The Fixed Charge Transportation Problem: A Strong Formulation Based - PowerPoint PPT Presentation

Introduction and background The fixed charge transportation problem Concluding comments The Fixed Charge Transportation Problem: A Strong Formulation Based On Lagrangian Decomposition and Column Generation Yixin Zhao, Torbj¨ orn Larsson and Elina R¨ onnberg Department of Mathematics, Link¨ oping University, Sweden Column generation 2016 Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments What is this talk about? Strong lower bounding for the fixed charge transportation problem by ◮ Lagrangian decomposition: supply and demand side copies of the shipping variables ◮ Dual cutting plane method (column generation) Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments What is this talk about? Strong lower bounding for the fixed charge transportation problem by ◮ Lagrangian decomposition: supply and demand side copies of the shipping variables ◮ Dual cutting plane method (column generation) Why? ◮ Lagrangian decomposition can give strong formulations ◮ Strong formulations of interest in column generation ◮ Combination not utilised in many papers ◮ Preliminary work to study the strength of the formulation: theoretically and empirically Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Outline Introduction and background The fixed charge transportation problem Concluding comments Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Lagrangian decomposition / Lagrangian relaxation Consider the problem (P) min { cx s.t. Ax ≤ b , Cx ≤ d , x ∈ X } = min { cx s.t. Ay ≤ b , Cx ≤ d , x = y , x ∈ X , y ∈ Y } , where Y is such that X ⊆ Y Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Lagrangian decomposition / Lagrangian relaxation Consider the problem (P) min { cx s.t. Ax ≤ b , Cx ≤ d , x ∈ X } = min { cx s.t. Ay ≤ b , Cx ≤ d , x = y , x ∈ X , y ∈ Y } , where Y is such that X ⊆ Y Lagrangian decomposition (LD) min { cx + u ( x − y ) s.t. Ay ≤ b , Cx ≤ d , x ∈ X , y ∈ Y } = min { ( c + u ) x s.t. Cx ≤ d , x ∈ X } + min {− uy s.t. Ay ≤ b , y ∈ Y } Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Lagrangian decomposition / Lagrangian relaxation Consider the problem (P) min { cx s.t. Ax ≤ b , Cx ≤ d , x ∈ X } = min { cx s.t. Ay ≤ b , Cx ≤ d , x = y , x ∈ X , y ∈ Y } , where Y is such that X ⊆ Y Lagrangian decomposition (LD) min { cx + u ( x − y ) s.t. Ay ≤ b , Cx ≤ d , x ∈ X , y ∈ Y } = min { ( c + u ) x s.t. Cx ≤ d , x ∈ X } + min {− uy s.t. Ay ≤ b , y ∈ Y } Lagrangian relaxation w.r.t. one of the constraint groups, for example Ax ≤ b (LR) min { cx + v ( Ax − b ) s.t. Cx ≤ d , x ∈ X } = min { ( c + vA ) x s.t. Cx ≤ d , x ∈ X } + vb , where v ≥ 0 Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Strength of bounds [ Guignard and Kim(1987) ]: The Lagrangian decomposition bound is as least as strong as the strongest of ◮ the Lagrangian relaxation bound when Ax ≤ b is relaxed ◮ the Lagrangian relaxation bound when Cx ≤ d is relaxed and there is a chance that it is stronger! Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Related work Very few papers on Lagrangian decomposition and column generation: ◮ [ Pimentel et al.(2010) ]: The multi-item capacitated lot sizing problem − Branch-and-price implementations for two types of Lagrangian relaxation and for Lagrangian decomposition − Lagrangian decomposition: No gain in bound compared to Lagrangian relaxation when capacity is relaxed ◮ [ Letocart et al.(2012) ]: The 0-1 bi-dimensional knapsack problem and the generalised assignment problem − Illustrates the concept − No full comparison of bounds, conclusions not possible Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Related work Very few papers on Lagrangian decomposition and column generation: ◮ [ Pimentel et al.(2010) ]: The multi-item capacitated lot sizing problem − Branch-and-price implementations for two types of Lagrangian relaxation and for Lagrangian decomposition − Lagrangian decomposition: No gain in bound compared to Lagrangian relaxation when capacity is relaxed ◮ [ Letocart et al.(2012) ]: The 0-1 bi-dimensional knapsack problem and the generalised assignment problem − Illustrates the concept − No full comparison of bounds, conclusions not possible Our work this far: Find an application where we gain in strength compared to the strongest obtainable from Lagrangian relaxation and investigate further ... Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Fixed charge transportation problem (FCTP) For each arc ( i , j ), i ∈ I , j ∈ J : u ij = min( s i , d j ) = upper bound c ij = unit cost for shipping f ij = fixed cost for shipping Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Fixed charge transportation problem (FCTP) For each arc ( i , j ), i ∈ I , j ∈ J : u ij = min( s i , d j ) = upper bound c ij = unit cost for shipping f ij = fixed cost for shipping Variables: x ij = amount shipped from source i to sink j , i ∈ I , j ∈ J Concave cost function: � f ij + c ij x ij if x ij > 0 g ij ( x ij ) = i ∈ I , j ∈ J 0 if x ij = 0 Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments Fixed charge transportation problem (FCTP) � � min g ij ( x ij ) i ∈ I j ∈ J � s.t. x ij = s i i ∈ I j ∈ J � x ij = d j j ∈ J i ∈ I x ij ≥ 0 i ∈ I , j ∈ J ◮ Polytope of feasible solutions, minimisation of concave objective ⇒ Optimal solution at an extreme point (can be non-global local optima at extreme points) ◮ MIP-formulation: A binary variable to indicate if there is flow on an arc or not Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments MIP-formulation of FCTP � � � � min c ij x ij + f ij y ij i ∈ I j ∈ J � s.t. x ij = s i i ∈ I j ∈ J � x ij = d j j ∈ J i ∈ I x ij ≤ u ij y ij i ∈ I , j ∈ J x ij ≥ 0 i ∈ I , j ∈ J y ij ∈ { 0 , 1 } i ∈ I , j ∈ J Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments The reformulation: variable splitting Supply and demand side duplicates of the shipping variables: x s ij and x d ij Introduce a parameter ν : 0 ≤ ν ≤ 1 � � � � g ij ( x s g ij ( x d min ν ij ) + (1 − ν ) ij ) i ∈ I j ∈ J j ∈ J i ∈ I x s ij = x d s.t. i ∈ I , j ∈ J ij � x s ij = s i i ∈ I j ∈ J � x d ij = d j j ∈ J i ∈ I x s ij , x d ij ≥ 0 i ∈ I , j ∈ J , Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments The reformulation: inner representation Let each column correspond to an extreme point of a set X s i = { x s j ∈ J x s ij = s i , 0 ≤ x s ij , j ∈ J | � ij ≤ u ij , j ∈ J } , i ∈ I , or of a set X d j = { x d i ∈ I x d ij = d j , 0 ≤ x d ij , i ∈ I | � ij ≤ u ij , i ∈ I } , j ∈ J The flow from one source / to one sink is a convex combination of extreme point flows, introduce: ip = convexity weight for extreme point p ∈ ˜ λ s P s i of set X s i , i ∈ I and jp = convexity weight for extreme point p ∈ ˜ λ d j of set X d P d j , j ∈ J Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments The reformulation: column oriented formulation     � � � � � � λ s ip x s  λ d jp x d  min ν g ij  + (1 − ν ) g ij   ijp  ijp     i ∈ I j ∈ J p ∈ ˜ j ∈ J i ∈ I p ∈ ˜ P s P d i j � � x s ijp λ s x d ijp λ d s.t. ip = i ∈ I , j ∈ J jp p ∈ ˜ P s p ∈ ˜ P d i j � λ s ip = 1 i ∈ I p ∈ ˜ P s i � λ d jp = 1 j ∈ J p ∈ ˜ P d j p ∈ ˜ λ s P s ip ≥ 0 i , i ∈ I p ∈ ˜ λ d P d jp ≥ 0 j , j ∈ J Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments The reformulation: approximating the objective The objective is bound below by its linearisation:     � � � � � � λ s ip x s  λ d jp x d  ν g ij  + (1 − ν ) g ij   ijp  ijp     i ∈ I j ∈ J p ∈ ˜ j ∈ J i ∈ I p ∈ ˜ P s P d i j     � � � � � � g ij ( x s  λ s g ij ( x d  λ d ≥ ν ijp ) ip + (1 − ν ) ijp ) jp i ∈ I p ∈ ˜ j ∈ J j ∈ J p ∈ ˜ i ∈ I P s P d i j What is lost by the linearisation? Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments The reformulation: arc cost True cost Elina R¨ onnberg

Introduction and background The fixed charge transportation problem Concluding comments The reformulation: arc cost True cost In LP-relaxation of MIP-formulation Elina R¨ onnberg