The Target Visitation Problem Achim Hildenbrandt 1 Olga Heismann 2 Gerhard Reinelt 1 1 Ruprecht-Karls-Universit¨ at Heidelberg 2 Zuse-Institut Berlin January 09, 2014 A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 1 / 25
Introduction 1 IP Model 2 Polyhedral Combinatorics 3 Extended Formulation 4 A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 2 / 25
Introduction The Problem A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 3 / 25
Introduction The Problem A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 4 / 25
Introduction The Problem For each two targets i and j we have a preference value p i , j to visit i some time before j . For each two targets i and j we have cost d i , j to travel from i to j . A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 5 / 25
Introduction The Problem For each two targets i and j we have a preference value p i , j to visit i some time before j . For each two targets i and j we have cost d i , j to travel from i to j . The objective is to find a tour which is optimal in thr sense that the sum of the met preferences (denoted by P) minus the distance (denote by D) cost is maximal. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 5 / 25
Introduction The Problem For each two targets i and j we have a preference value p i , j to visit i some time before j . For each two targets i and j we have cost d i , j to travel from i to j . The objective is to find a tour which is optimal in thr sense that the sum of the met preferences (denoted by P) minus the distance (denote by D) cost is maximal. Combination of the Traveling Salesman Problem (TSP) and the Linear Ordering Problem (LOP) or, in other words, a TSP with an additional preference matrix. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 5 / 25
Introduction The Problem For each two targets i and j we have a preference value p i , j to visit i some time before j . For each two targets i and j we have cost d i , j to travel from i to j . The objective is to find a tour which is optimal in thr sense that the sum of the met preferences (denoted by P) minus the distance (denote by D) cost is maximal. Combination of the Traveling Salesman Problem (TSP) and the Linear Ordering Problem (LOP) or, in other words, a TSP with an additional preference matrix. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 5 / 25
Introduction Applications Planning of missions in disaster areas to distribute food and medicine Snow clearance Town cleaning In general: positioning problems with additional preferences A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 6 / 25
Introduction Some Facts: Fairly new problem with few research results Grundel, Jeffcoat: Formulation and solution of the target visitation problem, 2004 A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 7 / 25
Introduction Some Facts: Fairly new problem with few research results Grundel, Jeffcoat: Formulation and solution of the target visitation problem, 2004 Arulselvan, Commander, Pardalos: A hybrid genetic algorithm for the target visitation problem, 2007 A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 7 / 25
Introduction Some Facts: Fairly new problem with few research results Grundel, Jeffcoat: Formulation and solution of the target visitation problem, 2004 Arulselvan, Commander, Pardalos: A hybrid genetic algorithm for the target visitation problem, 2007 No approach for solving the TVP to optimality has been implemented so far A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 7 / 25
Introduction Some Facts: Fairly new problem with few research results Grundel, Jeffcoat: Formulation and solution of the target visitation problem, 2004 Arulselvan, Commander, Pardalos: A hybrid genetic algorithm for the target visitation problem, 2007 No approach for solving the TVP to optimality has been implemented so far NP-hard A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 7 / 25
Introduction Some Facts: Fairly new problem with few research results Grundel, Jeffcoat: Formulation and solution of the target visitation problem, 2004 Arulselvan, Commander, Pardalos: A hybrid genetic algorithm for the target visitation problem, 2007 No approach for solving the TVP to optimality has been implemented so far NP-hard The problem is also hard in practice. Special methods are needed. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 7 / 25
Introduction Some Facts: Fairly new problem with few research results Grundel, Jeffcoat: Formulation and solution of the target visitation problem, 2004 Arulselvan, Commander, Pardalos: A hybrid genetic algorithm for the target visitation problem, 2007 No approach for solving the TVP to optimality has been implemented so far NP-hard The problem is also hard in practice. Special methods are needed. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 7 / 25
IP Model Variables: TSP Variables: 1 if ∃ k ∈ { 0 , n − 2 } so that i = π ( k ) and j = π ( k + 1) x i , j := 1 if i = π ( n − 1) and j = π (0) 0 otherwise LOP Variables: � 1 if ∃ k ∈ { 1 , n − 2 } so that i = π ( k ) and j = π ( l ) with k < l w i , j := 0 otherwise A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 8 / 25
IP Model Removal of the base node: Since the base node is always the first node in our tour we can remove it by adjusting the distance matrix and objective function in the following way. Adjust the distance matrix as follows: d ′ ij = d ij − d i 0 − d 0 j i , j ∈ { 1 . . . n } Adjust the objective function as follows: max � n � n p i , j w i , j − � n � n i , j x i , j − � n i =1 d i 0 − � n d ′ i =1 d 0 i j =1 j =1 i =1 i =1 i � = j i � = j The TVP is now a combination of the LOP and the Hamiltonian Path problem. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 9 / 25
IP Model IP model: max � n � n p i , j w i , j − � n � n i , j x i , j − � n i =1 d i 0 − � n d ′ i =1 d 0 i j =1 j =1 i =1 i =1 i � = j i � = j subject to n n � � x i , j = n − 1 , i =1 j =1 , i � = j n � x i , j ≤ 1 , j ∈ N i =1 n � x i , j ≤ 1 , i ∈ N j =1 � � x i , j ≤ | S | − 1 , ∀ S ⊂ V , 2 ≤ | S | ≤ n i ∈ S j ∈ S w i , j + w j , k + w k , i ≤ 2 , i , j , k ∈ N w i , j + w j , i = 1 , i , j ∈ N x i , j ≤ w i , j , i , j ∈ N x i , j , w i , j ∈ { 0 , 1 } , i , j ∈ N A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 10 / 25
Polyhedral Combinatorics Polyhedral Results The dimension of the TVP HP polytope is: 3 n 2 − 3 n − 2 for n ≥ 4 2 That means there exist no more equations then the ones we already have in the model The examination of the polytope for n = 4 yields 1280 facets in 46 classes. We were able to generalize some classes. For n = 5 there are more than 100 Million classes of facets. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 11 / 25
Polyhedral Combinatorics Lifting Results Node i is calles a free node of a facet if all x i , j , x j , i , w i , j , w j , i are zero. Theorem Let ax ≤ a 0 define a facet for the TVP HP (n) with at least one free node. Then the zero lifting of ax ≤ a 0 defines a facet for the TVP HP (k) with k ≥ n . With this theorem we can prove that 12 classes of the 46 classes are facets of TVP HP ( n ). A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 12 / 25
Polyhedral Combinatorics Name Facet C2 w il + w kl + w lj + x ji + x jk + x jl + x li + x lk ≤ 3 w il + w kj + w lk + x ji + x jk + x jl + x kl + x li ≤ 3 C7 w ij + 2 w jk + 2 w kl + w li + w lj + x il + x ji + x jl + x kj + x lk ≤ 5 C11 − x ki ≤ 0 C13 w il + w ji + w jk + w kl + w lj + x ij + x kj + x li + x lj + x lk ≤ 4 C14 w ij + 2 w jk + w ki + w kl + w lj + x ji + x jl + x kj ≤ 4 C20 w ij + w il + 2 w jk + w ki + w lj + x ji + x jl + x ki + 2 x kj + x kl + x li ≤ 5 C25 w il + w jk + w ki + w lj + x jl + x kj + x kl ≤ 3 C29 w jk + w kl + w lj + x kj ≤ 2 C30 2 w jk + 2 w kl + 2 w lj + x jl + x kj + x lk ≤ 4 C39 w ij + 2 w jk + w kl + x ji + x ki + 2 x kj + x li + x lj + x lk ≤ 4 C41 w jk + w kl + x kj + x lj + x lk ≤ 2 C46 w jk + x kj ≤ 1 Table : Some facets of the TVP HP (4) polytope for which zero lifting is possible A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 13 / 25
Polyhedral Combinatorics Observations The class of the extended three cycle inequalities w i , j + w j , k + w k , i + x j , i ≤ 2 can replace the normal three cycle inequalities. The extended three cycles imply the subtour elimination constraints. A. Hildenbrandt (Heidelberg) An extended formulation for the TVP January 09, 2014 14 / 25
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