E. GENEROSO Jr. LAUREN B. HITCHCOCK
Optimizing Two Cases n the field of chemical process economics, past opti-
I mization studies have dealt extensively with the opti-
mal design of new process equipment (77) or a new plant (3, 9, 70). Recently the optimization of a plant expansion was investigated (7). This problem is analogous to, but broader in scope than, the optimal design of a new plant. Two rypes of plant expansion were optimized: (1) a single-step expansion and (2) expansion in time for a growing market. This paper summarizes that investigation. Optimal Plant Dorian
Optimization, in general, quires evaluation of n alternatives. If, as frequently occurs, n is large, hand calculation becomes impractically laborious and timeconsuming. In the past, a relatively s m a l l number of alternatives were selected empirically on the basis of judgment and compared in the hope that the practical pwibilities had been bracketed. The optimum was choscn, again by judgment, with perhaps some simple graphical aids. But experience too often revealed later that better solutions existed. The development of the computer made available a tool which could greatly accelerate repetitive calcula-
12
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
tions, and made pwible a quantitative determination of the optimum condition as rigorous as permitted by the model of the phenomenon or process. Optimal design of a new plant has been no exception. Cochcan (3) appears to have been the first, as recently as 1960, to adapt computer techniques to the sdection of optimum conditions for a chemical plant. Fan and Wang (5) investigated the discrete maximum principle as an optimization technique. Mitten and Nemhauser (70) found dynamic prcgramming to be practical for optimizing plant design. Coleman and York (4) found the optimum initial design capacity for a n m plant where the product has a growing market. Each of these valuable contributions has advanced the capability for optimal design of a new plant. There is another situation of equal importance, and that is the optimal design for the expansion of an existing planta problem usually more complex. The seemingly obvious decisions--for example, to “add another line like the present one,” or to “substitute larger equipment”overlook many intermediate or combination altematives. As will be shown, there are normally many more alternatives to be evaluated in considering the optimization of the expansion of a multistage process. Unique applications of existing mathematical techniques are required if a reliable optimal solution is to be found expeditiously. These are set forth here; that they appear to be valid has been demonstrated by application to the actual completed expansion of an existing large chemical plant. It was phssible to compare venture worth and optimal routes as determined by the techniques of this paper with venture worth and routes actually experienced. Use of these techniques in planning the expansion would have been advantageous in chwsing the optimal route.
r W h a t is t h e best way t o arrange f o r e x t r a plant capacity when t h e demand
is either immediate
or spread over several years? aptimization procedures consider many alternatives
VOL 60
NO. 1
JANUARY 1968
13
Fundamental t o t h e plant expansion problem is t h e question : W h a t is a good measure of t h e attractiveness of t h e proposed expansion 7’
Plaht expansion problem. This problem may be investigated in two cases. Case I: “Given the size of a proposed expansion, what is.the best way to add the required capacity?” Case II: “Given the growing market demand forecast for a pr6duct, what is the optimum way to add the required capacity during the forecast period?” Fundamental to both is the question: .“What is a good measure of the attractiveness of the proposed expansion in each case cited above?” Take, for instance, the case in which management proposes to double existing plant capacity. Management may decide to add an entirely new plant equaling the capacity of the original. Or it may scrap the existing plant and build a new one of twice the capacity. Or, so far as possible in existing equipment, it may attempt to double present throughput. But is it making the optimum decision? And is the proposed expansion attractive? It is obvious that without the aid of a computer, engineers confronted with this problem could investigate only a few design alternatives. A more conclusive determination is essential. Likewise, take the second case in which the growing market forecast for a product is given, say, for the next four years. Equipment cost for a single-step expansion now to satisfy the demand for the next four years would be less than the equipment cost for a two- or three-step expansion in time. But this would be at the cost of idle money tied up in the overcapacity. Here again there is an optimum condition. It is believed that a computerized solution to this problem of optimal plant expansion can be found for both cases cited above. The solutions represent unique applications of existing mathematical techniques for optimization. Let u s now discuss and demonstrate the solutions to Cases I and 11. The generation of the solutions consists of (1) defining possible decisions, (2) finding the best measure of optimality, (3) applying a practical optimization method, and (4) writing a computer program that is characterized by the chosen objective, possible decisions, and appropriate optimization, tool. Typical Care I Problem
Let us take as an example the actual plant for which the simplified flow sheet is shown in Figure 1. Manage14
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
ment proposes to expand present capacity in one step from 25 to 45 million pounds per year. , What is the best way to add the required capacity? How attractive is the p r o p d expansion? The plant uses a continuous process. Raw materials are mixed with a chemical in two stirred-tankreactors in series, where an isothermal exothermic reaction &urs. The reaction product from the reactor flows countercurrently to a solvent through a series of colupns. A steam-heated evaporator vaporizes the treated product from the columns. The vapor. mixes with a preheated gas also introduced in the evaporator. The vapor and gas mixture from the evaporator are ,preheated and passed through a fixed bed of catalyst, where a second isothermal reaction occurs. The unreacted gas from the reactor is recycled back to the evaporator,’ while the reaction product proceeds to a stripping column for purification. A distillation column further purifies it to the final product. The plant has been in continuous opvation for a number of years. It has a continuous.processand a chemical product. Its estimated l i e was 10 years. Income tax rate has been 52% per year. Rate of interest on capital has been 10% per year. .The plant has been operating for an average of 8000 hours per year at 80% of rated capacity in accordance with plant policy. Plant policy for the minimum acceptable rate of, return on investment for expansion projects has been 20% per year. For the proposed expansion, the pospible decisions at each process unit in the plant are: (1) replace the existing unit with a new one of larger capacity; (2). use the existing unit at a higher throughput, if possible; and (3) add a new unit to the existing unit. The problem is to find the best combination of decisions for all the
AUTHORS E. Gen&roso, Jr., is with &so Standard Eastern, Inc., Manila, Philippines. Lauren B. Hiichcock is Professw of Engineering at the State Uniunsity of Ntw Ywk at Buffalo. The authors gratefully acknowledge the innoluablc counsel of the faculty committee which supnuiscd the dissntation on which this article is based, including Dr. H. T . Cullinon, Jr., and Dr. W . H . Thomas. Dr. W . A . Greancy assisted as refnec on the dissntation and in review of this papa.
mrunln
E‘
-
7
*
process units. A measure of optimality (more commonly known as the “objective function”) and an optimization tool are needed. Optimum solution. To arrive at the optimum solution, we can either (1) maximize profit from the investments for the expansion (or minimize cost, if gross receipts are fixed), or (2) maximize rate of return from the investments, or (3) maximize venture profit, defined (8)as the incremental return over the minimum acceptable return:
v = p - -m(I ’
+ 1,)
But profit and rate of return on investments do not take into account the variability of the required investment for a given expansion with the alternative chosen. For this reason, venture profit, the capacity of a venture to attract new capital, is chosen as the appropriate objective function. Of the known optimization methods, dynamic programming, a multistage optimization tool, would be practical for considering the possible decisions for expansion at all the process units in the existing plant. A single-step expansion problem is set up in multistages, as follows: (1) divide the existing plant flowsheet into “major” process steps, and consider each step as a stage; (2) classify each stage according to the noapihle deV V L
00
NO. l
JANUARY 1968
1.5
Figure 2. The p l d in mdtistagcr
.
'
cisions for the proposed expansion; and (3) draw the Case I problem in multistages. A process step is considered "major" when: (1) the installed cost of the existing equipment in that step contributes significantly to the total plant coat; and/or (2) there is a signiflcaht technological change.in that step. Each "major" process step may consist of one or more units of process equipment. The decision as to the significance of the cost or technology of a process step depends on the accuracy that can be built into the solution and on the availability of data. For instance, the plant flow sheet in Figure 1 can be set up in multistages, as shown in Figure 2. Table I compares the symbols of Figure 2 to those of Figure 1. Table I1 defines the "major" process steps, their comsponding equipment numbers in Figure 1, and their assigned stage numbers. The process steps of stages 1 to 4 were considered "major" on the cost basis, thoae of stages 5 and 6, on the technological basis. . As mentioned earlier, the third possible aecision for expansion is' "add a new unit to the existing one." The stages are n&t dassified according to this third poasible decision. There are two possible alternatives for clasSifying a stage according to this decision. One alternative is to combine the existing and the new-either in series or in parallel-in one stage, and let this new combined stage be called n'. The other alternative exists only when the new unit may be added in series. I n this case the new unit may be considered as a separate stage. The existing unit is then called stage n',. Stage n' .1 is the new unit added to it in series. For the multistage setup of Figure 2, only parallel, not series, additions could be considered in the third alternative decision for expansion, because of the limited availability of data. Hence, all the stages in Figure 2 are classified as stage n # n'. Finally a Case I problem is drawn in multistages, as illustrated by Figure 3 for the problem of expanding the plant in Figure 1. In the decision variable e,,,", j' = 1, 2, 3 designates one of the three possible decisions for expansion cited previously and j' = 1, 2 . . . ..k designates the operating and/or design variable for the nth stage corresponding to decision j. The venture profit at each stage n is expressed as follows:,
+
.
I
I
I
I
I
I
Figure 3. Tha Cast I c x q b problem in multistagts
where V,,(x"-',
e,,.")
= venture profit at stage n as a
function of the input state and decision variables R.(x"-', e,,,") = increase in annual gross return at stage n as a function of the input state and decision variables Is@,,,") = the cost of new units for stage n as a function of the decision variable e,,,") = working capital at stage n as a function of the input state and decision variables So.(x"-l, e,,.")= salvage value of new units at stage n as a function of the input state and decision variables
-
and where R.(x"-', e,,") = A& AOC.(x*-l, 0") IWs(x"-l, 0,,,") = 1/12 [2 A& AOC&"-',
+
e,,,")1
AS,, = increase in annual sal- at stage n AOC.(X"-~, e,,.") = increase in annual operating cost at stage n as a function of the input state and decision variables For example, for each stage in Figure 3, Equation 1 above may be used, wherein:
forn = 1,2, . . ., 5: A& = 0 n = 6: A& = new total sales
-
existing total sales n = 2, 3, ...6: AOC.(X~-~, e,.") = OCnmn X (x"-',
e,,.") - o.c,
e,,P) = (new - existing) raw materials cost plus [OCmm'(xnmo,
n = 1: AOC.(X*-~,
e,,,') - OC,'] where OCmW"(x*-',
e,,.")
= new annual operating a t
at stage n as a function of the input state and decision variables 16
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Optimization of a Case I plant expansion problem.. . Maximize venture profit, t h e capacity of a venture t o a t t r a c t new capital
OC-" = existing annual operating cost at stage n
The maximum venture profit for n stages, according to the principle of optimality of dynamic programming (z), is m x
fN-I+l(~n-:
XN)
= e,*
I
TABLE I. COMPARISON OF FLOW SHEET AND MULTISTAGE SYMBOLS
Symbol in Fig. 2
I
SymboI in Fig. I
+
[v,,(e,,.*lxn-l)
(2) where n = 1, 2 , . . ., N and f o ( x N ) = 0, and where, when n = n' 1 , j = 1, 2; fN--I(x", X")l
+
=
, v.(e,,+-l)
x"-l
=
o
1, j = 1,2,3,andwhenn = n'
andwhenn # n ' +
+ 1,
j = 3: X"
=
T,,(e,,+=-l)
Note that x N is also a parameter in Equation 2, because in Case I this is fixed. For instance, in Figure 3, x6 is the new throughput to which the plant in Figure 1 is to be expanded. The optimal policy for the expansion at each stage is the e,." that maximizes the venture profit in Equation 2, or 8,,."*, n = 1, 2, , , N . This policy tells at each stage what j decision number is optimum and what optimal j ' (operating and/or design condition) corresponds to this j decision. For example, the solution to the expansion problem in Figure 3 shows decision number 3 (adding a second parallel line to the existing) is the optimum at stages 1, 3, 4, 5, and 6. At stage 2, the optimal decision is to use only the existing equipment, or decision number 2. The maximum venture profit for the multistage setup of a single-step expansion problem is then fN(x0*, x"), where xo* is the feed rate corresponding to the maximum f N ( x o , x N ) . For the multistage setup in Figure 3, f@*, x6) is $362,000. O t h r costs. The maximum venture profit for the multistage setup does not necessarily hold true for the over-all plant expansion. Usually there are costs involved in the plant expansion aside from the costs directly related to the multistage setup. These costs, which generally cannot be optimized because of lack of data for applying an optimization method, are:
"*.
qo?'
procrrr
S@P
Reaction Extraction Evaporation Conversion Stripping Distillation
No. in Fig. 7
7-9 10-16 17-19 20-22
VOL 60
cost of equipment 8139,000 73,000 168,000 282,000 38,000 36,000
NO. I
JANUARY i 9 6 e
17
(1) capital for new installations not directly or solely connected in a continuous fashion to the existing p m (examples are new pumps for utilities, new boilers, new furnaces, new units for batch processes indirectly connected with the process, new buildings, new warehouses, and new storage tanks) ; (2) capital for new installations connected in a continuous fashion to the existing process, but which may not be an integral part of, or contributing directly to, any of the “major” process steps (examples are transfer pumps between “major” process steps, instrument panels, equipment supports, and tanks); (3) other production costs, aside from raw materials and operating costs at the process units, that may also increase with throughput, such as maintenance, labor, property taxes and insurance, power, packaging, management, and supervision. We shall call new capital for items (1) and (2) above “related fixed capital,” and the costs in (3) “related production costs.” As an example for the single-step expansion of the plant in Figure 1, the “related fixed capital” amounts to $800,000. About 40% of this capital is to be invested in new buildings, instrument panels, equipment supports, and foundations; about 15% in new pumps (which include those new transfer pumps between the “major” process steps in Figure 1); about 15% in tanks; the remaining 30% in miscellaneous equipment (such as a new mixer added to that in Figure 1). Prior to the expansion, the existing “related production costs” of the plant in Figure 1were $554,000, while those forecast for the expansion were $766,000. Measure of attractiveness. Now let In the ‘‘related fixed capital,” and let RF’C, and RPC- be the existing and new “related production costs,” respectively. The maximum over-all venture profit for a proposed single-step expansion would be
-
D Figure 4. Cam I cmnpuk? block diagram
!
44 If the above fN*(xo*, x”) has a positive value, the proposed expansion is attractive. By the definition of venture profit, this means that the actual rate of retllrn from the proposed investments is greater than the minimum acceptable rate of return. The p r o p a d expansion of the plant in Figure 1 has fN*(xp’, x”) equal to 8260,000; hence, it is attractive. Optimal polieics. The optimum solution shows what are the optimal operating policies after the expansion, and what is the optimal capital requirement. The optimal operating policies are the j ‘ decisions corresponding to the optimal policies at each stage (B,p*, n = 1, 2,. . ., N), and the optimal flow rates, 18
INDUSTRIAL AND ENGINEERING CHEMISTRY
36 d
t
d Z 4
bl 12 0
1
2 YEAR
3
Figure 5. Can II-routa I
4
TABLE 111
Stage -
Cost
1 reactor None 1 compressor, 1 heat exchanger, 1 evaporator 2 reactors, 1 preheater, 2 coolers, 1 condenser 1 column, 1 reboiler, 1 condenser 1 column, 1 reboiler, 1 condenser Fixed capital “Related fixed capital’’ Working capital Total capital
1 2 3
4 5 6
I
New Equipment
25,000 0
$
121,000
68,000 33,000
33,000 380,000 800,000
I
I
PRODUCT D E M A N D FORECAST FOR CASE I I ILLUSTRATION
TABLE IV.
Period, yr. Demand forecast, mmlb./yr.
1 12
2 20
I
4 36
3 28
months. Fictitious data were used for the trial runs, and computer results were checked by hand calculations. The authors recommend that during a debugging stage, a similar study should base computer runs on simple problems which can readily be checked by hand calculations. The time consumed for this study to gather and prepare the data for the actual test of the model was about one month. The actual running time on the IBM-7044 computer was only one minute. Data were actually salvaged from laboratory results and plant test runs that had been made prior to this study. In the application of this model to a proposed plant expansion, a plant may very well need to make more laboratory tests and plant test runs should there be not enough data available. Typical Case II Problem
, n = 1, 2,. . ., N . The optimal capital requirement for the proposed expansion is
Xn-~*
N
~
= *
t
+
C [In(ewn*> + L , ( x ~ - ~ * , ejj>,*)i I , n 11
(4)
For the single-step expansion problem in Figure 3, Table 111 shows the optimal capital required for new equipment at each of the six stages and the total optimal capital requirement. Computer Model. If hand-calculated, the solution of Case I, using Equations 1 to 4, is generally laborious, if not impractical. A computer model is needed, and is shown in terms of a block diagram in Figure 4. From the authors’ experience, the time consumed to write and debug the computer program was about three
9
44
44
36
36
-
\
Given a growing market forecast of a product such as that in Table IV, an existing plant has four alternatives for expanding during the forecast period, as shown in Figures 5 to 8. The numbers on the horizontal solid lines are the proposed capacities. The dotted lines correspond to the average production rates. The problem is to determine the best of the four possible routes. Note that in Figures 5 to 8, the forecast period is divided into four time intervals (where time n = 0 is the present time, and time n = 1 is the time for the first addition to capacity). Forecast periods commonly range from four to six years; beyond this time they are usually of questionable accuracy. Common experience is that it takes from at least a year to 18 months to design, build, and start up new
44 d
t
%! 28
5 28
%! 28
=
-1
320
=
4
20 12
12
0
1
2
3
YEAR Figure 6. Case 11- route 2
4
36
20 12
0
1
2
3
4
0
1
YEAR Figure 7. Case 11-route
2
3
4
YEAR 3
Figure 8. Case I I - r o u t e 4 VOL. 6 0
NO.
1
JANUARY
1968
19
TABLE V.
COMPARISON OF FOUR ALTERNATIVE ROUTES FOR EXPANSION FOR CASE I I ILLUSTRATION
I
Route X o . Venture worth Present worth of: Total fixed capital Total “related fixed capital” Total working capital Total capital
7
k=l
where W , m W,
(1
+w, + 2 i)n-+
n=n*-1
(1
$640,000
456,000 661 ,000 698,000 S1,815,000
473,000 661,000 695,000 $1,829,000
VTk+nk + i)”+’
1
the venture worth of mth route 1, 2 , 3, 4 the venture worth of the kth expansion in mth route K , = the total number of expansions in mth route n, = the time for the kth expansion TK= total time between kth and (k 1)th expansion VTX+,,,= maximum venture profit during time t = T , - 1 to timet = Ti, after kth expansion, based on V,, = 0 i = the nominal rate of interest on capital, compounded every N T / 4 years, where N , is the total years of demand forecast = = =
+
Before we can calculate Equation 6, we need to find W,, k = 1, . . ,K, , for each route as follows :
where t = 1, 2 , . . . , T , = nth number of N T / 4 time interval of time after kth expansion. Each kth expansion in any route (at time n,) is actually a fixed-size single-step expansion. Each step expands the plant to a new capacity designed to satisfy the throughput required at the time interval from t = T, - 1 to t = T,. Hence, applying the Case I com20
2
$587,000
capacity. Hence, for convenience, the six-year period is divided into four equal time intervals in this example. The total forecast period and time intervals could have any desired values without affecting the technique described. Optimum solution. Analogous to the reasoning in choosing the objective function for the solution of Case I, the most appropriate objective function for finding the best route for expansion is venture worth, the present worth of the venture profits to be earned during the forecast period. It is also the capacity of a chosen route to attract new capital during the forecast period. The optimum route has the maximum venture worth. The venture worth of each route is calculated as follows :
w , = q
I
INDUSTRIAL A N D ENGINEERING CHEMISTRY
I
3 $767,000 305,000 661,000 _698,000 ___ $1,664,000
I
4 ~~
$771 ,000 335,000 661,000 696,000 $1,692,000
puter model to the existing plant at the time of each kth expansion gives the optimum capital, I , , required for that expansion, and the venture profit V,+,, , where t = T , (based on V,, = 0). The venture profits at time t = 1 to t = T , - 1 are determined next by also applying the Case I computer model at time interval t = 0 to t = T , - 1. Here the alternative decisions in the model investigate only the operating variables of the new and existing process units that are established as optimum by the application of the model at time interval t = ‘ T , - 1 to t = T,. Table V compares the present worth of the total capital requirement and the venture worth of the four routes shown in Figures 5 to 8. From the practical point of view, Routes 3 and 4 are the optimal policies; their venture worths are not significantly different. The positive values of the venture worths of all the routes in Table V prove that any one of these policies is attractive. Summary
This investigation has shown successfully that the capital allocation can be optimized for either a singlestep (Case I) or a time-multistep (Case 11) expansion based on a growing market forecast for the product of an existing plant. The solutions to the two expansion problems (Cases I and 11) have been subjected to a practical demonstration and proved to be powerful tools for management decision. Venture profit, the capacity of a given venture or project to attract new capital, was made the measure for optimality for Case I. Venture worth, the present worth of the venture profits from a given project over a time period, was established as the basis for the optimum for Case 11. The alternative decisions for Case I (single-step expansion) considered (1) replacing the existing equipment with new units of larger capacity, ( 2 ) using only the existing units if possible, and (3) adding new equipment to the existing units. For Case I1 (multistep expansion), the alternative decisions examined four possible routes for expansion during the forecast period; each expansion in any route was considered a Case I problem. Dynamic programming was found to be the most practical tool for optimization in Case I. For instance, for the single-step expansion investigated, examining
all the possible designs individually would have required about 50,000 designs. The search technique of dynamic programming required only about 150 calculations. The solutions tell not only the best way to expand a plant, but also indicate whether the proposed expansion is attractive-Le., a single-step expansion with a positive venture profit, or a multistep expansion with a positive venture worth.
Nomenclature rl
i Zln
I I, In IR
OC
P r
Modifications for Special Cases
1. Limited capital. This study assumed unlimited capital availability. But, if capital is limited, instead of fixing the size of the expansion in Case I, different size expansions could be investigated, with a determination of capital requirement for each. 2. Branching and feedback systems. While the Case I method illustrated here was based on dividing a process flow sheet into “major” process steps and connecting these steps in series, branching and feedback systems can be included if enough data are available to orient the model with real life situations, and if the computer has sufficient storage space. The model modifications for these systems may be derived (7, 6, 72). 3. Discounted cash flow rate of return. Minimum acceptable rate of return was assumed to be known in Case 11, but where this cannot be readily fixed, the discounted cash flow method could be substituted for the venture worth approach. 4. “Related fixed capital” and “related production costs.” If data can be provided in sufficient detail (often not the case), these costs may also be optimized by analogous applications of the techniques illustrated here. Other modifications would accommodate discontinuous processes; in Case 11, if the market forecast had a definable probability distribution, adaptation to a probabilistic model should offer no obstacle. The two major shortcomings of dynamic programming are: (1) it is an impractical method when more than two “state” variables are involved (the dimensionality problem) and (2) it cannot cope with complex interaction between stages. Items 1 and 2 under “Modifications” above deal in part with these problems, but not explicitly as limitations. Rather, it has been shown how one can circumvent an additional state variable-producing situation such as limited capital by using a trial and error routine. In item 2, references are supplied concerning how one might circumvent branching and feedback problems which cause complex interaction. That dynamic programming does not solve every problem is a valid complaint but not insuperable. O n the contrary, it provides a framework for formulating a problem and offers a challenge to the engineer to interpret the problem suitably.
R
RPC S Sa
t T V
*
X xn-1 X”
9
= depreciation rate = rate of interest on capital = minimum acceptable rate of return after taxes = fixed capital investment = working capital = cost of new unit(s) for stage ‘W’ = related fixed capital = annual operating cost = net annual profit after taxes = estimated life of plant, years = increase in annual gross return = related production costs = annual sales = salvage value = income tax rate = transformation function = venture profit = value which maximizes V = state variable-e.g., flow rate or temperature = state variable of a n input stream to nth stage = state variable leaving nth stage = decision variable
Subscripts
i
= one of three possible decisions for
j’
=
1, 2 . . .n. . . N = n’
=
n = n’ n’ 1
+
= =
expansion method operating and/or design variable for nth stage corresponding to decision j designate stages in a multistage flow sheet; also used as superscripts new combined stage resulting from addition of new unit of equipment to existing one, either parallel or series existing unit when new unit added only in series new unit added to n’ in series
Venture Worth Notation = venture worth = nominal rate of interest compounded every N T / years ~ = number of expansion in mth route = total number of expansions = route 1, 2, 3, or 4 = time for kth expansion = total years of demand forecast = number of N T / time ~ intervals = total time between kth and k 1 expansion = venture profit during kth expansion
+
REFERENCES (1) Ark, R.,“Discrete Dynamic Programming,” Chap. 11, Blaisdell, New York, 1963. (2) Bellman, R., “Dynamic Programming,” Princeton University Press, Princeton, 1957. (3) Cochran, W. O., “Procedures for Selection of Optimum Conditions,” Chem. Eng. Progr. Symp. Ser. 5 6 (31), 88-94 (1960). (4) Coleman, J. R., York, R., “0 timum Plant Design for a Growing Market,” IND. ENG.CHEM.5 6 (l), 28-34 (f964). (5) Fan, L. T., Wang, C. S.,“The Discrete Maximum Principle,” Chap. 2, Wiley, New York, 1962. (6) Fan L T Wang C S “Optimization of Continuous Complex Processes by dectron. Control 17 (2), 199-209 (1964). Maxikum Pkinciple:” (7) Generoso E. E., Jr., “The Optimal Capital Allocation to the Expansion of a n Existing Cdemical Plant,” Ph.D. Thesis, State University of New York a t Buffalo (February 1966). (8) Happel, J., “Chemical Process Economics,” Chaps. 1 and 2, Wiley, New York, 1958. (9) Lee, E. S., “Optimum Design and Operation of Chemical Processes,” IND. END.CHEM.5 5 (8), 30-39 (1963). (10) Mitten, L. G., Nemhauser, G. L., “Multistage Optimization,” Ch8m. Eng. Progr. 59, 52-60 (1963). (11) Taborek, J. J., “Optimization of Process Equipment Design,” Ibid., 56 (E), 37-41 (1960). . . (12) Wilde, D.,paper on “Strategies for Optimizing Macrosystems,” presented a t AIChE Meeting, Las Vegas, Sept. 23, 1964.
i.
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