Planning of location and capacity of industrial plants in the construction industry

Build. Sci. Vol. 7, pp. 209-214. PergamonPress 1972. Printed in Great Britain

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Planning of Location and Capacity of Industrial Plants in the Construction Industry ABRAHAM WARSZAWSKI* Industrial plants are an important production factor in the building sector. A centralized planning of components prefabrication must indicate the production method to be employed, the number, location and capacity of plants, and the allocation of total demand to established plants. The influence of the planning decision on direct and indirect production costs, transportation cost of finished components, and the value of opening a plant to the local community must be considered in the search for an optimal solution.

INTRODUCTION ONE OF the most important factors of modern housing construction is the industrialized plant for prefabrication of building components. The plant may be designed for comprehensive prefabrication of dwelling units--usually within a framework of a closed building system--or for prefabrication of single components which may be used with different building systems. Examples of typical components in the latter class, are prefabricated stairways, partitions, curtain walls, building carpentry or complete installation units. Considering the fact that, a major part of housing projects is sponsored by government, and lhat system building is usually encouraged and aided by government--as the only way to cope with the shortage of labour and growing demand for housing--some sort of countrywide or regional planning of prefabrication seems possible, justified and even essential. The centralized prefabrication planning of a component or a system involves the following decisions: I. Production method to be employed. 2. Number and location of plants. 3. Output capacity in every established plant. 4. Supply pattern between the established plants and the demand concentrations--i.e, allocation of building sites and areas to established plants. * Senior Lecturer, Faculty of Civil Engineering, Senior Research Engineer, Building Research Station, Technion-Israel Institute of Technology. 209

The prefabrication planning must be based on a housing demand forecast which is also essential for planning of other production factors in construction, such as skilled labour, raw materials, etc. The forecasting of demand may be performed with the aid of some directly related socioeconomic indicators--the estimated population growth, formation of new households, disposable income, number of old buildings to be replaced and also with the aid of general indicators of economic growth like the Gross National Product, National Investment, etc. The different techniques for demand forecasting are presented in[7], [11] and others. A comprehensive study of demand for buildings in Denmark, which may be cited as an example of this type of work is presented in[8]. Another prerequisite for rational planning decision is a selection of a typical technological process to be employed in the new prefabrication plants. It is possible that the selection of the method will be influenced by the planned capacity of the plant. Thus, a method with a higher capital investment, but lower labour requirement, may be justified for a certain production level and uneconomical below this level. The analysis of production function--i.e, the labour, materials, and investment required per unit of finished product-in different prefabrication methods must be performed for several anticipated production levels. Any planned plant capacity should be therefore characterized for the purpose of further study by the appropriate production method to be employed. An example of analysis "of production function for different construction methods is given in[9]. Given the quantity (in terms of dwelling units for closed, or component units for open systems) and

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Abraham Warszawski

geographical dispersion of demand in the period of time for which the planning is performed, the criterion for effectivity of solution to the problem discussed here, will be the total cost affected by the planning decision. This cost includes the following: (a) direct production cost in plant. (b) indirect production cost in plant. (c) transportation cost between plant and construction sites. (d) the value of opening a plant to the local community. T H E DIRECT P R O D U C T I O N COST The direct cost includes all expenses which may be directly allocated to a production unit. This cost is usually linearly output-dependent. The main components of the direct cost will be in our case labour and materials. The labour cost will often depend on the plant location especially if the set of feasible sites includes some, near urban centers and others in rural unindustrialized areas. The labour cost might also depend upon the planned plant output. It was already stated, that a higher output might justify a different technology with a less labour consuming production process, but with a higher capital investment. The labour cost may be therefore omitted from analysis only if it is quite obviously unaffected by the plants location and capacity. The cost of raw materials at the plant depends upon their cost at the source and their transportation cost from the source to the plant. The choice of the source, the cost of materials at the source, and their transportation cost to the plant, will most obviously depend upon the choice of the plant location. The total direct cost per unit of output hk~, in the plant located at the site i, and operating at capacity k, may be therefore calculated as follows: hk~ = hi,+

~

figr,

(1)

r=l

with : the direct labour cost per unit of output. r = 1,..., w the different raw materials necessary for production. the quantity of raw material r per fr unit of output. the cost per unit of raw material r gri at plant location i, when brought to it from the most economical source (i.e. from the source from which it can be obtained at the least cost).

INDIRECT P R O D U C T I O N COS]' The indirect cost of production is related to a period of time--a month or a year and not to a unit of output. The main components of the indirect cost are the capitalized investment in plant, the maintenance, the indirect labour and other expenses. (a) I n v e s t m e n t cost The investment cost per period is obtained by capitalizing the investment in structures, equipment, land and working capital over the economic life of plant. For convenience of calculation it is worthwhile to distinguish between the investment in depreciable assets such as buildings and equipment and assets which are not subject to depreciation such as land and working capital. The cost per period of depreciable assets may be calculated from : A ~ = (P-L)

i(1 +i)" +Li (1 +i)"--1

(2)

with: A C investment cost per period P

initial investment

n

economic life of plant

i

cost of capital (desired rate of return)

L

value of investment at the end of its economic life

the cost per period of the undepreciable assets for which by definition P = L will be: A c = Pi

(31)

It was assumed here implicitly, that the major assets will be utilized over the total economic life of the plant, which is usually the case. If an asset has to be replaced earlier, the value n in (2) will refer to its own economic life. The economic life of plant is determined by the rate of physical deterioration of its main assets, the obsolescence of the production method employed, the regional development at the plant location and the stability of demand (this aspect will be discussed later). The theoretical background of (2), (3) and the various ways for evaluation of their parameters are presented in [3]. (b) M a i n t e n a n c e cost The maintenance cost includes all expenses necessary for maintaining all the production assets (structures, equipment, etc.) in working condition. The most dependable source for evaluation of maintenance cost is the records of maintenance expenses incurred in similar plants. In absence of

Planning of Location and Capacity o f Industrial Plants in the Construction Industry

such records it is customary to estimate the maintenance cost of equipment as a fixed fraction of the capitalized investment cost, as recommended in [14] or as a fixed fraction of the initial investment, as recommended in[10]. Thus: A me = k~A ~

(4)

A "e = k~P

(5)

or

Ovi

Q21

with A m ~ t h e equipment maintenance cost, Ac, P - the equipment capitalized cost and initial investment, respectively, and k o k~ characteristic constants for the type of equipment used. The maintenance cost of buildings and other fixed facilities may be assessed from the anticipated maintenance activities such as painting, whitewashing, cleaning, etc. (c) The indirect labour cost This cost includes salaries and wages of plant management, foremen, clerks, storekeepers and other occupations not included directly in the production process. This expense can be evaluated from the planned organization structure of the plant personnel. (d) Other indirect expenses These expenses include electricity, telephone, insurance taxes, small working tools etc., and must be estimated item by item. The indirect production costs do not vary linearly with output, however, they cannot be assumed as totally independent of output. It is evident that plants planned for different output capacities, even if employing the same technology will require larger facilities, more equipment, more working capital, more supervisory staff, etc. Since establishment of any continuous dependence function between indirect costs and output seems to be highly unrealistic, it was assumed here that the costs remain constant between certain "critical" production levels as shown in Fig. 1. The cost ak~ (with k = 1..... v) is the sum of all indirect cost components described in paragraphs (a)--(d) above. The dependence of this cost on location (by adding the index i) is explained by the labour and materials which compose a large part of the cost, and which depend on location, as was already shown in the discussion of the direct costs. Qk are the corresponding output levels, which if exceeded require larger staff and facilities or a different technology. The output for "closed" methods will be measured in dwellings, and for " o p e n " methods in appropriate units of manufactured components.

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Qi

Q• .

I .

.

.

Qv

Fig. 1

T H E T R A N S P O R T A T I O N COST The transportation cost depends upon the distance between the plant location and the building site, the transportation device and the quantities of product to be transferred. The cost was assumed here to be linear with the quantity. This assumption, although computationally convenient is not always economically sound. It is true either when the transportation is subcontracted on a unit of quantity-unit of distance basis (e.g. ton-miles), or when the transportation distances are very long. For analysis of transportation costs in other cases see[13]. T H E VALUE OF OPENING A PLANT The search for the optimal plant sites will be naturally confined to locations suitable for this purpose due to their proximity to demand concentrations and the existing conditions necessary for establishing of industry. It is well known, however, that opening of industrial plants in any place and especially in a small community has a significant influence on the economic development of the area. To encourage economic development of some places, the government or local authorities are ready to grant the entrepreneurs subsidies, loans, exemption from taxes, or other benefits. The economic value of such benefits should naturally be taken into account in the choice of plant locations. THE MATHEMATICAL MODEL In order to arrive at an optimal decision concerning the number, location and capacity of prefabrication plants it is necessary to define and evaluate the following parameters: i= 1,2,..,n the possible sites for plant location. j = 1, 2 . . . . m the demand concentrations i.e. single or groups of construction sites. blj the transportation cost per unit of quantity and, distance between locations i andj.

Abraham Warszawsk i

212

the direct production cost per unit with plant of planned capacity k located at i. the m a x i m u m output per year of a plant with a planned capacity k. the yearly demand for product at./. the indirect cost per year associated with operating a plant of capacity k, at location i. the value per year of opening a plant of capacity k at location i.

that the demand at every location will be supplied, and the constraint ( I I ) that the total output at any plant does not exceed the planned capacity (it also implies that if)'k~ = 0, for any combination of i and k, then Zki j will equal 0 for any /). The constraint (12) assigns to every plant one delinitc capacity. (8) and (12) result in the obvious condition:

Any decision, or solution to the problem will be defined by the variables:

Every solution to (10)-(14) may be defined by a r . n array of variables Y = IlYk,l[- From a given array Y it will be possible to tell if a plant is opened at any location i, and if it is, for which capacity/, it is planned, and thus answer the question of how m a n y plants, at which locations and with what capacities. The total n u m b e r of possible arrays Y, obtained from assigning the values 1 or 0 to the components Yk~ is 2 "v, however, not every array represents a feasible solution to our problem. In order to represent a feasible solution, the c o m p o n e n t s of an array must satisfy constraint (12) and also the condition:

h k;

Qk

Rj a~;

~lki

Yk~

a bivalent variable with a value of 1 if a plant of capacity k is established at i, and otherwise with a value of O. the quantity supplied per year from location i toj.

X~j

For convenience of presentation let us define the total direct cost Ckij as :

Ckij = bij+hki

(6)

the total indirect cost ak~ as :

aki = a~i--a~i

(7)

~ Zkij

:

~'ij

1 .....

i =

n'j

=

I ..... m

(15)

k=l

SOLUTION

TO

THE

MODEL

and a new variable Z~ij:

Zk~j = X~j)'~

(8)

The economic meaning of the variable Zkii is the quantity supplied from i to j from a plant with capacity k. It follows implicitly from this definition and also from (8), that Zki j will equal 0 for all j if there is no plant at i, or if there is a plant, but with a planned capacity different from k. The total decision-dependent cost C, which is now to be minimized may be obtained from:

C = ~ ~ akiYki~- ~ k=l

i=l

k=l

~, ~ i=1

ZkijCkij

(9)

j=l

under the following constraints :

~ k=t

Zki j = Rj

j = 1..... m

(10)

i=l

~'~ Qk)'kl

~Zkij

k = 1..... r ; i =

1..... n

(11)

j=t

• Yki ~ 1

i = 1,..., n

(12)

k=l

.t'kl

=

1 or 0

Zkli >----0

k = 1.... , v ; i = k = 1..... v ; i =

1..... n

(13)

1 ..... n ; j =

1..... rn (14)

The objective function (9) consists of two components: the indirect cost c o m p o n e n t and the direct cost component. The constraint (10) ensures

j=l

Rj <= ~ k=l

~, QkYki

(16)

i=l

The latter follows from (10) and ( 11 ) and specifies that the capacities of all activated plants must be sufficient to supply the total demand. For any feasible array Y it is possible to obtain the optimal transportation system by introducing the values of .Vki into (91-(14) and solution for variables Zkl i by means of linear p r o g r a m m i n g (the transportation method). The resulting total cost C will indicate the effectivity of the particular solution. One way of obtaining an optimal solution to the problem is by enumeration of all feasible solutions to (9)-(141, solving of linear p r o g r a m m i n g problem for Zkij in each of them, and the choice of solution resulting in m i n i m u m cost C. This method will be suitable however, only for small problems due to the large a m o u n t of computations involved. A problem with 9 possible sites and 3 production capacities to be considered (say 500, 750 and 1000 dwelling units per year) will have 204,630 solutions satisfying constraint (10). Even allowing for solutions which do not satisfy (16), and m a y therefore be discarded, the number of remaining alternatives to be evaluated by means of linear p r o g r a m m i n g is too large to be practical.

Planning of Location and Capacity of Industrial Plants in the Construction Industry' A more attractive approach calls for implicit enumeration of all feasible solutions with the aid of "Branch and Bound" technique as suggested in [1], [4] or [12], but even this method proves exceedingly time consuming in case of large, reallife problems. Some heuristic procedures employing the "steepest ascent" concept (of the type presented in [2], [5] or [6]) may be adapted for solution of the problem discussed here. Such procedures are more economical from calculation viewpoint, but promise only suboptimal solutions. The development and testing of an algorithm for solution of the particular problem discussed here is included in a comprehensive study on planning and location of industrial plants in conconstruction industry, which is being performed in the Technion-Israel Institute of Technology.

U T I L I Z A T I O N OF EXISTING PLANTS The discussion involved, until now, only the location and capacity planning of new plants. It has to be remembered, however, that the existing prefabrication plants must be also included in the general distribution pattern. There is no particular difficulty to include the existing plants with their respective technologies in the same manner as was done for new plants. The only difference will involve the investment cost which cannot be recouped even if the existing plant is shut down temporarily or permanently. This type of costs, referred to, usually as "sunk" costs are obviously irrelevant in a comparative analysis

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employed here, and must not be included in the indirect cost component of (9). CHANGES IN DEMAND W I T H TIME The problem becomes more complicated if the anticipated demand does not remain stable over time, but varies either with respect to locations or in magnitude. In this case it might be necessary to close plants with the years, to open new ones, or to change the output rate in others. The total planning period will have to be divided into stages, each stage with a characteristic demand pattern, and the solution to the problem will indicate the optimal location and capacity of plants in every stage. The analysis of this case must consider the cost of closing, opening and changing of capacity of plants in different stages and also possible variations in direct and indirect costs with time. An example of analysis of multistage location problem may be seen in[l 3]. CONCLUSIONS Planning of prefabrication systems in construction industry should be performed on a regional or countrywide basis. The planning decisions will involve the choice of technologies, the number of plants, their location and capacity. The decisions should be based on a quantitative analysis of the information about anticipated demand for product, the possible location sites, the production and the transportation costs. The planning must take into account the existing plants and provide for possible variations in demand pattern over time.

BIBLIOGRAPHY

1. M. A. EFROYMSONand T. L. RAY, A branch-bound algorithm for plant location, Ops Res., 14, No. 3. 2. E. FELDMAN,F. A. LEHRER, and T. L. RAY, Warehouse location under continuous economies of scale, Mgmt Sci., 12. 3. E. L. GRANT and W. G. IRESON, Principles of Engineering Economy Ronald Press, New York, (1970). 4. P. GRAY, Mixed integer programming algorithms for site selection and other fixed charge problems having capacity constraints, Technical Report No. 101, Stanford University, 1967. 5. A. A. KUEHN and M. J. HAMBERGER,A heuristic program for locating warehouses, Mgmt Sci., 11, No. 2. 6. A. S. MANNE, Plant location under economics of scale-decentralization and computation, Mgmt Sci. 11, 7. W.T. MORRIS,The Analysis of Management Decisions, Richard D. Irwin, Illinois, (1964). 8. O. O. PEDERSON,Perspective Planning of the Building Sector, Danish Building Research Institute, Copenhagen, (1971). 9. S. PEER and A. WARSZAWSKI,Economic analysis of housing construction methods, J. ASCE Co2, September 1972 10. R. L. PEURIFOY, Construction Planning Equipment and Methods, McGraw-Hill, New York, (1956). l l. M. H. SPENCERand L. SIEGELMAN,Managerial Economics, Richard D. Irwin, Illinois, (1964).

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Abraham Warszawski

12. K. SPIELBERG,On Solving P/ant Location Problems, Applications ot" Mathematical Programming Techniques, University of London Press, (1970). 13. A. WARSZAWSKI,Optimization of Location on the Building Site, unpublished D.Sc. Thesis, Technion-Israel Institute of Technology, 1971. 14. "Bauger~iteliste", Bauverlag GmbH, Wiesbaden. Berlin, 1960. Les entreprises industrielles sont un facteur de production important dans le secteur du b~timent. Un planning centralis6 de la pr6fabrication de composants doit donner une indication de la m6thode de production ~t utiliser, du nombre, de la location et de la capacit6 des entreprises, et de l'allocation de la demande totale aux entreprises en existence. L'influence de la d6cision du planning sur les frais de production directes et indirectes, les frais de transport des composants finis, et la valeur pour la communaut6 locale de l'ouverture d'une entreprise, doivent &re consid6r6s dans la recherche d'une solution optimale. lndustrielle Anlagen sind ein wichtiger Produktionsfaktor im Baufach. Eine zentralisierte Planung der vorherigen Herstellung von Teilen muss die zur Anwendung kommende Produktionsmethode, die Anzahl, Lage und Kapazit~it der Anlagen und das Kontingent des gesamten Bedarfs von den bestehenden Anlagen angeben. Es muss der Einfluss der Planungsentscheidung auf direkte und indirekte Produktionskosten, Transportkosten fiJr fertiggestellte Teile und der Wert der Er6ffnung der Anlage ffir die Ortsgemeinschaft in der Suche nach einer optimalen L6sung ber~icksichtigt werden.