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Unformatted text preview: Nonlinear Objectives Absolute Value Multiperiod Planning IE426: Optimization Models and Applications: Lecture 8 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University September 21, 2006 Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippys Fine Bourbon Skippys Fine Liquors makes smallbatch bourbon. Forecast demands d t . The demand (in numbers of barrels) of bourbon for each time period t T def = { 1 , 2 , . . . ,  T } . Costs ($/period/barrel) to carry inventory. Costs ($/period/barrel) to change the production level from one period to the next Hiring and Firing costs. Initial Inventory I Initial Production level P Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippys Model Decision Variables P t : Number of barrels of bourbon to make in period t T I t : Inventory of bourbon after time period t T Key Algebraic Quantities Inventory Cost:  T  t =1 I t Production Level Change Cost:  T  t =1  P t 1 P t  . Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Modeling Hire and Fire Costs Can we leave   in the objective function? No! (Not if you want to get full credit!) D t : Decrease in Production Level (Barrels) in period t U t : Increase in Production Level (Barrels) in period t U t D t = P t P t 1 , t = 1 , 2 , . . .  T  . Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model More of Skippys Fine Bourbon Our Objective Components 1 Minimize inventory holding costs from one period to the next 2 Minimize hiring and firing costs incurred by changing the production rate from one period to the next Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippys Data Four time periods. T = { 1 , 2 , 3 , 4 } (  T  = 4 ) Demands: 20, 30, 50, 60. Inventory carrying cost = $700/barrel/period Production Change Cost = $600/barrel Initial Inventory: I = 0 Initial Production Rate: P = 40 barrels/period Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippys Instance Objective min 700 4 t =1 I t + 600 4 t =1  P t P t 1  Recall that this is equivalent to min 700 4 t =1 I t + 600 4 t =1 ( U t + D t ) if we just make sure that ( P t P t 1 ) = U t D t t = 1 , 2 , 3 , 4 Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippys Instance Constraints I + P 1 = 20 + I 1 I 1 + P 2 = 30 + I 2 I 2 + P 3 = 50 + I 3 I 3 + P 4 = 60 + I 4 U 1 D 1 = P 1 P U 2 D 2 = P 2 P 1 U 3 D 3 = P 3 P 2 U 4 D 4 = P 4 P 3 I = 0 P = 40 Dont Forget!...
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This note was uploaded on 02/29/2008 for the course IE 426 taught by Professor Linderoth during the Spring '08 term at Lehigh University .
 Spring '08
 Linderoth
 Optimization

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