Question: 1. Decision Theory
There are two available diagnostic methods ¬- D1 and D2 - to predict if a tissue sample is benign or malignant. Ninety percent of the samples processed are benign. The performance of the diagnostic methods are shown in the tables below:
Diagnostic Method: D1
|
|
Predicted: Benign
|
Predicted: Malignant
|
|
|
Actual: Benign
|
0.60
|
0.30
|
0.90
|
|
Actual: Malignant
|
0.04
|
0.06
|
0.10
|
Diagnostic Method: D2
|
|
Predicted: Benign
|
Predicted: Malignant
|
|
|
Actual: Benign
|
0.50
|
0.40
|
0.90
|
|
Actual: Malignant
|
0.02
|
0.08
|
0.10
|
There is no cost associated with a correct prediction.
a. The cost of predicting a malignant sample as benign is $4,000 and the cost of predicting a benign sample as malignant is $1,000. Which diagnostic method should a risk-neutral rational decision maker choose? Why?
b. Under what ranges of the cost of a predicting a malignant sample as benign is diagnostic method D2 the preferred choice? Assume that the cost predicting a benign sample as malignant remains $1,000.
2. Bayesian Inference
A diagnostic method ¬is used to predict if a tissue sample is benign or malignant. The method correctly predicts 60% of the malignant samples as malignant. But it also incorrectly predicts 33% of the benign samples as malignant. Historical data suggests that 90 percent of the samples processed are benign.
Estimate the probability that a sample predicted by the method as malignant is actually malignant.
3. Optimization Models
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 unitsof 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) i. What is the minimum cost attainable under an optimal plan?
The minimum cost attainable is $ _______________
ii. How many jobs of each type should be processed in-house and out-sourced under this optimal plan?
|
Number of jobs processed
|
A
|
B
|
C
|
|
In-house
|
|
|
|
|
Out sourced
|
|
|
|
i. How many units of CPU and storage are used up under this optimal plan?
|
Resource
|
Units used
|
Available
|
|
CPU
|
|
1,600,000
|
|
Storage
|
|
2,400,000
|
b) 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.