数据库文化基础 (21).pdf

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1、Introduction to Mathematical Programming Models12thweek/Optimization(I)Objectives of This WeekUnderstand the definition of mathematical programming modelingExplore the components of mathematical programming models and relate them to business goals and constraints Optimization:process of finding the

2、best(optimal)way Recommend decisions which best meet its goal(s)3Optimization ModelProblem:Constraints defined by business rules,contracts,regulations,physicsUncontrollable Variables(Parameters)DecisionVariablesObjective function(s)Performance measuresKey performance indicators(KPI)4Components of Op

3、timization Model Decision variables Controllable variables(under control of the decision maker)Objective function(of the decision variables)Goal of the problem when deciding values of decision variables Measure(s)to evaluate system performance Constraints Restrictions on the decision variables Restr

4、icted by business or physical rules Parameters Uncontrollable variables Data5Mathematical Programming Models A family of tools designed to help solve managerial,planning&design problems in which the decision makermust allocate scarce(or limited)resources among variousactivities to optimize a measure

5、able goal6Mathematical Programming Models Find =(1,2,)to()MaximizeSubject toConstraint functionsConstraints =1,2,=1,2,=1,2,(or Minimize)(or)=1,2,Objective functionSet of decision variablesSingle vs.Multiple ObjectivesUnconstrained vs.ConstrainedContinuous,integersFunctions,Linear vs.non-linearConvex

6、 vs.non-convex*Feasible satisfiesall the constraintssimultaneously7Mathematical Programming Models Feasible solution =(1,2,)satisfies all the constraints Feasible set Set of all feasible solutions Optimal solution is a feasible solution ()for every feasible solution for minimization problem8Aside:Li

7、near function A linear function is a polynomial of degree one or less The graph of a linear function is a straight line =+1,.,=11+,1,are constants ,1,are variables9Mathematical Programming ModelsLinear ProgrammingInteger ProgrammingNon-linear Programming(1)All,are linear functionsAt least one of,is

8、NOT a linear function(2)All decision variable values are real numbersSome or all decision variablevalues are integer numbers-()MaximizeSubject to =1,2,=1,2,=1,2,(or Minimize)(or)=1,2,1212 Examples of non-linear programming10Mathematical Programming ModelsQuadratic ProgrammingQuadratic Constrained Pr

9、ogramming=1+=1=1=1=0=1,2,=1,2,MinimizeSubject to=1+=1=1=1+=1=1=0=1,2,=1,2,MinimizeSubject to,are parameters,and is a decision variable11Mathematical Programming Models Three types of constraints Less-than()constraints Representing availability of resources or capacities(e.g.)Number of products in pr

10、oduction production capacity Greater-than()constraints Representing thresholds or commitments(e.g.)Number of products produced number of products ordered Equal-to(=)constraints Representing balance or consistency(e.g.)Products manufactured+products purchased=products in stock12Mathematical Programmi

11、ng ModelsSubject toMinimize2+1Example:Support Vector Machine0215for =1,(1,2)(2,3)(3.2,1.5)(4,3.2)(5,4.5)(6,2.5)Example:Linear Regression13Mathematical Programming ModelsComponents of Optimization Models:Decision variable,Objective function,Constraint,ParameterSummaryMathematical programming model is

12、 used when the decision maker needs to allocate limited resources among various activities to optimize a measureable goalMathematical programming model includes linear programming,integer programming,non-linear programming and so onOptimization Modelling:A Practical Approach,R.A.Sarker and C.S.Newton,CRC ProcessReferences