Classification of Dantzig-Wolfe Reformulations for MIP s

Transcription

1 Classification of Dantzig-Wolfe Reformulations for MIP s Raf Jans Rotterdam School of Management HEC Montreal Workshop on Column Generation Aussois, June 2008

2 Outline and Motivation Dantzig-Wolfe reformulation for lotsizing CLSP (big bucket) : we cannot put binary conditions on the new columns. CSLP (small bucket): we can put binary conditions on the new columns Question: In which cases is it (not) allowed to put binary conditions on the new master variables Classification for Binary Mixed Integer Programs 2

3 Lot Sizing Problems Lot Sizing = determine the timing and level of production Production planning in a deterministic environment with a finite time horizon A classic OR problem (Wagner and Whitin 1958) Core substructure in many industrial planning problems A small example: Period Item 1 Demand Production Inventory Item 2 3 Item 3

4 4 The Uncapacitated Lot Sizing Problem

5 The Uncapacitated Lot Sizing Problem: A fixed charge network problem In an uncapacitated network with one source, the extreme flow can have at most one positive input in each node. (Zangwill 1969) 5 This is exactly the Wagner-Whitin property (1958): s x = 0 t 1 t

6 Capacitated Lotsizing with Setup Times Minimize total cost Min Satisfy demand ( sci yit + vci xit + hci sit ) i P t T s + x = d + i, t 1 it it s it i P, t T Set up forcing it { cap t st i sd it } y it x min, i P, t T 6 Limited capacity Integrality i P ( sti yit + vti xit ) capt s, 0 0 ; s, = 0; x, s 0; y i = i m it it it { 0,1 } t T

7 7 Capacitated Lot Sizing: Literature Review

8 Dantzig-Wolfe Decomposition Min Total Cost Item 1: Demand Set Up Item 2: Demand Set Up Item 3: Demand Set Up Capacity 8

9 CLST : Decomposition and CG approach Master Minimize total cost Min i P q Q Convexity constraint z = 1 iq q Q i Capacity constraint i P q Q r iqt Non-negativity z iq 0 i i z c iq iq z iq cap Structural Problem with textbook decomposition t μ i π t Subproblem Minimize reduced cost Min t T Satisfy demand Set up forcing Integrality ( sc y + vc x + hc s ) μ + ( st y + vt x )π i t T Wagner Whitin (1958): i it A solution (= column) consists of both a set up and production quantity decision i i it it i i it it s t 1 xt = 0 t 9

10 CLST : Key Observation Every feasible production schedule for a specific setup schedule can be written as a convex combination of the Wagner-Whitin production schedules, associated with the subset of the setup. Set up Set up Subset Production Subset Schedule S1 S2 S3 S4 P1 P2 P3 P4 Period d 1 +d 2 +d 3 0 d 1 +d 2 0 Period d 1 +d 2 +d 3 0 d 1 +d 2 Period Period d 3 d 3 10

11 CLST : New Extreme Point Formulation Minimize total cost Min cs zs + ij ij i P j S i P j S k S j cp ik zp ijk 11 Select one setup proposal j S Relationship setup and production Capacity constraint Integrality zs = 1 ij st zs ij = zp ijk k S j ss zs + i ijt ij i P j S i P j S k S zs ij { 0,1 } j vt i ps ikt zp ijk cap t i P i P, j S t T i P, j S

12 Another lotsizing problem Continuous Setup Lotsizing Problem Small bucket problem Single mode constraint Decomposition: single mode constraint is the complicating constraint (Vanderbeck 1998) We can put binary conditions on the new column variables to obtain an equivalent reformulation 12

13 Central question In which cases can we (not) put integrality constraints on the new master variables? 13

14 14 Generalizability: DW reformulations for MIP s

15 15 Some Notation

16 Convexification approach for MIP s In the convexification approach we keep the binary restrictions on the original variables (Barnhart et al. 1998, Vanderbeck 2000). The feasible MIP region can then be rewritten as follows: 16

17 17 Convexification approach

18 18 Generalizability: DW reformulations for MIP s

19 19 Overview of different BMIP models

20 20 Case 1: Generalized Linear Programming

21 21 Cases 2 and 4: independent subproblems

22 22 Case 7: No binary variables in the subproblem

23 23 Case 5: The pure binary problem

24 Case 5: The pure binary problem The binary constraint on the original variables can be replaced by binary conditions on the new master variables: There are no interior points and hence: 24

25 25 Case 8: Johnson (1989)

26 26 The interesting cases: BMIP subproblem

27 27 Case 3 and 9: The capacitated Lotsizing Problem (with setup times)

28 The capacitated Plant Location Problem The capacitated plant location problem s.t. Min j M c x + ij ij i N j M j M f j y j x = 1 i N ij dixij i N s j y j j M x i N, j M ij y j { 0,1 }, x 0 y i N, j M i ij 28

29 The capacitated Plant Location Problem One possible decomposition is to leave the capacity constraints in the master. The resulting subproblem is the Simple Plant Location Problem, which is NP-hard (Krarup and Pruzan 1983). An optimal solution to this simple Plant Location Problem has the Single Assignment Property. In the optimal solution for the capacitated case, however, it is highly likely that demand for some customers will be supplied from more than one open plant. By putting integrality constraints on the columns generated by the subproblem, we can never attain such a split supply. In this case, we will need to apply a similar approach as with the CLST, namely a separation of the binary location decision and the fractional supply decision. (Klose and Götz, 2007). 29

30 30 Case 6: CSLP Vehicle Routing with Time Windows

31 Case 6: Hence, we can impose the binary conditions directly on the new master variables. 31

32 32 Overview of different BMIP models

33 Remarks 1. If general integer variables are present, the optimum solution to the original MIP may be an interior point of conv(x s ). Imposing integrality constraints on the new master variables will hence not give an equivalent formulation. 2. Model formulation versus algorithmic solution approach: Even if you can impose integrality on the new master variables, this is probably not an efficient branching strategy because of the unbalanced tree and the possible difficulties in the pricing problem. 33

34 Conclusions Focus on correct reformulations, not on algorithms. Systematic analysis of the effect of DW-decomposition on the binary variables. General overview of when it is allowed to put integrality constraints on the new master variables and when not. 34