Or1 Winf 2004 Part-c

8/9/2019 Or1 Winf 2004 Part-c

1/70

1

Mathematical ProgrammingMathematical Programming

Linear Programming

8/9/2019 Or1 Winf 2004 Part-c

2/70

2

TThreehree Goals in this Chapter Goals in this Chapter

• Learn the principle of Simplex-Algorithm

• Learn how to formulate pbs as LP’s

– Like brainteasers: can be fun – Man problems that !o not look likestereotpical "pro!uct mix# problems can beformulate! as LP’s

– Spotting LP’s is an art

• Learn how to implement an! sol$e LP’sin %xcel

8/9/2019 Or1 Winf 2004 Part-c

3/70

3

Linear Linear ProgrammingProgramming

LP Models

8/9/2019 Or1 Winf 2004 Part-c

4/70

4

LP ModelLP Model

An optimi&ation mo!el is a linear

program 'or LP( if it has continuous$ariables) a single linear ob*ecti$efunction) an! all constraints are

linear e+ualities or ine+ualities,

8/9/2019 Or1 Winf 2004 Part-c

5/70

5

LP ModelLP Model

• Linearit

• i$isibilit '.ontinuous(

• Assumption of .ertaint

8/9/2019 Or1 Winf 2004 Part-c

6/70

6

LP ModelLP Model

21 2420: x x Maximize +

3224

6063..

21

21

≤+

≤+

x x

x xt s

;0;0: 21 ≥≥ x x NNC

8/9/2019 Or1 Winf 2004 Part-c

7/70

7

Standard FormStandard Form

4321 002420: x x x x Min ++−−−

3224

6063..

421

321

=++

=++

x x x

x x xt s

;0;0

;0;0:

43

21

≥≥

≥≥

x x

x x NNC

8/9/2019 Or1 Winf 2004 Part-c

8/70

8

Linear ProgrammingLinear Programming

For purposes of describing and analyzing algori!"s# !e

proble" is ofen saed in !e sandard for"

$0#:"in% ≥= xb Ax xcT

&!ere ' is !e (ecor of n un)no&ns# c is !e n di"ensional

cos (ecor# and * !e consrain "ari' +" ro&s and ncolu"ns,.

8/9/2019 Or1 Winf 2004 Part-c

9/70

-

Steps in Formulating a LinearSteps in Formulating a Linear

Programming ProblemProgramming Problem

• /n!erstan! the problem

• 0!entif the ecision Maker 'M(

• 0!entif the !ecision $ariables• State the ob*, function as a linear

combination of the !ecision $ariables

•State the constraints as a linearcombination of the !ecision $ariables

• 0!entif upper or lower boun!s on 1’s

8/9/2019 Or1 Winf 2004 Part-c

10/70

10


Solving and Sensitivity

8/9/2019 Or1 Winf 2004 Part-c

11/70

11

Linear ProgrammingLinear Programming

!e feasible region described by !e consrains

is a polytope, or simplex# and a leas one

"e"ber of !e soluion se lies a a (ere' of !is polyope

/ac! consrain +euaion, defines a sraig! line in

!e space of !e un)no&ns '

8/9/2019 Or1 Winf 2004 Part-c

12/70

12

Ways of Solving an LPWays of Solving an LP

• 2raphical Metho!

• %numerating all extreme points

• Simplex metho! in$ente! b 2, ant&ig – Speciali&e! software such as L034 – 2eneral software such as %xcel sol$er

• 0nterior point metho!s of the sort propose! b

5armarkar • Speciali&e! algorithms for special tpes of

LP’s

8/9/2019 Or1 Winf 2004 Part-c

13/70

13

Sensitivity Analysis ofSensitivity Analysis of

LPsLPs

• Simplex Metho! is a stan!ar! wa to sol$e linearprograms

• 0ts solution iel!s a "simplex tableau# with "!ual$ariables# which sol$e a closel relate! "!ual# problem

6 which contain sensiti$it analsis information fororiginal problem concerning – 7ow much coul! ob*, fn, coefficients change without changing

optimal solution8

– 7ow woul! changing 97S $alues affect $alue of optimal

solution8 – how much coul! one change the 97S $alues without

changing the pattern '"nature#( of optimal solution8

– ;oul! it be optimal to pro!uce a new pro!uct if it werea$ailable8

8/9/2019 Or1 Winf 2004 Part-c

14/70

14

Simple!"#ased SensitivitySimple!"#ased Sensitivity

Analysis is $ery LimitedAnalysis is $ery Limited

• .an’t see "aroun! corners#

• 4nl rele$ant for linear programs

• ;as more rele$ant before .P/ cclesbecame cheap

• 3ow more con$enient to sol$e

iterati$el an! plot results) e,g,) with – Spi!er

8/9/2019 Or1 Winf 2004 Part-c

15/70

15


Various types of LP Models

8/9/2019 Or1 Winf 2004 Part-c

16/70

16

%e!t Topic& Learning%e!t Topic& Learning

'o( to Formulate LPs'o( to Formulate LPs

• 2etting !ec $ars right is often the ke• on’t rein$ent the wheel= get familiar with

common tpes of LP’s

– Man problems are $ariants on thesecommon tpes

– Man other problems ha$e components whichare similar to one or more of the common

tpes•

8/9/2019 Or1 Winf 2004 Part-c

17/70

17

)evie( of Common)evie( of Common

Types of LP Pbs*Types of LP Pbs*

>( Pro!uct mix problems

?( Make $s, bu

@( 0n$estmentPortfolio allocation pbsB( Sche!uling

C(

8/9/2019 Or1 Winf 2004 Part-c

18/70

18



1) Product mix problems

?( Make $s, bu


C(

8/9/2019 Or1 Winf 2004 Part-c

19/70

1-

+,& Product Mi! -ecisions+,& Product Mi! -ecisions

.Li/e the 'o(ies Problem0.Li/e the 'o(ies Problem0

• ecision: 7ow man of each tpe of pro!uctshoul! be ma!e 'offere!(

• ecision $ariables – Gi H amount of pro!uct i to make 'offer(

•

8/9/2019 Or1 Winf 2004 Part-c

20/70

20




2) Make vs. buy


C(

8/9/2019 Or1 Winf 2004 Part-c

21/70

21

+1& Ma/e vs* #uy+1& Ma/e vs* #uy

-ecisions-ecisions

• Se$eral pro!ucts) each can be ma!e inhouse or purchase! from $en!ors

• ecision: 3ot *ust how much of each pro!uctto obtain but also how much to make an!how much to bu) so ,,,

• ecision $ariables

for each pro!uct i: – Mi H amount of pro!uct i to make in house

– i H amount of pro!uct i to purchase

8/9/2019 Or1 Winf 2004 Part-c

22/70

22

Ma/e vs* #uy -ecisionsMa/e vs* #uy -ecisions

• .onstraints – Meet !eman!

– Pro!uction capacit constraints

– 3onnegati$it

• 4b*ecti$e: Minimi&e cost 'or max profit(

• %xamples

– .7AMP/S '.i$ilian 7ealth an! Me!ical Programof the /niforme! Ser$ices(

– 4utsourcingpri$ati&ation

– Staffing courses with a!*uncts

8/9/2019 Or1 Winf 2004 Part-c

23/70

23

Ma/e vs* #uy -ecisionsMa/e vs* #uy -ecisions

More General PerspectiveMore General Perspective

• Se$eral pro!ucts) each can be obtaine!through one or more sources

• ecision $ariables – Gi* H amount of pro!uct i obtaine! from source *

• .onstraints – Suppl constraints on each source *

– eman! constraints on each pro!uct i – Pro!uction capacit constraints

– 3onnegati$it

8/9/2019 Or1 Winf 2004 Part-c

24/70

24




?( Make $s, bu

) !nvestment"Portfolio allocation pbsB( Sche!uling

C(

8/9/2019 Or1 Winf 2004 Part-c

25/70

25

+2& 3nvestment4Portfolio+2& 3nvestment4Portfolio

AllocationAllocation

• Pool of resources 'e,g,) mone orworkers( nee!s to be allocate! across anumber of a$ailable "instruments#

• ecision: how much to put 'e,g,) in$est(in each instrument) so ,,,


for each pro!uct i: – Gi H amount of in$este! in instrument i

3 t t4P tf li3 t t4P tf li

8/9/2019 Or1 Winf 2004 Part-c

26/70

26

3nvestment4Portfolio3nvestment4Portfolio

Allocation -ecisionsAllocation -ecisions

• .onstraints – All resources allocate!

– i$ersit constraints on amount in$este! in an

one instrument or tpe of instrument – 3onnegati$it

• 4b*ecti$e: Maximi&e returnbenefit

• %xamples – ollars to financial in$estments

– ollars to !e$elopment pro*ects

– Staffpersonnel to work pro*ects

8/9/2019 Or1 Winf 2004 Part-c

27/70

27




?( Make $s, bu

@( 0n$estmentPortfolio allocation pbs#) Sc$eduling

C(

8/9/2019 Or1 Winf 2004 Part-c

28/70

28

+5& Scheduling Personnel .or+5& Scheduling Personnel .or

6ther )esources0 to Shifts6ther )esources0 to Shifts

• 4utline of problem: eman! for ser$ices $arieso$er time 'time of !a or !a of week(, Iou canha$e emploees start at an time) but ou ha$e

less control o$er when the stop,

8/9/2019 Or1 Winf 2004 Part-c

29/70

2-

Personnel Scheduling %aturalPersonnel Scheduling %atural

Language -escriptionLanguage -escription

• ecisions: 7ow man people shoul! beassigne! to each shift 'or shift tpe( – 'Must explicitl i!entif shiftsJ(

• 4b*ecti$e: Minimi&e the cost of theassignment) which is the sum o$er all shifts ofthe number of people working that shift timesthe costperson assigne! to that shift

• .onstraints: 4ne for e$er time perio!, Meet!eman! in that perio! '6 nonneg(

8/9/2019 Or1 Winf 2004 Part-c

30/70

30

Scheduling 7!ample&Scheduling 7!ample&

Police Shift AssignmentPolice Shift Assignment

• eman! for police is !efine! for four hourblocks throughout a ?B hour !a – 'DF hour week an!

pa attention to K of consecuti$e !as officersworke! too,(

• 4fficers can work F or >? hour shifts

• Police are pai! !ouble time for workingbeon! F hours) so >? hour shift costs twiceas much as an F hour shift

8/9/2019 Or1 Winf 2004 Part-c

31/70

31

-emand -ata for Police-emand -ata for Police

Shift Assignment 7!ampleShift Assignment 7!ample

Perio! >> pm - @ am @

? @ am - E am >?

@ E am - >> am >D

B >> am - @ pm ?

C @ pm - E pm @D

D E pm - >> pm @B

8/9/2019 Or1 Winf 2004 Part-c

32/70

32

Algebraic Formulation ofAlgebraic Formulation of

Police Scheduling 7!amplePolice Scheduling 7!ample

• ecision 1ariables

– Gi H officers starting F hour shift in perio! i

– Ii H officers starting >? hour shift in perio! i• 4b*ecti$e: Minimi&e Labor .ost

– Assume base pa e+ual for all officers) an!

measure in terms of multiples of base pafor one shift

– Min H G> N ? I> N G? N ? I? N ,,,

Al b i F l ti

8/9/2019 Or1 Winf 2004 Part-c

33/70

33

Algebraic FormulationAlgebraic Formulation

.cont*0.cont*0

• .onstraintsGD N G> N IC N ID N I> OH @ 'Perio! >(

G> N G? N ID N I> N I? OH >? 'Perio! ?(G? N G@ N I> N I? N I@ OH >D 'Perio! @(

G@ N GB N I? N I@ N IB OH ? 'Perio! B(

GB N GC N I@ N IB N IC OH @D 'Perio! C(GC N GD N IB N IC N ID OH @B 'Perio! D(

Gi) Ii OH for all i

8/9/2019 Or1 Winf 2004 Part-c

34/70

34




?( Make $s, bu


%) &ransportation"'ssignment problems

D( len!ing

E( Multi-perio! planning

F( .utting stock problems

+ 4+8 T t ti 4

8/9/2019 Or1 Winf 2004 Part-c

35/70

35

+8& Transportation4+8& Transportation4

Assignment ProblemsAssignment Problems

• 7a$e $arious +uantities of a commo!it atmultiple sources, 3ee! to meet !eman! forthat commo!it at "sinks# '!estinations(, 7ow

much shoul! ou mo$e that from each sourceto each sink in or!er to minimi&e the cost ofmeeting !eman! at each !estination,


– Gi* H amount sent from source i to sink *• .ost parameters

– ci* H cost per unit of shipping from source i to sink *

8/9/2019 Or1 Winf 2004 Part-c

36/70

36




?( Make $s, bu


C(

8/9/2019 Or1 Winf 2004 Part-c

37/70

37

+9& #lending Problems+9& #lending Problems

• Se$eral ingre!ients 'fee!stocks( are mixe! tocreate !ifferent final pro!ucts, 7ow much ofeach ingre!ient shoul! go into each pro!uct

in or!er to minimi&e pro!uction costs whilesatisfing +ualit constraints on the pro!ucts8%,g) – oil refining

– pro!ucing animal fee! mixes – pro!uction of !air pro!ucts – allocation of coal to power plants

8/9/2019 Or1 Winf 2004 Part-c

38/70

38

#lending Problems#lending Problems

• ecision 1ariables

– Gi* Hunits of ingre!ient i use! in pro!uct *

– units coul! be poun!s) tons) gallons) etc,• 4b*ecti$e

– Minimi&e cost of pro!ucing re+uire!

amounts of each pro!uct 'tpicall !ri$enb cost of ingre!ients(

#l di P bl#l di P bl

8/9/2019 Or1 Winf 2004 Part-c

39/70

3-

#lending Problemslending Problems&

Constraint FormulationConstraint Formulation

• eman! constraint example – 3ee! at least F) poun!s of pro!uct >

– G>> N G?> OH F)• ualit constraint example

– 0ngre!ient > is ?Q corn, 0ngre!ient ? isCQ corn,

– Pro!uct > must be at least @Q corn,

– ,? G>> N ,C G?> OH ,@ 'G>> N G?>(

f C) i f C

8/9/2019 Or1 Winf 2004 Part-c

40/70

40




?( Make $s, bu


C(

8/9/2019 Or1 Winf 2004 Part-c

41/70

41

+:& Multi"period Planning+:& Multi"period Planning

ProblemsProblems

•

8/9/2019 Or1 Winf 2004 Part-c

42/70

42

+:a& Multi"period Financial+:a& Multi"period Financial

Planning ProblemsPlanning Problems

• 0n$est mone to maximi&e return)manage cash flow !uring construction

pro*ect) etc,• ecision $ariables – Gi*Hamount in$este! in instrument i at time *

•

8/9/2019 Or1 Winf 2004 Part-c

43/70

43

Multi"period PlanningMulti"period Planning

ConstraintsConstraints

• "Mass balance# constraint on mone for eachtime perio!

9e$enue at time t N R maturing at time t H

amount in$este! at time t N paments

!ue at time t 'for e$er perio! t(

• .onstraints on mix of in$estments

– o not excee! maximum risk threshol! – Maintain minimum le$el of li+ui!it

– %tc,

+:b M lti P i d Pl i

8/9/2019 Or1 Winf 2004 Part-c

44/70

44

+:b& Multi"Period Planning&+:b& Multi"Period Planning&

Trading 6ff 6T and 3nventoryTrading 6ff 6T and 3nventory

• Suppose that in some perio!s !eman!excee!s normal pro!uction capacit,

8/9/2019 Or1 Winf 2004 Part-c

45/70

45

%otation%otation


– Gi H regular pro!uction in perio! i

– Ii H 4< pro!uction in perio! i

– i H shortage in perio! i

– 0i H in$entor carrie! into perio! i

• Parameters

– i H !eman! in perio! i – 0> H initial in$entor

– .ost parameters

8/9/2019 Or1 Winf 2004 Part-c

46/70

46

FormulationFormulation

• Minimi&e weighte! sum of G’s) I’s) ’s)an! 0’s

• Sub*ect to – Gi TH regular pro!uction capacit

– Ii TH 4< pro!uction capacit

– 0i TH storage capacit

– 0i N Gi N Ii N i H i N 0iN>

mass balance constraint

+:c& Multi Period Planning&+:c& Multi Period Planning&

8/9/2019 Or1 Winf 2004 Part-c

47/70

47

+:c& Multi"Period Planning&+:c& Multi"Period Planning&

Wor/ Force SchedulingWor/ Force Scheduling

• Si&e of 'traine!( labor force re+uire! $arieso$er time 'e,g,) 09S staff(

• .an

– 7ire new emploees) but the nee! to be traine!b experience! workers 'which takes time awafrom primar task( an! ma not sta withorgani&ation

– 'Sometimes( can la-off emploees – 'Sometimes( can outsource or hire temps) buttpicall at a high cost

– 'or proacti$el balance the workloa!(

8/9/2019 Or1 Winf 2004 Part-c

48/70

48

%otation%otation


– ;i H K of contractors hire! in perio! i

– Gi H K emploees hire! 6 traine! in perio! i

– Ii H K of experience! workers in perio! i

– i H K lae! off at beginning of perio! i

• Parameters

– i H !eman! for exp, workers in perio! i

– I H initial number of experience! workers

– .ost parameters

8/9/2019 Or1 Winf 2004 Part-c

49/70

4-

FormulationFormulation

• Min weighte! sum of ;’s) G’s) I’s) 6’s

• Suppose – 4ne exp worker can train four new hires

– UCQ retention of experience! workers

– CQ retention of trainees

• N ,C Gi-> mass balance

– nonnegati$it

) i f C)evie( of Common

8/9/2019 Or1 Winf 2004 Part-c

50/70

50




?( Make $s, bu


C(

8/9/2019 Or1 Winf 2004 Part-c

51/70

51

+;& Cutting Stoc/ Problem+;& Cutting Stoc/ Problem

• Suppose ou pro!uce a wi!e)continuous sheet of material 'steel) film)

paper) fabric) etc,(, .ustomers !eman!$arious +uantities 'lengths( of thinnerstrips, 7ow shoul! ou cut the wi!esheet into strips to meet !eman! whileminimi&ing either amount of rawmaterial cut or amount waste!8

Cutting Stoc/ Problem&Cutting Stoc/ Problem&

8/9/2019 Or1 Winf 2004 Part-c

52/70

52

Cutting Stoc/ Problem&Cutting Stoc/ Problem&

-ecision $ariables-ecision $ariables

•

8/9/2019 Or1 Winf 2004 Part-c

53/70

53

7!ample of Patterns for 297!ample of Patterns for 29

Foot Wide )a( MaterialFoot Wide )a( Material

Pattern FV pieces >?V pieces >DV pieces ;aste

K> B L L B

K? @ > L LK@ ? L > B

KB > ? L B

KC > > > L

KD L @ L LKE L L ? B

Cutting Stoc/Cutting Stoc/

8/9/2019 Or1 Winf 2004 Part-c

54/70

54

Cutting Stoc/Cutting Stoc/

Problem ConstraintsProblem Constraints

• Meet !eman! for F foot stripsB G> N @ G? N ? G@ N GB N GC OH b>

• Meet !eman! for >? foot stripsG? N ? GB N GC N @ GD OH b?

• Meet !eman! for >D foot strips

G@ N GC N ? GE OH b@• 3onnegati$itGi OH for all i

8/9/2019 Or1 Winf 2004 Part-c

55/70

55


Solving LPs in MS /xcel

8/9/2019 Or1 Winf 2004 Part-c

56/70

56

Solving LPs in 7!celSolving LPs in 7!cel

• 4rgani&e !ata for mo!el in sprea!sheet

• 9eser$e cells for !ecision $ariables

• .reate formula for the ob*, function $al,• or each constraint) create

– algebraic expression for L7S $alue

– 97S constraint $alue

• Label e$erthingJ

) ll C Mi 7 l

8/9/2019 Or1 Winf 2004 Part-c

57/70

57

)ecall Course Mi! 7!amples)ecall Course Mi! 7!amples

Algebraic FormulationAlgebraic Formulation

• ecision $ariables – G> H K of seminars offere! – G? H K of lectures offere!

• Maximi&e H @ G> N ? G?s,t,

G> N ? G? TH @B 'facult time(

G? TH > 'lecture rooms(G> TH ? 'seminar rooms(

?C G> N > G? OH F 'enrollment(

G>) G? OH 'non-negati$it(

Course Mi! 7!ample&Course Mi! 7!ample&

8/9/2019 Or1 Winf 2004 Part-c

58/70

58


Spreadsheet FormulationSpreadsheet Formulation

G> H K of G? H K of

Seminars Lectures .ourse

K offere! 1ariet

@ ? /nit 1ariet Score 0n!ex

1ariet ? @B acult > Lecture 7alls

> ? Seminar 9ooms

OH constraints A$ailable 3ee!e!

?C > F %nrollment


8/9/2019 Or1 Winf 2004 Part-c

59/70

5-


Spreadsheet FormulasSpreadsheet Formulas

G> H K of G? H K of

Seminars Lectures .ourse

K offere! 1ariet

@ ? /nit 1ariet Score 0n!ex

HA@WAB H:@W:B 1ariet ? HAR@WAFN:R@W:F @B acult HAR@WAUN:R@W:U > Lecture 7alls

> HAR@WA>N:R@W:> ? Seminar 9ooms


?C > HAR@WA>@N:R@W:>@ F %nrollment

Spreadsheet Solution&Spreadsheet Solution&

8/9/2019 Or1 Winf 2004 Part-c

60/70

60


Cells Specified to Solver Cells Specified to Solver

G> H K of G? H K of Seminars Lectures .ourse

K offere! 1ariet@ ? /nit 1ariet Score 0n!ex 1ariet ? @B acult > Lecture 7alls

> ? Seminar 9ooms

OH constraints A$ailable 3ee!e!?C > F %nrollment

8/9/2019 Or1 Winf 2004 Part-c

61/70

61

The Solver -ialog #o!The Solver -ialog #o!

• %nter target cell) changing cells) an!constraint cells information – can *ust point to cells= no nee! to tpe

– can highlight sets of constraints at once

– nonnegati$it constraints are *ust like otherconstraints= for 97S *ust tpe in

• /n!er options – .heck "Assume linear mo!el#

– All other !efaults shoul! be fine


8/9/2019 Or1 Winf 2004 Part-c

62/70

62


6btained by Solver 6btained by Solver

G> H K of G? H K of Seminars Lectures .ourse

? E K offere! 1ariet@ ? /nit 1ariet Score 0n!ex

D >B 1ariet ? @B @B acult E > Lecture 7alls

> ? ? Seminar 9ooms

OH constraints A$ailable 3ee!e!?C > >? F %nrollment

Suppose 'ad 6ne MoreSuppose 'ad 6ne More

8/9/2019 Or1 Winf 2004 Part-c

63/70

63

Suppose 'ad 6ne MoreSuppose 'ad 6ne More

Faculty& 7nter B 1ariet ? @B % acult E > Lecture 7alls

> ? ? Seminar 9ooms


?C > >? F %nrollment

Pull -o(n Solver andPull -o(n Solver and

8/9/2019 Or1 Winf 2004 Part-c

64/70

64

Pull -o(n Solver andPull -o(n Solver and

Clic/ on C 1ariet ? @C @C acult E,C > Lecture 7alls> ? ? Seminar 9ooms

OH constraints A$ailable 3ee!e!?C > >?C F %nrollment

Spreadsheet -esignSpreadsheet -esign

8/9/2019 Or1 Winf 2004 Part-c

65/70

65

Spreadsheet -esignSpreadsheet -esign

GuidelinesGuidelines

• uil! mo!el aroun! !ispla of !ata

• on’t bur constants in formulas

• Logicall close +uantities shoul! bephsicall close

• esign so formulas can be copie!

• /se color) sha!ing) bor!ers) an!protection

• ocument with text boxes 6 cell notes

Some 7!cel Functions ThatSome 7!cel Functions That

8/9/2019 Or1 Winf 2004 Part-c

66/70

66

Some 7!cel Functions ThatSome 7!cel Functions That

Are =seful in LP FormulationAre =seful in LP Formulation

• S/MP94/.

8/9/2019 Or1 Winf 2004 Part-c

67/70

67

Summary of Solving LP sSummary of Solving LP s

(ith 7!cels Solver (ith 7!cels Solver

• %xcel facilitates entering 6 sol$ing LP’s

• %as to explore "what if# $ariations

•

8/9/2019 Or1 Winf 2004 Part-c

68/70

68

Mathematical ProgrammingMathematical Programming

Linear Programming

MS %xcel Sol$er

.ases

8/9/2019 Or1 Winf 2004 Part-c

69/70

69


7omework

8/9/2019 Or1 Winf 2004 Part-c

70/70

#rea/#rea/

Or1 Winf 2004 Part-c

Documents

Transcript of Or1 Winf 2004 Part-c