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# Determining Optimal Ranges

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Decision Theory Question

An organization uses a spam filtering software to block potentially spam messages. The spam filter can be set to one of three security modes: High-Security-Mode (H), Moderate-Security-Mode (M), or Low-Security-Mode (L).

Extensive experimentation using a benchmark corpus of spam messages yields the following performance statistics for the spam filter:
• 98% of the spam messages are blocked in the High-Security-Mode.
• 90% of the spam messages are blocked in the Moderate -Security-Mode.
• 80% of the spam messages are blocked in the Low-Security-Mode.

Extensive experimentation using a benchmark corpus of non-spam (legitimate) messages yields the following performance statistics for the spam filter:
• 12% of the non-spam messages are blocked in the High-Security-Mode
• 8% of the non-spam messages are blocked in the Moderate -Security-Mode
• 5% of the non-spam messages are blocked in the Low-Security-Mode

There are costs associated with not blocking spam messages and blocking non-spam messages.

(a)
The organization estimates that 75% of all messages it receives are spam messages. If the cost of not blocking a spam message is \$1, under what ranges of the cost of blocking a non-spam message is each security mode the optimal choice for a rational (risk neutral) information assurance manager? Fill up the ranges in the table below.

Optimal choice When cost of blocking a non-spam message is in the range
High-Security-Mode
Moderate-Security-Mode
Low-Security-Mode

(b)
The organization notices over time that the proportion of spam messages that it receives are declining (i.e., less that 75% of the messages are spam messages). It estimates the cost of blocking a non-spam message as \$20. How high must the proportion of spam messages be for a rational (risk neutral) information assurance manager to use the spam filter? Alternately, at least how low should the proportion of spam messages be for the information assurance manager to prefer the policy of not using the spam filter at all? Assume that all the other parameters remain the same as in part (a).

https://brainmass.com/math/linear-programming/determining-optimal-ranges-533920

#### Solution Summary

The expert determines the optimal ranges uses the spam filtering software to block potentially spam messages.

\$2.19

## Ranges of optimality

Answer Questions 2 and 3 based on the following LP problem.

Maximize 2X1 + 5X2 + 4X3 Total Profit
Subject to X1 + X2 + X3 > 150 At least a total of 150 units of all three products needed
X1 + 3X2 + 2X3 â?¤ 300 Resource 1
2X1 + X2 + 2X3 â?¤ 250 Resource 2
2X1 + 2X2 + 3X3 â?¤ 300 Resource 3
And X1, X2, X3 â?¥ 0

Where X1, X2, and X3 represent the number of units of Product 1, Product 2, and Product 3 to be manufactured.

The QM for Windows output for this problem is given below.

X1 X2 X3 RHS Dual
Maximize 2 5 4
Constraint 1 1 1 1 >= 150 -.5
Constraint 2 1 3 2 <= 300 1.5
Constraint 3 2 1 2 <= 250 0
Constraint 4 2 2 3 <= 300 .5
Solution-> 75 75 0 Optimal Z-> 525

Linear Programming Results:
Variable Status Value
X1 Basic 75
X2 Basic 75
X3 Basic 0
surplus 1 NONBasic 0
slack 2 NONBasic 0
slack 3 Basic 25
slack 4 NONBasic 0
Optimal Value (Z) 525

Ranging Results:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
X1 75 0 2 -Infinity 2.2
X2 75 0 5 2 6
X3 0 0 4 3.75 Infinity

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Constraint 1 -.5 0 150 120 150
Constraint 2 1.5 0 300 250 450
Constraint 3 0 25 250 225 Infinity
Constraint 4 .5 0 300 300 350

2. (a) Determine the optimal solution and optimal value and interpret their meanings.
(b) Determine the slack (or surplus) value for each constraint and interpret its meaning.

3. (a) What are the ranges of optimality for the profit of Product 1, Product 2, and Product 3?
(b) Find the dual prices of the four constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?
(c) If the profit contribution of Product 2 changes from \$5 per unit to \$5.50 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results. Do not solve the problem again).
(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above. Do not solve the problem again).

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