Problem A: (20 points) An online store receives orders throughout the day, but the order fulfillment process each day starts at 11:00am and ends until all the orders received in the last 24 hours (i.e. from 11:00am of the previous day to 11:00am of the current day) are fulfilled. The orders that arrive after 11:00am on a day will be fulfilled on the next day. There are three steps required to fulfill a customer order. The first step is to process the order from the order pool that keeps all the orders that have arrived in the last 24 hours. The second step is to pack the order and the last step is to ship the order. Orders go through these steps one by one. The manager wants to analyze the fulfillment process and has provided the process flow diagram shown below. The capacity of each step is shown in the process flow diagram. The storage buffers between consecutive steps are sufficiently large such that there is no blocking at any time. Order pool Process order 1 Pack order 2 Ship order 400 orders per hour 500 orders per hour 300 orders per hour Assume that there are a total of 3000 orders in the order pool at 11:00am on a specific day. The store needs to fulfill all these orders on this day. (a) (8 points) How long will the order processing operations have to work on this day? How long will the order packing operations have to work on this day? How long will the order shipping operations have to work on this day? What is the maximum number of orders waiting to be packed (i.e. at storage #1) at any moment? What is the maximum number of orders waiting to be shipped (i.e. at storage #2) at any moment? (b) (4 points) On average, how long does an order have to wait at storage #1? Similarly, on average, how long does an order have to wait at storage #2? (c) (8 points) Now suppose that storage #2 has a limited capacity and can only hold 600 orders whereas storage #1 still has unlimited capacity. When storage #2 is full, the order packing step has to slow down to the same processing rate as the order shipping step. When will storage #2 be full? How long will the order packing operations have to work on this day? Problem B: (20 points) The local Department of Motor Vehicles issues new licenses and renews licenses. (See the diagram below.) The office receives 100 customers per hour. All customers first see a receptionist. The receptionist directs them in one of three directions: (i) 75% go directly to Issue License (staffed by 9 workers) where a new photo and license are done; (ii) 15% are required to take an eye test (staffed by 1 worker); and (iii) 10% must first take a multiple-choice written test (on one of five computers). Only 80% of people pass the eye test and the remaining 20% exit. The customers who pass the eye test proceed on to the Written Test. 10% of the people who take the written test fail it, while 90% pass the test and then proceed to Issue License. 75% 15% 20%, Issue Reception Eye Test Exit Exit License 80% 10% Written Test 90% Exit 10% Data on each station are provided in the following table: Workers Activity time per customer (minutes) 0.4 (a) (20 points) Reception 1 Eye Test 1 5 Written Test 5 15 Issue License 9 6 What is the implied utilization of each step? Reception: Eye Test: Written Test: Issue License: (b) (5 points) Which step is the bottleneck? What is the capacity of this process? 4 Problem C: (18 points) Customers arrive at Best-Bank-in-Town's (BBT) sole ATM location at a rate of 21 customers per hour. It is assumed that the arrival process is random. BBT recently hired an intern who estimated that the average service time is 2 minutes per customer with a standard deviation of 1.2 minutes. [Assume that the arrival rate is the same throughout the day without peak and off-peak considerations and that there is only one ATM machine at that location] (a) (6 pts) Calculate the average waiting time for an ATM user at that BBT location (b) (3 pts) Calculate the average length of the queue at that ATM location (i.e average number of customers waiting for service) (c) (3 pts) What is the probability that an arriving customer has no waiting time to use the ATM machine? [Hint: think utilization] (d) (6 pts) BBT wants to cut the average waiting time in half without necessarily adding another ATM machine. They are in discussions with their ATM software provider who has told them that their new software is very consistent with information retrieval and can greatly minimize the standard deviation of the ATM machine. Assume then that the std dev for service time can be brought down to zero but that the mean service time is unchanged. Will BBT achieve its desired objective of cutting the average waiting time in half? Problem D: (9 points) Clips, a competitor of Staples, is a national chain that provides office supplies to small businesses. Maya, who is the store manager at the Bethesda store, is also responsible for the store's replenishment functions. Maya is interested in establishing a standard replenishment policy for some of her steady, non-seasonal products of which paperclips is one. Paperclips sell in small boxes that contain two hundred units. Maya estimates that Clips-Bethesda sells 100 boxes of paperclips in a typical week and that the purchase price for Clips per box of paperclips is $2.00. The fixed ordering cost any time an order is placed is $20. Maya also estimates a nominal holding cost of $0.10 per box per week. (a) (5 points) What is the optimal order quantity for paperclips at their Bethesda store? (b) (4 points) Assuming 50 weeks in a year, how many times a year should the paperclips be ordered?