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Lecture 6 MGMT 650 Simulation – Chapter 13 1 Announcements HW #4 solutions and grades posted in BB HW #4 average = 111.30 Final exam today Open book, open notes…. Proposed class structure for today Lecture – 6:00 – 7:50 Class evaluations – 7:50 – 8:00 Break – 8:00 – 8:30 Final – 8:30 – 9:45 2 Lecture 6 Simulation Chapter 13 3 Simulation Is … Simulation – very broad term methods and applications to imitate or mimic real systems, usually via computer Applies in many fields and industries Simulation models complex situations Models are simple to use and understand Models can play “what if” experiments Extensive software packages available ARENA, ProModel Very popular and powerful method 4 Applications Manufacturing facility Bank operation Airport operations (passengers, security, planes, crews, baggage, overbooking) Hospital facilities (emergency room, operating room, admissions) Traffic flow in a freeway system Waiting lines - fast-food restaurant, supermarkets Emergency-response system Military 5 Example – Simulating Machine Breakdowns The manager of a machine shop is concerned about machine breakdowns. Historical data of breakdowns over the last 100 days is as follows Number of Breakdowns Frequency 0 10 1 30 2 25 3 20 4 10 5 5 Simulate breakdowns for the manager for a 10-day period 6 Simulation Procedure Number of Breakdowns 0 1 2 3 4 5 Day 1 2 3 4 5 6 7 8 9 10 Frequency 10 30 25 20 10 5 100 Random Number 90 73 82 16 94 92 68 5 84 91 Probability Cum Prob 0.10 0.10 0.30 0.40 0.25 0.65 0.20 0.85 0.10 0.95 0.05 1.00 Corresponding Random Numbers 01 to 10 11 to 40 41 to 65 61 to 85 86 to 95 96 to 00 Simulated # of Breakdowns 4 3 3 1 4 4 3 0 3 4 19 Expected number of breakdowns = 1.9 per day 7 Statistical Analysis Day # Replication 1 Replication 2 Replication 3 Replication 4 Replication 5 Replication 6 Replication 7 Replication 8Replication 9 Replication 10 1 1 4 4 1 0 2 5 1 1 0 2 3 5 0 2 0 4 3 2 0 3 3 3 1 1 1 2 1 0 1 1 5 4 1 2 2 3 2 3 1 1 1 1 5 2 0 1 1 1 5 2 2 0 2 6 0 2 3 1 3 1 4 3 2 2 7 1 1 2 2 1 2 1 1 1 0 8 3 3 2 2 0 4 2 1 3 2 9 1 1 1 2 2 1 4 0 1 4 10 5 1 3 2 3 2 1 0 1 1 2.00 2.00 1.90 1.70 1.40 2.50 2.30 1.20 1.10 2.00 95 % confidence interval for mean breakdowns for the 10-day period is given by: xt n 1;1 2 s 0.458 1.81 2.262( ) [1.665,1.955] n 10 8 Monte Carlo Simulation Monte Carlo method: Probabilistic simulation technique used when a process has a random component Identify a probability distribution Setup intervals of random numbers to match probability distribution Obtain the random numbers Interpret the results 9 Example 2 – Simulating a Reorder Policy The manager of a truck dealership wants to acquire some insight into how a proposed policy for reordering trucks might affect order frequency Under the new policy, 2 trucks will be ordered every time the inventory of trucks is 5 or lower Due to proximity between the dealership and the local office, orders can be filled overnight The “historical” probability for daily demand is as follows Demand (x) P(x) 0 0.50 1 0.40 2 0.10 Simulate a reorder policy for the dealer for the next 10 days Assume a beginning inventory of 7 trucks 10 Example 2 Solutions x 0 1 2 P(x) 0.5 0.4 0.1 Cum P(x) RN 0.5 01 to 50 0.9 51 to 90 1.0 91 to 00 Day 1 2 3 4 5 6 7 8 9 10 RN 81 20 82 34 85 35 10 14 84 92 Demand Begin Inv End Inv Reorder 1 7 6 0 0 6 6 0 1 6 5 2 0 7 7 0 1 7 6 0 0 6 6 0 0 6 6 0 0 6 6 0 1 6 5 2 2 7 5 2 11 In-class Example using MS-Excel The time between mechanics’ requests for tools in a AAMCO facility is normally distributed with a mean of 10 minutes and a standard deviation of 1 minute. The time to fill requests is also normal with a mean of 9 minutes and a standard deviation of 1 minute. Mechanics’ waiting time represents a cost of $2 per minute. Servers represent a cost of $1 per minute. Simulate arrivals for the first 9 mechanic requests and determine Service time for each request Waiting time for each request Total cost in handling all requests Assume 1 server only 12 AAMCO Solutions Inter-request time Cum Int-req time Service Time Service Begins Service Ends Wait 11.70 11.70 8.80 11.70 20.50 7.08 18.78 10.28 20.50 30.78 5.38 24.16 8.98 24.16 33.14 5.92 30.08 8.26 30.08 38.34 6.17 36.25 8.49 36.25 44.74 7.10 43.35 8.61 43.35 51.96 6.58 49.93 8.76 49.93 58.69 7.52 57.45 8.09 57.45 65.54 5.71 63.16 8.91 63.16 72.08 Sum of wait Server cost/min Waiting cos/min Total cost Time 0.00 1.72 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.72 1 2 75.52 13 Discrete Event Simulation Example 1 - A Simple Processing System 14 Discrete Event Simulation Example 2 - Electronic Assembly/Test System Produce two different sealed elect. units (A, B) Arriving parts: cast metal cases machined to accept the electronic parts Part A, Part B – separate prep areas Both go to Sealer for assembly, testing – then to Shipping (out) if OK, or else to Rework Rework – Salvaged (and Shipped), or Scrapped 15 Part A Interarrivals: expo (5) minutes From arrival point, proceed immediately to Part A Prep area Process = (machine + deburr + clean) ~ tria (1,4,8) minutes Go immediately to Sealer Process = (assemble + test) ~ tria (1,3,4) min. 91% pass, go to Shipped; Else go to Rework Rework: (re-process + testing) ~ expo (45) 80% pass, go to Salvaged; Else go to Scrapped 16 Part B Interarrivals: batches of 4, expo (30) min. Upon arrival, batch separates into 4 individual parts From arrival point, proceed immediately to Part B Prep area Process = (machine + deburr +clean) ~ tria (3,5,10) Go to Sealer Process = (assemble + test) ~ weib (2.5, 5.3) min., different from Part A, though at same station 91% pass, go to Shipped; Else go to Rework Rework: (re-process + test) = expo (45) min. 80% pass, go to Salvaged; Else go to Scrapped 17 Run Conditions, Output Start empty & idle, run for four 8-hour shifts (1,920 minutes) Collect statistics for each work area on Resource utilization Number in queue Time in queue For each exit point (Shipped, Salvaged, Scrapped), collect total time in system (a.k.a. cycle time) 18 Simulation Models Are Beneficial Systematic approach to problem solving Increase understanding of the problem Enable “what if” questions Specific objectives Power of mathematics and statistics Standardized format Require users to organize 19 Simulation Process 1. Identify the problem 2. Develop the simulation model 3. Test the model 4. Develop the experiments 5. Run the simulation and evaluate results 6. Repeat 4 and 5 until results are satisfactory 20 Different Kinds of Simulation Static vs. Dynamic Continuous-change vs. Discrete-change Can the “state” change continuously or only at discrete points in time? Deterministic vs. Stochastic Does time have a role in the model? Is everything for sure or is there uncertainty? Most operational models: Dynamic, Discrete-change, Stochastic 21 Advantages of Simulation Solves problems that are difficult or impossible to solve mathematically Flexibility to model things as they are (even if messy and complicated) Allows experimentation without risk to actual system Ability to model long-term effects Serves as training tool for decision makers 22 Limitations of Simulation Does not produce optimum solution Model development may be difficult Computer run time may be substantial Monte Carlo simulation only applicable to random systems 23 Fitting Probability Distributions to Existing Data Data Summary Number of Data Points Min Data Value Max Data Value Sample Mean Sample Std Dev = 187 = 3.2 = 12.6 = 6.33 = 1.51 Histogram Summary Histogram Range Number of Intervals = 3 to 13 = 13 24 ARENA – Input Analyzer Distribution Summary Distribution:Gamma Expression: 3 + GAMM(0.775, 4.29) Square Error:0.003873 Chi Square Test Number of intervals =7 Degrees of freedom =4 Test Statistic = 4.68 Corresponding p-value = 0.337 Kolmogorov-Smirnov Test Test Statistic = 0.0727 Corresponding p-value > 0.15 Data Summary Number of Data Points Min Data Value Max Data Value Sample Mean Sample Std Dev = 187 = 3.2 = 12.6 = 6.33 = 1.51 Histogram Summary Histogram Range Number of Intervals = 3 to 13 = 13 25 Simulation in Industry 26 Course Conclusions Recognize that not every tool is the best fit for every problem Pay attention to variability Forecasting Inventory management - Deliveries from suppliers Build flexibility into models Pay careful attention to technology Opportunities Improvement in service and response times Risks Costs involved Difficult to integrate Need for periodic updates Requires training Garbage in, garbage out Results and recommendations you present are only as reliable as the model and its inputs Most decisions involve tradeoffs Not a good idea to make decisions to the exclusion of known information 27