Homework Assignment 1: The Geometry of Linear Programming Problems

Suppose a farm can produce either corn or soy beans. Both corn and soy beans require land and water which the farm has in limited amounts. Suppose the land available is 100 acres and the water is 200 acre-feet. The resource requirements for the production of one ton of corn and one ton of soy beans are as follows:

The prices of corn and soybeans are $200 and $300, respectively.

• 1. Draw a graph with production of corn on the horizontal axis and of soy beans on the vertical axis. Plot up all the combinations of corn and soybeans which do not use more land than is available. Shade this area in in some color.
• 2. Do the same for the water supply.
• 3. Indicate the combinations of corn and soy bean production that do not use more land or water than is available.
• 4. Plot the line that shows the combinations of corn and soy bean production which have a value of $20,0000; i.e., the isovalue line. Do the same for a value of $30,000. Are the lines parallel? Draw the isovalue line for $10,000.
• 4. Which point in the set of feasible productions is on the highest isovalue line? Read off from the graph the levels of production at that point. What is the value of the production at that point? This is the maximum value of output.
• 5. Now consider the effect of having more land. Suppose the amount of land is 110 acres. Repeat the above steps and determine the optimal combination of corn and soy beans and the maximum value of output. Compute the increase in output over the original problem. What is the increase in the value of output per unit increase in the availabiility of land; i.e., divide the increase in output by 10 acres, the increase in land availability. This is the marginal value of an acre of land.
• 6. Set the land back to 100 acres and increase the availability of water to 220 acre-feet. Determine the marginal value of another acre-foot of water by finding the increase in the value of production which can be achieved with the additional water. Now consider another linear programming problem that involves minimizing V= 100r+200s subject to the constraints that

  2r + 2s >= 200 and r + 4s >= 300

• 7. Indicate the set of r and s such that

  2r+2s>=200.

  Indicate the set such that r+4s>=300. Plot the values of r and s such that 100r+200s is equal to 20,000. Find the combination of r and s which minimizes 100r+200s. How do their values compare to the marginal values of land and water found in #5 above.

• 8. Consider the effect on the minimum of 100r+200s if the first constraint 2r+2s>=200 is replaced by 2r+2s>=210. What is the change in the minimum per unit change in the right hand constant of the constraint. How does this value compare to the optimal value of x in the original problem? Consider the change in the other constraint to r+4s>=330. What is the change per unit change in the constant? How does the result compare to the optimal value of y in the original problem?
• 9. Change the price of soybeans to $500 per ton and rework the entire problem up to this point.
• 10. Set the price of soybeans back to $300 and introduce a constraint of labor assuming there are 60 workers available and both corn and soybeans require 1 worker per ton of product. Rework the entire problem taking into account this additional constraint.

Homework Assignment 2: The Geometry of Integer and Mixed Linear Programming Problems

• 1. Integer Variables:

  Suppose a railroad can operate two types of trains on some route. Type A trains require one engine and three crew members whereas Type B requires two engines and a crew of five. The company has ten engines and twenty two crew members. Plot a graph which shows the numbers of Type A and Type B trains the railroad can provide. First construct the set of points satisfying the constraints ignoring the integer restrictions. Then indicate the points within this set which involve integral numbers of the two types of trains.

• 2. Mixed Integer and Continuous Variables

  Let x be a variable that can only take on nonnegative integral values. The variable y, on the other hand, can take on any real nonnegative value. If x and y must satisfy the constraints

  2x + y <= 10, x + 3y <= 12.

  Plot a graph showing the feasible combinations of x and y.

Homework Assignment 3: Exercises in Three Dimensional Geometry for Linear Programming Problems

• 0. On graph paper draw a representation of a three dimensional set of axes as follows. Draw an horizontal x axis to the right and label it. Draw a vertical axis perpendicular to the x axis and label it the z axis. Draw a line out from the origin, down and to the left, on the 45^o line. Label it the y axis.
• 1. Plot the plane given by the equation 2x+3y+4z=120. First find the intersection point of the plane with the x axis by taking y and z to zero. Then find the intersection of the plane with the y axis by taking x and z equal to zero. Likewise find the intersection of the plane with the z axis. Connect the points of intersection with the axes to show the intersection of the plane with the xy, xz, and yz planes. Use a colored pen or pencil to help visualize the plane.
• 2. Plot the plane given by the equation 5x+3y+4z=120. Draw in the intersection of this plane with the one involved in #1.
• 3. Draw another 3D graph to show the region bounded by

  2x+3y+4z <= 120

  5x+3y+4z<= 120.

• 4. Indicate by means of a set of planes the isovalue lines for the objective function

  V = 3x+3y+3z.

• 5. Suppose a region can produce corn, soybeans, and wheat. The input requirements are:

  RESOURCE	Corn (One ton)	Soybeans(One ton)	Wheat (One ton)
  Land	2 acres	1 acre	1.5 acre
  Water	2 ac-ft	4 ac-ft	3 ac-ft

  The amount of land is 100 acres and the amount of water 200 acre-feet.

  Draw a 3D graph to show the production possibilities.

• 6. The prices of corn, soybeans, and wheat are $200, $300, and 250, respectively. Show a set of iso-value lines.

Herman Weyl, a mathematical physicist, once said something to the effect that in any field of mathematics the angel of geometry is always fighting the demon of algebra.