Linear Programming (LP) Formulation ProblemsStaffing ProblemComputer Leasing ProblemBlending ProblemInvestment ProblemInventory ProblemLinear Programming (LP) Formulation ProblemsMath 364: Principles of Optimization, Lecture 6Haijun [email protected] of MathematicsWashington State UniversitySpring 2012Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 1 / 18Linear Programming (LP) Formulation ProblemsStaffing Problem:Linear Programming FormulationFigure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 2 / 18Linear Programming (LP) Formulation ProblemsFigure: Two consecutive shifts and constraintsHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 3 / 18Linear Programming (LP) Formulation ProblemsFigure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 4 / 18Linear Programming (LP) Formulation ProblemsComputer Leasing ProblemWV-IMP, Page 111, Problem 2:Figure: Graphical SolutionHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 5 / 18Linear Programming (LP) Formulation ProblemsFigure: Graphical SolutionHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 6 / 18Linear Programming (LP) Formulation ProblemsFigure: Graphical SolutionHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 7 / 18Linear Programming (LP) Formulation ProblemsBlending ProblemFigure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 8 / 18Linear Programming (LP) Formulation ProblemsConstraints:Figure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 9 / 18Linear Programming (LP) Formulation ProblemsObjective Function:xij= # barrels of oil i used to make product jyj= $ spent on advertisement for product jg = gas, h= heating oilFigure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 10 / 18Linear Programming (LP) Formulation ProblemsFigure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 11 / 18Linear Programming (LP) Formulation ProblemsNPV: Net Present ValueMoney sitting in a bank receives interest at an annual rate r > 0.Figure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 12 / 18Linear Programming (LP) Formulation ProblemsWhen making investment decisions, we compare variousoptions in terms of their NPV.Negative cash flow = $ moving out of investmentFigure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 13 / 18Linear Programming (LP) Formulation ProblemsMax the NPVNPV of Investment 1 = $2.00NPV of Investment 2 = $1.97.Figure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 14 / 18Linear Programming (LP) Formulation ProblemsFigure: Graphical SolutionHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 15 / 18Linear Programming (LP) Formulation ProblemsInventory Problem (WV-IMP, Page 104)Figure:Haijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 16 / 18Linear Programming (LP) Formulation ProblemsFigure: Inflow = OutflowHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 17 / 18Linear Programming (LP) Formulation ProblemsInflow = OutflowFigure: Graphical SolutionHaijun Li Math 364: Principles of Optimization, Lecture 6 Spring 2012 18 /