Question

Meeting Demands in a Fitness Center Individuals who become members of a fitness center do sọ for a variety of reasons, but one reason is often the availability of specialized facilities or equipment. Consequently, man- agement of the center is sensitive to the need for meeting member demand for equipment. At the same time, providing specific equipment requires funds to purchase and maintain it and space to house it. Indeed, because of financial and space constraints, making a decision to purchase a specific piece of equipment may mean that other equipment cannot be purchased. We consider a situation where management of a fitness center is evaluating whether it has enough stationary bicycles to meet the demand. There are currently five bicycles, and the demand in the early morning appears to exceed the capacity. Management believes members are frustated by the need to wait for bicycles in the period from 6:00 a.m. to 7:30 a.m.., add more bicycles. Data have been collected on equipment utilization between 6:00 a.m. and 7:30 a.m., and the data include arrival times and excercise times. The data represent observations of a stochastic process (actually several stochastic processes), and to simplify the modeling process, it is helpful to make assumptions about the underlying distributions for these random variables. Suppose that data are collected and reviewed with a goal of making reasonably simple assumptions about arrival rates. On the basis of the review, we assume that the rate of arrivals in the period 6:00 to 7:30 a.m. varies, with a peak arrival rate during the 30-minule period from 6:30 to 7:00 and lower rates before 6:30 and after 7:00. More specifically, we assume that arrivals have an exponential distribution with an average of 30 arrivals in the period from 6:00 to 6:30, an average of 45 arrivals in the period from 6:30 to 7:00, and an average of 20 arrivals in the period from 7:00 to 7:30, and the goal of the study is to determine whether it is economically desirable to Suppose that most of the members use the bicycles as part of an exercise program Let includes other activities, and we assume that the period of excercise time spent on the eycles is of fixed duration for all members: 8 minutes. Finally, we assume that if no bicycle DIe ailable when a member arrives, the member will wait for one to become available if IS e are three or fewer members waiting, but if there are four members already waiting, then the new arrival will leave. With these assumptions, we can simulate a single day as follows: We consider each ute of the 90-minute period, one minute after another, and we determine whether one amore of the members currently using a bike completes a session during that minute. If cach of the bicycles released becomes available for a new user. If there are members waiting, as many as possible are assigned bikes; those in line the longest are assigned first. Then, members arriving during that minute are considered, and if more than one member arrives in a minute, they are considered one after another. For cach arrival, we check whether there is a bicycle not being used by another person. If so, the member begins to use the hicycle. If there is no unused bicycle, then the member joins the queue of people waiting for a bicycle if there are three or fewer people already waiting. If there are four people waiting, then the new arrival leaves. Note that the protocol for this simulation is for those who finish using a bicycle during a particular minute to do so before newly arrived members look for a bicycle. This is, of course, only one of several such assumptions that could be made. With the original task in mind, it is appropriate to record the total number of members who use the equipment each day, the total number who do not use it because there are four others waiting when they arrive, and the average waiting time for members who use the equipment. We assume that no member tries to use the equipment more than once a day. We will refer to members who leave without using a bicycle as disappointed. We repeat the simulation for 1000 runs, and we have the following results: Average number who try to use a bicycle = 94.60 Average number who are disappointed = 37.15 = 38.71% Average percent who are disappointed = 4.1 minutes Average waiting time The average percentage is computed by computing the percent disappointed each day and then averaging these percentages over the 30 days. This is not (necessarily) the same as the ratio of the average number who are disappointed to the average number who tried to use a bicycle. For the particular goal of this discussion, it is the average percentage of disappointed members that is of interest. In other situations, what should be noted might be the average number of disappointed members or the number (or percentage) of members who succeed in riding a bicycle. The questions that the managers of the center ask are 'What are the economic impli- cations of these results, and how do these results change if more respond to these questions, we look at the economics of the situation. Suppose the yearly membership dues for the facility are $360, or $30 per month. Also, suppose that members Ho use the stationary bikes do so ten times a month and that half of those who are dis- appointed more than three times in a month will cancel their membership. Finally, assume hat the yearly costs for purchase and maintenance of each stationary bicycle is $1200, or ipment is added?' To $100 per monthnner Can 200 CHAPTER 4 Simulation Models With these assumptions, we can investigate the economics of the current situation, Recall that the member's decision criterion results in half of the disappointed members dropping their membership if bicycles are unavailable more than 30% of the time-that is, unavailable more that three times a month, Let s be the probability that a member wll be disappointed when trying to use a bicycle on a random visit. Then, viewing multiple attempts to ride a bicycle as independent repetitions of a binomial trial with disappointment probability s, we find that the probability of four or more disappointments in a month (ten visits to the fitness center) is 1- (1 – s)10 – 10s(1 – s)º – 45s²(1 –s)% – 120s (1 - s)? (4.10) Half of the members with four or more disappointments will cancel, and the probability of this event is .5 times the expression (4.10). The economic cost to the center is $30 per month for each membership cancelled. Using the simulation results for the current situation. we estimate s as .387, so the probability that a random member who attempts to use the bicycles will cancel is ,2923, Also, we have the average number of members who attend each day-approximately 94.5 in the simulation leading to the data in Table 4.19. Thus, in this situation, the cost of cancellations is about $830 per month. This substantially exceeds the cost of installing another bicycle, and the next step is to use simulations to determine how many additional bicycles should be installed to maximize net income-that is, dues from continuing members less new costs. We repe 12 bicycles. We assume that the arrival schedules do not depend on the number of bicycles, an assumption that could be changed if there were data on which to base other assumptions about arrivals. Because the arrivals are given by a stochastic process, the average number of members who try to use bicycles varies slightly from one simulation to another. For the eight simulations reported here, that number was between 93.94 and 95.14. Using data derived from the simulations, we have the results summarized in Table 4.19. The cost and the simulation with additional stationary bicycles, using a total of 6 through revenue data in the table are monthly figures. Judging on the basis of the results shown in Table 4.19, management would achieve higher net revenue with more bicycles. The highest net revenue would be achieved for eight bicycles, although seven, eight, or nine bicycles would all give improvements of about 20% or more above the current level, Table 4.19 Percent of Disappointed Members Expected Cost of Cancellation Expected Net Cost Number of Probability of Cancellation of New Bicycles Bicycles Revenue 5. 38.7 .292 $830 $2006 6. 30.1 22.1 .177 504 $100 2231 .0804 228 2407 200 8. 15.9 .0301 86 2449 300 11.2 .0095 27 2408 2330 2234 2135 400 10 7.0 .0018 500 11 4.6 .0004 600 2.9 12 000 700 4. Repeat the analysis of the fitness center model when the assumption that half of the mem- bers who are disappointed more than three times a month will cancel their memberships is replaced by the assumption that (a)a30% of such members will cancel. CS (bar0% of such members will cancel. उ