1、Basic Optimization Problem Notes for AGEC 641Bruce McCarl Regents Professor of Agricultural Economics Texas A&M UniversitySpring 2005,Basic Optimization Problem McCarl and Spreen Chapter 1Optimize F(X)Subject To (s.t.) G(X) S1X S2X is a vector of decision variables. X is chosen so that the objective
2、 F(X) is optimized.F(X) is called the objective function. It is what will be maximized or minimized. In choosing X, the choice is made, subject to a set of constraints, G(X) S1 and XS2 must be obeyed.,Basic Optimization ProblemOptimize F(X)Subject to (s.t.) G(X) S1X S2A program is a linear programmi
3、ng problem when F(X) and G(X) are linear and Xs 0 When X S2 requires Xs to take on integer values, you have an integer programming problem. It is a quadratic programming problem where G(X) is linear and F(X) is quadratic. It is a nonlinear programming problem when F(X) and G(X) are general nonlinear
4、 functions.,Decision VariablesThey tell us how much of something to do: acres of cropsnumber of animals by typetruckloads of oil to moveThey are generally assumed to be nonnegative.They are generally assumed to be continuous.Sometimes they are problematic. For example, when the items modeled can not
5、 have a fractional part and integer variables are neededThey are assumed to be manipulatable in response to the objective.This can be problematic also.,Constraints Restrict how much of a resource can be used what must be doneFor exampleacres of land availablehours of laborcontracts to deliverproduct
6、ion requirementsnutrient requirementsThey are generally assumed to be an inviolate limit.They can be combined with variables to allow the use of more resources at a specific price or a buy out at a specified level.,Nature of Objective functionA decision maker is assumed to be interested in optimizin
7、g a measure(s) of satisfaction by selecting values for the decision variables.This measure is assumed to be quantifiable and a single item. For example:Profit maximizationCost minimizationIt is the function that, when optimized, picks the best solution from the universe of possible solutions.Sometim
8、es, the objective function can be more complicated. For example, when dealing with profit, risk or leisure.,Example ApplicationsA firm wishes to develop a cattle feeding program.Objective - minimize the cost of feeding cattle Variables - quantity of each feedstuff to use Constraint- non negative lev
9、els of feedstuffs nutrient requirements so the animals dont starve.A firm wishes to manage its production facilities.Objective - maximize profits Variables - amount to produceinputs to buy Constraints- nonneg production and purchaseresources availableinputs on handminimum sales per agreements,Exampl
10、e ApplicationsA firm wishes to move goods most effectively.Objective - minimize transportation costs Variables - amount to move from here to there Constraints- nonnegative movementavailable supply by placeneeded demand by placeA firm is researching where to locate production facilities.Objective - m
11、inimize production + transport cost Variables - where to buildamount to move from here to thereamount to produce by location Constraints- nonnegative movement, construction, productionavailable resources by placeproducts available by placeneeded demand by placeThis mixes a transport and a production
12、 problem.,Approach of the CourseUsers generally know about the problem and are willing to use solvers as a “black box.”We will cover:appropriate problem formulationresults interpretation model use We will treat the solution processes as a “black box.“ Algorithmic details and explanations will be lef
13、t to other texts and courses such as industrial engineering.,Fundamental Types of UsesMathematical programming is way to develop the optimal values of decision variables. However, there are a considerable number of other potential usages of mathematical programming. Numerical usage is used to determ
14、ine exact levels of decision variables is probably the least common usage. Types of usage:problem insight construction numerical usages which find model solutionssolution algorithm development and investigationWe discuss the first two types of use.,Problem Insight ConstructionMathematical programmin
15、g usage requires a rigorous problem statement. One must define: the objective function the decision variablesthe constraintscomplementary, supplementary and competitive relationships among variables The data must be consistent. A decision maker must understand the problem interacting with the situat
16、ion thoroughly, discovering relevant decision variables and constraining factors in order to select the appropriate option. Frequently, resultant knowledge outweighs the value of any solutions.,Numerical Mathematical ProgrammingThree main subclasses: prescription of solutions prediction of consequen
17、ces demonstration of sensitivityIt usually involves the application of prescriptive or normative questions. For example: What decision should be made, given a particular specification of objectives, variables, and constraints? It is probably the model in least common usage over universe of models. D
18、o you think that many decision makers yield decision making power to a model?Very few circumstances deserve this kind of trust. Models are an abstraction of reality that will yield a solution suggesting a practical solution, not always one that should be implemented.,PredictionMost models are used f
19、or decision guidance or to predict the consequences of actions. They are assumed to adequately and accurately depict the entity being represented. They are used to predict in a conditional normative setting. In other words, if the firm wishes to maximize profits, then this is a prediction of what th
20、ey should do, given particular stimulus.In business settings models predict consequences of investments, acquisition of resources, drought management, and market price conditions. In government policy settings models predict the consequences of:policy changesregulationsactions by foreign trade partn
21、erspublic service provision (weather forecasting)environmental change (global warming),Sensitivity DemonstrationMany firms, researchers and policy makers would like to know what would happen if an event occurs. In these simulations, solutions are not always implemented. Likewise, the solutions may n
22、ot be used for predictions. Rather, the model is used to demonstrate what might happen if certain factors are changed. In such cases, the model is usually specified with a “realistic“ data set. It is then used to demonstrate the implications of alternative input parameters and constraint specifications.,
