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Unformatted text preview: Pro (Rania/J R)“ Hie/Am, blems M— 23 are estimated to be $6 per book. The publisher plans to sell the text to college and univer sity bookstores for $46 each. a. What is the breakeven point? b. What proﬁt or loss can be anticipated with a demand of 3800 copies? c. With a demand of 3800 copies, what is the minimum price per copy that the publisher
must charge to break even? I d. If the publisher believes that the price per copy could be increased to $50.95 and not
affect the anticipated demand of 3800 copies, what action would you recommend?
What proﬁt or loss can be anticipated? Preliminary plans are under way for the construction of a new stadium for a major league '1 , 16. 17. baseball team. City ofﬁcials have questioned the number and proﬁtability of the luxury cor— porate boxes planned for the upper deck of the stadium. Corporations and selected individu als may buy the boxes for $300,000 each. The ﬁxed construction cost for the upper—deck area
is estimated to be $4,500,000, with a variabie cost of $150,000 for each box constructed. a. What is the breakeven point for the number of luxury boxes in the new stadium? b. Preliminary drawings for the stadium show that space is available for the construction
of up to 50 luxury boxes. Promoters indicate that buyers are available and that all 50
could be sold if constructed. What is your recommendation concerning the construc—
tion of luxury boxes? What proﬁt is anticipated? Financial Analysts, Inc., is an investment ﬁrm that manages stock portfolios for a number
of clients. A new client is requesting that the ﬁrm handle an $800,000 portfolio. As an
initial investment strategy, the client would like to restrict the portfolio to a mix of the
following two stocks: Maximum Price] Estimated Annual Possible
Stock Share Return/Share Investment
Oil Alaska $50 $6 ‘ $500,000
S outhwest Petroleum $30 $4 $450,000 Let x = number of shares of Oil Alaska
y = number of shares of Southwest Petroleum 3. Develop the objective function, assuming that the client desires to maximize the total
annual return.
b. Show the mathematical] expression‘for each of the following three constraints:
(1) Total investment funds available are $800,000.
(2) Maximum Oil Alaska investment is $500,000.
(3) Maximum Southwest Petroleum investment is $450,000. Note: Adding the x 2 0 and y 2 0 constraints provides a linear programming model for the
investment problem. A solution procedure for this model will be discussed in Chapter 2. Models of inventory systems frequently consider the relatioliships among a beginning
inventory, a production quantity, a demand or sales, and an ending inventory. For a given
production period j, let Sjil = ending inventory from the previous period (beginning inventory for period j)
x; = production quantity in period j
d, = demand in period j
s; = ending inventory for period j Case problem Scheduling a Golf League 25 Carpentry costs $15 per hour, painting cests $12 per hour, and ﬁnishing costs $18 per
hour, and the weekly number of hours available in the processes is 3000 in carpentry,
1500 in painting, and 1500 in ﬁnishing. Brooklyn also has a contract that requires
the company to supply one of its customers with 500 contemporary cabinets and 650
farmhouse style cabinets each week. ' . Let x = the number of contemporary style cabinets produced each week
y = the number of farmhouse style cabindts produced each week a. Develop the objective function, assuming that Brooklyn Cabinets wants to maximize
the total weekly proﬁt. [1. Show the mathematical expression for each of the constraints on the three processes. c. Show the mathematical expression for each of Brooklyn Cabinets’ contractual agreements. . i.
= PromoTime, a local advertising agency, has been hired to promote the new adventure
 ‘ ’ ﬁlm Tomb Raiders starring Angie Harrison and Joe Lee Ford. The agency has been given
a $100,000 budget to spend on advertising for the movie in the week prior to its release,
and the movie‘s producers have dictated that only local television ads and locally tar—
geted Internet ads will be used. Each television ad costs $500 and reaches an estimated
i 7000 people, and each Internet ad costs $250 and reaches an estimated 4000 people. The
movie's producers have also dictated that, in order to avoid saturation, no more than
20 television ads will be placed. The producers have also stipulated that, in order to reach '
a critical mass, at least 50 Internet ads will be placed. Finally, the producers want at least
one—third of all ads to be placed on television.
Let IO :11
.st
1e or x = the number of television ads purchased y = the number of Internet ads purchased a. Develop the. objective function, assuming that the movie’s producers want to reach
the maximum number of people possible. b. Show the mathematical expression for the budget constraint. c. Show the mathematical expression for the maximum number of 20 television ads to
be used. ’ d. Show the mathematical expression for the minimum number of Internet ads to be _ used. , e. Show the mathematical expression for the stipulated ratio of television ads to Internet
ads.  f. Carefully review the constraints you created in part (b), part (c), and part (d). Does
any aspect of these constraints concern you? If so, why? tal iti 1n
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Ich Case Problem SCHEDULING A GOLF LEAGUE Chris Lane, the head professional at Royal Oak Country Club, must develop a schedule
of matches for the couples’ golf league that begins its season at 4:00 P.M. tomorrow.
Eighteen couples signed up for the league, and each couple must play every other couple
over the course of the 17W63k season. Chris thought it would be fairly easy to develop a
schedule, but after working on it for. a couple of hours, he has been unable to come up with
a schedule. Because Chris must have a schedule ready by tomorrow afternoon, he asked
you to help him. A possible complication is that one of the couples told Chris that they
may have to cancel for the season. They told Chris they will let him know by 1:00 P.M.
tomorrow whether they will be able to play this season. il I‘; I Problems 7i . 18. For the linear program
3, WXM+M “I. s.t.
10.4 + 23 s 30
3A + 2B 5 12
2A + 213 s 10
A320 a. Write this problem in standard form.
h. Solve the problem using the graphical solution procedure.
c. What are the values of the three slack variables at the optimal solution? 19. Given the linear program 'Max 3A + 43 , 5 s.t. ; 3' —1A + 23 s 8
' M+m$m
2A+lle6 A, B 2 0
I :I ‘ a. Write the problem in standard form.
b. Solve the problem usingthe graphical solution procedure.
c. What are the values of the three slack variables at the optimal Solution? 1 I. For the linear program 3 Max 3A + 23
‘ [Y _ St
1 , A + B 2 4
a f‘ M+4BSM
‘ , A 2 2
V _ A — B s 0
e ' A, B 2 0
. 3. Write the problem in standard form.
: b. Solve the problem. ' ‘
5 l
l c. What are the values of the slack and surplus variables at the optimal solution?
21. Consider the following linear program: Max 2A + 313
2. s.t. __ .
. ' 5A + SE S 400 Constraint 1 —1A + 13 S 10 Constrath
1A + 33 '2 90 Constraint 3
ABEO Figure 2.22 shows a graph of the constraint lines. 33 he .l m..___.._—. ...,... , a —'. 7.. A“ . Problems ' 127 .., .;__._ I “ :1 d. The dual value for constraint 2 is 3. Using this dual value and the righthand—side range
.g‘ information in part (0), what conclusion can be drawn about the effect of changes to . the righthand side of constraint 2?
.. 3 ® Refer to the Kelson Sporting Equipment problem (Chapter 2, Problem 24). Letting
‘ SELF tCSt , R = number of regular gloves
C = number of catcher's mitts : . leads to the following formulation:
Max 5R + 8C
_ at
H.“ ' R + 3/2C 5 900 Cutting and sewing
as + 1/8C 5 300 Finishing
1/812 + ‘AC 5 100 Packaging and shipping
R, C 2 0 J The computer solution is shown in Figure 3.13.
a. What is the optimal solution, and what is the value of the total proﬁt centribution? } b. Which constraints are binding?
 ' c. What are the deal values for the resources? Interpret each.
... ,. d. If overtime can be scheduled in one of the departments, where would you recommend
"I  doing so? FIGURE 3.13 THE SOLUTION FOR THE KELSON SPORTING EQUIPMENT PROBLEM Optimal Objective Value = 3700.00000
'1. Variable Value Reduced Cost
. " R 500.00000 0.00000
. C 150.00000 0.00000
" Constraint Slack/Surplus Dual Value
1 175.00000 0.00000
2 0.00000 3.00000
3 I 0.00000 28.00000
‘ Objective Allowable  Allowable
Variable Coefficient Increase Decrease
R 5.00000 7.00000 1.00000
C ' 8.00000 2.00000 _: . 4.66667
_‘ RHS Allowable Allowable
Constraint Value Increase Decrease
“ﬂ? 1 900.00000 Infinite 175.00000
 2 300.00000 100.00000 166.6666? 3 _ 100.00000 35.00000 25.00000 LU: m, ad: " numberﬂ Iooxa’. éﬂd.
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