The Pooling Problem - Applications, Complexity and Solution Methods

Transcription

1 The Pooling Problem - Applications, Complexity and Solution Methods Dag Haugland Dept. of Informatics University of Bergen Norway EECS Seminar on Optimization and Computing NTNU, September 30, 2014

2 Outline Introduction Illustration Applications Complexity Recognizing hard cases Cases solvable as linear programs Solution methods Global optimization: Exact (branch-and-bound) methods Local optimization: Inexact (heuristic) methods

3 Illustration: Bar keeping Suggested ratios of gin-to-tonic are 1:1, 2:3, 1:2, and 1:3 (Wikipedia) 20% % Gin % % Tonic 13% %

4 Illustration: Bar keeping % % 20 0% Gin Tonic % 13% % Profit: 62.4

5 Illustration: Bar keeping 20% % Gin % % Tonic Pool 13% %

6 Illustration: Bar keeping 20% % 20 0% Gin Tonic Pool % 13% % Profit: 56.7

7 Applications

8 Applications Oil refining Distiller Refined products Component tanks Gasoline blending

9 Applications Crude oil scheduling (Lee et al. 1996, Wu et al. 2005, Mouret 2010) Pipeline(s) Distiller Storage tanks Charging tanks

10 Applications Pipeline transportation of natural gas Network arcs: Subsea pipelines Sources: Gas fields, process units Pools: Junction points Terminals: Gas Markets Quality parameters: Combustion values, contamination levels

11 Applications More applications Food industry (Ruiz et al. 2013): Wastewater treatment (Galan and Grossmann 1998, Meyer & Floudas 2006) Environmental protection (Misener et al. 2010) Kallrath (2002): The pooling problem occurs in all multi-component network flow problems in which the conservation of both mass flow and composition is required.

12 Extensions Not considered in this talk: Networks with pool-to-pool connections: The generalized pooling problem (Audet et al. 2004, Meyer & Floudas 2006, Alfaki & H. 2013) Non-linear blending (octane numbers, gas combustion values) MILP models (e.g. for network design) (Meyer & Floudas 2006)

13 Problem definition Input Digraph: D = (S, P, T, A) = (sources, pools, terminals, arcs), where A (S P) (P T ) Quality attributes: K Node capacities: b i (i S P T ) Arc costs: c ij ((i, j) A) Source qualities: q k s (s S, k K) Quality bounds at terminals: u k t (t T, k K)

14 Problem definition Input Digraph: D = (S, P, T, A) = (sources, pools, terminals, arcs), where A (S P) (P T ) Quality attributes: K Node capacities: b i (i S P T ) Arc costs: c ij ((i, j) A) Source qualities: q k s (s S, k K) Quality bounds at terminals: u k t (t T, k K) Linear blending assumption: Pool qualities: w k p = Terminal qualities: s qk s fsp s fsp p w p kfpt p fpt u k t qs s f sp p w p

15 Problem definition Input Digraph: D = (S, P, T, A) = (sources, pools, terminals, arcs), where A (S P) (P T ) Quality attributes: K Node capacities: b i (i S P T ) Arc costs: c ij ((i, j) A) Source qualities: q k s (s S, k K) Quality bounds at terminals: u k t (t T, k K) Linear blending assumption: Pool qualities: w k p = Terminal qualities: s qk s fsp s fsp p w p kfpt p fpt u k t qs s f sp f p pt w p t u t

16 Problem definition Pooling Problem Find a flow allocation in D minimizing total costs while respecting capacity and quality bounds

17 Problem definition Pooling Problem Find a flow allocation in D minimizing total costs while respecting capacity and quality bounds Formulation: Flow and quality variables (Haverly 1978) Arc flow: f R A + Pool qualities: w R P K + Flow polytope: F (D, b) = { f R A + : f respects node caps and flow cons } min f,w (i,j) A c ijf ij f F (D, b) wp k s f sp = s qk s f sp p w p k f pt ut k p f pt p P, k K t T, k K

18 Complexity

19 Complexity Theorem (Alfaki & H. 2012) The Pooling Problem with P = 1 is NP-hard.

20 Complexity Theorem (Alfaki & H. 2012) The Pooling Problem with P = 1 is NP-hard. Reduction: Maximum Independent Set: Given G = (V, E), find V V, E(V ) =, V maximum.

21 Complexity Theorem (Alfaki & H. 2012) The Pooling Problem with P = 1 is NP-hard. Reduction: Maximum Independent Set: Given G = (V, E), find V V, E(V ) =, V maximum. e a b d c

22 Complexity Theorem (Alfaki & H. 2012) The Pooling Problem with P = 1 is NP-hard. Reduction: Maximum Independent Set: Given G = (V, E), find V V, E(V ) =, V maximum. e d a b c q a = (1,0,0,0,0) a b c d e q e =(0,0,0,0,1) l a =(1/5,0,0,0,0) u a =(1,0,1,0,0) a b c d e l e =(0,0,0,0,1/5) u e =(0,1,1,0,1)

23 Complexity Theorem (Alfaki & H. 2012) The Pooling Problem with P = 1 is NP-hard. Reduction: Maximum Independent Set: Given G = (V, E), find V V, E(V ) =, V maximum. e d a b c q a = (1,0,0,0,0) a b c d e q e =(0,0,0,0,1) l a =(1/5,0,0,0,0) u a =(1,0,1,0,0) a (1/2,0,1/2,0,0) b c (1/2,0,1/2,0,0) d e l e =(0,0,0,0,1/5) u e =(0,1,1,0,1)

24 Complexity Theorem The Pooling Problem with K = 1 is NP-hard.

25 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Reduction from Packing in two bins

26 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Reduction from Packing in two bins

27 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Reduction from Packing in two bins

28 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Reduction from Packing in two bins Do fit! 5 5 6

29 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Reduction from Packing in two bins

30 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Reduction from Packing in two bins Do not fit!

31 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Corresponding Pooling Problem network

32 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Corresponding Pooling Problem network and parameter values 1 3 q=0 c=3 4 c=-4 u=0 5 q=1 c=1 5 c=-2 u=1 6 8

33 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Corresponding Pooling Problem network and parameter values 1 3 q=0 c=3 4 c=-4 u=0 5 q=1 c=1 5 c=-2 u=1 6 8

34 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Infeasible flow 1 3 q=0 c=3 4 c=-4 u=0 5 q=1 c=1 5 c=-2 u=1 6 8

35 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Suboptimal flow 1 3 q=0 c=3 4 c=-4 u=0 5 q=1 c=1 5 c=-2 u=1 6 8

36 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Optimum: Each pool satisfies either of the flow patterns

37 Complexity Theorem The Pooling Problem with K = 1 is NP-hard. Optimum: Each pool satisfies either of the flow patterns Full pool capacity utilization Items fit in two bins

38 Complexity Theorem The Pooling Problem with K = 1 and S = T = 2 or S = P = 2 or P = T = 2 is NP-hard.

39 Complexity - Overview NP-hard instance classes of the Pooling Problem Class Hardness Reduction P = 1 Strong Max Independent Set K = 1, S = T = 2 Weak Packing In Two Bins K = 1, S = P = 2 Weak Packing In Two Bins K = 1, P = T = 2 Weak Packing In Two Bins K = 1 Strong Exact Cover By 3-sets δ + (i) 2 Strong Max 2-Satisfiability δ (i) 2 Strong Min 2-Satisfiability

40 Easy versions The Pooling Problem is an LP if S = 1 Remove all terminals requiring better quality than provided by the unique source Solve minimum cost flow problem in the reduced graph T = 1 All flow ends up in the unique terminal t Disregard pool qualities (do not introduce variable wp k ) Apply the quality constraints to the total flow: s qk s p f sp ut k p f pt

41 Easy versions More generally: The Pooling Problem is an LP if p P: min {δ (p), δ + (p)} =

42 Easy versions More generally: The Pooling Problem is an LP if p P: min {δ (p), δ + (p)} = 1 No need for pool quality variables! q1 kf 48 + q1 kf 15 + q2 kf 25 + q3 kf 35 u8 k(f 48 + f 58 )

43 Easy versions More generally: The Pooling Problem is an LP if p P: min {δ (p), δ + (p)} =

44 Possibly easy versions If P = 1 the Pooling Problem can be solved by solving ( T + 1) K LPs, or 2 T LPs Polytime algorithm when K or T is bounded.

45 Possibly easy versions If P = 1 the Pooling Problem can be solved by solving ( T + 1) K LPs, or 2 T LPs Polytime algorithm when K or T is bounded. But be aware of the contrast: Solvable in polynomial time when P = 1 and K = 2 NP-hard when P = 2 and K = 1

46 Strong formulations and global optimization

47 Formulations based on proportion variables Source proportions (Ben-Tal et al. 1994) New variables: Proportion of flow through p coming from s: y s p [0, 1] Flow on path (s, p, t): x spt min f,x,y (i,j) A c ijf ij f F (D, b) x spt = ypf s pt (s, p, t) s y p s = 1 p ( p s q k s ut k ) xspt 0 t, k

48 Formulations based on proportion variables Terminal proportions (Alfaki & H. 2012) New variables: Proportion of flow through p going to t: y t p [0, 1] Flow on path (s, p, t): x spt min f,x,y (i,j) A c ijf ij f F (D, b) x spt = ypf t sp (s, p, t) t y p t = 1 p ( p s q k s ut k ) xspt 0 t, k

49 Formulations based on proportion variables Combined source and terminal proportions (Alfaki & H. 2012) min f,x,y (i,j) A c ijf ij f F (D, b) x spt = ypf s pt = ypf t sp (s, p, t) s y p s = 1 p t y p t = 1 p ( p s q k s ut k ) xspt t, k

50 Nonlinearities: Bilinear terms Quality variables: p w k p f pt u k t Source proportions: x spt = y s pf pt p f pt Terminal proportions: x spt = y t pf sp Fixing w k p, y s p, y t p Linear program in f

51 Lower bounds: Relaxations McCormick s envelopes: Approximate x = fy by (assume y y ȳ and f f f ): x yf + f y yf x ȳf + f y ȳ f x yf + f y y f x ȳf + f y ȳf Valid inequalities: f sp b p yp s (s, p) f pt b p yp t (p, t) f pt = s x spt (p, t) f sp = t x spt (s, p)

52 Lower bounds: Relaxations Comparing proportion formulations Relaxed optimal objective function value, proportion formulations: z S = Only source proportions z T = Only terminal proportions z ST = Both source and terminal proportions Comparison: z S vs z T : Instance dependent z ST max { z S, z T } When pool capacities are tight (b p < min { s b s, t b t}), we can have z ST > max { z S, z T }

53 Global optimization by B&B Using the ST-formulation, BARON (Sahinidis 1996) solves instances where max { S, P, T, K } 8 in < 1 CPU-second K = 1 and max { S, P, T } in a few CPU-seconds, and some instances where K = 24 and max { S, P, T } 20 in 10 CPU-minutes, but for instances where K = 34, S = 35, P = 17, T = 21 the optimality gap is > 50% after 1 CPU-hour.

54 Fast computation: Heuristics

55 Fast computation: Improvement methods Fix y, optimize f (Audet et al. 2004, Kejriwal 2014) Algorithm 1 ImproveFlow f feasible (non-zero) flow vector, mode S repeat if mode = S then Compute source proportions: yp s fsp t fpt For fixed y, let f be the solution to the S-formulation else Compute terminal proportions: yp t fpt s f t p For fixed y, let f be the solution to the T -formulation end if switch mode until no change in f Produces a sequence of flows with non-increasing cost

56 Fast computation: Construction methods Recall: The Pooling Problem with T = 1 is an LP Kejriwal (2014): For all t T, compute z t = min cost in (S, P, {t}). Rank the terminals by increasing z t (t 1,..., t T ) For i = 1,..., T, send flow to t i while preserving qualities at the pools supporting t 1,..., t i 1 Constraint at supporting pools: No quality deterioration Assumption: Quality at supporting pools remains unchanged

57 Fast computation: Construction methods

58 Fast computation: Construction methods Quality constraint

59 Fast computation: Construction methods Quality constraint

60 Fast computation: Construction methods Quality constraint Fixed flow

61 Fast computation: Construction methods Quality constraint Fixed flow

62 Fast computation: Construction methods Quality constraint Fixed flow

63 Fast computation: Construction methods Quality constraint Fixed flow

64 Fast computation: Construction methods Quality constraint Fixed flow

65 Summary The Pooling Problem occurs in network flow models where flow composition must be traced from sources to terminals is NP-hard even if P = 1 or K = 1 is an LP if S = 1, T = 1, or min {δ + (p), δ (p)} = 1 for all p P Global optimization of small-scale instances: linear relaxations of bilinear constraints embedded in spatial B&B-algorithms Fast computations without optimality guarantee: improvement techniques exploiting bilinearity: Sequences of LPs greedy construction techniques: Introduce terminals one by one