- •1 Introduction to operations research
- •Explain the terms:
- •What is the matrix form of the system of linear equations?
- •X column vector with n entries;
- •Apply the Frobenius theorem in the solution of the system of linear equations.
- •Describe the algorithm of the Gauss-Jordan total elimination.
- •Describe the algorithm of finding the inverse matrix.
- •Which are the phases of the decision making process?
- •Describe the Anthony´s classification of the decision making. Draw the Anthony´s pyramid.
- •Which are the typical features of the strategic planning, tactic planning and operational control?
- •Compare programmed and non programmed decisions.
- •Compare the main kinds of the mathematical models (operational exercise, gaming, simulation, analytical model).
- •11. Which are the main types of the variables in the mathematical model?
- •11. Describe the process of formulation and application of the linear programming model.
- •What is the goal of the linear programming model?
- •2 Introduction to linear programming
- •14. Describe the constraints and the objective function in the linear programming model.
- •15.Which are the basic groups of applications of the linear programming model?
- •3 Applications of linear programming
- •18. Which are the phases of application of the linear programming model?
- •19.Describe the steps in the graphical solution of the linear programming model.
- •20.What is the graphical presentation of the feasible region?
- •21. Which solutions of the lp model are feasible, which are optimal and which are suboptimal?
- •23. What is a shadow price on a constraint? In graphical representation.
- •38. Describe the process of solution (steps) of the transportation problem.
- •39. What is the number of basic and decision variables in the transportation problem?
- •40.Describe the optimality test in the transportation problem. Which information it provides?
- •40. What is the purpose of the Dantzig loop (closed path)?
- •41.How do we solve the situation when the sum of sources is not equal to the sum of demands?
- •43. How do we solve the situation when one or more communication routes are not available?
- •44. How do we solve transportation problems with degeneration?
- •7 Graph theory
- •47. Describe the graph called Hamiltonian circuit. In which method it is applied?
- •8 Network models
- •50.What are the differences between the transportation problem and assignment problem?
- •51.What is total opportunity cost matrix? In which method it is used? How does it look?
- •52. What is the goal of the assignment problem? What are the prerequisites?
- •54. How can be applied the Vogel approximation for the travelling salesman problem?
- •55.Describe the maximal flow problem and its solution.
- •56.Describe the shortest path problem and its solution.
- •57. Describe the Dijkstra algorithm. For what it is applied?
1 Introduction to operations research
Explain the terms:
Unique matrix: determinant =1, all rows and columns are linearly independent,– square matrix nxn with 1 in diagonal
Reduced echelon form: a matrix is in this form if it satisfies conditions: a) It is row echelon form, b) every leading coefficient is 1 and is only nonzero entry in its column, May be computed by Gauss-Jordan elimination
Is unique and doesn’t depend on the algorithm used to compute it
Coefficient matrix: the matrix formed by the coefficients in a linear system of equations, for example:
2x-3y=8 →2 -3 4x+5y=1 4 5
Right hand side vector (RHS):values of conditions, has to be behind inequality sign( <=)created by for example maximal capacity of something, or minimum of something
Augmented matrix: a matrix form of a linear system of equations obtained from the coefficient matrix. It is created by adding an additional column for the constants on the right of the equal signs. The new column is set apart by a vertical line.
What is the matrix form of the system of linear equations?
Ax=b is A is [mxn] matrix;
X column vector with n entries;
b column vector with m entries
Apply the Frobenius theorem in the solution of the system of linear equations.
Frobenius theorem: System of linear equations has at least one solution if and only if the rank of matrix of system is equal to the rank of augmented matrix of system.
The rank of matrix is number of linearly independent rows (columns) in matrix.
The rank of matrix can be set as number of nonzero rows (columns) in row echelon form of matrix.
Describe the algorithm of the Gauss-Jordan total elimination.
GAUSSÐJORDAN ELIMINATION
Step 1. Choose the leftmost nonzero column and use appropriate row operations to get a 1 at the top.
Step 2. Use multiples of the row containing the 1 from step 1 to get zeros in all remaining places in the column containing this 1.
Step 3. Repeat step 1 with the submatrix formed by (mentally) deleting the row used in step 2 and all rows above this row.
Step 4. Repeat step 2 with the entire matrix, including the mentoly deleted rows. Continue this process until it is impossible to go further.
Goal of the elimination: one variable is isolated in each row.
Describe the algorithm of finding the inverse matrix.
FINDING A-I BY ROW REDUCTION - Algorithm of finding the inverse of an n x n matrix:
Form an augmented n x 2n matrix by writing the nxn identity matrix right of A
Performing row operations on the augmented matrix transform A to the identity matrix I;
The matrix I that we added will be automatically transformed to A-I; If it is impossible to transform A into identity, A is not invertible
Which are the phases of the decision making process?
-Intelligence, design, choice, implement
Describe the Anthony´s classification of the decision making. Draw the Anthony´s pyramid.
Robert N. Anthony classification: a) Strategic Planning(top) b) Tactical Planning(middle) c)Operations Control (bottom)