Socio-Econ. Plan. Sei., Vol. II, pp. 331-338. Pergamon Press 1977. Printed in Great Britain

COORDINATING PUBLIC UTILITY EXPANSION, INDUSTRIAL SITING AND POLLUTION CONTROL:

A WORKABLE DYNAMIC PROGRAMMING ALGORITHM

MICHAEL SHEEHAN and K. C. KOGIKU Department of Economics, University of California, Riverside, CA 92502, U.S.A.

(Received 22 November 1976; revised 20 June 1977)

Abstract-Many rapidly growing regions are faced with the dilemma of wanting to encourage growth in the number of industrial firms located within their boundaries so as to reap the benefits of increased employment and added tax revenue while at the same time being concerned that new industrial development will be costly in terms of air and water pollution and the services of utilities which will be required. This paper presents a procedure for choosing among the applicant firms those to be allowed to settle in the region over the horizon. This choice is made so as to maximize net social benefits over the planning period subject to the relevant air and water quality constraints. Mathematically our procedure is based on two separate dynamic programming models linked through an iterative process which avoids the computational difficulties usually associated with large scale dynamic programming formulations.

1. INTRODUCTION

Many municipalities in Southern California and else- where are confronted with an increasingly serious plan- ning problem. In order to keep unemployment low and tax revenues high it is necessary to attract industry. From the public perspective, however, industry is costly, both in terms of water and air pollution as well as in terms of the municipal services which must be provided. Yet surprisingly one has difficulty finding a simple and clear-cut approach for weighing these tradeoffs over time, and selecting that path of development which max- imizes the gains of the community given the relevant c0nstraints.t However, with the approach of the new standards deadlines contained in the 1970 Clean Air Amendments and the 1972 Federal Water Pollution Con- trol Act Amendments (FWPCA), along with the recent ascent to prominence of section 208 of the FWPCA (which delegates substantial new powers over land use planning to local, regional and federal authorities), municipal planners can anticipate an augmentation of State and Federal pressure towards greater rationaliza- tion of industrial planning.

An interesting current example of this increased pres- sure, albiet without any systematic effort toward coor- dinated medium-long term planning, is the U.S. Environmental Protection Agency’s somewhat awkward

tAn exception is the literature on regional water resource management which seems to deal with generally analogous (at least from a mathematical perspective) problems fairly routinely [ 11.

fWe deal below exclusively with industrial location though the analysis could also be used, with appropriate modifications, for commercial and residential location.

!For a readable analysis of the problems of uncertainty in public resource decision making, see Gordon C. Rausser and Gerald W. Dean, Uncertainty and decision making in water resources. In David Seckler [2].

TFor an interesting case study which points out some of the pitfalls involved in planning under these conditions see Henry Vaux, Jr., Desert land use and residential subdivision. Dry Lands Institute, University of California at Riverside, August 1974. NSF publication Number 1.

use of federal waste-treatment-plant grant funds, under 208 authority, to control land use decisions in the Chino Basin area of Southern California. EPA’s decision to limit further expansion of sewage treatment capacity was taken with an eye to reducing air pollution by closing the Chino area to Los Angeles-based-workers who would like to live outside that city and are willing to commute.

The result of this has been a great deal of confusion, initially approaching panic, on the regional-local level. Proposals have been forthcoming from various local sources for a general reversion to septic tanks for new housing developments on the one hand and for the use of separate (existing) industrial waste facilities (non- reclaimable waste lines) for residential use on the other. The impact of limiting the expansion of municipal sewer capacity on potential industrial users has not been dealt with in any systematic fashion, nor have the opportunity costs of allowing residential developments the use of current excess capacity in industrial waste treatment facilities, which involves an element of irreversibility, been considered beyond the very short run.

Although the Chino Basin case is perhaps an extreme example in that available water treatment capacity has been reduced to zero, it is indicative of the need to provide for a more timely and closer coordination of the medium-long range planning function over the large number of planning entities within the county or region possessing either direct or indirect incentive control over the locational plans of industrial firms.*

The current lack of coordination between the many agencies which have an impact on one or another of the determinants of industrial location will prove to be in- creasingly costly in an era of rising congestion and intensification of resource utilization due to the large amount of secondary uncertainty it engenders3 Unless a procedure can be devised for evaluating and weighting the plans and goals of the many overlapping agencies in terms of a more comprehensive regionwide “objective function” planning and/or reasonable growth is bound to be undermined.

On the local level there have been two analytical responses to the legal and economic symptoms of this

331

332 M. SHEEHAN and K. C. KOCIKLI

predicament. The first has been the cost-benefit and environmental analysis found in the environmental im- pact statements/reports required of many major projects and developments. The returns from this EIR-EIS process have been of notably uneven quality;? in addi- tion to having the obvious flaw of being written solely on a static, project-by-project basis these reports often embody assumptions of certain information which com- pletely ignore the costs of risk and uncertainty4

The second response has been the state-mandated general plan process. This was initiated with the idea of forcing municipalities to put their goals on paper and coordinate their day-to-day activities to reach those goals[S]. However, in most cases the general plan is either ignored or altered ex post to conform to short-run decisions already taken, and hence has been ineffective.

Overall the counties are seemingly unaware that this primitive decision making “procedure” is and will become increasingly expensive to the general welfare as the goals of employment expansion, increased local tax revenue, resource conservation and pollution control, as well as the public’s liability for utility and service expansion for industrial use, come into close conflict [6].

The purpose of this paper is to present a practical method for dealing prospectively with the problems we have outlined above. Our methodology facilitates the formation of a joint plan for utility capacity manage- ment, pollution control and the choice and staging of

industrial development. Data requirements are

reasonable, and the two models involved are mathema-

tically uncomplicated.

Below we develop an outline of a suitable planning procedure and then elaborate our models through a sim-

tFor documentation of the problems of many Federal Environmental Impact Statements see Refs. [3,4]. In California the California Environmental Protection Act, in addition to other state legislation, requires EIR’s on many projects. Personal inspection of some of these reports as well as our conversations with a large number of local planners indicates that many of these EIR’s are only written to comply with the letter of the law, while others are incomplete or are based on improper assump- tions. Some, of course, are quite well done.

fA good example of this is found in Imperial County’s study of Geothermal development. Since exact information is not available on the dangers of increased seismicity or potential ecological damage from blowouts, etc., these effects are simply assumed to be of zero magnitude.

5Many of the “costs” to the community generated by the firms are dependent upon outppt, technology, quality of inputs (high vs low sulfur coal, for example) etc. and are therefore negotiable. The advent of a permit system would automatically strengthen the municipality’s position. Firms would have an incentive to “bid” high in order to remain competitive since our procedure sorts out the high cost-low benefit firms.

Warious municipal information systems are available com- mercially which are capable of projecting a particular project’s (or group of projects) impact on variables of this sort[8].

“Our iterative procedure allows us to choose a tentative list of firms and from that list determine what the benefits of scale in utility are liable to be. Since we begin by assuming zero economies of scale we are able to work with a relatively short list of “chosen” firms and then expand it when the possibilities of scale economies become evident. On the other hand if we were to begin by assuming maximum economies of scale we would have had to work with a long list of initially chosen firms and then iteratively reduce it. This would be less efficient from a computational point of view.

ttFor an explanation of distribution-of-effort models in dynamic programming see Ref. [9].

plified example using a single (air quality) constraint. Problems involving two utilities, a single type of pollu- tion, and a reasonably small number of applicant firms lend themselves to solution using a desk calculator. If both pollution constraints (air and water) were to be included, then the problem is suitable for quick solution on a computer.

2. PLANNING PROCEDURE

Institutionally the application of our procedure as a tool for prospective planning would require little outside of the resources already available in the usual town- planning agency. On the other hand, if the procedure were to be made fully operational as a direct planning tool, it would require a county or regional decision to centralize all the long-range, pollution-capacity expan- sion-industrial location planning activities into one agency. This agency would have to be invested with some form of direct or indirect permit power over the ingress of industrial enterprises. The legality of ‘adopting a plan of this sort would be open to question from several angles; however, reinforcements in the form of regional land use planning through section 208 may remove some of these burdens.

Historically, one of the major difficulties in bringing even small large-scale models on line has been the paucity of the necessary data[7]. The procedure we are suggesting does require the collection and organization of certain classes of information that have not regularly been prepared in the past. For the most part this relates to a description of each applicant firm in terms of its average employment impact over time, the tax yield expected if the firm were issued a permit for a particular place and time, the firm’s utility requirements, peak air and water pollution generated, and the firm’s preferences relating to the time and place of location.8 The remainder of the necessary data relates to costs of utility expan- sion, which should already be available; the data on the “cost” of pollution emanating from a certain point with a certain intensity, at least some of which ought to be available from the local Air Resources Boards and Water Pollution Control Boards.

Using a point system managed by the local Planning Department (or by the Planning Commission or Board of Supervisors in the case of particularly large or important developments) in conjunction with data on the estimated impact of the project on municipal and special district revenues, employment (both direct and indirect effects). demand for housing, etc.,l/ a measure of gross benefits can be arrived at for each applicant firm. Gross benefits are then combined with the cost of providing the firm with utilities under the least favorable scale of opera- tions, to give us net benefits.”

Once data on net benefits as well as individual pollu- tion levels is available we then begin to sort the firms. First, all firms which would individually violate either of the pollution constraints would be permanently eliminated. Firms whose net benefits are initially nega- tive are temporarily eliminated. We then apply a dis- tribution-of-effort algorithm to choose from the remain- ing firms (the “feasible set”) those firms which maximize the net gains of the municipality subject to the absolute constraint that aggregate pollution does not exceed the amount of slack remaining in the pollution standards.tl

Having established a tentative list of firms to be per- mitted to locate within the controlled region in different Years, we can then aggregate their utility and service

Coordinating public utility expansion, industrial siting and pollution control 333

requirements and determine, using a finite-horizon regeneration model, the optimal capacity construction plan to meet this demand over the horizon. Since for simplicity we are assuming away production intercon- nections among the utilities, the expansion plan for each may be computed separately, thereby substantially sim- plifying the computations. Also, if it is reasonable to assume, as we tentatively do here, that utility capacity can be expanded under linear or increasing incremental returns to scale, then the number of our policy alter- natives at each state will be reduced to the set of multiples of the yearly demands.?

Having arrived at this point we investigate the optimal pattern of utility expansion to see if average costs have fallen over any period due to the construction of larger than minimum size plants. If average costs have fallen over some years, we return to our initial list of firms to see if any which were initially disqualified because of negative net benefits can now be included in a revised feasible set.*

If after again using our distribution-of-effort algorithm to choose the new optimal set of firms from among the revised feasible set we find that we have in fact retained one or more of the “new” firms, we then run the capacity-expansion algorithm through another iteration to see if it is necessary to change the time pattern of sizing of utility capacity expansion. This is an important element of the model as it may be the case that if the expansion program is altered then some of the “new” firms may be assigned to utility facilities which had previously been planned to embody large economies of scale but which now, because of the vicissitudes of the utility expansion model, are planned to be con- structed with a lesser scale of operation in mind. This difference in attributable costs may be sufficient to shift marginal “new” firms from the small-positive-net-benefit set back into the negative net benefits group. In this case such a firm would again be deleted. If only this one firm were “new” then we would revert to the prior solution as the overall optimum. On the other hand, if some of the new firms remained in while others were deleted, then a new utility expansion iteration would be run as the final step in the process. If all new firms were accepted after the second capacity run then again we have an optimum (see Fig. 1).

In the firm selection model (distribution-of-effort) the decision variable is the allocation of permits to firms (i), the state variables are the differences between current air and water quality (ambient values) and the maximum allowable levels, U(water) and V(air). The only exo- genous variable is the set of firms requesting permits each year. Initial values are the levels of slack in the air and water quality standards at the beginning of the planning period. However, the utility expansion model

tFor elaboration of this point see Ref. [lo], p. 36f; and Wagner, Principles of Operations Research, p. 303ff _ A large region around this and other associated practical problems owes much to the work of Donald Erlenkotterjll], much of those writings are conveniently available in the working paper series of the Western Management Science Institute, UCLA.

SLikely targets are firms with small-positive-net-benefits but with very low production of pollutants.

$In the example presented below we present two tables: the first with full demands and excess initial capacity represented exolicitlv. and then a second table entitled, “After ‘On Hand’ Depleted”, showing first period demand reduced by the amount of the excess.

requires that the system’s initial state be of zero excess capacity. To provide for this we simply reduce the first period’s demand by the amount of the surplus.§

In the capacity-expansion models the decision variables are the phasing of new capacity construction for each utility over the horizon (x& The state variable for each utility is the excess capacity, J!$, available at the beginning of each period. The exogenous variables are the demand for new capacity (r,J for each utility. The initial value for each model is given by the amount of excess capacity each utility has available at the beginning of the planning period. The capacity expansion models (one for each utility) are based on the following assumptions:

(1) Utilities do not consume significant amounts of each other’s output.

(2) Utilities are not consumers of air or water quality; alternatively, provision for their consumption has already been made as indirect consumption on the part of the firms they are to serve.

(3) Each utility, as noted above, can be expanded under a regime of constant oi increasing incremental returns to scale.

Assumptions one and two above have to be weighed against the facts of the particular situation being con- sidered. An alternative, albeit more complicated, way of handling this problem is to include a share of the costs and benefits of the new utility facilities on the balance sheets of the firms using their services. The third assumption seems more secure over the range relevant for most small to medium size towns and cities in a context of rapid growth[ll, 121.

Finn selection model (Distribution-of-e#oti involving two constraints) (see Refs. 9, 13).

Let us adopt the following notation: U = Amount of “excess” air quality, i.e. difference between the initial pollution level and the maximum allowable level; V= Amount of excess water quality; g,(u,, ui) =The im- mediate net return from selecting the ith firm given that air pollution ui and water pollution vi will be generated by that firm. i = i, . . , n. We then want to maximize the return function in 2n variables:

G(u,, uzr . . . , u,; 01, ~2,. . . , u,) = i: gi(ui, vi) (1) i=l

s.t.

+J ui20

n

&IV vi 2.0 i=,

This gives us the following recurrence relation:

fk(U V) = OSm,~~x,Op”~~” kk(&, Q) (2)

+ fk,l( rJ - Uk, v - &)I

(k=l,...,N-1)

where gt(.) is the stage summation of the net benefits from each firm due to allocating uk and vp in stage k. X+,(U - ah V- UJ is the optimal return from the

334 M. SHEEHAN and K. C. KOGIKU

Eliminate all firms which individually exceed total pollution constraints.

Compute net benefits for each remaining firm. Delete those with negative net benefits.

Least Cost Pattern of Q Construction

Fig. 1.

Coordinating public utility expansion, industrial siting and pollution control 335

remaining stages with U - ur and V- vl, remaining to be allocated.

Capacity expansion model Cfinite state regeneration) (see Refs. [9,131)

Let: r I, Z,...r r r, be the sequence of demands for utility capacity from the selected optimal firms, where r, is the new demand in tne kth period; x, be the new capacity added; Ek be the excess utility capacity at the end of the kth period; EO = c, i.e. initial capacity for each utility is known; and k = 1, . . . , N.

Since we required that capacity must always be sufficient to meet demand we have:

E,_, +x, 2 r, for all k.

A,(E,_,) = The cost incurred at the kth stage if E,_, is positive, a charge for idle capacity assessed at the beginning of the period, where A,, is a scalar constant; &(x~) = The cost of adding extra capacity at the kth stage, where &(.) is a concave scalar function.

Total cost is then:

C(x,, x2,. , x,) = i {A,@,-d+ Bk(Xk)l k=l

(3)

We want to choose the &, k = 1,2, . . . , N, subject to the demand satisfaction restriction, so as to minimize costs.

This gives us the following recurrence relations:

fk@k-1) = xk$i~k_, {A,@,-,) + Bk(Xk) + fk+,(Ed

(k=l,...,N-1)

fN = x,~~~EN_, hMG.-I)+ &(x~)I (4)

and the state transition equation:

Ek=Ek_,+xk-rk, k=l,..., N. (5)

4. APPLICATIONS

In our example we deal with a community with an air quality problem but we abstract away from water qual- ity. For simplicity the community has only two utilities,

tThe procedure elaborated here will handle any reasonable number of utilities with only a gradual increase in complexity.

$Given in terms of physical units which are initially priced at the unit cost of the plant size with the highest unit costs.

sewer and e1ectricity.t The planning department has accepted permit applications from 12 firms. Firms 1, 2 and 3 wish to locate in the controlled area in period 1; firms 4,5,6 and 7 in period 2; 8 and 9 in period 3; and 10, 11 and 12 in period 4.

Table 1 contains data on each applicant firm’s utility requirements,+ its air quality requirement (i.e. its esti- mated contribution to air pollution), and the amount the firm will contribute to community welfare.

Engineering Data V,, = available air quality, 6 ppm EO = available excess utility capacity, 0 units for elec-

tricity, and 2 units for sewer A(E) = excess capacity cost, $1 million per unit per

period for electricity, and $0.5 million per unit per period for sewer

B(x) = total costs for plants

For Electricity:

Size (units) 3 6 9 12 15 18 21 Cost ($ million) B;x) 6 10 15 20 24 27 30

For Sewer:

Size (units) Cost ($ million)

3 6 9 12 15 B;ix) 9 16 24 32 39

A vector of characteristics is given for each firm in Table 1.

Step 1. Eliminate all firms whose air quality requirement

exceeds that available. Thus firms 1, 2, 7 and 9 are permanently deleted.

Tentatively eliminate all firms with negative net benefits. Thus firm 12 is deleted.

The first-round feasible set is made up of firms 3,4, 5, 6, 8, 10 and 11.

Step 2. Use the distribution-of-effort algorithm to choose the

firms which maximize the net benefits within the limits of the 6 ppm air quality constraint. In this case the optimal set of firms is composed of firms 3, 5 and 11 which exhaust the air quality resource as shown in Table 2:

Period (k)

Firm (i)

1 I 2 3 4

12 3 4 5 6 7 8 9 IO 11 12

Air Quality Requirement (ui)

Electric Requirement

Sewer Requirement

Aggregate Benefits

781332858 5 2 0

384512257 9 5 9

817755134 5 3 2

8 52 35 57 62 45 64 21 60 99 51 17

Net Benefits (gi) 22 33 6 26 45 26 57 2 34 66 32 -7

Table 1. Firm data

336 M. SHEEHAN and K. C. KO~~KLI

Table 2. Determination of optimal firms by computing net benefits G

'eriod

Value of State Variable U Firms Firms (Units of air quality available)

Applying Admitted 0 1 2 3 4 5 6

3

/+J I I I i':* 6 :: :t

4, 6 _ _ _ _ - 52 58

5, 6 - 71* 77*

6 - 26* 32 32 32 32

Admit None 0* 6* 6 6 6 6 6

a bdmit\ 0' 6% 26* 45* m 71* 77*

a 2 8

IO, 11

Admit None - - - - - - 77

Step 3. Identify the pattern of utility demands for the selected

firms:

Period On Hand 1 2 3 4

Electric 0 4 1 0 5 Sewer 2 7 5 0 3

After “On Hand” Depleted

Electric 0 4 1 0 5 Sewer 0 5 5 0 3

Use the finite stage regeneration model to determine the optimal capacity expansion plan for each utility, as shown in Tables 3(a) and 3(b).

Step 4 Given that the optimal expansion patterns have been

determined and economies of scale attained over some segments of the planning horizon, we now return to check if the lower utility costs will move some of the initially tentatively non-feasible firms into the feasible column. Recompute net benefits (see Table 4). The result is that the feasible set now includes firm 12.

Step 2 Begin the process again by choosing the optimal set

from among firms 3, 4, 5, 8, 10, 11 and 12. The result is that the optimal firms are firms 3,5, 11 and 12. (Note that as soon as net benefits went positive, firm 12 was sure to come in.)

Step 3’ Identify the pattern of utility demand for the selected

firms.

Period On Hand 1 2 3 4

Electric 0 4 1 9 14 Sewer 2 7 5 0 5

After “On Hand” Depleted

Electric 0 4 1 0 14 Sewer 0 6 0 0 15

Determine the optimal capacity expansion plan for each utility.

Optimal expansion plan for electricity Period 1 2 3 4

Demand 4 1 0 14

Capacity added 6 0 0 15

Optimal expansion plan for sewer Period 1 2 3 4

Demand 5 5 0 5

Capacity added 6 6 0 3

Step 4’ Repeat of Step 4. No new firms enter the feasible set.

Coordinating public utility expansion, industrial siting snd pollution control 337

SEP.? Vol. II. No. 6-D

338 M. SHEEHAN and K. C. KIXIKU

Table 4. Recomputation of benefits

Period (k)

Firm (i)

Air Quality Requirement

Electric Requirement

SehWt Requirement

ABBreBate Benefits

Net Benefits

1

3

1

6.67 I

18.7

35

__---

10

2

4 5 6

3 3 2

3.35 1.67 3.34

18.7 13.4 13.4

57 62 45

._____________._.

30 47 28

Step 5 As a check repeat Step 3’. No change in optimal firms.

STOP

6. CONCLUSION

The institutions of land use planning in the United States and especially in California are entering a period of fundamental change. The government role in planning is expanding and with this expansion it becomes in- creasingly important that governments be furnished with the tools to do the job well and provide timely solutions to long range problems.

Many of these problems involve the planning of rapidly changing environments with high levels of both risk and uncertainty. We have attempted to show above that relatively simple procedures exist for assisting planners in dealing with these situations. Though the procedure we have suggested does require types of data not normally collected, it would probably be conceded that this information should be collected in any case. By using an interactive search procedure computation has become easily feasible for a large class of public sector problems whose solution would appear to generate sub- stantial benefits to the community.

RJTERENCES

1. Y. Y. Haimes and W. S. Nainis, Coordination of regional water resource supply and demand planning models. Water Resources Res., 10(6), 1051-59 (1974).

2. D. Seckler (Ed.), Cakfornia Water. University of California

3

8

5

8.35

8

21

_-___-

5

8

_

4

10 11 12

5 2 0

15 8.35 15

15 9 6

99 51 22

_________________.

69 34 1

3. F. A. Anderson, NEPA in the Courts: A Legal Analysis of the National Environmental Policy Act. Resources for the Future, Baltimore (1973).

4. R. L. Berkman and Kip Viscusi, Damming the West. Gross- man, New York (1973).

5. Anonymous, The significance of recently enacted state legislation: planning and zoning law, environmental impact reports. Duncan and Jones, Urban Planning Consultants, 2161 Shattuck Avenue, Berkeley, California. 1972.

6. R. C. Fellmeth, Politics of Land, Chap. 3. Grossman, New York (1973).

7. R. J. Struyk, Empirical foundations for modeling urban in- dustrial locations. JAIP 42(2), 165-173 (April 1976).

8. Project Economics/Management Group, Financial planning model-public sector version. Manhattan Beach, California, no date.

9. Harvey M. Wagner, Principles of Operations Research, pp. 332-339. Prentice-Hall, Englewood Cliffs NJ (1%9).

10. D. Erlenkotter, Capacity planning for large multilocational systems: approximate and incomplete dynamic programming approaches. Mangmt. Sci. 22(3), 274-285 (1915).

11. Allan S. Manne, Investments for Capacity Expansion. Allen and Unwin, London (1967).

12. Dermot Gately, Sharing the gains from regional cooperation: a game theoretic approach. Int. Econ. Reu. 15(l), 195-208 (1974).

13. R. E. Bellman and S. E. Dreyfus, Applied Dynamic Pro- gramming. Princeton University Press, Princeton (1962).

FURTZIER READING G. Mills, Public utility pricing for joint demand involving a

durable good. Bell 1., 7(l) 299 (1976). A. G. Wilson, Urban and Regional Models in Geography and

Press, Berkeley (1971). Planning. Wiley, New York (1973).