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Chapter 1
Problem Summary
Prob. #
Concepts Covered
1.1
1.2
1.3
1.4
The management science process
When to use simplistic vs. complex models
Building a simple mathematical model
Writing constraints and an objective function –
solving for an optimal solution.
Writing and objection function and a constraint
for a production problem
Model shells, linear and integer models
Different objectives for the same data
Identifying controllable and uncontrollable
inputs
Building a model shell
Substituting raw data into a model shell to build
a complete model
Spreadsheet functions: SUM, SUMPRODUCT,
MAX, IF, SUMIF, absolute cell references, F4
key, dragging, formula writing
Spreadsheet functions: normal probabilities,
NORMDIST, NORMINV
Spreadsheet Data Analysis functions:
DESCRIPTIVE STATISTICS, REGRESSION
Hypothesis testing (p-values using TDIST) and
confidence interval generation
Spreadsheet functions: RAND(), VLOOKUP
Difference between a parameter and a decision
variable
Use of spreadsheets in management science
Optimization and Prediction Models
Input data for profit maximization
Potential management science studies
Construction of a prediction model
Different models for the same problem
Evaluation of computer output; what-if analyses
Writing constraints in a more readable format
Development and solution of a one-variable nonlinear model; supply and demand
Reading and analyzing computer output, writing
a business memo
1.5
1.6
1.7
1.8
1.9
1.10
1.11-1.25
1.26-1.29
1.30-1.38
1.39-1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
Case 1.1
Chapter 1- 1
Level of
Difficulty
1
4
2
4
Notes
3
5
3
3
6
5
3
3
4
3
2
2
4
4
4
5
5
4
2
5
4
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Problem Solutions
1.1 The four steps include:
1) Problem Definition -- Problem definition should be coordinated with management
so that the right problem is solved.
2) Model Building -- The management scientist must determine the complexity of the
model that can be supported time, budget, and solution approach considerations.
3) Solution of the Model -- The management scientist can develop heuristics to give a
good solution, use an appropriate preprogrammed algorithm or develop software to
derive an optimal solution for the model. If the model results are not consistent with past
experience or common sense, the model should be reformulated.
4) Communication of Results -- Good business reports are essential to the success and
implementation of the model.
1.2
When solving for optimal policies at Ford, even small (a fraction of 1%)
improvements can affect Ford’s bottom line by millions of dollars, more than offsetting
the cost of the management science employees. At Villa Park Ford, such a percent
savings probably would not justify the cost of full-time management science employees.
1.3
MIN T
S.T.
W(T) 
It is an optimization model.
1.4 a.
X1 + X2
=1
X1
0
X1
1
X2
0
X2
1
X1, X2 are integers
b. The minimum expected profit outdoors = -$25,000 and indoors is $35,000
MAXIMIZE -25000X1 + 35000X2
Answer: X1 = 0, X2 = 1; hold the tournament indoors.
Chapter 1- 2
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1.5
X1 = the number of 2-drawer models produced
X2 = the number of 3-drawer models produced
X3 = the number of 4-drawer models produced
a. MAXIMIZE 4X1 + 6X2 + 10X3
b. 40X1 + 55X2 + 70X3  25000
1.6
Day workers work 10hrs./day x 3 days = 30 hours
Night workers work 14 hrs./day x 4 nights = 56 hours
a. MAXIMIZE
S.T.
B - 30DX1 -56EX2 - F
X1
 2 + (T/1000)
X2
N
X1, X2  0 and integer
b. The negotiated price, B, and fixed costs, F, are constants;
if we maximize -30DX1 -56EX2, this is equivalent to minimizing 30DX1 + 56EX2
c. MAXIMIZE 10000 -450X1 - 672X2 - 1000, or
MAXIMIZE 9000 -450X1 - 672X2
S.T.
X1
4
X2
2
X1, X2  0 and integer
d. MAXIMIZE 10000 -450X1 - 672X2 - 1000, or
MAXIMIZE 9000 -450X1 - 672X2
S.T.
X1
 4.4
X2
2
X1, X2  0 and integer
1.7
X1 = number of Compaq systems displayed
X2 = number of IBM systems displayed
a. MAX X1 + X2
b. MAX .02X1 + .01X2
c. MAX (.02)200X1 + (.01)3X2 = MAX 4X1 + 3X2
Chapter 1- 3
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1.8
a. Uncontrollable inputs: The selling prices (G,T), the demands (D1, D2), the amount of
availability of the three compounds (L1, L2, L3), the percent of nitrogen, phosphorous,
and potassium in each compound.
b. Controllable inputs: the number of pounds of compounds 1, 2, and 3 used in the
production of Beauty Green (X1, X2, X3); the number of pounds of compounds 1, 2, and 3
used in the production of Beauty Turf (Y1, Y2, Y3).
1.9
a.
(i) X1 + Y1
(ii) X2 + Y2
(iii) X3 + Y3
b. Each bag is 20 pounds; total pounds divided by 20 yields total bags. Thus,
Total bags of Beauty Green = .05X1 + .05X2 + .05X3
Total bags of Beauty Turf = .05Y1 + .05Y2 + .05Y3
c.
X1 +
Y1
X2
+
Y2
X3
.05X1 + .05X2 + .05X3
Y3
.05Y1 + .05Y2 + .05Y3
 L1
 L2
 L3
 D1
 D2
d. MAX G(.05X1 + .05X2 + .05X3) + T(.05Y1 + .05Y2 + .05Y3) - C1X1 - C2X2 - C3X3 C1Y1 - C2Y2 -C3Y3, or
MAX (.05G-C1)X1 + (.05G-C2)X2 + (.05G-C3)X3 + (.05T-C1)Y1 + (.05T-C2)Y2 +
(.05T-C3)Y3
1.10
X = total number of pounds of Beauty Green produced
Y = total number of pounds of Beauty Turf produced
We must then add constraints:
X = X1+ X2 + X3, or: X1 + X2 + X3 - X = 0
Y = Y1+ Y2 + Y3, or: Y1 + Y2 + Y3 - Y = 0
This simplifies the notation for the constraints for this problem.
a. (X1/X)N1 + (X2/X)N2 + (X3/X)N3  25
b. (Y1/Y)N1 + (Y2/Y)N2 + (Y3/Y)N3  10
Chapter 1- 4
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c. MAX .05X1 + .08X2 + .10X3 + .025Y1 + .055Y2 + .075Y3
S.T.
X1 + X2 + X3
- X
Y1 +
Y2 +
Y3
-Y
X1 +
Y1
Y2
X2 +
X3 +
Y3
.05X
.05Y
(30/X)X1 + (10/X)X2 + (40/X)X3
(20/X)X1 + (30/X)X2 + (10/X)X3
(10/X)X1 + (10/X)X2
(30/Y)Y1 + (10/Y)Y2 + (40/Y)Y3
(20/Y)Y1 + (30/Y)Y2 + (10/Y)Y3
(10/Y)Y1 + (10/Y)Y2
= 0
= 0
 15000
 20000
 30000
 1000

800

25

5

5

10

18

5
X1, X2, X3, Y1, Y2, Y3, X, Y  0
Chapter 1- 5
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1.11-1.25
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
See file Ch1.11-25.xls
Cell G6:
=SUM(B6:F6)
Dragging down gives =SUM(B7:F7) in cell G7, =SUM(B8:F8) in cell G8,
=SUM(B9:F9) in cell G9 and =SUM(B10:F10) in cell G10.
Cell B11:
=B6+B7+B8+B9+B10
Dragging gives the right values. =C6 +C7+C8+C9+C10 in cell C11, etc.
Cell B12:
=B4*B11
Dragging gives the right values. =C4*C11 in cell C12, etc.
Cell H6:
=SUMPRODUCT(B4:F4,B6:F6)
Dragging did not give the right formulas.
For example in cell H7 is =SUMPRODUCT(B5:F5,B7:F7)
Highlight B4:F4 in the formula in cell H6 and press the F4 key. The formula in
H6 is now =SUMPRODUCT($B$4:$F$4,B6:F6). Now dragging down DOES
give the correct formulas in cells H7:H10.
Cell B13:
=MAX(B6:B10)
Dragging cell B13 to C13:F13 did give the correct maximums.
Cell I6:
=IF(B6=$B$13,A6,"")
Dragging cell I6 to I7:I10 did give the correct formulas and did print out the
person with the most Christmas card sales in column I. (Ellen in cell I10.)
Cell C15:
=0.12*B6^2+0.02*C6^3+3.5*SQRT(D6)+0.05*E6*F6
Cell C16:
=IF(C15>H6,"NEW","OLD")
Cell C19:
=SUMIF(B6:B10,"<20",F6:F10)
Chapter 1- 6
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1.26-1.29
See file Ch1.26-29.xls
Note round-off errors when doing by hand.
1.26
a.
b.
Cell B1: =NORMDIST(30,47,5.5,TRUE) =
z = (30-47)/5.5 = -3.09 P(Z < -3.09) = .5 - .4990 =
.000998
.0010
1.27
a.
b.
Cell B2: =1-NORMDIST(60,47,5.5,TRUE) =
z = (60-47)/5.5 = 2.36 P(Z > 2.36) = .5 - .4909 =
.009048
.0091
1.28
a.
Cell B3: =NORMDIST(50,47,5.5,TRUE) - NORMDIST(40,47,5.5,TRUE) =
.605722
For x = 50, z = (50-47)/5.5 = .55, for x = 40, z = (40-47)/5.5 = -1.27
P(0<Z<.55) = .2088; P(-1.27<Z<0) = .3980
Thus, by hand, P(40<X<50) = .2088 + .3980 =
.6068
b.
1.29
a.
b.
Cell B4: =NORMINV(.95,47,5.5) =
56.04669
The z value that puts probability .95 below x is the z value that puts
probability .45 between the mean and x. This is z = 1.645.
Thus the x value = 47 + 1.645(5.5) =
56.0475
Arrival by 8:56AM
Chapter 1- 7
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1.30-1.38
See file Ch1.30-38.xls
1.30
See worksheet Scatterplot
Chapter 1- 8
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1.31
See worksheet Descriptive Stat
a. Mean = 6.2; Standard Deviation = 2.167948
_
b. x = (6 + 6 + 7 + 9 + 3)/5 = 6.2
s2 = ((6-6.2)2 + (6-6.2)2 + (7-6.2)2 + (9-6.2)2 + (3-6.2)2)/4 = 4.7
___
s = 4.7 = 2.167948
1.32
a. Cell G18:
Cell G19:
= G3-G16
= G3+G16
LCL = 3.508131
UCL = 8.891869
b. LCL = 6.2 - t.025,4(2.167948/5)
UCL = 6.2 + t.025,4(2.167948/5)
= 6.2 - 2.7765(2.167948/5) = 3.50808
= 6.2 + 2.7765(2.167948/5) = 8.89192
1.33
Cell B8:
t = (mean -5)/(standard error) = (G3-E21)/G4 = 1.237705
t.05,4 from book = 2.1318, Cannot conclude average study time > 5 hours.
1.34
Cell B9:
p-value = TDIST(t-value, degrees of freedom, 1) = TDIST(B8,4,1) =
.141755532 > .05 --- reach the same conclusion)
Chapter 1- 9
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1.35
See worksheet Grade vs. Test and Study
Grade = 35.12658 - 3.48101(# other tests) + 6.136456(Study Hours)
1.36
a. p-value for the F-test = Significance F = .041283 <  = .05. Can conclude at least
one is significant.
b. p-value for # other tests = .539542 >  = .05. Cannot conclude this is significant
in this model.
p-value for study hours = .034187 <  = .05. Can conclude this is significant in
this model.
Chapter 1- 10
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1.37
See Worksheet Grade vs. Study
a. Grade = 30.05319 + 6.797872(Study Hours)
b. Using regression formulas gives the same result.
1.38
a. Yes, p-value for study hours = .005172 <  = .05.
b. t = (6.797872 - 0)/.922838 = 7.36627. Critical value of t = t.025,3 = 3.1824.
7.36627 > 3.1824. There is enough evidence to conclude a linear relation exists.
Chapter 1- 11
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1.39-1.40
1.39
See file Ch1.39-40.xls
Cell A2:
Cell A3:
0
=A2+B2 ---- Drag to cells A4:A6
1.40
a.
b.
Cell A10:
Cell A10:
c.
Cell A11:
=RAND() --- Note your number will be different
Copy -- Paste Special -- Values
(fixes the number generated in part a)
=VLOOKUP(A10,A2:C6,3)
1.41 A parameter is an uncontrollable input value such as distance, cost, a mean value,
etc. A decision variable is a controllable input whose value can be set by the decision
maker such as production quantities, reorder points, routes traveled, etc.
1.42 A spreadsheet facilitates instantaneous changes to many cells simultaneously.
Much input information is already in spreadsheet format; output can be arrange in easily
readable and presentable fashion; charts and graphs are easily generated from spreadsheet
data; spreadsheets have many built-in algorithms and tools for performing management
science techniques; spreadsheets are the de facto "language of business" and are
understood by most professionals and decision makers.
Chapter 1- 12
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1.43 Optimization models could include minimizing fuel, minimizing customer
waiting time to purchase tickets, minimizing the percent of vacant seats, maximizing
profit, minimizing the total number of aircraft needed to cover a particular set of routes,
maximizing revenue from ticket prices, minimizing the percentage of delayed flights,
minimizing the number of agents, mechanics, etc. needed during peak periods, or
minimizing the wasted cost from food orders.
Prediction models could include predicting the amount of fuel needed in a given time
period, fuel prices, competition pricing policies, customer demand vs. ticket prices,
customer demand for new routes, aircraft landing times, on time performance, or inflation
(as a cost measure).
1.44
a. Product cost/demand relations with customers, quantity discounts/incentive programs
offered by vendors, delivery times of products from vendors, optimal service levels
(being able to fill customer orders) for products, inventory costs, and shortage costs.
b. Databases of potential orders, prices, orders filled, customer cancellations,
competitors in the area, population characteristics (census tract data including income and
spending characteristics) of neighboring areas, etc.
1.45 Forecast of orders vs. the number of commercials, times of the commercials, and
the channels on which the commercials are run; maximizing the number of commercials
run during each commercial spot given a particular budget; scheduling of production; the
number of operators required during the airing of infomercials and at other times, finding
the target population, etc.
1.46
a. This is a prediction model.
b. Baylor’s objective would be to determine whether or not to hire the collection agency.
c. Let T = total number of accounts not paid off after 120 days
B = .4A
C = .5B
D = .3C
T = .2D
A + B + C + D + T = 1000
A, B, C, D, T  0 and integer
The constraints define the relationship between the number of accounts in each category.
Assuming that the collection agency would be assigned to collect from the T accounts
past due by more than 120 days, the agency should be hired only if it generates more
income than the amount paid the collection agency.
Chapter 1- 13
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1.47 Expected Returns -- Topeka 8/42 = .1905, Crosswind = .20, Genserve 11/64 =
.1719, C/D = .05.
X1 = amount invested in Topeka
X2 = amount invested in Crosswind
X3 = amount invested in Genserve
X4 = amount invested in savings certificate
a. MAX .1095X1 + .20X2 + .1719X3 + .05X4
S.T.
X1 + X2 +
X3 + X4 = 4000
X2
 800
X4  1000
X1
 2000
X3
 2000
X1, X2, X3, X4  0
b. MAX 8Y1 + 6Y2 + 11Y3 + .05X4
S.T. 42Y1 + 30Y2 + 64Y3 + X4
30Y2
X4
42Y1
64Y3
Y1, Y2, Y3, X4  0
= 4000
 800
 1000
 2000
 2000
c. Should get the same answer if shares were converted to dollars by X1 = 42Y1, X2 =
30Y2, and X3 = 64Y3.
1.48
a. Yes -- only 2.2165 people ahead of an arrival and an average wait time of only
.0185(60) = 1.11 minutes before ordering; the customer receives food only .0518(60) =
3.11 minutes after arriving.
b. Adding or shutting down some checkout stands; adding more personnel in the
cooking area to reduce service time.
1.49
X1 = Standard rooms
X2 = Deluxe rooms
X3 = Two-bedroom suites
X4 = Luxury suites
a. X3  .6(900) or X3  540.
b. X3  .6(X1 + X2 + X3 + X4) or -.6X1 -.6X2 + .4X3 - .6X4  0.
c. (1) X1 + X2 + X3 + X4 = X5
or
X1 + X2 + X3 + X4 - X5
(2)
X3
 .6X5
or
X3
-.6X5
Chapter 1- 14
=0
0
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1.50
a.
Revenue = (54-2X)X = 54X - 2X2
b.
Cost = 11(54-2X) = 594 - 22X
Profit = -2X2 + 54X - (594-22X) = -2X2 + 76X - 594
Thus the model is:
MAX -2X2 + 76X - 594
S.T.
X
 11
X
 21
c. df/dx = -4X +76 = 0. Thus X = 19 ($19,000);
Number sold = 54-2(19) = 16
Monthly profit = -2(19)2 + 76(19) - 594 = 128 ($128,000)
Chapter 1- 15
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CASE 1.1
Any fractional value can be considered work in progress for August or values can be
rounded to integers.
1.
Toasters
Ovens
Mixers
Blenders
Food Processors
Steamers
Profit
Fractional Values
35.579
16.872
7.500
37.500
37.500
15.049
Integer Values*
36
17
8
37
37
15
$1,1134.42
$1131.00
(*could give an infeasible solution; must check resource, production time constraints)
2. From the Solver Answer Report solution,
1. All production time is used up
2. All of Material 2 is used up
3. Materials 1,3 and 4 are not used up
4. 150 total items are produced
5. Production Percentages
Actual
Toasters
23.72%
Microwaves
11.25%
Mixers
5.00%
Blenders
25.00%
Food Processors
25.00%
Steamers
10.03%
Toasters, Mixers, Blenders 53.72%
Chapter 1- 16
Rounded
24.00%
11.33%
5.33%
24.67%
24.67%
10.00%
54.00%
Limits
5% - 25%
5% - 25%
5% - 25%
5% - 25%
5% - 25%
5% - 25%
 50%
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