As weve previously seen, equations describing situations often contain uncertain parameters, that is, parameters that arent necessarily a single value but instead are associated with a probability distribution function. When more than one of the variables is unknown, the outcome is difficult to visualize. A common way to overcome this difficulty is to simulate the scenario many times and count the number of times different ranges of outcomes occur. One such popular simulation is called a Monte Carlo Simulation. In this problemsolving exercise you will develop a program that will perform a Monte Carlo simulation on a simple pro?t function.

Consider the following total pro?t function: P_T = nP_V Where P_T is the total pro?t, n is the number of vehicles sold and P_V is the pro?t per vehicle.

PART A Compute 5 iterations of a Monte Carlo simulation given the following information:

n follows a uniform distribution with minimum of 1 and maximum 10 P_V follows a normal distribution with a mean of $4500 and a standard deviation of $1000 Number of bins: 10

Recall that for all practical purposes we will use 3 std. deviations from the mean as the maximum value for parameters following a normal distribution. Obviously, 5 iterations are not very many. In fact, typically you would simulate 10,000 iterations or so to view meaningful results, but I ?gured that Id give you a break. If youd like to compute 10,000 iterations by hand for extra credit, go ahead...

i.) What are the ranges for the 10 bins? ii.) Fill in the table below:

iii.) Fill in the frequency of occurrences of each bin:

PART B

Write the following three methods:

public int GetRandomUniform(int min, int max) This method returns a random number from a uniform distribution between min and max.

public double GetRandomNormal(double mean, double stddev) This method returns a random number from a normal distribution with a mean of mean and standard deviation of stddev

public int GetBinIndex(double mini, double maxi, int numbins, double valuetobin) This method returns the Bin Index given an input minimum of mini, input maximum of maxi, numbins number of bins, and a value to bin of valuetobin

PART C

Include the methods created in part B to develop a Visual C# .NET program that will simulate the basic pro?t calculation, P_T = nP_V, where n follows a uniform distribution, P_V follows a normal distribution, and the user can input the number of bins and number of iterations. The user must also input the min and max for n and the mean and standard deviation for P_V. Finally, the user can click a button and the results will be graphed on a bar chart using the Microsoft Chart Control and the average total pro?t (P_T) will be displayed in a textbox. Turn in a screen shot of the resulting chart using:

1. Iterations: 10000 Bins: 5 n-min: 1 n-max: 10 P_V-mean: 6000 P_V-stddev: 750 2. Iterations: 10000 Bins: 10 n-min: 1 n-max: 10 P_V-mean: 6000 P_V-stddev: 750 3. Iterations: 10000 Bins: 10 n-min: 1 n-max: 10 P_V-mean: 5000 P_V-stddev: 1500

Thats three screen shots.

PART D

Extend the Visual C# .NET program developed in part C to simulate the basic pro?t calculation, P_T = nP_V, where the user can select either a uniform or normal distribution for n using radio buttons and then must input the appropriate parameters (min and max if they select uniform, mean and standard deviation if they select normal) and they can similarly select either a uniform or normal distribution for P_V with appropriate parameters depending on the selection. Of course, the user will input the number of bins and number of iterations. Finally, the user can click a button and the results will be graphed on a bar chart using the Microsoft Chart Control and the average total pro?t ( P_T) will be displayed in a textbox. Also include in the program any necessary input validation for all input values. Turn in a listing of the code and a screen shot of the resulting chart using:

1. Iterations: 10000 Bins: 5 n-min: 1 n-max: 10 P_V-mean: 8000 P_V-stddev: 2000 2. Iterations: 10000 Bins: 10 n-mean: 8 n-stddev: 2 P_V-min: 1000 P_V-max: 9000 3. Iterations: 10000 Bins: 10 n- mean: 14 n-stddev:3 P_V-min: 1500 P_V-max: 10000

Thats three screen shots and a listing of the code.

PART E

Youre going to go to a job interview for OU Car Co. Knowing that the field is highly competitive, you have run sales scenarios ahead of time experimenting with different numbers of customers and vehicle pro?ts given that in one month OU Car Co. sells between 3 and 10 cars uniformly distributed with profits of $5,500 on the average with stande deviation of $950. Which has a higher payoff, focusing on selling to a couple more customers or by increasing the average sale (with the same std. dev. of $950) by retraining your sales force or do they have basically the same effect on total sales? Support your answer.

Parameter Iteration Iteration 2 terion 3 Iteration 4 teion 5 7 $4500 4 $2100 7 $6500 $3000 $4000 Pr Bin # $ Range Parameter Iteration Iteration 2 terion 3 Iteration 4 teion 5 7 $4500 4 $2100 7 $6500 $3000 $4000 Pr Bin # $ Range