17.7 Case Study: Holiday Baskets
Case Study: Holiday Baskets
Holiday Baskets is a small seasonal business specializing in the preparation and delivery of nut baskets during the Christmas season. The company was founded by Helen Hemkey, who oversees all aspects of operations, from order intake to final delivery.
The business model is simple: customers place orders for baskets containing four types of nuts (walnuts, cashews, almonds, and peanuts), all of which are kept in stock. Helen receives the orders, picks the required quantities of nuts, and places them into bags along with shipping labels. These are then sent to the basketing workshop, where her four employees, Dianne, Nancy, Sue, and Mariam, assemble the final product.
- Dianne works independently at Table 1.
- Nancy and Sue collaborate at Table 2, with Nancy starting the basket and Sue completing it.
- Mariam works independently at Table 3.
Once assembled, the baskets are labelled and placed on the pick-up table for courier delivery.
Concern 1: Quality Assurance
Helen has built a strong reputation for the presentation quality of her baskets, specifically, that the nuts are neatly sectioned and do not mix. She aims to keep the defect rate below 2%. However, based on recent observations, she estimates that each employee has a 5% probability of producing a disarrayed basket.
Question: Can Helen maintain her desired quality standard of 98% or higher for the final baskets?
Analysis:
Let’s assume the probability that each basket is correctly assembled is:
Pcorrect=0.95
The probability that all three workstations (Dianne, Nancy, Sue, and Mariam) produce a correct basket is:
Psystem success=0.95×0.95×0.95=0.857375
However, we must consider the complement—the probability that at least one basket is disarrayed. The probability that all three produce a disarrayed basket is:
Pall fail=(1−0.95)3=0.000125
Thus, the probability that at least one basket is correct is:
1−Pall fail=1−0.000125=0.999875
Yes, Helen can maintain her quality standard. The probability that the baskets are correctly assembled exceeds 99.98%, which is well above her 98% threshold.
Concern 2: Delivery Reliability
Helen strives for next-day delivery and uses multiple courier companies to increase the likelihood of on-time delivery. Each courier has a 70% probability of delivering on time.
Question: She wants to know how many independent couriers, each carrying an identical set of orders, are needed to achieve at least 99% confidence in next-day delivery.
Analysis:
- P=0.70P=0.70 (probability of success per courier)
- n = number of couriers
- We want:
Pat least one success≥0.99
1−(1−P)n≥0.99⇒(1−P)n≤0.01
(1−0.70)n≤0.01⇒(0.30)n≤0.01
- n=1: 0.300.30
- n=2: 0.090.09
- n=3: 0.0270.027
- n=4: 0.00810.0081
Helen needs to use at least 4 courier companies, each carrying the same set of orders, to achieve 99% confidence in next-day delivery.
