Water Resources Development and Management Optimization (Linear Programming)

1. Water Resources Development and ManagementOptimization(Linear Programming) CVEN 5393 Mar 14, 2011

2. Acknowledgements Dr. Yicheng Wang (Visiting Researcher, CADSWES during Fall 2009 – early Spring 2010) for slides from his Optimization course during Fall 2009 Introduction to Operations Research by Hillier and Lieberman, McGraw Hill

3. Today’s Lecture • Simplex Method in Matrix form • Basics of Matrix Algebra • Revised Simplex • Dual Simplex • Network Flow Problems • R-resources / demonstration

4. Networkflow problems

5. Network Flow Models • Shortest-Path Problem • Transportation Problem • Transshipment Problem • Maximum Flow problem • Assignment Problem • Travelling Salesman Problem

6. Shortest Path Problem • Find the shortest path between points A and B

10. Dijkstra’s Algorithm for shortest path network-algorithm-demo.pdf

11. Transportation and assignment Problems The transportation problem received this name because many of its applications involve determining how to optimally transport goods. However, some of its important applications actually have nothing to do with transportation. The assignment problem involves such applications as assigning people to tasks. It can be viewed as a special type of transportation problem.

12. The Transportation Problem Prototype Example One of the main products of the P & T Company is canned peas. The peas are prepared at 3 canneries : Bellingham, Washington; Eugene, Oregon; Albert Lea, Minnesota and then shipped by truck to 4 distributing warehouses in the western United States (Sacramento, California; Salt Lake City, Utah; Rapid City, South Dakota; Albuquerque, New Mexico).

13. The Output from each cannery, the amount allocated to each warehouse, the shipping cost per truckload for each cannery-warehouse combination are given in the table. Because the shipping costs are a major expense, management objective is to reduce them as much as possible.

14. Table 8.3 shows the constraint coefficients.

15. The Transportation Model ∑ Supply = ∑ Demand Or ∑Shipments <= ∑ Supply ∑ Shipments >= ∑ Demand

16. The Assignment Problem Prototype Example An English document needs to be translated into Chinese, Japanese, German and Russian (denoted by C, J, G, R, respectively). 4 people are assigned to translate the document. Each of them can translate all the 4 languages, but everyone is assigned to translate the document into one language. The time required for each person to translate the document into each language is shown in the following table. The objective is to use the least total time to finish the translation.

17. The assignment problem is a special type of linear programming problem where assignees are being assigned to perform tasks. Assigning people to tasks is a common application of the assignment problem. However, the assignees need not be people. They also could be machines, or vehicles, or plants, or even time slots to be assigned tasks. To fit the definition of an assignment problem, these kinds of applications need to be formulated in a way that satisfy the following assumptions.

18. The Assignment Problem Model A feasible solution for the translation example is

19. Assignment Problem St. X11+X12+X13+X14=1 X21+X22+X24+X24=1 X31+X32+X33+X34=1 X41+X42+X43+X44=1 X11+X21+X31+X41=1 X12+X22+X32+X42=1 X13+X23+X33+X43=1 X14+X24+X34+X44=1 Min. 4X11+6X12+5X13+5X14 +7X21+4X22+5X23+6X24 +4X31+7X32+6X33+4X34 +5X41+3X42+4X43+7X44

20. Hungarian algorithm http://www.ee.oulu.fi/~mpa/matreng/eem1_2-1.htm

21. Example • We must determine how jobs should be assigned to machines to minimize setup times, which are given below:

22. Hungarian Algorithm • Step 1: (a) Find the minimum element in each row of the cost matrix. Form a new matrix by subtracting this cost from each row. (b) Find the minimum cost in each column of the new matrix, and subtract this from each column. This is the reduced cost matrix.

23. Example: Step 1(a) Row Reduction

24. Example: Step 1(b) Column Reduction

25. Hungarian Algorithm • Step 2: Draw the minimum number of lines that are needed to cover all the zeros in the reduced cost matrix. If m lines are required, then an optimal solution is available among the covered zeros in the matrix. Otherwise, continue to Step 3. How do we find the minimum number of lines?!

26. Example: Step 2 We need 3<4 lines, so continue to Step 3

27. Hungarian Algorithm • Step 3: Find the smallest nonzero element (say, k) in the reduced cost matrix that is uncovered by the lines. Subtract k from each uncovered element, and add k to each element that is covered by two lines. Return to Step 2.

28. Example: Step 3

29. Example: Step 2 (again) Zero Assignment Need 4 lines, so we have the optimal assignment and we stop

30. Example: Final Solution Optimal assignment How did we know which 0’s to choose?!

31. Transshipment Problem

34. Maximum Flow Problem • Flow enters at node 2 • Flow exits at node 13 • Cij are maximum flow • Limit in each arc

35. Maximum Flow Problem

37. Dijkstra’s Algorithm for shortest path network-programming.pdf