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The Optimal Design of a RefineryJames McD White aa Stevens Institute of TechnologyPublished online: 27 Apr 2007.

To cite this article: James McD White (1968) The Optimal Design of a Refinery, The Engineering Economist: A Journal Devotedto the Problems of Capital Investment, 13:4, 199-210, DOI: 10.1080/00137916808928783

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The Optimal Design of a Refinery

James McD. WhiteStevens Institute of Technology

The purpose of this paper is to explore the possibility of using the

concept of the production function and the technique of Monte Carlo simu­

lation in the design of optimal production processes. Optimal design in

t:his context means the determination of the production process and the

selection of components so as to maximize profits. It is not concerned

~rith hardware design. It is assumed that the technology of the production

process is well understood and is practiced effectively.

The problem considered is the fairly general one of determining the

best combination of the variable factors of production and the optimum

rate of output which will maximize profits. It is invariably solved in

practice by means of an extensive use of subjective determinations. This

paper is concerned with the possible use of objective determinations. The

first step in the analysis is to determine the effect on rate of output of

varying values of rates of inputs, all in physical terms; this involves

the concept of the production function. Then, by applying prices to physi­

cal quantities, costs and revenues are determined for each combination of

inputs thus making it possible to select that combination(s) for which net

profit is a maximum.

A simple example is given to illustrate this approach. In the example,

a model of the production system is set up as a basis for simulating the

input-output analysis. The physical quantities are then valued, with the

assumption that unit prices will not change over the relevant range of in­

puts and outputs. This assumption of "perfect markets" is often a close

enough approximation to the real world to give valid results. The problem

has been simplified in other respects as well, particularly in regard to

199

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200 The Engineering Economist

the number of variables and the complexity of operations. This has been

done for the purpose of clarity in the presentation. The" computational

procedures have also been chosen with clarity of presentation in view and

are not intended to be representative of the best methods of numerical

analysis.

PROPOSAL FOR AN OIL REFINERY COMPLEX1

The current operation of one of the facilities of an oil company is

as follows: Crude oil is produced in an oil field having a maximum output

of 200,000 bbls. per day. The oil from the wells flows through gathering

lines to a main pipeline and is then pumped to a storage tank at the ship­

ping facility. The maximum throughput is 150,000 bbls. per day, and the

storage tank has a capacity of 300,000 bbls. Tankers with a capacity of

100,000 bbls. arrive on the average of one a day. From a study of past

data of tanker arrivals, it has been determined that the frequency of ar­

rivals can be described by a Poisson density function. World market con­

ditions are now such that the price of refined oil at the oil field would

be $18 more per 1000 bbls. than crude oil loaded on tankers. It is desired

to investigate the feasibility of building a refinery at the oil field to

process excess oil not required for shipping operations.

THE INPUT-OUTPUT ANALYSIS

The first step is to set up a model for the operation of the system

complex so that the input-output relationships can be determined by simu­

lation. It has been decided that the following ground rules will govern

operations: (1) the terminal operation is to take precedence; (2)all incom­

ing tankers are to be loaded as soon as possible, assuming no limitations

on terminal facilities other than the amount of oil that can be made avail­

able in any given day; (3)a tanker is either loaded to full capacity or

not at all; (4)if a tanker cannot be loaded on the day of arrival, it must

be held over and loaded the following day; (5)after the terminal tank

has been filled with the maximum amount of crude oil at the end of the day,

as much as possible of the excess will be pumped to a storage tank at the

refinery.

lAdapted from a paper by S. E. Eubank and J. McD. White, "Use of theMonte Carlo Technique for the Engineering Economist," The EngineeringEaonomist, Volume 3, No.1 (Fall 1957), p.l.

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Volume 13 - Number 4 201

It is, of course, desired to maximize the output of the refinery (as­

suming unlimited facilities for storing refined oil), subject to the above

ground rules. It is believed that the mode of operation of the refinery

that will accomplish this, is to process daily the entire amount of oil in

the storage tank at the beginning of the day, subject to the production

capacity of the refinery. The simulation is facilitated if decision rules

are formulated for determining the values of stated variables, as they

change over an n-day period of simulation.

If a Poisson distribution is assumed, ship arrivals can be determined,

with the use of random numbers, by setting up a correspondence as in the

following table:

Random Number

00 - 36

37 - 73

74 - 91

92 - 97

98 - 99

Ship Arrivals

o1

2

3

4

The block of random numbers was assigned by calculating probabilities

of 0.37, 0.37, 0.18, 0.06, and 0.02 to correspond to arrivals of 0, 1, 2,

3, and 4 tankers, respectively, on any given day. Two-place accuracy was

deemed sufficient in this case. The probability of more than 4 arrivals

has been treated as being negligibly small.

The remaining decision rules are as follows:

Number of ships loaded

If 100(SA + SY) < 150 + YT, then SL = SA + SY.

If 100(SA + SY) > 150 + YT,then SL = largest integer in

The symbols are defined as follows:

SA - number of tankers arriving today

SL - number of tankers loaded today

SY - number of tankers left over from yesterday

YT - yesterday's closing inventory at the terminal

Amount of crude piped to terminal tank

YT + 150100

If 300 - YT + 100SL < 150, then PT

If 300 YT + 100SL > 150, then PT

300 YT + 100SL.

150.

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202 The Engineering Economist

The symbols are defined as follows:

PT - today's pipeline receipts through terminal line

Daily output of refinery

If YR < RCAP, then RA

If YR > RCAP, then 1M

YR.

RCAP.

The symbols are defined as follows:

RA - today's actual run at refinery

RCAP - refinery capacity

YR - yesterday's closing inventory at the refinery

Amount of crude piped to refinery tank

If PCAP < 200 PT and if TCAP - YR + RA> PCAP, then PR PCAP.

If PCAP < 200 - PT and if TCAP - YR + RA< PCAP,

then PR = TCAP - YR + RA.

If PCAP > 200 - PT and if 200 - PT > TCAP - YR + RA,

then PR = TCAP - YR + RA.

If PCAP > 200 - PT and if 200 - PT < TCAP - YR + RA,

then PR = 200 - PT.

The symbols are defined as follows:

PR - today's pipeline receipts through refinery line

PCAP - pipeline capacity

TCAP - tank capacity at refinery

In the above decision rules, all quantities of oil are expressed in

1000's of barrels. Simple computations, such as the balance in a tank

at the end of a day, have been omitted.

The results of a simulation are shown in Tables 1, 2, and 3. The

only portions of the tables which have been shown are those pertinent to

the remainder of the analysis. An examination of the tables reveals di­

minishing returns. In each of the tables, a point is reached beyond which

further increases in the size of the tank and/or the pipeline will not re­

sult in additional output. The next step is to apply prices to the inputs

and outputs and determine the optimal combination of the inputs.

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*T

AB

LE

1

Refi

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Cap

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80

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801

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40

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68

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*T

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* All

data

are

for

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1s.

per

day

20

4

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Volume 13 - Number 4

DETERMINING THE OPTIMAL COMBINATION OF INPUTS

20S

Additional estimates of cost and performance characteristics are as

follows: It is estimated that the oil field will be depleted in about

ten years if the rate of output is of the order of 200,000 bbls. per day

for a 7-day week. 2 The installed cost of a pipeline with a capacity of

80,000 bbls. per day is $200,000, with each increment of added capacity

of 10,000 bbls. per day costing $20,000. The cost of a 100,000 bbl. ca­

pacity storage tank is $60,000, and each additional 20,000 bbls. of ca­

pacity would cost $10,000. The cost of an 80,000 bbls.-per-day refinery

is $1,600,000, and each increment of added capacity of 10,000 bbls. per

day would cost $150,000. It is believed that the labor force and other

operating costs would be about constant over the relevant possible range

of output. All prices are assumed to be constant. Finally, a 50-percent

income tax rate and straight-line depreciation with zero residual value

will be used.

The problem is to determine the best size of each of the three refin­

ery components. The criterion of optimality is a combination of inputs

such that an additional investment in anyone factor or combination of

factors will increase the present value of the additional revenue, when

discounted at an appropriate rate of interest, by less than the amount of

the additional investment. This is in accordance with straightforward

marginal analysis in which profit maximization requires that capital in­

vestment be carried to the point at which the marginal value of the prod­

uct is equal to the cost of capital. In this example, an annual rate of

interest of ten percent has been arbitrarily selected.

The following computational procedure was employed: Proceeding row

by row in each table and starting with the figure in the first column, we

continue moving across a row until a point is reached beyond which further

increments of size are uneconomical. This can be determined on a step-by­

step basis. An incremental change in an input--an increase in size of

some component--is desirable if the added increment of present value, 6PV,

is positive. The change in present value of the cash flow results from

the increase in revenue resulting from the increased output, after income

2 I n the subsequent analysis, 10 years has been taken to be the esti­mated life of the project, regardless of the rate of output of the refin­ery.

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206 The Engineering Economist

and,

tax but before depreciation. A discounting model using continuous flows

and continuous discounting was used for convenience.

~PV = ~PV of cash flow - ~I.

The incremental profit--before income tax--resulting from an incremental

increase in output is 6~ x 365 x .018 - ~I/IO, recalling that straight-line

depreciation, 10-year life, and zero residual have been used. Assuming a

50-percent income-tax rate on additional income, the incremental profit,

after income tax, is 1/2(~~ x 365 x .018 - ~I/IO). Then:

~cash flow = 1/2(~~ x 365 x .018 - ~I/IO) + ~I/IO

1/2(~~ x 365 x .018 + ~I/IO).

Therefore,10 It

~PV of cash flow = f 1/2(~~ x 365 x .018 + ~I/IO)e' dt,o

I 0 1~PV = f 1/2(~~ x 365 x .018 + ~I/IO)e' t d t - ~I .

o

The optimal values in each row have been circled in the tables. These

values represent optimal tank sizes for given refinery and pipeline capaci­

ties. The symbols in the above equation are defined as follows:

~ - daily output of refinery, in bbls.

I - investment in refinery components.

As long as we are varying only tank capacity, all terms in the pres­

ent-value equation are constant except ~~. This suggests that a relative­

ly simple procedure is to calculate a "break-even value" of output. Once

this is done, the values in the table may be picked out by inspection. In

this particular problem, the following computation shows that an increase

in output of 506 bbls. per day warrants an incremental increase in tank

size. Letting ~~ = 500 and ~I = 10,000 in the equation, we get

~PV = 1/2 flo [(500 x 365 x .018) + 1,0001e-·ltdt - 10,000o

2142.5 floe-·ltdt - 10,000o

= $3,570

Since a casual inspection of the data reveals that this is the small­

est size of increased output, we move from left to right in each row until

we arrive at that tank size such that no further increase in output is

obtainable from the use of a larger tank. These are the values which have

been circled in the tables.

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Volume 13 - Number 4 207

The next step is to follow the same procedure and determine, in each

table, the optimum pipeline size. The above equation is again employed

for this purpose, but this time we check only the circled figures. To

illustrate, let us check the circled entries in rows 1 and 2 of Table 2.

An additional output, ~~, of 5500 bb1s. per day can be obtained by using

a larger tank and a larger pipeline. The increased investment in these

facilities, ~I, is $30,000. Therefore,

~PV = 1/2 flo [(5500 x 365 x .018)+ 3000le-· l tdt - 30,000o

~ $125,000 - $30,000

- $95,000

Since this quantity is greater than zero, the larger output is desir­

able. A square has been placed around the optimal output in each table.

The entries marked with squares denote the optimal tank and pipeline size

for a given refinery capacity. The last step calls for making the same

comparison among the squared boxes in each table, in order to select the

optimal-size refinery. The following result for the illustrative case is

a refinery capacity of 90,000 bbls., a pipeline capacity of 110,000 bbls.,

and a tank capacity of 260,000 bbls.

CONCLUSIONS

The output data in Tables 1, 2, and 3 clearly show diminishing margin­

al productivity. For any given capacity of the pipeline and refinery,

there is an absolute limit to the increase in output that can be obtained

by increasing the size of the storage tank. In this example, in which the

amount of oil required for the terminal operation varies considerably from

day to day, it is clear that the storage tank at the refinery serves to

maintain a larger and more uniform output at the refinery than would other­

wise be possible. If the size of this tank is increased to the point where

it is no longer a limiting factor, the limitations of the pipeline and re­

finery capacities would still limit the output of refined oil. More gen­

erally, output will be limited as long as there is a limitation on the

size of anyone input. It is true that there are other constraints on out­

put in this particular example, but the above statement would still be true

in the absence of these constraints.

Apart from possible errors due to imperfections in the model itself,

the final determination of the system components is only an approximation

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208 The Engineering Economist

because of the incremental changes in sizes of the components. The accu­

racy would be increased if smaller increments were used, if this is possi­

ble, although it may be that it is not desirable. The choice of the ap­

propriate size of the increments for a component depends upon the sensi­

tivity of that component, although the choice may be limited to standard

sizes in some cases. Conceptually, at least, it is possible to work with

continuous variables and employ methods of analysis suggested by economic

theory. It may well turn out that such methods would be more difficult,

particularly if a rather large number of variables was involved, because

of the complicated functional relationships that exist. There is another

source of error in the analysis due to the simulation process and sampling

methods, and this is discussed in the appendix.

There is no particular significance in the fact that the only vari­

able inputs in this illustration were fixed investment components. In

most situations, one also expects other factors to be variable, particu­

larly labor. The treatment of such factors would be similar to the one

in our illustration. There is, however, this consideration: - In the case

where labor is a variable factor, it is conceivable that labor could be a

complementary factor; i.e., that particular kinds ,and quantities of labor

would be associated with particular combinations of physical facilities.

In this case, the analysis would differ little from that of the illustra­

tive problem. If labor could be substituted for capital, however, the

analysis would be more complicated. Further consideration of this problem

is outside the scope of this paper.

It is interesting to observe that the operating efficiency--the ratio

of average output to capacity--for each component, as determined by the

analysis, is rather low. This criterion played no part in the analysis

and is no indication of the merit of the design. In fact, an increase in

operating efficiency, which could be achieved by using components of lesser

capacity, would result in reduced pr~fitability. The results of the study

also illustrate the economic principle of the conservation of scarce re­

sources. Thus it is observed that the operating efficiency of the refin­

ery, which is the most costly component of the system and is therefore,

in some sense, the most scarce resource, is much higher than that of the

tank, which is the least costly component. It pays to be wasteful of tank­

age space in order to achieve a higher operating efficiency for the refin­

ery.

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Volume 13 - Number 4 209

One final point is that the analysis, to the extent to which it has

been carried, does not answer the question: Should the refinery be built?

In this particular illustration, there is insufficient information avail­

able to answer the question, although a break-even point could be deter­

mined. A knowledge of all costs and investments would be necessary for

this purpose. Insofar as this particular question is concerned, what has

been determined is that the final selection is the best of all possible

alternatives, i.e., it would yield the maximum profit (or minimum loss) if

the refinery were built.

APPENDIX

The method of analysis followed in this paper was chosen more for

expositional purposes than for any other reason. This resulted in some

computational difficulties, particularly in the case of the data shown in

the input/output tables. The appropriate range of variables to work with

would not generally be known in advance. In the illustrative problem,

some of the data was eliminated so that only pertinent results are shown

in the tables.

The problem had to be rerun several times when it was discovered that

critical portions of the tables had been omitted. Thus, it may be seen

that, for a low refinery capacity, an upper limit to the output was reached

in a relatively few steps as the sizes of the pipeline and storage tank

were increased. But, as the size of the refinery increased, it became nec­

essary to increase the number of incremental changes for the other factors.

This suggests that the number of iterations should not be fixed in advance

but should be terminated at some point, which could be the point at which

the increase in output became zero, that is, the point at which row and

column vectors started to duplicate preceding values.

Actually, the input/output tables are not necessary in order to solve

the problem, although some of the data might be of interest in the neigh­

borhood of the optimal sizes. More efficient computer programs could eas­

ily be devised. One alternative that could have been used in the illus­

trative example is the following: Considering each table separately and

proceeding row by row, we could calculate for the first row the amount

of tank capacity which maximizes the present value of net worth and stop

at that value. The calculation for the following row would start at the

same value of tank capacity that was optimal for the precedin9 row and

the optimal tank size computed for that row. Then a comparison could be

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210 The Engineering Economist

made to determine if the larger pipeline size increases the present value

of net worth. If it does not, we would proceed to the next refinery size

and repeat the procedure. If it does, we would continue with the third

row, and so on. The last step consists of comparing the optimal values in

each table. It is an easy extension to treat any number of factors with

this method. The necessary conditions for this procedure to be valid are

rather obvious and may not necessarily hold in all cases.

The input/output data were obtained by running the problem on a com­

puter. It may have been noticed that the low-order digit in the data was

either a zero or a five. This is explained by the fact that only a twenty­

day simulation 'period was used and that each item of data was the average

of the output over the period; this, obviously, is much too small a sample

for a proper degree of accuracy. The random numbers were stored in memory,

and the same sequence was used for each period. Regardless of the modifi­

cations one would want to make in these procedures in practice, the use of

the same set of random numbers for each simulation run has some merit.

Although this may result in a bias in the value of the outputs, there

should be little or no bias in the differences, less 50 in fact than would

be the case if a different set of random numbers was used for each simu­

lation run.

REFERENCES

[1] S. B. Eubank and James McD. White, "Use of the Monte Carlo Tech­nique for the Engineering Economist," The Engineering Economist, Volume 3,No. 2 (Fall 1957).

[2] M. Frankel, "The Production Function: Allocation and Growth,"American Economic Review, Volume LII (December, 1962).

[3] M. Kurz and A. S. Manne, "Engineering Estimates of Capital-LaborSubstitution in Metal Machining," American Economic Review, Volume LIII(September, 1963).

[4] R. Meyer, Jr., and E. Kuh, The Investment Decision--An EmpiricalStudy (Cambridge, Mass.: Harvard University Press, 1957).

[5] T. H. Naylor, Computer Simulation Techniques (New York: JohnWiley and Sons, 1966).

[6) E. E. Nemmers, Managerial Economics (New York: John Wiley andSons, 1967).

[7] D. Orr, "Costs and Outputs: An Appraisal of Dynamic Aspects,"The Journal of Business (January, 1964).

[8] H. L. Timms, The Production Function in Business (Homewood, Ill.:Richard D. Irwin, Inc., 1966 rev.).

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