The design of nanobodies with high binding affinity for a given target is the primary objective in the therapeutics development. However, existing generative models often sample the sequence space of nanobody without explicit guidance, which leads to suboptimal affinity. To address this, we introduce DiffNB, a novel diffusion-based framework for controllable, high-affinity nanobody generation. DiffNB is the first diffusion model that leverages Direct Preference Optimization (DPO) to align a pre-trained generative prior with desired bio-physical properties. By fine-tuning the model on preference pairs of high- and low-affinity nanobodies, DiffNB learns to characterize and generate the variants with higher binding affinity and other bio-physical properties improved. During generation, the DPO-aligned DiffNB co-designs the sequence and structure of CDR regions to produce optimized and novel candidates. In our extensive experiments on three therapeutic antigens(HER2, IL-6, and CD45), we demonstrate that DiffNB can generate nanobodies exhibiting higher binding affinity and better structural diversity. Compared to the state-of-the-art baselines, our DPO-guided generation improves the “in silico” binding affinity by up to 35%, structural diversity by up to 46%, stability by up to 8%, and humanness by up to 25%. Our work establishes DPO as a powerful and efficient paradigm for steering generative models in nanobody design, paving the way for targeted and accelerated drug discovery.

This paper studies the reliable uncapacitated facility location problem in which facilities are subject to uncertain disruptions. A two-stage distributionally robust model is formulated, which optimizes the facility location decisions so as to minimize the fixed facility location cost and the expected transportation cost of serving customers under the worst-case disruption distribution. The model is formulated in a general form, where the uncertain joint distribution of disruptions is partially characterized and is allowed to have any prespecified dependency structure. This model extends several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. An efficient cutting plane algorithm is proposed to solve this model, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. Extensive numerical study shows that the proposed cutting plane algorithm not only outperforms the best-known algorithm in the literature for the stochastic problem under independent disruptions but also efficiently solves the robust problem under correlated disruptions. The practical performance of the robust models is verified in a simulation based on historical typhoon data in China. The numerical results further indicate that the robust model with even a small amount of information on disruption correlation can mitigate the conservativeness and improve the location decision significantly. Summary of Contribution: In this paper, we study the reliable uncapacitated facility location problem under uncertain facility disruptions. The problem is formulated as a two-stage distributionally robust model, which generalizes several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. To solve this generalized model, we propose a cutting plane algorithm, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. The efficiency and effectiveness of the proposed algorithm are validated through extensive numerical experiments. We also conduct a data-driven simulation based on historical typhoon data in China to verify the practical performance of the proposed robust model. The numerical results further reveal insights into the value of information on disruption correlation in improving the robust location decisions.

Most existing facility location models assume that the facility cost is either a fixed setup cost or made up of a fixed setup and a problem-specific concave or submodular cost term. This structural property plays a critical role in developing fast branch-and-price, Lagrangian relaxation, constant ratio approximation, and conic integer programming reformulation approaches for these NP-hard problems. Many practical considerations and complicating factors, however, can make the facility cost no longer concave or submodular. By removing this restrictive assumption, we study a new location model that considers general nonlinear costs to operate facilities in the facility location framework. The general model does not even admit any approximation algorithms unless P = NP because it takes the unsplittable hard-capacitated metric facility location problem as a special case. We first reformulate this general model as a set-partitioning model and then propose a branch-and-price approach. Although the corresponding pricing problem is NP-hard, we effectively analyze its structural properties and design an algorithm to solve it efficiently. The numerical results obtained from two implementation examples of the general model demonstrate the effectiveness of the solution approach, reveal the managerial implications, and validate the importance to study the general framework.

In this paper, we study a multisourcing supply network design problem, in which each retailer faces uncertain demand and can source products from more than one distribution center (DC). The decisions to be simultaneously optimized include DC locations and inventory levels, which set of DCs serves each retailer, and the amount of shipments from DCs to retailers. We propose a nonlinear mixed integer programming model with a joint chance constraint describing a certain service level. Two approaches—set-wise approximation and linear decision rule-based approximation—are constructed to robustly approximate the service level chance constraint with incomplete demand information. Both approaches yield sparse multisourcing distribution networks that effectively match uncertain demand using on-hand inventory, and hence successfully reach a high service level. We show through extensive numerical experiments that our approaches outperform other commonly adopted approximations of the chance constraint.

Containers are widely used in the shipping industry mainly because of their capability to facilitate multimodal transportation. How to effectively reposition the nonrevenue empty containers is the key to reduce the cost and improve the service in the liner shipping industry. In this paper, we propose a two-stage robust optimization model that takes into account the laden containers routing as well as the empty container repositioning, and define the robustness for this model with uncertainties in the supply and demand of the empty containers. Based on this definition, we present the robust formulations for the uncertainty sets corresponding to the ℓp-norm, where p = 1, 2, and ∞, and analyze the computational complexities for all of these formulations. The only polynomial-time solvable case corresponds to the ℓ1-norm, which we use to conduct the numerical study. We compare our approach with both the deterministic model and the stochastic model for the same problem in the rolling horizon simulation environment. The computational results establish the potential practical usefulness of the proposed approach.

In this paper, we study approximation algorithms for two supply chain network design problems, namely, the warehouse-retailer network design problem (WRND) and the stochastic transportation-inventory network design problem (STIND). These two problems generalize the classical uncapacitated facility location problem by incorporating, respectively, the warehouse-retailer echelon inventory cost and the warehouse cycle inventory together with the safety stock costs. The WRND and the STIND were initially studied, respectively, by Teo and Shu (Teo CP, Shu J (2004) Warehouse-retailer network design problem. Oper. Res. 52(3):396–408) and Shu et al. (Shu J, Teo CP, Shen ZJM (2005) Stochastic transportation-inventory network design problem. Oper. Res. 53(1):48–60), where they are formulated as set-covering problems, and column-generation algorithms were used to solve their linear programming relaxations. Both problems can be regarded as special cases of the so-called facility location with submodular facility costs proposed by Svitkina and Tardos (Svitkina Z, Tardos É (2010) Facility location with hierarchical facility costs. ACM Trans. Algorithms 6(2), Article No. 37), for which only a logarithmic-factor approximation algorithm is known. Our main contribution is to obtain efficient constant-factor approximation algorithms for the WRND and the STIND, which are capable of solving large-scale instances of these problems efficiently.