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IE 501: Optimization Models Autumn 2009 Vishnu Narayanan - - PowerPoint PPT Presentation

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IE 501: Optimization Models Autumn 2009 Vishnu Narayanan Industrial Engineering and Operations Research Indian Institute of Technology Bombay Lecture 1, 23rd July, 2009 1 Vishnu Narayanan IE 501: Optimization Models General Info Slot 4: M

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IE 501: Optimization Models

Autumn 2009 Vishnu Narayanan

Industrial Engineering and Operations Research Indian Institute of Technology Bombay

Lecture 1, 23rd July, 2009

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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General Info

Slot 4: M 1135–1230, Tu 0830–0925, Th 0930–1025 At your service: Vishnu Narayanan Email: vishnu@iitb.ac.in Phone: x-7878 Office: F20, Old CSE Please contact me in this order! Teaching Assistant: Virendra Patidar (virendra.patidar@iitb.ac.in) Course webpage: http://www.ieor.iitb.ac.in/∼vishnu/teaching/ie501/ Check this page and moodle regularly.

Vishnu Narayanan IE 501: Optimization Models

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Textbooks/References

1

Wayne L. Winston, Operations Research: Applications and Algorithms, 4th edition, Thomson Learning, 2004.

2

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

3

  • H. Paul Williams, Model Building in Mathematical

Programming, 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, 8th

edition, Prentice Hall India, 2002.

Vishnu Narayanan IE 501: Optimization Models

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Grading

Regular homework assignments (40%). Modeling, solution, theory, computation. Will be posted on the webpage every monday! Project (10%). Formulate and solve a real-world optimization problem. 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) for award of AU. Cheating ⇒ Severe penalty

Vishnu Narayanan IE 501: Optimization Models

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Grading

Regular homework assignments (40%). Modeling, solution, theory, computation. Will be posted on the webpage every monday! Project (10%). Formulate and solve a real-world optimization problem. 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) for award of AU. Cheating ⇒ Severe penalty

Vishnu Narayanan IE 501: Optimization Models

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Grading

Regular homework assignments (40%). Modeling, solution, theory, computation. Will be posted on the webpage every monday! Project (10%). Formulate and solve a real-world optimization problem. 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) for award of AU. Cheating ⇒ Severe penalty

Vishnu Narayanan IE 501: Optimization Models

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Grading

Regular homework assignments (40%). Modeling, solution, theory, computation. Will be posted on the webpage every monday! Project (10%). Formulate and solve a real-world optimization problem. 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) for award of AU. Cheating ⇒ Severe penalty

Vishnu Narayanan IE 501: Optimization Models

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Grading

Regular homework assignments (40%). Modeling, solution, theory, computation. Will be posted on the webpage every monday! Project (10%). Formulate and solve a real-world optimization problem. 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) for award of AU. Cheating ⇒ Severe penalty

Vishnu Narayanan IE 501: Optimization Models

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What we will learn

Modeling fundamentals Linear programming Network models Combinatorial optimization Mixed-integer programming

Vishnu Narayanan IE 501: Optimization Models

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What we will learn

Modeling fundamentals Linear programming Network models Combinatorial optimization Mixed-integer programming

Vishnu Narayanan IE 501: Optimization Models

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5

What we will learn

Modeling fundamentals Linear programming Network models Combinatorial optimization Mixed-integer programming

Vishnu Narayanan IE 501: Optimization Models

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5

What we will learn

Modeling fundamentals Linear programming Network models Combinatorial optimization Mixed-integer programming

Vishnu Narayanan IE 501: Optimization Models

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What we will learn

Modeling fundamentals Linear programming Network models Combinatorial optimization Mixed-integer programming

Vishnu Narayanan IE 501: Optimization Models

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Modeling fundamentals

Problems from science, engineering, business, and social sectors Continuous and discrete problems Constrained and unconstrained problems Single- and multi-stage models Formulations and equivalences

Vishnu Narayanan IE 501: Optimization Models

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Modeling fundamentals

Problems from science, engineering, business, and social sectors Continuous and discrete problems Constrained and unconstrained problems Single- and multi-stage models Formulations and equivalences

Vishnu Narayanan IE 501: Optimization Models

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Modeling fundamentals

Problems from science, engineering, business, and social sectors Continuous and discrete problems Constrained and unconstrained problems Single- and multi-stage models Formulations and equivalences

Vishnu Narayanan IE 501: Optimization Models

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Modeling fundamentals

Problems from science, engineering, business, and social sectors Continuous and discrete problems Constrained and unconstrained problems Single- and multi-stage models Formulations and equivalences

Vishnu Narayanan IE 501: Optimization Models

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Modeling fundamentals

Problems from science, engineering, business, and social sectors Continuous and discrete problems Constrained and unconstrained problems Single- and multi-stage models Formulations and equivalences

Vishnu Narayanan IE 501: Optimization Models

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Linear Programming

Geometry and algebra of Simplex method Duality and complementary slackness Sensitivity analysis Post-optimal analysis

Vishnu Narayanan IE 501: Optimization Models

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Linear Programming

Geometry and algebra of Simplex method Duality and complementary slackness Sensitivity analysis Post-optimal analysis

Vishnu Narayanan IE 501: Optimization Models

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Linear Programming

Geometry and algebra of Simplex method Duality and complementary slackness Sensitivity analysis Post-optimal analysis

Vishnu Narayanan IE 501: Optimization Models

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Linear Programming

Geometry and algebra of Simplex method Duality and complementary slackness Sensitivity analysis Post-optimal analysis

Vishnu Narayanan IE 501: Optimization Models

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Network models

Decision problems involving network flows (shortest path, . . . ) Modeling problems as shortest paths, maximum flow, minimum cost flow, etc. Integrality of solutions Matching, assignment, and transportation problems Multi-stage flows

Vishnu Narayanan IE 501: Optimization Models

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Network models

Decision problems involving network flows (shortest path, . . . ) Modeling problems as shortest paths, maximum flow, minimum cost flow, etc. Integrality of solutions Matching, assignment, and transportation problems Multi-stage flows

Vishnu Narayanan IE 501: Optimization Models

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Network models

Decision problems involving network flows (shortest path, . . . ) Modeling problems as shortest paths, maximum flow, minimum cost flow, etc. Integrality of solutions Matching, assignment, and transportation problems Multi-stage flows

Vishnu Narayanan IE 501: Optimization Models

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Network models

Decision problems involving network flows (shortest path, . . . ) Modeling problems as shortest paths, maximum flow, minimum cost flow, etc. Integrality of solutions Matching, assignment, and transportation problems Multi-stage flows

Vishnu Narayanan IE 501: Optimization Models

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Network models

Decision problems involving network flows (shortest path, . . . ) Modeling problems as shortest paths, maximum flow, minimum cost flow, etc. Integrality of solutions Matching, assignment, and transportation problems Multi-stage flows

Vishnu Narayanan IE 501: Optimization Models

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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

Vishnu Narayanan IE 501: Optimization Models

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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

Vishnu Narayanan IE 501: Optimization Models

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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

Vishnu Narayanan IE 501: Optimization Models

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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

Vishnu Narayanan IE 501: Optimization Models

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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

Vishnu Narayanan IE 501: Optimization Models

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Mixed-integer programming

Problems with integer variables Use of binary variables in modeling alternative decisions Difficulty of solution Formulations of combinatorial optimization problems as mixed-integer programs

Vishnu Narayanan IE 501: Optimization Models

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Mixed-integer programming

Problems with integer variables Use of binary variables in modeling alternative decisions Difficulty of solution Formulations of combinatorial optimization problems as mixed-integer programs

Vishnu Narayanan IE 501: Optimization Models

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Mixed-integer programming

Problems with integer variables Use of binary variables in modeling alternative decisions Difficulty of solution Formulations of combinatorial optimization problems as mixed-integer programs

Vishnu Narayanan IE 501: Optimization Models

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Mixed-integer programming

Problems with integer variables Use of binary variables in modeling alternative decisions Difficulty of solution Formulations of combinatorial optimization problems as mixed-integer programs

Vishnu Narayanan IE 501: Optimization Models

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IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITB faculty shuttle between BOM and AHD to teach courses. I fly out from BOM on Mondays and fly back from AHD on Thursdays 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 out Friday and return Thursday). How should I buy my tickets to minimize the money spent?

Vishnu Narayanan IE 501: Optimization Models

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IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITB faculty shuttle between BOM and AHD to teach courses. I fly out from BOM on Mondays and fly back from AHD on Thursdays 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 out Friday and return Thursday). How should I buy my tickets to minimize the money spent?

Vishnu Narayanan IE 501: Optimization Models

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IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITB faculty shuttle between BOM and AHD to teach courses. I fly out from BOM on Mondays and fly back from AHD on Thursdays 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 out Friday and return Thursday). How should I buy my tickets to minimize the money spent?

Vishnu Narayanan IE 501: Optimization Models

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IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITB faculty shuttle between BOM and AHD to teach courses. I fly out from BOM on Mondays and fly back from AHD on Thursdays 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 out Friday and return Thursday). How should I buy my tickets to minimize the money spent?

Vishnu Narayanan IE 501: Optimization Models

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11

IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITB faculty shuttle between BOM and AHD to teach courses. I fly out from BOM on Mondays and fly back from AHD on Thursdays 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 out Friday and return Thursday). How should I buy my tickets to minimize the money spent?

Vishnu Narayanan IE 501: Optimization Models

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11

IIT Gandhinagar example

New IIT at Gandhinagar, mentored by IIT Bombay. IITB faculty shuttle between BOM and AHD to teach courses. I fly out from BOM on Mondays and fly back from AHD on Thursdays 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 out Friday and return Thursday). How should I buy my tickets to minimize the money spent?

Vishnu Narayanan IE 501: Optimization Models

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Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondays and 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 returning in week 5, and four AHD–BOM–AHD tickets that span weekends. Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Vishnu Narayanan IE 501: Optimization Models

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12

Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondays and 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 returning in week 5, and four AHD–BOM–AHD tickets that span weekends. Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Vishnu Narayanan IE 501: Optimization Models

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12

Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondays and 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 returning in week 5, and four AHD–BOM–AHD tickets that span weekends. Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Vishnu Narayanan IE 501: Optimization Models

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12

Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondays and 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 returning in week 5, and four AHD–BOM–AHD tickets that span weekends. Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Vishnu Narayanan IE 501: Optimization Models

slide-53
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12

Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondays and 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 returning in week 5, and four AHD–BOM–AHD tickets that span weekends. Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Vishnu Narayanan IE 501: Optimization Models

slide-54
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12

Three possible alternatives

Buy five BOM–AHD–BOM tickets for departure on Mondays and 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 returning in week 5, and four AHD–BOM–AHD tickets that span weekends. Cost: Rs. 5 × 0.8 × 6,000 = Rs. 24,000.

Vishnu Narayanan IE 501: Optimization Models

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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 ℓ, how would one make a rectangle of maximum area with it? Alternatives, constraints? Note: As opposed to the flight problem, the possible alternatives are uncountably many!

Vishnu Narayanan IE 501: Optimization Models

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13

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 ℓ, how would one make a rectangle of maximum area with it? Alternatives, constraints? Note: As opposed to the flight problem, the possible alternatives are uncountably many!

Vishnu Narayanan IE 501: Optimization Models

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13

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 ℓ, how would one make a rectangle of maximum area with it? Alternatives, constraints? Note: As opposed to the flight problem, the possible alternatives are uncountably many!

Vishnu Narayanan IE 501: Optimization Models

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13

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 ℓ, how would one make a rectangle of maximum area with it? Alternatives, constraints? Note: As opposed to the flight problem, the possible alternatives are uncountably many!

Vishnu Narayanan IE 501: Optimization Models

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Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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SLIDE 62

14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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SLIDE 64

14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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SLIDE 65

14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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SLIDE 66

14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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SLIDE 67

14

Maximum-area rectangle

Let w and h be the dimensions of the rectangle. (decision variables) Constraints: 2(w + h) = ℓ, w ≥ 0, h ≥ 0. Objective function: area = wh Optimization problem: maximize z = wh subject to 2(w + h) = ℓ w, h ≥ 0.

Vishnu Narayanan IE 501: Optimization Models

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More terminology

problem: min {f(x) : x ∈ S} If x satisfies all constraints (i.e., x ∈ S), then it is a feasible

  • solution. Otherwise, it is infeasible.

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

Vishnu Narayanan IE 501: Optimization Models

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15

More terminology

problem: min {f(x) : x ∈ S} If x satisfies all constraints (i.e., x ∈ S), then it is a feasible

  • solution. Otherwise, it is infeasible.

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

Vishnu Narayanan IE 501: Optimization Models

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15

More terminology

problem: min {f(x) : x ∈ S} If x satisfies all constraints (i.e., x ∈ S), then it is a feasible

  • solution. Otherwise, it is infeasible.

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

Vishnu Narayanan IE 501: Optimization Models

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SLIDE 71

15

More terminology

problem: min {f(x) : x ∈ S} If x satisfies all constraints (i.e., x ∈ S), then it is a feasible

  • solution. Otherwise, it is infeasible.

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

Vishnu Narayanan IE 501: Optimization Models

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15

More terminology

problem: min {f(x) : x ∈ S} If x satisfies all constraints (i.e., x ∈ S), then it is a feasible

  • solution. Otherwise, it is infeasible.

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

Vishnu Narayanan IE 501: Optimization Models

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More terminology

problem: min {f(x) : x ∈ S} If x satisfies all constraints (i.e., x ∈ S), then it is a feasible

  • solution. Otherwise, it is infeasible.

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

Vishnu Narayanan IE 501: Optimization Models