2 Standard Minimization Problems Minimization with constraints Example Solve the linear programming problem minimize 4 2 2 10 4 12 xyz Standard Minimization Problems 1 Objective function is minimized 2 All variables are nonnegative 3 All constrai ID: 22462 Download Pdf

2 Standard Minimization Problems Minimization with constraints Example Solve the linear programming problem minimize 4 2 2 10 4 12 xyz Standard Minimization Problems 1 Objective function is minimized 2 All variables are nonnegative 3 All constrai

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Math 141-copyright Joe Kahlig, 10C Page 1 Section 4.2: Standard Minimization Problems Minimization with constraints: Example: Solve the linear programming problem. minimize + 4 + 2 + 2 10 + 4 12 x,y,z Standard Minimization Problems: 1) Objective function is minimized. 2) All variables are non-negative. 3) All constraints are in the form: ax by ...... constant Dual Problems: Every standard minimization linear programming problem is associated with a standard maximization problem (and vice versa). The origin al problem is called the primal prob- lem and the associated problem is

called the dual problem

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Math 141-copyright Joe Kahlig, 10C Page 2 The Fundamental Theorem of Duality: A primal problem has a solution if and only if the correspondi ng dual problem has a solution. Fur- thermore, if a solution exists, then: A) The objective function of both the primal and the dual prob lems attain the same optimal value. B) The optimal solution to the problem appears under the slac k variables in the simplex tableau associated with the dual problem. Example: Find the dual problem and give the solution to the mi nimization (primal) problem. Minimize = 40 + 12 +

40 + 5 20 30 x,y,z

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Math 141-copyright Joe Kahlig, 10C Page 3 Example: The shipping problem from the section 3.2 notes. Minimize = 16 + 20 + 22 + 18 + 16 + 14 800 600 500 400 450 x,y,z,u,v,w 1 0 0 0 800 0 0 0 600 1 0 0 1 0 0 500 0 1 0 0 1 0 400 0 0 1 0 0 1 450 16 20 22 18 16 14 1 0 1 0 0 16 1 0 0 1 0 20 1 0 0 0 1 22 1 1 0 0 18 1 0 1 0 16 1 0 0 1 14 800 600 500 400 450 1 0 1 0 0 1 0 0 0 0 0 0 16 1 0 0 1 0 0 1 0 0 0 0 0 20 1 0 0 0 1 0 0 1 0 0 0 0 22 1 1 0 0 0 0 0 1 0 0 0 18 1 0 1 0 0 0 0 0 1 0 0 16 1 0 0 1 0 0 0 0 0 1 0 14 800 600 500 400 450 0 0 0 0 0 0 1 Note: 4 steps thru the

simplex method:R1C3, R6C5, R5C4, R2C2 1 0 1 0 0 1 0 0 0 0 0 0 16 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 20 1 0 0 0 1 0 1 0 0 1 1 0 18 50 0 0 0 0 500 250 0 0 150 450 1 21700

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