Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Topics in Applied Probability » Cost with Price Breaks

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Rice Digital Scholarship

    This collection is included in aLens by: Digital Scholarship at Rice University

    Click the "Rice Digital Scholarship" link to see all content affiliated with them.

  • NSF Partnership display tagshide tags

    This collection is included inLens: NSF Partnership in Signal Processing
    By: Sidney Burrus

    Click the "NSF Partnership" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • UniqU content

    This collection is included inLens: UniqU's lens
    By: UniqU, LLC

    Click the "UniqU content" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Cost with Price Breaks

Module by: Paul E Pfeiffer. E-mail the author

Summary: A general pattern for costs with price breaks is set up and an application is made in the case demand D has a Poisson distribution.

Let D be the demand random variable. Suppose there is

  • A flat fee of C for D≤a1D≤a1
  • A cost of c1 per unit for a1<D≤a2a1<D≤a2
  • A cost of c2 per unit for a2<D≤a3a2<D≤a3
  • A cost of c3 per unit for a3<Da3<D

Then (see figure)

g ( t ) = C + c 1 I M 1 ( t ) ( t - a 1 ) + ( c 2 - c 1 ) I M 2 ( t ) ( t - a 2 ) + ( c 3 - c 2 ) I M 3 ( t ) ( t - a 3 ) where M i = [ a i , ∞ ) g ( t ) = C + c 1 I M 1 ( t ) ( t - a 1 ) + ( c 2 - c 1 ) I M 2 ( t ) ( t - a 2 ) + ( c 3 - c 2 ) I M 3 ( t ) ( t - a 3 ) where M i = [ a i , ∞ )
(1)

Note:

Since (t-ai)=0(t-ai)=0 at t=ait=ai, we could as well have Mi=(ai,∞)Mi=(ai,∞).

Special Case. If D is Poisson (μ)(μ), we may obtain an exact solution for E[g(D)]E[g(D)] by using the fact that E[I[m,∞)(D)]=P(D≥m)E[I[m,∞)(D)]=P(D≥m) and

E [ I [ m , ∞ ) ( D ) D ] = e - μ ∑ k = m ∞ k μ k k ! = μ e - μ ∑ k = m - 1 ∞ μ k k ! = μ P ( D ≥ m - 1 ) E [ I [ m , ∞ ) ( D ) D ] = e - μ ∑ k = m ∞ k μ k k ! = μ e - μ ∑ k = m - 1 ∞ μ k k ! = μ P ( D ≥ m - 1 )
(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). Arrangements for transportation and food have been made as follows:

  • If D≤4D≤4, a minimal flat fee of $450 will be charged.
  • If 4<D≤194<D≤19, the additional charge is $100 for each person more than 4 (but less than 20).
  • If 19<D19<D, the charge is $80 for each person more than 19.

Thus, C=450,a1=4,a2=19,c1=100C=450,a1=4,a2=19,c1=100, and c2=80c2=80. The distribution for D is Poisson (15). Then

Y = g ( D ) = 450 + 100 I [ 4 , ∞ ) ( D ) ( D - 4 ) + ( 80 - 100 ) I [ 19 , ∞ ) ( D ) ( D - 19 ) Y = g ( D ) = 450 + 100 I [ 4 , ∞ ) ( D ) ( D - 4 ) + ( 80 - 100 ) I [ 19 , ∞ ) ( D ) ( D - 19 )
(3)
= 450 + 100 I [ 4 , ∞ ) ( D ) D - 20 I [ 19 , ∞ ) ( D ) D - 400 I [ 4 , ∞ ) ( D ) + 380 I [ 19 , ∞ ) ( D ) = 450 + 100 I [ 4 , ∞ ) ( D ) D - 20 I [ 19 , ∞ ) ( D ) D - 400 I [ 4 , ∞ ) ( D ) + 380 I [ 19 , ∞ ) ( D )
(4)
  1. Obtain the “exact” solution to E[g(D)]E[g(D)].
  2. Approximate D by terms up to 40 and obtain E[g(D)]E[g(D)] and P(g(D)≥v)P(g(D)≥v) for v=v= 1000, 1500, 2000, 2500.

Solution

mu = 15;
% Part (a)
EG = 450 + 100*mu*cpoisson(mu,3) - 20*mu*cpoisson(mu,18)...
- 400*cpoisson(mu,4) + 380*cpoisson(mu,19)
EG = 1.5433e+03
% Part (b)
t = 0:40;
M1 = t >= 4;
M2 = t >= 19;
G = 450 + 100*M1.*(t - 4) - 20*M2.*(t - 19);
PD = ipoisson(mu,t);
EGa = dot(G,PD)
EGa = 1.5433e+03
P1000 = dot(G >= 1000,PD)
P1000 = 0.9301
P1500 = dot(G >= 1500,PD)
P1500 = 0.5343
P2000 = dot(G >= 2000,PD)
P2000 = 0.1248
P2500 = dot(G >= 2500,PD)
P2500 = 0.0062

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks