Paul Pfeiffer - Topics in Applied Probability

Topics in Applied Probability

Topics in Applied Probability is meant to supplement the collection Applied Probability with examples, further problems, and m-files. This collection is not meant to stand alone but rather serve as a miscellany of further content not included in the original collection. Topics discussed in this collection include Martingale Sequences and Markov Procedures as well as a few others. In addition, the collection of m-files contains both the user-defined programs and a collection of m-files for specific problems with properly formatted data.

These files can be entered into MATLAB by calling the appropriate file. These m-files come from a variety of sources ( e.g., exams or problem sets, hence the odd names) and may be useful for examples and exercises. In order to allow for easy access to the helpful m-files, a supplementary link has been placed in the upper right side of each module in the "Topics in Applied Probability" collection. Selecting this link will launch the download of a zip document containing a suite of m-files. Unzipping this file and saving its contents to your local system will allow for easy and reliable access when working with MATLAB.

Cost with Price Breaks

Let D be the demand random variable. Suppose there is

A flat fee of C for Da 1

A cost of c1 per unit for a 1 < Da 2

A cost of c2 per unit for a 2 < Da 3

A cost of c3 per unit for a 3 < D

Then (see figure)

(1)

Note

Since at t = ai , we could as well have .

Special Case. If D is Poisson () , we may obtain an exact solution for E [ g ( D )] by using thefact that and

(2)

As an alternate approach, we approximate D by an appropriate number of values.

Exercise 1. Example ()

A residential College is planning a camping trip over Spring Recess.The number D of persons planning to go is assumed to be Poisson (15). Arrangementsfor transportation and food have been made as follows:

If D 4 , a minimal flat fee of $450 will be charged.

If 4 < D 19 , the additional charge is $100 for each person more than 4 (butless than 20).

If 19 < D , the charge is $80 for each person more than 19.