ISM 206Optimization Theory and

Applications

Fall 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 559

• 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 Readi

ngScribe Assessment

1 Thu, Sep 22 Introduction and Modeling Ch 1&2

2 Tue, Sep 27 Intro to Linear Programming Ch 3, 4, 5

3 Thu, Sep 29 The simplex method Ch 6 Homework 1 assigned

4 Tue, Oct 4 Duality and Sensitivity Analysis

Ch 7

5 Thu, Oct 6 Other LP Methods.Transportation, Assignment

and Network Optimization Problems

Ch 8 &9

6 Tue, Oct 11 Unconstrained Nonlinear Optimization

Homework 1 dueHomework 2 assigned

7 Thu, Oct 13 Nonlinear Programming Ch 12

8 Tue, Oct 18 Nonlinear Programming 2

Class Schedule

Lecture Date Topic Reading Scribe Assessment9 Thu, Oct 20 Nonlinear

Programming 3

Homework 2 due Homework 3 assigned

10 Tue, Oct 25 Dynamic Programming

Ch 10

11 Thu, Oct 27 Integer Programming

Ch 11

12 Tue, Nov 1 Metaheuristics Ch 13 Homework 3 due.

13 Thu, Nov 3 Midterm Exam Midterm Exam

14 Tue, Nov 8 Game Theory Ch 14 Homework 4 assigned

15 Thu, Nov 10 Decision Analysis Ch 15

16 Tue, Nov 15 Markov Chains Ch 16 Homework 4 due. Homework 5 assigned

Class ScheduleLecture Date Topic Reading Scribe Assessment

17 Thu, Nov 17 Queueing Theory Ch 17

18 Tue, Nov 22 Inventory theory

Thu, Nov 24 No classThanksgiving Break.

Ch 18 Homework 5 due

19 Tue, Nov 29 Simulation Ch 20

20 Thu, Dec 1 Review

8-11am, Wed Dec 7 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

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 tutor and software