DSCI 3870.001: Management Science
Exam # 2
Version: Blue
November 8, 2012
Time allotted: 75 minutes
Name:
SID#:
Please read this carefully
The questions, which you attempt today, consist of True/False and multiplechoice questions worth 133 points total. Some of these questions are based on
descriptive cases. There are a total of 32 questions.
Please answer all the questions on the scantron sheet provided. After you are
done, please turn in the scantron and this question booklet.
Please note that you have to enter your name and Student ID Number (SID#) in
the above area and on the scantron. Failure to do so will result in a grade of zero
.
This is an open book exam. You are allowed to use a programmable calculator and/or
laptop computer. Besides the textbook, you are allowed to use only those notes, which
have been handed out in class, and notes, which you have recorded during, class
sessions. Be sure to allocate you time wisely between the multiple choice and T/F
questions.
Acronyms such as LP, LHS (left-hand side), RHS (right-hand side) etc. may have
been used. If you are unsure about an acronym, please ask me/the invigilator
immediately.
Best of luck!!
1
1. A negative dual price for a “>”constraint (with a positive RHS and all positive
technology coefficients) in a minimization problem (with all positive objective function
coefficients) means,
a. as the right-hand side increases, the objective function value will increase.
b. as the right-hand side decreases, the objective function value will increase.
c. as the right-hand side increases, the objective function value will decrease.
2.
3.
4.
In the game that we played in class, the cost of the tiles was a sunk cost.
a. True
b. False
An objective function reflects the relevant cost of labor hours used in production
rather than treating them as a sunk cost. The correct interpretation of the dual
price associated with the labor hours constraint is
a. the maximum premium (say for overtime) over the normal price that the
company would be willing to pay.
b. the upper limit on the total hourly wage the company would pay.
c. the number of hours by which the right-hand side can change before there is a
change in the solution point.
A section of output from MS Excel’s Solver Sensitivity Report is shown here.
Constraints
Name
Contraint#2
Final Shadow
Value Price
300
xxxx
Constraint
R.H. Side
300
Allowable
Increase
120
Allowable
Decrease
60
What will happen if the right-hand-side for Constraint#2 increases to 500?
a.
Nothing. The values of the decision variables, the dual prices, and the
objective function will all remain the same.
b.
The value of the objective function will change, but the values of the
decision variables and the dual prices will remain the same.
c.
The problem will need to be resolved to find the new optimal solution and
dual price.
5.
Which of the following is not a question answered by sensitivity analysis as typically
shown in an MS Excel output?
a.
If the right-hand side value of a constraint changes, will the objective
function value change?
b.
Over what range can a constraint’s right-hand side value change without
changing the constraint’s dual price?
c.
By how much will the objective function value change if the right-hand
side value of a constraint changes beyond the range of feasibility?
2
6.
A marketing research application uses the variable HD to represent the number of
homeowners interviewed during the day. The objective function minimizes the
cost of interviewing this and other categories and there is a constraint that HD >
100. The solution indicates that interviewing another homeowner during the day
will increase costs by 10.00. What do you know?
a. the objective function coefficient of HD is 10.
b. the objective function coefficient of HD is 10.
c. the dual price for the HD constraint is 10.
7.
If an LP problem is not correctly formulated, MS Excel will indicate where the
infeasibility occurs when trying to solve it.
a. True
b. False
8.
A binding constraint at optimality will always have a non-zero shadow price.
a. True
b. False
The next four questions refer to the following case:
The manager of McDonald’s schedules workers for 8-hour shifts. The beginning times
for the shifts are 7:00 am, 11:00am, 3:00 pm, 7:00 pm, 11:00pm and 3:00 am. A worker
beginning a shift at one of these times works for the next 8 hours. During normal
weekday operations, the number of workers needed varies depending on the time of the
day. The department staffing guidelines require the following minimum number of
workers on duty (see the table below). The manager would like to formulate a linear
program and determine the number of workers that should be scheduled to begin the 8hour shifts at each of the six times (7:00 am, 11:00am, 3:00 pm , 7:00 pm, 11:00pm and
3:00 am) to minimize the total number of workers required. Let X1= the number of
workers beginning work at 7:00 am, X2= the number of workers beginning work at
11:00am, and so on.
Time of Day
7:00 am – 11:00am
11:00am - 3:00 pm
3:00 pm - 7:00 pm
7:00 pm – 11:00pm
11:00pm - 3:00 am
3:00 am - 7:00 am
Minimum Officers on Duty
9
8
10
5
7
8
3
9.
An appropriate objective function would be:
a. Min: 9X1 + 8X2 + 10X3 + 5X4 + 7X5 + 8X6
b. Min: X1 + X2 + X3 + ……………..+ X22 + X23 + X24
c. Min: X1 + X2 + X3 + X4 + X5 + X6
d. Min: X1 + X2 + X3 + X4 + X5 + X6+ X7 + X8
e. Max: 9X1 + 8X2 + 10X3 + 5X4 + 7X5 + 8X6
10.
The constraint for 7:00 pm – 11:00 pm is given as:
a. X2 + X3 ≥ 5
b. X3 + X4 ≥ 5
c. X1 +X2 + X3 + X4 ≤ 5
d. X4 ≥ 5
e. X1 +X2 + X3 +X4≥ 5
11.
The constraint X1 + X2 ≥ 8 is for:
a. 11:00am - 3:00 pm
b. 3:00 pm - 7:00 pm
c. 7:00 pm -11:00pm
d. 11:00pm - 3:00 am
e. 3:00 am - 7:00 am
12.
Scheduling X1=9, X3 =10 and X5=8, is a feasible solution to the problem.
a. True
b. False
The next question is related to the formulation and graphical solution given below:
Max 6x1 + 5x2
.
s.t.
x1 + 3x2 < 360
4x1 + 5x2 < 670
3x1 + 2x2 < 400
x1 , x2 > 0
- (A)
- (B)
- (C)
4
A: 1.0 X1 + 3.0 X2 = 360.0
X2
120
114
108
102
96
90
84
78
72
66
60
54
48
42
36
30
24
18
12
6
0
Payoff: 6.0 X1 + 5.0 X2 = 858.5
B: 4.0 X1 + 5.0 X2 = 670.0
C: 3.0 X1 + 2.0 X2 = 400.0
0
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
102 108 114 120 126 132
Optimal Decisions(X1,X2): (94.3, 58.6)
A: 1.0X1 + 3.0X2 <= 360.0
B: 4.0X1 + 5.0X2 <= 670.0
C: 3.0X1 + 2.0X2 <= 400.0
13. Over what range can the coefficient (c1) of x1 vary before the current solution is
no longer optimal?
a. 4 < c1 < 6.5
b. 4.5 < c1 < 6.5
c. 4 < c1 < 7.5
d. 3.5 < c1 < 7
The next four questions are based on the following case:
Futurama Kitchen Appliances Ltd (FKA) produces trendy microwaves. Two of its
microwave models are the market leaders. FKA Genius is one of the models and is sold
for a price of $ 150 and FKA Miracle is sold for a price of $120. Producing FKA Genius
requires 4 standard heating coils and 2 IC boards and producing FKA Miracle requires 3
heavy duty heating coils and 3 IC board. There are presently 200 standard heating coils
and 120 heavy duty heating coils available. There are 165 IC boards available. Variable
definition for the linear programming problem, the feasible region and the sensitivity
analysis are provided below. You are required to answer the questions which follow
using this information, the GLP figure and the MS Excel analysis as needed.
Let, G = number of Genius models produced
M = number of Miracle models produced
5
X1
X1
M
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Payoff: 150.0 G + 120.0 M = 10100.0
Hvy. Duty Coil: 0.0 G + 3.0 M = 120.0
IC Boards.: 2.0 G + 3.0 M = 165.0
Std. Coil: 4.0 G + 0.0 M = 200.0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
Optimal Decisions(G,M): (50.0, 21.7)
Std. Coil: 4.0G + 0.0M <= 200.0
Hvy. Duty Coil: 0.0G + 3.0M <= 120.0
IC Boards.: 2.0G + 3.0M <= 165.0
Futurama.
A
Decision Variables
Quantity
Profit Contribution
B
Genius
50
150
1
2
3
4
Subject To
5
Stand Coil
4
6
Heavy Duty Coil
7
IC Boards
2
8
Microsoft Excel 12.0 Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 Quantity Genius
50
$C$3 Quantity Miracle
21.67
C
Miracle
21.67
120
D
E
F
<=
<=
<=
RHS
200
120
165
Profit
10100
3
3
LHS
200
65
165
Reduced
Cost
0
0
Objective
Coefficient
150
120
Allowable
Increase
1E+30
105
Allowable
Decrease
XXX
120
Shadow
Price
17.5
XXX
XXX
Constraint
R.H. Side
200
120
165
Allowable
Increase
130
1E+30
XXX
Allowable
Decrease
110
55
65
Constraints
Cell
$D$7
$D$8
$D$9
Name
Stand Coil LHS
Heavy Duty Coil LHS
IC Boards LHS
Final
Value
200
65
165
6
G
14.
What is the allowable increase for the RHS value of the constraint “IC Boards”?
a. 32
b. 65
c. 48
d. 55
e. 0
15. What is the shadow price for the “Heavy Duty Coil” constraint in the above
sensitivity analysis?
a. 20
b. 40
c. 48
d. 32
e. 0
16. What is the shadow price for the “IC board” constraint in the above sensitivity
analysis?
a. 32
b. 40
c. 48
d. 20
e. 0
17. In the above sensitivity analysis, what is the allowable decrease for the “Genius”
model objective coefficient?
a. 55
b. 105
c. 120
d. 70
e. ∞
The next five questions are based on the following case
FarmFresh Foods manufactures a snack mix called TrailTime by blending three
ingredients: a dried fruit mixture, a nut mixture, and a cereal mixture. Information about
the three ingredients (per ounce) is shown below.
Ingredient
Cost
Volume
Dried Fruit
Nut Mix
Cereal Mix
0.15
0.30
0.45
3/8 cup
1/4 cup
1/8 cup
7
Fat
Grams
3
4
1
Calories
150
200
120
The company needs to develop a linear programming model whose solution would tell
them how many ounces of each mix to put into the TrailTime blend. TrailTime is
packaged in boxes that will hold between three and four cups. The blend should contain
no more than 1200 calories and no more than 30 grams of fat. Cereal Mix must be no
more than 40% of the volume of the mixture, and dried fruit mix must be at most 30% of
the weight of the mixture. Develop a model that meets these restrictions and minimizes
the cost of the blend.
Let us define the variables as follows,
D = the number of ounces of dried fruit mix in the blend
N = the number of ounces of nut mix in the blend
C = the number of ounces of cereal mix in the blend
18. The objective function is given as:
a. Max. 0.85D + 0.70N + 0.55C
b. Max. 0.15D + 0.30N + 0.45C
c. Min. 0.85D + 0.70N + 0.55C
d. Min 0.15D + 0.30N + 0.45C
19. The constraint on fat grams is given as:
a. 3D + 4N + 1C < 30
b. 1D + 3N + 4C < 30
c. D + N + C < 30
d. 3D + 4N + 1C > 30
20. The constraint 0.375D + 0.25N + 0.125 C < 4 represents ________.
a. the constraint on calories.
b. the constraint on minimum volume that has to be packaged.
c. the constraint on nut weight
d. the constraint on maximum volume that can be packaged.
21. The restriction on dried fruit mix may be represented as:
a. 0.7D  0.3N – 0.3C > 0
b. 0.7D + 0.3N – 0.3C > 0
c. 0.7D + 0.3N + 0.3C > 0
d. 0.7D  0.3N + 0.3C < 0
22. The constraint on cereal mix volume may be represented as:
a. 0.15D + 0.1N  0.075 C < 0
b.  0.15D  0.1N + 0.075 C < 0
c. 0.15D + 0.1N + 0.05 C > 0
d. 0.15D  0.1N  0.05 C < 0
8
The next three questions are based on the following case
A&C Distributors is a company that represents many outdoor products companies
and schedules deliveries to discount stores, garden centers, and hardware stores.
Currently, scheduling needs to be done for two lawn sprinklers, the Water Wave
and Spring Shower models. Requirements for shipment to a warehouse for a
national chain of garden centers are shown below.
Month
March
Shipping
Capacity
8000
April
7000
May
6000
Minimum
Product
Requirement
Water Wave
3000
Spring Shower 1800
Water Wave
4000
Spring Shower 4000
Water Wave
5000
Spring Shower 2000
Unit Cost
to Ship
.30
.25
.40
.30
.50
.35
Per Unit
Inventory Cost
.06
.05
.09
.06
.12
.07
Let Xij be the number of units of sprinkler i shipped in month j and let Yij be the number
of sprinklers that are at the warehouse at the end of a month, in excess of the minimum
requirement, where i = 1, 2 for Water Wave and Spring Shower respectively, and j = 1, 2,
3 for March, April and May respectively.
23. The most appropriate objective function is given as:
a. Max .3X11 + .25X21 + .40X12 + .30X22 + .50X13 + .35X23
b. Min .06Y11 + .05Y21 + .09Y12 + .06Y22 + .12Y13 + .07Y23
c. Min 3000X11 + 1800X21 + 4000X12 + 4000X22 + 5000X13 + 2000X23
d. Max Y11 + Y21 + Y12 + Y22 + Y13 + Y23
e. Min 8000(X11 + X21) + 7000(X12+X22 ) + 6000(X13 + X23 )
24. The constraint on Shipping Capacity in May is given as:
a. X11 + X21  8000
b. X12 + X22  8000
c. X13 + X23  6000
d. X31 + X32 ≤ 6000
e. X13 − X32 ≤ 7000
25. The constraint on inventory of Spring Shower in April is given as:
a. Y11 + X12 – Y12 = 4000
b. Y21 + X22 – Y22 = 4000
c. Y21 + X22 = 4000
d. Y22 + X23 = 5000
e. Y11 + X21 – Y22 = 4000
9
The next five questions are based on the following case:
A large sporting goods store is placing an order for bicycles with its supplier. Four
models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite,
and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their
profits, respectively, are 48, 60, 20, and 38. The LP model should maximize profit. There
are several conditions that the store needs to worry about. One of these is space to hold
the inventory. An adult’s bike needs three feet, but a child's bike needs two feet. The
store has 800 feet of space. There are 1350 hours of assembly time available. The child's
bike need 3 hours of assembly time; the Open Trail needs 4 hours and the Cityscape
needs 5 hours. The store would like to place an order for at least 300 bikes.
Let, X1 = number of adult Open Trail bicycles ordered
X2 = number of adult Cityscape bicycles ordered
X3 = number of girl's Sea Sprite bicycles ordered
X4 = number of boy's Trail Blazer bicycles ordered
The algebraic LP formulation is given as:
Max.
48 X1 + 60 X2 + 20 X3 + 38 X4
Subject to:
3 X1 + 3 X2 + 2X3 + 2X4 < 800
4 X1 + 5 X2 + 3X3 + 3X4 < 1350
X1 + X2 + X3 + X4 > 300
X1, X2, X3, X4 > 0
The formulation in Excel with the solution and the sensitivity report are shown below.
Please answer the questions that follow these outputs.
A
6
7
8
9
10
Resource
11
B
C
D
E
Resource Usage per Unit of Each Bicycle Ordered
Open
Sea
Trail
Trail
CityScape
Sprite
Blazer
F
G
H
Resource
Totals
Available
Space
3
3
2
2
800
<
800
Assembly Time
4
5
3
3
1300
<
1350
Minimum No. of Bicycles
1
1
1
1
300
>
300
Unit Profit
48
60
20
38
10
Microsoft Excel 12.0 Sensitivity Report
Adjustable Cells
Cell
$B$4
$C$4
$D$4
$E$4
Name
Open Trail
CityScape
Sea Sprite
Trail Blazer
Final
Value
0
200
0
100
Reduced
Cost
-12
0
-18
0
Objective
Coefficient
48
60
20
38
Allowable
Increase
12
2.8E+12
18
2
Allowable
Decrease
1E+30
3.00
1E+30
18
Final
Value
800
1300
Shadow
Price
22
0
Constraint
R.H. Side
800
1350
Allowable
Increase
25.00
1E+30
Allowable
Decrease
199
50.00
300
-6.00
300
99.5
33.33
Constraints
Cell
$F$8
$F$9
$F$10
Name
Space Totals
Assembly Time Totals
Minimum No. of Bicycles
Totals
26.
How many of each kind of bike should be ordered and what will the resulting
profit be?
a. Order 48 Open Trails, 60 Cityscapes, 20 Sea Sprites, and 38 Trail Blazers.
Profit will be $7748.
b. Order 200 Open Trails, 0 Cityscapes, 100 Sea Sprites, and 0 Trail Blazers.
Profit will be $11600.
c. Order 0 Open Trails, 200 Cityscapes, 0 Sea Sprites, and 100 Trail Blazers.
Profit will be $15800.
27.
Would the optimal profit change if the store had 100 more feet of storage space?
a. Yes
b. No
c. Inconclusive
28.
How much should the company be willing to pay for some more assembly time?
a. $6
b. $18
c. $22
d. Nothing
29.
If we enforced the restriction that at least two Open Trail bicycles must be
ordered, the profit will:
a. increase by $48
b. decrease by $48
c. increase by $99.5
d. decrease by $24
11
30.
If we require 5 more bikes in inventory than the current minimum order of 300,
the profit will:
a. decrease by $30
b. increase by $30
c. decrease by $6
d. increase by $6
31.
In the CNN article related to the jailor that we saw in class, the direct analogy of
the problem faced by the jailor was to ________________.
a. the traveling salesman problem
b. the diet problem in linear programming
c. the media selection problem
d. the portfolio selection problem
32. Is the solution to the following problem (Chapter 4, Problem 8, that we solved in
class) optimal?
a. Yes
b. No
c. Cannot be said with certainty
############# END OF EXAM #############
12