1. Consider the following LP problem, in which X and Y denote the number of units of products X and Y to produce, respectively:
Maximize profit = $4X + $5Y
Subject to the constraints
X + 2Y 5- 10 (labor available, in hours)
6X + 6Y 5- 36 (material available, in pounds) 8X + 4Y s 40 (storage available, in square feet) X, Y 0 (nonnegativity)
Variable Cells
|
|
|
Final
|
Reduced
|
Objective
|
Allowable
|
Allowable
|
|
Cell
|
Name
|
Value
|
Cost
|
Coefficient
|
Increase
|
Decrease
|
|
WU
|
Solution value X
|
2.00
|
0 00
|
4.00
|
1.00
|
1.50
|
|
SC$4
|
Solution value Y
|
4.00
|
0 00
|
5.00
|
3.00
|
1.00
|
Constraints
|
Cell
|
Name
|
Final Value
|
Shadow Price
|
Constraint R.N. Side
|
Allowable Increase
|
Allowable Decrease
|
|
5DS7
|
Labor
|
10 00
|
1 .00
|
10
|
2.00
|
2.00
|
|
SOS8
|
Material
|
36.00
|
0.50
|
36.00
|
4.00
|
6.00
|
|
W39
|
Storage
|
32.00
|
0 .00
|
40.00
|
1E+30
|
8.00
|
Calculate and explain what happens to the optimal solution for each of the following situations. Each question is independent of the other questions.
(a) You acquire 2 additional pounds of material.
(b) You acquire 1.5 additional hours of labor.
(c) You give up 1 hour of labor and get 1.5 pounds of material.
(d) The profit contributions for both products X and Y are changed to $4.75 each.
(e) You decide to introduce a new product that has a profit contribution of $2. Each unit of this prod-uct will use 1 hour of labor, 1 pound of material, and 2 square feet of storage space.
2. The Tiger Catering Company is trying to determine the most economical combination of sandwiches to make for a tennis club. The club has asked Tiger to provide 70 sandwiches in a variety to include tuna, tuna and cheese, ham, ham and cheese, and cheese. The club has specified a minimum of 10 each of tuna and ham and 12 each of tuna/cheese and ham/ cheese. Tiger makes the sandwiches using the fol¬lowing resources: bread, tuna, ham, cheese, mayon¬naise, mustard, lettuce, tomato, packaging material, and labor hours.
|
|
Tuna
|
Tuna/Ch
|
Ham
|
Ham/Ch
|
Cheese
|
|
|
|
Number to make
|
10 00
|
30 00
|
10.00
|
12 00
|
8.00
|
|
Cost
|
$2.42
|
$2 12
|
S3.35
|
S3 02
|
52 36
|
$176.42
|
|
|
Constraints
|
|
|
|
|
|
|
|
|
|
Bread (slices)
|
2
|
2
|
2
|
2
|
2
|
140.00
|
a
|
140
|
|
Tuna (oz.)
|
4
|
3
|
|
|
|
130.00
|
<=
|
130
|
|
Ham (or.)
|
|
|
4
|
3
|
|
76.00
|
<=
|
100
|
|
Cheese (oz.)
|
|
1
|
|
1
|
4
|
74.00
|
<=
|
80
|
|
Mayo (oz.)
|
1.2
|
0.9
|
0.5
|
0.5
|
0.5
|
54.00
|
<=
|
72
|
|
Mustard (oz.)
|
|
|
0.2
|
0.2
|
|
4.40
|
<=
|
8
|
|
Lettuce (or)
|
0.25
|
0.25
|
0.25
|
0.25
|
0.25
|
17.50
|
<=
|
20
|
|
Tomato (oz.)
|
0.5
|
0.5
|
0.5
|
0.5
|
0.5
|
35.00
|
<=
|
40
|
|
Package (unit)
|
1
|
1
|
1
|
1
|
1
|
70.00
|
<=
|
72
|
|
Labor (hrs)
|
0.08
|
0.08
|
0.08
|
0.08
|
0.08
|
5.60
|
<=
|
8
|
|
Min total
|
1
|
1
|
1
|
1
|
1
|
70.00
|
>=
|
70
|
|
Min Tuna
|
1
|
|
|
|
|
10.00
|
>=
|
10
|
|
Min Tuna/Ch
|
|
1
|
|
|
|
30.00
|
›=
|
12
|
|
Min Ham
|
|
|
1
|
|
|
10 00
|
>=
|
10
|
|
Min Ham/Ch
|
|
|
|
1
|
|
12.00
|
>=
|
12
|
|
|
|
|
|
|
LHS
|
Sign
|
RHS
|
The objective funcntion coefficients in the screenshots refer to unit cost per item. Each of the following questions is independent of the others.
(a) What is the optimal cost represented by the objective function and what is the optimal sandwich-making plan?
(b) Which constraints are binding?
(c) What is the range over which the cost for cheese sandwiches could vary without changing the production plan?
(d) What is the range over which the quantity of tuna could vary without changing the combina¬tion of binding constraints?
(e) Does this Sensitivity Report indicate the presence of multiple optimal solutions? How do you know?
(f) After the sandwiches are made, how many labor hours remain?
3 Consider the Tiger Catering problem the impact on the sandwich-making plan and total cost? If it is possible to compute the new cost or sandwich-making plan, do so.
Variable Cells
|
Cell
|
Name
|
Final
Value
|
Reduced
Cost
|
Objective
Coefficient
|
Allowable
Increase
|
Allowable Decrease
|
|
$B$4
|
Number of units Oak tables
|
3.00
|
0.00
|
75.00
|
0.00
|
1E+30
|
|
$C$4
|
Number of units Oak chairs
|
51.67
|
0.00
|
35.00
|
1E+30
|
0.00
|
|
$D$4
|
Number of units Cherry tables
|
3.00
|
0.00
|
90.00
|
0.00
|
1E+30
|
|
$E$4
|
Number of units Cherry chairs
|
85.56
|
0.00
|
60.00
|
1E+30
|
0.00
|
|
$F$4
|
Number of units Pine tables
|
42.26
|
0.00
|
45.00
|
88.33
|
0.00
|
|
$G$4
|
Number of units Pine chairs
|
33.08
|
0.00
|
20.00
|
0.00
|
13.25
|
Constraints
|
Cell
|
Name
|
Final
Value
|
Shadow
Price
|
Constraint
R.H. Side
|
Allowable
Increase
|
Allowable Decrease
|
|
$G$7
|
Labor hours
|
1000.00
|
10.00
|
1000.00
|
373.30
|
37.21
|
|
$G$8
|
Oak (pounds)
|
2150.00
|
0.00
|
2150.00
|
318.93
|
1250.00
|
|
$G$9
|
Cherry (pounds)
|
3800.00
|
0.00
|
3800.00
|
223 25
|
2239.78
|
|
$G$10
|
Pine (pounds)
|
8500.00
|
0.00
|
8500.00
|
1488.33
|
5039.50
|
|
$G$11
|
Min oak tables
|
3.00
|
0.00
|
3.00
|
6.25
|
2.35
|
|
$G$12
|
Min cherry tables
|
3.00
|
0.00
|
3.00
|
11.33
|
1.20
|
|
$G$13
|
Min oak chairs
|
51.67
|
0.00
|
10.00
|
41.67
|
1E+30
|
|
$G$14
|
Min cherry chairs
|
85.56
|
0.00
|
10.00
|
75.56
|
1E+30
|
|
$G$15
|
Min pine chairs
|
33.08
|
0.00
|
5.00
|
28.08
|
1E+30
|
(a) The unit cost for tuna sandwiches decreases by $0.30.
(b) The unit cost for tuna and cheese sandwiches increases to $2.40.
(c) The unit cost for ham sandwiches increases to $3.75.
(d) The unit cost for ham and cheese sandwiches decreases by $0.70.
(e) The club does not want any more than 12 ham sandwiches.
(f) The unit cost for cheese sandwiches decreases to $2.05.
4) Consider the Tiger Catering problem. For each of the following situations, what would be the impact on the sandwich-making plan and total cost? If it is possible to compute the new cost or sandwich-making plan, do so.
(a) The quantity of tuna available decreases to 120 ounces.
(b) The quantity of ham available increases to 115 ounces.
(c) The quantity of cheese available decreases to 72 ounces.
(d) Tiger is required to deliver a minimum of 13 tuna sandwiches.
(e) Tiger is required to deliver only a minimum of 10 tuna and cheese sandwiches.
(t) Tiger is asked to bring a minimum of only 66 sandwiches.