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 costStep 2: Assign as much as allocation to this cell Step 3: Block those cells that cannot be allocatedStep 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 5Second 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

Introducea 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...