Example
- Locate p on a network of m models
- Locating two service facilities (p=2) for supplying
components to five manufacturers (towns) (m =5)
- The demand for the services, zi, is measured by the
number of units required by the i-th manufacturer
Spatial optimization Weighting Method for
Weighting Method Example location allocation
problem (example)
s
ude
- At least one set of spatially explicit decision cl
variables:
In The problem involves optimizing three
Weighting and constraint objective functions:
Example: location allocation for defining a set of methods
- The set of (non-dominated) solutions to the problem is 1. Total distance
spatial alternatives
generated by parametric variation of the weights 2. Total environmental impact associated
- Any locational alternative can be defined as a
- An approximation of the solution set can be generated with transportation of the components
binary vectorx = (x1, x2, ..., xm), where a decision
by systematically varying the weighting coefficients and (measured by an index assigned to links to
variable, xj, is defined as follows: xj = 1, if an activity
solving the associated single-objective model the network)
(e.g., health service facility) is located at the i-th
- Multi-objective problem is first transformed into a
Inclu
site; and xj = 0, otherwise 3. Total risk of accident
scalar problem and then solved as a single-objective
optimization problem
des
- Basic difference among the methods lies in how they
make the transformation from a multi- to single-
objective model
- The most often used methods for tackling spatial
Multi-objective Weighting and Constraint
multi-objective problems are the weighting and
optimization Constraint Method Method (dis)advantages
constraint methods
- The weighting method involves assigning a weight,
wk(k = 1, 2, ..., n), to each objective function, fk x
- The multi-objective function ,
- Multi-objective optimization methods, or multi- - Constraint method involves maximizing only one of Weighting and Constraint method advantages :
Includ
can then be converted into a single-objective form
objective decision analysis (MODA), define decision the objective functions while all others are converted - Reducing the multi-objective optimization problem to a scalar
through the linear combination of the objectives
alternatives in terms of a model consisting of a set of into inequality constraints valued function o vast body of algorithms, software, and
toghether with the corresponding weights:
- Multiple objective problem can be transformed to
es
objective functions and a set of constraints experience that exist for single-objective optimization models
imposed on the decision variables. Formally, MODA the following single-objective problem: can be directly applied to multi-objective problems
problems are formulated as follows: - Easily used and intuitively appealing
Weighting and Constraint method disadvantages:
- Computationally intensive:
The set of non-dominated solutions can be generated - Computational requirements for the weighting and
by solving the single-criterion problem with the constraint methods depend on the number of objective
parametric variation of the ck value functions and the number of weights or constraints
- Exponential relationship between the number of objective
functions and computational burden
Conventional
optimization
approaches in GIS- Inclu
MCDA des Compromise
Distance-based methods Includes
programming
- Aim at minimizing a function of the distance between the - Based on the assumption that the performance of decision Compromise programming advantages :
desired (usually unachievable) and achieved solutions alternatives can be evaluated with respect to a point of - Simple conceptual structure
- The desired solution (target values) can be defined as an reference
ideal point, some reference point, or a set of goals - A point of reference is the ideal solution (or ideal point), which Compromise programming disadvantages:
- The most often used distance metric approaches include: defines the optimal value for each objective considered - No clear interpretation of the various values of the parameter p
- Goal programming separately (except for the two extremes (that is, when p=0 and )
- Compromise programming
Includes
- Reference point method -The method identifies the non-dominated solution closest to
des
- These methods are also the most popular distance metric the ideal point using various weighted Lp norms as follows:
Inclu
procedures implemented in the GIS environment
- Also referred to as the Lp-norm approaches
- Definition of distance metric is the main procedural
difference between the different types of those methods
- Generic form of the distance metric model:
Goal programming
Interactive methods
- The goal programming methods require the decision maker
Goal programming advantages :
to specify the most desirable value (goal) for each objective
- Computational efficiency
(criterion) as the aspiration level or target value
- While dealing with the multi-objective decision problems,
- The objective functions
goal programming approaches allow us to stay within an efficient
are then transformed into goals as follows:
- Determine the best (compromise or satisficing) decision outcome among the linear programming computational environment
set of efficient solutions by means of a progressive communication process
between the decision maker and the computer based system Goal programming disadvantages:
- Require the decision maker to specify fairly detailed a priori
An interactive procedure consists of two phases: information about his/her aspiration levels, and the importance
of goals in the form of weights
1. Dialogue phase: the decision maker analyzes and evaluates information Two types of variables are part of any goals programming - Difficult (or even impossible) in complex spatial situation
provided by a computer-based system and articulates his/her preferences formulation:
2. Computational phase: a solution (or a group of solutions) that meets the - Decision variables,
decision maker’s requirements specified in the dialogue phase, is generated - Deviational variables,
Measures of multidimensional deviations (achievement
This interactive exchange of information is continued until an outcome is functions) can be formulated in terms of the weighted Lp-norm
deemed acceptable to the decision maker as follows:
- Locate p on a network of m models
- Locating two service facilities (p=2) for supplying
components to five manufacturers (towns) (m =5)
- The demand for the services, zi, is measured by the
number of units required by the i-th manufacturer
Spatial optimization Weighting Method for
Weighting Method Example location allocation
problem (example)
s
ude
- At least one set of spatially explicit decision cl
variables:
In The problem involves optimizing three
Weighting and constraint objective functions:
Example: location allocation for defining a set of methods
- The set of (non-dominated) solutions to the problem is 1. Total distance
spatial alternatives
generated by parametric variation of the weights 2. Total environmental impact associated
- Any locational alternative can be defined as a
- An approximation of the solution set can be generated with transportation of the components
binary vectorx = (x1, x2, ..., xm), where a decision
by systematically varying the weighting coefficients and (measured by an index assigned to links to
variable, xj, is defined as follows: xj = 1, if an activity
solving the associated single-objective model the network)
(e.g., health service facility) is located at the i-th
- Multi-objective problem is first transformed into a
Inclu
site; and xj = 0, otherwise 3. Total risk of accident
scalar problem and then solved as a single-objective
optimization problem
des
- Basic difference among the methods lies in how they
make the transformation from a multi- to single-
objective model
- The most often used methods for tackling spatial
Multi-objective Weighting and Constraint
multi-objective problems are the weighting and
optimization Constraint Method Method (dis)advantages
constraint methods
- The weighting method involves assigning a weight,
wk(k = 1, 2, ..., n), to each objective function, fk x
- The multi-objective function ,
- Multi-objective optimization methods, or multi- - Constraint method involves maximizing only one of Weighting and Constraint method advantages :
Includ
can then be converted into a single-objective form
objective decision analysis (MODA), define decision the objective functions while all others are converted - Reducing the multi-objective optimization problem to a scalar
through the linear combination of the objectives
alternatives in terms of a model consisting of a set of into inequality constraints valued function o vast body of algorithms, software, and
toghether with the corresponding weights:
- Multiple objective problem can be transformed to
es
objective functions and a set of constraints experience that exist for single-objective optimization models
imposed on the decision variables. Formally, MODA the following single-objective problem: can be directly applied to multi-objective problems
problems are formulated as follows: - Easily used and intuitively appealing
Weighting and Constraint method disadvantages:
- Computationally intensive:
The set of non-dominated solutions can be generated - Computational requirements for the weighting and
by solving the single-criterion problem with the constraint methods depend on the number of objective
parametric variation of the ck value functions and the number of weights or constraints
- Exponential relationship between the number of objective
functions and computational burden
Conventional
optimization
approaches in GIS- Inclu
MCDA des Compromise
Distance-based methods Includes
programming
- Aim at minimizing a function of the distance between the - Based on the assumption that the performance of decision Compromise programming advantages :
desired (usually unachievable) and achieved solutions alternatives can be evaluated with respect to a point of - Simple conceptual structure
- The desired solution (target values) can be defined as an reference
ideal point, some reference point, or a set of goals - A point of reference is the ideal solution (or ideal point), which Compromise programming disadvantages:
- The most often used distance metric approaches include: defines the optimal value for each objective considered - No clear interpretation of the various values of the parameter p
- Goal programming separately (except for the two extremes (that is, when p=0 and )
- Compromise programming
Includes
- Reference point method -The method identifies the non-dominated solution closest to
des
- These methods are also the most popular distance metric the ideal point using various weighted Lp norms as follows:
Inclu
procedures implemented in the GIS environment
- Also referred to as the Lp-norm approaches
- Definition of distance metric is the main procedural
difference between the different types of those methods
- Generic form of the distance metric model:
Goal programming
Interactive methods
- The goal programming methods require the decision maker
Goal programming advantages :
to specify the most desirable value (goal) for each objective
- Computational efficiency
(criterion) as the aspiration level or target value
- While dealing with the multi-objective decision problems,
- The objective functions
goal programming approaches allow us to stay within an efficient
are then transformed into goals as follows:
- Determine the best (compromise or satisficing) decision outcome among the linear programming computational environment
set of efficient solutions by means of a progressive communication process
between the decision maker and the computer based system Goal programming disadvantages:
- Require the decision maker to specify fairly detailed a priori
An interactive procedure consists of two phases: information about his/her aspiration levels, and the importance
of goals in the form of weights
1. Dialogue phase: the decision maker analyzes and evaluates information Two types of variables are part of any goals programming - Difficult (or even impossible) in complex spatial situation
provided by a computer-based system and articulates his/her preferences formulation:
2. Computational phase: a solution (or a group of solutions) that meets the - Decision variables,
decision maker’s requirements specified in the dialogue phase, is generated - Deviational variables,
Measures of multidimensional deviations (achievement
This interactive exchange of information is continued until an outcome is functions) can be formulated in terms of the weighted Lp-norm
deemed acceptable to the decision maker as follows:
