This document presents an algorithm for solving the combined distribution and assignment problem using generalized Benders' decomposition. The algorithm formulates the problem as a modified distribution problem with a minimax objective function instead of a linear one. It solves this master problem using the Newton-Kantorovich method for nonlinear concave programming problems with linear constraints. The algorithm iterates between solving the assignment problem given a distribution and solving the modified distribution problem subject to optimality constraints from the assignment problem. When the solution converges, it provides the optimal traffic flows for both distribution and assignment.

1. Trun~pn Rt'~. Vol. 15B, pp. 21 33 004 -1647,'8 /0201-0021/$02.00/0
Pergamon Press Ltd., 1981. Printed in Grcat Britain
AN ALGORITHM FOR THE COMBINED
DISTRIBUTION AND ASSIGNMENT
PROBLEMt
KURT O. JORNSTEN
Link6ping Institute of Technology, Department of Mathematics, S-581 83 Link6ping, Sweden
(Received 18 July 1979; in revisedform 22 October 1979)
Abstract--Much interest has recently been shown in the combination of the distribution and
assignment models. In this paper we adopt a generalized Benders' decomposition to solve this
combined problem for a system optimized assignment with linear link costs and explicit
capacity constraints on link flows. The master problem which is generated is used to show that
the combined problem can be viewed as a modified distribution problem, of gravity form,
with a minimax instead of a linear objective function. An algorithm for solving the master
problem is discussed, and some computational results presented.
INTRODUCTION
The traffic assignment and traffic distribution problems have been studied in many
different ways. First as if they were two completely independent problems and more
recently incorporated into each other, forming a single problem, to take into account
the interaction between the two problems. Tomlin (1971) formulated the combined
traffic distribution and assignment problem as a mathematical program and studied the
behaviour of the model using Dantzig-Wolfe decomposition as solution method.
The same subject was studied by Florian-Nguyen-Ferland (1975) with a model differing
from Tomlin's in the formulation of the assignment problem. In their paper Florian
et al. also adapt Benders' partitioning to their model. Scheele (1977) has made use of
Tomlin's work in her Ph.D. Thesis, but has employed an alternative formulation for
the distribution problem. The following paper is a study of the combined distribution
and assignment problem using a combination of the formulations indicated above and
using Benders' decomposition as a suggested solution method.
1. FORMULATION
For the assignment problem we can choose between a user optimized formulation
adopted by Florian et al. (1974) or the system optimized assignment employed by
Tomlin (1971). In addition, the network flow equations associated with the assignment
program may be expressed in either link-path or node-link form. Here we will give
the mathematical formulation of the system optimized link-path traffic assignment
problem with fixed linear link costs and explicit capacity restrictions on the link-flows.
Assignment: Assume that the flow from p to q, fpq is given.
Let
Ai =linki i= 1,...,M
bi = capacity of link i
c~ = cost per unit flow on link i
{p,q,j} = path No. jbetweenpandq j= 1,...,Npq
ai~] = {; if Ai~{p'q'j}
otherwise
x~~ = flow along path j from p to q
fvq = number ofjourneys between origin p and destination q
f = [fpq] the journey vector
fThis Research was supported by grants from the Swedish Transport Research Delegation.
21

2. 22 KURT O. JI)RNSTEN
The traffic assignment problem can then be formulated as
man ~~e.amYPq
~E~.,. af]x~q <- bi i= 1.... ,M
P q J
subject to ~. x~q = f~q Vp, q
J
x~q > 0 j= 1, ...,Npq p= 1..... m, q = 1..... n.
For the distribution part of the problem we can also choose between different
formulations. Tomlin (1971) and Florian et al. (1974) use a distribution formulation
where the objective function consists of two parts, a cost function and a weighted entropy
function. Erlander (1976) and SchOele (1977) use a formulation of the distribution
problem with a cost objective and an entropy constraint. The third alternative would
be to use Wilson's (1970) classical gravity distribution model with the entropy
function as an objective and the cost function as a constraint. In the following we will
concentrate our attention on the combined distribution and assignment problem, where
we use a distribution problem with an entropy constraint.
min Z ~q ~ ~ c apqvpq
subject to
~ ~ aeqym < bl i= 1, M
• ~ J " J __ • ..,
P q J
~. x~q = f pq Vp, q
.I
~fpq = Sp p = l,...,m
q
Zf,, = T, q = 1,...,,,
P
P q
x~q > 0 W,P,q
fpq > 0 Vp, q
where
F=~.fpq
p q
Sp = number of journeys from origin p
Tq = number of journeys to destination q
H0 = required entropy/interactivity.
2. SOLUTION BY GENERALIZED BENDERS' DECOMPOSITION
One way of solving the combined distribution and assignment problem is by applying
Dantzig-Wolfe decomposition (Tomlin 0971), SchOele (1977)). Another way would be to
solve the problem by using generalized Benders' decomposition. From now on we
will concentrate our attention on the formulation given above although the same
arguments can easily be used on the other possible formulations. As can be seen the
variables in the problem can be partitioned into two sets Sx and 82; $2 containing
all fpq variables and $1 covering the remaining variables.

3. Thecombineddistributionandassignmentproblem 23
The problem is in the family of problems described by
min c(x)
t
Ax < b
subject to Bx = f
x > O,f~F
where
F = {f~fpq = Yq q=l ..... n, ~fpqq = Sp p=l .... ,m,
-~f-~-log~>Ho,
f p q > O p
q Vpq}
Remark: F is a convex and closed region.
For a given fe F. our combined distribution and assignment problem reduces to an
ordinary assignment problem, we get
min ~ ~ ~ ~ r.ae.%c~q
~t--tJ ~j
p q i j
{ ~ ~ V aeq.xeq< h.
p q J
subject to . x~q = fpq Yp, q
xtjq >_0 Vj, p, q
with associated dual
i=1 ..... M
max 2b,u, + 2ZL. ,.
i p q
subject to
~ ai~ul + vpq <_~ ciai~ Vj, p, q
ui<O i=1 ..... M.
Adding an extra link between each origin~destination pair will eliminate any
infeasible f for the primal linear program. In addition the extra links have a natural
interpretation as walking links. Consequently the dual has a feasible finite solution
and by the duality theorem we know that
min E E E E ,~.,~eq~pq
~,_,j_, = max Z bifi, + E E f pqbpq.
p q j i i p q
This means that our original combined distribution and assignment problem can be
solved by solving the problem
minrmax ~ biui
fcF L i
Let
P --- {u,
+ ~ ~ f ~ vpq ~Vaeq,
j ui q-1)pq~ ~iciai~ Vj, p,q ul <Oi = 1,...,M 1
)
• auu i -Jr- Vpq _ . claij Vj, p,q, ui < 0 i = 1, . .,M .
This polyhedron, where u and v are the dual variable vectors respectively, has a
finite number of generators and the maximum of this inner maximization is achieved

4. 24 KURT 0. JORNSTEN
at an extreme point of P. That is, the problem can be put into the form
min z
z 2b,ur + 2Es,
P q
~fpq = Sp p = 1..... m
q
subject to ~ fpq = Tq q = 1..... n
P
-ZZglo _.o
p q
fpq ~ 0 gp, q.
r : 1, . . .~Ylp
Where np is the number of generators associated with the polyhedron P. Thus the
combined distribution and assignment problem can be solved as a modified distribution
problem.
Remark." This means that the CDA-model is in fact a modified distribution problem
(gravity model) with a minimax objective function instead of a linear objective function
as in the ordinary gravity model. This gives us the following algorithm :
Step 0 (i)Choose an initial distribution matrix f°e F (for example, f0 can be
taken as the solution to the modified distribution problem without the
constraints of the type z > ...). Let the upper bound UBD--oo,
k=l.
Step k (ii) Solve the assignment problem with righthand side equal tof k- 1.The solution
to the LP problem gives us optimal path flows ~qk and optimal dual
variables Ak "k
gl i~ Vpq.
SetUBD =min[UBD,~~ci'a~qJ~q].
(iii) Solve the modified distribution problem (including the newly generated
constraint)
rain
subject to
Z
z > y', b,fi} + E E fpqb[,q
i p q
F,fpq=- 8p p= l,...,m
q
Efpq-= Tq q= 1..... n
P
p q
fpq >_ 0 Vp, q
r=l ..... k
if z >__UBD - E STOP. Otherwise the solution to the modified distribution
problem gives usfk Let k = k + 1, go to (ii).

5. Thecombineddistributionand assignmentproblem 25
3. OPTIMALITY CONDITIONS
In the preceding section we found that the use of Benders' partitioning leads to
modified distribution problems of the form
min z
z >_E b, 7 + Z Zf, qv;q
i
Zfpq=Sp
q
subject to Z f , q = Tq
P
p q
p = 1,...,m
q = 1, ...,n
r--1 ..... R
fpq > 0 Vp, q.
Suppose we have chosen Sp, Tq and Ho such that a feasible solution {fpq} exists.
We then have a problem with linear constraints except for the entropy constraint
which is convex. The function
P q
is strictly convex in the region defined by the linear constraints. The objective function
and the constraints are all differentiable. This guarantees that a solution to the
Kuhn-Tucker stationary point problem also is a solution to the above minimization
problem.
The Lagrangian function for the problem above is
The Kuhn-Tucker stationary point problem is formulated as:
Find fpq, ap, flq, rlpq, 7, re, such that
P
(2/
n, Vpq + flq + c9 + 1 +log > 0 p 1.... ,m
, q = 1,...,n
f pq grZ)pq q = 1, ..., n
7(~ ~ ~ logf~ + Ho) = 0 (4,
biu~ + ~V~qfpq- z <_ 0 r = 1..... R (6)
i p q

6. 26 KURT 0. JORNSTEN
fpq- Sp = 0 p= 1..... m (7)
q
T,, : o
P
q=l,.. ,n (8)
~ ~ f~log f~ + Ho_<0 (9)
q q
fpq > O, 7r, > 0, ? > 0. p = 1..... m q= 1.... ,n r = 1..... R.
(lO)
Theorem: If in the Kuhn-Tucker stationary point problem the entropy constraint
is active and 7 > 0 then fpq > 0 for all p and q and fpq takes the form
fPq - exp - zr, Vpq + ap + flq - 1 .
F
Proof.. Suppose we have fp,,q, = 0 for some p~, q~. The K-T condition (2) is
/
0
r F F /
where r/pq is the Lagrangian multiplier corresponding to the non-negativity restriction
on fpq.
That is for Pl, ql we have
0 -- fPlq' -- expIFIyIPlq' -- ~7~rUrp'qt - O~P'-- flq'l -- 1
I F r
as tlplq I is non-negative we must have that either ep, = + ~ (or /~ql = -~- ,3()). But then
fp,q = 0 q= 1.... ,n (orfpq, -- 0 p= 1,...,m) whichimplies
and the conditions (7) (or (8)) is violated which contradicts tile assumption that
fplq, belongs to the solution or
r
E T~rl)pq = ~- OC,
r
which means that the condition (1) is violated and fp,q, cannot belong to the solution.
That is fpq > 0 Vp, q if the entropy constraint is active and 7 > 0 and fpq can be
written
~- - ~rVpq + ~p + flq - 1 .

7. The combined distribution and assignmentproblem 27
4. SOLUTION METHOD FOR THE MODIFIED
DISTRIBUTION PROBLEM
The master problems, which are of modified distribution form, are thus expressible
as convex nonlinear programs.
min z
subject to
i p q
~ f~ = S~
~p f pq = Tq
- EE fvq-logfvq > Ho
p q /~ F -
fvq >_ O.
rE1 .... ,R
In Section 3 we have shown that the solution to this problem has the form
f Pq - exp - ~,Vpq + OCp+ flq - 1 .
F
If we make use of this fact we will get the following Lagrangian problem.
max min ~(z,f, 7, ~z,~, fl) -- max min ~<e(z,7, ~, o¢,fl) : z - z ~ n, + ~ ~ ~ b,u~
y>0 z,f y>0 z r r i
~z>0 =>0
-ZflqTq-ZaPSv-~q 7exp -7 n,v~,,+av+flv -1 + 7H o.
q P
This maximin problem can be reduced even further by using the
condition (1).
The resulting dual problem is
Kuhn-Tucker
max
~,,> 0,~> 0
:t,fl, 7~
~(7'~Z'O~'fl) = Z 7~r~i biu~ -- E flqTq -- E o~pSp
P
subject to ~ zt, = 1.
r
This is a nonlinear concave programming problem with linear constraints. The problem
has the same set of Kuhn-Tucker conditions as the modified distribution problem.
Using this observation and the suggestion of Erlander (1977) that the Newton-
Kantorovich method be used to solve linear programs with entropy constraints, we
obtain an effective method for solving the master problem. (Alternative ways to solve
modified distribution problem are further discussed in J6rnsten (1979)). The solution
of maximum entropy problems with linear constraints using the Newton-Kantorovich
method has been further developed by Eriksson (1978, 1979).

8. 28 KURT O. JORNSTEN
L
C6o
160:20
424:9
160:20
424:12
296:22
1050:13
160:20
424:12
tOo'~
160:13
424:6 //~
~0
424 : 6
160:33
424:14
160:20
424:9
292:22
io5o: 12
143:25
5o~: 14
~z
_mo
160:20
424:9
296:22
1050:13
Fig. 1.
Destination
node
18
15
4
16
11
3
8
14
2
I
5
Table 1.
Origin nodes
I 2 3 ~ 5 7 8 11 12 14 15 16
3 58.2 89.2 174.4 222.6 379.6
0.4 65 133.8 0.1 0.1 32.6
0.2 211.5 147 5.3 13.0
3.3 0.9 194.5 248.3
T2.1 12.5 417.1 0.3 54
o.I 91.6 2.3
17.3 546.9 11,7
5.~ 105.2 402.4
~07.6 0.4
364.9 1. I
321 .4 0.6
79
6?
Attrac
tion
1006
299
377
447
556
94
576
513
4O8
366
322
Production 480 345 147 65 782 1042 236 369 928
[Lower bound, Upper bound]
Iteration I [83192.1, 86902.2]
Iteration 2 [86378.0, 86401.9]
Iteration 3 [86399.6, 86399.6]
f~umber of' diEferent paths generated: 103.
Required entropy: 3.00
42k 79 67

9. The combined distribution and assignment problem 29
4.0
H 3.5
3.0
)3(21-
. ~ 5(31
4)
6121
f
7(3)
5(21
,(I)
i
90'
i i i l l i
I00' II0' 120' 130' 140' I@0'
Fig. 2.
1 2
Table 2.
3 4 5 7 8 11 12 14 15 16
18
15
4
16
1t
3
8
14
2
1
5
3.8 22.7
1.8 12.5
16.3 112.9 140.9
3.8 22.8 4
99 18.4 2.1
8 55.3
9.4 65
5.9 35.4
332. I
7.8 12.3 91.2 131.5 175 198.1 264.8 79
57.2 1.1 9.6 64.1 15.8 17.8 56.1
4.5 55 38.4 1.8 2 1.2
12.4 23.7 1.3 176.5 199.7 2.8
320.8 12.1 0.4 102.5 0.8
2.2 27 0.4 I
30.3 368.1 24.9 78,3
19.2 I42.7 309.7
28.8 39.2 7.9
350.3 12,7 3,0
260.7 61 .3
63.1
3.9
[Lower bound, Upper bound]
Iteration I [93878, 98213.9]
Iteration 2 [96844.1, 97809.3]
Iteration 3 [97094.2, 97107.3]
Iteration h [97097.5, 97107.3]
Iteration 5 [97102.h, 97102.4]
Number of different paths generated: 123.
Required entropy: 3.50.

10. 30 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
KURT 0. J~RNSTEN
Table 3.
1 2 3 4 5 7 8 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML
11 12 14 15 16
I
la 33.3 47 38 54.9 111.6 122.3 177 143 199.8 79
15 13.4 la.9 27 12.5 28.6 37 40.2 32.5 45.4 43.6
4 34.4 48.6 82.3 24.1 59.3 40.4 26.1 21.1 17.5 23.4
16 23.7 33.4 30.5 39 51.7 15.7 125.7 101.6 25.7
11 100.5 45.3 34.2 165.5 58 14.6 11L 23.8
3 15.6 22 10.9 26.9 6 9.5 3.1
a 48 67.0 79.2 194.5 77.8 108.7
14 43.9 61.9 72.3 146.8 188.2
2 167.2 88.1 92 60.7
1 235.5 78.7 51.9
5 194.1 127.9
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 5
[Lower bound, Upper bound]
[12019-f, 125751.81
c124388.9 125697.31
r125101.5 1252051
l125145.a 125163.71
L125154.8 125154.81
Number of different paths generated: 150
Required entropy: L .00.
5. NUMERICAL EXAMPLE
To illustrate the solution method described in the preceding sections the combined
distribution and assignment problem has been solved for a model city. We have chosen
to use the same example as Tomlin (1971). The structure of the network is given in
Fig. 1. The links are assumed to be undirected and a two step flow/travel time
relation is used. The numbers below the links in Fig. 1 give the maximum flow
that can be attained and the travel time per unit flow. The numbers above the link
give the maximum fl’ow that can be achieved at the larger travel time. The productions
and attractions are given in Table 1.
The model has been solved for different requirements of interactivity. For a discussion
on what the requirements on entropy/interactivity means to the solution confer
Erlander (1977). This yielded us the relation between total cost and entropy given
in Fig. 2. The numbers beside the curve give the number of cuts generated and the
numbers in brackets give the number of active cuts at the optimal solution. The
main computational effort consists in solving the subproblems. For further computa-
tional results see Hlglund-Jiirnsten (1980). In Tables l-3 we give the optimal O-D
trip matrix for the entropy levels 3.00, 3.50 and 4.00 and Figs. 3-5 show the
corresponding optimal assignments. Note: When Ho approaches its lower limit the
distribution problem tends to the transportation problem of linear programming. See
Evans (1973).

11. The combined distribution and assignment problem 31
421,6
321.7 (
423.6 (~
303.9 (~
.
o
:564.4
/ '4°'424 6) ,o6.
O
630,6 (
2444
b 580.2
,¢
458.1
tt3
~b
178.8 (
)
ea Ho=3.00
COPT
=86400
b
Fig. 3.
CONCLUSIONS
In this paper we adapt generalized Benders' decomposition to solve the combined
distribution and assignment problem. We have developed our ideas using a system
optimized model with linear link cost and explicit capacity constraints on the link
/424 351.6
ID
301.2
I~ 609.7
t 206.7 I
13
(') 629, ~b
2
I
~b 28o.8
,158.7
/ 424
t",/
6~o9 ~b
5O3
121
2329
~b
H
o=3.50
~ COPT:97102
Fig. 4.

12. 32 KURT O. Jt)RNSTEN
()
QO
~6
h-
/
424
388
~1050
325.8
0
774.4
333.5 (
551.4
/
" 424
tO
o5
505
tO
~f
tO
,)
161.4
2552
tt)
m
HO=400
COp T =125155
Fig. 5.
flows, although almost the same development can be done for a user optimized model
with nonlinear link costs.
By use of the generalized Benders' decomposition we get as a result that the combined
distribution and assignment problem (model) can be viewed as a modified distribution
problem which is a gravity model with a convex instead of a linear cost function.
Benders' decomposition leads to a sequence of ordinary assignment problems and
the master problems are modified distribution problems. What makes Benders'
decomposition attractive is the fact that the structure of the underlying assignment
problem is kept and as a result we can use special purpose methods for problems with
this structure. Contrary to the remark made by Florian et al. we claim that the
master problems are nonlinear programming problems that are not too hard to solve
(see Eriksson, 1979; H~iglund-J~Jrnsten, 1979). In contrast to this we know that
Dantzig Wolfe decomposition leads to sub-problems which are ordinary distribution
problems and restricted master problems to which, in each iteration, we add a
completely dense column. This means that, after a few iterations, the restricted master
problems in the Dantzig-Wolfe approach have lost the special traffic assignment
structure and sparseness.
REFERENCES
Erikssen J. (1978) Solution of Large Sparse Maximum Entropy Problems with Linear Equality Constraints.
LiTH-MAT-R-78-02, Linktiping Institute of Technology.
Eriksson J. (1979) Entropy problems with equality and inequality constraints. Private communication.
Erlander S. (1977) Accessibility, entropy and the distribution and assignment of traffic. Transpn Res.
11, 149-153.
Erlander S. (1977) The Newton-Kantorovich Method for Soh~ing the Gravity Model in Traffic Phmninq.
LiTH-MAT-R-77-1, Link/Sping Institute of Technology.
Erlander S. (1977) Entropy in Linear Programs an Approach to Planning. LiTH-MAT-R-77-3, Link~Sping
Institute of Technology.
Evans S. P. 0973) A relationship between the gravity model for trip distribution and the transportation
problem in linear programming. Transpn Res. 7, 39 61.
Florian-Nguyen-Ferland (1975) On the combined distribution and assignment of traffic. Transpn Sci. 9(1), 43-53.

13. The combined distribution and assignment problem 33
Geoffrion A. M. (1972) Generalized Benders' decomposition. J. Optimization Theory and Applic. 10(4),
237-260.
H~iglund-J/Srnsten (1980) Computational Experience of BCDA. Working paper. (A generalized Benders'
decomposition code for the combined distribution and assignment problem). Link6ping Institute of
Technology. Forthcoming.
J0rnsten K. O. (1979) A Modified Distribution Problem. LiTH-MAT-R-1979-31, Link6ping Institute of
Technology.
Lasdon L. S. (1970) Optimization Theoryfor Large Systems.Collier-MacMillan, New York.
Sch6ele S. (1977) A Mathematical Programming Algorithmfor Optimal Bus Frequencies. Link6ping Studies
in Science Technology, Dissertation No. 12.
Tomlin J. A. (1971,) A mathematical programming model for the combined distribution-assignment of traffic.
Transp Sci.5(2).
Wilson A. G. (1970) Entropy in Urbanand RegionalModelling. Pion Ltd., London.
TR.B Vol. 15B,No. I--C