We prove characterization theorems for Weibull distributions based on invariance of hazard rate under scale transformations in a countable dense set near origin. Similar characterizations are also obtained for two different types of discrete Weibull distributions, when logarithm of survival function / hazard rate are scale invariant on the set of non negative integers. Thus scale invariance of survival function / hazard rate of a variable on a small domain is equivalent to Weibull distribution. This assumption on reliability function (or hazard rate) when satisfied, leads to an appropriate model selection. Modified Weibull Distribution (MWD) with bathtub hazard rate are characterised and its discrete versions are also discussed. Survival function and hazard rate of yam plant lifetime, related to harvest scenario are examined under Weibull model in forecasting market supply of the crop and in analysing time lag in supply. The proposed analysis may be adopted for similar situations in production and marketing of other products. Observed growth curve of yam plant lifetime based on field experiment data has a good match with simulated growth curves.