This paper deals with the systematic and quantitative robustness analysis of solutions to the volume-to-point (VP) heat conduction optimization problem, including solutions by constructal theory (Cr), bionic optimization (BO), simulated annealing (SA) as well as random solutions. The impacts of both load-induced and random failures are examined. All the solutions behave similarly during a failure. A tiny high-heat-flux-induced failure causes huge performance loss, which is the worst possible case. While the best case is a failure at positions of low heat flux, a large amount of which only causes a small performance loss. Random failure generally results in a loss roughly linear with failure amount, but sudden loss increase happens. Both intuitive and practical definitions of robustness are considered. Intuitively, a performance-robustness trade-off is seen for all the solutions, with poorer solutions being robuster. Practically things are different, with SA solutions both the best in performance and the robustest during a failure in the worst case. However, Cr and BO solutions turn out to be robuster than SA solution in the best case, and Cr solution may be more preferable in practice due to its regular geometry. With robustness taken into consideration, different solutions could be evaluated more comprehensively, the critical aspects concerning the practical realization of the solutions are also revealed, and furthermore the significance of robustness analysis in general VP problems is accentuated. (C) 2011 Elsevier Ltd. All rights reserved.