MODULE 7 â&#x20AC;&#x201C; SIMULATION PROBLEMS 7.1

Good-Day Tires Company has a workshop at the centre of a town. Besides selling and fitting tires, the workshop provides wheel alignment and balancing services for customers. Inside the workshop Good-Day has one service bay with all the necessary equipment and space to park one additional car. There is further parking outside which will accommodate only one car. Parking are prohibited on the access road, so that any customer who arrives when the service bay is in used and there are two cars parked, cannot stop and is lost to the workshop. Customers arrive at the workshop with the following inter-arrival times, based on 100 observations. Inter-arrival time (min.) : 5 10 15 20 25 30 Probability : 0.14 0.12 0.16 0.29 0.18 0.11 The service times taken to examine and replace tires and provide the necessary services have the following distribution: Service Time (min.) : 10 15 20 25 30 Probability : 0.16 0.23 0.20 0.27 0.14 (a) Use the following random numbers and the information provided; simulate the first 15 customers arriving at the workshop after the workshop opens at 9.00 am. Inter-arrival Times : 65 13 29 37 81 54 36 28 22 83 08 12 15 99 46 75 Service Times : 17 94 76 79 98 01 42 52 34 07 98 33 93 27 (b) From your simulation, estimate (i) the total time to serve the 15 customers, (ii) the proportion of time the service bay is idle, (iii) the proportion of the time there are one or two customers waiting, (iv) the proportion of customers who cannot wait because of insufficient parking space.

7.2

A barber runs a one-man shop. He does not make appointments but attends the customers on a first-come first-served basis. He can deal with one customer at a time but he has two chairs for other customers to wait. However, it is noticed that when customers arrive and the two chairs are occupied, virtually no one waits but goes to another shop a few blocks away. Based on 100 observations, his customers arrive at the shop with the following inter-arrival pattern: Inter-arrival time (minutes) : 5 10 15 20 25 30 Frequency : 12 15 30 20 13 10 His service times for a sample of 100 customers have the following distribution: Service time (minutes) : 10 15 20 25 30 Frequency : 15 25 32 15 13 (a) Using the random digits given below, simulate the arrival of 15 customers at the barbershop. Assume that the simulation clock begins at 9.00 a.m. Random Numbers Inter-arrival times : 09 06 51 62 83 61 59 20 82 68 69 33 30 27 44 Service times : 39 60 50 31 02 04 83 90 71 16 72 19 88 19 66 (b) Use your simulation to estimate (i) the average number of customers in the shop at any given time; (ii) the expected time a customer must wait for service; (iii) the proportion of customers who refuse to wait.

7.3

A retailer sells an item for which the weekly demand can be described by the following probability distribution. Demand per week Probability 30 0.03 35 0.04 40 0.09 45 0.15 50 0.38 55 0.15 60 0.09 65 0.04 70 0.03 There is an initial stock of 200 units of the item and the retailer orders in batches of 250 units whenever stock level falls below 100 units. Past experience indicates that the lead-time varies as follows: Lead Time (weeks) : 1 2 3 Probability : 0.25 0.50 0.25 The unit cost of holding is \$0.50 per week, applied to the total stock held at each week ending. The cost of placing an order is \$50 and if there is no stock to satisfy a customer, the retailer suffers a loss of \$5 for each unit of unsatisfied demand. (a) Using the following random numbers, simulate the demand over a period of 15 weeks. You may assume that all accounting is done at the end of the week and that all ordering and deliveries occur at the beginning of the week. Demand: 87 55 81 22 04 62 21 45 81 82 43 96 17 70 61 Lead time: 53 17 73 61 99 (b) Use your simulation to estimate (i) Mean stock level at the end of each week. (ii) Mean shortage per week. (iii) Mean cost per week of the above policy. (c) Describe how the simulation can be extended to trying to determine a policy of stock holding which keeps the mean cost per week to a minimum.

7.4

The inter-arrival time (in weeks) of orders (from customers) for an item is assumed to follow the probability distribution shown in the first table. The probability distribution of the size of order (in units) is shown in the second table. The initial inventory level of the item is 200 units. Time between orders (weeks) Probability Size of order (units) Probability 1 0.30 10 0.20 2 0.25 20 0.30 3 0.25 30 0.35 4 0.20 40 0.15 Orders are processed on a FIFO basis (first in first out). Those that cannot be filled immediately are usually backlogged. Every five weeks (end of the week) the stock is reviewed. If the inventory position (on hand plus on order minus backlogged) drops below 100 units, a replenishment order that brings the inventory position back to 200 units is placed. Delivery of the stock takes place a week later (arrives at the end of the week). Using the following random numbers, simulate the process for 10 orders. Time between orders : 52 37 82 69 98 96 33 50 88 90 Size of order : 88 02 28 49 36 87 21 95 50 24

Determine (a) the average safety stock level at the time a replenishment takes place; (b) the average stock replenishment size; and (c) the average inventory position. 7.5

A consumer association is keen to know the reactions of customers on a Bill on Consumersâ&#x20AC;&#x2122; Right which is to be tabled in the Parliament. It plans to interview passers-by at a strategic spot at the town centre. Only one interviewer will be stationed at that point to approach passers-by for interview. However, no interview will take places unless the passer-by has consented to the interview. Any one, who passes by when the interviewer is busy, is lost. A pilot study of 100 passers-by has generated the following information: Time between arrivals at the interviewer (minutes): 0 1 2 3 4 5 Frequency: 25 35 18 10 8 4 Of the 100 people who were asked, 75 were willing to be interviewed. Interview time (minutes) 1 to 3 3 to 5 5 to 7 Number of interviews 40 45 15 Suppose the following random numbers were extracted: Time between arrivals: 03 47 43 73 86 97 74 24 67 62 16 76 62 27 66 12 56 85 99 Willingness to be interviewed: 64 16 22 77 94 39 84 42 17 53 31 63 Interview times: 57 29 34 12 21 33 59 78 63 01 56 59 55 (a) Using the random numbers given, perform a simulation to obtain 10 interviews. (b) From the result of your simulation, (i) estimate how long it takes to complete 10 interviews; (ii) find what proportion of the time the interviewer is idle; (iii) find how many people passes by before 10 interviews were obtained.

7.6

A company dealing in factory equipment wants to determine the levels of inventory it should carry for the items in its range. The demand and the lead-time for stock replenishment are not certain and follow the following probability distributions. Demand (units/day) Probability Lead-time (days) Probability 8 0.14 1 0.12 9 0.22 3 0.68 10 0.35 4 0.20 11 0.19 12 0.10 The companyâ&#x20AC;&#x2122;s inventory policy is to order 35 units from a supplier when the present inventory plus any outstanding order falls below 20 units. Order is made at the end of the day when the inventory level falls below 20 units. If the lead-time is one day, it means that the order does not arrive the next morning, but at the beginning of the following day. Each order costs \$50.00; carrying cost is \$2.00 per unit per day based on the closing stock at the end of the day. The company estimates that it will cost \$1.00 per unit if shortages occur. (a) Carry out a simulation run over a period of 12 days with the objective of obtaining the total inventory cost for each day, assuming the starting inventory is 30 units. Use the following random numbers; Demand: 91 66 53 37 88 99 50 47 06 10 52 30 Lead time: 37 63 02 14 35 24 (b) Hence give your comment and suggestion about the above inventory system.

7.7

The time between arrivals of customers at Alice restaurant is given by the following probability distribution:  x / 50, 0  x  10 f ( x)   otherwise  0, Using the random numbers 50, 32, 78, 20 and 10, simulate the time between arrivals for five customers and determine the average time between arrivals.