Description
This model helps a farmer to decide how to allocate his or her land. The yields are uncertain.
Small Model of Types : DECIS lp
Category : GAMS Model library
Main file : farm.gms
$title The Farmer's Problem formulated for GAMS/DECIS (FARM,SEQ=199)
$onText
This model helps a farmer to decide how to allocate
his or her land. The yields are uncertain.
Birge, R, and Louveaux, F V, Introduction to Stochastic Programming.
Springer, 1997.
Keywords: linear programming, stochastic programming, agricultural cultivation,
farming, cropping
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$if not set decisalg $set decisalg decism
Set
crop / wheat, corn, sugarbeets /
cropr(crop) 'crops required for feeding cattle' / wheat, corn /
cropx / wheat
corn
beets1 'up to 6000 ton'
beets2 'in excess of 6000 ton' /;
Parameter
yield(crop) 'tons per acre' / wheat 2.5
corn 3
sugarbeets 20 /
plantcost(crop) 'dollars per acre' / wheat 150
corn 230
sugarbeets 260 /
sellprice(cropx) 'dollars per ton' / wheat 170
corn 150
beets1 36
beets2 10 /
purchprice(cropr) 'dollars per ton' / wheat 238
corn 210 /
minreq(cropr) 'minimum requirements in ton' / wheat 200
corn 240 /;
Scalar
land 'available land' / 500 /
maxbeets1 'max allowed' / 6000 /;
*--------------------------------------------------------------------------
* First a non-stochastic version
*--------------------------------------------------------------------------
Variable
x(crop) 'acres of land'
w(cropx) 'crops sold'
y(cropr) 'crops purchased'
yld(crop) 'yield'
profit 'objective variable';
Positive Variable x, w, y;
Equation
profitdef 'objective function'
landuse 'capacity'
req(cropr) 'crop requirements for cattle feed'
ylddef 'calc yields'
beets 'total beet production';
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The YLD variable and YLDDEF equation isolate the stochastic
YIELD parameter into one equation, making the DECIS setup
somewhat easier than if we would substitute YLD out of
the model.
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profitdef.. profit =e= - sum(crop, plantcost(crop)*x(crop))
- sum(cropr, purchprice(cropr)*y(cropr))
+ sum(cropx, sellprice(cropx)*w(cropx));
landuse.. sum(crop, x(crop)) =l= land;
ylddef(crop).. yld(crop) =e= yield(crop)*x(crop);
req(cropr).. yld(cropr) + y(cropr) - sum(sameas(cropx,cropr),w(cropx)) =g= minreq(cropr);
beets.. w('beets1') + w('beets2') =l= yld('sugarbeets');
w.up('beets1') = maxbeets1;
Model simple / profitdef, landuse, req, beets, ylddef /;
solve simple using lp maximizing profit;
*--------------------------------------------------------------------------
* Extensive form stochastic model
* This is a standard LP.
*--------------------------------------------------------------------------
Set s 'scenarios' / above, avg, below /;
Variable
ws(cropx, s) 'crops sold under scenario s'
ys(cropr, s) 'crops purchased under scenario s';
Positive Variable ws, ys;
Parameter p(s) 'probability';
p(s) = 1/3;
abort$(abs(sum(s,p(s)) - 1.0) > 0.001) "probabilities don't add up";
Parameter syield(crop,s);
syield(crop,'below') = 0.8*yield(crop);
syield(crop,'avg') = yield(crop);
syield(crop,'above') = 1.2*yield(crop);
Equation
sprofitdef 'objective function extensive form stochastic model'
sreq(cropr,s)
sbeets(s);
sprofitdef.. profit =e= - sum(crop, plantcost(crop)*x(crop))
+ sum(s, p(s)*(- sum(cropr, purchprice(cropr)*ys(cropr,s))
+ sum(cropx, sellprice(cropx)*ws(cropx,s))));
sreq(cropr,s).. syield(cropr,s)*x(cropr) + ys(cropr,s)
- sum(sameas(cropx,cropr),ws(cropx,s))
=g= minreq(cropr);
sbeets(s).. ws('beets1',s) + ws('beets2',s) =l= syield('sugarbeets',s)*x('sugarbeets');
ws.up('beets1',s) = maxbeets1;
Model extform / sprofitdef, landuse, sreq, sbeets /;
solve extform using lp maximizing profit;
* collect results for x for different runs
Set runs / extform 'extensive form'
decisapprox 'default decis'
decisexact 'stochastic universe option' /;
Parameter px(runs,crop) 'results for stage 1 variables';
* store stage 1 results
px('extform',crop) = x.l(crop);
*--------------------------------------------------------------------------
* Default DECIS setup
* Based upon the non-stochastic (core) model.
*--------------------------------------------------------------------------
* output the stochastic file
File stg / MODEL.STG /;
stg.nd = 8;
put stg;
put 'BLOCKS DISCRETE'/;
loop(s,
put 'BL BLOCK1 PERIOD2 ',p(s)/;
loop(crop, put 'x ', crop.tl,' ylddef ', crop.tl,' ', (-syield(crop,s))/;);
);
putClose;
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Notice the (-yield) expression in for the coefficient for x in
equation ylddef. The row listing for YLDDEF shows that the expression
yield(crop)*x(crop) is moved by GAMS to the left-hand side causing a
minus sign. The parentheses are needed as the PUT syntax does
not allow expressions there.
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* output a MINOS option file
File mopt / MINOS.SPC /;
put mopt;
put "begin"/;
put "rows 250"/;
put "columns 250"/;
put "elements 10000"/;
put "end"/;
putClose;
* assign stages
x.stage(cropr) = 1;
y.stage(cropr) = 2;
w.stage(cropx) = 2;
yld.stage(crop) = 2;
landuse.stage = 1;
ylddef.stage(crop) = 2;
req.stage(cropr) = 2;
beets.stage = 2;
option lp = %decisalg%;
solve simple using lp maximizing profit;
* store stage 1 results
px('decisapprox',crop) = x.l(crop);
*--------------------------------------------------------------------------
* Let DECIS solve the model exactly
* Stochastic Universe option: 4 "ISTRAT"
*--------------------------------------------------------------------------
File decopt / %decisalg%.opt /;
put decopt;
put '4 "ISTRAT"'/;
putClose;
simple.optFile = 1;
solve simple using lp maximizing profit;
* store stage 1 results
px('decisexact',crop) = x.l(crop);
display px;