The authors study multiple stopping problems in a general context and with applications to finance. More precisely, the payoff is considered to be some abstract functional of an ordered sequence of stopping times. The goal of the paper is to find pure martingale dual representations for such generalized multiple stopping problems in a discrete time setting. It is shown that such representations can be constructed even in a most general setting. However, for practical implementation these general representations unfold their full strength only, if applied to some more specifically structured cashflows. In this respect, the authors study generic payoffs with both multiplicative and additive structure that incorporate integer valued volume constraints and refraction periods given by stopping times. An explicit Monte Carlo-based algorithm is provided and a detailed numerical study exemplifying the pricing of swing options is given. Moreover, a numerical example is presented which involves swing options subject to both volume constraints and refraction periods and tight confidence intervals for the respective option prices are calculated.