Questions 27 & 28 apply to this information: Quill Manufacturing Business makes two models of marking pens. An unlabeled graph for this problem and the requirements for each lot of pens in the three manufacturing departments are given below. All three departments are necessary in the production of both types of pens. The profit for either kind of pen is $1000 per lot.  An unlabeled graph for this problem is given below. The dotted line represents the objective function line.

Fliptop Model Tiptop Model Available production hrs.
Ink Assembly 3 4 36
Molding Time 5 4 40
Plastic 5 2 30

27. What is the optimal production quantity of the Fliptop model?
A. 5 lots
B. 4 lots
C. 2 lots
D. 7 lots
E. 6 lots

28. If all the constraint inequalities in the original problem were ≥, then the following is true:

A. The value of the objective function at the optimum solution is zero
B. There will be multiple optimal solutions
C. The problem will become unbounded
D. The problem has a unique solution
E. None of the above is true

 

29. Let M be the number of units to make and B be the number of units to buy of a certain  product. If it costs $2 to make a unit and $3 to buy a unit and 4000 units are needed, the  objective function of the LP model to minimize the cost of  production would be,

A. Min 4000 (M + B)
B. Max 8000M + 12000B
C. Min 2M + 3B
D. Max 2M + 3B
E. Min 4000U -2M-3B

30.   The Quiet Meadow Studio sells photographs and prints.  It cost $20 to purchase each photograph and it takes 2 hours to frame it.  It costs $25 to purchase each print and it takes 5 hours to frame it.  The store has at most $400 to spend and  at most 60 hours to frame.
It makes $30 profit on each photograph and $50 profit on each print. Determine the maximum profit.

A. 360
B. 600
C. 700
D. 740
E. 800

Questions 31 & 32 apply to this information:  Quality Bike Maps has produced four map designs for the local area. A limited amount of time (in minutes) is allocated to the printing, cutting and folding of each map. Additionally, at least one thousand of map designs A, B, and C must be printed.  The profit per map is $1 for A and B and $2 for C and D.  The Excel output is provided below.
Max Profit = A + B + 2 C + 2 D
s.t.
A + 2 B + 3 C + 3 D <   15000 Print
2 A + 4 B +   C + 3 D <   20000  Cut
3 A + 2 B + 5 C + 3 D <   20000 Fold
A >   1000    Print A
B >   1000    Print B
C >   1000    Print C

Microsoft Excel 14.0 Sensitivity Report
Objective Function Value  $10,166.67

Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$18 Map A 1500 0 1 1 0.333333333
$C$18 Map B 1000 0 1 0.333333333 1E+30
$D$18 Map C 1000 0 2 0.333333333 1E+30
$E$18 Map D 2833.333333 0 2 1 0.5

Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$B$24     Print 15000 0.5 15000 1000 5666.666667
$B$25     Cut  16500 0 20000 1E+30 3500
$B$26     Fold 20000 0.166666667 20000 7000 1000
$B$27     Print A 1500 0 1000 500 1E+30
$B$28     Print B 1000 -0.33333333 1000 1750 1000
$B$29     Print C 1000 -0.33333333 1000 500 1000

31. Answer the following question using the Excel output above. Which constraint(s) are binding?
A. Print and Fold
B. Cut and Print A
C. Print B and Print C
D. Print and Cut
E. Print, Fold, Print B and Print C

32. Answer the following question using the Excel output above. Keeping within the confines of the problem, the profit on Map A has increased by one dollar.  Determine the new objective function value.

A. 1,500
B. 11,166.67
C. 11,500.67
D. 11,666.67
E. 12,566.67

33. Quentin Magic Brown manufactures sports shoes and wants to maximize the company’s profits. The company makes two types of sport shoe, Airwalkers and Bouncy Basketball shoes.  The company earns $10 profit on each pair of Airwalkers and $18 profit on each pair of Bouncy Basketball shoes.
The manufacturing process includes cutting the materials on a machine and having workers assemble the pieces.  Each pair of Airwalkers requires 3 minutes of cutting time and the Bouncy Basketball shoes require 2 minutes. The machines that cut the material can run at most 1200 minutes a week.
Each worker takes 7 hours to assemble a pair of Airwalkers and 8 hours to assemble a pair of Bouncy Basketball shoes; the maximum number of hours available is 3500 per week.
Determine the maximum profit for this problem?

A. $3200
B. $4000
C. $4280
D. $6295
E. $7875

Questions 34-37 apply to the Excel output for the Quantum Mo-Botics model is below. The company makes three types of machines and has limitations with regards to the amount of skilled and unskilled labor hours available and time on the assembly line.

MAX Profit = 800SemiAuto+1000Robotic + 500Manual
s.t.
30SemiAuto + 100Robotic + 45Manual < 4500  Skilled Labor
100SemiAuto + 70Robotic + 90Manual  < 9000  Unskilled Labor
15SemiAuto +  20Robotic + 10Manual  < 2000  Assembly Line

Microsoft Excel 14.0 Sensitivity Report
Objective Function Value  $ 82,025.32

Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$18  SemiAuto 74.05063291 0 800 628.5714286 432.4786325
$C$18  Robotic 22.78481013 0 1000 1666.666667 440
$D$18  Manual 0 -320.253164 500 320.2531646 1E+30

Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$B$24 Skilled labor  4500 5.569620253 4500 3605.263158 1800
$B$25 Unskilled labor  9000 6.329113924 9000 3805.555556 5850
$B$26 Assembly line  1566.455696 0 2000 1E+30 433.5443038

34.  Answer the following question using the Excel output above, determine the new objective function value if the profit on the second variable, Robotic, increases by $1000?

A. $22,785.00
B. $19,125.64
C. $33,800.00
D. $102,805.32
E. $104,810.32

35.  Answer the following question using the Excel output above. Keeping within the confines of the problem, you are required to hire a full time (40 hours) person who is qualified to work in any department. Select the constraint where you will gain the most profit and determine the additional profit to be gained?

A. $253.16
B. $222.80
C. $341.00
D. $129.50
E. $119.50

36.  Answer the following question using the Excel output above. Keeping within the confines of the problem, how many more hours of skilled workers could you add to the  department?

A. 2875.33
B. 1280.56
C. 1550.56
D. 3805.56
E. 3605.26

37.  Using the Excel output above, how much is each additional unit of unskilled labor worth?

A. $74.500
B. $22.790
C. $5.570
D. $6.329
E. $3.250

38. An ice cream plant make’s Chocolate and Strawberry ice cream.
There is $40 profit for a case of Chocolate and $32 for a case of Strawberry and has the following constraints:

32C + 8S   < 4,800 Flavoring
28C + 32S < 14,000 Coloring

a. What is the optimal solution?
b. Now add a constraint: demand for Strawberry is always less than 200 cases and determine the optimal solution.
c. Add another constraint: demand for Chocolate is always less than 400 cases and determine the optimal solution.

Solution-
38a ( 52,392)  $14,624
38b (100,200) $10,400
38c  redundant constraint; no change