Simplex method algebraic interpretation 1 Add slack variables

Simplex method (algebraic interpretation) (1) Add slack variables(여유변수) to each constraint to convert them to equations. (We may refer it as an augmented LP) (2) OR-Opt. 2019 1

q q Every feasible solution of (1) can be extended, in the unique way as given above, into a feasible solution of (2). Every feasible solution of (2) can be restricted, by deleting slack variables, into a feasible solution of (1). The correspondence between feasible solutions of (1) and (2) carries optimal solutions of (1) onto optimal solutions of (2), and vice versa. So solve (2) instead of (1) and disregard the values of slack variables to obtain an optimal solution to the original problem. OR-Opt. 2019 2

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q. Next let Then find solution to the following system which maximizes z (tableau form) (zeroth equation) (first equation) (second equation) (third equation) (nonnegativity constraints) In the text, dictionary form is used, i. e. each dependent variable (including z) (called basic variable) is expressed as linear combinations of indep. var. (called nonbasic variable). (Note that, unlike the text, we place the objective function at the top. Such presentation style is used more widely and we follow that convention) OR-Opt. 2019 4

q From previous lectures, we know that if the polyhedron P has at least one extreme point and the LP over P has a finite optimal value, the LP has an extreme point optimal solution. Also an extreme point of P for our problem is a basic feasible solution algebraically. We obtain a basic solution by setting x 1 = x 2 = x 3 = 0 and finding the values of x 4, x 5, and x 6 , which can be read directly from the dictionary. (also z values can be read. ) If all values of x 4, x 5, and x 6 are nonnegative, we obtain a basic feasible solution. OR-Opt. 2019 5

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q Change the dictionary so that the new solution can be directly read off x 1 : 0 (5/2), x 4 : 5 0 So change the role of x 1 and x 4 becomes nonbasic variable and x 1 becomes basic variable. Why could we find a basic feasible solution easily? 1) all nonbasic variables appear at the right of equality (have value 0) 2) each basic variable appears in only one equation 3) each equation has exactly one basic variable appearing ( z variable may be interpreted as a basic variable, but usually it can be treated separately since it always remains basic in the simplex iterations and it is irrelevant to the description of the feasible solutions) So change the dictionary so that it satisfies the above properties. OR-Opt. 2019 9

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Equivalent to performing row operations OR-Opt. 2019 12

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q Moving directions in Rn in the example x 1 = (5/2), x 2, x 3 = 0, x 4 = 0, x 5 = 1, x 6 = (1/2), z = 25/2 OR-Opt. 2019 16

Geometric meaning of an iteration q Notation x 2=0 x 1=0 x 3=0 x 1 x 2 OR-Opt. 2019 17

q Our example : assume x 2 does not exist. It makes the polyhedron 2 dimensional since we have 5 variables and 3 equations (except nonnegativity and zeroth equation) x 3=0 A x 1=0 x 4=0 OR-Opt. 2019 x 6=0 d B 18