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404 RICHARD BELLMAN

processes with almost-independent stages. Here the weakness of couplingand the almost-independence is measured by the number of state variablesat one stage which depend on the variables of the preceding stage.

By means of the functional equation technique of dynamic programming, weshall show that the computational solution can be reduced to that of the com-putation of a sequence of functions of one variable, in the particular problemwe treat.

In another paper, [2], we have illustrated the application of the same idea tothe study of Jacobi matrices, and to the study of weakly-coupled mechanicaand electrical systems.

2. A Weakly-Coupled SystemConsider the problem of maximizing the linear form

over all x.- satisfying the constraints(2) anXi+ anX +

+ 032 2-f a-i-iXs + biXi C3,44X4+ 45X5+ aaXt Ct54X4+ O55X5+ 56X6 d0*4X4+ o5X6-f OseXe + b-iXjS Ce

and(3) X, ^ 0.

It is assumed throughout that b, > 0, a.y ^ 0, with a sufficient number positive so that the maximum ofLix is not infinite.We wish to attack this problem, arising from the study of weakly coupled


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SOLUTION OF LINEAR PROGRAMMING PROBLEMS 405It will be clear that a number of similar problems involving almost block-

diagonal matrices can be treated by means of the same general method.3 . Dynamic Programming Formulation

Let us define the sequence of functions ofz,(1) Mz = Max L^{xwhere the a;, are subject to the constraints given above, with the exception thatthe last constraint is now( 2 ) a3 f .HN-'iXzif-i + a^fi ,3K-lX3N-l + aiK,itlX3K = 2.

Employingtheprinciple ofo pt imal i ty,cf. [1],wesee tha t thesequence {/ (satisfies the recurrence relationft,iz = Max [x3f/-2 a; Ar_l X K fN-l{C3N-3- b

withthe variables X3Ar^2, x^nî, X3Nsubjectto the constraints(4) flajV 2,3JV 2a;3A 2 ~l a3K~2.3N~1^3Hl ~\~ a3N2,iKX3K C3iV2

a ; 3 ^ f - 2 , X3X-1, X3N ^ 0 .The function /o(z) is identically zero.

4 . S impl i f i ca t i onIêtuswrite the recurrence relation of (3.3) in the formfîz) = M a x f Max []]

= MaxTMax {X3NwhereRnis the region in (xjw-i, X3K) space defined by( 2 ) a3if-i,3K-lX3K-l + a3K- ,3NX3N C3N-2 flsw


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4 0 6 RICHARD BELLMANThus we can write

(3 ) fs{z = Max[g'Ar(x3Ar-2,2) + fs iicm i AT iXiN~2 ]whereX3f,-2is constrained by4) 0

The function gN^y, z is readily determined, since the maximum over Rn iattained at a vertex of the region.

B i b l i o g r a p h y1. R. BELLMAN,TheTheory of Dynamic Programming, Bull . Amer. Math.Soc, vol. (1954),pp. 503-516.2. , Some Applications of Dynamic Progranmiing toM atrix T heory , Illinois Jour,oM a t h , 1957 to appear) .For a further discussion of dynamic programming, werefer to our book Dynam iProg ram m ing, Princeton University Press, 1957.


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APPLICATIONS OF LINEAR PROGRAMMING IN THEOIL INDUSTRY*'

W. W. GARVINS H. W. CRANDALL', J . B. JOHN*, AND R . A. SPELLiMAN'umm ry

This paper is the result of a survey made during the summer of 1956. It is a

1 . I n t r o d u c t i o nPla ns were m ad e during early 1956 for a symp osium on indu strial application

them e of th at me eting was A Progress R ep or t and some of the earliest

We were requested by George Dantzig to present such a review and to include

f discussing the ir work a nd ours in the linear program ming field.The oil industry became aware of linear programming through the pioneeringond s (1953). W e owe a great deal to these gentlemen an d to Alan M an ne (1956)for pointing out to us that linear programming has a place in our business. Aew years ago, there were few people indeed in the oil industry who had evereard of such things as basic solution or convex set . Tod ay, these term sre much more familiar and as a result much less frightening to some. What is

ow. It is am azing, therefore, to see how much has been done in such a com para-


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GARV IN CRANDALL JOHN ANO SPELLMAN

As technology advances and improves, problems become more interwoven ancomplex. The problems of the oil industry are no exception. They can logicallbe grouped into categories according to the different phases of our business ashown in Figure 1. An integra ted oil company must first of all carry out exploration activities to determine the spots where oil is most likely to be found. Thland must then be acquired or leased and an exploratory well or w ildcat as iis called is drilled. If luck is with us, we hit oil. Additional wells are drilled to develop the field and production gets underway. The oil is transported by varioumeans to the refinery where a variety of products are manufactured from itThe products in turn leave the refinery, enter the distribution system and armarketed.Needless to say, each of th e areas shown in Figure is full of unanswered quetions and problems. Different methods exist for exploring the oil potentialities oa region. How should they be combined for maximum effectiveness? An oil fiecan be produced in many different ways. Which is best? The complexity of amodern refinery is staggering. What is the best operating plan? And whaprecisely do we mean by best ? Of course, not all the problems in these arealend themselves to linear programming but some of them do. W hat we would likto do is to pick out a few representative P type problems from each area, showhow they were formulated and in some cases, discuss the results that were

obtained.We had hoped to find applications in all four of the areas shown in Figure 1Unfortunately, we were successful only in three. We did not find any nonconfidential applications in the field of exploration. Exploration is one of the mosconfidential phases of our business and it is for that reason that oil companies arnot very explicit about their studies in this field. We can state, however, frompersonal experience, that a number of apphcations to exploration are under investigation.

ExplorationLand a Lease

Drilling aProduction

>Manufacturing


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LINEAR PROGRAMMING IN OIL INDUSTRY 409

ExplorationLand a Lease

Drilling aProduction

Model of 0 producing complex

Manufacturing Increme ntal product costs Non lineor effects of tet ra eth yt leodNon hneo r effects of vorioble cut points Cost coefficien ts

Distributiona Marketing Distrib ution to an exponding morketServ ice stotton de liveries long rongeService stotion deliveries stiort range

Km 2

Let us therefore turn our attention to the remaining three areas of Drillingand Produ ct ion M anufactur ing and Dist r ibut ion and M arket ing. Figure 2shows an outline of the applications that will be discussed. Out of the Drillingand Production area the problem of devising a model for a producing complexwas selecte d. In the case of M an ufa ctu ring th e selection was difficult b ecause his-torically this was the first area of application and much work has been done inthis field. The problems shown were selected because they either illustrate animportant concept or because they i l lustrate a peculiar twist in mathematicalformulation. The problem of incremental product costs illustrates the techniqueof para m etric prog ramm ing an d also shows wh at can happe n if too m any sim-plifications are introduced. The methods developed for handling tetra-ethyl leadand variab le cut poin ts illustrate how und er certain conditions nonlinearitiescan be introduced into the system. The problem of cost coefficients will illustrateth e need for realistic refinery c osts. Fin ally th ree pro blem s ou t of th e area ofDistribution and Marketing were selecteda bulk plant distribution problemhaving to do with the shipment of products from refineries to bulk plant.s in anexpanding market and the problem of devising long-range and short-range de-livery schedules from bulk plants to service stations.

2 Model of a Producing Complex


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410 GARVIN, CRANDALL, JOHN AND SPELLMANoil field s or reservoirs (f = 1, 2 A ) as shown in Figure 3 which are producat rates Q,(0 wheret is the time. The to tal production of the A^ reservoirs is toadjusted to meet a commitment Qdt) (such as keeping a pipe line full orrefinery supplied). An outside source of crude oil is also available. I^et the prorealizable per barrel be c,(0 and consider that the operation is to be run on thbasis for a period of T years. Production limitations exist which require that tQi{t do not exceed certain values and that the pressures in the reservoirs do fall below certain values. These limits may be functions of the time. We shconsider the case where these fields are relatively young so that developmedrilling activity will occur during the time period under consideration. Tproblem is to determine a schedule ofQi{t such that the profit over T years imaximum.By splitting up the period T into time intervals (k = 1,2 K) and bringin the physics of the problem, it can be shown that the condition that the fipressures are not to fall below certain minimum values assumes the form:

for all i and A;. The / s describe the characteristics of the fields and are knowThe righthand side is the difference between the initial and th e minimum permsible pressure of the i th field. T he variable is Qo which is the production ratethe i th field during the j th period. Additional constraints on the Qa s are ththe total production for any time period plus the crude oil possibly purchasfrom the outside source, Q,, be equal to the commitment for that time perio

= 1,2Furthermore, production limitations exist such that:

Q.y ^ Q.ymax which are simple under bound constraints. The objective function expressiprofit over the time period considered is:

NE Z c.;Q.; + c,Q, = max i=2

Q , t) Q , t)


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LINEAR PROGRAMMING IN OIL INDUSTRY 4which completes the formulation of the linear programming problem. The coeffi-cientsdj and cy are the profit per barrel of the i th reservoir at tim e j and cor-respondingly for purchased crude oil.Thus far, everything has been rather straightforward. But now, the time hascome to clutter up the theory with facts. Let us take a closer look at the coeffi-cients dj. If we plot revenue vs a particular production ra te Q.y, we get astraight line passing through the origin as shown in Figure 4. Cost vs Q,, isalso more or less a straight line which, however, does not pass through theorigin. The cost function is discontinuous at the origin, corresponding to a set-upcharge such as building a road, a pipe line or harbor facilities or installing a gas-oil separator. It drops to zero when Q,y = 0 because this corresponds to not yetdeveloping the field. Also shown on Figure 4 is profit vs Q,, which is the dif-ference between revenue and cost. The profit function thus is the straight lineshown plus the origin. Hence, we can say that profit from Q,, production isdiQij ~ Sijwhere s,y is zero if Q.y is zero and s,y is a constant if ^j > 0.Thisis a particularly difficult constraint. No general methods are available for han-dling this except a cut-and-try approach. This type of fixed set-up charge con-strain t occurs in many practical problems and we shall meet it again later on.One other complicating feature should be mentioned. Consider that during acertain time period, Q,y was at level A as shown in Figure 4 and tha t in thesucceeding tim e periodQi, y ihas dropped to level B . The profit at level Bis not obtained by following the profit line to operating level B but rather byfollowing a line as shown which is parallel to the revenue line. The reductionin level from A to B involves merely turn ing a few valves and essentiallydoes not entail any reduction in operating costs. If, on the other hand, we gofrom A to C in succeeding time periods, then we do follow the profit linebecause an increase in production necessitates drilling additional wells assumingthat all the wells at A are producing at maximum economic capacity. If weshould go from A to B to C in succeeding time periods and if A wasthe maximum field development up to that time, then in going from B to C we would follow the broken path as shown in Figure 4.Th is sta te of affairs can be handled by building the concept of productioncap acity into the model and requiring that production capacity never decreaseswith time. But this can be done only at the expense of enlarging the system ap-preciably.


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412 ;AIIVIX, CRANDALL, JOHN AND SPELLMANThere exist other factors and additional constraints which must be taken intaccount. As is so often the case, we are dealing here with a system which on thsurface looks rather simple but which becomes considerably more compleas we get deeper into it to make it more realistic. Nevertheless, the simpsystem or modest extensions of it enables an entire producing complex to bstudied thus providing a good basis upon which to build more realistic model

3 . Incrementa l P roduc t Cos t sLet us now leave the problems of petroleum production behind us and ventuinto th e petroleum refinery. As was indicated before, a great deal of work has beedone in this area. The few problems we shall discuss will be illustrative of what

going on in this field.We shall consider at first a simple but nevertheless instructive example. Ware indebted to Atlantic Refining Company for contributing this applicatio Birkhahn, Ramser and Wrigley, 1956). A refinery produces gasoline, furnacoil and other products as shown in Figure 5. The refinery can be supplied withfairly large number of crude oils. The available crude oils have different propeties and yield different volumes of finished products. Some of these crudes mube refined because of long-term minimum volume commitments or because orequirements for specialty products. These crudes are considered fixed and jdelgasoline and furnace oil volumes Voand Vr respectively. From the remainincrudes and from those crudes which are available in volumes greater than theminimum volume commitment must be selected those which can supply threquired products most economically. These are the incremental crudes. Denothe gasoline and furnace oil volumes which result from the incremental crudeby Vgand A F , and the total volumes fixed plus incremental) by For and VF TThe problem is to determine the minimum incremental cost of furnace oil as function of incremental furnace oil production keeping gasoline production angeneral refinery operations fixed.

The formulation of this problem is straightforward:oiF . = - Vo= 5

F, = - F. = AF, 6

Fixedcrudes

6 T Gasoline - Fur nac e O i l

-Other Products


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LINEA R PROGRAMMING IN OIL INDUSTRY 4 1 3F. ^ V t'7\

t _ ,mai \^()NS c,Vi = min (8)

here aoian d o^, are th e gasoline an d furnace oil yields of the t'th crud e, F , i'th. incremental crude and c,

arrel of the t ' th crude. This cost is made up of the cost of crude at the refinery,

Th e proced ure now consists of assuming a value for A F , and obtaining an op-

ed , how eve r, is valid o nly over rang es of variation of A F , which are suffi-e range of A F , the b asis m ust b e changed w ith a resulting change in t heFo r problems of this typ e, the so-called param etric pro gram m ing

I M 704

s also op tim um and repe ats this process until a term ination is reached.An actual problem was run with the model shown on P'igure 5. Thirteen

xed a t 14,600 barrels daily. T he results are shown in Figure 6 which shows the

eedom to pick the cheape st crude comb ination. Figure 7 shows the increm ental


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414 GARVIN CRANDALL JOHN AND SPELLMAN

I 95 000o

h. 90 000oc

Z 8^000o

80P00 Incrementai Furnace Oil Production BPD)7000 8000 9000 IQOOO 11 000F I G .6

2 0

- 10O8c

S-20

-3 0

7 0 0 0 8|000 9000 10 000 11 000Incremental Furnace Oil Productian BPD)

F I G .7


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LINEAR PROGR.\MMING IN OIL INDUSTRY 415fuel th a t is m ad e can b e sold, the n et cost of the furnace oil over-production wouldbethenegat ive of thevalue of heavy fuel indicating a credit we receive for in-creasing heavy fuel production.W e are temp ted, therefore, to trythe form ulation shown inFigure 8where wepermi tthediversion of some furnace oil toheav y fuel. T he eq uation forgasolineproduct ion remains unchangedbut thefurnace oilequation nowreads:

and the objective form is:=mm

(9)

(10)wheresi is as lack variable indicating th e volume offurnace oil div erted toheavyfuel andVH is thevalue perbarrel ofheav y fuel.It is notpossible, how ever, todivert unl imited amountsoffurnace oil into heavy fuel without violating heavyfuel s specifications. T he upp er limitonhow m uch furnace oil canbe mixed intoheavy fuel dep)endson thevolume of heavy fuel produced which in turn is re-lated to th e crud e slate, and would depen d also on the specificationsofhea vy fuel.Fur the rmore , if webring h eav y fuel in to the picture explicitly, the cost coeffi-cients used before must bem odified. Theproblem isbeginning to become morecomplex.Totak e these effects in to acc oun t would form thebasis of an entirelynew study. For purposesofth e presen t illustration, how ever, the situation can behandled roughlyasfollows. It t u rns outfrom experience and byconsidering thevolumes involved that the excess furnace oil production should be less than or atmost equaltoabou t 15 percent of theincrem ental furnace oilproduction if allthe excessis to go to heavy fuel andspecifications onheav y fuelare to be met.Therefore, the additional constraint

+ S2= 1.15AF (11)was addedtothe system w heres is a slack variable. This constraint insures thatno undue advantage is taken of thefreedom introduced by excess furnace oilproduct ion.Th e resu lts for this second formulationofth e problem are shown by the dashedlinesinFigures 6 and 7. The abscissa now refers tot ha t pa r t ofincremental fur-

ixedcrudesGasoline urnoce ilHeavy ueland


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416 GAHVIN, CRANDALL, JOHN AND SPELLMANnace oil production which leaves the refinery as furnace oil. Excess furnace ois produced below incremental furnace oil production of about 8600 bpd. Aboth at level, it is not economic to produce more furnace oil tha n required and cosequently, there is no difference between the two formulations of the problemConstraint (11) is limiting for incremental furnace oil production below abo7500 bpd . Figure 9 shows the composition of the optimum crude slate for thsecond formulation as a function of incremental furnace oil production. This useful information to have on hand. Note that no changes occur in the rangof incremental furnace oil production from 7500 to 8600 bpd. In this range, atual incremental furnace oil production remains fixed at 8600 bpd with any excess going into heavy fuel.

The modem refinery is a complicated system with strong interdependenamong the activities within it. The example just described illustrates this poiand shows the importance of the refiners experience in correctly isolating potions of the refinery which can be separately considered.4. Nonlinear Effect of Tetra ethyl Lead

The next two applications are concerned with partially nonhnear systemOne of the most conunon types of nonlinearity encountered in refinery operationis connected with the effect of tetra-ethyl lead (TEL). TEL is added to gasolinto increase the gasoline s octane number. The increase in octane number, however, is not a linear function of the TEL concentration. The first cc of TEL ha pronounced effect on octane number, the second cc, however, has a smalleeffect, and for the third cc the effect will be still smaller. The maximum concetration permitted in motor gasoline is cc per gallon.

35000


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LINEAR PKOGRAMMING IN OIL INDUSTRY 4 7

A great deal of work has been done in the past few years on gasoline blendingby linear programming. The problem is to blend the different stocks coming outof the refinery into gasolines having specification properties and to do it at mini-m um cost. In addition to octane nu mb er, other properties such as vapor pressureand various distillation points must be considered. All the important propertiesblend linearly on a volume basis except for the eilect of TEL. To get around theTEL difficulty, it was usually assumed in setting up the linear programmingmodel th a t th e gasoline was shipped out a t maxim um T E L level of 3 cc per gallonor the T E L level was arbitra ri ly set at some lower value. In any event, TE L didnot enter the system as a variable and thus wa.s not permitted to seek its ownlevel as determ ined by m inim um cost. To get a feeling for the order of m agn itudeof money involved here, consider an average TEL concentration of 2 cc per gal-lon. At a price of TEL of about $2 per liter, a TEL bill of about $180,000 resultsfor each million barrels of gasoline produced. Many companies produce of theorder of tens of millions of barrels per year. Thus, even a reduction of only a fewper cent in lead concentration begins to look big when translated into moneysavings.Consider now the general blending problem shown in Figure 10. The streamscoming out of the refinery are split three waysto Premium grade gasoline, toRegular grade gasoline or to temporary storage. Additional stocks may be pur-chased from outside sources to go into gasoline. TEL is one such stock. Thegasoline blends must satisfy a variety of quality specifications such as vaporpressure, dist i l lat ion points and octane number.In setting up this problem in linear programming language, we have first of allthe usual tjTses of linear constraints which relate the properties of the stocksand the fraction of their volumes to the desired properties of the blended gaso-line. Ther e is no difficulty here u ntil we get to the octan e cond ition. T he relatione have i s tha t :

^ ^ z ^ *

here ONd is the clea r octane num ber of the t ' th stock (its octane num berno T E L in the stock ), F< is i ts volume,ON is the specified m inimum octan e AON is the octane increase due to lead. The first term on the leftnts the clea r octane num ber of the blend under the assumption of l inear

so-called ble nd ing octane num bers instead of actu al ones, a sufficientlyLe t us now tak e a closer look at th eAON term. If, for a specified octane numberf the b lend, we plot the difference between t he clear octan e numb er of the b lend


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418 GARV IN CKANDALiL JOH N AND SPELLMAN

PREMIUM

REGULAR

EXCESSF i o . 10

to 15 F I G . 11

From past experience it is usually possible to estim ate within reasonablimits what the lead susceptibility of the blend is going to be. We can then construct curves as shown in Figure 11 for the estimated lead susceptibility and fosusceptibilities deviating from tha t value by say 1 0 per cent. Data are avaidothis.Letus nowimagine tha t into


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LINE R PROGR MMINGIN OILINDUSTRY 4 9

be parallel. The bands can be interpreted in the following manner. Instead ofhaving only one type of TEL, we have, in this case, five fictitious typesTEL 1through TEL 5. Each band corresponds to one type of lead. These fictitious leadshave two important properties: they do not saturate, i. e., their efifect on octanenumber is linearly related to the amount of each lead present, and the effect oflead susceptibiHty onAON is independent of the TEL concentration. They areot all equally effective, however, as far as increasing the octane number isoncerned. In view of the concavity of the function, lead1is much more effective

^ i ^ ^ ^ 13)hereSi is the lead susceptibility of the i th stock, L, is the amount of lead of

j present in the blend and a, b,and m, are constants determined from the

w y < m , . 14)

availability restrictions on the L/s for otherwise we would satisfy the Li because octane wise it is cheapest and, as a result,

linear function of susceptibility, corresponding to the bounding straight lines

dj and ey again are constants determined from the curves. SubstitutingAONinto equation 12) and multiplying through by ^ F , , we

We are not yet quite through, however. Each grade of gasoline has two octane

11 and all four must be represented by the procedure justrent TEL types but also between TEL going into Premium or Regular to


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4 2 0 GARVIN, CRANDALL, JOHN AND SPELLMANThey were separated in the matrix for mathematical reasons only. The samtype of cons traint applies to Regular. Hence, we must stipula te:

/ .,Li j Premium ^-1 = 2 - i Premium r-/ Lij Regular F-l 2-i ^i . Lij Regular F l 2 i ^i egulsr f ^

Finally, the objective will be of the form: + CLY, (^ i premium + Lj Regular + ' = min (1

whereCiis the un it cost of TE L and the dots indicate other terms whatever thmay be.It is clear that the optimum solution will make physical sense only if the fictitious leads for the limiting octanes are involved in the solution in a physicallrealizable way. Consider the situation where, let us say, lead and 2 are at theupper bound, while lead3and 4 deviate from their upper bound and lead5is zerThis is not a physically realizable situation because of the gap existing betweelead 3 and 4. This, however, could never occur in an optimal solution because othe concavity of the TEL response curve and because we are aiming to use alittle TEL as possible. If a gap exists between L, and Lj+i,it will always be moeconomic to reduce the level ofLj i and pushLj up to its upper bound becauLjis more effective octane-wise thanLj+i.Therefore, we have the assurance ththe fictitious leads for the limiting octanes always will be involved in the optimsolution in a physically realizable way. This will not necessarily happen, howevefor those leads which belong to the nonlimiting octane specifications. As we havtwo octane specifications for each grade, there will in general be one octane ieach grade which is limiting while there is give-away on the other two. The computer will have no incentive to meet or exceed physically realizably the octanrating for which there is give-away. It cannot make any money by it becausthe to ta l amoxmt of TEL already is fixed by th e octane ra ting which is limiting arequired by constraints (16) and (17). The computer simply picks that octanwhich is limiting, works on it to meet it most economically and lets the chips fawhere they m ay, as far as the o ther octane rating is concerned. The optimal solution will still be perfectly satisfactory because the exact value of the give-awafor the nonlimiting octane does not affect the solution.Let us now briefly discuss the results for a case where this approach to the T Eproblem was tried. The da ta were based on an actual situation that existed in onof our refineries a few years ago. In this case, gasoline production was fixed atgiven level. The objective was to minimize cost of TEL minus credit for excestocks. Two solutions of the same problem were available to which the lineprogramming solution could be compared. One was the solution tha t was actual


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LINEAR PHOGRAMMING IN OIL INDUSTRY 421levels between these two solutions and the one obtained by linear programmingThe reduction in TE L is clearly eviden t. The solutions should, of course not becompared m erely on the basis of lead savings. As can be seen from the objectivefunction, the credit for excess stocks must also be considered. The net savings ofthe linear programm ing solution still were substantial.

5. Nonlinear EflFects of Variable Cut PointsIn the blending problem just discussed, the volumes and properties of thestocks coming out of the refinery were given and the problem was to blend thesestocks to make certain end-products in the most economical way. The refineryas a whole was fixed and the optimum blending solution gave us little or no in-

formation about what the optimum refinery operation should be. This, of course,is a tremendous problem because the refinery abounds with nonlinearities andalltypes of mathematically peculiar constraints. One interesting step toward theover-all refinery optimization was discussed recently by Schrage (1956) wherelinear programming was combined with the method of steepest ascents.The next application we would like to discuss is an attempt to reach back intothe refinery just a litt le way and optimize with respect to gasoline blending a fewof the operating conditions. The conditions we shall consider are the re-run stillut points. A re-run still is a unit within the refinery which separates a stock intoight and heavy components. The operating temperature of the unit determinescut poin t between the two components. The volumes and the propertiesf the cu ts are nonlinear functions of the cut point. The cut point can be

One way of handhng this problem is to introduce fictitious stocks as shown Ti Tj, andT3

TABLE ITEL Content in cc/gal. of BlendsHand Blend No. 1

2.960.31

Hand Blend No . 22.650.56

Lin eu Prog. Blend1.510.72

B I


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422 GARVIN, CRANDALL, JOHN AND SPELLMANwhich yield small fictitious cuts B and C and major segregations A and D .The volumes and properties of the fictitious cuts are determined such thawhen they are combined linearly with the major segregations, correct volumeand properties result. Consequently, the fictitious cuts sometimes have abnormaproperties when considered by them selves.The major segregations and the fictitious cuts are now made available to thegasoline blend just as if they were actual stocks coming out of the refinery. Thresulting optimal solution then is examined to see what happened to the fictitioucuts B and C in the shuffle. A number of thingsc noccur as shown in Figur13.Because of the natural variation in properties with distillation temperature ofthe stock, it usually happens th at in the optimum solution A goes entirely toPremium and D goes entirely to Regular. If B goes to Premium and Cgoes to Regular, we can conclude that the cut point should be at T2 If both B and C go to Premium, the cut point should be at Tzor at a higher temperature; of they both go to Regular, it should be at Ti or at a lower temperature . There is nothing in the program that prevents B and C from splittingIf B splits and C goes to Regular, the cut point should be between Ti andT2 If C splits and B goes to Premium , the cut point should be between Tand Tz These five situations are the normal ones encountered most of the timbecause of the normal progression to higher sulfur content and lower octane athe cuts get heavier. Occasionally, however, it may happen tha t both B an C split in such a way th at the fraction of C going into Premium is greatethan the corresponding fraction of B or th at B goes entirely to Regulaand C goes entirely to Premium. These are situations which are not reahzablin practice becausewehave only one cut point in reality. To prevent such situtions from occurring, additional constraints are imposed on the system whicstipulate tha t the percentage of B going into Premium should be greater thanthe corresponding percentage of C . These constraints will insure th at th e optimal solution will be physically realizable without too much trouble.

AB_c_D IL

ABD

ApPcD

1 _Ap

, cD

^ Ap

D


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LINEAR PROGRAMMING IN OIL INDUSTRY 4236. Cost Coefficients

Before leaving the field of refining, let us consider the effect of cost coefficientson optimum gasoline blending. If the objectiveisan economic one, costs or valuesave to be determined for some of the stocks that are produced. This can be aomplicated problem. In the case of the blending example discussed previously,he objective was to minimize lead costs minus credit for the excess stocks. Thisto be used up. The situation was complicated further by the fact that some ex-ess stocks were earmarked for shipment to another refinery. This meant that

mined but they must be realistic for the solution to have meaning and aAs an illustration of the effect of the cost coefficients on the optimal solution,

ll quality requirem ents. Tw o cases were run which were identical in all respectseased by a small amoun t. StocksAthrough L were available for blending. The

shown on Figures 14 and 5where the composition, volume, and TELtent of the gasolines are compared for the two cases.Asexpected, the optimum

enter Premium for case 1. Regular loses its B content and part of C

r mium

1 . ^K1GDBA

i

LK1GDCB

A


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424 QAKVIN CRANDALL JOHN AND BPELLMANRe

LJ1HDCB

- ~ ; ; ~ ^

~ ^ _ ^

J1

F

Dc

Cose I C o s e ;Fio . 15

Premium and extensive utilization of F in Regular. As the change in gasolinevalue was not drastic, it can be seen that we are dealing here with a systemwhich is rather sensitive to the price structure.7. Distribution to an Expanding MarketLeaving the refinery with all its problems behind us we shall now turn to thearea of marketing and distribution which has problems of its own. The classicexample of a problem in th is area is the transportation problem. A great deal ofwork has been done on this, particularly by oil companies. The first applicationolinear programming that we would like to discuss in this area is a type of trans-portation problem which, however, has some complicating features. We are in-debted to Atlantic Refining Company for contributing this application.Consider refineries (i = 1, 2 m) and n bulk terminals or distributio

centers (j = 1, 2 n) as shown in Figure 16. At the present tim e, the refinerare producing at levels P , and the demands at the bulk plants are D , . We mayconsider the sum of the P i s to be equal to the sum of the Dy s so that all thedemands are met. Assume now that we find ourselves in an expanding marketProjections are available for what the demand at the different bulk plants isgoing to be, say, five years from now. Denote these projected demands by DjTo try to meet the increased demands, we must expand refining capacity. D enotethe increased production by P i + e where e, is a variable denoting the am ountoexpansion. We must also increase the capacity of our bulk plants. The expansionof refining and bulk plant capacity costs money and an upper bound exists onhow m uch can be spent on over-all expansion. This upper bound is such that it i


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IilNEAB PBOGRAMMING IN OIL INDUSTBT 4 2 5

Fia 16

This problem can be formulated as follows. The total production leaving theh refinery must be equal to the old production plus the expansion. Hence:nE Xii= p .- } - e,, t = 1, 2, m (19)

i to j The amount received at the j tht must be less than or can a t most be equal to the projected demand inmZxij ^ D /, J = 1, 2, , n (20)

iR is the unit cost of expanding refinery capacity at i, then the total cost ofinery expansion is ^ c.Be.. In considering the cost of bulk plant expansion,ally be reduced while others expand so as to be able to take full advantage of

a bulk plant does require capital but a contraction does not because itmply means that shipments to the bulk plant are reduced. To handle this situa-

m1 ^ X ij - D j = s y * - s ~ j = 1 2 n 21 )

ystem where s / and sf are non-negative variables. We also stipulatem n

ya is the unit cost of bulk plant expansion and M is the maximum ex-


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QARVIN, CRANDALL, JOHN AND SPELLMANlast constraint. If j con tracts , thenSj~will be in the basis and there will be expansion cost.

Finally, the objective function is:m m n23 23 ijXij 23 c.ei - 13CjBSj ^ = max i i 1 iwhere Ci, is the profit per barrel shipped from i toj.This formulation is satisfactory as long as the new shipments to the contrac ted bulk terminals do not fall below a certain value (they may even go zero).This is a situation which is analogous to the one encountered in discussinthe model of a producing complex. The plot of profit at thej th bulk plant asfunction of shipments to the bulk plant is again a straight line displaced from thorigin because of a fixed overhead. The actu al profit function is again the stra ighline plus the origin. Thus, our objective function should really be :

n / m \ m n2 3 ( 2 3 dj^ii - ay) - 23 c.fte, - 23 CJBS,^ = max (2where:

[0 if 23 .Tu = 0' ^ (2const, if 2^Xij> 0

but no general method exists for handling situations of this type. However, if turns out in the optimal solution that none of the bulk plant volumes contract bsubstantial amounts the solutions will be useful.8. Service Station DeliveriesLong Range

Having considered the link of refinery to bulk terminal, let us now considthe last link in the chainthe flow of products from bulk terminal to servicstations. Consider the situation shown in Figure 17. We are given the location service stations and the roads connecting them. The small circles are the servistations, while the large circle denotes the bulk plant which supplies them btruck. Each service station, k, requires a delivery of Dt gallons of gasoline (fsimplicity, let us assume only one grade of gasohne). Different truck types, dnoted by the index s, are available for making the deliveries. The trucks diffin regard to carrying capacity and operating characteristics. We have a numbof trucks of each type available for the operation. The problem is to devise delivery schedule such that the transportation cost is minimized.We are actually dealing here with two different types of problems dependinon whetherw look at this operation from the long-range or the short-range poi


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LINEAR PROGRAMMING IN OIL INDUSTRY 427

F I G 17

be made to each station during that period in order to meet the demand. Underthese conditions, the problem becomes a transpo rtation type problem with trans -shipment of goods. The trans-shipment feature comes about through the factth a t if a truck leaves the bulk p lant and makes deliveries to, say, service stations1, 2, and 3 in that order, then the gasoline destine for station 2 is trans-shippedvia station and the gasoline for station 3 is trans-shipped via stations and 2.Some work has been done on the trans-shipment problem, Manne, 1954,Kalaba and Juncosa, 1956, Dwyer and Galler, 1956, Orden, 1956) in connectionwith aircraft scheduling and communication networks. Our problem here isslightly different but the general approach is the same. The key to the mathe-matical formulation lies in the use of triple indices. Adopt the convention thatthe first index refers to the point of departure, the second index to the inter-mediate destination and the third index to the ultimate destination. If i/.y*denotes the number of gallons shipped from i toj destined for k then we canwrite:all j k b u t j 9^ k

allk] C 12 yojk=

262728

The left side of 26) is the sum of what arrives atj from all points but destinedfor kwhile the right side is the sum of what leavesj for all points destined for k.These two must be equal because we do not wish to accumulate anjrthing at jdestined for k.Equation 27) states tha t the sum of what arrives at k from all


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