FACILITY LOCATION

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FACILITY LOCATION
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Agenda

• Objectives in Location Models
• Continuous Location Problems
• Competitive Facility Location in the Plane
• Multifacility Retail Networks

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• OBJECTIVES IN LOCATION PROBLEMS

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INTRODUCTION

• Modern location theory is usually said to have
  begun with Webers treatise on the locations of
  industries.
• Webers treatment of location problems in general
  and objective functions in particular has
  dominated the literature.
• Hakimi, whose landmark contribution first
  distinguished between cost minimizing minisum
  objectives and minimax objectives. This is the
  first major attempt by an operations researcher
  to solve location models.
• 1960s- early 1970s minisum objectives were
  applied to private sector applications, whereas
  minimax objectives were suggested for public
  sector applications.
• These two types of problems reigned for about a
  decade

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• In the mid-1970s Goldman, Church and Garfinkel
  suggested obnoxious location models.
• Models in which customers consider the facility
  undesirable and avoid the facility and stay away
  from it.
• Exp Nuclear reactors, garbage dumps, water
  purification plants.
• Another line of investigations were based on
  fairness and equity.

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• We will survey the most important contributions
  to location models with different objectives but
  we also purpose a framework for them.
• Our discussion in the framework of attracting and
  repelling forces, which move facilities into
  certain directions, similar to those of a magnet.
  
• Customers or users may either try to attract (or
  pull) desirable facilities closer to them, or
  repel (or push) undesirable facilities from them.

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Agenda

• Location Models major components and describing
  a framework of decision making.
• The location of desirable facilities
• The location of undesirable facilities
• Objectives that try to keep a balance while
  locating facilities
• Results and a number of potential future research
  directions.

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Elements of Location Models

• Space
• The number of new facilities
• The number of existing facilities
• The decision makers objective
• Customer

• Continuous Location Models
• Discrete Location Problems
• Network Location Models
• Forbidden Zones

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Elements of Location Models
The decision maker has p N
facilities to locate. The establishment of
each facility carries a given fixed cost. In case
of p 1 facilities, you should also decide how
to allocate customer to facilities.

• Space
• The number of new facilities
• The number of existing facilities
• The decision makers objective
• Customer

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Elements of Location Modelsc
Customers patronize Exp School
boards Customers who decide where to shop or
obtain a service. When locating multiple new
facilities, interactions between them may have to
be considered. Terms in the objective include
distances between new facilities Military
installations, franchises of gas stations, fast
food restaurants will disperse because of
competition

• Space
• The number of new facilities
• The number of existing facilities
• The decision makers objective
• Customer

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Elements of Location Models

• Space
• The number of new facilities
• The number of existing facilities
• The decision makers objective
• Customer

Two different types The number of facilities to
be located by the decision maker is fixed. The
number of facilities to be located by the
decision maker is variable.
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Elements of Location Models

• Space
• The number of new facilities
• The number of existing facilities
• The decision makers objective
• Customer

Minimization of a function of distance Maximizatio
n of a function of distance Balancing of
functions of distance
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Elements of Location Models

• Space
• The number of new facilities
• The number of existing facilities
• The decision makers objective
• Customer

Distribution customers are either distributed
uniformly over a given a set, or they are located
at specific points in space. Actual demands
Inelastic or elasticfunctions of distance or
other externalities. Customer behavior
Customers may be assigned to a facility or may be
free to choose. Exp if we have to locate a
facility that sells required good for which there
is no substitute, then customers have to buy the
good, regardless of how far or undesirable the
facility is, and thus their demand is fixed. Most
goods or services are not of that type.
Retaurants If the nearest restaurant is too far
away, there is usually the choice of eating at
home
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Main Categories of Objectives

• The major elements of location models are
  discussed, we are now able to derive the main
  categories of objectives.
• We can classify objectives in many ways. One of
  these ways is the difference between private and
  public facilities.
• Private Minisum locate a facility so as to
  minimize the sum of transport costs. That is best
  for the planner.
• Public Minimax Library determine a location
  that is close to even the most distant potential
  user. If we use minisum, we will find a solution
  that minimizes the total distance travelled by
  all library users. It can be create an
  experession of collective welfare. In th case
  of public facilites , it means what to planners
  judge to be best for the users.

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Main Categories of Objectives

• Considering any existing or planned facility, a
  user has essentialy four choices find it
  desirable, undesirable, partly desirable and
  partly undesirable, or be indifferent.
• If a user considers a facility desirable, then he
  would like to have it as close as possible to his
  own location. In such a case, the user would like
  to exert a certain pull on the facility.
  Restaurants (Pull Objectives)
• If the facility is judged undesirable, the user
  would like to push the facility away. Nuclear
  centre, garbage dump.(Push Objectives)
• The case of judging a facility partly desirable
  and partly undesirable means customer wanting a
  facility close, but not too close. Supermarket.
  Close convenience not too close noise,
  traffic. push-pull objectives. (Balancing
  Objectives)

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• A number of well known scenarios
• Inelastic demand and no competition. Users who
  consider the facilities desirable will patronize
  any one of the planners facilities, most likely
  the closest facility.
• If the planner has to pay for transporting the
  goods to the customers, he will open as many
  facilities as necessary in order to minimize the
  sum of facility costs and transport costs the
  standart simple facility location problem
  results.
• If the facilities are franchises,the planner may
  wish to balance business between franchises, a
  balancing objective.
• If demand is elastic, the number of facilities to
  be located is variable, and the planner various a
  cost minimization objective, we obtain a min-cost
  covering problem. If the number of facilites is
  fixed, the planner may follow a max covering
  objective.

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PULL OBJECTIVES

• The planner wishes to locate close to the
  customers.
• Minimizing the sum of weighted distances with so
  called minisum objectives
• Consider the location of a private facility under
  profit maximization. Assume for now that
  delivered pricing is applied, the decision maker
  pays for the transport of the goods to the
  customers.

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• pi price charged at location i (p pi for all
  i)
• di demand at location i
• t i j unit transport cost between a customer
  located at i and the facility located at j.
• Suppose that the demand will be satisfied
  exactly, and let F be the fixed and c be the
  variable costs in a linear cost function. The
  profit function
• Where the first sum denotes the revenue, the
  second specifies the variable production costs,
  and the last sum defines the transport cost.

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• For any set of fixed prices pi, revenue and
  production costs are fixed, so that profit is
  maximized whenever transport costs are minimized.
• For mill pricing customers pay their own
  transport costs. Profit
• For any fixed price p, the function simplifies to
  a maximization of sales. The problem thus becomes
  one of maximum capture. Maximize revenue, sales,
  the number of customers reached.

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• Facilities x1, x2, xp
• Customers v1, v2,..vn
• Locations of customers and facilities can either
  be in some space (Rd) or on a network.
• Weights wi,i 1,2,,n to the customers the
  minisum objective is

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• Single facility location problems with minisum
  objective in the plane are also referred as Weber
  problems or, sometimes Steiner- Weber problems.
• The objective of te problem is convex. The
  optimal solution equals the center of gravity.
  As far as forces are concerned, squared Euclidean
  introduce a higher penalty for far away customers
  than standart Euclidean distances, thus resulting
  in locations that avoid distant customers, even
  those with smaller weights.
• In case p 1facilities are to be located, the
  problem is frequently referred to as the
  multi-Weber problem. Two version exist
• In the first, shipments between facilities
  as well as individual facilities and customers
  are known.
• In the second version it is not known which
  customers are going to be served by an individual
  facility until facilities are actually located.

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• The latter version gives rise to two major
  issues locating facilities and allocating
  customers to them.
• Coopers alternate location- allocation
  heurastic strating with any intial location of p
  facilities, allocate each customer to its closest
  facility, and then optimally relocate a facility
  among the customers allocated to it. The
  procedure is repeated until it converges. One of
  the difficulties related to multi-Weber problems
  is that its objective function is neither convex
  nor concave.

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• The first results regarding minisum location
  problems on networks are found in Hakimis
  seminal paper. One of the major contributions of
  this paper is the characterization of the set of
  solutions for minisum problems. Hakimi proves
  that for problems on general graphs with customer
  demands occuring at vertices, the minisum
  objective is concave and thus its minimum is
  attained at one of its boundaries( a vertex of
  the graph). This Hakimi property also holds for
  p 1 facilities.

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• Another major class of problems attempts to
  locate a facility so as to make the longest
  customer- facility distance as short as
  possibleminimax objectives Formally, a
  p-facility problem with minimax objective can be
  written as
• Locations satisfying the above relation are
  usually referred to as p-centers. In the plane,
  locating one center can be accomplished by
  geometrically inspired methods.

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• Some researchers have tried to combine the
  efficiency of the median and the fairness of the
  center objective. One way of accomplishing this
  is by way of a constrained p-median problem. A
  formulation with a minisum(median) objective
  that has upper bound constraints on all
  facility-customer distances.
• dmax maximal distance between any customer and
  its closest facility.

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• Cent-dians whose objective is a linear convex
  combination of minisum and minimax objectives.
• Viz.f fsum (1- )f max with
  01

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Objectives in Covering Problems

• A different type of objective is used in covering
  problems. The idea is to locate facilities so as
  to cover customers and thus capture their
  demand. There are two versions of covering
  problems, one in which the number of facilities
  is fixed and another in which it is variable.
• With a fixed number of facilities, we obtain a
  max cover promlem in which we locate facilities
  so as to maximize the demand captured by the
  facilities.
• With a variable number of facilities, we may wish
  to cover the entire demand with the smallest
  number of facilities, given that no customer is
  farther than a prespecified distance d from its
  closest facility. This is the min(cost) cover
  problem.

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Supply and Demand Curves

• In case of public facilities is to maximize the
  gain to users, the value of the service minus the
  cost of travel minus the cost of supplying the
  service. This gain to users equals consumers
  surplus, the difference between the aggregate
  price consumers are prepared to pay for a product
  and the amount they are actually paying. The
  concept is illustrated in the quantity price
  graph of Figure.1. The resulting equilibrium
  price is at which
• units of the good will be sold. Suppose
  now that all potential customers are ordered
  according to their willingness to pay, starting
  with those willing to pay the highest price for
  the good. Then all customers to the left of
  are prepared to pay the corresponding prices on
  the demand curve, but actually pay only the lower
  price .

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Supply and Demand Curves

• Their agglomerated benefit is the difference
  between what they are prepared to pay and what
  they actually pay. This is the consumers
  surplus, shown as the diagonally hatched area in
  Figure. Similarly, we can order producers
  according to their willingness to sell for a
  given price starting with the one who would sell
  for the lowest price. Then the producers surplus
  is the agglomerated difference between what
  producers would sell for and the price they
  actually receive. This is the horizantally
  hatched area in figure 1.

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Supply and Demand Curves
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DEMAND FUNCTIONS

• Three types of demand functions
• Linear
• Modified constant elacticity
• Modified exponential
• Pearn and Ho prove that consumer surplus is
• convex under any nondecreasing demand
• function. As the Hakimi property holds, only
• nodes have to be considerd as potential
• solutions.

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• The computational part of the study considers
  four problems
• Locate a prespecified number of facilities so as
  to maximize consumers surplus
• Maximize net consumer benefit with a variable
  number of facilities.
• The usual p median
• Minimize total cost with variable p.
• 1-2 and 3-4, differences in locational behavior
  increase as the size of network increases.

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• The steeper the slope of the demand function, the
  larger the difference between fixed and inelastic
  demand models. This is not surprising at all as
  inelastic demand has zero slope, so all this says
  is that the results are getting more different
  the more different the problems are. Also, the
  maximization of net benefit tends to result in
  fewer facilities than the minimization of total
  cost.

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PUSH OBJECTIVES

• Location problems involving undesirable
  facilities have been discussed since the early
  1970s.
• To visualize the difference between minisum and
  maxisum objectives
• EXP Four equally weighted customers be located
  at the vertices of a square in R2 with Euclidean
  distances. The unique minisum location of a
  single new facility is at the intersection of the
  diagonals.

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• The maxisum objective, on the other hand, is
  optimized by locating the facility as far away as
  possible towards infinity, clearly showing the
  forces that push the facility. In order to avoid
  such unbounded optimal solutions, it is necessary
  to define a set within which the facility can be
  located, something unnecessary with minisum
  objectives for which it is a elementary exercise
  to prove that it is always better to locate the
  facility inside the convex hull spanned by the
  customers.
• If we were to restrict in our above exp the
  location of a new facility to the convex hull of
  customers, the maxisum objective would be
  optimized by locating at any of the four
  customers locations.

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• A facility provides services and disservices,
  work and pollutants, at the same time called
  semiobnoxious
• The authors delineate an algorithm and give an
  example of the location of a toxic dump.

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P- dispersion Problem

• A new type of problem which is quite distinct
  from other location models. It does not include
  customers and focuses exclusively on the
  facilities. The goal is to locate p new
  facilities on a network so as to maximize the
  minimum interfacility distance, where distances
  are measured as the length of the shortest path
  between facilities. The model is referred o as a
  p-dispersion problem. (maximinmin problem)
  military and civil applications exist for this
  model.

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P-defense Problem

• P-defense problems objective is to maximize the
  sum of distaces between each facility and its
  nearest neighbor (maxsummin problem)

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BALANCING OBJECTIVES

• The main goal of balancing is to achieve equity
  and fair. Balancing objectives attempt to
  balance distances between facilities and
  customers.
• Most applications that employ equity objectives
  are part of public decision making where the
  objective is to serve the population fairly.
  There are, however, apllications of balancing
  objectives in the private sector as well.
  Workload among employees.

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• If we consider the distribution of all facility-
  customer distances for any given solution, push
  and pull objectives optimize some function of the
  mean. In contrast, most equity objectives attempt
  to minimize the variability of the distribution
  of distances. It is by no means clear why that
  should be the case. Surely, it seems to fit the
  objective of equity if all users have the same
  distance to walk or drive in order to reach a
  public facility.

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Efficiency-EffectivenessEquality-Equity

• Efficiency deals with a simple measure of output
  such as profit or cost, effectiveness puts
  efficiency in relation to some expressionor
  perceived need.
• A very large number of poorly positioned
  ambulances may effectively serve the population
  they may, however, not be very efficient in doing
  so.
• Equality, an objective that similar to
  efficiency, tries to achieve balance without any
  reference to need.
• Equity, similar to effectiveness, relates
  equality to need in some way
• Efficiency/effectivenessequality/equity

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Main Categories of Balancing Objectives

• There are only two different categories of
  balancing objectives
• Minimize the spread of the distriution
  minimization of the maximum.
• Minimize the deviation from a central point
  minimization of the range.

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• Four substitute criteria for equity is
  identified
• Equal payments
• Equal outputs
• Equal inputs(per area or capital)
• Equal satisfaction of demand( equal input per
  unit of demand)
• In location analysis, interest would appear to
  focus on the last two criteria with weights
  suitably defined as population or potential
  demand.

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• A covering objective with a properly defined
  covering distance d could solve the equal
  input/satisfaction of demand problem in the sense
  that no potential customer would be farther away
  than d from the nearest facility such as
  ambulance or fire station. Most importantly, the
  choice of the substitute criterion is a matter of
  value.

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Axioms To Evaluate Balancing Objectives

• Reasonable axioms have been used to evaluate
  balancing objectives.
• The first axiom isScale invariance is satisfied
  if the degree of equality does not change with
  the type of measure applied to the problem. In
  other words, an income distribution does not
  become any more unequal because incomes are
  measured in pounds Sterling rather than in
  dollars. In terms of location problems, the
  distribution of facility- customer distances is
  no more unequal if we decide to measure distances
  in feet rather meters.

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• The second popular axiom is the principle of
  transfers (Pigou-Dalton).The principle of
  transfers was originally desgined for income
  distributions where it states that an income
  distribution becomes less unequal if one dolar
  from an above- average income is transferred to
  an income that is below average. (income-
  facility-customer distance)
• There are some criteria for the selection of
  balancing criteria
• Analytic tractability
• Normalization of measures (in the interval 0,1
• Impartiality (nodes are numbered arbitrarily)
• Pareto optimality

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• Equality objectives whose function is to locate
  facilities so as to equalize all
  customer-facility distances. The first seven
  objectives are non-normalized.
• Min f(x) minimize measures of inequality or
  inequity
• di distance between the facility and the i-th
  customer
• average customer-facility distance.

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• Extreme values
• Center
• (center)
• (range)

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• (MAD Mean absolute
  deviation)
• Variance
• Maximum deviation
• (variance of logarithms)

Maximum deviation could possibly be redefined to
only consider distances longer than average(for
desirable facilities) or shorter than average(for
undesirable facilities).
3,4 and 5 satisfy neither scale invariance nor
the principle of transfers, and 6 does satisfies
scale invariance, but not the principle of
transfers.
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• The last class of non-normalized equality
  objectives measure differences between all pairs
  of distances.
• The next class consist of normalized equality
  objectives. In essence, they are similar to the
  above objectives, but divided by a measure of
  central tendency, typically the mean. One such
  measure is
• Schutzs Index

The index does satisfy scale invariance, but
violates the principle of transfers.
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• Another popular measure is
• (coefficient of variation)

The coefficient of variation is easy to use and
compute as it is derived directly from the
variance. Among the most popular measures of
equality is the Gini index. It is defined as
twice the area between the Lorenz curve and the
45 line, the latter denoting total equality.
(Gini coefficient)
The Gini coefficient is described via the Lorenz
curve known from economics. It first orders the
customers in increasing order of their distance
to the facility that serves them and then plots
the cumulative proportion of customer against the
cumulative proportion of distance.
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(No Transcript)
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• A point (a,b) on the Lorenz Curve then means that
  the first 100a percent of customers are a total
  of 100b percent of the total distance away from
  the facility. Clearly, the straight line segment
  between (0,0) and (1,1) includes points of
  complete equality. The Gini index then expresses
  the area between the Lorenz curve and the line of
  perfect equality in relation to the area below
  the line of perfect equality.

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Another measure in this set derives from
information theory. It is
Theils entropy coefficient
Finally, there is Atkinsons coefficient which
was designed for social welfare functions. It can
be written as
Atkinsons coefficient
Among the indices of this class,schutzs
coefficient does satisfy scale invariance but not
the principle of transfers. 9-12 coefficients
satify both scale invariance as well as the
principle of transfers. All indices discussed
above measure inequality none of them is
designed to be used as an expression for
fairness. To do so, we require another parameter.
This parameter is designed to put the distance
into relation to need. We can called this
parameter as an attribute, and denote it by ai.
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• i1,2,,n
• Examples for attributes could be population size
  or demand for a spesific service. All location
  models with balancing objectives use equality
  rather than equity objectives.
• The simplest equity objective is a simple measure
  comparing distances with attributes.

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• (13) is reminiscent of the mean absolute
  deviation in fact, it reduces to MAD if all
  attributes are equal to average distance.
• A variance-like measure is
• Hoovers concentration index expresses the
  average sum of absolute differences of the
  proportions of distance and need.

Hoovers concentration index
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• Coulters coefficient squares the differences,
• A pairwise comparison between the
  distance-to-attribute ratios similar to (7) is
• Adams coefficient is written as
• Finally, there is the sociospatial version of
  Schutzs coefficient

Adams coefficient
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Result

• Three different types of objectives were
  identified pull, push and balance objectives.
• Applying the concept of pulling, pushing, or
  balancing forces to any of the objectives in the
  individual contributions mentioned.

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Continuous location problems

• Frank Plastria
• Vrije Universiteit Brussel

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CONTINUOUS LOCATION PROBLEMS
The central concern in location problems is
determining sites for one or more new facilities
with respect to a set of fixed points (existing
facilities, markets, sources) with which it
should interact. Such problems are termed
continuous when the underlying space both for
facility sites and given points is a continuous
one because of all points under discussion are
determined by way of one or more variables which
may vary continuously. The special case of a
linear or circular network may be considered as
both continuous and will in continuous framework
be described as one dimensional. But most
continuous location problems will be considered
in a space of dimension at least two. Two
dimensional problems are the most popular for
evident reasons of geographical nature but higher
dimensional problems also appear within a
multiple floor building or underwater for three
dimensions.
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FORM
The concept of distance is central of continuous
location. Once the distance measures to be used
has been determined, one may start comparing
different sites on the basis of quality of the
involved interactions. It is traditional to
represent objectives as costs and therefore the
aim will be minimization. Thus the initial
paradigm is multiobjective. Since the qualities
directly depend on distance, we face a vector
optimization problem where each component is
represented by an individual distance. Such
problems are usuallly solved by the determination
of the pareto-optimal. In practise, solution
often need a more precise description of the
actual aim of the problem described by a
globalising function which combines all
individual distances into a global value. So
optimization problem has become single objective
and well defined.
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FORM
As a main classification factor we will
distinguish between pure location problems where
the only aim is the determination of an optimal
site and location allocation problems in which
other aspects are an important part of the
solution to the problem at hand. This distinction
is not always wright, as the globalising function
consisting of the largest among all distances may
be either viewed as a pure function problem when
the knowledge of which distance is the largest is
not relevant, or as a location-allocation problem
when this knowledge is important. The second
classification factor is the number of facilities
to be located single facility and multifacility
location problems.
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DISTANCE
Assumptions A distance measure on a space X
attributes to each pair of points x,y a real
value d(x,y). The most popular properties
connected with a suitable distance measure are
-nonnegativity d(x,y)0
for all x,y
X -stationarity d(x,y)0
for all x
X -definiteness d(x,y) 0
for all x y
X -symmetry d(x,y)d(y,x)
for all x,y X -triangle
inequality d(x,y)d(y,z)d(x,z)
for all x,y,z X. Two first properties
are generally accepted, third relates to the
discussion of what is a point,when two points
are at zero distance, redefine them as a point,
recovering definiteness. Forth property, symmetry
has been attacked unrealistic, since some of the
movements are not reversible without change.
64
DISTANCE
For optimisation purposes some more properties of
the distance measure are often required
-convexity d(x,.)is a
convex function for all x X. -analytic
expression d(x,y) is given by a closed
analytical formula, only involving the
coordinates of z and y. -linearity
d(x,y) is a piece-wise linear function
of the coordinates of z and y. The convexity
assumption is needed in order to enable the use
of convex analysis, the main analytical tool in
nonlinear optimisation. An analytical expression
of distance is useful since it allows for
application of classical tools of calculus, while
linearity leads to linear programming problems
for which optimisation techniques and programs
abound.
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DISTANCE
With respect to the convexity assumption, its
combination with the basic properties was shown
to imply that d is derived from a gauge. This is
a realvalued function g defined on a real
vectorspace X satisfying -nonnegativity
g(x)0 -definiteness
g(x)0 if x0 -positive homogeneity
g(x,y)rg(x) when r0 -sublinearity
g(xy)g(x)g(y).
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NORMS and GAUGES
The most familiar and used distance measure is
the euclidean. It is derived from the euclidean
norm where
(i1,2,..,n) is the i-th coordinate of x in
. The second tapper is the rectangular distance
derived from rectangular norm .
Its use is mostly rationalized by reference to
movement in a perffectly rectangular network,
being the lenght of a shortest path consisting
of pieces parallel to either one of the
coordinate axes. The third tapper is the maxi
distance derived from the norm
. In the plane the maxi-distance may
be seen as rectangular after a change of
coordinates, by explaining why it did not receive
more attention.
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NORMS and GAUGES
All of the mentioned distances are part of the
family of distances, derived from the
norms g(x) (x). ,for 1pFor p1, we obtain rectangular form, denoted by
. The euclidean norm is . The maxi form
is .
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DOMINANCE, EFFICIENCY, VOTING
Pareto For a destination a in a space X, to
which distance is measured by , a site x
X is no worse than some other site y if
(x,y) (y,a). Two points are always
comparable for this destination. Asking that
comparison comes out the same for all
destinations, this is the traditional Pareto view
on such multi-objective problems. With respect to
some set of destinations A X, often supposed
to be finite, and for each a A a distance
measure up to a. -x dominates y if x is no
worse than y for all a A. -x strictly
dominates y if x dominates y and is better than y
for at least one destination. -x strongly
dominates y if x is better than y for all a
A.
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CONDERCET, SIMPSON, et al.
Second way of deciding that one site is better
than another from the theory of voting and has
direct applications to situations with
competition. Assigning a normalized weight to
each destination, we say that dominates y
if the total weight of all destinations for which
x better than y at least .1 dominance is seen
strong dominance, 1 efficience is seen week
efficience. ½ efficient point is known as a
Condercet point, is seen as an optimal site in
terms of simple majority voting is prefered to
it by a majority of no other site voters. In most
cases no condercet points exists, and one has to
choose 1/2 in order to obtain
efficient points. Taking for the smallest
of such fractions, the corresponding
efficient points are called Simpson points. These
are sites a minimal number of voters will be
against when compared to any other site, the
points at which a first firm should locate in
order to loose as little as possible to a
competitor locating anywhere next, assuming
nearness only determines the market.
70
SINGLE FACILITY LOCATION PROBLEMS
General general single facility location
problems are obtained as follows -Let X denote
the space of sites and A be a finite family of
destinations, each having a location in X. In
some cases, we may have two destinations at the
same location, we denote location by the same
notation as the destination. When the set of
destinations A is infinite, it is described by
continuous distribution. For each destination a
A, denotes the distance measure up to
a. To any site x X a vector of distances
D(X) is associated, with components the distances
of x to each destination
a A These
distances are combined by a globalizing function
G ?R t ?G(t) into a global objective
value assumed to be a disutility, hence to be
minimized within a given feasible region S
X.
71
SINGLE FACILITY LOCATION PROBLEMS

• In short we face the problem MIN G(D(X))I x
  S, any optimal solution of which will be called
  optimal site.
• The most studied globalising functions are of
  following two kinds
• G decomposes into a sum of one-dimensional
  functions, i.e G/t) in which case is
  called minisum problem,
• G is a maximum of one-dimensional functions,
  G(t) and we obtain a minimax
  problem.

72
MINISUM PROBLEMS
Fermat-Steiner-Weber The simplest problem in this
class is obtained when G is taken as linear,
G(T) , and shortly called WP. The
coefficient of are traditionally called
weights, and are usually assumed to be positive.
The original WP is in the plane, using euclidean
distance, has a history back to Fermat. Even in
this simple case, direct analytic solution is
impossible, except for some particular
situations up to four destinations with equal
weights, up to three destinations with general
weights, when one destination comes sufficiently
close to holding a majority and when destinations
are colinear, and the problem reduces to the
simple one-dimensional case.
73
MINISUM PROBLEMS
Fermat-Steiner-Weber One of the main difficulties
of the WP is due to its nondifferentiability,
occuring at least at each destination point, due
to the fact that a norm is never smooth at the
origin. WP may be solved by an analog method and
any convex programming method. By far the most
popular technique of this kind is the method
which may be viewed as a calculated step gradient
method. In this method there is a step-length
parameter into the algoritm, which may be choosen
freely between 1 and 2, possibly different at
each step. The sum of powers of euclidean
distances should be minimized. In case of squared
euclidean distance one obtains a minimal sum of
squares model classical in statics, which is
directly solved by the centre of gravity and
whose level sets are circles around it.
74
MINISUM PROBLEMS
Fermat-Steiner-Weber When the rectangular
distance is used the Weber problem, the objective
function decomposes into two independent
one-dimensional instances. The Weber problem on a
line is simply solved by any median point, a
property still holding for a continuous weight
distribution. Discrete medians can be calculated
in time linear in the number of destinations.
Similar simplicity is obtained when polyhedral
gauges are used as distance measure, since the WP
may be formulated as a linear program. The set of
optimal solutions is an elementary convex set,
even mixed gauges and in any dimension, when
there always exists an optimal solution at an
intersection point. Following the study of
approximated distance measures, WP has also been
extensively studied for norms, often with
the being convex power functions. This
method and results are extended to the more
general case of any globalising function which is
nondecreasing, differentiable and has a property
related to quasiconvexity.
75
MINISUM PROBLEMS
Fermat-Steiner-Weber One of the practical
diffuculties with such iterative methods is to
determine the precision obtained in order to know
when to stop the calculations and several lower
bounds on the objective value are developed. The
more generally appliciable and cutting plane
method generate such lower bounds. The case of
asymetric distance measures studied theoritically
has been studied using a Weiszfeld-type
algorithm. Then conditions under minisum and
minimax single facility problems with general
cost functions having a unique optimal solution
were studied. It appers that the cost functions
claimed by economics have received attention.
They propose a finite descent method yielding a
local minimum, while discussing a global
optimization method. And now it is seen that only
solution technique is global optimisation.
76
MINISUM PROBLEMS
Uncertainty Uncertainty in the destinations is
modelled by a continuous distribution of one or
more destinations. Minisum objective corresponds
to minimising the expected value of the distance
of the facility to the random destinations. The
analytic evaluation of such an objective is only
possible in some simple situations like
rectangular and circular regions. Therefore they
propose the use of approximate average distances.
It optimises without evaluating, by the ellipsoid
method. This method handle the location of a
facility of area type instead of a point
type. Another way to handle uncertainity is to
circumvent the difficulties inherent to the
stochastic approach by replacing each region by
some representative point and solve the resulting
locaiton problem.
77
MINISUM PROBLEMS
Uncertainity The most general question is to
determine all the points which may be optimal for
any choice of positive weights. Such points are
called properly efficient. A related problem of
finding a minimal set containing an optimal site
for any choice of nonnegative weights is taken up
for distances in the plane. Instead of
considering any combination of weights one may
constrain the weights by bounds or inequalities
and ask for the possible optimal sites for the
corresponding Weber problems. This is directly
connected to the sensivity analysis of Weber
point to a change in weights. More recently study
a block norm situation with weights independently
normally distributed and consider the
minimisation of some given percentile of Weber
objective. If the facility is allowed to move in
order to service random demand, questions arise
about if it is best to use stationary positions
and where, or better to move? After researches,
the result is stationary points are better.
78
MINISUM PROBLEMS
Constraints An early research about constraints
developes a special method for solving WP with
distance with in a finite union of convex
polygons. This is followed where mixed norms
and nonlinear cost functions are allowed in a
similar environment, introducing the big square
small square method of global optimisation. The
types of constraints were mainly restricted to
maximal distance constraints around destinations,
indicating that service is only possible or
useful within a certain distance. In restricted
minisum problems the site must be outside some
convex forbidden regions and take additional
barriers to travel into account.
79
MINISUM PROBLEMS
Sphere Researches give that the first
theoritically convergent scheme, based on an
iterative reduction to a spherical minimax
problem. Only a few results were published for
constrained problems on the sphere. Other metrics
on the sphere such as asymetric distance for high
altitude flight in presence of jet stream could
be incorporated in the models. Repulsion For
modeling noxious facilities, the objective is to
minimise the total polution load on a set of
destinations due to the location in a given
region of a gas expelling plant. The objective
function is obtained by considering Gaussian
dispersion models for the gas plume for several
main wind directions and summing the effects on
all destinations. The effects does not only
depend only distance but also on direction and
Big square small square method can be used for
optimisation.
80
MINIMAX PROBLEMS
Centers The simplest minimax problems are
obtained by choosing linear
with positive weights ,yielding
weighted center problems the cases with 1
and 0 for all destinations a A are
usually called center problems(Rawls problems). A
repelling version of this problem, locating a
circle or rectangle of fixed dimension to cover
a minimum weight of given weighted points in the
plane. In case of fixed euclidean distance in the
plane, the center problem asks for the center of
the circle with smallest radius covering all
destinations, with clear applications to location
of emergency facilities, radio emitters. A more
efficient method increases the radius until all
destinations are covered, the worst case
complexity of which was shown to be at least 0,
but reaching an approximate solution in 0 time.
81
MINIMAX PROBLEMS
Centers The higher dimensional euclidean center
problem was shown to be solvable by a finite
series of convex quadratic programs. The
weighted center problem can be solved with
rectangular norm. Maxi-norm problems are same
with the rectangular norm in the plane and
indicates that simple analytical solution extends
to higher dimensions contrary to the rectangular
norm. Probabilistic versions of the center
problem, in which demand arises at random points
in the plane according to some distribution and
the expected maximum distance is taken as an
objective. Any minimax problem with gauges and
nondecreasing s yields a quasiconvex
objective, the ellipsoid method or cutting plane
method may also be used when the feasible region
is convex.
82
MINISUM PROBLEMS
Anticenters Many installations which are either
polluting and involve a risk to the environment
have an effect which spreads out in all
directions and depends on distance. For example,
one may cite the location of storage tanks for
highly inflammable substances, there is a study
of a peak shaving installation for liquid natural
gas in Netherlands. Therefore the site of such
facilities should be choosen as far away as
possible of population centers, leading to
maximising the minimal distance.
83
OTHER OBJECTIVES
Cent-dian A combination of the economic aspect of
the Weber objective and the social aspect of the
Rawls objective is obtained by taking a convex
combination of both minisum and minimax
globalising functions. With linear or convex cost
functions this objective is convex and may be
minimised by convex optimisation. Inequality A
site may be considered ideally distributionally
just when all destinations are at equal distance.
This is unfeasible in most cases, and should be
relaxed into minimising some measure of deviation
among distances. This leads to considering the
range in distances, as the objective to be
minimized. The variant in distances to the
destinations may be used. Other deviation type
objectives like lenght of confidence interval,
coefficient of variation etc. may be useful and
they are considered as inequality measures.
84
OTHER OBJECTIVES
Generalized minimax This method generalize both
minisum and minimax objectives discussed before.
It shows how these kinds of problems may be
solved by an ellipsoid method, when the
objectives are quasiconvex fuctions or gauge
distances. A particular case is the round trip
location problem introduced for rectangular
distance, asking to minimize the maximum distance
around pairs of given destinations. The
multiobjective problem where each criterion is
either of minisum or of minimax type with respect
to fixed destinations is investigated. They
determine the efficient set for such
multiobjective problems with block norms.
85
OTHER OBJECTIVES
Queuing Taking congestion effects into account
leads to new types of single facility location
models. Basic congestion type models are derived
from the classical M/G/1/8 and M/G/c/c queuing
models, for which closed formula are available.
The servers are located at some home location to
be determined. Total service time of a call
consists of random on scene and off scene service
times amd travel time component which is a
function of distance between home location and
calling destination. The M/G/1/8 based model
assumes a unique server, any call occuring while
the server is busy being placed in a queue. The
stochastic queue median problem aims at finding a
home location minimising expected response time.
It shows convexity of this objective function and
derive some location results in the euclidean
distance case. The M/G/c/c based model assumes c
servers, all with the same home location, no
queue being allowed any call occuring when all
servers are busy is lost. The stochastic loss
median problem asks for the home location which
minimizes the fraction of rejected calls.
86
SINGLE FACILITY LOCATION-ALLOCATION PROBLEMS
Minisum Location-allocation problems include
other aspects to be determined than just the site
of the new facility. This means that evaluation
of the globalising function G at some site is an
optimisation problem. More complicated models
arise when recognising that location and design
of a production plant are interrelated, leading
to production-location models. One particular
aspect is optimal price setting for products,
leading to profit-maximising location problems
which may be equivalent to transport cost
minimisation. A different type of models,
classified as minisum location-allocation are
location-routing problems or travelling salesman
location problems, in which the destinations are
visited along a tour of minimal length, instead
of each individually.
87
SINGLE FACILITY LOCATION-ALLOCATION PROBLEMS
Minimax The objective is to find a site
minimising the distance within which a given
fraction of the destinations lie, which are
considered to be weighted. Such location problems
are closely related to maxcovering location
problems asking to determine the site from which
a maximal number of destinations lie within a
given distance. Competition A related class
market maximising problems in which a new
facility is to be located in a competitive
environment where other competitors have their
own market shares. The objective is capturing, by
a good choice of the site, the biggest possible
market share. The captured market share is
expressed in closed analytical form in terms of
distance.
88
MULTIFACILITY LOCATION PROBLEMS
General Let X be the space of sites, A family of
destinations considered as points in X. Let V
denote a finite set of new facilities to be
located in X, which will interact with
destinations and among themselves. The structure
of the interactions may be represented by an
undirected graph H(AUV, E) where nodes indicate
destinations or facilities and edges represent
interactions. Any choice of sites for the new
facilities is an embedding of H in X with fixed
points indicated by destinations, to be viewed as
some mapping x A U V ?X f ? with a
for all a A. With each iteration e(u,v) E
between nodes u and v one associates a distance
measure use to calculate the distance
between the sites and choosen for u and
v, which we will denote as . This
associates to any embedding x a vector odf
distances . These
distances are then combined using a globalising
function into a global value to
be minimised. Locational constraints may be
considered of two main kinds Constraint on nodes
of type saying that
only embeddings with are
admissible.
89
MULTIFACILITY LOCATION PROBLEMS
General Constraints on edges of type
saying that only embeddings with
are admissable. It is clear
that multifacility location problem is much more
complicated than the single facility one it
generalizes. In view of higher complexity it is
not surprising that multifacility location
problems have only been studied for some of the
simpler situations described for single
location. Most of the literature is concerned
with minisum problems, while some attention went
to minimax versions. Minisum When G is taken
as linear, with positive weights and distances
are measured by norms, we obtain the extension of
the Weber problem to multiple facilities. This
model has a high degree of nondifferentiability,
appearing at least two or more facilities have
the same location. It is considered a hard
nonlinear optimisation problem and used as test
problem for nondifferential optimisation
techniques.
90
MULTIFACILITY LOCATION PROBLEMS
Minisum The easiest version is obtained for the
rectangular norm, establishes that there always
exists an optimal solution with new facilities at
intersection points inside the convex hull of the
destination points like in single. The problem
decomposes into one dimensional ones like in
single, solvable as a network problem, a linear
problem. In general, the simple question to
check optimality of a given embedding is far from
trivial, due to nondifferentiability, a situation
often occuring at the optimal solution.
Sufficient conditions for coincidence at optimum
involving the weights only, have been derived in
general setting of distances measured by
metrics Exact optimality conditions were obtained
for coincidences involving at most twp facilities
and for general case for fixed euclidean norm and
mixed norms.
91
MULTIFACILITY LOCATION PROBLEMS
Minisum Contrary to the single facility Weber
problem, the euclidean multifacility location
problem may have several optimal solutions.
Solution methods of many different kinds have
been applied mainly to euclidean norm case and
fixed norm case. Then they proposed to adapt
Weiszfelds method to the facility problem and
showed that this method yields decreasing
objective values as long as coincidences do not
appear. Other techniques have been tried such as
approximation methods, a trajectory method amd
subgradient method, but with limited success.
Then much more powerful methods were developed
which include special treatment in case of
coincidences. All these methods being iterative
ones, using local information as derived from
subgradients, shoud include stopping rule. In
order to measure the quality of the solution
reached the lowerbound, applicable to fixed
norm models may prove useful.
92
MULTIFACILITY LOCATION PROBLEMS
Minisum Up to date the most comprehensive method
for solving minisum multifacility location
problems seems the primal dual technique, which
applies to arbitrary mixed norms problems
including arbitrary convex constraints. The main
advantage of this method is it avoids all
problems of nondifferentiability and
automatically generates lowerbounds enabling
rational stopping rules, while its main
disadvantege is convergence may be rather slow
and therefore it may not be competitive. Similar
multifacility problems have been investigated on
the sphere wth great-circle distance, proposing a
Weiszfeld-like method without convergence proof.
It is clear that minisum multifacilty location
problems still offer great opportunities for
research, extending what has been ansd still may
be done for single facility models. The
difficulties are much larger, in particular when
repelling destinations would be considered, since
thede imply loss of convexity, thus all of the
techniques above become inoperable.
93
MULTIFACILITY LOCATION PROBLEMS
Minimax Researches have paid much less attention
to the minimax multifacilty problems. It is
obtained by taking for
. We consider only linear functions on
each individual interaction distance and then
only for positive and nonnegative The
rectangular norm version may be reduced to a
linear program or solved by a network flow
method. Euclidean or fixed norm problems
have mainly been attacked by several nonlinear
programing approaches. The subgradient technique
may handle convex constraints. The most
applicable technique, accounting for mixed norms,
fixed costs and any type of convex constraints is
an adapted version of the primal-dual method. A
different type of minimax is in which the aim is
to locate a given number of independent detection
stations, detecting the occurance of some event
with probability that decreases with distance, so
as to maximise the probability of detecting an
event occuring anywhere within a given region.
94
MULTIFACILITY LOCATION-ALLOCATION PROBLEMS
Minisum The traditional minisum version is
usually called the p-median or multi-Weber
problem each destination will be served by one
of the new facilities and the sum of costs af all
these services is to be minimised. This problem
turns out to be very hard to solve, consisting
simultaneously of a nontrivial combinatorial part
and nonlinear part. The objective is neither
concave nor convex and not everywhere
differentiable. Popular alternating heuristics
consists of alternating between an allocation
phase in which sites of facilities are kept fixed
and a location phase in which allocation is kept
fixed, the whole process stopping when no new
improvement is found. This tecnique choose
starting locations, quickly find good solutions
and an optimal one. The one dimensional
multi-Weber problem may be solved by a dynamic
programming. In the plane with rectangular
distances we have again an intersection point
property, where attention may be restricted to
those intersection points inside the convex hull
of the destinations.
95
MULTIFACILITY LOCATION-ALLOCATION PROBLEMS
Minisum The euclidean norm case has attracted
most attention. The first exact solution method
was developed as a branch and bound method with
rather weak lower bound calculations. It is
proved that the number of disjoint convex hull
partitions of n points in the plane into p groups
is polynomial in n for fixed p. A more recent
approach using the disjoint convex hull property
consists of generating the list of all possible
convex hulls, together with the value of the
corresponding Weber problem, and use this as data
of a discrete set-covering problem. A similar
but much improved technique is applicable to
norms problems. These methods find local minima
which often seem to be global ones on small scale
problems. Much larger problems may be handled
with these methods. In location allocation
problems this is often necessary to reduce the
number of destinations in order to obtain
solvable problem. This leads to different types
of aggregation errors. These effects of
aggregation were sufficiently investigated,
especially in continuous setting.
96
MULTIFACILITY LOCATION-ALLOCATION PROBLEMS
Minimax The minimax version, known under the
name p-center problem, in euclidean distances, as
covering all destinations by p-equal identical
circles of minimum radius. There are several
researchs about rectangular and euclidean
distance p-center problems, and solution methods.
Several heuristics methods exists applicable to
general norms in any dimension. Variants An
alternative to the p-median problem arises when
the facilities to be located have limited
produciton or supply capacity.. The allocation
part of the problem involves the solution of a
transportation problem. Then a new variant is
proposed which may be considered as a contiuous
version of the simple plant location problem.
97
MULTIFACILITY LOCATION-ALLOCATION PROBLEMS
Variants One of the difficulties of the p-center
problem is that usually m