Transcript 幻灯片 1 - Shandong University

Optimization Methods
LI Xiaolei
[email protected]
Syllabus
Instructor: LI Xiaolei
Office: Main Bldg. 604
E-mail: [email protected]
(For correspondence, please include OM2015 in the SUBJECT)
Office Hours: workhours on weekdays
Course Objectives:
The aim of this course is to,
– Give methods, algorithms, and solutions to solve mathematical
programming problems.
– Expose students to some specialized software for the solution of such
problems.
– Formulate and develop mathematical models for the solution of real
world problems.
Syllabus
Course Outcomes:
– Formulate engineering problems as mathematical programming problems,
such as Linear Programming problems, Integer Programming problems,
Nonlinear Programming problems, and so on.
– Use the simplex algorithm to find the optimal solution for LP. Students can
look insight of the LP by sensitivity analysis and duality.
– Solve IP/MIP by the Branch-and-Bound Method.
– Use numerical optimization methods to find the optimal solution for
unconstrained NLP - functions of single variable.
– Know the necessary and sufficient optimality criteria for unconstrained and
constrained non-linear programming problems.
– Use Largrange multipliers to find the optimal solution for nonlinear programs
constrained by equations.
– Know some modern optimization methods like Genetic Algorithm, Ant
Colony Algorithm, Particle Swarm Optimization Algorithms and Artificial
Fish-school Algorithm.
– Know some of the available solution packages such as MATLAB, Excel
Solver, LINGO, etc. and know how to use them to solve mathematical
programming problems.
Syllabus
References
– Lecture notes
– WAYNE L. WINSTON. OPERATIONS RESEARCH: Mathematical
Programming (Third Edition), 2003.
– R.Fletcher. Practical Methods of Optimization (Second Edition), John
Wiley & Sons, 2008.
– Scott Kirkpatrick. Optimization by Simulated Annealing: Quantitative
Studies Journal of Statistical Physics, Vol. 34, Nos. 5/6, 1984:975-986.
– John H. Holland. Genetic Algorithms. Science American, 1992 July: 4450.
– Kennedy, J., Eberhart, R. Particle Swarm Optimization. Proceedings of
IEEE International Conference on Neural Networks. IV. 1995: 1942–
1948.
– M. Dorigo, V. Maniezzo & A. Colorni. Ant System: Optimization by a
Colony of Cooperating Agents, IEEE Transactions on Systems, Man,
and Cybernetics–Part B, 26 (1), 1996: 29–41.
Syllabus
Mark Distribution
– Attendance
– Tasks
– Final exam
10%
40%
50%
Chapter 1
Introduction
Introduction
• Encyclopedia of Mathematics
Optimization Theory
See Operations Research
• During World War II, British military leaders asked
scientists and engineers to analyze several military
problems. The application of mathematics and the
scientific method to military operations was called
operations research.
• Today, the term operations research means a scientific
approach to decision making, which seeks to determine
how best to design and operate a system, usually under
conditions requiring the allocation of scarce resources.
The Methodology of operations research
Seven-step procedure
• Step1. Formulate the problem
– Specify the organization’s objectives and the parts of the
system that must be studied before the problem can be solved.
• Step2. Observe the system
– The analyst collects data to estimate the values of parameters
that affect the organization’s problem.
• Step3. Formulate a mathematical model of the problem
– The analyst develops a mathematical model of the problem.
The Methodology of operations research
• Step4. Verify the model and use the model for
prediction
– The analyst now tries to determine if the mathematical model is
an accurate representation of reality.
• Step5. Select a suitable alternative
– Given a model and a set of alternatives, the analyst now
choose the alternative that best meets the organization’s
objectives.
The Methodology of operations research
• Step6. Present the results and conclusions of the study
to the organization
– The analyst presents the model and recommendations from
step5 to the decision making individual or group. Let the
organization choose the one that best meets its needs.
• Step7. Implement and evaluate recommendations
– If the organization has accepted the study, the analyst aids in
implementing the recommendations.
– The system must be constantly monitored to ensure that the
recommendations are enabling the organization to meet its
objectives.
MODEL CLASSIFICATION
• Operational Exercise
–
This modeling approach operates directly with the real
environment in which the decision under study is going to take
place. The modeling effort merely involves designing a set of
experiments to be conducted in that environment, and
measuring and interpreting the results of those experiments.
MODEL CLASSIFICATION
• Operational Exercise
–
–
–
Operational exercises contain the highest degree of realism of
any form of modeling approach, however, external
abstractions or oversimplifications are introduced.
The method is exceedingly, usually prohibitively, expensive to
implement. Moreover, in most cases it is impossible to
exhaustively analyze the alternatives available to the decisionmaker. This can lead to severe suboptimization in the final
conclusions.
For these reasons, operational exercises seldom are used as
a pure form of modeling practice.
MODEL CLASSIFICATION
• Gaming
–
–
–
This model provides a responsive mechanism to evaluate the
effectiveness of proposed alternatives, which the decisionmaker must supply in an organized and sequential fashion.
The model should reflect, with an acceptable degree of
accuracy, the relationships between the inputs and outputs of
the process.
Subsequently, all the personnel who participate in structuring
the decision process in the management of the process would
be allowed to interact with the model.
MODEL CLASSIFICATION
• Simulation
–
–
Simulation models are similar to gaming models except that all
human decision-makers are removed from the modeling
process. The model provides the means to evaluate the
performance of a number of alternatives, supplied externally to
the model by the decision-maker, without allowing for human
interactions at intermediate stages of the model computation.
Many simulation models take the form of computer programs,
where logical arithmetic operations are performed in a
prearranged sequence. This provides an added flexibility in
model formulation and permits a high degree of realism to be
achieved, which is particularly useful when uncertainties are
an important aspect of the decision.
MODEL CLASSIFICATION
• Analytical Model
–
–
The problem is represented completely in mathematical terms,
normally by means of a criterion or objective, which we seek
to maximize or minimize, subject to a set of mathematical
constraints that portray the conditions under which the
decisions have to be made. The model computes an optimal
solution, that is, one that satisfies all the constraints and gives
the best possible value of the objective function.
Most of the work undertaken by management scientists has
been oriented toward the development and implementation of
analytical models.
Optimization Tree
Successful applications of optimization methods
• Police patrol officer scheduling in San Francisco.
• Reducing fuel cost in electric power industry.
• Designing an ingot mold stripping facility at Bethlehem
Steel.
• Gasoline blending at Texaco.
• Scheduling trucks at north American van lines.
• Inventory management at Blue Bell.
• ……
Math Programming and Radiation
Therapy
• High doses of radiation (energy/unit mass) can kill
cells and/or prevent them from growing and dividing
– True for cancer cells and normal cells
• Radiation is attractive because the repair
mechanisms for cancer cells is less efficient than
for normal cells
Use of Multi-leaf Collimaters
• multi-leaf collimator
– blocks radiation
– turns a large beam into a focused beam
Conventional Radiotherapy
Conventional Radiotherapy
• In conventional radiotherapy
– 3 to 7 beams of radiation
– radiation oncologist and physicist work together to
determine a set of beam angles and beam intensities
– determined by manual “trial-and-error” process
Goal: maximize the dose to the tumor
while minimizing dose to the critical area
Radiation Therapy: Problem Statement
• For a given tumor and given critical areas z For a
given set of possible beamlet origins and angles
• Determine the weight of each beamlet such that:
– dosage over the tumor area will be at least a target level γL.
– dosage over the critical area will be at most a target level
γU.
Display of radiation levels
Optimal Solution for the LP
An Optimal Solution to an NLP
Your time
To describe an optimization
problem that you learn from
your supervisors or your
senior brothers/sisters.