**Unformatted text preview: **MGT 355 - Practice Exam (1) with Study Hints 2016
Make-up exams are not allowed, except in cases of emergency (You need to provide
supporting documents. Also, the exam format will be different – e.g. using the Solver to find
solutions). Low class participation (e.g. using cell phones in the classroom) combined with several
absences can lead to a failing grade. Simulation
____1. The daily demand for newspapers at the subway station shop is 35, 40, 45, 50, or 55 newspapers,
with probabilities of 0.1, 0.1, 0.3, 0.3, 0.2, respectively. Assume the following random numbers have been
generated: .10, .47, .99, .37, .66, .91, .35, .20, .00, and .84. Simulate the daily demand for the next 10 days.
The average of simulated sales is a) 46, b) 47, c) 48, d) 49, e) none of the above. Hint:
Prob. Cumulative Prob. R.N.I .1
.1
.3
.3
.2 .1
.2
.5
.8
1.0 .00-.1
.1-.2
.2-.5
.5-.8
.8-1.0 Da
y
1
2
3
4
5 R
N
.1
.47
.99
.37
.66 Simulated Daily
Demand Daily Demand Da
y
6
7
8
9
10 R
N
.91
.35
.20
.00
.84 Simulated Daily
Demand ____3. The probability that the value of a standard normal variable (i.e. Z) is less than 3.0 equals a) .9772, b) .
0287, c) .9987, d) .8413, e).0919. Hint: _____9. A meteorologist was simulating the number of days that rain would occur in a month. The random number interval from .00 to .30 was used to indicate that rain occurred on a particular day, and
the interval .30-1.00 indicated that rain did not occur. What is the probability that rain did occur?
a. 0.30, b. 0.31, c. 1.00, d. 0.70. 1 _____19. The daily demand for newspapers at the subway station shop is 35, 40, 45, 50, or 55
newspapers, with probabilities of 0.1, 0.1, 0.3, 0.3, 0.2, respectively. Simulate the daily demand for
the next 10 days. Simulate the daily demand for the next 10 days using the RAND function. Enter
the formula ____________ in cell C11 (referring to the following printout).
a) =vlookup(B11,B3:D7,3) or =vlookup(B11,$B$3:$D$7,3), b) =rand(), c) none of the above. _____20. In cell C17, we need to enter a) =VLOOKUP(B17,$B$6:$D$12,3 ), b) =VLOOKUP(B17,$B$6:$D$12,2), c) =RAND( ), d) =VLOOKUP(B17,$A$6:$D$12,3), e) none of the
above. 21. You have the following data. Establish your probability distribution for this variable. 2 Water Heater Sales Per Week Number of Weeks this number was sold 4
5
6
7
8
9
10 6
5
9
12
8
7
3 Probabilit
y _____31. Currently the three-section SAT is calibrated to have a mean total score of 1500 with a
standard deviation of 300. What is the simulated SAT score with the random number .8413? a) 1800, b)
1900, c) 2000, d) 2100, e) 2200, f) none of the above.
_____32. Please show how we can replicate the RP simulation by assuming that the returns follow
normal distribution. In cell H13, for example, we need to enter the following formula (with the random
number .61 or RAND()): a) =VLOOKUP(G13,$F$6:$H$9,3), b) =NORMINV(G13,$K$6,$K$7), c)
=NORMINV(RAND(),$K$6,$K$7), d) both b and c, e) none of the above. 1 (b), 3 (c), 9 (a), 19 (a), 20 (a), 21 Ans: Total: 50; 0.12, 0.1, 0.18, 0.24, 0.16, 0.14, 0.06. 31 (a) (Z=1),
32 (d) LP 1
_____1. Hong Securities has $300,000 to invest in four stocks and three bonds. X1, X2, X3, and X4
denote the amounts invested in each of the stocks, and Y1, Y2, and Y3 equal the amounts invested
in each of the three bonds. Which of the following shows that at least 40% of the investment in
stocks must be in stock 1?
a) X1 <= 120,000
b) X1 - .4X2 -.4X3 - .4X4 0 3 c) .6X1 - .4X2 - .4X3 - .4X4 0
d) X1 .4(X2 + X3 + X4 + Y1 + Y2 + Y3) The following business story pertains to questions 2 to 4 The Lego Furniture Company manufactures inexpensive tables and chairs. The firm’s daily LP
formulation is given as
maximize profits $7 X 1 $5 X 2
subject to
4 X 1 3 X 2 240 hours of carpentry time available
2 X 1 1 X 2 100 hours of painting time available
In addition, Lego finds that three more constraints are in order. First, each table and chair must be
inspected and may need rework. The following constraint describes the time required on the average for
each:
1 X 1 3 X 2 36 hours of inspection /rework time availabe
2
5
Second, Lego faces a resource constraint relating to the lumber needed for each table or chair and the
amount available each day:
32 X 1 10 X 2 1,248 linear feet of lumber available for production
Finally, the demand for tables is found to be a maximum of 40 daily. There are no similar constraints
regarding chairs.
X 1 40 maximum table production daily
______2. Referring to the following Solver model, fill out the constraint box (also, assign integer
values to all variables):
(a) $B$18: $B$22 <=$D$5:$D$9; $B$14:$C$14 int
(b) $B$18: $B$22 <=$D$5:$D$9
(c) $B$18: $B$23 <=$D$5:$D$10; $B$14:$C$14 int
(d) $B$18: $B$23 <=$D$5:$D$10
(e) none of the above
Referring
to the
following
computer
outputs,
answer the
questions. 4 _____3. What is the profit generated by this solution? (a) 374, (b) 0, (c) 27, (d) 37, (e) none of the
above
_____4. Lego’s owner has been approached by a friend whose company would like to use several hours
in the painting facility every day. Should Lego sell time to the other firm? If so, how much? (a) Yes; 9
hours, (b) Yes; 21 hours, (c) Yes; 374 hours, (d) Yes; 27 hours, (e) none of the above. 5 The following business story pertains to question 5
Electronic Communications manufactures portable radio systems that can be used for two-way
communications. There are four distribution channels available for their new radio. The profitability,
the advertising cost, and the personal sales effort required will vary with the distribution channels. The
table below summarizes the relevant data:
Distribution
Channel
Marine distributors
Business distributors
National retail stores
Direct mail Profit per
Unit Sold
$90
$84
$70
$60 Advertising
Cost per
Unit Sold
$10
$8
$9
$15 Effort
per
Unit Sold
2 hours
3 hours
3 hours
None The advertising budget is set at $5000, and a maximum of 1800 hours of sales force time is available
for allocation to the sales effort. Management also decided to produce 600 units for the current
production period. Finally, an ongoing contract with the national chain of retail stores requires that at
least 150 units be distributed through this distribution channel.
Establish a distribution strategy for the radios that will maximize overall profitability of the new
radio production.
Referring to the following computer output, answer the question.
_____5. What is the profit generated by this solution? (a) $38,460, (b) $49,050, (c) $27,380, (d)
$48,450, (e) none of the above. (1) C , (2) A, (3) A, (4) A, (5) D 6 LP 2 (I) Brooke has $70,000 to divide among several investments. The alternative investments are municipal bonds
with an 8.5% annual return, certificates of deposit with a 5% return, treasury bills with a 6.5% return, and a
growth stock fund with a 13% annual return (let Xi=$ invested in option i, i=1,2,3, and 4). The investments are
all evaluated after 1 year. However, each investment alternative has a different perceived risk to the investor;
thus, it is advisable to diversity. Brooke wants to know how much to invest in each alternative in order to
maximize the return. The following guidelines have been established for diversifying the investments and
lessening the risk perceived by the investor:
1. No more than 20% of the total investment should be in municipal bonds. How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4 0
2. The amount invested in certificates of deposit should not exceed the amount invested in the other three
alternatives. How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4 0
3. At least 30% of the investment should be in treasury bills and certificates of deposit. How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4____ 0
Note: the objective function: Max Z = .085X1+.05X2+.065X3+.13X4 Ans: (1) .8, -.2, -.2, -.2; (2) -1,+1,-1,-1; (3) -.3, +.7,+.7, -.3, ≥
(II) The Investment Club at Bell Labs has solicited and obtained $50,000 from its members.
Collectively, the members have selected the three stocks, two bond funds, and a tax-deferred annuity
shown in the following table as possible investments (let Xi=$ invested in option i, i=1, 2, 3, 4, 5, and
6). Formulate and solve a linear program that will maximize the total projected annual return subject to
the conditions set forth by the Investment Club members.
Decision Variable
Investment Option
Risk
Projected Annual
Return
Stock – EAL
High
15%
X1
Stock
–
BRU
Moderate
12%
X2
Stock – TAT
Low
9%
X3
Bonds – long term
11%
X4
7 X5
X6 Bonds – short term
Tax-deferred
annuity 8%
6% The club members have decided on the following strategies for investment: All $50,000 is to be invested.
How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4_______X5 _______X6 ___________
ANS: X1+X2+X3+X4+X5+X650,000 At least $10,000 is to be invested in the tax-deferred annuity. How should this constraint be written? Please fill out the following (you need to be aware of the
positive /negative signs, and , =, or ≥ )
:________ X1 _______X2 _______X3 _______X4_______X5 _______X6___________
Ans: X6≥10,000 At least 25% of the funds invested in stocks are to be in the low-risk stock (i.e. TAT).
How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4_______X5 _______X6≥ 0
ANS: -.25X1-.25X2+.75X3≥0 (X3≥.25(X1+X2+X3) No more than $12,500 of the total investment is to be placed in investments with projected
annual returns of less than 10% (i.e. TAT, Short term Bonds, and Tax-deferred annuity)
How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4_______X5 _______X6 ___________
ANS: X3+X5+X612,500
Referring to the following computer outputs, answer the question.
X1 X2 X3 X4 X5 X6 $7,50
0 $0 $2,50
0 $30,00
0 $0 $10,00
0 What is the projected rate of return of this portfolio? Ans:____________________.
ANS: $5,250 (=$7,500*0.15+$2,500*0.09+$3,000*0.11+$10,000*0.06) At least as much is to be invested in bonds as stocks.
How should this constraint be written? Please fill out the following (you need to be aware of the
positive and negative signs)
:________ X1 _______X2 _______X3 _______X4_______X5 _______X6≥ 0
ANS: -X1-X2-X3+X4+X5≥0 (X4+X5≥X1+X2+X3)
8 9 Transportation Problem 10 ______7. Formulate the linear program to determine the minimum cost shipping schedule for the
following problem (Let Xij = Number of tons to ship from i to j). Which of the following is correct? a)
Xhk<= Xbh + Xjh (Transshipment point: Hamburg), b) Xrk<= Xbr + Xjr (Transshipment point: Rotterdam),
c) Xnk<= Xbn + Xjn (Transshipment point: Napoli), d) Xbk<= Xbb + Xjb (Transshipment point: Berlin), e)
Xlk<= Xbl + Xjl (Transshipment point: London), f) Xik<= Xbi + Xji (Transshipment point: Istanbul ), g) all
of the above. 11 Ans: G (all of the above)
Truck Path 12 8. The distribution system for the Herman Company consists of 3 plants, 2 warehouses, and 4
customers. Plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as
follows:
Warehouse
Plan
1
2
Capacit
t
y
1
$4
$7
450
2
$8
$5
600
3
$5
$6
380
Customer demand and shipping costs per unit (in $) from each warehouse to each customer are as
follows:
Customer
Warehouse 1
2
3
4
1
$6 $4 $8 $4
2
$3 $6 $7 $7
Demand
30 30 30 40
0
0
0
0
a) Develop a network representation of this problem.
b) Formulate a linear programming (LP) model of the problem.
c) Solve the LP to determine the optimal shipping plan.
d) Suppose that shipments between the two warehouses are permitted at $2 per unit and that direct
shipments can be made from plant 3 to customer 4 at a cost of $7 per unit. Formulate a LP model of this
problem. Solve the LP to determine the optimal shipping plan.
Problem 8 (a: developing a network representation) – Answer: 13 Problem 8 (b: formulating a linear programming (LP) model) – Answer:
Min Z=$4 X14 + $7 X15 +…….+$7 X59
s.t.
X14+X15<=450; X24+X25<=600; X34+X35<=380;
(node 4) X46+X47+X48+X49- X14-X24-X34=0 (Flow Out = Flow In);
(node 5) X56+X57+X58+X59- X15-X25-X35=0 (Flow Out = Flow In);
X46+X56=300; X47+X57=300; X48+X58=300; X49+X59=400; Xij>=0.
Problem 8 (c: solving the LP to determine the optimal shipping plan) – Answer: There is an excess capacity of 130 units at plant 3.
Problem 8 (d: what if……solving the LP to determine the optimal shipping plan) – Answer: 14 The LP formulation and optimal solution is shown below:
Min Z=$4 X14 + $7 X15 +…….+$7 X59+$7 X39 + $2 X45+ $7 X54
s.t.
X14+X15<=450; X24+X25<=600; X34+X35 + X39<=380;
(node 4) X45+X46+X47+X48+X49- X14-X24-X34-X54=0 (Flow Out = Flow In);
(node 5) X54+X56+X57+X58+X59- X15-X25-X35- X45=0 (Flow Out = Flow In);
X46+X56=300; X47+X57=300; X48+X58=300; X39+X49+X59=400; Xij>=0. 15 More Sample Problems (Source: Fall 2015 Midterm Exam)
______1. The following are two constraints:
X 1 +2 X 2 ≤12(Constraint 1) ; 2 X 1+ 3 X 2 ≤ 30( constraint 2)
If X 1=2
and X 2=4 , what are the values for Slack 1∧Slack2 ?
(a) S 1=¿ 12, S 2=30
(b) S 1=¿ 6, S 2=24
(c) S 1=¿ 2, S 2=14
(d) S 1=¿ 0, S 2=0
(e) of the above
¿
_____2. What is the total cost represented by the solution shown in the following Table?
(a) 600
(b) 2500
(c) 2600
(d) 560
(e) None of the above
To 1 2 3 3 6 3 Supply 3. The Investment
Club at Bell Labs has
From A
20
30
50
solicited and obtained
$50,000 from its
4
4
3
members.
Collectively, the
B
members have
selected the three
40
40
stocks, two bond
funds, and a tax7
6
5
deferred annuity
shown in the
C
10
15
25
following table as
possible investments
(let Xi=$ invested in
option i, i=1, 2, 3, 4,
Demand 20
80
15
5, and 6). Formulate
and solve a linear
program that will maximize the total projected annual return subject to the conditions set forth by the
Investment Club members.
Decision Variable
Investment Option
Risk
Projected Annual
Return
Stock – EAL
High
15%
X1
Stock – BRU
Moderate
12%
X2
Stock – TAT
Low
9%
X3
Bonds – long term
11%
X4
Bonds – short term
8%
X5
Tax-deferred
6%
X6
annuity
The club members have decided on the following strategies for investment: All $50,000 is to be invested.
At least $10,000 is to be invested in the tax-deferred annuity.
At least 25% of the funds invested in stocks are to be in the low-risk stock (i.e. TAT).
16 No more than $12,500 of the total investment is to be placed in investments with projected
annual returns of less than 10% (i.e. TAT, Short term Bonds, and Tax-deferred annuity)
At least as much is to be invested in bonds as stocks. _____3. One typo can be found in (a) cell D7, (b) cell D8, (c) cell D9, (d) cell D10, (e) none of the
above. _____4. In cells C17 and D17, we need to enter a) =VLOOKUP(B17,$A$6:$D$12,3) in cell C17,
=D15+C17 in cell D17, b) =VLOOKUP(B17,$B$6:$D$12,2) in cell C17, =D15+C17 in cell D17, c)
=RAND( ) in cell C17, =D15+C17 in cell D17, d) =VLOOKUP(B17,$B$6:$D$12,3) in cell C17,
=RAND( ) in cell D17, e) none of the above. 17 The following business story pertains to question 5
Consider the following network representation of a transportation problem: The supplies, demands, and transportation costs per unit are shown on the network. Give the linear
programming model’s objective function for the transportation decision. (New York City =1, Newark
=2, EB=3; Atlantic City = 1, Long Branch =2, Edison =3)
_____5.
a) Min
14X11 + 9X12 + 7X13 + 8X21 + 10X22 + 10X23 + 8X31 + 3X32 + 1X33
b) Min
30X11 + 30X12 + 30X13 + 20X21 + 20X22 + 20X23 + 8X31 + 3X32 + 1X33
c) Min
25X11 + 15X12 + 10X13 + 25X21 + 15X22 + 10X23+ 8X31 + 3X32 + 1X33
d) Min
14X11 + 9X12 + 7X13 + 10X21 + 8X22 + 4X23+ 8X31 + 3X32 + 1X33
e) None of the above
____6. Use the following random numbers to simulate yes and no answers to 10 questions by starting in the first row and letting the double-digit numbers .00-.50 represent yes, and .50-1.00 represent no.
Random numbers: .12 .06 .50 .88 .53 .30 .10 .47 .99 .37 .66 .91 .35 .32 .00 .84 .57 .00
a. no yes no no yes yes yes yes no yes
b. no yes no no no yes yes yes no yes
c. no yes yes no no yes yes yes no yes
d. no yes no no no yes yes yes yes yes
e. none of the above would be appropriate 18 7. A meteorologist was simulating the number of days that rain would occur in a month. The random
number interval from .00 to .30 was used to indicate that rain occurred on a particular day, and the
interval .30-1.00 indicated that rain did not occur. Use the following five random numbers to generate
5-day weather forecast: .95 .18 .71 .56 .90.
Fill out the following:
Prob Cumulative
.
Prob. da
y
1
2
3
4
5 random
value
.95
.18
.71
.56
.90 random number
interval Rai
n Simulated Forecast (i.e.
Rain) 19 Z table 20 QM in Action QM in Action 21 ...

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- Fall '08
- Management, optimal shipping plan, Daily demand, NEGATIVE SIGNS, Simulated Daily, following business story