Topic7 logicalConstFormulations

Topic7 logicalConstFormulations - Algebraic formulation of...

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Algebraic formulation of logical Constraints Operations Management MSIS 33:623:386:02 Fall 2008 Farid Alizadeh Last updated November 1, 2010 Thus far, all constraints we have encountered are quantitative in nature, that is there are some numerical limit for certain quantities. For example, Total cost should not exceed budget; Total time used shad not exceed so many days; We must have at least so much of some product. There are however other kinds of constraint that are qualitative in nature. For example: You must select a subset from a set 9 choices You cannot have both choice A and choice B If You choose A you must choose B too. Such constraints often can be formulated algebraically using linear functions plus the additional condition that decision variables involved must be binary, that is either zero or one. We will review some examples here. Notice that the critical path problem and the assignment problem already contain such qualitative constraints. We now examine some common examples. 1 The Binary Knapsack Problem Study the Stocko problem on page 105 of the course pack. Here we have four investment choices, and for each one we can either choose that investment or not choose it. 1
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Decision Variables: Each investment i we create a decsion variable x i which will be zero if the i th investment is not chosen, and one, if it is; x i cannot be assigned any other value that a zero or a one. Remember that such variables are called binary . Objective function Here the objective is simply: max 19 x 1 + 21 x 2 + 14 x 3 + 10 x 4 Constraints: We have a budget of $2100, thus 7 x 1 +9 x 2 +6 x 3 +5 x 4 21 In addition we must have that x 1 ,x 2 ,x 3 , andx 4 equal zero or one. In general, a problem which has only one constraint, with the additional con- dition that all of its decision variables are binary are called the binary knapsack problem . See the Stocko.xls Fle. 2 Set Covering Problem Study the airline hub problem on page 127 of the course pack. Problems like this are called set covering problems. This problem is depicted in the picture below: We can model the airline hub problem in the form of an undirected graph : 2
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For each airport create a node . If two airports are within 1000 miles of each other connect their corre- sponding nodes with a line called an edge or arc . PIT SE SF LA SL DE HOU NO CH AT NY BOS Notice that this graph represents a “relationship” among nodes; in this case two nodes are related if the airports are within 1000 miles of each other. Also observe that this relationship is symmetric : If airport A is within 1000 miles of airport B, then airport B is also with 1000 miles of airport A. Compare this
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Topic7 logicalConstFormulations - Algebraic formulation of...

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