With Vegetable Farm Example

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With Vegetable Farm Example
Extra LP Notes With Vegetable Farm Example

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?

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

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.

Graph of PPC enterprise 2 enterprise 1

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

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

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

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

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

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

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

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

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.

Decision Rule Physical Substitution Ratio = Price Ratio

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

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

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.

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

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

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.

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.

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

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.

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

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.

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

Restrictions Y1 ≤ 16 (marketing restriction for tomatoes)
Y2 ≤ 8 (marketing restriction for lettuce) Y1 + 2Y2 ≤ (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.

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

Graphing the constraints
lettuce (mktg 1) 12 (labor) 8 (mktg 2) 24 16 tomatoes

Creating The "PPC" lettuce (mktg 1) 12 (labor) 8 (mktg 2) 24 16
Feasible Region 24 16 tomatoes

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

Optimizing: Max profit \$760
lettuce isoprofit lines slope = -30/40 Profit-Max Combination (16,4) 8 Feasible Region 16 tomatoes

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.

Equation Form Again Objective max 40Y1 + 30 Y2 =  Subject to: Y1 ≤ 16

The LP "tableau" Y1 Y2 Type RHS OBJ 40 30 MT1 1 0 LE 16 MT2 0 1 LE 8
LBR LE 24 Where LE means less than or equal to and RHS stands for "right hand side"

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

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

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.

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.