Monte Carlo Simulation

Transcription

1 Monte Carlo Simulation The fish tank modeling problem based on Math 381 course notes University of Washington presentation by Tim Chartier and Sara Billey

2 Modeling the Stinky Fish Store The Stinky Fish Store has the following dilemma sells 150 gallon fish tanks sells about 1 tank each week the tanks are a high profit item 5 days are needed for a new one to be delivered Goal: Find a strategy for ordering new fishtanks which maximizes profit.

3 Modeling the Stinky Fish Store Strategies for maximizing profits Strategy 1: Order a new tank each time one is sold Strategy 2: Order a new tank once a week. Question: Which is a better strategy?

4 Modeling the Stinky Fish Store Additional information is needed to evaluate such a question. 1. Profit of fish tank vs. other items in the store 2. Cost of having additional tanks in stock 3. What does approx. 1 sale/week mean?

5 Modeling the Stinky Fish Store Simplifying assumptions Assume uniform sales over the year. Take approximately 1 sale/week to mean there is a 1/7 probability of a customer on a given day. Notice, we are introducing probability into our simulation!

6 Simulation and Maple We will simulate the competing strategies on a computer using R. The command runif(1) returns a number Uniformly distributed between 0 and 1 We will take < 1/7 to imply a customer comes We will take > 1/7 to imply no customer Code on p. 25 simulates Strategy 1 (order a tank only after one is purchased) Let us review this code.

7 Modeling the Stinky Fish Store General rule of modeling: Best to start with a simple model and refine it incrementally. Starting with a completely realistic model is usually hopeless.

8 First pass at R program a <- 1/3; # probability of a customer each day days_for_delivery <- 2; # days from order # to delivery of new tanks stock <- 1; # number of tanks in stock deliv <- -1; # number of days until order delivered # -1 means none on order

9 Program Output Sample Simulation week day stock cust sold lost Summary of Data total over entire simulation: customers sold lost ect.

10 Sample Simulation Remember we are using a random number generator! customers sold lost We also need a larger sample size. We will run the problem for many more weeks!

11 Modeling the Stinky Fish Store General rule of computing: Check every step of your program on small data sets before running big computations.

12 Accumulating information Run simulation several times, each for 104 weeks = 2 yrs Results of 10 different simulations, each for 104 weeks: customers sold lost fraction served expect 104*7/3 = 243 customers (due to rate of 1 every 3 days) 60% served, might be useful in comparison (can be predicted using the theory of Markov chains)

13 A more robust program The program fishtank.advanced.r is more robust allowing comparison of strategies. It contains 2 variables order_when_out - determines whether to order a new tank whenever we run out. fixed_delivery - determines whether there is a fixed schedule of deliveries, once ever so many days. Strategy 1 corresponds to: order_when_out := 1; fixed_delivery := 0; # no standing order Strategy 2 corresponds to: order_when_out := 0; fixed_delivery := 7; # arrival every 7 th day The program allows comparison of strategies. What do we compare?

14 Comparing Strategies Percentage of customers served is one important number to compare. More importantly, profit over time! This depends not only on how many customers we serve or lose, but also on the cost of any overstock. Big difference between methods!

15 Comparing Strategies Run simulation several times, each for 104 weeks = 2 yrs Results of 100 different simulations, each for 104 weeks. For the 104 simulations profit is computed for each simulation and the average profit over 100 simulations with each strategy. This profit is based on the assumed profit per sale, cost of losing a customer, and cost per night of overstock as illustrated in the program.

16 Comparing Strategies Strategy 1: customers sold lost fraction_served overstock end_stock profit etc average_profit = Strategy 2: customers sold lost fraction_served overstock end_stock profit etc average_profit =

17 Comparing Strategies Interesting features of simulations: Percentage of customers served is generally larger with Strategy 2, (>90%), Strategy 1 (~60\%) Average profit is larger with Strategy 1 due to overstock in Strategy 1 The results for Strategy 1 have much smaller variance than for Strategy 2. This is seen more strikingly by plotting histograms of the profits over each set of 100 simulations!

18 Comparing Strategies Histograms of profit observed over 100 simulations with two different strategies. Strategy 2 takes great risk at gains and losses!

19 Comparing Strategies Strategy 2 as implemented has no mechanism to stop automatic delivery if the stock goes too high. Customers arrive at the same average rate as new tanks. So, a high stock will tend to stay high. By the end of two years the stock has grown to 20 tanks. History of number of tanks in stock each day for one particular simulation with Strategy 2.

20 Statistical Analysis Statistics can aid in comparing the methods. Sample mean Sample variance Standard deviation y 1 N N i 1 y i s 2 1 N 1 N i 1 ( y i y) 2 s 2 s With a large enough sample, we can gain information about the underlying distribution. (How big is big enough depends on the distribution.)

21 Statistical Analysis N = 5,000 Histograms of profit observed over 5,000 simulations with different strategies Strategy 1 y 783 s 83 Strategy 2 y 439 s 1020

22 Hybrid Strategy standing order for a tank to arrive every N days, (N > 7) But, order a new tank whenever we run out. In the program, N is denoted by fixed_delivery. Histograms of profit observed over 100 simulations with hybrid strategies

23 Try It! Recommended Problem:Find the best hybrid strategy which maximizes profit by adjusting the parameters: order.when.out <- 1; # = 1 ==> order a new tank when stock==0 # = 0 ==> don't order when out of tanks fixed.delivery <- 12; # >0 ==> standing order for a new tank # every so many days

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