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The SIMPLEX Method - Development
Every LP is in exactly one of the following states:
- Feasible with a unique optimum solution.
- Feasible with infinitely many optima.
- Feasible, with no optimum solution (because the objective is unbounded ).
- Infeasible , and hence with no optimum solution.
Assume we are in States 1 or 2. Then the following are true:
A. The LP has at least one optimal corner point (or extreme point ). 1) If in State 1, exactly one extreme point is optimal. 2) If in State 2, at least two adjacent (neighboring) extreme points are optimal.
B. The number of extreme points is finite.
C. If the objective function at some extreme point is as good as or better than at all of its adjacent extreme point, then this extreme point is optimal for the LP.
It follows then that we can use the following iterative algorithmic procedure:
STEP 0 ( Initialization ): Find an initial extreme point and make it the current candidate (if one cannot be found the LP is in state 4, i.e. it is infeasible – so STOP).
STEP 1 ( Stopping Criterion Check ): Is the objective at the current extreme point at least as good or better than it is at all of its adjacent (neighboring) extreme points? If so this must be the optimal extreme point (via C ) – so STOP. If not, go to Step 2.
STEP 2 ( Iterative Step ): At least one of the adjacent extreme points is better – so make it the current candidate and go to Step 1.
QUESTIONS:
- How to find an initial extreme point?
- What is the algebraic characterization of an “extreme point?”
- What is the algebraic characterization of adjacent extreme points, i.e., how to move from an extreme point to one of its neighbors (in Step 2)?
Then the LP may be restated as Min (or Max) cx , st Ax = b , x ≥ 0 , where b ≥ 0.
This final system of constraint equations of the LP in standard form is called the AUGMENTED SYSTEM.
Consider such an augmented system of n variables in m equations, where m ≤ n. Note that the augmented system does not include the nonnegativity conditions!
Also note that this system has three possible states: (1) no solution, (2) exactly one solution, or (3) infinitely many solutions.
Feasible solution : Any solution to the augmented system that also satisfies nonnegativity is called a feasible solution
E.g. Consider the following constraints and the corresponding augmented system:
x 1 + 2x 2 ≤ 120 x 1 + 2x 2 + S 1 = 120 x 1 + x 2 ≤ 90 x 1 + x 2 + S 2 = 90 x 1 ≤ 70 x 1 + S 3 = 70 x 2 ≤ 50 x 2 + S 4 = 50
Note that n =6 and m =4.
An (infeasible) solution is [x 1 =70, x 2 =50, S 1 =-50, S 2 =-30, S 3 =0, S 4 =0]
A feasible solution is [x 1 =20, x 2 =20, S 1 =60, S 2 =50, S 3 =50, S 4 =30] Basic Solution: Suppose we fix ( n-m ) out of the n variables at zero, and try to solve the system of m equations in the remaining m variables. If a solution to this exists, then it is called a basic solution. Solution (1) on the previous page is an example of a basic solution.
Basic Feasible Solution: A basic solution that also satisfies nonnegativity is called a basic feasible solution (BFS). An example of a BFS is [x 1 =20, x 2 =50, S 1 =0 , S 2 =20, S 3 =50, S 4 =0 ]. Note that (1) is not a BFS even though it is a basic solution!
Original LP has exactly Original LP is infeasible one feasible solution – it Feasible Infeasible must also be the optimal one!
Augmented System
One Solution
No Solution Infinitely many solutions
BasicBasic NonbasicNonbasic BasicBasic NonbasicNonbasic
X 2 Max 20X 1 +10X 2 S 3 =0, X 1 =70 st X 1 +2X 2 ≤ 120 (Constr. 1, S 1 ) X 1 +X 2 ≤ 90 (Constr. 2, S 2 ) 90 X 1 ≤ 70 (Constr. 3, S 3 ) S 2 =0, X 1 +X 2 =90 X 2 ≤ 50 (Constr. 4, S 4 ) X 1 , X 2 ≥ 0
50 S^1 =S^4 =0^ S^2 =S^4 =0^ S^3 =S^4 =0^ S^4 =0, X^2 = II III
IV^ S^1 =S^2 =0S^ X^1 *= Feasible Region^1 =S^3 =0^ XOpt. Val. =1600^2 *= V S 2 =S 3 = S 1 =0, X 1 +2X 2 = (^10) I VI X 1 10 20 30 40 50 60 70 80 90 100 110 120
1. , : BASIC SOLUTIONS (13 of them). Note that there should be ( 46 )=15 of
these, but only 13 exist because X 1 =70 is parallel to the X 2 -axis and X 2 =50 is parallel to the X 1 -axis.
- : BASIC FEASIBLE SOLUTION: (6 of the above 13, numbered I, II, III, IV, V, VI )
- : Contour of the objective function corresponding to a value of 1600
- : Set of all Feasible solutions to the LP
