Linear Programming
CHAPTER 6S
LINEAR PROGRAMMING
PROBLEMS
P1.
Given this LP problem:
Maximize
4
3
1
2
x
x
+
s.t.:
A)
14
6
84
1
2
x
x
+
≤
B)
7
12
84
1
2
x
x
+
≤
C)
x
x
1
2
0
,
≥
Solve the problem using the graphical method:
a.
Plot and label the constraints.
b.
Indicate the feasible solution space.
c.
Find the optimal solution using the enumeration method.
d.
Plot the objective function and use it to identify the optimum point of the graph.
e.
Use simultaneous equations to determine the optimal values of
x
1
, and
x
2
f.
Compute the optimum value of the objective function.
P2.
Solve problem P1 by using an EXCEL spreadsheet.
a.
How do you enter the objective function?
b.
How do you enter the constraints?
c.
What are the optimal values of
x
1
and
x
2
?
d. What is the value of the objective function?
e.
How much slack does each constraint, A and B, have?
P3.
Given this LP problem:
Minimize
5
8
1
2
x
x
+
s.t.:
A)
2 2 100
1
2
x x
+ ≥
B)
10
35
700
1
2
x
x
+
≥
C)
x
2
15
≥
D)
x
x
1
2
0
,
≥
Solve this problem using the graphical method.
a.
Plot and label the constraints.
b.
Indicate the feasible solution space.
c.
Plot the objective function and use it to identify the optimum vertex.
d.
Use simultaneous equations to determine the optimal values of
x
1
and
x
2
.
e.
Compute the optimal value of the objective function.
P4.
Solve Problem P3 by using an EXCEL spreadsheet.
a.
How do you enter the objective function?
b.
How do you enter the constraints?
c.
What are the optimal values of
x
1
and
x
2
?
1
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d. What is the value of the objective function?
e.
How much surplus does each constraint, A , B, and C have?
P5.
A dietitian in a hospital is required to devise a recipe for a food which will provide at least the
following amounts of vitamins: 500 units of vitamin A. 500 units of vitamin B. and 700 units of
vitamin C. The dietitian may use three ingredients, P, Q, and R in the recipe which are described
below.
At least one ounce of ingredient R must be used in the recipe.
Units per Ounce:
Ingredient
A
B
C
Cost per Ounce
P
20
30
60
$0.30
Q
60
30
0
$0.20
R
10
50
30
$0.15
a.
Is this a problem in maximization or minimization?
b. What are the decision variables?
c.
What is the objective function?
d. What are the constraints?
P6.
Solve problem P6 on an EXCEL spreadsheet.
a.
How do you enter the objective function?
b.
How do you enter the constraints?
c.
How much of each ingredient should be used in the recipe?
d.
How large is one serving of this food, in kilograms?
In ounces? (I kg. = 2.2 lbs., or 35.2 oz.)
e.
What is the cost of one serving?
f.
How much extra of each vitamin does this recipe provide?
P7.
Refer to your EXCEL spreadsheet for Problem P6.
a.
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 Spring '08
 WILLIAMCOSGROVE
 Operations Research, Linear Programming, Optimization, objective function, hockey, RHS

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