4120_lecture18-F10 - Computational Methods for Management...

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Computational Methods for Management and Economics Carla Gomes Lecture 18 Reading: Section 11.1-11.3 of text book.
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Outline Introduction to Integer Programming
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Examples of Applications of Binary Variables Making “yes-or-no” type decisions Build a factory? Manufacture a product? Do a project? Assign a person to a task? Logical constraints Alternative constraints Conditional constraints Representing non-linear functions Fixed Charge Problem If a product is produced, must incur a fixed setup cost. If a warehouse is operated, must incur a fixed cost. Piecewise linear representation Diseconomies of scale Approximation of nonlinear functions Set-covering, and set partitioning Make a set of assignments that “cover” a set of requirements. Partition a set into subsets meeting given requirements
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StockCompany Example Capital Budgeting Allocation Problem StockCompany is considering 6 investments. The cash required from each investment as well as the NPV of the investment is given next. The cash available for the investments is $14,000. Stockco wants to maximize its NPV. What is the optimal strategy? An investment can be selected or not. One cannot select a fraction of an investment.
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Data for the StockCompany Problem Investment 1 2 3 4 5 6 Cash Required (1000s) $5 $7 $4 $3 $4 $6 NPV added (1000s) $16 $22 $12 $8 $11 $19 Investment budget = $14,000
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Integer Programming Formulation Max 16x 1 + 22x 2 + 12x 3 + 8x 4 + 11x 5 + 19x 6 1 0 , if we invest in i 1,. ..,6, , else i x = = What are the decision variables? Objective and Constraints?
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Capital Budgeting Allocation Problem (one resource) Knapsack Problem Why is a problem with the characteristics of the previous problem called the Knapsack Problem? It is an abstraction, considering the simple problem: A hiker trying to fill her knapsack to maximum total value. Each item she considers taking with her has a certain value and a certain weight. An overall weight limitation gives the single constraint. Practical applications: Project selection and capital budgeting allocation problems Storing a warehouse to maximum value given the indivisibility of goods and space limitations Sub-problem of other problems e.g., generation of columns for a given model in the course of optimization – cutting stock problem (beyond the scope of this course)
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The previous constraints represent “economic indivisibilities”, either a project is selected, or it is not. There is no selecting of a fraction of a project. Similarly, integer variables can model logical requirements (e.g., if stock 2 is selected, then so is stock 1.)
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How to model “logical” constraints Exactly 3 stocks are selected. If stock 2 is selected, then so is stock 1.
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4120_lecture18-F10 - Computational Methods for Management...

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