem
A Transportation Example (2 of 3) Model Summary and Computer Solution with Excel Transportation Tableau
We know how to formulate it using LP technique
Refer to lecture 2 note
Here, we study its solution by firstly attempting to determine its initial tableau Just like the first simplex tableau!

Solution to a transportation problem

Initial tableau

Optimal solution

Important Notes

Tutorials
initial tableau
Three different ways:
Northwest corner method
The Minimum cell cost method
Vogel’s approximation method (VAM)

Now, are these initial tableaus given us an
Optimal solution?
Northwest corner method
Steps:
assign largest possible allocation to the cell in the upper left-hand corner of the tableau Repeat step 1 until all allocations have been assigned
Stop. Initial tableau is obtained

Example
Northeast corner
Initial tableau of NW corner method
Repeat the above steps, we have the following tableau.
Stop. Since all allocated have been assigned
The Minimum cell cost method
Here, we use the following steps:
Steps:
Step 1 Find the cell that has the least cost
Step 2: Assign as much as allocation to this cell
Step 3: Block those cells that cannot be allocated
Step 4: Repeat above steps until all allocation have been assigned.

Example:
Step 1 Find the cell that has the least cost Step 2: Assign as much as allocation to this cell Step 3: Block those cells that cannot be allocated Step 4: Repeat above steps until all allocation have been assigned. The initial solution

Stop. The above tableau is an initial tableau because all allocations have been assigned

Vogel’s approximation method
Operational steps:
Step 1: for each column and row, determine its
penalty cost by subtracting their two of their least cost
Step 2: select row/column that has the highest penalty cost
in step 1
Step 3: assign as much as allocation to the
selected row/column that has the least cost
Step 4: Block those cells that cannot be further allocated
Step 5: Repeat above steps until all allocations have been
assigned

Example
subtracting their two of their least cost Steps 2 & 3
Step 4
Step 5 Second Iteration
3rd Iteration of VAM
Initial tableau for VAM
Optimal solution?
Initial solution from:
Northeast cost, total cost =$5,925
The min cost, total cost =$4,550
VAM, total cost = $5,125
(note: here, we are not saying the second one always better!)
It shows that the second one has the min cost, but is it the optimal solution? Solution methods
We need a method, like the simplex method, to check and obtain the optimal solution Two methods:

Stepping-stone method
Modified distributed method (MODI)
Stepping-stone method
Introduce a non-basic variable into basic variables
Here, we can select any non-basic variable as an entry and then using the “+ and –” steps to form a closed loop as follows: Stepping stone
Stepping stone
Getting optimal solution
In such, we introducing the next algorithm called Modified Distribution (MODI)

Modified distributed method (MODI)
It is a modified version of stepping stone method
MODI has two important elements:
It determines if a tableau is the optimal one
It tells you which non-basic variable should be firstly considered as an entry variable It makes use of stepping-stone to get its answer of next iteration

How it works?
Procedure (MODI)
Step 0: let ui, v , cij...
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