Math Problem Statement

The local Department of Motor Vehicles issues new licenses and renews licenses. (See the diagram below.) The office receives 100 customers per hours. All customers first see a receptionist. The receptionist directs them in one of three directions. 75% go directly to Issue License (staffed by 9 workers) where a new photo and license are done. 15% are required to take an eye test (staffed by one worker) and 10% must first take a multiple-choice electronic written test (on one of three computers). Only 85% of people pass the eye test and the remaining 15% exit. The customers who pass the eye test proceed on to the Written Test. 10% of the people who take the written test fail it, while 90% pass the test and then proceed to Issue License.

Workers Activity time per worker (min) Reception 1 0.4 Eye Test 1 5 Written Test 3 15 Issue License 9 6 What is the implied utilization of the issue license (as a %)?

Solution

To determine the implied utilization of the Issue License station, follow these steps:

Step 1: Calculate the number of customers arriving at Issue License

The Issue License station handles customers from two sources:

  1. Customers going directly to Issue License from Reception (75%).
  2. Customers who pass the Written Test (via Eye Test if applicable).

Customers directly from Reception:

75% of 100 customers = 0.75×100=750.75 \times 100 = 75 customers/hour.

Customers from the Written Test:

  • Eye Test Path:

    • 15% of 100 customers go to Eye Test = 0.15×100=150.15 \times 100 = 15 customers/hour.
    • Only 85% pass the Eye Test = 0.85×15=12.750.85 \times 15 = 12.75 customers/hour.
    • These 12.75 customers proceed to the Written Test.

  • Written Test Outcomes:

    • 10% fail the Written Test, so 90% pass = 0.9×12.75=11.4750.9 \times 12.75 = 11.475 customers/hour.

  • Total from Written Test: 11.475 customers/hour.

Total customers arriving at Issue License:

75+11.475=86.47575 + 11.475 = 86.475 customers/hour.

Step 2: Calculate the capacity of the Issue License station

  • There are 9 workers.
  • Each worker processes customers in 66 minutes per customer.
  • Number of customers handled by 1 worker in 1 hour: 606=10\frac{60}{6} = 10 customers/hour.
  • Total capacity of the Issue License station: 9×10=909 \times 10 = 90 customers/hour.

Step 3: Calculate implied utilization

Utilization is the ratio of demand to capacity: Utilization=DemandCapacity×100\text{Utilization} = \frac{\text{Demand}}{\text{Capacity}} \times 100 Substitute the values: Utilization=86.47590×100=96.08%\text{Utilization} = \frac{86.475}{90} \times 100 = 96.08\%

Final Answer:

The implied utilization of the Issue License station is 96.08%.

Would you like a breakdown of other processes or further clarification? Here are some related questions:

  1. What is the utilization of the Eye Test station?
  2. How many customers fail each part of the process (Eye Test and Written Test)?
  3. What is the overall capacity of the system (bottleneck analysis)?
  4. How can the DMV improve its utilization and throughput?
  5. What is the implied utilization of the receptionist?

Tip: Always identify bottlenecks in systems like this to improve efficiency and reduce wait times.

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Math Problem Analysis

Mathematical Concepts

Percentage Calculation
Utilization Analysis
Workflow Optimization

Formulas

Utilization = (Demand / Capacity) × 100
Capacity = Number of Workers × (60 / Activity Time per Worker)
Demand = Sum of customer flows to a station

Theorems

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Suitable Grade Level

Grades 10-12, College Level

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