运营管理第14版蔡斯ipptchap010

WAITING LINE ANALYSIS AND SIMULATION Chapter Ten Copyright 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin Learning Objectives LO101: Understand what a waiting line problem is. LO102: Analyze waiting line problems. LO103: Analyze complex waiting lines using simulation. 10-2 Economics of the Waiting Line Problem A central problem in many service settings is the management of waiting time. Reducing waiting time costs money. When people waiting are employees, it is easy to value their time. When people waiting are customers, it is more difficult to value their time. Lost sales is one (low) value. 10-3 The Practical View of Waiting Lines 10-4 More on Waiting Lines One important variable is the number of arrivals over the hours that the service system is open. Customers demand varying amounts of service, often exceeding normal capacity. We can control arrivals. Short lines Specific hours for specific customers Specials We can affect service time by using faster or slower servers. 10-5 The Queuing System 1.Source population and the way customers arrive at the system 2.The servicing system 3.The condition of the customers exiting the system Do they go back to source population or not? 10-6 Components of the Queuing System Visually Customers come Customers come inin Customers are Customers are servedserved Customers leaveCustomers leave 10-7 Customer Arrivals Finite population: limited-size customer pool that will use the service and, at times, form a line When a customer leaves his/her position as a member of the population, the size of the user group is reduced by one. Infinite population: population large enough so that the population size caused by subtractions or additions to the population does not significantly affect the system probabilities 10-8 Distribution of Arrivals Arrival rate: the number of units arriving per period Constant arrival distribution: periodic, with exactly the same time between successive arrivals Variable (random) arrival distributions: arrival probabilities described statistically Exponential distribution Poisson distribution 10-9 Distributions Exponential distribution: when arrivals at a service facility occur in a purely random fashion The probability function is f(t) = e-t Poisson distribution: where one is interested in the number of arrivals during some time period T The probability function is 10- 10 Customer Arrivals in Queues 10-11 Other Arrival Characteristics Arrival patterns Size of arrival units Degree of patience Balking Reneging 10- 12 The Queuing System Length Infinite potential length Limited line capacity Number of lines Queue discipline: a priority rule or set of rules for determining the order of service to customers in a waiting line 10- 13 Service Time Distribution Constant Service provided by automation Variable Service provided by humans Described using exponential distribution 10- 14 Line Structure 10-15 Exiting the Queuing System 10-16 Properties of Some Specific Waiting Line Models 10-17 Notation for Equations 10-18 Equations for Solving Three Model Problems 10-19 Example 10.1: Customers in Line Western National Bank is considering opening a drive-through window for customer service. Management estimates that customers will arrive at the rate of 15 per hour. The teller who will staff the window can service customers at the rate of one every three minutes. Part 1 Assuming Poisson arrivals and exponential service, find 1.Utilization of the teller 2.Average number in line 3.Average number in system 4.Average waiting time in line 5.Average waiting time in system, including service 10- 20 Example 10.1: Solution 10-21 Example 10.1: No More Than Three Cars 10-22 Example 10.1: 95 Percent Service Level 10-23 Example 10.2: Equipment Selection The Robot Company franchises combination gas and car wash stations throughout the United States. Robot gives a free car wash for a gasoline fill-up or, for a wash alone, charges $0.50. Past experience shows that the number of customers who have car washes following fill-ups is about the same as for a wash alone. The average profit on a gasoline fill-up is about $0.70, and the cost of the car wash to Robot is $0.10. Robot stays open 14 hours per day. Robot has three power units and drive assemblies, and a franchisee must select the unit preferred. Free with fill-up 50 for wash alone 50/50 mixture Average profit on fill-up is 70 Wash costs 10 Three units under consideration 1.Unit I washes one per 5 minutes and costs $12 per day 2.Unit II washes one per 4 minutes and costs $16 per day 3.Unit III washes one per 3 minutes and costs $22 per day 10- 24 Example 10.2: Basic Calculations 10-25 Example 10.2: Profits 10-26 Example 10.3: Determining the Number of Servers In the service department of the Glenn-Mark Auto Agency, mechanics requiring parts for auto repair or service present their request forms at the parts department counter. The parts clerk fills a request while the mechanic waits. Mechanics arrive in a random (Poisson) fashion at the rate of 40 per hour, and a clerk can fill requests at the rate of 20 per hour (exponential). If the cost for a parts clerk is $6 per hour and the cost for a mechanic is $12 per hour, determine the optimum number of clerks to staff the counter. (Because of the high arrival rate, an infinite source may be assumed.) Arrivals of 40 per hour Service at 20 per hour Clerk costs $30 per hour Mechanic costs $60 per hour 10- 27 Example 10.3: Solution 10-28 Example 10.4: Finite Population Source Studies of a bank of four weaving machines at the Loose Knit textile mill have shown that, on average, each machine needs adjusting every hour and that the current servicer averages 7.5 minutes per adjustment. Assuming Poisson arrivals, exponential service, and a machine idle time cost of $40 per hour, determine if a second servicer (who also averages 7.5 minutes per adjustment) should be hired at a rate of $7 per hour. Each machine needs adjusting every hour. Service averages 7.5 minutes per adjustment. Poisson arrival and exponential service. Machine idle cost is $40 per hour. Should we hire a second service repairer? Costs $7 per hour 10- 29 Example 10.4: Terms N = Number of machines in the population S = Number of repairers T = Time required to service a machine U = Average time a machine runs before requiring service X = Service factor L = Average number of machines waiting H = Average number of machines being serviced D = Probability machine needing service will wait F = Efficiency factor 10- 30 Example 10.4: Case 1One Repairer N = 4 S = 1 T = 7.5 U = 60 10- 31 Example 10.4: Finding F for Case 1 10-32 Example 10.4: Case 1 Cost L = N(1 - F) = 4(1 - 0.957) = 0.172 machine H = NFX = 0.957(4)(0.111) = 0.425 machine Number of machines down L + H = 0.172 + 0.425 = 0.597 Cost for machines being down L + H x $40 = 0.597 x $40 = $23.88 Total cost = machine cost + labor = $23.88 + $7.00 = $30.88 10- 33 Example 10.4: Case 2 Cost L = N(1 - F) = 4(1 - 0.998) = 0.008 machine H = NFX = 0.998(4)(0.111) = 0.443 machine Number of machines down L + H = 0.008 + 0.443 = 0.451 Cost for machines being down L + H x $40 = 0.451 x $40 = $18.04 Total cost = machine cost + labor = $18.04 + $14.00 = $32.04 10- 34 Approximating Customer Waiting Time All you need is the mean and standard deviation to approximate waiting time. “Quick and dirty” mathematical approximation No assumption needed about a particular arrival rate or service distribution. Need data on service times. Need data on time between arrivals. Interarrival time 10- 35 Equations 10-36 Example 10.5: Waiting Line Approximation 10- 37 Bu Example 10.5: Step 1: Calculate Arrival Rate, Service Rate, and Coefficients of Variation 10-38 Example 10.5: Step 2: Calculate Server Utilization and Step 3: Calculate Wait Information 10-39 Computer Simulation of Waiting Lines Some waiting line problems are very complex. Assumed waiting lines are independent. When a services is becomes the input to the next, we can no longer use the simple formulas. This is also true for any problem where conditions do not meet the requirements of the equations. Here, computer simulation must be used. 10- 40 Simulating Waiting Lines Waiting lines that occur in series and parallel cannot be solved mathematically. Assembly lines Work centers These waiting lines are easily simulated on a computer. Will simulate a two-stage assembly line as an example. 10- 41 Example 10.6: A Two-Stage Assembly Line Consider an assembly line that makes a large product. Because the product is large, the workstations are dependent on each other. Ray cannot work faster than Bob. 10- 42 Example 10.6: Objective of the Study What is the average performance time of each worker? What is the output rate of product through this line? How much time does Bob wait for Ray? How much time does Ray wait for Bob? Would output rates increase if storage space between the workers were added? 10- 43 Example 10.6: Data Collection 10-44 Example 10.6: Simulation of Bob and RayTwo-Stage Assembly Line 10-45 Example 10.6: Results The output time averages 60 seconds per unit. Utilization of Bob is 470/530 = 88.7 percent. Utilization of Ray is 430/550 = 78.2 percent. Ignores initial startup wait Average performance time for Bob is 470/10 = 47 seconds. Average performance time for Ray is 430/10 = 43 seconds. 10- 46 Example 10.6: Spreadsheet Simulation 10-47 Example 10.6: Average Time per Unit of Output 10-48 Example 10.6: Average Time the Product Spends in the System 10-49 Example 10.6: Results of Simulating 1,200 Units with a Spreadsheet 10-50 Simulation Programs and Languages Continuous Based on mathematical equations Used for simulating continuous values for all points in time Example: The amount of time a person spends in a queue Discrete Used for simulating specific values or specific points Example: Number of people in a queue 10- 51 Types of Simulation Programs General-purpose: allows programmers to build their own models SLAM II SIMSCRIPT II.5 SIMAN GPSS/H GPSS/PC PC-MODEL RESQ Special-purpose: specially built to simulate specific applications Extend SIMFACTORY 10- 52 Desirable Features of Simulation Software 1.Be capable of being used interactively as well as allowing complete runs. 2.Be user-friendly and easy to understand. 3.Allow modules to be built and then connected . 4.Allow users to write and incorporate their own routines. 5.Have building blocks that contain built-in commands. 6.Have macro capability. 10- 53 Desirable Features of Simulation Software Continued 7.Have material-flow capability. 8.Have output standard statistics such as cycle times, utilization, and wait times. 9.Allow a variety of data analysis alternatives for both input and output data. 10.Have animation capabilities to display graphically the product flow through the system. 11.Permit interac