THE CAR WASH PROBLEM

A Discrete-Event Analysis by Vitalis Okafor

Introduction : The Car Wash Problem

Simulation Model

Performance Analysis

Operating Cost Analysis

Observations

Questions: #1 to #4

Outline

Processes: Markovian (exponential distribution)

◦ Arrival rate, λ (average incoming cars): 12 cars per hour

◦ Service rate, μ (serving average) – 6 cars per hour

◦ Mean Time to Failure (MTTF) – 100 hours

◦ Mean Time to Repair (MTTR) – 4 hours

Resources: 2 Car Wash Machines

Queue capacity (parking space/ line) – 4 cars

Operation:

◦ 8 hours per day; 300 days per year

Maintenance Costs: $16,000 per year per machine

Customer time value: $20 per hour

Loss in Customer Good will: $50 per lost sale

Repair costs: $500 per repair

Introduction: The Car Wash Problem

The Car Wash problem was modeled using Arena as shown below

Modules:◦ Create (Dirty Cars): Random (Expo) w/value 1; 12 entities per arrival◦ Decide (Is there Parking Space?): 2-way by Condition; NQ(Washing Dirty Cars.Queue) < 4)◦ Process (Washing Dirty Cars): Seize Delay Release Action; Resource, Car Wash Machines, 1; Delay: EXPO

(1/6) Hours◦ Record (Lost Sales): Count; Value: 1◦ 2 Dispose Modules: ‘Clean Exit’ and ‘Dirty Exit’ ( = lost sales)◦ Resource: Car Wash Machines; Fixed Capacity; Capacity = 2; Failures 1 Row (Car Wash Machine Failure)◦ Failure: Car Wash Machine Failure; Time; Up Time – EXPO(100) hours; Down Time – EXPO(4) hours

Replication Parameters: Length – 2400 hours; Hours Per Day – 8 hours

Simulation Model

Simulation Model (cont’d) Results:

Total number of cars through carwash in 1 year (300 days): 28,932

Total number of cars washed: 10,366 (~36%) Total number of lost sales: 18,566 (~64%)

System Metric

Value Half-width Minimum Maximum

Average Total Processing Time (TIS) 0.1319 0.00642345 0 14.159

Average WIP 1.5902 0.089335264 0 18

Average Total Washing Time per Car 0.3682 0.025650106 0.00007157 14.159

Accumulated VA Time 1728.27

Accumulated Wait Time 2088.27

Average Wait Time in Parking lot (queue) 0.2015 0.023967654 0 14.0586

Average Number Waiting in Parking lot (queue) 0.8701 0.084888612 0 4

Average Machine Instantaneous Utilization 0.3601 0.015533451 0 1

Avg Machine Number Busy 0.7201 0.031066901 0 2

Avg Time Machine Failed 5.5827

Number of Failure Observations 24

% Busy 40.47

%Failed 5.58

% Idle 53.95

Performance Analysis

Machine Maintenance:

$16,000/yr/machine × 2 machines × 1 yr = $32,000

Total Customer Time Valuation (for washed cars):

$20/hour × Avg Washing Time per car × Cars Washed

=$20/hr × 0.3682 hr × 10,366 = $76,335.22

OR

$20/hour × (Accum VA Time + Accum Wait Time)

= $20/hr × (1728.27 + 2088.27) hours = $76,330.80

Total Loss in Customer Good Will (for lost sales):

$50/lost sale × Lost Sales

=$50/lost sale × 18,566 lost sales = $928,300

Repair Costs:

$500/repair × Number of Failure Observations

=$500/repair × 24 = $12,000

Operating Cost Analysis

After several iterations, some common trends were noted:◦ Increase in number of car wash machines (resource capacity) leads to a significant increase in percentage of

washed cars.

◦ Increase in parking lot (queue) size leads to a gradual increase in percentage of washed cars.

Observations

2 3 4 50

20

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Percentage of Cars Washed vs Number of Machines

Percentage of Cars Washed vs Number of Machines

No. of machines

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Percentage of Cars Washed vs Parking Lot Size

Percentage of Cars Washed vs Parking Lot Size

Parking Lot Size

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After several iterations, some common trends were observed:◦ The wait time (or time in parking lot/queue) significantly decreases with and increase in the number of

machines.

◦ The Average Machine Instant Utilization decreases with addition of car wash machines and a greater

percentage of idle time.

◦ With the constant number of machines (resource capacity), increase in queue size led to an increase in wait

times but also an increase in Value Added (VA) Time (and Customer Time Valuation). However, simultaneous

increase in both factors led to an overall improvement in system performance.

◦ Cost of machine maintenance is proportional to the number of machines available.

◦ VA Time and Customer time valuation decreased with an increase of number of machines but was maximized

at 5 machines.

◦ Loss in customer good will decreases significantly with an increase in number of machines and gradually with

an increase in the number of parking lot spaces (queue size).

◦ Number of Failure Observations were random amongst the iterations. However, the repair costs did not weigh

much compared to the other values in the cost analysis.

Observations (cont’d)

Question1:a) What number of car wash bays do you recommend?

In order to meet the loading demands of 12 cars per hour, the car wash would require at least 4 car wash machines. Any less than 4 machines, there will be a greater percentage of lost sales than cars that make it through to get washed, which is an unfavorable outcome in a competitive market. In general, the more machines available, the more cars can be washed.

b) What are the performance characteristics of your recommended design?My recommended design will be a ‘5 car wash machines – 5 car parking lot’ system. This recommendation is made due to cost and resource utilization constraints as shown previously. The performance analysis and cost analysis are attached to the next couple of slides

c) How will performance be impacted if your recommendations are not followed?As mentioned in part a, the more machines available, the more cars can be washed. If the system is left as it currently is, the car wash would be loosing almost $1 million a year to lost sales, or ‘loss in customer good will’, as it was aptly termed. This net loss will most likely go to the competition and this would lead to loss in more customers – as a more successful business is always more attractive to the customer - and consequently, a possible business closure.

Questions

5 car wash machines and 5 car parking lot

Recommended Design (1b)•Total number of cars through

carwash in 1 year (300 days):

28,344

•Total number of cars washed:

18,793 (~66%)

•Total number of lost sales:

9,551 (~34%)

Cost Analysis

Machine Maintenance $ 80,000.00

Total Customer Time Valuation (Cars Washed) $ 91,709.84

Total Loss in Customer Good Will $ 477,550.00

Repair Costs $ 13,000.00

System Metric

Value Half-width Minimum Maximum

Average Total Processing Time (TIS) 0.1594 Correlated 0 12.5685

Average WIP 1.8825 0.07794983 0 22

Average Total Washing Time per Car 0.244 0.012308638 0.00003178 12.5685

Accum VA Time 3171.11

Accum Wait Time 1346.23

Average Wait Time in Parking lot (queue) 0.07163448 0.012017539 0 11.8991

Average Number Waiting in Parking lot (queue) 0.5609 Correlated 0 5

Avg Machine Instant Utilization 0.2643 0.010831435 0 1

Avg Machine Number Busy 1.3215 0.054157176 0 5

Avg Time Machine Failed 2.9943

Number of Failure Observations 26

% Busy 39.92

%Failed 3.24

% Idle 56.84

Recommended Design (1b)

Question 2a) What is the economic value of an additional parking space for the

queue?This value was calculated by considering the difference in total customer time valuation and total loss in customer good will in the iterations with 4 car parking lot – 2 machines and 5 car parking lot – 2 machines, as shown below:

Economic value: $(928,300 – 841,650) + $(93,638.52 – 76,335.22) = $103,953.30

b) What is the value of the last parking space in the default design (4th space)?Applying the same logic as in part a, the economic value of the 4th space we calculated to be $50,383.31.

c) Shouldn’t the 4th and 5th spaces have the same value?No they should not, because the improvement in system performance with additional parking spaces was not a linear progression. The improvements cause a gradual increase in customer time valuation and a significant reduction in loss in customer good will which add up to make a very impactful net gain.

Questions (cont’d)

Question 3Car wash bays tend to break down with a MTTF of 100 hours of use time and take 4 hours (MTTR) to repair for $500. MTTF and MTTR are exponential distributed. How does this feature impact your optimal design?

This feature led to 26 failure observations in the 300 day-1year cycle. This led to a failed time percentage of 3.24%; with the machines busy 39.92% of the time, and idle 56.84% of the time. These failures cost a total of $13,000 to repair for the year.

Question 4How sensitive are your recommendations to the current load estimate of 12 cars per hour? Our demand may go up if we successfully prepare for a competitor.

The recommended design was selected especially with the anticipated increase in load demands in mind. As is seen in the question above, the resources still experience a bit of idle time which means they have the capacity to handle more load. To verify this, the system was run with an average incoming of up to 15 cars per hour and was still able to output favorable system performance values. (i.e. Cars washed > Lost sales)

Questions (cont’d)