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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
2
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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Chapter 4 - Linear Programming:Modeling Examples
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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Chapter 4 - Linear Programming:Modeling Examples
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A Product Mix Example (! of 8
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Chapter 4 - Linear Programming:Modeling Examples
!
A Product Mix Example
Data (" of 8
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Chapter 4 - Linear Programming:Modeling Examples
"
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
A Product Mix Example
Model Con#truction (4 of 8
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Chapter 4 - Linear Programming:Modeling Examples
7Exhibit 4*1
A Product Mix Example
Computer Solution +ith Excel (, of 8
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Chapter 4 - Linear Programming:Modeling Examples
#
Exhibit 4*!
A Product Mix Example
Solution +ith Excel Soler -indo+ (. of 8
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Chapter 4 - Linear Programming:Modeling Examples
$
Exhibit 4*"
A Product Mix Example
Solution +ith /M for -indo+# (0 of 8
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Chapter 4 - Linear Programming:Modeling Examples
10
Exhibit 4*4
A Product Mix Example
Solution +ith /M for -indo+# (8 of 8
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Chapter 4 - Linear Programming:Modeling Examples
11
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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Chapter 4 - Linear Programming:Modeling Examples
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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 $ariable# (! of ,
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Chapter 4 - Linear Programming:Modeling Examples
1
Minimi1e ; - 0.>x< 0.,,x, < 0.0x< 0.,x< 0.0x4< 0.09x=
< 0.0x+< 0.=x>< 0.40x9< 0.0+x0
s!b?ect to:90x< 0x, < 00x< 90x< +4x4< 4x=< =4x+
< 00x>< ,0x9< =4x0 ,0
,x, < ,x< ,x< 4x4< x=< x>< x0 ,0
,+0x4< >x=< ,x> 0
=x< x, < ,x< x< x4< x+< x0 4
,0x< >x, < ,x< >x< 0x4< 4,x+< ,40x>
< x9< ,=x0 00
x< x, < 4x< =x< +x4< ,x=< x+< 9x>< x9< x0 ,0
4x< ,x,< x< x< x+< x0 ,
A Diet Example
Model Summar2 (" of ,
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Chapter 4 - Linear Programming:Modeling Examples
14Exhibit 4*,
A Diet Example
Computer Solution +ith Excel (4 of ,
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Chapter 4 - Linear Programming:Modeling Examples
1!
Exhibit 4*.
A Diet Example
Solution +ith Excel Soler -indo+ (, of ,
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Chapter 4 - Linear Programming:Modeling Examples
1"
Maximi1e ; - 30.0>4x< 0.04x,< 0.0=4 x< 0.0xs!b?ect to:
x/000
x,% x% x% x0
x,< x,/000
%.,x< x,< x % ., x0
x< x,< x< x- +0/000
x/ x,/ x/ x0
)here x- amo!nt in#ested in m!nicipal bonds 637
x,- amo!nt in#ested in certi*icates o* deposit 637
x- amo!nt in#ested in treas!ry bills 637
x- amo!nt in#ested in gro)th stoc$ *!nd637
An Ine#tment Example
Model Summar2 (1 of 4
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Chapter 4 - Linear Programming:Modeling Examples
17Exhibit 4*0
An Ine#tment Example
Computer Solution +ith Excel (! of 4
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Chapter 4 - Linear Programming:Modeling Examples
1#
Exhibit 4*8
An Ine#tment Example
Solution +ith Excel Soler -indo+ (" of 4
A I t t E l
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Chapter 4 - Linear Programming:Modeling Examples
1$Exhibit 4*
An Ine#tment Example
Sen#itiit2 7eport (4 of 4
A M 3 ti E l
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Chapter 4 - Linear Programming:Modeling Examples
20
B!dget limit 300/000
Tele#ision time *or *o!r commercials @adio time *or 0 commercials
Ae)spaper space *or + ads
@eso!rces *or no more than 4 commercials and(or ads
A Mar3eting Example
Data and Problem Definition (1 of .
A M 3 ti E l
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Chapter 4 - Linear Programming:Modeling Examples
21
Maximi1e ; - ,0/000x< ,/000x,< 9/000xs!b?ect to:
4/000x< =/000x ,< /000x00/000
x
x,0 x+
x< x,< x4
x/ x,/ x0
)here x- Expos!re *rom Tele#ision Commercial 6people7
x,- Expos!re *rom @adio Commercial 6people7
x- Expos!re *rom Ae)spaper d 6people7
A Mar3eting Example
Model Summar2 (! of .
A M 3 ti E l
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Chapter 4 - Linear Programming:Modeling Examples
22Exhibit 4*15
A Mar3eting Example
Solution +ith Excel (" of .
A M 3 ti E l
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Chapter 4 - Linear Programming:Modeling Examples
2
Exhibit 4*11
A Mar3eting Example
Solution +ith Excel Soler -indo+ (4 of .
A M 3 ti E l
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Chapter 4 - Linear Programming:Modeling Examples
24
Exhibit 4*1!
Exhibit 4*1"
A Mar3eting Example
Integer Solution +ith Excel (, of .
A Mar3eting E ample
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Chapter 4 - Linear Programming:Modeling Examples
2!Exhibit 4*14
A Mar3eting Example
IntegerSolution +ith Excel (. of .
A ran#portation Example
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Chapter 4 - Linear Programming:Modeling Examples
2"
-arehou#e #uppl2 of 7etail #tore demand
elei#ion Set#% for telei#ion #et#%
% Cincinnati 00 % Ae) or$ 40
, % tlanta ,00 B % "allas ,40
% Pittsb!rgh ,00 C % "etroit ,00
Total +00 Total =00
A ran#portation Example
Problem Definition and Data (1 of "
A ran#portation Example
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Chapter 4 - Linear Programming:Modeling Examples
27
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
Example Problem Solution
Model )ormulation (! of ,
Example Problem Solution
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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
Example Problem Solution
Model Summar2 (4 of ,
Example Problem Solution
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p
Solution +ith /M for -indo+# (, of ,