SWP- 93
This paper is prepared for staff
use and is not for publication.
The views expressed are those of
the author and not necessarily
those of the Bank.
INTEFtIATIONAL BANK FOR RECONSTRUCTION AND DEVELOPMENT
INTERNATIONAL DEVELOPMENT ASSOCIATION
Economics Department Working Paper No. 93
Sudan Transport Study Model
November 9, 1970
This is the seventh in the series of Transport Planning Models Study
papers. The Study, directed by Messrs. Jan de Weille and Leon H. Miller, Jr.
is a continuing investigation of mathematical models developed for transport
planning. Existing transport models are being analyzed, and revised and
extended where practical. New models will be developed where needed.
Eventually, the Study will include cases of models' application in specific
transport planning studies and a critical review of the methodology.
This paper is based on the three-volume consultant's report Transportation
Development Through Systems Analysis prepared in 1968 for Sudan by Lockheed
Aircraft International, Inc. under contract to the U.S. Agency for Inter-
national Development. The Lockheed model places transportation sector and
project planning in a general planning framework which facilitates comparison
between transport and nontransport projects. Transport projects and projects
in other sectors are ranked heuristically according to multiple objectives
and criteria, political and social as well as economic. A regional economic
model is used to generate surplus and deficit zones for each major transport
commodity, and three transport submodels are used to determine (1) the costs
and performance on each link, (2) the optimum route for each origin-destination
flow, and (3) the optimum interzonal distribution of traffic flows. The
implementation timing of potential transport projects may be tentatively
determined by a mixed linear and integer programming routine in the route
optimization model, but the implementation ultimately recommended by the model
depends on the pro ct's rank vis-a-vis projects in other sectors and on
available funds. Qongestion costs are not considered in the model, but there
is a fixed capacity constraint for each segment of the transport network in
the route assignment model.
The paper was prepared by Clell G. Harral and Suzy Henneman and draws
upon an earlier review by Iona Isaac. It is expository in nature; a critical
evaluation of the model will be given in a subsequent paper. Bank staff
members are invited to make comments and suggestions.
Sector and Projects Studies Division
TABLE OF CONTEMTS
Page No.
I. Introduction... ... . 1
II. Project Ranking and Balance of
Required and Available Funds over Time . . . . 5
Project Ranking (DEPICT) . . . . . . . . . . . .-. . . 5
The Availability of Investment Funds (MEMO). . . . . . 10
Project Timing (PRITI) . . . . . . . . . . . . . . . . 12
III. Zonal Production and Consumption (DANSE) . . . . . . . 14
IV. The Transport Models . . . . . . . . . . . . . . . . 16
Interzonal Distribution of Traffic Flows (TRADE) . . - 17
Link-Cost Performance Model (COMPAC) . . . . . . . . . 18
Vehicle Assignment and Route
Optimization (VARO) . . . . . . . . . . . . . . . 22
FIGURES
1. Structure of Lockheed Model for the Sudan . . 2
2. Project Ranking and Investment Timing Models . . . 7
3. Sudan Transport Models . . . . . . . . . . . . . . 16
TABLES
1. Calculating Criteria Weights for One Region . . . 8
2. Pair-Wise Preference of Projects . . . . . . . . . 9
3. Priority Summary Matrix . . . . . . . . . . . . . 10
tHE LOCKHEED TRANSPORTATIO\T 1iDDEL FOR THE SUDAN
I. INTRODUCTION
1. This paper briefly describes the methodological framework of the
Lockheed transportation study for the Sudan. Tne overall system,cr
what we shall call "the Lockheed model"t, consists in a number of parts
which< m.ay be grouped ir different ways. Here we find it convenient tc
distinguiEh three major components: (i) the project identification and
ranking models, DEPICT and PRITI, and the macroeconomic model, MEMO,
which generates the capital budget constraint; (ii) the regional income
and consumption mcdel, DArSE; and (iii) the family of transport cost
and traffic simulation models, CONPAC, TRADE, and VARO. *Figure 1 presents
in flow char form the Lockheed model and the interactions among the
various components.
2. A primary characteristic of the Lockheed Model is its emphasis
from the beginning on project identification and ranking. The planning
process is initiated by assembling a list for each region of all poten-
tial projects in all economic sectors, which is entered in the DEPICT
routine. In practice this list would be compiled by asking government
departments, foreign consultants, major private firms, and knowledgeable
individuals what their investment expectations are for the immediate
future. Relatively little need be known about the investments at this
stage beyond the type of product or service they will create and the
regional location. Government officials are then asked to go through a
rather elaborate rarking process by which each protect is compared with
each other project according to each of several weighted social,
political and ecolnomic objectives and criteria, such as political
stability, income redistribution and income growth. The result of this
stage of the analysis is a list of projects for each region ranked accor-
ding to oruer of priority as viewed from.a multidimensional, rather than
strictly economically oriented, objective function of the government.
3. The slnm of the investment requirements generated by the pro-
posed projects ir any given time period cannot exceed the available
investment funds, i.e., domestic savings plus foreign investments and
grants. It is the function of the m3croeconomic model, M0IIO, to
generate the me ureas of domestic savings and foreign investment funds
expected to b. vailable for each time period. Detailed information
specifying an estimated schedule of expected capital outlays for each
project on the DEPICT list is then submitted to the PRITI routines
which calculate the time profile of total investment requirements for
1/ Lockheed Aircraft International, Inc., Final Report U.S. AID Contract
AID/afr. 359 (Los Angeles, 1968): Vol. I, Transportation Develop-
ment Plan Sudan; 'vlol. II, Iransportation Development Through
Systems Analysis, Part 1: Concepts, Part 2, Applications.
Acknowledgement is made to the Agency for International Development
for permission to use this material.
Figure 1. Structuire of the Lockheed Model for the Sudan
_ _ DEPICT r --
Project identification Macroeconomic Model
and ranking| It + X - M + CH + GC = GDPt
For each region ___PRiIT _ Investment Investment
list of projects Timing of requirement BALANCE availabilities
ranked by priority project im- by period s by period
_plem nt.ation
L--__ DAN~ SE _
Regional production and consumption
Regional Surplus (+)
Output - Consumption = or
L_____ Regional Deficit (-)
G A ~~~~~~~~~~Potential transport!
demands by zone, |
period and commodity
CAPAC '
Cost-perform- Transport cost
ance model Lon each link
VARO Minimum bution model
Vehicle cost
assignment routing
and route nimum Compile Capacity of'
optimizatior cost - all traffic on - transport syat T
flows each link in BALANCE for each lir*
each period |e in each pericli
Project require--
_________ _ I [Iments for new
RETURN TO DEPICT; transport capacity
-__ - ___REPEAT UNTIL in each period
!CAONQTQqTENT SEWT 0-
LPROJECTS FOUNDj
-3-
domestic and foreign currencies implied by the list of projects, and
compare this total with the MEMO estimate of total local and foreign
funds available. Initially the total proposed investments may
greatly exceed the availability of funds in one or more time periods,
so that some projects must be eliminated or postponed and the
implementation of others is stretched out over time. It is assumed
that, ultimately, a set of investment projects over time will be
found which is consistent with the level of investment availabilities.
4. Once such a feasible set of investment projects is found, the
effects of these investments on regional output are estimated in the
regional economic model, DANSE. DANSE calculates production and
consumption for each of some 20 commodities in each of 66 regions for
each time period. Since the Sudan is predominantly a primary economy
with quite limited demands for intermediate and manufactured products,
the DANSE model concentrates on agriculture.l/ In each region,
consumption of each good is subtracted from production to determine
the excess supplies or excess demands, which constitute the internodal
transport demands.2/
5. A family of three models is used to develop a transport plan
to meet these demands. Highway, railway, river and air transport modes
are considered. The CCMPAC model first calculates transport costs
(inclu d.ing vehicle operating costs, route maintenance and construction
costs-, the cost of time losses and expected cargo casualty) for each
link in the transport network. The VARO model then uses this information
in a linear programming routine to determine the minimum cost route
between each pair of demand-supply nodes for a specified network, and the
related vehicle requirements. An integer programming routine in the
VARO model makes it possible to consider the effect on minimum cost
routing of introducing as many as 20 new network links in a given year.
6. Using the minimum cost mode and route determined in VARO, the
TRADE model (a version of the classical Hitchcok-Koopmans linear prog-
ramming model) determines the pattern of internodal origin-destination
commodity flows which minimizes transportation costs.
1/ Apparently for this reason, no input-output analysis of intersectoral
flows (intermediate demands) was attempted.
2/ Intranodal transport demand (assumed to be carried by trucks) is
calculated by multiplying the tonnage requiring transport within
the given region by the average length of haul.
3/ Construction outlays are annualized by application of an amortization
(capital recovery) factor.
Thus, transport investment, operating and maintenance costs are
minimized for the given development plan.
7. After the minimum cost traffic flows have been determined for
each commodity, the traffic flows on each link are sumrmed and
compared with the link's capacity. 'Where the demand on a given link
exceeds the fixed capacity of a link, a "rbottleneckl is identified
and a potential transport investment is specified. The list of all
Suc'I projects constitutes a new estimate of the transport investment
program required to meet demands implied by the original plan of
development, given by the DEPICT routine.
8. The new program for the transport sector is entered into a new
DEPICT "scenario" and the entire procedure we have just described is
repeated until a national investment program in all sectors is found
which yields the minimum transport costs associated with the maximum
attainable national objectives consistent with investment availabilit-
ies. Other scenarios may be performed to examine as many alternative
assumptions and policies as the government and its planners may wish
to consider.
II. PROJECT RANKING AND BALANCE OF REQUIRED AND AVAILABLE FWNDS OVERTIME
9. Three models are used together to translate the available infor-
mation on proposed projects, on social, economic and political objectives,
and on the availability of funds into a list of projects by year which
constitutes the series of investments most valued in terms of several
public objectives, consistent with the available funds.
10. The DEPICT model (DEvelopment Projects Interleafed by Criteria
Technique) embodies a procedure for ranking projects within a single region
according to national objectives and criteria. The PRITI model (PRoject
Implementation TIming) combines detailed information on the investment
requirements of development projects specified by DEPICT with a projection
of investment availabilities as provided by MEMO (Macro Economid MOdel) to
determine the year-by-year timing of all projects. The set of projects thus
chosen provides a picture of the investment stream. on which the calculation
by DANSE of zonal production and consumption for each period, described in
section III, can be based. For example, since the Sudan is primarily
an agricultural country, investments to improve agricultural yields or
acreage will affect zonal production.
Project Ranking (DEPICT2./
11. The DEPICT model can handle eight objectives, fifteen criteria
and fifteen projects. The model is applied separately to each region.
Since the objectives and criteria may be weighted differently in each region,
the final DEPICT output is a list of ranked projects, for each region.
12. No formal mechanism is provided for combining the regional lists
into a country-wide list. Since such a ranking is needed as an input to
PRITI, the DEPICT output is very simply regrouped: the projects with priority
rank (1), of which there is one for each of the six regions, are the elements
of the first priority group, called Priority I; projects with priority rank (2)
constitute Priority II, and so forth, down to Priority XIII. The ranking of
the projects within a priority group is not specified by the DEPICT model,
that is, there is no explicit ranking or weighting of the relative importance
of the different regions' projects.
13. To implemerrn ne DEPICT model, the regions, objectives and criteria
must first be specified. The country is divided into five regions: the
central area, in which the country's most modern economic activities take
place; and four less developed areas pivoting around it, each with
relatively similar resources and geography. A sixth t"superregion", which
refers to activities which are important to the country as a whole but are
not of great importance in any single region, is also defined. For each of these
1/ Throughout our presentation of the individual models, we shall use the
same notation and acronyms as in the consultant's report; however, the
equations and figures in this paper are numbered differently from those
in the report.
-6
regions, a list of candidate development projects is prepared, drawing on
the government's Ten-Year Plan (1961-1971), on analysis of uniquely
regional problems and resources, and on new suggestions made by individuals
and agencies. The lists for the country's five regions specify projects
(ports, roads, airports, rail links, bridges, plantations, etc.), while
the list for the superregion comprises what might better be called programs
(e.g., animal resource development, industrial development, urban deve-
lopment).
The qualitative objectives according to which candidate projects
lrn every region are judged consist in attracting foreign investment,
.maintaining political and economic stability, broadeining the base of
op oduction and consumption, increasing per capita income, and improving
the country's physical and cultural accessibility. The criteria are more
concrete than the objectives: growth in the export sector, creditworthiness,
disposable income, availability and distribution of goods and services,
diffusion of technology, capital-output ratio, value added, debt service
ratio, and internal rate of return.
15. We summarize in the next two paragraphs the procedure which is
carried out separately for each of the six regions. In the subsequent sub-
sections, we describe in more detail each of the steps, which are schematized
in Figure 2.
16c A weighting exercise must first be performed, comprised of three
steps: (i) weights are assigned to each of the 15 criteria to measure its
contribution to each of the eight objectives; (ii) weights are assigned to each
obJective to measure its importance to the region, and (iii) (i) is weighted
by (ii), that is, the weight attached to each criterion to measure its
contribution to each objective (a 15 by 8 matrix) is weighted by the regional
sweight of each objective (an 8 by 1 matrix) to yield the overall weight
attached to each criterion in that region (a 15 by 1 matrix).
17? Then, the project ranking exercise may begin. Each pair of projects
in the region's list is compared, taking one criterion at a time. Whichever
member of the pair will contribute more according to that criterion receives
as its score the whole value of the criterion's overall weight as determined
in the exercise desc -3ed above. The other member of the pair scores zero
for that criterion. if there is no preference for one member over the other,
the two members of the pair split equally the value of the weight. One
project's score vis-a-vis one other project is the sum of all the criteria
weights credited to that project. The sum of a project's scores vis-a-vis
all other projects is its rank value; the project with the highest rank value
receives rank (1), the one with the next highest, rank (2), and so on, until
all the projects in that region are ranked.
Calculating Criteria Weights
18 For brevity's sake, we will limit ourselves to five criteria and
four objectives, and we will illustrate the ranking of three projects in a
given region. The weighting exercise can be summarized in Table 1.
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Figure 2 ': Project Ranking and Investment Timing Models
DEPICT
FOR Define 7
FLACH candidate
k REGION projects
Define Pair-wise
set of comparison of
f ! criteria IDI D projects for
I. __ . l Weight criteria Weight each criterion
for each D criteria
objective with
weights on
objectives
Development
] Define set Give weights D projects
of objectives to objectives ranked by
Define set Give weights ~~~~ priority
Yearly cash flow schedule
(investment requirements) A ranked list
for each project of projects
for each region
PRI TI
In each year, schedule the sectoral investmient needs of projects
in the following order of :uiority until available funds are MEMO
depleted: A
1. T'ransport projects. Available
2. Projects already begun. ment
funds
3. Projects receiving foreign loans.
4. Projects in highest DEPICT priority group. BALANCE
Fund projects in lower priority groups first only when: /
F Funds are inadequate for large projects of higher priority, _
but are sufficient for smaller projects of lower priority. Yearly invest-
\ ment requiremqents
6. A higher priority project must wait for implementation of V for all projects
another project not yet funded.
-8-
Table 1: Calculating Criteria Weights for One Region
Weights of Objectives (V:. £v. 1)
0.5 0.2 0.2 0.1
Weights of
Objectives (O0) Criteria (-(i)
01 02 03 04
Criteria (C-)
C1 0.1 0.5 0.1 0 0.17
C2 0.3 0.1 0 0.1 0.18
C3 0.2 0.4 0.1 0 0.20
C 0.4 0 0.2 0 0.24
[j5 o 0 o.6 0.9 0.21
Total 1.0 1.0 1.0 1.0 1.00
19. The matrix of the Ci by the O., whose cells we call Wi. relates the
criteria and objectives. The weighting factors (Wi2) are assi qd on a best-
judgment basis to every criterion for every objecti4e so that: _/
Wij = 1, 0o W.j,l.
The resulting matrix reflects the fact that the accomplishment of one objective
is very often measured by more than one criterion and that a given criterion
may give an indication of the accomplishment of more than one objective. This
matrix will be the same for every region.
20. The weights of objectives (Vj) are assigned to each objective according
to a procedure which formalizes decision-makers' views on the outlook and
potential of the re aon at hand. 2/ The (Wij) are then weighted by the (Vj)
1/ The notation in this subsection has been simplified considerably from the
consultant's version.
2/ The "minimum discernible difference methodt" for weighting objectives
devised by Lockheed is not described here, but is given in the Final Report,
Vol. II, section 4.2, pp. 12-16.
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to give the (Xi). For example, the weight of criterion C1 in the region is:
Xl = VlWJl + V2 W12 + V3 W13 + V4W
or, numerically:
X= (°-5) (0.1) + (0.2) (0.5) + (0.2) (0.1) + (0.1) (0)
= 0.05 + 0.10 + 0.2 + 0
= 0.17.
Generally, then:
(1) XI Vj Wij, and Xi = 1.
Ranking Projects
21. Comparing now the region's three projects (1), (2) and (3), the
decision maker (e.g. government planning board) considers them two by two
and criterion by criterion. The judgments as to their relative merits are that:
for criterion Cl = (3) is superior to (2) and (2) to (1);
C2 = (1) is superior to (2) and (2) to (3);
C3 = (2) is superior to both (1) and (3), which are
tied in preference;
C = (3) is superior to (2) and (1), which are tied
in preference; and
C5 = (3) is superior to (2) and (2) to (1).
These preferences tell us how to assign the Xi in the Table 2 below:
Table 2: Pair-Wise Preference of Projects
Project Pairs
Ci Xi (1) (2) (1) (3) (2) 3
C1 0.17 0 0.17 0 0.17 0 0.17
C2 0.18 0.18 0 0.18 0 0.18 0
C3 0.20 0 0.20 0.10 0.10 0.20 0
C4 0.24 0.12 0.12 0 0.24 0 0.24
C5 0.21 0 0.21 0 0.21 0 02I
Total 1.00 0.30 0.70 0.28 0.72 0.38 0.62
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22. -What we have s_rlier called the score of (1) vis-a-vis (2) is, then,
thle sum of (1)'s share of the criteria weights, or, in our example, 0.30.
We enter tnis in the matrix -which is our ultimate interest:
Table 3: Priority Snruiary MIatrix
Project Total
Project H1) (2) (3) Rank Value Rank
(1) 0.30 0.23 0.53 Third
(2) 0.70 * 0.38 1.08 Second
(3) 0.72 .62 * 1.3 First
The totals of each project's scores versus the other projects then tell us the
project ranking in the region we are analyzing, as shown in the last column
of Table 3.
23. Mathematically, let us call Pmn the score of the (m)th project vis-a-vts
the (n)th project, that is:
(2) P = X. E(m,n)
-here E(m) is an existernce function such that
E(m,n) = 1 -when (m) is prefered over (n);
E(m,n) = 0 when (n) is prefered over (m);
arnd E(m,n) = 3-2 when (m) and (n) are tied in preference.
Then t.e raank -ialue R! of the (m)th of I-I projects wi7l be:
M
(3) R = S
m -1=n L
arii rhe nihhst Ist takes the highest rank.
The 4ailabilitv of Investment Funds (M2I.0)
24. ZYe basic O lction in sector planning of the macroeconomic model, EgN10,
t r_ ,ict t mount of domestic and foreign funds which wi1l be available
.or investment in each time period. The measure thus estimated then becomes
the cac al bi- a] e constraint in the urolect ranking and time staging process
of t' ais total pro action of good (c) in zone (n) in period (t). Subscripts (i)
ano9(r) refer throughout to irrigated and rain acreage, respectively. The
in ,e terms denote acrea,ge of crop (c) taken over by modernization projects
in ile (k) periods follo-ing the irdtial period (o); acreage in the initial
period devoted to (c) is Anco* Yield on traditionally cultivated acreage is
Y; on acreage cultivated by modern techniques, Y'. The goods transported (r)
are the same as those produced (c), but only about half of these 20 goods are
specified as consumption goods (g). The 66 zones (n) are the areas influenced
by the 66 specified transport nodes througiaccessibility, geography, or soil
type.
- 15 -
35. The production of nine goods - tea, coffee, kerosene, gasoline,
salt, industrial and craft iteins, personal (body) oil, oils and fats, and
sugar products -- cannot be related to acreage and yields. These goods are
all imported via Port Sudan either as raw materials or finished products.
The model sets tons of production equal to tons of consumption (discussed
next) for each of tnese goods, and assigns either Port Sudan or Ihartoum
as the production center.
36. For all goods except one - industrial and craft items (manufactures)
the basic consumption equation defines total consumption as the sum of
three components: consumption of farm dwellers who farm irrigated landc7, of
those >7Tno farm rainland, and of city dwellers. Each component is the
product of the populafon in the designated group and a per capita consumption
norim, with the consumption of the two farm-dwelling groups being weighzed
by a measure derived from the prooortion of land under modern cultivation.
Ihe baic consumption equation is:
lng- = iLng 1ni ft 9nift Erfng nrft nrft hng nht
venere C t is total consum;ption in zone (n) of good (v) in period (t), E is
the per capita consumption norm, and P is population. The (W) are the
weighting factors. The subscript (if) refers to d-aellers on irrigated farm
land; (rf), to dwellers on rain-fed land, and (h), to city dwellers. For
the goods category (12), industrial and craft items, consumption is calculated
as a fraction of the total consumption of all other goods.
37. The ecuations for calculating net demand or supply are of the form:
(10 ) D3r:= F C c
u nct ngt'
that is, in zone (n), the tonnage of a good available for interzonal transpor'
(10 in oneriod (t) equals the difference between production of that -ood (Fc
ann conseuntio-n of the. good (C ), in the same zone in the same period.
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IV. THE TRANSPORT MIODELS
38. Tne transportation sector is simulated by three interrelated
models, TRAWE, C0ITPAC, and VARO, and an information organizing
routine, REGENT. Before discussing these models individually, we
-would like to describe briefly how they are related. TRADE (TRans-
port Allocation DEvice) uses the information on net regional demands
and supplies from DANSE to find the set of flows (origin-destination
distribution) which satisfies all demands at the lowest total
transport cost. The model is based on the classical linear prog-
ramming formulation of the transportation problem. CCMIPAC (COnput-
ation of Matrix Productivities And Costs) is the link cost-performance
model, which calculates average vehicle productivities and operating
and maintenance cost for each link or segment of every transport mode.
VARO (Vehicle Assignment and Route Optimization) uses the output from
COMPAC to search all possible routes (combinations of links) and
select that route which, for each required shipment (O-D flow),
minimizes the costs.
39, Tn actual usage, an initial solution for the TRADE model is
found by assuming the interzonal transport demands of DANSE are
distributed over the shortest distance route. Once the C14PAC and
-JARO model calculations, which specify rcuting by the minimum cost
path, are complete, REGENT (REport GEnerator on National Transportation)
organizes the VARC and COMPAC outputs to be fed back into TRADE. The
resulting new picture of interzonal transport flows which TRADE provides
is now based on minimizing total transport costs, the desired criterion.
The figure below summarizes these interrelations.
Figure 3: Sudan Transport Models
_ ______> DANF_
STAT~ Zonal net/T
demands by
_ ~~~~~ ~~END_I
TRDEAv Organizes
Inezona~ ___ >\ Final transport X O&CAC I
dintributional flows: minimum outputs
distribution :~
iof net demands c o
VARO
Initial transport / V !as -In
Iflows: minimum I cle assign
.,distance routes ment and route
---i j loptimiz aon ty
Cost and perform-
IVehicle, mo3 7-7 _gs\|ance by link, node'
Land link data and commodity
- 17 -
40. COMPAC, VARD, and TRADE are individually described in the next
three subsections.
Interzonal Distribution of Traffic Flows (TRADE)
41. Once DANSE has determined nodal zones and their respective
production and consumption patterns, it is necessary to calculate
the pattern of interzonal (more precisely, internodal) flows --
how much of each good flows from which supply source to which demand
"sink". It is desirable to be able to distribute the traffic
according to any one of a number of criteria, such as cost, distance
or time.
42. Thus, the TRADE model solves transportation problems in which
an optimal O-D distribution must be determined for specific commodities
for which:
(i) a fixed amount is available at sources (production
centers);
(ii) fixed amounts are sent directly (without transshipment)
to various "sinks" (consumption centers);
(iii) the total supply is equal to the total demand;
(iv) the cost of shipment is directly proportional to the
amount shipped.
43. In mathematical terms, we seek to minimize:
n m
(11) OBJ = e E ij xij
subject to:
i n
2 a, = b.
i=1 j j1 J
n
~ x . = s. i = 1, 2, ..., m
j=l 1
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m
x xi= d. j =1, 2, ,n
th
where s. = the supply at the i source;
th
d; = the demand at the j "sink";
xij = the amount shipped from i to j;
cij = the cost in terms of time, money, distance, or
other measure, of shipping from i to j.
Available computer programs can simulate very large networks extremely
efficiently while staying well within today's computer limitations.
Lockheed used the IBM SOTRC program.
Link-Cost Performance Model (CCMPAC)
44. CCMPAC calculates the costs of transport separately for each
link and time period for each of four modes: highway, rail, river,
and a-r. Two cost concepts are employed throughout so that two
separate estimates are derived: system cost (SYSC) and shippers' cost
(SHPC). The system cost consists of facility construction and maintenance
costs, eouipment ownership, and operating costs. Shippers' cost is the
sum of transport rates charged to the shipper by the carriers, and service
quality costs associated with speed, dependability and safety of service.l/
The model also calculates the productivity of vehicle classes for each
mode, which is essential to the estimation of vehicle requirements by the
VARO model. Since the equations for the different modes are structured
in essentially the same way, we will consider the highway mode for purposes
of illustration.
Transport System Cost (SYSC)
L5. Facility Construction Cost. The equation below expresses total
road construction r-st as the product of per kilometer improvement cost
on each segment che road link, length of the segment, on an annual
basis through application of an amortization factor.
1/ Neither concept (SYSC or SHPC) constitutes an estimate of economic
(or social) costs as commonly defined by economic planners. If
transport charges were subtracted from shippers' cost and the
remainder added to system costs, the resulting measure would estimate
economic (or social) costs as commonly employed. This could be
approximated by a linear combination of SYSC and SHPC as discussed
under the VARO model below.
1,~ ~ ~ ~ ~ ~~~~~1
- 19 -
N
(12) RPCjk=( = IRInjk NDSnj (AF)
n=l n
where RPCjk = total improvement cost for road link (j) in period (k);
RInjk = improvement cost per kilometer for the given surface
uype (specified in the code for each link) on segment
(n) of link (j) in period (k);
NDS length in kilometers of segment (n);
AF = amortization (capital recovery) factor.
46. For both road and rail, the improvement cost per kilometer
(RInjk in the case of road) is a function of soil type, proximity to
water and to crushed rock aggregate, length of haul, manpower, and
equipment.
47. Vehicle-Associated Costs. The equation used to calculate cost
associated with operating and owning road vehicles states that the total
cost is the sum of four components: operating cost, fixed vehicle
maintenance cost, cost of terminal usage adjusted for the commodity
carried, and variable road maintenance cost.
(13) ROCtJk = (V1 tjk) (RKCtjk) + VPCtjk + (RRTtJk) (RTFCjk) (RTCtjk)
+ (RRTtjk) (RDSj) (RMCtjk)
where ROCtjk = total vehicle operating, ownership, and road maintenance
costs for vehicle (t) on road link (j) during period (k);
KMltjk = kilometers traveled by vehicle (t) on link (j) during (k);
RKCtjk = per kilometer operating cost for vehicle (t) on link (j)
during (k);
VPCtjk = periodic ownership cost for vehicle (t) on link (j) during (k);
RRTtjk = number of round trips for vehicle (t) on link (j) during (k);
RTCt.k = terminal cost per round trip for vehicle (t) on link (j) during (k);
RTFcjk = cost adjustment factor for commodity (c) carried on link (j)
c ljring (k);
RDS. = length of link (j);
RMCtjk = road maintenance variable cost per kilometer per round trip
for vehicle (t) on link (j) during (k).
- 20 -
L8. Vehicle operating cost (RKC), which includes the cost1yf fuel,
oil, tires and repairs, is drawn from WXinfrey and de Weille _
with adjustments for the effect of different surface types on fuel
consumption and tire wear based on records of three survey vehicles
operated for approximately 20,000 miles. The periodic vehicle
ownership cost for the existing fleet (V`PC) contains allowances for
depreciation, insurance, crew pay and maintenance. Tables of these
costs by vehicle type and road class are generated and stored for
reference in the calculation of the vehicle coefficients.
Shippers' Cost (SHPC)
49. The four major categories of shippers' service cost included
in the model are: interest cost on goods in transit (Sl), inventory
cost due to variability in transit time (53), transport rates or carrier
charges (S), and losses caused by spoilage, breakage, and pilferage
during loading, unloading and transib (32 and S ). The model also
includes a measure of passenger time lost in tFansit and of cost due
to transfer operations.
50. The opportunity cost of capital tied up in goods in transit is
expressed as a function of a shipment's value, the interest rate and the
transit time.
(1a) S1 C /exp(kt) - 17,
where S1 = interest cost due to transit time;
C per ton value of the cargo; x
exp the exponential function (exp (x) = e , e = 2.718..);
k = dally rate of interest;
t = travel time in days.
,1. A significant variability in transit time is likely to create a
cost for the shipper whether he ships prematurely and incurs storage
charges o. ships late and risks penalty for late delivery. The following
method estimates this cost (S3) under the assumption that the shipper
seels to minimize it. We also assume that transit time is normally
distributed. The general expression for expected cost due to storage and
delay penalties w be:
1/ The consultants used a preliminary draft of Rob-ey Winfrey's Economic
Analysis for Highways, Scranton, Pennsylvania: International Textbook Co.,
19S9; Jan de Weille, Quantification of Road User Savings, World Bank
Occasional Paper No. 2, Baltimore: The Johns Hopkins Press, 1966.
- 21 -
T
(15) C(x) = C j(T - z) N(z; x+t+,C) dz
-o0
+ C2 (z - T) N(z; x+t, 5 ) dz,
T
where C storage cost, incurred by premature arrival;
C2 = late delivery penalty;
T = date shipment was promised;
z =date shipment actually arrives;
w = weight of shipment;
x = shipping date;
t = average transit time;
J-= standard deviation of transit time;
y total transit time;
N(z; x+t, )= normal distfibution with mean at x+t and
variance cr .
52. This expression for C(x) is differentiated and solved explicitly
to determine the optimum time of shipment and the minimum inventory cost
in a given case. If we assume that there is a minimum transit time
a nd that after that arrival time is governed by the exponential
distribution, we can derive
(16) S3 = C1O ln L l ' j
53. Cargo casualty may occur while in transit (S ) or vAhen the cargo
is being loaded or unloaded (S5). The former is esiimated as a function
of transit ti- , while the latter is directly related to the number
of transfers.
(17) S2 =CO /1 - exp (-pt)_7
(18) 35 obnC
where CO = value of the cargo;
spoilage loss per day-
average rate oT loss 'ue to breakage and pilferage;
n = number of transfers.
- 22 -
Vehicle Assigrment and Route Optimization (VARO)
54. The VARO model is used to determine optimum vehicle routing
(including intermodal allocation), vehicle recuiremeIts, and possible
construction of new links inthe transport network To serve a knowr
pantern of interzonal shipments. The model is a linear progran
combined with a mixed integer subroutine. The O-D distiribution patterQ
is supplied by the TRADE model (or can be exogenously specified),
while the Ca'PAC model provides the cost and vehlcle produc-ivitv
coefficients for each link of the transport system upon whi1-1 ch the cost
mininization is based. Up to 20 new links may be (exogenously) pro-
posed for consideratiorn in each time period by specifying construction
costs; the computations of the mixed integer subrouti4e can determine
whether or not each new linkXaay is to be constructed.i/
55. The planner can optimize either system cost (SYSC) or shipper's
cost (SHPC), or, by introducing a linear factor, alpha (OC ), any
linear combination of the two. As previously described, COM'PAC
calculates all components of SYSC and SHPC; these cost coefficients
are inputs for the VARO model. Thus, the possible objective functions
are given by (19), (20) and (21).
I J K
(19) SYSC= X E Z x, V
i=l j=l k=l jk ijk
I K
+ _ E DbV IDC
i=l k=l ik ik
I K J K
+ V VC PV + A E
i=l k=l ik ik j=l k=l jk jk
where SYSC = system cost;
SC = c-erating and maintenance cost of vehicle type (i) on
ijk -nk (j) during period (k);
1/ Apparently, the operational version of the ARO model, wihich handles
up to 17 commodities, 5 transfers and 20 new routes, relates to
the base period only. A growth version of the model, which would
determine the time period in which new construction is to take place
is referred to, but computational requirements for a realistic
problem would exceed the capacity of the TRl 7094 with which the
Lockheed team was working, e.g. for 50 internodal flows and five
commodities, a maximum of three time periods could be considered;
for more than ten commodity classes the model is restricted to one
period. See Lockheed Final Report, Exhibit 6.3-11.
-23 -
V; k= nunber of vehicles of type (i) operating on link (j) during (k);
ID1'1ik =number of idle vehicles of type (i in period (k);
IDCik = cost of an idle vehicle of type (i) during (k);
VCik= cost of purchasing a vehicle of type (i) in (k);
PJik = cumulative number of vehicles of type (i) purchased up to period (k);
Aa = cumulative amortized costs of link (n) established in period (k);
Enk = existence factor: equals unity if link (n) exists in (k) and
zero if it does not.
I N K 5
(20) Alternately, SHPC = K E E L SmVi4k
i=l n=1 k=1 m=l
where
SHPC = shippers' cost;
SI = capital loss due to transit time;
S2 = expected value of loss due to spoilage,
breakage, and theft while in transit;
S3 = charges due to late or early delivery;
Sh = tariff charges
S5 = estimate of loss during transfers.
(21) Or, finally, OBJ =CX SYSC + (1 - Ot ) SHPC
56. The specified objective function is solved for the value of two
variables: (Vijk), the number of vehicles on each route (i.e. route
assignment and vehicle requirements), and (E. k), the existence or
nonexistence of proposed new routes linking Aodal pair (j) in period
(k). The solution is found subject to constraints concerning (i) the
fulfillment of each transport demand, (ii) the (fixed) capacity of each
transport link, and (iii) the capital budget available for equipment
investment.
Demand Satisfaction-Vehicle Utilization Constraint
(22) t V
i =-- kijk -jk
57. The number of vehicles of type (i) operating between the node-
pair (j) in period (k), or (Vijk), multiplied by its productive capacity
in tons carried by each vehicle type, (PD. k) should meet the demand (D)
for each commodity that must be moved on iink (j) in period (k).
- 24i -
Link Capacity Limit
58. Capacity is treated simply as a fixed limit or discrete interval;
if traffic volumle is below the fixed capacity limit, then there is assumed
to be no congestion. Thus, for highways the capacity constraint for
segment (n) can be expressed very simply.
J I
(23) Vzjjf - CnGZ for all n,
*j =1 k n
j=1 i=l
where Cn is the capacity of the link in numbers of vehicles per time period,
exogenously determined from engineering studies.
59. For railways, the capacity, which is expressed in numbers of trains
per day, is dictated by the number of sidings and the length of time neces-
sary for a train to traverse the given segment:
(24) 1 ' PTink . - °
/l J Tijk VTjkn
where PTink = traveling time for vehicle type (i)
on segment (n) in period (k);
Tijk = traveling time for vehicle type (i)
on node-pair (j) in period (k);
V.jkn = number of rail vehicles of type (i) operating
between node-pair (j) on segment (n) in period (k);
VTjkn = number of rail vehicles per train traveling
between node-pair (j) on segment (n) in period (k);
IMnk = number of sidings.
Vehicle Availability Constraints
60. The number of vehicles of type (i) required to be purchased in time
period (k), (TT I, is equal to the total number of vehicles required to handle the
traffic, less tne total number of vehicles available at the beginning of the
time period, (Vik), net of vehicles out of service, (IDVik). Thus:
J
(25) Fik = Vi - (Vik - IDVik)
j =1
The funds available for the purchase of new vehicles may, however, be cons-
trained by a fixed capital budget, (CAPk), or
I Z
(26) VCik PVik - CAPk
where VCik is the purchase price of a vehicle of type (i).