Agricultural Economics

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AgEc 301 Agricultural Economics I
Slide Set 14 Chapter 9
Linear Programming - Introduction
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Text

• If you have the 11th edition of the text, this
  chapter is missing.
• I have posted the chapter as it will appear in
  the 12th edition online.
• http//courses.ag.uidaho.edu/aers/agecon301/ME12CH
  09.pdf

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Linear Programming

• Linear programming can be used to find the best
  answer to an assortment of questions expressed in
  terms of functional relationships.
• LP can accommodate multiple constraints in
  maximizing or minimizing a function of interest.

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Linear Programming

• Many types of LP software (or solvers) are
  available.
• It is my preference to show you how to use the
  solver included in Microsoft Excel.
• Software designed specifically for solving LPs
  may be easier to use.

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LP Assumptions

• Many production or resource constraints faced by
  managers are appropriately expressed as
  inequalities.
• Constraints often limit resources to be less than
  or equal some fixed amount.

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LP Assumptions

• Inequality constraints may also take the form of
  greater than or equal to when minimum
  production runs or a basic level of demand must
  be met.

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LP Assumptions

• A typical linear program might maximize profit
  subject to constraints on available labor,
  capital and other resources.
• With inequality constraints, not all constraints
  will be binding.

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LP Assumptions

• Linearity LP can be applied only in situations
  where the relevant objective function and
  constraint set can be expressed as linear
  equations.

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LP Assumptions

• When we say an equation is linear, we mean it is
  linear in parametersnot that it is only capable
  of modeling a straight line.

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LP Assumptions

• Most profit, revenue or cost functions (typical
  objectives to be minimized or maximized) can be
  expressed in linear form.
• Remember, an arbitrary function can be accurately
  approximated by a polynomial of sufficiently high
  order.

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LP Assumptions

• For revenue to be a linear function of output,
  output prices must be constant.
• For cost to be a linear function of output, input
  prices and returns to scale must be constant.

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LP Assumptions

• If both output prices and input costs are
  constant, them profit contribution and profits
  will also rise in a linear fashion with output.

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LP Assumptions

• To summarize
• Linear objective function
• Linear constraints
• Constant output prices
• Constant input prices
• Constant returns to scale

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LP Assumptions

• Product and input prices are assumed constant
  under perfect competition.
• In many cases it can be assumed that prices will
  be constant over the relevant range for decision
  making.

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LP Assumptions

• For example, an oil refinery may wish to examine
  their output mix from a production of 150,000
  barrels a day.
• Within the relevant range of decisions, they are
  justified in basing their analysis on prevailing
  prices.

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LP Assumptions

• Up to capacity limits, even for a business as
  large as an oil refinery that clearly doesnt
  operate under perfect competition, it is unlikely
  that marginal changes in their output mix will
  affect prices.

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LP Assumptions

• When the objective function and/or constraints
  are nonlinear such as constraints that would
  incorporate risk, LP cannot be used.
• However, mathematical programming methods exist
  for these problems.

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Single Output Planning

• One of the simplest problems that can be
  addressed using LP is the single output, multiple
  input planning problem.
• We will start by looking at a 1-output, 2-input
  decision.

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Single Output Planning

• Assume a firm produces a single product Q, using
  two inputs, L and K. Or, Q f(L,K)
• Assume further that four different processes
  exist to produce Q, each of which uses a slightly
  different combination of K and L

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Single Output Planning

• This could represent 4 different plants, each
  with its own fixed asset configuration and
  production requirements.

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Single Output Planning

• The mapping of points of equal production from
  the different processes gives us isoquants.
• Least cost combinations can be found at the point
  of tangency between a budget line and the
  isoquant.

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Optimal Input Combinations

• When resources are limited in a less than or
  equal to sort of way, the constraints can be
  illustrated on our single output graph.

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Optimal Input Combinations

• The feasible region defined by the constraints is
  found between the limits imposed by the
  constraints, the origin, and the X and Y axis.
• The feasible region defines the area where none
  of the constraints have been exceeded.

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Multiple Products

• Many production operations either produce
  multiple outputs or have the option of producing
  more than one output.
• Consider a firm that produces goods X and Y,
  using inputs A, B and C

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Multiple Products

• Suppose that the firm wishes to maximize total
  profits from the two products. The per-unit
  profit contribution for X is 12, and for Y is 9.

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Objective Function

• An equation that expresses the goal of a linear
  programming problem is called the objective
  function.
• In this example, that function would be

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Objective Function
In this function, again, the dollar value
represent profit contribution, or depending on
how you formulate it, they could also represent
per unit prices.
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Constraint Equations

• In linear programming, constraints are specified
  according to the number of units of each
  resourced needed to produce one unit of output,
  and relative to the resource limit.

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Constraint Equations
This equation says it takes 4 units of this
resource to make 1 unit of QX and 2 units to make
1 unit of QY. 32 units of this resources is
the total available for production
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Graphing Constraints

• If it takes 4 units of resource A to make one
  unit of QX then if all of resource A were
  utilized, 8 units could be produced.
• Likewise, if all was used in production of QY 16
  units could be produced.

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Graphing Constraints

• By marking the points 8 and 16 on the X and Y
  axis and drawing a straight line between those
  points, we can visually represent the constraint
  imposed by resource A.

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The Constraint Set

• In this problem, three resources are utilized (A,
  B, and C) in the following quantities

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The Constraint Set

• These constraints can also be written
  mathematically as

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Nonnegativity Constraint

• Implied in each linear programming problem is a
  constraint requiring the quantities to be
  positive.
• To be technically correct, we should add the
  nonnegativity constraints to our constraint set.

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Nonnegativity Constraint

• These constraints are
• Note that when we solve LPs in excel, we do not
  have to specify this constraint, it is assumed.

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Graphing Constraints

• Now that all important constraints have been
  specified, we can graph all of them to determine
  the feasible region for production.

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The Feasible Region

• The feasible region for a profit maximization
  problem is defined as the area bounded by the
  axis (nonnegativity) and the intersection of the
  resource constraints.

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Graphing the Objective Function

• The objective function can be represented
  graphically in much the same way as the
  constraints.
• When you do this, you are graphing isoprofit
  lines.

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Graphing the Objective Function

• The easiest way to find an isoprofit line is to
  determine a level of profit, say, 36 for our
  example, and determine how many units of X and Y
  must be sold to achieve this.
• Plot these on each axis, and draw a straight line
  between them.

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Graphing the Objective Function

• Since the slope of the isoprofit line will not
  change (by assumption) all other isoprofit lines
  will be parallel to the one you have drawn.

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The Graphic Solution

• Graphically, the profit maximizing combination of
  X and Y will be determined by the highest
  isoprofit line that intersects the feasible
  region.

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Multiple Solutions

• Occasionally an LP problem will find multiple
  alternative solutions.
• Graphically, this happens when the isoprofit line
  is coincidental with the boundary of the feasible
  region.

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