### Table 3: Qualitative effects of abstraction: general trends.

2006

"... In PAGE 16: ... We arbitrar- ily choose three diverse LRTS parameter combinations of lookahead d, optimality weight , and learning quota T : (1; 1:0; 1); (3; 0:2; 100); (9; 0:4; 1). The plots are in Figure 14, a qualitative summary is in Table3 , and an analysis of the trends is below. Convergence planning decreases with abstraction level.... ..."

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### Table 3: Qualitative effects of abstraction: general trends.

2006

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### Table 2Lagrangian and Linear Programming Relaxations Lagrangian LB Strong LP

2000

"... In PAGE 12: ... For these instances, the Lagrangian approach requires a maximal computation time of less than 9 minutes (537 seconds), which is in sharp contrast to the 72 hours needed by Brucker and Knust (2000). Table2 compares the Lagrangian approach to the corresponding linear programming relaxation (2), (3), (4), (8), and (9), again based on the ProGen instances Table 1 Comparison of Quality and Computation Times for Lower Bounds Lagrangian LB+(14) Brucker and Knust # jobs # inst. Dev.... In PAGE 13: ... It turns out that the primal simplex method solves this linear programming relaxation much faster than the barrier method does. With the barrier method, the instances with 120 jobs could not be solved in reason- able time, hence the data is missing in Table2 . More importantly, these computation times are, in fact, dras- tically higher than the computation times required to (approximately) solve the Lagrangian dual.... ..."

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### Table 7. Linear programming relaxation

in Statistical Disclosure Limitation with Released Marginals and Conditionals for Contingency Tables

### Table 2. Relaxed planning graph for the example problem.

### Table 1. Computation times for solving the linear relaxation.

"... In PAGE 3: ... Computation times for solving the linear programming relaxation by different algorithms available with the CPLEX 8.0 package are given in Table1 . The problem is very degenerated and requires the use of perturbations, leading to large computation times.... In PAGE 3: ... The interior point algorithm was the best solution strategy. We also can see in Table1 that the new formulation considerably reduced the computation time of the interior points algorithm, making possible an efficient implementation of the cutting plane algorithm. Table 1.... ..."

### Table 2: Linear Fitness Combination

"... In PAGE 16: ... This technique can be used in tness proportional, tournament, or rank-based selection. Table2 lists the known linear combinatoric MOEA techniques. Table 2: Linear Fitness Combination... In PAGE 17: ...A PRIORI TECHNIQUES 3.3 Nonlinear Fitness Combination Techniques Table2 : continued Approach Description Application Objectives (#) Chromosome Multi-Sexual GA [92] (1997) Individuals are \sex quot; coded (one for each function); Recombina- tion uses one parent from each sex; Individ- uals evaluated by their sex apos;s function None (2) Numeric opti- mization Binary string; \Sex quot; marker at end GA [149] (1998) Integrated two GAs with electromagnetic evaluation code; Fit- ness mapping (scaling) Wire antenna ge- ometry design (4) Antenna gain; Radiation symme- try; Resistance; Re- actance Real values; Gene triplets represent wire endpoints in 3- D space GA [104] (1998) Weights are functions of objective functions apos; max and min values yet found Computer aided process planning (2) Processing and transportation time; Workstation load variation Integer string; Genes are plans producing certain parts GA [164] (1998) Steady-state GA; Specialized EVOPs and population re- initialization Telephone operator scheduling (2) Operator short- age; Operator sur- plus Integer string; Genes are par- tial schedules composed of shift time, and number and time of rest breaks GA [23] (1998) Specialized crossover; 3 4 population: tour- nament selection, 1 4 roulette wheel and tness scaling Non-chromatic rectangle boards (4) Distribution of colors; # Red, white, and blue chromatic rectan- gles 2-D array of integer values; Genes are colored squares 3.3 Nonlinear Fitness Combination Techniques Nonlinear tness combination is also a scalar aggregative method; several EA- based variants have been implemented.... In PAGE 35: ... This is probably due to its simplicity. Table2 re ects its application to many real-world problems, although often incorporated with \vari- ations on a theme. quot; The basic weighted sum MOEA is both easy to understand and implement; the technique is also computationally e cient.... In PAGE 70: ...MOP TEST FUNCTIONS 10.3 MOEA Experimental Methodology Table2 0: Possible Multiobjective NP -Complete Functions NP -Complete Problem Examples 0/1 Knapsack - Bin Packing Max pro t; Min weight (Multiple Knapsacks [167]) Coloring Min # colors, # of each color Layout Min space, overlap, costs Maximum Independent Set (Clique) Max set size; Min geometry Scheduling Min time, missed deadlines, waiting time, resources used Set/Vertex Covering Min total cost, over-covering Traveling Salesperson Min energy, time, and/or distance; Max expansion Vehicle Routing Min time, energy, and/or geometry NP -Complete Problem Combinations Vehicle scheduling and routing 10.3.... ..."

### Table 1. Computation times for solving the linear relaxation.

"... In PAGE 3: ... Computation times for solving the linear programming relaxation by different algorithms available with the CPLEX 8.0 package are given in Table1 . The problem is very degenerated and requires the use of perturbations, leading to large computation times.... In PAGE 3: ... The interior point algorithm was the best solution strategy. We also can see in Table1 that the new formulation considerably reduced the computation time of the interior points algorithm, making possible an efficient implementation of the cutting plane algorithm. Results obtained with algorithms B amp;B-ANFP and B amp;C-ANFP are given in Ta- ble 2.... ..."

### Table 1. Comparisons of the relaxations for example 3. Relaxation

2002

"... In PAGE 14: ... For this example, only one cutting plane yields the same tightness of the relaxation as the convex hull. The numerical results are shown in Table1 . Note that the big-M relaxation yields the lowest objective value to the optimal solution, 4.... In PAGE 14: ... As shown in Figure 15, the cutting plane is a facet of the convex hull. From Table1 it can be seen that the big-M relaxation with a cutting plane yields a competitive relaxation compared with the convex hull. Cutting planes in x-y space: example 2 Let us revisit example 2.... In PAGE 26: ...List of Tables Table1 . Comparisons of the relaxations for example 3.... ..."

### Table 1. Relaxed propositional heuristic.

2003

"... In PAGE 10: ... Solving relaxed plans can efficiently be done by building a relaxed problem graph, followed by a greedy plan generation process. Table1 depicts the implementation of the plan construction and the solution extraction phase. The first phase constructs the layered plan graph identical to the first phase of Graphplan [2].... ..."

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