Investment planning for urban roads

European Journal of Operational Research 71 (1993) 257-268 257 North-Holland

Investment planning for urban roads *

Marielle Christiansen Division of economics, The Norwegian Institute of Technology, 7034 Trondheim, Norway

Received September 1991; revised February 1992

Abstract: The planning of house building, technical infrastructure and roads is the responsibility of local authorities. The purpose of this paper is to describe a planning model for investments in housing areas, technical infrastructure and a road network for connecting housing areas to work zones. The model is a mixed integer programming model, and the objective is to minimise the investment and operating costs for the housing areas, the technical infrastructure and the roads. The model presented here is based on an existing model which has been used in the planning of several councils and cities in Norway. In this paper, we expand the existing model and focus on the development of urban roads in a network. The model is solved by means of a standard mathematical programming code. A simplified version of the model is also given and tested together with the original model on a test example. The computational results for this example are given.

Keywords: Urban planning; Mixed integer programming

I. Introduction

Today the local councils in Norway are not just providing services for the population, but they are also required to plan and organize future eco- nomical and physical developments. The trend seems to be that the councils get increasing re- sponsibilities, and because of this the need for systematic resource management has grown through the years. A group at our division has worked with such problems for a long period of time, St01an (1983), R0nvik (1984) and Gran (1985). They have among other things taken the following into account: land use, housing developments, roads, waterworks, sewage systems and schools. This existing model has been used as a tool in the planning of several councils and cities in Norway with satisfying results. Now there is new interest from the local councils to work further with these problems. Here I extend the

Correspondence to: M. Christiansen, Division of economics, The Norwegian Institute of Technology, 7034 Trondheim, Norway. * Paper presented at 'Seventh Euro Summer Institute', Ce-

traro Calabria, Italy, 21.6-7.7, 1991.

existing model to include the road network. Also some minor changes compared with the existing model are done.

The model presented here is a multiperiod inuestment model that has similarities with the model classes in Weingartner (1963). The similar- ity is clear, even though the model has a transportation network as its major component. Net- work design models are described thoroughly in the literature, see for instance Ben-Ayed et al. (1988), Current et al. (1986), LeBlanc (1975, 1976), Magnanti and Wong (1984), Nemhauser and Wolsey (1988), Pirkul et al. (1991), Poorzahedy and Turnquist (1982) and Steenbrink (1974). The combination of network design models and models with other discrete choice components are used in many different types of applications. An example is the development of petroleum fields with connecting transport system (Aboudi et al., 1989). The basic idea of the petroleum model is that the pipeline network can be extended over time (within a given planning horison) as new fields are brought into production. The discrete choices are if and when a pipeline segment and a field should be operative, which are comparable

0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

258 M. Christiansen / The Norwegian Institute of Technology

with our choices for the road segments and technical infrastructure projects.

LeBlanc (1975) presents an algorithm for the discrete network design problem. That model has similarities with the model presented in this paper. Both models consider which new road segments should be built into the network, and which existing ones should be expanded. LeBlanc re- gards the transportation demand for the different road segments as exogenously given, but this is not the case here. Opposed to the model presented here, LeBlanc's model is based on a traffic assignment approach and uses network equilib- rium analyses. This is the case for many of the papers concerning transportation systems and urban land use/ t ransporta t ion models. Numerous authors, e.g. Boyce (1984), Hutchinson (1984) and Mackett (1991), have recognized the important linkage between urban land use and the transportation system. Though the use of quantitative methods in transportation planning environment is described thoroughly in the literature, which e.g. Bianco (1989), Magnanti and Wong (1984) and Putman (1975) give a survey of, I have not found a model which is based on the same assumptions as made here. Although separately most assumptions are recognized in different applications. The assumptions are as follows:

The model described here is a cost minimiza- tion model for long term planning for urban roads, housing areas and necessary technical infrastructure. At the beginning of the planning period some roads, housing areas and infrastructure already exist. During the planning period a development program for house building is stated, and the demand for labour is given. These are the driving forces in the model.

In every time period the local council is facing an investment budget, and also there exists a budget for the total planning period. These budgets may limit the investment activity and development plans.

The total technical network may curtail the building activity and transportation of workers. The building of houses may be restricted due to a limitation on the number of houses that can be built in each time period and during the total planning period. In addition new housing areas need infrastructure like water, road and sewage connections before the houses can be built. Also sufficient capacity of the technical infrastructure

has to be assured when new houses are built on a housing area. For some technical infrastructures the order of the technical systems put into opera- tion is of importance. The roads have to satisfy several purposes, but it is sufficient to dimension the roads from the work traveling between the housing areas and the work zones. The demand for labour from the different work zones is then considered to determine the necessary capacity of the roads. If extra road capacity is needed to particular destinations, artificial work zones can be defined. The destination of trips are therefore by a common word called work zones. The capacity of old and new roads may limit the flow of people. The model will tell which existing roads should be expanded and which new roads built according to minimum cost. Note that this model does not take account of congestions on the roads, as discussed in Wardrop (1952). We view each road as having a given capacity, and if the flow exceeds this capacity, part of the flow will have to be rerouted to another road. Just for capacity control, we need to know the traffic flow.

In the next section the mixed integer programming model will be described and explained. In section 3 some simplifications of the model are presented. Section 4 gives information about the code used in the implementation. Then a test example is given for the original and the simplified model. Finally some concluding remarks fol- low.

2 . T h e u r b a n m o d e l

In this section a discussion of the individual parts of the technical network and the budget constraints follows. The following individual parts are discussed and described:

- the bounds for the house building activity; - the technical infrastructures and road net-

work; - the relationships between housing areas and

technical projects; - the capacity constraints for the roads; - some common constraints for the technical

projects and the roads; - the transport flow balance; - the budget constraints; and - the objective function, which is a cost mini-

mization function.

M. Christiansen / The Norwegian Institute of Technology 259

The constraints and the new constants and variables will be declared consecutively. The no- tation chosen in this paper is based on the use of lower case letters to represent subscripts and decision variables, and capital letters to represent constants. In order to avoid an overwhelming number of mathematical relations, a simplified version of the implemented model is presented. Normally variables and constraints are only defined for some combinations of the subscripts. Generally we are excluding sets representing the combinations of no interest, and to facilitate the reading, the actual combination of subscripts are here explained by words.

of the following constraints and avoid a dense matrix, a variable for the total number of houses in an area and period is introduced. The balance relations then become

t~l,t--bm--l'h~, l )=O Vh, t, (2)

where: t,m~ = N~, Vh. t~'ht = The volume (total number) of houses in housing area h in time period t. N h = The number of houses in area h at the beginning of the planning period.

The number of houses in housing area h cannot be greater than its capacity:

2.1. Housing areas t'hT < V h Vh, (3)

Within a council, there are several existing housing areas. In a planning period it could be of interest to expand these areas. In addition possible new housing areas are taken into account. The constraints for the pure house building activity are concerned with the limitation on the number of houses that can be built in each time period and during the total planning period, and the lower bounds on the house building due to the development program stated.

For large housing areas it is necessary to ensure that the number of houses built within the area in a given time period cannot be greater than a specified maximum. This is formulated as upper bounds in the following way:

where: V h = Maximum volume (in number) of houses in housing area h. T = Last time period in the total planning period.

The council is specifying a minimal house building plan for each time period, and the number of houses built cannot be less than this political lower bound:

~ b m > L t Vt, (4) h

where L, is the political lower bound on the number of houses to be built in time period t.

bm < B h Vh, t, (1)

where: b m = The number of houses built in housing area h in time period t. B h = Maximum number of houses which can be built in housing area h per time period.

For existing housing areas the number of houses at the beginning of the planning period is given. Now it is possible to formulate a 'house balance ' for each time period. For each housing area the total number of houses in a particular period is the sum of the total number of houses in the previous time period and the number of houses built in that time period. Here we do not take into account those houses which are vacated or demolished. In order to facilitate the reading

2.2. The projects

In order to expand the housing areas and transport people between the housing areas and work zones, new technical infrastructures and roads are needed. These technical infrastructures and roads are commonly called projects. The two main types of projects are technical projects like schools, service utilities, waterworks, sewage systems and pipes, and road projects. For each project p of type d and time period t a variable is defined as follows: Spt a = 1 if project p which is of type d starts in time period t; and = 0 otherwise, where d = 1 indicates a technical project and d = 2 indicates a road project.


2.3. Relationships between housing areas and technical projects

For building new housing areas and expanding others, some technical infrastructure is needed as mentioned. Capacity constraints for new and possible extended housing areas are described below.

Some new housing areas need a technical opening project in order to start building the area. This technical project cannot start after the house building starts. For the new housing area h which needs technical project p in order to start building houses the constraints become:

t

bht -- ~ BhSpu 1< 0 Vh, t. (5) u = l

Note that constraints (5) are only defined for the new housing area h which needs one technical project p as described above. For these actual housing areas the upper bounds (1) become re- dundant and are excluded in the implemented version.

In section 2.2 we divided the projects into two main types: technical and road projects. The technical projects can be further divided into several groups. Each of these groups gives different types of capacity as for instance student places for school projects and pipe dimensions and pump capacities for water supply projects. In the model these capacities are converted into capacities in number of houses. For instance if the average number of pupils per house is 0.8 and the school has a capacity of 400 pupil places, then the capacity of the school project is 500 houses. We introduce an index c which refers to a combination of capacity type and capacity at different geographi- cal places.

It is necessary to ensure that the total number of houses in the areas which need capacity type c is not greater than the capacity of type c. The capacity of a particular type consists of the existing capacity at the beginning of the planning period, and the capacities for technical projects of capacity type c introduced during the planning period. The capacity constraints are as follows:

t

Cht-- E E C~sm,1 <Ec Vc, t, (6) h p u - 1

where: Cp = The capacity (in number of houses) of tech-

nical project p; maximum number of houses to be served from p. E c = Existing capacity (in number of houses) of type c for the technical infrastructure at time zero. The summation over h is just for the housing areas which need capacity of type c, and the summation over p is just for the technical projects which give capacity of type c.

2.4. The roads

This model emphasizes the road network compared with the earlier work at our division. The existing model (St01an, 1983) looks at the roads as we do with the technical infrastructure. No network is included and therefore no need to declare the traffic flow and the origin and destination of trips. Here origins and destinations of trips are defined in locational terms. The places of residence (housing areas) are considered as the origins, and the destinations are the places of employment (work zones). As mentioned in the introduction, if extra road capacity is needed to particular destinations, artificial work zones can be defined.

So the main purpose of the roads in this model is to transport workers living in the different housing areas to work zones, and the work travelling determines the necessary capacity of the roads. In this representation the capacity of the roads is assumed equal in both directions, but with a slight change in the data inputs we can allow roads with reversible lanes and one-way roads.

The roads taken into account are the main roads in a network. At the beginning of the planning period, there may exist some roads with a given capacity, or decisions have been made to build some roads in the future with given capacities. This given flow capacity from node i to node j at time t is represented by a constant, and here this constant is also the capacity from node j to node i. The nodes in the road network are logical nodes and traffic centres, and the road capacities are for the road segment between two such nodes.

Since we expect a continuous house building activity, more people have to be transported through the road network. There will then be a need for expanding existing roads o r / a n d build a new road between two nodes. In section 2.2 we

M. Christiansen / The Norwegian Institute of Technology. 261

introduced the starting project variable for new and extended roads. Capacities for these road projects in each time period are exogenously determined.

In order to force the flow between node i and j at time t to be less than or equal to the total capacity, the capacity constraints are formulated as follows:

l

Y'~fijw, - E E Fpt.Sp.2 < Gi j t Vi, j , t, (7) w p u - 1

where: fijwt = The traffic flow of people from node i to node j working in work zone w in time period t. Fpt . = The transport flow capacity of people at time t for road project p when built in period u (new and extended roads). Gij, = The decided and existing transport flow capacity of people at time t from node i to node j. The summation over p is just for road projects starting in node i and terminating in j.

2.5. Some common constraints for the projects

A project p can at most start once. In order to formulate this condition, an explicit slack variable is introduced, and this variable is equal to 1 if project p of type d never starts. The constraints now become:

E S p t d -k npd = 1 Vp, d, (8) t

be built before project p can be put into regular use:

t

Spt d - ~ ~ S q , a<_O Vp, t ,d . (9) q u 1

The summation over q is for project predecessors in the precedence constraint. Constraint (9) is declared for project p, if p is the successor for the actual precedence constraint. The constraint is formulated such that a project can only be a successor in one precedence constraint. In the implemented version I allow a project to be a successor in several precedence constraints.

A particular project can usually be built in different ways. For instance different road alternatives for a road section may have different capacities, investments, operating costs and so on. I give the opportunity to introduce alternative projects, which are defined as a group of projects where at most one can start. The constraints representing these conditions are formulated as follows:

~npd--Xad =Xad-- 1 Va, d, (10) P

where: Xad = 1 if no project of type d in alternative a starts, and = 0 otherwise. g a d =Number of projects in alternative a of type d. The summation over p is just for the projects of type d in alternative a.

2. 6. The transport flow balance

w h e r e npd = i if project p of type d never starts, and = 0 otherwise.

In addition to constraint (8) for each project, we may have precedence relations among the projects. Some roads cannot be built before others, or at least be operating before others. By introducing precedence constraints between some roads, the continuous feasible region will reduce and make the optimization easier when having large road networks. Also some technical projects may depend on other technical projects. For instance, the sewage systems cannot be operating without the pipes connecting the system to the housing areas. For each precedence constraint, at least one out of a specified set of projects must

Before defining the transport flow balance, we have to make some assumptions about the way people are travelling:

- The demand for labour (number of workers) in each work zone is known in each time period.

- The demand for houses is determined from the demand for labour. In order to make the model more flexible, we allow the number of houses to exceed the demand for houses.

- The average number of workers per house may vary from housing area to housing area. This average number is exogenously determined and specified for each time period and housing area.

- People are moving, so where people live in relation to their work is changing. For each time


period a constant share of workers from a housing area working in a zone w is given as input.

- In this model version the traffic in the road network is spread due to the minimisation of investments and operating costs. All possible routes between a housing area and a work zone are candidates before the optimisation.

The total flow into a node must equal the total flow from the node. The flow into a node j can be due to the flow of people from one or several housing areas working in any zone w, and the flow from other nodes i. The flow from a node j can go to other nodes in the transport network, and the work zones with a specified demand for labour.

The node flow balances are formulated as follows:

E S h w t A h t ( V h t - - e h t ) + E f i j w t - E f j iw t=Ojwt h i i

Vj', w, t, (11)

where: eht = The excess number of houses in housing area h in time period t. Ant = The average number of workers per house in housing area h and period t. Djw t =Demand for labour (in number of people) in work zone w connected to node j in time period t. Shw t =Share of workers, living in housing area h, which works in zone w in time period t. The first summation is restricted to the housing areas directly connected to node j.

2. 7. The investment budget

We are faced with an overall investment budget for the total planning period. This budget is split and allocated to each time period. Within each time period it is allowed to exceed the 'normalized ' periodical budget by a specified rate as long as the total investment budget is not exceeded. The investments and budgets are those concerned by the local authorities. In order to make the con.~.raints as clear as possible, a variable representing the total investment in time period t is introduced. This results in a reduction of the computational time by avoiding a dense

matrix. Possible investments are house, technical project and road building.

EYhbm + E E Z p a s p m - m , =0 Vt, (12) h p d

where: m, = The investment in time period t. Yh = Investment cost for housing area h per house. Zpd = Investment cost for project p of type d.

The total investment in a time period cannot be greater than a specified exceeding of the investment budget in the same time period:

mt <Mt( l + K ) Vt, (13)

where: K = The maximum rate for exceeding the investment budget in a single time period. M, = The 'normalized ' investment budget in time period t.

In addition, the total investment volume in the planning period cannot be greater than the total investment budget:

~ m t < Y'.M t. (14) t t

2.8. Objective function

Under the conditions stated in section 2, the decision makers (local councils) want to find the minimal net present value of the total cost. The objective of the problem is therefore to minimise this net present value. The objective takes the following into account:

- Investment costs for all the houses, technical projects and roads. In order to get the present value of the investment costs, these costs will be discounted by an appropriate discounting factor.

- Operating costs for all the houses, technical projects and roads. The operating costs start to run when the investments are done, and termi- nate after a specified number of time periods. Just the discounting factor makes the operating costs time dependent. Note the assumption that the operating costs for a road project are inde- pendent of the real traffic flow on the road.


In a compact way the objective function be-

comes:

Min E EPmbh, + E E EQp.lSp,a, (15) h t p t d

where: Ph, Calculated net present value of the total costs for a house built in housing area h in time period t. Both the investments and operating costs are incorporated. Qptd Calculated net present value of the total costs for project p of type d started in time period t. Both the investments and operating costs are incorporated.

3. A simplified model

Some test have shown that for small road networks the model can be solved within reasonable time, but for larger road networks simplifications of the original formulation are necessary. Just the number of traffic flow variables and thereby the number of constraints (11), will imply a large contribution to the total number of columns and rows in the coefficient matrix. In the original model all possible routes between a housing area and a work zone are candidates before the optimization. Normally some routes are very unrealistic, so just a selection of routes is reasonable between a housing area and a work zone.

A possible simplification can therefore be ob- tained by fixing a set of possible routes between a housing area and a work zone. The flow from h to w can then choose one or several of the routes defined. A new traffic flow variable gives the flow from h to w along route r in period t.

The following two constraints, (16) and (17), substitute equations (11). The balance equations for the people travelling from housing area h to work zone w become:

Shw, Ah,(Vh,--eht)-- Y'~g,hwt=O Vh, w, t, r

(16)

where grhwt is the traffic flow of people from housing area h to work zone w along route r in time period t (replacing variable fi/wr).

In addition we have to ensure that the total number of workers travelling to work zone w must equal the demand for labour:

Y'Sh, Aht(t'm--eh, )=Djw , Vw, t. (17) h

The last term is for the j which work zone w is directly connected to. For the other (j, w)-combinations the demand constant is zero.

The capacity constraints (7) now become:

t

E grhwt-- E E Fpt,Sp,d<Gij;, ( r , h , t ) p u -1

d = 2 , Vi, j , t . (18)

The summation over (r, h, w) is for the combination of r, h, and w, where route r from area h to zone w passes through node i and then directly to node j. The summation over p in this relation is just for roads starting in node i and terminating in node j.

This simplification reduces the feasible region compared with the original model (if not all possible routes are defined). We may therefore expect that the optimal value of the objective function is more unfavourable than the optimal value of the original model but more realistic if all reasonable routes are defined.

The differences between the original model and the simplified one are the changes in the transport flow variables and the substitution of (11) by (16) and (17). In the comparison of the two models the following constants have to be declared: H = Number of housing areas. I = Mean number of neighbour nodes to a node. J = Number of nodes in the road network. R = Mean number of routes from a housing area to a work zone. W = Number of work zones.

There are at most J . W. T equations of type (11), while the number of constraints for (16) and (17) is at most ( H + 1). W. T. So when

J > H + I ,

the number of constraints is normally reduced with the simplified model. The original traffic flow variable f~jw~ is replaced by grhw,. So when

I . J > R . H ,


also the number of variables is normally reduced. When the road network is large, the problem is reduced considerably by this simplification.

4. Implementation

The model has been implemented by use of the Matrix Generator Generator known as MGG from Scicon (Scicon, 1991), which takes the mathematical formulation of the problem as input and produces a Fortran matrix generator. SCICONIC 2.11 (Scicon, 1990) has been used to solve the planning problems. The computations have been done on a VAX 3100 under VAX/VMS 5.1. Both the original and simplified formulation have been implemented.

In addition to the possible problem reduction discussed, the model allows the user to reduce the problem dimension by forcing some roads,

technical projects and road alternatives to start. For each of the roads, technical projects, and alternatives the user can limit the possible start period.

Most of the variables declared are continuous except Sptd, npd and Xad, which can be declared as binary variables. But SCICONIC allows sets of variables to be jointly restricted in special ways, defined by special ordered sets, as introduced by Beale and Tomlin (1970). The condition that a project p can at most start once is represented by (8), and this is best formulated by use of special ordered sets of type 1 ($1). It is preferable to declare the start and no-start project variables (spt d and npd) as members of $1 sets instead of binary variables, due to the branching in the branch and bound algorithm. Equations (8) are then the convexity rows. SCICONIC requires an explicit reference row for the special ordered sets. This row is unnecessary in order to specify

h=l 1

h=5

h=7

w=l

j=l

p=l ~ h=2

, I

I 1

j=4 I)=9

I j=2 I

p=4 p = 5 l P =6

I j=5

p=2, p=3 h=3 h=4

~=10± ~=11

] j=3 I

p:7 I p:8 I

j:6 T

~ - ~ I 3=7 w=3

p=14

h=8 F----l-, j-10

F q h=9

T j=8 I I p=16 [ p:17 I

w=5

p=15

_pEI9,._p=20

Figure 1. The road network (d = 2) with housing areas and work zones

p=12

j=9

p=13

W ~ ' X

p=18


the model, and was therefore left out in the presentation of the model.

5. Test example

At this point it will be illustrative to present a small planning problem. Figure 1 shows the road network for the main roads, and in Figure 2 we can see the relationships between the housing areas and technical projects.

Housing areas 1, 2, 5, 7, 8 and 10 exist, whereas the others (3, 4, 6, 9 and 11) represent potential new areas. The new areas need the introduction of the technical opening projects 1 to 6 in order to develop, while the rest of the technical projects (7-12) give capacity of type c (c = 1, 2 and 3). For each housing area the number of existing houses, the number of houses which can be built

in every time period and total are given. In addition the capacities of the technical infrastructure are given. The demand for labour and the housing policy determine the development of house building.

The roads which do not exist initially are marked by dotted lines. Between two such nodes we may start one or several road projects. In addition some of these road projects are defined as an alternative. The existing roads are marked by solid lines. Some of these roads can be expanded, and these expansions are represented by road projects.

In this example ten time periods are used. Two versions of the problem have been run.

For problem 1 the possible start period for each of the projects and alternatives is smaller than for problem 2, making problem 2 more difficult to solve than 1. Both these problem versions are run

h=l D Flh=2

h=5 ":1

p=7

p-8

p=9 ~ - -

p=l

h=

p=lO

h=4

h=6

h=7

h=8 U-

c=3

I

1-

11

p=12

. ~ h=ll

Figure 2. The relationships between housing areas and technical infrastructure (d = 1)

p=4


Table 1 Problem information

Information Problem 1 Problem 2

Simpl. Orig. Simpl. Orig.

No. of rows 1210 1230 1210 1230 No. of columns 1138 2148 1159 2169 No. of set members 102 102 123 123 Value of:

a) LP opt imum 2059 2015 2044 1999 b) IP opt imum 2215 2139 2172 2105

with the original and simplified formulation. At most 4 routes between an (h, w)-pair are used.

The problem information is shown in Table 1. 'Simpl.' means the simplified model, and 'Orig.' means the original model. First the information about the number of rows and columns in the coefficient matrix is given. Then comes the information about the total number of set members. Finally the values of the continuous (LP) solution and the integer solution are given.

SCICONIC uses a branch and bound algorithm to solve (mixed) integer programs, and the user has the possibility to give branching priorities to the integer variables and the special ordered sets (Scicon, 1990). Since the road network is emphasized, high piority is given to the set members which determine if and when a road

should be built. Rp-Tp in Table 2 means that highest priority is given to the road projects and then the technical projects. In addition I have tested if it is favourable to know the starting condition of the alternatives early in the branching. In the Ap-Rp-Tp run the priority ranking is: alternatives, roads and technical projects. The results from these tests are shown in Table 2. First Table 2 gives CPU times for obtaining the continuous solution, the optimal integer solution and when the search for a better solution is completed. This is followed by the number of nodes explored in the branch and bound tree when the optimal solution is found and when the search is completed. This information is given for both problems 1 and 2.

From these results we see that in general there is no gain in giving priorities. In most cases the computational time is minimal without any priorities at all. I have also tested other branching priorities, as giving highest priority to the technical projects and then to the roads. This strategy resulted in a very high running time compared with the other ones. From these tests the standard estimation code in SCICONIC can be considered to be effective.

The same priority is given here to all the roads, but it would also be possible to give individual priority to the roads.

Table 2 Test problems run on a VAX 3100 under VMS 5.1

Information Simplified model

No prior.

Rp-Tp Ap-Rp-Tp

Original model

No Rp-Tp Ap-Rp-Tp prior.

Problem 1 CPU time in sec:

a) LP solution 97 97 97 384 384 384 b) Opt. solution 365 282 1127 2713 3214 4655 c) Search complet. 1051 911 1355 4949 8339 8917

No. nodes optimal 69 41 327 198 243 544 No. nodes search

completed 294 246 425 368 908 998

Problem 2 CPU time in sec:

a) LP solution 83 83 83 312 312 312 b) Opt. solution 676 361 747 2342 3553 3008 c) Search complet. 918 1078 1428 11141 17080 13650

No. nodes optimal 178 107 251 192 438 416 No. nodes search

completed 270 377 522 1061 2143 1880


From Table 2 we see that the computational time for running the simplified model is almost equal for the two problems (1 and 2), while the computational time for the original model increased markedly with increased complexity. As we have seen, for difficult problems the effect on the running time of giving pre-determined routes may be great, and the effect on the objective value is rather small.

In Section 3 we justified the use of the simplified model with given routes. Nevertheless for certain networks the original model is realistic, as when all routes between a housing area and a working zone are possible. For complex planning structures it may be impossible to solve the original model directly. It is then possible to combine the two model versions. First the model with given routes is solved. From the solution we get the values of the integer variables and set members: road alternative, road project, and technical project variables. For the data input to the original model we fix the integer variables and set members at their optimal values from the simplified model. The original model with the new data input is optimised, and the results are shown in Table 3. The original model is here a pure linear programming model, without any integer variables or special ordered sets.

Table 3 should be compared with Tables 1 and 2. Comparing the simplified model and the com- bined models, the objective value is more favourable when the original model is solved after the simplified one, and the increase in running time is almost negligible. An almost optimal solution to the original model can therefore be

Table 3 Results from the original model, after solving the simplified model

Information Problem 1 Problem 2

Rows 1230 1230 Columns 2074 2074 No. of set members 0 0 Value of:

a) LP/IP optimum 2184 2140 CPU time in sec:

a) LP solution 152 " 126 h

a The total CPU time for solving the two models for problem 1 is 1051 + 152 = 1203.

b The total CPU time for solving the two models for problem 2 is 918+126 = 1044.

found with dramatically less computational time than by solving the original model directly.

6. Concluding remarks

Two models have been run for several planning problems. The original model is very time consuming to solve, except for small road network. The simplified model is acceptable with regard to the computational time. With increased complexity, running time is not increasing as much with the simplified model as with the original one. The simplified model is acceptable for some networks depending on the number of possible routes given, and thereby the deviation from the optimal solution for the original formulation. For some networks the simplified model is even more correct, as allowing all routes between an origin and a destination is not realistic.

But for the networks where the original model is preferred, it is possible to combine the two models. The objective value will with the com- bined models normally be more favourable than just solving the simplified model, without being too time consuming.

The possibility of giving branching priorities to integer variables and special ordered sets has not shown promising results. Nevertheless individual priorities for the roads and technical projects may reduce running time. This can be done by a user with knowledge of the different projects, but will be difficult for a mathematical programming modeller without this actual problem knowledge.

The existing model without a road network has been used in the planning of several councils and cities in Norway with satisfying results (see for instance Gran, 1985). The runs done indicate that it is possible to expand the existing model and include the road network when the simplified model or the combination of the two versions is used.

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Investment planning for urban roads

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