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Can a linear programming problem have two optimal solutions?
No. However, a special subset of such problems: integer programming, can have two optimal solutions.
What is infeasibility in linear programming?
In linear programming, infeasibility refers to a situation where no feasible solution exists for a given set of constraints and objective function. This can occur when the constraints are contradictory or when the feasible region is empty. Infeasibility can be detected by solving the linear programming problem and finding that no solution satisfies all the constraints simultaneously. In such cases, the linear programming problem is said to be infeasible.
What is the difference between linear and integer programming?
Integer programming is a method of mathematical programming that restricts some or all of the variables to integers. A subset of Integer programming is Linear programming. This is a form of mathematical programming which seeks to find the best outcome in such a way that the requirements are linear relationships.
Who invented linear programming?
George B. Dantzig
Why is it important the linear equations and inequalities?
There are many simple questions in everyday life that can be modelled by linear equations and solved using linear programming.
Is it possible for an linear programming model to have exact two optimal solutions?
Yes, but only if the solution must be integral. There is a segment of a straight line joining the two optimal solutions. Since the two solutions are in the feasible region part of that line must lie inside the convex simplex. Therefore any solution on the straight line joining the two optimal solutions would also be an optimal solution.
What is optimal feasible solution?
It is usually the answer in linear programming. The objective of linear programming is to find the optimum solution (maximum or minimum) of an objective function under a number of linear constraints. The constraints should generate a feasible region: a region in which all the constraints are satisfied. The optimal feasible solution is a solution that lies in this region and also optimises the obective function.
What is simplex method of linear programming?
The simplex method is an algorithm used for solving linear programming problems, which aim to maximize or minimize a linear objective function subject to linear constraints. It operates on a feasible region defined by these constraints, moving along the edges of the feasible polytope to find the optimal vertex. The method iteratively improves the solution by pivoting between basic feasible solutions until no further improvements can be made. It's widely used due to its efficiency and effectiveness in handling large-scale linear optimization problems.
What is the strong duality proof for linear programming problems?
The strong duality proof for linear programming problems states that if a linear programming problem has a feasible solution, then its dual problem also has a feasible solution, and the optimal values of both problems are equal. This proof helps to show the relationship between the primal and dual problems in linear programming.
Can a linear programming problem have multiple optimal solutions?
When solving linear prog. problems, we base our solutions on assumptions.one of these assumptions is that there is only one optimal solution to the problem.so in short NO. BY HADI It is possible to have more than one optimal solution point in a linear programming model. This may occur when the objective function has the same slope as one its binding constraints.
What is the difference between simplex and dual simplex problem?
The simplex method is an algorithm used to solve linear programming problems, typically starting from a feasible solution and moving toward optimality by improving the objective function. In contrast, the dual simplex method begins with a feasible solution to the dual problem and iteratively adjusts the primal solution to maintain feasibility while improving the objective. The dual simplex is particularly useful when the primal solution is altered due to changes in constraints, allowing for efficient updates without reverting to a complete re-solution. Both methods ultimately aim to find the optimal solution but operate from different starting points and conditions.
Similarities between graphical and simplex methods?
both are used to solve linear programming problems
What are the 2 major computational method of linear programming?
Simplex Method and Interior Point Methods
Can a linear programming problem have two optimal solutions?
No. However, a special subset of such problems: integer programming, can have two optimal solutions.
What is the difference between linear programming and nonlinear programming?
LPP deals with solving problems which are linear . ex: simlpex method, big m method, revised simplex, dual simplex. NLPP deals with non linear equations ex: newton's method, powells method, steepest decent method
Why are integer programming problems more difficult to solve than linear programming problems?
In both cases the constraints are used to produce an n-dimensional simplex which represents the "feasible region". In the case of linear programming this is the feasible region. But that is not the case for integer programming since only those points within the region for which the variables are integer are feasible.The objective function is then used to find the maximum or minimum - as required. In the case of a linear programming problem, the solution must lie on one of the vertices (or along one line in 2-d, plane in 3-d etc) of the simplex and so is easy to find. In the case of integer programming, the optimal solution so found may contain one or more variables that are not integer and so it is necessary to examine all the points in the immediate neighbourhood and evaluate the objective function at each of these points. This last requirement makes integer programming solutions more difficult to find.
What is primal simplex method?
II. SIMPLEX ALGORITHM A. Primal Simplex Algorithm If the unconstrained solution space is defined in n dimensions (each dimension assumed to be infinite), each inequality constraint in the linear programming formulation divides the solution space into two halves. The convex shape defined in n-dimensional space after m bisections represents the feasible area for the problem, and all points which lie inside this space are feasible solutions to the problem. Figure 1 shows the feasible region for a problem defined in two variables, n = 2, and three constraints, m = 3. Note that in linear programming, there is an implicit non-negativity constraints for the variables. The linearity of the objective function implies that the the optimal solution cannot lie within the interior of the feasible region and must lie at the intersection of at least n constraint boundaries. These intersections are known as corner- point feasible (CPF) solutions. In any linear programming problem with n decision variables, two CPF solutions are said to be adjacent if they share n − 1 common constraint boundaries. When interpreted geometrically, the Simplex algorithm moves from one corner-point feasible solution to a better corner-point-feasible solution along one of the constraint boundaries. There are only a finite number of CPF solutions, although this number is potentially exponential in n, however it is not necessary to visit all of them to determine the optimal solution to the problem. The convex nature of linear programming means that there are no local maxima present in the problem which are not also global maxima. Hence if at some CPF solution, no improvement is made by a move to another adjacent CPF then the algorithm terminates and we can be confident that the optimal solution has been found.
