# Decision Analysis, Constrained Optimization Models, Prediction using machine learning techniques, and Inductive learning

- Please answer all questions and submit ONE consolidated
- The last three pages of this document explain how you should organize your solutions.
- You may use reference material, but your submission should strictly reflect your individual effort.
- Please present your work clearly and concisely so that I can follow your arguments easily. Irrelevant and incorrect assertions will be penalized.

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**Question 1. Decision Theory [6 Points]**

An organization uses a spam filtering software to block email messages that may potentially be spam messages. The spam filter can be set to one of two security modes: High-Security-Mode or Low-Security-Mode.

Extensive evaluation using a benchmark corpus consisting exclusively of *spam* messages yields the following performance statistics for the spam filter:

- 98% of the
*spam*messages are blocked in the High-Security-Mode. - 96% of the
*spam*messages are blocked in the Low-Security-Mode.

Extensive evaluation using a benchmark corpus consisting exclusively of *non-spam* (legitimate) messages yields the following performance statistics for the spam filter:

- 4% of the
*non-spam*messages are blocked in the High-Security-Mode - 2% of the
*non-spam*messages are blocked in the Low-Security-Mode

It is estimated that 75% of the messages are *spam* messages.

(a) Let denote the conditional probability that a message that is NOT blocked by the spam filter operating in the High-Security-Mode is actually a spam message. Estimate . [** 1 point**]

(b) Let denote the conditional probability that a message that is NOT blocked by the spam filter operating in the Low-Security-Mode is actually NOT a *spam* message. Estimate [** 1 points**]

(c) There are costs associated with not blocking *spam* messages and blocking *non-spam* messages. Let the cost of not blocking a spam message be $1. At least how high should the cost of blocking a non-spam message be for a risk-neutral rational decision maker to prefer operating the spam filter in the Low-Security-Mode? [** 2 points**]

(d) The organization estimates that the cost of blocking a *non-spam* (legitimate) message is $10 and the cost of not blocking a *spam* message is $1. Let be the amortized cost per message of operating the spam filter. At most how high can be for a risk-neutral rational decision maker to use the spam filter at all? [** 2 points**]

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**Question 2. Constrained Optimization [6 points]**

A data-processing company processes three types of jobs – A, B, and C – for clients. In-house processing costs per job are estimated to be $1, $2, and $3 for job types A, B, and C, respectively. Each job requires two types of computing resources – CPU and storage. It requires:

- 1 unit of CPU, and 2 unit of storage to process each job of type A;
- 2 unit of CPU, and 2 units of storage to process each job of type B; and
- 2 unit of CPU, 3 units of storage to process each job of type C.

Over this time period it has 1,600,000 units of CPU, and 2,400,000 units of storage available. Because of contractual obligations the company must process 800,000 jobs of type A, 500,000 jobs of type B, and 400,000 jobs of type C in the upcoming week. Limited resource availability prevents the company from meeting the entire demands for all job types through in-house processing alone. The company has can out-source some jobs to an external supplier. No in-house computing resources are used for out-sourced jobs. The external supplier charges $2 for each job of type A, $3 for each job of type B, and $4 for each job of type C.

For your convenience, the information presented above is summarized in the table below:

Job Type |
A |
B |
C |
Available |

CPU units per job | 1 | 2 | 2 | 1,600,000 |

Storage units per job | 2 | 2 | 3 | 2,400,000 |

Number of jobs to process | 800,000 | 500,000 | 400,000 | |

In-house cost per job | $1.00 | $2.00 | $3.00 | |

Out sourcing cost per job | $2.00 | $3.00 | $4.00 |

The company uses a linear programming model to determine an optimal job processing plan so as to meet their contractual obligations in the upcoming week at minimum cost. Formulate the problem as a linear program (LP), solve the LP, and perform sensitivity analysis to answer the following questions:

- a)
**[3 points]** - What is the minimum cost attainable under an optimal plan?
- How many jobs of each type should be processed in-house and out-sourced under this optimal plan?

iii. How many units of CPU and storage are used up under this optimal plan?

- b)
**[3 points]**

The company has located an alternate business partner (*New-Partner*) that can only process jobs of type A. It can process at most 100,000 jobs of type A in the upcoming week, but the price per job is subject to negotiations. What is the maximum amount that the company should be willing to pay *New-Partner* for processing each unit of job type A? Justify your answer by explaining your approach.

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**Question 3: Inductive learning [6 Points]**

The file “*q3_train.csv*” contains 751 lines with data for Question 3. The first line contains column headers that may be interpreted as follows:

** t1**: measurement on test 1;

**: measurement on test 2.**

**t2**** t3**: measurement on test 3;

**: measurement on test 4.**

**t4**** t5**: measurement on test 5;

**: measurement on test 6.**

**t6**** t7**: measurement on test 7;

**: measurement on test 8.**

**t8**** d**: binary output variable set to 1 if product is defective and 0 otherwise.

The next lines contain examples, for which the values of the above variables are specified. The table below reproduces the first 2 observations.

t1 |
t2 |
t3 |
t4 |
t5 |
t6 |
t7 |
t8 |
d |

72 | 26 | 1 | 67 | 65 | 81 | 48 | 37 | 0 |

59 | 82 | 81 | 76 | 26 | 30 | 55 | 36 | 1 |

Use the 750 examples in “*q3_train.csv*” to come up with a small set of rules that correctly classify the output variable “** d**” based on input variable values (t1, t2, t3, t4, t5, t6, t7, and t8).

(a) Specify the rules in IF … THEN form. **[2 points]**

(b) Now use these rules to predict the output** d** based on the inputs for the 250 examples provided in the file “

*q3_validation.csv*”. Present the confusion matrix obtained for these 250 examples.

**[2 points]**(c) Then use the rules to predict the output class ** d** for the test cases (presented in the file “

*q3_test.csv*”):

**[2 points]**

Test_case |
t1 |
t2 |
t3 |
t4 |
t5 |
t6 |
t7 |
t8 |
d |

1 | 35 | 89 | 95 | 97 | 84 | 32 | 64 | 16 | |

2 | 36 | 25 | 42 | 96 | 94 | 13 | 86 | 52 | |

3 | 20 | 82 | 76 | 82 | 61 | 86 | 2 | 6 | |

4 | 34 | 21 | 9 | 50 | 3 | 85 | 60 | 82 | |

5 | 68 | 40 | 48 | 75 | 14 | 71 | 36 | 93 | |

6 | 15 | 87 | 60 | 11 | 60 | 55 | 66 | 10 | |

7 | 96 | 88 | 3 | 7 | 37 | 25 | 67 | 35 | |

8 | 53 | 75 | 13 | 34 | 85 | 91 | 5 | 86 | |

9 | 92 | 43 | 58 | 26 | 76 | 42 | 26 | 92 | |

10 | 2 | 7 | 63 | 28 | 52 | 6 | 62 | 1 | |

11 | 26 | 74 | 73 | 62 | 36 | 37 | 84 | 99 | |

12 | 2 | 37 | 8 | 8 | 99 | 6 | 27 | 47 | |

13 | 61 | 65 | 57 | 86 | 93 | 5 | 72 | 22 | |

14 | 93 | 21 | 51 | 65 | 45 | 64 | 97 | 79 | |

15 | 70 | 80 | 86 | 74 | 12 | 85 | 86 | 23 | |

16 | 33 | 71 | 19 | 27 | 1 | 78 | 98 | 83 | |

17 | 45 | 23 | 14 | 68 | 52 | 24 | 5 | 21 | |

18 | 1 | 25 | 69 | 19 | 80 | 35 | 87 | 29 | |

19 | 7 | 2 | 83 | 43 | 84 | 37 | 97 | 57 | |

20 | 16 | 99 | 8 | 26 | 15 | 66 | 76 | 42 |

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**Question 4. On the Aral & Walker (2014) article [4 points]**

__Article__: Aral, S., & Walker, D. (2014). Tie Strength, Embeddedness, and Social Influence: A Large-Scale Networked Experiment. *Management Science*, 60(6), 1352–1370.

Consider the parameter estimates from the single failure proportional hazards model presented in Table 2 (page 1362). The authors explain that: “Attending the same college as one’s friend is associated with a 1,355% increase in influence (p < 0.01) compared to attending different colleges.” Estimate the percentage increase in influence if the coefficient estimate for College (same) was 10.0 (instead of 8.554). Assume that all other coefficients remain unchanged.

Briefly explain your reasoning.

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**Question 5. On the article Ghose & Han (2014). [4 points]**

__Article__: Ghose, A., & Han, S. P. (2014). Estimating demand for mobile applications in the new economy. *Management Science*, 60(6), 1470-1488.

This paper uses a random-coefficients nested logit model (**NLM**) to estimate the effect of price discounts on app demand and reports the following in Figures 4 and 5:

Price discount (%) | 10 | 20 | 30 | 40 | 50 |

Sales change (%) using NLM |
5 | 11 | 18 | 29 | 52 |

Sales change (%) using M |

Consider a simple model **M**: , where is the Sales, is a constant, and is Price. By running ordinary least square regression of on using available data, we obtain estimate for the parameters: and . Based on these coefficient estimates, by what percentage do you expect Sales to increase for the same price discounts? Fill out the last row of the table above with your estimates. Briefly explain your reasoning.

**Question 6. On the article Gallino & Moreno (2014). [4 points]**

__Article__: Gallino, S. & Moreno, A. (2014). Integration of Online and Offline Channels in Retail: The impact of Sharing Reliable Inventory Availability Information, *Management Science*, 60(6), 1434–1451.

Consider the model presented in equation 4 to study the impact of BOPS implementation on Brick-and-Mortar store sales and the coefficient estimates presented in Table 4 (page 1440). Based on these results, what is the expected sales of a Brick-and-Mortar store in US after BOPS implementation if its sales before BOPS was $200,000? Assume that *TRAFFIC* is used as part of the *CONTROLS*. Briefly explain your reasoning.

** Format for presenting solutions**.

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** Name**: ______________________________

**Question 1. Decision Theory [6 Points]**

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- …… (
*round to 4 decimal places*) - …… (
*round to 4 decimal places*) - Cost ≥ …… (
*round to the nearest cent*) - $ …… (
*round to the nearest cent*).

*Explanations:*

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**Question 2. Constrained Optimization Models [6 points]**

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- a)
**[3 points]**

- The minimum cost attainable is $ …….. (
*round to nearest dollar*).

- Number of jobs of processed under optimal solution (
*round to nearest integer*):

Number of jobs processed | A | B | C |

In-house | |||

Out sourced |

iii. Resources used under optimal solution:

Resource | Units used | Available |

CPU | 1,600,000 | |

Storage | 2,400,000 |

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- b)
**[3 points]**

The maximum amount that the company should be willing to pay *New-Partner* per job for processing job type A is ……. (*round to nearest cent*).

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*Explanations:*

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**Question 3: Inductive learning [6 points]**

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** (a) **Specify the rules that you obtain using the 750 examples in “

*q3_train.csv*”.

Rule 1: IF ……… THEN d =

…

Rule k:

** (b) **Specify the confusion matrix obtained with the 250 examples from “

*q3_validation.csv*”.

Actual: d=0 | Actual: d=1 | |

Predicted: d = 0 | ||

Predicted: d = 1 |

** (c) **Specify the predicted value for the output

**for the test cases in “**

**d***q3_test.csv*”:

Test_case |
t1 |
t2 |
t3 |
t4 |
t5 |
t6 |
t7 |
t8 |
d |

1 | 35 | 89 | 95 | 97 | 84 | 32 | 64 | 16 | |

2 | 36 | 25 | 42 | 96 | 94 | 13 | 86 | 52 | |

3 | 20 | 82 | 76 | 82 | 61 | 86 | 2 | 6 | |

4 | 34 | 21 | 9 | 50 | 3 | 85 | 60 | 82 | |

5 | 68 | 40 | 48 | 75 | 14 | 71 | 36 | 93 | |

6 | 15 | 87 | 60 | 11 | 60 | 55 | 66 | 10 | |

7 | 96 | 88 | 3 | 7 | 37 | 25 | 67 | 35 | |

8 | 53 | 75 | 13 | 34 | 85 | 91 | 5 | 86 | |

9 | 92 | 43 | 58 | 26 | 76 | 42 | 26 | 92 | |

10 | 2 | 7 | 63 | 28 | 52 | 6 | 62 | 1 | |

11 | 26 | 74 | 73 | 62 | 36 | 37 | 84 | 99 | |

12 | 2 | 37 | 8 | 8 | 99 | 6 | 27 | 47 | |

13 | 61 | 65 | 57 | 86 | 93 | 5 | 72 | 22 | |

14 | 93 | 21 | 51 | 65 | 45 | 64 | 97 | 79 | |

15 | 70 | 80 | 86 | 74 | 12 | 85 | 86 | 23 | |

16 | 33 | 71 | 19 | 27 | 1 | 78 | 98 | 83 | |

17 | 45 | 23 | 14 | 68 | 52 | 24 | 5 | 21 | |

18 | 1 | 25 | 69 | 19 | 80 | 35 | 87 | 29 | |

19 | 7 | 2 | 83 | 43 | 84 | 37 | 97 | 57 | |

20 | 16 | 99 | 8 | 26 | 15 | 66 | 76 | 42 |

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*Explanations:*

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**Question 4. On the Aral & Walker (2014) article [4 points]**

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“If the coefficient estimate for College (same) is 10.0, attending the same college as one’s friend is associated with a _____% increase in influence compared to attending different colleges.” (*round to nearest integer*)

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*Explanations:*

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**Question 5. On the article Ghose & Han (2014). [4 points]**

Fill out the last row of the table below with your estimates for the percentage by which you expect Sales to increase (*round to nearest integer*).

Price discount (%) | 10 | 20 | 30 | 40 | 50 |

Sales change (%) using M |

*Explanations:*

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**Question 6. On the article Gallino & Moreno (2014). [4 points]**

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Expected sales of a Brick-and-Mortar store in US after BOPS implementation is ………..$ if its sales before BOPS was $200,000 (*round to nearest dollar*).

*Explanations:*