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DADSS

DADSS. Optimization: Integer Programming. Administrative Detals. Final presentations on Wednesday Homework due. Integer Restrictions . Can you actually produce 0.6629 television sets? Sometimes… But not always.

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DADSS

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  1. DADSS Optimization: Integer Programming

  2. Administrative Detals • Final presentations on Wednesday • Homework due

  3. Integer Restrictions • Can you actually produce 0.6629 television sets? • Sometimes… • But not always. • Problems in which the variables/constraints in question must be whole numbers require different techniques • Integer problems are everywhere… • Scheduling • Transportation/Logistics • Project Management (PERT/CPM) • Portfolio Management

  4. Worker Assignment • Speaking of projects… • Assume you have 4 task and 4 people to do them, with each person requiring a different amount of time to do a task. To be fair, each person only does one task but all tasks need to be completed. What’s the shortest amount of time required?

  5. Branch and Bound Alg • Several methods to solve optimization problems with integer restrictions. • “Branch and Bound:” intelligently enumerate the possible solutions. • Find and upper and lower bound to the problem and only work on subproblems that could improve upon the bound. • Best case (infeasible): • A-> 2; B-> 1; C->2; D->2 = 10 • Worse case (feasible): • A-> 1; B-> 2; C->3; D->4 = 15

  6. Branch and Bound Alg • Once you have upper and lower limits, start assigning and calculate best possible solutions • A->1: B->3; C->2; D->2 = 12 • A->2: B->1; C->3; D->4 = 12 • A->3: B->1; C->2; D->2 = 12 • A->4: B->1; C->2; D->2 = 11 • Branch on the best solution

  7. Branch and Bound Alg • Branch on the best solution • A->4 + B->1 • C->3; D->4 = 12 • A->4 + B->2 • C->3; D->1 = 17 • A->4 + B->3 • C->1;D->2 = 17 • SOLUTION. Why is this assignment the best • versus having A assigned to another task? • The secret to solving branch and bound • quickly is finding good bounds.

  8. Capital budgeting Suppose we wish to invest $15,000. • Seven investment opportunities.Obviously no partial investments. • Requires an investment of $5,000 and has a present value (a time-discounted value) of $16,000; • Requires $2,500 and has a value of $8,000; • Requires $3,500 and has a value of $10,000; • Requires $6,000 and has a value of $19,500. • Requires $7,000 and has a value of $22,000. • Requires $4,500 and has a value of $12,000. • Requires $3,000 and has a value of $7,500. Where should we place our money so as to maximize our total present value?

  9. Knapsack • You have a knapsack with a finite amount of space to take with you on a long trip. You want to take as much as possible – particularly things you value highly – but you have limited space. How do you fill the pack? • Knapsack Shell.xls • 20 objects with 20 weights and 20 “desirabilities”

  10. Scheduling Back to our scheduling problem • Scheduling shell.xls

  11. Set Covering Transportation hubs • Minimum set of airline hubs to serve a set of cities • Locating Hubs 1.xlsx

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