Operations Research Problems: Statements and Solutions

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Each extreme point is called a basic solution , but only 0, 65 , 0, 0 , 72, 0 , and An optimum basic feasible solution for the primal problem maximizes the objective function P on the feasible set.


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The feasible set of a linear programming problem is called a convex set , because the feasible set includes the line connecting any two points in the feasible set, including points on the boundary of the feasible set. The above linear programming problem is called feasible because the feasible sets of the primal and dual programs were non-empty. In the next example, the primal problem is unbounded , and the dual problem is unfeasible. If the primal program is unbounded, then the dual program is unfeasible, and vice versa.

However, if the primal program is unfeasible, then the dual program is either unbounded or unfeasible, and vice versa. Using matrix and vector notation with the superscript T indicating the transpose of a matrix or vector, let :. This statement of the linear programming problem generalizes to A m by n matrix, x and b 1 by n , and y and c 1 by m matrices. Adding a constraint to the primal program adds a variable to the dual program.

If the new constraint binds, the optimal programs shift, and the optimal value of each objective function is diminished. This solution can be verified by using the online linear programming solver. If one proceeds this way, the results page displays as:. The three dimensional dual graph shows this situation. In the diagram below, the objective function, D, and the constraints, G1 and G2, are planes in Y1, Y2, Y3 space, while the G1 and G2 planes intersect along a line:. Using the online linear programming solver is easier than using a 3-dimensional graph to find the solution when the problem has three or more variables.


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  • One can obtain the solution to the dual problem directly using the online linear programming solver. Mulitply the dual objective function by -1 to change it to a maximization problem; similarily change the primal objective function to a minimization problem.

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    Also, switch the x and y variable labels. Using the online linear programming solver , the results page displays as:. In the problem above:. With slack variables, the constraints of this linear programming problem become the following systems of equations:. The primal problem's system of equations has five variables and three equations. The diagram of this simultaneous system of linear equations for the constraints L1, L2, and L3 follows:.

    For example, at the point In a basic feasible solution for this system, the values of exactly 2 variables must be zero, the remaining 3 variables must be positive. The optimum basic feasible solution for this system obtains at the point Here the system's variables take the values:. The basic feasible solution at the origin deserves special mention. The primal simplex algorithm used to solve linear programming problems starts at the basic feasible solution at the origin , and proceeds to the optimum basic feasible solution along the vertices of the feasible set.

    With slack variables, the constraints of the linear programming problem can be stated in general as the systems of equations:.

    Egwald Operations Research - Linear Programming - Graphical Statement and Solutions

    Many algorithms are available for solving linear programming problems. An early step in using a specific algorithm requires one to convert the mathematical statement of the linear programming problem to the standard form for that algorithm. However, it insists that the RHS right hand side of the constraints be non-negative.

    Statements and Solutions

    Many text books introduce students to the simplex algorithm , a computational method that one can use to solve small l. In the general linear programming problem, A is an m by n matrix, I m is the m dimensional identity matrix; x, t, and b are n-dimensional column vectors; I n is the n dimensional identity matrix; and y, s and c are m-dimensional column vectors.

    The objective of this book is to provide a valuable compendium of problems as a reference for undergraduate and graduate students, faculty, researchers and practitioners of operations research and management science. These problems can serve as a basis for the development or study of assignments and exams. Also, they can be useful as a guide for the first stage of the model formulation, i.

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    The book is divided into 11 chapters that address the following topics: Linear programming, integer programming, non linear programming, network modeling, inventory theory, queue theory, tree decision, game theory, dynamic programming and markov processes. Readers are going to find a considerable number of statements of operations research applications for management decision-making. The solutions of these problems are provided in a concise way although all topics start with a more developed resolution.

    The proposed problems are based on the research experience of the authors in real-world companies so much as on the teaching experience of the authors in order to develop exam problems for industrial engineering and business administration studies.

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