Joint Operation of Multiunit Complexes for Minimum Cost

the joint effort of various plants at minimum cost,. The product in question,say, a particular resin, may be made in part of the original plant, in ne...
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT

Joint Operation of Multiunit Complexes for Minimum Cost KARL M. MAYER U. S. Atomic Energy Commission, Washington 25, D. C.

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A RECENT article ( 2 )it was stressed that the important economic goal of a company is the attainment of the maximum over-all profit and not the operation of all or particular production units a t minimum cost. In fact, it was brought out that individual productive units should be assigned output quotas consistent with over-all company objectives even if particular plants appeared t o operate a t uneconomic levels of output (at levels other than minimum unit cost).

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COYBIHATIOH I

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Classical Approach to Multiunit Operations Assumes increasing Marginal Costs

If it is possible to make a given product in a number of plants or production units, the problem immediately arises as to how much should be produced a t each facility to attain a given production goal. What is the best combination of plants? If a particular department or division of a corporation is charged with the responsibility of producing aspirin, resin, caustic, or aluminum it will attempt to meet the division quota through the joint effort of various plants a t minimum cost. The product in question, say, a particular resin, may be made in part of the original plant, in new plants that are in a high labor area, or in other plants that suffer from high power costs or high maintenance charges. For this situation it will suffice to note that the resin division can make its product to the same specifications using various plants or processes, each with a different oost characteristic.

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L1

4, 0 0

z c v)

0

"1

xi

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OUTPUl

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Figure 1.

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Cost of Production of Various Plant Cornbinations

The purpose of the article is to aid individual production supervisors, once assigned production quotas, to determine that they are obtaining the desired output from processes, pot lines, mills or plants a t minimum cost to their companies. Whereas top management, strongly influenced by over-all company policies, market conditions, etc., assigns quotas to various department heads, it is up to the department head and engineer to determine how to meet desired quotas a t minimum cost regardless of whether more or less could be produced even a t less expense. This situation is illustrated in Figure 1. Whereas, the desired output quota might correspond to Y1 (at an average cost of Yz), the plant output a t absolute minimum cost, Xg,can be attained only a t output XI. It is quite probable that the plant was originally designed to operate a t X I but, for example, market conditions may have dictated a cutback. It is then the responsibility of the engineer in charge to operate his department with the given quota at an average unit cost of Y2 and not any other cost such as Ys,which will always be higher. June 1954

OUTPUT I N U N I T S

Figure 2.

Single-Plant Total Cost Characteristic

Before a definitive analysis of joint operation can be made, cost characteristics must be determined for each production unit available. It will first be necessary to plot, for each machine, pot line, or plant, total cost against output. The shape of the curve might be as indicated in Figure 2. The total cost curve begins to rise sharply as nominal capacity, output N , is exceeded because of overtime premiums, excessive maintenance, or high chemical cost resulting from incomplete reactions as more batches are pushed through the equipment faster. At the other end of the scale, the ordinate intercept is not zero, indicating that certain fixed charges must be incurred regardless of output

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ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT level. (For the time being the problem of standby charges is not considered.) The same data that are plotted in Figure 2 can be recast into various forms for different analytical purposes. For instance, the total cost can he divided by various outputs to give an average unit cost (AUC) curve or the slope of the curve can be plotted against output to yield a marginal unit cost (MUC) curve. The latter two cost characteristics are illustrated in Figure 3 which

cost C is reached. At this point-where the niarginal costs in both plants are equal-no further “profitable” trading can take place, and the combined total output of X (A Y iB Y ) is being met a t minimum total cost to the division. The conclusion that minimum cost operation is ensured vi hen both plants are operating a t the same marginal cost can also be reached through a graphical analysis. Figure 5 s h o w that the total cost of CA CB, when summed, is less than CBX, In other words, the cost of running two plants at roughly one third and two thirds capacity is less than runuing one plant, even though it is the “lovv cost” facility. In Figure 5$ the slopes a t outputs AY and BY-that is, d c / d o or the marginal costs-are equal. Although it is cheaper to run both plants or processes under capacity than to run only one plant, it can be shown graphically that it is cheapest to iun a t only one point. It is cheapest to produce X when d Y

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BY

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=x.

I n Figure 6 various combinations of plants A and B were selected and the resultant total costs plotted. If the curves were all drawn consistently to scale, the mininium cost would again result when plant B is operating a t two thirds and A a t one third capacity provided the data correctly reflect plant operations However. regardless of the shapes of the curves the general truth remains that multiple plants, pot lines, or proceses will pioduce a given joint output a t minimum cost only when all units ale operating a t equal marginal costs. M

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OUTPUT IN U N I T S

Figure 3.

Single-Plant Marginal and Average Cost Characteristics

shows that the $UC curve drops as fixed charges are spread over more units of output, goes through a minimum, and rises as premium costs are incurred. Perhaps of more significance, however, is the fact that the marginal cost curve ( d c l d o or the rate of change of total cost n i t h respect to output plotted against output) rises and passes thiough the minimum of ACC. Minimum Cost Results When All Units Operate at Equal Marginal Costs

I n the simplest case, illustrated by Figure 4, two plants are available for production; one operates consistently a t higher cost than the other. MC.4 represents a plot of the marginal cost of plant A, which is older and less effective, while MCB represents the marginal cost curve ( a s . output) for the newer, more efficient plant. For purposes of this paper it is sufficient to define physically the marginal cost as “cost of producing the last unit a t any given output” or “cost of incremental production a t any particular level of output.” A quick look a t Figure 4 might lead to the conclusion that if X units of reein is the needed output or quota of the division, then it would be best to close the older plant (A) and operate the newer plant (B) a t near capacity. However, this would be a wrong conclusion. The correct solution to the problem is that for minimum cost, the particular plant characteristics depicted are such that plant A should be run a t one third of capacity ( A Y ) and plant B a t two thirds of capacity ( B Y ) . Output AY B Y = X , the desired output or quota of resin, and no other combination of output nil1 give the desired output quota a t minimum cost to the division. First, as shown in Figure if plant B were used alone to produce the quota, X , then the last unit made in plant B would have cost D dollars. However, there are many units that could be produced in plant A for less than D dollars. By cutting back high cost production in plant B and substituting lorn cost production in plant A the resin division can make a “profit” until

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OLTPUT

Figure 4.

BY

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IN U N I T S

Two-Plant Complex Characteristics

Marginal Cost

These conclusions can also be verified by mcans of algebraic and simple differential techniques or m-ith the help of methods using Lagrangian multipliers ( 1 ) . Here too, in order to produce a given output, joint facilities operate a t minimum cost 1% hen incurring equal individual marginal costs. This rule holds regardless of whether there are two plants or twenty in the complex, regardless of the shapes of their respective cost curves, and regardless of whether the various-shaped curves are parallel, converging, diverging, or intersecting. The relationship of in& vidual unit capacities to the combined quota does not have t o be unique nor do the capacities of the producing units have to he the same. An example showing that all production units within a niultiunit complex should be operated a t equal marginal cost for minimum over-all cost might involve a complex with three units whose cost characteristics are as shown in Figure 7 .

INDUSTRIAL AND ENGINEERING CHEMISTRY

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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT is the desired total output, then BC Since must equal BX, and

BHXC

+ Sui + B

~ Z

+ B T , YU1 + BT2ZUz = total cost

Since the objective is to minimize total cost, the proof resolves into expressing each of the areas as a function of output, differentiating, and setting thLdifferStia1 equal to zero. Such a method will show X C = YU1 = ZUz. The i n e i d u a l C p u t s z a y be & u l a t e e follows: Since h=+ BH = tl BUl B T , z tz B s + ' B z = E, = conBU1 BUi = B A stant marginal unit cost and BC

+

If the total operating cost of an individual plant or process is directly proportional to output or as shown in Figure 8, then the marginal cost over the same output range will be a straight line with a zero slope. If a corporation has a number of plants available for the joint production of a uniform product employing a process which has a constant marginal cost characteristic, then a new method of analysis must be adopted to determine optimum joint operation.

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TCA

tim+m--m h

TCB

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AY

BY

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OUTPUT I N UNITS

Figure 5.

Two-Plant Complex Total Characteristics

Then find BT1 and B C by substituting in previous expressions, where

BA

marginal cost characteristic of first plant marginal cost characteristic of second plant marginal cost characteristic of third plant optimum output of first plant optimum output of second plant optimum output of third plant E B T l RT2 = optimum output of the firm for mnxinium profit

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Constant or Decreasing Costs Typify Anomalous Nonclassical Cases

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In the classical method of solution of joint operation problems the cost functions were assumed to be known and continuous; standby charges were neglected and, as is usually the case, plants or processes were pictured as exhibiting increasing marginal cost curves. In this section anomalous cases of constant or decreasing marginal cost functions are considered. Constant Marginal Cost. Although classical theory generally works with increasing marginal costs, a practical example can be developed on the basis of constant marginal costs. In other words, it is possible to visualize two or more plants, processes, or departments producing identical products a t substantially constant marginal costs. As shown in Figure 8, a substantial portion of the operating cost of a typical plant, process, or department on stream is attributable to fixed charges. Contained in the fixed portions are supervisors' salaries, an allotment of the general corporation overhead, guard service, and even operating labor if the characteristics of the process are such that a full crew is needed whether the throughput is large or small. The general variable charge might contain power, fuel, and process steam which could conceivably be proportional to output, just as is the chemicals charge. For certain types of plants and processes, over a wide range of throughputs, the straight line could be a good description of the operating cost characteristics. June 1954

Figure 9 shows that of the three plants or processes available A has the lowest fixed charge but high variable charges. Plants B and C have different fixed charges but the same variable charges over identical output ranges. Since the slopes of B and C are small and both the same, the marginal cost lines are coincident a t a low level. Because of the high variable charges, the marginal cost of plant A is high but again constant since the slope of the total cost curve is constant. The department head faced with a situation similar to that shown in Figure 9 cannot use the techniques outlined for dealing with increasing marginal costs since the curves in Figure 9 are flat. Neglecting for a moment the influence of start-up and standby costs, i t becomes apparent that total costs rather than marginal costs take on a new significance in this anomalous case. Without consideration of start-up and standby costs, the first

Y units of output will best be obtained from factory A since the

total cost is always below total costs of other facilities. In fact, it is not until output Z (Figure 9) that it becomes immaterial whether plant A or C is used. From output Z to X (ultimate capacity) factory C is the cheapest. A t output X M it would be best to use factories C and A, but beyond X $1 factories C and B would be cheaper up to an output of 2X. In this very simplified example factory A would be the last to come on stream for outputs in excess of X M and the first to be cut back because of the higher total cost a t all but very low outputs. If (Figure 9) one assumes that fixed charges exist only in factories which operate, then the simplified conclusion developed in the previous paragraph will hold. If, on the other hand, one assumes that the fixed charges in all plants must be borne in any event, then i t becomes obvious that factories B and C will be run a t full capacity before one unit of production is authorized for factory A since B and C have lower marginal costs for all levels of output. I n many cases, however, it is not a matter of bearing all the fixed charges or none, it is a matter of bearing a given portion of the fixed charges a t zero output; this situation is considered in a later section dealing with the role of start-up and standby costs in joint operation for minimum over-all cost.

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Decreasing Marginal Cost. I n this section processes of decreasing marginal costs are examined-processes where cost

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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT increases are disproportionately small when compared to associated output increases. A chemical process that has a decreasing marginal cost characteristic must likewise yield a concave-downward total cost curve. The sum of the variable and fixed charges must increase at a decreasing rate when plotted against output. although it is possible to draw a graph such as Figure 10 on paper and although many engineers at first glance vi11 insist thab they know of chemical processes having such characteristics, it is hard to find such a plant or process in actual practice. h precess with a decreasing marginal cost curve is a rare one, and this is particularly true when engineers are pushing a plant toward ultimate capacity. The exceptions nil1 probably be found most often in plants that are far overdesigned or those vioiliing far below nominal capacity. The bases for classifying decreasing marginal cost processes as b'exceptions'' are both theoretical and practical. I n the first place, as the total cost curve levels off, it indicates that the cost

of making the last unit is approaching zero. (Any negativc slope would indicate "negative costs of product,ion.") Since the slope is decreasing, the marginal cost is decreasing-the last unit alLyays costs less than the next-to-last unit. A process exhibiting a decreasing marginal cost is hard to visualize from an economic point of view but is equally difficult from a chemical or unit operations viewpoint. As output is pushed beyond design levels toward the ultimate capacity of the plant and equipment, the efficiency of many of the diffusion processes drops: 4 s more batches are pushed through per shift, the timee of digestion are cut a litt,le shorter and lese ext,raction is realized per unit of acid added; as more and more slurry is passed over a bank of filters, less and less yield must be expected per unit of feed; as greater and greater amount,s of liquor are pushed through the columns, less and less product will be got'ten per unit, of extractant, etc. I n short, chemical cads per unit of output may be determined by stoichiometric relat,ionships if reactions are permitted to reach equilibrium, but generally as the output is pushed, increasing costs are incurred. K h a t has been said for cheinical costs can be repeated for maintenance, povier consumption, and process steam usage. Whether it is a tolyer that is pushed to the flooding point or a n-hole plant that is being strained to the utmost, it is likely from hot,h theoretical economic and practical chemical viewpoints that the tot,al cost curve vi11 be concave upward and the marginal cost increasing since nonchemical costs are constant or increasing. The average cost curve or unit cost curve, so often Ivatched n-ith great care by engineers, ie not generally a helpful analytical tool and may conceivably at this point have a, negative, zero: or positive slope. The average unit cost plot may have great advertising value in the front office, but it gives little guidance in optimizing plant operation. But Decreasing Marginal Costs Occur Only in Special Situations

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I t is conceivable that a plant operating a t a small fraction of its noniinal capacity might actually be operating in a region of decreasing marginal costs. I n fact, converted World \Tar I1 plank bought for a small fract,ion of the original cost of construction (or reproduction cost' Tvhen new) may have, in some ca'es, operated a t very low throughputs with greatly oversized equipment in the immediate postn-ar period. K i t h these unusual

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Figure 6. Two-Plant Complex Simple Optimum Combination

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OUTPUT IN U N I T S

Figure 7.

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Three-Plant Complex Operating a t Equalized Marginal Cost

FIXED

OUTPUT

Figure 8.

OPERh.IING

IN

CHLRGES

UNITS

General Analysis of Single-Plant Total Operating Charges Constant marginal costs

INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY

Val. 46, No. 6

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT conditions, chemical unit opcration efficiencies could increase sufficiently to produce decreasing marginal cost curves. At very low throughputs it is certainly possible that the marginal unit pumped, scrubbed, filtered, agitated, etc., could actually cost less than the next-to-last unit. This might be called the “benefit of proximity” of the material being treated or reacted. I n other words, the presence o€ material being pumped may help other material to be pumped; the presence of material in a scrub tower in unproductive locations increases the probability of a neTv material unit to occupy a productive position; the presence of some slurry or cake on a filter may help other material to be filtered better; or the presence of Some material in an agitator or grinder may help to agitate or grind marginal units added. Although decreasing marginal costs are conceivable a t very low throughputs, they become less likely as capacity is a i p poached. I n fact, if plants do exhibit decreasing marginal costs it is to be expected that the curves will pass through a minimum and begin to rise long before capacity is approached. Since QP erations in general are expected to be carried out somewhere near capacity, the previous analysis and most classical examples consider increasing marginal cost ranges of output only. However, Figure 11 presents a simplified case in which a conmpkx consists of two plants with concave downward total cost curves that can be translated into marginal cost curves with negative slopes. Again, if start-up and standby costs are neglected and if it is assumed that fixed charges must be borne by only those plants which are operated, it is quite obvious that plant B will be selected for all outputs below Y . Beyond output Y it will be better to operate only plant A since TCA falls below TCB. If, on the other hand, it is assumed that all (in this case both) fixed changes must be borne regardless of the level of output, then an entirely different method of approach is necessary to determine how to produce a t minimum cost. Data taken from the total cost curves of Figure 11 should first be recast and marginal cost curves drawn as shown in Figure 12. It then becomes obvious that the first production should not come from plant B and should not come from the joint operation of the two plants a t equalized marginal cost but should come solely from plant A. Production should continue from plant A since production costs decrease throughout the whole range of outputs. Anomalous decreasing marginal costs, therefore, must be given special treatment in any analysis for minimum cost joint output. I t should be stressed again that although such cases can be met

in practice they are very unusual. However, no treatment of the subject would have been complete without a thorough consideration of the unusual as well as the expected. A General Method of Finding Minimum Cost of Operation for Multiunits Is Based on Variable Marginal Costs

I n the example of general variable marginal costs used here, the method of solution will yield correct results in any conceivable practical case whether it involves increasing, decreasing, or constant marginal costs or any combination thereof. I n addition, the general method of solution is adaptable to all plants whether %heydo or do not involve start-up and standby charges. In the classical approach it was implicitly assumed that etartup and standby costs either do not exist or have no influence on the conclusions. As a practical matter, most industrial chemical

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OUTPUT IN UNITS

Figure

10. Simplified Analysis of SinglePlant Cost Characteristic Decreasing marginal costs

TCB

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OUTPUT I N U N I T S

Figure 9. Three-Plant Complex Total and Marginal Cost Characteristics Constant marginal costs

June 1954

OUTPUT I N U N I T S

Figure 1 1.

Two-Plant Complex Cost Characteristics

Decreasing marainal costs

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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT enterprises must contend with both start-up costs, when a new plant comes on stream, and with standby costs, when an existing plant stops operation, The existence of both these very real factors makes it necessary to develop a new scheme of analysis. The problem of finding the minimum cost of joint operation of multiple facilities can be rephrased as the problem of determining the optimum supply or cost of production schedule. That is, a knowledge of the cost locus of minima for each rate of production from zero to the capacity of all plants operating simultaneously would be very helpful in determining how the facilities

Production in All Units Is Cut Back Simultaneously to Ensure Equal Marginal Coots

In times such as the present (1953-54), with outputs a t record levels and rumblings of impending recession or adjustment periods, it appears appropriate to consider a general example of production in joint facilities operating a t capacity and the method used to reduce production to some lower level with assurance that the correct amount of output will come from each facility and the over-all total cost for the newer, reduced output will be a minimum. If all plants are operating a t capacity, then any contemplated reductions in output should be made “at the margin” as described earlier. In other words, factories with the highest marginal (not total, unit, or average) costs should be reduced incrementally first. Production in all plants should be cut back simultaneously to ensure that all are incurring equal marginal costs. I n this v a y the integrated marginal cost (total saving) for any increment of production cutback will be mayimized. This first step is illustrated in Figure 13 for three plants of different capacities and different cost characteristics selected for joint output analysis. When all plants are operating at B C. As capacity the joint output of the complex is A production is incrementally reduced from capacity, plant A with a marginal cost (LICA) of AA will be cut back first to a marginal cost of BB. At that point output from both plants A and B should be reduced simultaneously until cost CC is reached. Thereafter, all three should be cut back together. By j$ay of illustration, if it is desired to cut back total production from

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MCA

OUTPUT IN UNITS

Figure 12.

Two-Plant Complex

Decreasing marginal costs

should be combined for minimum cost a t any desired output. Such a predetermined schedule can be calculated and used to establish which plants should operate and which should be placed in standby; i t will also set up operating rates for each plant for any desired joint output. The problem in practice is usually much more complex than running the good plants and shutting down the bad ones. If i t is assumed that each subprocess and process (or plant) is individually optimized and the cost characteristics have been determined as suggested by the author in an earlier article ( 2 ) , then i t follows that the minimum cost of near maximum production is the sum of costs incurred by all plants operating optimally a t capacity. Similarly, the minimum total cost is incurred by the corporation or department when all plants are in standby (and production is at zero) provided that it costs less to keep a plant or process in standby than a t any level of operation. With the terminal points determined, it becomes necessary to calculate intermediate points to complete the schedule. I n order to calculate the minimum cost of producing intermediate outputs between zero and joint capacity, either one of two methods can be selected: It can be assumed that none of the plants is in operation and production is to be built up from zero to capacity, or it can be assumed that all the plants are a t capacity and production is to be cut back, always a t minimum cost, to zero output. Although the latter method is given in more detail here, the principles underlying both analyses are the same. A thorough treatment of the “cutback” method is followed by a briefer illustration of the “build-up” method of calculating costs for joint intermediate outputs.

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OUTPUT

Figure 13,

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mN U N ! T S

Production Cutback for Minimum Cost

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B C to A’ B’ C’, all plants should be operating a t a marginal cost of DD,and the desired output will be ensured a t minimum cost. In conjunction with each incremental cutback, the total cost of operation should be read from the plot of total cost against output from which the marginal curves were derived. The total cost derived thereby for each intermediate level of production below but near capacity can then be plotted as a function of output. -4s production is cut back substantially below capacity, toward zero, it becomes necessary in practice to calculate break-even points which will indicate how long a facility or process should be allowed to operate a t equalized marginal cost and when it should be placed in standby. At some point it will pay to put

A

INDUSTRIAL A N D ENGINEERING CHEMISTRY

Vol. 46, No. 6

ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT one of a series of plants in standby and increase production in the remaining plants to cover the output loss of the quieted facility. I n Figure 13, standby costs are shown as SA, SC, and SB, respectively, and are illustrated as extensions of the basic marginal cost curves. (The ordinate intercepts have no absolute or even relative significance since standby costs are logicallv extensions of total cost curves to zero output. SA, SB, and SC are meant to serve only as reminders of the existence of standby charges.) Standby costs may even be lower than indicated, under certain circumstances, but the general method illustrated in this paper is applicable regardless of the standby cost level. It is obvious that production in all plants will not be reduced to levels close to zero because a point will be reached where it will be profitable to shut down a plant, such as A, and still incur a etandby cost of S A . The profitableness of such a move is illustrated by debits and credits, as follows: Credit Debit Saving resulting from not I . Standby cost of A: SA operating plant A 2. Incremental cost of increasing production in plant B LIathematically, output Mathematically, (MC.4) do S = (MCB) do where MCA is expressed as a where 1 and 2 correspond to function of output 0 outputs of B before and after placing plant A in standby 3. Incremental cost of increasing production in plant C to help make up loss of plant A Mathematically,

a. Total cost of output (step 1)incurred by producing in all facilities a t capacity after putting- the “smallest capacity” - -plant instandby. b. Total cost of producing output (step 1) by producing in all facilities (none in standby) a t equalized marginal cost. 3. If more than one facilitv has a aroductive caaacitv eoual to the smallest capacity unit “then the first plant eiimin’ated in step 2a should be the one with the highest average cost at the smallest capacity output,

CAPACITY OF COMPLEX

-

A U P U N T S OPERATINO

1.

.=x

S

=

BEP

JOINT OUTPUT

Figure 14.

OF

0tC

A L L OPERATING P L A N T S IN UNITS

Graphical Solution for Breakeven Point

(MCC) do

As production is incrementally reduced, as shown in Figure 13, the credits and debits should be calculated and plotted to determine the break-even point-that is, the point where it just pays to cease cutting back all plants simultaneously and to put one plant in standby. This procedure for determining the breakeven point (BEP) is illustrated in Figure 14. I n the example chosen, it is necessary to operate plant A for C for sheer physical reasons. Furall outputs in excess of B ther, all plants should be operated for outputs even a t rates less C for economic reasons. In fact, all plants should be than B kept operating a t equalized marginal cost for all outputs greater than BEP. Only a t a level of BEP does it pay to switch from three-plant to two-plant operation. However, if it is physically possible to shut down plant ,4 or B or C the analyst should be sure that the economically prudent one is shut down first by testing the cost situations of various combinations. Exactly the same procedure is used in determining the point at which a firm or department should switch from two-plant operation to one-plant operation. In this manner any desired level of output is attained with the correct plants operating a t correct levels in relation to one another to yield the production quota a t minimum total cost. A detailed, step-by-step method of calculating all points (capacity cost, standby cost, intermediate costs, and break-even points) in the schedule or minimum cost locus is given in the following example: A method of solution for finding a combined cost of production schedule for operating several units engaged in the production of a uniform product is as follows (see also Figure 15):

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1. Cut back all facilities a t e ualized marginal costs until it is feasible, production-wise, to efiminate the smallest capacity facility by running all facilities remaining to capacity. 2. At the output represented by step 1, calculate two quantities: lune 1954

0

4. If step 2 shows that the quantity calculated under 2a is smaller than under 2b, then the highest cost-smallest capacity unit should be put into standby and subsequent increments of output curtailment should be made on the basis of equalized marginal cost production in the remaining units. Skip to step 12 if step 4 was appropriate 5. If step 2 shows the quantity calculated under 2b smaller than under 2a then continue reducing the joint production of all the facilities (none in standby) a t equalized marginal costs to a break-even level of production provided that the joint production level is not reduced to point where the next-to-smallest capacity plant can be eliminated. Skip to step 7 6. If output is reduced to a point of full production less the capacity of the next-to-smallest unit, then skip to step 12 below. 7. The break-even level of production can be calculated by repeating steps 2a and 2b at a level of joint production below that assumed in 2. 8. Plot the a and b points calculated in step 2 and step 6 against output and straight lines drawn to a point of crossover. 9. The marginal cost corresponding t o the point of crossover should be selected to calculate new over-all production costs corresponding to steps 2a and 2b. 10. If step 9 shows that total costs are equal, the leastcapacity, high-cost plant should be put in standby and further curtailments in output should be made from the point of crossover a t equalized marginal costs.

Skip to step 12 11. If step 9 shows that total costs are not equal, the curves plotted under step 8 should be corrected and the procedure for step 9 should be repeated. This process should be repeated until step 10 becomes feasible.

Return step 9 12. The reduction of joint production started under step 4 or step 10 should be continued a t equalized marginal costs until the reduction in over-all production is equal t o a or b, whichever is smaller : a. The capacity of the next-to-lowest unit

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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT LEVEL OF PRODUTION

Return to step 13 for method if appropriate . 18. If 121, is smaller t h m 12a, then carry out' method of calculation in a manner analogous t,o steps 13 through 17. Be careful, however, to select the two smallest capacity plants that have average cost of product at the smallest capacity output.

FULL PRODUCTION

FULL PRODUCTION LESS SMALLEST UNIT

Ii , 1-)5(

Return to step 1.3 for method if appropriate

(6)

.smallest in standb'y

The n-hole procedure from capacity t o zero output essentially balances the

1

costs and benefits of putt,ing plants into standby with all incremental 1'0dcctions being carried out at equaiized marginal cost. The method of 'ILt' solution is illustrated graphically in Figure 16. This p.ocedure is specifically designed to deal with sit,uations in which major cutbacks are neemsari- for indefinite periods. m'herc cutbacks are intended to be minor and temporary it) will pay to delay the putting of a plant into standby beyond the BEP to a level of output, a t which the annual penalty of FULL PRODUCTiON keeping the plant in operation just LESS SMALLequals the annual cost of plant ESTAND NEXTTO-SMALLEST startup. UNITS I n Figure 16 two plants are shown being o p e r a t e d j o i n t l y , b u t t h e Figure 15. System for Calculating Least-Cost Production Schedule method is general so that it could be looked upon as a description of b. Twice the capacity of the smallest capacity plant (preperformance of two or more plants. As the joint output of viously put in standby under step 4) all plants approaches O D from higher levels, a break-even plot 13. If 12a is smaller than 12b, calculate the total cost of production through t'he use of: a . A11 facilities at equalized marginal cost except the smallest capacity unit (which is considered in standby) b. ill1 facilities at equalized marginal cost, except t'he nestto-smallest unit (which is considered in standby) c. All facilities at equalized marginal cost (none in standbj-) ran,

i

Proceed to step 14 or step 15 as appropriate 14. If the calculated dollar figure in step 13a is smaller than 13b and 13c, then proceed to calculate the break-even point keeping only the smallest' capacity plant in standby and reducing production a t equalized marginal costs in a met,hod analogous to that outlined in steps 5 , i, 3, 9, 10, and 11. z w

Return to step 5 if appropriate 0

15. If t,he calculated dollar figure in step 13b is smaller than 13a and 13c, then put the larger facility in standby and put the smallest capacity unit back on stream. 16. With the next-to-smallest unit in standby, continue to reduce production at equalized marginal costs until the total reduction is equal t'o the capacity of the smallest and next-tosmallest unit,s, and repeat the calculation process in a manner analogous t o that outlined in steps 2 through 4. Hon-ever, the two quantit,ies calculated in this case are: a . Total cost holding smallest and nest-to-smallest unit in standby b. Total cost' holding only nest-to-smallest unit in standby. Then continue calculat'ion as before. Return to step 2 for general method if appropriate

17. If 13c is smaller than 13a and 13b, continue output reductions at equalized marginal cost to a point of full production less smallest plus next-to-smallest capacity. At intermediate points, break-even test calculations can be performed, but they are not necessary if "all plants" can continue t o be cheapest a t the lower level of production. For further reductions in output continue calculations in a manner analogous to that outlined. 1254

L1 c

8

0

a

B

OJTPII'

Figure 16. MCA.

MCB. ACA. ACB. OA.

OB.

OD. AUS.

0

IN U N I T S

Standby Cost Balance

Marginal cost of plant A Marginal cost of plant B Average cost of piant A Average cost of plant B Output of plant A Output of plant B Desired joint output which corresponds t o O A Average unit standby charge

INDUSTRIAL AND ENGINEERING CHEMISTRY

f OB

Vol. 46, No. 6

ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT fQUR PLbNTS

uininw

COST LOCUS

P R Q O V C T I Q H BUILD-UP

THREE

/A\

m.Anrs

TWO PLANTS

ii

0

CAPACITY

OUTPUT

Figure 17.

IN UNITS

Production Build-up and Cutback Loci

will show that a t OD the amount of dollars saved by shutting down plant A (area A A ) just equals the additional out-ofpocket outlay for running other production units from OB to OD (area BB). -4t any point greater than joint output OD, area BB would be larger than A A thereby encouraging the continued operation of plant A. At any joint output less than OD it is cheaper to operate with A in standby. This procedure can be repeated until the last plant itself is run back to zero after all others have been eliminated. In this way a complete schedule of minimum costs for all outputs can be developed. Build-up Procedure Follows Same Principles Used to Develop the Cutback Method

In the previous sections a method was outlined for cutting back the joint output of a number of plants producing a homo-

geneous product from capacity of the complex to zero output. I t was mentioned earlier that present business conditions (195354) made the cutback method distinctly more timely. However, the same principles can be used to develop a minimum cost locus from zero production to the capacity of all plants operating simultaneously. During a depression when most or all plants of a complex are in standby it is necessary to develop a plan for increasing production (as business conditions permit) a t minimum over-all cost. If all of the plants are in standby then the solution of the problem begins by considering which of the plants in standby has the least net annual cost of start-up (annual start-up minus annual standby charge). However, just as soon as or as long as any plant in the complex is in operation, a balance must be struck b e h e e n increasing production from a going unit and bringing another plant on stream. Output in the plant with the least net start-up cost is increased from zero to a point where the saving in cutting back an increment of production just equals the cost of bringing the second plant on stream. From this point on both plants are operated jointly a t equalized marginal cost until it pays to take the third plant out of standby. This process is repeated until all plants are on stream and operating simultaneously a t capacitv. The principles underlying the build-up method are exactly the same as those outlined in detail for the cutback method. The schedules developed through the uee the catback and buildup methods (see Figure 17) \Till be different because they result from two different situations. The cost X - Y (exaggerated for emphasis) is the “inertia” cost of start-up -the penalty which must be paid for allowing a complex of plants to grind to a halt. However, in either event, if the method selected truly reflects the circumstances of joint output (build-up or cutback), then it will yield the best procedure for adjusting output either up or down and ensure that any desired level of output will be produced a t the minimum over-all cost t q the multiplant complex. Literature Cited (1) Allen, R. G. D., “llathematical Analysis for Economists,” p. 366, London, Macmillan, 1950. ( 2 ) Lfayer, K. &I., Chem. Eng.,60, 214-16 (1953). R E ~ E I V E D for revieiv january 29, 1953. ACCEPTED April 7, 1954

Mechanism of Solute Transfer from Droplets LIQUID-LIQUID EXTRACTION F.

H. GARNER AND A. H. P. SKELLAND

Deparfmenf o f Chemical Engineering, The Universify o f Birmingham, Birmingham 75, England

I

N ORDER to improve the efficiency of the operation of liquid-

liquid extraction for the separation of components of solutions, a more complete knowledge is required of the factors affecting its basic mechanism-solute transfer between two liquid phases, one of which is dispersed in the other. Relatively few papers on transfer t o or from single drops have appeared. In the earlier work of Sherwood, Evans, and Longcor ( I S ) the size and number of drops were known, giving the area available for transfer, and hence the variation in the transfer coefficient, K , could be determined. They extracted acetic acid from methyl isobutyl ketone and benzene droplets with water and found that about 40% of the solute was extracted before the drop

June 1954

left the nozzle. Comparison between experimental and theoretical transfer coefficients during rise led them t o postulate a degree of circulation within the drops which assisted transfer. West, Robinson, Morgenthaler, Beck, and MeGregor ( 1 6 ) repeated some of this work on the benzene system, but found only 14 to 20% extraction during formation and concluded that the drops were internally stagnant. In a later paper, West, Herrman, Chong, and Thomas (16) explained the discrepancy in terms of contamination of the benzene-acetic acid solution due to its passage through Tygon tubing. The rate of transfer in the system benzene-acetic acid-water was also substantially altered by addition of various alcohols to the organic phase.

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