MATH451-Unit 2 Intellipath
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MATH451 – Unit 2 Intellipath
Alternative Formulations
Sometimes it becomes necessary to rescale the numbers used in the objective function and constraints equations to handle the application more efficiently.
In this section, you will be presented an original scenario. You will then revise
the model based on the new scenario. If you are already familiar with the original scenario, you may move onto the section called
Revised Scenario
.
Original Scenario
A bank has decided to promote four different certificates of deposit (CD) with
higher rate of return compared to nearby banks to its new depositors to attract more customers.
The first CD (A) offers a 3% interest rate for amount greater than $10,000.
The second CD (B) offers a 4% interest rate for amount greater than $25,000.
The third CD (C) offers a 4.5% interest rate for amount greater than $30,000.
The fourth CD (D) offers a 5% interest rate for amount greater than $35,000.
A customer has $120,000 to deposit. She wishes to deposit no more than 1/3
of her money in any of the CDs. She likes to deposit exactly twice of her money in CD#2 than CD#1. She wishes the sum of the first and the second CD’s to be more than the third CD.
Setting Up the Linear Programming Model
Here are the objective function and the constraints she is facing with:
Maximizing interest = 0.03A + 0.04B + 0.045C + 0.05D
subject to
A + B + C + D ≤ $120,000
A ≤ (1/3)(A + B + C + D)
B ≤ (1/3)(A + B + C + D)
C ≤ (1/3)(A + B + C + D)
D ≤ (1/3)(A + B + C + D)
B = 2A
A + B ≤ C
A, B, C, and D ≥ 0 (nonnegativity)
Original Scenario Results
Entering these equations into Excel’s Solver program and the program solved the problem with a feasible solution maximizing her interest to $5,266,67 utilizing her entire money in four different CDs satisfying all the constraints.
Figure 1 shows the results of original scenario.
Figure 1:
Results of the Application Using Excel’s Solver
Revised Scenario
Revise the scenario and reprogram it using a different scale. Instead of dealing with dollars, you will deal with thousands of dollars. For example, you
can change the number $1,000 to $1Kilo or $1K. You enter 1 in the program instead of 1,000 understanding all the associated calculations with this problem are now in thousands.
Now the modified objective function and the constraints equations are as follows:
Maximizing interest = 0.03A + 0.04B + 0.045C + 0.05D
subject to
A + B + C + D ≤ $120
A ≤ (1/3)(A + B + C + D)
B ≤ (1/3)(A + B + C + D)
C ≤ (1/3)(A + B + C + D)
D ≤ (1/3)(A + B + C + D)
B = 2A
A + B ≤ C
A, B, C, and D ≥ 0 (nonnegativity)
Figure 2 shows the same program containing the numerical values of the decision-making variables and the amount of resources consumed. At first glance, this modification does not seem to be an important issue. You can see that $120,000 was changed to $120K.
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Figure 2:
Results of the Application Using Excel’s Solver for Revised Scenario
However, when you look at the results shown in Figure 1 for the original scenario, all the calculations are performed in thousands of dollars. For example, the balance of $13.33333 in CD#1 really means $13333.33 and the balance for CD#4 is $40 which must be interpreted as $40,000.
Figure 3:
Results of the Application Using Excel’s Solver for Revised Scenario
Understanding the Results
You could also scale all the numbers based on 100% or its equivalent of 1. In this case, $120,000 can be considered as 1 or 100% amount without giving the actual numerical value of the dollar amount. This way the problem can be solved based on percentage and after the results are obtained you can assign the actual value and recalculate each result from the percentages to the actual value.
This way you have generalized the problem for any dollar value to be deposited in those 4 CDs. Figure 3 shows the results. Selecting the percentage for CD#, 0.111111 and multiplying it by the total dollar amount of $120,000 in this case provides the numerical value of $13,333.33 which is indeed the actual value if the amount deposited in CD#1 as obtained earlier.
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are lowering all amounts by a factor or scale of 1,000.
How much actual money does the objective function have after running the problem through Excel’s Solver?
(OR worded this way: How much money does the program show for the objective function after running the problem through Excel’s Solver?)
A:
$6,825
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are using percentage scale for all amounts.
What value does the program show for the objective function after running the problem through Excel’s Solver?
A: 6,825
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are using percentage scale for all amounts.
What would you replace $150,000 within the constraints equations?
A:
1
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are lowering all amounts by a factor or scale of 1,000.
What would you replace $150,000 within the constraints equations?
A:
$150
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are using percentage scale for all amounts.
What value does the program show for each account after running the problem through Excel’s Solver?
A:
0.083333, 0.25, 0.333333, and 0.333333, respectively
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
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for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are lowering all amounts by a factor or scale of 1,000.
How much actual money does each account have after running the problem through Excel’s Solver?
A:
$12,500, $37,500, $50,000, and $50,000, respectively
Q:
Fifth National bank offers four different certificates of deposit (CD). The first CD (A) offers a 4% interest rate for amount greater than $10,000. The second CD (B) offers a 4.2% interest rate
for amount greater than $25,000. The third CD (C) offers a 4.5% interest rate for amount greater than $30,000. Finally, the fourth CD (D) offers a 5% interest rate for amount greater than $35,000. A customer has $150,000 to deposit. She wishes to deposit not more than 1/3 of her money in any of the CDs. She likes to deposit exactly three times of her money in CD#2 than CD#1. She wishes the sum of the first and the second CDs to be less than or equal to the third CD. You are lowering all amounts by a factor or scale of 1,000.
What is the objective function?
A:
Maximize interest for $0.04A + $0.042B + $0.045C + $0.05D
Q:
Sometimes, the constraints in optimization problems look like
, assuming n decision variables. In general the questions lools like “could the
be replaced with a strict inequality
?” In order to get a feel
for it, we'll look at a specific case.
Consider the problem to maximize
, subject to
. Does this problem have a solution?
A: No. If x,10, it is always possible to find some y with x<y<10
Q: Sometimes,
the constraints in optimization problems look like
, assuming n decision variables. Is it enough to
only
consider
, or do
you
also have to consider
when analyzing these problems in general? Why?
A:
It is enough to only consider because you can multiply the constraint by -1 to flip the inequality, but that does not change the set of feasible solutions
.
Q:
Sometimes, the constraints in optimization problems look like , assuming n decision variables.
To clarify, suppose that the s represent dollars, and for some reason, they seem to work better with hundreds of dollars. Let the s be the new variables representing thousands of dollars.
If after solving, you find that and , what are the actual dollar amounts?
A: $983,743 and $57,767 respectively
Q:
Suppose that you want to maximize the objective
, subject to constraint 1:
, constraint 2:
, and constraint 3:
. You can eliminate the variable
and constraint 3, but that requires changes to the objective and constraints 1 and 2.
Which of the following is correct?
A:
Replace constraint 2 with
. OR Replace the objective with x + 2z.
Q:
Suppose that you want to maximize the objective
, subject to constraint 1:
, constraint 2:
, and constraint 3:
. You can eliminate the variable
and constraint 3, but that requires changes to the objective and constraints 1 and 2.
Which of the following is correct?
A: Replace constraint 1 with Q:
When purchasing shares of stock, you are subject to many risks, including adverse business results and share price fluctuations. If you have stocks whose returns are independent, the variability of a portfolio will be less than the variability of the individual securities.
Consider the following:
The variance is the square of the standard deviation.
The variance of a sum of independent random variables is the sum of the variances.
If
X
is a random variable, then
. Thus, for example, if
a $1 investment in stock X has variance V, then a $10 investment will have variance 100V.
Stock A has a return of 5% and a standard deviation of .012 per dollar invested.
Stock B has a return of 7% and a standard deviation of .01 per dollar invested.
If
a
is the amount invested in A and
b
is the amount invested in B, then
the variance of the portfolio is
.
The standard deviation of the portfolio is
.
The return is
.
Use Solver to find the fraction that you would invest in A and the fraction in B that maximizes risk adjusted return, that is
.
Hint: Use the generalized reduced gradient (GRG) nonlinear method.
Suppose that the investor has $170,000. How much should be invested in stock B? Round it off to the nearest whole dollar.
A:
$113,634
Q:
When purchasing shares of stock, you are subject to many risks, including adverse business results and share price fluctuations. If you have stocks whose returns are independent, the variability of a portfolio will be less than the variability of the individual securities.
Consider the following:
The variance is the square of the standard deviation.
The variance of a sum of independent random variables is the sum of the variances.
If
X
is a random variable, then
. Thus, for example, if
a $1 investment in stock X has variance V, then a $10 investment will have variance 100V.
Stock A has a return of 5% and a standard deviation of .012 per dollar invested.
Stock B has a return of 7% and a standard deviation of .01 per dollar invested.
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If
a
is the amount invested in A and
b
is the amount invested in B, then
the variance of the portfolio is
.
The standard deviation of the portfolio is
.
The return is
.
Use Solver to find the fraction that you would invest in A and the fraction in B that maximizes risk adjusted return, that is
.
Hint: Use the generalized reduced gradient (GRG) nonlinear method.
What percent should be invested in stock A? Round it off to the nearest whole percentage.
A:
33%
Q:
Replacing a variable that represents dollars with one that represents thousands of dollars is a change of variables.
Suppose that you want to maximize
, subject to (constraint 1)
and (constraint 2)
. You can replace the variable with x' and y', representing hundreds, and replace the constraints with
and
.
Another change would be to set
and
. Which of the following is correct?
A: The objective becomes a.
Q:
Replacing a variable that represents dollars with one that represents thousands of dollars is a change of variables.
Suppose that you want to maximize
, subject to (constraint 1)
and (constraint 2)
. You can replace the variable with x' and y', representing hundreds, and replace the constraints with
and
.
Another change would be to set
and
. Which of the following is correct?
A:
Constraint 2 becomes
Marketing
Linear programming model can be used in marketing as well. Companies can expand their operations from smaller size to larger size in volume or increase their stores in numbers by advertising their reputations. The power of advertising whether by word of mouth, radio, newspaper, Internet, or other medias can indeed make or break a company existence. You look at a minimization problem in this section regarding advertising a new product for a company.
These days you see or hear advertisements of all kinds for larger business that are willing to spend large amount of money to target more customers while minimizing the advertisement prices. Assume that a company is selecting Television (TV), Radio, Newspaper, and Magazine ads to increase the volume of their customers.
The company has $16,000 for advertisement purposes. The cost per advertisement is as follows:
TV
$220
Radio
$50
Newspaper $50
Magazine
$30
You do not want to spend more than $13,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 320 and not
more than 350 and the maximum amount spend on TV ads should be less $2,500. You should have a minimum of 10 TV ads.
Setting Up the Linear Programming Model
The objective function and the constraints are as follows:
Minimizing cost =$220T + $50R +$50N +$30M
subject to
T + R + N + M ≥ 320
N ≥ 2M
N - 2M ≥ 0
T ≥ 10
T + R + N + M ≤ 350
$50N +$30M ≤ $13,000
$220T < $2,500
T, R, N, and M ≥ 0 (nonnegativity)
Figure 1 shows the Excel’s Solver program used to solve the linear programming model containing all of the required excel functions.
Figure 1:
Excel’s Solver Program Used to Solve the Linear Programming Model
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Figure 2:
Results of the Application Using Excel’s Solver
Understanding the Results
Figure 2 shows the same program containing the numerical values of the decision-making variables and the amount of resources consumed. Depending on the values used in the objective function or the constraints, the units of advertisements may become a fraction. In this case, you may want to round down or up some of the values and see their effect on the overall performance. In this case, unit price is extremely large; you need to round down the number of units presented by the solution.
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
What is the value of the objective function after running the problem through Excel’s Solver?
A:
$26,000
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
How much money was left in the advertising budget after the conclusion of all ads?
A:
$0
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
To what amount can they lower the advertising price allocated for TV ads without changing the number of ads after running the problem through Excel’s Solver?
A:
≤ $1,500
Q: Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
To what amount can they lower the advertising price per TV ad without changing the number of ads after running the problem through Excel’s Solver?
A:
$100
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
Which resource do they run out first after running the problem through Excel’s Solver?
A:
Money spent on Newspaper and Magazine ads
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should
not be more than 360 and the maximum amount spend on TV
ads should be less $1,600. They should have a minimum of 10 TV ads.
How many constraint equations other than nonnegativity do you have?
A: 6
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
How do you address the “The number of Newspaper ads should be at least twice the Magazine ads” constraint?
A:
N ≥ 2M OR N − 2M ≥ 0
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and $40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
How do you address the “They do not want to spend more than $15,000 on Newspaper and Magazine ads” constraint?
A:
80N + 40M ≤ $15,000
Q:
Assume that XYZ company is selecting TV, Radio, Newspaper, and Magazine ads to increase the volume of their customers. The company has $26,000 for advertisement purposes. The cost per advertisement for TV, Radio, Newspaper, and Magazine are $150, $100, $80, and
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$40, respectively. They do not want to spend more than $15,000 on Newspaper and Magazine ads. The number of Newspaper ads should be at least twice the Magazine ads. The total number of ads all together should not be less than 330 and not more than 360 and the maximum amount spend on TV ads should be less $1,600. They should have a minimum of 10 TV ads.
How many TV, Radio, Newspaper, and Magazine ads do they receive after running the problem through Excel’s Solver?
A:
10, 95, 150, and 75, respectively
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
180
Radio
90
Newspaper
75
Magazine
44
The head of the marketing department has attended a seminar on the importance of diversity and as a result has decided that each type of ad should get at least 15% of the spending amount but no more than 45%. A minimum of 400 ads is required. How much will this cost?
A:
$24,905.66
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
180
Radio
90
Newspaper
75
Magazine
44
The head of the marketing department has attended a seminar on the importance of diversity and as a result has decided that each type of ad should get at least 15% of the spending but no more than 45%. A minimum of 400 ads is required.
If this is run naively, Solver will find a solution that involves fractional amounts. That does not make sense because each item must be purchased in integral amounts. Add a
constraint to Solver to require that the decision variables be integers.
By how much does this increase the total cost? Round your answer to the nearest dollar.
A:
$55
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
$180
Radio
$90
Newspaper
$75
Magazine
$44
You are in the computer age. It turns out that you can buy Internet ads for $15 apiece. Keeping with the requirement that at least 15% and no more than 45% of the spending goes to each category and a requirement of at least 500 ads, what is the minimum budget for this? Require that the numbers be integral when you run Solver.
A:
$14,799
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
180
Radio
90
Newspaper
75
Magazine
44
You are in the computer age. It turns out that you can buy Internet ads for $15 apiece. Keeping with the requirement that at least 15% and no more than 45% of the spending goes to each category and a requirement of at least 500 ads, how many Internet ads will be purchased? Require that the numbers be integral when you run Solver.
A:
381
Q: Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
180
Radio
90
Newspaper
66
Magazine
44
Internet
9
Optimizing the spending amount relative to various constraints is a way to allocate the budget, but this is not the only way. One thing might be to maximize the number of customers. The following table shows how many customers are expected of each ad:
Channel
Customers per ad
TV
22.5
Radio
14.7
Newspaper
6.4
Magazine
6.7
Internet
.9
Subject to spending a minimum of 10% and a maximum of 45% on each category, a maximum spending of $40,000 brings the most customers. How many total customers are expected? Do not require integral answers. Round to the nearest whole number.
A: 5,751
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
225
Radio
90
Newspaper
66
Magazine
44
Internet
15
Optimizing the spending amount relative to various constraints is a way to allocate the budget, but this is not the only way. One thing might be to maximize the number of customers. The following table shows how many customers are expected of each ad:
Channel
Customers per ad
TV
22.5
Radio
14.7
Newspaper
6.4
Magazine
6.7
Internet
2.1
Due to experience with the Internet, its effectiveness has improved to 2.1. Subject to spending a minimum of 10% and a maximum of 45% on each category, a maximum spending of $40,000 brings the most customers. How many radio ads will be purchased? Do not require integral answers. Round to the nearest integer.
A:
200
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
225
Radio
90
Newspaper
66
Magazine
44
Internet
9
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Optimizing the spending amount relative to various constraints is a way to allocate the budget, but this is not the only way. One thing might be to maximize the number of customers. The following table shows how many customers are expected of each ad:
Channel
Customers per ad
TV
22.5
Radio
14.7
Newspaper
6.4
Magazine
6.7
Internet
2.1
Due to experience with the Internet, its effectiveness has improved to 2.1. Subject to spending a minimum of 10% and a maximum of 45% on each category, a maximum spending of $40,000 brings the most customers. How many customers are expected? Do not require integral answers. Round to the nearest whole number.
A:
7,230
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
$170
Radio
$100
Newspaper
$80
Magazine
$40
After reading an article, the head of a marketing department has decided on the following advertising campaign:
The maximum amount spent on newspaper and magazine ads should be $13,000.
The number of newspaper ads should be at least two and a half times the number of magazine ads.
The total number of ads should not be less than 320 and not more than
350.
The maximum amount spent on TV ads should be $3,000.
The number of TV ads should be at least 10.
How much will this cost? (Make sure to force the decision variables, the number of ads,
to be integers.)
A:
$26,760
Q:
Staying with the advertising example, consider the following cost per ad:
Channel
Cost per ad
TV
170
Radio
100
Newspaper
80
Magazine
40
The head of the marketing department has read an article in an inflight magazine and has come up with the following ideas:
The maximum spend on newspaper and magazine ads should be $13,000.
The number of newspaper ads should be at least two and a half times the number of magazine ads.
The total number of ads all together should not be less than 320 and not more than 350.
The maximum spending amount on TV ads should be less $3,000.
You should have a minimum of 10 TV ads.
Suppose the head of the marketing department has seen a video extoling the virtues of simplicity and as a result has decided to drop the requirement that the number of newspaper ads exceed 250% of the number of magazine ads.
How much will this change the total cost?
A:
It will decrease by $11,900.
Employee Staffing
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
Which constraint equation does Mary need to change to have at least one part-time employee remains to 11:00 PM?
A:
F + P3 ≥ 4
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
What is the constraints equation for number of employee from 6:00 to 7:00 PM?
A:
F + P1 ≥ 6
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
What is the constraints equation for number of employee from 8:00 to 9:00 PM?
A: F + P1 + P2 + P3 ≥ 8
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
In which interval does Mary use the most of the part-time employees?
A: 7:00 to 10:00 PM
Q: Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
What is the objective function?
A:
Minimize total Friday’s personnel cost = $140F + $50(P1 + P2 + P3)
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the
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different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
What is the value for the objective function after running the problem through Excel’s Solver?
A: $1,150
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for three hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
By how much the value of the objective function changes if Mary changes the requirement of having at least 4 employees from 9:00 to 10:00 to at least 5 employees?
A: $0
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
How many part-time employees report to work at 6:00 PM if Mary changes the requirement of having at least 4 employees from 9:00 to 10:00 PM to at least 5 employees?
A: 6
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
How many part-time employees report to work at 7:00 PM if Mary changes the requirement of having at least 4 employees from 9:00 to 10:00 to at least 5 employees?
A: 0
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the
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different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
How many part-time employees report to work at 8:00 PM if Mary changes the requirement of having at least 4 employees from 9:00 to 10:00 PM to at least 5 employees?
A: 3
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
How many of the full-time employees must be in the restaurant on Friday night?
A: 5
Q:
Mary’s restaurant has 5 full-time cooks each getting $140 on Friday night and work from 5:00 to 11:00 PM You also have some part-time cooks that can start any time from 6:00 and work for 3 hours for $50 a night. You use the character F for full-time cooks and P for the different cooks starting in 1 hour intervals. You need to have at least 3 cooks from 5:00 to 6:00, 6 from 6:00 to 7:00, 8 from 7:00 to 8:00, 8 from 8:00 to 9:00, 4 from 9:00 to 10:00, and 5 from 10:00 to 11:00. You also do not want more than 9 part-time employees.
How many part-time employees do they report to work at 8:00 PM?
A: 0
Q:
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m., and there are 3 classes of employee. The first are your experts, who work a 5-hour shift,
with an hour break in the middle. If one of your experts starts at 8 a.m., he or she can get a break from 10 to 11 a.m. and work from 11 a.m. to 1 p.m. The cost per shift is $100. The second are your beginners, who work a 4-hour shift with an hour break after the first 2 hours, costing $45 per shift. Finally, you have your senior experts, who work a
2 hour shift, at the cost of $70 per shift.
Shifts that end early due to the end of the day are still paid in full.
You need to arrange for at least 1 expert or senior expert to be assigned throughout the day. Also, you need to assign at least 3 people from 8 to 10 a.m., 4 people from 12 to 1 p.m., and 2 people from 4 to 5 p.m. For the rest of the day, 1 person is sufficient. The help desk is staffed from 6 a.m. to 10 p.m.
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When setting up the problem in Excel Solver, the expressions to compute the coverage at any hour are complicated. Allowing starts that are earlier than 6 a.m. will allow you to use the same formula with relative reference for all the coverage computations.
For example, the expert coverage at 11 a.m. is given by E7 + E8 + E10 + E11, where the En variables represent the number of experts starting at that hour. Excel will automatically adjust the reference when this formula is copied and pasted.
What is the earliest start and latest start that make sense?
A:
2 a.m., 8 p.m.
Q:
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m., and there are 3 classes of employee. The first are your experts, who work a 5-hour shift,
with an hour break in the middle. If one of your experts starts at 8 a.m., he or she can get a break from 10 to 11 a.m. and work from 11 a.m. to 1 p.m., and the cost per shift is $100. The second are your beginners, who work a 4-hour shift with an hour break after the first 2 hours, costing $45 per shift. Finally, you have your senior experts, who work a
2-hour shift, at the cost of $70 per shift.
Shifts that end early due to the end of the day are still paid in full.
You need to arrange for at least 1 expert or senior expert to be assigned throughout the day. Also, you need to assign at least 3 people from 8 to 10 a.m., 4 people from 12 to 1 p.m., and 2 people from 4 to 5 p.m. For the rest of the day, 1 person is sufficient. The help desk is staffed from 6 a.m. to 10 p.m.
Suppose that the cost of senior experts can be reduced from $70 to $40, but only during
the hours of 9 a.m. to 3 p.m.
How will the number of beginners change?
A:
Decrease by 3
Q:
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m. You can make assignment from two groups of employees—experts and beginners. There must be at least one expert on the desk at all times. Shifts are 4 hours.
You can start employees at any hour. Experts cost $100 per shift; beginners are $60 per shift.
You need to have at least 2 people in the morning from 8 to 9 a.m., at least 3 people from 11 a.m. to 1 p.m., and at least 2 people from 4 to 5 p.m.
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How much will the staffing cost?
A:
540
Q: Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m., and there are 3 classes of employee. The first are your experts, who work a 5-hour shift,
with an hour break in the middle. If one of your experts starts at 8 a.m., he or she can get a break from 10 to 11 a.m. and work from 11 a.m. to 1 p.m. The cost per shift is $100. The second are your beginners, who work a 4-hour shift with an hour break after the first 2 hours, costing $45 per shift. Finally, you have your senior experts, who work a
2-hour shift, at the cost of $70 per shift.
Shifts that end early due to the end of the day are still paid in full.
You need to arrange for at least 1 expert or senior expert to be assigned throughout the day. Also, you need to assign at least 3 people from 8 to 10 a.m., 4 people from 12 to 1 p.m., and 2 people from 4 to 5 p.m. For the rest of the day, 1 person is sufficient. The help desk is staffed from 6 a.m. to 10 p.m.
What is the minimum cost solution? Assume that you can start fractional people, and let
variables be integers (int).
A:
695
Q:
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m., and there are 3 classes of employee. The first are your experts, who work a 5-hour shift,
with an hour break in the middle. If one of the experts starts at 8 a.m., he or she can get
a break from 10 to 11 a.m. and work from 11 a.m. to 1 p.m. The cost per shift is $100. The second are your beginners, who work a 4-hour shift with an hour break after the first 2 hours, costing $45 per shift. Finally, you have your senior experts, who work a 2-
hour shift, at the cost of $70 per shift.
Shifts that end early due to the end of the day are still paid in full.
You need to arrange for at least 1 expert or senior expert to be assigned throughout the day. Also, you need to assign at least 3 people from 8 to 10 a.m., 4 people from 12 to 1 p.m., and 2 people from 4 to 5 p.m. For the rest of the day, 1 person is sufficient. The help desk is staffed from 6 a.m. through 10 p.m.
Suppose that the cost of senior experts can be reduced from $70 to $40, but only during
the hours from 9 a.m. to 3 p.m.
What is the minimum cost solution? Insist that only whole people can start by adding the
constraint that all the variables must be integers (int).
A:
680
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Q:
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m., and there are 3 classes of employee. The first are your experts, who work a 5-hour shift,
with an hour break in the middle. If one of your experts starts at 8 a.m., he or she can get a break from 10 to 11 a.m. and work from 11 a.m. to 1 p.m. The cost per shift is 100.
The second are your beginners, who work a 4-hour shift with an hour break after the first 2 hours, costing 45 per shift. Finally, you have your senior experts, who work a 2-
hour shift, at the cost of 70 per shift.
Shifts that end early due to the end of the day are still paid in full.
You need to arrange for at least one expert or senior expert to be assigned throughout the day. Also, you need to assign at least 3 people from 8 to 10 a.m., 4 people from 12 to 1 p.m., and 2 people from 4 to 5 p.m. For the rest of the day, one person is sufficient. The help desk is staffed from 6 a.m. to 10 p.m.
Suppose that the cost of senior experts can be reduced from 70 to 40, but only during the hours of 9 a.m. to 3 p.m.
How many additional senior experts can now be used with this change?
A: 3
Q:
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m. You can make assignment from two groups of employees—experts and beginners. There must be at least one expert on the desk at all times. Shifts are 4 hours.
You can start employees at any hour. Experts cost $100 per shift; beginners are $60 per shift.
You need to have at least 2 people in the morning from 8 to 9 a.m., at least 3 people from 11 a.m. to 1 p.m., and at least 2 people from 4 to 5 p.m.
Sometimes, a little logic can help simplify the problem. You must have an expert employee on the help desk at all times, shifts are 4 hours, and there are 10 hours to cover. Which of the following expressions tells you the number of expert employees you
will need?
Hint:
or floor(
x
) is the largest integer less than or equal to
x
, so
.
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or ceiling(
x
) is the smallest integer greater than or equal to
x
, so
.
is the remainder when
n
is divided by
d
.
A:
Q: Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m. You can make assignment from two groups of employees—experts and beginners. There must be at least one expert on the desk at all times. Shifts are 4 hours.
You can start employees at any hour. Experts cost $100 per shift; beginners are $60 per shift.
You need to have at least 2 people in the morning from 8 to 9 a.m., at least 3
people from 11 a.m. to 1 p.m., and at least 2 people from 4 to 5 p.m.
How many different solutions can you find?
A:
At least 6
Q: The requirements are shown in the following table:
From
To
#
5
6
2
6
7
5
7
8
8
8
9
7
9
10
6
10
11
4
Full-time cooks work from 5 to 11 hours and cost $150 each; part-time cooks work for 3 hours, can start at 6, 7, or 8, and cost $45 each; and the total number of part-time cooks
must be 8 or less.
How many different solutions can you find?
A: 2
Q:
The requirements are shown in the following table:
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From
To
#
5
6
2
6
7
5
7
8
8
8
9
7
9
10
6
10
11
4
Suppose that you need to staff a help desk, which has hours from 8 a.m. to 6 p.m. You can make assignment from two groups of employees—experts and beginners. There must be at least one expert on the desk at all times. Shifts are 4 hours.
You can start employees at any hour. Experts cost $100 per shift; beginners are $60 per shift.
You need to have at least 2 people in the morning from 8 to 9 a.m., at least 3 people from 11 a.m. to 1 p.m., and at least 2 people from 4 to 5 p.m.
How many decision variables do you have?
A: 14
Multiperiod
Q:
A:
Q:
A:
Q:
A:
Q:
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A:
Q:
A:
Q:
A:
Q:
A:
Q:
A:
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