Glossary

9.4 Facility location problems

9 min read•august 21, 2024

Facility location problems are a key area of computational geometry, focusing on optimally placing resources in space. These problems blend concepts like distance calculations and spatial partitioning to solve real-world challenges in resource allocation and service provision.

From emergency services to retail outlets, facility location problems have wide-ranging applications. By understanding different problem types, objective functions, and solution techniques, we can tackle complex spatial optimization challenges across various industries and public services.

Types of facility location

  • Facility location problems form a crucial subset of computational geometry, focusing on optimal placement of resources in spatial contexts
  • These problems intersect with various geometric concepts, including distance calculations, spatial partitioning, and optimization techniques
  • Understanding different types of facility location problems provides a foundation for solving complex real-world spatial optimization challenges

Point vs area facilities

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  • Point facilities represent discrete locations with no spatial extent (ATMs, cell towers)
  • Area facilities cover a defined region or service area (parks, shopping malls)
  • Point facilities often use center-based models, while area facilities may employ coverage-based approaches
  • Geometric representation differs significantly between point and area facilities, impacting problem formulation and solution methods

Single vs multiple facilities

  • Single facility problems focus on finding the optimal location for one resource (emergency response center)
  • Multiple facility problems involve placing several facilities to serve a given area or population (chain of retail stores)
  • Multiple facility problems introduce additional complexity, including facility interactions and service area partitioning
  • Geometric techniques like Voronoi diagrams often play a crucial role in solving multiple facility problems

Capacitated vs uncapacitated facilities

  • Uncapacitated facilities assume unlimited service capacity, simplifying problem formulation
  • Capacitated facilities have limits on the demand they can serve, adding constraints to the optimization problem
  • often lead to more complex geometric partitioning of service areas
  • Solving capacitated problems may require advanced techniques like network flow algorithms or

Objective functions

  • Objective functions in facility location problems define the optimization goals, guiding the solution process
  • These functions often incorporate geometric concepts such as distances, areas, or spatial relationships
  • Understanding different objective functions is crucial for formulating and solving various types of facility location problems

Minisum problems

  • Aim to minimize the total sum of distances between facilities and demand points
  • Also known as the in the context of multiple facility location
  • Often used in applications where transportation costs or travel times are primary concerns (warehouse locations)
  • Geometric median serves as the optimal solution for single facility minisum problems in continuous space

Minimax problems

  • Focus on minimizing the between any demand point and its nearest facility
  • Also referred to as the p-center problem for multiple facility locations
  • Commonly applied in emergency service placement to ensure quick response times for all locations
  • Solving minimax problems often involves geometric concepts like farthest-point Voronoi diagrams

Covering problems

  • Seek to maximize the coverage of demand points within a specified distance or time threshold
  • Include set covering problems (cover all points) and maximal covering problems (cover as many points as possible)
  • Often used in applications with service radius constraints (wireless network coverage)
  • Geometric techniques like circle intersection and union calculations play a key role in solving covering problems

Distance metrics

  • Distance metrics define how spatial separation between points is measured in facility location problems
  • Choice of distance metric significantly impacts problem formulation, solution methods, and resulting facility placements
  • Understanding different distance metrics is crucial for accurately modeling real-world spatial relationships

Euclidean distance

  • Measures the straight-line distance between two points in a plane or higher-dimensional space
  • Calculated using the Pythagorean theorem: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  • Commonly used in open-space applications or when movement is unrestricted (drone delivery services)
  • leads to circular service areas in single facility problems

Manhattan distance

  • Also known as L1 distance or city block distance
  • Calculated as the sum of absolute differences in coordinates: d=∣x2−x1∣+∣y2−y1∣d = |x_2 - x_1| + |y_2 - y_1|
  • Appropriate for grid-like environments or when movement is restricted to orthogonal directions (urban navigation)
  • results in diamond-shaped service areas in single facility problems

Network distance

  • Measures distance along a predefined network of paths or roads
  • Requires a graph representation of the transportation network
  • More accurately models real-world travel distances in many applications (emergency vehicle routing)
  • Solving problems often involves graph algorithms like Dijkstra's shortest path

Solution techniques

  • Solution techniques for facility location problems encompass a wide range of computational and geometric approaches
  • The choice of technique depends on problem complexity, desired solution quality, and computational resources available
  • Understanding various solution methods is essential for tackling different types of facility location problems effectively

Exact methods

  • Provide guaranteed optimal solutions but may be computationally intensive for large problems
  • Include techniques like mixed-integer linear programming and branch-and-bound algorithms
  • Often rely on geometric properties to reduce the search space (convex hull for minisum problems)
  • Suitable for small to medium-sized problems or when optimality is crucial (critical infrastructure placement)

Heuristic approaches

  • Offer approximate solutions with reasonable computational efficiency
  • Include methods like greedy algorithms, local search, and constructive heuristics
  • Often leverage geometric intuitions to guide the search process (incremental Voronoi diagrams)
  • Useful for large-scale problems or when quick solutions are needed (real-time facility relocation)

Metaheuristics

  • Advanced search techniques that combine multiple heuristics or search strategies
  • Include methods like genetic algorithms, simulated annealing, and tabu search
  • Often incorporate problem-specific geometric knowledge to improve search efficiency
  • Effective for complex, multi-objective facility location problems (supply chain network design)

Geometric considerations

  • Geometric concepts and structures play a fundamental role in formulating and solving facility location problems
  • Understanding these geometric considerations is crucial for developing efficient algorithms and gaining insights into problem properties
  • Many geometric structures used in facility location have broader applications in computational geometry

Voronoi diagrams

  • Partition space into regions based on the nearest facility, defining service areas
  • Dual to the , providing a geometric relationship between facilities
  • Efficient construction algorithms exist, such as Fortune's sweepline algorithm
  • Applications include modeling market areas in competitive facility location problems

Delaunay triangulation

  • Maximizes the minimum angle of all triangles in the triangulation
  • Dual to the , connecting nearest neighbor facilities
  • Used in interpolation and proximity analysis for facility location problems
  • Efficient algorithms exist for construction, including incremental and divide-and-conquer approaches

Convex hull applications

  • Defines the smallest convex set containing all points in a given set
  • Used to reduce the search space in some facility location problems (single facility minisum)
  • Efficient algorithms for construction include Graham scan and Quickhull
  • Applications in identifying potential facility locations on the boundary of a service area

Constraints and variations

  • Facility location problems often include additional constraints or variations to model real-world complexities
  • These constraints can significantly impact problem formulation, solution methods, and computational complexity
  • Understanding common constraints and variations is crucial for addressing practical facility location challenges

Capacity constraints

  • Limit the maximum demand that can be served by each facility
  • Lead to more complex service area partitioning, often requiring flow-based optimization techniques
  • May result in overlapping service areas, unlike uncapacitated problems
  • Often modeled using mixed- or network flow algorithms

Budget constraints

  • Restrict the total cost or number of facilities that can be established
  • Introduce trade-offs between facility coverage and resource limitations
  • May lead to multi-objective optimization problems balancing cost and service quality
  • Often addressed using knapsack-like algorithms or Lagrangian relaxation techniques

Time-dependent problems

  • Consider variations in demand, travel times, or other parameters over time
  • Require dynamic modeling approaches to capture temporal changes in the system
  • May involve predictive elements to anticipate future conditions (traffic patterns)
  • Often solved using time-expanded networks or rolling horizon approaches

Computational complexity

  • Understanding the computational complexity of facility location problems is crucial for developing efficient solution strategies
  • Many facility location problems are NP-hard, requiring careful consideration of problem size and solution quality trade-offs
  • Complexity analysis guides the choice between exact and approximate solution methods

NP-hardness of problems

  • Many facility location problems, including p-median and p-center, are NP-hard
  • Implies that no known polynomial-time algorithm exists for finding optimal solutions
  • Complexity often arises from the combinatorial nature of facility placement decisions
  • Understanding NP-hardness motivates the development of and heuristics

Approximation algorithms

  • Provide solutions with guaranteed performance bounds relative to the optimal solution
  • Often leverage geometric properties to achieve good approximation ratios
  • Include techniques like local search with swaps for p-median problems
  • Trade off optimality for computational efficiency in large-scale problems

Polynomial-time special cases

  • Certain facility location problems admit polynomial-time exact solutions under specific conditions
  • Include single facility problems in continuous space with convex objective functions
  • Exploiting problem structure and geometric properties can lead to efficient algorithms
  • Understanding these special cases provides insights for developing heuristics for more general problems

Real-world applications

  • Facility location problems have numerous practical applications across various industries and public services
  • Understanding these applications helps in formulating relevant problem instances and selecting appropriate solution techniques
  • Real-world problems often require considering multiple objectives and constraints simultaneously

Emergency services location

  • Involves placing facilities like fire stations, hospitals, or ambulance depots
  • Often uses minimax objectives to minimize worst-case response times
  • May incorporate backup coverage requirements for system reliability
  • Geometric considerations include road network distances and population density distributions

Retail outlet placement

  • Aims to optimize store locations for maximum market coverage and profitability
  • Often considers competitor locations and market share capture
  • May use gravitational models to account for customer attraction to different store sizes
  • Geometric analysis includes trade area delineation and spatial interaction modeling

Warehouse distribution centers

  • Focuses on optimizing the location of warehouses in a supply chain network
  • Often uses minisum objectives to minimize total transportation costs
  • May consider multi-echelon systems with intermediate distribution centers
  • Geometric aspects include service territory partitioning and transportation network analysis

Modeling approaches

  • Different modeling approaches capture various aspects of facility location problems
  • The choice of modeling approach impacts problem formulation, solution methods, and computational requirements
  • Understanding these approaches is crucial for selecting the most appropriate model for a given problem

Continuous vs discrete models

  • Continuous models allow facilities to be placed anywhere in the feasible region
  • Discrete models restrict facility locations to a predefined set of candidate sites
  • Continuous models often leverage geometric properties for efficient solution methods
  • Discrete models may be more computationally tractable but can miss optimal locations between candidate sites

Deterministic vs stochastic models

  • Deterministic models assume all parameters are known with certainty
  • Stochastic models incorporate uncertainty in demand, travel times, or other factors
  • Deterministic models are often simpler to solve but may not capture real-world variability
  • Stochastic models use techniques like scenario analysis or chance-constrained programming

Static vs dynamic models

  • Static models assume fixed parameters over the planning horizon
  • Dynamic models account for changes in the system over time (population growth)
  • Static models are simpler but may not capture long-term trends or seasonal variations
  • Dynamic models often use time-expanded networks or rolling horizon approaches

Evaluation metrics

  • Evaluation metrics quantify the performance of facility location solutions
  • These metrics help compare different solutions and assess their effectiveness in meeting objectives
  • Understanding various metrics is crucial for selecting appropriate objective functions and validating solutions

Coverage ratio

  • Measures the proportion of demand points served within a specified distance or time threshold
  • Calculated as the number of covered points divided by the total number of demand points
  • Often used in set covering and maximal covering problems
  • Geometric interpretation involves calculating the union of facility service areas

Average distance

  • Computes the mean distance between demand points and their nearest facilities
  • Used to assess overall system efficiency in minisum problems
  • Can be weighted by demand to account for varying importance of different points
  • Geometric calculation involves determining nearest facility assignments and distance summation

Maximum distance

  • Identifies the largest distance between any demand point and its nearest facility
  • Key metric for minimax problems and emergency service applications
  • Represents the worst-case scenario for service accessibility
  • Geometric interpretation involves finding the farthest point from any facility

Key Terms to Review (41)

Approximation Algorithms: Approximation algorithms are strategies used to find solutions that are close to the best possible answer for complex optimization problems, particularly when finding exact solutions is computationally infeasible. These algorithms provide guarantees on how close the solution is to the optimal one, often expressed as a ratio or a factor of the optimal solution. They are particularly valuable in scenarios where decision-making needs to happen quickly and accurately despite high computational complexity.
Average distance: Average distance refers to the mean distance between a set of points, often calculated in the context of optimizing locations for facilities. This measure is crucial when assessing how well a facility serves its users by minimizing the average travel distance for all users to reach the facility. Average distance helps inform decisions about where to place services and resources efficiently.
Budget constraints: Budget constraints refer to the limitations that consumers or decision-makers face in terms of their available resources when making choices about facility locations. These constraints play a crucial role in determining how to allocate limited funds efficiently to maximize benefits, particularly in scenarios where multiple locations need to be evaluated for cost-effectiveness and accessibility. Understanding budget constraints helps in optimizing decisions regarding investments in infrastructure and services within facility location problems.
Capacity Constraints: Capacity constraints refer to the limitations on the ability of a facility to produce or serve based on its physical, financial, or operational resources. In facility location problems, these constraints can influence decisions on where to place facilities in order to optimize service delivery while considering the maximum capacity each facility can handle. This plays a crucial role in ensuring that the demands of the market are met efficiently without overloading any single facility.
Christos H. Papadimitriou: Christos H. Papadimitriou is a prominent computer scientist known for his significant contributions to the field of theoretical computer science, particularly in the areas of computational complexity and algorithms. He has authored influential texts and research papers that have shaped the understanding of algorithmic efficiency and problem-solving, with applications in areas such as facility location problems. His work emphasizes the importance of designing algorithms that not only solve problems but also consider their computational feasibility.
Continuous vs Discrete Models: Continuous and discrete models are two fundamental approaches used to represent and analyze mathematical problems. Continuous models deal with data that can take any value within a given range, often represented by real numbers, while discrete models handle distinct, separate values, usually integers. Understanding the distinction between these models is essential for effectively addressing various problems, including those related to facility location.
Convex Hull Applications: Convex hull applications refer to the various practical uses of the convex hull, which is the smallest convex polygon that can enclose a set of points in a plane. This concept is crucial in solving facility location problems, as it helps in determining optimal locations for facilities based on the distribution of demand points. The ability to quickly compute the convex hull allows for efficient decision-making in various scenarios, such as minimizing distances and optimizing resource allocation.
Coverage ratio: The coverage ratio is a measure used in facility location problems that indicates the proportion of demand points that can be served by a set of facility locations. This metric is essential for assessing how well the selected facilities can meet the needs of the surrounding population or demand regions, thereby influencing decisions about where to place new facilities. A higher coverage ratio signifies better service provision and efficiency in reaching more customers or demand points.
David S. Johnson: David S. Johnson is a prominent figure in the field of computer science, particularly known for his contributions to algorithm design and optimization, as well as computational geometry. His work has significantly influenced the study of facility location problems and geometric set cover, which involve optimizing resource allocation and covering geometric spaces efficiently.
Delaunay triangulation: Delaunay triangulation is a method for creating a triangulation of a set of points in a plane, ensuring that no point is inside the circumcircle of any triangle in the triangulation. This property maximizes the minimum angle of the triangles, helping to avoid skinny triangles and producing well-shaped triangles that are useful in various applications.
Deterministic vs Stochastic Models: Deterministic models provide precise outputs based on defined inputs and parameters, leading to consistent results each time the model is run. In contrast, stochastic models incorporate randomness and uncertainty, producing different outcomes even with the same set of inputs, which is essential in scenarios where variability is a key factor. Both types of models play crucial roles in decision-making processes, especially in analyzing complex systems like facility location problems where factors such as demand fluctuations and transportation costs can vary.
Emergency Service Location: Emergency service location refers to the strategic placement of facilities, such as fire stations, hospitals, and police stations, to ensure optimal response times and service coverage for emergency situations. Effective emergency service location is critical for minimizing response times during crises, enhancing public safety, and improving resource allocation. This concept is integral in facility location problems, where the objective is to find the best sites for such services based on various factors, including population density, accessibility, and demand.
Emergency services location: Emergency services location refers to the strategic placement of emergency response facilities, such as fire stations, ambulances, and police departments, to optimize response times and coverage for communities. The main goal is to ensure that help arrives as quickly as possible during critical situations, which involves analyzing population density, traffic patterns, and geographic barriers. Effective facility location can significantly impact the efficiency of emergency services and ultimately save lives.
Euclidean distance: Euclidean distance is a measure of the straight-line distance between two points in Euclidean space. It's calculated using the Pythagorean theorem and is crucial in various applications, particularly in identifying how far apart points are from one another in multi-dimensional spaces. Understanding this concept is essential when analyzing spatial relationships, especially in tasks such as finding the nearest neighbor or determining optimal facility locations.
Exact Methods: Exact methods are algorithms used in optimization and computational geometry that guarantee finding the best possible solution to a problem within a finite amount of time. These methods often involve rigorous mathematical formulations and exhaustive search techniques, ensuring that the solutions derived are optimal and not merely approximations. They are particularly vital in facility location problems where precision is critical for determining the best locations for services or resources to minimize costs or maximize efficiency.
Facility Location Theory: Facility location theory is a field of optimization that focuses on determining the best locations for facilities to minimize costs and maximize service efficiency. It plays a crucial role in logistics and supply chain management, where selecting optimal site locations can significantly impact overall performance and profitability.
Greedy algorithm: A greedy algorithm is a problem-solving strategy that makes the best possible choice at each small step, aiming for a locally optimal solution in hopes that these local solutions will lead to a global optimal solution. This approach is efficient for many optimization problems, as it focuses on immediate gains without reconsidering previous choices. Greedy algorithms are particularly useful in scenarios where making locally optimal choices leads to an acceptable overall solution, often seen in various geometric and combinatorial optimization contexts.
Heuristic approaches: Heuristic approaches are problem-solving methods that utilize practical techniques and shortcuts to produce solutions that may not be optimal but are sufficient for reaching immediate goals. These methods are especially useful in complex problems where traditional algorithms may be too slow or complicated. They rely on experience-based techniques to guide decision-making and simplify the process of finding solutions.
Integer Programming: Integer programming is a mathematical optimization technique where some or all of the variables are constrained to take on integer values. This method is particularly useful in situations where decisions are discrete, such as in facility location problems, where you might need to decide how many facilities to open or which locations to select while minimizing costs and meeting specific constraints.
K-means clustering: K-means clustering is an unsupervised machine learning algorithm used to partition data into k distinct clusters based on feature similarities. This technique iteratively assigns data points to the nearest cluster centroid and updates the centroids based on the current assignments, ultimately leading to a well-defined grouping of the data. The effectiveness of k-means clustering in organizing data makes it applicable in various fields such as data mining, image processing, and market segmentation.
Linear programming: Linear programming is a mathematical method used for optimizing a linear objective function, subject to linear equality and inequality constraints. It is extensively utilized in various fields to find the best possible outcome, whether maximizing profits or minimizing costs. The relationships between variables in linear programming are represented as convex sets, allowing for efficient solutions through geometrical interpretation, especially in contexts involving facility location and optimization problems.
Manhattan Distance: Manhattan distance, also known as taxicab or city block distance, measures the distance between two points in a grid-based system. It is calculated as the sum of the absolute differences of their Cartesian coordinates, which reflects the total distance traveled when moving along grid lines rather than in a straight line. This concept is particularly important in algorithms that require measuring proximity, such as finding the nearest neighbor or determining optimal locations for facilities.
Maximum distance: Maximum distance refers to the farthest possible distance between a set of points in a geometric space, often applied in various optimization problems. In the context of facility location problems, it helps determine the best placement of facilities so that the maximum distance any client must travel to reach a facility is minimized. This concept is crucial for ensuring accessibility and efficiency in service delivery.
Metaheuristics: Metaheuristics are advanced problem-solving frameworks designed to find approximate solutions for complex optimization problems, especially when traditional methods are inefficient. These techniques employ strategies that guide the search process toward better solutions while balancing exploration and exploitation of the solution space. They are particularly useful in facility location problems where the goal is to optimally position facilities to minimize costs and improve service efficiency.
Minimax problem: The minimax problem is a decision-making strategy used in various fields, particularly in optimization and game theory. It involves minimizing the maximum possible loss or cost in a worst-case scenario. This concept is crucial in facility location problems, where one seeks to find optimal sites for facilities that minimize the maximum distance to the farthest customer or client, ensuring equitable service distribution.
Minisum problem: The minisum problem is a type of facility location problem that aims to minimize the total distance or cost associated with serving a set of demand points from one or more facility locations. This problem is particularly relevant in logistics and operations research, where the objective is to find optimal placement for facilities such as warehouses or service centers to best serve customer needs while reducing operational costs.
Network Distance: Network distance refers to the measurement of distance within a network, often used to evaluate the shortest path between two points considering the connectivity and structure of the network. This concept is crucial for optimizing various facility location problems where determining the most efficient placement of facilities relative to demand points can significantly impact logistics and service efficiency.
Network flow model: A network flow model is a mathematical representation used to optimize the flow of resources through a network, consisting of nodes and directed edges. In the context of facility location problems, this model helps determine the most efficient way to transport goods from multiple supply points to various demand locations, minimizing costs and maximizing service levels. This approach is particularly beneficial in logistics and supply chain management.
Np-hardness of problems: NP-hardness refers to a class of problems for which no known polynomial-time algorithm exists, meaning that solving them efficiently in all cases is believed to be impossible. These problems are at least as hard as the hardest problems in NP, and they can be used to demonstrate the complexity of various computational challenges. In the context of facility location problems, understanding NP-hardness is crucial because it indicates that finding optimal solutions might require significant computational resources, thus affecting decision-making in logistical and operational contexts.
P-median problem: The p-median problem is a mathematical optimization issue that focuses on selecting p locations for facilities (medians) in order to minimize the total distance or cost associated with serving a set of demand points. This problem arises in various fields such as logistics, urban planning, and telecommunications, where efficient placement of resources is essential for maximizing service while minimizing costs.
Polynomial-time special cases: Polynomial-time special cases refer to specific instances of computational problems that can be solved efficiently in polynomial time, even though the general problem may be NP-hard. These special cases arise in various optimization problems, including facility location problems, where certain constraints or structures allow for more efficient algorithms to be applied.
Retail outlet placement: Retail outlet placement refers to the strategic positioning of stores and shops within a geographical area to maximize visibility, accessibility, and sales potential. This involves analyzing various factors such as customer demographics, competition, traffic patterns, and local regulations to determine the best locations for retail establishments. Effective retail outlet placement can significantly enhance a business's reach and influence in the market.
Service Level: Service level refers to the performance metric that measures the degree to which a facility meets customer expectations regarding service delivery, typically in terms of speed, accuracy, and availability. It serves as a critical benchmark in facility location problems, influencing decisions about where to establish services based on factors like proximity to customers and the ability to meet their needs efficiently.
Set Covering Problem: The set covering problem is a classic optimization issue where the goal is to select the smallest number of subsets from a given collection so that their union covers all elements in a specified universal set. This problem is significant in various applications, especially in facility location problems, where it helps determine the optimal placement of facilities to ensure complete service coverage for clients or locations.
Static vs Dynamic Models: Static models are representations of systems or scenarios that do not change over time, while dynamic models account for changes and evolution in systems, incorporating the element of time. In the context of facility location problems, these models help in analyzing and determining the best locations for facilities based on various factors like demand, costs, and distances, with static models focusing on a fixed set of parameters and dynamic models adjusting these parameters as they evolve.
Time-dependent problems: Time-dependent problems refer to issues where the conditions or parameters change over time, affecting the optimal solutions in various applications. In the context of facility location problems, these scenarios require considering factors such as travel times, demand fluctuations, or service availability that can vary throughout the day or week. Addressing these problems often involves dynamic optimization techniques to ensure that decisions adapt to changing circumstances.
Total distance minimization: Total distance minimization refers to the process of finding a location for a facility such that the overall distance between the facility and a set of demand points is minimized. This concept is crucial in facility location problems, where the objective is to optimize the placement of facilities to serve customers effectively while reducing transportation costs and improving service efficiency.
Voronoi Diagram: A Voronoi diagram is a partitioning of a plane into regions based on the distance to a specific set of points, called seeds or sites. Each region consists of all points closer to one seed than to any other, which makes Voronoi diagrams essential for spatial analysis, nearest neighbor search, and various applications in computational geometry.
Warehouse distribution centers: Warehouse distribution centers are large facilities used to store and manage inventory, designed to facilitate the efficient distribution of goods to retailers or customers. These centers play a crucial role in the supply chain by optimizing storage, order fulfillment, and transportation processes, ensuring that products are delivered promptly and accurately.
Warehouse placement: Warehouse placement involves determining the optimal locations for warehouses within a supply chain to minimize costs and maximize efficiency in delivering goods to customers. This concept connects to various logistics and distribution strategies, impacting the overall performance of a business by influencing transportation costs, service levels, and inventory management.
Weighted graph: A weighted graph is a type of graph where each edge has a numerical value, or weight, assigned to it, representing the cost, distance, or some other measure associated with traveling between two vertices. These weights allow for more complex analysis of relationships between nodes, making them crucial in various applications such as network design and optimization problems. In the context of facility location problems, weighted graphs help in determining optimal placements of facilities by considering not just distances but also the significance of those distances based on the weights assigned.
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