Agricultural Economics

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AgEc 301 Agricultural Economics I
Slide Set 15 Chapter 9
Linear Programming Part 2
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Our Previous LP Example
We wish to maximize profit from the production of
two goods, X and Y, where X has a profit
contribution of 12 per unit and Y 9 per
unit. In equation form this is
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Our Previous LP Example
The constraints can be illustrated in table form
as follows
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Our Previous LP Example

• Re-writing the constraints in equation form gives
  us

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Constraint on resource A
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Solution

• Note, the graphic solution occurs at the corner
  where constraints on inputs B and C are binding.
• Thus, we could rewrite these constraints as
  equalities and solve for QX and QY.

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Solution

• 4QX 2QY 32
• Minus 2QX 2QY 20
• Equals 2QX 12
• Or QX 6
• And it follows that QY 4

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Algebraic Specification

• As you can probably guess, it doesnt take a very
  complicated LP problem to be too large to try and
  solve graphically.
• Thus, we must also know how to set up and solve
  these problems algebraically (and using computer
  software).

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Slack Variables

• In order to solve an LP problem algebraically, we
  must introduce the concept of slack variables.
• Slack variables represent the unused portion of a
  resource (excess capacity).

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Slack Variables

• When slack variables are used, we can rewrite the
  constraints as equalities. For example, allowing
  SA to represent the unused units of resource A,
  our first constraint can be rewritten
• 4QX 2QY SA 32

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Slack Variables

• SA then will take on whatever value needed to
  convert the inequality to an equality.
• If the constraint is binding then SA 0

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Slack Variables

• Rewriting our constraint set we obtain
• 4QX 2QY SA 32
• 1QX 1QY SB 10
• 3QY SC 21

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Slack Variables

• The addition of slack variables simplifies
  algebraic analysis and also provide us with
  important information for planning.
• A value of zero indicates a binding constraint,
  while a positive value indicates excess capacity.

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Algebraic Solution

• The complete specification is now
• Subject to
• 4QX 2QY SA 32
• 1QX 1QY SB 10
• 3QY SC 21

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Algebraic Solution

• The implied nonnegativity constraints should be
  included for the complete specification
• The slack variables are included in the
  nonnegativity constraints
• Why?

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Algebraic Solution

• The problem is to solve this system for values of
  the Q and S variables (5 unknowns).
  Unfortunately, we have only 3 constraint
  equations, so we cannot solve this without
  additional information.

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Algebraic Solution

• In order to solve this problem, we can exploit
  the fact that LP solutions occur at corners of
  the feasible region.
• At each corner, the number of unknowns is exactly
  equal to the number of known constraint
  conditions.

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Algebraic Solution

• A solution can be obtained for each corner of the
  feasible region.

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Algebraic Solution

• At each corner, the constraints can be expressed
  as a system of three equations with three
  unknowns. For example at point N
• SA 0
• QY 0, and
• 4QX 2QY SA 32

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Algebraic Solution

• So, this system can be solved as
• 4QX 2QY SA 32
• 4QX 2(0) (0) 32
• 4QX 32
• QX 8

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Algebraic Solution

• This procedure can be repeated at each of the
  corner points, and the values for the Qs plugged
  into the objective function to calculate profit
  at each corner.

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Algebraic Solution

• This iterative process is followed in what is
  called the simplex solution method.
• We will not go through the simplex method here
  (not by hand anyway). Solvers use the simplex
  method to quickly solve LP problems.

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Solving the LP with Excel

• In order to solve LP problems in Excel, you need
  to adopt a very consistent method for
  constructing the problem.
• Consistency avoids confusion in this case because
  the solver requires you to do some of the work.

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Solving the LP with Excel

• Step 1. Write the objective function. Label
  adjacent cells (in different columns) as the
  prices or profit contributions.
• Under these enter the numeric price or profit
  contribution for each production alternative

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Solving the LP with Excel
For example
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Solving the LP with Excel

• Then, in similar fashion, set up two cells for
  the quantities of X and Y to be produced. For
  the time being set these equal to 1.

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Solving the LP with Excel

• In this example cells A4 and B4 are important,
  all of our formulas will refer to these cells.
• To finish Step 1, we must put the formula for the
  objective function into the spreadsheet.
• A2A4 B2B4

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Solving the LP with Excel
It will look like this
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Solving the LP with Excel

• Step 2. In the same manner as the objective
  function, construct your constraint set.
• It is helpful to write the problem out in
  equation form before doing this.

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Solving the LP with Excel

• One additional thing is required in Step 2 for
  setting the problem up for Excel.
• You must add a cell that contains the total units
  of that constraint used.

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Solving the LP with Excel
Here cell D6 contains the formula A6A4
B6B4
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Solving the LP with Excel

• Step 3. Next to the cells which calculate the
  total of each resource used, enter the resource
  constraints.

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Solving the LP with Excel
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Solving the LP with Excel

• Step 4. Now that we have all the pieces
  assembled, we can use the Excel solver.
• To do this, from the Tools menu, select solver.

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Solving the LP with Excel

• Set Target Cell will be E2
• Equal to will be Max
• By changing cells will be A4B4
• Each constraint will be entered as
• D6 lt E6

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Solving the LP with Excel
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Solving the LP with Excel

• Now, simply click Solve and the program will
  automatically find the optimal solution for you.
• Once the solution is found you can have Excel
  print it to a new spreadsheet ply by selecting
  answer and clicking OK

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Solving the LP with Excel
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Solving the LP with Excel