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1Chapter 5 - Integer

Chapter 5

Integer Programming

Introduction to Management Science

9thEdition

by Bernard W. Taylor III

2007 Pearson Education

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Chapter TopicsChapter Topics

Integer Programming (IP) ModelsInteger Programming (IP) Models

Integer Programming Graphical SoltionInteger Programming Graphical Soltion

!ompter Soltion o" Integer Programming Problems!ompter Soltion o" Integer Programming Problems

With E#cel and $M "or Windo%sWith E#cel and $M "or Windo%s

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Integer Programming Models

Types of Models

Total Integer Model& 'll decision ariables reired to haeinteger soltion ales.

*+, Integer Model& 'll decision ariables reired to haeinteger ales o" -ero or one.

Mi#ed Integer Model& Some o" the decision ariables (bt

not all) reired to hae integer ales.

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A Total Integer Model (1 of 2)

Machine shop obtaining ne% presses and lathes.

Marginal pro"itability& each press ,**/day0 each lathe,1*/day.

2esorce constraints& 3*4***4 5** s. "t. "loor space.

Machine

prchase prices and space reirements&

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A Total Integer Model (2 of 2)

Integer Programming Model

Ma#imi-e 6 7 ,**#,8 ,1*#5

sbect to& :4***#,8 34***#53*4***

,1#,8 ;*#55** "t5

#,4 #5

* and integer #,7 nmber o" presses

#57 nmber o" lathes

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2ecreation "acilities selection to ma#imi-e daily sage byresidents.

2esorce constraints& ,5*4*** bdget0 ,5 acres o" land.

Selection constraint& either s%imming pool or tennis center(not both).

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7/467Chapter 5 - Integer


Ma#imi-e 6 7 ;**#,8 9*#58 3**#;8 ,1*#3

sbect to&

;14***#,8 ,*4***#58 514***#;8 9*4***#3,5*4***

3#,8 5#58 =#;8 ;#3,5 acres

#,8 #5, "acility

#,

4 #5

4 #;

4 #3

7 * or ,

#,7 constrction o" a s%imming pool

#57 constrction o" a tennis center

#;7 constrction o" an athletic "ield

#37 constrction o" a gymnasim

A ! " 1 Integer Model (2 of 2)

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A Mi#ed Integer Model (1 of 2)

51*4*** aailable "or inestments proiding greatestretrn a"ter one year.

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Ma#imi-e 6 7 94***#,8 ,41**#5 8 ,4***#;

sbect to&

1*4***#,8 ,54***#5 8 :4***#;51*4***

#,3 condominims

#5,1 acres

#;5* bonds

#5*

#,4 #;* and integer

#,7 condominims prchased

#5 7 acres o" land prchased

#;7 bonds prchased

A Mi#ed Integer Model (2 of 2)

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2onding non+integer soltion ales p to the nearestinteger ale can reslt in an in"easible soltion

' "easible soltion is ensred by ronding do%n non+integer soltion ales bt may reslt in a less than optimal(sb+optimal) soltion.

Integer Programming $raphical Solution

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Integer Programming %#ample

$raphical Solution of Ma#imi&ation Model

Ma#imi-e 6 7 ,**#,8 ,1*#5sbect to& :4***#,8 34***#53*4***

,1#,8 ;*#55** "t5

#,4 #5* and integer

?ptimal Soltion&6 7 ,4*11.1@

#,7 5.55 presses#57 1.11 lathes

'igure 51 'easile Solution Space *ith Integer Solution Point

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+ranch and +ound Method

Traditional approach to soling integer programmingproblems.

Based on principle that total set o" "easible soltions can bepartitioned into smaller sbsets o" soltions.

Smaller sbsets ealated ntil best soltion is "ond.Method is a tedios and comple# mathematical process.

E#cel and $M "or Windo%s sed in this booA.

See !

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,ecreational 'acilities %#ample

Ma#imi-e 6 7 ;**#,8 9*#58 3**#;8 ,1*#3

sbect to&

;14***#,8 ,*4***#58 514***#;8 9*4***#3,5*4***

3#,8 5#58 =#;8 ;#3,5 acres

#,8 #5, "acility

#,4 #54 #;4 #3 7 * or ,

Computer Solution of IP Prolems

! - 1 Model *ith %#cel (1 of 5)

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%#hiit 52



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%#hiit 5.


! - 1 Model *ith %#cel (. of 5)

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%#hiit 55



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! - 1 Model *ith 0M for indo*s (1 of .)

,ecreational 'acilities %#ample

Ma#imi-e 6 7 ;**#,8 9*#58 3**#;8 ,1*#3

sbect to&

;14***#,8 ,*4***#58 514***#;8 9*4***#3,5*4***

3#,8 5#58 =#;8 ;#3,5 acres

#,8 #5, "acility

#,4 #54 #;4 #3 7 * or ,

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19Chapter 5 - Integer%#hiit 5


! - 1 Model *ith 0M for indo*s (2 of .)

C S l i f IP P l

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%#hiit 53


! - 1 Model *ith 0M for indo*s (. of .)

C t S l ti f IP P l

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Total Integer Model *ith %#cel (1 of 5)


Ma#imi-e 6 7 ,**#,8 ,1*#5

sbect to& :4***#,8 34***#53*4***

,1#,8 ;*#55** "t5

#,4 #5* and integer

C t S l ti f IP P l

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C t S l ti f IP P l

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%#hiit 5


Total Integer Model *ith %#cel (/ of 5)

C t S l ti f IP P l

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%#hiit 51!


Total Integer Model *ith %#cel (. of 5)

C t S l ti f IP P l

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Ma#imi-e 6 7 94***#,8 ,41**#5 8 ,4***#;

sbect to&

1*4***#,8 ,54***#5 8 :4***#;51*4*** #,3 condominims

#5,1 acres

#;5* bonds

#5* #,4 #;* and integer


Mi#ed Integer Model *ith %#cel (1 of .)


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Total Integer Model *ith %#cel (2 of .)


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%#hiit 51.


Solution of Total Integer Model *ith %#cel (. of .)


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29Chapter 5 - Integer%#hiit 51/


Mi#ed Integer Model *ith 0M for indo*s (1 of 2)


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%#hiit 515


Mi#ed Integer Model *ith 0M for indo*s (2 of 2)

! 1 Integer Programming Modeling %#amples

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niersity booAstore e#pansion proect.Fot enogh space aailable "or both a compter departmentand a clothing department.

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#,7 selection o" %eb site proect#57 selection o" %arehose proect

#;7 selection clothing department proect

#37 selection o" compter department proect

#17 selection o" 'TM proect#i 7 , i" proect CiD is selected4 * i" proect CiD is not selected

Ma#imi-e 6 7 ,5*#,8 :1#58 ,*1#;8 ,3*#38 =*#1

sbect to&11#,8 31#58 @*#;8 1*#38 ;*#1,1*

3*#,8 ;1#58 51#;8 ;1#38 ;*#1,,*

51#,8 5*#58 ;*#3@*

#;8 #3

,# 7 * or ,

! - 1 Integer Programming Modeling %#amples

Capital +udgeting %#ample (2 of /)


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Capital +udgeting %#ample (. of /)


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%#hiit 513


Capital +udgeting %#ample (/ of /)


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'i#ed Charge and 'acility %#ample (1 of /)

Which o" si# "arms shold be prchased that %ill meet

crrent prodction capacity at minimm total cost4 incldingannal "i#ed costs and shipping costs

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yi7 * i" "arm i is not selected4 and , i" "arm i is selected4 i 7 ,454;43414@

#i7 potatoes (tons4 ,***s) shipped "rom "arm i4 i 7 ,454;43414@ to plant 4 7'4B4!.

Minimi-e 6 7 ,:#,'8 ,1#,B8 ,5#,!8 ,;#5'8 ,*#5B8 ,=#5!8 ,@#;'8

,3#;B8 ,:#;!8 ,9#3'8 ,1#3b8 ,@#3!8 ,=#1'8 ,9#1B8

,5#1!8 ,3#@'8 ,@#@B8 ,5#@!8 3*1y,8 ;9*y5 8 31*y;8

;@:y38 15*y18 3@1y@sbect to& #,'8 #,B8 #,B+ ,,.5y,H * #5'8 #5B8 #5!+,*.1y5*

#;'8 #;'8 #;!+ ,5.:y;* #3'8 #3b8 #3!+ 9.;y3*

#1'8 #1B8 #1B+ ,*.:y1

* #@'8 #@B8 @!+ 9.@y@

*

#,'8 #5'8 #;'8 #3'8 #1'8 #@' 7 ,5

#,B8 #5B8 #;B8 #3B8 #1B8 #@B 7 ,*

#,!8 #5!8 #;! 8 #3!8 #1!8 #@! 7 ,3

# J * 7 * or ,


'i#ed Charge and 'acility %#ample (2 of /)


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'i#ed Charge and 'acility %#ample (. of /)


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%#hiit 51


'i#ed Charge and 'acility %#ample (/ of /)


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!ities !ities %ithin ;** miles ,. 'tlanta 'tlanta4 !harlotte4 Fashille 5. Boston Boston4 Fe% KorA

;. !harlotte 'tlanta4 !harlotte4 2ichmond 3. !incinnati !incinnati4

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#i7 city i4 i 7 , to ,54 #i7 * i" city is not selected as a hb and #i7 ,i" it is.

Minimi-e 6 7 #,8 #5 8 #;8 #38 #18 #@8 #=8 #:8 #98 #,*8 #,,8 #,5

sbect to&

'tlanta& #,8 #;8 #:,

Boston& #5 8 #,* ,

!harlotte& #,8 #;8 #,,,

!incinnati& #38 #18 #@ 8 #:8 #,*,

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41Chapter 5 - Integer%#hiit 52!


Set Co6ering %#ample (. of /)


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%#hiit 521


Set Co6ering %#ample (/ of /)

Total Integer Programming Modeling %#ample

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Prolem Statement (1 of .)

Te#tbooA company deeloping t%o ne% regions.Planning to trans"er some o" its ,* salespeople into ne% regions.

'erage annal e#penses "or sales person&

2egion , + ,*4***/salesperson

2egion 5 + =41**/salespersonTotal annal e#pense bdget is =54***.

Sales generated each year&

2egion , + :14***/salesperson

2egion 5 + @*4***/salesperson

Lo% many salespeople shold be trans"erred into each region inorder to ma#imi-e increased sales


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Step 1ormlate the Integer Programming Model

Ma#imi-e 6 7 :14***#,8 @*4***#5

sbect to,8 #5,* salespeople

,*4***#,8 =4***#5=54*** e#pense bdget

#,4 #

5* or integer

Step 2

Sole the Model sing $M "or Windo%s

g g g g p

Model 'ormulation (2 of .)


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g g g g p

Solution *ith 0M for indo*s (. of .)

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End of chapter

chap05-9th

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