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# Hw6s - ENGRI 1101 Engineering Applications of OR Fall 2008...

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ENGRI 1101 Engineering Applications of OR Fall 2008 Homework 6 Homework 6 Solutions 1. (a) We’re looking at a transportation problem with the additional twist that the demand of the stores does not necessarily have to be satisfied if one wants to account for shortfall cost ( k j per unit shortfall for store j ) instead. We wish to determine the amount to ship from each warehouse i to each store j ; let x ij be a variable that will specify the amount shipped for each i = 1 , . . . , m , j = 1 , . . . , n . Let’s start off with the constraints that we can use from the general transportation problem: each warehouse can only ship as much as it has in stock n X j =1 x ij s i for all i = 1 , . . . , m (1) and all shipping quantities have to be nonnegative (this is important and should not be forgotten!): x ij 0 for all i, j (2) The most elegant (and the simplest) way of formulating this problem as a mathematical program is to introduce a new variable y j , the shortfall for store j . Note that each y j is a variable , so it’s not known beforehand (like the input parameters c ij and k j for example) but instead it will be treated in a similar manner as the x ij ’s, as part of the solution. First of all we set each y j to be nonnegative since shortfall is either there or not, it doesn’t make sense for it to be negative: y j 0 for all j = 1 , . . . , n (3) The constraint involving the demand in each store now has the following shape: m X i =1 x ij + y j d j for all j = 1 , . . . , n (4) This says that the sum of all the shipments plus the shortfall y j has to be at least the demand. Note that it does not make any explicit statements about the size of the shipments: if nothing is shipped ( x ij = 0 for all i ) then the whole demand is shortfall ( y j d j ), and if we ship at least as much as demanded then the shortfall variable can be zero. The cost function we want to minimize

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