By O'Quigley J.

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Example text

Making xi and X2 the subjects of their respective equations gives: Setting ui and U2 ('non-basic' variables) to zero gives the fractional solution, we have already obtained, to the LP Relaxation. 80). Clearly since xi and X2 can only take integer values this is true for ui and U2 as well. 92) can be rewritten separating out their integer and fractional parts and also ensuring that the expression on the right is non-negative. e. e. 8. 8. An integer programme with Gomory cuts can be generated systematically if the Simplex Algorithm is used to solve the LP Relaxation.

We have modelled constraints (ii). However we have predefined some solutions to constraints (i). e. we use 23 of pattern 1 and 4 of pattern 3 making a total of 27 standard lengths. However, we have not incorporated some patterns in the model. g. the column for X2 has coefficients 2,1,0 since pattern 2 has 2 of length 4, 1 of length 5 and 0 of length 5. If the above model is solved as a Linear Programme (LP) we obtain dual values for the constraints (this concept is explained in any standard text on LP).

8. 8. An integer programme with Gomory cuts can be generated systematically if the Simplex Algorithm is used to solve the LP Relaxation. 98) to the model and resolve. If both cuts are appended we obtain another fractional solution at E. 11 o3 ^, . ^14 ^1 =4—,X2 ==3-, Objective - 7— This solution can be used to generate further cuts. 2). 88) at the first stage we could have obtained the optimal integer solution in one step. As we have emphasised before, however, there is no known systematic way of generating Chvatal cuts.

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