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lecture8 - Nonlinear Objectives Absolute Value Multiperiod...

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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 Skippy’s Fine Bourbon Skippy’s 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 α ($/period/barrel) to carry inventory. Costs β ($/period/barrel) to change the production level from one period to the next “Hiring and Firing” costs. Initial Inventory I 0 Initial Production level P 0 Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippy’s 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: α | T | t =1 I t Production Level Change Cost: β | 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
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Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model More of Skippy’s 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 Skippy’s Data Four time periods. T = { 1 , 2 , 3 , 4 } ( | T | = 4 ) Demands: 20, 30, 50, 60. Inventory carrying cost α = $700/barrel/period Production Change Cost β = $600/barrel Initial Inventory: I 0 = 0 Initial Production Rate: P 0 = 40 barrels/period Jeff Linderoth IE426:Lecture 8 Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippy’s 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 Skippy’s Instance – Constraints I 0 + 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 0 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 = 0 P 0 = 40 Don’t Forget! All vars 0! Jeff Linderoth IE426:Lecture 8
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Nonlinear Objectives Absolute Value Multiperiod Planning Example Changing the Model Skippy Likes Mosel Important Point The implementation “trick” of getting these multi-period models to work (especially for a general number of time periods) is to get the boundary conditions right skippy.mos
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