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.
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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