Operations and Supply Chain Management The Core

222 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
SOLUTION—PART 2
The present level of service for three or fewer cars is the probability that there are 0, 1, 2, or 3 cars in
the system. From Model 1, Exhibit 7.12,
n
P n = ​( 1 − ​ _ λ λ__
μ)( μ ​ ​ )
at n = 0, P 0 = (1 − 15/20) (15/20) 0 = 0.250
at n = 1, P 1 = (1/4) (15/20) 1 = 0.188
at n = 2, P 2 = (1/4) (15/20) 2 = 0.141
at n = 3, P 3 = (1/4) (15/20) 3 = 0.105
0.684 or 68.4 percent
The probability of having more than three cars in the system is 1.0 minus the probability of three
or fewer cars (100 − 68.4 = 31.6 percent).
For a 95 percent service level of three or fewer cars, this states that P 0 + P 1 + P 2 + P 3 = 95 percent.
0
0.95 = ​ 1 − ​ _ λ
( μ ​ λ__
)( μ) ​ ​ + ​( 1 − ​ _ λ μ ​ λ__
)( μ) ​ ​ + ​( 1 − ​ _ λ μ ​ λ__
)( μ) ​ ​ + ​( 1 − ​ _ λ μ ​ λ__
)( μ) ​ ​
​
0.95 = ​ 1 − ​ λ 2 3
_
( μ ​ )[ 1 + ​ _ λ
μ + ​ λ__
( μ) ​ ​ + ​
λ__
( μ) ​ ​ ]
We can solve this by trial and error for values of λ/μ. If λ/μ = 0.50,
With λ/μ = 0.45,
With λ/μ = 0.47,
0.95 ​=​
? ​ 0.5(1 + 0.5 + 0.25 + 0.125) ​
0.95 ≠ 0.9375
1
0.95 ​=​
? ​ (1 − 0.45)(1 + 0.45 + 0.203 + 0.091) ​
0.95 ≠ 0.96
0.95 ​=​
? ​ (1 − 0.47)(1 + 0.47 + 0.221 + 0.104) = 0.9512 ​
0.95 ≈ 0.95135
Therefore, with the utilization ρ = λ/μ of 47 percent, the probability of three or fewer cars in the
system is 95 percent.
To find the rate of service required to attain this 95 percent service level, we simply solve the
equation λ/μ = 0.47, where λ = number of arrivals per hour. This gives μ = 32 per hour. That is, the
teller must serve approximately 32 people per hour (a 60 percent increase over the original 20-perhour
capability) for 95 percent confidence that not more than three cars will be in the system. Perhaps
service may be speeded up by modifying the method of service, adding another teller, or limiting the
types of transactions available at the drive-through window. Note that with the condition of 95 percent
confidence that three or fewer cars will be in the system, the teller will be idle 53 percent of the time.
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Example 7.2: Equipment Selection
The Robot Company franchises combination gas and car wash stations throughout the United States.
Robot gives a free car wash for a gasoline fill-up or, for a wash alone, charges $5. Past experience
shows that the number of customers that have car washes following fill-ups is about the same as for a
wash alone. The average profit on a gasoline fill-up is about $7, and the cost of the car wash to Robot
is $1. Robot stays open 14 hours per day.