lecture5 - Convexity of Functions Production Planning...

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Convexity of Functions Production Planning Separating Models and Data Yummy IE426: Optimization Models and Applications: Lecture 5 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University September 12, 2006 Jeff Linderoth IE426:Lecture 5 Convexity of Functions Production Planning Separating Models and Data Yummy Intro Stuff Did Everyone get my email? Bad News! I need to be out of town next Tuesday. Class will not be canceled. You can view last year’s lecture #7 (Check with Brian/???) Questions on the homework? (5 min max.) Jeff Linderoth IE426:Lecture 5 Convexity of Functions Production Planning Separating Models and Data Yummy Definitions Example Pictures Another example – Yummy Yummy Foods is planning on creating a new snack product that is manufactured by blending a combination of vegetable and non-vegetable oils. (Sounds Delicious!!!!!) Analysis has determined the following: 1 The two types of oils are produced separately. 2 In any month, it is not possible to produce more than 200 tons of vegetable oil and more than 250 tons of non-vegetable oil. 3 According to food experts, the “yummy coefficient” of the final product must be between 3 and 6 in order to meet quality standards. Jeff Linderoth IE426:Lecture 5 Convexity of Functions Production Planning Separating Models and Data Yummy Definitions Example Pictures Yummy Example, Cont. Market analysis reveals that the final product can be sold for $15,000 per ton. The cost and yummy coefficient for each oil is shown below: Veg1 Veg2 NVeg1 NVeg2 NVeg3 Cost ($1000/ton) 11 12 13 11 11.5 Yummyness 8.8 6.1 2.0 4.2 5.0 Jeff Linderoth IE426:Lecture 5
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Convexity of Functions Production Planning Separating Models and Data Yummy Definitions Example Pictures Decision Variables Clearly define the decision variables. x 1 : Number of tons of Veg1 used in a month x 2 : Number of tons of Veg2 used in a month x 3 : Number of tons of NVeg1 used in a month x 4 : Number of tons of NVeg2 used in a month x 5 : Number of tons of NVeg3 used in a month Jeff Linderoth IE426:Lecture 5 Convexity of Functions Production Planning Separating Models and Data Yummy Definitions Example Pictures Constraints—Key Physical Quantities Tons of YummySnack produced. x 1 + x 2 + x 3 + x 4 + x 5 What assumption did I make here? Number of Tons of Vegetable Oil Per Month x 1 + x 2 ( 200) Number of Tons of Non-Vegetable Oil Limited Per Month x 3 + x 4 + x 5 ( 250) Yummy Coefficient : weighted average of the yummy coefficients of the individual oil components. 8 . 8 x 1 + 6 . 1 x 2 + 2 x 3 + 4 . 2 x 4 + 5 x 5 / 5 i =1 x i What assumption did I make here? Jeff Linderoth IE426:Lecture 5 Convexity of Functions Production Planning Separating Models and Data Yummy Definitions Example Pictures Yummy Problem Maximize Net profit: 15( x 1 + x 2 + x 3 + x 4 + x 5 ) - 11 x 1 - 12 x 2 - 13 x 3 - 11 x 4 - 11 . 5 x 5 subject to x 1 + x 2 200 x 3 + x 4 + x 5 250 8 . 8 x 1 + 6 . 1 x 2 + 2 x 3 + 4 . 2 x 4 + 5 x 5 5 i =1 x i 6 8 . 8 x 1 + 6 . 1 x 2 + 2 x 3 + 4 . 2 x 4 + 5 x 5 5 i =1 x i 3 x i 0 i = 1 , 2 , . . . 5 Jeff Linderoth IE426:Lecture 5 Convexity of Functions Production Planning
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