1
Integer Programming
1.1
Introduction
1.1.1
Historical comments
•
pre1947: K¨onig  Egerv´
ary (1920’s,30’s), Kantorovitch (1930’s), Hitchcock (1940’s)
•
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).
•
earlymid 1960s: Gomory’s finite algorithms  cutting planes for solving IPs.
•
earlylate 1960s: Enumerative/branchandbound methods for solving IPs.
•
late 1960s: Clear that large IPs are computationally intractable using either Gomory’s methods
or branchandbound.
•
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 NPcomplete = difficult w.r.t.
worstcase analysis; and so are many other “hard”
combinatorial optimization models (e.g., traveling salesman problem).
•
(early) 1970s: Improvements via “Lagrangian relaxation”, “Bender’s 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” – problemspecific cutting planes in general
branchandbound framework:
–
Crowder, Johnson, Padberg (1983): (0,1)IP models;
–
Barahona, Gr¨
otschel, J¨unger, Reinelt (1987): VLSI, maxcut;
–
Gr¨
otschel et al., Padberg et al. (1980–present)
–
Applegate, Bixby, Chv´atal, Cook (1990’s–present)
–
Dantzig, Fulkerson, Johnson (1954)
TSP
•
late 1980s – present:
–
Exploitation of results from geometry of numbers: Lenstra, L. Lov´asz, 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 “largescale”
•
Comments on DFJ (1954) technical report
•
Comments from Held & Karp (1970)
•
Examples from Padberg & Rinaldi (1991)
1
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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.
n
summationdisplay
j
=1
a
ij
x
j
≤
b
i
1
≤
i
≤
m
x
j
∈{
0
,
1
}
1
≤
j
≤
n
where
x
j
is a binary decision variable with the following interpretation
x
j
=
braceleftBigg
1
...
adopt project
j
;
0
...
do not do project
j
.
Since the decision variable is binary, the capital budgeting problem is a binary optimization problem
.
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 Spring '08
 BLAND
 Operations Research, Linear Programming, Optimization, cij xij, Function problem

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