Download AGEC 622 * Overhead 1

AGEC 622 – Overhead 1
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INTRO TO
MATHEMATICAL
PROGRAMMING
Definition
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 What is Mathematical Programming?
 Refers to a set of procedures dealing with the analysis of
optimization problems.
 Optimization of an objective function subject to a set of
constraints.
 How is Linear Programming different?
 Optimization of a linear objective function subject to a set of
linear constraints.
Basic Optimization Problem
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𝑂𝑝𝑡𝑖𝑚𝑖𝑧𝑒
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑠. 𝑡.
𝐹(𝑋)
𝐺 𝑋
𝜀 𝑆1
𝑋
𝜀 𝑆2
 X is a vector of decision variables. X is chosen such
that an objective 𝐹(𝑋) is optimized.
 𝐹(𝑋) is called the objective function, which will be
maximized or minimized.
 In choosing 𝑋, the choice is made subject to a set of
constraints (𝐺 𝑋 𝜀 𝑆1 and 𝑋 𝜀 𝑆2 must be obeyed).
Types of Basic Optimization Problem
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 Linear Programming: 𝐹(𝑋) and 𝐺 𝑋 are linear and
𝑋’s ≥ 0
 Integer Programming: 𝑋 must take on integer values

Mixed Integer Programming is where some 𝑋 take on integer values
 Quadratic: 𝐺 𝑋 is linear and 𝐹(𝑋) is quadratic
 Nonlinear: Both 𝐺 𝑋 and 𝐹(𝑋) are nonlinear functions
Nature of Decision Variables
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 Decide how much of something to do:
 Acres of crops to plant
 Number of animals by type to buy
 Truckloads of grains to move
 Economically assumed to be nonnegative
 Whether continuous or integer depends on the
problem.
Nature of Constraints
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 Constraints
 How much of a resource can be used
 What level of items must be supplied
 Common examples
 Acres of land available
 Hours of labor
 Production requirements
 Nutrient requirements
 Generally assumed to be an inviolate limit
Nature of the Objective Function
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 A decision maker is assumed to be interested in
optimizing a measure(s) of satisfaction by selecting
values for the decision variables
 This measure is assumed quantifiable and a single
item

Ex.: Profit Maximization, Cost Minimization
 The function that when optimized, picks the best
solution out of the set of all possible solutions

Can be more complicated (ex.: Inclusion of risk)
Example Applications
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 A firm wishes to develop a cattle feeding program.
 Objective:
 Variables:
 Constraints:
 A firm wishes to manage its production facilities.
 Objective:
 Variables:
 Constraints:
Example Applications
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 A firm wishes to develop a cattle feeding program.
 Objective: Minimize the cost of feeding cattle
 Variables: Quantity of each feedstuff to use
 Constraints: Nonnegative levels of feedstuffs, minimum
nutrient requirements
 A firm wishes to manage its production facilities.
 Objective:
 Variables:
 Constraints:
Example Applications
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 A firm wishes to develop a cattle feeding program.
 Objective: Minimize the cost of feeding cattle
 Variables: Quantity of each feedstuff to use
 Constraints: Nonnegative levels of feedstuffs, minimum
nutrient requirements
 A firm wishes to manage its production facilities.
 Objective: Maximize profits
 Variables: Amount to produce and inputs to buy
 Constraints: Nonnegative production and purchase, resource
availability, inputs on hand, minimum sales agreements
Example Applications
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 A firm may wish to best move goods.
 Objective:
 Variables:
 Constraints:
 A firm may wish to locate production facilities in a
distribution and production network
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
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Objective:
Variables:
Constraints:
Example Applications
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 A firm may wish to best move goods.
 Objective: Minimize the transportation cost
 Variables: Amount of goods to move from here to there
 Constraints: Nonnegative movement, available supply by
place, needed demand by place
 A firm may wish to locate production facilities in a
distribution and production network



Objective:
Variables:
Constraints:
Example Applications
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 A firm may wish to best move goods.


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Objective: Minimize the transportation cost
Variables: Amount of goods to move from here to there
Constraints: Nonnegative movement, available supply by place,
needed demand by place
 A firm may wish to locate production facilities in a
distribution and production network


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Objective: Minimize production and transportation costs
Variables: Where to build, amount to move from here to there,
amount to produce by location
Constraints: Nonneg transport, nonneg production, nonneg
building, resources available by place, needed demand by place
Problem Insights
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 The decision maker must deeply understand the problem
 One must define:




Decision Variables
Constraints
Objective Function
Linkages between variables and constraints


Must reflect complementary, supplementary, and competitive
relationships among variables
Consistent Data
 Use your knowledge of the problem when checking if
solutions make sense!
Numerical Mathematical Programming
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 Three main Numerical Usage subclasses
 Prescription of solutions
 Prediction of consequences
 Demonstration of sensitivity
Numerical Usage – Prescription of Solution
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 It usually involves prescriptive or normative
questions:

What decision should be made given a particular
specification of objectives, variables, and constraints?
 Not a common usage in the “real world”
 Do you think many decision makers yield decision making
power to a model?
 Most models used for decision guidance and predict
the consequence of actions
Numerical Usage – Prediction of Consequences
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 The model is used to predict in a conditional
normative setting
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In a Business Setting: Models predict consequences caused by
investments, acquisition of resources, market price conditions,
etc.
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In a Government Policy Setting: Models predict consequences
of policy changes, actions of foreign trade partners, etc.
Numerical Usage – Demonstration of Sensitivity
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 Many firms, researchers, and policy makers would
like to know what would happen if an event were to
occur?
 In these simulations, solutions are not always
implemented
 Rather, the model is used to demonstrate what might
happen if certain factors (parameters in the model)
are changed.