Download Multi-Objective Simplex: Maximize Linear Objectives with Constraints - Prof. Steven A. Gab and more Study notes Civil Engineering in PDF only on Docsity! Multi-Objective Simplex Method Algorithm Multi-Objective Simplex Method Algorithm Michel Santos Purpose Maximize multiple linear objectives subject to multiple linear constraints x 1 x 2 x 1 x 2 Z 1 Z 2 Definitions Variables: x 1 , x 2 , ..., x n Basic variable #2: x B2 Non-basic variable #4: x NB4 Objectives: Z 1 , Z 2 , ..., Z k Constraints: a 1 , a 2 , ..., a m Reduced cost gradients: f 1 , f 2 , ..., f m One Dimensional (1D) Example Multi-Objective Simplex Method Algorithm Michel Santos x 1 x 1 <= 5x 1 >= 0 x 1 x 1 Slack Z 1 Z 1 = 3 x 1 x 1 x 1 = 0 a 1 : x 1 + x 1 Slack = 5 Variables T=[x1x1Slack] zT=cT=[30] x 1 = 0 x 1 Slack = 5 x B1 = x 1 x NB1 = x 1 Slack x={ x1x1Slack}={05}-f 1T x B x B1 −f xB 1 =zB−cB⋅ da1 dxB1 =0−0⋅1=0 xB T=[x1Slack] cB T=[0] da1 dxB1 = da1 dx1Slack =1 x NB1 -f1 T x NB xNB T =[x1] cNB T =[3] −f xNB 1 =zNB−cB⋅ da1 dxNB1 =3−0⋅1=3 da1 dxNB1 = da1 dx1 =1 Step 4a Is the current solution obviously noninferior? (Are the reduced gradients all non-negative for a particular objective?) Multi-Objective Simplex Method Algorithm Michel Santos -f1 T x NB x 1 x 2 Not noninferior x 1 x 2 Noninferior -f2 T x NB Step 4b and 4c Is the current solution uniquely noninferior? (Do any of the non-basic variables have reduced cost equal to zero?) Multi-Objective Simplex Method Algorithm Michel Santos x NB1 x NB2 Not Uniquely Noninferior Step 4C Find the other points that this leads to x NB1 x NB2 Uniquely Noninferior -f2 T x NB -f2 T x NB Step 5 Is the current solution obviously inferior? (Will the introduction of a nonbasic variable lead to an increase in all objectives?) Multi-Objective Simplex Method Algorithm Michel Santos -f1 T x NB x NB1 x NB1 Variable x NB1 Not clearly inferior x NB2 x NB2 Variable x NB2 Current solution is clearly inferior x NB2 will now become basic Which currently basic variable will it replace? Answer comes in Step 12 -f2 T x NB -f1 T x NB -f2 T x NB Step 9a Any reduced costs for any objective not zero? If so, it may lead to unexplored bases (to be checked in Step 12) Multi-Objective Simplex Method Algorithm Michel Santos x 1 x 2 No x 1 x 2 Yes x 1 x 2 Yes x 1 x 2 Yes -f1 T x NB -f2 T x NB -f3 T x NB -f4 T x NB Step 12 Would the introduction of a nonbasic variable lead to an unexplored basis? Form the basis, and then check whether it is new. Multi-Objective Simplex Method Algorithm Michel Santos Introduce the nonbasic variable x 4 and check each basic variables to determine whether the constraints impose a maximum increment for the incoming nonbasic variable x 4 x 1 Current Basic Variable New Basic Variable x 4 x 1 Current Basic Variable New Basic Variable 1= x1 −{dx4dx1 } 2= x1 −{dx4dx2 } 1 2 All of the thetas represent the maximum allowable step size before a currently basic variable becomes zero. Therefore, choose the smallest maximum step size. Step 13 Introduce a nonbasic variable and remove a basic variable to form a new basis Multi-Objective Simplex Method Algorithm Michel Santos x 1 x 2 x 3 Old solution New solution Introduce the nonbasic variable x 3 and remove the basic variable x 1 x 3 Objectives z 1 z 2 Old solution New solution
