solutions to the optimal network design problem 'with ... - Science Direct

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Rex-B.

Vol. 13B, pp. 65-80.

Per&mm

Press 1979.

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SOLUTIONS TO THE OPTIMAL NETWORK DESIGN PROBLEM ‘WITH SHIPMENTS RELATGD TO TRANSPORTATION COST D. E. BOYCE Departmentof Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801,U.S.A. and

J. L. SOBERANES Ministryof Human Settlements,Mexico City, Mexico (Received I8 July 1978) Abstract-Two versions of an optimal network design problem with shipments proportional to transportation costs are formulated. Extensions of an algorithm developed in prior research for solving these problems are proposed and tested. The performance of the algorithms is found to improve substantially as the dependence of shipments on costs is increased. Moreover, the optimal solutions obtained are unexpectedly robust with respect to a wide range of transportation cost assumptions. These findings could have important computational and policy implications if applicable to larger networks. INTRODUCTION

Algorithms

proposed

in recent

years for the optimal network problem (or network design

problem) typically have been evaluated on test networks with all shipments equal to unity; e.g. see Scott (1%9), Hoang (1973), Boyce, Farhi and Weischedel (1973), Los (1976), and Dionne and Florian (1977). The problem solved by these algorithms may be stated briefly as follows: given a set of nodes, the transportation cost and construction cost of the links between all pairs of nodes, or between a specified subset of all pairs, and the shipments between all pairs of nodes, find the subset of links that minimizes total transportation cost subject to a budget constraint on total construction cost. Los (1976) includes construction costs in the objective function and finds the subset of links that minimizes the sum of transportation and construction cost. Setting shipments equal to unity in test networks somewhat simplifies the computations and facilitates comparisons of results; however, it also results in the study of network flows which are rather atypical as compared with those observed in the real world. Thus, once an algorithm is working reasonably well, it is appropriate to consider test networks which have more realistic shipments, or equivalently, trip interchanges. There is another important reason to perform such tests in addition to the obvious one of evaluating the algorithm’s performance. Testing a set of carefully specified networks may reveal new insights regarding the structure of transportation networks that could not be obtained from either theoretical analysis or algorithm construction. A provocative result of this type is obtained in the tests reported in this paper. In addition, the algorithm’s performance is shown to depend on the shipments’ relation to intemodal costs. Two sets of results are reported in this paper in which shipments are defined as a function of internodal cost. In the first set, intemodal shipments are a function of fixed intemodal costs, Previous tests with shipments equal to unity form a subset of these results. In the second set of tests, shipments are permitted to vary with the network configuration under consideration. A doubly-constrained gravity model is used to specify these shipments. This formulation leads to two complications which are considered in some detail below. First, the shipments are different for each network. Second, the objective function of the algorithm (total shipment cost) is a constraint on the gravity model used to specify the shipments. Consideration of these aspects led to the specification of an algorithm and calculation of results with properties that are rather different from those considered elsewhere. In the paper, the first set of tests are referred to as the fixed shipment problem; the seco.nd set is called the variable shipment problem. In so doing, the use of the term demand is 65

D. E. BOYCEand J. L. SOBERANES

66

consciously avoided since these shipments are not demands in the usual economic sense. Moreover, the term elastic demand has been used in a different way elsewhere; see, for example, Florian and Nguyen (1974) and Nguyen (1977). Since the gravity model employed here takes a rather different view of demand, care is exercised so as not to confuse the two concepts. The paper begins with a brief review of concepts in the Boyce-Farhi-Weischedel (BFW) algorithm, and then considers the findings with fixed shipments. Then the formulation based on variable shipments is presented together with the findings of the test problems. These are followed by a reexamination of the problem formulation and its interpretation. A section on future research concludes the paper. OVERVIEW

OF THE ALGORITHM

The optimal network algorithm of BFW is an extension of the tree-search algorithm developed by Beale, Kendall and Mann (1%7) to problems with continuous budget constraints. The objective function is min C(N) = 2 2 i=l

SijCij(N)

j=1

where shipments (trips) from node i to node j minimum path transportation cost from node i to node j over the network N; cii is assumed to be constant and therefore independent of Sik N a network comprised of a subset of links from L, the set of all nondirectional links connecting pairs of nodes i and j. n number of nodes represented by i, j. The budget constraint is defined as

Sii c,(N)

where K(N) =

E

k(Ah) = construction

of cost network N which is the sum of the construction

costs,

LEN

k(hh) of each link Ah contained

in N

KO = budget constraint. The problem is to find the network which minimizes C and is feasible in terms of the budget constraint. The algorithm examines, in principle, all subsets of L which form connected networks, and therefore have finite values of C, and which are feasible. In conducting the search, the algorithm makes repeated use of the following monotonicity property: if network N, includes all of the links in network Nz, then C(Ni) 5 C(N3. This condition is satisfied in the case of uncongested networks, since adding a link can never lengthen the shortest path cost between two nodes. The monotonicity property is used at the outset of the algorithm to compute the unconditional threshold of each link, A: U(A) = C(L -{A}). If at any stage of the search, U(A) 1 C*, where C* is the lowest value of C found so far, then all networks which do not include A may be eliminated from the search, since A must be the reason for U(A) exceeding C*. Thus one useful measure of the difficulty of a network problem is the number of links whose unconditional thresholds exceed the objective function for the optimal network. As this number approaches the number of links in the network, the size of the search is sharply reduced. The threshold concept is used in a second form, the conditional threshold, to identify links which must be in the best solution of a subproblem defined by excluding

Solutions to the optimal network design problem

61

a specified subset of links. Further details of the bounding rules and the search procedure are given in Boyce et al. (1973, 1974). The tree search algorithm systematically considers all possible subsets of links which satisfy the budget constraint, K,, except that those links which pass the unconditional threshold are permanently in the solution. Accordingly, the computational effort may be reduced if a good starting solution is found, thus reducing the value of C* and possibly increasing the number of links with U(A) > C*. All links in the initial solution which do not pass the unconditional threshold are removed in turn from the current solution to determine if a better solution exists. Therefore, the final result is guaranteed by the algorithm to be independent of the initial solution; however, the computational effort involved does depend on the quality of the initial solution. Experience has shown that the minimum spanning tree (MST) defined on construction costs is a good starting point for constructing the initial solution. Moreover, the MST is generally, if not always, included in the optimal solution. This observation is a research result for the networks tested to date, not a property of the algorithm.

SOLUTIONS

TO PROBLEMS

WITH

FIXED

SHIPMENTS

As an initial exercise in analyzing the role of size of shipments in the optimal network problem, the simple gravity-type relationship shown in Fig. 1 was defined. The definition of internodal cost chosen for this exercise was the node-to-node straight-line distance; this cost and the corresponding shipment are independent of the network under consideration. The shipment function was specified by drawing a straight line through the coordinate of the

Fig. 1. Fixed shipments related to intemodal cost.

D. E. BOYCE and J. L. SOBERANES

68

Table 1. Fixed shipments for Network Link

cost

1

Shipments O0

1,2 1.3 I,4 1,s I,6 1,7 1x8 1,9 1,lO 233

470 519 236 369 94 383 371 300 547 982

1 1 1 1 1 1 1 :

2,4 2,s 2,6 237 238 299 2,lO 334

30’

45’

60’

1 :

3 3 8 4 PO 4 4 6 3 1

2

2 2 3 2

1

; 2 2 1 1

! 2 3 2 :

281 143 488 851 98 296 393 750

; 1 1 1 1 1 1

2 1 1 3 2 2 1

4 2 1 5 ; 1

1: 3 1 19 6 4 2

3,5 376 3,7 3,a 3,9 3,lO 4,5 436

864 539 204 835 801 934 246 220

1 1 1 1 1 1 1

: 2 1 1 1 2 2

: 4 1 1 1 3 3

: Y 1 1 1 7 9

4,7 4,8 4,4 4,lO 5,6 5,7 5,8 5,9 5,lO 637 6,8 6,9 6,lO 798

594 191 81 523 410 !50 95 298 297 374 391 262 622 752 633

1 1 1 1 1 :

1

21 3 1 2 i 3 2 2 2 2 2 1 1 1

2 4 6 2 2 1 6 3 3 2 2 3 2 1 1

819 X0 8,lO 9,lO

881 224 389 588

1 1 1

2 1 2 1

3 1 2 2

1’0 23 3 4 2 20 6 6 4 4 7 2 2 2 1 8 4 2

Total

Shipnmts

45

74

107

255

1 1

1 1 :

graph with shipment equal to 1, and the internodal cost equals 1000. Then the angle of this line with the x-axis, 8, was set equal to O”,30”,45”, and 60”.The shipments corresponding to each value were read from the graph for each internodal distance, and rounded to the nearest integer. Of course, for 0 = 0”, all shipments equal 1. The internodal distances and corresponding shipments are shown in Table 1. The optimal network algorithm described in Boyce et al. (1974) was modified slightly to accept unequal shipments; this modified algorithm, called ONC, was used to find optimal solutions to Network 1 (Boyce et al. 1973, 1974) for five budget constraints: 2000, 4000, 6000, 8000, 10000. These budget levels represent increments of about 10% in the total length of all 45 links in the network (20,619), the units of length being completely arbitrary. These five budget constraints were solved for shipments corresponding to 0 equal to 30”, 45” and 60”). In most cases the algorithm was not run to completion in order to conserve computer time. Computation time for these tests was about 35 iterations per set on the IBM 370/168 with the Fortran H compiler. The results of these computations together with the previous computations for Network 1 with shipments of unity (0 = 0”) are shown in Table 2. To facilitate comparison of the solutions for different values of S, the objective function is stated in terms of the mean shipment cost, F = Z X Sircij(N)/XIc S+ The mean shipment costs are also reported i

j

i

j

Solutions to the optimal network design problem

69

for the minimum spanning tree defined on link length and the network with all 45 links constructed. Although the incompleteness of the computations somewhat restricts the conclusions that can be drawn from Table 2, two findings are worth noting. First, as the dependence of shipments on cost increases (increasing values of 0), the number of iterations required to complete the search decreases sharply. Between values of 8 equal to 0” and 60”, improvements of more than one order of magnitude occur in the number of iterations for both low and high budget levels. This effect may also be seen in the number of links passing the unconditional threshold: as B increases, the number of such links also increases sharply, thereby substantially reducing the search. A similar result no doubt holds for the conditional threshold, but the number of such links was not tabulated. Thus, use of shipments which are more comparable to observed flows substantially improves the performance of the algorithm. The extent of the improvement depends on the magnitude of shipments between nearby nodes as compared with shipments between more distant nodes. A second result of these experiments is the following: for a given budget constraint, shipments with different relationships to internodal distance (0 = 30”, 45”, 60% and therefore different mean shipment costs, tend to have the same best network; this network is usually different from the solution for equal weights (0 = 0”). In the case of the first three budget levels (2000, 4000, 6000), the best solution found is the same for each set of unequal shipments (0# 0’). For budget levels of 8000 and 10000, two of the three networks are identical. As expected at the outset of the experiment, the introduction of unequal shipments tends to increase the number of links in the solution; that is, shorter links tend to be preferred over longer ones. The solutions, however, are much more robust with respect to the magnitude of the shipments than anticipated. This result suggested on interesting conjecture, which is examined in the remainder of this paper: Table 2. Solutions for Network

c) .’ zz :: z 23 MST

2000

4000

6000

8000

10000

all links

Y_ CY

1with four sets of shipments 4 .:: --1 ‘c 02.:

.I 0 ‘c10 65 g; cI F‘i g .a$

_= PQ *Ln GFS

cu g;

43: 60

1654 1654 1654 1654

1086. 879. 817. 623.

;

-

;

-

0 30 45 60

1900 1890 1890 1890

1060. 858. 797. 60b.

:z 10 10

0 30 45 60

3929 3927 3927 3927

962. 786. 731. 560.

15 16 16 16

0 30 45 60

5888 5958 5958 5958

937, 762. 709. 545.

i2” 22 22

0 30 45 60

7919 7976 7976 7862

925. 753. 702. 540.

0 30 45 60

9919 9919 9941 9941

920. 749. 699. 538.

g, 2; 0

0 30 43 60

20619 20619 20619 20619

*algorithm completed iteration limit.

the

916. 747.

697. 537. search;

f!ZilZS c c’;; 2“ ”

0 0

2 .z 2 5-5 P’ l%

r0.E 6: ‘2:: 2, ,“i

8000 500 500 *605

; 1 1

k? 4

6000 500 500 1000

3”4j8 114 1

1 3 5 7

6000 500 500 *780

1 7 2 1

25

3

E 27

; 13

*1329 250 250 *95

2 1 61 1

29

12 17

*138 *54 *42 *18

1 1 2 1

:; 30

i 0

:t

;;

:

45 45

-

otherwise,

algorithm

stopped

by

D.E.Bom

70

~~~J.L.SOBERANES

The slope of the shipment function is the best network for connecting a set of nodes at a given budget level and is relatively independent of the mean shipment cost over the network, inversely proportional to the mean shipment cost: the steeper the function, the smaller the mean shipment cost. As the mean shipment cost increases, one might expect more longer links to be found in the best solutions. Such results only seem to occur, however, at the extreme case when shipments between all pairs of nodes are equal. Otherwise, for a wide range of shipments (0 = 30”-60”), the solutions are quite robust, especially for budget levels corresponding to surface networks, or networks which are essentially planar. For example, the third set of solutions with a budget level of 6000 has degree (link ends per node) of 4.4. This corresponds to a rather dense planar network. The first set of solutions, of course, corresponds to a very sparse network. In order to pursue the above conjecture, the shipment function was reformulated to a doubly-constrained spatial interaction model with a deterrence parameter associated with a constraint on mean shipment cost. Since the internodal distances in this model were defined to be network path costs, rather than direct internodal co&s, a substantial adaption of the algorithm was necessary. The results of this modified algorithm (OND) are described next. SOLUTIONS

TO PROBLEMS

WITH

VARIABLE

SHIPMENTS

A model for variable shipments Wilson (1%7, 1970) has proposed a formulation of the gravity or spatial interaction that is well-suited for the specification of variable shipments: Tij = AiOiBjDj exp

model

(-DC,)

Ai =

(7 B$j exp(-BCij))-’

Bj =

(7 AiOi exp (-@ij))-’

where

Kj = number of units shipped from node i to node i; Oi = number of units leaving node i; Dj = number of units arriving at node j; cij = minimum path cost from node i to node j. This model is derived by maximizing the entropy function:

7 rj=Dj,

j=l,...n

where

E = mean shipment cost zjiO,i,j=l,... n. Given the vectors of origins and destinations, (Oi) and (Dj) and C, Wilson’s model generates a matrix of shipments. Since the network algorithm is formulated for nondirectional shipments, let

Solutions to the optimal network design problem

71

The values of Sij are scaled to values substantially larger than one and rounded to integers; the summation in the objective function is modified to reflect the two-way flows between i and j. The algorithm, as noted in the above section, seeks to minimize

7

xi SjCij(N).

This is equivalent to minimizing

which is the constraint in Wilson’s model. Thus, the objection function of the network algorithm is equivalent to the constraint of the gravity model which determines the shipments. Therefore, specifying the mean shipment cost, E, in effect determines the value of the objective function, introducing an interesting circularity into the problem. Given the origins, destinations and mean shipment cost, a solution to the network problem is desired with transportation costs, (cij), which give rise to a set of flows (Tij), and nondirectional shipments (Sij), which has the mean shipment cost specified. What does it mean for such a network to be optimal, since the value of the objective function is determined by the specification of E? In the adapted algorithm which follows, the network is optimal in the following sense: given the shipments, Sij(N) corresponding to network N, no other network N’ has a lower total transportation cost; that is, T T Sij(N)cij(N) < F 5: Sij(N)c,(N’)

for all N’ Z N.

If such a network N’ is found, then it is substituted for N and the computation of shipments repeated. The adaption of the algorithm to this concept is considered in the next section. First, the question of the calibration of the deterrence parameter p is examined. Calibration of p, given I?,has been studied by Hyman (1%9), A. W. Evans (1971) and more recently by Williams (1976). Evans’ method does not require iteration, whereas Hyman’s does. For this reason Evans’ method was selected. Subsequently, Williams showed that his method, which is based on Hyman’s, is both faster and mote accurate. For the present purposes, this is not a crucial problem. What is troublesome, however, is that Evans’ method requires that intranodal flows be permitted. In principle, these flows can be driven to zero by setting the cii equal to a large number; unfortunately, Evans’ algorithm fails to converge in this case. The calibration problem can be solved by setting Cii equal to a small cost consistent with an in&modal shipment. In this case, however, the number of trips allocated to intranodal flows would vary as a function of c. To avoid this undesirable aspect, set cii = c. In this formulation, then, the mean transportation cost for the internodal flows (Tii, i# j), is always equal to c. To see this point, rewrite the mean transportation cost constraint, 2 = + 7 5: Tijcij= + (]i: 2 Tiicij+ 7 7’$); rearranging terms,

Let T’ = T - x Tii = number of internodal trips. 1

Then, t = $, x 2 Tijcib I

jfi

Now, revise the definition of shipments to

D. E. BOYCE and I. L. SOBERANES

72

Thus the objective function of the network algorithm is

T’ varies slightly for different values of F; however, this variation does not appear to affect the results. T’/T is reported in the findings to enable the reader to examine its fluctuation. Adapting the network algotithm

The shipments defined above depend in part upon the minimum cost over a given network, N. The deterrence parameter p depends on 17,(Oi), (Dj) and (cij(N)). Thus, for every connected network, a value of p can be determined such that the objective function of the network algorithm has the same value. Given a budget constraint, which of these networks is optimal, and what is meant by optimal in this situation? Consideration of this formulation led to an algorithm which produces an “optimal network,” N, in the sense stated above: given the shipment weights, (Sij(N)), no other network N’ has a lower transportation cost. if such a network N’ is found, then compute (Sij(N’)), and search for a network N” with

This process continues until N’ or its successor is determined to be the “optimal network.” The adapted algorithm is described by the flowchart in Fig. 2. The following notes may be useful in understanding the flowchart: (1) The minimum spanning tree (MST) defined on construction costs is taken as the starting point of the algorithm; all shipments are defined to be unity. (2) ONC adds individual links to the MST up to the budget constraint so as to maximize the reduction in the objective function for each link added; a check is made to insure that adding a combination of two links does not provide a greater reduction than one link. (3) Using the shipments for the best current solution, the objective function for the network found in Step 2 is compared with the best current solution.

1. Find the minimum

equal to one, and

4 7. Initialize ONC for a new search with tha new shipments

Fig. 2. Flowchart for OND.

Solutions to the optimal network design problem

73

(4) ONC checks whether the best current solution is optimal; in other words, has the search been completed? (5) The value of the parameter p is calculated given the new network, the specified value of E and the origins and destinations. (6) The new values of Sij are calculated from the Kj matrix, and the value of the objective function is revised. (7) ONC is initialized for a new search using the shipments based on the new solution, taking the new solution as the initial solution for ONC. One might ask why the shipments are not calculated for the minimum spanning tree instead of setting the shipments equal to unity in step 1. Calculation of shipments based on the MST led to difficulties because of the MST’s high internodal costs as compared with the solutions for typical budget constraints. Experience indicated, therefore, that the procedure shown in the flowchart worked more smoothly. In the case of very low budget constraints, an improved solution may not be found in the first iteration of ONC. In this event, the shipments are then calculated based on the MST, and the search proceeds. In the computational results which follow, the term cycle is used to denote the number of times the algorithm completes steps 5-7, that is the number of times shipments are calculated and the search restarted. The term iteration roughly means the selection and evaluation of one vertex on the search tree; a more detailed definition is given in Boyce et al. (1973).

Computational

tests

The properties and performance of the algorithm were explored with extensive tests on Networks 1 and 2 defined in Boyce et al. (1974, p. 115). Each network consists of ten nodes defined by pairs of randomly chosen coordinates on a 1000x 1000 grid. The shipment cost and construction cost for each link are defined to be the straight-line distance between pairs of coordinates. All 45 links were included in the tests. Each network was tested on two sets of origin and destination totals shown in Table 3. The totals were drawn from a table of random numbers and normalized so that

7

Oi = 7 Dj

= T.

Budget constraints were chosen corresponding to 10% increments in the total construction cost. For Network 1, the first four budget constraints in Table 2 were used; for Network 2, budget constraints corresponding to 20, 30 and 40% of the total were used. A range of mean travel costs from very low to rather high values were selected based on early computational experience with the algorithm. The value of /3 reported for the final solution is useful in judging the relative value of c. Values of p greater than three correspond to low values of I?,approaching the minimum possible value. Negative values of p (i.e. a positive

Table 3. Shipment origin-destination

totals

1 :

660 170 110

750 870 260

130 620 720

520 790 40

4

620

560

920

430

: 7

750 620 870

790 510 430

370 180 310

260 310 990

i 10

970 980 510

810 740 260

670 760 350

840 650 590

6260

5980

5030

5420

Total

D. E. BOYCEand J. L. SOBERANES

Fig. 3. Relationship of /3 to C.

exponential deterrence function) mean that higher transportation costs are preferred to lower costs in selecting destinations. Since the converse is usually postulated, negative /3 values signify an upper limit on reasonable values of ?. The general relationship between p and c shown in Fig. 3 may be useful in understanding the relationship of the values of E and p in the tests. Initially, 500 iterations of the algorithm were computed. If the search appeared to be nearly completed, an additional 500 iterations were performed in some cases. Computation time was about 45 iterations per set on the IBM 370/168 using the Fortran H compiler. Findings

The results for algorithm OND applied to Networks 1 and 2 with Origin and Destination Totals 1 and 2 are shown in Table 4. Generally, the mean transportation costs chosen for the two O-D Totals are the same, but there are a few differences. Following the budget constraint and mean cost, the table shows the final value of p and the construction cost and number of links for the best network found during the search. The number of links passing the unconditional threshold is also shown here. The value of the objective function is not shown as it is equivalent to the mean cost. Next, the number of cycles, the iteration number of the best solution and the total number of iterations over all cycles are shown. Note that in several cases, the number of cycles and the iteration number of the best solution are equal. For example, if this number is two, this indicates that the algorithm proceeded as follows: (1) Found MST. (2) Found an improved solution-iteration 1. (3) Calculated new shipments-cycle 1. (4) Found an improved solution using the new shipments-iteration 2. (5) Calculated new shipments-cycle 2. (6) Continued iterating (searching for improved solution) with the second set of shipments until the search was completed or the iteration limit reached. If the iteration number of the best solution is greater than the number of cycles, this means that an improved solution was found with a new set of shipments, but not on the first iteration after the shipments were computed. The final column of each section of the table gives the proportion of shipments that were internodal with the intranodal cost equal to the mean transportation cost. In four cases, multiple solutions were found. When a multiple solution was encountered, the algorithm behaved as follows: (1) Found MST. (2) Found improved solution A. (3) Calculated shipments based on network A. (4) Found improved solution B.

-

-

_

--

-

2

1

4.76 2.32 0.98 -0.43 -3.75

4.28 2.27 0.89 -0.53 -3.94

0.30 0.35 0.40 0.45 0.55

0.30 0.35 0.40 0.45 0.55

6000

3.92 2.62 1.35 0.05

-1.33 -1.33

0.35 0.40 0.45 0.50 0.55

0.35 0.40 0.45 0.50

0.55a 0.55b

6900

9200

9150 8954

8954 9165 9165 9165

6658 6763 6763 6763 6763

4564 4564

7826 7826 7826 7826 7826

5962 5962 5962 5962 5962

3921 3932 3932

Eao

;44

24 24

::

22: 20

::

16 16

1 1

2 3

2 3 :

2 2 2

4 ;

17 15 12 7

;

:

::

:

: 2

z 25

2 2 2 ;

:

:

::

2 :

3 ;

1 1

9 45

3 :

00

:55

22 22 22 22 22

16 15 15

lo 10

:

2 3 1:

2 2 2

:

:

G 3

z

2 9 47 63 12

3 12 29

:

na na

*I36 *175 *213 *341

l57 *129 *320 *995 500

500 500

*45 *a2 *178 *2!39 1000

*96 *496 l1168 1000 500

500 500 500

500 500

0.89 0.89

0.84 0.87 0.89 0.90

0.84 0.87 0.89 0.90 0.89

0.86 0.88

0.87 0.89 0.90 0.89 0.81

0.87 0.89 0.90 0.89 0.81

0.89 0.90 0.90

0.89 0.90

I

0.40 0.45 0.50 0.55a 0.55b 0.65a 0.65b

0.40 0.45 0.50 0.55 n.65a 0.65b

0.45 0.50

0.35 0.40 0.45 0.50 0.60

0.35 0.40 0.45 0.50 0.60

0.35 0.40 0.45

0.40 0.45

3.57 2.12 0.78 -0.64 -0.64 -4.13 -4.13

3.58 2.16 0.86 -0.50 -3.73 -3.06

2.51 1.29

-0.08 -2.84

9173 9173 9173 9173 9145 8937 9060

6763 6763 6763 6763 6763 6806

4535 4535

7995 7937 7937 7999 7997

3.23 2.03 0.98

5886 5886 5888 5888 5888

3940 3940 3934

1900

3.26 2.08 1.07 0.03 -2.65

3.30 2.20 1.23

2.53 1.66

;: 23

15 15

;: 24

E

21 21 20 20 20

16 16 15

10

17 14 10 7 7 D 0

0

0

12 6 3 1

2 0

: 1

10 6

; 1

:

2 2 1

0

: 1: 15 2:

3" 4 :

: 2 3

:

2 2

; 2

:

11 97 1 1 1

3

;

33

2 2 2 2 2 3

2 2

: 2

3 3

2 2 1 1 1

3 3 2

;

_ -

_^ -

-

._. -

-

*Algorithm completed the search; otherwise, algorithm stopped by iteration limit. a, b - multiple optimal solutions; na - not applicable

3.92 2.65 1.42 0.17 -1.19

0.40 0.45

4600

2.76 1.67

2.50 1.25 -0.01

0.35 0.40 0.45

4000

1.87 0.88

0.40 0.45

2000

T

Table 4. Summary of findings for Networks I and 2 Origin - Destination Totals 1

*89

*266 na na na na

l166

*34 *55 *184 *750 na na

500 500

*485 l622 *1037 500 500

500 500

1000

1000 1000

500 500 500

500 500

0.86 0.88 0.90 0.90 0.90 0.85 0.85

0.86 0.88 0.89 0.90 0.85 0.85

0.88 0.89

0.86 0.88 0.89 0.90 0.87

0.86 0.88 0.89 0.90 0.87

0.86 0.88 0.89

0.87 0.88

76

D. E. BOYCEand J. L. SOBERANES

Calculated shipments based on network B. (6) Found improved solution, which was network A. (7) Calculated shipments based on A. (8) Again, found network B, and continued in an infinite loop consisting of steps 4-7. To explore the properties of these networks, one of the links included in network B, but not in network A, was excluded from consideration. The algorithm was rerun and in each case network A was found as the unique solution. Then, a link in network A, but not in B, was excluded; in each case, the algorithm then found network B as the unique solution. In these four cases, the dual results are reported in Table 4. The total number of iterations is not shown since it is not comparable with the other tests. The networks found by the algorithm are presented in Tables 5 and 6 for Networks 1 and 2. The cost of the minimum spanning trees is 1654 and 2141, respectively. (5)

Interpretation

The two main conclusions observed for fixed shipments are also found in these tests. As the mean shipment cost decreases, the algorithm’s performance improves. For values of p between zero and three, which correspond to reasonable values for these networks, the algorithm performs quite well. For negative values of /3, the algorithm performs poorly, but such values are rather unrealistic. The algorithm performs very well for values of #I greater than three, but these values are also rather unrealistic since they tend toward cost minimization in the choice of destinations. The number of links passing the unconditional threshold provides further support for these conclusions, especially when the algorithm was not permitted to complete the search. The second conclusion is that the same network is found to be the best solution for a wide range of mean transportation costs. The main exceptions to this statement are for outlier values of E, as indicated by negative values of p or /? > 3. This result suggests, for these small networks at least, that the same network is optimal for a wide range of travel behavior. From the viewpoint of transportation policy, this is a rather tantalizing result. It is also reassuring that for some values of c, the networks selected conform to our a priori expectations. For example, four different solutions were found for Network 1 with a budget constraint of 8000 and O-D Totals 2. The one corresponding to the lowest mean cost has 26 links, shorter links being favored. Two different solutions with 24 links each were found corresponding to two values of high mean costs, meaning longer links were emphasized. The solutions for two intermediate mean cost values had 25 links. This type of result was expected to the general case; however, in most of the tests performed, a single network dominates for most values of mean cost. As the size of the network (number of links) is increased, and the number of solutions increases, this finding may not be so prevalent. A final conclusion is warranted regarding the stability of the solutions using the cyclic approach to calculating shipments. It is an interesting finding that the algorithm tends to identify quickly a solution which is best, given the shipments defined on that solution. One could imagine that the proposed algorithm would simply generate on ongoing sequence of solutions, or that multiple solutions of the type encountered in four out of 54 cases would be common. Clearly, neither is the case. The number of cycles needed to converge on the solution was normally two or three, and never more than four. The number of iterations to find the best solution was generally less than ten. These results indicate stability that is worthy of further study. Reexamination of the formulation of shipments

The formulation proposed for variable shipments involves a circularity with the objective function of the minimization problem. This circularity may be resolved somewhat by reformulating the gravity model in a manner discussed by Erlander (1977) and SchCele (1977). Recall that the objective function of the optimal network algorithm is Min z z 1

‘Z’ijC,(N).

I

This summation, constrained to a specified value, appears as a constraint in the trip distribution

O-D

2

1

Total

*

0.50 0.60

0.35 0.40,0.45

0.40,0.45 0.35 0.40,0.45 0.30-0.55 0.30-0.55 0.40,0.45 0.35,0.40 0.45 0.35.0.40 D.45-0.60

Mean Cost

A B

1 3 2 23456789103456789104567891056789106 * * * * **** * * * * * * * * * ******** * * ******** * * * * * * * * * * * ** *** * * * * * * * * * * * * * * * * * * * * * ******** ** *** ****** * * *** ****** * * ** ****** * * * * *

* : * ** ** ** **

**

* * * * * ** ** ** **

*

*

* : ***** ***** * ** ***** *****

*

** **

4

in solutions to Network 1

denotes a link that is constructed between nodes shown on lines A and B, MST.

0.4

0.3

0.2

0":;

0.3

0.2

Budget Level MST 0.1

Table 5. Links constructed

6

7891078910891091l * ** * ** * ** * * ** * *** * * * *** *** * ** * *t * ** * * * * * ** * *** * * *** * * *** * * *** ***

5

minimum spanning tree.

***

** ** ** ** ** ** ** ** ** * *

7

* *

* * * * * * * *

* * * *

8

!

0.2

1

2

Level

0.4

0.3

0.2

0.4

0.3

Budgei

O-D

-otal

A B

1 2 3 ! 3 4 5 6 7 8 9 10 3 4 5 6 7 8 9 11 456;891( * * * * * * * ** * * * * * * * * * * ** * * * * ** * * ** * * * * *** ** ** * * * * ** * * ** * * * * *** * * ** * * * * *** * * *t * * * ** * ** * * * * * * * * ** * * * * ** * t ** * *** *** * * * ** * * * *** ** * ** * * t* * * * * *** ** ** * * * * *** ** ** ** ** ** ** ** ** ** ** ** ** ** ** **

4 5 5 6 7 8 9 1( 6 7 8 9 10 * * ** ** * ** ** * *** **** ** **** *** l ***** *** * **** ***** * * * * *** * ** * ** **** * * **x** ** **SC** * * * ***** * * ***** ** * ***** **

in solutions to Network 2

Notes: *. denotes a link that is constructed between nodes shown on lines A and B MST: minimurn spanning tree O-D: shipment origin-destination totals a, b. These tests with multiple solutions correspond to thbse shown in Table 4.

o.6#

0.65a

o.55”

0.40-0.55a

0.40,0.45 0.35 0.40-0.55 0.35 0.40-0.50 0.553 0.55b 0.45.0.50 0.40-0.65a o.6+

Mean Cost

Table 6. Links constructed

**

**

** **

* *

* * *

*

* * *

*

6 7 8 9 I(

3

3 1t 2 l(

* * * * -

* * * * * -* *

lo

a

Solutions to the optimal network design problem

19

model used to generate the shipment matrix. The Kuhn-Tucker conditions obtained in solving the Lagrangian equation for the entropy maximizing formulation are Tij = exp (-

1 -

PCij -

Ai -

/Lj),

all i, j.

In the entropy maximizing formulation, hi and pi are dual variables associated with the origin and destination constraints, and p is associated with the mean shipment cost constraint. The same Kuhn-Tucker conditions are obtained for the following minimization problem if E and H are related in the proper way. Min + 2 C Tiicij 1 I

s.t.

7

Tij = Oi

for all i

7 Tij = Dj

for all j

Here H is a specified level of entropy. Erlander (1977) interprets H as the accessibility in the transportation system; in view of the various meanings attributed to that term, perhaps interacfion would be a better choice for describing H. Alternately, Senior and Wilson (1974) interpret H as the degree of dispersal in the shipment matrix. The value of I-Zcorresponding to Tij = Oaj/T is the maximum interaction attainable; in this case transportation cost has no effect on travel (/3 = 0). Viewed from this perspective, our formulation seeks to find the network that minimizes shipment cost subject to a constraint on interaction, H. For a given network, the resulting shipments are identical to those found by maximizing entropy subject to the shipment cost. As different networks are considered, however, the value of /3 corresponding to a constant value of H does vary. Thus, in implementing this approach, /3 must be recalibrated for each network. Such an approach is considered to be superior over one which adopts a constant value of P, since the latter implicitly assumes that p is independent of transportation costs. This line of research has been pursued in a subsequent paper (Boyce and Janson, 1977). CONCLUSIONS

Three directions for future research are indicated by these findings: (a) further tests with an improved version of OND: (b) reconsideration of the objective function; (c) consideration of congested networks. The formulation reported here for variable shipments could be improved by using Williams (1976) algorithm for calibrating the deterrence parameter p to eliminate intranodal shipments. It would be useful to apply the algorithm to somewhat larger test networks (e.g. n = 15, 20) in order to determine whether the robustness of solutions over a range of values of E prevails. By excluding very long links from the set of links considered, such computations should be practicable. Pearman (1977) has suggested for variable shipments that the appropriate objective function is to maximize consumer surplus rather than minimize cost. In some situations these solutions may be equivalent, but in others they are not. This matter needs to be carefully explored. It is especially pertinent in use of the algorithm to find optimal additions to existing networks. The assumption that internodal costs are independent of size of shipments may be suitable for rural networks. It is inappropriate, however, for urban areas or congested groups of cities. The extension of the approach of this paper to congested networks may provide some additional insights into the structure of that problem which have not been revealed by past research; see LeBlanc (1975) and Dantzig, Maier and Lansdowne (1976, Ch. 4).

80 Acknowledgement-We reported in this paper.

D. E. BOYCEand J. L. SOBERANES are grateful to the University of Pennsylvania

for the use of computational

facilities for the tests

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