Start-up firms play an important role in the economy. Statistics show that a large percent of start-up firms fail after few years of establishment. Raising capital, which is crucial to success, is one of the difficulties start-up firms face. This Ph. D thesis aims to draw suggestions for start-up firm survival from mathematical models and numerical investigations. Instead of the commonly held profi t maximizing objective, this thesis assumes that a start-up firm aims to maximize its survival probability during the planning horizon. A firm fails if it runs out of capital at a solvency check. Inventory management in manufacturing start-up firms is discussed further with mathematical theories and numerical illustrations, to gain insight of the policies for start-up firms. These models consider specific inventory problems with total lost sales, partial backorders and joint inventory-advertising decisions. The models consider general cost functions and stochastic demand, with both lead time zero and one cases. The research in this thesis provides quantitative analysis on start-up firm survival, which is new to the literature. From the results, a threshold exists on the initial capital requirement to start-up firms, above which the increase of capital has little effect on survival probability. Start-up firms are often risk-averse and cautious about spending. Entering the right niche market increases their chance of survival, where the demand is more predictable, and start-ups can obtain higher backorder rates and product price. Sensitivity tests show that selling price, purchasing price and overhead cost have the most impact on survival probability. Lead time has a negative effect on start-up firms, which can be offset by increasing the order frequent. Advertising, as an investment in goodwill, can increase start-up firms' survival. The advertising strategies vary according to both goodwill and inventory levels, and the policy is more flexible in start-up firms. Externally, a slightly less frequency solvency check gives start-up firms more room for fund raising and/or operation adjustment, and can increase the survival probability. The problems are modelled using Markov decision processes, and numerical illustrations are implemented in Java.