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Extra LP Notes With Vegetable Farm Example

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Whole Farm Planning Whole-farm planning is largely a matter of enterprise selection. What crops and livestock enterprises will be produced on this farm in the next year?

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Background: Enterprise Combinations Economic theory behind whole-farm planning.

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Production Possibility Curve Definition: A Production Possibility Curve (PPC) is the geometric representation of the combination of products that can be produced with a given set of inputs. It can be defined for an entire economy or for a single firm.

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Graph of PPC enterprise 1 enterprise 2

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Types of Enterprise Relationships n Competitive with constant substitution n Competitive with increasing substitution n Supplementary n Complementary

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Competitive with Constant Substitution enterprise 1 enterprise 2 These enterprises use the same inputs, in the same ratios.

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Competitive with Increasing Substitution enterprise 1 enterprise 2 The enterprises use different ratios of inputs and inputs experience diminishing marginal returns in each case.

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Supplementary enterprise 1 enterprise 2 supplementary range enterprise 1 makes use of some inputs that are not needed for enterprise 2

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Complementary enterprise 1 enterprise 2 as we produce more of enterprise 1, we can also produce more of enterprise 2

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Examples Competitive Constant Sub: Competitive Increasing Sub: Supplementary: Complementary corn and milo corn and cotton soybeans and winter stockers broilers and cattle

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Terms n Physical substitution ratio: n Profit Ratio Quantity of Output Lost Quantity of Output Gained Profit per unit of gained output Profit per unit of lost output

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Physical Substitution Ratio The physical substitution ratio is the slope of the Production Possibility Curve.

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Profit Ratio Profit Ratio is the slope of the isoprofit line: = 1* Y1 + 2 *Y2 where 1 is profit per unit of enterprise 1, Y1 is the number of units (e.g. acres) produced, 2 is the profit per unit of enterprise 2 and Y2 is the number of units produced.

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Decision Rule Physical Substitution Ratio = Price Ratio

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Graph: Point of Tangency enterprise 1 enterprise 2 isoprofit lines and PPC

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In real life We don't know the PPC. We are going to approximate this process using a technique called "Linear Programming."

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Linear Programming Linear programming maximizes or minimizes a particular linear objective function, subject to linear restrictions. Here our objective function is to maximize the returns over variable costs. This is a one-year or short-run plan.

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Returns over variable costs The returns over variable costs come from the enterprise budgets.

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Farm Planning Process n Inventory available resources n Select enterprises to be considered. n Obtain appropriate Enterprise Budgets. n Figure out the "technical coefficients" and "RHS" (limits) n Develop linear programming tableau. n Find optimal enterprise combination.

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Resource Inventory The resource inventory tells you how much of each resource (e.g. land, labor, other inputs) you have on the farm. Labor resources is usually calculated for several periods of the year. Land may be of several different types.

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Technical Coefficients Technical Coefficients tell you how much of each resource you need to produce one unit of a given enterprise. For example, it takes one acre of row-crop land to produce one acre of cotton.

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Restrictions in LP Each limited resource requires one linear restriction in the LP model. They are normally "inequality constraints."

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Consider a simple example: Vegetable production in Zaire. Possible enterprises: Lettuce and tomatoes. Each bed of lettuce makes a profit of 30 "Zaires" (local currency). Each bed of tomatoes makes a profit of 40 Zaires.

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Marketing Restrictions Marketing: The local market will absorb no more than the output of: 16 beds of tomatoes 8 beds of lettuce

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Labor Restriction The student who wants to grow vegetables can work up to 24 hours per week on his garden. Tomatoes require 1 hour per week. Lettuce requires 2 hours per week.

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Setting up the LP: Objective Function = 1 Y1 + 2 Y2 Y1 is the number of tomato beds Y2 is the number of lettuce beds = 40 Y Y2

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Restrictions Y1 ≤ 16 (marketing restriction for tomatoes) Y2 ≤ 8 (marketing restriction for lettuce) Y1 + 2Y2 ≤ 24 (labor) So we can produce no more than 16 beds of tomatoes and 8 beds of lettuce. And we must limit our labor so that the amount expended is less than 24 hours per week.

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All Together in Equation Form Objective max 40 Y Y2 = Subject to: Y1 ≤ 16 Y2 ≤ 8 Y1 + 2Y2 ≤ 24

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Graphing the constraints (mktg 2) (mktg 1) 16 8 (labor) lettuce tomatoes

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Creating The "PPC" (mktg 2) (mktg 1) 16 8 (labor) lettuce tomatoes Feasible Region

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16 8 lettuce tomatoes Feasible Region (8,8) (16,4)

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Optimizing: Max profit $ lettuce tomatoes Feasible Region isoprofit lines slope = -30/40 Profit-Max Combination (16,4)

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With more enterprises With more than two enterprises, we can't graph the solution. We will use some software to find our answer. First we must put the problem in proper form.

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Equation Form Again Objective max 40 Y Y2 = Subject to: Y1 ≤ 16 Y2 ≤ 8 Y1 + 2Y2 ≤ 24

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The LP "tableau" Y1 Y2 Type RHS OBJ MT1 1 0 LE 16 MT2 0 1 LE 8 LBR 1 2 LE 24 Where LE means less than or equal to and RHS stands for "right hand side"

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The RHS The RHS (right-hand side) contains the amount of the constrained resource you have available.

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Technical Coefficients The numerical values in the constraint rows, other than the RHS entries, are the technical coefficients.

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Objective Function The values in the OBJ row are the amount of profit per unit of enterprise produced. In your farm plan, you will get these values from the Enterprise Budgets. For your OBJ values: Use Returns above Variable Costs.

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Using Excel to Solve the LP I took the tableau for the vegetable example, and solved it using Excel Solver (a tool in Excel). I get the answers 16 beds of tomatoes, 4 beds of lettuce, and profit of $760. If you pop this page open, you can see the formulas I used. You'll learn how to use Solver in the following slides.

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