DantzigWolfe bound and DantzigWolfe cookbook


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1 DantzigWolfe bound and DantzigWolfe cookbook DTUManagement Technical University of Denmark 1
2 Outline LP strength of the DantzigWolfe The exercise from last week... The DantzigWolfe cookbook 2
3 DantzigWolfe bound strength I have previously stated that the main reason for using DZ is speed. This is not entirely true: The LP bound can be strengthened using DantzigWolfe! 3
4 Example from last time 4 Objective 3 2 A1 A
5 Example from last time: MIP version 4 Objective 3 2 A1 A2 1 Optimum
6 Example from last time: MIP version 4 LP optimum IP optimum Objective 3 2 A1 A2 1 Convex hull optimum
7 Condition for bound improvement So, besides improving the speed of LP solution, we can get an improved bound if: The subproblem (A 2 ) has nonintegral solution property So we have to solve an integer problem which is hard in the subproblem... We may need specialized algorithms... We may be able to improve solution speed if a number of smaller subproblems are created. 7
8 Job planning under budget constraint i: Jobs to perform j: Persons to perform job C i,j : Cost if person j performs job i T i,j : Training cost to train j for job i 8
9 Direct model Min: s.t.: i,j C i,j x i,j i,j x i,j = 1 j i j x i,j = 1 i T i,j x i,j 18 x i,j [0, 1] 9
10 The possibilities of decomposition Which constraint should be removed? The assignment constraints? The subproblem creates full schedules assigning all jobs to all persons The master problem selects combinations of the schedules. The budget constraint??? 10
11 1 Subproblem (only the constraints) Min: c r s =??? s.t.: x i,j = 1 j i x i,j {0,1} Notice: α 0 and β R j x i,j = 1 i 11
12 1 Master problem Min: C s y s s.t.: T s y s 18 (α 0) s s y s = 1 (β R) s y s [0, 1] 12
13 1 Subproblem Min: c r s = i,j C i,j x i,j α i,j T i,j x i,j β = i,j (C i,j α T i,j )x i,j β s.t.: x i,j = 1 j i j x i,j {0,1} x i,j = 1 i Notice: α 0 and β R 13
14 2 decomposition The budget constraint: The schedules are now job selections which obeys the training budget constraint. The master problem job is then to mix (fractionally) the different schedules The subproblem is then a knapsack problem... 14
15 2 Subproblem (without objective function) Min: c i,j =??? s.t.: T i,j x i,j 18 i,j x i,j {0,1} Notice: α j R and β i R 15
16 2 Master problem Min: C s y s s.t.: A s jy s = 1 j (α j R) s s s B s iy s = 1 i (β i R) y s = 1 (γ R) s y s [0, 1] 16
17 2 Subproblem Min: c i,j = C i,j x i,j i,j j α j β i x i,j γ x i,j i i j = i,j (C i,j α j β i )x i,j γ s.t.: T i,j x i,j 18 i,j x i,j {0,1} Notice: α j R and β i R 17
18 DantzigWolfe cookbook Write up the original MIP problem (carefully!) Write up dual variables for each type of constraints and set dual variable bounds Move set of constraints (A 2 ) to subproblem (ignore the subproblem objective for now). WHAT IS THE SUBPROBLEM INTUITIVELY??? What does one solution correspond to? 18
19 DantzigWolfe cookbook Write up the master problem, with the new variables, both constraints and objective function Finally, write up the subproblem objective: minc red =(c π A 1 )x α, i.e. the original cost minus the column times the dual variables. Open question: How should the subproblem be solved? 19
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