Optimization Modeling Using GAMS
Problem Statement:
Several forms of gasoline are produced during the petroleum refining process, and a last step
combines them to obtain market products with specified quality measures. Suppose the following
four gasoline products are available:
Type Quality Index-1 Quality Index-2 Cost($/Barrel)
1
99
210
48
2
70
335
43
3
78
280
58
4
91
265
46
Determine the minimum cost blend which has quality index-1 between 85 and 90 and quality index2 between 270 and 280
Mathematical Formulation:
Min 48 x1  43x 2  58 x 3 46 x 4
(Cost of blend )
Subject to
85  99 x1  70 x 2  78 x3  91x 4  90 (Quality Index  1 Specificat ion )
270  210 x1  335 x 2  280 x3  265 x 4  280 (Quality Index  2 Specificat ion )
x1  x 2  x3  x 4  1
0  x1  1 0  x 2  1 0  x3  1 0  x 4  1
( Solution : x1*  0.1765, x 2*  0.3529, x3*  0.0000, x 4*  0.4706, Cost  45.2941)
GAMS solution (blend_model.gms)
sets
i
j
set of gasoline
set of quality indices
/G1,G2,G3,G4/
/Q1,Q2/;
parameters
cost(i) cost of gasoline set i
/G1 48
G2 43
G3 58
G4 46/
quality_spec_min(j) quality specifications -minimum
1
/Q1 85
Q2 270/
quality_spec_max(j) quality specifications-maximum
/Q1 90
Q2 280/;
Table quality_val(i,j) quality indices
Q1
Q2
G1
99
210
G2
70
335
G3
78
280
G4
91
265;
variables
tot_cost total cost in $
x(i)
fraction of each Gasoline set;
equations
objective
min_spec_cons(j)
max_spec_cons(j)
tot_frac_cons
objective or cost function
minimum Quality index constraint
maximum Quality index constraint
total fraction=1 constraint ;
***********Model Definition********************
objective ..
tot_cost =e= sum(i,cost(i)*x(i));
min_spec_cons(j) ..
sum(i,quality_val(i,j)*x(i)) =g= quality_spec_min(j);
max_spec_cons(j) ..
sum(i,quality_val(i,j)*x(i)) =l= quality_spec_max(j);
tot_frac_cons ..
sum(i,x(i)) =e= 1;
x.lo(i) = 0; x.up(i) = 1;
***********************************************
model blend /
objective,min_spec_cons,max_spec_cons,tot_frac_cons/;
***********Solve Statement*********************
solve blend using lp minimizing tot_cost;
***********************************************
display x.l;
***writing to a file***********************
file res /blend2.res/;
put res;
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put "Total Cost = ",tot_cost.l:8:4/;
loop(i,
put "x(",i.tl:1:0,") = ",x.l(i):8:4/;
);
MATLAB Solution:
f = [48;43;58;46];
A = [ 99 70 78 91
-99 -70 -78 -91
210 335 280 265
-210 -335 -280 -265];
b = [90; -85; 280; -270];
lb = [0; 0; 0; 0];
ub = [1; 1; 1; 1];
Aeq = [1 1 1 1];
beq = [1];
options=optimset('Display','iter');
[x,objective,exitflag,output,lambda]=
linprog(f,A,b,Aeq,beq,lb,ub,[],options)
Mathematica Solution (Algebraic Form):
NMinimize[{48*x1 + 43*x2 + 58*x3 + 46*x4,
85 <= 99*x1 + 70*x2 + 78*x3 + 91*x4 <= 90 &&
270 <= 210*x1 + 335*x2 + 280*x3 + 265*x4 <= 280 &&
x1 + x2 + x3 + x4 == 1 &&
0 <= x1 <= 1 && 0 <= x2 <= 1 &&0 <= x3 <= 1&& 0 <= x4 <= 1},
{x1, x2, x3, x4}]
Nonlinear programming example using GAMS (Qp.gms)
option
solprint = off
reslim = 50000
iterlim = 100000
domlim = 10
limcol = 0
limrow = 0
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optca = 0
optcr = 0
lp = cplex
nlp = snopt
mip = cplex
minlp = sbb ;
sets
i
/1*3/;
variables
z
x(i);
equations
objective
cons1
cons2;
***********Model Definition********************
objective ..
z =e= x('1') + 2*x('2') + 3*x('3') +
(-31*power(x('1'),2) + 12*x('1')*x('2') - 22*power(x('2'),2)
+ 23*x('2')*x('3') - 11*power(x('3'),2) )/2 ;
cons1 ..
-x('1') + x('2') + x('3') =l= 20;
cons2 ..
x('1') - 3*x('2') + x('3') =l= 30;
x.lo('1') = 0;
x.up('1') = 40;
***********************************************
model quad /
objective
cons1
cons2 /;
***********Solve Statement*********************
x.l('1') = uniform(0,1);
x.l('2') = uniform(0,1);
x.l('3') = uniform(0,1);
solve quad using nlp maximizing z;
display x.l, x.m;
Prepared by
Vikas Argod (vikasargod@psu.edu)
Research Computing and Cyberinfrastructure
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GAMS Standard functions
Function Description
Functions
errorf(x)
exp(x)
log(x)
log10(x)
normal(x,y)
Description
Integral of the standard normal distribution form from -∞ to x
Exponential, ex
National logarithm, loge x
Common logarithm, log10 x
Random number normally distributed with mean x and standard
deviation y
Random number with uniform distribution between x and y
Absolute value of x, i.e |x|
Ceiling of x. Smallest integer ≥ x
Floor of x. Largest integer ≤ x
Mapping function. Assign unique numbers to special values
Largest value among all arguments
Smallest value among all arguments
Remainder. x – y*trunc(x/y)
Integer power. xy,where y must be an integer
round x to the nearest integer
Rounds x to y decimal places right(+) or left(-) to the decimal point
Rounds to 1 if x > 0, -1 if x < 0, and 0 if x = 0
Square of x. x2
Square root of x. √𝑥
sign(x)*floor(abs(x))
tan-1 x. Result in radians
cos x. x in Radians
sin x. x in Radians
uniform(x,y)
abs(x)
ceil(x)
floor(x)
mapval(x)
max(x,y,..)
min(x,y,..)
mod(x,y)
power(x,y)
round(x)
round(x,y)
sign(x)
sqr(x)
sqrt(x)
trunc(x)
arctan(x)
cos(x)
sin(x)
Symbols from reference maps:
The symbol column is an all inclusive alphabetical list of all named items in the GAMS program. The type column
gives a categorical classification of the basic nature of the symbol using abbreviation codes as follows
Type
Type of GAMS item
ACRNM
Acronym
EQU
Equation
FILE
Put file
MODEL
Model
PARAM
Parameter (also Table, Scalar)
SET
Set
VAR
Variable
The symbol map also contains a line number indexed list of references to the items, grouped by type of reference
which are abbreviated as below
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Reference
Description
DECLARED
This identifies lines where the named item is declared in a Set, Parameter, Table, Scalar,
Variable, Equation, Acronym, File or Model command. This will be the first appearance.
DEFINED
This identifies the line number where explicit data are entered (in a table or a data list
between slashes) or in the case of an equation where the .. specification begins.
ASSIGNED
This identifies lines where this item has data put into it by being on the left hand side of
an assignment statement.
IMPL-ASN
This identifies lines where an equation or variable has data put into it by the results of a
solve statement.
CONTROL
This identifies lines where this set is used as the driving index either in an assignment,
equation, loop or other indexed operation (sum, prod, smin or smax).
REF
This identifies lines where this item is used either on the right hand side of an
assignment, in a display, in an equation, in a model, in a put statement or in a solve
statement.
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