Transcript Linear Programming (Optimization)

Operations Research - 1
Spring 2012
 Instructor: Sungsoo Park ([email protected]), Building E2-2, room 4112,
Tel: 3121
Office hour: Tue, Thrs, 14:30 – 17:00 or by appointment
 TA: Kiho Seo ([email protected]), Office hour: Mon, Wed, 15:00 –
17:00 or by appointment
Ilhoon Cho ([email protected]), Office hour: Mon, Wed, 15:00 –
17:00 or by appointment
Building E2-2, room 4113, Tel: 3161
 Homepage: http://solab.kaist.ac.kr/, download class notes, homeworks.
 Text: “Linear Programming”, Vasek Chvatal, Freeman, 1983 and class
handouts
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 Grading: Midterm 30 - 40%, Final 40 - 50%, Homework 10 - 20% (Including
computer assignments, CPLEX/Xpress-MP)
 Class codes
No copying of homeworks. You may consult with others on homework problems,
but writing should be your own. Copying will be penalized.
No passing grade for 5 or more classes missing. Random check of attendance. Give
advance notices for unavoidable situations.
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Origins of OR
 Contributions of scientists and engineers during world war II.
Battle of Britain (radar site selection and control), Submarine warfare, Design of
B-29, ..
 Battle of Britain: integration of radar(hardware) and warning and control
system. Addition of radar sites causes problems.
(the name operational research (research in (military) operations)
 Maintenance of aircraft: For 350 flying hours, need 7 minor inspections( 2-5
days each) and a major inspection (14 days). Each aircraft had a devoted
aircrew and a ground crew
 change to central garage system.
: Flying hour increased by 61% over previous best record.
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 Submarine warfare:
Used patrol airplanes to detect surfaced U-boats and attack.
Needed 170 man-hours by maintenance and ground staff to produce one hour
of operational flying.
More than 200 hours of flying to produce one attack on a surfaced U-boat.
(34,000 man-hours for an attack)
In 1941, attack kill probability was 2% - 3%
 1.1M  1.7M man-hours needed to destroy one U-boat. (needed
improvements)
 Important decision variables:
1.
2.
3.
4.
5.
6.
Depth (time) setting for depth charge explosion (30/45m -> 8m)
Lethal radius (Use large or small bombs?)
Aiming errors in dropping the stick (aim ahead?)
Orientation of the stick with respect to the U-boat. (along U-boat track)
Spacing between successive depth charges in the stick (12m -> 33m)
Low level bombsights
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 Overall effect: By 1945, the attack kill probability had risen to over 40%.
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 After the war, methodologies used by the scientists adopted by government,
industry.
Called Operations Research (US), Operational Research(UK, Europe) (운용과학),
Management Science (경영과학)
 Characteristics: Use of mathematical models to solve decision problems arising
in management of industry, government, military, ….
E=mC2, F=ma, …
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Nature of OR
 “research on operations”
Applied mathematics + computer science + management
 Models : Deterministic models (확정적 모형, OR-I)
Stochastic models (확률적 모형, OR-II)
 Needed background: Algebra, calculus, discrete mathematics, probability,
statistics, data structure, algorithm, data base, programming skills, …)
 Important thrusts in early stages
1. Technical progress (Simplex method for linear programming, Dantzig, 1947)
2. Invention of computer and PC
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Study areas
 Deterministic models
Linear programming(선형계획법, linear optimization):1975, Nobel prize,
Kantorovich, Koopmans (efficient allocation of resources)
min x1  2 x2
3x1  2 x2  10
x1  2 x2  3
x1  0, x2  0
Nonlinear programming(비선형계획법):1990 Nobel prize, Markowitz (portfolio
selection)
max x1x2 x3
2 x1x2  2 x2 x3  2 x1x3  a
x1  0, x2  0, x3  0
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Integer Programming(정수계획법), Combinatorial optimization (조합최적화)
• Knapsack problem
max x1  2 x2  5x3  3x4
2 x1  6 x2  3x3  2 x4  6
xi {0,1} for all i
• Traveling Salesman Problem (외판원문제)
Given n cities, and distances cij between city i and j. What is the shortest
sequence to visit each city exactly once and return to the starting city?
( Applications: PCB assembly, Off-shore drilling, vehicle routing (delivery/pickup problem), bio, …)
web site: http://www.tsp.gatech.edu/
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Networks and graphs
Inchon
Seoul
Kangnung
Daejeon
Daegu
Kwangju
Pusan
• Shortest path to move from Inchon to Kangnung? (Shortest path problem)
– Logistics, Telecommunication routing, …
• Connect the cities with roads (or communication lines) in a cheapest way.
(Minimum spanning tree problem)
• How much commodities (or packets) can we send from Kwangju to Daegu if
edges have limited capacities? (maximum flow problem)
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Dynamic programming
• If a system changes in time and the status of the system in the next period
depends on the current status and decisions made, what is the best decision in
each stage to optimize our goal in the end?
• Not the formalized problems, but refer the structured steps (methods) used to
solve problems involving many stages.
Game theory
• Investigate the best strategy when the outcome of cooperation and/or
competition between people or groups depends on the collective decisions
made by individual person/group.
• Economics, Marketing (1994, Nobel prize, Nash, Harsanyi, Selten)
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 Computational complexity: Theory that investigate the inherent difficulty of problems.
Turing machine model of computation.
NP-completeness.
• NP-complete (NP-hard) problems: Knapsack problem, Traveling salesman problem,
…
• Easily solvable problems: shortest path problem, minimum spanning tree problem,
…
– Problems for which polynomial running time algorithms exist.
• A little bit of changes in the problem structure may make the problem hard.
– Minimum spanning tree problem vs. Steiner tree problem
• Useful tool when we try to solve some new problems.
 Note that the basic models may appear as subproblems in a big problem. Also the
models may be hidden in the real problem in some unexpected way. Identifying the
hidden model may be crucial.
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 Stochastic models (OR-II)
Markov chain
Queueing theory
Decision analysis
Simulation
Reliability
 Recently, uncertainties of data are reflected in the optimization models
Stochastic Programming
Robust Optimization
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