We consider the optimization problem related to the sugar beet transportation when supplying sugar mills in the sugar production. The sugar beet transportation comprises of loading the beet collected at storage piles and then delivering it to sugar mills. An essential prerequisite to guarantee a viability of the considered sugar mill, is to transport the required quantities of sugar beet while maximizing technological quality and minimizing transportation costs. Some growers may be privileged to conduct collection activities in days of a planning period when sugar beet is fresh and contains larger amount of sucrose, while others do not. This unfair collect scheduling plan should be avoided to provide equal treatment of growers. We propose an Integer Linear Programming (ILP) model with the aim of simultaneously maximizing the amount of collected sucrose during the planning period while minimizing the number of vehicles of a homogenous vehicle fleet, including constraints that provide equal opportunities for growers in sugar beet collection. The problem is denoted by the Sugar Beet Transportation Problem under Growers’ Equity Regulations (SBT-GER). By applying the weighted sum method, the two objective functions are combined to transform the bi-objective problem into a single-objective one. Equity regulations are expressed through the requirement that the minimum percentage of the quantity of sugar beet is guaranteed to be collected from each grower on the day of harvest. For real-sized instances, we propose two metaheuristic algorithms, based on Variable Neighborhood Search (VNS) and Greedy Randomized Adaptive Search Procedure (GRASP), respectively. The developed mathematical model and the proposed metaheuristic approaches are evaluated on a set of randomly generated test instances. The obtained results show that VNS outperforms exact solver and GRASP for the majority of examples.

The separate delivery and collection services of goods through different routes is an issue of current interest for some transportation companies by the need to avoid the reorganization of the loads inside the vehicles, to reduce the return of the vehicles with empty load and to give greater priority to the delivery customers. In the vehicle routing problem with backhauls (VRPB), the customers are partitioned into two subsets: linehaul (delivery) and backhaul (pickup) customers. Additionally, a precedence constraint is established: the backhaul customers in a route should be visited after all the linehaul customers. The VRPB is presented in the literature as an extension of the capacitated vehicle routing problem and is NP-hard in the strong sense. In this paper, we propose a mixed integer linear programming formulation for the VRPB, based on the generalization of the open vehicle routing problem; that eliminates the possibility of generating solutions formed by subtours using a set of new constraints focused on obtaining valid solutions formed by Hamiltonian paths and connected by tie-arcs. The proposed formulation is a general purpose model in the sense that it does not deserve specifically tailored algorithmic approaches for their effective solution. The computational results show that the proposed compact formulation is competitive against state-of-the-art exact methods for VRPB instances from the literature.

Hospital departments are continuously challenged to allocate their limited resources—beds, nursing staff, and operating room (OR) time—among patients with different clinical priorities on the waiting list. This paper presents a solution to this intricate planning problem by constructing a novel Integer Linear Programming (ILP) model to schedule elective patient admissions optimally. The goal is to increase the total priority score of the patients admitted, where the score indicates clinical urgency and waiting time, while at the same time honoring the restrictions on bed capacity, nursing hours, and OR availability through a multi-day planning horizon. The model uses binary decision variables for patient admission as well as pre-processed parameters to map the individual patient resource consumption over the time they are expected to stay in the hospital. We showcase the model's application through a realistic scenario with synthetic data. The outcomes reveal that the suggested framework not only effectively creates an optimum admission timetable but also considerably enhances the use of vital resources when compared to a first-come-first-served standard. The ILP model is not only potent but also clear and just decision-support tool for hospital management, according to the study's findings. By assigning priority to clinical criteria and making patient access more equitable, it provides a data-guided way to improve operational productivity, as well as to decide the trade-offs that come with the limitations of healthcare facilities in terms of resources. Besides, the model is computationally capable of meeting the needs for the inclusion of extra real-world constraints and adjustments.

Modern healthcare's growing intricacy, which is a result of an aging population and increased use of technology, necessitates the use of very advanced operational management. A solution is provided to the issue of hospital optimization through the use of a unified mathematical programming method that combines two main areas, resource allocation and workforce management. We present a new multi-objective Linear Programming (LP) and Mixed-Integer Linear Programming (MILP) framework made especially for the "smart hospital" setting. The model is able to do several things at once; it is able to come up with the best patient admission scheduling, use the beds and operating rooms in the most efficient way possible, and schedule the nurse staff all while being subjected to the real-world constraints of demand fluctuations, staff preferences, skill mixes, and labor regulations. A detailed case study is presented which clearly shows the effectiveness of the model, revealing a 15% boost in resource utilization, a 12% decrease in operational costs, and increased nurse schedule satisfaction, all whilst still being able to offer high standards of patient care coverage. The findings point out the great opportunity of using integrated optimization models to make healthcare delivery systems more efficient, cheaper, and more adaptive, thus giving hospital administrators a strong decision-support tool.

The Quadratic Assignment Problem (QAP) is still considered to be one of the most difficult and widely used models in combinatorial optimization. The layout of healthcare facilities has been its most significant application area since the 1970s, representing a crucial field of study for increasing operational efficiency, patient safety, and staff flow. The QAP context has been continually altered and supplemented to cover the particular intricacies of the healthcare sector. After Elshafei's groundbreaking paper in 1977, the QAP framework was reinvented and extended to the point where it gained acceptance in the healthcare facility location planning area. This review offers a synthesis of the existing literature from 1977 to 2025 and classifies the research into ten different methodological streams: Exact Solution Methods, Classical Heuristics, Metaheuristics, Hybrid Approaches, Robust Optimization, Fuzzy QAP, Stochastic Programming, Multi-Objective QAP, Special Structure Exploitation, and Parallel & Dis-tributed Computing. The critical assessment of the transition of solution procedures and how the techniques for handling uncertainty have been developed shows how the research has progressed from modeling with one deterministic objective to a sophisticated data-driven approach where multiple objectives are characterized as well as the inherent uncertainties of the system. The analysis indicates the integration and hybridization trend—in the case of algorithms, objectives, and data sources is quite strong—pointing out the future lines of research in areas such as real-time adaptive layouts, deep learning integration, and pandemic-responsive design.