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Unformatted text preview: 1. Which of the following mathematical relationships could be found in a linear progrannningj
model, and which could not? For the relationships that are unacceptable for linear pro
grams, state why.
a. —lx1 + 2x2 — 1x3 5 70
b. 2x1 — 2x3 = 50
c. 1.171  235% + 4153 E 10 5:
{1. 3x51 + 2x2 — 1x3 2 15 g i 9. 1x1 + 1x2 + 'lx3 = 6
f. 2x1 + 5362 + 1):le “£ 25 2. Find the feasible solution points for the following constraints:
a. 4x1 + 2x2 5 16
b. 43c1 + 2x2 2 16
8. 4x1  2x2 = 16 3. Show a separate graph of the constraint lines and feasible solutions for each of the fol— lowing constraints:
a. 3x1 + 2x2 5 18
b. 12x1 + 81172 a 480
«3. 5x1 + 10562 = 200 Show a separate graph of the constraint lines and feasible solutions for each of the fol
lowing constraints: a. 3x1 — 4x2 3: 60 b. #6331 + 5x2 5 60 C. 53cI — 2x2 5 O Show a separate graph of the constraint lines and feasible solutions for each of the fol
lowing constraints: a. x1 2 0.25 (x1 + x2) b. 352 S {110(Jr1 + x2) c. x1 E 0.50031 + x2)  Three objective functions for linear programming problems are 7x1 + 10:52, 61:1 + 4x2, and
— 4x1 + 7x2. Determine the 510pe of each objective function. Show the graph of each for
objective function values equal to 420. Identify the feasible region for the following set of constraints:
1f2):1 + 14x: 3 30
1x1 + 5x2 33: 250
143:1 + 1/2352 5 50 x1, x2 3 0 8. Identify the feasible region for the following set of constraints: 2x1  hr2 i: 0
“le + 1.5a:2 :3 200
x1, x2 3 0 9. Identify the feasible region for the following set of constraints: 3x1 * 2x2 33 0
2x1 — 1.7462 1: 200
1.:rl E 150
x1, x2 3: 0 10. For the linear program 10. 11. For the linear program Max 2x1 + 3x2 SI.
1x1 + 2x2 5 6
5161 + 3x2 5 15 3'
xv):2 _ 0 ﬁnd the optimal solution using the graphical solution procedure. What is the value of the
objective function at the optimal solution? Solve the following linear program using the graphical solution procedure. Max 5x1 + 53c2
S.t.
1x1 5 100
1x2 5 80
2x1 + 4x2 5 400 xhxz E 0 12. Consider the following linear programming model: PF.” 13. Consider the following linear program: i 14. Consider the following linear program: Max 3x1 + 3x2 S.t.
2x1 + 4x2 5 12
6x1 + 4x2 5 24 :1"
x1,x2 _ 0 Find the optimal solution using the graphical solution procedure. If the objective function is changed to 2x1 + 6x2, what will the optimal solution be? How many extreme points are there? What are the values of x1 and x2 at each ex Ti
'3‘ treme point? Max 3x] + 2x2
S.t.
2x1 + 2):2 E 8
3x1 + 2x2 5 12
1x1 + 0.5x2 E 3
xvxz E 0 Find the optimal solution using the graphical solution procedure. What is the value of
the objective function?
Does this linear program have a redundant constraint? If so, what is it? Does the so“ if
lution change if the redundant constraint is removed from the model? Explain.  '3': 14. Consider the following linear program: Max 1.1:] + 2.122
SJ.
1x1 3: 5 I
1.1::2 E 4
2x1 + 2172 = 12 x1,x2 E 0 a. Show the feasible region.
I}. What are the extreme points of the feasible region?
c. Find the optimal solution using the graphical procedure. 16. Refer to the feasible region for the Par, Inc., problem'in Figure 2.13.
a. Develop an objective function that will make extreme point @ the optimal extreme
point.
I). What is the optimal solution using the objective function you selected in part (a)?
c. What are the values of the slack variables associated with this solution? 17. Write the following linear program in standard form: Max 5x1 + 2x2 + 8x3 S.t.
1x] + 2x2 + 1/5173 S 420
2x1 + 3x2 — 1x3 5 610
6x1 _ lx2 + 3x3 5 125
x1, x2,x3 2': 0 13. For the linear program Max 4x1 + 1x2
8.1:.
10x1 + 2352 E 30
3x1 + 2x2 *5 12
2361 + 2x2 10
x1, x2 2 0 [AI a. Write this linear program in standard form.
b. Find the optimal solution using the graphical solution procedure.
c. What are the values of the three slack variables at the optimal solution?  19. Given the linear program / 19. 20. Given the linear program a. Write the linear program in standard form. b. Find the optimal solution using the graphical solution procedure.
c. What are the values of the three slack variables at the optimal solution? Embassy Motorcycles (EM) manufactures two lightweight motorcycles designed for easy
handling and safety. The EZ—Rider model has a new engine and a low proﬁle that make it
easy to balance. The LadySport model is slightly larger, uses a more traditional engine,
and is speciﬁcally designed to appeal to women riders. Embassy produces the engines
for both models at its Des Moines, Iowa, plant. Each EZ—Rider engine requires 6 hours of
manufacturing time and each LadySport engine requires 3 hours of manufacturing time.
The Des Moines plant has 2100 hours of engine manufacturing time available for the next
production period. Embassy’s motorcycle frame supplier can supply as many EZRider
frames as needed. However, the LadySport frame is more complex and the supplier can
provide only up to 280 LadySport frames for the next production period. Final assembly
and testing requires 2 hours for each EZ—Rider model and 2.5 hours for each LadySport model. A maximum of 1000 hours of assembly and testing time are available for the next
production period. The company’s accounting department projects a proﬁt contribution of $2400 for each EZRider produced and $1800 for each Lady—Sport produced. a. Formulate a linear programming model that can be used to determine the number of
units of each model that shOuld be produced in order to maximize the total contribu
tion to profit. b. Find the optimal solution using the graphical solution procedure. Which constraints are binding? 21. RMC, lnc., is a small ﬁrm that produces a variety of chemical products. In a particular pro— duction process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 2/2, ton of mate rial 1 and 3/2, of material 3. A ton of solvent base is a mixture of 1/2 ton of material 1, 1/5 ton of material 2, and 9/10 ton of material 3. After deducting relevant costs, the proﬁt contribution is $40 for every ton of fuel additive produced and $30 for every ton of solvent base produced RMC’s production is constrained by a limited availability of the three raw materials. For the current production period, RMC has available the following quantities of each raw
material: Raw Material Amount Available for Production Material 1 20 tons
Material 2 5 tons
Material 3 21 tons Assuming that RMC is interested in maximizing the total proﬁt contribution, answer the following: a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many tons of
each product should be produced, and what is the projected total proﬁt contribution? c. Is there any unused material? If so, how much? (I. Are there any redundant constraints? If so, which ones? 22. Kclson Sporting Equipment, Inc, makes two different types of baseball gloves: a regular
model and a catcher’s model. The ﬁrm has 900 hours of production time available in its cut
ting and sewing department, 300 hours available in its ﬁnishing department, and 100 hours
available in its packaging and shipping department. The production time requirements and
the proﬁt contribution per glove are given in the following table. Production Time (hours) Cutting ' Packaging
Model ' and Sewing Finishing and Shipping Profit/Glove
Regular model 1 1/2; 1/3 $5 Catcher’s model 3/3 1/3 ‘ 1/4 $8 Assuming that the company is interested in maximizing the total proﬁt contribution, an
swer the following:
a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many gloves
of each model should Kelson manufacture? c. What is the total proﬁt contribution Kelson can earn with the listed production quantities? 3:9 1:2, .2" .'?“‘
rig?” "i\ d. How many hours of production time will be scheduled in each department?
e. What is the slack time in each department? George Johnson recently inherited a large sum of money; he wants to use a portion of this
money to set up a trust fund for his two children. The trust fund has two investment op—
tions: (1) a bond fund and (2) a stock fund. The projected returns over the life of the in—
vestments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the
inheritance he ﬁnally decides to commit to the trust fund, he wants to invest at least 30%
of that amount in the bond fund. In addition, he wants to select a mix that will enable him
to obtain a total return of at least 7.5%. a. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investrnent alternatives.
b. Find the optimal solution using the graphical solution procedure. The Sea Wharf Restaurant would like to determine the best way to allocate a monthly adver—
tising budget of $1000 between newspaper advertising and radio advertising. Management
has decided that at least 25% of the budget must be spent on each type of media, and that the
amount of money spent on local newspaper advertising must be at least twice the amount spent
on radio advertising. A marketing consultant has developed an index that measures audience
exposure per dollar of advertising on a scale from 0 to 100, with higher values implying
greater audience exposure. If the value of the index for local newspaper advertising is 50 and
the value of the index for spot radio advertising is 80, how should the restaurant allocate its
advertising budget in order to maximize the value of total audience exposure? a. Formulate a linear programming model that can be used to determine how the restau
rant should allocate its advertising budget in order to maximize the value of total au—
dience exposure. [1. Find the optimal Solution using the graphical solution procedure. {’25. Blair & Rosen, Inc. (B&R), is a brokerage firm that specializes in investment portfolios de
signed to meet the Specific risk tolerances of its clients. A client who contacted B&R this
past week has a maximum of $50,000 to invest. B&R’s investment advisor has decided to
recommend a portfolio consisting of two investment funds: an Internet fund and a Blue
Chip fund. The Internet fund has a projected annual return of 12%, while the Blue Chip.
fund has a projected annual return of 9%. The investment adviser requires that at most
$35,000 of the client’s funds should be invested in the Internet fund. B&R services include
a risk rating for each investment alternative. The Internet fund, which is the more risky of
the two investment alternatives, has a risk rating of 6 per thousand dollars invested. The
Blue Chip fund has a risk rating of 4 per thousand dollars invested. For example, if $10,000
is invested in each of the two investment funds, B&R’s risk rating for the portfolio would
be 6(10) + 4(10) = 100. Finally, B&R has developed a questionnaire to measure each
client’s risk tolerance. Based on the responses, each client is classified as a conservative,
moderate, or aggressive investor. Suppose that the questionnaireresults have classiﬁed the
current client as a moderate investor. B&R recommends that a client who is a moderate in
vestor limit his or her portfolio to a maximum risk rating of 240. a. What is the recommended investment portfolio for this client? What is the annual re turn for the portfolio? _ . b. Suppose that a second client with $50,000 to invest has been classified as an aggres sive investor. B&R recommends that the maximum portfolio risk rating for an aggres sive investor is 320. What is the recommended investment portfolio for this aggressive investor? Discuss what happens to the portfolio under the aggressive investor strategy. c. Suppose that a third client with $50,000 to invest has been classiﬁed as a conservative ' investor. B&R recommends that the maximum portfolio risk rating for a conservative in— vestor is 1 60. Develop the recommended inve strnent portfolio for the conservative investor.
Discuss the interpretation of the slack variable for the total investment fund constraint. 26. 27. Tom’s, Inc., produces various Mexican food products and sells them to Western Foods, a
chain of grocery stores located in Texas and New Mexico. Tom’s, Inc., makes two salsa
products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have
different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods
Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20%. tomato paste. The
Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole
tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs
10 ounces. For the current production period Tom’s, Inc., can purchase up to 280 pounds
of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price
per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the
spices and the other ingredients is approximately $0.10 per jar. Tom’s, Inc., buys empty
glass jars for $0.02 each, and labeling and ﬁlling costs are estimated to be $0.03 for each
jar of salsa produced. Tom’s contract with Western Foods results in sales revenue of $1.64
for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa. 3. Develop a linear programming model that will enable Tom’s to determine the mix of salsa products that will maximize the total proﬁt contribution.
I), Find the optimal solution. Autoignite produces electronic ignition systems for automobiles at a plant in Cleveland,
Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s
plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units
of component 1, 1000 units of component 2, or any combination of the two components
each day. For instance, 60%. of Buffalo’s production time could be used to produce com—
ponent 1 and 40% of Buffalo’s production time could be used to produce component 2; in
this case, the Buffalo plant would be able to produce 0.6(2000) = 1200 units of compo
nent 1 each day and 0.4(1000) = 400 units of component 2 each day. The Dayton plant
can produce 600 units of component 1, 1400 units of component 2, or any combination of
the two components each day. At the end of each day, the component production at Buf—
falo and Dayton is sent to "Cleveland for assembly of the ignition systems on the follow—
ing work day.
a. Formulate a linear programming model that can be used to develop a daily production
schedule for the Buffalo and Dayton plants that will maximize daily production of ig
nition systems at Cleveland. 0&1} b. Find the optimal solution. '_X"+!. '31 A ﬁnancial advisor at Diehl Investments identiﬁed two companies that are likely candi
"'" dates for a takeover in the near future. Eastern Cable is a leading manufacturer of ﬂexible
cable systems used in the construction industry and ComSwitch is a new firm specializing
in digital switching systems. Eastern Cable is currently trading for $40 per share and Com
Switch is currently trading for $25 per share. If the takeovers occur, the ﬁnancial adviser
} t; estimates that the price of Eastern Cable. will go to $55 per share and ComSwitch will go
I to $43 per share. At this point in time, the ﬁnancial adviser identified CemSwitch as the
f 53' higher risk alternative. Assume that a client who indicated a willingness to invest a maxi—
‘ mum of $50,000 in the two companies wants to invest at least $15,000 in Eastern Cable
1.  %:.‘aiid at least 33 10,000 in ComSwitch. Because of the higher risk associated with ComSwitch,
the ﬁnancial adviser recommends that at most $25,000 should be invested in ComSwitch.
a. Formulate a linear programming model that can be used to determine the number of
shares of Eastern Cable and the number of shares of CemSwitch that will meet the in
vestment constraints and maximize the total return for the investment.
b. Graph the feasible region.
c. Determine the coordinates of each extreme point.
d Find the Optimal solution. 1 £33 . /.i
:L_~" a 29. Consider the following linear program: Min 3x1 + 4x2 8.1:.
1x1 + 3x2 2 6
1x1 + 1x2 2 4
x1, x2 3 0 Identify the feasible region and ﬁnd the optimal solution using the graphical solution pro—
cedure. What is the value of the objective function? 30. Identify the three extremepoint solutions for the M&D Chemicals problem (see Section 2.5).
Identify the value of the objective function and the values of the slack and surplus variables
at each extreme point. 32. Consider the following linear program: Min 2x1 + 2.752 8.1:.
1x1 + 3x2 E 12
3x1 + 1x2_13
1x x1 — Li:2 = 3
xv):2 2 0 a. Shevv the feasible region.
b. What are the extreme points of the feasible region?
c. Find the Optimal solution using the graphical solution procedure. 33. For the linear program Min 6x] + 4952
31.
2x1 + 1x2 2 12
1x1 + 1x2 2 10
1x2 5 4
x1, x2 2 0 3. Write the linear program in standard form. b. Find the optimal solution using the graphical solution procedure.
c. What are the values of the slack and surplus variables? As part of a quality improvement initiative, Consolidated Electronics employees complete
a three—day training program on teaming and a two—day training program on problem solv
ing. The manager of quality improvement requested that at least 8 training programs on
teaming and at least 10 training programs on problem solving be offered during the next
six months. In addition, senior—level management speciﬁed that at least 25 training pro
grams must be offered during this period. Consolidated Electronics uses a consultant to
teach the training programs. During the next six months, the consultant has 84 days of
training time available. Each training program on teaming costs $10,000 and each training
program on problem solving costs $8,000. a. Formulate a linear programming model that can be used to determine the number of
training programs on teaming and the number of training programs on problem solv
ing that should be offered in order to minimize total cost. 1}. Graph the feasible region. c. Determine the coordinates of each extreme point.
{1. Solve for the minimum cost solution ...
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 Spring '09
 shakroh
 Math

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