This problem is concerned with the following linear program: 2x1 + 4x2 Maximize Subject to x1...

 

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This problem is concerned with the following linear program: 2x1 + 4x2 Maximize Subject to x1 + x < X1 4x2 - 5x1 2x1 + x2 < X1 - 3 8 X2-1 5 0 X2 > 0 You are welcome to use AMPL to help check your answers. a: Graph the feasible region corresponding to the above constraints. Be sure to clearly indicate, by shading or some other means, the part of the graph that represents the set of feasible solutions. b: How many different vertex solutions does this linear program have? Draw a small circle around each vertex solution in your graph. c: A basic solution is defined by setting two of the constraints to equality and solving the resulting two equations for the two variables. If the variables also satisfy the other four constraints, then the solution is feasible. Basic feasible solutions are exactly the vertex solutions. Find a point on your graph that represents a basic solution that is infeasible. Draw a small square around this solution. d: On your drawing, add a dashed line that represents the objective function and that passes through the optimal solution. What are the optimal values of and 2, and of the objective? e: The point (0,0) is an integer feasible solution for the constraints. What are all of the other integer feasible solutions? What are the optimal integer values of and 2 for the objective, subject to the constraints? What is the resulting value of the objective function? f: Is the objective value in (e) larger or smaller than the objective value in (d)? Explain in one or two sentences why this is the result that you would expect. This problem is concerned with the following linear program: 2x1 + 4x2 Maximize Subject to x1 + x < X1 4x2 - 5x1 2x1 + x2 < X1 - 3 8 X2-1 5 0 X2 > 0 You are welcome to use AMPL to help check your answers. a: Graph the feasible region corresponding to the above constraints. Be sure to clearly indicate, by shading or some other means, the part of the graph that represents the set of feasible solutions. b: How many different vertex solutions does this linear program have? Draw a small circle around each vertex solution in your graph. c: A basic solution is defined by setting two of the constraints to equality and solving the resulting two equations for the two variables. If the variables also satisfy the other four constraints, then the solution is feasible. Basic feasible solutions are exactly the vertex solutions. Find a point on your graph that represents a basic solution that is infeasible. Draw a small square around this solution. d: On your drawing, add a dashed line that represents the objective function and that passes through the optimal solution. What are the optimal values of and 2, and of the objective? e: The point (0,0) is an integer feasible solution for the constraints. What are all of the other integer feasible solutions? What are the optimal integer values of and 2 for the objective, subject to the constraints? What is the resulting value of the objective function? f: Is the objective value in (e) larger or smaller than the objective value in (d)? Explain in one or two sentences why this is the result that you would expect.