Fast Food Nation

In: Science

Submitted By qasdina
Words 1101
Pages 5
25 3.3. CUTTING PLANE METHODS Consider a pure integer linear programming problem in which all parameters are integer. This can be accomplished by multipying the constraint by a suitable constant. Because of this assumption, also the objective function value and all the "slack" variables of the problem must have integer values. We start by solving the LP-relaxation to get a lower bound for the minimum objective value. We assume the final simplex tableau is given, the basic variables having columns with coefficient 1 in one constraint row and 0 in other rows. The solution can be read from this form: when the nonbasic variables are 0, the basic varibles have the values on right hand side (RHS) The objective function row is of the same form, with its basic variable f. If the LP-solution is fractional i.e. not integer, at least one of the RHS values is fractional. We proceed by appending to the model a constraint that cuts away a part of the feasible set so that no integer solutions are lost. Take a row i from the final simplex tableau, with a fractional RHS d. Denote by xjo the basic variable of this row and N the index set of nonbasic variables. Row i as an equation: xjo + =d

Denote by ldm the largest integer that is #d (the whole part of d, if d is positive). Because all variables are nonnegative,
# Y #d

xjo + Left hand side is integer
Y

xjo +

# ldm

From the first and last formula it follows that d - ldm #

If we denote the fractional parts by symbols r = d - ldm fij = wij - lwijm, we get a cut constraint or a cutting plane in the solution space:
$ r.

Equation form, using a slack variable si :

26 + si = -r

This equation is of basic form, with basic variable si = -r. The resulting simplex tableau is optimal but infeasible, and we apply the dual simplex method until all variables are nonnegative. The cut constraints do not cut out any feasible integer points and…...

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