optimal solution example

Each basic feasible solution has 2 nonbasic variables and 4 basic variables. Which 2 are nonbasic variables? www.utdallas.edu/~metin. 21 

Second the simplex method provides much more than just optimal solutions. In the example above

common linear optimization software packages and to the reports they generate for optimal solutions and for sensitivity analysis. A Transportation Example.

Now suppose the company in the previous example used linear programming to minimize costs and found that their optimal solution was to skip traditional 

Mar 17 2015 A feasible solution is optimal if its objective function value is ... For example

N Variables M constraints. • U = Set of all feasible solutions Example: Convex combination of two ... x is a basic feasible solution to a LP

Example Dominance Test Graphical Depiction of. Pareto Optimal Solution feasible objective ... ?It cannot find certain Pareto-optimal solutions in.

The following example shows how an operational problem can be represented rules of thumb which can result in less than optimal solutions. Optimization.

Problems where the corresponding random functions are not everywhere differentiable appear naturally for example

For this example S=(2

The rst example is an ordinary linear program with optimal solution (4; 3 2) The second example is a (mixed) integer program where (4; 3 2) is still the optimal solution In fact here all vertices of the feasible region have x 2Z; if we know this ahead of time we can solve the integer program as a linear program

Here are two examples: (goal: minimization)f2 f1 The blue point minimizes both f1and f2 There is only one Pareto-optimal solution (goal: maximization)f2 f1 Although orange is on the Pareto front moving to purple costs very little f2for huge gains in f1 If there is an obvious solution identify it by color If not write “no obvious solution”

This has optimal solution (x;y) = (4;1:5) with 4x + 5y = 23:5 Branch-and-bound methods Example The general method Branch-and-bound example We will use branch and bound to solve the following linear program: maximize x;y2Z 4x + 5y subject to x + 4y 10 3x 4y 6 x;y 0 Step 1: solve the LP relaxation

and so no feasible solution has cost higher than 2 3 so the solution x 1:= 1 3 x 2:= 1 3 is optimal As we will see in the next lecture this trick of summing inequalities to verify the optimality of a solution is part of the very general theory of duality of linear programming

2 Applications and an example We can use complementary slackness to do two things: • Go from the optimal primal solution to the optimal dual solution and vice versa This will become more and more useful as we learn new uses for duality • Verify that a solution is optimal by checking if there’s a dual solution that goes with it

• A candidate optimal solution x to the integer program and its objective value z (Initially there is no candidate x and we set z = 1 ) A step in the algorithm examines a single node in L: we solve the linear program associated with that node and remove that node from L Let x be the optimal solution we get and let z be its objective

The normal form of an optimal solution allows one to describe the entire set of optimal solutions and derive the formula for the dimension of this set in terms 

The flow chart indicates how the algorithm is used to show that the problem is infeasible to find an optimal solution or to show that the objective function

Basic Feasible Solutions: A Quick Introduction N Variables M constraints • U = Set of all feasible solutions Example: Convex combination of two

Construct an initial basic feasible solution Each basic feasible solution has 2 nonbasic variables and 4 basic variables Which 2 are nonbasic variables? www

Degeneracy Convergence Multiple Optimal Solutions Warm up Example Suppose we are using the simplex method to solve the following canonical form LP:

Example Suppose we are using the simplex method to solve the following canonical form LP: maximize x + y subject to

An optimal solution for the model is the best solution as measured by that The following example shows how an operational problem can be represented

Now reconsider the example with the modification that tables sell for $35 instead of $30 10 Page 11 Alternate Optimal Solutions 11

Now reconsider the example with the modification that tables sell for $35 instead of $30 10 Page 11 Alternate Optimal Solutions 11

4 1 Multiple Optimal Solution Example 1 Solve by using graphical method Max Z = 4x1 + 3x2 Subject to 4x1+ 3x2 ? 24 x1 ? 4 5 x2 ? 6 x1 ? 0 x2 ? 0