Test case 3: IMRT problem

As IMRT is one of the common application areas, we also include a test case from this field to compare the algorithms. The IMRT optimization problem used in this comparison is a 2D phantom pancreas case. Figure 7.13 shows the tumor and the five nearby organs. The high-energy photon beams used in radiation therapy to treat cancer tumors have

to pass through surrounding tissue to reach the tumor. To reduce the risk of damaging healthy tissue, the radiation dose delivered to this tissue should be minimized. The five organs indicated in Figure 7.13 are especially sensitive to radiation and are therefore referred to as the organs-at-risk (OARs). The radiation dose delivered to these OARs should be limited, while enough radiation should be delivered to the tumor to eradicate it. To calculate these doses, the relevant part of the body of the patient is discretized by dividing it into voxels. Using a dose influence matrix, the radiation dose delivered to each voxel can be calculated for a specific radiation plan. The objectives are often formulated in terms of the mean and maximum dose delivered to all voxels belonging to a tumor or OAR.

Figure 7.13: 2D pancreatic phantom case.

Dose in Gy Min. Max. Tumor 50 75 Left kidney 70 Right kidney 70 Spinal cord 45 Small bowel 75 Liver 75 Other tissue 75

Table 7.1: Minimal and maximal dose allowed for the tumor, OARs and other tissue.

In our problem, we consider four objectives. The first objective is aimed at delivering a dose of 60 Gray (Gy) to the tumor. Any voxel of the tumor that receives less than 60 Gy is penalized; the mean of these penalties forms the first objective. The three other objectives measure the maximum dose delivered to any of the voxels belonging to the left kidney, right kidney, or spinal cord, respectively. Furthermore, constraints are added to ensure a minimal or maximal dose for the OARs and other tissue. The values for these bounds can be found in Table 7.1. As all constraint and objectives are linear, this IMRT problem is a multi-objective LP problem and thus convex.

Figures 7.14 and 7.15 again contain the values of α(P S, IP S) and α(OP S, IP S). KLAMROTH≤ _{and KLAMROTH}= _{are not used in this test case as they performed con-}

siderably worse than the other algorithms in the previous two test cases. For this IMRT case, CRAFT performs better than SOLANKI for all nopt_{, but ENHANCED still performs}

better than the former two algorithms. The quality guarantee of α(OP S, IP S) ≤ 0.1 is reached by CRAFT after 42 optimizations whereas ENHANCED needs only 14 op- timizations. After 24 optimizations, ENHANCED even gives a quality guarantee of α(OP S, IP S) ≤ 0.05.

0 10 20 30 40 50 60 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 nopt α (PS,IPS) ENHANCED SOLANKI CRAFT

Figure 7.14: α(P S, IP S) of test case 3 as function of nopt_. 0 10 20 30 40 50 60 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 nopt α (OPS,IPS) ENHANCED SOLANKI CRAFT

Figure 7.15: α(OP S, IP S) of test case 3 as function of nopt_.

7.8.5

Test case 4: geometric programming problem

In the fourth and final test case, we consider the effect of transforming convex objectives through a geometric programming example. For a general introduction to geometric programming, we refer to Boyd et al. (2007). Although we use only one example to show the effects of transforming the objectives, the transformation described below can be applied to any geometric program.

In this test case, we consider the following geometric programming problem: f1(x) = e−x1−x2−x3 f2(x) = ex4 f3(x) = ex5 2e−x4+x1_(ex2 _{+ e}x3_{) ≤ 1} e−x5+x2+x3 _{≤ 1} e|x(2)−x(1)| _{≤ 2} e|x(2)−x(3)| _{≤ 2} f (x) ≤ [e3 e3 e3]>

This problem corresponds to Example 5 in Boyd et al. (2007) after applying the logarith- mic change of variables and adding the upper bound zub _{= [e}3_e3 _e3_]>_{. We can easily show}

that this MOP is convex. When applying the transformation function h(z) = log(z), the function h ◦ f becomes a linear function and thus remains convex. As h(z) = log(z) is a strictly increasing and concave transformation, we can apply the results attained in Section 7.6. In this test case we did not use the transformation for the selection of facets. Furthermore, as this test case is intended to show the benefits of applying transformations and not to compare different algorithms, we used only the ENHANCED-algorithm.

0 5 10 15 20 25 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 nopt α α(OPS−1,IPS−1) α(P S, I P S) α(P S,IPS−1) α(OP S, I P S)

Figure 1: α-values of test case 4 as function of nopt_.

1

Figure 7.16: α-values of test case 4 as function of nopt_.

Figure 7.16 shows the values of α(OP S, IP S), α(P S, IP S), α(

OPS

_IPS

α(P S,

IPS

_OPS

_IPS

and α(P S, IP S) with α(P S,

IPS

racy improves considerably by applying the transformation. At nopt _{= 4, the real accuracy}

α(P S,

IPS

α(

OPS

_IPS

more than 25 optimizations are needed to reach the same level of accuracy.

7.9

Conclusions and future research

In this chapter, we have introduced several enhancements to improve sandwich algorithms for approximating multi-dimensional convex Pareto sets. Firstly, dummy points were introduced to overcome the problem of ‘undesirable’ normals of IP S-facets. Adding these dummy points, we can determine also the set IP SnId_{of non-IP S-dominated points more}

easily. Secondly, we introduced α(P S, IP S) and α(OP S, IP S), which can be used to determine when a set IP SnId_{is both an ²-approximation and ε-optimal. As both concepts}

of ²-approximation and ε-optimal have a clear interpretation, these two measures provide quality guarantees that are easy to understand by a decision maker. Furthermore, we introduced a method to calculate α(P S, IP S) and α(OP S, IP S). This method simplifies the calculation of α(OP S, IP S) to solving a finite number of simple LP problems and thus improves the practical applicability of this error measure. Thirdly, we showed that transformations of the objective functions can improve OP S and IP S for certain convex MOPs and to extend the application of sandwich algorithm to certain non-convex MOPs. The calculation of the error measure when using these transformations was also discussed. To test the benefits of these enhancements, we constructed the algorithm ENHANCED

by enhancing SOLANKI with dummy points and the error measure α(OP S, IP S). Three test cases showed a considerable efficiency improvement of ENHANCED compared with the other four tested methods. A fourth test case shows that using suitable transforma- tions can still further improve the efficiency.

A limitation of these comparisons is that they consider only sandwich algorithms. Generally non-sandwich algorithms for approximating convex Pareto sets cannot provide quality guarantees, but they still can provide good approximations of the Pareto set. Therefore, it would be interesting to perform more extensive comparisons among different methods for approximating multi-dimensional convex Pareto sets.

Another interesting subject for further research would be to determine if the efficiency could be further improved by using a more interactive approach. Klamroth and Miettinen (2008) describe an approach where decision makers can refine their preferences to identify regions of P S where the approximation should be improved. A similar refinement might also be incorporated into the sandwich approaches by allowing the decision maker to change zub_{. However, more research is necessary to determine the effects and benefits of}

Aarts, E.H.L. and J. Korst (1989). Simulated annealing and Boltzmann machines: A stochastic approach to combinatorial optimization and neural computing. New York, NY, USA: John Wiley & Sons, Inc.

Agca, S., B. Eksioglu, and J.B. Ghosh (2000). Lagrangian solution of maximum dis- persion problems. Naval Research Logistics, 47(2), 97–114.

Alam, F.M., K.R. McNaught, and T.J. Ringrose (2004). A comparison of experimental designs in the development of a neural network simulation metamodel. Simulation Modelling: Practice and Theory, 12(7-8), 559–578.

Alexandrov, N.M., J.E. Dennis, R.M. Lewis, and V. Torczon (1998). A trust-region framework for managing the use of approximation models in optimization. Struc- tural and Multidisciplinary Optimization, 15(1), 16–23.

Audze, P. and V. Eglais (1977). New approach for planning out of experiments. Prob- lems of Dynamics and Strengths, 35, 104–107.

Baer, D. (1992). Punktverteilungen in W¨urfeln beliebiger Dimension bez¨uglich der Maximum-norm. Wissenschaftliche Zeitschrift der P¨adagogischen Hochschule Er- furt/M¨uhlhausen, Mathematisch-Naturwissenschaftliche Reihe, 28, 87–92.

Banzhaf, W., F.D. Francone, R.E. Keller, and P. Nordin (1998). Genetic Program- ming: An introduction on the automatic evolution of computer programs and its applications. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.

Barber, C.B., D.P. Dobkin, and H. Huhdanpaa (1996). The quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software (TOMS), 22(4), 469– 483.

Barthelemy, J.F.M. and R.T. Haftka (1993). Approximation concepts for optimum structural design – A review. Structural and Multidisciplinary Optimization, 5(3), 129–144.

Bates, R.A., R.J. Buck, E. Riccomagno, and H.P. Wynn (1996). Experimental design and observation for large systems. Journal of the Royal Statistical Society: Series B, 58, 77–94.

Bates, S.J., J. Sienz, and V.V. Toropov (2004). Formulation of the optimal Latin hypercube design of experiments using a permutation genetic algorithm. In 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 1–7.

Beers, W.C.M. van and J.P.C. Kleijnen (2008). Customized sequential designs for ran- dom simulation experiments: Kriging metamodeling and bootstrapping. European Journal of Operational Research, 186(3), 1099–1113.

Bermond, J.C. and D. Sotteau (1976). Graph decompositions and G-designs. In Nash- Williams and Sheehan (Eds.), Proceedings of the 5th British Combinatorial Con- ference 1975, Winnipeg, Canada, 53–72. Utilitas Mathematica Publishing Inc. Bisschop, J. and R. Entriken (1993). AIMMS: The Modeling System. Haarlem, The

Netherlands: Paragon Decision Technology.

Booker, A.J., J.E. Dennis, P.D. Frank, D.B. Serafini, V. Torczon, and M.W. Trosset (1999). A rigorous framework for optimization of expensive functions by surrogates. Structural and Multidisciplinary Optimization, 17(1), 1–13.

Boyd, S., S.J. Kim, L. Vandenberghe, and A. Hassibi (2007). A tutorial on geometric programming. Optimization and Engineering, 8(1), 67–127.

Branke, J., K. Deb, K. Miettinen, and R. Slowinski (2008). Multiobjective Optimization - Interactive and Evolutionary Approaches. New York, NY, USA: Springer-Verlag. Bulik, M., M. Liefvendahl, R. Stocki, and C. Wauquiez (2004). Stochastic simulation

for crashworthiness. Advances in Engineering Software, 35(12), 791–803.

Bursztyn, D. and D.M. Steinberg (2006). Comparison of designs for computer experi- ments. Journal of Statistical Planning and Inference, 136(3), 1103–1119.

Campos, T.P.R. (2006). Computational simulations in medical radiation - a new ap- proach to improve therapy. Boletim da Sociedade Brasileira de Matem´atica VII , 2, 7–20.

Carlyle, W.M., J.W. Fowler, E.S. Gel, and B. Kim (2003). Quantitative comparison of approximate solution sets for bi-criteria optimization problems. Decision Sci- ences, 34(1), 63–82.

Chen, V.C.P., K.L. Tsui, R.R. Barton, and M. Meckesheimer (2006). A review on de- sign, modeling and applications of computer experiments. IIE Transactions, 38(4), 273–291.

Cherkassky, V. and F. Mulier (1998). Learning from data : Concepts, theory, and methods. New York, NY, USA: John Wiley & Sons, Inc.