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    Transportation Problem: - Transportation Problem - A "Special Case" of LP - Reasons?

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    Fun Toosh345
    The document discusses different methods for solving transportation problems: 1. The Northwest corner method assigns supply to demand starting from the top left cell moving right and down. 2. The Minimum cell cost method assigns supply to the cell with the lowest cost, blocks that cell, and repeats until all supply is assigned. 3. Vogel's approximation method calculates penalty costs for rows and columns, assigns supply to the row/column with the highest penalty cost and lowest cell cost, and repeats. 4. The stepping-stone and modified distributed methods can be used to check if the initial solutions are optimal by introducing non-basic variables as basics and forming closed loops. Dummy rows and columns address when total demand does

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    Transportation Problem: - Transportation Problem - A "Special Case" of LP - Reasons?

    Uploaded by

    Fun Toosh345
    46 views26 pages
    The document discusses different methods for solving transportation problems: 1. The Northwest corner method assigns supply to demand starting from the top left cell moving right and down. 2. The Minimum cell cost method assigns supply to the cell with the lowest cost, blocks that cell, and repeats until all supply is assigned. 3. Vogel's approximation method calculates penalty costs for rows and columns, assigns supply to the row/column with the highest penalty cost and lowest cell cost, and repeats. 4. The stepping-stone and modified distributed methods can be used to check if the initial solutions are optimal by introducing non-basic variables as basics and forming closed loops. Dummy rows and columns address when total demand does

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    Original Title

    Modelling Transportation Problem

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    The document discusses different methods for solving transportation problems: 1. The Northwest corner method assigns supply to demand starting from the top left cell moving right and down. 2. The Minimum cell cost method assigns supply to the cell with the lowest cost, blocks that cell, and repeats until all supply is assigned. 3. Vogel's approximation method calculates penalty costs for rows and columns, assigns supply to the row/column with the highest penalty cost and lowest cell cost, and repeats. 4. The stepping-stone and modified distributed methods can be used to check if the initial solutions are optimal by introducing non-basic variables as basics and forming closed loops. Dummy rows and columns address when total demand does

    Copyright:

    © All Rights Reserved
    Download as PPT, PDF, TXT or read online from Scribd
    Download as ppt, pdf, or txt
    46 views26 pages

    Transportation Problem: - Transportation Problem - A "Special Case" of LP - Reasons?

    Uploaded by

    Fun Toosh345
    The document discusses different methods for solving transportation problems: 1. The Northwest corner method assigns supply to demand starting from the top left cell moving right and down. 2. The Minimum cell cost method assigns supply to the cell with the lowest cost, blocks that cell, and repeats until all supply is assigned. 3. Vogel's approximation method calculates penalty costs for rows and columns, assigns supply to the row/column with the highest penalty cost and lowest cell cost, and repeats. 4. The stepping-stone and modified distributed methods can be used to check if the initial solutions are optimal by introducing non-basic variables as basics and forming closed loops. Dummy rows and columns address when total demand does

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    © All Rights Reserved
    Download as PPT, PDF, TXT or read online from Scribd
    Download as ppt, pdf, or txt
    of 26

    Transportation Problem

    • Transportation problem - a “special case”


    of LP

    • Reasons?
    – it can be formulated & solved using LP
    technique
    Review of Transportation Problem

    Warehouse supply of televisions sets: Retail store demand for television sets:

    1- Cincinnati 300 A - New York 200


    2- Atlanta 200 B - Dallas 300
    3- Pittsburgh 200 C - Detroit 200
    total 700 total 600

    From To Store
    Warehouse
    A B C
    1 $16 $18 $11
    2 14 12 13
    3 13 15 17

    2
    A Transportation Example (2 of 3)
    Model Summary and Computer Solution with Excel
    Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 13x3A + 15x3B + 17x3C

    subject to
    x1A + x1B+ x1C = 300
    x2A+ x2B + x2C = 200
    x3A+ x3B + x3C =300
    x1A + x2A + x3A = 200
    x1B + x2B + x3B = 250
    x1C + x2C + x3C = 200
    xij  0

    3
    Transportation Tableau

    4
    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?

    5
    Northeast corner

    150
    150

    50

    --

    6
    Initial tableau of NW corner method
    • Repeat the above steps, we have the following
    tableau.
    • Stop. Since all allocated have been assigned

    Ensure that all columns and rows added up to its respective totals. 7
    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.

    8
    Step 1 Find the cell that has the least cost
    Step 2: Assign as much as allocation to this cell

    Step 1:
    Step 2:

    200

    The min cost, so allocate as much resource as possible here

    9
    Step 3: Block those cells that cannot be allocated
    Step 4: Repeat above steps until all allocation
    have been assigned.

    Second iteration, step 4


    Step 3:

    --

    --

    200 75

    10
    The initial solution

    • Stop. The above tableau is an initial tableau


    because all allocations have been assigned

    (to p8)
    11
    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

    12
    subtracting their two of their
    least cost

    Step 1

    (8-6)

    (11-7)

    (5-4)

    (6-4) (8-5) (11-10)

    13
    Steps 2 & 3
    Step 2:
    Highest penalty
    cost

    Step 3: this has the least cost


    14
    Step 4

    --- ---

    15
    Step 5
    Second Iteration

    ---

    --- ---

    16
    3 Iteration of VAM
    rd

    --- ---

    --- ---

    17
    Initial tableau for VAM

    18
    Solution methods
    • We need a method, like the simplex
    method, to check and obtain the optimal
    solution
    • Two methods:

    1. Stepping-stone method
    2. Modified distributed method (MODI)

    19
    Stepping-stone method
    Let consider the following initial tableau from the Min Cost algorithm

    There are
    Non-basic variables

    These are basic


    variables
    Question: How can we introduce
    a non-basic variable into basic variable?
    20
    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:

    let consider this non


    basic variable

    21
    Stepping stone

    + -

    - +

    The above saying that, we add min value of all –ve cells into cell that has “+” sign, and subtracts
    the same value to the “-ve” cells 22
    Thus, max –ve is min (200,25) = 25, and we add 25 to cell A1 and A3, and subtract it from B1 and A3
    Stepping stone

    The above tableaus give min cost = 25*6 + 120*10 + 175*11


    175*4 + 100* 5 = $4525

    We can repeat this process to all possible non-basic cells in that above
    tableau until one has the min cost! NOT a Good solution method 23
    Total demand ≠ total supply

    Note that, total demand=650, and total supply = 600

    How to solve it?

    We need to add a dummy row and assign o cost to each cell as such .. 24
    Dd≠ss

    Extra row, since Demand > supply


    25
    Dd≠ss

    Extra column is added


    26

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