56
Chapter 4
Linear Programming: The Simplex Algorithm
4.18
a. 
∞
<
∆
C
1
≤

∆
Z
1
; so 
∞
< C
1
≤
23
No change in optimum solution, since X
1
is nonbasic.
b. Since X
2
is a basic variable, we compute
2
2j
Z
a
∆
for each nonbasic
variable.
These values (15, and +9, ignoring the artificial
variables) tell us the amounts by which C
2
can change before bringing
a new variable into the solution.
15
≤
∆
C
2
≤
+9, or 10
≤
C
2
≤
+14
is the allowable range.
If C
2
is reduced, Z
OPT
is reduced, if C
2
is
increased, so is Z
OPT
.
c. S
1
≤
∆
b
1
< +
∞
; so 8.167
≤
b
1
< +
∞
, but since b
i
cannot be negative, 0
≤
b
1
< +
∞
is the range.
The only change is
1 for 1 reduction of S
1
if b
1
is reduced and a similar increase in S
1
as b
1
is increased.
d.
Constraint #2 is binding, so the limits on the allowable change in b
2
are found by computing
i
i2
X
a
.
The positive a
i2
values determine the
lower bound and the negative a
i2
values determine the upper bound.
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 Spring '10
 Koslov
 Operations Research, Linear Programming, Optimization, slack

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