14. Simulation and Factory Physics @Risk: Harriet Hotel and Overbooking Problems Introduction to Simulation Methods of modeling uncertainty Monte Carlo simulation Managing production lines without variability Managing production lines with variability Throughput rate, flow time and inventory levels Line balancing Single machine stations vs. parallel machines

Overbooking Problem Common practice in Difference between overbooking problems and newsvendor problems

Harriet Hotel Note: HarrietHotel.xls is available on CourseInfo 100 125 30 0.95 200 Note: HarrietHotel.xls is available on CourseInfo =B2-B3  try different values =RiskBinomial(B7, B4) =MIN(B1,B8) =B8-B9 =RiskOutput() +B6*B9-B5*B10 Sheet 3: Cell A7: put a number there first and go through the rest of the sells. Actual arrivals: Put in 103 and 98 to validate the accuracy Put in RiskBinomial(B7,B4): random variable. To make sure it gives a random number, go to @Risk, Simulation, Setting, Sampling tab, select Monte Carlo under Standard Recalc (see slide 12 on page 6) Follow the instruction on the slide. First, start @Risk. Excel will start automatically. Make “Actual Arrivals" a Binomial Random Variable: Menu: Insert, Function, @RiskDistribution, RiskBinomial Make "Nightly Profit" an Output: Menu: @Risk, Model, Add Output Set Iterations & Sampling: Menu: @Risk, Simulation, Settings Run the Simulation: Menu: @Risk, Simulation, Start Analyze the Results: Results Window: Results, Results Settings or Quick Report

@Risk Simulation Settings. Menu: @Risk, Simulation, Settings Number of times the simulation is repeated for each scenario. Number of Scenarios. use 1 if you enter a single value for “Number of Reservations Accepted", or use 7 if use RiskSimTable with seven different values. “Monte Carlo” causes @Risk to show randomly generated values when you press function key F9. A "fixed seed" causes @Risk to use the same random numbers every time a run is repeated. This means that all simulations will face the same “Actual Arrivals”. You may want to choose Multiple Simulation use different seed values.

@Risk Report Settings Results Window: Results, Report Settings Specify the reports you are interested in. For example, you can put these results on an Excel spreadsheet. You may also want to choose Summary and Input Graphs. Generate the selective reposts.

Using the @Risk Simulation Add-in for Excel Open @Risk. (Excel will be opened for you.) Create your model and think about what are the Performance measures (output cells) Decision variables (under your control) Random variables (input cells) Use probability functions to represent your random variables Go to Insert | Function and select @Risk Distribution, or go to @Risk | Model | Define Distributions Identify the performance measures you wish to gather data on Go to @Risk | Add Output, or simply type in Riskoutput() You can see the list of your input and output cells by going to @Risk | Model | List of Outputs and Inputs Specify simulation settings: @Risk | Simulation | Settings Iterations: # iterations and # simulations sampling Start the simulation (@Risk | Simulation | Start) Analyze results Ask students to open @Risk. Enable macro when asked. Open Harriet.xle If @Risk does not work, it is likely that you need to do the following: Tools, macro, security, select Medium.

Selecting a Distribution (p. 550) Quantifying Uncertainty Mean and Standard Deviation Shape (skewness) Min, mostlikely, max Discrete Probability Distributions: RiskIntUniform (x,y) RiskDuniform({x1,x2,…,xn}) RiskDiscrete ({x1,x2,…,xn}, {p1,p2,…,pn}) RiskBinomial(n,p) Continuous Probability Distributions: RiskUniform(x,y) RiskNormal(m,s) RiskLogNorm(m,s) RiskTriang(min, most likely, max) You can also go to @Risk | Model | Define Distributions, which is helpful in choosing among the different probability distributions. If you will ever be a simulation practitioner, you have to think long and hard on what distribution you are going to use. That is what you tell what the world is like. It is not a trivial exercise. Many distribution has the same mean and variance. Normal distribution has a strong central tendency and you have to be sure that the demand is not uniform. Normal can also be negative. If you truncate somewhere, the whole distribution is different. Quantify uncertainty: specify 1) mean and std, 2) shape (skewness, not symmetric, have long tail many times, what distribution? E.g., demand for fashion goods, cost, time), and 3) min, most likely and max. Using data, std is harder than mean. If no histogram, experience, judgment.

Analyzing Simulation Results After the simulation runs, the Results Window will automatically open, showing summary statistics for the output cells and the input cells if you’ve chosen to collect them in the Sampling tab of Simulation Settings You can move back and forth between the results and your spreadsheet through the “Show Excel Window” button and the “Show Result Window” button (or through @Risk | Results). From the Results Window: Copy the simulation results to an Excel worksheet for further analysis and safekeeping by going to Results | Report Setting or Quick Report To generate a graph, right-click on an output (or input) cell and then choose the type of graph you want (histogram or cumulative). Right-click on any graph to change its format or to copy it into a standard Excel graph. To simulate for different values of a decision variable (One variable at a time!): Use RiskSimTable({x1,x2,…xn}); x1, x2, … xn can be cells or numbers. type n in “# of simulations” under @Risk | Simulation | Settings Reports: mean, std, percentiles

Some Tips @Risk will run all the models that are open. If you are only interested in results from one model, close all other models. @Risk can handle multiple random variables. @Risk allows formulas such as: B1*(1+RiskNormal(10,9))+B3 @Risk can handle multiple output cells Year Return 1998 =Riskuniform(0.07,0.15) 1999 =Riskuniform(0.03,0.10) Annual profit RiskOutput()+formula Service level Inventory cost

Introduction to Simulation Approaches to analyzing uncertainty: Monte Carlo simulation using computers Why important? Disadvantages Why simulation? Cheap. Cheap than a pilot and crash a plane. Quick. Advantages: (done in the first slide) can be used for virtually any problem (unlike analytical models), less costly and less time consuming than changing the real system, usefulness is proportional to computer power, able to calculate any and all performance measures, forces you to specify your view of the system (important! Interaction, what’s important, main effects) Disadvantages: Only provides estimates of the performance measures (accuracy?), doesn’t optimize (doesn’t specify what to do), need to simulate every combination of decisions separately, curse of dimensionality (# of configurations is exponential in the # of decisions), output is very precise but may be inaccurate (over confidence in results?) Examples? Beer Game, no beer and 50 weeks in an hour. Other kind of simulation: physical simulation. Talk about the examples in next slide. Example of venture capital and deal terms. Joe Bartlett, web site VCexperts.com, talked about how founders don’t understand the outcome of their venture and ended up with little. Most money goes to VC if the venture is not too successful and need second round VC. Mock interview is a simulation; Biosphere is a simulation. They put people in a totally closed environment and see if people can live in the moon. They don’t die if it doesn’t work; DOD war games (red team, blue team); Flight simulators; Computer games; Test market; Estimate value of customer loyalty at a credit card co., low initial “tester” rate; Insurance; NPD at a pharmaceutical co: multiple stages, uncertain times, costs, revenues When you have options to use mathematical models and simulation, which one do you choose? Analytical models: sometimes the problem is still tractable, and we can find the best decision, or at least expected values. (e.g., Newsvendor: different upside and downside risks; queueing theory for average wait; inventory models. Math models are more precise.). When there are several uncertain factors, mathematical analysis falls apart. You can treat them as deterministic. What-if analysis: pick one branch of a decision tree. What is the effect of changing an assumption? (unsophisticated, simple, sensitivity analysis) Best/Worst: two special “what if” cases, how unlikely are these cases (e.g., what’s your best case evening? Long term capita; markets), to what degree should your decisions be based on these unlikely cases? People are no good at what best/worst cases are. Quote someone “the worst case is always optimistic”. So use probabilistic terms to describe best and worst cases as well, although people are no good at estimating those probabilities either. Useless because they cases are not likely to happen. Scenario analysis may help. Choose one outcome from every random element. Slow. That is, follow one path and say what will happen along the string (if these happen, what will happen. Build good intuition this way.) Simulation: cheap and easier than testing the real system, exceptionally applicable, very easy to learn, compared to other quantitative tools. Key is random number generation! Monte Carlo: random number generation (dice, cards, machines), from gambling. When I get an idea, I can generate 30 years of demand. You know people don’t generate numbers randomly, few would pick 1 and 9. Culture is an issue, In US, people are more likely to generate 3 and 7. Asians don’t think 3 and 7 are lucky numbers. We will not have you do it. We will have a physical device (computer, number draw machine to decide winning lottery) using mathematical methods. Passed test for randomness. In the long run, 1/10 each.

Factory Physics Managing production lines without variability Managing production lines with variability Throughput rate, flow time and inventory levels Line balancing Single machine stations vs. parallel machines Sources of variability

punch press cuts penny blanks places a rim on the penny The Penny Fab punch press cuts penny blanks stamps Lincoln’s face places a rim on the penny cleans away any burrs A production line that makes giant one-cent pieces. The line consists of four machines in sequence. Capacity of each machine is one penny every two hours. (A balanced line with no variability.) Theoretical flow time (hours) T0 = Bottleneck rate per hour R0 = To achieve R0 , Inventory needed is: I0 =

T = Flow Time for each Penny R = Throughput Rate for System Penny Fab One 2 hrs 2 hrs 2 hrs 2 hrs T = Flow Time for each Penny R = Throughput Rate for System Critical WIP level I0

Throughput and Flow Time vs. Inventory .5 20 Throughput Rate (Jobs/hr) .4 16 Flow Time (Hours) .3 12 Throughput (Jobs/hr) Flow time (Hours) .2 8 .1 4 0 2 4 6 8 10 12 14 Inventory (jobs) To achieve the Theoretical Throughput Rate R0 = The minimum Inventory needed is I0 = 0.5 jobs per hour 4.0 jobs

Penny Fab Two 2 hr 5 hr 3 hr 10 hr Station 1 Station 2 Station 3

Penny Fab Two Station 1 Station 2 Station 3 Station 4 # of jobs R0 = T0 = _______________ I0 = ________________ Station 1 Station 2 Station 3 Station 4 # of jobs Utilization Where the inventories are? 0.8+ 2+ 4+1.2=8.

=> Extra capacity at the first station! Line Balancing 5 jobs/shift 2.5 jobs/shift on each machine => Extra capacity at the first station! Processing rate at station 1: 1 job/shift 50% of the time, 4 jobs/shift 50% of the time; avg 2.5 jobs/shift Processing rate at station 2: 2 jobs/shift 50% of the time, 8 jobs/shift 50% of the time; avg 5 jobs/shift

The expected output rate = jobs/shift Line Balancing (cont.) The expected output rate = jobs/shift

What is the capacity of a line with variability? Line Balancing (cont.) If we shut down one of the machines at station 1, the expected output rate jobs/shift. What is the capacity of a line with variability?

Penny Fab Two Throughput Rate Inventory, I 0.1 0.2 0.3 0.4 0.5 2 4 6 8 10 12 14 16 18 20 22 24 26 with variability without variability Throughput Rate, R With variability Simulation is the tool to find R(I) and T(I) !! To get close to the bottleneck rate R0 you might need a huge inventory!!

Penny Fab Two Flow Time With variability without variability 80 70 60 With variability 50 Flow Time, T T 40 30 without variability 20 10 2 4 6 8 10 12 14 16 18 20 22 24 26 Inventory, I

Penny Fab One Single Machine Stations Parallel Machines c = 1 2 hrs

Internal Benchmarking Example Large Panel Line:

Internal Benchmarking Best inventory level without variability = 126.5  33.1 = 4,187 Actual Values: • Benchmark: T = 34 days = 816 hours – Theoretical FT = 33.1 hours I = 37,400 panels – “best inv level” = 4,187 panels R = 45.8 panels/hour – Bottleneck rate = 126.5 panels/hr Conclusions: Throughput is 36% of capacity WIP is 8.9 times the “best inventory level” Flow Time is 24.6 times theoretical flow time Why?

Takeaways @Risk Simulation Spreadsheet model: performance measures, decision variables, random variables Probability functions for representing random variables, e.g., RiskNormal(m,s) Decision variables: RiskSimTable (one variable at a time!) Output cells for performance measures (Riskoutput) Simulation settings (run length, desired reports) Reports (mean, std., percentiles) Simulation Methods of modeling uncertainty Monte Carlo simulation: random number generation

Takeaways Managing production lines without variability There exists an optimal Inventory level = bottleneck rate  theoretical flow time Managing production lines with variability Throughput rate increases as inventory increases Throughput rate < bottleneck rate Unbalance Lines Single machine stations vs. parallel machines