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ORIE 3300/5300
Fall 2011
Prof. Bland
Problem Set 7 Solutions
2.(a). An algebraic model for this problem is:
max
2200
x
1
+ 2695
x
2
+ 3190
x
3

ρT
subject to:
1
.
75
x
1
+ 1
.
9
x
2
+ 2
x
3
≤
1500
2
x
1
+ 2
x
2
+ 4
x
3
≤
3000
14
x
1
+ 16
x
2
+ 23
x
3
≤
14000
9
x
1
+ 15
x
2
+ 15
x
3
≤
10500
x
1
+
x
2
+
x
3
≤
700
14
x
1
+ 16
x
2
+ 23
x
3

T
≤
Θ
x
1
, x
2
, x
3
, T
≥
0
There are feasible solutions (
x
1
,x
2
,x
3
,T
) with
T >
max
{
0
,
14
x
1
+ 16
x
2
+ 23
x
3

Θ
}
.
(1)
However, these solutions cannot be optimal. For any solution (
x
1
,x
2
,x
3
,T
) such that
(1) holds, the solution (
x
1
,x
2
,x
3
,
max
{
0
,
14
x
1
+ 16
x
2
+ 23
x
3

Θ
}
) is also feasible and
has higher objective value, because
ρ >
0 and
T
has decreased. So, all optimal solutions
(
x
*
,T
*
) have
T
*
= max
{
0
,
14
x
*
1
+ 16
x
*
2
+ 23
x
*
3

Θ
}
.
2.(b).
The AMPL model ﬁle is:
set PROD;
#products
set RESOURCES;
#ingredients in scarce supply
param avail{RESOURCES};
#amount of each ingredient (in appropriate units) available
param revenue{PROD};
#revenue per unit
param recipe{RESOURCES,PROD};
#amount (in appropriate units) of each resource
#
needed to produce one unit of product
set TAXEDRESOURCE;
#*****************
1
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View Full Documentparam voltaxthreshold;
# carbon tax paid for MWH used in excess of taxthreshold
param voltaxrate;
# carbon tax rate (in dollars per MWH)
var Taxedpower >= 0;
# power usage in excess of taxthreshold (in MWH)
var Make{p in PROD}>=0;
#units produced
maximize net_revenue:
sum{p in PROD} revenue[p]*Make[p]  voltaxrate*Taxedpower;
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 '10
 TODD

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