- 18th Feb 2024
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**EuroWatch Case Study Question**

EuroWatch Company assembles expensive wristwatches and then sells them to retailers throughout Europe. The watches are assembled at a plant with two assembly lines. These lines are intended to be identical, but line 1 uses somewhat older equipment than line 2 and is typically less reliable. Historical data have shown that each watch coming offline 1, independently of the others, is free of defects with a probability of 0.98. The similar probability for line 2 is 0.99. Each line produces 500 watches per hour. The production manager has asked you to answer the following questions.

1. She wants to know how many defect-free watches each line is likely to produce in a given hour. Specifically, find the smallest integer k (for each line separately) such that you can be 99% sure that the line will not produce more than k defective watches in a given hour.

2. EuroWatch currently has an order for 500 watches from an important customer. The company plans to fill this order by packing slightly more than 500 watches, all from line 2, and sending this package off to the customer. EuroWatch wants to send as few watches as possible, but it wants to be 99% sure that when the customer opens the package, there are at least 500 defect-free watches. How many watches should be packed?

3. EuroWatch has another order for 1000 watches. Now it plans to fill this order by packing slightly more than one hour’s production from each line. This package will contain the same number of watches from each line. As in the previous question, EuroWatch wants to send as few watches as possible, but it again wants to be 99% sure that when the customer opens the package, there are at least 1000 defect-free watches. The question of how many watches to pack is unfortunately quite difficult because the total number of defect-free watches is not binomially distributed. (Why not?) Therefore, the manager asks you to solve the problem with simulation (and some trial and error).

4. Finally, EuroWatch has a third order for 100 watches. The customer has agreed to pay $50,000 for the order—that is, $500 per watch. If EuroWatch sends more than 100 watches to the customer, its revenue doesn’t increase; it can never exceed $50,000. Its unit cost of producing a watch is $450, regardless of which line it is assembled on. The order will be filled entirely from a single line, and EuroWatch plans to send slightly more than 100 watches to the customer. If the customer opens the shipment and finds that there are fewer than 100 defect-free watches (which we assume the customer can do), then he will pay only for the defect-free watches—EuroWatch’s revenue will decrease by $500 per watch short of the 100 required—and on top of this, EuroWatch will be required to make up the difference at an expedited cost of $1000 per watch. The customer won’t pay a dime for these expedited watches. (If expediting is required, EuroWatch will make sure that the expedited watches are defect-free. It doesn’t want to lose this customer entirely.) You have been asked to develop a spreadsheet model to find EuroWatch’s expected profit for any number of watches it sends to the customer. You should develop it so that it responds correctly, regardless of which assembly line is used to fill the order and what the shipment quantity is.

**EuroWatch Case Study Solution**

The EuroWatch Company, a pioneer in the European wristwatch industry, is renowned for its focus on producing high-quality, luxury wristwatches. The company has carved a niche for itself in the market by mastering the art of assembling and selling premier wristwatches that cater to an elite clientele. This clientele demands not only exquisite timepieces but also outstanding customer service, which includes defect-free products. Consequently, the company is continually seeking ways to optimize its operations, with a particular focus on improving assembly line reliability. This in-depth study intends to shed light on Euro Watch’s current operations and the potential pathways to increased efficiency and productivity.

**Question 1**

The assembly plant operates two parallel lines: Line 1 and Line 2. Both lines, despite employing different generations of machinery, are designed to produce 500 wristwatches per hour. However, it's important to note that due to the inherent intricacies and inevitable human involvement in the assembly process, occasional defects in the watches are an inevitable part of production (Montgomery, 2020). Each assembly line has a distinct defect rate. Line 1, with slightly outdated machinery, exhibits a 98% success rate, implying a 2% chance of manufacturing a defective watch. Simultaneously, Line 2, equipped with more advanced and reliable machinery, has a slightly lower defect rate, exhibiting a 99% success rate.

One of the tasks posed by the production manager revolves around finding the smallest integer 'k' for each assembly line, such that with 99% certainty, one can say that the line won't produce more than 'k' defective watches in any given hour. Essentially, this calculation helps establish a threshold for the number of defective watches produced that the company can expect with a high level of confidence.

To compute “k”, the Binomial distribution was used, which models the number of successes in a fixed number of trials, each with the same probability of success (Albright & Winston, 2017). In this context, assembling a watch is a binomial trial, with "success" being a defect-free watch and "failure" being a defective watch. The parameters of the binomial distribution are the number of trials (in this case, the number of watches produced per hour, which is 500), and the probability of success on each trial (which is the probability of a watch being defect-free, 0.98 for Line 1, and 0.99 for Line 2). Using the cumulative distribution function of the binomial distribution, we can find the number 'k' for which the cumulative probability is closest to but does not exceed 0.01. This 'k' is the number of defective watches for which there is 99% certainty that will not be exceed in a given hour. Using the Microsoft Excel formula = BINOM.INV (), “k” was found to be 18 for Line 1 and 11 for Line 2 (see Appendix I)

**Question 2**

As a company that places its customers at the helm of its operations, EuroWatch aims to not just meet but exceed customer expectations. To this end, EuroWatch plans to send slightly more than 500 watches, all from Line 2, to the customer to assure that there are at least 500 defect-free watches upon the package's opening. The task of finding the optimal number of watches to send entailed application of binomial probability distribution. Line 2 is known for its high reliability, with each watch independently being free of defects with a 0.99 probability. Given this information, one approach to solving this problem involves determining the smallest integer 'n' for which the probability of having more than 'n' - 500 defective watches is less than 1%, given that 'n' watches are made in total. This task involves using the =BINOM.DIST function, a statistical function that returns the probability of a trial result using a binomial distribution, with 'n' - 500 as the number of successful trials, 'n' as the number of trials, 0.01 as the probability of success on each trial, and TRUE as the cumulative parameter. a range of potential 'n' values were evaluated, starting from 500 and increasing incrementally.

For each 'n', the probability of getting more than 'n' - 500 defective watches was calculated using the formula =1-BINOM.DIST(). Based on this calculation, it was determined that EuroWatch needs to send at least 511 watches. This figure corresponds to a calculated probability of approximately 0.9938, which is close to the 99% certainty level EuroWatch aims to achieve (see Appendix II)

**Question 3**

The company aims to fill an order of 1000 watches by packaging a quantity slightly exceeding one hour's production from each assembly line, with the watches evenly split between Line 1 and Line 2. Consistent with Euro Watch’s reputation for outstanding quality and customer satisfaction, the company strives to ensure, with 99% certainty, that the package contains at least 1000 defect-free watches when the customer opens it.

Several simulations for varying numbers of watches were made and calculated the probability of getting 1000 or more defect-free watches. The simulations were executed numerous times to ensure the statistical significance and reliability of the results. The simulations were conducted using the =CRITBINOM () and =RAND () functions, as well as, as What-If Analysis. The findings suggested that when 513 watches are produced from each line, the probability of having 1000 or more defect-free watches surpasses the 99% certainty threshold (see Appendix II). Thus, to fulfill this substantial order with the desired level of confidence, EuroWatch should package a total of 1026 watches, evenly divided between the two lines.

**Question 4**

EuroWatch has secured a significant order for 100 high-quality wristwatches from a valued customer, which is projected to generate $50,000 in revenue, considering the agreed-upon price of $500 per watch. Any watches that exceed the order's threshold of 100 units will not contribute to EuroWatch's revenue, placing a cap of $50,000 on the income from this order. Nevertheless, it is advantageous to send slightly more than 100 watches to the customer to ensure that at least 100 defect-free watches are included. This contingency caters to the probability of defects and shields the company from the high cost of rectifying a shortfall in defect-free watches, even though it increases production costs marginally.

If the customer receives fewer than 100 defect-free watches, EuroWatch's revenue will decrease by $500 for each watch short of the required 100. In such an event, EuroWatch would also be obligated to provide additional watches at an expedited cost of $1000 per unit, bearing all the additional costs. This scenario is clearly detrimental to EuroWatch's financial position, emphasizing the importance of delivering at least 100 defect-free watches in the initial shipment.

This order is to be fulfilled entirely from one of the assembly lines, where the unit cost of producing a watch is $450, irrespective of which line is used. The reliability of the two assembly lines varies, with Line 1 producing defect-free watches with a probability of 0.98, and Line 2 with a probability of 0.99. In light of this, a spreadsheet model was developed to calculate EuroWatch's expected profit for any quantity of watches sent to the customer. It is designed to respond correctly regardless of the assembly line used and the shipment quantity, providing a versatile tool for decision-making.

The BINOM.DIST function was utilized to calculate the probabilities of each possible number of defective watches, with the last argument set to 0 for a cumulative distribution. Alongside these probabilities, EuroWatch's profit was calculated for each possible scenario. By using the SUMPRODUCT function, these two sets of data were combined to yield the expected profit. analysis indicated that EuroWatch would never find it optimal to send more than 110 watches, and in fact, the optimal number of watches to send is likely significantly lower than this upper limit. After a series of calculations, it was determined that sending 103 watches from Line 1, which produces defect-free watches with a 0.98 probability, would yield an expected profit of approximately $3301.54 (see Appendix IV).

**References**

Albright, C., & Winston, W. (2017). *Business analytics: Data analysis and decision making. *

Boston, MA: Cengage Learning.

Montgomery, D. (2020). *Introduction to statistical quality control.* New York, NY: Wiley.

**Appendix I**

**Question 1**

**Appendix II**

**Question 2**

**Appendix III**

**Question 3**

**Appendix IV**

**Question 4**