Presentation on theme: "By: Abby Almonte Crystal Chea Anthony Yoohanna"— Presentation transcript:
1By: Abby Almonte Crystal Chea Anthony Yoohanna IE 417: Operations ResearchHOT DOG KING RESTAURANTExtra CreditChapter 20 section 5, #2By: Abby AlmonteCrystal CheaAnthony Yoohanna
2Overview Problem Statement Assumptions M/M/1/GD/C/∞ Manual Solutions WinQSB SolutionsExtra QuestionsSensitivity AnalysisReport to Manager
3Problem StatementAn average of 40 cars per hour which are exponentially distributed want to use the drive-in at the Hot Dog King Restaurant. If more than 4 cars are in the line, including the car at the window, a car will not enter the line. It takes an average of 4 minutes to serve a car.a) What is the average number of cars waiting for the drive- in window? (not including the car at the window)b) On the average, how many cars will be served per hour?c) If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?
4Assumptions The information in the problem is accurate The queuing problem is a M/M/1/GD/C/∞ problem
5Type of System M/M/S/GD/C/∞ ArrivalProcessServiceProcessGeneralQueueDisciplineCustomerPopulationExponentialDistributionNumber ofServersLimitCapacity
6FOR EXAMPLE… Type of System M/M/1/GD/4/∞ Exponential Distribution GeneralQueueDisciplineCustomerPopulationNumber ofServersLimitCapacityFOR EXAMPLE…
7Manual Solutions Make pre-calculations Identify Arrival Rate (λ) Calculate Effective Arrival Rate (λe)Identify Service Rate (μ)Calculate rho (ρ) = λ/μ
8Pre-Calculations λ = 40 cars per hour λe = λ ( 1 – π4) Given that there are an average of 40 cars per hour to use the drive-inλ = 40 cars per hourBUT there can only be 4 cars in the drive-in, therefore the effectivearrival rate is:λe =λ ( 1 – π4)= 40 (1 – )= 14.9 cars per hourThen, we calculate service rate by using the given average servicetime which is 4 minutes per car.μ = minute = 15 cars per hour4 minutes per carUsing λ & μ values, we find ρλ = cars per hour = 2.667μ cars per hour
9with respect to an M/M/1/GD/∞ system Part Aa) What is the average number of cars waiting for the drive- in window? (not including the car at the window)Lq= L – LsWe use this equation which represents the number of customers in the queuewith respect to an M/M/1/GD/∞ system
11Part A continued. Lq = L – Ls = 3.437 – 0.0988 = 2.449 Finally Lq Solution: About 2.5 cars
12Part B π4= (ρ4)(π0) Π4 = ρ4 π0 Π4 = 2.6674 (0.012452) = 0.62967 b) On the average, how many cars will be served per hour?First we find the probability of having 4 cars in the drive-in, whichis noted as π4π4= (ρ4)(π0)Π4 = ρ4 π0Π4 = ( ) =Refer to Part A
13Solution: About 14.8 cars per hour Part B continued.Thenλ ( 1 – π4)= 40 (1 – )=Solution: About 14.8 cars per hour
14Solution: About 13.9 minutes Part Cc) If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?C = 4π4W = ___L___ λ(1 – πc)Refer to Part BWe use this equation which represents time a customer spends in the system.W = ___ ___ = ( )hoursSolution: About 13.9 minutes
15Using WinQSBThe red box indicates the cost values we decided to apply in WinQSB tocalculate hourly cost.
17Probability of Customers This table represents the probability of having the number of customers in the line. Since there is a capacity of 4 customers and an arrival rate of 40 customers per hour, the table shows that having 4 customers in line has a higher probability compared to having zero.
18Manual Solutions vs. WinQSB QuestionManualWinQSBa)What is the average number of cars waiting for the drive-in window?2.449 carscarsb)On the average, how many cars will be served per hour?carsc) If a customer just joined the line to the drive-in window, on average how long will it be until he or she has received their food?minuteshours
19Questions SENSITIVITY ANALYSIS What happens if the average service time changes?What happens if the arrival rate changes?What happens if the number of servers changes?What can be done to reduce the hourly cost?What is the best option for reducing the hourly cost?InitiateSENSITIVITY ANALYSIS
20Sensitivity Analysis: Number of Servers Average balked customers drops to nearly zeroTotal hourly cost of the system drops to $116.44From 1 server to 2 the cost drops the most.
21Sensitivity Analysis: Number of Servers More than 5 servers results in a gain of cost.
22Sensitivity Analysis: Service Rate From increasing your service rate from 15 cars to 25 cars per hour, the total cost is still higher than increasing the number of servers.
23Sensitivity Analysis: Arrival Rate From increasing your service rate from 15 cars to 25 cars per hour, the total cost is still higher due to the amount of customers balking.
24Report to Manager Original Data The total hourly cost is $529.69 The total hourly cost from balking is $453.36Sensitivity AnalysisAdd up to 4 servers to reduce total hourly cost significantlyIncreased service rate has small effect on total hourly costIf arrival rate decreases, so will total hourly cost (not ideal)