1 / 16

Lecture 18: Topics Integer Program/Goal Program

Lecture 18: Topics Integer Program/Goal Program. AGEC 352 Spring 2012 – April 2 R. Keeney. Assumptions of Classical Linear Programming. There are numerous assumptions that are in place when you solve an LP Proportionality – straight line behavior

tasha
Download Presentation

Lecture 18: Topics Integer Program/Goal Program

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 18: TopicsInteger Program/Goal Program AGEC 352 Spring 2012 – April 2 R. Keeney

  2. Assumptions of Classical Linear Programming • There are numerous assumptions that are in place when you solve an LP • Proportionality – straight line behavior • Optimization – choose a single measure and make it best • Divisibility – agents can choose any real number for decision variables • We will break the first one next week • We will break 2 and 3 today

  3. Optimization • A single indicator variable (often measured in dollars) is used to determine well-being of the decision maker • The decision maker wants to maximize or minimize this indicator • Tradeoff: Almost everyone enters into a management decision with multiple goals • Earn highest profits • Maximize sales • What if these conflict? • Can we estimate the profit equivalent of a unit of sales?

  4. Goal Programming • Use no objective variable • Instead design a loss variable that keeps track of how far short of a number of goals you fall • Forces you to set targets for a number of objectives • You may need to establish the tradeoff rates for each of these • Sales vs. Profits • Target Sales = $1,000,000 • Target Profits = $125,000

  5. Goal Programming for Competing Objectives • Objective variable = V • V = (125,000 – Profits) + (1,000,000 – Sales) • Minimize this variable • What does it mean if: • V < 0? • V > 0? • V = 0? • Do the units make sense? • What if profits are twice as important as sales?

  6. Example: Diet Problem • McDonalds Food Example

  7. Diet Problem as a Goal Program • Units are a problem (as always) • Convert to percentages • Set a maximum cost = $10 • Eat ten Double Cheeseburgers This is not a desirable result. Over consume some nutrients to change the objective. Need to add caps to all of the nutrients.

  8. Diet Problem as Goal ProgramWith Nutrient Caps • No excess of any nutrient • Minimize the percentage loss of our target daily nutrition needs • Spend at most $10 • Eat 2.4 Hamburgers and 0.35 Big Tasty • Cost is $2.80 • Total Loss = 549.82 (max is 1000) • Hit the target for daily fat (30gm) and sodium (1500mg) still need everything else • What do we learn from this model? • Anyone see an additional problem?

  9. McDonalds won’t sell me 2.4 hamburgers • Do I buy 2 or 3? • Do I buy a Big Tasty or not? • Integer constraints • Any number can be written as: • X+a/b • If a can be simplified to zero then we have an integer (nothing after the decimal) • In Solver: choose int

  10. Diet Problem w Goals and Integer Constraints • Instruct Solver to find integer values for all decision variables with integer (int) constraints • Solve the same model as before adding only these constraints • How does it compare to rounding off the original solution?

  11. Completely different situation • 1 Cheeseburger • 1 Filet o’ fish • Total loss = 601.05 (max 1000) • Total cost = $2.94 • We are worse off because McDonald’s will not sell us parts of a hamburger and Big N Tasty • Paying more for less nutrition

  12. Rounding to an integer solution • In many instances analysts solve the relaxed integer program and just round the solution • When is that appropriate? • In this case: • Round down (otherwise violate the boundaries) • 2 hamburgers, no Big Tasty • Total Loss = 703.97 / Total Cost = 1.60

  13. Integer Programming Fact • Objective variable • Linear Program = VL • Rounded off linear program = VR • Integer Program = VI • For a min: • VL <= VI <= VR • For a max: • VL >= VI >= VR

  14. Other issues in integer programming • No sensitivity analysis or shadow prices are calculated • Can’t find them via the simplex or calculus methods • Have to resolve the model with a one unit change to the RHS of the constraint you are interested in • Complex mathematics to solve • Large models (e.g. Sudoku) can take a long time to solve because many combinations must be checked

  15. Integer Solution Algorithms • Allow us to relax the divisibility assumption of linear programming • Search program in the neighborhood of the relaxed solution • Active area in the fields of math programming, operations research, and applied mathematics

  16. Lab/Project • Diet problem • Lab and today’s lecture • Example question: • Given a common set of food information and cost min objective • Student A solves linear program • Student B solves linear program with rounded solution • Student C solves integer program • Who has the lowest objective variable?

More Related