2Linear Programming - Review Graphical Method:What is the feasible region?Where was optimal solution found?What is primary limitation of graphical method?Conversion to Standard Form:-
3Linear Programming – Review Solving Systems of Linear Equations:What is a basic solution?How did we obtain a basic solution?What is a basic feasible solution?Relationship between graphical and algebraicrepresentation of the feasible region:corner point basic solution
4Linear Programming – Review Fundamental insight – the optimal solution to a linearprogram, if it exists, is also a basic feasible solution.Naïve approach – solve for all basic solutions and findthe feasible solution with the largest value (maximizationproblem).What is the problem with this approach? – there arepossible basic solutions, where m is the number ofconstraints and n is the number of variables.
5Linear Programming – Simplex Algorithm Step 1 Convert the LP to standard form.Step 2 Obtain a bfs (if possible) from the standard form.Step 3 Determine whether the current bfs is optimal.Step 4 If the current bfs is not optimal, then determine which nonbasic basic variable should become a basic variable and which basic variable should become a nonbasic variable to find a new bfs with a better objective function value. (pivot operation)Step 5 Use EROs to find the new bfs with the better objective function value. Go back to step 3.Operations Research, Wayne L. Winston
8Linear Programming – Simplex Method: Computational Problems Breaking Ties in Selection of Non-Basic Variable – if tie fornon-basic variable with largest relative profit ( ), arbitrarilyselect incoming variable.Ties in Minimum Ratio Rule (Degeneracy) – if more than onebasic variable have same minimum ratio, select either variableto leave the basis. This will result in a basic variable taking ona value of 0. When this occurs, the solution is referred to asa degenerate basic feasible solution.When this occurs, you may transition through more than onesimplex tableau with the same objective (Z) value.
9Linear Programming – Simplex Method: Computational Problems Unbounded Solutions – if when performing the minimum ratiorule, none of the ratios are positive, then the solution isunbounded (e.g Max Z = or Min = ).
10Simplex Method – Finding an Initial Basic Feasible Solution Min Z = -3x1 + x2 + x3s.t x1 – 2x2 + x3 <= 11-4x1 + x2 +2x3 >= 32x x3 = -1x1, x2, x3 >= 0Standard Form:(-) Max Z = 3x1 - x2 - x3s.t x1 – 2x2 + x3 + x = 11-4x1 + x2 +2x x5 = 3-2x x = 1x1, x2, x3, x4, x5 >= 0
11Simplex Method – Finding an Initial Basic Feasible Solution (-) Max Z = 3x1 - x2 - x3s.t x1 – 2x2 + x3 + x = 11-4x1 + x2 +2x x5 = 3-2x x = 1x1, x2, x3, x4, x5 >= 0Only x4 is basic.Introduce artificial variables.s.t x1 – 2x2 + x3 + x = 11-4x1 + x2 +2x x5 + x = 3-2x x x7 = 1x1, x2, x3, x4, x5, x6, x7 >= 0
12Simplex Method – Solve Using Big-M Method Let M be an arbitrarily large number, then:(-) Max Z = 3x1 - x2 - x3 + 0x4 + 0x5 – Mx6 – Mx7s.t x1 – 2x2 + x3 + x = 11-4x1 + x2 +2x x x = 3-2x x x7 = 1x1, x2, x3, x4, x5, x6, x7 >= 0Note: If the simplex algorithm terminates with one of the artificial variables as a basic variable, then the original problem has no feasible solution.