PROBLEMS
.'
~,.
ti:
~
ff
1.
Recall the RMC problem (Chapter 2, Problem 21). Letting
F
=
tons of fuel additive
S
=
tons of solvent base
leads to the formulation
Max
40F
+
30S
s.t.
%F
+
%S:$
20
¥SS :$ 5
%F
+ %oS :S 21
,
~S~O
~
Use the graphical sensitivity analysis approach to determine the range of optimality for the ~
..
'
objectivefunctioncoefficients.
,I
2.
For Problem 1 use the graphical sensitivity approach to determine what happens if an ad
,
ditional3
tons of material
3 tIec,?me available.
What
is the corresponding
dual price for the
1
oons~a
~
3.
Considerthe followinglinearprogram:
I
~
Material 1
Material 2
Material 3
s.t.
XI
+
X2:S 10
2x1 +
X2
~ 4
XI
+
3X2
:S 24
2x1 +
X2:$
16
XI' X2.~
0
a.
Solve this problem using the graphical solution procedure.
b.
Compute
therangeofoptimality
fortheobjectivefunctioncoefficientof
XI'
'
c.
Computethe rangeof optimalityfor the objectivefunctioncoefficientof
X2'
'1
d.
Suppos~ the obj~ve
function coefficient of
XI
is increased from 2 to 2.5. What is the
.
new Optimalsolution?
~
e.
Suppose the objective function coefficient of
X2
is decreased from 3 to 1. What is the 1
newoptimalsolution'?
.
.
4.
Referto Problem3. Computethe dual pricesfor co~ts
1and 2 andinterpretthem.
!
,~
:5.
,Consider the following linear program:
'
Min
s.t.
XI
+ 2x2 ~ 7
2x1 +
x2
~
5
xI
+
6x2
~
11
xI,X2
~
0
a.
Solve this problem using the graphical solution procedure.
b.
Compute the range of optimality for the objective function ooefficient of
XI.
'c.
Compute the range of optimality for the objective function coefficient of
X2'
d.
Suppose the objective function coefficient of
xI
is increased to 1.5. Find the new op
timal solution.
e.
Supposethe objectivefunctioncoefficientof
X2
is decreasedto
VS.Find the new opti
mal solution.
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Chapter 3
Linear Programming:
Sensitivity Analysis and Interpretation
of Solution
131
6.
Refer to Problem 5. Compute and interpret the dual prices for the constraints.
7.
Consider the following linear program:
Max
s.t.
2xI
+
X2
~ 3
I

XI
+
5X2
~
4
1
2xI

3X2
:S
6
;
3Xl
+
2X2
:S 35 if
%Xl
+
X2:S
10 =S
Xl,X2
~
0
a.
Solve this problem using the graphical solution procedure.
b.
Compute the range of optimality for the objective function coefficient of
XI'
c.
Compute the range of optimality for the objective function coefficient of
X2'
d.
Suppose the objective function coefficient of
Xl
is decreased to 2. What is the new op
timal solution?
e.
Suppose the objective function coefficient of
X2
is increased to 10. What is the new
optimal solution?
8.
Refer to Problem 7. Suppose that the objective function coefficient of
X2
is reduced to 3.
a.
Resolve using the graphical solution procedure.
b.
Compute the dual prices for constraints 2 and 3.
9.
Refer again to Problem 3.
a.
Suppose the objective function coefficient of
XI
is increased to 3 and the objective
function coefficient of
X2
is increased to 4. Find the new optimal solution.
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 Spring '09
 shakroh
 Math, Operations Research, Linear Programming, Optimization, optimal solution

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