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Mathematical Programming Model

Mathematical Programming Model. Factory Automation Lab. SNU. Jul. 30. 1999 Min, Dai ki. Contents. Recent development in mathematical programming modeling systems Introduction Modeling language extensions Modeling interface extensions Online optimization services

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Mathematical Programming Model

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  1. Mathematical Programming Model Factory Automation Lab. SNU. Jul. 30. 1999 Min, Dai ki

  2. Contents • Recent development in mathematical programming modeling systems • Introduction • Modeling language extensions • Modeling interface extensions • Online optimization services • Steel industry example • New Directions in Algebraic Modeling Languages • Paper review • Database structure for mathematical programming models SNU IEFAL SEMINAR

  3. Recent development in mathematical programming modeling systems Introduction • Modeler’ form - Algebraic • Characteristics Minimize (function of variables) Subject to (function of variables)  (constant) Standard “mathematical” notation • Advantages • Familiar to everyone • Applicable to a broad variety of models • Extendible to nonlinearities, networks, activities… • Examples • AMPL, GAMS, CPLEX, ILOG Opt Suite, OSL... Conception of Problem Modeler’ s Form of Problem Computer Algorithm’ s Form of Problem Algorithm’ s Form of Results Modeler’ s Form of Results Understanding of Results Using an optimization model SNU IEFAL SEMINAR

  4. Recent development in mathematical programming modeling systems Introduction • Design objectives • Minimize disruption • Introduce few new features • Make new features follow established conventions • Impose few new rules • Maximize naturalness • Describe optimization problems in the way people think of them and people think in many ways SNU IEFAL SEMINAR

  5. Recent development in mathematical programming modeling systems Modeling languages extensions • Complementarity problems • Collections of complementarity conditions • Express complementarity conditions in a natural and convenient way • Stochastic programming • Combinatorial optimization • Algebraic modeling languages have not been particularly successful for modeling combinatorial optimization problems • Solver strategies • Convert to integer program • Extend branch-and-bound approach • Extend logic programming approach... SNU IEFAL SEMINAR

  6. Recent development in mathematical programming modeling systems Modeling interface extensions • Database features • Database links • Map optimization system’s data to relational data tables • Use ODBC • Database integration • Use relational database for a model’s data in place of the usual text files • Extending the database paradigm • Model base : Generalize the idea of a database to store whole cases - models and data • Model management : Maintain a series of scenario, or library of a models … scenarios in MathPro SNU IEFAL SEMINAR

  7. Recent development in mathematical programming modeling systems Modeling interface extensions • Model analysis support • Principles & practice • Access to input and output values • Access to analysis • Examples • Warm start after data change • Analysis of nonlinear functions • Interactive analysis • In analysis system’s environment : MProbe, ANALYZE • In modeling system’s environment : MIMI, AIMMS SNU IEFAL SEMINAR

  8. Recent development in mathematical programming modeling systems Modeling interface extensions • Application development • Model development cycle • Prototyping of model • Construction of user application • Maintenance and updating • System design approaches • Use modeling system to build application • Embed modeling system in application SNU IEFAL SEMINAR

  9. Recent development in mathematical programming modeling systems Online optimization services SNU IEFAL SEMINAR

  10. Recent development in mathematical programming modeling systems Steel industry exampleOptimization Approaches to Production Planning in the Steel Industry • Continuous deterministic & easy to solve • Spreadsheet Tools for Planning • Reliance on rules of thumb to “maximize” profits • Optimization Tools for Planning • Complex steel-making configurations • Applications at Tata Steel in India • Generalized network flow linear program • 4th Dimension 1.0 database software, Apple Macintosh II,XMP linear programming library SNU IEFAL SEMINAR

  11. Recent development in mathematical programming modeling systems New Directions in Algebraic Modeling Languages • Integration • Model analysis support • Database connections • Application development tools • Combinatorial optimization • Online optimization services SNU IEFAL SEMINAR

  12. Database structure formathematical programming models Robert Fourer Department of Industrial Engineering and Management Sciences, Northwestern University, IL, USA DSS, Vol. 20, 1997, pp.317-344. SNU IEFAL SEMINAR

  13. Paper review Contents • Introduction • Formulations • Database structures • Relational structures • Hierarchical structures • Comparisons • Conclusions SNU IEFAL SEMINAR

  14. Paper review Introduction • What is the problem? • In the design and use of large-scale mathematical programming systems, a substantial portion of the effort has no direct relation to the variable and constraints. • Goal of this paper • Codify some of the principles of database construction for LP • Steel optimization model example • This paper is based on a generic model SNU IEFAL SEMINAR

  15. Paper review Formulations • General formulation • Specific model formulation • Generic continuous-flow production process 1 2 3 A 4 B C 6 6 7 SNU IEFAL SEMINAR

  16. Paper review Formulations • First formulation ; m : set of materials t : set of facilities SNU IEFAL SEMINAR

  17. Paper review Formulations • First formulation (con’t) SNU IEFAL SEMINAR

  18. Paper review Formulations • First formulation (con’t) SNU IEFAL SEMINAR

  19. Paper review Formulations • Second formulation SNU IEFAL SEMINAR

  20. Paper review Formulations • Second formulation (con’t) SNU IEFAL SEMINAR

  21. Paper review Formulations • Second formulation (con’t) SNU IEFAL SEMINAR

  22. Paper review Database structures- Relational structures • General model ; α : set of constraints, β : set of variables VARIABLES col_name col_profit col_min col_optimal col_max CONSTRAINTS COEFFICIENTS row_name row_min row_max coeff_row -> CONSTRAINTS coeff_col -> VARIABLES coeff_value SNU IEFAL SEMINAR

  23. Paper review Database structures- Relational structures • Principles of relational structure • Rule 1 (files) • For each set in the model, there is a corresponding file in the database • Rule 2 (key fields) • The -file has a number of key fields equal to the dimension of • Rule 3 (data fields) • The -file has an additional data field for each model entity indexed over • Rule 4 (records) • The -file has a record corresponding to each member of • Rule 5 (many-to-one relationships) • For each containment restriction of the form the dth key in the -file has a many-to-one relationship to the -file SNU IEFAL SEMINAR

  24. Paper review Database structures- Relational structures • Multi-facility production model MATERIALS FACILITIES MATERIAL_CONVERSIONS ACTIVITIES mat_name buy_min buy_opt buy_cost sell_min sell_opt sell_max sell_cost fac_name cap_min cap_max from_mat -> MATERIALS to_mat -> MATERIALS conv_yield conv_cost conv_opt act_fac -> FACILITIES act_name act_min act_opt act_max act_cost act_cap_rate FACILITY_OUTPUTS ACTIVITY_OUTPUTS FACILITY_INPUTS ACTIVITY_INPUTS out_fac -> FACILITIES out_mat -> MATERIALS out_min out_opt out_max act_out_fac -> FACILITIES act_out_mat -> MATERIALS act_out act_out_rate in_fac -> FACILITIES in_mat -> MATERIALS in_min in_opt in_max act_in_fac -> FACILITIES act_in_mat -> MATERIALS act_in act_in_rate SNU IEFAL SEMINAR

  25. CONSTRAINTS row_name row_min row_max VARIABLES col_name col_profit col_min col_optimal col_max COEFFICIENTS coeff_row -> CONSTRAINTS coeff_value Paper review Database structures- Hierarchical structures • General model SNU IEFAL SEMINAR

  26. Paper review Database structures- Hierarchical structures • Principles of hierarchical structure • Rule 1a (files) • For each unindexed set in the model, there is a corresponding file in DB • Rule 1b (subfiles) • For each collection of sets indexed over the -file has a subfile corresponding to the collection • Rule 2 (key fields) • Rule 3 (data fields) • An additional data field for each model entity indexed over (or ) • Rule 4 (records) • Rule 5 (many-to-one relationships) • For each containment restriction of the form , the key record in the -subfile has a many-to-one relationship to the -file SNU IEFAL SEMINAR

  27. Paper review Database structures- Hierarchical structures • Multi-facility production model MATERIALS FACILITEIS mat_name buy_min buy_opt buy_max buy_cost sell_min sell_opt sell_max sell_cost fac_name cap_min cap_max INPUTS ACTIVITES act_name act_min act_opt act_max act_cost act_cap_rate in_mat -> MATERIALS in_min in_opt in_max CONVERSIONS ACT_INPUTS to_mat -> MATERIALS conv_yield conv_cost conv_opt act_in_mat -> FACILITIES.INPUTS act_in_rate OUTPUTS ACT_OUTPUTS out_mat -> MATERIALS out_min out_opt out_max act_out_mat -> FACILITIES.OUTPUTS act_out_rate SNU IEFAL SEMINAR

  28. Paper review Comparisons SNU IEFAL SEMINAR

  29. Paper review Conclusions • The principles of this paper shows the database structures can be derived in a systematic way from sets and indexing that are characteristics of mathematical programming data. • The choice of index sets for the formulation of an optimization problem is observed to involve certain tradeoffs in convenience and efficiency of data access. • Other indexing structure • Use modeling languages : MPL, AMPL, AIMMS… • Other database type • Multidimensional database : On-Line Analytical Processing(OLAP)... SNU IEFAL SEMINAR

  30. References • Database structures for mathematical programming models , Robert Fourer, DSS. vol. 20(1997), pp317-344. • Recent development in mathematical programming modeling systems, Robert Fourer, Conference of the Operational Research Society, Lancaster, September 8-10, 1998. • AMPL new standard database features, Robert Fourer, Informs Meeting, Cincinnati, May 2, 1999. • Optimization Software • http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/index.html • Optimization FAQ on Linear and Nonlinear Programming. • http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html • Modeling solution ; AMPL • http://www.modeling.com • http://lionhrtpub.com/orms/surveys/LP/LP-survey.html SNU IEFAL SEMINAR

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