ISM 206 Optimization Theory and Applications Spring 2005 Lecture 1: Introduction.

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