COMBINATORIAL FRAMEWORK

Haryo Tomo

adopted from Process Synthesis Lectureby Univ. Panonia Veszprem, Hungary1COMBINATORIAL FRAMEWORK FOR PROCESS DESIGN AND SYNTHESISOUTLINE2General IntroductionApproaches to Solve Process Design and Operations ProblemsComponent Problems of Process Design and OperationsProcess Design ProblemsCombinatorial Tools for Process Design P-graph representationAxioms of Combinatorially Feasible StructuresCombinatorial AlgorithmsMSG for Generating the SuperstructureStructural MappingsSSG for Generating Combinatorially Feasible StructuresDecision MappingGeneral Philosophyin Solving Process Design and Operations ProblemsConventional Approach Conventional Mathematical ProgrammingAlgorithmic Process SynthesisSuperstructure ConceptRigorous SuperstructureConventional Approach to Solve Process Design and Operations ProblemsFormulation as a general mathematical programming problem (e.g., LP, MILP, MINLP, NLP)Application of a general-purpose solver (e.g., GAMS)Conventional Mathematical ProgrammingMathematical programming problem(objective function, constraints)General purpose solverOptimal solutionDifficultyProcess synthesis problems are not specified as standard optimization problemsProcess Synthesis ProblemGiven:set of products,set of raw materials,mathematical models of the operating unitsGenerate:optimal process or n-best processes or every feasible processOptimality criteria:cost, waste generation, controllability, risk, orcombinations of themAlgorithmic Process SynthesisCost function and constraints for the operating units, raw materials, and productsMathematical programming modelModel generation: Synthesis(Generating LP, MILP, MINLP, or NLP model)Optimal solutionSolution: Analysis(Mathematical programming method)QuestionHow to generate and how to solve the mathematical programming model?Algorithmic Process SynthesisCost function and constraints for the operating units, raw materials, and productsMathematical programming modelOptimal solutionSolution of the modelModel generationGeneration of the SuperstructureGeneration of the mathematical model based on the SuperstructureQuestionHow to generate the superstructure?Rigorous SuperstructureSuppose that systematic procedure is available so that a valid mathematical programming model can be generated for a network of the given operating unitsA network of operating units is defined to be a rigorous super-structure if the optimality of the resultant solution cannot be improved for any instance of the class of problems by any other procedure for network and model generationMultiscale OptimizationClasses of Process Synthesis ProblemsMacroscopicMezoscopicMicroscopicSystematic Hierarchical ApproachAlgorithmic Synthesis of Supply Chains Production Planning and SchedulingLevels of AbstractionIndustrial problems because of their complexity are examined in multiple levels of abstraction:Macroscopic level: Elementary step: operating unitsMesoscopic level:Elementary step: equipments(Multiple equipments forms an operating unit.)Microscopic level:Elementary step: physical / chemical / biochemical transformation(Inside one equipment.)-+Levels of Abstraction: Component ProblemsMacroscopic level: Conceptual designReaction-Network SynthesisTotal-Flowsheet SynthesisMesoscopic level:Separation-Network SynthesisHeat-Exchanger-Network Synthesis SchedulingProcess ControlMicroscopic level:Azetropic DistillationReaction Pathway Identification

-+Macroscopic Level: Operating UnitsComponent problem: Total-Flowsheet SynthesisBuilding blocks: operating units

Example:

direct chlorinating unitseparating unitethylene oxygenethylene dichloride + waterhydrocloric acid ethylene dichloridewaterethylene dichloride + water-Macroscopic Level: Network of Operating UnitsExample:

direct chlorinating unitethyleneoxygenhydrocloric acidvinyl chloride oxychlorinatingunitseparatingunitpyrolyzingunitchlorineseparatingunitwater-Mesoscopic level: EquipmentsComponent problem: Separation-Network SyntehsisBuilding blocks: equipments

Example: distillation column

vinyl chloride +hydrocloric acid +ethylene dichloride hydrocloric acidvinyl chloride +ethylene dichlorideMesoscopic level: Network of EquipmentsExample: implementing separating unit by two distillation columns

vinyl chloride +hydrocloric acid +ethylene dichloridehydrocloric acidethylene dichloridevinyl chlorideseparating unitMicroscopic Level: Physical TransformationsComponent problem: Heat-Exchanger-Network Synthesis Building blocks: physical transformations

Example:

heatingcooling +Microscopic Level: Network of Physical TransformationsExample: distillation column with three distillation steps

+distillation column 22COMBINATORIAL TECHNIQUE IN PROCESS DESIGN AND SYNTHESIS23INTRODUCTIONMINLPmin g(x,y)s.t.f(x, y)0xn , y{0, 1}m

Most MINLP model can not represent a practical problem.Additional information is embedded implicitly in the model of a practical problem.Idea: this information can effectively control the procedure.24ILLUSTRATIVE EXAMPLE PNS 1Operating units:

Product: ARaw materials: E, G, J, K, L1CFA1CFA2DAB2DAB3CEF3CEF6JF6JFFeasible flowsheet7KLH7KLH5GHD4CDFG4CDFG34DEFGC1FA25EXAMPLE PNS 1Product: ARaw materials: E, G, J, K, LPlausible operating unitsTypeInputsOutputs1CA, F2DA, B3E, FC4F, GC, D5G, HD6JF7K, LH26Number of operating units:7binary variables:7combinations:127(=27-1)27SYNTHESIS OF AN INDUSTRIAL PROCESS (EXAMPLE PNS 2)Product: A61Raw materials: A1, A2, A3, A4, A6, A7, A8, A11, A15, A17, A18, A19, A20, A23, A27, A28, A29, A30, A34, A43, A47, A49, A52, A5428PLAUSIBLE OPERATING UNITSNo.TypeInputsOutputs1FeederA1A52ReactorA2, A3, A4A93ReactorA3, A4, A6, A11A104ReactorA3, A4, A5A125ReactorA3, A4, A5A136ReactorA7, A8, A14A167ReactorA8, A14, A18A168SeparatorA9, A11A21, A22, A249SeparatorA10, A11A22, A24, A3710SeparatorA12A25, A2611SeparatorA13A25, A3112DissolverA15, A16A3229PLAUSIBLE OPERATING UNITS (Contd)No.TypeInputsOutputs13ReactorA14, A17, A18, A19, A20A3314ReactorA6, A21A3515WasherA22, A23A4816WasherA5, A24A3617SeparatorA5, A11, A25A37, A38, A3918SeparatorA11, A26A40, A4219ReactorA14, A27, A28, A29, A30A4120SeparatorA11, A31A40, A4221CentrifugeA32A44, A4522WasherA33, A34A4623SeparatorA36A14, A4824SeparatorA38A14, A4830PLAUSIBLE OPERATING UNITS (Contd)No.TypeInputsOutputs25FilterA41A50, A5126WasherA43, A44A5327FilterA46A55, A5628SeparatorA47, A48A5, A5729SeparatorA48, A49A5, A5830SeparatorA50A59, A6031DryerA51, A54A6132DryerA52, A53A6133DryerA54, A55A6134DistillationA59A62, A6335SeparatorA60A64, A6531Number ofoperating unit:35binary variables:35combinations:34billionsubproblems at a B&B (worst case):130million32SOURCE OF COMPLEXITYCombinatorial nature of the problem33COMBINATORIAL TOOLSOur rigorous technique is based on combinatorics, especially,on the following items.P-graphNew structure representation.AxiomsThe fundamental properties of combinatorially feasible process structures (e.g., every operating unit has at least one path leading to a product).AlgorithmsEffective and rigorous combinatorial algorithms for process synthesis.34STRUCTURAL REPRESENTATIONSimple directed graphs are incapable of providing an unambiguous representation in process synthesis.Process graphs or P-graphs are introduced for structural representation in process synthesis.35AMBIGUOUS GRAPHICAL REPRESENTATION: Digraph

Case (1.1). Two different materials are produced separately, one by operating unit 02 and the other by operating unit 03. Moreover, it is necessary to feed both of these materials to operating unit 01 to generate the final product.

Case (1.2). One material is produced by both operating units 02 and 03. This material is subsequently fed to operating unit Ol to generate the final product.

36AMBIGUOUS GRAPHICAL REPRESENTATION: Signal-flow graphCase (2.1). Two separate operating units, one receiving material B as its input and the other receiving material C as its input, produce the same material which is subsequently fed to another operating unit where material A (product) is generated.

Case (2.2). A single operating unit, receiving materials B and C as its inputs, produces a material which is subsequently fed to another operating unit where material A (product) is generated.

37CONVENTIONAL AND P-GRAPH REPRESENTATIONreactordistillation columnreactordistillation columnP-graphFormal definition38UNAMBIGUOUS GRAPHICAL REPRESENTATION: P-graphP-graphs uniquely representing cases (1.2) and (2.1), case (2.2), case (1.1).

39P-GRAPH REPRESENTATION OF A SYNTHESIS PROBLEM PNS 2Notation: material operating unitA44A44A55A55A6A3329A2481523104A416A14A1867A15A1612A17A19A20A27A28A29A301319A41A332225A46A54A51A503127A563332A61A53A5226A45A4321A325A1311A31A26A25A38A37A11A12A10A2A9A21A22A23A47A48A492928A1A58A5A57A3924A3617A6A3329A2481523104A416A6A3329A2481523104A416A14A1867A15A1612A17A19A20A27A28A29A301319A41A33A342225A46A54A51A503127A563332A61A53A5226A45A4321A32A1311A31A26A25A38A37A11A12A10A2A9A21A22A23A47A48A492928A1A58A5A57A3924A3617511A7A8A7A840AXIOMS OF COMBINATORIALY FEASIBLE PROCESS STRUCTURESFor given process synthesis problem, a P-graph satisfying the following five axioms is a combinatorially feasible structure.(S1)Every final product is represented in the structure.(S2)A material represented in the structure is a raw material if and only if it is not an output of any operating unit represented in the structure.(S3)Every operating unit represented in the structure is defined in the synthesis problem.(S4)Any operating unit represented in the structure has at least one path leading to a product.(S5)If a material belongs to the structure, it must be an input to or output from at least one operating unit represented in the structure.41REDUCTION OF THE SEARCH SPACEFeasible StructuresSearch SpaceCombinatorially Feasible Structures41Since the combinatorically properties are the necessary properties of the feasible pathways, we do not lose any of the feasible pathways if we reduce the search space to the combinatorically feasible solutions.42ILLUSTRATIVE EXAMPLE FOR THE COMBINATORIALLY FEASIBLE STRUCTURES:EXAMPLE PNS 12DAB1CFAEFC34CFGDGHD5KLH7JF6Operating units given:Available raw materials: E, G, J, K, LProduct: A43COMBINATORIALLY FEASIBLE STRUCTURES OF EXAMPLE PNS 1ECFA3ECFA13ECF136ECJFA1364CFDA14CFGDA14CJFGDA16ECFDA4ECFGDA13Solution #1Solution #2Solution #3Solution #4Solution #54ECFDA1364ECJFGDA13642CFDAB642CJFGDAB652HDAB752KLGHDAB742AB67452CJKLFGHDAB67Solution #6 Solution #7Solution #8 Solution #944COMBINATORIALLY FEASIBLE STRUCTURES OF EXAMPLE PNS 1 (Contd)2AB137Solution #10Solution #11 Solution #12Solution #152AB1375E2CKLFGHDAB137E2CFHDAB1675E2CJKLFGHDAB136742CFDAB142CFGDAB1Solution #10Solution #11 Solution #1242CFDAB1642CJFGDAB16452CFHDAB17452CKLFGHDAB17452CFHDAB167452CJKLFGHDAB1674E2CFDAB134E2CFGDAB13Solution #13 Solution #14Solution #16Solution #1545COMBINATORIALLY FEASIBLE STRUCTURES OF EXAMPLE PNS 1 (Contd)Solution #194E2CFDAB1364E2CJFGDAB13645E2CFHDAB13745E2CKLFGHDAB137E2CFHDAB16745E2CJKLFGHDAB1367Solution #17 Solution #18Solution #1946SYNTHESIS OF AN INDUSTRIAL PROCESS(EXAMPLE PNS 2)Product: A61Raw materials: A1, A2, A3, A4, A6, A7, A8, A11, A15, A17, A18, A19, A20, A23, A27, A28, A29, A30, A34, A43, A47, A49, A52, A5447PLAUSIBLE OPERATING UNITSNo.TypeInputsOutputs1FeederA1A52ReactorA2, A3, A4A93ReactorA3, A4, A6, A11A104ReactorA3, A4, A5A125ReactorA3, A4, A5A136ReactorA7, A8, A14A167ReactorA8, A14, A18A168SeparatorA9, A11A21, A22, A249SeparatorA10, A11A22, A24, A3710SeparatorA12A25, A2611SeparatorA13A25, A3112DissolverA15, A16A3248PLAUSIBLE OPERATING UNITS (Contd)No.TypeInputsOutputs13ReactorA14, A17, A18, A19, A20A3314ReactorA6, A21A3515WasherA22, A23A4816WasherA5, A24A3617SeparatorA5, A11, A25A37, A38, A3918SeparatorA11, A26A40, A4219ReactorA14, A27, A28, A29, A30A4120SeparatorA11, A31A40, A4221CentrifugeA32A44, A4522WasherA33, A34A4623SeparatorA36A14, A4824SeparatorA38A14, A4849PLAUSIBLE OPERATING UNITS (Contd)No.TypeInputsOutputs25FilterA41A50, A5126WasherA43, A44A5327FilterA46A55, A5628SeparatorA47, A48A5, A5729SeparatorA48, A49A5, A5830SeparatorA50A59, A6031DryerA51, A54A6132DryerA52, A53A6133DryerA54, A55A6134DistillationA59A62, A6335SeparatorA60A64, A6550The five axioms reduce the

34 billion combinations of the operating units to3,465 combinatorially feasible structures.

The optimal solution is included in the set of 3465 feasible structures.51ILLUSTRATION OF THE REDUCTION IN THE SEARCH SPACE52ILLUSTRATION OF THE REDUCTION IN THE SEARCH SPACE10.000 x