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Unformatted text preview: 1 Integer Programming 1.1 Introduction 1.1.1 Historical comments pre-1947: Konig - Egerv ary (1920s,30s), Kantorovitch (1930s), Hitchcock (1940s) 1947 present: Computational and modeling success of LP: Dantzig, Koopmans, Kantorovitch. mid 1950s mid 1960s: Formulation of many OR models as IPs in areas such as production, allocation, distribution, location, scheduling (e.g. capital budgeting, plant location). early-mid 1960s: Gomorys finite algorithms - cutting planes for solving IPs. early-late 1960s: Enumerative/branch-and-bound methods for solving IPs. late 1960s: Clear that large IPs are computationally intractable using either Gomorys methods or branch-and-bound. mid 1960s present: Emphasis on efficient combinatorial algorithms for some IPs (Edmonds, Fulkerson), such as assignment problem, max flow, matching, shortest path, etc. 1970s: IP is NP-complete = difficult w.r.t. worst-case analysis; and so are many other hard combinatorial optimization models (e.g., traveling salesman problem). (early) 1970s: Improvements via Lagrangian relaxation, Benders decomposition in IP solution methodology. 1970s: Developments of specific classes of facets/cutting planes for integral convex hull in certain optimization problems (e.g. subtour elimination cuts for TSP). 1980s present: Using polyhedral combinatorics problem-specific cutting planes in general branch-and-bound framework: Crowder, Johnson, Padberg (1983): (0,1)-IP models; Barahona, Gr otschel, Junger, Reinelt (1987): VLSI, max-cut; Gr otschel et al., Padberg et al. (1980present) Applegate, Bixby, Chvatal, Cook (1990spresent) Dantzig, Fulkerson, Johnson (1954) TSP late 1980s present: Exploitation of results from geometry of numbers: Lenstra, L.Lovasz, R. Kannan, H.Scarf; Gr obner bases and test sets: B.Sturmfels, D. Bayer, H. Scarf. 1.1.2 Comments on evolution of solution technology and meaning of large-scale Comments on DFJ (1954) technical report Comments from Held & Karp (1970) Examples from Padberg & Rinaldi (1991) 1 1.2 Some examples (1) Capital budgeting is a resource allocation problem. For projects/activities j = 1 ,...,n and resources i = 1 ,...,m , we assume the following data: c j = unit profit of activity j b i = the amount of resource i available a ij = the amount of resource i consumed by project j The goal is to decide which projects to undertake in order to maximize the total profit. Hence, the decision maker wants to solve the following problem: max n summationdisplay j =1 c j x j s.t....
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This note was uploaded on 02/09/2012 for the course ORIE 3310 taught by Professor Bland during the Spring '08 term at Cornell University (Engineering School).
- Spring '08