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Transportation Problem & Degeneracy Presented By : Abhishek Upadhyay , Akhil Bhadrecha , Anshul K. Singh , Avijit Das , Kesharpu Sekhar & Saurabh Tamrakar
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Transportation Problem Aim: To find out optimum transportation schedule keeping in mind cost of transportation to be minimized.
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Transportation Problem <ul><li>The transportation problem is a special type of LPP where the objective is to minimize the cost of distributing a product from a number of sources or origins to a number of destinations. </li></ul><ul><li>Because of its special structure the usual simplex method is not suitable for solving transportation problems. These problems require special method of solution. </li></ul>
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<ul><li>The origin of a transportation problem is the location from which shipments are dispatched. The destination of a transportation problem is the location to which shipments are transported. </li></ul><ul><li>The unit transportation cost is the cost of transporting one unit of the consignment from an origin to a destination. </li></ul>
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Methods of solving Transportation Problem <ul><li>North West Corner Method (NWCM) </li></ul><ul><li>Least Cost Method (LCM) </li></ul><ul><li>VOGEL’S APPROXIMATION METHOD (VAM) </li></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step I – Check whether the given problem is a balanced problem (i.e. Requirement = Capacity). In case not then add dummy row (origin) or column (destination) with zero cost to make an unbalanced transportation problem balanced. </li></ul><ul><li>Step II – Reproduce the squares and inset squares and in the inset squares copy down the problem. </li></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step III – Find a solution using VAM method </li></ul><ul><ul><li>Calculate penalties for each row and each column. </li></ul></ul><ul><ul><li>By “penalties” we mean the difference between two best possibilities. </li></ul></ul><ul><ul><li>Give priority to the largest penalty. It belongs to a row or a column. </li></ul></ul><ul><ul><li>In that row or column make allocation to the smallest cost cell. </li></ul></ul><ul><ul><li>Cut off that row or column which is exhausted. </li></ul></ul><ul><ul><li>Continue in the same way with the rest of the table until two squares are left. </li></ul></ul><ul><ul><li>Then fill up the smallest cost square out of the remaining two. </li></ul></ul><ul><ul><li>Finally fill up the last square. </li></ul></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step IV – Check for Basic </li></ul><ul><ul><li>Initial Basic Solution = m + (n – 1) = Number of allocation </li></ul></ul><ul><ul><li>where, m is Row and n is Column </li></ul></ul><ul><ul><li>If Basic then go to next step. </li></ul></ul><ul><ul><li>Else - Covert first the Non-Basic into Basic by making an artificial allocation with zero units in the smallest cost cell. </li></ul></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step V – MODI Check Procedure </li></ul><ul><ul><li>Copy down the VAM solution along with cost matrix. </li></ul></ul><ul><ul><li>Put a ‘0’ (zero) on the top right corner in the first row. </li></ul></ul><ul><ul><li>For each filled in square adjust in such a way that: Row Number + Column Number = Cost </li></ul></ul><ul><ul><li>After filling all the row and column number look at the unused routes (open cells) </li></ul></ul><ul><ul><li>For each such cell find: Row Number + Column Number – Cost. This is called check number for that cell. </li></ul></ul><ul><ul><li>Write this number in that cell and encircle it. </li></ul></ul><ul><ul><li>If all check number are smaller than or equal to zero (<=0 ) then that solution is the best solution (i.e. solution obtained is an initial basic feasible solution). </li></ul></ul><ul><ul><li>In case any check number is positive then revision is required. Hence go to next step. </li></ul></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step VI – Procedure for Revision </li></ul><ul><ul><li>To revise the transportation table a loop is to be drawn and loop is made up of horizontal and vertical lines travelling through the cost matrix by starting and ending at the same point with the following conditions. </li></ul></ul><ul><ul><ul><li>Condition 1 – The starting point should be an open square / cell. </li></ul></ul></ul><ul><ul><ul><li>Condition 2 – All the other squares travelled should be closed squares or allocated squares. </li></ul></ul></ul><ul><ul><ul><li>Condition 3 – In horizontal or vertical travel exactly two squares can be travelled including the starting cell. </li></ul></ul></ul><ul><ul><ul><li>Condition 4 – Jumping of a square is permitted </li></ul></ul></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step VI – Procedure for Revision </li></ul><ul><ul><li>Step for Revision: </li></ul></ul><ul><ul><ul><li>Step 1 – Of all positive check number choose the largest check number and put a cross mark. </li></ul></ul></ul><ul><ul><ul><li>Step 2 – Form the loop for this cross mark square. </li></ul></ul></ul><ul><ul><ul><li>Step 3 – Along the loop write down the number in order. </li></ul></ul></ul><ul><ul><ul><li>Step 4 – Underline the alternate number. </li></ul></ul></ul><ul><ul><ul><li>Step 5 – Encircle the smallest underlined number. </li></ul></ul></ul><ul><ul><ul><li>Step 6 – Prepare the revised cell that is put in the cross mark cell the smallest number from Step 4. Continue along the loop with minus, plus and minus signs (- + -). Do not disturb other number not on loop. </li></ul></ul></ul>
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Steps involved in solving Transportation Problem by VAM method: <ul><li>Step VII – Prepare the Optimum Transportation Schedule. </li></ul><ul><li>Step VIII – Find the minimum Transportation Cost. </li></ul>
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<ul><li>A manager has three factories (i.e. origins) and four warehouses (i.e. destinations). The quantities of goods available in each factory, the requirements of goods in each warehouse and the costs of transportation of a product from each factory to each warehouse are given below. </li></ul><ul><li>Determine the optimum transportation schedule and transportation cost using: </li></ul><ul><ul><ul><li>North West Corner Method (NWCM) </li></ul></ul></ul><ul><ul><ul><li>Least Cost Method (LCM) </li></ul></ul></ul><ul><ul><ul><li>VOGEL’S APPROXIMATION METHOD (VAM) </li></ul></ul></ul>D1 D2 D3 D4 Supply P1 19 30 50 12 7 P2 70 30 40 60 10 P3 40 10 60 20 18 Demand 5 8 7 15
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