srkandw.gms : Stochastic Programming Scenario Reduction

**Description**

A refinery can blend raw materials into different products. At present, the management is trying to decide how much of each of the raw materials to purchase and stock, so that they can be blended to satisfy the demand for the products in future time periods. The demand has to be completely satisfied, and in case of raw material shortage the products can be outsourced at a higher cost. There is an inventory constraint on how much raw material can be stocked in total. Kall, P, and Wallace, S W, Stochastic Programming. John Wiley and Sons, Chichester, England, 1994. Keywords: linear programming, stochastic programming, blending problem

**Reference**

- Kall, P, and Wallace, S W, Stochastic Programming. John Wiley and Sons, Chichester, England, 1994.

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** srkandw.gms

```
$title Stochastic Programming Scenario Reduction (SRKANDW,SEQ=248)
$onText
A refinery can blend raw materials into different products. At present,
the management is trying to decide how much of each of the raw materials
to purchase and stock, so that they can be blended to satisfy the demand
for the products in future time periods. The demand has to be completely
satisfied, and in case of raw material shortage the products can be
outsourced at a higher cost. There is an inventory constraint on how much
raw material can be stocked in total.
Kall, P, and Wallace, S W, Stochastic Programming. John Wiley and Sons,
Chichester, England, 1994.
Keywords: linear programming, stochastic programming, blending problem
$offText
Set
i 'raw materials' / raw-1, raw-2 /
j 'products' / p-1,p-2 /
t 'time periods' / time-1, time-2 /
n 'nodes' / n-0*n-12 /;
Alias (n,nn);
Parameters c(i) 'present cost of raw materials' / raw-1 2, raw-2 3/;
Table a(j,i) 'yields'
raw-1 raw-2
p-1 2 6
p-2 3 3.4;
Table f(j,t) 'cost of outsourcing'
time-1 time-2
p-1 7 10
p-2 12 15;
Scalar b 'inventory capacity' / 50 /;
Table stdat(n,*) 'scenario tree data'
prob p-1 p-2
n-1 .3 200 180
n-2 .4 180 160
n-3 .3 160 140
n-4 .2 200 180
n-5 .5 180 160
n-6 .3 160 140
n-7 .3 200 180
n-8 .4 180 160
n-9 .3 160 140
n-10 .4 200 180
n-11 .4 180 160
n-12 .2 160 140;
Set
tn(t,n) 'time node mapping' / time-1.(n-1*n-3), time-2.(n-4*n-12) /
tree(n,n) / n-0.(n-1*n-3), n-1.(n-4*n-6), n-2.(n-7*n-9), n-3.(n-10*n-12) /
sn(n) 'subset of nodes in reduced subtree'
leaf(n) 'leaf nodes in original tree';
leaf(n)$(sum(tn('time-2',n), 1)) = yes;
display leaf;
Parameter
dem(j,n) 'stochastic demand'
prob(n) 'node probability'
sprob(n) 'node probability in reduced tree';
dem(j,n) = stdat(n,j);
prob('n-0') = 1;
prob(n)$tn('time-1',n) = stdat(n,'prob');
prob(n)$tn('time-2',n) = sum(tree(nn,n), stdat(nn,'prob')*stdat(n,'prob'));
display prob;
Variable
x(i,t) 'raw material purchased for use in time t'
y(j,t,n) 'outsourced products'
cost;
Positive Variable x, y;
Equation
obj 'total cost definition'
bal 'purchase limit'
dembal(j,t,n) 'demand balance';
obj.. cost =e= sum((i,t), c(i)*x(i,t)) + sum((j,tn(t,sn)), sprob(sn)*f(j,t)*y(j,tn));
bal.. sum((i,t), x(i,t)) =l= b;
dembal(j,tn(t,sn)).. sum(i, a(j,i)*x(i,t)) + y(j,tn) =g= dem(j,sn);
* In order to use the SPOSL system we need to insert some dummy links
* between stages two and three. Without these links, the SPOSL system will
* identify only a two stage problem with different subproblem structures.
* The value EPS is used to insert a constraint entry with numerical values
* of zero.
Equation dembalx(j,t,n) 'demand balance modified to include back link';
dembalx(j,tn(t,sn)).. sum(i, a(j,i)*x(i,t)) + y(j,tn) =g= dem(j,sn) + eps*sum(tree(nn,sn), y(j,t-1,nn));
Model srkandw / obj, bal, dembalx /;
$if set noscenred $goto noscenreduction
* now we prepare to run ScenRed
* this includes some sets & parameters used for scenred I/O
$libInclude scenred2.gms
Scalar psum, rc, runCount, runMax;
Set
run / run1*run9 /
method 'reduction method used' / '0-default', '1-fastback'
'2-fastback+forw', '3-fastback+back' /;
Parameter report(method,run,*);
Set rleaf(method,run,n) 'leaf set of reduced tree';
runMax = inf;
$if set runmax runMax = %runmax%;
ScenRedParms('report_level') = 0;
runCount = 0;
loop(method$(runCount < runMax),
ScenRedParms('reduction_method') = ord(method) - 1;
loop(run$(runCount < runMax),
* these parms control the tree output from ScenRed
* at least one of the following two parameters is required
ScenRedParms('red_num_leaves') = ord(run);
* ScenRedParms('red_percentage') = 0.5;
$ libInclude runscenred2 kandw scen_red n tree prob na sprob dem
sn(n) = sprob(n);
display ScenRedParms, ScenRedReport, sprob, sn;
psum = sum{leaf(sn), sprob(sn)};
abort$[abs(psum-1) > 1e-8] "Error in reduced tree: leaf probabilities do not sum to 1";
solve srkandw min cost using lp;
runCount = runCount + 1;
report(method,run, 'obj') = srkandw.objval;
report(method,run, 'red_percentage') = ScenRedReport('red_percentage');
report(method,run, 'reduction_method') = ScenRedReport('reduction_method');
report(method,run, 'run_time') = ScenRedReport('run_time');
rleaf (method,run, leaf(sn)) = yes;
);
);
display report, rleaf;
$onText
* experiment with a different strategy here
* set the desired reduction in terms of distance, not the size
* also, there's really no need to dump the original tree more than once
ScenRedParms('red_num_leaves') = 0;
Parameter report2(method,run,*);
Set rleaf2(method,run,n) 'leaf set of reduced tree';
runCount = 0;
loop(method$(runCount < runMax),
ScenRedParms('reduction_method') = ord(method) - 1;
loop(run$((runCount < runMax) and (ord(run) <= 5)),
ScenRedParms('red_percentage') = ord(run)/4;
$ libInclude runscenred2 kandw scen_red n tree prob na sprob dem
sn(n) = sprob(n);
display ScenRedParms, ScenRedReport, display sprob, sn;
psum = sum {leaf(sn), sprob(sn)};
abort$[abs(psum-1) > 1e-8] "Error in reduced tree: leaf probabilities do not sum to 1";
solve srkandw min cost using lp;
runCount = runCount + 1;
report2(method,run, 'obj') = srkandw.objval;
report2(method,run, 'red_percentage') = ScenRedReport('red_percentage');
report2(method,run, 'red_leaves') = ScenRedReport('red_leaves');
report2(method,run, 'reduction_method') = ScenRedReport('reduction_method');
report2(method,run, 'run_time') = ScenRedReport('run_time');
rleaf2 (method,run, leaf(sn)) = yes;
);
);
display report2, rleaf2;
$offText
$goto alldone
$label NoScenReduction
* set "reduced tree" to be the whole tree
sn(n) = yes;
sprob(n) = prob(n);
solve srkandw min cost using lp;
$label alldone
```