This paper investigates two-machine flow shop and open shop scheduling problems under a knapsack constraint. In these problems, a set of jobs is available at time zero. Each job can either be accepted and processed on both machines, or rejected. A rejected job incurs a rejection penalty and occupies a certain rejection size. The objective is to minimize the makespan of the accepted jobs plus the total rejection penalty of the rejected jobs, subject to the constraint that the total rejection size of the rejected jobs does not exceed a given upper bound. For both the two-machine flow shop and the open shop settings, we develop pseudo-polynomial-time dynamic programming algorithms that yield exact solutions, as well as 3-approximation algorithms. Furthermore, based on these exact dynamic programming algorithms and the 3-approximation algorithms, we design fully polynomial-time approximation schemes (FPTASs) for these two problems.
Online retailers face a coupled recommendation-display decision: they must infer user preferences from sparse, attribute-level, and linguistically uncertain evaluations, and then allocate scarce display slots under conflicting user-utility and platform-revenue objectives. This study proposes a two-stage hybrid decision-support model. Stage I transforms linguistic evaluations, or normalized star ratings when ratings are numerical, into interval type-2 fuzzy sets. It then uses item- and user-based collaborative filtering to complete missing attribute ratings and applies TOPSIS to obtain interpretable product preference scores. Stage II embeds these scores in a bi-objective integer programming model that selects products, assigns display slots, and allocates impressions. A big-M mixed-integer linear reformulation and an equivalence lemma are provided to strengthen mathematical reliability. An illustrative example, comparative analysis, sensitivity analysis, and random-instance robustness analysis show that the proposed method can handle sparse uncertain evaluations and quantify the trade-off between recommendation relevance and platform income.
This study addresses the shift-scheduling problem for nursery teachers. We propose two integer programming models: the shift preference maximization model and shift preference and fairness maximization model. The primary aim is to develop a scheduling framework that not only accommodates individual shift preferences but also upholds fairness, while adhering to stringent constraints regarding labor and facility resources. A key contribution of this research is the introduction of a method to quantify the degree of preference using integer values. The shift preference and fairness maximization model is designed to generate schedules that respect the shift preferences of full-time nursery teachers, considering both the intensity of their preferences and the fairness of these preferences among nursery teachers. A numerical analysis reveals that the proposed models significantly enhance the overall satisfaction levels among nursery teachers, outperforming previous schedules in both short-term and long-term scenarios. In addition, this study elucidates the correlation between the satisfaction of full-time nursery teachers and the associated costs of hiring part-time nursery teachers.
Cross-efficiency-based Malmquist productivity measurement involves both the selection of DEA-optimal weights across Malmquist states and the aggregation of heterogeneous peer assessments. This study develops the Inertial Cross-Efficiency Malmquist Productivity Index (I-CEMPI). The method retains the state-specific primary-optimal weighting sets and coordinates the four Malmquist states through minimum intertemporal adjustment, yielding a unique and reproducible weight representation while preserving primary DEA optimality. Peer Cross-Malmquist assessments are subsequently combined through strategic-perspective-balanced geometric aggregation, which preserves the multiplicative decomposition into efficiency change and technical change. Applying I-CEMPI to 440 listed Chinese SRDI “Little Giant” enterprises over 2019–2024 shows substantial scope for intertemporal selection within the empirical optimal sets. Perspective balancing produces larger changes in aggregate productivity assessments than intertemporal coordination, while comparisons with the standard MPI and a published Cross-GMPI show high overall agreement together with differences in a subset of firm-level classifications. Innovation productivity contracts over most horizons, with technical change accounting for the larger share of cumulative contraction. Strategic perspectives differ more clearly in their efficiency-change and technical-change compositions than in persistent aggregate productivity rankings.
With the development of e-commerce, there are many scenarios of trade-in in online channels. According to customers’ heterogeneous preferences, platforms can choose different combinations among timing of rebates (instant, deferred, and mixed rebates) and forms of rebates (cash and gift-card rebates) to maximize their profits. We derive the platform’s optimal rebates and profits for five different scenarios: instant rebate, deferred cash rebate, mixed cash rebate, deferred gift-card rebate, mixed gift-card rebate. Furthermore, we show that the platform should choose the deferred rebate strategy when the waiting cost of impatient consumers is relatively low, otherwise, they should choose the mixed rebate strategy. When the cross-selling profitability is relatively low, the cash rebate strategy is dominant, otherwise the gift-card rebate strategy is dominant. Finally, the platform is more inclined to choose the mixed rebate strategy under the cash rebate strategy, and the deferred strategy under the gift-card rebate strategy. The findings provide guidance to platforms on how best to choose the trade-in programs.
