MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |1
The following questions are based on this problem and accompanying Excel windows.
Jack's distillery blends scotches for local bars and saloons. One of his customers has requested a special
blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products,
call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer
wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50
1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B.
The customer also specified that the blend have an alcohol content of at least 85%. Product A contains
95% alcohol while product B contains 78%. The blend is sold for $12.50 per bottle. Product A costs $7
per liter and product B costs $3 per liter. The company wants to determine the blend that will meet the
customer's requirements and maximize profit.
Let
X1 = Number of liters of product A in total blend delivered
X2 = Number of liters of product B in total blend delivered
MIN:
Subject to:
7 X1 + 3 X 2
X1 + X2 = 1.5 * 50 (Total liters of mix)
X1  0.48 * 1.5 * 50 (X1 minimum)
X2  0.5 * 50 (X2 minimum)
.0.95 X1 + 0.78 X2  0.85 * 1.5 * 50 (85% alcohol minimum)
X1, X2  0
A
1
2
3
4
5
6
7
8
9
10
11
B
C
D
E
Jacks' Distillery
Liters to use
Unit cost:
Constraints:
Total Liters
A required
B required
85% alcohol
A
B
10.5
4.5
1
1
1
Total Cost:
Supplied
0.95
1
0.78
Requirement
75
36
25
63.75
1a. Refer to the problem statement . What formula should be entered in cell E5 in the accompanying Excel
spreadsheet to compute total cost?
a. =B4*C4+B5*C5
b. =SUMPRODUCT(B4:C4,B5:C5)
c. =SUM(B5:C5)
d. =SUM(E8:E10)
ANS: B
1b. Refer to the problem statement. What formula should be entered in cell D11 in the accompanying Excel
spreadsheet to compute the total liters of alcohol supplied?
a. =B4*B5+C4*C5
b. =SUMPRODUCT(B11:C11,$B$4:$C$4)
c. =SUM(B5:C5)
d. =SUM(E8:E10)
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |2
ANS: B
1c. Refer to the problem statement. Which cells should be changing cells in this problem?
a. B4:C4
b. E5
c. D8:D10
d. E8:E10
ANS: A
1d. Refer to the problem statement. Which cells should be the constraint cells in this problem?
a. B4:C4
b. E5
c. D8:D11
d. E8:E10
ANS: C
1e. Refer to the problem statement. Which of the following statements could represent a constraint in this
problem?
a. B4:C4  B5:C5
b. E5  0
c. D8 = E8
d. E8:E11  D8:D11
ANS: C
1. The following questions are based on this problem and accompanying Excel windows.
A financial planner wants to design a portfolio of investments for a client. The client has $300,000 to
invest and the planner has identified four investment options for the money. The following requirements
have been placed on the planner. No more than 25% of the money in any one investment, at least one
third should be invested in long-term bonds which mature in seven or more years, and no more than 25%
of the total money should be invested in C or D since they are riskier investments. The planner has
developed the following LP model based on the data in this table and the requirements of the client. The
objective is to maximize the total return of the portfolio.
Investment
A
B
C
D
Return
6.45%
7.10%
8.20%
9.00%
Years to Maturity
9
8
5
8
Let
X1 = Dollars invested in A
X2 = Dollars invested in B
X3 = Dollars invested in C
X4 = Dollars invested in D
MAX:
Subject to:
.0645 X1 + .071 X2 + .082 X3 + .09 X4
X1 + X2 + X3 + X4  300000
Rating
1-Excellent
2-Very Good
4-Fair
3-Good
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |3
X1  75000
X2  75000
X3  75000
X4  75000
X1 + X2 + X4  100000
X3 + X4  75000
X1, X2, X3, X4  0
A
1
2
3
4
5
6
7
8
Bond
A
B
C
D
Total Invested:
Total Available:
<
<
<
<
<
<
<
<
<
1
2
3
4
5
6
7
8
E
Years to
Maturity
9
8
5
8
Total:
Required:
B
Amount
Invested
$0
$0
$0
$0
$0
$300,000
F
7+ years?
(1-yes, 0-no)
1
1
0
1
$0
$100,000
C
Maximum
25.0%
$75,000
$75,000
$75,000
$75,000
Total:
G
Rating
1-Excellent
2-Very Good
4-Fair
3-Good
Total:
Allowed:
D
Return
6.45%
7.10%
8.20%
9.00%
$0
>
>
>
>
>
>
>
>
>
H
Good or worse?
(1-yes, 0-no)
0
0
1
1
$0
$75,000
2a. Refer to the problem statement. What formula should be entered in cell B7 in the accompanying Excel
spreadsheet to compute total dollars invested?
a. =ADD(B3:B6)
b. =SUM(B3:B6)
c. =TOTAL(B3:B6)
d. =TALLY(B3:B6)
ANS: B
2b. Refer to the problem statement. What formula should be entered in cell D7 in the accompanying Excel
spreadsheet to compute the total return?
a. =B7*SUM(D3:D6)
b. =SUMPRODUCT(B3:B6,D3:D6)
c. =SUM(B3:B6)
d. =SUMPRODUCT(B3:E3,B6:E6)
ANS: B
2c. Refer to the problem statement. Which cells are changing cells in the accompanying Excel spreadsheet?
a. B3:B6
b. B7:I7
c. C7
d. E7
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |4
ANS: A
3.
A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8
consecutive hours but can start work at the start of 6 different shifts. They are paid different wages
depending on when they start their shifts. The number of nurses required per 4-hour time period and their
wages are shown in the following table.
Time period
12 am  4 am
4 am  8 am
8 am  12 pm
12 pm  4 pm
4 pm  8 pm
8 pm  12 am
Required # of Nurses
20
30
40
50
40
30
Wage ($/hr)
15
16
13
13
14
15
Enter the numbers in the appropriate cells of ranges B6:G11 and B13:G13 in the Excel spreadsheet to
solve this problem based on the following formulation.
Let
Xi = number of nurses working in time period i; i = 1,6
MIN:
Subject to:
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6
1X1 + 1X2  30
1X2 + 1X3  40
1X3 + 1X4  50
1X4 + 1X5  40
1X5 + 1X6  30
1X1 + 1X6  20
Xi  0
A
1
2
3
4
5
6
7
8
9
10
11
12
13
Shift
1
2
3
4
5
6
Available:
Required:
B
C
Mid
4am
4am
8am
D
Nurse
E
Hiring
F
G
Hours for each shift
8am
Noon
4pm
Noon
4pm
8pm
8pm
Mid
H
I
Nurses
Scheduled
Wages per
Nurse
$15
$16
$13
$13
$14
$15
Total Wages:
ANS:
A
1
2
B
C
D
Nurse
E
Hiring
F
G
H
I
MA/OR 504
3
4
5
6
7
8
9
10
11
12
13
4.
PRACTICE PROBLEMS FOR MIDTERM EXAM
Shift
1
2
3
4
5
6
Available:
Required:
Mid
4am
1
0
0
0
0
1
4am
8am
1
1
0
0
0
0
20
30
Hours for each shift
8am
Noon
4pm
Noon
4pm
8pm
0
0
0
1
0
0
1
1
0
0
1
1
0
0
1
0
0
0
8pm
Mid
0
0
0
0
1
1
Page |5
Nurses
Scheduled
Wages per
Nurse
$15
$16
$13
$13
$14
$15
Total Wages:
40
50
40
30
The hospital administrators at New Hope, County General, and City East recently received notice of an
impending state inspection of their facilities. Under new guidelines established to improve the overall
health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New
Hope has suggested a mutual assistance program in preparation for the inspections and have proposed
using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized
in the following table. All data reflects averages compiled over the past six months.
New Hope
Hospital
County General
City East
83.0
123.8
225.0
105.0
162.3
200.0
104.1
154.0
231.0
105.0
253.0
125.0
98.0
71.0
92.0
45.0
88.0
82.7
175.0
65.0
83.0
Input Measures
Bed days unused (1000s)
Supply expense ($1000s)
Full-time staff
Output Measures
Patient-days (1000s)
Nurses qualified
Assistants on staff
Customer satisfaction
Enter the numbers in the appropriate cells of ranges B4:H6 in the Excel spreadsheet to solve this problem
based on the following formulation.
Let
wi = weight assigned to output j, j = 1, ..., 4
vi = weight assigned to input i, i = 1,...,3
MAX:
82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4
Subject to:
105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4  83.0 v1  123.8 v2  225.0 v3  0
71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4  105.0 v1  162.3 v2  200 v3  0
82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4  104.1 v1  154.0 v2  231.0 v3  0
104.1 v1 + 154.0 v2 + 231.0 v3 = 1
w1, w2, w3, w4, v1, v2, v3  0
A
1
2
3
4
5
Hospital
New Hope
Cnty. General
B
Patient
Days
(1000s)
C
Nurses
Qual.
D
Asst
on
Staff
E
Cust
Sat.
>
>
>
>
>
>
MA/OR 504
6
7
8
9
10
11
12
<
<
<
<
<
<
<
<
<
<
<
<
<
PRACTICE PROBLEMS FOR MIDTERM EXAM
City East
0
Weights
0
>
>
>
>
>
>
>
0
3
0.81
1.0
UNIT
Output
Input
1
2
3
4
5
6
7
8
9
10
11
12
0
F
Bed-Days
Unused
(1000s)
G
Supply
Expense
($1000s)
H
Full
Time
Staff
0
0
0
Page |6
I
J
K
Wgt.
Output
97%
83%
81%
Wgt.
Input
97%
87%
100%
Diff
0.0000
0.0381
0.1877
ANS:
A
1
2
3
4
5
6
7
8
9
10
11
12
<
<
<
<
<
<
<
<
<
<
<
<
<
Hospital
New Hope
Cnty. General
City East
Weights
0.002009
C
Nurses
Qual.
253.00
92.00
175.00
D
Asst
on
Staff
125.0
45.0
65.0
E
Cust
Sat.
98
88
83
0
0
0.00778
3
0.812259
1
UNIT
Output
Input
1
2
3
4
5
6
7
8
9
10
11
12
B
Patient
Days
(1000s)
105.10
71.00
82.70
F
Bed-Days
Unused
(1000s)
83.00
105.00
104.10
G
Supply
Expense
($1000s)
123.80
162.30
154.00
H
Full
Time
Staff
225.00
200.00
231.00
0
0
0.004329
>
>
>
>
>
>
>
>
>
>
>
>
>
I
J
K
Wgt.
Output
97%
83%
81%
Wgt.
Input
97%
87%
100%
Diff
0.0000
0.0381
0.1877
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |7
5. The following questions correspond to the problem below and associated Solver sensitivity report.
Robert Hope received a welcome surprise in this management science class; the instructor has decided to
let each person define the percentage contribution to their grade for each of the graded instruments used
in the class. These instruments were: homework, an individual project, a mid-term exam, and a final
exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor
complicated Robert's task somewhat by adding the following stipulations:




homework can account for up to 25% of the grade, but must be at least 5% of the grade;
the project can account for up to 25% of the grade, but must be at least 5% of the grade;
the mid-term and final must each account for between 10% and 40% of the grade but
cannot account for more than 70% of the grade when the percentages are combined; and
the project and final exam grades may not collectively constitute more than 50% of the
grade.
The following LP model allows Robert to maximize his numerical grade.
Let
W1 = weight assigned to homework
W2 = weight assigned to the project
W3 = weight assigned to the mid-term
W4 = weight assigned to the final
MAX:
Subject to:
75W1 + 94W2 + 85W3 + 92W4
W1 + W 2 + W 3 + W4 = 1
W3 + W4  0.70
W2 + W4 ≤ 0.50
0.05  W1  0.25
0.05  W2  0.25
0.10  W3  0.40
0.10  W4  0.40
Adjustable Cells
Cell
Name
$F$5
Mid Term to grade
$F$6
Final to grade
$F$7
$F$8
Project to grade
Homework to grade
Final
Value
Reduced
Cost
Objective
Coefficient
Allowable
Increase
Allowable
Decrease
0.40
0.25
0.25
0.10
10.00
0.00
2.00
0.00
85
92
94
75
1E+30
2
1E+30
10
10
17
2
1E+30
Final
Value
Shadow
Price
Constraint
R.H. Side
Allowable
Increase
Allowable
Decrease
0.65
0.5
1.00
0
17
75.00
0.7
0.5
1
1E+30
0.05
0.15
0.05
0.15
0.05
Constraints
Cell
Name
$E$14
$E$15
$F$9
Both Exams Total
Final & Project Total
100% to grade
5a. Refer to the problem statement. Constraint cell F9 corresponds to the constraint, W1 + W2 + W3 + W4 = 1,
and has a shadow price of 75. Armed with this information, what can Robert request of his instructor
regarding this constraint?
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |8
ANS:
Nothing. The constraint has the largest shadow price but enforces the total percentages to equal 1, thus
nothing can be changed.
5b. Refer to the problem statement. Based on the Solver sensitivity report information, is there anything
Robert can request of his instructor to improve his final grade?
ANS:
Robert can request an increase in the total weight allowed for the project and final exam combined since
this has a positive shadow price.
5c. Refer to the problem statement. Based on the Solver sensitivity report information, Robert has been
approved by his instructor to increase the total weight allowed for the project and final exam to 0.50 plus
the allowable increase. When Robert re-solves his model, what will his new final grade score be?
ANS:
88.85 since shadow price of 17 and increase of 0.05 equates to 0.85.
6. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources
of crude oil. The following network representation depicts this problem.
Write out the LP formulation for this problem.
ANS:
MIN:
Subject to:
15X13 + 13X14 + 9X23 + 11X24 + 4X35 + 7X36 + 8X37 + 3X45 + 9X46 + 6X47
X13  X14 = 100
X23  X24 = 50
0.80X13 + 0.95X23  X35  X36  X37 = 0
0.85X14 + 0.85X24  X45  X46  X47 = 0
0.95X35 + 0.90X45 = 50
0.90X36 + 0.95X46 = 25
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
Page |9
0.90X37 + 0.95X47 = 75
Xij  0
7. A small town wants to build some new recreational facilities. The proposed facilities include a swimming
pool, recreation center, basketball court and baseball field. The town council wants to provide the
facilities which will be used by the most people, but faces budget and land limitations. The town has
$400,000 and 14 acres of land. The pool requires locker facilities which would be in the recreation center,
so if the swimming pool is built the recreation center must also be built. Also the council has only enough
flat land to build the basketball court or the baseball field. The daily usage and cost of the facilities (in
$1,000) are shown below.
Variable
X1
X2
X3
X4
Facility
Swimming pool
Recreation center
Basketball court
Baseball field
Usage
400
500
300
200
Cost ($1,000)
100
200
150
100
Land
2
3
4
5
Based on this ILP formulation of the problem and the indicated optimal solution what values should go in
cells B5:G12 of the following Excel spreadsheet?
MAX:
Subject to:
400 X1 + 500 X2 + 300 X3 + 200 X4
100 X1 + 200 X2 + 150 X3 + 100 X4  400
2 X1 + 3 X2 + 4 X3 + 5 X4  14
X1  X2  0
X3 + X4  1
Xi = 0, 1
budget
land
pool and recreation center
basketball and baseball
Solution: (X1, X2, X3, X4) = (1, 1, 0, 1)
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
F
Pool
Facilities
Rec center
Basketball
Baseball
Total usage:
G
Select (0=no, 1=yes)
Usage
Resources
Cost
Land
Pool & Rec center
Basket or Baseball
Used
Available
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
P a g e | 10
ANS:
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
F
Select (0=no, 1=yes)
Usage
Pool
1
400
Facilities
Rec center
1
500
Basketball
0
300
Baseball
1
200
Total usage:
1100
Resources
Cost
Land
Pool & Rec center
Basket or Baseball
100
2
1
0
200
3
1
0
150
4
0
1
100
5
0
1
Used
400
10
0
1
G
Available
400
14
0
1
8. The following questions are based on the problem below.
A company wants to advertise on TV and radio. The company wants to produce about 6 TV ads and 12
radio ads. Each TV ad costs $20,000 and is viewed by 10 million people. Radio ads cost $10,000 and are
heard by 7 million people. The company wants to reach about 140 million people, and spend about
$200,000 for all the ads. The problem has been set up in the following Excel spreadsheet.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
Problem Data
Cost
Coverage
B
TV
20
10
C
Radio
10
7
D
E
Goal Constraints
Actual Amount
+Under
 Over
= Goal
Target Value
TV
0
0
0
0
6
Radio
0
0
0
0
12
Cost
Coverage
0
0
0
200
0
0
0
140
1
0
1
0
1
0
1
0
Percentage Deviation:
Under
Over
Weights
Under
Over
Objective
0
8a. Refer to the problem statement. What formula goes in cell D6?
a. =SUMPRODUCT(B2:B3,B6:B7)
b. =B2*C2+B6*C6
MA/OR 504
PRACTICE PROBLEMS FOR MIDTERM EXAM
P a g e | 11
c. =SUMPRODUCT(B2:C2,B10:C10)
d. =SUMPRODUCT(B2:C2,B6:C6)
ANS: D
8b. Refer the problem statement. What formula goes in cell B9?
a. =SUM(B6:B8)
b. =B6+B7-B8
c. =B6-B7+B8
d. =B10-B8
ANS: B
8c. Refer to the problem statement. Which of the following is a constraint specified to Solver for this model?
a. $B$9:$E$9=$B$6:$E$6
b. $B$9:$E$9<$B$10:$E$10
c. $B$9:$E$9=$B$10:$E$10
d. $B$9:$E$9>$B$10:$E$10
ANS: C
8d. Refer the problem statement. Which cells are the variable cells in this model?
a. $B$6:$C$6, $B$7:$E$8
b. $B$6:$C$6
c. $B$9:$E$9
d. $B$6:$E$8
ANS: A
8e. Refer to the problem statement. Which cell(s) is(are) the objective cell(s) in this model?
a. $B$20
b. $D$6
c. $E$6
d. $B$13:$E$14, $B$9:$E$9
ANS: A
8f. Refer to the problem statement. If the company is very concerned about going over the $200,000 budget,
which cell value should change and how should it change?
a. D13, increase
b. D13, decrease
c. D14, increase
d. D14, decrease
ANS: C