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# heizer10e_tut4 - Online Tutorial The MODI and VAM Methods...

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Online Tutorial 4 The MODI and VAM Methods of Solving Transportation Problems Tutorial Outline MODI METHOD How to Use the MODI Method Solving the Arizona Plumbing Problem with MODI VOGEL’S APPROXIMATION METHOD: ANOTHER WAY TO FIND AN INITIAL SOLUTION D ISCUSSION Q UESTIONS P ROBLEMS

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T4-2 CD T UTORIAL 4 T HE MODI AND VAM M ETHODS OF S OLVING T RANSPORTATION P ROBLEMS This tutorial deals with two techniques for solving transportation problems: the MODI method and Vogel’s Approximation Method (VAM). MODI METHOD The MODI ( modified distribution ) method allows us to compute improvement indices quickly for each unused square without drawing all of the closed paths. Because of this, it can often provide considerable time savings over other methods for solving transportation problems. MODI provides a new means of finding the unused route with the largest negative improvement index. Once the largest index is identified, we are required to trace only one closed path. This path helps determine the maximum number of units that can be shipped via the best unused route. How to Use the MODI Method In applying the MODI method, we begin with an initial solution obtained by using the northwest cor- ner rule or any other rule. But now we must compute a value for each row (call the values R 1 , R 2 , R 3 if there are three rows) and for each column ( K 1 , K 2 , K 3 ) in the transportation table. In general, we let The MODI method then requires five steps: 1. To compute the values for each row and column, set R i + K j = C ij but only for those squares that are currently used or occupied . For example, if the square at the intersection of row 2 and column 1 is occupied, we set R 2 + K 1 = C 21 . 2. After all equations have been written, set R 1 = 0. 3. Solve the system of equations for all R and K values. 4. Compute the improvement index for each unused square by the formula improvement index ( I ij ) = C ij R i K j . 5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method. Solving the Arizona Plumbing Problem with MODI Let us try out these rules on the Arizona Plumbing problem. The initial northwest corner solution is shown in Table T4.1. MODI will be used to compute an improvement index for each unused square. Note that the only change in the transportation table is the border labeling the R i s (rows) and K j s (columns). We first set up an equation for each occupied square: 1. R 1 + K 1 = 5 2. R 2 + K 1 = 8 3. R 2 + K 2 = 4 4. R 3 + K 2 = 7 5. R 3 + K 3 = 5 Letting R 1 = 0, we can easily solve, step by step, for K 1 , R 2 , K 2 , R 3 , and K 3 . 1. R 1 + K 1 = 5 0 + K 1 = 5 K 1 = 5 2. R 2 + K 1 = 8 R 2 + 5 = 8 R 2 = 3 3. R 2 + K 2 = 4 3 + K 2 = 4 K 2 = 1 R i K j C ij i j i j ij = = = value assigned to row value assigned to column cost in square (cost of shipping from source to destination )
MODI M ETHOD T4-3 TABLE T4.1 Initial Solution to Arizona Plumbing Problem in the MODI Format FROM TO ALBUQUERQUE BOSTON CLEVELAND FACTORY CAPACITY DES MOINES EVANSVILLE FORT LAUDERDALE WAREHOUSE REQUIREMENTS 5 8 4 3 100 K j R i R 1 R 2 R 3 K 1 K 2 K 3 200 200 300 100 100 300 100 4 3 9 7 5 700 200

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heizer10e_tut4 - Online Tutorial The MODI and VAM Methods...

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