Optical transfer function - Revision history

en>Srleffler: Rm useless link to SPIE homepage. Restore link to SPIE-hosted article, and to a useful-looking corporate whitepaper based on published sources.

2014-12-02T04:36:05Z

Rm useless link to SPIE homepage. Restore link to SPIE-hosted article, and to a useful-looking corporate whitepaper based on published sources.

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	~~The '''newsvendor''' (or '''newsboy''' or '''single-period'''<ref name=stevenson>William J. Stevenson~~, ~~Operations Management~~. ~~10th edition, 2009; page 581</ref>) '''model'''~~ is ~~a mathematical model in [[operations management]] and [[applied economics]] used~~ to ~~determine [[Optimization (mathematics)\|optimal]] [[inventory]] levels.~~ ~~It is (typically) characterized by fixed prices~~ and ~~uncertain demand for a perishable product. If the inventory level~~ is <math>q</math>, each unit of demand above <math>q</math> is lost. This model is also known as the ''Newsvendor Problem'' or ''Newsboy Problem'' by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to ~~stock in the face of uncertain demand and knowing that unsold copies will be worthless~~ at ~~the end of the day.~~		Nice to meet you, my title is Refugia. One of the issues she loves most is to do aerobics std testing at home and now she is trying to std testing at home make [http://www.animecontent.com/blog/124234 home std test kit] cash with it. South [http://www.neweracinema.com/tube/blog/181510 http://www.neweracinema.com/tube/blog/181510] Dakota is home std test kit where me and my husband reside and my family members loves it. He used to be unemployed but now he is a pc operator but his [http://www.itsyoursexlife.com/gyt/std-and-testing-faqs/ marketing] never [http://En.wikipedia.org/wiki/Genital_wart arrives].<br><br>Feel free to surf to my web page: [http://flashtrue.com/stay-yeast-infection-free-using-these-useful-tips/ http://flashtrue.com/stay-yeast-infection-free-using-these-useful-tips]

	~~== History ==~~

	~~The mathematical problem appears to date from 1888<ref>{{cite jstor\|2979084}}</ref> where [[Francis Ysidro Edgeworth\|Edgeworth]] used the~~ [~~[central limit theorem]] to determine the optimal cash reserves to satisfy random withdrawals from depositors.<ref>{{cite web\|url=~~http://www.~~columbia~~.~~edu/~gmg2/4000/pdf~~/~~lect_07.pdf\|title=IEOR 4000 Production Management Lecture 7\|author=Guillermo Gallego\|publisher=[[Columbia University]]\|date=18 Jan 2005\|accessdate=30 May 2012}}<~~/~~ref>~~

	~~== Profit function ==~~
	~~The standard newsvendor [[Profit (economics)\|profit]] function is~~

	~~: <math>\pi =E\left[p\min (q,D)\right]-cq</math>~~

	~~where <math>D</math> is a [[random variable]~~] with [[probability distribution]] <math>F</math> representing demand, each unit is sold for price <math>p</math> and purchased for price <math>c</math>, <math>q</math> is the number of units stocked, and <math>E</math> is the [[expectation operator]]. ~~The solution to the optimal stocking quantity of the newsvendor which maximizes expected profit is:~~

	: ~~<math>q=F^{-1}\left( \frac{p-c}{p}\right)</math>~~

	~~where <math>F^{-1}<~~/~~math> denotes the [[Inverse function\|inverse]] [[cumulative distribution function]] of <math>D<~~/~~math>~~.

	~~Intuitively, this ratio, referred to as the '''critical fractile''', balances the cost of being understocked (a lost sale worth <math>(p-c)<~~/~~math>) and the total costs of being either overstocked or understocked (where the cost of being overstocked is the inventory cost, or <math>c<~~/~~math> so total cost is simply <math>p<~~/~~math>).~~

	~~== Numerical Examples ==~~
	~~=== Uniform Distribution ===~~

	~~Assume that~~: ~~retail price is <math>p=7<~~/~~math> [$~~/~~unit] and purchase price is <math>c=5</math> [$/unit]~~. ~~Furthermore the <math>D</math> demand follows a [[uniform distribution (continuous)]] between <math>D_\min = 50</math> and <math>D_\max = 80</math>~~.

	~~: <math>q_\text{opt}=F^{-1}\left( \frac{7-5}{7}\right)=F^{-1}\left( 0.285 \right) = D_\min+(D_\max-D_\min) \cdot 0.285 = 58.55\approx59.<~~/~~math>~~

	~~Therefore optimal inventory level is approximately 59 units.~~

	~~=== Normal Distribution ===~~

	~~Assume that: retail price is <math>p=7<~~/~~math> [$~~/~~unit~~] and ~~purchase price is <math>c=5</math> [$/unit]. Furthermore the <math>D</math> demand follows a [[normal distribution]] with a mean, <math>\mu</math>, demand of 50~~ and ~~a [[standard deviation]], <math>\sigma</math>, of 20~~.

	~~: <math>q_\text{opt}=F^{-1}\left( \frac{7-5}{7}\right)=\mu + \sigma Z^{-1}\left( 0.285 \right) = 50 + 20 \cdot -0.56595 = 38.68\approx 39.</math>~~

	~~Therefore optimal inventory level is approximately 39 units.~~

	~~=== Lognormal Distribution ===~~

	~~Assume that: retail price~~ is ~~<math>p=7</math> [$/unit] and purchase price is <math>c=5</math> [$/unit]. Furthermore the <math>D</math> demand follows~~ a [~~[lognormal distribution]] with a mean demand of 50, <math>\mu<~~/~~math>, and a [[standard deviation]], <math>\sigma<~~/~~math>, of 0~~.2.

	~~: <math>q_\text{opt}=F^{-1}\left(\frac{7-5}{7}\right)=\mu e^{Z\left(0.285\right) \sigma} = 50 e^{\left(0.2 \cdot -0.56595 \right)} = 44.64\approx 45.<~~/~~math>~~

	~~Therefore optimal inventory level is approximately 45 units.~~

	~~=== Extreme situation ===~~
	~~If <math>p<c<~~/~~math> (i.e. the retail price is less than the purchase price), the numerator becomes negative. In this situation, it isn't worth keeping any item in the inventory.~~

	~~== Cost based optimization of inventory level ==~~
	Assuming that the 'newsvendor' is in fact a small company who wants to produce goods to an uncertain market. In this more general situation the cost function of the newsvendor (company) can be formulated in the following manner:

	~~: <math>K(q) = c_f + c_v (q~~-~~x) + p E\left[\max(D~~-~~q,0)\right] + h E\left[\max(q~~-~~D,0)\right]<~~/~~math>~~

	~~where the individual parameters are the following:~~
	* <math>c_f</math> – fixed cost. This cost always exists when the production of a series is started. [$/production]
	* <math>c_v</math> – variable cost. This cost type expresses the production cost of one product. [$/~~product]~~
	* <math>q</math> – The product quantity in the inventory. The decision of the inventory control policy concerns the product quantity in the inventory after the product decision. This parameter includes the initial inventory as well. ~~If nothing is produced, then this quantity is equal to the initial quantity, i.e. concerning the existing inventory~~.
	* <math>x</~~math> – Initial inventory level. We assume that the supplier possesses <math>x<~~/~~math> products in the inventory at the beginning of the demand of the delivery period~~.
	* <~~math~~>p<~~/math~~> ~~– penalty cost (or back order cost). If there is less raw material in the inventory than needed~~ to ~~satisfy the demands, this is the penalty cost of the unsatisfied orders.~~ [$/~~product]~~
	* <math>E[D]</~~math> – Expected value of the <math>D</math> stochastic variable~~.
	* <math>D</~~math> – This means the demand from the receiver for the product, which is an optional probability variable. [unit]~~
	* <math>h</math> – inventory and stock holding cost. [$ / product]

	~~On the basis of the cost function the determination of the optimal inventory level is a minimization problem. So in the long run the amount of cost~~-~~optimal end~~-~~product can be calculated on the basis of the following relation:<ref name=stevenson/>~~

	~~: <math>q_\text{opt} = F^{~~-~~1}\left( \frac{p~~-~~c_v}{p+h}\right)</math>~~

	~~== See also ==~~
	*[[Economic order quantity]]
	*[[Inventory control system]]
	*[[Extended newsvendor model]]

	~~== References ==~~
	~~<references~~/>

	~~== Further reading ==~~
	* Ayhan, Hayriye, Dai, Jim, Foley, R. D., Wu, Joe, 2004: Newsvendor Notes, ISyE 3232 Stochastic Manufacturing & Service Systems. [http://~~www2~~.~~isye.gatech.edu~~/~~people/faculty/Hayriye_Ayhan/newsvendor825.pdf]~~

	~~[[Category:Operations research]]~~
	~~[[Category:Mathematical optimization]~~]

en>WereSpielChequers: posses possess

2014-01-19T21:00:43Z

posses possess

← Older revision		Revision as of 23:00, 19 January 2014
Line 1:		Line 1:
	~~Myrtle Benny~~ is how I'~~m called~~ and ~~I feel comfortable when individuals use~~ the ~~complete name~~. ~~One~~ of the ~~things she loves most~~ is to ~~do aerobics~~ and ~~now she~~ is ~~trying to make money~~ with it. ~~Her family life~~ in ~~Minnesota~~. ~~For many years I've been operating~~ as a ~~payroll clerk~~.<br><br>~~My blog post;~~ [http://~~www~~.~~neweracinema~~.~~com~~/~~tube~~/~~blog~~/~~181510 over the counter std test~~]		The '''newsvendor''' (or '''newsboy''' or '''single-period'''<ref name=stevenson>William J. Stevenson, Operations Management. 10th edition, 2009; page 581</ref>) '''model''' is a mathematical model in [[operations management]] and [[applied economics]] used to determine [[Optimization (mathematics)\|optimal]] [[inventory]] levels. It is (typically) characterized by fixed prices and uncertain demand for a perishable product. If the inventory level is <math>q</math>, each unit of demand above <math>q</math> is lost. This model is also known as the ''Newsvendor Problem'' or ''Newsboy Problem'' by analogy with the situation faced by a newspaper vendor who must decide how many copies of the day's paper to stock in the face of uncertain demand and knowing that unsold copies will be worthless at the end of the day.

			== History ==

			The mathematical problem appears to date from 1888<ref>{{cite jstor\|2979084}}</ref> where [[Francis Ysidro Edgeworth\|Edgeworth]] used the [[central limit theorem]] to determine the optimal cash reserves to satisfy random withdrawals from depositors.<ref>{{cite web\|url=http://www.columbia.edu/~gmg2/4000/pdf/lect_07.pdf\|title=IEOR 4000 Production Management Lecture 7\|author=Guillermo Gallego\|publisher=[[Columbia University]]\|date=18 Jan 2005\|accessdate=30 May 2012}}</ref>

			== Profit function ==
			The standard newsvendor [[Profit (economics)\|profit]] function is

			: <math>\pi =E\left[p\min (q,D)\right]-cq</math>

			where <math>D</math> is a [[random variable]] with [[probability distribution]] <math>F</math> representing demand, each unit is sold for price <math>p</math> and purchased for price <math>c</math>, <math>q</math> is the number of units stocked, and <math>E</math> is the [[expectation operator]]. The solution to the optimal stocking quantity of the newsvendor which maximizes expected profit is:

			: <math>q=F^{-1}\left( \frac{p-c}{p}\right)</math>

			where <math>F^{-1}</math> denotes the [[Inverse function\|inverse]] [[cumulative distribution function]] of <math>D</math>.

			Intuitively, this ratio, referred to as the '''critical fractile''', balances the cost of being understocked (a lost sale worth <math>(p-c)</math>) and the total costs of being either overstocked or understocked (where the cost of being overstocked is the inventory cost, or <math>c</math> so total cost is simply <math>p</math>).

			== Numerical Examples ==
			=== Uniform Distribution ===

			Assume that: retail price is <math>p=7</math> [$/unit] and purchase price is <math>c=5</math> [$/unit]. Furthermore the <math>D</math> demand follows a [[uniform distribution (continuous)]] between <math>D_\min = 50</math> and <math>D_\max = 80</math>.

			: <math>q_\text{opt}=F^{-1}\left( \frac{7-5}{7}\right)=F^{-1}\left( 0.285 \right) = D_\min+(D_\max-D_\min) \cdot 0.285 = 58.55\approx59.</math>

			Therefore optimal inventory level is approximately 59 units.

			=== Normal Distribution ===

			Assume that: retail price is <math>p=7</math> [$/unit] and purchase price is <math>c=5</math> [$/unit]. Furthermore the <math>D</math> demand follows a [[normal distribution]] with a mean, <math>\mu</math>, demand of 50 and a [[standard deviation]], <math>\sigma</math>, of 20.

			: <math>q_\text{opt}=F^{-1}\left( \frac{7-5}{7}\right)=\mu + \sigma Z^{-1}\left( 0.285 \right) = 50 + 20 \cdot -0.56595 = 38.68\approx 39.</math>

			Therefore optimal inventory level is approximately 39 units.

			=== Lognormal Distribution ===

			Assume that: retail price is <math>p=7</math> [$/unit] and purchase price is <math>c=5</math> [$/unit]. Furthermore the <math>D</math> demand follows a [[lognormal distribution]] with a mean demand of 50, <math>\mu</math>, and a [[standard deviation]], <math>\sigma</math>, of 0.2.

			: <math>q_\text{opt}=F^{-1}\left(\frac{7-5}{7}\right)=\mu e^{Z\left(0.285\right) \sigma} = 50 e^{\left(0.2 \cdot -0.56595 \right)} = 44.64\approx 45.</math>

			Therefore optimal inventory level is approximately 45 units.

			=== Extreme situation ===
			If <math>p<c</math> (i.e. the retail price is less than the purchase price), the numerator becomes negative. In this situation, it isn't worth keeping any item in the inventory.

			== Cost based optimization of inventory level ==
			Assuming that the 'newsvendor' is in fact a small company who wants to produce goods to an uncertain market. In this more general situation the cost function of the newsvendor (company) can be formulated in the following manner:

			: <math>K(q) = c_f + c_v (q-x) + p E\left[\max(D-q,0)\right] + h E\left[\max(q-D,0)\right]</math>

			where the individual parameters are the following:
			* <math>c_f</math> – fixed cost. This cost always exists when the production of a series is started. [$/production]
			* <math>c_v</math> – variable cost. This cost type expresses the production cost of one product. [$/product]
			* <math>q</math> – The product quantity in the inventory. The decision of the inventory control policy concerns the product quantity in the inventory after the product decision. This parameter includes the initial inventory as well. If nothing is produced, then this quantity is equal to the initial quantity, i.e. concerning the existing inventory.
			* <math>x</math> – Initial inventory level. We assume that the supplier possesses <math>x</math> products in the inventory at the beginning of the demand of the delivery period.
			* <math>p</math> – penalty cost (or back order cost). If there is less raw material in the inventory than needed to satisfy the demands, this is the penalty cost of the unsatisfied orders. [$/product]
			* <math>E[D]</math> – Expected value of the <math>D</math> stochastic variable.
			* <math>D</math> – This means the demand from the receiver for the product, which is an optional probability variable. [unit]
			* <math>h</math> – inventory and stock holding cost. [$ / product]

			On the basis of the cost function the determination of the optimal inventory level is a minimization problem. So in the long run the amount of cost-optimal end-product can be calculated on the basis of the following relation:<ref name=stevenson/>

			: <math>q_\text{opt} = F^{-1}\left( \frac{p-c_v}{p+h}\right)</math>

			== See also ==
			*[[Economic order quantity]]
			*[[Inventory control system]]
			*[[Extended newsvendor model]]

			== References ==
			<references/>

			== Further reading ==
			* Ayhan, Hayriye, Dai, Jim, Foley, R. D., Wu, Joe, 2004: Newsvendor Notes, ISyE 3232 Stochastic Manufacturing & Service Systems. [http://www2.isye.gatech.edu/people/faculty/Hayriye_Ayhan/newsvendor825.pdf]

			[[Category:Operations research]]
			[[Category:Mathematical optimization]]

164.67.196.59 at 21:17, 1 September 2012

2012-09-01T21:17:37Z

New page

Myrtle Benny is how I'm called and I feel comfortable when individuals use the complete name. One of the things she loves most is to do aerobics and now she is trying to make money with it. Her family life in Minnesota. For many years I've been operating as a payroll clerk.<br><br>My blog post; [http://www.neweracinema.com/tube/blog/181510 over the counter std test]