# Case 4.2 Pittsburg Lighting

Question 1

Based on the information provided, the purchase cost for Altman Electronics would be \$(3.00 X 100,000) = \$300,000. On the other hand, for Crestwell Lights, it would be \$(2.70 X 100,000) = \$270,000.

Question 2

In spite of the above, there would be about 5% defect rate among the bulbs purchased from Altman Electronics. Accordingly, the warranty replacement cost for this supplier would be;

\$(0.05 X 100,000 X 3.0) = \$15,000

On the other hand, the warranty replacement cost for Crestwell Lights would not be determined at this point because the defect rate is not available. For this reason, if Altman Electronics is selected, then the company would incur about \$15,000 replacing bulbs that would spoil. In total thereby it would incur \$(300,000 + 15,000) = \$315,000 to purchase and replace bulbs from Atman Electronics.

Question 3

Based on the available data from the market, if Crestwell Lights would be selected, then the total cost for each possible defect level would be as Table 1 depicts. On the other hand, it would be as Table 2 depicts based on the 100 bulbs that were purchased for testing purposes.

Question 4

Based on the 100 bulbs that were purchased from Crestwell Lights, then Tony’s revised subjective probabilities would be as provided in Table 3 depict. As a result, the joint probability that a bulb purchased from the supplier would be defective would be 0.1275 suggesting that the defect rate from Crestwell Lights would be 12.75%.

Question 5

Given that Crestwell Lights’ defect rate would be 12.75 percent meaning that it would be significantly higher than that from Altman Electronics then the weighted total cost for Crestwell would be as follows.

Purchase cost would be \$(2.7 X 100,000) = \$270,000

However, since the defect rate is at 12.72 percent, then it would be expected that about (12.75/100 X 100,000) = 12,750 bulbs would be purchased from the supplier would be defective. Accordingly, the warranty repair cost would be \$(12,750 X 2.7) = \$34,425 (Groebner et al., 2011). Therefore, the weighted total cost for Crestwell Lights would be \$(270,000 + 34,425) = \$304,425, which is significantly lower than that for Altman Electronics despite the variations in defect rates.

Recommendation

Based on the above analysis, it is evident that even though Crestwell Lights is offering its bulbs at \$2.70, which is slightly lower than \$3.0 that Altman Electronics is offering, there is about 12.75% chance that part of those bulbs would be defective. Accordingly, they would require to be replaced if the company would purchase bulbs from Crestwell Lights. Assuming that this was the case then the company would incur about \$304,425 to purchase and replace bulbs. In contrast, the company would incur about \$(300,000 + 15,000) = \$315,000 to purchase and replace the defective bulbs if it would contract Altman Electronics to supply the bulbs. Based on the above analysis, then the company should purchase bulbs from Crestwell Lights because even if there are higher chances that part of those bulbs would be defective, the company would incur less purchasing and replacing defective bulbs from this supplier.

To begin with, the number of trucks arriving at the corporation is discrete random variable because it can only take a numerical value that varies from one hour to the other. In spite of this, it has a mean rate that can be determined with some level of certainty as the manager has done from the data he has collected. Nonetheless, it is not possible for the manager to know the exact number of trucks that would arrive at any given hour. For this reason, it would not be possible for the manager to develop a frequency table for a given hour to be able to determine the probability distribution and use it in making decision (Groebner et al., 2011). In spite of this, the manager has been able to determine that on average only 12 trucks can arrive at the corporation. In addition, he has been able to determine that each scale station can handle a maximum of six trucks (10 minutes each). Based on this information, then the manager should do the following to determine the number of scale stations that should be opened to serve the trucks cost effectively.

• Firstly, based on the information provided, the manager should utilize poison distribution to determine the probable probability distribution for the number of trucks that may arrive at any given time.
• Secondly, once the probability distribution has been determined, then the manager should go a step further to calculate the mean for that probability distribution. Since this is a poison distribution, then the mean would be the 12 trucks that arrive at any given hour. Because each scale station can serve utmost 6 trucks in an hour time, then two scale stations would be sufficient to serve the 12 trucks that would arrive at any given hour (Groebner et al., 2011). However, since the number of trucks that arrive at the corporation at any given hour may vary from one hour to the other, then it would be important to determine the standard deviation for the distribution to establish the maximum number of trucks that are likely to arrive at any given hour. Once again, since this is a poison distribution, then the standard deviation would be calculated as follows;

σ = square root (12*1/6) = square root (2) = 1.414

This means that about 14 trucks may arrive at any given time. As a result, to address this challenge, then the manager should add an extra scale station to serve the additional trucks that may arrive beyond the 12 expected to arrive at any given hour. This means that a maximum of three scale stations would be enough to serve the trucks that would arrive at the corporation. It would provide the corporation with a capacity of serving a maximum of 18 trucks at any given hour of the day.

Case 6.2 Credit Data, Inc

Question 1

Since the company is dealing with time that can be expressed in decimal points, then it is dealing with a continuous distribution. For this reason, any fraction of time would be acceptable in making decision. In the first case, the company wishes to know the level to which mean call time would need to be reduced to meet the 10% requirement assuming that standard deviation would not be changed by training. To do this, then the company should do the following.

Firstly, it should utilize the standard normal distribution table to approximate the probability at which the intended decision would occur. As Figure 1 depicts, the probability on either side should be 0.1 meaning that the allowable mean should lay within 80 percent mark. However, since the table is divided into 0.5 equal portions, then the probability on each side would be 0.4.

Secondly, upon determining the above, then the person making the decision should find the z value that corresponds to 0.4 from the standard normal table. This value would be -1.28 because it is on the left hand side.

Thirdly, the person should proceed to substitute the z value in the following formula to find the µ value as follows.

Z = (x-µ)/σ

That is, 1.28 = (8 – µ)/2.5 . . . . . .(i)

Therefore, µ = 8 – (1.28)(2.5) = 4.8

Accordingly, for the company to meet the 10% requirement, then the mean call rate should be reduced to 4.8 minutes.

Question 2

Assuming that the mean time would remain at 8 minutes, then the company should do the following to meet the 10% requirement. Firstly, it will be necessary to ensure that the standard deviation among the mean call time would reduce significantly. Since this would affect the shape of the current normal distribution, then the new distribution will resemble that of an exponential distribution. Accordingly, its standard deviation will be equal to the inverse of its mean call time. That is, 1/8 = 0.125 meaning that the standard deviation should be reduced to 0.125.

Question 3

If nothing is done, then the percentage of the calls that would be expected to require more than 10.5 minutes would be as follows.

Firstly, a normal distribution has a probability density function of the form;

f(x) = (1/σ√22∏)e-(x -µ)2/2σ2 (Groebner et al., 2011)

Inserting the mean (8), x = 10.5 and standard deviation (2.5) values into the equation gives a probability distribution of 0.05045 suggesting that about 5 percent of all calls would take more than 10.5 minutes. Therefore, the company should train the employees to reduce the number of calls that would exceed 10 minutes.

Reference

Groebner, D. et al. (2011). Business statistics: A decision-making approach. Upper Saddle River, New Jersey: Prentice hall.

Appendices

Table 1: Total Cost for Crestwell Lights using market data

Table 2: Total costs for Crestwell Lights using the 100 bulbs

Table 3: Conditional probabilities

Figure 1: A normal distribution

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