lecture8 - Nonlinear Objectives Absolute Value Multiperiod...

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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 small-batch 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 .

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lecture8 - Nonlinear Objectives Absolute Value Multiperiod...

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