Dennis L. ricker Dept of Mechanical & Industrial Engineering The University of Iowa & Dept of usiness Lithuania hristian ollege Transportation Simplex: Initial FS 0/0/0 page of
Obtaining an Initial asic Feasible Solution ommonly-used methods: Northwest-orner Method Least-ost Method Vogel s pproximation Method (VM) Russell s Method Some of these give better quality initial solutions (which often requires fewer iterations of the simplex method in optimization) than others. Transportation Simplex: Initial FS 0/0/0 page of
ommon to all methods: Select a cell of the current tableau (i.e., a shipment route) ssign to that route the smaller of the available supply for the row the unsatisfied demand for the column Update the available supply & unsatisfied demand accordingly ancel row with no remaining supply or column with no remaining demand but never both! Repeat these steps until no rows & columns remain. The different methods differ only in their choice of a cell in the first step. Transportation Simplex: Initial FS 0/0/0 page of
Northwest-orner Method simplest to explain & use lower quality starting solution more appropriately called Upper-left orner Method Rule for selecting next cell to be assigned a shipment: hoose the upper-left corner of the remaining tableau ecause costs are ignored in the selection process, the resulting solution is generally of lower quality. Transportation Simplex: Initial FS 0/0/0 page of
Example (NW-orner Method) 7 6 NW- orner Min{S, D } = Min{,}= 7 6 0 The cell is selected, and the largest shipment possible () is assigned. This satisfies the demand for destination #, so the first column will now be ignored. Transportation Simplex: Initial FS 0/0/0 page of
NW- orner Min{S, D } = Min{,}= 7 6 0 The northwest corner of the remaining tableau is cell. 0 We assign it a shipment which is the smaller of (the remaining supply for ) and (the demand for dstn#). We will now ignore row, since the supply is now exhausted. etc. Transportation Simplex: Initial FS 0/0/0 page 6 of
Least-ost Method Rule for selecting next cell to be assigned a shipment: hoose the cell in the remaining tableau which has the lowest shipping cost Note: ignore any row or column with all costs identical, e.g., a column for a dummy destination with zero costs. Transportation Simplex: Initial FS 0/0/0 page 7 of
Example: The lowest cost in the tableau is in cell, namely. 7 6 Lowestcost cell Min{S, D } = Min{,}= 7 We assign the smaller of the available supply () and the unsatisfied 6 0 demand (). Since the supply of has been exhausted, that row will now be ignored. Transportation Simplex: Initial FS 0/0/0 page 8 of
Vogel s pproximation Method (VM) Rule for selecting next cell to be assigned a shipment: ) associate with each row a penalty for not selecting the lowest cost in that row: ) namely, the difference between the lowest and nd -lowest costs ) associate with each column a penalty for not selecting the lowest column in that row. ) Identify the row or column with the largest penalty. ) Select the lowest-cost cell in that row or column. The motivation is that, by means of this choice, the largest penalty will be avoided! Transportation Simplex: Initial FS 0/0/0 page 9 of
Example: The penalty for a row or column is the difference between the two smallest costs. 7 6 Row penalties = = = olumn penalties: = = 6 = = 7 6 0 The largest penalty is in the column for destination #. The smallest cost in that column is in cell. y choosing this cell, we avoid having to use cell with a cost of 6 (an increase of )! Minimum cost in column Min{S, D } = Min{,}= Transportation Simplex: Initial FS 0/0/0 page 0 of
Russell s Method Motivation: approximate the dual variables Ui for each source i & V for each destination j j in order to approximate the reduced cost ( ) = U + V of each cell (i,j). ij ij i j The cell with the most negative ij is then selected. The approximations : U i = largest cost in row i (ignoring those columns whose demands are already satisfied) V = largest cost in column j (ignoring those rows whose supplies are already j exhausted) Transportation Simplex: Initial FS 0/0/0 page of
Example: We first compute the estimates of the dual variables, and then the estimated reduced cost. 7 6 V j 7 The smallest (i.e., most U i 7 6 = (7+)= 7 = (6+)= 9 = (7+)= 7 = (6+)= 6 =7 (7+7)= 7 7 =6 (6+7)= 6 6 = (7+)= 8 = (6+)= 8 negative ) ij is 9, in both cells and. (In this example, we = (+)= 7 = (+)= 8 = (+7)= 9 = (+)= have arbitrarily chosen 0 Minimum cost in column Min{S, D } = Min{,}= cell.) Transportation Simplex: Initial FS 0/0/0 page of