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Linear programming and comprehensive planning: the context of site development
So&-Econ.
Plan. Sci. Vol. 6, pp.
227-240
(1972).
LINEAR PROGRAMMING
Pergamon
Press.
Printed
in Great
Butam
AND COMPREHENSIVE PLANNING:
THE CONTEXT OF SITE DEVELOPMENT STEPHEN S. SKJEI
Department
of Environmental
Sciences, University of Virginia, Charlottesville,
Virginia 22903
(Keceived 18 Decenrber 1971) Linear programming can be successfully applied in the comprehensive analysis of a wide range of planning problems. As part of a general planning process it facilitates the synthesis and evaluation of alternatives. This potential is demonstrated in the context of the site development problem. A set of linear relations characterizing one approach to this problem is presented and extensions are noted. The approach taken assumes maximization of discounted cash flow as the developmental objective and restricts the ways of obtaining this objective by economic, social and physical design constraints. INTRODUCTION THOUGH a proclaimed
functional goal of most public planning agencies, comprehensive planning is rarely defined in the same manner by different agencies and often appears to exist more in name than in fact. In the United States different metropolitan planning agencies frequently construct comprehensive plans from different constituent elements (basic studies and analyses) and approach the task of integrating these constituent elements with different degrees of rigor. Many agencies synthesize and evaluate but one alternative. Others may initially consider several alternatives but terminate the planning process with the evaluation of a single alternative [l]. With few exceptions the analytical use of basic studies and goals statements is an intuitive process enshrouded in imprecise professional jargon. Ideally the task of comprehensive planning requires that the solution to any problem be obtained from an analytical process which integrates all relevant information. The findings of basic studies and goals analyses should be combined to synthesize and evaluate alternatives. Relevant information should not be ignored nor should it be used as a partial basis for structuring the problem or a solution to it. In the synthesis and evaluation of alternative land use plans, for example, socio-economic considerations should be integrated with more traditional physical considerations. They should not be treated separately from physical considerations nor should they, because of inadequate integrating mechanisms, dominate or be subordinate to these considerations. True comprehensive planning requires an operational framework for combining the information relevant to any planning problem. That is, it requires techniques and tools which permit the planner to order or structure complexity. ORDERING
COMPLEXITY
Several techniques for ordering complexity exist and have been successfully used. “efficient land use patterns” and “integrated Admirable plans for “balanced communities”, 221
228
STEPHENSS. SKJEI
developments” have been constructed with only the most general notion of a planning process as a guide. The various steps of a planning process (e.g. goals analysis, basic studies, synthesis, evaluation) provide a broad framework for analysis. They constitute a general pattern of checks and balances, of feedback loops and attendant reformulations of objectives, design standards and alternatives. Within the context of a planning process, rules of thumb, the lessons of experience, tenets of accepted practice and other organizing ideas may be used to structure complexity and to facilitate evaluation. However by itself the paradigm of a planning process provides at best an intuitive, subjective guide to the analysis and evaluation of alternatives. More rigorous analytical tools can be employed. Mathematical models of the interaction between land use and transportation systems have been used since the early 1960% to judge the efhcacy and consequences of new public investment in highways i2]. Following the publication of AIexander’s The Synthesis of Form [3], the decomposition of design problems was intensively studied and a number of computer codes became available. Recently developed computer codes permit evaluation of the environmental quality of building designs [43. Though frequently viewed in isolation from the concept of a planning process, these tools can also be ~ornbin~ with it to provide a basis for organizing complexity. Each of the procedures has its applications. As approaches to the task of comprehensive planning, however, they suffer from several deficiencies. Many are difficult and costly to employ. Computer time is always expensive, but in addition the costs of “debugging” a program, developing relevant parameters for a model and structuring a model so that it reffects the salient characteristics of a bulling or transpo~ation net may be high. Some of the modefs cannot deal with a task the size of the typical site design problem. Others provide only a starting point for anaIysis. They indicate how a problem statement can be organized but beyond this contribute little to the actual task of analysis. Still others, primarily the land use transportation models, are predictive rather than evaluative devices. They generate a solution but do not analyze it. These models may be useful in predicting the physical consequences of alternative public policies. But their usefulness in analyzing the economic feasibility of a project to be constructed over a time horizon or for generating a design which is consistent with social considerations or the ecological and other environmental characteristics of the site has not been demonstrated. This paper explores the application of linear pro~ammjng, a simple though rigorous quantitative tool, in the task of comprehensive planning. The site design problem is used to demonstrate how linear programming can provide a structure for integrating socio-economic, financial and traditional design considerations in the evaluation of alternatives. This problem was chosen to explore the potentials of linear programming for several reasons. First, it is an important planning problem, frequently confronted by professionals. Second, it has many characteristics in common with larger scale planning problems such as metropohtan land use planning, regional development, new towns deveIopment and urban renewal. Third, as a by-product of this focus operational definitions of such commonly used terms as “balanced community”, “integrated whole”, “holding capacity” and “efficient land use pattern” are obtained. Each of these terms has normative content and unless each is linked with a procedure for analysis which insures that a plan or proposal is in fact “balanced”, ~‘integrate~’ or iiefhcient”, it tends to have, like %omprehensive planning”, little operational significance.
Linear Programming LINEAR
and Comprehensive
PROGRAMMING
Planning
229
MODELS
As a planning tool, linear programming is not without its shortcomings. It requires, for example, that relevant information be expressed quantitatively in the form of a linear relation. This may preclude its application in certain circumstances or limit its usefulness in analysis. Yet because of its general flexibility, rigor and capacity to integrate a broad range of planning considerations linear programming would seem to be a tool which planners could frequently use to advantage. Moreover because of the widespread availability of both computers and computer codes it would also seem to be a tool which planners could readily employ. And as a technique of analysis applied within the framework of a planning process, linear programming would appear to be a profitable alternative to subjectivity and intuition in plan evaluation. To construct the linear programming formulation of the site development problem two sets of decision variables must be distinguished: that which represents different types of residential structures and that which represents different types of dwelling units. The variable =1...j* t =l...t*
j
Yjfd
d =l...d* d” > t” indicates a structure of type j erected in year t and to be sold in year d. For d < t, the variable obviously has no meaning and can be omitted. The notion of a structural type
can be based on a very simple classificatory scheme or on a detailed architectonic interpretation. A set of simple prototypal forms such as single family detached, townhouse, garden apartment, low rise, medium rise, high rise constitutes one approach to the differentiation of structures. On the other hand a detailed notion of structural type might imply not only buildings of different heights and shapes but also buildings of different sizes, operating characteristics, facades, interior configurations and other architectural and environmental qualities. Such detailed specification greatly increases the number of decision variables in the problem. Since each structural type is differentiated by year of construction and year of intended sale, increasing the number of structural types distinguished by one, for example, results in more than t* additional variables. This is by no means a problem and in many ways is a benefit. Most linear programming codes can readily handle large numbers of variables. Greater detail in the specification of structural types increases the planner’s ability to control complexity and increases the usefulness and rigor of a linear programming analysis. Like structures, dwelling units can be differentiated in a number of ways. The variable xijktd
i=l...i* j==1...j* k=l.. . k” and k” = f (j) t=l...t* d=l...d*
represents a dwelling unit of type i on the kth floor of a structure of type j built in year t and to be sold in year d. The number of rooms, architectural quality (e.g. nature of floor and wall materials, quality of fixtures, appointments and built in facilities), total floor area, configuration and size of rooms, operating and other characteristics of service
230
STEPHENS.
SKJE~
systems and related factors can be used to distinguish dwelling unit types. The subscript k has an upper limit k* which is determined for any variable by j, the structural type within which dwelling unit i is to be located.
A solution to the linear programming model of the site development problem is obtained by assigning a non-negative value to each of the variables xijktd and Yjtd. In other words a solution consists of a construction schedule indicating (a) the quantity of each type of structure to be built, (b) when each structural type is to be built, (c) the quantity of each type of dwelling unit every structure should contain, (d) the distribution by floor of dwelling units within each structure and (e) the time period in which every structure should be sold. Obviously a vast number of construction schedules can be proposed. Through linear programming it is possible to determine that schedule which is consistent with all design criteria and which yields the greatest contribution to an objective such as the maximization of net profits or discounted cash flow. The discussion of these capabilities of linear programming will be facilitated by a notational reformulation of the decision variables xijktd and yjtd. Because these variables are extensively subscripted, notational simplicity and brevity can be obtained by defining cumulative variables which are based on them. A useful set of cumulatives based on the variables yjrd is Yjt*d*= c c Yjtd r
j=
I..
,j*
d
where by convention the unspecified summations cover the entire range of the subscript. The cumulative variable YjfY* simply sums all of the structures of type j to be built. Similarly the cumulatives
1.. .i*
i=
j = 1 . . .j* i=
I...["
j=
1.. .j*
t=l...t* may be used to aggregate various combinations of dwelling units. The first, Xij*k*t*d*, simply sums all the dwelling units of type i. The second and third cumulatives, Xijk*f*d*and Xijk*ld*,represent, respectively, all the dwelling units of type i located in structures of typej and all the dwelling units of type i located in structures of typej and built in year t. With these cumulatives and the original variables design requirements reflecting physical, social, economic and financial considerations may be expressed in linear form. An obvious physical restriction is that dwelling units must be assigned to the floors of all structures exactly to the extent that space permits. In quantitative form this becomes
2 fiXi,iktd
=fjk>!jtd
j
=
1
. * .,j”
k=l...k” t=l...t*
d=l...d*
(1)
Linear Programming
and Comprehensive
Planning
231
wheref, represents the floor area of dwelling unit i and fjk is the area on floor k of structurej which can be allocated to residential uses. Because structures do not need to be seen solely as rectangular blocks fjk
=
or # fjk'
for any pair of floors, k and k’, defined for j, and k $1 k’. Another obvious developmental restriction is the total demand expected in any time period for housing of a particular type. For financial and social welfare reasons construction should not proceed ahead of demand. This requirement can be acknowledged by the relation Xijk*fd*
<
Dijr
i=l...i* . . .j* 1 . ..t*
j=l
t=
(2)
The constant B ijr is the expected upper limit on the demand in period t for new dwelling units of type i in structures of type j. It must be estimated prior to analysis on the basis of expected and past housing market conditions and the rent to be charged for or the price of dwelling unit-structure combination (ij) in period t. This relation assumes that all dwelling units constructed prior to period t will remain occupied. Depending on the available data about the demand for dwelling units, (2) can be reformulated in a number of ways. With extensive data about demand, it might be possible to disaggregate (2) further to reflect relative preferences for the same dwelling units on different floors within the same structure. With less extensive data about demand, the relation can be aggregated to reflect (a) only total demand for new dwelling units in any time period or (b) demand for general types of dwelling units. Finally the relation could be modified so that the impact of development prior to period t on Bij, is recognized. The force of (2) should be relaxed, i.e. Bijt should be increased, as the number and types of dwelling units, public services and amenities provided in previous periods increase. Attendant with the demand side restrictions (2) may be supply side or productive constraints limiting both the total amount of residentia1 space and the total number of structures of any type that can be provided in any period or set of periods. This design consideration may be expressed as j =
1 . . ..j*
(3)
d
t=l...t*
C C FjYjtdg Ft j d
(4)
where isj, and F, are, respectively, the maximum number of structures of type j and the maximum total residential floor area that can be produced in any period t. The coefficient Fj is the total floor area of structural type j. Relations (3) and (4) combine to impose an upper limit on the total number of new structures that can be built and thus the number of residents that can be accommodated in any period. For design, policy or program reasons the minimum total population which any
STEPHEN S. SKJEI
232
development on the site must accommodate fications may be presented in linear form as
may be specified in advance.
7 PiXij*k*f*d*2 lp
These speci(9
where is is the minimum total desired residential population and the coefficients Pi represent the expected number of residents of dwelling unit type i. By modifying (5) certain demographic considerations can be incorporated into the linear programming statement of the site development problem. To illustrate, suppose that just four basic dwelling unit types are distinguished, each thought to appeal to a different socio-economic class or, more simply, to be appropriate for a family of a different size or maturity. Then (5) might be reformulated as PiXij*k*t***2 Pi
i=1...4
(5’)
where Pi is the desired number of residents in the ith socio-economic class or the desired number of families of a given size. The resident population accommodated on any site must be supported by nonresidential activities, public services and amenities. It is always desirable and it is often necessary that these be provided simultaneously with dwelling units. To express this and to reflect the fact that some of these activities, services and amenities have a market determined value, two sets of relations are needed: 7 F &
ZijkqXijkrd-Zgfd
=
q
O
=
1.. .q*
(6)
t==l...t* d=l...d”
and
C C C C zijkgXijkfd-CZ*fd i
=
jkd
d
O
g = 1.. .g*
(7)
t==l...t”
The first of these sets of relations (6) establishes the amount of non-marketable service or amenity q needed in t for dwelling units provided in that period and intended to be sold in d. Because these services have no market value, control of them is assumed to be relinquished when the dwelling units for which they were built are sold. The second set of relations establishes the total amount, xzgtd, of marketable activity g which needs to be d
provided in t. The variable z,,, represents the portion of this total which is to be sold in period d. The coefficients Zijkq and Zijkg indicate the quantity of services q and g which must be provided for each dwelling unit of type i located on the kth floor of a structure of type j. These coefficients are formed by multiplying per person requirements for services 4 and g by the expected number of residents of dwelling unit i and adjusting the product to reflect whatever differential needs for these services might be imposed by the location of dwelling unit i on the kth floor of structure j. Establishing the appropriate values of the coefficients Zijkq and Zijkg may be a difficult task. Per person requirements for non-residential activities such as commerce, education and recreation often decline with total population while those for transportation, police services or fire services may be invariant over large increments in population. One solution to this problem is to base the value of the coefficients Zijkq and Zijkg initially on B, the minimum desired population. Depending on the extent to which actual residential
Linear Programming
and Comprehensive
233
Planning
population exceeds P, some of these coefficients may need to be adjusted after a first solution is obtained. Making these adjustments does not necessarily require that the site development problem be resolved. Through sensitivity and parametric analysis the consequences of some of these adjustments can readily be ascertained.* Should the quantity of a service or amenity that can be provided in any time period be limited, upper bounds may be placed on the variables z,,, and zstd. These bounds would impose another constraint on the residential population that could be accommodated in any period. Also for f 2 d, the variables zgfd and zstd are like the variables and yjYjrd.They have no meaning and in effect do not exist. This means of course that no relation exists for t 2 d in (6). The size of the site always restricts developmental possibilities. This may be expressed by relation xijk*d
cGj
&d*+x
F
.i
G8!fSdi~
c
T
2:
CgZgtd
s
G
(9
4
where % is the total ground area of the site. The coefficients G,, G, and Gg are the ground areas associated with a structure of typej, a service of type q and a service of type g. To reflect some of the traditional physical concerns in site development, residential density and residential floor area restrictions may be incorporated into the linear programming formulation. The residential density constraint may be inte~reted as establishing an upper limit on the “activity” in all residential areas of the site. Such a constraint may be expressed as 4
L
f
d
9
f
d
where R is the upper limit on desired net residential density. The summations inside the parentheses represent the land needed for non-residential land uses. The floor area ratio restriction can be used to limit general building mass to a human scale and may be written as
C F’Yj,*e I T(C GjYjf*d*) j
j
where T is the desired floor area ratio and Fj is total floor area of structural type j. Because expressions (5) and (10) are suflicient to establish a lower and an upper limit on R the inequality is quite likely to hold in at least one of (9) or (10). While this suggests that one of them can be eliminated, prior to analysis it is difficult, if not impossible, to determine which. Expressions (S)-(10) place general limitations on the physical solution to the planning problem. They require that the final distribution of dwelling units and structures be consistent with the size of the site and with design standards determined prior to analysis. Additional restrictions can be developed to constrain the location of dwelling unit types within structures and to define further the desirable distribution of structures. The final plan for a new community may, for example, be required from the beginning to indicate specified percentages of singIe family detached homes, townhouses and garden apartments. Multi-family structures may be required to contain a fixed percentage of dwelling units of a given type and the dwelling units of any type within such a structure may be restricted to higher or lower floors. These and related limitations can be based on sociological * See Hadley [S] for a discussion of sensitivity and parametric analysis. B
STEPHEN S. SKJEI
234
research, established public policy or on concepts of good planning and design. A linear programming analysis will quickly indicate whether all of these requirements can be met and how variations in them will affect the solution to the planning problem. Financial constraints can be added to (l)-(10) in a number of ways. Restrictions on the quantity of external funds employed can be introduced as can relations requiring a minimum net profit. The following discussion assumes that the relevant financial objective is maximization of discounted cash flow. In any period two measures of cash flow can be distinguished, one which is calculated before and one which is calculated after the assessment of taxes. Pre tax cash flow is gross revenues net of operating costs and debt service. After tax cash flow, the measure usually used in determining discounted cash flow, is defined as pre tax cash flow minus capital gains and income tax liabilities. Taxable income is defined as gross revenues net of operating costs, depreciation and interest payments. The pre tax cash flow in any period h can be expressed in linear form as
-Cgtdh)Zgtd+cc 2 (bqtdh-Cqtd/&qtd +cT 7 (b,tdh q
B
-u;+u,
=0
h=
t
d
l...d*
(11)
The coefficients Cjtdh,cqf&,and c&h represent the financial costs of capital expenditures. They indicate the interest and amortization (debt service) payments for period h on a mortgage obtained in period one and used in period t to finance the construction of a structure of type j, a unit of service 4 and a unit of service g. The coefficient cijkrdhreflects the debt service payment on a mortgage obtained in period one to finance the distinguishable or incremental capital costs which arise when a dwelling unit of type i is located on the kth floor of structure j in period t. Aggregating these financial costs for all structures, dwelling units and services yields the debt service payment which must be made in h on a mortgage obtained in period one to cover all costs of development. In any period h, revenues may be obtained from three sources: the interest earned on unused funds, the sale of structures and marketable services, and the rental of dwelling units and marketable services. With careful definition each of these sources can be indicated by the COeffiCientSbqfdh, bgtdh, bjtdh and bijktdh. When t > h, these coefficients indicate the interest earned on funds borrowed in period one to finance construction in t. Thus for t > h, b,,,, and bgfdhrepresent the interest earned by borrowed funds which have not been used to finance the construction of a unit of service q or g. Similarly bjrdh and bijkrdh represent the interest on funds which will be used in t to cover the costs of erecting a structure j and of placing a dwelling unit of type i on the kth floor of that structure. When t < h and d = h each of the four coefficients may be interpreted as a sales price. The Saks price bqfhh is of magnitude zero, however, since by definition service type q has no market value. But the coefficient bgrhhof marketable service g does have a positive value. The coefficients bjtdk and bijktdh represent, for d = h, different portions of the sale price of structure j. Since the final sales price of any structure depends on the dwelling units it contains, bjthh is most rigorously interpreted as a base sales price for a structure of type j built in year t. Adjustments of this base sales price to reflect the actual distribution of dwelling units in a structure j are accomplished through the coefficients bijkrhh. These
Linear Programmingand ComprehensivePlanning
235
coefficients can be viewed as the increment to the base sales price of j attributable to the presence of one more dwelling unit of type i on floor k. Rental revenues net of operating costs from dwelling units and services are indicated by the coefficients bijktdh and bgrdh for t < h and d > h. But the coefficients bjtdh and bqtdh are both zero for these subscript values. Structures are not assumed to yield rent independent of dwelling units and non-marketable services by definition can provide no monetary return. Finally, each of the coefficients must be assigned a value of zero for t = h or d < h. Revenues cannot be generated by funds, structures or facilities in the year of construction and, once a structure or service is sold, it yields no revenue. In addition recall that for t > d the decision variables and thus their coefficients are not defined. Under certain circumstances distinguishing between the various determinants of the selling price or the capital costs of a structure may not be necessary. If all dwelling units which might be located in a structure yield the same net rent per square foot or require the same capital expenditure per square foot of floor space in all time periods then neither sales price nor construction cost should be affected by the types of dwelling units in the structure. In this case all the coefficients bijkthh and cijktd in (11) would be zero. The variables Ui and U, are the magnitudes of a positive or negative pre tax cash flow in h. Expression (11) sums all revenues obtained in period h and subtracts from them the debt service on a mortgage obtained in period one to cover all construction costs. Revenues from sales, the rental of dwelling units and the lease of services are defined net of operating and transactions costs. Because of the structure of the linear programming algorithm only one of Ui and U, can have a positive value in any period. The other must take on a value of zero. By subtracting the tax liability in any period h from the pre tax cash flow, Ui or U,, the after tax cash flow can be obtained. The discounted value of this quantity when summed for all periods is the discounted cash flow of a project. * In linear form the tax liability in h may be expressed as
-;
T iy (~qfdh+C'gldh)~qfdl
-v;+v,
=0
h = 1 . . . d*
(12)
Expression (12) sums the capital gains taxes from the sale of structures and marketable services and adds to this the taxes on rental revenues net of depreciation and interest charges. The coefficients ejthr and egthhare the product of the factor l/E,, and the capital gains tax on a structure or marketable service sold in period d = h. The constant E,, is the tax rate applicable to earned income in h and its inverse is involved in the coefficients * In symbolic form the formula used here for calculating cash flow is (GR- DS)-g(GRD-IP) where GR is gross revenue, DS is total debt service, D is depreciation, IP is interest payment and g is the expected rate of taxation. The first expression above is pretax cash flow and the second is tax liability. Subtracted they yield cash flow.
STEPHEN S. SKJEI
236
only because all quantities in (12) are multiplied by it. For the same reason ejtdh and egtdh represents the product of l/Eh and any distinguishable capital gains the coefficient tax liability generated because dwelling unit type i is located on the kth floor of structure ,j built in year t and sold in year d. For d < h, d > h or t 2 h each of the parameters ejtdh, has a value of zero. A capital gains tax liability is generated only by the egtdh and eijktdh sale of structures and services in period h.” The coefficients b’ijktdh= bijkrdhand the coefficients b’gtdh= bstdhfor all i, j, k, t, d except d = h where they are equal to zero. That is, the coefficients b’ijktdhand b’gtdhindicate either positive rental revenues from dwelling units and services (built prior to h and not sold in that period) or the interest earned or unused construction funds. They do not reflect sales. represent only the interest charges in h on the funds The coefficients C’gtdh and borrowed in year one to finance the construction of one unit of marketable services q and the distinguishable costs of one more dwelling unit i. The coefficients cYjtdh and ctgtd,,indicate the interest charges in h on funds obtained to cover the capital expenditure in t for a structure of type j and one unit of service q minus any interest which might be earned in h on unused funds. Thus for t I h, these coefficients indicate interest charges on funds used to finance development; but for t > h they indicate interest charges on the unused portions of a mortgage obtained in year one minus whatever interest can be earned on those unused funds. The depreciated value of structures and services in period h is given by the factors Any distinguishable depreciation of a dwelling unit of type i on the Wjtdhv Wgtdh and Wqtdh, Each of these coefficients has a value kth floor of structure j is given by the factor of zero for t 2 h or d I h. That is, depreciation occurs only on those structures and services built prior to h and not sold in or before that period. The only sources of revenues in (12) are rents (net of operating costs) and interest earned on unused funds. From these revenues depreciation charges on structures, dwelling units and services and the interest charges on the loan needed to cover all construction costs are subtracted to obtain taxable income. Multiplying taxable income by Eh yields the income tax liability. By adding to this the product of Eh and eijktdh,ejtdh, egrdh(which products are the capital gains tax liability from the sale of structures, the dwelling units they contain and marketable services), the total tax liability in h is obtained. The variables Vi and V; reflect the magnitude of this liability and whether it is, respectively, positive or negative. Because of the structure of the linear programming algorithm only one of these two variables can have a positive value in any period; the other must take a value of zero. After tax cash flow in any period h can be defined as eijktdh
C’ijktdh
Wijktdh.
u;-u,-v;+v, This definition treats any tax shelter created by the project, i.e. V,, as a benefit to be applied against other non-project income in the year of occurrence. To maximize discounted cash flow over all time periods, the expression
c r,(U,+ -u,-v;+v,)
(13)
h
* The parameters Wijktdh, “jt&, ejjkfh and e,th may not be needed in Certain inStanCeS. Where all dwelling units have the same per square foot capital costs and yield the same net per square foot rents it is not necessary to distinguish between the two depreciation parameters or between the two tax parameters.
Linear Programming
and Comprehensive
Planning
231
where r, is the present value of a dollar in period h must be used as the objective function of the linear programming model of the site development problem. As a model of the site development problem, expressions (l)-(13) provide a solution which is a construction schedule for residential and non-residential facilities. The unique characteristics of this schedule are that it maximizes discounted cash flow and is consistent with social, physical economic and other design restrictions. Different solutions to the site development problem can easily be obtained by modifying this model. For example to determine the maximum holding capacity of the site subject to the constraints (l)-(12) it is only necessary to replace (13) with
and add the restrictions h=l...p
where Qh is a minimum desired cash flow in period h. Given a different objective function and a specification of control variables which acknowledged site characteristics it would also be possible to determine an “efficient land use” pattern. Through linear programming it is possible to analyze a planning problem from a variety of perspectives and thereby obtain a better understanding of that problem. DATA
REQUIREMENTS
AND
EXTENSIONS
The data required to use linear programming in site development analysis do not in general appear to be difficult to obtain. Except for those in relations (11) and (12) most of the parameters needed are either design and amenity standards or physical standards for structures. The former can be based on accepted practice, tradition or the results of analysis while the latter can be obtained from the drawing boards or the models of prototypal forms. The parameters in relations (11) and (12) may require some extensive preliminary cost and financial analysis, which may initially appear to obviate the gain obtained by adding these sets of relations to the others. However multi-family structures are rarely developed without planning analyses of their financial aspects. Net revenues and capital costs for structures and individual dwelling units within any structure are often estimated in the project planning stage. Once capital costs have been estimated, amoritization, interest and depreciation charges can be calculated. Similarly once sales price at any point in time has been estimated, potential capital gains tax can be determined. Thus (I 1) and (12) require estimates that are often made in other types of design analyses and calculations which are based on these estimates. With appropriate computer codes it is possible not only to have these calculations performed automatically but also to have relations (11) and (12) formulated as inputs to a computerized linear programming algorithm. As formulated (11) and (12) provide great flexibility in analysis. The consequences of interest rate subsidies, construction cost subsidies, rent subsidies, rent distribution schemes and other policies can readily be analyzed with them. They afford analytical advantages beyond their very obvious function of permitting discounted cash flow analysis. Numerous extensions of the discounted cash flow model of the site development problem are possible. Sociological criteria reflecting the interaction between man and his environment can be added almost without limit. The only restriction is that all criteria
STEPHEN S. SKJEI
238
must be formulated on an interval or ratio scale and must be expressed as linear functions only hints at of the control variables xijkrd, yjtd, zgtd and z,,,. The preceding formulation the potential of linear programming to integrate social considerations into traditional planning analysis. Another extension might involve improving the relation used to express the costs of non-residential facilities. The cost curve for these facilities is normally thought to demonstrate economies of scale in a step fashion. Through mixed integer programming or the imposition of upper bounds on variables this characteristic may be more fully incorporated into the model. Similarly environmental considerations may be more explicitly represented. The preceding formulation assumed the site was a homogeneous plain, primarily for purposes of notational simplicity and brevity. By relaxing this assumption and differentiating among areas of the site, environmental factors could have been introduced quite readily. In addition differentiating among areas of the site would have provided a better basis for assessing the costs of public services and amenities and would have allowed the introduction of relations which modelled transportation requirements within the site. In turn this would have allowed inclusion of relations reflecting standards for propinquity between residences, public services and employment nodes. The location of manufacturing facilities and other employment nodes could thus have been analyzed and propinquity standards could have been incorporated to restrict the location of services and amenities such as schools to specified distances from residences. It is also possible through parametric and sensitivity analysis to analyze the effect of The consequences of miscalculation in errors in the values assigned to coefficients. estimating financial parameters can be established and the effect of variations in the design constraints, standards and criteria specified prior to analysis can be evaluated. Consequently the implications of subjectivity and intuition for discounted cash flow can be determined. Finally the model presented here can be easily extended to analyze many other planning problems. Its application to redevelopment and renewal problems should be evident. With the inclusion of site characteristics it can be used to analyze zoning patterns from a Applications to smaller scale planning problems clearly defined normative standpoint. such as the provision of housing for the aged or to larger scale problems such as regional land use utilization should also be possible without too much difficulty.* EXAMPLE
AND
CONCLUSION
To illustrate an extension, which is however not as complex as the preceding formulation, consider the following problem. Given (1) a site of 400 acres, (2) a set of standards (modifications of those promulgated by the American Public Health Association) [9] for non-residential uses, (3) a set of dwelling unit types and structural forms and (4) a design population of 14,500, what is the maximum amount of land which can be allocated to A linear programming analysis provided the following single family detached structures. results : 1. 2. 3. 4. 5.
154 164 158 420 139
acres could be devoted to single family detached residential structures; one bedroom single family detached residences were required; three bedroom single family detached residences were required; four bedroom single family detached residences were required; three story multi-family structures were required;
* Recent applications of linear programming
are [6-S].
239
Linear Programming and Comprehensive Planning 6. 7. 8. 9. 10: 11.
60 ten story multi-family structures were required; 15 acres of land were needed for commercial activities; 62 acres of land were needed for school facilities; 39 acres of land were needed for open space; 34 acres of land were needed for parking facilities; in addition a specification of the type and number of dwelling each floor of each multi-family structure was obtained.
units
needed
on
These figures are merely illustrative; they suggest the type of information that can be obtained through the use of linear programming. With sensitivity and parametric analysis the consequences of providing, say, more variety in residential structures could have been ascertained. Given the objectives and constraints used, reducing the required design population resulted in less structural variety in the final solution. The analysis of the site development problem suggests that linear programming can facilitate the task of comprehensive planning. It is a technique of analysis which should It is rigorous yet flexible and may be accessible to many if not all planning agencies. provide significant quantities of information about the nature of a given planning problem. It can be used to give operational significance to the terms “balanced community”, “efficient land use pattern” and “integrated whole” and to determine the implications of the set of design parameters by which notions of efficiency, wholeness and balance are to be judged. A linear programming formulation will indicate clearly how the operational meaning of many planning terms depends on a set of design criteria and amenity standards. Linear programming is nevertheless only a planning aid. While it can be used to reduce uncertainty about a given planning problem, creativity will still remain an important part of the process by which solutions are developed. The combination of prototypal forms and linear programming suggested here only provides a macro-scale solution to the site development problem for example. Many micro-scale problems remain and solutions for them require ingenuity and creativity. With other planning problems the same may be true. Linear programming may suggest general characteristics of a solution but will not provide a detailed solution. Moreover in some instance linear programming may contribute only a partial solution to a problem or may not be an appropriate analytical tool. In the analysis of dynamic processes other programming tools, e.g. dynamic programming, may be more useful. The potential of these tools cannot however be explored here. Finally it is important to note that linear programming may be most effectively employed within the context of a planning process. The very nature of such a process tends to insure that this rigorous quantitative tool will not be narrowly applied. REFERENCES The Metropolitan Planning Process: An Exploratory Study, Appendix 11,2, Department of Housing and Urban Development, Urban Planning Research and Demonstration Program, Washington D.C. (1970). 2. See BRITTONH-IS, (Ed.), Urban development models: new tools for planning, J. Am. Inst. Pkznners XxX1, 90-184 (1965). 3. CHRISTOPHER ALEXANDER,Notes on the Synthesis of Form. Harvard University Press, Cambridge (1964). 4. DEAN HAWKES, A History of Models of the Environment in Buildings, Land Use and 1. MAYNARD M. HUFSCHMIDT et al.,
Built Form Studies, Working Paper 34, University of Cambridge (1970).
5. GEORGEHADLEY,Linear Programming. Addison Wesley (1963). 6. DARWIN G. STUART,Urban improvement programming models, Socio-Econ. Plan. Sci. 4, 217-238 (1970).
240
SI-EPHEN S. SKJE~
7. ROBERT C. MEII.H. Programming of recreational land acquisition. Suck+tion. Ph. Sri. 2, 15-24(1968). 8. KENNETH J. SCI-ILAGER, A land use plan design model, J. ,4/u. fn.rr. Planners 31, 103-110 (1965). 9. American Public Health Association, Committee on the Hygiene of Housing, Planning the Neighborhood, pp. I-50, Chicago (1960).
Plan. Sci. Vol. 6, pp.
227-240
(1972).
LINEAR PROGRAMMING
Pergamon
Press.
Printed
in Great
Butam
AND COMPREHENSIVE PLANNING:
THE CONTEXT OF SITE DEVELOPMENT STEPHEN S. SKJEI
Department
of Environmental
Sciences, University of Virginia, Charlottesville,
Virginia 22903
(Keceived 18 Decenrber 1971) Linear programming can be successfully applied in the comprehensive analysis of a wide range of planning problems. As part of a general planning process it facilitates the synthesis and evaluation of alternatives. This potential is demonstrated in the context of the site development problem. A set of linear relations characterizing one approach to this problem is presented and extensions are noted. The approach taken assumes maximization of discounted cash flow as the developmental objective and restricts the ways of obtaining this objective by economic, social and physical design constraints. INTRODUCTION THOUGH a proclaimed
functional goal of most public planning agencies, comprehensive planning is rarely defined in the same manner by different agencies and often appears to exist more in name than in fact. In the United States different metropolitan planning agencies frequently construct comprehensive plans from different constituent elements (basic studies and analyses) and approach the task of integrating these constituent elements with different degrees of rigor. Many agencies synthesize and evaluate but one alternative. Others may initially consider several alternatives but terminate the planning process with the evaluation of a single alternative [l]. With few exceptions the analytical use of basic studies and goals statements is an intuitive process enshrouded in imprecise professional jargon. Ideally the task of comprehensive planning requires that the solution to any problem be obtained from an analytical process which integrates all relevant information. The findings of basic studies and goals analyses should be combined to synthesize and evaluate alternatives. Relevant information should not be ignored nor should it be used as a partial basis for structuring the problem or a solution to it. In the synthesis and evaluation of alternative land use plans, for example, socio-economic considerations should be integrated with more traditional physical considerations. They should not be treated separately from physical considerations nor should they, because of inadequate integrating mechanisms, dominate or be subordinate to these considerations. True comprehensive planning requires an operational framework for combining the information relevant to any planning problem. That is, it requires techniques and tools which permit the planner to order or structure complexity. ORDERING
COMPLEXITY
Several techniques for ordering complexity exist and have been successfully used. “efficient land use patterns” and “integrated Admirable plans for “balanced communities”, 221
228
STEPHENSS. SKJEI
developments” have been constructed with only the most general notion of a planning process as a guide. The various steps of a planning process (e.g. goals analysis, basic studies, synthesis, evaluation) provide a broad framework for analysis. They constitute a general pattern of checks and balances, of feedback loops and attendant reformulations of objectives, design standards and alternatives. Within the context of a planning process, rules of thumb, the lessons of experience, tenets of accepted practice and other organizing ideas may be used to structure complexity and to facilitate evaluation. However by itself the paradigm of a planning process provides at best an intuitive, subjective guide to the analysis and evaluation of alternatives. More rigorous analytical tools can be employed. Mathematical models of the interaction between land use and transportation systems have been used since the early 1960% to judge the efhcacy and consequences of new public investment in highways i2]. Following the publication of AIexander’s The Synthesis of Form [3], the decomposition of design problems was intensively studied and a number of computer codes became available. Recently developed computer codes permit evaluation of the environmental quality of building designs [43. Though frequently viewed in isolation from the concept of a planning process, these tools can also be ~ornbin~ with it to provide a basis for organizing complexity. Each of the procedures has its applications. As approaches to the task of comprehensive planning, however, they suffer from several deficiencies. Many are difficult and costly to employ. Computer time is always expensive, but in addition the costs of “debugging” a program, developing relevant parameters for a model and structuring a model so that it reffects the salient characteristics of a bulling or transpo~ation net may be high. Some of the modefs cannot deal with a task the size of the typical site design problem. Others provide only a starting point for anaIysis. They indicate how a problem statement can be organized but beyond this contribute little to the actual task of analysis. Still others, primarily the land use transportation models, are predictive rather than evaluative devices. They generate a solution but do not analyze it. These models may be useful in predicting the physical consequences of alternative public policies. But their usefulness in analyzing the economic feasibility of a project to be constructed over a time horizon or for generating a design which is consistent with social considerations or the ecological and other environmental characteristics of the site has not been demonstrated. This paper explores the application of linear pro~ammjng, a simple though rigorous quantitative tool, in the task of comprehensive planning. The site design problem is used to demonstrate how linear programming can provide a structure for integrating socio-economic, financial and traditional design considerations in the evaluation of alternatives. This problem was chosen to explore the potentials of linear programming for several reasons. First, it is an important planning problem, frequently confronted by professionals. Second, it has many characteristics in common with larger scale planning problems such as metropohtan land use planning, regional development, new towns deveIopment and urban renewal. Third, as a by-product of this focus operational definitions of such commonly used terms as “balanced community”, “integrated whole”, “holding capacity” and “efficient land use pattern” are obtained. Each of these terms has normative content and unless each is linked with a procedure for analysis which insures that a plan or proposal is in fact “balanced”, ~‘integrate~’ or iiefhcient”, it tends to have, like %omprehensive planning”, little operational significance.
Linear Programming LINEAR
and Comprehensive
PROGRAMMING
Planning
229
MODELS
As a planning tool, linear programming is not without its shortcomings. It requires, for example, that relevant information be expressed quantitatively in the form of a linear relation. This may preclude its application in certain circumstances or limit its usefulness in analysis. Yet because of its general flexibility, rigor and capacity to integrate a broad range of planning considerations linear programming would seem to be a tool which planners could frequently use to advantage. Moreover because of the widespread availability of both computers and computer codes it would also seem to be a tool which planners could readily employ. And as a technique of analysis applied within the framework of a planning process, linear programming would appear to be a profitable alternative to subjectivity and intuition in plan evaluation. To construct the linear programming formulation of the site development problem two sets of decision variables must be distinguished: that which represents different types of residential structures and that which represents different types of dwelling units. The variable =1...j* t =l...t*
j
Yjfd
d =l...d* d” > t” indicates a structure of type j erected in year t and to be sold in year d. For d < t, the variable obviously has no meaning and can be omitted. The notion of a structural type
can be based on a very simple classificatory scheme or on a detailed architectonic interpretation. A set of simple prototypal forms such as single family detached, townhouse, garden apartment, low rise, medium rise, high rise constitutes one approach to the differentiation of structures. On the other hand a detailed notion of structural type might imply not only buildings of different heights and shapes but also buildings of different sizes, operating characteristics, facades, interior configurations and other architectural and environmental qualities. Such detailed specification greatly increases the number of decision variables in the problem. Since each structural type is differentiated by year of construction and year of intended sale, increasing the number of structural types distinguished by one, for example, results in more than t* additional variables. This is by no means a problem and in many ways is a benefit. Most linear programming codes can readily handle large numbers of variables. Greater detail in the specification of structural types increases the planner’s ability to control complexity and increases the usefulness and rigor of a linear programming analysis. Like structures, dwelling units can be differentiated in a number of ways. The variable xijktd
i=l...i* j==1...j* k=l.. . k” and k” = f (j) t=l...t* d=l...d*
represents a dwelling unit of type i on the kth floor of a structure of type j built in year t and to be sold in year d. The number of rooms, architectural quality (e.g. nature of floor and wall materials, quality of fixtures, appointments and built in facilities), total floor area, configuration and size of rooms, operating and other characteristics of service
230
STEPHENS.
SKJE~
systems and related factors can be used to distinguish dwelling unit types. The subscript k has an upper limit k* which is determined for any variable by j, the structural type within which dwelling unit i is to be located.
A solution to the linear programming model of the site development problem is obtained by assigning a non-negative value to each of the variables xijktd and Yjtd. In other words a solution consists of a construction schedule indicating (a) the quantity of each type of structure to be built, (b) when each structural type is to be built, (c) the quantity of each type of dwelling unit every structure should contain, (d) the distribution by floor of dwelling units within each structure and (e) the time period in which every structure should be sold. Obviously a vast number of construction schedules can be proposed. Through linear programming it is possible to determine that schedule which is consistent with all design criteria and which yields the greatest contribution to an objective such as the maximization of net profits or discounted cash flow. The discussion of these capabilities of linear programming will be facilitated by a notational reformulation of the decision variables xijktd and yjtd. Because these variables are extensively subscripted, notational simplicity and brevity can be obtained by defining cumulative variables which are based on them. A useful set of cumulatives based on the variables yjrd is Yjt*d*= c c Yjtd r
j=
I..
,j*
d
where by convention the unspecified summations cover the entire range of the subscript. The cumulative variable YjfY* simply sums all of the structures of type j to be built. Similarly the cumulatives
1.. .i*
i=
j = 1 . . .j* i=
I...["
j=
1.. .j*
t=l...t* may be used to aggregate various combinations of dwelling units. The first, Xij*k*t*d*, simply sums all the dwelling units of type i. The second and third cumulatives, Xijk*f*d*and Xijk*ld*,represent, respectively, all the dwelling units of type i located in structures of typej and all the dwelling units of type i located in structures of typej and built in year t. With these cumulatives and the original variables design requirements reflecting physical, social, economic and financial considerations may be expressed in linear form. An obvious physical restriction is that dwelling units must be assigned to the floors of all structures exactly to the extent that space permits. In quantitative form this becomes
2 fiXi,iktd
=fjk>!jtd
j
=
1
. * .,j”
k=l...k” t=l...t*
d=l...d*
(1)
Linear Programming
and Comprehensive
Planning
231
wheref, represents the floor area of dwelling unit i and fjk is the area on floor k of structurej which can be allocated to residential uses. Because structures do not need to be seen solely as rectangular blocks fjk
=
or # fjk'
for any pair of floors, k and k’, defined for j, and k $1 k’. Another obvious developmental restriction is the total demand expected in any time period for housing of a particular type. For financial and social welfare reasons construction should not proceed ahead of demand. This requirement can be acknowledged by the relation Xijk*fd*
<
Dijr
i=l...i* . . .j* 1 . ..t*
j=l
t=
(2)
The constant B ijr is the expected upper limit on the demand in period t for new dwelling units of type i in structures of type j. It must be estimated prior to analysis on the basis of expected and past housing market conditions and the rent to be charged for or the price of dwelling unit-structure combination (ij) in period t. This relation assumes that all dwelling units constructed prior to period t will remain occupied. Depending on the available data about the demand for dwelling units, (2) can be reformulated in a number of ways. With extensive data about demand, it might be possible to disaggregate (2) further to reflect relative preferences for the same dwelling units on different floors within the same structure. With less extensive data about demand, the relation can be aggregated to reflect (a) only total demand for new dwelling units in any time period or (b) demand for general types of dwelling units. Finally the relation could be modified so that the impact of development prior to period t on Bij, is recognized. The force of (2) should be relaxed, i.e. Bijt should be increased, as the number and types of dwelling units, public services and amenities provided in previous periods increase. Attendant with the demand side restrictions (2) may be supply side or productive constraints limiting both the total amount of residentia1 space and the total number of structures of any type that can be provided in any period or set of periods. This design consideration may be expressed as j =
1 . . ..j*
(3)
d
t=l...t*
C C FjYjtdg Ft j d
(4)
where isj, and F, are, respectively, the maximum number of structures of type j and the maximum total residential floor area that can be produced in any period t. The coefficient Fj is the total floor area of structural type j. Relations (3) and (4) combine to impose an upper limit on the total number of new structures that can be built and thus the number of residents that can be accommodated in any period. For design, policy or program reasons the minimum total population which any
STEPHEN S. SKJEI
232
development on the site must accommodate fications may be presented in linear form as
may be specified in advance.
7 PiXij*k*f*d*2 lp
These speci(9
where is is the minimum total desired residential population and the coefficients Pi represent the expected number of residents of dwelling unit type i. By modifying (5) certain demographic considerations can be incorporated into the linear programming statement of the site development problem. To illustrate, suppose that just four basic dwelling unit types are distinguished, each thought to appeal to a different socio-economic class or, more simply, to be appropriate for a family of a different size or maturity. Then (5) might be reformulated as PiXij*k*t***2 Pi
i=1...4
(5’)
where Pi is the desired number of residents in the ith socio-economic class or the desired number of families of a given size. The resident population accommodated on any site must be supported by nonresidential activities, public services and amenities. It is always desirable and it is often necessary that these be provided simultaneously with dwelling units. To express this and to reflect the fact that some of these activities, services and amenities have a market determined value, two sets of relations are needed: 7 F &
ZijkqXijkrd-Zgfd
=
q
O
=
1.. .q*
(6)
t==l...t* d=l...d”
and
C C C C zijkgXijkfd-CZ*fd i
=
jkd
d
O
g = 1.. .g*
(7)
t==l...t”
The first of these sets of relations (6) establishes the amount of non-marketable service or amenity q needed in t for dwelling units provided in that period and intended to be sold in d. Because these services have no market value, control of them is assumed to be relinquished when the dwelling units for which they were built are sold. The second set of relations establishes the total amount, xzgtd, of marketable activity g which needs to be d
provided in t. The variable z,,, represents the portion of this total which is to be sold in period d. The coefficients Zijkq and Zijkg indicate the quantity of services q and g which must be provided for each dwelling unit of type i located on the kth floor of a structure of type j. These coefficients are formed by multiplying per person requirements for services 4 and g by the expected number of residents of dwelling unit i and adjusting the product to reflect whatever differential needs for these services might be imposed by the location of dwelling unit i on the kth floor of structure j. Establishing the appropriate values of the coefficients Zijkq and Zijkg may be a difficult task. Per person requirements for non-residential activities such as commerce, education and recreation often decline with total population while those for transportation, police services or fire services may be invariant over large increments in population. One solution to this problem is to base the value of the coefficients Zijkq and Zijkg initially on B, the minimum desired population. Depending on the extent to which actual residential
Linear Programming
and Comprehensive
233
Planning
population exceeds P, some of these coefficients may need to be adjusted after a first solution is obtained. Making these adjustments does not necessarily require that the site development problem be resolved. Through sensitivity and parametric analysis the consequences of some of these adjustments can readily be ascertained.* Should the quantity of a service or amenity that can be provided in any time period be limited, upper bounds may be placed on the variables z,,, and zstd. These bounds would impose another constraint on the residential population that could be accommodated in any period. Also for f 2 d, the variables zgfd and zstd are like the variables and yjYjrd.They have no meaning and in effect do not exist. This means of course that no relation exists for t 2 d in (6). The size of the site always restricts developmental possibilities. This may be expressed by relation xijk*d
cGj
&d*+x
F
.i
G8!fSdi~
c
T
2:
CgZgtd
s
G
(9
4
where % is the total ground area of the site. The coefficients G,, G, and Gg are the ground areas associated with a structure of typej, a service of type q and a service of type g. To reflect some of the traditional physical concerns in site development, residential density and residential floor area restrictions may be incorporated into the linear programming formulation. The residential density constraint may be inte~reted as establishing an upper limit on the “activity” in all residential areas of the site. Such a constraint may be expressed as 4
L
f
d
9
f
d
where R is the upper limit on desired net residential density. The summations inside the parentheses represent the land needed for non-residential land uses. The floor area ratio restriction can be used to limit general building mass to a human scale and may be written as
C F’Yj,*e I T(C GjYjf*d*) j
j
where T is the desired floor area ratio and Fj is total floor area of structural type j. Because expressions (5) and (10) are suflicient to establish a lower and an upper limit on R the inequality is quite likely to hold in at least one of (9) or (10). While this suggests that one of them can be eliminated, prior to analysis it is difficult, if not impossible, to determine which. Expressions (S)-(10) place general limitations on the physical solution to the planning problem. They require that the final distribution of dwelling units and structures be consistent with the size of the site and with design standards determined prior to analysis. Additional restrictions can be developed to constrain the location of dwelling unit types within structures and to define further the desirable distribution of structures. The final plan for a new community may, for example, be required from the beginning to indicate specified percentages of singIe family detached homes, townhouses and garden apartments. Multi-family structures may be required to contain a fixed percentage of dwelling units of a given type and the dwelling units of any type within such a structure may be restricted to higher or lower floors. These and related limitations can be based on sociological * See Hadley [S] for a discussion of sensitivity and parametric analysis. B
STEPHEN S. SKJEI
234
research, established public policy or on concepts of good planning and design. A linear programming analysis will quickly indicate whether all of these requirements can be met and how variations in them will affect the solution to the planning problem. Financial constraints can be added to (l)-(10) in a number of ways. Restrictions on the quantity of external funds employed can be introduced as can relations requiring a minimum net profit. The following discussion assumes that the relevant financial objective is maximization of discounted cash flow. In any period two measures of cash flow can be distinguished, one which is calculated before and one which is calculated after the assessment of taxes. Pre tax cash flow is gross revenues net of operating costs and debt service. After tax cash flow, the measure usually used in determining discounted cash flow, is defined as pre tax cash flow minus capital gains and income tax liabilities. Taxable income is defined as gross revenues net of operating costs, depreciation and interest payments. The pre tax cash flow in any period h can be expressed in linear form as
-Cgtdh)Zgtd+cc 2 (bqtdh-Cqtd/&qtd +cT 7 (b,tdh q
B
-u;+u,
=0
h=
t
d
l...d*
(11)
The coefficients Cjtdh,cqf&,and c&h represent the financial costs of capital expenditures. They indicate the interest and amortization (debt service) payments for period h on a mortgage obtained in period one and used in period t to finance the construction of a structure of type j, a unit of service 4 and a unit of service g. The coefficient cijkrdhreflects the debt service payment on a mortgage obtained in period one to finance the distinguishable or incremental capital costs which arise when a dwelling unit of type i is located on the kth floor of structure j in period t. Aggregating these financial costs for all structures, dwelling units and services yields the debt service payment which must be made in h on a mortgage obtained in period one to cover all costs of development. In any period h, revenues may be obtained from three sources: the interest earned on unused funds, the sale of structures and marketable services, and the rental of dwelling units and marketable services. With careful definition each of these sources can be indicated by the COeffiCientSbqfdh, bgtdh, bjtdh and bijktdh. When t > h, these coefficients indicate the interest earned on funds borrowed in period one to finance construction in t. Thus for t > h, b,,,, and bgfdhrepresent the interest earned by borrowed funds which have not been used to finance the construction of a unit of service q or g. Similarly bjrdh and bijkrdh represent the interest on funds which will be used in t to cover the costs of erecting a structure j and of placing a dwelling unit of type i on the kth floor of that structure. When t < h and d = h each of the four coefficients may be interpreted as a sales price. The Saks price bqfhh is of magnitude zero, however, since by definition service type q has no market value. But the coefficient bgrhhof marketable service g does have a positive value. The coefficients bjtdk and bijktdh represent, for d = h, different portions of the sale price of structure j. Since the final sales price of any structure depends on the dwelling units it contains, bjthh is most rigorously interpreted as a base sales price for a structure of type j built in year t. Adjustments of this base sales price to reflect the actual distribution of dwelling units in a structure j are accomplished through the coefficients bijkrhh. These
Linear Programmingand ComprehensivePlanning
235
coefficients can be viewed as the increment to the base sales price of j attributable to the presence of one more dwelling unit of type i on floor k. Rental revenues net of operating costs from dwelling units and services are indicated by the coefficients bijktdh and bgrdh for t < h and d > h. But the coefficients bjtdh and bqtdh are both zero for these subscript values. Structures are not assumed to yield rent independent of dwelling units and non-marketable services by definition can provide no monetary return. Finally, each of the coefficients must be assigned a value of zero for t = h or d < h. Revenues cannot be generated by funds, structures or facilities in the year of construction and, once a structure or service is sold, it yields no revenue. In addition recall that for t > d the decision variables and thus their coefficients are not defined. Under certain circumstances distinguishing between the various determinants of the selling price or the capital costs of a structure may not be necessary. If all dwelling units which might be located in a structure yield the same net rent per square foot or require the same capital expenditure per square foot of floor space in all time periods then neither sales price nor construction cost should be affected by the types of dwelling units in the structure. In this case all the coefficients bijkthh and cijktd in (11) would be zero. The variables Ui and U, are the magnitudes of a positive or negative pre tax cash flow in h. Expression (11) sums all revenues obtained in period h and subtracts from them the debt service on a mortgage obtained in period one to cover all construction costs. Revenues from sales, the rental of dwelling units and the lease of services are defined net of operating and transactions costs. Because of the structure of the linear programming algorithm only one of Ui and U, can have a positive value in any period. The other must take on a value of zero. By subtracting the tax liability in any period h from the pre tax cash flow, Ui or U,, the after tax cash flow can be obtained. The discounted value of this quantity when summed for all periods is the discounted cash flow of a project. * In linear form the tax liability in h may be expressed as
-;
T iy (~qfdh+C'gldh)~qfdl
-v;+v,
=0
h = 1 . . . d*
(12)
Expression (12) sums the capital gains taxes from the sale of structures and marketable services and adds to this the taxes on rental revenues net of depreciation and interest charges. The coefficients ejthr and egthhare the product of the factor l/E,, and the capital gains tax on a structure or marketable service sold in period d = h. The constant E,, is the tax rate applicable to earned income in h and its inverse is involved in the coefficients * In symbolic form the formula used here for calculating cash flow is (GR- DS)-g(GRD-IP) where GR is gross revenue, DS is total debt service, D is depreciation, IP is interest payment and g is the expected rate of taxation. The first expression above is pretax cash flow and the second is tax liability. Subtracted they yield cash flow.
STEPHEN S. SKJEI
236
only because all quantities in (12) are multiplied by it. For the same reason ejtdh and egtdh represents the product of l/Eh and any distinguishable capital gains the coefficient tax liability generated because dwelling unit type i is located on the kth floor of structure ,j built in year t and sold in year d. For d < h, d > h or t 2 h each of the parameters ejtdh, has a value of zero. A capital gains tax liability is generated only by the egtdh and eijktdh sale of structures and services in period h.” The coefficients b’ijktdh= bijkrdhand the coefficients b’gtdh= bstdhfor all i, j, k, t, d except d = h where they are equal to zero. That is, the coefficients b’ijktdhand b’gtdhindicate either positive rental revenues from dwelling units and services (built prior to h and not sold in that period) or the interest earned or unused construction funds. They do not reflect sales. represent only the interest charges in h on the funds The coefficients C’gtdh and borrowed in year one to finance the construction of one unit of marketable services q and the distinguishable costs of one more dwelling unit i. The coefficients cYjtdh and ctgtd,,indicate the interest charges in h on funds obtained to cover the capital expenditure in t for a structure of type j and one unit of service q minus any interest which might be earned in h on unused funds. Thus for t I h, these coefficients indicate interest charges on funds used to finance development; but for t > h they indicate interest charges on the unused portions of a mortgage obtained in year one minus whatever interest can be earned on those unused funds. The depreciated value of structures and services in period h is given by the factors Any distinguishable depreciation of a dwelling unit of type i on the Wjtdhv Wgtdh and Wqtdh, Each of these coefficients has a value kth floor of structure j is given by the factor of zero for t 2 h or d I h. That is, depreciation occurs only on those structures and services built prior to h and not sold in or before that period. The only sources of revenues in (12) are rents (net of operating costs) and interest earned on unused funds. From these revenues depreciation charges on structures, dwelling units and services and the interest charges on the loan needed to cover all construction costs are subtracted to obtain taxable income. Multiplying taxable income by Eh yields the income tax liability. By adding to this the product of Eh and eijktdh,ejtdh, egrdh(which products are the capital gains tax liability from the sale of structures, the dwelling units they contain and marketable services), the total tax liability in h is obtained. The variables Vi and V; reflect the magnitude of this liability and whether it is, respectively, positive or negative. Because of the structure of the linear programming algorithm only one of these two variables can have a positive value in any period; the other must take a value of zero. After tax cash flow in any period h can be defined as eijktdh
C’ijktdh
Wijktdh.
u;-u,-v;+v, This definition treats any tax shelter created by the project, i.e. V,, as a benefit to be applied against other non-project income in the year of occurrence. To maximize discounted cash flow over all time periods, the expression
c r,(U,+ -u,-v;+v,)
(13)
h
* The parameters Wijktdh, “jt&, ejjkfh and e,th may not be needed in Certain inStanCeS. Where all dwelling units have the same per square foot capital costs and yield the same net per square foot rents it is not necessary to distinguish between the two depreciation parameters or between the two tax parameters.
Linear Programming
and Comprehensive
Planning
231
where r, is the present value of a dollar in period h must be used as the objective function of the linear programming model of the site development problem. As a model of the site development problem, expressions (l)-(13) provide a solution which is a construction schedule for residential and non-residential facilities. The unique characteristics of this schedule are that it maximizes discounted cash flow and is consistent with social, physical economic and other design restrictions. Different solutions to the site development problem can easily be obtained by modifying this model. For example to determine the maximum holding capacity of the site subject to the constraints (l)-(12) it is only necessary to replace (13) with
and add the restrictions h=l...p
where Qh is a minimum desired cash flow in period h. Given a different objective function and a specification of control variables which acknowledged site characteristics it would also be possible to determine an “efficient land use” pattern. Through linear programming it is possible to analyze a planning problem from a variety of perspectives and thereby obtain a better understanding of that problem. DATA
REQUIREMENTS
AND
EXTENSIONS
The data required to use linear programming in site development analysis do not in general appear to be difficult to obtain. Except for those in relations (11) and (12) most of the parameters needed are either design and amenity standards or physical standards for structures. The former can be based on accepted practice, tradition or the results of analysis while the latter can be obtained from the drawing boards or the models of prototypal forms. The parameters in relations (11) and (12) may require some extensive preliminary cost and financial analysis, which may initially appear to obviate the gain obtained by adding these sets of relations to the others. However multi-family structures are rarely developed without planning analyses of their financial aspects. Net revenues and capital costs for structures and individual dwelling units within any structure are often estimated in the project planning stage. Once capital costs have been estimated, amoritization, interest and depreciation charges can be calculated. Similarly once sales price at any point in time has been estimated, potential capital gains tax can be determined. Thus (I 1) and (12) require estimates that are often made in other types of design analyses and calculations which are based on these estimates. With appropriate computer codes it is possible not only to have these calculations performed automatically but also to have relations (11) and (12) formulated as inputs to a computerized linear programming algorithm. As formulated (11) and (12) provide great flexibility in analysis. The consequences of interest rate subsidies, construction cost subsidies, rent subsidies, rent distribution schemes and other policies can readily be analyzed with them. They afford analytical advantages beyond their very obvious function of permitting discounted cash flow analysis. Numerous extensions of the discounted cash flow model of the site development problem are possible. Sociological criteria reflecting the interaction between man and his environment can be added almost without limit. The only restriction is that all criteria
STEPHEN S. SKJEI
238
must be formulated on an interval or ratio scale and must be expressed as linear functions only hints at of the control variables xijkrd, yjtd, zgtd and z,,,. The preceding formulation the potential of linear programming to integrate social considerations into traditional planning analysis. Another extension might involve improving the relation used to express the costs of non-residential facilities. The cost curve for these facilities is normally thought to demonstrate economies of scale in a step fashion. Through mixed integer programming or the imposition of upper bounds on variables this characteristic may be more fully incorporated into the model. Similarly environmental considerations may be more explicitly represented. The preceding formulation assumed the site was a homogeneous plain, primarily for purposes of notational simplicity and brevity. By relaxing this assumption and differentiating among areas of the site, environmental factors could have been introduced quite readily. In addition differentiating among areas of the site would have provided a better basis for assessing the costs of public services and amenities and would have allowed the introduction of relations which modelled transportation requirements within the site. In turn this would have allowed inclusion of relations reflecting standards for propinquity between residences, public services and employment nodes. The location of manufacturing facilities and other employment nodes could thus have been analyzed and propinquity standards could have been incorporated to restrict the location of services and amenities such as schools to specified distances from residences. It is also possible through parametric and sensitivity analysis to analyze the effect of The consequences of miscalculation in errors in the values assigned to coefficients. estimating financial parameters can be established and the effect of variations in the design constraints, standards and criteria specified prior to analysis can be evaluated. Consequently the implications of subjectivity and intuition for discounted cash flow can be determined. Finally the model presented here can be easily extended to analyze many other planning problems. Its application to redevelopment and renewal problems should be evident. With the inclusion of site characteristics it can be used to analyze zoning patterns from a Applications to smaller scale planning problems clearly defined normative standpoint. such as the provision of housing for the aged or to larger scale problems such as regional land use utilization should also be possible without too much difficulty.* EXAMPLE
AND
CONCLUSION
To illustrate an extension, which is however not as complex as the preceding formulation, consider the following problem. Given (1) a site of 400 acres, (2) a set of standards (modifications of those promulgated by the American Public Health Association) [9] for non-residential uses, (3) a set of dwelling unit types and structural forms and (4) a design population of 14,500, what is the maximum amount of land which can be allocated to A linear programming analysis provided the following single family detached structures. results : 1. 2. 3. 4. 5.
154 164 158 420 139
acres could be devoted to single family detached residential structures; one bedroom single family detached residences were required; three bedroom single family detached residences were required; four bedroom single family detached residences were required; three story multi-family structures were required;
* Recent applications of linear programming
are [6-S].
239
Linear Programming and Comprehensive Planning 6. 7. 8. 9. 10: 11.
60 ten story multi-family structures were required; 15 acres of land were needed for commercial activities; 62 acres of land were needed for school facilities; 39 acres of land were needed for open space; 34 acres of land were needed for parking facilities; in addition a specification of the type and number of dwelling each floor of each multi-family structure was obtained.
units
needed
on
These figures are merely illustrative; they suggest the type of information that can be obtained through the use of linear programming. With sensitivity and parametric analysis the consequences of providing, say, more variety in residential structures could have been ascertained. Given the objectives and constraints used, reducing the required design population resulted in less structural variety in the final solution. The analysis of the site development problem suggests that linear programming can facilitate the task of comprehensive planning. It is a technique of analysis which should It is rigorous yet flexible and may be accessible to many if not all planning agencies. provide significant quantities of information about the nature of a given planning problem. It can be used to give operational significance to the terms “balanced community”, “efficient land use pattern” and “integrated whole” and to determine the implications of the set of design parameters by which notions of efficiency, wholeness and balance are to be judged. A linear programming formulation will indicate clearly how the operational meaning of many planning terms depends on a set of design criteria and amenity standards. Linear programming is nevertheless only a planning aid. While it can be used to reduce uncertainty about a given planning problem, creativity will still remain an important part of the process by which solutions are developed. The combination of prototypal forms and linear programming suggested here only provides a macro-scale solution to the site development problem for example. Many micro-scale problems remain and solutions for them require ingenuity and creativity. With other planning problems the same may be true. Linear programming may suggest general characteristics of a solution but will not provide a detailed solution. Moreover in some instance linear programming may contribute only a partial solution to a problem or may not be an appropriate analytical tool. In the analysis of dynamic processes other programming tools, e.g. dynamic programming, may be more useful. The potential of these tools cannot however be explored here. Finally it is important to note that linear programming may be most effectively employed within the context of a planning process. The very nature of such a process tends to insure that this rigorous quantitative tool will not be narrowly applied. REFERENCES The Metropolitan Planning Process: An Exploratory Study, Appendix 11,2, Department of Housing and Urban Development, Urban Planning Research and Demonstration Program, Washington D.C. (1970). 2. See BRITTONH-IS, (Ed.), Urban development models: new tools for planning, J. Am. Inst. Pkznners XxX1, 90-184 (1965). 3. CHRISTOPHER ALEXANDER,Notes on the Synthesis of Form. Harvard University Press, Cambridge (1964). 4. DEAN HAWKES, A History of Models of the Environment in Buildings, Land Use and 1. MAYNARD M. HUFSCHMIDT et al.,
Built Form Studies, Working Paper 34, University of Cambridge (1970).
5. GEORGEHADLEY,Linear Programming. Addison Wesley (1963). 6. DARWIN G. STUART,Urban improvement programming models, Socio-Econ. Plan. Sci. 4, 217-238 (1970).
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7. ROBERT C. MEII.H. Programming of recreational land acquisition. Suck+tion. Ph. Sri. 2, 15-24(1968). 8. KENNETH J. SCI-ILAGER, A land use plan design model, J. ,4/u. fn.rr. Planners 31, 103-110 (1965). 9. American Public Health Association, Committee on the Hygiene of Housing, Planning the Neighborhood, pp. I-50, Chicago (1960).