1/15/18 TALK 1 LINEAR PROGRAMMING Agenda What is Linear - - PDF document

SLIDE 1

1/15/18 1

LINEAR PROGRAMMING

J E S S I C A B E R G A N

TALK 1

§ Agenda

• What is Linear Programming?
• History
• Applications
• Solving

WHAT IS LINEAR PROGRAMMING

Forma mal De l Defini nition: n: Line near p programmi mming ng i is a a ma mathe hema matical me l metho hod o

• f

_____________________( _____________________(such a h as t the he a allo llocation o n of resources) b by me y means ns o

• f li

line near f func nctions ns w whe here t the he variable les i involv lved a are s subje ject t to c cons nstraint nts. .

• M
• Merriam-

m-Webster Di Dictiona nary y

IN OTHER WORDS

• Line

near P Programmi mming ng i is a a s speciali lized a area o

• f ma

mathe hema matical o l often u n used b by t y the he ____________ a ____________ and nd o

• the

her v various i ind ndustries t to he help lp t the hem ma m make d decisions ns. .

• Aids i

in a n addressing ng t the he _________ o _________ of ho how t thi hing ngs w work o k or s sho hould ld w work. .

• The

he g graphs hs f forme med t thr hrough li h line near p programmi mming ng a als lso a aid i in _____________ d n _____________ data colle llected. .

• The o

e over erall g goal w when en u using l linea ear p programming i is t to f find t the e ______________________________. ______________________________.

Starting ng i in t n the he e early 1 ly 18000s

• _________________, a

_________________, a F Frenc nch ma h mathe hema matician a n and nd phys ysicist, f , formu mula lated t the he li line near p programmi mming ng p proble lem. m. 19 1900s

• __________________, a

__________________, a R Russian ma n mathe hema matician a n and nd econo nomi mist, d , develo loped t the he p proble lem m

• The

he g goal w l was t to i improve ____________ p ____________ pla lanni nning ng i in t n the he US USSR. .

EARLY LINEAR PROGRAMMING

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_______ p _______ provided t the he s spark, , urgenc ncy a y and nd f fund nding ng ne needed t to b begin r n research. h.

Why? y?

_________________________ _________________________ _____________. _____________.

• Ex. L

. Large s scale le mi mili litary y pla lanni nning ng f for f fle leets o

• f

cargo s shi hips w with s h suppli lies and nd s sold ldiers. .

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THE CATALYST

SLIDE 2

1/15/18 2

19 1947 – _______________a _______________and nd associates a at t the he U U.S .S. . De Departme ment nt o

• f t

the he A Air F Force develo loped t the he _________________. _________________. 19 1951 – __________________ __________________ develo loped a a s special li l line near programmi mming ng s solu lution u n used to p pla lan t n the he o

• ptima

mal l mo moveme ment nt o

• f s

shi hips b back a k and nd forth a h across t the he A Atla lant ntic during ng t the he w war

IN AMERICA

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______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________ ______________

APPLICATIONS

1. 1. ___________ c ___________ cost w whi hile le me meeting ng p product specifications ns 2. 2. ___________ p ___________ profit w with h

• ptima

mal p l production n processes o

• r p

products 3. 3. ____________ c ____________ cost i in n trans nsportation r n routes 4. 4. De Determi mine ne b best s sche hedule les for p production a n and nd s sale les

COMMON GOALS PROBLEM #1

Dr Dr. . Te Teek, t , the he e emi mine nent nt me medical s l speciali list, c , cla laims ms t tha hat he he c can c n cure c cold lds w with h hi his r revolu lutiona nary 3 y 3-la

• layer p

pills

• lls. T

. The hese c come me i in t n two s sizes: : reg egular s size c cont ntaini ning ng 2 g grains ns o

• f a

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 g grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns o

• f c

codeine ne. . Now Dr Dr. . Teek’s k’s r research ha h has c convinc nced hi him t m tha hat i it r requires a at le least 1 12 g grains ns

• f a

aspirin, 7 n, 74 g grains ns o

• f b

bicarbona nate, a , and nd 2 28 g grains ns o

• f c

codeine ne t to e effect hi his rema markable le c

• cure. De

. Determi mine ne t the he le least nu numb mber o

• f p

pills lls he he s sho hould ld p prescribe i in n

• rder t

to me meet t the hese r requireme ment nts. . reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, , and nd 1 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of aspirin, 8 n, 8 g grains ns o

• f b

bicarbona nate, , and nd 6 6 g grains ns o

• f c

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns

• f b

bicarbona nate, a , and nd 2 28 g grains ns o

• f

codeine ne

Let x x = = _________ _________ _____________ _____________ Let y = y = _________ _________ _____________ _____________ Let m = m = ________ ________ _____________ _____________ PROBLEM #1

reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns o

• f

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns o

• f

bicarbona nate, a , and nd 2 28 g grains ns o

• f

codeine ne

x ____a ____and nd y _____ y _____ Thi his i is c cons nsidered a a ________ ________ o

• r r

restriction. n.

PROBLEM #1

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1/15/18 3

reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns o

• f

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns o

• f

bicarbona nate, a , and nd 2 28 g grains ns o

• f

codeine ne Form a m an e n equation f n for A Aspirin n in t n terms ms o

• f r

regula lar a and nd ki king ng size p pills lls. . Regula lar s size = = ___ ___ King ng s size = = ___ ___ Go Goal: l: ≥ ____ ____

____________ ____________

PROBLEM #1

reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns o

• f

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns o

• f

bicarbona nate, a , and nd 2 28 g grains ns o

• f

codeine ne Form a m an e n equation f n for Bicarbona nate i in t n terms ms o

• f

regula lar a and nd ki king ng s size p pills lls. . Regula lar s size = = ____ ____ King ng s size = = ____ ____ Go Goal: l: ≥ ____ ____

______________ ______________

PROBLEM #1

reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns

• f c

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns o

• f

bicarbona nate, a , and nd 2 28 g grains ns o

• f

codeine ne Form a m an e n equation f n for C Codeine ne in t n terms ms o

• f r

regula lar a and nd ki king ng size p pills lls. . Regula lar s size = = ___ ___ King ng s size = = ___ ___ Go Goal: l: ≥ ___ ___

_______________ _______________

PROBLEM #1

reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 g grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns o

• f

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns o

• f

bicarbona nate, a , and nd 2 28 g grains ns o

• f

codeine ne Typ ypical F l Form Us m Used: :

De Determi mine ne: x : x ≥ 0 0 a and nd y y ≥ 0 So t tha hat: : 2x + + y y ≥ 1 12 5x + + 8 8y y ≥ 7 74 1x + + 6 6y y ≥ 2 24 And nd s so t tha hat: : ________________ ________________ …is a a _____________ _____________

PROBLEM #1

{

reg egular s size c cont ntaini ning ng 2 2 g grains ns o

• f

aspirin, 5 n, 5 g grains ns o

• f b

bicarbona nate, a , and nd 1 1 grain o n of c codeine ne; ; King s size e cont ntaini ning ng 1 1 g grain o n of a aspirin, 8 n, 8 grains ns o

• f b

bicarbona nate, a , and nd 6 6 g grains ns o

• f

codeine ne. . Goal: : 12 g grains ns o

• f a

aspirin, 7 n, 74 g grains ns o

• f

bicarbona nate, a , and nd 2 28 g grains ns o

• f c

codeine ne

Wha hat Q Quadrant nt A Are W We Worki king ng In? In?

PROBLEM #1

De Determi mine ne: x : x ≥ 0 0 a and nd y y ≥ 0 So t tha hat: : 2x + + y y ≥ 1 12 5x + + 8 8y y ≥ 7 74 1x + + 6 6y y ≥ 2 24 And nd s so t tha hat: : m = m = x x + + y y …is a a Mini nimu mum m

Cha hang nge t to p point nt- slo lope f form. m.

_________________ _________________ _________________ _________________ _________________ _________________

SLIDE 4

1/15/18 4

Y Y ≥ - 2 X + 1 2

• 2 X + 1 2

Y Y ≥ - 5 X / 8 + 3

• 5 X / 8 + 3 7 / 4

7 / 4

Intersection points: (__,___), (___,___) Intersection points: (__,___), (___,___) y ≥ -x/6 + 4 Intersection points: (__,___), (___,___) ___________________________ (polygonal region) ______________ ___________(__________) Now we will need to graph lines with slopes of ____________. For example: y = -x +8, y = -x + 10, y = -x + 15

OPTIMAL FEASIBLE SOLUTION

So w we a arrived a at t the he v valu lue o

• f _______ t

_______ total p l pills lls Specifically: _____ lly: _____regula lar s size p pills lls a and nd _______ ki _______ king ng s size p pills lls

Thi his yi yield lds Aspirin: ______________ n: ______________ Bicarbona nate: ______________ : ______________ Codeine ne: _________________ : _________________

PROBLEM #2

The he B Broad M Motor C Company, M , Manu nufacturer o

• f t

the he Bulks lkswagon ( (the he la latest w word i in c n compact c cars f for no not-s

• so-c
• compact p

people le), d , decides t to s spons nsor a a o

• ne

ne-ha

• half

lf ho hour t tele levision s n sho how f featuring ng a a c come median n and nd a a b band

• nd. T

. The he C Company i y ins nsists t tha hat t the here mu must b be a at le least 3 3 mi minu nutes o

• f c

comme mmercials

• ls. T

. The he T T.V .V. . ne network r k requires t tha hat t the he t time me a allo llotted t to c comme mmercials ls mu must no not e exceed 1 12 mi minu nutes, a , and nd u und nder no no circums mstanc nces ma may i y it e exceed t the he t time me a allo llotted t to t the he c come

• median. T
• n. The

he c come median i n is r relu luctant nt t to w work k mo more t tha han 2 n 20 mi minu nutes o

• f t

the he ha half lf-ho

• hour s

sho how, s , so t the he b band nd i is t to b be u used t to f fill i ll in a n any r y rema maini ning ng t time me. . The he c come median c n costs t the he s spons nsor $ $150 p per mi minu nute, t , the he b band nd $ $100 p per mi minu nute a and nd t the he c comme mmercials ls $50 p per mi minu

• nute. E

. Experienc nce i ind ndicates t tha hat f for e every mi y minu nute t the he c come median i n is o

• n t

n the he a air, 4 , 4000 additiona nal v l viewers t tune ne i in; f n; for e every mi y minu nute o

• f b

band nd t time me, 2 , 2000 ne new v viewers ma may b y be e expected; b ; but for e every mi y minu nute o

• f c

comme mmercial t l time me, 1 , 1000 p people le w will t ll tune ne O OUT UT! H How s sha hall t ll the he a availa lable le t time me b be allo llotted i if t the he s spons nsor i is i int nterested i in n a.

• a. Obtaini

ning ng t the he ma maximu mum nu m numb mber o

• f v

viewers; ;

• b. P

. Producing ng t the he s sho how a at mi mini nimu mum c m cost. .

SLIDE 5

1/15/18 5

Le Let t x = = nu numb mber o

• f mi

minu nutes a allo llotted t to t the he __________ __________ Le Let t y = y = nu numb mber o

• f mi

minu nutes a allo llotted t to _____________ _____________ And nd 30mi minu nutes – x x – y = y = nu numb mber o

• f mi

minu nutes a allo llotted t to the he _________ _________

RESTRICTIONS

_______ _______ _______ _______ ___________ ___________ _____ _____ restriction i

n imposed b by t y the he s spons nsor

______ ______ ______ ______ ______ ______ restrictions

ns i imposed b by t y the he c come median n

}

Because the time allotted to each portion of the program can not be negative.

} Restrictions imposed by the TV network

VIEWERS/COST

Le Let t n n = t the he nu numb mber o

• f _____________________________, t

_____________________________, the hen t n the he v valu lue o

• f n i

n is g given b n by: y: n = n = 4 4000x + + 2 2000(30 – x x – y) y) – 1 1000y y Or Or n = n = 1 1000(2x 2x – 3 3y y + 6 60) Le Let t c = = _________________________ t _________________________ to t the he s spons nsor, t , the hen t n the he v valu lue o

• f c

c i is g given b n by: y: c = = 1 150x + + 1 100(30 – x x – y) y) + + 5 50y y Or c = = 5 50(x x – y y + 6 60) Go Goal: De l: Determi mine ne t the he v valu lues f for x x a and nd y w y whi hich s h sha hall e ll eithe her ______________ n ______________ n or

• r __________________ c

__________________ c. .

Let u us d deno note ______ ______as t the he M MAXIM XIMUM UM and nd ______ a ______ as t the he M MIN INIM IMUM UM

De Determi mine ne: : x ≥ 0 0 a and nd y y ≥ 0 x + + y y ≤30 So t tha hat: : y y ≥ 3 3 y y ≤ 1 12 x x – y y ≥ 0 x ≤ 2 20 And nd s so t tha hat: : eithe her M M = = 2 2x – 3 3y i y is a a ______________ ______________

• r m =

m = x x – y i y is a a ______________ ______________

{

{

Maximu mum i m is a attaine ned a at p point nt ( (____, ___) ____, ___). . i.e .e ma maximu mum # m # o

• f v

viewers n = n = 1 1000 ( (2(_____) _____) – 3 3(_____) _____) + + 6 60) = = 1 1000(_____) _____) = = __________ V __________ Viewers c = = 5 50 ( (____ ____ – ___ + ___ + 6 60) = = 5 50(____) ____) = = $ $ ___________ ___________ Mini nimu mum i m is a attaine ned a at p point nt ( (____, ___) ____, ___). i.e .e mi mini nimu mum c m cost n = n = 1 1000 ( (2(___) ___) – 3 3(___) ___) + + 6 60) = = 1 1000(_____) _____) = = __________ V __________ Viewers c = = 5 50 ( (____ ____ – ____ + ____ + 6 60) = = 5 50(_____) _____) = = $ ____________ ____________ For ma maximu mum v m viewers: : _____mi _____min f n for t the he c come median, ______mi n, ______min t n to c comme mmercials ls, a , and nd ______mi ______min n to t the he b band nd For mi mini nimu mum c m cost: : _______mi _______min f n for t the he c come median, ______mi n, ______min t n to c comme mmercials ls, a , and nd ______mi ______min n to t the he b band nd

SLIDE 6

1/15/18 6

PART TWO

CONVEX SETS IN THE CARTESIAN PLANE

Straight ht li line nes a and nd ha half lf p pla lane nes i in R n R2 p possess a an i n important nt p property c y calle lled ______________. ______________. A s set o

• f p

point nts S S, i , is s said t to b be ____________ ____________, i , if a and nd o

• nly i

nly if, t , the he s set c cont ntains ns t the he e ent ntire ___________ ___________ jo joini ning ng a any t y two o

• f i

its p point nts. . All o ll of t the he e example les b belo low a are c convex s sets. .

CONVEX SETS IN THE CARTESIAN PLANE

Straight ht li line nes a and nd ha half lf p pla lane nes i in R n R2 p possess a an i n important nt p property c y calle lled C CONVEXIT XITY. . A s set o

• f p

point nts S S, i , is s said t to b be co convex, i , if a and nd o

• nly i

nly if, t , the he s set c cont ntains ns t the he e ent ntire seg egmen ent jo joini ning ng a any t y two o

• f i

its p point nts. . All o ll of t the he e example les b belo low a are N NOT scon sconvex s sets. .

LET US DEFINE A “SEGMENT” USING THE MIDPOINT FORMULA

Cons nsider t two d distinc nct p point nts A A = = ( (x1, y , y1) a and nd B B = = ( (x2, y , y2). . If If M M = = ( (x , y) , y) i is t the he mi midpoint nt o

• f s

segme ment nt A AB, t , the hen t n the he c coordina nates o

• f M

M a are given b n by y

SIMILAR TRIANGLES SIMILAR TRIANGLES

SLIDE 7

1/15/18 7

USING SIMILAR TRIANGLES

More g gene nerally lly, i , if M M d divides t the he s segme ment nt A AB i in t n the he r ratio p p : q : q ( (i.e .e. i . if ) ), , the hen b n by u y using ng s simi mila lar t triang ngle les w we s see t tha hat

= =

AM MB = p q

SOLVING THE FIRST EQUATION FOR X YIELDS: SOLVING THE OTHER EQUATION FOR Y YIELDS: PARAMETER

Now, i , if w we le let t i is a a Parame meter De Defini nition A A p parame meter i is a a q quant ntity t y tha hat i inf nflu luenc nces t the he o

• utput
• r b

beha havior o

• f a

a ma mathe hema matical o l obje ject. Can t n t b be a any v y valu lue?

Now w we c can e n express x x a and nd y a y as…

THEOREM 1

If If C C i is a any p y point nt o

• n li

n line ne l, , with p h parame metric r represent ntation g n given b n by y And nd i if C C c correspond nds t to t the he p parame meter v valu lue t t, t , the hen A n AC = = | |t|A |AB. .

SLIDE 8

1/15/18 8

PROOF THEOREM 1 C LIES BETWEEN A AND B

Now s sinc nce 0 0 < < t t < < 1 1 t the hen w n we kno know 0 0 < < ( (1-t

• t) <

< 1 1. . Applyi lying ng T The heorem 1 m 1 w we ha have A AC = = t t A AB And nd i if w we i int ntercha hang nge A AB a and nd u use C CB w we g get C CB = = ( (1-t

• t) B

BA Add A AC a and nd C CB t togethe her w we g get. .

DEFINITIONS

De Defini nition: n: Point nt C C i is b between p n point nts A A a and nd B B, i , if a and nd o

• nly i

nly if A AC + + C CB = = A AB. . De Defini nition: n: If If A A = = ( (x1, y , y1) a and nd B B = = ( (x2, y , y2) a are p point nts i in R n R2 t the hen w n we d define ne (a (a) A c clo losed s segme ment nt: _______ i : _______ is t the he s set c cons nsisting ng o

• f p

point nts A A a and nd B B, t , togethe her w with h all t ll the he p point nts b between A n A a and nd B B. . (b) (b) A An o n open s n segme ment nt: ________ i : ________ is t the he s set c cons nsisting ng o

• f p

point nts B BETWEEN A A a and nd B B. .

Using parametric representation, we may express the closed segment as follows: Open segment:

DEFINITION

A s set o

• f p

point nts i is c calle lled Co Convex, i , if a and nd

• nly i

nly if, w , whe hene never i it c cont ntains ns p point nts A A and nd B B, t , the hen i n it a als lso c cont ntains ns a all ll point nts b between A n A a and nd B B. .

19 1947 – Ge George Da Dant ntzig a and nd associates a at t the he U.S .S. De . Departme ment nt

• f t

the he A Air F Force develo loped t the he simple lex me metho hod. . RECALL ADVANTAGES OF SIMPLEX METHOD

1.

• 1. R

Recogni nized a at t the he mo most e effective g gene neral me l metho hod f for ha hand ndli ling ng li line near p programmi mming ng p proble lems ms. . 2.

• 2. M

More p practical f l for p proble lems ms i involv lving ng mo more t tha han 2 n 2 variable les 3.

• 3. R

Readily p ly programma mmable le f for a a c computer

SLIDE 9

1/15/18 9

SIMPLEX METHOD The he u und nderlyi lying ng me metho hod i is t the he Ga Gauss-J

• Jordan

n comple lete e eli limi mina nation p n procedure f for s solv lving ng sys ystems ms o

• f s

simu mult ltane neous li line near e equations ns. . EXAMPLE 1 - REDUCED ROW ECHELON FORM

1 1 1 2 3 5 4 5 5 8 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 1 1 1 __ __ __ 4 5 5 __ 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥

R2 − 2R

1

u r uuuuuuu R3 − 4R

1

u r uuuuuuu

1 1 1 __ __ __ __ __ __ 5 __ __ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥

1 1 1 __ __ __ 4 5 5 __ 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥