Reference no: EM132362030

Assignment -

Overview - This assignment relates to Chapter 2 of the text and will make use of two PMF models (i.e., families): the geometric PMF and the Poisson PMF. The objectives of this assignment are:

1. To apply discrete probability models into a problem scenario that involves making a decision between two service plans.

2. To determine the conditions upon which one plan dominates another plan in terms of cost.

3. To increase proficiency in MATLAB skills, including using a function program.

In this assignment, the goal will be to compare the costs associated with using two different home service plans. A type of home service plan that many consumers sign up for is one that covers all of a home's appliances (e.g., refrigerator, oven) and AC/heating systems. We will consider two different service plans: SP1 and SP2. The costs that will be considered will only pertain to service charges, which are specified below. Costs for replacing parts will not be addressed as these depend on the particular part that may need to be replaced.

Problem Details - SP1 charges an annual (fixed cost) fee of $400. This means that even if nothing goes wrong, the client must pay this annual fee. SP2 charges an annual fee of $300. In addition to these fixed annual charges, both plans charge a service call fee each time someone needs to be sent to the client.

For SP1, this service call fee is $25, but only if the service responds within the 24-hour period the service request is made (this period will be referred to as a day). For each day SP1 is late in responding, their service call fee is reduced by $5 until, if they are sufficiently late, there is no service call charge. The probability that SP1 responds within the first, second, third, and so on, day, will be modeled by the geometric distribution with parameter p = 0.2. Therefore, there is a probability of 0.2 that the "first success" (i.e., that a service technician representing SP1 shows up) occurs on day 1.

For SP2, the service call fee is $40, but only if the service responds within the 24-hour period the service request is made. For each day SP2 is late in responding, their service call fee is reduced by $10 until, if they are sufficiently late, there is no service call charge. The probability that SP2 responds within the first, second, third, and so on, day, will be modeled by the geometric distribution with parameter p = 0.9. This implies that SP2 is more reliable than SP1 in responding to service calls.

Finally, we will consider the possibility that a client will need between zero and 10 service calls a year. The probability of needing any of these numbers of calls for service will be modeled by the Poisson distribution; therefore, we are assuming the problems/failures that occur in a client's home are independent of one another. The parameter λ will be 5 (i.e., we are assuming on average the need for making five service calls per year (T = 1 so α = λT = λ).

Files and Variables - You will need two MATLAB script programs for this assignment:

1. A main or "calling" program, which will be called assignment1.m.

2. A function program that should be called serviceplan.m. This function program will be called by assignment1.m and will return an array.

Assignment1.m will call the function twice: once to compute an array associated with SP1 and a second time to compute a corresponding array associated with SP2. The format for the function call in assignment1.m is [cost] = serviceplan(a,b,c,d,e), where cost is the output variable-in this case an array-that will be computed in the function program. The variables a, b, c, d, and e are the 5 input variables (that you will name appropriately), which will be passed on to the function in your calling program.

The format of the first line of code in serviceplan.m, which contains the computation of the function, should be function [cost] = serviceplan(a, b, c, d, e). (In reality you can have several output variables computed in your function and passed back to the calling program; we will have just one.). The input variables within the function program can have different names than the corresponding variable names used in the calling program. This allows you to call the function multiple times with different values for these input variables. In this assignment you will call it twice: once for computing the output array for SP1 and a second time for computing the output array for SP2.

The output array that your function program will be computing should be an 11 × 4 array (11 rows and 4 columns). The columns should contain the following data:

• Column 1 should contain the annual number of service calls (i.e., failures) being considered, which is 0-10.
• Column 2 should contain the probability of each of the number of failures contained in column 1. You can use the MATLAB function factorial(n), which computes n!, to help generate these probabilities.
• Column 3 should contain the "variable costs" associated with each of the number of failures contained in column 1. Variable costs are those associated with service call fees. Note that for each service plan, the total variable cost associated with a given number of failures will be the variable cost associated with any single service call multiplied by the number of service calls.
• Column 4 should contain the total costs associated with each of the number of failures in column 1, which is simply the variables costs + the fixed cost (the annual service charge).

The 5 input variables in the function call in assignment1.m should be:

• The annual (fixed) service plan charge.
• The service plan's service fee per call.
• The decrement in service fee charge for each day the service is late.
• The parameter of the geometric model (p).
• The parameter (λ) of the Poisson distribution (which is the same for SP1 and SP2).

Plots

Once you have computed the cost array for each for each service plan, you need to generate the following plots, using the MATLAB plot function. Be sure to label your axes and provide an appropriate legend.

• On the x-axis, plot the number of service calls/failures, 0-10 (column 1 from the cost array of either SP1 or SP2). On the y-axis, plot column 4 from the cost array for SP1 and column 4 from the cost array from SP2-that is, plot the total costs of the plan versus number of service calls needed.
• On the x-axis, plot the number of service calls/failures (0-10). On the y-axis, plot the complimentary cumulative probability associated with these numbers of failures (column 2 from the cost array of either SP1 or SP2). You will first need to generate a vector of cumulative probabilities of failures, cumprob, using the MATLAB cumsum function applied to column two of either plan's cost array. Then, you will need to convert this vector into a vector of complimentary cumulative probabilities, complprob, where the complimentary cumulative function is one minus the cumulative distribution function. To ensure you have positive values, you will need to apply the MATLAB abs function to this computation.
• On the x-axis, plot the total costs for SP1 and for SP2 (column 4 from the SP1 and SP2 cost arrays). On the y-axis, plot the complprob vector.

Modifications

In this assignment, you will also explore the impact of the following modifications:

• SP1 has decided to become more reliable in the speed with which they respond to service calls. This will be reflected in a change in it parameter p from .2 to .7.
• SP2 will reduce its reliability in responding to service calls. This will be reflected in a change in its parameter p from .9 to .4. It will reduce its annual fixed fee from $300 to $20, but will increase its service fee cost from $40 to $65.

Overall Program Plan

1. Specify the input argument values for SP1. Be sure to use comment statements to adequately document the meanings of variables.

2. Call the service plan function in order to derive the 11 × 4 cost array for SP1.

3. In the service plan function program, use the geometric PMF to determine the variable costs and use the Poisson PMF to generate the failure probabilities for 0-10 failures. Then fill in the entire 11 × 4 cost array for SP1.

4. Upon returning to the main program you may want to rename the cost array to make sure that you differentiate the SP1 and SP2 cost arrays.

5. Repeat the process above for SP2, by first specifying the input argument values for SP2 and then calling the service plan function once again.

6. Generate the 3 plots specified above.

7. Make the changes to the parameters as indicated in the "Modifications" section and regenerate the same plots.

Answer the following questions:

1. Is there a specific value (a "crossover point") for the number of service calls at which one plan becomes superior, with respect to total costs, to the other? Or does one plan dominate the other-that is, is superior for all values that the number of annual service calls can take on?

2. What is the probability (as estimated from the plot) of requiring more than 2 service calls? Of requiring more than 8 service calls?

3. What is the probability (as estimated from the plot) of requiring more than approximately $450 of total costs for each plan?

4. Answer questions 1-3 based on the plots generated for the modifications that were made.