Q 1 Furgon Van Hire rents out trucks and vans. One service they offer is a sameday rental deal under which account customers can call in the morning to hire a van for the day. Five vehicles are available for hire on these terms. The demand for the service varies according to the following distribution: Simulate the demand for this service over a period of ten days and using your results work out the average number of disappointed customers per day. Use random numbers given below 0.46 0.84 0.43 0.18 0.61 0.10 0.11 0.37 0.07 0.77 Use the following headings day Demand Unsatisfied customers Q 2 Orders for the Potchtar Mail Order Company are sent in batches of 50 by the Post Office. The number of orders arriving each working day varies according to the following probability distribution: These orders are opened and keyed into the company s system by a data entry assistant who, following the departure of a colleague, will handle this task by himself. The rate per day at which he processes orders varies as follows:
Any orders that are not processed by the end of one day are held over and dealt with first on the following day. (a) Simulate the operation of this system for ten working days using two streams of random numbers given below. From your results work out the average number of orders held over per day. for orders for order processed 0.25 0.02 0.19 0.21 0.97 0.62 0.03 0.4 0.28 0.52 0.01 0.52 0.26 0.76 0.16 0.23 0.34 0.17 0.22 0.02 Use the following headings Days orders processed heldover (b) A new assistant may be appointed to help out. If this happened the distribution of processing s is expected to be: Simulate the operation of the new arrangement for ten days using the same sets of random numbers you used in part (a). Work out the average number of orders held over per day and compare your result to those you obtained for part (a). Q 3 A restaurant serves fresh locally grown strawberries as part of its Wimbledon working lunch menu for ten working days in the early summer. The supplier provides 100 kg per day and sells to the hotel at a price that varies according to the probability distribution: The number of portions sold per day at 3 per 250 g per portion varies according to the probability distribution:
Any strawberries not sold during the lunch are sold to a jammaker the same day for 0.50 per kilogram. Using simulation, estimate the profit that the restaurant can expect to make over the ten days using the random numbers below. for prices for orders 0.77 0.34 0.97 0.09 0.33 0.18 0.52 0.17 0.62 0.34 0.74 0.11 0.22 0.32 0.88 0.78 0.53 0.52 0.91 0.07 Use the following heading days prices Cost of ordering 100 orders order* $3 revenue Total gram ordered =100*1000 total grams used( portions * 250 unused unused kg profit form unused total revenue total cost profit Q 4 A book trader has taken out a 13week lease on a small unit in a shopping mall prior to it being fitted out as a new outlet by a major retailer. The trader intends to sell remaindered books, coffee table classics selling for 3 each and children s books selling at 1 each. She has a stock consisting of 2000 of the former and 5000 of the latter. The demand for the books per week is expected to vary as follows: Simulate the sales and use them to estimate the total revenue over the 13 weeks the shop will be open using the random numbers below.] for 3.00 for 1.00
0.45 0.14 0.19 0.26 0.83 0.71 0.72 0.56 0.21 0.76 0.31 0.51 0.13 0 0.13 0.42 0.98 0.95 0.52 0.37 0.69 0.25 0.96 0.73 0.85 0.74 Use the following headings weeks O/ stock demand $3. closing stock revenue O/stock demand for 1.00 C/stock revenue Q 5 A builder completes a large project for a property company who, due to cash flow problems, offer him a small seaside hotel in lieu of payment. The builder, who has no desire to enter the hotel business, has found a buyer for the hotel, but this deal will not be completed until 12 weeks after the hotel is due to open for the summer season. The builder has decided to operate the hotel himself for these 12 weeks. To keep this simple he has decided to accept only oneweek bookings. There are 8 double rooms and 3 single rooms in the hotel. The cost of a double room for one week will be 300, and the cost of a single room for one week, 200. The numbers of rooms of each type booked per week varies according to the following distributions: Simulate the operation of the hotel for these 12 weeks and work out the total revenue. for double rm for single rm
0.22 0.82 0.13 0.69 0.88 0.52 0.13 0.55 0.75 0.31 0.75 0.87 0.42 0.25 0.69 0.89 0.32 0.07 0.15 0.97 0.8 0.93 0.02 0.59 headings weeks D/room demand revenue S/room demand revenue Q6 a) The manager of a machine shop is concerned about machine breakdowns and wants to simulate breakdowns for a 10 day period Historical breakdowns over last 100 days are given in following table No of breakdowns Q 6b) The number of jobs received by a small shop is to e simulated for an 8 day period the manager has collected the following data: no of jobs frequency 2 0 3 10 4 50 5 80 6 40 7 16 8 4 9+ 0 200 100
a. Simulate the number of 8 day periods starting the mapping with 00 read 2 digit from table below at column 3 and reading down Compute the average number of machines that will break down per week. Q 7 Jack M sells insurance on a part basis. His records on the number of policies sold per week over 50 week period are as follows no of jobs frequency 0 8 1 15 2 17 3 7 4 3 50 a. Simulate three 5 day periods. Use table A column 6 for first simulation, column 7 for 2nd simulation and column 8 for 3rd. In each case read 2 digit number beginning at bottom and going up. determine the percentage of days on which 2 or more polices are sold. 100 Q 8 The manager at the Roseau credit union is attempting to determine how many tellers are needed at the drive in window during peak s. As a general policy, the manager wishes to offer service such that average customer waiting does not exceed 2 minutes. Given the existing service level as shown in the data below does the drive in window meet the criteria.?
between arrivals prob service prob 0.00 0.10 0.00 0.00 1 0.35 1 0.25 2 0.25 2 0.2 3 0.15 3 0.4 4 0.1 4 0.15 5 0.05 Service 0.52 0.37 0.82 0.69 0.98 0.96 0.33 0.50 0.88 0.90 Time between successive customer arrivals 0.50 0.28 0.68 0.36 0.90 0.62 0.27 0.50 0.18 0.36 Construct the appropriate random number mappings for the random variables. (3 marks)
Simulate 10 customers arriving for service using the random numbers given below. (12 marks) What is the average a customer waits for service? (1 mark) What is the average a customer is in the system (wait plus service ) (2 marks) What is the percent of the server is busy with customers? (2 marks) Use the following headings customer TBA Arrival Begin service wait for service random service end service in system Q 9 Resimulate the number of stockouts incurred over a 20week period (assuming Carl maintains a constant supply of 8 heaters). Use the following random numbers: 10,24,03,32,23,59,95,34,51,08,48,66,97,03,96,46, 74, 77,44,30 What is the new expected number of sales per week? week random number simulated demand Q 10 The Dynaco Manufacturing Company produces a product in a process consisting of operations of five machines. The probability distribution of the number of machines that will break down in a week follows: Machine Breakdowns per Week Probability 0.10 1.10 2.20 3.25
Machine Breakdowns per Week Probability 4.30 5.05 1.00 a. Simulate the machine breakdowns per week for 20 weeks. b. Compute the average number of machines that will break down per week Use following headings weeks breakdowns Use the following random numbers for breakdown 39 73 72 75 37 2 87 98 10 47 93 21 95 97 69 41 91 80 67 59 Q 11 Every a machine breaks down at the Dynaco Manufacturing Company above, either 1, 2, or 3 hours are required to fix it, according to the following probability distribution: Repair Time (hr.) Probability 1.30 2.50 3.20 1.00
a. Simulate the repair for 20 weeks and then compute the average weekly repair. b. If the random numbers that are used to simulate breakdowns per week are also used to simulate repair per breakdown, will the results be affected in any way? Explain. c. If it costs $50 per hour to repair a machine when it breaks down (including lost productivity), determine the average weekly breakdown cost. d. The Dynaco Company is considering a preventive maintenance program that would alter the probabilities of machine breakdowns per week as shown in the following table: Machine Breakdowns per Week Probability 0.20 1.30 2.20 3.15 4.10 5.05 1.00 The weekly cost of the preventive maintenance program is $150. Using simulation, determine whether the company should institute the preventive maintenance program Use the following heading for part a and b weeks repair breakdowns Total repair repair total repair Use the following random numbers for breakdown and repair respectively 39 73 72 75 37 2 87 98 10 47 93 21 95 97 69 41 91 80 67 59 65 71 18 12 17 48 89 18 83 8 90 5 89 18 8 26 47 94 6 72
Use the following heading for part d weeks breakdowns repair Total Repair Q 12 First American Bank is trying to determine whether it should install one or two drivethrough teller windows. The following probability distributions for arrival intervals and service s have been developed from historical data: Time Between Automobile Arrivals (TBA) (min.) Probability 1.20 2.60 3.10 4.10 1.00 Service Time (min.) Probability 2.10 3.40 4.20 5.20 6.10 1.00 Construct the appropriate random number mappings for the random variables.
Simulate 10 customers arriving for service using the random numbers given below. ( What is the average a customer waits for service? What is the average a customer is in the system (wait plus service ) (2 marks) What is the percent of the server is busy with customers? (2 marks) for for TBA service 0.65 0.19 0.70 0.12 0.92 0.99 0.41 0.64 0.45 0.44 0.43 0.85 0.14 0.32 0.19 0.45 0.73 0.61 0.57 0.28 Use the following headings customer TBA Arrival Begin service wait for service random service end service in system Q 13 Video Works is a retail establishment that sells DVD players to its customers. Video Works orders 15 DVD players (Q) from their supplier when their inventory reaches 5 units (R). Daily demand for DVD players varies according to the probability distribution shown below. There is no lead since orders made are received the next day in for sale. The cost to hold one unit in inventory for one day is $0.50. The cost to place an order to replenish their inventory is $100. Stockout costs per unit are $40. Initial inventory is 15 units. Simulate this inventory policy for 10 days and calculate average daily costs Daily Demand (units) Probability 0 0.05 1 0.05 2 0.2
3 0.3 4 0.3 5 0.1 Construct the appropriate random number mapping for the random variable. Simulate 10 days by using the random numbers given below. What is the average daily cost to manage this inventory policy?= 250/10=25 Random Numbers To Be Used in the Simulation Day 1 2 3 4 5 6 7 8 9 10 Demand 0.57 0.59 0.8 0.2 0.82 0.9 0.02 0.11 0.81 0.84 Use the following headings Day OI U R A I D D F EI SO P.O IC SOC OC Day, open inventory (OI), units received (UR), Available inventory (AI),, demand (D), demand filled(df), ending inventory (EI), stock outs (SO), place order(po), inventory cost(ic), stock out cost (SOC), order cost (OC) Simulation using Theoretical distributions Q 14 The number of customers who arrive at a transmission repair shop can be described by a poisson distribution that has a mean of 3 per hour. Assuming the distribution hold for an entire 8 hour day, simulate customer arrivals for the first 4 hours of a day. Read random numbers from the table B columns 4 and 5 going down. Obtain cumulative probabilities from table C
Q15 Jobs arrive at a workstation at fixed intervals of one hour. Processing is approximately normal and has a mean of 56 minutes per job and a standard deviation of 4 minutes per job. Using the 5 th row of the table A of normally distributed random numbers simulate the processing s for 4 jobs and determine the amount of the operator idle and job waiting Q16 The between mechanics request for tools in large plant is distributed with a mean of 10 minutes and a standard deviation of 1 minute. The to fill requests is also normal with a mean of 9 minutes per request and a standard deviation of 1 minute. Mechanics waiting represents a cost of 2 per minute and servers represents a cost of 1 $ per minutes. simulate the arrivals for the first 9 mechanics requests and their service s and determine the mechanics waiting assuming one server. Would it be economical to add another server? From table A Use column 8 for request and column 9 for service Q17 The number of loss accidents at a logging firm can be described using a poisson distribution that has a mean of 4 accidents per month. using the last 2 columns of table B simulate the accidents for 12 month period Q18 The a physician spend with patients can be modeled using a normal distribution that has a mean of 20 minutes and a standard deviation of 2 minutes. using the table A of normally distributed random numbers, simulate the s doctors might spend with next 7 patients ( use column 4 start at bottom of column and read up\ Q 19 Jobs are delivered to a workstation at random intervals. The between arrivals tend to be normally distributed with a mean of 15 minutes and a standard deviation of 1 minute. Job processing is normally distributed with a mean of 14 minutes per job and a standard deviation of 2 minutes Using table A simulate the arrival and processing of 5 jobs. use column 4 of table A for job arrival and column 3 for processing. start each column at row 4 find the total s jobs wait for processing. The company is considering the use of new equipment that would result in processing that is normal with a mean of 13 minutes and a standard deviation of 1 minute. job waiting cost $3.00 per minute and the new equipment would represent ad additional cost of.50 per minute. Would the cost be justified? use the same arrival s and same random number as processing
Table A normally distributed random numbers 1 2 3 4 5 6 7 8 9 10 1 1.46 0.09 0.59 0.19 0.52 1.82 0.53 1.12 1.36 0.44 2 1.05 0.56 0.67 0.16 1.39 1.21 0.45 0.62 0.95 0.27 3 0.15 0.02 0.41 0.09 0.61 0.18 0.63 1.2 0.27 0.5 4 0.81 1.87 0.51 0.33 0.32 1.19 2.18 2.17 1.1 0.7 5 0.74 0.44 1.53 1.76 0.01 0.47 0.07 0.22 0.59 1.03 6 0.39 0.35 0.37 0.52 1.14 0.27 1.78 0.43 1.15 0.31 7 0.45 0.23 0.26 0.31 0.19 0.03 0.92 0.38 0.04 0.16 8 2.4 0.38 0.15 1.04 0.76 1.12 0.37 0.71 1.11 0.25 9 0.59 0.7 0.04 0.12 1.6 0.34 0.05 0.27 0.41 0.8 10 0.06 0.83 1.6 0.28 0.28 0.15 0.73 0.13 0.75 1.49 TABLE B Random numbers 1 2 3 4 5 6 7 8 9 10 11 12 1 18 20 84 29 91 73 64 33 15 67 54 7 2 25 19 5 64 26 41 20 9 88 40 73 34 3 73 57 80 35 4 52 81 48 57 61 29 35 4 12 48 37 9 17 63 94 8 28 78 51 23 5 54 92 27 61 58 39 25 16 10 46 87 17 6 96 40 65 75 16 49 3 82 38 33 51 20 7 23 55 93 83 2 19 67 89 80 44 99 72 8 31 96 81 65 60 93 75 64 26 90 18 59 9 45 49 70 10 13 79 32 17 98 63 30 5 10 01 78 32 17 24 54 52 44 28 50 27 68 11 41 62 57 31 90 18 24 15 43 85 31 97 12 22 07 38 72 69 66 14 85 36 71 41 58
Table C