Question

a. The current checkout format is individual lines. For example, on Tuesday evenings, there is one line and one cashier. On average, a customer arrives to the checkout line every 90 seconds, exponentially distributed. The cashier is fast, and can on average checkout one customer in 65 seconds, also exponentially distributed.

b. The third option is to use customer self-service. Half Foods would give each customer a scanner, then for checkout, the customer would simply scan a barcode at the single register and the items purchased would be downloaded to the register. The customer then simply needs to pay. This would decrease the mean service time per customer to 50 seconds, with a standard deviation of 10 seconds. However, arrivals are not expected to change. The opportunity cost of waiting to pay at the register remains $8 per hour of waiting. The cost to implement and use this system is $10 per hour operated, but no cashiers are needed.

Should half Foods implement this system or keep the existing checkout format (part a)? Please justify you answer by calculating the expected benefit or loss from implementing this system.

Answer 1

(a)

Average arrival rate, λ = 1 customer in 90 seconds = (3600/90) per hr. = 40 per hr.
Average service rate, μ = 1 customer in 65 seconds = (3600/65) per hr. = 55.38 per hr.

The average number of customers waiting, L_q = λ² / {μ.(μ - λ)} = (40^2)/(55.38*(55.38 - 40)) = 1.88

Overall cost of waiting = L_q * $8 per hour = $15.02 per hour

Cost of servers = 1 cashier * $12 pr hour per cashier = $12 per hour

So, the total cost per hour = 15.02 + 12 = $27.02 -------------------(1)

(b)

Average arrival rate, λ = 40 per hr.
Average service rate, μ = 1 customer in 50 seconds = (3600/50) per hr. = 72 per hr.

L_q = λ² / {μ.(μ - λ)} = (40^2)/(72*(72 - 40)) = 0.694

Overall cost of waiting = L_q * $8 per hour = $5.56 per hour

Cost of the system = $10 per hour

So, the total cost per hour = 5.56 + 10 = $15.56 ---------------------(2)

Comparing (1) and (2), we recommend the new system should be implemented.