Note that, the list of decision variables, which are BV's for an optimal solution is readily available in the QSB optimal solution tableau. In every basic solution, the variables, which you set equal to zero are called the Non-Basic Variables (NBV), all other variables you compute by using the system of equations are called Basic Variables (BV). Those BS, which are feasible are called Basic Feasible Solutions (BFS). Remember that: A linear system AX=b has a solution if and only if the rank of A is equal to the rank of the augmented matrix (A,B).Įach solution to any system of equations is called a Basic Solution (BS). There might be multiple optimal solutions. Selecting the Optimal Corner Point: Among all feasible solutions (i.e., feasible corner points), find the optimal one (if any) by evaluating the objective function.Each feasible solution is called a Basic Feasible Solution, which is a corner point of the feasible region. Check For Feasibility: All slack and surplus must be non-negate and check for restricted condition on each variable, if any. Note also that when any Xi is an unrestricted in sign, it does not add a constraint. Notice if Xi ³ 0, then it could be written as Xi - Si = 0, setting the corresponding Si to zero means Xi = 0, therefore, it is not necessary to explicitly introduce addition slack and surplus variables. After setting these many variables to zero, then find the other variables by solving the resulting squared system of equations. The variables to set to zero are the slack, surplus, and restricted variables (any Xi's ³ 0, or Xi's £ 0) only. Finding All the Vertices: If the number of variables (including slack and surplus) is more than the number of equations, then set the following number of variables to zero:.Construction of the boundary of the restricted variables is included in the next step. Construction of the Boundaries of the Constraints Set: Transform all inequalities (except the restricted condition on each variable, if any) to equalities by adding or subtracting slack and surplus variables.Here is a four-step procedure for the algebraic solution algorithm: The customary notation is used: C = is used for the decision variables. With possibly some restricted variables: some Xi's ³ 0, some Xi's £ 0, and all others unrestricted in sign. We are interested in solving the following general problem: LP models with multi-dimensional decision variables The Algebraic Method is designed to extend the graphical method results to multi-dimensional LP problem. Now, by using the Analytical Geometry concepts, we will overcome this limitation of human vision. Therefore, what is needed to be done is to find all the intersection points (vertices) and then examine which one among all feasible vertices, provides the optimal solution. For example, we learned that: if a linear program has a non-empty, bounded feasible region, then the optimal solution is always one the vertices of its feasible region (a corner point). Having a visual understanding of the problem helps with a more rational thought process. However, it provides a clear illustration of where the feasible and non-feasible regions are, as well as, vertices. The Graphical Method is limited in solving LP problems having one or two decision variables. Collection of JavaScript E-labs Learning Objects.From Linear to Nonlinear Optimization with Business Applications.
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