A Lagrangian bound for many-to-many assignment problems - Additional Results

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1 Noname manuscript No. (will be inserted by the editor) A Lagrangian bound for many-to-many assignment problems - Additional Results Igor Litvinchev Socorro Rangel Jania Saucedo May, 2008 / last updated: November, 2008 Abstract A simple procedure to tighten the Lagrangian bounds is proposed. The approach is interpreted in two ways. First, it can be seen as a reformulation of the original problem aimed to split the resulting Lagrangian problem into two subproblems. Second, it can be considered as a search for a tighter estimation of the penalty term arising in the Lagrangian problem. The new bounds are illustrated by a small example and studied numerically for a class of the generalized assignment problems. Keywords Lagrangian bounds, integer programming, many-to-many-assignment problem Test bed and results The objective of the numerical study is to compare the relative quality of the bounds presented in Litvinchev et al. (2008b) as well as their proximity to the optimal objective. We numerically compare the Lagrangian bounds, standard and modified, for four sets of instances of (MMAP): small instances with sizes m n for m {5, 8, 10} and n = 50 (Two sets, Set1 and Set2), and large instances with m {5, 10, 20} and n = 100 (Two sets, Set3 and Set4). The data were random integers generated as follows: with: c ij U[10, 50], a ij U[5, 25], d ij U[3, 20] b i = α( a ij 1), d j = α( d ij 1), 0 < α 1 j and divided in three classes (a, b, and c) with respect to the values of α: a (α = 1), b (α = 0.9), c (α = 0.8) (Martelo and Toth, 1990). Igor S. Litvinchev and Jania Saucedo Department of Mechanical and Electrical Engineering, UANL, Mexico address: igorlitvinchev@gmail.com, jania@yalma.fime.uanl.mx Socorro Rangel UNESP - São Paulo State University Rua Cristóvão Colombo, 2265, S. J. do Rio Preto, Brazil address: socorro@ibilce.unesp.br, Phone: j

2 2 All Lagrangian-type bounds were calculated by the subgradient method. We used K = 5 and if (ϕ k ϕ k+1 )/ϕ min δ(= 0.005) for 5 consecutive iterations with fixed ε k, this parameter was modified to ε k = ε k /2. Here ϕ min is the best current objective value. The optimization models and the algorithms were coded in the syntax of the modeling language AMPL (Fourer et al., 1993). All optimization subproblems associated with subgradient algorithm were solved by the system CPLEX 10.0 (ILOG, 2001). The runs were executed on a machine AMD Athlon 64X2 Dual Core, 2.8Ghz and 2048MB RAM. For all problem instances we have calculated: - optimal objective of the original integer problem, z lp - optimal objective of the LP relaxation, z lag - classical Lagrangian bound w D, z md - modified Lagrangian bound w L MD(W 1 ) calculated for π = 0, z mdπ - modified Lagrangian bound w L MD(W 1 ) thus obtaining four upper bounds for. The relative quality of the bounds was measured by: rel0 = z mdπ z md rel1 = z md z lag rel2 = z md z lp and rel3 = z lag z lp where each indicator compares two subsequent (in terms of number of multipliers used) bounds. Here rel0 indicates improvement obtained by introducing the objective copy constraint (cx = cy), rel1 represents improvement of the modified bound with π = 0 over classical, rel2 shows the strength of the modified bound over LP relaxation, and rel3 compares the quality of classical bound with LP-bound. The proximity to the optimal integer solution was represented by: gap0 = z mdπ gap1 = z md gap2 = z lag 100% and gap3 = z lp 100%. The results for the small instances are reported in Table 1a (Set1 - s1*.dat) and Table 1b (Set2 - s2*.dat), while Table 2a and 2b presents results the larger instances (Set3 and Set4, respectively). For each problem instance it is shown the values of m,n, its class (cl = a,b, or c), the relative quality of the bounds (rel0-rel3), and the proximity to the optimal integer solution (gap0-gap3). The instances in Set1 and Set3 were used in the computational study presented in Livinchev et al. (2008a, 2008b).

3 3 Table 1a - Relative quality of the bounds - Results for small problems - Set a b c a b c a b c Table 1b - Relative quality of the bounds - Results for small problems - Set a b c a b c a b c

4 4 Table 2a - Relative quality of the bounds - Results for large problems - Set a b c a b c a b c Table 2b - Relative quality of the bounds - Results for large problems - Set a b c a b c The solution of the Lagrangian dual problem, classical or modified, is frequently used as a starting or reference point to produce a feasible suboptimal solution by various heuristics and approximate techniques. Therefore, it is often of interest to see the feasibility and suboptimality characteristics of the Lagrangian solutions. In our case we have three Lagrangian-type bounds and Tables 3a, 3b, 4a and 4b show the constraints violations for x and y corresponding to the bounds, z lag, z md, and z mdπ, obtained within the given time limit. The following indicators were used: v rel(σ) = max{0, max i,j {( m i=1 d ijσ ij d j )/d j, ( n j=1 a ijσ ij b i )/b i }}, gap4(σ) = m i=1 n j=1 c ijσ ij 100% and nv(σ) for the number of violated constraints.

5 5 Table 3a - Constraints violation - Results for small problems - Set a b c a b c a b c Table 3b - Constraints violation - Results for small problems - Set a b c a b c a b c

6 6 Table 4a - Constraints violation - Results for large problems - Set a b c a b c a b c Table 4b - Constraints violation - Results for large problems - Set a b c a b c References I. Litvinchev, S. Rangel, J. Saucedo, Modifying Lagrangian bounds, Graduate Program in Systems Engineering, UANL, Mexico, Technical Report , 2008a. I. Litvinchev, S. Rangel, J. Saucedo, A Lagrangian bound for many-to-many assignment problems, 2008b.