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On the dynamics of a class of optimum routing problems
On the Dynamics of a Class of Optimum Routing Problems* H.
by MARLIN
and WILLIAM
MICKLE
G. VOGT
Department of Electrical Engineering University of Pittsburgh, Pittsburgh, Pennsylvania A
ABSTRACT:
transportation
particular systems
class of routing
as a part of the more general by methods presented
routing
problem.
other than classical minimum
model for proceaaing
problems
associated
with
require8the repeated regeneration of minimum
demands
The generation
route solutions
in a demand
actuated
demand
actuated
or optimum
or regeneration
is discussed.
transportation
A general
or routing
routes
of routes system
Byetern is
and, disczLesed.
I. Introduction
The minimum route problem (1) is interesting and has a number of important applications. Considerable progress has been made in reducing the time (normally computer time) required to obtain the minimum route between two points, 0 and D, origin and destination respectively, in a network composed of N vertices, V, and a number of edges, E, and associated costs, C. The similar problems of finding all minimum routes in a network or all routes from a fixed origin have also been considered. For a single (0, D) pair, this computation time is relatively small although it is a function of the size of the network, the type of computer, etc. There exist applications (2, 3) where, as a part of a larger problem, normally some type of routing or traffic control, it is required to obtain the minimum route, r, between 0 and D. Such problems typically have thousands of vertices (4) or nodes as a part of the network and require the minimum routes, on demand, possibly hundreds or thousands of times/hr. It is important to point out that the bounds on network size and demand rate are in reality a function of the state-of-the-art of routing algorithms and the minimum route problem. The network sizes and demand rates typically mentioned represent a sort of minimum for practical applications of a number of automatic control schemes for demand actuated systems. The typical dimensions of the problem to be formulated and discussed in this paper represent somewhat of a fuzzy point or threshold at which these applications become practical. As practicality is achieved, the sizes of systems to be considered as desirable for implementation will increase by orders of magnitude. * The resemch reported in this paper w&s supported in part by National Foundation Grant GK 32237.
Science
Marlin H. Mickle and William G. Vogt There are two limiting factors to the basic problem. The first is the network size in terms of the number of vertices characterized by N. The larger the value of N, the larger storage and data handling capabilities required to obtain a solution. The second factor is the rate at which solutions are required. This factor can be characterized by T* which may represent the minimum time? between which any two solutions are required. It would be premature to discuss queues at this time, and consequently the time T* will be considered as the minimum time between solutions over a complete demand profile. The application of the minimum route type problem to the more general routing problem under formulation is thus one of being able to obtain a minimum route, r, for any (0, D) pair in time T* for a network of size N. As T* decreases, the allowable computation time for the route r must also decrease. However, the general nature of the application is such that it encourages the use of larger networks consequently requiring more time to obtain r for the now larger N. The upper bound on N is a function of the application. If the application is transportation, the upper bound is likely to be a function of geographical area to be covered. The time required to obtain a solution by any particular method will be noted as T which will vary as the method and associated details to obtain r for (0, D) vary. For any method to be satisfactory, it is required that for the corresponding T, T < T *, for any (0, D)E(V, V). An obvious alternative to computing r each time a demand occurs is to store all of the routes and to look up (retrieve) r from a storage device or devices with some appropriate scheme. Such an alternative is reasonable in that all possible routes can, for example, be stored in a matrix called the su,ccessor matrix, S, which essentially requires the same amount of storage as the cost matrix, C, which describes the system. Any element of S, sij, is the next vertex in the route having i as an origin and j as a destination, i.e. (0,D) = (Cj). All routes are computed “off line”, and the success or matrix is formed. In this process, the minimum route techniques are important to obtain S. Once S is generated, the minimum route techniques are no longer needed, and a route can be obtained in some time, T+. The problem of obtaining minimum routes from S is one of table look-up. However, for general applications, N will frequently be large enough to render this method of obtaining the routes infeasible. As N becomes large, the available storage media is normally slower, and T+ becomes too large. The problem to be solved is one of obtaining a method which will produce the minimum route, r, on demand for any (0, D) pair in time T+< T* requiring a minimum amount of storage. Conventional methods of solving for the minimum route are ineffective because the associated time, T, is greater than the minimum time T*. Route t Variations on this time will be discussed later in this paper.
Journal OFThe Franklin Institute
Problems storage techniques such as the successor matrix are ineffective because the storage requirement is too large although T+ may be less than many practical T”. ZZ. Previous Related Work
It is generally agreed that some cost and route information is contained in geographic or other composition of many practical applications as in transportation for example. One of the more frequent attempts to take advantage of this situation is the “cost proportional to distance” assumption (5). However, there are definite problems in this assumption which are sufficient to make the routing system completely ineffective. In this discussion, ineffective implies that incorrect decisions are made on the basis of any assumption which involves the data for some model of the system. While it is important to employ techniques to take advantage of any information which can reduce the amount of data required, such techniques are only considered acceptable if they do not result in any incorrect decisions made by the system using them. Thus a technique will be considered ineffective if its application has a non-zero pobability of causing at least one incorrect decision. The data storage technique must thus be a perfect filter with the ability to exactly regenerate input data on demand. Obviously “garbage in garbage out” applies to any supplied data. The type of data storage technique as conceived here is one which accepts input data in a specified form and converts it to some form which minimizes the amount of storage required for the new form according to a specified objective or objective function. Using the successor matrix as an example, one data storage technique is a technique which may require the cost and successor information for a given network and will produce a successor element, Sij, on demand. The successor matrix can then be used to produce the desired element on demand with a minimum amount of programming. Given the cost matrix, a minimum route method can be coded to generate the successor element. Any other method which accepts the cost and successor information and produces su on demand is an example of a data storage technique. The above examples are further illustrated in Fig. 1. One method of storing the route information is based on a cost proportional to distance assumption and involves the cost from the origin, 0, to the destination, D. An old adage, “ . . . a long trip always starts with a first step”, provides the foundation for the technique of Neighborhood Storage (5). The basic reasoning of this technique will now be summarized. The successor vertex, sij, will be noted as v(i,j). For any origin, 0 E 7, there exists a number, n(O), of vertices which are connected to 0 through an edge eoj E E for the graph characterization of a network, G = (V, E). The set of vertices connected to any vertex, i, will be
Vol.300,No.1, July 1975
9
Marlin H. Mickle and William G. Vogt
a. successor Matrix Implementation
c
-
*
b.
MINIMUM ROUTE PROGRAM
sij
Minimum Route Implementation
C. Data Storage Technique FIG. 1. (a) Successor matrix implementation.
implementation.
(b) Minimum route (c) Data storage technique.
as J’(i) and will be termed the set of feasible intermediate vertices of i, i.e., “The possible first steps”. Thus given an origin, destination combination, (0, D), the first step in the route generation is one of finding the v EF(O) such that ver(0, D) where r(0, D) is the ordered set of vertices making up the optimum route from 0 to D. It is a simple matter to prove that for positive cij and that for any vertex v lr(0, D) such that v is connected to 0, vrv(0, D). Therefore, for demand (;,j), sij = v(i,j) Er(i,j). Note that ~(0, D) = F(0) I-Ir(0, D) = soI) where n is the set intersection. Assume all vertices are located in R x R = R2, a two-dimensional Euclidean space. Each vertex has a two-dimensional representation (x1,x2). For any vertex i, this representation is noted as x(i). Consider now the following functional : identified
f(i,j, k) = d[x(i), x(j)1 +d[x(jL WI, 10
Journal of The
(1)
FranklinInstitute
Dynamics of a Class of Optimum Routing Problems where d[u, v] is the Euclidean distance from u to v. The functional, (l), can now be the sum of the straight line distance segment from the origin, 0, to a feasible intermediate vertex, WET(O), and the straight line distance segment from v to the destination, D, e.g.,
f(0, v,D)
= d[x(O), x(w)] +d[x(v), x(D)] : v~F(0).
(2)
The functional, (2), is a rather good approximation for a number of transportation systems (5-7). f(0, v, D) is t ermed an ideal policy equation. If for any (0, D) combination, the ideal policy equation works, it can be used as a part of the route generation system discussed previously. If the ideal policy equation does not work for some (0, D) combinations, the exceptions can be noted and stored. Given any demand (0, D) the process is one of determining if (0, D) E A or (0, D) EA’ where A is the set of all (0, D) combinations for which (2) works and correspondingly A’ is the set of all (0, D) combinations for which (2) does not work. Obviously the problem is to choose an ideal policy equation and process for which the order of A is larger than the order of A’. Another obvious possibility is to formulate a step decision process A,, A,, . . . ; A; AL, . . . . However, the fundamental formulation of the individual A,, A,, . . ., etc. is the same as for A with some appropriate partitioning scheme. Therefore we will only be concerned with a single step process at this time noting its obvious extensions. Given a demand, (i,j), the determination of the membership of (i,j) in A or A’ requires a certain amount of storage (in general of various means and types) which will be symbolically described as X(A) or #(A’), respectively, with no particular quantification specified. The “goodness” of any method of route generation can thus be described as one where S(A) u B(A’) is relatively small and the associated time, T+, is also small. The present method of obtaining the routes is simple. Given (i,j) ; if (i,j)E& (3) if (i, j) $ A, some other method (including table look-up) is used to obtain w(i,j). Thus the object of the method is to maximize the membership of A and minimize #(A’) and its associated details. Other ideal policy equations can be used (5,7) and the basic idea is independent of the equation although the order of A, o(A), is obviously not. Using the above technique, the storage, S(A), is primarily dependent on two types of information, the connection information in order to define P(i) and the location of each vertex in R x R. Some work has been done in this area with relatively good results (4). The storage, X(A’), has recently been considered (8) again with relatively good results. In &‘(A’), the problem is one of finding an individual optimum covering identified with each (i, j) E A’. Each covering is then tagged with (i, j) and So,. The retrieval process is consequently one of locating which set covers a given (;,j).
Vol. 300, No. 1, July 1975
11
Marlin H. Mickle and William G. Vogt Obviously, the intent of the above work is not to compute minimum routes but rather to retrieve minimum routes in a time T+ which is less than T* and hopefully considerably less than T* so as to implement higher demand rates and larger networks. ZZZ. A Generalization
The previous discussion can be characterized as consisting of two parts or problems. The first is given a demand, (i, j), find a vehicle?, k, from some assigned set of vehicles to transfer the demand initiate from vertex i to vertex j, according to a schedule p(i,j, k) which follows an optimum1 route, r*(Q), while satisfying a set of constraints and maximizing or minimizing an objective function in a time T+ < T *. This problem is one of algorithms, objective functions, constraints, etc. The second problem is one of given an origin-destination combination, (;,j), find an optimum route r(i,j). The above characterization is illustrated in Fig. 2. 7-*
,
(i,j)
r(i,j),
(i,j)
I(i,j) ,
FIG.
p(i.j;k)
T”
2. Problem characterization.
This paper is primarily concerned with problem two. The solution of the problem may be achieved by a number of means, three of which will be mentioned. The first is the direct computation of r(i,j) using any minimum route the size of the network and the technique. For many applications, corresponding T* may be such that this straightforward method is the most appropriate. A second method is the table look-up of r(Q)) using a successor matrix. This method is entirely appropriate for situations where T* is large and storage is not a problem. t In this oontext, a vehicle is any physical means by which the specified route will be traversed using up some capacity and requiring an associated non-zero time. $ While the optimum route is normally the minimum route, other criteria may be specified. Thus while the term minimum may be used, the technique is more general in application.
12
Journal of The Franklin Institute
Dynamics of a Class of Optimum Routing Problems A third method is the method discussed in the previous section where T* is small and storage is a problem. Other methods may exist which are applicable to specific problem classes such as those of street networks which display very regular characteristics such as intersection location, all costs equal, etc. However, it is the purpose of this paper to focus on techniques of the type previously discussed which are applicable to general systems although acceptability may vary according to different objectives such as time, storage, regularity, etc. IV.
A Dynamic
System
Model
The raison d’&e for the current problem formulation is in the minimum amount of time in which a solution must be obtained. The time, T*, is the minimum time in which a route, r*, must be determined, and the time, T*, is the minimum amount of time in which a route, r, must be determined. It was pointed out in the beginning of this paper that queueing would not be discussed. However, it can be noted that the high rates which dictate T* and T* may be of short duration and infrequent occurrence and if taken as an absolute limit would place an inordinate demand on the corresponding techniques. Thus it is possible that a certain amount of queueing may be acceptable. The following discussion is based on this assumption. Consider a system of the type discussed in earlier sections with a demand rate (demands/time), d(t), and an assignment rate (optimum assignments/ time), s(t). Any given technique, computing system, algorithm, etc. will be assumed to have an associated response time T which is a variable and is a function both of the given characteristics (technique, computing system, algorithm, etc.) and also of the characteristics of the demands such as number of vertices in the minimum path, vehicles available at the time a demand occurs, etc. The time constant T* is a function of the particular network being used in terms of connectivity, E, dimension (which can be expressed in terms of V) and cost, C, i.e., T* =f*(V, C,E). Given any particular origin-destination pair, (0, D), the computation time, T is a function of (0, D), i.e. T = f(0, D). In terms of Fig. 2, the discussion is with the T* of problem two. Consider now the T* of problem one. The time constant is a function of the algorithm being used (denote this by a), the time constant of problem two, T*, the number of vehicles (which will be denoted by v), computer and computer program (denoted by c), i.e., T* = g*(or, v, T*, c). Once the algorithm is introduced, the factors making up any particular 7 must be accounted for by the state of the system, x; at the time, t, the demand, d, to go from i to j occurs; (i,j; t). Thus 7 = g(a,v, T*,c,x; t). At this point, the problem begins to become overly complicated by the general nature of the unknown functions, f, f *, g and g*. It is conjectured that it is possible to express g* in terms of T* as shown in Eq. (4), T* = g*(ar, v, T*, c) = g+(cll,v, c) T*.
Vol.300.N&l, July1975
(4
13
Marlin H. Mickle and William G. Vogt T* can be determined without much difficulty, and for a fixed ~11 and v,g+ can be determined experimentally. Thus it is not unrealistic to talk about a r* for a given processor (c), LXand v. Consider now the dynamics of the total routing problem as for example in the case of demand actuated transportation systems. Demands come in at a rate, d(t), in terms of demands per unit time. The demands are in turn to be satisfied at some rate, call it s(t). It is important to elaborate on the point at which a demand is considered satisfied. Assume for the moment that time is of no importance to the customer, a demand may be considered as being satisfied when the customer is scheduled on a particular vehicle, i.e. s(t) is a mea’sure of people being assigned. If time is important to the customer, a demand is considered as satisfied when a person is delivered within the time constraint specified by the customer, the system or both. Otherwise, another category is necessary to include those persons who have not been delivered and whose time constraints have been violated. In this paper, the measure of demands satisfied is taken as the point in time at which the person is delivered to the specified destination. Therefore, there is a time delay between when a demand rate appears and when the demands are actually satisfied. In terms of per cent of demands satisfied, a graphical description is given in Fig. 3(a) where one time constant, h, is measured as the time required to satisfy 63 per cent of a step change in demand as illustrated.
FIG.
3. Dynamic
response characteristics.
In Fig. 3(a), the time constant is a function of a number of factors including the standard travel time from the origin to the destination assuming no outside influences on the tour such as picking up additional passengers, delays in pick-up, etc. Thus the standard travel time is a physical delay which introduces an unfair comparison penalty on the ability of the system to respond to changes in demand. Therefore, in computing the time constant and describing the system operation dynamics, it will be assumed that the demand representation is taken as the actual demand rate delayed by the standard travel time (or appropriate statistically determined constant delay). It is tempting to account for the demand as being satisfied earlier as for
14
Journal of The Franklii Institute
Dynamics of a Class of Optimum Routing Problems example the time at which it is assigned to a vehicle, but this leads to a number of anticipation difficulties. The system dynamics will be represented as shown in Fig. 3(b). In order to avoid unnecessary complication in notation, it will be assumed that the time axis and demand rate have been shifted such that t = 0 begins at the start of the step, d(t), in Fig. 3(b). The time constant, A, of Fig. 3(b) is a function of the state of the system at t = 0, x(0); the processor time constant, T*; and the algorithm, 8, for routing or rerouting vehicles to which passengers have been assigned for t
x = @x(O), 7*,
fq.
(5)
V. Multiprocessing
As discussed thus far, the dynamic model formulation has been a single delivery system and a single computing system. The ways of introducing multiprocessing include (1) the use of multiprocessor computers with a single dispatching system, (2) multidispatching systems with multiprocessing computers, (3) multidispatching with a single processor computer, (4) multidispatching with a single multiprocessor computer, etc. Of these, (1) and (2) will be discussed. The other variations follow trivially from this discussion. The multiprocessor computer with a single dispatching system is used to speed up the computation time and directly affects T* and T* through c and affects X through 8. The effect on X through 8 is more of a policy or a decision matter than a hardware matter and will be discussed later. The effects on T* and T* are actually brought about by distributing the demands and consequently the computation tasks among the individual processors. The distribution of demands among processors can be represented as shown in Eq. (6) where f3 is a decision vector giving an appropriate distribution, i.e. s(t) = p(t).
(6)
The vector p is to be determined by some appropriate strategy involving the particular computer configurations considered with the appropriate reduction in T* and r*. Consider now the multidispatching system. In this case, some partition, e.g., geographic boundaries, determines the distribution of demands, d(t), where d(t) is an n-element vector representing the individual demand rates arriving out of the n-partitioned sectors. Each of the demand elements, d,(t), is satisfied at some rate, s&t), giving the n-element vector, s(t). Thus it is possible to measure the demands not yet satisfied or “error” as shown in Eq. (7) e(t) = d(t) -s(t).
(7)
It is this error which provides the “drive” for the dynamic routing system. The routing system is able to respond to an error (unsatisfied demand) with some time constant, h, or set, {h), of time constants (accounting for a
Vol. 300, No. 1, July 1975
15
Marlin H. Mickle and William G. Vogt separate time constant for each of the sectors). In a dynamic sense, the drive for the system is provided by the error through a matrix, F, of inverse time constants as shown in Eq. (8) where viris the time rate of change of the rate at which demands can be processed, G(t) = Fe = F[d(t) -s(t)].
(8)
In processing the demands for one sector, some of the processing capability of other sectors may be required such as checking for vehicles which may be operating between sectors, shared computer facilities, data access for intersector travel, etc. Thus the total processing capability of the sector cannot be devoted to satisfying demands in that sector, d,(t). Based on past operating experience, it would be possible to determine, on an aggregated basis, the amount of processing capability from the ith sector required to process a demand in the jth sector. Thus define aij as the rate at which the ith processing capability is required to process one demand in the jth sector per unit time. Based on the definitions for w, s and (aij ) = A, it is possible to define a set of static production equations which are very similar to the Leontief input-output formulation for economic systems (9), i.e., wi = gaijwi+si. j=I
The matrix form of (9) is w = Aw+s, (I-A)w The matrix (Iviable (10). The term Aw interactions. The dynamic constant value, actuating error.
A) is known
(10)
= s.
as the Leontief
(11) matrix if the matrix
represents the loss in processing capability
A is
to the intersector
model is formed assuming that if the actuating error is a ultimately the total rate of produ&ion will equal the The dynamics are given by G(t) = -Fw(t)
+ Fe(t).
(12)
F is a diagonal matrix formed from the set of time constants where each fii is the inverse of the corresponding element of {A>. Under this formulation, the dynamic syst,em model is analogous to a dynamic Leontief formulation (9) and yields the model described by Eqs. (13) and (14) and illustrated in Fig. 4 :
2
‘-‘:32
-(I-FIG. 4. The dynamic
16
-A)-‘4 multisector
model.
Journal of The Franklin Institute
Dynamics of a Class of Optimum Routing Problems G(t) = - F(I-
A) w(t) + Fd(t),
s(t) = (I-A)
(13)
w(t).
(14)
The use of a multiprocessor for a single sector is essentially the same principle as the multisector situation. In this case, there is a single demand input stream, d(t), which in some way is distributed among the n processors. The distribution of demands will be according to a decision vector p which in general may be a function of time, geography, state of the system, etc. Using the decision vector p, the model of the system is described by Eqs. (15) and (16) and illustrated in Fig. 5, -n :
>&
<**9x
-(I-nP!X FIG.
5. The dynamic multiprocessor
k(t) = -F(I-A)w(t)+FPd(t), s(t) = (I- A) w(t).
model.
(15) (16)
The model of a multisector, multiprocessor system is now a natural extension of the previous two models. Each sector of d(t), d,(t), can be processed through a multiprocessor with associated decision vector pi. It is important to note at this point that two different types of A matrices are involved. The A matrix of the model in Fig. 4 describes the loss in processing capability due to the sector interaction for route assignments, information transfer, etc. This particular matrix will be designated A,. The A matrix of Fig. 5 describes the loss of processing capability due to the shared facilities such as core memory, input-output channels, etc. This particular matrix will be designated A,. The model of Fig. 5 represents one sector of the model of Fig. 4. The combined model is illustrated in Fig. 6. The total aggregated demand is given by a(t) which is divided into n sectors by a giving d(t) = aa( Each sector is divided among m processors by pi giving x,(t) = Piwi(t). Assume that each pi in Fig. 6 is an m element column vector, p = col(Lj92, . . . . /3J and a is an n element column vector, a = co1 (acl,ffz, . . . . a,). Now e, ti, w, z and s are n element column vectors; and xi and yi are m element column vectors. F and K are n x n matrices with F = diag (fi,fi,
. . . ,f,J
The matrix A, becomes an m x m matrix, and A, becomes an n x n matrix. The vector 1 becomes 1 = col(l,l, . . . . 1). The model of Fig. 6 can now be redrawn as shown in Fig. 7 taking advantage of the above definitions. It is
Vol. 300, No. 1, July 1975
17
Marlin H. Mickle and William G. Vogt
a
FIG. 6. Sectors of the combined dynamic model.
-(r-n,iln,[e,(I-npl/~]-’ FIG. 7. The combined dynamic model.
to note that care must be exercised to insure appropriate multiplication or postmultiplication of the matrices and vectors. The equations for the combined model are developed below:
important
Substitution
pre-
fi = -Fw+Fe,
(1’)
e = ~-KS,
(13)
d = a&
(13)
s = (I-A,)wp’(I-A,)%
(20)
yields fi = -Fw-F(I-A,)wP’(I-A,)‘l+Faa,
(21)
5 = 1’s.
(22)
The closed loop equations for the model can now be obtained: G = -Fw-FK(I-A,)w@‘(I-A,)‘l+Faa, 5 = l’s
(23)
= l’(I-A,)wp’(I-A,)‘l.
(24)
Now K is to be chosen such that at equilibrium at vi = 0, s, = (I-A,)w,Y(I-AJ’l,
(ti = 0), s, = d, = aa. Thus (25)
w, = e, = d,-Ks,,
(26)
giving s, = (I-A,)
18
(a,-K,)
p’(I-A,)‘l.
(27)
Journalof The Franklin
Institute
Dynamics of a Class of Optimum Routing Problems Now
(27) can be manipulated
to yield
(I - A,)-1 [I + (I - A,) KfJ’(I- A,)‘11 s, = d,. Applying
(28)
d, = s,, K can be solved for with the following result: K = - (I - A,)-1 A#‘(1
-A,)’ 11-l.
(29)
The condition d, = s, is not a condition on the input and output a and 5. However, consider the condition on d and s and its relationship to a and 5. At equilibrium, the supply is to be equal to the demand, i.e. d, = s,. From Fig. 7, d=
aa
and
{=
1’s.
(30)
and
&,=l’se
(31)
Thus, d,=&, and 5, = 1'aa,.
(32)
From the definition of a, gap=1
(33)
i=l
and thus, [e=I’016e
2iSai
(i=l
)
a,=a,
which assures that 5, = a, when the condition VI. Summary
(34)
d, = s, is satisfied.
of The Dynumic Model
Equations (23) and (24) can be manipulated algebraically to yield the following simpler form for the dynamic model incorporating the K matrix previously determined, G = - FII - (I - A,)-1 A,(1 - A&] w 5=
+
Fat?,
(35)
l’(I-AJwfY(I-A,)‘1
(36)
which can be reduced to ti = -F(I-A,)w+Faa, 5=
(37)
l’(I-As)wp’(I-A,)‘l.
(38)
The solution of (35) for w is given in (39). 5 can be obtained directly from (36) and (39), w = exp{-Ffl-(A,]t}w(O)+ The method of obtaining including (11).
Vol. 300, No. 1. July 1976
fexp { - FII - A,] 7) Fa d7. s0
exp{ } is discussed
in a number
(39)
of references
19
Marlin H. Mickle and William G. Vogt VII.
A Numerical
Example
Consider a demand actuated transportation system which uses the model of Fig. 7 and is described by Eqs. (37) and (38). The system will be divided into two sectors with an accompanying a = (0-2,0-S) and will be processed on a two processor computer where the task of routing is divided according to p = (0.5,0.5) between two processors. The Ap matrix is given below in (40):
Ap =
L1 0.0
o-1
o-1
o-0
(40)
.
It is assumed that processing a demand in any processor reduces by 10 per cent the capability of processing a demand in any other processor except the given processor. The second sector has the majority of the demands as can be seen from the given a. For example purposes, the following A, matrix will be assumed: 0.0
0.3
A, = i 0.3
1
(41)
0.0 *
From the definitions of a and p, m = n. The unit of time will be taken as an hour with the following matrix of inverse time constants : F=
Equation
11
*=-[5;6
= [:
4.
(42)
[;,“,
;“o”]w+[I
;]
(43)
[J%
Jw+[Js.
(44)
(38) becomes
<=[I
11 [‘b”,
l”o”] wP.5 5=
20
I:RS]
(37) thus becomes
*--[I
Equation
r/I2
[0.63
o-51 [ y1 0.631~.
;J
[:I,
(45) (46)
Journal of The Franklin Institute
Dynamics of a Class of Optimum Routing Problems The solution for w with i?(t) taken as a unit step function is given in (47) : w = exp[-[
“,“,
i15]t)w(OJ
+/;exP(-[ (
r 5.0
- 0.6
;I, -1.5 2.0
;;5](t-r))
[::I
t w(O) I)
dr,
(47)
exp(-[;;6 ;;]‘)[::I]+ 1 0.48
(48)
[ 0.95 *
Substitution of the result for w obtained from (48)can be substituted (47)to obtain the final result for 5 as shown in (49) : 5 = [O-63 0*63]w(t).
into
(49)
VIII. Discussion
The general problem of satisfying demands over a routing network has been discussed in the first part of this paper as it applies to a method of solution which is also presented in a somewhat general context. The basic approach to the method of dealing with route generation or regeneration has been presented along more specific lines. Although some work has been done in the techniques of storing and regenerating routes, this is essentially a new area with a considerable number of problems especially as the range commonly termed large scale is extended (the large becomes larger) and the demand rates become higher. It was specifically pointed out earlier that in dealing with route regeneration techniques only those which regenerated routes without a mistake would be considered. However, in the method of analysis of how these route regeneration techniques are a part of the dynamic system of demands and routes, such a precise structural model may not be required although the model must accurately describe the dynamics of the process. The model of the previous section is considered to be an accurate model of the dynamic demand and assignment problem. However, it is the authors’ opinion that the assumptions avoid unnecessary complication without detracting from the overall validity of the model. The methods for taking these assumptions into account is straightforward and will not be included here. The model Fig. 7 is a classical type of linear system model. Thus all the analysis using this model can be done using the many results of linear system
Vol.300,No.l.July1975
21
Marlin H. Mickle and William G. Vogt theory including stability, sensitivit,y, etc., and the reader is referred to the extensive literature in this area for the corresponding developments. Among the most notable references in this area is Zadeh and Desoer (12). IX.
Operational
Analysis
Three items which are of particular significance in the model of Fig. 7 are the matrices a, B, A, and A,. The elements of the F matrix represent the time constants of the system or systems processing the demands. The analysis of scheduling techniques, routing algorithms, etc. for given demands provides a basis for the evaluation of alternatives. The p vector is a decision vector which allocates demands among the various possible processors which may make up the complete routing system. While a considerable amount of work of this type has been done for such things as job shop analysis and data processing schemes, very little work has been done in this area of respect to demand actuated systems. The A, matrix, while it can be determined from known data processing information, provides a basis for evaluating alternative strategies in the assignment of available facilities for handling the route demands. It is the authors’ opinion that this model will provide the basis for the operational analysis for demand actuated transportation or routing systems with multisectors using multiprocessor computers. X.
Summary
The paper is concerned primarily with the regeneration of minimum or optimum routes over a network and with the incorporation of such methods of route regeneration as a part of a larger routing and scheduling system. A particular method of route regeneration is discussed and illustrates the kinds of route generation which can be used for this type of problem as opposed to classical minimum or optimum routing methods. The problem is subsequently generalized, and a dynamic system model is presented which adds to the generalization and provides a system formulation to the general type of demand actuated transportation and routing systems. References
(1) M. Pollack and W.
Wiebenson, “Solutions of the short.est route problem-a review”, Operat. Res., Vol. 8, pp. 866468, 1960. (2) N. H. M. Wilson and D. Roos, “CARS : computer aided routing system”, Eighth Annual Meeting, Trans. Res. Forum, 1967. (3) D. A. Rose, F. J. Mammo and R. Fevout, “An electronic guidance system for highway vehicles”, IEEE Trans. Vehicular Technology, Vol. 19, No. 1, pp. 143152, 1970. (4) R. G. Hoelzeman and RI. H. Mickle, “An efficient data model for large scale transportation systems”, Kybernetes: Int. J. of Cybernetics and Genera.1 Systems, Vol. 3, pp. 73-79, 1974.
22
Journal
of The Franklin
Inst,itute
Dynamics of a Class of Optimum Routing Problems (5) M. H. Mickle, T. W. Sze and D. E. Rathbone, “Neighborhood storage”, IEEE Trans. Systems Science Cybernetics, Vol. 4, No. 2, pp. 138-144, 1968. (6) M. H. Mickle, “On a policy equation and contours with neighborhood storage”, Proc. E”irst Hou8ton Conf. on Circuits, System and Computers, University of Houston, 1969. (7) M. H. Mickle, R. G. Hoelzeman and J. T. Cain, “A class of system estimators for routing problems”, Record Seventh Allerton. Conf. on Circuits Systems, University of Illinois, 1969. (8) J. C. Wang, M. H. Mickle and R. G. Hoelzeman, “A set covering applicat,ion in minimum route generation”, Int. J. Cybernetics and General System, Vol. 3, No. 4, pp. 205-217, 1974. (9) W. W. Leontief, “The structure of the American Economy 1919-1939”, Oxford University Press, London, 1951. (10) S. Chakravarty, “Capital and Development Planning”, MIT Press, Cambridge, Mass., 1969. (11) J. L. Melsa and D. Schultz, “Computer Programs for Computational Assistance in the Study of Linear Control Theory”, McGraw-Hill, New York, 1970. (12) L. A. Zadeh and C. A. Desoer, “Linear System Theory”, McGraw-Hill, New York, 1963.
Yol. 300, No. 1, July 1975
23
by MARLIN
and WILLIAM
MICKLE
G. VOGT
Department of Electrical Engineering University of Pittsburgh, Pittsburgh, Pennsylvania A
ABSTRACT:
transportation
particular systems
class of routing
as a part of the more general by methods presented
routing
problem.
other than classical minimum
model for proceaaing
problems
associated
with
require8the repeated regeneration of minimum
demands
The generation
route solutions
in a demand
actuated
demand
actuated
or optimum
or regeneration
is discussed.
transportation
A general
or routing
routes
of routes system
Byetern is
and, disczLesed.
I. Introduction
The minimum route problem (1) is interesting and has a number of important applications. Considerable progress has been made in reducing the time (normally computer time) required to obtain the minimum route between two points, 0 and D, origin and destination respectively, in a network composed of N vertices, V, and a number of edges, E, and associated costs, C. The similar problems of finding all minimum routes in a network or all routes from a fixed origin have also been considered. For a single (0, D) pair, this computation time is relatively small although it is a function of the size of the network, the type of computer, etc. There exist applications (2, 3) where, as a part of a larger problem, normally some type of routing or traffic control, it is required to obtain the minimum route, r, between 0 and D. Such problems typically have thousands of vertices (4) or nodes as a part of the network and require the minimum routes, on demand, possibly hundreds or thousands of times/hr. It is important to point out that the bounds on network size and demand rate are in reality a function of the state-of-the-art of routing algorithms and the minimum route problem. The network sizes and demand rates typically mentioned represent a sort of minimum for practical applications of a number of automatic control schemes for demand actuated systems. The typical dimensions of the problem to be formulated and discussed in this paper represent somewhat of a fuzzy point or threshold at which these applications become practical. As practicality is achieved, the sizes of systems to be considered as desirable for implementation will increase by orders of magnitude. * The resemch reported in this paper w&s supported in part by National Foundation Grant GK 32237.
Science
Marlin H. Mickle and William G. Vogt There are two limiting factors to the basic problem. The first is the network size in terms of the number of vertices characterized by N. The larger the value of N, the larger storage and data handling capabilities required to obtain a solution. The second factor is the rate at which solutions are required. This factor can be characterized by T* which may represent the minimum time? between which any two solutions are required. It would be premature to discuss queues at this time, and consequently the time T* will be considered as the minimum time between solutions over a complete demand profile. The application of the minimum route type problem to the more general routing problem under formulation is thus one of being able to obtain a minimum route, r, for any (0, D) pair in time T* for a network of size N. As T* decreases, the allowable computation time for the route r must also decrease. However, the general nature of the application is such that it encourages the use of larger networks consequently requiring more time to obtain r for the now larger N. The upper bound on N is a function of the application. If the application is transportation, the upper bound is likely to be a function of geographical area to be covered. The time required to obtain a solution by any particular method will be noted as T which will vary as the method and associated details to obtain r for (0, D) vary. For any method to be satisfactory, it is required that for the corresponding T, T < T *, for any (0, D)E(V, V). An obvious alternative to computing r each time a demand occurs is to store all of the routes and to look up (retrieve) r from a storage device or devices with some appropriate scheme. Such an alternative is reasonable in that all possible routes can, for example, be stored in a matrix called the su,ccessor matrix, S, which essentially requires the same amount of storage as the cost matrix, C, which describes the system. Any element of S, sij, is the next vertex in the route having i as an origin and j as a destination, i.e. (0,D) = (Cj). All routes are computed “off line”, and the success or matrix is formed. In this process, the minimum route techniques are important to obtain S. Once S is generated, the minimum route techniques are no longer needed, and a route can be obtained in some time, T+. The problem of obtaining minimum routes from S is one of table look-up. However, for general applications, N will frequently be large enough to render this method of obtaining the routes infeasible. As N becomes large, the available storage media is normally slower, and T+ becomes too large. The problem to be solved is one of obtaining a method which will produce the minimum route, r, on demand for any (0, D) pair in time T+< T* requiring a minimum amount of storage. Conventional methods of solving for the minimum route are ineffective because the associated time, T, is greater than the minimum time T*. Route t Variations on this time will be discussed later in this paper.
Journal OFThe Franklin Institute
Problems storage techniques such as the successor matrix are ineffective because the storage requirement is too large although T+ may be less than many practical T”. ZZ. Previous Related Work
It is generally agreed that some cost and route information is contained in geographic or other composition of many practical applications as in transportation for example. One of the more frequent attempts to take advantage of this situation is the “cost proportional to distance” assumption (5). However, there are definite problems in this assumption which are sufficient to make the routing system completely ineffective. In this discussion, ineffective implies that incorrect decisions are made on the basis of any assumption which involves the data for some model of the system. While it is important to employ techniques to take advantage of any information which can reduce the amount of data required, such techniques are only considered acceptable if they do not result in any incorrect decisions made by the system using them. Thus a technique will be considered ineffective if its application has a non-zero pobability of causing at least one incorrect decision. The data storage technique must thus be a perfect filter with the ability to exactly regenerate input data on demand. Obviously “garbage in garbage out” applies to any supplied data. The type of data storage technique as conceived here is one which accepts input data in a specified form and converts it to some form which minimizes the amount of storage required for the new form according to a specified objective or objective function. Using the successor matrix as an example, one data storage technique is a technique which may require the cost and successor information for a given network and will produce a successor element, Sij, on demand. The successor matrix can then be used to produce the desired element on demand with a minimum amount of programming. Given the cost matrix, a minimum route method can be coded to generate the successor element. Any other method which accepts the cost and successor information and produces su on demand is an example of a data storage technique. The above examples are further illustrated in Fig. 1. One method of storing the route information is based on a cost proportional to distance assumption and involves the cost from the origin, 0, to the destination, D. An old adage, “ . . . a long trip always starts with a first step”, provides the foundation for the technique of Neighborhood Storage (5). The basic reasoning of this technique will now be summarized. The successor vertex, sij, will be noted as v(i,j). For any origin, 0 E 7, there exists a number, n(O), of vertices which are connected to 0 through an edge eoj E E for the graph characterization of a network, G = (V, E). The set of vertices connected to any vertex, i, will be
Vol.300,No.1, July 1975
9
Marlin H. Mickle and William G. Vogt
a. successor Matrix Implementation
c
-
*
b.
MINIMUM ROUTE PROGRAM
sij
Minimum Route Implementation
C. Data Storage Technique FIG. 1. (a) Successor matrix implementation.
implementation.
(b) Minimum route (c) Data storage technique.
as J’(i) and will be termed the set of feasible intermediate vertices of i, i.e., “The possible first steps”. Thus given an origin, destination combination, (0, D), the first step in the route generation is one of finding the v EF(O) such that ver(0, D) where r(0, D) is the ordered set of vertices making up the optimum route from 0 to D. It is a simple matter to prove that for positive cij and that for any vertex v lr(0, D) such that v is connected to 0, vrv(0, D). Therefore, for demand (;,j), sij = v(i,j) Er(i,j). Note that ~(0, D) = F(0) I-Ir(0, D) = soI) where n is the set intersection. Assume all vertices are located in R x R = R2, a two-dimensional Euclidean space. Each vertex has a two-dimensional representation (x1,x2). For any vertex i, this representation is noted as x(i). Consider now the following functional : identified
f(i,j, k) = d[x(i), x(j)1 +d[x(jL WI, 10
Journal of The
(1)
FranklinInstitute
Dynamics of a Class of Optimum Routing Problems where d[u, v] is the Euclidean distance from u to v. The functional, (l), can now be the sum of the straight line distance segment from the origin, 0, to a feasible intermediate vertex, WET(O), and the straight line distance segment from v to the destination, D, e.g.,
f(0, v,D)
= d[x(O), x(w)] +d[x(v), x(D)] : v~F(0).
(2)
The functional, (2), is a rather good approximation for a number of transportation systems (5-7). f(0, v, D) is t ermed an ideal policy equation. If for any (0, D) combination, the ideal policy equation works, it can be used as a part of the route generation system discussed previously. If the ideal policy equation does not work for some (0, D) combinations, the exceptions can be noted and stored. Given any demand (0, D) the process is one of determining if (0, D) E A or (0, D) EA’ where A is the set of all (0, D) combinations for which (2) works and correspondingly A’ is the set of all (0, D) combinations for which (2) does not work. Obviously the problem is to choose an ideal policy equation and process for which the order of A is larger than the order of A’. Another obvious possibility is to formulate a step decision process A,, A,, . . . ; A; AL, . . . . However, the fundamental formulation of the individual A,, A,, . . ., etc. is the same as for A with some appropriate partitioning scheme. Therefore we will only be concerned with a single step process at this time noting its obvious extensions. Given a demand, (i,j), the determination of the membership of (i,j) in A or A’ requires a certain amount of storage (in general of various means and types) which will be symbolically described as X(A) or #(A’), respectively, with no particular quantification specified. The “goodness” of any method of route generation can thus be described as one where S(A) u B(A’) is relatively small and the associated time, T+, is also small. The present method of obtaining the routes is simple. Given (i,j) ; if (i,j)E& (3) if (i, j) $ A, some other method (including table look-up) is used to obtain w(i,j). Thus the object of the method is to maximize the membership of A and minimize #(A’) and its associated details. Other ideal policy equations can be used (5,7) and the basic idea is independent of the equation although the order of A, o(A), is obviously not. Using the above technique, the storage, S(A), is primarily dependent on two types of information, the connection information in order to define P(i) and the location of each vertex in R x R. Some work has been done in this area with relatively good results (4). The storage, X(A’), has recently been considered (8) again with relatively good results. In &‘(A’), the problem is one of finding an individual optimum covering identified with each (i, j) E A’. Each covering is then tagged with (i, j) and So,. The retrieval process is consequently one of locating which set covers a given (;,j).
Vol. 300, No. 1, July 1975
11
Marlin H. Mickle and William G. Vogt Obviously, the intent of the above work is not to compute minimum routes but rather to retrieve minimum routes in a time T+ which is less than T* and hopefully considerably less than T* so as to implement higher demand rates and larger networks. ZZZ. A Generalization
The previous discussion can be characterized as consisting of two parts or problems. The first is given a demand, (i, j), find a vehicle?, k, from some assigned set of vehicles to transfer the demand initiate from vertex i to vertex j, according to a schedule p(i,j, k) which follows an optimum1 route, r*(Q), while satisfying a set of constraints and maximizing or minimizing an objective function in a time T+ < T *. This problem is one of algorithms, objective functions, constraints, etc. The second problem is one of given an origin-destination combination, (;,j), find an optimum route r(i,j). The above characterization is illustrated in Fig. 2. 7-*
,
(i,j)
r(i,j),
(i,j)
I(i,j) ,
FIG.
p(i.j;k)
T”
2. Problem characterization.
This paper is primarily concerned with problem two. The solution of the problem may be achieved by a number of means, three of which will be mentioned. The first is the direct computation of r(i,j) using any minimum route the size of the network and the technique. For many applications, corresponding T* may be such that this straightforward method is the most appropriate. A second method is the table look-up of r(Q)) using a successor matrix. This method is entirely appropriate for situations where T* is large and storage is not a problem. t In this oontext, a vehicle is any physical means by which the specified route will be traversed using up some capacity and requiring an associated non-zero time. $ While the optimum route is normally the minimum route, other criteria may be specified. Thus while the term minimum may be used, the technique is more general in application.
12
Journal of The Franklin Institute
Dynamics of a Class of Optimum Routing Problems A third method is the method discussed in the previous section where T* is small and storage is a problem. Other methods may exist which are applicable to specific problem classes such as those of street networks which display very regular characteristics such as intersection location, all costs equal, etc. However, it is the purpose of this paper to focus on techniques of the type previously discussed which are applicable to general systems although acceptability may vary according to different objectives such as time, storage, regularity, etc. IV.
A Dynamic
System
Model
The raison d’&e for the current problem formulation is in the minimum amount of time in which a solution must be obtained. The time, T*, is the minimum time in which a route, r*, must be determined, and the time, T*, is the minimum amount of time in which a route, r, must be determined. It was pointed out in the beginning of this paper that queueing would not be discussed. However, it can be noted that the high rates which dictate T* and T* may be of short duration and infrequent occurrence and if taken as an absolute limit would place an inordinate demand on the corresponding techniques. Thus it is possible that a certain amount of queueing may be acceptable. The following discussion is based on this assumption. Consider a system of the type discussed in earlier sections with a demand rate (demands/time), d(t), and an assignment rate (optimum assignments/ time), s(t). Any given technique, computing system, algorithm, etc. will be assumed to have an associated response time T which is a variable and is a function both of the given characteristics (technique, computing system, algorithm, etc.) and also of the characteristics of the demands such as number of vertices in the minimum path, vehicles available at the time a demand occurs, etc. The time constant T* is a function of the particular network being used in terms of connectivity, E, dimension (which can be expressed in terms of V) and cost, C, i.e., T* =f*(V, C,E). Given any particular origin-destination pair, (0, D), the computation time, T is a function of (0, D), i.e. T = f(0, D). In terms of Fig. 2, the discussion is with the T* of problem two. Consider now the T* of problem one. The time constant is a function of the algorithm being used (denote this by a), the time constant of problem two, T*, the number of vehicles (which will be denoted by v), computer and computer program (denoted by c), i.e., T* = g*(or, v, T*, c). Once the algorithm is introduced, the factors making up any particular 7 must be accounted for by the state of the system, x; at the time, t, the demand, d, to go from i to j occurs; (i,j; t). Thus 7 = g(a,v, T*,c,x; t). At this point, the problem begins to become overly complicated by the general nature of the unknown functions, f, f *, g and g*. It is conjectured that it is possible to express g* in terms of T* as shown in Eq. (4), T* = g*(ar, v, T*, c) = g+(cll,v, c) T*.
Vol.300.N&l, July1975
(4
13
Marlin H. Mickle and William G. Vogt T* can be determined without much difficulty, and for a fixed ~11 and v,g+ can be determined experimentally. Thus it is not unrealistic to talk about a r* for a given processor (c), LXand v. Consider now the dynamics of the total routing problem as for example in the case of demand actuated transportation systems. Demands come in at a rate, d(t), in terms of demands per unit time. The demands are in turn to be satisfied at some rate, call it s(t). It is important to elaborate on the point at which a demand is considered satisfied. Assume for the moment that time is of no importance to the customer, a demand may be considered as being satisfied when the customer is scheduled on a particular vehicle, i.e. s(t) is a mea’sure of people being assigned. If time is important to the customer, a demand is considered as satisfied when a person is delivered within the time constraint specified by the customer, the system or both. Otherwise, another category is necessary to include those persons who have not been delivered and whose time constraints have been violated. In this paper, the measure of demands satisfied is taken as the point in time at which the person is delivered to the specified destination. Therefore, there is a time delay between when a demand rate appears and when the demands are actually satisfied. In terms of per cent of demands satisfied, a graphical description is given in Fig. 3(a) where one time constant, h, is measured as the time required to satisfy 63 per cent of a step change in demand as illustrated.
FIG.
3. Dynamic
response characteristics.
In Fig. 3(a), the time constant is a function of a number of factors including the standard travel time from the origin to the destination assuming no outside influences on the tour such as picking up additional passengers, delays in pick-up, etc. Thus the standard travel time is a physical delay which introduces an unfair comparison penalty on the ability of the system to respond to changes in demand. Therefore, in computing the time constant and describing the system operation dynamics, it will be assumed that the demand representation is taken as the actual demand rate delayed by the standard travel time (or appropriate statistically determined constant delay). It is tempting to account for the demand as being satisfied earlier as for
14
Journal of The Franklii Institute
Dynamics of a Class of Optimum Routing Problems example the time at which it is assigned to a vehicle, but this leads to a number of anticipation difficulties. The system dynamics will be represented as shown in Fig. 3(b). In order to avoid unnecessary complication in notation, it will be assumed that the time axis and demand rate have been shifted such that t = 0 begins at the start of the step, d(t), in Fig. 3(b). The time constant, A, of Fig. 3(b) is a function of the state of the system at t = 0, x(0); the processor time constant, T*; and the algorithm, 8, for routing or rerouting vehicles to which passengers have been assigned for t
x = @x(O), 7*,
fq.
(5)
V. Multiprocessing
As discussed thus far, the dynamic model formulation has been a single delivery system and a single computing system. The ways of introducing multiprocessing include (1) the use of multiprocessor computers with a single dispatching system, (2) multidispatching systems with multiprocessing computers, (3) multidispatching with a single processor computer, (4) multidispatching with a single multiprocessor computer, etc. Of these, (1) and (2) will be discussed. The other variations follow trivially from this discussion. The multiprocessor computer with a single dispatching system is used to speed up the computation time and directly affects T* and T* through c and affects X through 8. The effect on X through 8 is more of a policy or a decision matter than a hardware matter and will be discussed later. The effects on T* and T* are actually brought about by distributing the demands and consequently the computation tasks among the individual processors. The distribution of demands among processors can be represented as shown in Eq. (6) where f3 is a decision vector giving an appropriate distribution, i.e. s(t) = p(t).
(6)
The vector p is to be determined by some appropriate strategy involving the particular computer configurations considered with the appropriate reduction in T* and r*. Consider now the multidispatching system. In this case, some partition, e.g., geographic boundaries, determines the distribution of demands, d(t), where d(t) is an n-element vector representing the individual demand rates arriving out of the n-partitioned sectors. Each of the demand elements, d,(t), is satisfied at some rate, s&t), giving the n-element vector, s(t). Thus it is possible to measure the demands not yet satisfied or “error” as shown in Eq. (7) e(t) = d(t) -s(t).
(7)
It is this error which provides the “drive” for the dynamic routing system. The routing system is able to respond to an error (unsatisfied demand) with some time constant, h, or set, {h), of time constants (accounting for a
Vol. 300, No. 1, July 1975
15
Marlin H. Mickle and William G. Vogt separate time constant for each of the sectors). In a dynamic sense, the drive for the system is provided by the error through a matrix, F, of inverse time constants as shown in Eq. (8) where viris the time rate of change of the rate at which demands can be processed, G(t) = Fe = F[d(t) -s(t)].
(8)
In processing the demands for one sector, some of the processing capability of other sectors may be required such as checking for vehicles which may be operating between sectors, shared computer facilities, data access for intersector travel, etc. Thus the total processing capability of the sector cannot be devoted to satisfying demands in that sector, d,(t). Based on past operating experience, it would be possible to determine, on an aggregated basis, the amount of processing capability from the ith sector required to process a demand in the jth sector. Thus define aij as the rate at which the ith processing capability is required to process one demand in the jth sector per unit time. Based on the definitions for w, s and (aij ) = A, it is possible to define a set of static production equations which are very similar to the Leontief input-output formulation for economic systems (9), i.e., wi = gaijwi+si. j=I
The matrix form of (9) is w = Aw+s, (I-A)w The matrix (Iviable (10). The term Aw interactions. The dynamic constant value, actuating error.
A) is known
(10)
= s.
as the Leontief
(11) matrix if the matrix
represents the loss in processing capability
A is
to the intersector
model is formed assuming that if the actuating error is a ultimately the total rate of produ&ion will equal the The dynamics are given by G(t) = -Fw(t)
+ Fe(t).
(12)
F is a diagonal matrix formed from the set of time constants where each fii is the inverse of the corresponding element of {A>. Under this formulation, the dynamic syst,em model is analogous to a dynamic Leontief formulation (9) and yields the model described by Eqs. (13) and (14) and illustrated in Fig. 4 :
2
‘-‘:32
-(I-FIG. 4. The dynamic
16
-A)-‘4 multisector
model.
Journal of The Franklin Institute
Dynamics of a Class of Optimum Routing Problems G(t) = - F(I-
A) w(t) + Fd(t),
s(t) = (I-A)
(13)
w(t).
(14)
The use of a multiprocessor for a single sector is essentially the same principle as the multisector situation. In this case, there is a single demand input stream, d(t), which in some way is distributed among the n processors. The distribution of demands will be according to a decision vector p which in general may be a function of time, geography, state of the system, etc. Using the decision vector p, the model of the system is described by Eqs. (15) and (16) and illustrated in Fig. 5, -n :
>&
<**9x
-(I-nP!X FIG.
5. The dynamic multiprocessor
k(t) = -F(I-A)w(t)+FPd(t), s(t) = (I- A) w(t).
model.
(15) (16)
The model of a multisector, multiprocessor system is now a natural extension of the previous two models. Each sector of d(t), d,(t), can be processed through a multiprocessor with associated decision vector pi. It is important to note at this point that two different types of A matrices are involved. The A matrix of the model in Fig. 4 describes the loss in processing capability due to the sector interaction for route assignments, information transfer, etc. This particular matrix will be designated A,. The A matrix of Fig. 5 describes the loss of processing capability due to the shared facilities such as core memory, input-output channels, etc. This particular matrix will be designated A,. The model of Fig. 5 represents one sector of the model of Fig. 4. The combined model is illustrated in Fig. 6. The total aggregated demand is given by a(t) which is divided into n sectors by a giving d(t) = aa( Each sector is divided among m processors by pi giving x,(t) = Piwi(t). Assume that each pi in Fig. 6 is an m element column vector, p = col(Lj92, . . . . /3J and a is an n element column vector, a = co1 (acl,ffz, . . . . a,). Now e, ti, w, z and s are n element column vectors; and xi and yi are m element column vectors. F and K are n x n matrices with F = diag (fi,fi,
. . . ,f,J
The matrix A, becomes an m x m matrix, and A, becomes an n x n matrix. The vector 1 becomes 1 = col(l,l, . . . . 1). The model of Fig. 6 can now be redrawn as shown in Fig. 7 taking advantage of the above definitions. It is
Vol. 300, No. 1, July 1975
17
Marlin H. Mickle and William G. Vogt
a
FIG. 6. Sectors of the combined dynamic model.
-(r-n,iln,[e,(I-npl/~]-’ FIG. 7. The combined dynamic model.
to note that care must be exercised to insure appropriate multiplication or postmultiplication of the matrices and vectors. The equations for the combined model are developed below:
important
Substitution
pre-
fi = -Fw+Fe,
(1’)
e = ~-KS,
(13)
d = a&
(13)
s = (I-A,)wp’(I-A,)%
(20)
yields fi = -Fw-F(I-A,)wP’(I-A,)‘l+Faa,
(21)
5 = 1’s.
(22)
The closed loop equations for the model can now be obtained: G = -Fw-FK(I-A,)w@‘(I-A,)‘l+Faa, 5 = l’s
(23)
= l’(I-A,)wp’(I-A,)‘l.
(24)
Now K is to be chosen such that at equilibrium at vi = 0, s, = (I-A,)w,Y(I-AJ’l,
(ti = 0), s, = d, = aa. Thus (25)
w, = e, = d,-Ks,,
(26)
giving s, = (I-A,)
18
(a,-K,)
p’(I-A,)‘l.
(27)
Journalof The Franklin
Institute
Dynamics of a Class of Optimum Routing Problems Now
(27) can be manipulated
to yield
(I - A,)-1 [I + (I - A,) KfJ’(I- A,)‘11 s, = d,. Applying
(28)
d, = s,, K can be solved for with the following result: K = - (I - A,)-1 A#‘(1
-A,)’ 11-l.
(29)
The condition d, = s, is not a condition on the input and output a and 5. However, consider the condition on d and s and its relationship to a and 5. At equilibrium, the supply is to be equal to the demand, i.e. d, = s,. From Fig. 7, d=
aa
and
{=
1’s.
(30)
and
&,=l’se
(31)
Thus, d,=&, and 5, = 1'aa,.
(32)
From the definition of a, gap=1
(33)
i=l
and thus, [e=I’016e
2iSai
(i=l
)
a,=a,
which assures that 5, = a, when the condition VI. Summary
(34)
d, = s, is satisfied.
of The Dynumic Model
Equations (23) and (24) can be manipulated algebraically to yield the following simpler form for the dynamic model incorporating the K matrix previously determined, G = - FII - (I - A,)-1 A,(1 - A&] w 5=
+
Fat?,
(35)
l’(I-AJwfY(I-A,)‘1
(36)
which can be reduced to ti = -F(I-A,)w+Faa, 5=
(37)
l’(I-As)wp’(I-A,)‘l.
(38)
The solution of (35) for w is given in (39). 5 can be obtained directly from (36) and (39), w = exp{-Ffl-(A,]t}w(O)+ The method of obtaining including (11).
Vol. 300, No. 1. July 1976
fexp { - FII - A,] 7) Fa d7. s0
exp{ } is discussed
in a number
(39)
of references
19
Marlin H. Mickle and William G. Vogt VII.
A Numerical
Example
Consider a demand actuated transportation system which uses the model of Fig. 7 and is described by Eqs. (37) and (38). The system will be divided into two sectors with an accompanying a = (0-2,0-S) and will be processed on a two processor computer where the task of routing is divided according to p = (0.5,0.5) between two processors. The Ap matrix is given below in (40):
Ap =
L1 0.0
o-1
o-1
o-0
(40)
.
It is assumed that processing a demand in any processor reduces by 10 per cent the capability of processing a demand in any other processor except the given processor. The second sector has the majority of the demands as can be seen from the given a. For example purposes, the following A, matrix will be assumed: 0.0
0.3
A, = i 0.3
1
(41)
0.0 *
From the definitions of a and p, m = n. The unit of time will be taken as an hour with the following matrix of inverse time constants : F=
Equation
11
*=-[5;6
= [:
4.
(42)
[;,“,
;“o”]w+[I
;]
(43)
[J%
Jw+[Js.
(44)
(38) becomes
<=[I
11 [‘b”,
l”o”] wP.5 5=
20
I:RS]
(37) thus becomes
*--[I
Equation
r/I2
[0.63
o-51 [ y1 0.631~.
;J
[:I,
(45) (46)
Journal of The Franklin Institute
Dynamics of a Class of Optimum Routing Problems The solution for w with i?(t) taken as a unit step function is given in (47) : w = exp[-[
“,“,
i15]t)w(OJ
+/;exP(-[ (
r 5.0
- 0.6
;I, -1.5 2.0
;;5](t-r))
[::I
t w(O) I)
dr,
(47)
exp(-[;;6 ;;]‘)[::I]+ 1 0.48
(48)
[ 0.95 *
Substitution of the result for w obtained from (48)can be substituted (47)to obtain the final result for 5 as shown in (49) : 5 = [O-63 0*63]w(t).
into
(49)
VIII. Discussion
The general problem of satisfying demands over a routing network has been discussed in the first part of this paper as it applies to a method of solution which is also presented in a somewhat general context. The basic approach to the method of dealing with route generation or regeneration has been presented along more specific lines. Although some work has been done in the techniques of storing and regenerating routes, this is essentially a new area with a considerable number of problems especially as the range commonly termed large scale is extended (the large becomes larger) and the demand rates become higher. It was specifically pointed out earlier that in dealing with route regeneration techniques only those which regenerated routes without a mistake would be considered. However, in the method of analysis of how these route regeneration techniques are a part of the dynamic system of demands and routes, such a precise structural model may not be required although the model must accurately describe the dynamics of the process. The model of the previous section is considered to be an accurate model of the dynamic demand and assignment problem. However, it is the authors’ opinion that the assumptions avoid unnecessary complication without detracting from the overall validity of the model. The methods for taking these assumptions into account is straightforward and will not be included here. The model Fig. 7 is a classical type of linear system model. Thus all the analysis using this model can be done using the many results of linear system
Vol.300,No.l.July1975
21
Marlin H. Mickle and William G. Vogt theory including stability, sensitivit,y, etc., and the reader is referred to the extensive literature in this area for the corresponding developments. Among the most notable references in this area is Zadeh and Desoer (12). IX.
Operational
Analysis
Three items which are of particular significance in the model of Fig. 7 are the matrices a, B, A, and A,. The elements of the F matrix represent the time constants of the system or systems processing the demands. The analysis of scheduling techniques, routing algorithms, etc. for given demands provides a basis for the evaluation of alternatives. The p vector is a decision vector which allocates demands among the various possible processors which may make up the complete routing system. While a considerable amount of work of this type has been done for such things as job shop analysis and data processing schemes, very little work has been done in this area of respect to demand actuated systems. The A, matrix, while it can be determined from known data processing information, provides a basis for evaluating alternative strategies in the assignment of available facilities for handling the route demands. It is the authors’ opinion that this model will provide the basis for the operational analysis for demand actuated transportation or routing systems with multisectors using multiprocessor computers. X.
Summary
The paper is concerned primarily with the regeneration of minimum or optimum routes over a network and with the incorporation of such methods of route regeneration as a part of a larger routing and scheduling system. A particular method of route regeneration is discussed and illustrates the kinds of route generation which can be used for this type of problem as opposed to classical minimum or optimum routing methods. The problem is subsequently generalized, and a dynamic system model is presented which adds to the generalization and provides a system formulation to the general type of demand actuated transportation and routing systems. References
(1) M. Pollack and W.
Wiebenson, “Solutions of the short.est route problem-a review”, Operat. Res., Vol. 8, pp. 866468, 1960. (2) N. H. M. Wilson and D. Roos, “CARS : computer aided routing system”, Eighth Annual Meeting, Trans. Res. Forum, 1967. (3) D. A. Rose, F. J. Mammo and R. Fevout, “An electronic guidance system for highway vehicles”, IEEE Trans. Vehicular Technology, Vol. 19, No. 1, pp. 143152, 1970. (4) R. G. Hoelzeman and RI. H. Mickle, “An efficient data model for large scale transportation systems”, Kybernetes: Int. J. of Cybernetics and Genera.1 Systems, Vol. 3, pp. 73-79, 1974.
22
Journal
of The Franklin
Inst,itute
Dynamics of a Class of Optimum Routing Problems (5) M. H. Mickle, T. W. Sze and D. E. Rathbone, “Neighborhood storage”, IEEE Trans. Systems Science Cybernetics, Vol. 4, No. 2, pp. 138-144, 1968. (6) M. H. Mickle, “On a policy equation and contours with neighborhood storage”, Proc. E”irst Hou8ton Conf. on Circuits, System and Computers, University of Houston, 1969. (7) M. H. Mickle, R. G. Hoelzeman and J. T. Cain, “A class of system estimators for routing problems”, Record Seventh Allerton. Conf. on Circuits Systems, University of Illinois, 1969. (8) J. C. Wang, M. H. Mickle and R. G. Hoelzeman, “A set covering applicat,ion in minimum route generation”, Int. J. Cybernetics and General System, Vol. 3, No. 4, pp. 205-217, 1974. (9) W. W. Leontief, “The structure of the American Economy 1919-1939”, Oxford University Press, London, 1951. (10) S. Chakravarty, “Capital and Development Planning”, MIT Press, Cambridge, Mass., 1969. (11) J. L. Melsa and D. Schultz, “Computer Programs for Computational Assistance in the Study of Linear Control Theory”, McGraw-Hill, New York, 1970. (12) L. A. Zadeh and C. A. Desoer, “Linear System Theory”, McGraw-Hill, New York, 1963.
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