chap04-lp modeling examples

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1Chapter 4 - Linear Programming: Modeling Examples

Chapter 4

Linear Programming: Modeling

Examples

Introduction to Management Science

9thEdition

by Bernard W. Taylor III

2007 Pearson Education


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Chapter Topics

Chapter 4 - Linear Programming:Modeling Examples

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Prod!ct Mix Example Prod!ct Mix Example "iet Example "iet Example

n In#estment Examplen In#estment Example

Mar$eting Example Mar$eting Example Transportation Example Transportation Example

Blend Example Blend Example

M!lti%Period &ched!ling Example M!lti%Period &ched!ling Example "ata En#elopment nalysis Example "ata En#elopment nalysis Example


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A Product Mix Example

Problem Definition (1 of 8

'o!r%prod!ct T%shirt(s)eatshirt man!*act!ring company.'o!r%prod!ct T%shirt(s)eatshirt man!*act!ring company.M!st complete prod!ction )ithin +, ho!rsM!st complete prod!ction )ithin +, ho!rs

Tr!c$ capacity - /,00 standard si1ed boxes.Tr!c$ capacity - /,00 standard si1ed boxes.

&tandard si1e box holds, T%shirts.&tandard si1e box holds, T%shirts.2ne%do1en s)eatshirts box is three times si1e o* standard2ne%do1en s)eatshirts box is three times si1e o* standardbox.box.

3,4/000 a#ailable *or a prod!ction r!n.3,4/000 a#ailable *or a prod!ction r!n.

400 do1en blan$ T%shirts and s)eatshirts in stoc$.400 do1en blan$ T%shirts and s)eatshirts in stoc$.

5o) many do1ens 6boxes7 o* each type o* shirt to prod!ce85o) many do1ens 6boxes7 o* each type o* shirt to prod!ce8


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A Product Mix Example (! of 8


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Data (" of 8


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Deci#ion $ariable#%x- s)eatshirts/ *ront printing

x,- s)eatshirts/ bac$ and *ront printing

x- T%shirts/ *ront printing

x- T%shirts/ bac$ and *ront printing&b'ectie )unction%

Maximi1e ; - 390x< 3,4x,< 34x< 3=4x

Model Con#traint#%

0.0x< 0.,4x,< 0.0>x< 0.,x+, hr

x< x,< x< x/,00 boxes

3=x< 3>x,< 3,4x< 34x3,4/000

x< x,400 do1en s)eatshirts

x < x 400 do1en T%shirts


Model Con#truction (4 of 8


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7Exhibit 4*1


Computer Solution +ith Excel (, of 8


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#

Exhibit 4*!


Solution +ith Excel Soler -indo+ (. of 8


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$

Exhibit 4*"


Solution +ith /M for -indo+# (0 of 8


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Exhibit 4*4


Solution +ith /M for -indo+# (8 of 8


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Brea$*ast to incl!de at least ,0 calories/ 4 milligrams o*iron/ 00 milligrams o* calci!m/ ,0 grams o* protein/ ,grams o* *iber/ and m!st ha#e no more than ,0 grams o* *at

and 0 milligrams o* cholesterol.

A Diet Example

Data and Problem Definition (1 of ,


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x1 cup# of bran cereal

x! cup# of dr2 cereal

x" cup# of oatmeal

x4 cup# of oat bran

x, egg#

x. #lice# of bacon

x0 orange#

x8 cup# of mil3

x cup# of orange 'uice

x15 #lice# of +heat toa#t

A Diet Example

Model Con#truction 6 Deci#ion $ariable# (! of ,


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Minimi1e ; - 0.>x< 0.,,x, < 0.0x< 0.,x< 0.0x4< 0.09x=

< 0.0x+< 0.=x>< 0.40x9< 0.0+x0

s!b?ect to:90x< 0x, < 00x< 90x< +4x4< 4x=< =4x+

< 00x>< ,0x9< =4x0 ,0

,x, < ,x< ,x< 4x4< x=< x>< x0 ,0

,+0x4< >x=< ,x> 0

=x< x, < ,x< x< x4< x+< x0 4

,0x< >x, < ,x< >x< 0x4< 4,x+< ,40x>

< x9< ,=x0 00

x< x, < 4x< =x< +x4< ,x=< x+< 9x>< x9< x0 ,0

4x< ,x,< x< x< x+< x0 ,

A Diet Example

Model Summar2 (" of ,


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14Exhibit 4*,

A Diet Example

Computer Solution +ith Excel (4 of ,


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1!

Exhibit 4*.

A Diet Example

Solution +ith Excel Soler -indo+ (, of ,


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Maximi1e ; - 30.0>4x< 0.04x,< 0.0=4 x< 0.0xs!b?ect to:

x/000

x,% x% x% x0

x,< x,/000

%.,x< x,< x % ., x0

x< x,< x< x- +0/000

x/ x,/ x/ x0

)here x- amo!nt in#ested in m!nicipal bonds 637

x,- amo!nt in#ested in certi*icates o* deposit 637

x- amo!nt in#ested in treas!ry bills 637

x- amo!nt in#ested in gro)th stoc$ *!nd637

An Ine#tment Example

Model Summar2 (1 of 4


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17Exhibit 4*0


Computer Solution +ith Excel (! of 4


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1#

Exhibit 4*8


Solution +ith Excel Soler -indo+ (" of 4

A I t t E l


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1$Exhibit 4*


Sen#itiit2 7eport (4 of 4

A M 3 ti E l


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B!dget limit 300/000

Tele#ision time *or *o!r commercials @adio time *or 0 commercials

Ae)spaper space *or + ads

@eso!rces *or no more than 4 commercials and(or ads

A Mar3eting Example

Data and Problem Definition (1 of .

A M 3 ti E l


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Maximi1e ; - ,0/000x< ,/000x,< 9/000xs!b?ect to:

4/000x< =/000x ,< /000x00/000

x

x,0 x+

x< x,< x4

x/ x,/ x0

)here x- Expos!re *rom Tele#ision Commercial 6people7

x,- Expos!re *rom @adio Commercial 6people7

x- Expos!re *rom Ae)spaper d 6people7

A Mar3eting Example

Model Summar2 (! of .

A M 3 ti E l


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22Exhibit 4*15

A Mar3eting Example

Solution +ith Excel (" of .

A M 3 ti E l


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2

Exhibit 4*11

A Mar3eting Example

Solution +ith Excel Soler -indo+ (4 of .

A M 3 ti E l


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Exhibit 4*1!

Exhibit 4*1"

A Mar3eting Example

Integer Solution +ith Excel (, of .

A Mar3eting E ample


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2!Exhibit 4*14

A Mar3eting Example

IntegerSolution +ith Excel (. of .

A ran#portation Example


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-arehou#e #uppl2 of 7etail #tore demand

elei#ion Set#% for telei#ion #et#%

% Cincinnati 00 % Ae) or$ 40

, % tlanta ,00 B % "allas ,40

% Pittsb!rgh ,00 C % "etroit ,00

Total +00 Total =00


Problem Definition and Data (1 of "



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Minimi1e ; - 3=x< >xB< xC< x,< ,x,B< x,C00 lb *ishD 000 lbIngredients()ee$: =00 lb horse meatD >00 lb *ishD 000 lbcereal.cereal.

@ecipe re!irement: Meo) Cho) at least hal* *ishD Bo)@ecipe re!irement: Meo) Cho) at least hal* *ishD Bo)

Cho) at least hal* horse meat.Cho) at least hal* horse meat.,/,40 sixteen%o!nce cans a#ailable each )ee$.,/,40 sixteen%o!nce cans a#ailable each )ee$.

Pro*it (can: Meo) Cho) 30.>0D Bo) Cho) 30.9=.Pro*it (can: Meo) Cho) 30.>0D Bo) Cho) 30.9=.

5o) many cans o* Bo) Cho) and Meo) Cho) sho!ld be5o) many cans o* Bo) Cho) and Meo) Cho) sho!ld beprod!ced each )ee$ in order to maximi1e pro*it8prod!ced each )ee$ in order to maximi1e pro*it8

Example Problem Solution


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Chapter 4 - Linear Programming:Modeling Examples 47

Step 1%Define the Deci#ion $ariable#

xi?- o!nces o* ingredient i in pet *ood ? per )ee$/ )here i - h

6horse meat7/ * 6*ish7 and c 6cereal7/ and ? - m 6Meo) cho)7and b 6Bo) Cho)7.

Step !%)ormulate the &b'ectie )unction

Maximi1e ; - 30.046xhm< x*m< xcm7 < 0.0=6xhb< x*b< xcb7


Model )ormulation (! of ,



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Chapter 4 - Linear Programming:Modeling Examples 4#

Step "%)ormulate the Model Con#traint#Amount of each ingredient aailable each +ee3%

xhm< xhb9/=00 o!nces o* horse meat

x*m< x*b ,/>00 o!nces o* *ish

xcm< xcb=/000 o!nces o* cereal additi#e

7ecipe re


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Chapter 4 - Linear Programming:Modeling Examples 4$

Step 4%Model Summar2Maximi1e ; - 30.04xhm< 30.04x*m< 30.04xcm< 30.0=xhb

< 0.0=x*b< 0.0=xcbs!b?ect to:

xhm< xhb9/=00 o!nces o* horse meatx*m< x*b ,/>00 o!nces o* *ish

xcm< xcb=/000 o!nces o* cereal additi#e

% xhm< x*m% xcm0

xhb% x*b% xcb0

xhm< x*m< xcm< xhb< x*b< xcb=/000 o!nces

xi?0


Model Summar2 (4 of ,



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p

Solution +ith /M for -indo+# (, of ,

chap04-lp modeling examples

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