ism 206 optimization theory and applications spring 2005 lecture 1: introduction
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ISM 206Optimization Theory and
Applications
Spring 2005
Lecture 1: Introduction
ISM 206 Lecture 1 Overview
• Some Optimization problem examples
• Topics in this class
• Logistics
Names
Kevin Ross
• Assistant Professor, Information Systems and Technology Management
• Interests in queueing theory, optimization, scheduling, networks
• E2 room 367
• Office hours: Tuesday 2-4
Problem 1: Transportation
• P&T Company makes canned peas• Peas are prepared in 3 canneries
– Washington, Oregon, Minnesota
• Shipped to 4 distributing warehouses– California, Utah, South Dakota, New Mexico
• How much should we ship from each cannery to each warehouse?– Transportation costs are different between each pair
of locations– There is a limit on capacity at each plant
Unit Cost Destination (Warehouse) Range Name CellsSacramento Salt Lake City Rapid City Albuquerque Demand D17:G17
Source Bellingham $464 $513 $654 $867 ShipmentQuantity D12:G14(Cannery) Eugene $352 $416 $690 $791 Supply J12:J14
Albert Lea $995 $682 $388 $685 TotalCost J17TotalReceived D15:G15TotalShipped H12:H14
Shipment Quantity Destination (Warehouse) UnitCost D5:G7(Truckloads) Sacramento Salt Lake City Rapid City Albuquerque Total Shipped SupplySource Bellingham 0 0 0 0 0 = 75
(Cannery) Eugene 0 0 0 0 0 = 125Albert Lea 0 0 0 0 0 = 100
Total Received 0 0 0 0= = = = Total Cost
Demand 80 65 70 85 $0
Problem 2: Engineering Design Problem
• Consider lighting a large area with a number of lamps:
• Each lamp has a total power limit
• Several points in the room have a ‘desired illumination level’
How much power should be applied to each lamp to get the room as close as possible to desired level?
Problem 2: Engineering Design Problem
Now add two more constraints:
1. No more than half the total power goes to any five lamps
2. No more than 15 lamps are turned on
What effect do (1) and (2) have on the original problem?
Problem 3: Medical Team Distribution
• World Health Council is devoted to improving health care in underdeveloped countries:
• Need to allocate five teams to three different countries
• Each team added gains more person-years of life saved in the country
• You cannot assign partial teams or partial people
Thousand person-years gained
1 2 3
0 0 0 0
1 45 20 50
2 70 45 70
3 90 75 80
4 105 110 100
5 120 150 130
country
No.
of
team
s
Problem 4: Inventory Levels
• A wholesale Bicycle Distributor:– Purchases bikes from manufacturer and supplies to
many shops– Demand to each shop is uncertain
How many bikes should the distributor order from the manufacturer?
• Costs:– Ordering cost to manufacturer– Holding cost in factory– Shortage cost due to lack of sales
Course Overview
• First graduate class in optimization
• Main topics:– Linear Programming– Nonlinear programming– Heuristic Methods– Integer programming– Dynamic programming– Inventory Theory
Class ScheduleLecture Date Topic Reading Assessment1 Tue, 29 March Introduction and Modeling Ch 1&2
2 Thu, 31 March Intro to Linear Programming and the Simplex Method
Ch 3,4,5 Homework 1 assigned
3 Tue, 5 April Duality and Sensitivity Analysis
Ch 6
4 Thu, 7 April Other Algorithms for Linear Programming
Ch 7
5 Thu, 7 April 4-6pm
Transportation, Assignment and Network Optimization
Ch 8 & 9 Homework 1 dueHomework 2 assigned
6 Tue, 12 April Nonlinear Optimization Ch 12
7 Thu, 14 April Nonlinear Optimization ctd.
8 Thu, 14 April 4-6pm
Unconstrained Optimization Homework 2 due Homework 3 assigned
- Week 19,21 April
No class. Instructor away
9 Tue, 26 April Midterm Exam Midterm Exam
Class Schedule
Lecture Date Topic Reading Assessment10 Thu, 28 April Dynamic Programming Ch 10 Homework 3 due.
Homework 4 assigned
11 Thu, 28 April
4-6pm
Integer Programming Ch 11
12 Tue, 3 May Metaheuristics Ch 13
13 Thu, 5 May Metaheuristics 2
14 Thu, 5 May 4-6pm
Game Theory Ch 14
15 Tue, 10 May Decision Analysis Ch 15 Homework 4 due. Homework 5 assigned
16 Thu, 12 May Markov Chains Ch 16
- Week of 17, 19 May
No class. Instructor away
Class Schedule
Lecture Date Topic Reading Assessment
17 Tue, 24 May
Queueing Theory Ch 17
18 Thu, 26 May
Inventory theory Ch 18
19 Tue, 31 May
Simulation Ch 20 Homework 5 due
20 Thu, 2 June Final Class - Review
Tue, 7 June4:00 – 7:00
pm
FINAL Final Exam
Assessment
• Five homework sets, assigned approximately every two weeks.• Late assignments will lose 10% per day.
Lecture Notes• Each lecture one student will act as a scribe for everyone.• They are responsible for typing up the lecture notes using Latex.• The notes are due 1 week after the assigned lecture.• Depending on class size, you will be assigned two or three lectures to write up.
Exams• Exams will be open book and open notes.• You may bring a basic calculator but not a computer.
Homework 35%
Lecture Notes 5%
Midterm Exam 20%
Final Exam 40%
Lecture Notes Schedule
• Volunteers for today and Thursday– Each lecture one student will act as a scribe
for everyone.– They are responsible for typing up the lecture
notes using Latex.– The notes are due 1 week after the assigned
lecture.
• Schedule to be announced Thursday
Off weeks
• Instructor away 2 weeks of this quarter
• Need to agree on time for make-up classes
• Suggestion: Thursday afternoons. Time?
My request…
• Feedback!
• This class is for you
Optimization Overview
• Variables:
• Objective:
• Subject to Constraints:
• Sometimes additional constraints:– Binary– Integer
• Sometimes uncertainty in parameters (stochastic optimization)
)(min xf
ixc
ixc
i
i
,0)(
,0)(
),...,,( 21 Nxxxx
Types of Optimization Problems
• Linear: Linear functions for objective and constraints
• Nonlinear: Nonlinear functions…• Convex• Integer• Mixed-Integer• Combinatorial• Unconstrained: No constraints• Dynamic: Solved in stages
Optimization terms and Concepts
• Variable• Feasible region• Solution (feasible point)• Optimal solution (best point)• Global and local optimality• Optimality conditions• Duality• Direct methods• Numerical methods• Heuristics
Modeling and Optimization Stages
1. Define problem and gather data• Feasibility check
2. Formulate mathematical model3. Develop computer-based method for finding optimal
solution• Design and Software implementation
4. Test and refine model• Validation
5. Prepare for ongoing model utilization• Training, installation
6. Implement• Maintenance, updates, reviews, documentation, dissemination
Software with Text
• Link to software