Answer the following questions: Q1: Choose the correct answer ( 20 Points ):

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1 Benha University Final Exam. (ختلفات) Class: 2 rd Year Students Subject: Operations Research Faculty of Computers & Informatics Date: - / 5 / 2017 Time: 3 hours Examiner: Dr. El-Sayed Badr Answer the following questions: Q1: Choose the correct answer ( 20 Points ): 1. The maximization or minimization of a quantity is the a. goal of management science. b. decision for decision analysis. c. constraint of operations research. d. objective of linear programming. 2. Decision variables a. tell how much or how many of something to produce, invest, purchase, hire, etc. b. represent the values of the constraints. c. measure the objective function. d. must exist for each constraint. 3. Which of the following is a valid objective function for a linear programming problem? a. Max 5xy b. Min 4x + 3y + (2/3)z c. Max 5x 2 + 6y 2 d. Min (x1 + x2)/x3 4. Which of the following statements is NOT true? a. A feasible solution satisfies all constraints. b. An optimal solution satisfies all constraints. c. An infeasible solution violates all constraints. d. A feasible solution point does not have to lie on the boundary of the feasible region. 5. A solution that satisfies all the constraints of a linear programming problem except the nonnegativity constraints is called a. optimal. b. feasible. c. infeasible. d. semi-feasible. 6. Slack a. is the difference between the left and right sides of a constraint. b. is the amount by which the left side of a < constraint is smaller than the right side. c. is the amount by which the left side of a > constraint is larger than the right side. d. exists for each variable in a linear programming problem.

2 7. To find the optimal solution to a linear programming problem using the graphical method a. find the feasible point that is the farthest away from the origin. b. find the feasible point that is at the highest location. c. find the feasible point that is closest to the origin. d. None of the alternatives is correct. 8. Which of the following special cases does not require reformulation of the problem in order to obtain a solution? a. alternate optimality b. infeasibility c. unboundedness d. each case requires a reformulation. 9. The improvement in the value of the objective function per unit increase in a right-hand side is the a. sensitivity value. b. dual price. c. constraint coefficient. d. slack value. 10. As long as the slope of the objective function stays between the slopes of the binding constraints a. the value of the objective function won t change. b. there will be alternative optimal solutions. c. the values of the dual variables won t change. d. there will be no slack in the solution. Q2: TRUE/FALSE ( 20 Points ): 1. Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution. 2. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables. 3. In a feasible problem, an equal-to constraint cannot be nonbinding. 4. Only binding constraints form the shape (boundaries) of the feasible region. 5. The constraint 5x1-2x2 < 0 passes through the point (20, 50). 6. A redundant constraint is a binding constraint. 7. Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given positive coefficients in the objective function. 8. Alternative optimal solutions occur when there is no feasible solution to the problem. 9. A range of optimality is applicable only if the other coefficient remains at its original value. 10. Because the dual price represents the improvement in the value of the optimal solution per unit increase in right-hand-side, a dual price cannot be negative.

3 Q 3: ( 18 Points) a) The followin is a tableau of minimization problem. State conditions on a1, a2, b, c1, c5, c6 that are required to make the following true: z x1 x2 x3 x4 x5 x6 RHS 1 c c5 c a a b (a) The current solution is feasible and optimal. (b) The current solution is not feasible. (c) The current solution is degenerate. (d) The current solution is feasible and the LP is unbounded. b) Solve the following primal problem and its duality graphically : minimization z 240x 100x 1 Subject to : 2x x 28 4x 3x 64 x 13 x 12 2 ; x 0 where i 1,2 i Q4: ( 10 Points ) Solve the above problem using two-phase method and big M-method and which of them is the best generally. max z 8x 10x Subject to : x x x x x1 x2 4; xi 0 where i 1, 2 2

4 Q5: ( 22 Points ) a) Complete the following : 1- The general formula of Klee-Minty problem is.. 2- The number of iterations for Klee-Minty problem is.. 3- The optimal solution of Klee-Minty problem is. 4- The exponential algorithm is defined as. 5- The polynomial algorithm is defined as. b) SunRay Transport Company ships truckloads of grain from three silos to four mills. The supply (in truckloads) and the demand (also in truckloads) together with the unit transportation costs per truckload on the different routes are summarized in the following table. The unit transportation costs, cij (shown in the northeast corner of each box) are in hundreds of dollars. Supply x11 x12 x13 x x21 x22 x23 x x31 x32 x33 x34 Demand Find the minimum-cost shipping schedule between the silos and the mills using Northwest-corner, Leastcost and Vogel approximation methods? Good Luck

Benha University Final Exam. (تخلفات) Class: 2 rd Year Students Subject: Operations Research Faculty of Computers & Informatics Date: - / 5 / 2017 Time: 3 hours Examiner: Dr.

decision for decision analysis. c. constraint of operations research. d. objective of linear programming. 2. Decision variables a.

5 Benha University Final Exam. (تخلفات) Class: 2 rd Year Students Subject: Operations Research Faculty of Computers & Informatics Date: - / 5 / 2017 Time: 3 hours Examiner: Dr. El-Sayed Badr Answer the following questions: Q1: Choose the correct answer ( 20 Points ): 1. The maximization or minimization of a quantity is the a. goal of management science. b. decision for decision analysis. c. constraint of operations research. d. objective of linear programming. 2. Decision variables a. tell how much or how many of something to produce, invest, purchase, hire, etc. b. represent the values of the constraints. c. measure the objective function. d. must exist for each constraint. 3. Which of the following is a valid objective function for a linear programming problem? a. Max 5xy b. Min 4x + 3y + (2/3)z c. Max 5x 2 + 6y 2 d. Min (x 1 + x 2 )/x 3 4. Which of the following statements is NOT true? a. A feasible solution satisfies all constraints. b. An optimal solution satisfies all constraints. c. An infeasible solution violates all constraints. d. A feasible solution point does not have to lie on the boundary of the feasible region. 5. A solution that satisfies all the constraints of a linear programming problem except the nonnegativity constraints is called a. optimal. b. feasible. c. infeasible. d. semi-feasible. 6. Slack a. is the difference between the left and right sides of a constraint. b. is the amount by which the left side of a < constraint is smaller than the right side. c. is the amount by which the left side of a > constraint is larger than the right side. d. exists for each variable in a linear programming problem.

6 7. To find the optimal solution to a linear programming problem using the graphical method a. find the feasible point that is the farthest away from the origin. b. find the feasible point that is at the highest location. c. find the feasible point that is closest to the origin. d. None of the alternatives is correct. 8. Which of the following special cases does not require reformulation of the problem in order to obtain a solution? a. alternate optimality b. infeasibility c. unboundedness d. each case requires a reformulation. 9. The improvement in the value of the objective function per unit increase in a right-hand side is the a. sensitivity value. b. dual price. c. constraint coefficient. d. slack value. 10. As long as the slope of the objective function stays between the slopes of the binding constraints a. the value of the objective function won t change. b. there will be alternative optimal solutions. c. the values of the dual variables won t change. d. there will be no slack in the solution. Q2: TRUE/FALSE ( 20 Points ): 1. Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution. F 2. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables. T 3. In a feasible problem, an equal-to constraint cannot be nonbinding. T 4. Only binding constraints form the shape (boundaries) of the feasible region. F 5. The constraint 5x 1-2x 2 < 0 passes through the point (20, 50). T 6. A redundant constraint is a binding constraint. F 7. Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given positive coefficients in the objective function. F 8. Alternative optimal solutions occur when there is no feasible solution to the problem. F 9. A range of optimality is applicable only if the other coefficient remains at its original value. T 10. Because the dual price represents the improvement in the value of the optimal solution per unit increase in right-hand-side, a dual price cannot be negative. F

7 Q 3: ( 18 Points) a) The followin is a tableau of minimization problem. State conditions on a 1, a 2, b, c 1, c 5, c 6 that are required to make the following true: Solution: a) b > = 0 b) b < 0 z x 1 x 2 x 3 x 4 x 5 x 6 RHS 1 c c 5 c a a b (a) The current solution is feasible and optimal. (b) The current solution is not feasible. (c) The current solution is degenerate. (d) The current solution is feasible and the LP is unbounded. c) x5 is an entering variable ; a 1 = 1 and b = 1 or x6 is an entering variable and b = 2 d) b > 0 and a2 < = 0 b) Solve the following primal problem and its duality graphically : minimization z 240x 100x 1 Subject to : 2x x 28 4x 3x 64 x 13 x 12 2 ; x 0 where i 1,2 i

8 Point X coordinate (X1) Y coordinate (X2) Value of the objetive function (Z) O A B C D E F G H I J K L Q4: ( 10 Points ) Solve the above problem using two-phase method and big M-method and which of them is the best generally. max z 8x 10x Subject to : x1 x2 4; xi 0 where i 1, 2 2 Solution: x x x x The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate (show/hide details) As the constraint 1 is of type '=' we should add the artificial variable X5. As the constraint 2 is of type ' ' we should add the slack variable X3. As the constraint 3 is of type ' ' we should add the surplus variable X4 and the artificial variable X6.

9 MAZIMIZE: 8 X X2 MAZIMIZE: 8 X X2 + 0 X3 + 0 X4 + 0 X5 + 0 X6 1 X1-1 X2 = 1 1 X1-1 X2 + 1 X5 = 1 1 X1 + 1 X2 9 1 X1 + 1 X2 + 1 X3 = 9 1 X1 + 0,5 X2 4 1 X X2-1 X4 + 1 X6 = 4 X1, X2 0 X1, X2, X3, X4, X5, X6 0 We'll build the first tableau of Phase I from Two Phase Simplex method. Tableau Base Cb P0 P1 P2 P3 P4 P5 P6 P P P Z Intermediate operations (show/hide details) Tableau Base Cb P0 P1 P2 P3 P4 P5 P6 P P P Z Intermediate operations (show/hide details) Tableau Base Cb P0 P1 P2 P3 P4 P5 P6 P P

10 P Z Intermediate operations (show/hide details) Tableau Base Cb P0 P1 P2 P3 P4 P P P Z Intermediate operations (show/hide details) Tableau Base Cb P0 P1 P2 P3 P4 P P P Z The optimal solution value is Z = 80 X1 = 5 X2 = 4 Q5: ( 22 Points ) a) Complete the following : 1- The general formula of Klee-Minty problem is.. 2- The number of iterations for Klee-Minty problem is.. 3- The optimal solution of Klee-Minty problem is. 4- The exponential algorithm is defined as. 5- The polynomial algorithm is defined as.

11 b) SunRay Transport Company ships truckloads of grain from three silos to four mills. The supply (in truckloads) and the demand (also in truckloads) together with the unit transportation costs per truckload on the different routes are summarized in the following table. The unit transportation costs, c ij (shown in the northeast corner of each box) are in hundreds of dollars. Supply x 11 x 12 x 13 x x 21 x 22 x 23 x x 31 x 32 x 33 x 34 Demand Find the minimum-cost shipping schedule between the silos and the mills using Northwest-corner, Leastcost and Vogel approximation methods? Solution : a)

13 Good Luck

Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta

Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau