Linear Programming Solves Gasoline Refining and Blending Problems The refining of petroleum i s fairly complex and results in production of a number of separate gasoline streams from several distinct refinery processes. The components must thereafter be treated and blended to produce salable products. It i s common practice to determine the cost of manufacturing each of the gasoline components so that the refining and blending operations may be related to the ultimate costs of the finished products. Calculation of the most economical refining and blending operations, as illustrated here, is very well suited to solution by the method of linear programming.
GIFFORD H. SYMONDS Esso Standard Oil Co., Linden, N.
refiniug of crude petroleum the production of FpecificaTHE tion products is complex process. simplified of the operations for fuel products is sho~vn Figure for
a
; i
thpram in
usmil refiiiwy 1. The crude oil is fractionated by primary distillacion into a number of intermediate products requiring further processin( The light nnphtha requires only treating (not sbown in diagran and is available as part of the gasoline pool for blending int premium or regular grades of gasoline. The heavy nnphtha may also he tre:ited and blended or it ma:-, in whole or in part, he sent t o reforniing where its octane rating is improved. The reformate is then available for blending. ;iheating oil stream is also taken from the primary distillation, coml)itictl n-ith heatiiig oil streams from cracking and visbreaking, and trcjated t o produre heating oil product. The production of heating oil may be r c y i lated to some extent by cracking part or all of the heating oil along v i t h the gas oil. The gas oil cracking produces light and heavy cracked naphthas of high octane rating. These are treated and then blended. The distillation residuum is sent t o vislxeaking producing visbreaker naphtha and residual fuel oil. T h e refined products from the crude oil then are premium arid regular gasolines, heating oil, and fuel oil. Our stndy of t h r rrfinery processing :md gasoline blending pro1,leni is coricerircd irith maximizing profit. T h e profit is composed of the following items: Profit = sale of products - Cost of crude oil and 'LEIJ operating rsprnue
Furl oil is the loveet priced product, wiling belox the cost of crudc oil. The requirements for fuel oil are generally not dr,finite, and it is assumed that components not included in o t h r products may be included in fuel oil. 111ordrlr t o siniplif,v tlic. problem the f o l l o ~ ~ i ~rcntrttrigement ig in m n d ~of the profit function. Profit
=
sale of protlucta :ibove fuel oil price - cost of TEI, cost of crude oil above fuel oil price - openting expense lew fuel consunrption
J.
eration without shutting down and tlir ni:isimum operation as limited by equipment capacity. The rrgular and premium grades of gaaolirie are limited by their maximum recjuiremrilts. Thi. heating oil production will equal the mnsiniun~requirement if the availability and profit are sufficient. I n t.his problem i t is assumed that the availability of heating oil will exceed the maximum requircment for heating oil and that it is profitable to mect the maximum requirement. I n addition, the producation of regular nncl premium grades of gasoline is deperitlent on blending the gasoline components t o meet certain quality specifications. In some cases it might not he feasible t o use all the available components arid meet the specifications. I t might also be more prcifitnble to discard part of some Components to fuel oil. I t is d s o not certain that the crude rate may be increased profitably to meet all the requirements. Because of these reasons, in this problem, the gasoline production will he allowed to vary hplow niitl up to the maximum requirements. The quality specifications take the follo\ving form for a minimuni (not less than tL-pe) Fpecification: Sum[(hlending quality of each component) X (quantity of each component used in blend)] 2 (qiecification)(quantityblended) ttnd the following form for a maximum (not more th:m type) spccificntion : Sum[(Bl. Qual. Comp. ) ( g u a n . Comp. Bl.)] ixxificutiori i o r i n ~niay iw reduced
5 (Spec.)(Quan. B1.1 to tlir
follotyiiig sim-
plificd spwificatioii form$:
St;m[(lIiri. Spcc. - 131. (Jual. Comp.)(Qunn. Comp. El.)]
I0
and
Sum[~,U1.(&d. C'omp. - I I a x . Spec)(Qii:in, Comp. 131.)] I 0
The limitations to 0 . x oper:Ltions are set 11)- tiic requirements for products and the capacity for pr Minimum capacity (. crude rate _< I Regular gasoline 5 niaxiniuni req Premium gasoline 5 ma-iirrium requirement Heating oil = maximum requirement T h e crudc rate ia limited between the iniiiiniiini practicul op394
100,000
5
crude r:tte, k)bl./day 5 12.5,000 Regular gasoline I38,000 Premiiini gasoline 5 30,000 Heating oil = 27,000
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 3
ODerations Research T h e amount of reformate which can he produced is limited, in this problem, by the reforming capacity.
and the follon-ing constraints for specification qualities: for regular gasoline
Reformer feed bbl./daj-
-
lO)(Quan. Comp. SI.)] I0 Z((l3l. Vapor Pressure Comp. Z[(:30 - B1. Volatility 70/200" F. Conip.)(Quan.Comp. SI.)] 5 0 .z[(;O - El. Volatility %/300° F. Comp.)(Quan. Comp. Bl.)] 5 0 Z[f!KI - 131. Octane Rating Comp.)(Quan. Comp. SI.)] 50 and for pwmium gasoline Z[(Bl. T-xpor Pressure Comp. - lO)(Quan. Comp. Bl.)] 2[(12.5 - B1. 1'01. %/200" F. Comp.) X
50
(Quan. Comp. SI.)]I 0 Z[(Bl. Volatility 70/2000 F. Comp. - 53)(Quan. Comp. S I . ) ] 5 0 2[(86 - B1. Volatility %/300 Comp.)(Quan. Comp. SI.)] 5 0 Z[(94 - BI. Octane Rating Comp.j(Quan. Comp. S I . ) ] 50
5
15,000
We have now developed all the technical information required t o write the equations for the constraints in this problem. These equations cover the material balances for availabilities, requirements, and capacities and the quality balances for specifications. The material balances for availabilities include the following: Availability of Light Naphtha :
+
+
L S to fuel oil L?j t o regular gasoline L N t o premium L S from additional operagasoline = L N from base operation tion in algebraic form. Xi
+
+ Xzo + Xzl = 16,900 + 0.169X19
where Since i t is always profitable to blend t o maximum vapor pressure and since the specifications for both regular and premium grades are the same, it is possible t o eliminate these constraints by expressing t h e component yields and qualities as they would be if preblended t o specification 10 vapor pressure. T h e yields and qualities of gasoline components (preblended to 10 vapor pressure) from the minimum (base) crude oil operation are:
Yield,
70
Blending Octane Blend, Blend, In In %/200° ~ 0 / 3 0 0 " regupreF. F. Iar mium
Light naphtha
(LS)
Heavy naphtha (HS) Light cracked naphtha(LCS) Heavy cracked naphtha(HCS) Visbrenker naphtha ( I T )
16.9
73.5
96.0
87.3
91.1
14.5
10.5
46.0
81.0
81.1
15.8
72.5
98.0
96.6
97.3
12.0
92.3
92.7
56.0
86.5
87.4
5.8 2.0
-5.5 26.5
Xzo and Xzl represent bbl./day for each indicated disposition of LN X18represents bbl./day of additional crude oil above minimum XIrepresents amount of LX component not used in gasoline blending and is, in fact, a "slack" variable This equation m a y be restated as a constraint: 16,900
XI
=
- 0.169Xi9 4- X z o
+ Xu
(1)
Availability of Heavy Naphtha :
+
+ +
H S t o fuel oil H?;to regular gasoline H S to premium gasoline H S t o mild reforming for regular H N t o mild reH S t o severe reforming for regular HN forming for premium t o severe reforming for premium = HK from base operation H S from additional operation in algebraic form.
+
+
+
+
1XZ7= 14,500 + 0.145X10
0.807 where
T h e yields and qualities of gasoline cornponcnts (preblended t o 10 vapor pressure) from additional crude oil operation above the minimum are: Yield,
70
LN HN
LCS HCN
16 9
14 5 41 1 143 2 0
Blend, Blend, Blending Octane ?70/2000 ~ 0 / 3 0 0 " In In F. F. regular premium 73 5 87 3 91 1 96 0 10 5 46 0 81 0 81 1 44 5 84 0 95 5 95 8 -75 9 0 911 905 265 560 865 874
The much higher j-ields of cracked naphtha from the additional crude oil operation compared n-ith the base crude oil operation are due t o the additional cracking of heating oil in order t o maintain the over-all production of heating oil constant. Thus the additional crude oil will not produce any additional heating oil. Figure 1 shows the optional process of reforming which may be used to increase the octane rating of heavy naphtha, if justified. T h e yields and qualities of reformate a t two different levels of operation are:
Tield,
%
Mild operation reformate Severe operation reformate
March 1956
Blending OctaneBlend, Blendjo I n In 70/2000 %/300 regupreF. F. lar mium
XZ,X s e , and
X23 represent bbl./day for each of the direct dispositions of heavy HN X2, and X Zrepresent ~ hbl./day disposition of the mild reformate X26 and Xzi represent bbl./day disposition of t h e severe reformate X18,as before, represents bbl./day of additional crude Coefficients 1/0.862 and 1/0.807 are bbl./day of H S to reforming per bbl./day of mild and severe reformate, respectively
This equation may be restated as a constraint: 14,500
+ +
+
+
Xz - 0.145Xig X22 Xz8 1.16OXpa 1.16OXzj 1.239Xz6
+
+ 1.239Xa
(2)
Availabilities of Cracked Naphthas from Base Operation : 15,800
XB
5,800 =
xc
+
+
Xs8
+ Xzg
+xu
x30
(3) (4)
where
Xa and Xa are bbl./day of LCN and HCK, respectively, from base crude operation sent t o fuel oil X 2 8and X3o are bbl./day of L C N and H C T sent, t o regular gasoline Xz5and x31 are bbl./day of L C S and I l C S sent t o premium gasoline Availabilities of Cracked Naphthas from Additional Operation :
- 0.411Xls
86.2
10.5
62.0
92.9
94.4
0 = Xj
80.7
15.5
62.0
94.3
96 0
0 = X K - 0.143Xis
INDUSTRIAL AND ENGINEERING CHEMISTRY
+
3-32
4- X34
+ X,, +X?a
(5)
(6) 395
where Xs and X p are bbl./day of L C S and HCN, respectively, from additional crude operation sent to fuel oil X32 and Xt4 are bbl./day of LCN and H C N sent t o regular gasoline XS3and X85 are bbl./day of LCN and H C N sent t o premium gasoline
Capacity for Additional Crude Operation :
Availability of Visbreaker Naphtha : 2000 =
x 7
+ X36 +
- 0.020Xl0
octane rating multiplied b y the bbl./day blended, and the coefficients are derived from the octane blending values and the limiting specifications. The capacity restrictions in this problem are concerned with the additional crude run and with the reformer feed rate a8 follorvs:
25,000 = Xi7
(7)
x 3 7
where
where X7, X3k and X3?are bbl./day of VN sent to fuel oil, regular, and premium gasolines, respectively T h e only product requirements affecting this problem are those for regular and premium grades of gasoline as follows: Requirements for Regular and Premium Gasolines : 38,000 = XS
+ XZO +
x 2 2
f XlC xzs
30,000 = X Q f XZl f x23
+
x 2 5
x 2 9
+
f
1 2 6
xa +
f x30f f
x Z 7
+ X3l +
x34
$x 3 3
f XIS
f X3e (8)
+
x 3 7
(9)
where
Xs and X g represent bbl./day of requirements for regular and premium gasolines, respectively, which will not be supplied Other variables represent bbl./day of components which are actually blended
x 1 7
is the bbI./day of unused crude capacity.
Capacity for Reforming Operation : 15,000 = xis
+ 1.16Oxz4 $- 1.16oxz6 f 1 . 2 3 9 X ~f~ 1.239Xzr (18)
where xi6 is the bbl /day of unused reforming capacity. The foregoing 18 equations represent the set of constraints for this problem These 18 equations involve 37 variables, so that an infinite number of solutions to the problem should be possible. K e shall expect the method of linear programming t o select the best solution for us according to our objective. I t should be observed that the variables X I to X37 are quite special since negative values for these variables would be meaningless in our problem. We cannot run negative crude oil or blend negative components. We shall expect the method of linear programming t o protect this condition of nonnegativity. The general linear programming method may be expressed algebraically as follows including the constraints: Z,a,,X,
It is necessary t o include equations for the product quality specifications as follows:
with X j
Volatility Specifications :
a n d i = 1 , 2 , . . m; j = 1 , 2
+
- 43.5xzo 19.5x22 f 19.5Xu f 14.5x2.5 42.5x28 f 35.5x30 14.5Xaz 37.5x34 3.5X36 (10)
0 =
x i 0
0 =
x i 1
- 26x20
0 =
x i 2
- 31x21 f 32x23 f 32x25
+
+
f 24x22 -t8x24 8x26 28x2s 58x30 - 1 4 x 3 2 61x34
+
30x2, 0 =
+
-
+
+
f 27x27 48x81 - 2x33 f 50Xas
(13)
Development of the Objective Function
+ 30X37 (14)
Octane Rating Specifications : =
0
XIS x i 0
+ 2.7Xzo + 9.oxz2 - 2.9X’na - 4.3Xze 6.6X2a - 2.3X1c - 5.5X3z - 1.1Xar + 3.5Xis + 2.9x21 + 12.9Xza - 0.Ly25 - 2.0x27 3.3Xzg + 1.3xai - 1.8X33 + 3.5X35 T 6.6X37
where XI6 and
396
Xi6
and the objective 2,C,Xj = maximum (or minimum)
(12)
where each Xl0 t o XI, represents the “giveaway” from the limiting specification multiplied b y the bbl./day blended. Thus, for example, if XIO= 60,000 and the actual amount blended is 30,000 bbl./day then the giveanay is 2$70/2000 F. above the minimum specification of 30%/200° F. for the regular grade. Thcse giveaway variables allow the limiting specifications to be approached economically but not necessarily met exactly. The coefficients for the components blended, of course, are derived from the volatility blending values and the limiting specifications.
0
. . . n; n > rn
+ 16Xp7
+
+
20
(11)
f 20.5x21 - 42.5X23 - 42.5x25 - 37.5Xn 19.5X~y- 58.5X31 - 8.5X33 - 60.5x36 - 26.5Xz
+
b,
+ 14x36
1 1 3
+
=
I n this problem there are 18 equations so that i runs from 1 t o 18; there are 37 variables so that j runs from 1 t o 37. I t is next necessary to develop the objective function, In the linear prcgramming problem the C j values are the unit profits associated with each Xi. These values should be determined for this problem as follows:
Xla - 10Xzl f 40x23 f ~ U Z24-Xz7 S 12X2, T u 3 1 2x33 7iXa5
0
+ Xis
(15)
(16)
represent the giveanay values in units of
I t is advisable to set up a profit or loss balance in accordance n i t h the general procedure already given: Sale of products above fuel oil price less cost of 3.0 cc. of TELIgal.
Plup saving in T E L due to octane giveaway Less cost of base crude operation including crude cost above fuel oil price Less cost of additional crude operation including crude cost above fuel oil price Less cost of reforming operation without fuel Equils the profit
(38,000 - X,) X 2.550 for regular gasoline (30,000 - Xg) X 3.373 for premium gasoline 127,000 X 1.740 for heating oil XI6 X 0.310 Xis X 0.264
+
+
+
- 100,000 X 1.476 Xi9
X 1.954
-
Iri this expression, the gross iricome from the sale of product5 is composed of the income from the sale of maximum requirements less the potential but unrealized income from the sales deficiencies Xgand X S . It is assumed that the maximum T E L
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 3
Operations Research
- -
LfGHT NAPHTHA
HEAVY NAPHTHA
I
CRUDE O I L
-I
I
I
I
I
0 0
c n
REFORMING
PREMIUM
GASOLINE
w
z
g
:
REGULAR
CRACKING
GASOLINE RESID.
t VISBREAKING
J
I
I Figure 1.
HEATING OIL b
Typical refinery operations for fuel products
formate for the mild and severe operations, respectively. T h e profit function may now be rearranged in terms of the problem variables:
content of 3.0 cc. igal. has been used in the gross income calculation. If it is possible t o blend the gasolines to higher than specification octane rat.ings at 3.0 cc. of TEL/gal. content, it is then permissible t o reduce the T E L content until the limiting octane rating specification has been reached. This will not affect the gross income from the sale of products, but it will decrease the purchase cost of T E L . I n this way, the octane improvement operations of reforming and running additional crude oil (with a high cracking ratio) can be evaluated economically against the cost of marginal T E L content. S o w the relationship between octane rating and T E L content is distinctly nonlinear, but in the range 1.5 t o 3.0 cc./gal. a linear approximation may be used nith good accuracy. The values of 0.310 and 0.264 dollars/bbl./unit of octane rating have been obtained from these linear approsimations for T E L savings due t o octane givea\\-ay for regular and premium gasolines, respectively. The unit costs of $1.416 and $1.954 per bbl. of crude oil for the base and additional crude oil operations, respectively, include the cost of the crude oil above fuel oil price plus the operating cost for all refinery operations less fuel consumption except the optical process of reforming. T h e unit costs of reforming are 48.3 and 52.0 cents per bbl. of re-
bi 16,900 XI 14,500 = 15,800 = 5,800 =
o= o=
2,000 = 38,000 = 30,000 =
O =
o =
FUEL O I L
Z
=
+
-
38,000 x 2.550 30,000 X 3.373 f 27,000 X 1.740 100,000 X 1.476 - 2.550X~ - 3 . 3 7 3 x ~4- 0.310x15 f 0.264X16 - 1.954Xi~- 0.483Xzr - 0.483Xx - O.52OXzc 0.520x2,
In this expression the coefficients of the variables X , t o X I , are the C, values in the objective function for our problem. Four of these coefficients are for the slack variables X,, X P , X I 5 ,and X16.
A4distinguishing feature of linear programming is that a feasible solution t o the problem is always apparent a t every completed stage of calculation. This solution consists of the evaluation of a group of the variables equal in number t o the constraining equations. Thus, 18 of the 37 variables I d 1 be “in the basis” a t every completed stage of calculation. Each of these basic variables will appear in only one equation. All other variables Kill not be in the basis and mill have zero values. T h e arrangement of the basic feasible solution ~v-111be:
Xz X3
X4 x5
XB XI
XS x9
x! 0
o= o= o =
o=
O = 25,000 = 15,000 =
March 1956
INDUSTRIAL AND ENGINEERING CHEMISTRY
397
Table I . PO 16,900 14,500 15,800 5,800 0 0 2,000 38,000 30,000 0 0 0 0 0 0 0 25,000 15,000 - 100,620
PI 1
Problem Matrix for Refinery Processing and Gasoline Blending Problem P4 P5 PE Pi Pa PS PI2 PI1 P12 P13 Pl4
P?
P3
pi6
1 1 1 1 1
1 1
1 1 1 1
1 1 1
........................................................................................................................ c,
-2.550
-3.373
+-0.310
Table I (Continued) PlE
PI7
PI 8
P1o -0.169 -0.145
P7Q 1
P3
P22
P23
P24
P25
P2 6
P27
1
1
1.160
1.160
1.239
1.239
1
-0.411 -0,143 -0.020 1
1
1
19.5 24
-43.5 - 26
-2.9
9.0
1
-4.3 -2.0
-0.4
12.9
2.9
27 -37.5 24
32 -42.5 24
32 -42.5 40
2.7
1 14.5 8
19.5 8
-31 20.5 - 10 1
1 1
1
1
1 1 +1.954
... ...
-1.717
-2.607
....................
t0.240
. . .. . . .
+0.033
......
1.160 -2.996
II .
-0.483
- 1,954
+0.264
1.160 -2.966
- 0.483
1.239 -3.363
1.239 -3.381
-0,520
-0.520
.....................
Table I (Continued) P28
Pzo
1
1
PdO
P31
1
1
Pat
Pa
1
1
P37
P35
1
1 1
1
1
1 1
-42.5 - 28
- 30
-4.244
-3.263
-3.030
50 60.5 77
-2 -8.5 2
-4.255
-3.848
16 -26.5 30 3.5
-1.1 3.5
-1.8
1.3
-3.3
1
3.5 14
37.5 61
-5.5
-2.3
-6.6
-4.596
- 14.5 - 14
48 -58.5 74
19.5 - 12
1
1
1
35.5 58
-2.891
1
1
1
-2.449
6.6
-1.465
-1.631
+Z
.......................................................................................................................
398
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 3
Operations Research This t : t h I e a u rc~;irrs:~nts t h r li:i$ith feasihle solution for the proliwhich are the slack variables i t 1 Iem. The variables SI t o Sly the constrniniiig equiition. are ill the basis. -111 other w r i :ihles (XISto Xj;i havc zero values at this stage. In this solution, vtiritible 9! h:ia the v:due of lti,- of heating oil in addition to fuel oil. This is obviouslliiot a ver?- profitalde solution. and it is desirable t o change tht, solution I ) > - suhstitutirig different variahles into the Ixisis. 111 ’ linear progr:iinining the suhstitution of variables into the b ip regulated 1))- the olijective function, n-hich selects variables on of thixir :ihiIity t o improvc thc functional value-in this c;isc>!the profit. I n order t o do this, the ohjective function miid dwai-s lie expre3sed in terms of only those variables riot in the Our derived ohjectivr functional can be rearranged ‘37,470
=
Z
+ 2.55OSs + 3.37359 - 0.31O-Ti5 - O.26LYia + - 0.48;3-Y:4 f 0.483Sia + 0.520x26 + 0.520x’i7
1 ,(I&ylp
(19) I n order t o re-esprees this functional in terms of variables escludirig the basis variables, XI t o XI,, it n-ill be necessary t o combine the functional with Equations 8. 9, 15$ and 16 which are suit:itilv eculed to eliminate variables X 8 to XIS:
4.500 Jrhich is the maximum unit profit improvement poa.djle :it this stage. T h e variable t o he removcd from the 18-rqu:ition Insis is the oiie v-liich n-ill result in the grctteit v s l u e for 1 2 3 without ni:dw variable to comc itito the hnsis.
Problem Matrix Tahle I sho..vs tlicx problem matrix :is tlcriverl in t h iorcqiiiig discussion. I h e h of the columnq iq hcntled liy B syni!>ol, l’, t,o Ply:v-hich are the vectors reprrscliting the colninns of coefficients of the varinhles XI t o Sa;. The vector Po represents t l i o coluniii of Ixrsis d u e s : h,. Each row then is act,ii:dly n n equition n i t h the vnri:rhles understood !,ut not \vittP!i iii placc. Tlic objective functional is shon-n below the 18 con~tr:iiniiigequations. Although Z is not restricted by nonnrgitivity i t m:i)- also tw considered to he in the basis. T h e C, values are ,*howii on the lowtlst row for referprice and use in checking the profit in :my feasible solution.
Solution Matrix T h e solution of this problem W\ P, carried out by the Siml)les l l e t h o d of Liriesr Programming formulated by George Daiitzig of Rand Corporation and programmed h y II3M on their 701 computer. References t o the development of these methods :ire given (1-6). The optimal solution of this prohlcni is shown in Table 11. T h e diq~ositionof the gasoline components in the optimal solution is: ~~~~~
Regular
IThen these four equations are subtracted from Equation 10 the resulting espresjion will not contain any variahles, XI t o S18, because of cancellation of these terms:
LS HS Mild reformate Severe reformate LCS HC4 Incremental L C S Incremental I I C S VY
8,592
March 1956
~-
11,006 2587
6,344 12,391 5,800
2,713 3,222 3,400 8.750
598
2,447 2,126
38,000
T h e ohjertivr iuiiction:il for the hasic frasihle solution slioivs t h a t the over-all profit a t this stage is minus $100,620 per da>-for runniiig 100:000 bbl. Ida?- of base crude oil producing onlj- 27,000 hbl./day of heating oil in addition t o fuel oil. Under this condition it is not profitable to run additional crude XIS,since a loss of S1.95-i per bhl. will be incurred (AZ/’1X19 = - 1.954). However, it is profitable t o blend all components into gasoline except heavy naphtha ( A Z / A X 2 ? = -0.240 and AZ/AA-23 = -0.033). The use of reformates and cracked naphthas for hlending gasolines appears t o be most profitable. I n fact, the method of linear programming d l select X2alight cracked naphtha from the base crude operation t o Ije hlentlrti into regular g:rsoline :is the preferential varisl,le t o he substituted into the basis siiice 1 Z !AXi8 =
Barrels per Day Fuel Preinium oil
Ern0
3186
Total 20,408 2,587 2,713 9,566 15,800 5 800 8,750 :3 045 2,426 71,185 ~
A411the requirements for regular and premium grades are met, and some excesses of hear)- naphtha and h r a r y cracked naphtha are sent to fuel oil. T h e regular grade i3 blended from most of the components. T h e prrmium grade is blended from the light coniponents and the higher octane components. T h e incremental crude in the o p t i m d solution is -YIg = 21.2!)0 bbl./day leaving an unused crude cap:icity of SI; = 3710 bhl,/&y. There is no uriused reforming capacity (XI, = 0). The heavy naphtha balance is: 2587 bbl./day t o fuel oil t o mild reforming t o severe reforming
+ 1.160 X 2713 bbl./day + 1.239 X ‘3566 blil./day = 17,387 hhl./tlay available
Exactly met are the minimum yc’c/300” F. volatility specification on the regular grade (Xll = 0) :md the mnximuiii 70/2000 F. (X13= 0) and the minimum 70:3000 F. (X14= 0) volatility specifications and the minimum octane rating x i t h :3.0 cc. of
INDUSTRIAL AND ENGINEERING CHEMISTRY
399
Table II. Solution Matrix for Refinery Processing and Gasoline Blending Problem PI P? Ps P4 Ps PS PS P7 Po PlO
Po 2,447 2,587 12,392 8,592 8,750 3,222 598 2,426 21,290 502,274 5,800 315,000 3,408 2,713 93,407 11,906 3,710 6,344 78,540
0.1062 -0.2804 -0.5735 0.7661 -0.7949 -0.5357 -0.3828 -0.0387 -1.934 8.871 0 0 0.5735 0.8499 -6.720 -0.0929 1.934 -0.2601 1.700
0.1317 -0,2869 0.4133 - 0.2393 -0.8132 - 0.5480 -0.4146 - 0.0396 - 1.979 6.213 0
1
0
0
a . . . . . . . . . . . . . . . . . . . . . . . . . ,
0.5867 0.8695 2.513 - 0.0951 1.979 -0,2661 4.649
- 0.9619 - 0.0097
- 0.0241 - 0.2301
-0.0198 -0.0081 - 0.0274 -0,0185 0.9523 -0.0013 - 0.0667 -0.4872 1.000 0 0.0198 0.0293 1.099 -0.0032 0.0667 - 0.0090 0.4712
0.2509 -0.3089 0.3478 0.2343 -0.2028 - 0.0317 - 1.587 - 3.406 0 0 -0.2509 - 0.3718 3.064 0.0407 1.587 0.1138 4.049
- 0.4024
-0.1515
- 0.3098 -0.1264 -0.4294
- 0.2894 1
0.2530 0.9791 - 1.045 -4.966 0 0 0.3098 0.4592 6.177 - 0.0502 1.045 -0.1405 0.1286
0
0.2244 0.1966 0.4021 0.1640 0.5573 0.3756 - 0.0304 0.0271 1.356 13.07 0 0 -0.4021 - 0.5959 3.147 0.0652 - 1.356 0.1824 0.8733
0.0125 0.2438 0.1746 -0.1594 0.6911 0.3766 0.2279 0.0336 1.682 0.7894 0 10.5 -0.1746 - 0.3367 1.215 0.4436 - 1.682 - 0.0614 0.4622
-2.550
-3.373
pi1
0
0.0127 -0.0032 - 0.0066 -0.0027 -0.0091 - 0.0062 -0.0159 - 0.0004 - 0.0222 -0.8291 0 0 0.0066 0.0098 - 0.0336 -0.0011 0.0222 - 0.0030 0.0331
Pzz
p23
P?l
0.5295 1.119 0.2439 0.0995 0.3381 0.2278 -0,4119 0.0165 0.8225 12.68 0 0 -0.2439 - 0.3615 11.34 0.0395 -0.8225 0.1106 1.907
0.5657 1.138 3.364 -2.676 0.3918 3.294 - 0.4294 0.0191 0.9533 23.70 0 0 -3.364 -2.159 24.51 2.837 -0.9533 -1.272 5.727
0.0208 0.0109 0.0223 0.0091 0.0309 0.0208 -0.0100 0.0015 0.0751 6.335 0 0 -0.0223 -0.0330 1.309 0.0036 -0.0751 0,9463 0.2551
1
,............................................ ,.......................,,..........................
Ci
Table 1 I (Continued)
P12
1
0
Pi8
Pl4
0,0034 0.0018 0.0442 - 0.0414 0.0051 0.1653 -0.0017 0.0002 0.0123 0.0406 0 1.000 - 0,0442 -0.1697 0.3785 0.0435 -0.0123 - 0.0064 0.0925
0.0182 - 0.0003 0.0866 - 0.0634 - 0 * 0010 0.2877 -0,0185 0 - 0.0024 0.8441 0 0 - 0.0866 - 0.2631 0.5849 0.0630 0,0024 - 0.0414 0.1849
..............
P16
PI 6
p17
- 0.0023 - 0.0012 0.1241 -0.1350 - 0.0034 -0.1215 0.0011 - 0.0002 - 0.0084 - 0.7076 0 0 -0.1241 0.1154 1.239 0.1336 0.0084 0.0134 0.1371
1
0 0.310
1 0
P1a
PIS
-0.2632 - 1.138 - 0.2820 -0.1150 - 0.3908 -0.2634 0.1272 -0.0190 -0.9509 - 0.1690 0 0 0.2820 0.4179 1.147 - 0.0457 0.9509 0.6792 1.796
P?O
1
1
1
0
0
............................
0
.......................
- 1.954
0.264
P?l
*............. . . . . . . . - 0.483
Table II (Continued)
pZ6
P26
P27
P28
P:Q
P30
1 1
1 1 1
1
0
0
0
0
0
0
p3 1 0.1952 0.1023 4.143 -2.610 0.2899 11.82 -0,0943 0.0141 0.7052 59.47 1.000 0 -4.143 -9.697 25.06 2.729 - 0.7052 -2.743 6.350
p32 0.0222 0.0116 0.7453 -0.1072 1.033 0.6961 -0.0107 0.0016 0.0803 6.774 0 0 -0.7453 -1.105 1.181 0.1208 - 0.0803 0.3380 0.2043
Pa3
400
-0.520
P35
1
1.200 0.1046 4.608 -3.006 0.2966 12.11 0.0965 0.0144 0.7216 60.85 0 0 -4.608 -9.923 27.69 3.128 - 0.7216 -2.816 7.131
1
P3 8
PS7
1
0.0505 0.0265 2.112 - 1.981 0.0751 3.537 - 0.0244 1.004 0.1826 15.40 0 0 -2.112 -2.511 10.73
-
0
................................................................................... -0.483
P31
0
2.012
- 0.1826
- 1.186 0
*................
2.959
tZ
-0.520
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 3
Operations Research TEL'gal. (XI6= 0) on the premium grade. The optimal solution indicates t h a t it is not economical t o meet the minimum %/200" F. volatility specifications for the regular and premium grades. The giveaway volatilities are:
Xla
502,271 38,000
13.2%/200 for regular
=
T h e octane giveaxyay for the regular grade is:
xl5 -- 93'407 38,000
Operating costs Base crude operation Additional crude operation Mild reformine SeGere reforml'ng
= 2.46 octane units
This higher octane rating a t 3.0 cc. of TEL/gal. on the regular grade permits a decrease in the T E L content t o 1.4 cc./gal., and still meets the minimum octane specification (Figure 2).
This calculated profit checks the profit derived by the Simplex solution of t,he linear programming problem. In the optimal solution all coefficients in the objective function are either zero or positive. This means that the profit cannot be increased by replacing any variable (changing the basis). I n fact, the effects on profit of any variable replacement are shown by these coefficients, summarized as follovs: Dollars per Bbl. AZ/AXl = - 1 , 7 0 0 AZ/AXs = -4.649 4Z/AX4 = - 0.471 AZ/AXj = - 4.049 4Z/AX7 = -0.129 &?/AXs = - 0.873 A % / A X s = -0.162 >Z/AX1, = - 0 , 0 3 3 AZ/LSIa
=
-0.092
b
A Z / A X ,= ~ -0.185
I
0,
I
a
1I
I
I
1.40
3.0
CCIGAL. T E L NEEDED TO MEET SPECIFICATION
Figure 2 A summarj- of the blended qualities of the regular and premium grades is as follows:
Vapor pressure Volatility 7?0/200 Volatility 70i300 Octane rating T E L content, cc./gal.
Regular 10 43.2 70 90 1 '1
Premium 10 53 86
91 3 0
These qualities meet all of the specifications. T h e givearray values are chosen t o maximize the profit functional.
100,000 x 1 47ti = 147,600 21,290 X 1.954 = 41,601 2.713 X 0 483 = 1.310 9;566 X 0.520 = 41971 19%5,485 S e t Profit 78,540
AZ/AXlS = - 0 , 1 3 7 AZ/AXI8 = - 1.796 AZ/AX?p = - 1.907 AZ/AX23 = - 5 , 7 2 7 AZ/AX24 = -0.253 AZ/AX= ~ ~- 6,350 LIZ/AX,~ = -!.204 M / A X a j = - 1 ,131 AZ/AXi7 = - 2,939
L N sent t o fuel oil LCN sent to fuel oil HCN sent t o fuel oil .-ldditional LCX sent t o fuel oil VN sent t o fuel oil Regular gasoline requirements not met Premium gasoline requirements not met P r r 7,300 min. volatility giveaway on regular Per %200 mas. volatility givenray on premium Per %300 min. volatility giveaway on premium Per octane unit giveaway on premium Reforming capacity not used H S eent t o regular H N sent t o premium Mild reformate sent to regular HCN sent t o premium Sdditional L C N sent t o regular lldditional H C N sent t o premium VN sent t o premium
These 19 coefficients show the penalties which would be incurred by changing the optimal solution in the direction shown. All these changes would decrease profit thus proving t h a t this solution is the maximum profit case. Some of these coefficients may also be interpreted as benefits which mould occur b y relaxing the restrictions of the problem. Thus, increasing requirements would increase profit for regular gasoline by 87.3 cents/hhl. and for premium gasoline by 46.2 cents/bbl. Increasing capacity of reforming would increase profit by $1.796/bbl. I n addition, relaxing some of the specifications rrould result in increased profit provided the requirements r e r e not changed.
literature Cited Objective Functional in Optimal Solution
(1)
The profit for the optimal case can be calculated as follows:
(2)
Income and credits Regular gasoline Premium gasoline Heating oil TEL saving in regular gasoline
March 1956
(3)
38,000 X 2.550 = 96,900 30,000 X 3.373 = 101,190 27,000 X 1.740 = 46,980 2.458 X 38,000 X 0.310 = 28,955 274,025
(4) (5) (6)
Charnes, A., Cooper, W. ITr., Henderson, A , "Introduction to Linear Programming," Wiley, Kew York, 1953. Charnes, b., Cooper, 'AT. W., lfellon, B., Econometrica, 20, No 2, 135-59 (dpril 1952). Ibid., pp. 193-217 (April 1954). Eisemann, Kurt. Proc. I B l I Petroleum Conference, October 1953. Koopmans, Tjalling C., "Activity Analysis of Production and Allocation," Wiley, New York, 1951. Symonds, G. H.. Proc. IBM Petroleum Conference, October 1953.
RECEIVED for reviem April 14, 1955.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTEDJanuary 19, 1956.
401
