In decision analysis, the expected value of perfect information (EVPI) is a commonly used evaluation tool. We focus on the concept of maximum possible EVPI (MaxVPI) and demonstrate its relationship with one of the decision analysis criteria, minimax regret. The maximum possible EVPI is easy to evaluate because there is no need to estimate the probabilities of the states of the world. The MaxVPI is an achievable upper bound, meaning that there is a set of probabilities for which EVPI=MaxVPI, and the EVPI for any set of probabilities cannot exceed MaxVPI. We demonstrate this approach on two stochastic facility location models, in the context of both finite and infinite states of the world as well as available options. We also consider a situation in which only partial information about the probabilities is given.

We propose a Disc-Weber (DW) problem in which demand is represented as discs rather than points. We show exact formulations of the new model, study its properties, and propose a very close approximation. The new DW problem is strictly convex, and its analytical continuous derivative exists for the entire domain when all areas (radii) are positive. Therefore, unlike the original Weber problem, it lacks non-continuous derivatives at fixed points. We solve several illustrative examples to global optimality using the exact and approximate formulations. Our analysis and experiments show that the original point-based Weber formulation underestimates the objective function when area demand is present. Furthermore, the optimal solution for point and disc representations of demand can be substantially different. We also show that the DW problem approximation results are virtually identical to its exact formulation.

This paper investigates and extends the Generalized binomial distribution (GBD) as a novel extension of the classical binomial distribution designed to model dynamic feedback dependencies commonly encountered in decision-making contexts. By adjusting success probabilities based on prior outcomes, the GBD addresses critical limitations of traditional binomial and beta-binomial models, such as underestimating variance and failing to capture complex behaviors like bi-modality. Through extensive simulation studies and real-world applications, including marketing optimization and stock performance analysis, we demonstrate the versatility of the GBD in capturing feedback-driven phenomena. We also provide interactive tools (dashboards) to facilitate parameter exploration and practical implementation. These findings establish GBD as a valuable analytical tool for decision-making in management and operations offering both enhanced interpretability and actionable insights while retaining the simplicity of classical binomial frameworks.

In operations research and management science (ORMS), binomial modeling and Poisson distribution are routinely used to summarizethe number of ''successes'' in a list of $n$ events (wins in a season, stocks up-days in a window, defective units in a batch,customers reached by repeated advertising, etc.). The classical binomial and Poisson distributions assume (i) a constant success rateand (ii) independence between trials. In many ORMS computational statistics settings, neither assumption is fully credible: successes may be positivelyreinforced (heterogeneous skill, clustering, persistence) or negatively reinforced (sampling without replacement, mean reversion,capacity constraints). The purpose of this paper is to introduce to the ORMS community the Generalized Binomial (GBD) and the Correlated Poisson (CPD) distributions, which assume correlated events. They are useful tools for analyzing many ORMS stochastic models. Many applications apply the binomial or Poisson distributions even though the GBD or CPD may be more appropriate for such analyses because events are interdependent. If events are not correlated, the GBD and CPD will not be significantly different from the binomial and Poisson distributions because the binomial and Poisson distributions are special cases of the GBD and CPD. It is a good idea for researchers and practitioners to analyze a model by the GBD or CPD, and revert to the uncorrelated distributions when appropriate.An easy to use Excel file for calculating the GBD and CPD, find their optimal parameters, and the statistical analysis, is provided.

Determining an appropriate sequence of interrelated activities is one of the keys to developing a complex product. One of the approaches used to sequence activities consists of solving the feedback length minimization problem (FLMP). Several metaheuristic algorithms for this problem have been reported in the literature. However, they suffer from high computational costs when dealing with large-scale problem instances. To address this research gap, we propose a fast hybrid heuristic for the FLMP, which integrates the simulated annealing (SA) technique with the variable neighborhood search (VNS) method. The local search component of VNS relies on a fast insertion neighborhood exploration procedure performing only O(1) operations per move. Using rigorous statistical tests, we show that the SA-VNS hybrid is superior to both SA and VNS applied individually. We experimentally compare SA-VNS against the insertion-based simulated annealing (ISA) heuristic, which is the state-of-the-art algorithm for the FLMP. The results demonstrate the clear superiority of SA-VNS over ISA. The SA-VNS hybrid technique produces equally good or better results across all tested problem instances. In particular, SA-VNS is able to find better solutions than ISA on all instances of size 150 or more. Moreover, SA-VNS requires two orders of magnitude less CPU time than the ISA algorithm. Thus, SA-VNS achieves excellent performance regarding solution quality and running time.

Most location papers assume that demand is generated at points. This is convenient for designing models and solution methods. In many real applications, demand is generated in an area. For example, when planning to locate service facilities such as public schools in a large area, there may be a very large number of students in the area, and it is impractical to represent them as individual demand points. Also, students graduate every year and new students start schooling. It is reasonable to model the problem as demand generated at every point in the area. A recommended practical approach suggested in this paper, when very little information is known and future demand is unpredictable, is to assume a uniform demand density and to create a configuration of facilities, using the model and methods presented here. The size of each facility, for example, the number of teachers and classrooms, would be determined every year once there is more demand information. In this paper we derive explicit expressions for calculating the exact value of the objective function, the average distance between all the points in the area to their closest facility, when locating multiple facilities inside a convex polygon. Based on these formulations, heuristic procedures for finding the best locations of facilities are developed and tested with good results.

We formulate a planar obnoxious p-center problem with circular forbidden regions, such as a minimum required distance from some or all the demand points. We use geometric computation to develop a tight lower bound on the optimal value of the objective function and propose solution techniques which yield optimal solutions to several multiple large-scale tsplib instances in less than few hours’ time. We observe that, for larger forbidden regions, the obnoxious p-center problem reaches the best possible overall objective for a relatively small number of facilities. Our proposed solution approach is particularly useful in instances with multiple overlapping forbidden regions.

Generalizing the familiar two-correlation comparison, this paper presents a dependence-robust omnibus test to evaluate whether an outcome is equally correlated with multiple predictors. By accounting for shared sampling variation, the test simultaneously avoids false alarms and missed discoveries. The test also nests the pairwise test as a special case. Monte Carlo studies show near-nominal size ( ≈ 5% at α=0.05 ) for n≥50 across diverse dependence structures and under moderate non-normality (e.g., t_5 errors) together with high power for moderate departures from equality. We illustrate the method on publicly available educational data and provide an interactive web app (size/power simulator and point-and-click analysis) to facilitate adoption. Collectively, the results support the omnibus test as a practical default when assessing equality of outcome–predictor correlations to be augmented by pairwise contrasts for succinct context rather than primary inference.

The problem of locating facilities, such as garbage dumps or noisy factories, that have a negative impact on surrounding communities was initiated about 50 years ago, and intensively analyzed in the literature. In the models proposed in the literature, it is assumed that all facilities have the same capacity (size), and thus inflict the same negative impact. In this article, we propose to construct facilities of different capacities with a total given capacity. Facilities located in open areas are far from communities and can be larger without impacting the most affected community. This extra capacity allows facilities close to communities to be smaller without changing the total capacity and significantly reduces the negative impact on the most affected community. Such an important extension leads to a more complicated formulation that we were able to simplify and get much better solutions in shorter run times. In extensive computational experiments, we found in some cases that the negative impact on the most affected community is reduced by more than 50% when facilities of different capacities are established.

In this paper we propose a new and effective meta-heuristic that improves the performance of multi-start improvement algorithms (or local searches). It is a general follow-up (or post-optimizing) procedure that attempts to find a better final solution than the best one found by any multi-start approach. The additional required effort and run time is negligible. The output of the multi-start improvement algorithm is a single best-found solution. To apply the post-optimization step, we need to retain during parts of the the multi-start phase, a set of best-found solutions of specified cardinality (we use 5). Several offspring are created for each pair in this elite set. Each of the offspring are then improved by the improvement algorithm. If a better final solution is found, the process is repeated on the current generation of offspring. The process stops when the elite set of the current generation fails to find a better solution. Generating offsprings is an essential component of genetic algorithms. The meta-heuristic is tested on the planar p-median multi-facility location problem which is known to have many (may be millions for the instances tested in this paper) local optima. Forty eight well researched instances were tested, and the meta-heuristic improved the best known solution for all of them, in about the same run time required by the multi-start approach.

The competitive facilities location problem is to find the locations of one or more new facilities among existing competing facilities that maximize the captured market share by the new facilities. In the leader-follower models, the leader locates his facility first and anticipates a competing follower to locate his facility optimally, knowing the location selected by the leader. The leader's objective is to maximize his captured market share following the follower's action. In this paper we consider the case that the leader is not sure whether there will be a follower or not. We investigate and test the minimax regret and the expected value rules. Algorithms for locating the competing facilities anywhere in the plane, which are more difficult to solve, were designed. The follower's problem and the leader's problem when there is no follower are solved to optimality within a given relative accuracy by available algorithms. For solving the leader's problem when a follower will react knowing the leader's move, a special heuristic algorithm that can be applied to other location problems is constructed. The leader's location problem with the objective of minimax regret or expected value decision rules, and 20,000 demand points, were solved in less than 4 min of computer time.

The Weber problem requires finding the location of a new facility that minimizes a sum of weighted Euclidean distances to a set of given demand points in the plane. An open question relates to the probability that the optimal location coincides with a demand point. This question is not only of theoretical interest, but also of practical importance, since the convergence rate of popular algorithms used to solve the Weber problem may be sub-linear when the optimal solution occurs at a demand point, while the rate is linear if this does not occur. It has been shown by simulation that for the unweighted problem which we consider in this paper, and demand points uniformly distributed in a disc, this probability approaches 1/n as the number of demand points n becomes large. In practical problems where several new facilities need to be located, the average number of demand points assigned to each one can be relatively small. For this reason, it is also important to consider this probability for small values of n. Using a geometric proof for n=3 and 4, we show here that when demand points can be located anywhere in the plane (i.e., a region with unspecified boundaries), the probabilities are, respectively, 1/3 and 1/4 . For larger values of n, there is no known geometric argument. However, using a novel approach to describe a uniform distribution of the demand points anywhere in the plane, we are able to apply known properties of the planar random walk to prove that the probability is exactly 1/n for all n≥ 3 .

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The problem of locating facilities that have a negative impact on surrounding communities was initiated in the 1970’s, and intensively analyzed in the literature. In the models proposed in the literature, it is assumed that all facilities have the same capacity (size). In this paper we propose to construct facilities of different capacities. Such an important extension leads to a more complicated formulation that we were able to simplify. The simplified formulations resulted in much better solutions in shorter run times than the more complicated formulations.

The Gray Pattern Problem (GPP) requires selecting a given number (p) of pixels (or dots) within a central rectangle to be colored black (the remaining pixels remain white) to form a repeating pattern covering the plane that is as uniformly gray as possible. Taillard proposed the original model in 1995, which is based on the principle of minimization of entropy taken from physics. The aim of this paper is to present alternate formulations of the GPP inspired by dispersion models studied in location theory. We show that these variants have much greater tractability than the original model, while producing patterns of equivalent or better quality.

In a competitive multi-purpose (MP) trips model, it is assumed that the proportion of customers that do MP trips is given. In this paper, we investigate a model with a stochastic proportion of MP trips. Five decision analysis criteria are analyzed for finding the best location for a new facility under these circumstances. We first design a general approach that can be applied to many location problems. We then prove that by the optimistic rule, the solution is one of the extreme cases and construct solution algorithms to solve problems based on the other four rules optimally. The approach is demonstrated on the MP trips model. We performed computational experiments on instances between 100 and 20,000 demand points on five different models and two distance decay functions. The largest problem was solved within a given accuracy in about 8 h for the most time-consuming model. Our general approach can be applied to other location models (not necessarily competitive) when a parameter of such models is stochastic.

The most basic location problem is the Weber problem, that is a basis to many advanced location models. It is finding the location of a facility which minimizes the sum of weighted distances to a set of demand points. Solution approaches have convergence issues when the optimal solution is at a demand point because the derivatives of the objective function do not exist on a demand point and are discontinuous near it. In this paper we investigate the probability that the optimal location is on a demand point, create example problems that may take millions of iterations to converge to the optimal location, and suggest a simple improvement to the Weiszfeld solution algorithm. One would expect that if the number of demand points increases to infinity, the probability that the optimal location is on a demand point converges to 1 because there is no “space" left to locate the facility not on a demand point. Consequently, we may experience convergence issues for relatively large problems. However, it was shown that for randomly generated points in a circle the probability converges to zero, which is counter intuitive. In this paper we further investigate this probability. Another interesting result of our experiments is that FORTRAN is much faster than Python for such simulations. Researchers are advised to apply old fashioned programming languages rather than newer software for simulations of this type.

The inverse-square law states that a source's effect is inversely proportional to the distance from that source squared. In continuous location problems, objective functions often obey the inverse-square law. This work shows that for any region in D dimensions, the maximum inverse-square effect is on the region's boundary if D<4, the minimum is on the boundary if D>4, and both the maximum and minimum are on the boundary if D=4.

There are many planar multiple facilities location problems for which the optimal locations tend to be spread out. The most popular of these is the planar p-median problem. With this in mind, we propose several procedures to generate sparse configurations as starting solutions. The proposed procedures are easy to implement, and can be used as modules combined in different sequences within heuristics such as a recent trajectory-based procedure that we tested in this paper.The procedures are tested experimentally on a set of 24 large problem instances with up to 10,000 demand points and 100 facilities. We are able to demonstrate that the sparse starting solutions generated by the new procedures lead to significant improvements of final p-median solutions.

One of the classic foundational constructs of Location Science was proposed by Alfred Weber in 1909. His construct involved finding the location for a single production facility which minimized the sum of weighted distances of transporting the needed raw materials from localized sources along with the sum of weighted distances in delivering the final product to one or more markets.The first objective of this paper is to review the major advancements in this simple classic single facility location problem and its variations. One can find in the literature a very large number of algorithms to solve the standard Weber problem. Some are iterative and others are finite even for geometric Euclidean and rectilinear spaces. Moreover, some schemes are efficient (theoretically) and others are practically quite fast.The second goal of this paper is to show that many extensions of the standard Weber problem can be solved by solving a polynomial number of standard Weber problems. This unifying result implies, in particular, that all these extensions are polynomially solvable since the standard Weber problem can be solved in polynomial time. In addition, with this unifying approach we solve some important planar non-convex Euclidean location problems in polynomial time.