2.2.1 “Programa Cultura”
2.3 Consumo cultural en México
To examine whether the current approach reduces the solution time we conducted a series of simulations. To define the utility and cost values, we generate 100 random samples with a uniform distribution in the range
1 − 10 and no correlation between the utility and cost. Then we take a fraction of the summed cost of all nodes and use this as the given budget, e. g. 40% of total price. The following table (Table 5.3) shows more detail about the simulation procedure. Some notation used in the table needs to be explained. When a generated sample, e. g. instanci, is solved originally without any additional condition, by a given start node, or by a start solution, we call it Solve Original, Solve StartP oint, or Solve Initial, respectively.
5.4 Results
The main goal of this work is to show how our proposed model can be suit-able for large problems by attempting to reduce the solution time. The ex-perimental results show the proposed approach, solving the problem using a carefully chosen initial solution, is remarkably successful. It significantly reduces the solution time on average in the range of 38%− 51% (see Table 5.4). We summarize the solution time results of these hundred instances for three different methods in Table 5.4 and in figure 5.3.
To graphically represent these results and to compare the three different
Table 5.3: Simulation Procedure: Initialization of a solution to save running time For each case of finding M ax j(), j ={RB, NB, V B}
For i = 1,· · · , MaxIterations Let n be a given number of nodes
Generate a uniformly random instancei(U tility[n], Cost[n]) Find Sum(Cost[n])
Compute Budget = P ercentage∗ Sum, for a given P ercentage Solve Original(instancei, Budget)
Find M ax j(U tility[n]) and replace as a Start Point
Solve StartP oint(instancei, Budget); then W rite(Solution) Read(Solution); then Solve Initial(instancei, Budget) Increment i
Next Case For each Case
Compute Mean ”Solution Time” of Solve Original, Solve StartP oint, Solve Initial Analyze the ratio of mean solution time of Solve Initial to Solve Original
Investigate more about Solve StartP oint
methods we have found boxplots very useful. As Figure 5.3 shows there are five groups of data. The first group shows the solution time of solving 100 instances with the original model. The next four groups are the solution time of solving 100 instances by initialization using the RB, N B, and V B methods to reach the start solution. The boxes are significantly shifted to the low end. They are positively skewed. The boxes also are very thin
Figure 5.3: Box plot of data from the solution time of 100 samples solved originally and initially by application of three methods to find the best start node to initial a solution.
Each box represents the middle 50% of the data which lies between first quartile (Q1) and third quartile (Q3) and the remaining 50% is in the whiskers. The inset shows the zoomed in on first subset of the graph.
because a high number of cases are contained within a very small segment (refer to the peak of the distribution). Overall, we see that the boxes of all three solutions which use initialization are smaller and shorter than the result of the original solution. It means the solution time has been consistently decreased.
To assess the effectiveness of the three different methods of finding the
Table 5.4: Mean and 95% Confidence Interval of solution time of applying different meth-ods for 100 random samples
The name of solution Mean Solution Time and Confidence Interval(95%CI) sec
Solve Original 6.06 (0, 39.04)
Comparison of two solution strategies: solving originally and solving by initialization
The name of Mean Solution Time Ratio to Saved Time
Solution and 95%CI Solve Original (sec)
Solve Initial
M ax RB() 3.68 (0, 45.31) 0.61 39%
M ax N B() 3.78 (0, 49.56) 0.62 38%
M ax V B() 3.00 0, 24.59) 0.49 51%
Comparison of two solution strategies: solving originally and solving by a start node
The name of Mean Solution Time Saved Time Chance of How Close
Solution and 95%CI (sec) Optimality to be optimal
Solve StartP oint
M ax RB() 1.47 (1.14, 1.81) 76% 100% 100
M ax N B() 1.17 (0.87, 1.47) 81% 95% 99.75
M ax V B() 1.32 (0.87, 1.77) 78% 100% 100
start node (RB, N B, and V B), it is probably best to say their performance is more or less similar. In other words, they all imply a reasonable reduction of computational cost of the model, as we aimed. If the treatment of each method is dissected carefully, we should be able to decide which method is more recommendable. Looking at the results in Table 5.4, it is evident that using V alue Best or V B method to find the best start node drives the model to the largest amount of reduction in computation time. It is quite easy to understand the reason why this method is preferred. To justify this choice, consider the following example from one of the 100 random generated instances (see Figure 5.4).
When the aim is to find the best node having the highest utility value without concerning the cost, such as the two methods, RB and N B, some nodes have the highest utility but are too expensive to purchase. In con-trast, a node high in utility and low in cost is a much better preposition as a good start node for an optimal solution. Therefore, it would be sensible to consider the utility per cost of each cell then decide about the high value node, as the method V alue Best suggests and Figure 5.4 shows.
In addition to the discussion about saving solution time by applying the approach of initialization, we would like to draw attention to the result from solving the problem by a start node, (Solve StartP oint). The saved time is significant, about 80%, although it is not always an optimal solu-tion. However it is worth considering the question: ”If the solution is not optimal, How close is it to be optimal?”. The answer, as Table 5.4 shows, in the best case is 100% optimal and in the worst case it would be 95%
optimal , i.e. when the solution of Solve StartP oint is not optimal it is within 5%. And a more interesting result is in a small percentage of non-optimal cases, the solution is nearly non-optimal, only less than 1% far from the optimal solution. Therefore, it seems the trade-off between ”optimality”
Figure 5.4: In this picture, on the left hand, each cell has two values, the utility and the cost. While on the right hand, the value associated with each cell is the utility over cost. The shaded cells represent cells which have the highest values. Using three methods named Rand Best, N eighb Best, and V alue Best the start node was identified 92, 18, and 27 respectively.
and ”speed” in terms of running the solution is not a big deal. Knowing the chance of getting a near optimal solution might provide enough evi-dence to decision makers to rely on the solution of Solve StartP oint and sacrifice a bit to profit computationally.
Chapter 6 Conclusion
In this thesis we addressed the problem of post-hoc matching in paral-lel group RCTs by formulating it as a multi-objective constrained integer programming problem that explicitly takes into account the multiple ob-jective nature of the underlying decision making situation. We used ex-act optimization techniques, rather than heuristic methods, thus providing theoretically grounded assurance of optimality of the obtained solutions.
Using acute stroke trials as a context, we explored the relationship between the overall degree of individual post-hoc matching and the meaningful tol-erances on variables and investigated the association between the overall degree of individual post-hoc matching and the sample size of the respec-tive RCT, as well as between the overall degree of individual post-hoc matching and the respective dispersion in age and baseline stroke severity.
matching can only be achieved when the differences in prognostic base-line variables between individually matched subjects within the same pair are sufficiently large and that the unmatched subjects are qualitatively different to the matched ones. For larger RCTs the absolute numbers of unmatched patients remain large.
Another main focus of this work was to develop a new solution method for the network design problem. An innovative formulation of the ”Asym-metric Travelling Salesman Problem” (ATSP) which yields a mixed integer programming model based on the concept of the transshipment problem was proposed. The formulation was extended to the solution for the other transportation scheduling problems which are related to the TSPs such as the ”Multi-Travelling Salesman problem” (m-TSP) and the ”Selective Travelling Salesman Problem” (STSP). The major advantage of our model is that it eased difficulties which most AP based formulations have with solving real world instances. The proposed model is not restricted to the symmetric data i.e. it can be built on either a directed graph or an undi-rected graph.
The reserve network design problem is a variation of the STSP which
maximizes some utilities subject to various constraints. These constraints include a budget limitation and spatial attributes such as connectivity and compactness. The proposed model achieved selecting a fully connected set of sites and to some extent compactness. It does this avoiding any sub-tours and requiring any regular shape assumptions. Furthermore, it is easy to explore the trade-off between the objective function and the number of contiguous regions.
To improve efficiency of the model for large scale problems involving thousands of nodes much effort has been made. The solution by an initial solution successfully reduced the computational cost of the model. An initial solution was obtained by choosing the node with the highest utility value as the start node.
The travelling salesman problem with all its varieties has many applica-tions in real life. Many of these real problems cannot be solved easily and work continues on finding better methods for solving these problems. In this thesis we have added a new approach for solving these problems using the concept of the transshipment problem. The approach has produced some promising results.
Bibliography
R.K. Ahuja, T.L. Magnanti, J.B. Orlin, M.T. L., Network Flows: Theory, Algorithms, and Applications, Prentice Hall, 1993.
D.G. Altman, Practical Statistics for Medical Research, Chapman and Hall/CRC, 1990.
D.G. Altman, J.M. Bland, Treatment allocation in controlled trials: why randomise?, BMJ 318 (1999) 1209–1209.
D. Applegate, R. Bixby, V. Chvatal, W. Cook, The Traveling Salesman Problem: A Computational Study, Princeton University Press, Prince-ton, 2007.
P.C. Austin, A critical appraisal of propensity-score matching in the med-ical literature between 1996 and 2003, Statistics in Medicine 27 (2008) 20372049.
E. Balas, S. Ceria, G. Cornuejols, A lift-and-project cutting plane algo-rithm for mixed 0-1 programs, Mathematical Programming 58 (1993) 295–324.
F. Barahona, R. Anbil, The volume algorithm: producing primal solutions with a subgradient method, Mathematical Programming 87 (2000) 385–
399.
A.A. Batabyal, G.J. DeAngelo, To match or not to match: Aspects of marital matchmaking under uncertainty, Operations Research Letters 36 (2008) 94 – 98.
P. Bath, C. Hogg, M. Tracy, S. Pocock, Calculation of numbers-needed-to-treat in parallel group trials assessing ordinal outcomes: case examples from acute stroke and stroke prevention, International Journal of Stroke 6 (2011) 472–479.
M.S. Bazaraa, J.J. Goode, The traveling salesman problem: A duality approach, Mathematical Programming 13 (1977) 221–237.
T. Bektas, The multiple traveling salesman problem: an overview of for-mulations and solution procedures, Omega 34 (2006) 209 – 219.
M. Bellmore, G.L. Nemhauser, The traveling salesman problem: A survey, Operations Research 16 (1968) 538–558.
J.M. Bland, D.G. Altman, Statistics notes: Matching, BMJ 309 (1994) 1128.
E. Bonomi, J.L. Lutton, The n-city travelling salesman problem: Statistical mechanics and the metropolis algorithm, SIAM Review 26 (1984) 551–
568.
C.F. Breslin, C.H. Gladwin, D. Borsoi, J.A. Cunningham, Defacto client-treatment matching: how clinicians make referrals to outpatient treat-ments for substance use, Evaluation and Program Planning 23 (2000) 281 – 291.
R.A. Briers, Incorporating connectivity into reserve selection procedures, Biological Conservation 103 (2002) 77 – 83.
M. Caliendo, S. Kopeinig, Some practical guidance for the implementation of propensity score matching, Journal of Economic Surveys 22 (2008) 31–72.
J.D. Camm, S. Polasky, A. Solow, B. Csuti, A note on optimal algorithms for reserve site selection, Biological Conservation 78 (1996) 353 – 355.
V. Cerny, Thermodynamical approach to the traveling salesman problem:
An efficient simulation algorithm, Journal of Optimization Theory and Applications 45 (1985) 41–51.
R. Church, D. Stoms, F. Davis, Reserve selection as a maximal covering location problem, Biological Conservation 76 (1996) 105–112.
J.M. Conrad, C.P. Gomes, W.J. van Hoeve, A. Sabharwal, J.F. Suter, Wildlife corridors as a connected subgraph problem, Journal of Environ-mental Economics and Management 63 (2012) 1 – 18.
I. Contreras, E. Fernndez, General network design: A unified view of com-bined location and network design problems, European Journal of Oper-ational Research 219 (2012) 680 – 697.
A. Costa, M. Ramakrishnan, P. Taylor, A distributed approach to capacity allocation in logical networks, European Journal of Operational Research 203 (2010) 737 – 748.
A.R. Kiester, B. Downs, R. Hamilton, M. Huso, K. Sahr, A comparison of reserve selection algorithms using data on terrestrial vertebrates in oregon, Biological Conservation 80 (1997) 83 – 97.
G.B. Dantzig, S. Fulkerson, Rand Johnson, Solution of a large-scale traveling-salesman problem, Operations Research 2 (1954) 393–410.
S.M. Davis, G.A. Donnan, M.W. Parsons, C. Levi, K.S. Butcher, A. Peeters, P.A. Barber, C. Bladin, D.A. De Silva, G. Byrnes, J.B. Chalk, J.N. Fink, T.E. Kimber, D. Schultz, P.J. Hand, J. Frayne, G. Hankey, K. Muir, R. Gerraty, B.M. Tress, P.M. Desmond, Effects of alteplase beyond 3 h after stroke in the echoplanar imaging thrombolytic evalu-ation trial (epithet): a placebo-controlled randomised trial, The Lancet Neurology 7 (2008) 299 – 309.
E. Dettmann, C. Becker, C. Schmeiber, Distance functions for matching in small samples, Computational Statistics and Data Analysis 55 (2011) 1942 – 1960.
J.M. Diamond, The island dilemma: lessons of modern biogeographic
stud-ies for the design of nature reserves, Biological Conservation 7 (1975) 129–146.
G. Donnan, M. Fisher, M. Macleod, S. Davis, Stroke, Lancet 371 (2008) 1612–1623.
M. Dorigo, L. Gambardella, Ant colony system: a cooperative learning approach to the traveling salesman problem, Evolutionary Computation, IEEE Transactions on 1 (1997) 53 –66.
A. Drigas, S. Kouremenos, S. Vrettos, J. Vrettaros, D. Kouremenos, An expert system for job matching of the unemployed, Expert Systems with Applications 26 (2004) 217 – 224.
EC, Biodiversity is life, The LIFE Unit, 2006.
G. Erdogan, J.F. Cordeau, G. Laporte, The attractive traveling salesman problem, European Journal of Operational Research 203 (2010) 59–69.
C. Faloutsos, K.S. McCurley, A. Tomkins, Fast discovery of connection subgraphs, in: Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, New York,
D. Feillet, P. Dejax, M. Gendreau, Traveling salesman problems with prof-its: An overview, Transportation Science 39 (2005) 188–205.
D. Fischer, R. Church, Clustering and compactness in reserve site selection:
An extension of the biodiversity management area selection model, Forest Science 49 (2003) 555–565.
M. Fischetti, P. Toth, A polyhedral approach to the asymmetric traveling salesman problem, Management Science 43 (1997) 1520–1536.
M.L. Fisher, The lagrangian relaxation method for solving integer pro-gramming problems, Management Science 27 (1981) 1–18.
FSA, Conservation reserve program: Summary and enrollment statistics fy 2006, Farm Service Agency, 2006.
D. Gale, L.S. Shapley, College admissions and the stability of marriage, The American Mathematical Monthly 69 (1962) 915.
B. Gavish, S. Graves, The travelling salesman problem and related prob-lems, Working Paper OR-078-78, Operations Research Center (1978).
M. Gendreau, A. Hertz, G. Laporte, New insertion and postoptimization
procedures for the traveling salesman problem, Operations Research 40 (1992) 1086–1094.
A.M. Geoffrion, Lagrangean relaxation for integer programming, in: Ap-proaches to Integer Programming, volume 2 of Mathematical Program-ming Studies, Springer Berlin Heidelberg, 1974, pp. 82–114.
F. Glover, Tabu searchpart i, ORSA Journal on Computing 1 (1989) 190–
206.
F. Glover, Tabu searchpart ii, ORSA Journal on Computing 2 (1990) 4–32.
A. Government, Budget paper no. 2: Budget measures 2007-08, 2007.
J.C. Gower, A general coefficient of similarity and some of its properties, Biometrics 27 (1971) 857–871.
R. Greevy, B. Lu, S.J. H., P.R. Rosenbaum, Optimal multivariate matching before randomization, Biostatistics 5 (2004) 263–275.
M. Grotschel, L. Lovasz, Combinatorial optimization: A survey, 1993.
A. Gupta, J. Kleinberg, A. Kumar, R. Rastogi, B. Yener, Provisioning a virtual private network: A network design problem for multicommodity
flow, in: In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 389–398.
G. Gutin, A.P. Punnen (Eds.), The Traveling Salesman Problem and Its Variations, Springer US, 2004.
G.H. Guyatt, E.F. Juniper, S.D. Walter, L.E. Griffith, R.S. Goldstein, Interpreting treatment effects in randomised trials, BMJ 316 (1998) 690–
693.
R.G. Haight, S.A. Snyder, Integer programming methods for reserve se-lection and design, in: Spatial Conservation Prioritization: Quantitative Methods and Computational Tools, Oxford University Press, 2009.
S. Hajkowicz, A. Higgins, C. Miller, O. Marinoni, Targeting conserva-tion payments to achieve multiple outcomes, Biological Conservaconserva-tion 141 (2008) 2368 – 2375.
A. Haviland, D.S. Nagin, P.R. Rosenbaum, Combining propensity score matching and group-based trajectory analysis in an observational study, Psychological Methods 12 (2007) 247–267.
M. Held, R. Karp, The traveling-salesman problem and minimum spanning trees: Part ii, Mathematical Programming 1 (1971) 6–25.
M. Held, R.M. Karp, The traveling-salesman problem and minimum span-ning trees, Operations Research 18 (1970) 1138–1162.
S. Hollis, F. Campbell, What is meant by intention to treat analysis? sur-vey of published randomised controlled trials, BMJ 319 (1999) 670–674.
J.B., Kirkpatrick, An iterative method for establishing priorities for the selection of nature reserves: An example from tasmania, Biological Con-servation 25 (1983) 127 – 134.
D. Johnson, Local optimization and the traveling salesman problem, in:
Automata, Languages and Programming, volume 443, Springer Berlin Heidelberg, 1990, pp. 446–461.
B. Jovanovic, Job matching and the theory of turnover, Political Economy 87 (1979) 972–90.
M. Kershaw, P.H. Williams, G.M. Mace, Conservation of afrotropical an-telopes: consequences and efficiency of using different site selection
meth-ods and diversity criteria, Biodiversity and Conservation 3 (1994) 354–
372.
M. Knoflach, B. Matosevic, M. Rcker, M. Furtner, A. Mair, G. Wille, A. Zangerle, P. Werner, J. Ferrari, C. Schmidauer, L. Seyfang, S. Kiechl, J. Willeit, A.S.U.R. Collaborators., Functional recovery after ischemic stroke–a matter of age: data from the austrian stroke unit registry, Neu-rology 78 (2012) 279–85.
G. Laporte, The vehicle routing problem: An overview of exact and approx-imate algorithms, European Journal of Operational Research 59 (1992) 345 – 358.
G. Laporte, What you should know about the vehicle routing problem, Naval Research Logistics (NRL) 54 (2007) 811–819.
G. Laporte, S. Martello, The selective travelling salesman problem, Dis-crete Applied Mathematics 26 (1990) 193–207.
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, D.B. Shmoys, The Trav-eling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley, Chichester, 1985.
E. Lerner, Relationship between matching and assignment problems, Rus-sian Mathematics (Iz VUZ) 55 (2011) 27–32.
B.A. Liang, R. Lew, J.A. Zivin, Review of tissue plasminogen activator, ischemic stroke, and potential legal issues, Arch Neurol 65 (2008) 1429–
33.
B. Lu, E. Zanutto, R. Hornik, P.R. Rosenbaum, Matching with doses in an observational study of a media campaign against drug abuse, Journal of the American Statistical Association 96 (2001) 1245–1253.
J. Lysgaard, A.N. Letchford, R.W. Eglese, A new branch-and-cut algo-rithm for the capacitated vehicle routing problem, Mathematical Pro-gramming 100 (2003) 2004.
P. Mahalanobis, On the generalized distance in statistics, for the classifi-cation problem, in: the National Institute of Sciences of India II(1), pp.
49–55.
L. Margolin, On the convergence of the cross-entropy method, Annals of Operations Research 134 (2005) 201–214.
C.R. Margules, A. Nicholls, R. Pressey, Selecting networks of reserves to maximise biological diversity, Biological Conservation 43 (1988) 63–76.
The National Institute of Neurological Disorders and Stroke rt-PA Stroke Study Group, Tissue plasminogen activator for acute ischemic stroke, New England
Journal of Medicine 333 (1995) 1581–1588.
D.P. Memtsas, Multiobjective programming methods in the reserve selec-tion problem, European Journal of Operaselec-tional Research 150 (2003) 640 – 652.
C.E. Miller, A.W. Tucker, R.A. Zemlin, Integer programming formulation of traveling salesman problems, J. ACM 7 (1960) 326–329.
K. Ming, P. Rosenbaum, A note on optimal matching with variable controls using the assignment algorithm, Journal of Computational and Graphical Statistics 10 (2001) 455–463.
A. Moilanen, Reserve selection using nonlinear species distribution models., The American naturalist 165 (2005) 695–706.
D.J. Nalle, J.L. Arthur, J. Sessions, Designing compact and contiguous
reserve networks with a hybrid heuristic algorithm, Forest Science 48 (2002) 59–68.
A. Nedic, A. Ozdaglar, Subgradient methods in network resource alloca-tion: Rate analysis, in: Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference on, pp. 1189 –1194.
G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons Ltd, 1999.
H. ¨Onal, R. Briers, Incorporating spatial criteria in optimum reserve net-work selection, in: Proceedings of the Royal Society of London Series B-Biological Sciences, volume 269, pp. 2437–2441.
H. ¨Onal, R. Briers, Selection of a minimum-boundary reserve network using integer programming, in: Proceedings of the Royal Society of London Series B-Biological Sciences, volume 270, pp. 1487–1491.
H. ¨Onal, R. Briers, Designing a conservation reserve network with minimal fragmentation: A linear integer programming approach, Environmental Modeling and Assessment 10 (2005) 193–202.
H. ¨Onal, R.A. Briers, Optimal selection of a connected reserve network, Operations Research 54 (2006) pp. 379–388.
H. ¨Onal, Y. Wang, A graph theory approach for designing conservation reserve networks with minimal fragmentation, Netw. 51 (2008) 142–152.
T. Oncan, I.K. Altinel, G. Laporte, A comparative analysis of several asym-metric traveling salesman problem formulations, Computers & Opera-tions Research 36 (2009) 637 – 654.
A.J. Orman, H.P. Williams, A survey of different integer programming for-mulations of the travelling salesman problem, in: Optimisation, econo-metric and financial analysis. Advances in computational management science, Springer, Berlin, Germany, 2006, pp. 93–108.
J.Y. Potvin, Genetic algorithms for the traveling salesman problem, Annals of Operations Research 63 (1996) 337–370.
R. Pressey, C. Humphries, C. Margules, R. Vane-Wright, P. Williams, Beyond opportunism: Key principles for systematic reserve selection, Trends in Ecology and Evolution 8 (1993) 124 – 128.
G. Reinelt, Tsplib@ONLINE, 1991.
G. Reinelt, The Traveling Salesman: Computational Solutions for TSP Ap-plications (Lecture Notes in Computer Science), Springer, Berlin, 1994.
C.S. ReVelle, J.C. Williams, J.J. Boland, Counterpart models in facility
C.S. ReVelle, J.C. Williams, J.J. Boland, Counterpart models in facility