Business Decision Modeling - Assignment Example

Course Work Business Decision Modeling Name: Course: Tutor: February 16, 2011 a) Introduction to the formulae applicable for the simulated result and how to apply them for the two options. I order for the university to investigate the total number of till to be operating with during lunch time. It is necessary to evaluate the current system through simulation,1 and use it to compare with the performance provided while using the two tills. Therefore, having 100 arrivals during every lunch time for each week, simulation results could give an outcome that can be independently relied upon to make an appropriate decision. In the case of single channel queuing, such calling population is always infinite. This is because if any unit, leave the population to enter service time or join the population, there is always no change in the rate of arrivals. Arrival for each service occurs consecutively one at a time in a random fashion since arrivals are defined by using the distribution of time between customer arrivals. Immediately the join the waiting line, 2 serving begins immediately. The service times are assigned in some random numbers according to the given probability distribution which doest change for quite some time until further investigations were made. Based on the results obtained through simulation of the two tills option, the following formulae are used to interpret the outcome of the operation. Data analysis according to the table computed in the Excel spreadsheet3 will help the management in making decision to improve the entire performance of the business as well as determining which scenario is the best to operate within, either to use one till or both the two tills. b) Comparison of the alternative options based on the performance measures as per the mean queuing time obtained for each case and the assumptions made; According to the formulas bellow, it is possible to explain the outcome of each scenario based on the allocation of the callers between the two tills as shown in the simulation tables 1. Which gives high degree of confidence reliance on the information generated? Considering one till in operation whose total duration has a probability destiny function (p.d.f) = f(T) = u exp (wT), T > 0, where u represents service rate i.e. the average number it takes to complete service per each unit of time whenever the server is busy. 1/u represents the mean time of service. Therefore since, customers arrived at the system in a systematic manner, then each customer probability destiny distribution would be4; Fp (p;L, f) = fl ‘[F(L)]-' -exp(- fl), a >1 ,> 0 ,> 0 Formulae used to interpret the simulated data; 1) The average time customers will wait in the system before being served; =Total time customers wait/ the total number of customers = 3109 Minutes/100Customers=31.9 minutes 2) The estimated time that the server is idle; = total idle time of the server / total time of the simulation = 0 minutes / 8152 minutes = 0 minutes Thus, the probability that the server is busy is 1 - 0 = 1 This means the server is utilized 100% of the time. 3) The estimated time that the server will be idle; = total idle time of the server / total time of the simulation = 0 minutes/8152 minutes = 0 minutes 4) Average service time is determined as; =The total service time in minutes /total number of customers = 8152Minutes/100 Customers = 81.52 minutes This entire result is compared with the expected service time by determining the Mean of the service time distribution which is as follows5. =Which is the sum of the service time s multiplied with the probability of service time (s) This yields (8150*1=8150) when the simulation is run longer, then, the average and expected value will get closer together. 5) To determine the average time between each arrivals = sum of all times between arrivals in minutes divide by number of arrivals =236 minutes /99 =2.3 minutes 6) To determine the average time customers spends in the system = total time customers spend in the system over the total number of customers =3520 minutes/100 Customers =35.2 minutes Finally,= average time customer spends waiting in the queue + Average time customer spends in service =31.9 minutes + 35.2 minutes = 67.1 minutes Since the results obtained shows that the average time a customer will spend in the system is 35.2 minutes, which clearly indicates that, there is an improvement in the operations as compared to one till whose performance is lower as compared to the two tills. The mean queuing time which ought to be achieved, has not yet been attained thus giving a low degree of confidence of reliance on the results obtained. Although the number of arrivals actually reneged wasn’t so significant, the result of the model is fairly well. In order to provide a more defined evaluation, then simulation is statistically analyzed to construct a confidence interval according to the outcome. Some of the assumption made are that; the arrivals are very systematic and each customer is served in the sequence of their arrival i.e. first in first out (FIFO) method.6 . 2) Summary of the results obtained through simulation, the findings as well as the recommendations. According to the results obtained, the management of the university is interested in increasing the total revenues collected through reducing the costs associated with, idle time and waiting time to obtain an expected impact on the revenue collected due to additional till. The estimated time that the server will be idle is totally reduced to zero thus, giving a probability of 1 that the server is busy based on the results obtained from simulation table 1 analysis. In addition to the reduced idle time, the average time between arrivals are minimized to 2.3 minutes. Though this is the case, there is greater need to reduce the time customers spend in the system. Table 1 clearly shows that the time taken for customers to be served is 35.2 minutes giving an average of 67.1minutes after adding the time taken for customers waiting in the queue (31.9 minutes), thus it has to be reduced so as to increase performance and efficiency of the entire system. 3) The final analysis of the results after making the necessary modification on the initial simulation results. After collecting additional data, the pattern of customer’s arrival changed from first hour to the second hour according to table 2 and 3 respectively. This simulation tables results adjusted give the much more information for the management to make a better decision on the options available .The no of till needed for every hour should be at least two if there is need to maximize the out based on the result are analyzed. 1) The average time customers will wait in the system before being served; =Total time customers wait/ the total number of customers = 351 Minutes/100Customers=3.5 minutes For the send hour=3065/100=30.65 minutes 2) The estimated time that the server is idle; = total idle time of the server / total time of the simulation = 5 minutes / 8017 minutes = 0.0006 3minutes Thus, the probability that the server is busy is 1 – 0.00063 = 0.9 This means the server is utilized 90% of the time. For the second hour=355/8095=0.045 minute 3) The estimated time that the server will be idle; = total idle time of the server / total time of the simulation = 0 minutes/8017minutes = 0 minutes For the second hour=355/8095=0.045 minutes 4) Average service time is determined as; =The total service time in minutes /total number of customers = 8017 minutes/100 Customers = 80.17 minutes For the second hour=8095/100=80.95minute This entire result is compared with the expected service time by determining the Mean of the service time distribution which is as follows =Which is the sum of the service time s multiplied with the probability of service time (s) This yields (8017*0.9=7215.3) when the simulation is run longer, then, the average and expected value will get closer together. For the second hour=8095*0.045=364.7 minutes 5) To determine the average time between each arrivals = sum of all times between arrivals in minutes divide by number of arrivals =295 minutes /99 =2.98 minutes For the second hour=255/99=2.58 minutes 6) To determine the average time customers spends in the system = total time customers spend in the system over the total number of customers =399 minutes/100 Customers =3.99 minutes For the second hour=3419/100=34.9 minutes Finally, = average time customer spends waiting in the queue + Average time customer spends in service =3.99 minutes + 0 minutes = 3.99 minutes For the second hour=34.9+30.65=65.55 minutes There was massive improvement in the performance of the business due to the change in the inter arrival time, especially for the first hour where the average time the customers spent in the queue as well as being served reduced very much. This would directly increase the revenue collected due to reduced costs associated with idle time as well as waiting time. The comparison result shows 90% degree of confidence, therefore it can be confidently relied upon in making the best option. Reference; Gafarian A. & J. Ancker (1999) Queuing with impatient customers who leave at random. Law, A., M., &. Kelton, D., (1982) Simulation Modeling and Analysis. McGraw-Hill, New York Mesut Gunes (2003) Simulation discrete event systematic simulation analysis. Parkan, C., (1987). Simulation of a Fast-Food Operation where Dissatisfied Customer Renege. The Journal of the Operational Research Society, Vol. 38, No. 2, pp. 137- 14 Warren E., & Patrkan, C., (2006) Optional reneging decisions in queue. Decision, Sc 9.107-119 Read More