the economical planning period for engineering works

The Economical Planning Period for Engineering WorksAuthor(s): Ian McDowellSource: Operations Research, Vol. 8, No. 4 (Jul. - Aug., 1960), pp. 533-542Published by: INFORMSStable URL: http://www.jstor.org/stable/167295 .

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THE ECONOMICAL PLANNING PERIOD FOR

ENGINEERING WORKS

Ian McDowell*

Engineering Division, Postmaster-General's Department, Victoria, A.Iustralia

(Received February 1,1960)

In certain important situations, the observed nature of the growth of demand for service that must be met by engineering capital works is exponential in form. Plant to meet this demand is provided in instalments, each of which is usually kept in service for a much longer period than that which elapses before the next instalment is required. The problem is, what is the optimum length of the first and subsequent planning periods for various types of plant under different rates of interest and growth? Solutions are presented in this paper. Even though in practical cases the requirements of expediency must be met, it is still helpful to know what is best.

T HE ECONOMICAL planning period problem arises when engineering plant must be provided such that a proportion lies idle pending utili-

zation in accordance with demand. The cost of providing this initially idle plant should therefore be compared with the fixed cost associated with the provision of each individual instalment.

THE NATURE OF THE PROBLEM

Two EXAMPLES of situations of this nature are as follows. In a telephone exchange reticulation area, a certain cable is full. It is proposed to re- lieve it by installing an additional cable. Development forecasts of future demand for telephones in the area served by the cable follow an exponential law (approximately). What size should the relief cable be? In other words, for what planning period should the relief cable be provided? Agaii, a highway is to be constructed. Traffic forecasts support the assumption that the volume of traffic will increase exponentially (with population) from year to year. How many lanes should be provided at the time of construction?

The problem is significant for the following reason. Infrequent instalments that meet demand for service result in costly plant lying idle for excessively long periods. Over-frequent instalments introduce an over-all excessive expenditure in meeting the fixed costs of each installa- tion. If the optimum is not known for a given set of conditions, substantial

* Now with Arthur Andersen & Co., Melbourne, Australia. 533



534 Ian McDowell

sums of money may be wasted, for the costs of major engineering works are themselves very large.

Exponential growth of demand for service is the type commonly observed. This appears to be reasonable when population in an unsaturated area is considered. P'opulation increase is proportional to population, which gives rise to the exponential law. This is borne out in practice- e.g., the observed growth rate of demand for telephone service in large Australian cities is 6 to 7 per cent per year.

There are other possible types of growth, but these are considered less significant than the exponential case. The simplest is that of linear growth, in which demand increases by a fixed amount each year. Analysis of this case is relatively simple, and is given in the Appendix. The result indicates generally the result to be expected in more complex cases-slow growth rate, long economical planning period; and fast growth rate, shorter economical planning period.

Of course, there are other factors that determine the planning period to be applied in a particular instance. Some are discussed below. In many instances these factors are controlling; but to every growth situation corresponds an economical planning period that, if known, can be preferred wherever possible.

The solutions to the problem are unaffected by considerations of in- come to be obtained from the service to be provided as long as unsatisfied demand is to be avoided. This is usual with large public works, and will often be true in commercial situations as well.

THE PROBLEM IN MATHEMATICAL FORM

THE FOLLOWING assumptions underlie the analysis. The rate of growth of demand for service is k per annum such that the total demand at the nr-ith year is proportional to exp(kn,ri_). This demand at the nr-ith year will have been met by provision of instalments of plant at years 0, n1, n2, * , nr-2. Demand for service between years nril and nr will be met by the irnstalment at year nr-1, the extent of which will be proportional to [exp(knr) - exp(knri1)]. This is illustrated for years n2 and n3 in Fig. 1.

The interest rate is c per annum such that the sum of money to be set aside now in order to provide funds at the nr-ith year is proportional to (1+ c)nr-1. At values of c encountered in practice, this function equals exp(-cnrf1) approximately, and the latter form is preferred throughout for convenience.

There is a linear relation between the number of units of plant to be provided and the present value of all costs of providing and maintaining this plant. This assumption may be shown from studies of cost estimates in actual cases to be a valid approximation within reasonable limits. Thus at year nr-1 the cost of the instalment to be provided may be written



Economical Planning Period 535

a [exp((knr) - exp(knri)] + b, where a and b are constants, and the sum of money to be set aside at year 0 in order to finainee this instalment is

[a exp(knr) - a exp(knr-) + b] exp( - cnr,-).

Then, assuming an over-all planningg period nT years that may be made as large as necessary, the optimizing conditions may be written down. This

N

Size of instalmentnt +[nt exp[kn3) - exp[kn2 ) +

+ /[aexpkn N exp [k n]

ni n2 n3

Fig. 1. Growth curve showing instalcnedits of plant.

is that the present value of afll costs of all instalments of plant required in a given area be mitoimized. Thus:

F = [a exp(kn)-a+b] + [a exp(kCna exp ((k n) +b] exp- CUn)

+ + [a exp(knr? +) - a exp(knr ) + b] exp(-c Ur) (1)

+ *+[a exp(knT)-a= exp(- - (k/nTc-) exp(-cnk b,()

is required to be a minimum given b/a, c, k, s.nd nT, the variables being the values of nr that obey the obvious restri:tion that O<n <n2< <n, < .< = (nTk

This leads by partial differeiitiatioii with respect to the individual planning periods n,, etc., to the following set of minimiziillg coinditions:

aFl an, = ka exp (kn,.- cn,-,) -ka exp (kn - cn,) (2) - [a exp (kn,+,) -a exp (kn,) +b] c exp (-cn,) = 0,

etc., which may be rearranged in convenient form:

exp (kn,+,) = (klc) exp (kn + cn - cn,,) - (klc -1) exp (kn,) -b/a, (3)

etc. The first equation of this set is:

exp(kn2) =(k1c) exp (kn + cni) -(klc -1) exp (kni) -b/a, (4)



536 alai McDowell

and the final equatioln of the set is:

exp(knT) = exp(cnT-l). (5)

It can be readily shown that any set of results nr definied by the equations (3) minimizes this cost function F with respect to the set itself. It remains to find which set minimizes F absolutely for each combination of the parameters. This may be attempted by assuming values for n1, find- ing all nr<nT corresponding to them, substituting in the expression (1), and examining the results for the minimum value.

At first inspection it appears that the full set nr can be obtainied from calculations commencing with equation (5) and an assumed value of nl.

F nT cannot be reached with a start at n1 in this region

Fmin occurs with ni near

its own min. ~-v a Lu e

ni

Fig. 2. Behavior of cost function.

But while nT is a convenient limit when used as explained in subsequent paragraphs, it is not valid to make it a starting point for calculations, and indeed equation (5) leads to the anomaly that if k> c, then nTl> nT. It is theoretically possible to commence with assumed values of nT and nT-1

quite close together alnd calculate from this starting point, but it is found that as nT is approached, the periods nf-nr-- become shorter and more numerous and requiring a very large amount of computation.

The behavior of F and nr were studied at length from equations (1) and (3) using a desk calculator. In general terms, it was found that if a trial value of n1 was selected too low, then nr-i>nr before nT was reached, and a meaningful set of nr'S to substitute in F could not therefore be obtained; and that if n1 is found so that nr-i <nT< nr but only just, then F is minimized at a value of ni only very slightly higher. This effect is illustrated in Fig. 2. In the viciniity of nT (at the values used) the successive nr's became very close; but with a small iDerease in n1, this convergent ef-



Economical Planning Periodl .537

feet disappeared, and was replaced with a divergent effect and corresponding sharp increases in F. At values of nT of the order of ten times the value of n1, it was found that the solutions for n1 were not noticeably af- fected by variatiolns in nT. Thus it was concluded that there is reasonable assurance that a true minimum value of F and the corresponding ni, n2, n3, etc. can be found in this way.

Because of the large number of combinations of parameters involved, and the large number of n,.'s required to reach a given nT, solutions proved beyond the scope of the desk calculator, and a program was designed by

kni ~ ~~~~~~b/a= 1 00

4

b/a 1 0

b/a= 0-1 - - W'a= 0 - 01

0 025 050 10 20 4 0 k/c

Fig. 3. kn1 vs. k/c for various values of b/a.

the writer for presentation of the problem to the electronic digital computer C.S.I.R.A.C. at the UJniversity of Melbournle. The procedure was to obtain sets of nr, substitute terms progressively to obtain F, reject if nr< nr 1f<nT, increase n1, repeat until nr_i <fnT<nf., print F, repeat four times -for each combination of parameters c, k, b/a, nT, ni and its increment. Repeat for a more precise n1 and a smaller increment until the required degree of refinement is obtained. Inspect the values of F printed out, for the minimum.

THE SOLUTIONS AND THEIR USE

THE BASIC solutions (using values of nT from 25 to 300 years depending on the starting point n1) that were obtained are displayed in the curves given in Fig. 3. These cover the ranges of b/a and k/c likely to be en-



538 lain McDowell

countered in practice. It is not necessary to consider the absolute value of c, for to each combinatioln of b/a and k/c corresponds a value of knm and, from this, the solution n1 is obtained immediately. The differing

n 100

60--

4 0

20 -_ -- 2

0 025 0 050 0 075 0 100 c

Fig. 4. n vs. c for k=0.05 anid b/a=10.

values of nT were applied in order to save computer time, as they did not materially affect the results.

Figures 4, 5, and 6 display the effects of the variation of the individual parameters on ni, n2, n3, and n4 in typical cases. Increases in the interest

n 160_

140 - _ =

120

10 0 -

80 -__ \_

4 0 - - - _

20

0.025 0 050 0 075 0 100 k

Fig. 5. n vs. k for c=0.05 and b/a=10.

rate c and the growth rate k each increase the economical planning period, but the effect of the growth rate is proportionately higher. Also, the economical planning period rises sharply with the lost component of cost of an instalment of plant, expressed by the ratio b/a.



Economical Planning Period 539

The results have been applied to the determination of the economical planning period for telephone cable provision in the followilng maimier. The present value of all costs per unlit of extenit may be expressed as a function of the number of pairs of wires in the cable: C=a'N-b. With exponential growth: C a'No exp(lkn1) - a'No+-b. Thus a= a'No anld the curves in Fig. 3 are entered at the value of b/a = b/a'No where No is the present figure for development just met by existing plant. The value of c is known and k is deduced by fitting an exponential curve to historical data. This will be possible in many cases.

14 --~~~~~~n

140--

120 n Ot

Fig. 6. n vs. b/a for c=1=O.05.

For a 300-pair cable that is just full, and requires to be relieved by a 6j1/ lb/ml local type cable in existing conduit, b/a= 1093/10.1 X300=0.36. For growth rate k=7 per cent per annum (a commonly observed figure) and interest rate=5 per cent per annum, knl can be estimated from Fig. 3 at a value of 0.56. The first economical period is thus 0.56X100/7=8 years, and the size of cable to be installed is 300 [exp(0.56) - 11 = 225 pairs at least-i.e., a 200/6k1 cable in practice subject to considerations out- lined in the next section.

Similarly, a 28-pair trunk cable requiring relief by means of a cable with 20 lb/ml conductors laid directly in the ground, yields b/a= 1910/ 31 X28 = 2.20, using typical cost figures. For a growth rate k =4 per cent per annum and interest rate 5 per cent per annum, k/c = 0.8 and kn1 = 1 approximately. The economical planning period is therefore 1 X 100/4 = 25 years, implying 48 cable pairs. A 54/20 cable would be appropriate to this situation.



5440 Iat McDovell

PRACTICAL CONSIDERATIONS

LIKE ALL results in operations-research studies, the solutions of the economical planning period problem are intended to be used in the process of decision making, and not applied rigidly without consideration of other relevant factors.

In the case of underground telephone cable, the need to make best use of available accommodation in underground conduits may require sub- stantially larger cable than the solutions indicate. Again, new cable should be joined to existing cable in such a way as to obtain maximum probable over-all utilization of pairs, and this implies conformity with the planning period of existing cable. Obviously the length of the economical planning period rises with distance from the exchange as cable sizes dimninish; and, in practice wide divergence from the economical planning period may be avoided by use of cross-connecting cabinets betwteen sections of cable that will meet demand for service for differing periods. Small variations from the economical planning period do not result in large increases in over-all expenditure, and scope remains for the application of engineering skill supported by the best analysis of the facts of the case.

In the second example in the introduction, there may be insufficient finance to provide for more than a certain number of lanes in the highway; but mathematical support would, perhaps, lend confidence to those pro- posing the optimum number.

Values of the economical planning period below about two years would not usually be followed, for the time taken to plan and execute the work could absorb this period. Very large values lose their significance in relation to changes in the economy and the development of new techniques, so that schemes with sufficient flexibility to meet unforeseen long-term changes are preferable. Thus the main usefulness of the results appears to be in the range 2 to 20 years.

It is noteworthy that the solutions are unaffected by the starting point, i.e., by setting aside money before the first instalment is required, since all terms in equation (1) are multiplied by exp(-ct) where t is the length of time before the first instalment is required.

CONCLUSION

THE BASIC economical planning period problem for conditions of exponential growth has been solved and useful results obtained. Of necessity the statement of the problem has been simplified to a reasonable extent, but the solutions are a clear guide to the requirements of practical situations and can be an aid to decisiol-making when brought into relation with other factors in these situations.



Econontical Platnning Period 541

Poiints in common with queuing theory will be noticed. The rising demand represents large numbers of customers requiring service, and the instalments of plant represent the action taken to provide it. The tradi- tional problem of equilibrium between service cost and loss through result- ing inconveenience to customers relates to the individual instalments and could be included if necessary by a calculationi along the usual lines with the result expressed as a variation in the growth rate k. This was not done because thc size of the queues inherent in the problem would be such that the effect on k would be too small to vary the practical result. The maini problein of relation betweeni instalments over a lonig period remains, and the treatment given lhas therefore beeni confinled to it.

APPENDIX. ECONOMICAL PLANNING PERIOD UNDER CONDI'TIONS

OF LINEAR GROWTII

SOLUTIONS FOR linear growth are not as significant as those for the exponential case, for linear growth is not often observed, but are indicative of the type of result to be expected and may be applied occasionally.

It is assumed that the planning periods will be equal. This is reasonable since, at whatever point in time the growth situation is entered conditions are the same

1,400_

1,200- - _

1.00o l ---!-1- -

1, 000002

60?0 008 0 j 1

400. 6 00 ' - _=_

200-

0 4 8 12 16 20 24 28 n

Fig. 7. ak/bc vs. n for linear growth.

as at any other point, the initial economical planning period could be expected to be the same in each case. If this is so, any given period will be of constant length whether viewed from the present or from some time in the past. Thus all economical planning periods viewed from the present will be equal in length, and are written n in the analysis.

For a constant growth rate k units per annum (not a ratio, as used in the main problem), interest rate c per annum, and present value of all costs at time of pro-



542 Ian McDowell

vision per unit of extent of kn units of plant= (akn +b) ulnits of cost, the present value of all charges in connection with all future inlstalments of plant to meet demand for service is:

F = (akn+b)[1 +exp(-cn) +exp(-2cn) +exp(-3cn) +etc.]

= (aknm+b)/[1 -exp( -cn)],

which is to be iminimized by varying the planning period n; thus:

dF/dn = ak[l -exp( -cn) -c(akn +b)exp ( -cn)] /[1 -exp -cn)]2 (6) = 0 when ak/bc = 1 /[exp (cn) - cn - 1],

and it can be shown readily that solutions of this equation minimize F1. Use of an infinite over-all planning period is conisidered an allowable strategem

in this case. Curves of this equation (6) are given in Fig. 7. In general terms, these support

the expectancy that in more comiplex cases the result will be: smaller growth rate, longer economical planning period; larger growth rate, shorter economical planning period.



the economical planning period for engineering works

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