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# Sequential Framework for HENS @ IITM

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### Sequential Framework for HENS @ IITM

1. 1. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY T HE S EQUENTIAL F RAMEWORK FOR H EAT E XCHANGER N ETWORK S YNTHESIS Rahul Anantharaman rahul.anantharaman@ntnu.no Department of Energy & Process Engineering Norwegian University of Science and Technology IIT Madras Chennai, 18.12.2009
2. 2. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
3. 3. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
4. 4. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS For a given set of hot and cold process streams as well as external utilities, design a heat exchanger network that minimizes Total Annualized Cost (TAC). TAC = Capital Cost + Energy Cost Sequential Framework Engine
5. 5. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS S OLUTION METHODS Evolutionary methods such as Pinch Design Method Sequential synthesis methods Simultaneous synthesis methods Stochastic optimization methods
6. 6. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
7. 7. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
8. 8. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
9. 9. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
10. 10. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
11. 11. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
12. 12. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
13. 13. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY H EAT E XCHANGER N ETWORK S YNTHESIS T IMELINE
14. 14. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
15. 15. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY HENS IN THE 21 ST CENTURY R EVIEW 224 references published from 2000-2008 216 journal papers 48 jounals 43 countries 4 conference proceedings 10 Ph.D. theses 4 texts
16. 16. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY HENS IN THE 21 ST CENTURY R EVIEW 45 40 35 30 25 20 15 10 5 0 2000 2001 2002 2003 2004 2005 2006 2007 2008
17. 17. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY HENS IN THE 21 ST CENTURY R EVIEW Computers & Chem Eng 43 77 Applied Thermal Eng Industrial & Eng Chem  Research Chem Eng Research & Design 39 Latin American Appl Research Heat Transfer Eng 7 7 Chem Eng Science 7 23 7 7 10 Chem Eng and Processing Ch E dP i 7
18. 18. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY HENS IN THE 21 ST CENTURY R EVIEW 50 45 40 35 30 25 20 15 10 5 0
19. 19. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY HENS IN THE 21 ST CENTURY R EVIEW
20. 20. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY HENS IN THE 21 ST CENTURY R EVIEW HENS still an active area of research interest Over 25% of references devoted to case studies Pinch Analysis based evolutionary methods dominate Sustained interest in simultaneous MINLP methods Yee and Grossmann (1990) superstructure Pressure drop and detailed HX design considerations Small test problems Number of references related to genetic programming and other meta-heuristic methods increasing in frequency Though there has been signiﬁcant developments in HENS using mathematical programming methods, synthesis of large scale HENS problems without simpliﬁcations and heuristics have been lacking. This is an area that requires more research for mathematical programming based approaches to be used in the industry
21. 21. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
22. 22. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M OTIVATION FOR THE S EQUENTIAL F RAMEWORK Pinch based methods for network design Improper trade-off handling Time consuming Several topological traps MINLP methods for network design Severe numerical problems Difﬁcult user interaction Fail to solve large scale problems Stochastic optimization methods for network design Non-rigorous algorithms Quality of solution depends on time spent on search
23. 23. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M OTIVATION FOR THE S EQUENTIAL F RAMEWORK HENS TECHNIQUES DECOMPOSE THE MAIN PROBLEM Pinch Design Method is sequential and evolutionary Simultaneous MINLP methods let math considerations deﬁne the decomposition The Sequential Framework decomposes the problem into subproblems based on knowledge of the HENS problem Engineer acts as optimizer at the top level Quantitative and qualitative considerations included
24. 24. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY U LTIMATE G OAL Solve Industrial Size Problems Deﬁned to involve 30 or more streams Include Industrial Realism Multiple and ``Complex´´Utilities Constraints in Heat Utilization (Forbidden matches) Heat exchanger models beyond pure countercurrent Avoid Heuristics and Simpliﬁcations No global or ﬁxed ∆ Tmin No Pinch Decomposition Develop a Semi-Automatic Design Tool EXCEL/VBA (preprocessing and front end) MATLAB (mathematical processing) GAMS (core optimization engine) Allow signiﬁcant user interaction and control Identify near optimal and practical networks
25. 25. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UR ENGINE 3 way trade-off Compromise between Pinch Design and MINLP methods
26. 26. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
27. 27. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 1 Stream Tin Tout mCp ∆H h K K kW/K kW kW/m2 K H1 626 586 9.802 392.08 1.25 H2 620 519 2.931 296.03 0.05 H3 528 353 6.161 1078.18 3.20 C1 497 613 7.179 832.76 0.65 C2 389 576 0.641 119.87 0.25 C3 326 386 7.627 457.62 0.33 C4 313 566 1.69 427.57 3.20 ST 650 650 - - 3.50 CW 293 308 - - 3.50 Exchanger cost (\$) = 8,600 + 670A0.83 (A is in m2 )
28. 28. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 1 L OOPING TO THE SOLUTION HRAT ﬁxed at 20K (Qh,min = 244.1 kW & Qc,min = 172.6) Umin = 8 units Soln. No U EMAT (K) HLD TAC (\$) 1 8 2.5 A 199,914 2 8 5 A 199,914 3 8 7.5 - No Soln 4 9 2.5 A 147,861 5 9 2.5 B 151,477 6 9 5 A 147,867 7 9 5 B 151,508 8 9 7.5 A 149,025 9 9 7.5 B 149,224 10 10 2.5 A 164,381 11 10 5 A 167,111 12 10 7.5 A 164,764
29. 29. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 1 B EST SOLUTION
30. 30. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 1 C OMPARISON No. of units Area (m2 ) Cost (\$) Colberg and Morari (1990) 22 173.6 Colberg and Morari (1990) 12 188.9 177,385 Yee and Grossmann (1990) 9 217.8 150,998 Sequential Framework 9 189.7 147, 861
31. 31. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 2 Stream Tin Tout mCp ∆H h (°C) (°C) (kW/°C) (kW) (kW/m2 °C) H1 180 75 30 3150 2 H2 280 120 60 9600 1 H3 180 75 30 3150 2 H4 140 40 30 3000 1 H5 220 120 50 5000 1 H6 180 55 35 4375 2 H7 200 60 30 4200 0.4 H8 120 40 100 8000 0.5 C1 40 230 20 3800 1 C2 100 220 60 7200 1 C3 40 290 35 8750 2 C4 50 290 30 7200 2 C5 50 250 60 12000 2 C6 90 190 50 5000 1 C7 160 250 60 5400 3 ST 325 325 1 CW 25 40 2 Exchanger cost (\$) = 8,000 + 500A0.75 (A is in m2 )
32. 32. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 2 L OOPING TO THE SOLUTION HRAT ﬁxed at 20.35°C (Qh,min = 11539.25 kW & Qc,min = 9164.25 kW) Umin = 14 units Soln. No U EMAT (C) HLD TAC (\$) 1 14 2.5 A 1,545,375 2 15 2.5 A 1,532,148 3 15 2.5 B 1,536,900 4 15 5 A 1,529,968 5 15 5 B 1,533,261 6 16 2.5 A 1,547,353
33. 33. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 2 B EST SOLUTION
34. 34. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY E XAMPLE 2 C OMPARISON The solution given here with a TAC of \$1,529,968, about the same cost as the solution presented in the original paper by Björk and Nordman (2005) (TAC \$1,530,063) When only one match was allowed between a pair of streams the TAC reported by Björk & Nordman (2005) was \$1,568,745 The Sequential Framework allows only 1 match between a pair of streams Unable to compare the solutions apart from cost as the paper did not present the networks in their work
35. 35. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
36. 36. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY C HALLENGES C OMBINATORIAL E XPLOSION Reason: Binary Variables in MILP models Physical and engineering insights will mitigate, not remove, the problem MILP models are the bottlenecks that limit problem size due to computational time L OCAL OPTIMA Reason: Non-convexities in the NLP model Convex estimators developed for MINLP models are computationally intensive Time to solve the basic NLP is not a problem Sequence of MILP and NLP problems considerably easier to solve than MINLP formulations
37. 37. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
38. 38. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M INIMUM NUMBER OF UNITS SUB - PROBLEM D EFINITION Given: a set H of hot process streams to be cooled, a set C of cold process streams to be heated, start and target temperatures, heat capacities and ﬂow rates of the hot and cold process streams, a set of utilities, temperatures or temperature ranges and minimum requirement of the utilities and Exchanger Minimum Approach Temperature (EMAT) set to zero, calculate the minimum number of matches between hot process streams and utilities and cold process streams and utilities such that the heating and cooling requirements for each stream are met.
39. 39. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M INIMUM NUMBER OF UNITS SUB - PROBLEM F ORMULATION X X min z = yij (P1) i∈H j∈C s.t. H X Ri,k − Ri,k −1 + Qijk = Qik ∀ i ∈ Hk , k ∈ TI j∈Ck C X Qijk = Qjk ∀ j ∈ Ck , k ∈ TI i∈Hk X Qijk − Uij yij ≤ 0 ∀ i ∈ H, j ∈ C k ∈TI Rik ≥ 0 ∀ i ∈ Hk , k ∈ TI Ri0 = RiK = 0 ∀i ∈H Qijk ≥ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI yij = {0, 1} ∀ i ∈ H, j ∈ C
40. 40. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
41. 41. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M INIMUM NUMBER OF UNITS SUB - PROBLEM C HALLENGES Combinatorial explosion Problem is proved to be N P-hard in the strong sense
42. 42. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M INIMUM NUMBER OF UNITS SUB - PROBLEM C HALLENGES Combinatorial explosion Problem is proved to be N P-hard in the strong sense
43. 43. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY A LLEVIATING COMBINATORIAL EXPLOSION The three major ways to improve the model solution time are: 1 Pre-processing to reduce model size using insight and heuristics 2 Model modiﬁcation/reformulation 3 Improving efﬁciency of the B&B method
44. 44. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
45. 45. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S HARPENING LP RELAXATION BY DECREASING BIG M The gap, i.e. the difference between the LP relaxation and the actual binary solution, is dependent on the value of Uij , the upper limit on the amount of heat transfer between streams i and j. A smaller value of Uij corresponds to a smaller gap and thus reduced computing times.
46. 46. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S HARPENING LP RELAXATION BY DECREASING BIG M O RIGINAL U 8 9 <X = H C X Uij = min Qik , Qjk : ; k k M ODIFIED U 8 9 <X h “ ” “ ” i= H C H C H C X Uij = min Qik , Qjk , max min mCpi , mCpj · Tsi − Tsj − EMAT , 0 : ; k k
47. 47. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S HARPENING LP RELAXATION BY DECREASING BIG M L OCAL U L OCAL U: Deﬁne maximum amount of heat exchanged between streams on a tempertaure interval basis 80 1 9 < X = HA C Uijk = min @ Qi k , Qjk ¯ : ; ¯ k ≤k Logical constraint utilizing the local U Qijk − Uijk yij ≤ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI This constraint will reduce the feasible region as compared to the earlier one - thus leading to a tighter formulation.
48. 48. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY I NTEGER C UTS The gap between the LP relaxation based lower bound and the optimal integer solution. This gap can be reduced by employing integer cuts to the model. A potential drawback of adding such cuts is the increase in model size and hence computation time. 2 types of integer cuts applied to the model 1 Compulsory matches 2 Minimum number of matches per stream
49. 49. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL MODIFICATION T EST PROBLEM 22TP1 ROOT NODE LP RELAXATION VALUES FOR TEST PROBLEM 22TP1 WITH IP SOLUTION 23 U deﬁnition No integer Compulsory Minimum matches Both cuts cuts matches per stream Eq 1 12.21 - - - Global Eq 2 15.17 16.27 16.82 18.62 Local Eqs 3,4 15.78 16.56 17.63 18.62
50. 50. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL MODIFICATION T EST PROBLEM 21TP1 ROOT NODE LP RELAXATION VALUES FOR TEST PROBLEM 21TP1 WITH IP SOLUTION 22 U deﬁnition No integer Compulsory Minimum matches Both cuts cuts matches per stream Eq 1 11.93 - - - Global Eq 2 14.30 15.14 14.49 15.21 Local Eqs 3,4 14.87 15.39 14.95 15.40
51. 51. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL MODIFICATION T EST PROBLEM 21TP2 ROOT NODE LP RELAXATION VALUES AND TOTAL SOLUTION TIMES FOR TEST PROBLEM 21TP2 WITH IP SOLUTION 22 U deﬁnition No integer Compulsory Minimum matches Both cuts cuts matches per stream Eq 1 16 - - - 20 s - - - Global Eq 2 17.67 18.80 18.13 18.83 19 s 27 s 23 s 21 s Local Eqs 3,4 18.80 18.80 18.81 18.83 36 s 46 s 45 s 42 s
52. 52. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL MODIFICATION D ISCUSSION Modiﬁed U and Local U deﬁnitions give tighter lower bounds than original U deﬁnition Compulsory matches integer cuts always improve the lower bound Minimum number of matches integer cuts have varying results 21TP1 and 22TP1 still do not solve in less than 12 hours Gap is not the only measure of the complexity of an integer problem. Two of the main issues in this particular integer problem are the number of feasible solutions and the number of multiple optima.
53. 53. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
54. 54. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S ET PARTITIONING FORMULATION SSis be deﬁned to represent all feasible sets of matches, s ∈ Si , between a hot stream i and all cold streams: SSis = { j | j ∈ C if stream j is in set of matches s for stream i} s 1 2 3 4 5 6 7 C1 1 0 0 1 1 0 1 C2 0 1 0 1 0 1 1 CW 0 0 1 0 1 1 1 ( 1, if set of matches s is chosen for hot process stream i λis = 0, otherwise S ET PARTITIONING CONSTRAINT X λis = 1 ∀i∈H s∈Si
55. 55. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S ET PARTITIONING FORMULATION Maximum number of set of matches for a hot stream = 2nc − 1 Potentially large number of binary variables will be introduced Use thermodynamics and physical insight to reduced the number of binary variables The set partitioning constraint is expected to have more efﬁcient branching characteristics
56. 56. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S ET PARTITIONING FORMULATION The constraints for deﬁning the set of feasible matches are: 1 At least one cold process stream or utility in the set of matches must have a supply temperature lower than or equal to the hot process stream’s target temperature and satisfy the hot process stream’s duty in this temperature range. 2 The total heat demand for the set of cold process streams or utilities below the hot process stream’s supply temperature must be greater than or equal to the hot process stream’s total heat duty. 3 The total number of cold process streams and utilities in a set of matches should not exceed a user-speciﬁed maximum value. 4 For cases with streams having large duties, the number of streams in a set of matches for a utility must be larger than one. Constraints 1 and 2 are thermodynamically based while constraints 3 and 4 are heuristics based on insights gained from testing various problems. Constraint 3 is user speciﬁed to allow the user to impart their knowledge about the problem on hand to reduce the problem size.
57. 57. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S ET PARTITIONING FORMULATION X X min z = cis λis (P2) i∈H s∈Si s.t. H X Ri,k − Ri,k −1 + Qijk = Qik ∀ i ∈ Hk , k ∈ TI j∈Ck C X Qijk = Qjk ∀ j ∈ Ck , k ∈ TI i∈Hk X X Qijk − Uij λis ≤ 0 ∀ i ∈ H, j ∈ C k ∈TI s∈Pij X λis = 1 ∀i ∈H s∈Si X X λis ≥ 1 ∀j ∈C i∈H s∈Pij X X λis ≤ max value ∀j ∈C i∈H s∈Pij cis = card (SSis ) ∀ i ∈ H, s ∈ Si Rik ≥ 0 ∀ i ∈ Hk , k ∈ TI Ri0 = RiK = 0 ∀i ∈H Qijk ≥ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI λis = {0, 1} ∀ i ∈ H, s ∈ Si
58. 58. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S ET PARTITIONING FORMULATION XX min z = yij (P3) i j s.t. H X Ri,k − Ri,k −1 + Qijk = Qik ∀ i ∈ Hk , k ∈ TI j∈Ck C X Qijk = Qjk ∀ j ∈ Ck , k ∈ TI i∈Hk X Qijk − Uij yij ≤ 0 ∀ i ∈ H, j ∈ C k ∈TI X λis = 1 ∀i ∈H s∈Si X X λis ≥ 1 ∀j ∈C i∈H s∈Pij X X λis ≤ max value ∀j ∈C i∈H s∈Pij X λis = yij ∀ i ∈ H, j ∈ C s∈Pij Rik ≥ 0 ∀ i ∈ Hk , k ∈ TI Ri0 = RiK = 0 ∀i ∈H Qijk ≥ 0 ∀ i ∈ Hk , j ∈ Ck , k ∈ TI
59. 59. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL REFORMULATION T EST PROBLEM 22TP1 ROOT NODE LP RELAXATION VALUES FOR TEST PROBLEM 22TP1 WITH IP SOLUTION 23 Model Binary U model Additional LP relaxation variables constraints value P1 143 Global Eq-2 Eqs 5,6 18.62 Local Eqs 3,4 Eqs 5,6 18.62 P3 2500 Global Eq-2 Eqs 5,6 18.62 with λis Local Eqs 3,4 Eqs 5,6 18.62 P4 2625 Global Eq-2 Eqs 5,6 18.67 with µjt Local Eqs 3,4 Eqs 5,6 18.67 P5 4999 Global Eq-2 None 18.67 with λis & µjt Local Eqs 3,4 18.67
60. 60. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL REFORMULATION T EST PROBLEM 21TP1 ROOT NODE LP RELAXATION VALUES FOR TEST PROBLEM 21TP1 WITH IP SOLUTION 22 Model Binary U model Additional LP relaxation variables constraints value P1 131 Global Eq-2 Eqs 5,6 15.21 Local Eqs 3,4 Eqs 5,6 15.40 P3 4645 Global Eq-2 Eqs 5,6 15.74 with λis Local Eqs 3,4 Eqs 5,6 15.82 P4 5435 Global Eq-2 Eqs 5,6 16.33 with µjt Local Eqs 3,4 Eqs 5,6 16.45 P5 9879 Global Eq-2 None 16.86 with λis & µjt Local Eqs 3,4 16.93
61. 61. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY M ODEL REFORMULATION D ISCUSSION Reformulated models may results in strengthening the LP relaxation Contain more information regarding thermodynamics of the matches than the basic model Reformulated models for 21TP1 and 22TP2 cannot be solved within 12 hours These models are larger and with more binary variables counteracting any potential beneﬁt Reduction in gap not signiﬁcant enough
62. 62. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY O UTLINE 1 HENS Background HENS in the 21st century 2 S EQUENTIAL F RAMEWORK Introduction Examples Challenges 3 M IN U NITS SUB - PROBLEM Background Challenges Model modiﬁcation Model reformulation Further work 4 S UMMARY
63. 63. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY F URTHER WORK Optimum value is reached early in the solution process and most of the effort is expended in proving optimality. Develop heuristics to stop the search after an appropriate solution time. Identifying subnetworks by relaxing stream temperatures and ﬂow rates it may be possible to get a good initial bound on the minimum number of units using U = N + L − S. This value could be used by CPLEX as the initial lower bound thus tightening the gap.
64. 64. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY F URTHER WORK All NP-complete problems have a phase separation i.e. both hard and easy regions, with a sharp boundary between them. On crossing that frontier, the problem undergoes a phase transition, analogous to the boiling or freezing of water. Identifying the phase transition of the minimum units problem would enable the user to decide on using deterministic methods for solving it for other non-deterministic methods depending on which phase the problem happens to be in.
65. 65. HENS S EQUENTIAL F RAMEWORK M IN U NITS SUB - PROBLEM S UMMARY S UMMARY Sequential Framework has many advantages Automates the design process Allows signiﬁcant User interaction Progress EMAT identiﬁed as an optimizing ‘area variable´ Improved HLDs from Stream match generator subproblem Signiﬁcantly better and automated starting values for NLP subproblem Global optimum for NLP ensured using a modiﬁed GBD scheme Limiting elements Stream match generator Transportation Model for promising HLDs Signiﬁcant improvements required to ﬁght combinatorial explosion MILP Transhipment model for minimum number of units Similar combinatorial problems as the Transportation model Reliability of NLP solutions is no longer limiting