Mathematical Modeling and Applications
IE 456 MATHEMATICAL MODELING AND APPLICATIONS
Homework 1
1. [15 points] Consider the following linear programming problem:
Min z = 3x12x23x3
Subject to x1+2x2 +x3 14 x1+2x2+4x3 12 x1 x2 +x3 = 2 x3 3 x1; x2 unrestricted
(a) [6 points] Reformulate the problem so it is in standard format.
(b) [6 points] Reformulate the problem so it is in canonical format.
(c) [3 points] Convert the problem into a maximization problem.
2. [20 points] A lathe is used to reduce the diameter of a steel shaft whose length is 36 inches (in.) from 14 in. to 12 in. The speed x1 (in revolutions per minute), the depth feed x2 (in inches per minute), and the length feed x3 (in inches per minute) must be determined. The duration of the cut is given by 36=x2x3. The compression and side stresses exerted on the cutting tool are given by
30x1 + 4000x2 and 40x1 + 6000x2 + 6000x3 pounds per square inch respectively. The temperature
(in degrees Fahrenheit) at the tip of the cutting tool is 200 + 0:5x1 + 150(x2 + x3). The maximum compression stress, side stress, and temperature allowed are 150,000 psi, 100,000 psi, and 800F. It is desired to determine the speed (which must be in the range from 600 to 800 rpm), the depth feed, and the length feed such that the duration of the cut is minimized.
(a) [10 points] In order to use a linear model the following approximation is made: Since 36=x2x3 is minimized if and only if (x2 x3) is maximized, it was decided to replace the objective by the maximization of the minimum of x2 and x3. Formulate the problem as a linear model using this approximation.
(b) [10 points] Examine the LP in part (a) and comment on the model considering the solution.
Hint: Can you eliminate a variable and/or constraint(s)? Examine the model to nd out any simpli cations that can be done to reduce the problem size and help solving the problem.
3. [15 points] A government has allocated $1.5 billion if its budget for military purposes. 60% of the
Homework 1
1. [15 points] Consider the following linear programming problem:
Min z = 3x12x23x3
Subject to x1+2x2 +x3 14 x1+2x2+4x3 12 x1 x2 +x3 = 2 x3 3 x1; x2 unrestricted
(a) [6 points] Reformulate the problem so it is in standard format.
(b) [6 points] Reformulate the problem so it is in canonical format.
(c) [3 points] Convert the problem into a maximization problem.
2. [20 points] A lathe is used to reduce the diameter of a steel shaft whose length is 36 inches (in.) from 14 in. to 12 in. The speed x1 (in revolutions per minute), the depth feed x2 (in inches per minute), and the length feed x3 (in inches per minute) must be determined. The duration of the cut is given by 36=x2x3. The compression and side stresses exerted on the cutting tool are given by
30x1 + 4000x2 and 40x1 + 6000x2 + 6000x3 pounds per square inch respectively. The temperature
(in degrees Fahrenheit) at the tip of the cutting tool is 200 + 0:5x1 + 150(x2 + x3). The maximum compression stress, side stress, and temperature allowed are 150,000 psi, 100,000 psi, and 800F. It is desired to determine the speed (which must be in the range from 600 to 800 rpm), the depth feed, and the length feed such that the duration of the cut is minimized.
(a) [10 points] In order to use a linear model the following approximation is made: Since 36=x2x3 is minimized if and only if (x2 x3) is maximized, it was decided to replace the objective by the maximization of the minimum of x2 and x3. Formulate the problem as a linear model using this approximation.
(b) [10 points] Examine the LP in part (a) and comment on the model considering the solution.
Hint: Can you eliminate a variable and/or constraint(s)? Examine the model to nd out any simpli cations that can be done to reduce the problem size and help solving the problem.
3. [15 points] A government has allocated $1.5 billion if its budget for military purposes. 60% of the