In this paper, we investigate FRB, which is the single facility Euclidean location problem in the presence of a (non-)convex polygonal forbidden region where travel and location are not permitted. We search for a new facility's location that minimizes the weighted Euclidean distances to existing ones. To overcome the non-convexity and non-differentiability of the problem's objective function, we propose an equivalent reformulation (RFRB) whose objective is linear. Using RFRB, we limit the search space to regions of a set of non-overlapping candidate domains that may contain the optimum; thus we reduce RFRB to a finite series of tight mixed integer convex programming subproblems. Each sub-problem has a linear objective function and both linear and quadratic constraints that are defined on a candidate domain. Based on these sub-problems, we propose an efficient bounding-based algorithm (BA) that converges to a (near-)optimum. Within BA, we use two lower and four upper bounds for the solution value of FRB. The two lower and two upper bounds are solution values of relaxations of the restricted problem. The third upper bound is the near-optimum of a nested partitioning heuristic. The fourth upper bound is the outcome of a divide and conquer technique that solves a smooth sub-problem for each sub-region. We reveal via our computational investigation that BA matches an existing upper bound and improves two more.
This paper proposes a literature review and a mathematical goal program for the design of timetables for radiologists. The goal program converts the tedious monthly tasks of the head of the radiology department of a leading hospital to a simple goal optimization problem that abides to the regulations of the Ministry of Health and avoids conflicting issues that may arise among coworkers. The optimization problem which is designed for the tactical level can also be used at the strategic level (i.e., account for a long time horizon) to plan for longer term constraints such vacations, medical and study leaves, recruitment, retirement, etc. Despite its large size, the problem is herein solved using an off-the-shelf solver (CPLEX). Empirical tests on the design of timetables for the case study prove the efficiency of the obtained schedule and highlights the time gain and utility of the developed model. They reflect the practical aspects of timetabling and radiologists' availability. Specifically, not only does the model and its solution reduce the effort of the Department head in this design stage, but it also promotes social peace among the technicians and a sense of fairness / unbiasedness. In addition, the designed model can be used at the operational level as a rescheduling tool by those technicians wishing to trade their shifts, and as a sensitivity analysis tool by managers wishing to study the effect of some phenomena such as absenteeism, increasing or decreasing the workforce, and extending work hours on the welfare of patients.
Minimizing the total cost of transportation of a homogeneous product from multiple sources to multiple destinations when demand at each source and supply at each destination are deterministic and constant is commonly addressed in the literature. However, in practice, the demands and supplies may fluctuate within a certain range in a period due to variations of the global economy. Subsequently, finding the upper bound of the minimal total cost of this transportation problem with varying demands and supplies (TPVDS) is NP hard. The upper and lower bounds are of prime importance for financial sustainability. Although the lower bound of the minimal total cost can be methodologically attained, determining the exact upper bound is challenging. Herein, we demonstrate that existing methods may in some instances underestimate this upper minimal total cost bound. We further propose an alternative efficient and robust method for the purpose, provide theoretical evidence of its good performance in terms of solution quality, and undertake a theoretical analysis to prove its superiority in comparison to existing techniques. We further validate its performance on benchmark and newly generated instances. Finally, we exemplify its utility on a real-world TPVDS.
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