1 – (Long-term averages – Little’s Law) Hotel The construction planner of a large hotel complex needs to determine the size of the guests' parking lot. The complex attracts both business and leisure travellers so that occupancy rates are constant over time (days of the week and months). Given the size of the complex, demand for rooms is estimated at 300 rooms per day. Guests stay in the complex for 2 nights on average and about 2/3 of occupied rooms require a parking space. In addition, the planner wishes to maintain an average parking occupancy of no more than 75%. 1.1- Using the above information, provide an estimate of the minimal number of parking spaces needed in this parking lot? 2.2 - After the complex opening, parking spaces are scarce due to a higher than anticipated guest flow. A year later, management agrees that the higher average demand of 390 rooms per day is expected to continue (is not a novelty effect) and plans to extend the parking lot. What is the minimal number of parking spaces that need to be added to guarantee an average occupancy no more than 75%? Assume that the predictions about the average length of stay in the complex and the fraction of guests arriving by cars materialized. Note that the number of parking spaces should be integer. 2 - (Build-up diagram) Retailing Warehouse Consider an online retailing warehouse preparing for its Christmas peak demand season: based on experience from previous years, it anticipates the following (simplified) customer demand profile: • beginning of January to end of October: 50,000 cases per month • beginning of November to end of December: 200,000 cases per month In addition the VP of Operations has determined that in order to guarantee an acceptable level of product availability during the Christmas season (beginning of November to end of December), the minimum inventory on hand must be equal to 250,000 cases from January 1st to end of October and to 400,000 cases from November 1st to end of December. The warehouse receiving team has 30 permanent employees with a processing capacity of 2,000 cases per employee per month (i.e. 60,000 cases per month). From the beginning of September to the end of December it can also hire additional temporary employees by the month (i.e. 1-month contracts) with a processing capacity of 1,000 cases per employee per month, at a cost of 1,000 euros per employee per month. For simplicity you can also assume that there is no absenteeism and that hired temporary workers are immediately operational at the processing rate stated. Assume that as of the end of August the inventory level is 250,000 cases and no temporary workers have been hired. Now the VP of Operations wants to determine how many temporary workers to hire for the final four months of the year. 2.1 Assume a budget limit of 300 000 euros to hire temporary workers i. What is the maximum number of workers that can be hired with this budget? ii. Calculate the inventory available at the end of each month, taking into account the temporary workers hired with the 300 000 euro budget. iii. What will be the maximum difference between the inventory required and the inventory available at any time during the period? 2.2 Now assume that the VP wants to hire as many temporary workers as are needed to meet the minimum inventory level constraints at all times. i. How many cases need to be produced to ensure the minimum inventory level is met? How many cases can be produced by permanent workers? How many by temporary workers? ii. What is the minimum temporary worker budget that should be planned for the final four months of the year? 3 – (Capacity and Queues) Supermarket A large supermarket experiences stable demand on Saturday afternoons. During this period the store has 4 staffed registers and 5 self-checkout machines located in two separate check-out areas. Each register requires an average of 4 minutes to serve a customer and self-checkout takes 5.5 minutes on average. However, this activity time is subject to unpredictable variability. Specifically, both checkout types have a coefficient of variation of 0.75. The arrival rate to the self-checkout area is 0.5 customers every 1 minutewith coefficient of variation equal to 1. The arrival rate to the staffed registers is 0.6 customers per minute, also with a coefficient of variation equal to 1. Customers entering the self-checkout area stand in a single queue operating on a first-in-first-out basis for any of the 5 self-checkout machines (i.e., a single pooled queue). 2.1 – What is the utilization for the two check-out types? 2.2 – What is the average number of customers waiting in the self-checkout area? 2.3 – What is the average waiting time in the self-checkout area? 2.4 – What is the average customer waiting time for a staffed register? 2.5 – If there were 4 separate queues for each one of the staffed registers, each seeing 9 customers each hour on average (0.15 customers per minute on average), with a coefficient of variation equal to 1 (i.e. separate dedicated waiting lines), what would be the average waiting time per customer in each queue?