Introduction to Operations Research

1. Foundation of the Simplex Method Introduction to Operations Research

2. Constraints Boundary Equations • Graphical approach is very limited based on number of variables. The simplex method overcomes this obstacle • Optimal solutions are on the boundaries of the feasible region. Multiple variables

3. Corner-Point Feasible (CPF) solution is a feasible solution that does not lie on any line segment connection to other feasible solution. • For any linear programming problem with n decision variables, each CPF solution lies at the intersection of n constraint boundaries. • CPF solution is the simultaneous solution of a system if n constraint boundaries equations. • We call these constraint equations Defining Equations. CPF Solution Definition

4. n-decision variables (n non-negativity constraints) • m functional constraint • Total of n + m constraints CPF Solution CPF solutions Corner-point non feasible solutions Set of equations Solve simultaneously No solution

5. Simplex method moves form the Current CPF solution to an Adjacent CPF solution! • What is the path followed in the process? • What does the adjacent CPF solution mean? Adjacent CPF solutions

6. Adjacent CPF solution n=2

7. x1 + x2 ≤ 6 -x1 + 2x3 ≤ 4 x1 ≤ 4 x3 ≤ 4 x1 ≥ 0 x2 ≥ 0 x3 ≥ 0 Adjacent CPF solution n=3

8. A CPF solution lies at the intersection of n constraint boundaries • CPF solution satisfies the other constraint as well • An edge is a line segment that lies at the intersection of n-1 constraint boundaries • 2 CPF solutions are adjacent if the line segment connecting them is an edge of the feasible region • Emanating from each CPF are n edges which lead us to n adjacent CPF solutions • In any iteration of simplex method we are moving from current CPF solution to an adjacent one along with on of the edges. Adjacent CPF Solutions n>3

9. Property 1: • When there is only one Optimal Solution it should be a CPF solution • When there is multiple Optimal solutions at least two must be adjacent CPF solutions. • It suggests: • we just need to search the CPF solutions to find the optimal solution. Properties of CPF Solutions

10. Property 2: • There are only a finite number of CPF solutions. • n number of decision variables • m number of functional constraints • number of different sets of defining equations Properties of CPF Solutions

11. Property 3: • If no adjacent CPF solution is better than the current CPF solution, then the Optimal Solution is found Properties of CPF Solutions

12. Indicating Variable

13. 1) Deleting one non basic variable, entering basic variable • the variable was an indicating variable in current solution • it was used to define one of the constraints as defining constraint • deleting it from non-basics removes that constraint form the defining constraints Simplex Method

14. 2) Moving away from this current solution by increasing this one variable, while keeping the rest (n – 1) non basic variables at 0 • other non basic variables are indicating variables. • we keep them at 0 • which means, we are keeping n-1 other defining constraint as defining constraint at this stage Simplex Method

15. 3) Stopping when the first of the basic variables (leaving basic variable) reaches 0 • when a basic variable reaches 0 it will become an indicating variable. • so it defines a new constraint as the defining constraint. Simplex Method

16. In each iteration we are changing one of our defining constraints, which means that we are moving from one CPF solution to an adjacent one. Simplex Method