Denis McQuail y las racionalidades de una empresa de medios

We now combine the elements discussed in Chapters 3 through 5 to obtain a general routing problem, which we call the Maximum Collection Problem (MCP) . We look at what are the important features of the problems, and we consider a generic solu­ tion method for any type of problem. Our underlying problem is the Orienteering Problem (OP) , to which we add various additional features.

We define the MCP to have the following properties:

• there are a fixed number of vehicles available, each with a given capacity, a

given origin location, and a time period over which it can be used

• there are a number of customers available for servicing. Each customer has

an associated reward that is received for servicing the customer, where the value of the reward received is dependent upon certain attributes of the time at which the customer is serviced, e.g. , the time at which service is started and/or finished.

• each customer may be comprised of a number of points to be visited with loads to be transported between the points, and there may be a time window

(or time windows) during which each customer is available for servicing.

• there are penalties for non-service, which may relate to the individual cus­

tomers that are not serviced or to the total set of unserviced customers.

• the objective is to service a given subset of the customers such that the net

1 10 The Combined Problem

reward received is maximized, where the net reward is the reward collected less the penalties incurred and the costs of servicing.

The MCP, as we have defined it, consists of the components of Chapters 3, 4 and 5, i.e. , subset selection, precedences and capacities, time windows and time­ dependent rewards. Although the time window problem may be viewed as a special case of time-dependent rewards, we deal with these separately, as the time windows cause feasibility concerns while the time-dependent rewards influence the viability of the routing.

6 .1 A Classification Scheme for the MCP

We now consider the features that comprise the MCP, and use these t o create a classification scheme for subset selection problems. The classification scheme we are creating is based upon one that was used by Desrochers, Lenstra and Savels­ bergh [35] to describe vehicle routing and scheduling problems. This scheme was later used by Desrochers, Jones, Lenstra, Savelsbergh and Stougie [33] within a decision support system which attempted to use appropriate solution techniques for a problem instance of unknown type, through comparing the problem's repre­ sentation, via this classification scheme, with that for a known problem for which solution methods are available.

The scheme gives a classification of the form:

(classification) :: = (addresses)

(vehicle attributes)

(problem characteristics)

(objectives)

A fifth category may also be added, which gives some additional information about the practical aspects of the particular problem instance, and we will consider this category in more detail later. For the other categories, which are known as fields,

the scheme breaks these into subfields which correspond to the attributes being con­ sidered. For our notation, we surround each field and subfield by angular brackets

(,). The subfields are comprised of tokens, which are tokens that stand for the at­

tribute in the problem instance. We use the symbol

V

(the exclusive or) to separate the different possible values of the tokens, and we use the symbol 0 for the default

attribute. The representation of the classification scheme involves showing each of the relevant tokens from each of the subfields, where we separate tokens from the same field by commas, and we separate the different fields through using the vertical bar, I . A subfield being empty indicates that the default token is present in this problem.

We will now consider each of the fields, using the subfields from the original paper and noting the changes that we use for our classification scheme, where required. We adapt the scheme so that different attributes are only considered if they invoke different behaviour from the solution methods.

6 . 1 . 1 Addresses

The addresses field refers to the customer and depot attributes and these are given as:

(addresses) ::=

(number of depots) (type of demand)

(address scheduling constraints) (address selection constraints) (customer rewards)

(non-service)

(number of depots) : := 0

V l

o [one depot]

l

[specified number of depots] (type of demand) ::= (al) (a2)

(al) : := 0

V EDGE V MIXED V

Tfix

V

Trtg

V

Tsub

V

Tdec

V

Tpre

o [node routing]

EDGE [edge routing]

MIXED [mixed node and edge routing]

Tfix [task routing; tasks have a fixed order]

Trtg [task routing; routing allowed within tasks]

Tsub [task routing; subset of addresses within each task must be visited]

Tdec [task routing; decomposable tasks]

1 12 The Combined Problem

(a2) : : =

oV rv

o

[deterministic demand]

rv [stochastic demand]

(address scheduling constraints) : : = 0

V f Sj V tWj V mWj

o

[no time window constraints]

f Sj

[fixed schedule]

tWj

[single time window for each customer]

mWj

[multiple time windows for each customer]

(address selection constraints) : : = 0

V subset V choice V period

o

[single plan; all addresses must be visited]

subset

[single plan; only a given subset of addresses needs to be visited]

choice

[single plan; at least one address in each subset of a given partition must be visited]

period

[a number of plans over a given time period is to be made]

(customer rewards) : : = 0

V K V

TDI4nc

V

TDRdec

o

[no rewards]

K

[

constant rewards]

T D I4nc [time-dependent rewards, may increase]

T DRdec [time-dependent rewards, non-increasing]

(non-service) : : = 0

V

N P

V

00

V

M

o

[non-service is not allowed or there are no penalties for non-service]

pen

[there are finite, constant penalties for non-service]

00 [some penalties for non-service are infinite]

M [all customers must be serviced in some manner; costs are incurred for those customers that are not in the solution route]

For the address attributes we have added two subfields to those given in the original classification scheme; customer rewards and non-service. We consider the case of time-dependent rewards that may increase over time to be different from the strictly non-increasing case, as there may be advantages in delaying the time of servicing a customer with increasing rewards. For non-service, we consider dif­ ferent cases for where there are finite penalties for non-inclusion (which affects the

attractiveness of the customers ) , infinite penalties (which need to be explicitly con­ sidered in order to obtain feasible solutions) , and where the customers that are not considered to be in the solution route may contribute to the costs of the system.

We have also adapted the 'type of demand' subfield, in order to allow for the more general cases of tasks that we have considered previously. The case Tdec refers to decomposable tasks, where this incorporates the idea of split tasks, and also for cases where tasks may be partially serviced.

6 . 1 . 2 Vehicles

The next field considered relates to the vehicles used in the problem. The attributes considered here are very similar to those used in [35] , so we cover them quite briefly.

(vehicles) : : =

(number of vehicles)

(capacity constraints) (commodi ty constraints)

(vehicle scheduling constraints)

(number of vehicles) : : =

c V m

c

[there are a constant number of vehicles,

c

that are available for servicing]

m

[multiple vehicles are available, with the number used being a decision variable]

(capacity constraints) : : = 0

V cap V mcap V capi V mcapi

o [no capacity constraints]

cap

[common capacity for all vehicles]

mcap

[common, multiple capacities for all vehicles]

capi

[different capacities for the different vehicles]

mcapi

[different multiple, capacities for the different vehicles] (commodity constraints) : : = 0

V sep V ded

o [no capacity constraints]

sep

[vehicles have interchangeable compartments]

ded

[vehicles have dedicated compartments]

(vehicle scheduling constraints) : : = 0

V

T

V 7i

o [no scheduling constraints]

T [a common time of availability for all vehicles]

1 14 The Combined Problem

The differences we are making to the classification scheme are to not consider the case where all available vehicles have to be used, and to not consider there to be a distinction between time windows of availability and route duration constraints. The 'commodity constraints' subfield, enables different commodities to be handled in different manners. The inclusion of dedicated compartments would imply that there are multiple capacities of the vehicle, which gives additional features to be concerned with. Because of this possible occurrence, we introduce the additional tokens within the 'capacity constraint' field to deal with multiple capacity vehicles. For example, in the handicapped transportation application considered by Toth and Vi go [175] , the vehicles considered are able to carry various numbers of passengers with differing care requirements, and so the mix of the passengers assigned is an important feasibility consideration.

6 . 1 . 3 Problem Characteristics

The field on the problem characteristics contains a number of features that enable the classification scheme to be used for general routing problems. These allow for restrictions on the assignment of customers to vehicles, of customers to depots and vehicles to depots, and they also consider the restrictions of customers so that incompatible customers can not be carried together. We acknowledge that these type of restrictions may be useful in the classification of general routing problems, but since these are not relevant for the types of problems we are currently considering, we do not include these features within our scheme. The classification, in terms of problem characteristics, that we consider is then:

(problem characteristics) ::= (restrictions) (type of network) (type of strategy) (costs incurred) (restrictions) : : = 0

V rew

o [no target reward restriction on the routing]

('I) ::= 0

V

9

o [Euclidean distances]

cap

[general distances]

(,2) ::= 0

V dir

o [undirected network]

dir

[directed network]

(type of strategy) ::= (61)

V

(62)

(61 ) ::= 0

V back V full

o [no restrictions on the service order of the loads carried]

back

[node routing; only backhauling allowed]

full

[task routing; only one load at a time is allowed in the vehicle]

(62) : : = 0

V MD V path

o [each route starts and ends at the same location]

M D

[multiple-depot routes are allowed]

path

[each route is a path starting at a given depot]

We create a new subfield which considers the constraints on the problem. These

are the resource limitations that may be used within the subset selection problem to restrict the routing or to ensure that certain goodness of routing properties are met. We are also allowing different forms of the solution, through enabling the solution routes to consist of open paths. This case involves the service of customers being the important feature, with service allowed until customers are no longer available. The final location of the vehicle is assumed to be a far less important consideration than the customer service, in this situation.

6 . 1 .4 Objectives

The field containing the objectives, needs to be adapted from the original classifi­ cation scheme in order to handle subset selection problems. The original objectives considered only the minimization of costs, where either the minimum of the sum of costs or the minimum of the maximum cost of the vehicles was sought. For subset selection problems, the standard objective is either to maximize the net rewards or to minimize the costs of servicing a subset of the available customers. We also consider different cases concerning which customers affect the objective function, as we make the distinction between pure subset selection (where only the customers in

116 The Combined Problem

the solution route will affect the objective function) and restricted subset selection (where some or all of the customers will affect the objective function even if they are not included in the solution) .

This field is therefore given as: (objectives) : : =

(objective)

V

(objective) (objectives) ( consideration)

(objective) : : = (operator)

V

(function) (operator) : : = max

V

min

V

minmax

max [maximize the sum of the values]

mm [minimize the sum of the values]

mmmax [minimize the maximum value]

(function) ::=

duri V

NV

V

I4 ± Ci ± Cj ± NS ± Pi ( (vehicle constraints) )

±Pj ( (address constraints) )

duri

[route duration] NV [number of vehicles employed]

�

[reward of vehicle] Ci Cj [address costs] NS Pi ( (vehicle constraints) ) Pj ( (address constraints) ) [vehicle cost] [costs of non-service] [vehicle penalty] [address penalty]

(vehicle constraints) :: = (vehicle constraint)

V

(vehicle constraint) (vehicle constraints)

(vehicle constraint) : : =

cap V capi V mcap V mcapi V T V

Ti

(address constraints) : : =

tWj V mWj

(consideration) : : = 0

V sub V part

o [all customers are considered equally in the objective]

sub

[only the customers in the solution routes are considered]

part

[all customers are considered in the objective, with those in the solution contributing differently from those who aren't]

The objectives field therefore describes the possible objectives of the problem, where we drop the vehicle subscripts if we only have a single vehicle. Of note here is that we have included arithmetic operations within the 'function' subfield, in order to deal with cases where there are a number of factors in the objective function. In this case the original classification scheme lists all the components of

the objective function in decreasing order of importance, but this approach seems more appropriate for multiple objective problems than for composite objective functions. Our notation for a multiple objective problem is to list all the objectives, with these separated by colons if the functions are dealt with in an hierarchical fashion, or separated by semi-colons if all the objectives are considered.

6 . 1 .5 Examples

We now demonstrate how the above classification scheme may be used to describe a wide variety of routing problems, particularly with regard to subset selection problems.

In document La tensión entre racionalidades periodística y político económica en medios gráficos argentinos : los casos de Clarín y La Nación (página 61-65)