ibm1.gms : Aluminum Alloy Smelter Sample Problem

**Description**

This simple alloy smelter blending problem is used as an introductory example in several mpsx manuals.

**Reference**

- IBM, MPSX/370 Primer. Tech. rep., IBM, 1979.

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** ibm1.gms

```
$title Aluminum Alloy Smelter Sample Problem (IBM1,SEQ=79)
$onText
This simple alloy smelter blending problem is used as an introductory
example in several mpsx manuals.
IBM, MPSX/370 Primer. Tech. rep., IBM, 1979.
Keywords: linear programming, blending problem, scenario analysis, chemical engineering
$offText
Set
s 'scrap metals for blending' / bin-1*bin-5, aluminum, silicon /
sl(s) 'locally available blends' / bin-1*bin-5 /
e 'chemical elements' / iron, copper, manganese, magnesium
aluminum, silicon /
Table bspec(e,*) 'blending specs (lb for 2000 lb ingot)'
maximum minimum
iron 60
copper 100
manganese 40
magnesium 30
aluminum inf 1500
silicon 300 250;
Table prop(e,s) 'chemical properties (proportions)'
bin-1 bin-2 bin-3 bin-4 bin-5 aluminum silicon
iron .15 .04 .02 .04 .02 .01 .03
copper .03 .05 .08 .02 .06 .01
manganese .02 .04 .01 .02 .02
magnesium .02 .03 .01
aluminum .70 .75 .80 .75 .80 .97
silicon .02 .06 .08 .12 .02 .01 .97;
Parameter dcheck(s) 'other elements in prop (prop)';
dcheck(s) = 1 - sum(e, prop(e,s));
display dcheck;
Table sup(s,*) 'supply and cost data'
inventory min-use cost
* (lb) (lb) ($/lb)
bin-1 200 .03
bin-2 750 .08
bin-3 800 400 .17
bin-4 700 100 .12
bin-5 1500 .15
aluminum inf .21
silicon inf .38;
Variable
x(s) 'blending components (lb)'
bc(e) 'elements in blend (lb)'
cost 'total material cost ($)';
Equation
yield 'final blend requirements (lb)'
ebal(e) 'element balance (lb)'
cdef 'cost definition ($)';
yield.. sum(s, x(s)) =e= 2000;
ebal(e).. bc(e) =e= sum(s, prop(e,s)*x(s));
cdef.. cost =e= sum(s, sup(s,"cost")*x(s));
Model alloy 'alloy blending model' / all /;
x.lo(s) = sup(s,"min-use");
x.up(s) = sup(s,"inventory");
bc.lo(e) = bspec(e,"minimum");
bc.up(e) = bspec(e,"maximum");
Parameter report(s,*) 'blending results';
solve alloy minimizing cost using lp;
report(s,"run-1") = x.l(s);
option solPrint = off, limCol = 0, limRow = 0;
prop("iron","silicon") = .02;
prop("silicon","silicon") = .98;
solve alloy minimizing cost using lp;
report(s,"run-2") = x.l(s);
prop("iron","silicon") = .01;
prop("silicon","silicon") = .99;
solve alloy minimizing cost using lp;
report(s,"run-3") = x.l(s);
prop("iron","silicon") = 0;
prop("silicon","silicon") = 1.0;
option solPrint = on;
solve alloy minimizing cost using lp;
report(s,"run-4") = x.l(s);
display report;
Parameter costrep 'example cost report for 2000lb batch';
costrep(s ,"cost ") = sup(s,"cost");
costrep(s ,"quantity") = x.l(s);
costrep(s ,"c-cost ") = costrep(s,"cost")*costrep(s,"quantity");
costrep("**total**","c-cost ") = sum(s, costrep(s,"c-cost"));
display costrep;
* demonstrates GAMS' rounding capabilities according to ecl manual
Parameter
xr(s) 'rounded solution value'
xr3(s) 'numerical error rounding at third decimal'
num 'rounded solution counter';
xr(s) = round(x.l(s));
xr3(s) = round(x.l(s),3);
num = 0;
loop(s$(xr(s) <> xr3(s) and num < 2),
x.lo(s)$(xr(s) > xr3(s)) = xr(s);
x.up(s)$(xr(s) < xr3(s)) = xr(s);
num = num + 1;
);
solve alloy minimizing cost using lp;
```