ie 501: optimization models

IE 501: Optimization Models

Autumn 2009

Vishnu Narayanan

Industrial Engineering and Operations ResearchIndian Institute of Technology Bombay

Lecture 1, 23rd July, 2009

Vishnu Narayanan IE 501: Optimization Models

General InfoSlot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025

At your service: Vishnu Narayanan

Email: [email protected]: x-7878Office: F20, Old CSEPlease contact me in this order!

Teaching Assistant: Virendra Patidar([email protected])

Course webpage:http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/Check this page and moodle regularly.

Textbooks/References

1 Wayne L. Winston, Operations Research: Applicationsand Algorithms, 4th edition, Thomson Learning, 2004.

2 Wayne L. Winston, Introduction to MathematicalProgramming: Applications and Algorithms, 4th edition,Duxbury, 2003.

3 H. Paul Williams, Model Building in MathematicalProgramming, 4th edition, John Wiley and Sons, 1999.

4 Ashok D. Belegundu and Tirupathi R. Chandrupatla,Optimization Concepts and Applications in Engineering,Pearson Education India, 1999.

5 H. Taha, Operations Research: An Introduction, 8thedition, Prentice Hall India, 2002.

Grading

Regular homework assignments (40%). Modeling, solution,theory, computation. Will be posted on the webpage everymonday!

Project (10%). Formulate and solve a real-world optimizationproblem.

Mid-sem exam (20%) and Final exam (30%).

Audit students: do all of the above and get a passing grade(undergrads: DD, PhD/MSc/M.Tech/Dual degree/rest: CC) foraward of AU.

Cheating⇒ Severe penalty

Grading

What we will learn

Modeling fundamentals

Linear programming

Network models

Combinatorial optimization

Mixed-integer programming

What we will learn

Linear programming

Network models

What we will learn

Linear programming

Network models

What we will learn

Linear programming

Network models

What we will learn

Linear programming

Network models

Problems from science, engineering, business, and socialsectors

Continuous and discrete problems

Constrained and unconstrained problems

Single- and multi-stage models

Formulations and equivalences

Linear Programming

Geometry and algebra of Simplex method

Duality and complementary slackness

Sensitivity analysis

Post-optimal analysis

Linear Programming

Network models

Decision problems involving network flows (shortest path, . . . )

Modeling problems as shortest paths, maximum flow, minimumcost flow, etc.

Integrality of solutions

Matching, assignment, and transportation problems

Multi-stage flows

Network models

Multi-stage flows

Network models

Multi-stage flows

Network models

Multi-stage flows

Network models

Multi-stage flows

Combinatorial optimization models

Examples: knapsack, set cover, set packing, . . .

Large feasible space and neighbourhood solutions

Representation of solution space

Search tree

Search techniques, branch-and-bound

Search tree

Problems with integer variables

Use of binary variables in modeling alternative decisions

Difficulty of solution

Formulations of combinatorial optimization problems asmixed-integer programs

IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITBfaculty shuttle between BOM and AHD to teach courses.

I fly out from BOM on Mondays and fly back from AHD onThursdays for the next five weeks.

Regular round trip fare is Rs. 6,000. One-way ticket costs Rs.4,500.

20% discount if the ticket dates span a weekend (e.g., if I fly outFriday and return Thursday).

How should I buy my tickets to minimize the money spent?

Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondaysand return on Thursdays.Cost: Rs. 5 × 6,000 = Rs. 30,000.

Buy one BOM–AHD for week 1, one AHD–BOM for week 5,and four AHD–BOM–AHD tickets that span weekends.Cost: Rs. 2 × 4,500 + 4 × 0.8 × 6,000 = Rs. 28,200.

Buy one BOM–AHD–BOM flying out in week 1 and returningin week 5, and four AHD–BOM–AHD tickets that spanweekends.Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Terminology

In the above situation,

what are the decision alternatives?what are the constraints?what is a criterion for evaluating the alternatives?how many decision alternatives are there?

Another problem: Given a wire of length `, how would onemake a rectangle of maximum area with it?

Alternatives, constraints?

Note: As opposed to the flight problem, the possiblealternatives are uncountably many!

Terminology

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decisionvariables)

Constraints: 2(w + h) = `, w ≥ 0, h ≥ 0.

Objective function: area = wh

Optimization problem:

maximize z = whsubject to 2(w + h) = `

w, h ≥ 0.

More terminology

problem: min{f(x) : x ∈ S}

If x satisfies all constraints (i.e., x ∈ S), then it is a feasiblesolution. Otherwise, it is infeasible.

The set of all feasible solutions (in this case, S) is called thefeasible region.

x is called optimal if x ∈ S, and f(y) ≥ f(x) for all y ∈ S.

If x is feasible but not optimal, it is called suboptimal.

More terminology

ie 501: optimization models

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