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 simplications 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 military budget will be used to purchase tanks, planes and missile batteries. These can be acquired at a unit cost of $600,000, $2 million, and $800,000 respectively. It is decided that at least 200 tanks and 200 planes must be acquired. Because of the shortage of experienced pilots, it is also decided not to purchase more than 300 planes. For strategic disposes the ratio of the missile batteries to the planes purchased must fall in the range from 1

4 to 1

2 . The objective is to maximize the overall utility

of these weapons where the individual utilities are given as 1, 3, and 2 (i.e., 1 per tank, 3 per plane, and 2 per missile battery).

(a) [12 points] Formulate the given problem as an LP.

(b) [3 points] Use scaling to reduce the disparity between the coecients in your model. 1. [15 points] Let x1 = x+

1 x

1 , x2 = x+

2 x

2 , x3 = x+

3 3.

(a) [6 points] All constraints should be of \=" type and all variables should be nonnegative. Min z = 3(x+

1 x

1 )2(x+

2 x

2 )+3x+

3

Subject to

x+

1 x

1 +2(x+

2 x

2 ) x+

3 +x4 = 17

x+

1 x

1 +2(x+

2 x

2 )4x+

3 x5 = 24

x+

1 x

1 (x+

2 x

2 ) x+

3 = 5

x+

i ; x

j 0; 8i = 1; 2; 3; j = 1; 2; x4; x5 0

(b) [6 points] Since LP is of minimization type, all constraints should be of \" type and all decision variables should be nonnegative.

Min z = 3(x+

1 x

1 )2(x+

2 x

2 )+3x+

3

Subject to

x+

1 + x

1 2(x+

2 x

2 ) +x+

3 17

x+

1 x

1 +2(x+

2 x

2 )4x+

3 24

x+

1 x

1 (x+

2 x

2 ) x+

3 5

x+

1 + x

1 +(x+

2 x

2 ) +x+

3 5

x+

i ; x

j 0; 8i = 1; 2; 3; j = 1; 2

(c) [3 points] Same constraints with negative of z:

Max 3x1 + 2x2 + 3x3 or Max 3(x+

1 x

1 ) + 2(x+

2 x

2 ) 3x+

3

2. [20 points] Decision variables are already given in the question as: x1: The speed (in revolutions per minute),

x2: The depth feed (in inches per minute),

x3: The length feed (in inches per minute).

(a) [10 points]

Max z (1)

s.t.

z x2 0 (2)

z x3 0 (3)

30x1+4000x2 150; 000 (4)...