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FACILITY LOCATION

Agenda

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

- OBJECTIVES IN LOCATION PROBLEMS

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

- 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.

- 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.

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.

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

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

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

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.

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

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

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.

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)

- 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.

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.

- 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.

- 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.

- 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

- 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.

- 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.

- 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.

- 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.

- 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.

- Cent-dians whose objective is a linear convex

combination of minisum and minimax objectives. - Viz.f fsum (1- )f max with

01

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.

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 .

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.

Supply and Demand Curves

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.

- 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.

- 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.

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.

- 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.

- 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.

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.

P-defense Problem

- P-defense problems objective is to maximize the

sum of distaces between each facility and its

nearest neighbor (maxsummin problem)

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.

- 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.

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

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.

- 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.

- 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.

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.

- 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

- 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.

- Extreme values
- Center
- (center)
- (range)

- (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.

- 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.

- 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.

(No Transcript)

- 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.

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.

- 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.

- (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

- 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

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.

Continuous location problems

- Frank Plastria
- Vrije Universiteit Brussel

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.

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.

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.

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.

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.

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).

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.

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

MULTIFACILITY LOCATION-ALLOCATION PROBLEMS

Variants One of the difficulties of the p-center

problem is that usually m

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